Electrical Engineering
Suhas Bhandare
Application of Lithium Niobate-based Integrated
Optical Circuits to Optical Communication
PH.D. Dissertation
Paderborn, December 2003
DISSERTATION
ON
Application of Lithium Niobate-based
Integrated Optical Circuits to Optical
Communication
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN ELECTRICAL ENGINEERING
(DOKTOR-INGENIEUR)
TO
DEPARTMENT OF ELECTRICAL ENGINEERING
UNIVERSITY OF PADERBORN
WARBURGER STR. 100, 33098 PADERBORN
GERMANY
BY
M.Sc., M.Phil. Suhas Bhandare
from Pune, INDIA
Reviewers:
1. Prof. Dr. Reinhold No`
e
2. Prof. Dr. Wolfgang Sohler
Date of Thesis Submission: June 10, 2003
Date of Defense Examination: December 3, 2003
Paderborn, December 2003
Diss. 14–196
Dedicated
To
My Parents,
My wife Kranti, and
My dearest kids Mrunmayi and Saumitra
Contents
Contents V
List of Publications IX
ABSTRACT XI
1 Introduction 1
1.1 Background.................................. 1
1.2 Motivation................................... 2
1.2.1 An Integrated Optical Network Analyzer . . . . . . . . . . . . . . 3
1.2.2 An Integrated Optical PMD Compensator . . . . . . . . . . . . . . 5
1.3 OrganizationOfThesis............................ 5
2 Measurement Setup: Integrated Optical Network Analyzer 7
2.1 MeasurementSetup.............................. 7
2.2 TheDFBLaser ................................ 8
2.3 Design and Development of LiNbO3-basedIOC............... 9
2.3.1 Polarizer ............................... 13
2.3.2 PhaseModulator ........................... 17
2.3.3 TE↔TMModeConverter ...................... 20
2.3.4 PI Temperature Controller . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The3x3FiberCoupler ............................ 28
2.4.1 Analysis of 3x3 Fiber Coupler . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Basic MZI Using 3x3 Fiber Coupler . . . . . . . . . . . . . . . . . 30
2.4.3 Advanced MZI Using LiNbO3-based IOC and 3x3 Fiber Coupler . 32
2.5 Conclusion .................................. 34
3 Longitudinal Structure Characterization of FBG 37
3.1 Fiber Bragg Gratings: An Introduction . . . . . . . . . . . . . . . . . . . . 37
3.2 Longitudinal Structure Characterization of FBG . . . . . . . . . . . . . . . 40
3.2.1 Measurement of Complex Reflection Coefficient . . . . . . . . . . 40
3.2.2 Calculation of Impulse Response . . . . . . . . . . . . . . . . . . . 41
3.2.3 Calculation of Longitudinal Grating Structure . . . . . . . . . . . . 42
V
3.2.4 Scalar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Vectorial Structure Characterization of FBG . . . . . . . . . . . . . . . . . 47
3.3.1 Concept of Birefringence and Dichroism . . . . . . . . . . . . . . 47
3.3.2 Birefringence and Dichroism in Optical FBG . . . . . . . . . . . . 49
3.3.3 Polarization Mode Coupling . . . . . . . . . . . . . . . . . . . . . 51
3.3.4 Measurement of Complex Reflectance Jones Matrix . . . . . . . . 52
3.3.5 Polarization Orthogonalization . . . . . . . . . . . . . . . . . . . . 53
3.3.6 Calculation of Impulse Response Matrix . . . . . . . . . . . . . . . 56
3.3.7 Calculation of Vectorial Grating Structure . . . . . . . . . . . . . . 56
3.3.8 Vectorial Measurements . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Conclusion .................................. 61
4 Integrated Optical PMD Compensator 63
4.1 Design Issues for Integrated Optical PMDC . . . . . . . . . . . . . . . . . 63
4.1.1 Operational Principle . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Two vs. Three Phases . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.3 New Proposals for High-Bit Rate PMD Compensators . . . . . . . 66
4.1.4 Conclusion .............................. 69
4.2 DGD Profile Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 MeasurementSetup.......................... 71
4.2.2 Inverse Scattering Algorithm . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 DGD Profiles in Fibers . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.4 DGD Profiles of Distributed PMD Compensator in LiNbO3. . . . 77
4.2.5 Conclusion .............................. 77
4.3 Conclusion .................................. 78
5 Result Discussion and Future Scope 79
5.1 Characterization of LiNbO3-basedIOC ................... 79
5.2 Longitudinal Structure Characterization . . . . . . . . . . . . . . . . . . . 81
5.3 Integrated Optical PMD Compensator . . . . . . . . . . . . . . . . . . . . 82
5.4 Conclusion .................................. 83
A Point Matching Method 85
A.1 Formulation.................................. 85
A.2 Satisfying Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 88
A.3 Computation ................................. 90
B Generation of Optical Fields 91
C Derivation of Coupled Differential Equations for FBG 95
C.1 MaxwellEquations.............................. 95
C.2 ConstitutiveRelations............................. 96
C.3 WaveEquation ................................ 97
VI
C.4 Coupled Differential Equations for FBG . . . . . . . . . . . . . . . . . . . 97
D Derivation of Discrete Equations for FBG 101
D.1 S-Matrix Characterization of Optical Components . . . . . . . . . . . . . . 101
D.2 Calculation of S-matrix for FBG . . . . . . . . . . . . . . . . . . . . . . . 102
D.3 Down–Up Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . 106
References 111
Acknowledgements 117
VII
List of Publications
(1) S. Bhandare and R. No´
e, “Optimization of TE–TM Mode Converters on X–Cut, Y–
Propagation LiNbO3used for PMD Compensation”, Proc. ECIO’01 Paderborn, Germany
(2001), 172–174.
(2) D. Sandel, S. Bhandare, R. No´
e, “Reciprocal Circular Birefringence in X–Cut, Z–
Propagation LiNbO3Polarization Transformers”, Proc. ECIO’01 Paderborn, Germany,
(2001), 193–196.
(3) S. Bhandare and R. No´
e, “Optimization of TE–TM Mode Converters on X–Cut, Y–
Propagation LiNbO3used for PMD Compensation”, J. Appl. Phys. B: Lasers and Optics
(Special Issue on Integrated Optics) 73 (2001), 481–483.
(4) S. Bhandare, R. No´
e, and D. Sandel, “Origin of Reciprocal Circular Birefringence in
X–Cut, Z–Propagation LiNbO3Polarization Transformers”, J. Appl. Phys. B: Lasers and
Optics (Special Issue on Integrated Optics) 73 (2001), 549–553.
(5) S. Bhandare, D. Sandel, R. No`
e, H. Hermann, H. Suche, and W. Sohler, “Measurement
ofDifferential Group Delay Profiles in Fibers and LiNbO3DistributedPMDCompensator”,
Tech. Dig.: Symposium on Optical Fiber Measurements NIST (Boulder) Colorado, USA
(2002), 209–212.
(6) D. Sandel, V. Mirvoda, S. Bhandare, F. W¨
ust, and R. No`
e, “Some Enabling Techniques
for Polarization Mode Dispersion Compensation”, IEEE J. of Light. Technol.,21 (2003),
1198-1210.
(7) S. Bhandare, D. Sandel, R. No`
e, R. Ricken, H. Suche, and W. Sohler, “LiNbO3–
based Integrated Optical Network Analyzer for Vectorial Structure Characterization of
Fiber Bragg Gratings”, Proc. Int. Conf. on Fiber Optics and Photonics Mumbai, India
(2002), 97.
(8) S. Bhandare and R. No´
e, “Pushing Distributed PMD Compensator Performance To-
wards Highest Bit Rates Using Lithium–Niobate–Tantalate or Lithium Tantalate Crystals”,
Proc. ECIO’03 Prague, CZ (2003), 91–94.
(9) S. Bhandare, D. Sandel, R. No`
e, R. Ricken, H. Suche, and W. Sohler, “LiNbO3–
based Integrated Optical Network Analyzer for Vectorial Structure Characterization of
Fiber Bragg Gratings”, IEE Proc.: Circuits, Devices, and Systems (Special Issue on Fiber
Optics and Photonics) 150 (2003), 490–494.
IX
ABSTRACT
A lithium niobate-based, integrated optical network analyzer for the vectorial structure
characterization of optical fiber Bragg gratings is reported. Frequency-dependent complex
reflectance Jones matrix is measured by interferometry and transformed into time domain.
From the impulse response matrix, the vectorial grating structure is determined by inverse
scattering. Local dichroic reflectivity and birefringence are derived from this data. Know-
ledge of the vectorial nature of the refractive index modulation depth and phase should
allow an improvement of the ultra-violet (UV) light illumination process and to effectively
correct the phase mask stitch errors by longitudinally selective UV light post-processing,
in order to fabricate chirped and/or apodized gratings which require the highest fabrication
accuracy.
Especially, for vectorial structure characterization, integrated optical circuit based on
X-cut, Z-propagation lithium niobate was proposed and developed in-house. It consists of
a single-mode 1x2 optical power splitter and is integrated with a set of 3 TE-pass polarizers,
a set of 2 phase shifters, and a TE-TM mode converter on each arm of the power splitter.
Topological details of this integrated optical circuit are given in the dissertation. This inte-
grated optical circuit along with a 3x3 fiber coupler forms a Mach-Zehnder interferometer.
A 3x3 fiber coupler with three photodiodes at its output was chosen because it allows the
most accurate phase measurement. Real and imaginary parts of the frequency-dependent
complex reflection coefficient are calculated from two linear combinations of three (three-
phase) photocurrents. The integrated optic phase shifters allow digital phase shifting and
perform AC rather than DC measurement in order to increase sensitivity. On-chip TE-
pass polarizers maintain a reference polarization and ensure that only phase-modulated
light with a single polarization enters the mode converter sections of this integrated optical
circuit. The integrated optical TE-TM mode converters act as calibrated polarization trans-
formers and are used to generate the required polarizations for vectorial measurement. The
reflecting devices under test are connected to the measurement interferometer by means of
an additional coupler. Measurement setup also includes a wavelength meter for frequency
correction. A cleaved bare fiber end has a small reflectivity, which is independent of the fre-
quency and polarization, is used for calibrating the measurement setup. A uniform optical
fiber Bragg grating at 1548.25 nm with a reflectivity >95% and a 0.2 nm bandwidth was
characterized. From this the vectorial grating structure was obtained and these results are
summarized in the dissertation.
TE–TM mode converters with endlessly adjustable coupling phases on the X-cut, Y-
propagation lithium niobate are optimized by simulation studies for the highest electrooptic
efficiency for distributed PMD compensation. Two-phase and three-phase TE–TM mode
converters are compared, and the latter are found to have a slightly better electrooptic
efficiency. If just a little differential group delay needs to be compensated then the com-
pensation performance can be drastically improved if the compensator is realized in mixed
ferroelectric crystals like lithium–niobate–tantalate where the birefringence can be tuned
by varying the Ta content yin LiNb1−yTayO3. A Ta content yof up to 0.5 is good to realize
a PMD compensator for about 160 Gbit/s. This solution is particularly advisable at data
XI
rates of 40 Gbit/s and beyond. The above in-house developed integrated optical network
analyzer is also used to measure the frequency-dependent reduced M¨
uller matrix of this
in-house developed distributed PMD compensator. Such a reduced M¨
uller matrix measure-
ment allows to calculate the corresponding Jones matrix and hence the impulse response
of the devices with polarization mode dispersion. From the latter, differential group delay
profiles are determined by the inverse scattering technique. Some interesting DGD profiles
are obtained and are summarized in the dissertation.
XII
Chapter 1
Introduction
1.1 Background
The history of integrated optics began in 1969 with the paper published in Bell Systems
Technical Journal [1] where S. E. Miller discussed the dream of Integrated Optical Circuits
(IOC’s) which would use photons instead of electrons. Today’s high-speed electronics
is possible due to technological advances in integrating several electronic components on
to monolithic substrates in an IC form. Similarly, the future of optics could be seen in
an integration of bulky optical components into complicated planner circuits which could
perform various optical functions.
As in the optical fibers, light guiding in integrated optical waveguides is based on the
phenomenon of total internal reflection. Light is confined to an area of high refractive
index, which acts as core, with respect to its surroundings, called as clad. The integrated
optics has not only the potential of generation, manipulation, and detection of optical waves
but also has the capability to couple optical waves into and out of the integrated optical
circuit’s [2,3].
The IOC’s offers many significant advantages over their bulk optical counterparts, since
their features are based on guided wave optics. There main advantages are as follows:
(1) Compact and light weight
(2) Stable alignment by integration
(3) Low operating voltages and short interaction lengths
(4) Easy control of guided waves
(5) High-speed operation
(6) Inherent parallel processing capability
The IOC’s, in principle, consist of optical waveguides and waveguide-based devices.
Passive optical devices includes optical power splitters and combiners [4], polarizers, wave-
length division multiplexers and demultiplexers are developed for passive optical networks.
The IOC’s which could modulate the most important properties of light such as intensity,
phase, polarization, and frequency are realized and are commercially available today as
1
2Chapter 1 Introduction
intensity, phase, and polarization modulators, switches, and wavelength converters which
utilizes either electrooptic or acoustooptic or thermooptic or nonlinear optic effects. The
optically active IOC’s such as waveguide amplifiers, lasers, and even the optical parametric
oscillators have also been demonstrated very recently [5].
The materials used for realizing IOC’s should have very low propagation losses at the
operating wavelengths. Many kind of materials and the device fabrication techniques have
been tried for realizing the integrated optical waveguides. The most important techniques
include thin film deposition, ion exchange, thermal diffusion, ion implantation, and the
epitaxial growth. The materials are chosen depending on whether the device is active or
passive. If it is active then it further depends on which effect the device is based on. Based
on design objectives, the material is chosen and, then, the most appropriate technique based
on the waveguide specifications is selected for fabrication.
The principle materials for integrated optics are glass, lithium niobate, and gallium-
arsenide. For glass based devices, ion exchange is the low cost technology. Ti in-diffusion
is the most promising but expensive technology for lithium niobate based devices (a crystal
with high electro-optic coefficients), while MBE and MOCVD is rather the most expensive
technology for the devices which are based on gallium arsenide. The Table 1.1 summarizes
the propagation losses and technology used for these three material systems.
Table 1.1: Materials for integrated optics
Material Glass Lithium Niobate Gallium Arsenide
Technology Ion-exchange Ti-Indiffusion MBE
CVD Proton exchange MOCVD
Loss (dB/cm) 0.01 0.1 1
1.2 Motivation
The main objective of this entire work is to demonstrate the ability of the lithium niobate
based integrated optical circuits to integrate very basic integrated optical components such
as optical power splitters, polarizers, phase shifters, TE–TM mode converters etc. into
one complex circuit which provides more functionality for optical instrumentation and lat-
ter, their applications to optical communication. Two different application of the lithium
niobate-based integrated optical circuits that are demonstrated in this dissertation includes:
(1) An integrated optical network analyzer in X–cut, Z–propagation lithium
niobate for vectorial structure characterization of optical fiber Bragg gratings
(2) An integrated optical polarization mode dispersion compensator in X–cut,
Y–propagation lithium niobate for the polarization mode dispersion (PMD)
compensation at 40 Gbit/s and beyond
1.2 Motivation 3
1.2.1 An Integrated Optical Network Analyzer
Both microwave and optical networks are similar and consist of an assembly of components
which are interconnected using waveguides, and, the signals are very narrowband quasi-
monochromatic electromagnetic fields. Many microwave components have optical analogs
and the list is growing every year. The most striking differences between them are that:
(1) there are only square-law detectors, i.e. the photodiodes
(2) the optical networks offer bandwidths orders of magnitude higher than the
microwave networks
(3) the optical sources may have non-negligible linewidths which influences
the measurement contrast
(4) the optical waveguides are dielectric in nature
(5) there are linear and nonlinear field-matter interactions such as Rayleigh and
Brillouin scattering
(6) the common “single-mode” optical waveguides are in fact two-moded
In Table 1.2, the frequencies of the most popular bands used for optical network applica-
tions are listed.
Table 1.2: The frequencies of the optical carriers in three most popular optical bands
Wavelength (µm) Frequency (THz)
0.85 353
1.31 229
1.55 194
The linewidths of several common optical sources are listed in Table 1.3 because this
influences the signal-to-noise ratio.
Table 1.3: The linewidths and the coherence lengths of optical sources that are used in the optical network
applications
Type of Source Linewidth Coherence Length
Light emitting diode ∼20 THz ∼15 µm
Superluminescent diode ∼4THz ∼75 µm
Single mode laser diode ∼1GHz ∼30 cm
Distributed feedback laser diode ∼50 MHz ∼6m
External cavity laser diode ∼50 KHz ∼6km
As has been mentioned in the above differences that the two propagating modes that the
single-mode optical waveguide supports are nearly degenerate and differ mainly by their
polarization. Most optical components are sensitive to the state of polarization, and, there-
fore, a full account of its evolution is very much essential in optical network analyses [6].
This implies that the optical network algebra is a matrix algebra. In particular, the transfer
4Chapter 1 Introduction
function in optics becomes a transfer matrix, and the product of such transfer matrices is in
general noncommutative.
We understand optical network analysis to be direct analogon of the electrical network
analysis, which means that optical field transfer functions could be measured directly. It
is suitable for characterizing the linear and time independent optical components such as
Bragg gratings in fibers.
Fiber Bragg gratings are comparatively simple devices and in their most basic form
consist of a periodic modulation of the index of refraction along the fiber core [7,8]. There
are various types of fiber Bragg gratings that differ by application and hence by refractive
index modulation. Of great interest are chirped gratings for dispersion compensation which
require the highest fabrication accuracy [9]. For polarization insensitive operation not only
the desired refractive index modulation must be met precisely but the grating must also be
free from birefringence and dichroism. The fabrication process has, therefore, to be opti-
mized in all these respects. To what extent the specific Bragg grating design and the actual
fabricated device correspond to each other has to be checked by appropriate measurements.
Various interferometric methods such as optical low-coherence interferometry, optical
frequency-domain interferometry and even noninterferometric techniques which includes
time-domain reflectometry or the one that rely on modulation could be denominated as
optical network analysis. Especially, the noninterferometric techniques rely on modulation.
In fact, any direct detection experiment with analog intensity modulation and an electrical
network analyzer connected to the transmitter input and the receiver output yields the elec-
trical transfer function from which magnitude and delay of the optical transfer function
can be obtained. This scheme can be expanded to include different polarizations [10]. Of
course, the sensitivity is limited to the direct-detection process. The phase of the optical
transfer function could be accessible by integrating the group delay over frequency. This
not only limits the accuracy but also does not allow the determination of correct phase
relationship between transfer functions obtained from different polarizations.
Interferometric technique with fixed polarizations is the simplest, one-port form of
optical network analysis [11]. Low-coherence interferometry directly yields the impulse
response of a device with an excellent ∼100 µm spatial resolution. In contrast, frequency-
domain interferometry [12] which was used in this work, delivers the frequency-dependent
complex reflection coefficients from which the impulse response is calculated. The laser
tuning range limits spatial resolution, but large grating lengths may be investigated due
to high coherence of a single-mode laser. Impulse response allows us to determine the
complex coupling coefficients and hence longitudinal grating structure [13,14]. Acquired
knowledge about the vectorial nature of refractive index modulation depth and phase should
allow for a correction of aberrations from the desired structure by effectively correcting the
phase mask errors by longitudinally selective UV light post-processing [15,16].
Therefore, a lithium niobate-based integrated optical network analyzer is proposed,
designed, and built for longitudinal structure characterization of fiber Bragg gratings. The
functionality of this in-house developed optical network analyzer is demonstrated:
1.3 Organization Of Thesis 5
(1) by measuring the vectorial grating structure of a uniform optical fiber Bragg
grating under test [17] and
(2) by measuring differential group delay profiles of an integrated optical PMD
compensator [18] which is introduced in the next subsection.
1.2.2 An Integrated Optical PMD Compensator
Polarization mode dispersion is caused by the noncircular fiber cores and poses a serious
problem for transmitting 10 Gbit/s over older fiber and 40 Gbit/s over any type of fiber.
It can be conveniently modelled as a concatenation of different differential group delay
(DGD) sections connected by variable polarization transformers. It can be compensated if
an appropriately oriented birefringence is added at the receiver side in reverse order [19].
Therefore, a perfect PMD compensator would consist of a large number of short DGD
sections separated by variable polarization transformers. These polarization transform-
ers would be adjusted so that the vectorial DGD profile of the PMD compensator would
follow the DGD profile of the transmission line in reverse order. Implementation of such
polarization transformers with an endlessly adjustable coupling phase was proposed many
years ago by Heismann and Ulrich [20]. Since these polarization transformers require
X-cut, Y-propagation lithium niobate; the natural birefringence (0.22 ps/mm) of this bire-
fringent crystal cut can be used to compensate the DGD at the same time [21]. Since
the function has already been demonstrated by R. No´
e [22], the motive behind this work
is to particularly optimize the electrode design for this type of integrated optical PMD
compensator for the highest electro-optic efficiency by simulation studies [23]. This in-
house developed integrated optical PMD compensator is characterized for different DGD
profiles which were measured by the integrated optical network analyzer using inverse
scattering technique [18]. The performance of this distributed PMD compensator can be
pushed towards the highest bit rates if they are implemented in lithium–niobate–tantalate
crystals. A Ta content yin LiNb1−yTayO3is good to realize a perfect PMD compensator at
160 Gbit/s [24].
1.3 Organization Of Thesis
The dissertation is organized into five chapters. Chapter 1 is an introduction. Chapter
2 explains the design of measurement setup and development of lithium niobate based
integrated optical network analyzer.
Chapter 3 deals with the longitudinal structure characterization of fiber Bragg gratings
by inverse scattering. It also emphasizes on the theory and experimentally obtained results
on scalar as well as vectorial structure characterization of FBG.
Chapter 4 is devoted to design and development integrated optical polarization mode
dispersion compensator whose different differential group delay profiles are determined by
6Chapter 1 Introduction
inverse scattering technique using the above in-house developed integrated optical network
analyzer.
Chapter 5 in short summarizes these two different application of lithium niobate-based
integrated optical circuits to optical communication.
Chapter 2
Measurement Setup: Integrated Optical
Network Analyzer
2.1 Measurement Setup
The measurement setup of our integrated optical network analyzer consist of optics and
electronics. A 10 dB coupler is connected to a tunable laser in order to monitor the
laser power. A 3 dB coupler splits the remaining power between two Mach-Zehnder
Interferometers (MZI). The first MZI is a reference interferometer whose optical path
length is adjusted to 0.5 m and acts as a high-resolution wavelength meter for frequency
correction. The second MZI is a hybrid of the fiber- and integrated- optics and serves for
measurement. It consist of an integrated optical network analyzer (ONWA) circuit based
on X-cut, Z-propagation lithium niobate (LiNbO3), and a 3x3 fiber coupler. The reflecting
devices under test are connected to this MZI by means of an additional coupler. A 3x3
fiber coupler with 3 photodiodes at its outputs is used because it allows a more accurate
phase measurement than a standard 2x2 coupler [25–28]. The real and imaginary parts
of either frequency-dependent complex reflection coefficient or transmission coefficient of
the device under test (DUT) are calculated from two linear combinations of the three-phase
photocurrents obtained from the three photodiodes connected to the outputs of the respec-
tive 3x3 fiber couplers. Optical isolators are used to avoid any stray reflections from the
photodiodes back into the measurement MZI.
The electronic hardware includes in-house developed 8-channel data acquisition system
and 16-channel digital-to-analog converters. The data acquisition system consist of gain
programmable transimpedance amplifiers, inverting voltage amplifiers for additional gain,
integrators, and a high performance analog-to-digital converter (MAX 183) with 12-bit
resolution for each channel. A 16-channel digital-to-analog converter uses high perfor-
mance voltage output digital to analog converter (MAX 547) with 13-bit resolution with
built-in high voltage amplifiers having an output voltage swing of ±70 V. These are used
to drive phase shifter and TE–to–TM mode converter electrodes under computer control.
The in-house developed proportional-integral temperature controller is used to keep the
7
8Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
LiNbO3-based IOC temperature constant. Figure 2.1 shows the measurement setup of an
integrated optical network analyzer.
Figure 2.1: Measurement setup of an integrated optical network analyzer
2.2 The DFB Laser
In our transportable integrated optical network analyzer, a distributed feedback (DFB) laser
diode is used as a tunable laser source. Of course, tunable twin–guide (TTG) laser diode
can also be used. TTG has an advantage of having a higher tuning range say 4-5 nm as
opposed to DFB laser, which could be tuned electronically over 100 GHz (0.8–1 nm) at
1550 nm . TTG laser require two currents to operate: one for pump and another for tuning.
On the other hand DFB laser needs only pump current to operate. Only 50 GHz tuning
range is used around the center wavelength of the Bragg grating, λB, in order to keep
the measurement time short. Of course, thermal tuning is also possible to some extent.
Figure 2.2 shows power and wavelength tuning characteristic response of the DFB laser
diode at 1548 nm as a function of pump current.
This DFB laser diode is driven by the programmable constant current source which is
under computer control. It uses high-performance 16-bit digital-to-analog converter from
Analog Devices (AD669). Advantage of this technique is that one can have very high-
resolution wavelength steps within the linewidth of the single-mode DFB laser diode and
at the same time it allows us to do high-speed measurements, the major advantage of using
electronic tuning. This high-speed measurement is a key to the success in such interfero-
metric measurements where thermal noise in fiber limits the signal-to-noise ratio and the
measurement accuracy.
2.3 Design and Development of LiNbO3-based IOC 9
0 50 100 150 200 250 300
0.0
0.1
0.2
0.3
0.4
0.5
Optical Power [dBm]
∆λ [nm]
Pump Current [mA]
0
3
6
9
12
15
1548 nm
Figure 2.2: Power and tuning characteristic response of DFB LASER at 1548 nm
2.3 Design and Development of LiNbO3-based IOC
The electrooptic control of guided modes is performed by the use of ferroelectric crystals
such as lithium niobate and lithium tantalate. Lithium niobate is certainly one of the most
important material exhibiting the largest Pockel’s effect. It is very often used in integrated
optics to implement intensity, phase, and polarization modulators and switches. A detailed
study of it is thus necessary. It belongs to a crystalline class 3m (noncentrosymmetric) and
has hexagonally closed packed crystal structure. It is a negative uniaxial crystal having
extraordinary refractive index nefor the light which is polarized along its optic axis and
ordinary refractive index nofor the light which is polarized along both the axes which are
perpendicular to the optic axis.
Waveguide Fabrication
Three different techniques have been used to form waveguides in lithium niobate. Initially,
waveguides are formed by thermal outdiffusion of Li2O which results in an increased
refractive index for the extraordinary index ne. In addition to being limited to guiding
light in only one polarization, achievable index change is very small and therefore provides
waveguide modes whose confinement is relatively weak. In addition, channel waveguides
cannot be formed conveniently except by etching ridge waveguides. These problems can
be overcome using waveguides created by indiffusion of a dopant – almost exclusively
titanium – to raise the refractive index. More recently, waveguides have been formed by
an exchange process similar to that used for glass substrates. More specific is the proton
exchange using benzoic acid.
10 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
Based uponthepublishedresults, titaniumindiffusedwaveguides are currentlypreferred
for general device development. This has become a well known and the most standardized
technology in last two decades. One major advantage is that it can guide both polarizations:
ordinary and extraordinary. The fabrication of titanium indiffused waveguides is quite
straightforward. The titanium metal film is deposited onto optical grade lithium niobate
substrates using electron-beam evaporation since the melting point of Ti is as high as
1725◦C. This metal film is patterned by photolithography followed by wet etching of Ti
through positive phtotoresist to get Ti stripes of width equal to 7 µm. These Ti stripes are
diffused thermally into lithium niobate at 1060◦C for about 8 hours in moist argon ambient.
Cool down after diffusion is performed in oxygen to allow reoxidation of the crystal to
compensate for the oxygen loss during diffusion. The water vapor treatment was initially
employed to reduce the photorefractive effect but later it was found out that it also helps to
reduce Li2O outdiffusion which can cause unwanted planar guiding for the extraordinary
polarization. Steadily this technique is mastered to fabricate very high quality Ti-indiffused
channel waveguides in lithium niobate with propagation losses as low as 0.1 dB/cm.
Post-Waveguide Processing
If an electrode is to be placed on top of the waveguide, an intermediate buffer layer is
needed to reduce propagation losses for TM-polarized modes due to direct metal cladding.
SiO2is frequently employed. 0.2 µm thick layer of SiO2usually eliminates the measurable
loading loss.
The choice of electrode material depends on the application. For relatively low-speed
application (modulation frequencies below 100 MHz), an evaporated aluminum electrode
(∼0.5µm thick), with a flash of chrome for adhesion, is sufficient. Initially, aluminum
was employed. Sometimes, it used to get converted to colorless aluminum oxide due to
heating. On the other hand, the most preferred material for very-high-speed devices is the
gold. Gold is a noble metal and has noble properties. Therefore, gold was used in the
fabrication of electrodes, with a flash of titanium for adhesion.
The substrate endface is prepared for either end-fire lens coupling or fiber butt-coupling
by careful lapping and polishing. To eliminate Fresnel back reflections, multilayer anti-
reflection coating made of series of quarter wave optical thickness, or QWOT for short, of
SiO2and TiO2were used on the first IOC which was fiber pigtailed and packaged with
straight end faces. This antireflection coating were not of very good quality, and, there-
fore, this used to introduce undesirable Fabry-Perot noise into measurement system. When
this IOC was particulary used to evaluate the cleaved bare fiber end, which has very small
reflectivity, the Fresnel back reflections became very critical. In fact, it was not possible
to characterize the cleaved bare fiber end. Therefore, the next IOC was fiber pigtailed and
packaged with angled endfaces so as to reduce Fresnel back reflections. This has indeed
improved the overall performance of the integrated optical network analyzer.
2.3 Design and Development of LiNbO3-based IOC 11
Electro-Optic Effect
The linear electrooptic (Pockel’s effect) effect, which is the basis for active waveguide
device control, provides a change in the refractive index which is proportional to the applied
electrostatic field. A voltage Vapplied to the electrodes placed over or alongside the
waveguide, creates an internal electric field of approximate magnitude |E| ≃ V/G, G
being the gap between the electrodes. Since the Pockel’s effect is found in crystals with-
out an inversion symmetry (noncentrosymmetric), such as lithium niobate, the sign of the
induced refractive index change depends on the polarity of the voltage applied to it. On
other hand, in a centrosymmetric crystals, linear electrooptic effect does not exist, while
quadratic electrooptic effect (Kerr effect) is observed where the induced refractive index
change is proportional to the square of the applied electric field intensity.
The equation of the index ellipsoid in the presence of an applied electric field can be
written as
(1
n2
x
+r1jEj)x2+ ( 1
n2
y
+r2jEj)y2+ ( 1
n2
z
+r3jEj)z2
+2yzr4jEj+ 2xzr5jEj+ 2xyr6jEj= 1
(2.1)
where Ej(j= 1,2,3) is a component of the applied electric field and summation over
repeated indices jis assumed. Here 1, 2, and 3 corresponds to the principal dielectric axes
x, y, z and nx, ny, nzare the principal refractive indices. This new index ellipsoid reduces
to the unperturbed index ellipsoid when Ej= 0. In general, principal axes of the perturbed
ellipsoid do not coincide with the unperturbed axes (x, y, z). A new set of principal axes
can always be found out by a coordinate rotation, which is know as principal-axis transfor-
mation of a quadratic form. The dimensions and orientation of the index ellipsoid (2.1) are,
of course, dependent on the direction of the applied electrostatic field as well as 18 matrix
elements rij. The linear change in the coefficients of the index ellipsoid due to applied
electric field Ejalong the principal axes is
(∆n)i=−n3
2
j=3
X
j=1
rijEj,(2.2)
where i= 1,2, . . . , 6and rij is the 6×3electrooptic tensor. The form, but not the
magnitude, of the electrooptic tensor rij can be derived from symmetry considerations,
which dictate which of the 18 electrooptic coefficients rij are zero, as well as the relation-
ships that exist between the remaining coefficients. For lithium niobate, the coefficients of
the electrooptic tensor rij are in the form
0−r22 r13
0r22 r13
0 0 r33
0r51 0
r51 0 0
−r22 0 0
(2.3)
12 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
The Table 2.1 tabulates the values of the rij coefficients which are not zero for the lithium
niobate.
Table 2.1: Values of the electrooptic coefficients for lithium niobate
rij coefficients Low-frequency value High-frequency value
r33 30.9 pm/V 30.8 pm/V
r51 32.6 pm/V 28 pm/V
r13 9.6 pm/V 8.6 pm/V
r22 6.8 pm/V 3.4 pm/V
By inserting the electrooptic tensor rij, the values of ∆ncan be written as the elements
of a symmetric 3×3matrix. For lithium niobate
(∆n)ij =−n3
2
−r22Ey+r13Ez−r22Exr51Ex
−r22Exr22Ey+r13Ezr51Ey
r51Exr51Eyr33Ez
(2.4)
where nis either the ordinary index noor the extraordinary index nevalue. By carefully
looking at the above electrooptic tensor of lithium niobate, certain useful conclusions can
be drawn [29]:
[1] r11,r21,r31,r41 are all zero. There will be no stretching of the index
ellipsoid along the principle axes under applied electrostatic field Ex.
[2] r43,r53,r63 are all zero. There will be no stretching of the index ellipsoid
along the principle axes under applied electrostatic field Ez.
[3] An applied electric field Eywill cause both stretching along the principal-
axes as well as the rotation of the index ellipsoid cross-section in the Y Z-plane.
Perturbation of the index ellipsoid due to electrooptic effect depends on the relative
orientation of the polarization state of the input optical signal, the axis of propagation,
the crystal cut, and the magnitude and direction or sign of the applied electrostatic field.
Utilization of the diagonal elements 11, 22, and 33 of the perturbed refractive index matrix
results in an index change and therefore, a phase change, for an incident optical field
polarized along the crystallographic x-, y- and z-axes, respectively. The diagonal elements
causes an index change, which are essential for modulators and switches, for the optical
field polarized along the crystallographic j-axis, for the given electric field which is applied
in the appropriate direction. The off-diagonal elements, on the other hand, represent an
electrooptically induced mode mixing or conversion between the otherwise orthogonal
polarization components. It represents a rotation of the index ellipsoid that causes a mode
coupling which is proportional to the relevant electrooptic coefficient due to the applica-
tion of electrostatic fields. Utilization of off-diagonal electrooptic elements is necessary to
induce polarization change in Ti:LiNbO3waveguides, i.e. TE–TM mode converters.
2.3 Design and Development of LiNbO3-based IOC 13
Design Considerations: 1x2 Optical Power Splitter
Single-mode 1x2 optical power splitter is realized in X-cut, Z-propagation LiNbO3using
standard Ti-indiffused waveguide technology [30–33]. Scaling laws for the design of
such waveguide based components are derived in [34]. Waveguides are fabricated by in-
diffusing ∼100 nm thick, 7 µm wide, Ti-stripes for 7.5 hours in the moist argon ambient at
1060◦C. Waveguides propagation losses are measured using the Low-Finesse Method [35]
and are of the order of ≈0.1dB/cm for both TE and TM polarized modes. Power splitting
uniformity of this 3 dB power splitter is within ≈0.5dB for the TE-polarized modes.
2.3.1 Polarizer
A linear polarizer is an optical device, birefringent or not, that only transmits one linear
state of polarization and suppresses any transmission of the orthogonal state of polarization.
This is of course the definition of an ideal polarizer; a real component always lets through
a fraction of the orthogonal state. An extinction ratio must therefore be defined in order
to characterize such components. A linear analyzer is fundamentally identical to a linear
polarizer. Its particular name stems from the fact that it enables the emerging state of
polarization to be analyzed.
These devices are characterized by a Jones matrix JPwhich is expressed with respect
to a reference coordinate system Oxy. If the phase factor, which simply renders the propa-
gation of the light in the material medium making up the device, is not taken into account,
the Jones matrices JPxand JPyof the polarizers whose principal axes are respectively the
axes Ox and Oy are given by:
JPx=1 0
0 0 JPy=0 0
0 1 (2.5)
From a mathematical point of view, the Jones matrices of the polarizers are the matrices
associated with projection operators (J2
P=JP) whose eigenvalues are 1 and 0. An impor-
tant property of projection operation is that the intensity transmitted by a polarizer is always
less than or at best equal to the incident intensity it receives. The Jones matrices of linear
polarizers cannot be associated with unitary transformation because det(JP) = 0.
Let us consider a linear polarizer whose principal axis is the axis Ox. Whenever this
polarizer is turning in front of the a linearly polarized light having an intensity of 1, its
transmission Tvaries between two values, Tmax and Tmin, according to relation
T= (Tmax −Tmin) cos2θ+Tmin (2.6)
where θis the angle between the axis Ox and the azimuth of the polarized light. This law
of transmission is known as Malus’ law.
In the case of an ideal polarizer, Tmin is equal to 0 and Tmax is equal to 1, according
to Malus’ law. However, the imperfection of the device and the presence of the parasitic
14 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
light imply that Malus’ law is only partially obeyed. As a general rule, the principal trans-
missions (θ= 0 and θ=π/2)Tmax and Tmin fulfill the relation: Tmin Tmax and the
polarization extinction ratio (PER) is defined as
PER = Tmax
Tmin
.(2.7)
The value of Tmax depends on the nature of the material and also on the physical polarizing
effect used by the polarizer. There are several physical effects that allow polarizers to be
made. Among these metallic thin film polarizer that suppresses the TM-polarized mode
by a suitable dielectric/metal overlay due to the resonant coupling with a surface plasmon
mode offers the best compatibility with fabrication process of Ti:LiNbO3waveguide-based
devices and is a promising option for guided wave optics.
The general principle upon which polarizers in isotropic waveguides are built is based
on a greater loss of a given state (TE- or TM-polarized) with reference to the other during
propagation. The loss of power is not due to a dichroic phenomenon, but to the inability of
a mode to propagate. We know that a mode can only propagate if its normalized frequency
Vis larger than its cut-off frequency. The separation of the two fundamental modes TE0
and TM0may be done by using a guiding structure containing, for instance, a metal.
Polarizers are used to set a reference polarization and to ensure that phase modulated
light having a single polarization enters the mode converter sections of the IOC. This indeed
suppress the second undesired polarization mode, if any, from being interfered in 3x3 fused
fiber coupler in order to do polarization resolved grating characterization. It is easy to
fabricate TE-pass polarizers than TM-pass polarizers and, therefore, TE-pass metal-clad
polarizers based on the surface plasmon effect are used to suppress the TM-polarized mode
in Ti in-diffused LiNbO3waveguides.
Figure 2.3: Cross section of a metal-clad polarizer on X-cut, Z-propagation Ti:LiNbO3optical waveguide
2.3 Design and Development of LiNbO3-based IOC 15
Device structure
Figure 2.3 shows the cross section of polarizer based on X-cut, Z-propagation lithium
niobate using the thin metals films. It simply consist of metal-cladded optical waveguide
but with a intermediate dielectric buffer layer of right thickness and dielectric constant
to enhance the extinction of the TM-polarized mode due to the resonant coupling with a
surface plasmon mode and to reduce the excess propagation losses for the TE-polarized
mode.
Principle of Operation
Metal behaves as a high loss dielectric material with a ‘-’ve dielectric constant over the
entire frequency range of light, because the inertial effect of the carriers (electrons) inside
the metal becomes dominant when the frequency exceeds the plasma frequency of the
metal itself. Therefore, metal cladding on top of the optical waveguide provides significant
attenuation to the TM-polarized modes. This results from the fact that the TM-polarized
mode penetrates more deeply into metal than does the TE-polarized mode. Metal-clad
waveguide exhibits complex propagation constant for the TM-polarized modes and real
propagation constant for the TE-polarized modes. TM-polarized mode is highly attenuated
while TE-polarized mode pass through the polarizer.
For the design of metal-clad polarizers, it is essential to use an intermediate dielectric
buffer layer with a rightdielectric constant and a right thickness between the cladding metal
and the waveguide; and this layer helps to fulfil the phase matching condition between the
TM-polarized mode and the surface plasmon mode. An attenuation coefficient is at it’s
peak value and can be as high as ∼100 dB/cm. This completely blocks the TM-polarized
mode and passes TE-polarized mode without a significant attenuation.
Design Considerations
Aluminumisa natural choice as a cladding metal since it has alarge ‘-’ve dielectric constant
(−114−37) and due to the ease of fabrication. There are several choices of buffer materials
including Si3N4(nb∼1.89), Ta2O5(nb∼2.1), and Y2O3(nb∼1.795), MgO(nb∼1.75),
Nb2O5(nb∼1.95) etc. These materials are chosen on the basis that they have a high
refractive index at 1.55 µmwavelength and is comparable to that of the lithium niobate
refractive index of 2.2.
Stock [36] realized TE-pass polarizers on Ti:LiNbO3waveguides with >55 dB of PER
using Au, Al, and Ti metal cladding at 780 nm wavelength and has used MgO as a buffer
material. Thyagarajan et al [37] has done numerical modelling of single-mode metal-clad
graded-index waveguides with a dielectric buffer layer of Y2O3.ˇ
Ctyrok´
y and Henning [38]
have realized 2 mm long TE-pass polarizer at 1.3 µm using 45 nm thick Si3N4and 200 nm
thick Al with an extinction ratio >35 dB. We want to have TE-pass polarizers at 1.55 µm
wavelength and this wavelength is nearer to the polarizers realized by ˇ
Ctyrok´
y at 1.3 µm,
and therefore, it was planned to use the Si3N4as buffer layer and aluminum as a cladding
16 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
metal with a layer thicknesses of 20–30 nm and 500 nm, respectively. The silicon nitride
is an ideal choice since it is the most stable material from technological and physical point
of view. It is decided to optimize the buffer thickness for a 3 mm long and 50 µm wide
polarizer so that it at least gives 10 dB extinction ratio for TM-polarized modes as compared
to TE-polarized modes.
Realization and Characterization
Initially, in the development of this integrated optical network analyzer circuit, the near-
Z-axis propagation was used to cancel the modal birefringence of the Ti-indiffused optical
channel waveguides in X-cut, Z-propagation LiNbO3in order to make on-chip polarization
transformers more efficient. But this off-axis propagation makes these TE-pass polarizers
nonideal. This is because the TM-polarized mode is regenerated in the waveguide after
the polarizer. This regenerated TM-polarized mode effectively degrades the PER of TE-
pass polarizers. Details of this phenomenon are explained in Section 2.3.4. Therefore, it
was decided to fabricate several polarizers on the X-cut, Z-propagation Ti:LiNbO3straight
channel waveguides with varying buffer materials and thicknesses in order to optimize the
PER. Thickness of Al was chosen to be sufficiently thick (500 nm) to have a higher at-
tenuation coefficient for a 3 mm long polarizers. Polarization extinction ratio is measured
for these metal-clad polarizers using a pair of Glan-Thompson polarizers having an ex-
tinction ratio >55 dB. The measured results of the PER for various buffer materials and
thicknesses are summarized in Table 2.2.
Table 2.2: PER of TE-pass polarizers for various buffer materials
Material Thickness Length PER Problems
no buffer 0 nm 1.5 mm 3 dB/mm Poor PER
Si3N430 nm 4 mm 2.75 dB/mm In technology
Ta2O535 nm 4 mm 2.50 dB/mm Poor PER
Y2O327 nm 3.8 mm 9.5 dB/mm Good PER
Unfortunately polarizers did not work satisfactorily, even after several trials with Si3N4.
Main problem was in the fabrication of stoichiometric Si3N4buffer layer. Next highest
refractive-index material Ta2O5was tried but performance was more or less the same. Next
obvious and possible choice was to use Y2O3as a buffer material. Y2O3based polarizers
fortunately worked with an extinction ratio of 9.5 dB/mm. It is very difficult to fabricate
polarizers with very good repeatability because of the stringent requirement on the unifor-
mity of the buffer thickness, buffer stoichiometry and hence on the buffer refractive index.
These parameters are very difficult to control during the fabrication of buffer layer and of
course the buffer layer plays a key role in the realization of integrated optic metal-clad
polarizers with a very high extinction ratio.
2.3 Design and Development of LiNbO3-based IOC 17
Conclusion
The yttrium oxide based metal-clad polarizers giving PER of ≈30 dB for 3 mm long
polarizers with 27 nm thick Y2O3and 500 nm thick aluminum were optimized for our
application.
2.3.2 Phase Modulator
Perhaps, the simplest waveguide electrooptic device is the phase modulator where electro-
optically induced refractive index change causes a phase shift in the guided light. Figure 2.4
shows the Ti:LiNbO3waveguide based phase modulator with an applied voltage, Vπ, as
defined later, is used to characterize the phase modulators.
Figure 2.4: Cross section of phase modulator on X-cut, Z-propagation LiNbO3with an applied voltage Vπ
Device Structure
Phase modulator simply consist of a Ti in-diffused optical channel waveguide placed in
between the set of uniform electrodes of length, L, separated by a gap, G. The horizontal
y-directed electrostatic field (Ey) is produced by applying a voltage, V, across a gap, G, of
uniform electrodes which gives rise to the local electrooptically induced index change and
hence a corresponding phase change.
Principle of Operation
The local electrooptically induced refractive index change for this particular crystal cut and
propagation direction is given by
∆n22 =−n3
o
2r22Ey.(2.8)
18 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
However, neither the applied electrostatic field nor the optical field is uniform. It is con-
venient to model the effective applied electrostatic field inside the waveguide by that of
a simple parallel plate capacitor as in the bulk case modulator where the lithium niobate
crystal is sandwiched between two electrodes and both optical and electrostatic fields are
assumed to be uniform. The correction factor from this simple model is given by the field
overlap integral Γ. The effective electrooptically induced refractive index change is given
by
∆n22(V) = −n3
or22
2
V
GΓ,(2.9)
where nois the ordinary refractive index of LiNbO3, an interelectrode gap G, and Γis the
overlap integral between the applied electrostatic field and the optical mode. The quantity
Γis given by
Γ = G
VRR|Eo(x, y)|2Ey(x, y, z)dxdy
RR|Eo(x, y)|2dxdy .(2.10)
The total phase shift over the interaction length Lis then
φ= ∆βL =−πn3
or22
λ
V
GLΓ.(2.11)
Phase modulators are generally characterized by the voltage Vπwhich is defined as the
voltage required to obtain a phase shift of πrad. However, voltage–length product is more
useful to compare the performance of different phase modulators and is defined as
VπL=λG
πn3
or22Γ.(2.12)
Phase modulator is generally represented by a Jones matrix,
JPS = eφ
20
0e−φ
2!.(2.13)
It is simply modelled as a retarder with a phase angle φgiven by (2.11). It is very important
to note that, the Jones matrices of phase modulators are unitary matrices i.e. the product
JJ†=I. Mathematically speaking, the polarization transformation done by the phase
modulator is a unitary transformation. The norm and orthogonality of the vectors repre-
senting the incident states of polarization are thus conserved when the device is passed i.e.
the emerging state of polarization is same as the the incident state of polarization.
Design Considerations
Ordinary refractive index of the LiNbO3,no, equals 2.2125 at λ= 1.55 µm. Relevant
electrooptic coefficient, r22, is 6.8pm/V. Length and gap of phase modulator electrodes
are 15 mm and 15 µm, respectively. The unknown optical and electrostatic field overlap
2.3 Design and Development of LiNbO3-based IOC 19
integral factor, Γ, is used in the calculation of φand hence the voltage Vπ, which character-
izes the phase modulators. Once the overlap integral factor, Γ, is known, the voltage, Vπ,
required to give a phase shift of πrad using this retarder can be calculated.
Electrostatic field Eyis only required to compute the field overlap integral factor, Γ,
as defined by Kim and Ramaswamy [39] in (2.10). Software based on the Point Matching
Method (PMM) formulated by Marcuse [40] is developed in Borland C++ to calculate the
electrostatic fields in the LiNbO3. PMM is described in Appendix A. Optical mode is
assumed to Gaussian and Hermite-Gaussian along the width and depth of the single-mode
Ti in-diffused optical waveguide in LiNbO3with mode field diameters matched to our
experimental values. These experimental values of MFD are used to generate polarization
dependent optical intensity profiles for evaluating the field overlap integrals. Appendix B
gives details of this.
TE-polarized optical mode with full-width-half-maximum of 7µmand 4µmalong the
width and depth of the Ti in-diffused waveguide is assumed in the simulation. 0.3µm
thick buffer layer of SiO2is used to eliminate the measurable loading looses, due to metal
cladding, especially for the mode converter electrodes. Phase modulator electrodes has
width of 100 µm. If the interelectrode gap, G= 15 µm, then Γ≈0.51. Voltage Vπ≈40 V
for 15 mm long electrodes.
Realization and Characterization
Realized phase modulators could be conveniently characterized for the measurement of
voltage Vπusing the Fabry-Perot technique. Low–Finesse Method proposed by Regener
and Sohler [35] for the measurement of propagation losses in optical waveguides is also
based on the Fabry-Perot technique, where the optical straight channel waveguide with its
end faces polished perpendicular to the waveguide axis forms a optical resonator (Fabry-
Perot etalon). The transmitted intensity of the resonator varies periodically with the optical
phase difference, φ, which can be tuned by the temperature for simplicity, or, in principle,
it can also be tuned by means of an applied electrostatic field. For this particular measure-
ment, it is necessary to use single frequency laser otherwise it is difficult to monitor the
interference fringes. Phase modulator is used to tune the Fabry-Perot cavity and is driven
by a triangular wave obtained from the signal generator. The output intensity is monitored
using the photodiode and an oscilloscope. It is to be ensured that either the two maxima or
two minima of the Fabry-Perot fringes are within the single ramp, either going up or down.
Voltage difference between these two points on ramp corresponding to these two maxima
or minima directly yields the voltage Vπ.
Figure 2.5 shows the response of the phase modulator’s output intensity for the given
input voltage. As it can be seen from the graph, Vπ= 40.0V and this agrees very well with
the theoretically estimated value of Vπ. Phase modulator with 20 mm long electrodes is
also fabricated to verify the voltage-length product. Vπ= 30.0V for this phase modulator.
Therefore, the voltage-length product (Vπ×L) equals 60 V-cm for both the devices and is
found to be a constant.
20 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
-100 1020304050
0.7
0.8
0.9
1.0
Intensity [a.u.]
Vπ
Voltage [V]
Figure 2.5: Characteristic response of the phase shifter as a function of input voltage
Conclusion
15 mm long phaser modulator with a gap of 15 µmis realized and characterized for the
voltage Vπ(= 40.0 V) using the Fabry-Perot technique. Voltage-length product equals
60 V-cm for this device and is found to be a constant.
2.3.3 TE↔TM Mode Converter
An electrooptic polarization transformer or converter is a compact device that can provide
an electrical control over the polarization state and can be expected to serve a variety of use-
ful functions. Such devices have been demonstrated with Ti:LiNbO3waveguides. Optical
polarization state can be defined in a number of bases, for our purpose it is convenient to
define the polarization state in terms of the polarization angle, θand the phase angle φ. The
normalized TE and TM amplitudes can be written as
ATE
ATM =cos θ
eφ sin θ,(2.14)
where θspecifies the relative TE/TM amplitudes while φis the phase difference between
the TE and TM components. Light is linearly polarized at angle θif φ= 0;θ= 0 repre-
sent purely TE polarized, while θ=π/2is purely TM. Right circularly polarized light, for
example, is represented by θ=π/4and φ=π/2. In passive Ti:LiNbO3waveguides, light
that is linearly polarized along a principal axis has its polarization maintained for propaga-
tion along a principal axis. Thus, for example, light incident as TE (TM) polarization to
waveguides in z(or x, or y)-cut lithium niobate exits in the same state.
2.3 Design and Development of LiNbO3-based IOC 21
In order to implement broadband TE↔TM mode converters, a natural and the best
choice is to use X-cut, Z-propagation LiNbO3crystal [41–44]. Advantage of this particular
crystal cut is that one is free from the material birefringence and can obviously make use
of the wavelength independent uniform electrodes to perform mode conversion. Moreover,
they feature a high optical damage threshold, low temperature dependence and hence little
DC drift as demonstrated by Taniyavaran [45,46].
Device Structure
Figure 2.6 shows the cross section of uniform electrodes of length Lused for the imple-
mentation of broadband TE↔TM mode converters on X-cut, Z-propagation LiNbO3with
the applied voltages. A center electrode of width, W, is directly placed on top of the Ti
in-diffused optical channel waveguide, which may or may not be grounded. Side electrodes
are separated by gap, G, and are sometimes segmented for the implementation of several
electrooptic waveplates.
Figure 2.6: Cross-section of TE–TM mode converter on X-Cut, Z-propagation LiNbO3with an applied
voltages for compensating modal birefringence and mode conversion
Principle of Operation
If the voltages ±(VB/2) and VMC are applied to two side electrodes and the center electrode
of the mode converter, respectively, then its Jones matrix is given by
JMC =cos ψ+cos γsin ψ sin γsin ψ
sin γsin ψcos ψ−cos γsin ψ.(2.15)
The electrostatic field along y-axis (horizontal field), Ey, generated by applying the bias
voltage VBto two side electrodes compensates for the modal birefringence by addition-
ally providing the electrooptically induced phase shifts between the TE and TM polarized
modes via r12 (= −r22) and r22 coefficients, and r22 = 6.8pm/V. The electrostatic field
22 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
along x-axis (vertical field), Ex, generated by applying a voltage VMC to central electrode
leads to electrooptically induced TE–TM mode conversion via r61 (=−r22) coefficient.
ψ=√κ2+δ2L,tan γ=κ/δ,κis the coupling constant, and δis the phase mismatch
parameter between TE and TM polarized modes, and Lis the length of mode converter
electrodes. κand δare expressed as κ=πn3
or61ExΓ/λ and δ= (∆β)/2 + δ0, where
δ0=πn3
or22EyΓ/λ.∆βis the modal birefringence and δ0is the difference in the propaga-
tion constants of the TE and TM polarized modes induced by the external electric field to
compensate for ∆β/2.
Ti in-diffused waveguide sees the ordinary refractive index noin both the x(or y)-
direction, since the waveguide is fabricated on X-cut, Z-propagation LiNbO3. Ideally,
the waveguide is isotropic. However, the residual TE–TM waveguide birefringence (also
known as modal birefringence) is found to be harmful for the operation of some of the
devices such as mode converters. Order of this modal birefringence is ∆n≈1.0×10−4
at 1.55 µmwavelength and hence ∆β(= (2π/λ)∆n) is of the order of 4–5 rad/cm. This
residual modal birefringence, in fact, ruins the phase matching condition between the two
polarization modes and therefore limits the efficiency of TE↔TM mode conversion.
One option is to use near-Z-axis propagation, say at an angle of 2◦to regain the phase
matching condition [47]. This happens in Ti in-diffused channel waveguides because the
effective index for a TE-polarized mode, NTE, is greater than that of a TM mode, NTM.
Rotation of propagation direction in the Y Z-plane at an angle θlowers the value of NTE to
NTE(θ)≃none
pn2
ecos2θ+n2
osin2θ
NTM(θ)≃no
,(2.16)
while leaving NTM unchanged. Second option is to apply high bias voltages to two side
electrodes to compensate for ∆β[48], as explained above.
Using the coupled mode theory, one can immediately write the expressions for the
power in TE and TM polarization modes as
PTE = 1 −κ2
κ2+δ2sin2(√κ2+δ2L)
PTM =κ2
κ2+δ2sin2(√κ2+δ2L)
.(2.17)
For complete TE to TM mode conversion, boundary condition becomes PTE = 0 and
PTM = 1 for the given TE input. If δ= 0, then this boundary condition simplifies to
sin2(κL) = 1. The voltage required for full TE to TM mode conversion and vice-versa
which characterizes the mode converters is given by
VMC =λG
2n3
or61ΓL.(2.18)
2.3 Design and Development of LiNbO3-based IOC 23
Design Consideration
Ordinary refractive index of LiNbO3,no, equals 2.2125 at λ= 1.55 µm. The relevant
electrooptic coefficient r61 =−r22 =−6.8pm/V. The only unknown field overlap integral
factor Γis used in calculation of coupling constant κand hence in the calculation of voltage
required for full mode conversion, VMC.
Mode converters with 25 mm long electrodes, 10 µmwide central electrode, and an in-
terelectrode gap of 6 and 8 µmare designed. The electrostatic field, Ex, is calculated using
the PMM to calculate the unknown field overlap integral factor, Γ. In the simulation, same
parameters for FWHM of the TE-polarized optical mode were used. Table 2.3 summarizes
the simulation results for the calculation of overlap integral factor Γand VMC for 25 mm
long mode converters with different interelectrode gaps.
Table 2.3: Simulation results for the calculation of Γand VMC
GMC [µm]ΓVMC [V]
6≈0.26 9.75
8≈0.33 10.25
Realization and Characterization
Therefore, two devices each having a 25 mm long electrooptic mode converter as shown in
Figure 2.6, with a center electrode width of 10 µm, gap of 6µm, and 0.3µmthick buffer
layer of SiO2were fabricated in house. The first device used 2◦off-axis propagation to
cancel the modal birefringence, with a device length of ≈47 mm and the second device
used principal axis propagation.
No operational voltages were applied to the mode converter on the first device. This
device by default gave >8 % TE to TM static mode conversion for pure input TE mode at
1550 nm. Very similar results were also reported in [47,49]. This phenomenon was later
found to be length–dependent.
The second device needed bias voltages to operate as it needs to cancel the modal or
TE–TM waveguide birefringence. When no voltages were applied either to compensate
for the modal birefringence or to operate the mode converter, there was no static TE–TM
polarization mode conversion, as was observed in the first device. Bias voltages whichwere
applied to the two side electrodes were varied so as to maintain zero waveguide birefrin-
gence and to obtain the smallest possible voltage required for full TE to TM mode con-
version. This phase matching condition must be fulfilled in addition to mode coupling due
to the principal axis rotation of the index ellipsoid in order to reach 100 % mode conver-
sion. Bias voltages were ±VB/2 = ±38 V. The voltage required for full TE to TM mode
conversion was VMC = 9.75 V, as seen from the mode converter switching curves shown
in Figure 2.7 and it agrees well with the theoretically estimated value of 9.75 V. When
the mode conversion voltage was switched off (remember that the bias voltages were still
switched on) the device gave >5 % TE to TM static polarization mode conversion for the
24 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
input TE mode. When the bias voltages were varied, the coupling was found to vary and
it was found that this time static mode coupling was dependent on the electrostatic field
strengths which were used to compensate the modal birefringence. When bias voltages
were also switched off the entire power was present in the input TE mode as expected.
-5 0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
TE/TM Intensity [a.u.]
VMC
PTE
PTM
Voltage [V]
Figure 2.7: Characteristic response of the mode converter as a function of input voltage
Result Discussion
These experiments confirms that there is a change in the state of polarization of light and
this was interpreted as a rotation of the polarization plane [50]. After compensating the
modal birefringence which is linear, the circular birefringence may become the next hurdle
to make mode converters ideal. This may be called as a form of static polarization mode
coupling and is observed in both devices using either off-axis or principal-axis propagation.
In the off-axis case, static mode coupling is length-dependent and in the principal-axis case
it depends on the applied electrostatic field strengths which were used to compensate the
modal birefringence. Both experiments reveal that there exists a phenomenon of static
polarization mode conversion.
A symmetric 3x3 dielectric tensor generally characterizes the optical properties of
anisotropic crystals such as LiNbO3and is given by
ε(0) =
ε11 0 0
0ε22 0
0 0 ε33
.(2.19)
In the off-axis propagation case, we rotate the direction of propagation in the Y Z-plane
about the X-axis in order to compensate for the modal birefringence. So the off-diagonal
2.3 Design and Development of LiNbO3-based IOC 25
terms involving Yand Zcomponents of electromagnetic fields are responsible for this
static mode coupling. To find these off-diagonal terms, one needs to carry out the simple
coordinate transformation R(θ)ε(0)R(−θ)using a rotation matrix
R(θ) =
1 0 0
0 cos θ−sin θ
0 sin θcos θ
,(2.20)
where θ= 2◦. The rotated dielectric tensor is given by
ε(θ) =
ε11 0 0
0ε22cos2θ+ε33sin2θ(ε22 −ε33)cos θsin θ
0 (ε22 −ε33)cos θsin θ ε22sin2θ+ε33cos2θ
.(2.21)
The perturbation is ∆ε= (ε22 −ε33)cos θsin θ. The input is TE mode with electric field
distribution (0, Ey, 0) and the output is the TM mode with electric field distribution (Ex, 0,
Ez). Once we know ∆ε, we can immediately calculate the coupling coefficient
κ=κt
µ,ν +κz
µ,ν,(2.22)
which characterizes this static polarization mode coupling, where µand νare TE and TM
polarized modes involved in the coupling process, respectively. The tangential coupling
coefficient κt
µ,ν is zero. The longitudinal coupling coefficient κz
µ,ν is found to be responsible
for this type of static mode coupling. This is the rare case of transverse–longitudinal
coupling and the power is coupled to the output TM mode from the input TE mode via
longitudinal component Ezof the TM mode. Equation (46) in [51] tells which of the
electromagnetic field components are involved in the coupling process if we insert our
dielectric tensor given in (2.21) having off-diagonal terms. The coupling coefficient κz
µ,ν is
κz
µ,ν =ωε0
4ZEt†
µ∆ε(x, y, z)Ez
νdxdy, (2.23)
which is formally derived by Marcuse in [51]. Above equation simplifies to
κ=κz
µ,ν =ωε0
4∆ετ
τ=ZEt†
µEz
νdxdy
.(2.24)
In the simplified case of pure TM mode the EZcomponent is ∂Ex/∂x. The normalized
field overlap integral, τ, as defined in (2.24) between the transverse component of TE
mode and the longitudinal component of TM mode is not zero. Once we know κ, we can
quickly calculate the power in TE and TM modes as PTE = cos2κL and PTM = sin2κL,
respectively using the well known “Coupled Mode Theory”, assuming that there is no
residual modal birefringence. Lis the length of the IOC in the off-axis propagation case or
the mode converter electrode length in the principal-axis propagation case. Remember that
26 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
there is a phase shift of π/2between the transverse and the longitudinal components and
hence the coupling coefficient κis not real as in the case of transverse–transverse coupling,
but it is imaginary in the case of transverse–longitudinal coupling.
For example, ∆ε= 0.0024,τ= 0.248, and κ= 595.2 m−1is obtained for no=
√ε11 =√ε22 = 2.2125 and ne=√ε33 = 2.1446 at λ= 1550 nm, θ= 2◦, and L=
47 mm. The respective power in TE and TM mode comes out to be 0.9128 and 0.0872.
This well confirms the experimental results of 9–10 % static mode coupling that occurs
between the TE and TM modes in the off-axis propagation case.
For the principal-axis propagation case, we use the basic equation of index ellipsoid
(1
n2
x−r22Ee
y)X2+ ( 1
n2
y
+r22Ee
y)Y2+ (Z2
n2
z
)+2r51Ee
yY Z = 1 (2.25)
under the applied electrostatic field. The electric field is applied along Y-axis to compen-
sate for the modal birefringence using r12 =−r22. Indeed this equation also contains the
term involving Y Z and X,Y,Zare no longer the principal axes. To find the new principal
axes, one need to eliminate the Y Z term appearing in (2.25). This can be done by rotating
the index ellipsoid by an angle
θe=1
2tan−12r51Ee
y
(n−2
e−n−2
o)(2.26)
in the Y Z-plane using the rotation matrix given in (2.20).
If we apply, say, V= (38 −(−38)) V = 76 V to compensate for the modal birefrin-
gence across a gap of 22 µm(G= (10 + 6 + 6) µm), then we know the field strength
Ee
y=V/G causes θe= 0.4254◦. Angle θereplaces θin (2.21) to get off-diagonal term ∆ε,
and ∆ε= 0.0005.κsimply equals 114 m−1as explained above for the off-axis propaga-
tion case, and hence we can calculate the amount of static polarization mode coupling in
principal-axis propagation case.
The coupled modes µand νobey the coupled mode equations
∂Eµ(z)
∂z =ıκEµe−2ı∆βz
∂Eν(z)
∂z =ıκ∗Eνe2ı∆βz
,(2.27)
where κis the coupling coefficient between the two modes and ∆β= (βµ−βν)/2is phase
mismatch parameter. Above coupled differential equation (2.27) reduces to
∂E
∂z =˜
ME
E(z) =Eµe−βµz
Eνe−βνz,(2.28)
with matrix ˜
Mgiven by
˜
M=ıβµκ∗
κ βν.(2.29)
2.3 Design and Development of LiNbO3-based IOC 27
A solution of the above coupled differential equations is assumed to be of the form
E(z) =E(0)eıγz (2.30)
and results in ˜
ME =ıγE(2.31)
This is a standard matrix algebra eigenvalue problem, where Eis the eigenvector and ıγ
is the eigenvalue of the matrix ˜
M. Since the coupling coefficient is imaginary, the super-
modes of the waveguide or the eigensolution of the above coupled differential equations
are therefore circularly polarized modes [52]
E+
E−=1
√21
±ı,(2.32)
and this confirms the speculations of ˇ
Ctyrok´
y [49].
Conclusion
The only option to reduce this undesired mode coupling is to fabricate optical waveguides
along the principal axis. It is to be ensured that the waveguide is exactly parallel to the Z–
axis during photolithography. Otherwise, the LiNbO3anisotropy will play the undesired
role. Of course a relatively high VBwill result.
Whether one physically rotates the direction of propagation in the Y Z-plane globally or
rotates the index ellipsoid electrooptically in the Y Z-plane locally, the net modal birefrin-
gence (typically 1.1×10−4) compensation has the same effect. In this compensation pro-
cess, one always endsupwiththeundesiredimaginarycoupling coefficientbetweenthetwo
orthogonally polarized modes, and this makes mode converters nonideal. If one compen-
sates the modal birefringence globally then phenomenon is length-dependent and beyond
any physical control due to the inherent LiNbO3anisotropy. If one compensates it locally
then it is field–dependent. If the waveguide is exactly parallel to the Z–axis then the volt-
ages required for the compensation are ideally symmetric, otherwise they are asymmetric
in practice. This effect resembles more like a optical activity even though there are no
optical active chiral molecules present in LiNbO3because it gives rise to an apparent circu-
lar birefringence when such devices are characterized using polarimeter. This phenomenon
is found to be reciprocal because the waveguide itself is reciprocal.
25 mm long TE↔TM mode converter with a gap of 6 µmare designed, realized, and
characterized for the mode conversion efficiency and voltage required for full mode con-
version. TE-to-TM mode conversion efficiency is >99 % and voltage required for full
mode conversion is ≈10 V. Needed bias voltages were ±38 V.
2.3.4 PI Temperature Controller
Temperature of the LiNbO3-based IOC was kept constant during measurements in order to
get repeatable and reproducible results, especially from phase shifters and mode converters.
28 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
This is because the LiNbO3has small but finite thermooptic coefficients (∂n/∂t = +5.3×
10−5for neand ∂n/∂t = +0.56 ×10−5for no) and it can cause the considerable refractive
index change say 5×10−3when the crystal temperature changes from room temperature
to 100◦C.
Therefore, a simple temperature controller based on proportional and integral control
(PI) is designed and developed in-house, and is used to keep the temperature of the LiNbO3-
based IOC constant to say 25◦Cwithin ±0.1◦C. It uses thermistor from Siemens which
has a resistance of 10 kΩat 25◦Cand a set of two Peltier elements from Melcor to either
heat or cool the copper block on which the 80 mm long LiNbO3-based IOC is mounted.
Figure 2.8 shows the photograph of the fiber pigtailed and packaged lithium niobate-
based IOC. This device was fabricated by the Applied Physics group of Prof. W. Sohler,
here at the University of Paderborn.
Figure 2.8: Photograph of the fiber pigtailed and packaged LiNbO3-based IOC
2.4 The 3x3 Fiber Coupler
Use of 3x3 fused fiber optic coupler (FOC) as an interfering device, in the field of fiber sen-
sorsis becoming more popular. The advantageof 3x3 coupler is that the three output signals
are available. The three output intensities are sinusoidal function of the optical path length
difference between the any two arms of the interferometer. With the proper design of the
3x3 FOC, there exist a 120◦phase difference between any two of the three output sinusoids.
This phase difference can be exploited in electronic signal processing for avoiding the DC
thermal drift and to generate two signals which are 90◦out of phase with each other by
taking the linear combinations of these three 3-phase output sinusoids. This in principle
2.4 The 3x3 Fiber Coupler 29
allows us to perform more accurate phase measurement than a standard 2x2 coupler. This
is because the conventional 2x2 coupler needs a nonreciprocal phase bias (π/2rad), other-
wise the sensitivity is zero at initial operating point. Another reason is as the differential
phase φfluctuates in the conventional 2x2 coupler causes corresponding fluctuations in the
device sensitivity. Therefore 3x3 coupler is recommended over 2x2 coupler. Of course
the performance of the interferometer depends sensitively on the characteristics of the 3x3
FOC. In principle the operation of the 3x3 FOC can be predicted using the “Coupled Mode
Theory”. In practice, the coupling coefficients and the required geometric parameters are
neither known nor can be easily determined.
2.4.1 Analysis of 3x3 Fiber Coupler
Optical power transfer properties of the 3x3 FOC are characterized by the transfer matrix
M=
a b c
d e f
g h k
,(2.33)
which itself is 3x3 [26]. There is a reference point for each input fiber lead and the each
output fiber lead. The positions of these reference points can be so adjusted that the matrix
elements a, b, c, d, and gare all real and positive numbers while e, f, h, and kare the
complex numbers. Energy conservation principle results in
MM+=M+M=I.(2.34)
On expanding above equation into elemental equations, it is found that there are 9 con-
ditions on the 13 parameters (one each for the real components, 2 each for the others).
Therefore, 4 independent parameters e, f, g and hcharacterizes the optical power transfer
properties of the 3x3 FOC. The convenient choices are the modulus of the complex pa-
rameter e, f, g and h. The elemental equations derived from above equation yields the real
parameters as
a2=e2+f2+h2+k2−1
b2= 1 −e2−h2
c2= 1 −k2−f2
d2= 1 −e2−f2
g2= 1 −k2−h2
(2.35)
30 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
in terms of the modulus of e, f, g and h. The phase of the complex elements
cos(φe) = c2f2−a2d2−b2e2
2adbe , φe>0
cos(φf) = b2e2−a2d2−c2f2
2adcf , φf<0
cos(φh) = c2k2−a2g2−b2h2
2agbh , φh<0
cos(φk) = b2h2−a2g2−c2k2
2agck , φk>0
(2.36)
can also be derived from the elemental equations. Note that these phases depend only on
the modulus of the matrix elements of M. The modulus of the matrix elements are directly
related to the splitting ratios. For example, if the power is applied to the input arm 1 only,
then powers measured at the three output arms will be in the ratio a2:b2:c2. The modulus of
all the elements of Mcan be determined in principle from the splitting ratio data.
In the case of the completely symmetric 3x3 coupler, the square of the modulus of each
elements of Mis equal to 1/3. For this special case of symmetric coupler, φf−φe= 120◦
and φk−φh= 120◦. In summary, it has been shown that only 4 parameters are required to
characterize the power transfer properties of the lossless 3x3 coupler. These parameters can
be very easily obtained from the measurement of the splitting ratio data, and there exists
a phase difference of 120◦between any two outputs of the symmetric 3x3 coupler. It can
also be shown in the next section 2.4.2 that how the performance of a fiber interferometer
incorporating a 3x3 fiber coupler can be predicted from this splitting ratio data.
2.4.2 Basic MZI Using 3x3 Fiber Coupler
A typical all-fiber interferometer configuration is shown in the Figure 2.9. The interfero-
meter is of a Mach-Zehnder type and is the fiber analog of the classical, bulk optic version
of the Mach-Zehnder interferometer. It simply consist of the input 2x2 fiber coupler and
the output 3x3 fiber coupler. The coherent single-mode laser source (both transverse and
longitudinal modes) is used as a source in the MZI. The input light is then divided into two
beams with nominal equal intensity by the input 2x2 fiber coupler (referred to as divider),
part being sent through the measurement arm, the remainder through the reference arm.
These two outputs, after passing through the measurement and reference arms, are recom-
bined by the output 3x3 fiber coupler (referred to as recombiner). An interference signal
between the two beams is then formed which, after propagating the length of the output
fiber, is detected by the photodiodes.
Mathematically, the operation of such a interferometer can be described in terms of the
product of transfer matrices (2.37) of each of the component used to construct the interfero-
meter. First, 3x3 transfer matrix characterizes the output 3x3 fiber coupler. Second, transfer
matrix characterizes both the reference and measurement arm of the interferometer. The
ρis the magnitude and can be set to 1 for this simplified case, and φis phase difference
2.4 The 3x3 Fiber Coupler 31
Figure 2.9: Basic Mach-Zehnder interferometer utilizing 3x3 fiber coupler
between the two arms of the interferometer. Third, transfer matrix characterizes the input
2x2 fiber coupler. Fourth, column vector characterizes the input signal Pin applied to this
interferometer. Input power Pin can be set to 1 without the loss of generality.
α
β
γ
=
a b c
d e f
g h k
1 0 0
0 0 0
0 0 ρeıφ
A0B
0 0 0
B0−A
√Pin
0
0
a=b=c=d=e=f=g=h=k=1
√3and A=B=1
√2
(2.37)
α, β, γ characterizes the output amplitudes of the interferometer
α=pPin(aA +cBρeıφ)
β=pPin(dA +fBρeıφ)
γ=pPin(gA +kBρeıφ)
.(2.38)
The output intensities I1, I2,and I3are given by
I1=αα∗=Pin[a2A2+c2B2+ 4aAcBρ cos(φ)]
I2=ββ∗=Pin[d2A2+f2B2+ 4dAfBρ cos(φ+2π
3)]
I3=γγ∗=Pin[g2A2+k2B2+ 4gAkBρ cos(φ−2π
3)]
.(2.39)
±2π/3phase shifts are added due to the fact that the symmetric 3x3 fiber coupler has any
two of its three output signals phase shifted by 120◦with respect to each other.
Iref =Imes =a2A2=c2B2=d2A2=f2B2=g2A2=k2B2=1
3·1
2=0.1666 (2.40)
Therefore, the output intensities now become
I1=Pin[Iref +Imes + 4IrefImesρcos(φ)]
I2=Pin[Iref +Imes + 4IrefImesρcos(φ+2π
3)]
I3=Pin[Iref +Imes + 4IrefImesρcos(φ−2π
3)].
(2.41)
32 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
The output fringe visibility of the interferometer is given by
V=Imax −Imin
Imax +Imin
.(2.42)
It should be noted that two effects have been ignored in this calculations; the polarization
effects and the effects of the finite source coherence length. Here it is assumed that only
co-polarized signals interfere in output coupler. If one assumes a Lorentzian line shape, the
normalized self-coherence function γ(τ)is given by γ(τ) = exp(−|τ|/τs), where τis the
differential propagation time delay between the measurement and reference optical paths
and τsis the source coherence time. Thus the actual value of Vis a factor γ(τ)smaller
than Vitself. For this reason, the differential propagation delay between the measurement
and reference fiber arms is usually adjusted to be much less than the source coherence time
(i.e. ττs); then γ(τ)→1and output fringe visibility equals V.
Two signals which are 90◦out of phase with each other can be easily generated from
the linear combinations of the three output intensities I1, I2,and I3as 2I1−I2−I3and
I2−I3. For example, these two linear combinations yields
Re(ρ)
Im(ρ)=2
6−1
6−1
6
01
2√3−1
2√3
I1
I2
I3
(2.43)
the real and imaginary part of the measured complex reflection coefficient ρof the DUT’s
such as FBG’s.
2.4.3 AdvancedMZIUsingLiNbO3-based IOCand3x3FiberCoupler
Figure 2.10 shows the advanced Mach-Zehnder interferometer configuration using inte-
grated optical circuit based on X-cut, Z-propagation LiNbO3instead of the input 2x2 fiber
coupler as shown in Figure 2.9 and the output 3x3 fiber coupler. This IOC utilizes 1x2
optical power splitter to split the input power in the ratio of 1:1 between the reference and
measurement arms of the hybrid MZI. It also has two sets of electrooptic phase modulators
on each arm of the interferometer which facilitates the digital phase shifting of the input
signal in both arms of the MZI with respect to each other and a TE–TM mode converter
which facilitates polarization change, under computer control.
The power transfer matrix of the input 2x2 fiber coupler is modified to take into account
the facility of digital phase shifting. The signal in the reference arm may be phase shifted
by the phase φrwhile the signal in the measurement arm may be phase shifted by the
phase φm. The operation of this hybrid MZI can be described once again in terms of the
product of the transfer matrices (2.44) of each of the component used to construct this
hybrid interferometer with the modified matrix for the input 2x2 fiber coupler which now
2.4 The 3x3 Fiber Coupler 33
Figure 2.10: Measurement set-up, integrated optical network analyzer using LiNbO3-based IOC and 3x3
fiber coupler
takes into account the possibility of digital phase shifting.
α
β
γ
=
a b c
d e f
g h k
1 0 0
0 0 0
0 0 ρeıφ
Aeıφr0Beıφm
0 0 0
Beıφm0−Aeıφr
√Pin
0
0
a=b=c=d=e=f=g=h=k=1
√3and A=B=1
√2
(2.44)
α, β, γ characterizes the output amplitudes of the interferometer
α=pPin(aAeıφr+cBeıφmρeıφ)
β=pPin(dAeıφr+fBeıφmρeıφ)
α=pPin(gAeıφr+kBeıφmρeıφ)
.(2.45)
The output intensities I1, I2,and I3are given by
I1=αα∗
=Pin[a2A2+c2B2+ 4aAcBρ cos(φr−φm−φ)]
I2=ββ∗
=Pin[d2A2+f2B2+ 4dAfBρ cos(φr−φm−φ+2π
3)]
I3=γγ∗
=Pin[g2A2+k2B2+ 4gAkBρ cos(φr−φm−φ−2π
3)]
.(2.46)
±2π/3phase shifts are added due to the fact that the symmetric 3x3 coupler has any two
of its three output signals phase shifted by 120◦with respect to each other.
Iref =Imes =a2A2=c2B2=d2A2=f2B2=g2A2=k2B2=1
3·1
2=0.1666 (2.47)
34 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
Therefore, the output intensities now become
I1=Pin[Iref +Imes + 4IrefImesρcos(φr−φm−φ)]
I2=Pin[Iref +Imes + 4IrefImesρcos(φr−φm−φ+2π
3)]
I3=Pin[Iref +Imes + 4IrefImesρcos(φr−φm−φ−2π
3)]
.(2.48)
Phase angle φris set to −π/2,0,π/2,πwhile phase angle φmis set to −3π/4−π/4π/4,
3π/4, respectively. There are 16 linear combinations of these 4 phase shifts φrand φm,
which are successively applied to interferometer. This fact can be exploited to generate
two signals which are 90◦out of phase with each other using
Re(ρ) = 3
8
4
X
k=1
4
X
j=1
3
X
i=1
Iicos(φci+φrj−φmk−φ)
Im(ρ) = 3
8
4
X
k=1
4
X
j=1
3
X
i=1
Iicos(φci+φrj−φmk−φ−π
2)
.(2.49)
The phase angle φcin (2.49) takes into account the phase difference introduced by the out-
put 3x3 fiber coupler and is either set to 0or 2π/3or −2π/3. Index irepresents the intensi-
ties at the output of the 3x3 coupler and goes from 1 to 3 while index jand krepresents the
phase angles φrjand φmk, and both the indexes goes from 1 to 4. For example, this again
gives the real part and imaginary part of the measured quantity using this interferometer
such as a complex reflection coefficient ρof the DUT’s such as FBG’s.
The advantage of this technique is that the DC thermal drift in the fiber interferometers
could be avoided to some extent. This is because the digital phase shifting is done elec-
trooptically using the LiNbO3-based IOC which has typically a response time of 1 ns and
the very fast measurements are possible in principle. This is a kind of AC measurements
where we change the optical path lengths in both arms of the interferometer in a pre-
determined manner and sample the data so that the effects of power fluctuations could
be averaged out.
2.5 Conclusion
In summary, the measurement setup of an integrated optical network analyzer is developed
using fiber and the lithium niobate based integrated optical circuit. The basic integrated
optical components such as polarizers, phase modulators, and TE↔TM mode converters
are all integrated on to a single IOC to get more functionality. Each of these basic integrated
optical components are designed, fabricated, and characterized independently so as to get
anoptimum performance before integration. Theintegrated optical network analyzerisalso
interfacedtoacomputersystem. Analog–to–digitalconverters are used for data acquisition.
Digital–to–analog converters are used to drive phase modulator, TE↔TM mode converter,
2.5 Conclusion 35
and programmable constant current source which is interfaced to tuneable DFB laser diode,
under computer control. Proportional integral temperature controller is implemented to
keep the temperature constant of the integrated optical network analyzer circuit. Finally, the
operation of basic and advanced Mach-Zehnder interferometer using fiber and integrated
optics is explained. Moreover, the use of digital phase shifters to implement AC rather than
DC measurement is described.
36 Chapter 2 Measurement Setup: Integrated Optical Network Analyzer
Chapter 3
Longitudinal Structure Characterization
of FBG
3.1 Fiber Bragg Gratings: An Introduction
Fiber Bragg gratings represent a key element in the established fields of optical communi-
cations, optoelectronics, and optical sensors. Bragg Gratings allows to implement various
primary functions such as reflection, diffraction, and filtering in a highly efficient, low-loss
manner in single-mode optical fibers. Their unique filtering properties and versatility as in-
fiber component is illustrated by their use in a wavelength stabilized semiconductor lasers,
fiber lasers, tuneable wavelength filters, optical fiber polarization mode converters, gain
equalizers for Erbium Doped Fiber Amplifiers (EDFA), to improve the pump efficiency of
EDFA’s, dispersion compensators, wavelength division multiplexers and demultiplexers,
add or drop multiplexers, and optical sensors etc. These are comparatively simple device
and in their most basic form consists of a periodic modulation of the index of refraction
along the fiber core. Different types of fiber Bragg gratings are proposed and realized de-
pending on the function and hence by refractive index modulation. These includes uniform,
nonuniform, chirped, apodized, and blazed gratings.
Nobel Laureates Sir W. H. Bragg (1862–1942) and his son Sir W. L. Bragg (1890–
1971) discovered the well known Bragg diffraction condition (2Λ sin Θ = nλ), in 1913, for
the X-ray diffraction from the periodic crystal lattice. Diffraction can be considered as a
reflection of the incident X-ray beam from a series of lattice planes. In the Bragg condition,
Λis the periodicity of the atomic planes, Θis the angle of incidence, λis the wavelength
of X-rays, and nis the order of diffraction. For Θ = 90◦and n= 1, the Bragg diffraction
condition simplifies to 2Λ = λ. The periode Λof the refractive index modulation of an
optical fiber Bragg grating is set equal to half of the wavelength λof the light propagating
in the fiber. It is this phase matching condition between the grating planes and incident
light that results in coherent back reflection. Reflectivities approaching 100 % are possible
with grating bandwidth (∆λ) tailored from 0.1 nm to in excess of 100 nm.
Figure 3.1 shows the uniform fiber Bragg grating along with the incident, diffracted,
37
38 Chapter 3 Longitudinal Structure Characterization of FBG
Figure 3.1: Illustration of a uniform grating with constant amplitude of refractive index modulation and
grating period
and grating wavevectors. The Bragg condition is simply the requirement that satisfies both
energy and momentum conservation principles. The energy conservation principle (¯hωi=
¯hωr) requires that the frequency of the incident and the reflected light must be the same.
The momentum conservation principle requires that the incident wavevector, ki, plus the
grating wavevector, K, equal the wavevector of the scattered radiation, kr. This would
mean
ki+K=kr,(3.1)
where the grating wavevector, K, has a direction normal to the grating planes with a
magnitude2π/Λ. The diffracted wavevector isequalinmagnitude, but oppositeindirection
to the incident wavevector. Therefore, the momentum conservation principle
2(2πn0
λB
) = 2π
Λ,(3.2)
simplifies to first order Bragg diffraction condition
λB= 2n0Λ,(3.3)
where the Bragg wavelength, λB, is the free space center wavelength of the input light that
will be reflected back from the Bragg grating, and n0is the effective refractive index of the
fiber core at the free space center wavelength.
Figure 3.2 shows the grating structure and physical grating parameters that determines
the spectral response of the optical fiber Bragg gratings. The refractive index modulation
n(z)of the Bragg grating is written as
n(z) = n0+1
2∆npp(z) cos(2π
Λz+φ(z)),(3.4)
3.1 Fiber Bragg Gratings: An Introduction 39
Figure 3.2: Illustration of peak to peak refractive index modulation ∆npp and grating period Λ
where ∆npp(z)is the gratings peak to peak refractive index modulation amplitude (typical
values 10−5to 10−3), and φ(z)is the grating phase. Lis the grating length, and zis
the distance along the fiber longitudinal-axis. Using the “Coupled Mode Theory” of Lam
and Garside [53] that described the reflection properties of Bragg grating, reflectivity and
transmissivity of a uniform fiber Bragg grating with constant refractive index modulation
amplitude and period is given by the following expressions:
R(L, λ) = κ2sinh2(sL)
∆k2sinh2(sL) + s2cosh2(sL)
T(L, λ) = κ2
∆k2sinh2(sL) + s2cosh2(sL)
,(3.5)
where R(L, λ)and T(L, λ)is reflectivity and transmissivity as a function of grating length
Land the incident wavelength λ.κis the coupling coefficient, ∆k=k−π/λ is the
detuning wavevector, k= (2π/λ)n0is the propagation constant, and s2=κ2−∆k2.
Gratings local coupling coefficient, κ(z), is directly proportional to the peak refractive
index modulation amplitude, ∆n, and may be written as
κ(z) = π∆n(z)
λB
MP,(3.6)
where Mpis the fraction of the fiber mode power contained by the fibre core. On the basis
that the grating is uniformly written through the fibre core, Mpcan be approximated by
1−V−2, where Vis the normalized frequency of the fiber. V= (2π/λ)dpn2
co −n2
cl
where dis radius of the fiber core, nco and ncl are the core and cladding refractive indices,
respectively. The product κL gives the grating strength. At the Bragg gratings center
wavelength, λB, there is no wavevector detuning and ∆k= 0; therefore, the expressions
for peak reflection and peak transmission of the uniform fiber gratings simplifies to
R(L, λB) = tanh2κL
T(L, λB) = sech2κL .(3.7)
Depending on the application, fiber Bragg gratings with varying periods and as well as
depths of refractive index modulation have been proposed and fabricated. To what extent
40 Chapter 3 Longitudinal Structure Characterization of FBG
the specific fiber Bragg grating design and the actual fabricated device correspond to each
other has to be checked by the suitable characterization methods. So far very few studies
have been done in an area of reconstructing the longitudinal grating structure from the
experimental data. Here we would like to apply the concepts of optical network analysis
in order to efficiently characterize the fiber Bragg gratings and at the same time we would
like to demonstrate the versatility and functionality of the lithium niobate-based integrated
optical network analyzer.
3.2 Longitudinal Structure Characterization of FBG
Scalar characterization is the fastest and simplest method to characterizeoptical fiber Bragg
gratings for longitudinal structure characterization. To perform this, we need to correctly
measure either the frequency-dependent complex reflection or transmission coefficients of
the FBG in the frequency domain and then use the inverse Fourier transform to get the time
domain impulse response of the FBG. From the latter, the longitudinal grating structure
could be determined either by using the inverse scattering algorithms or by using layer-
peeling methods [54,55].
3.2.1 Measurement of Complex Reflection Coefficient
Optical frequency domain interferometry was used to measure the frequency-dependent
complex reflection coefficient ρ(f). The optical frequency is scanned symmetrically over
≈50 GHz optical bandwidth with the Bragg wavelength (λB= 1548.25 nm) being at the
scan center, of a given FBG under test. The DFB laser diode was used as a tuneable laser
source with a resolution of ≈25 MHz. Number of measurement points nwere equal to the
mth power of 2and mwas set to 11, i.e 2048 points.
Thermal noise ultimately represents the fundamental noise limit in fiber-optic interfero-
metric sensors which affects the signal-to-noise ratio (SNR) of the measurement system.
Thermal processes in the fiber modulate the refractive index of the fiber which again causes
phase modulation of the signals propagating in the fiber. There are two basic causes of
refractive index modulation, namely, temperature and density fluctuations. The former
dominate at low frequencies while later dominate at high frequencies with a cross-over
point of 1MHz. The frequency range of interest for most fiber sensor applications is be-
low 1MHz and therefore, in that frequency interval, it is the temperature fluctuations that
ultimately limits the signal-to-noise ratio of the system [56, 57]. In a MZI, the signals
in the two arms are exposed to two different random processes, and both arms contribute
to the thermally induced phase noise. Therefore, very fast measurements must be done
in order to combat the thermal drift in the fiber-optic interferometers. The measurements
are done using the programmable constant current source and the DFB laser diode which
is used as a narrowband tuneable laser source. Typical tuning slope is of the order of
≈10 pm/mA of pump current. This static wavelength shift also affects the laser power.
The optical power received by all photodiodes in the measurement setup, therefore, must
3.2 Longitudinal Structure Characterization of FBG 41
be normalized. This power normalization, for example, can be achieved by monitoring the
laser power as function of frequency by means of an additional 10 dB coupler connected
to the power monitor photodiode. The transmission coefficient τ(f)of the FBG is given
by pI4/I8. Two linear combinations of the three three-phase photocurrents yields the real
and imaginary parts of the frequency-dependent complex reflection coefficient ρ(f)of the
FBG under test as 2I1−I2−I3and I2−I3, respectively, as described in section 2.4.2.
3.2.2 Calculation of Impulse Response
The frequency-dependent complex reflection coefficient ρ(f)and the real transmission
coefficient τ(f)of FBG under test must satisfy the lossless relation
R+T≤1(3.8)
in frequency domain where the reflectivity and the transmissivity of the FBG is given by
R=|ρ(f)|2and T=|τ(f)|2, respectively.
The frequencies obtained from the reference interferometer were used to resample
(linearly interpolate) both magnitude and argument of the measured frequency-dependent
complex reflection coefficient, independently, so that, this frequency correction creates an
accurate and equidistant frequency grid for inverse Fourier transform which enables us to
calculate the impulse response,
h(t)⇐⇒ ρ(f),(3.9)
of the DUT. This inverse Fourier transform is calculated using the standard Inverse Fast
Fourier Transform algorithm (IFFT). For the inverse Fourier transform, the number of
Fourier steps N= 212 = 4096. Therefore, we shift ρ(f)so that it is at the center in the
frequency domain. The First and the last 1024 points are padded with 0’s, so that the total
number of Fourier steps Nequals 4096 together with ρ(f)sampled at 2048 points. The
frequency domain results were multiplied by a cos2window to suppress errors introduced
by the discontinuous borders. This scheme is implemented due to the fact that 62.5 GHz
(≈0.5nm at 1550 nm) is scanned in the frequency domain and we wanted to convert it to
125 GHz, so that, the corresponding time step becomes
∆t=1
N·∆f=1
125 GHz = 8 ps.(3.10)
This time step is equal to the reciprocal of the product of the number of Fourier steps N
and the corresponding frequency step ∆fused during the measurement of ρ(f), in the
frequency domain. Once the impulse response of the Bragg grating under test is known
accurately in the time domain, then the job of calculating the longitudinal grating structure
is straightforward.
42 Chapter 3 Longitudinal Structure Characterization of FBG
3.2.3 Calculation of Longitudinal Grating Structure
A second order differential equation governing two independent state variables one dimen-
sional (1-D) space coordinate, z, and time, t,
∂2E
∂z2=1
c2
∂2E
∂t2,(3.11)
generally, characterizes the propagation of electromagnetic waves, in the medium with
constant material parameters. The solutions of (3.11) are the waves having electric field E
propagating with speed calong the ±z-direction. Our aim is to know what happens when an
electromagnetic wave propagates in a medium (single mode fiber) with constant material
parameters (refractive index) and then the wave suddenly encounters an abrupt change,
or a discontinuity in the material parameters of the medium such as a periodic refractive
index change (fiber Bragg grating) as a function of z. This is an ideal case for inverse
problems. In inverse problems, the unknown properties of the medium are found out by
applying some input (say a delta function δ(t)) to the medium and measuring some output
(impulse response h(t)). A pair of first-order coupled differential equations, as derived in
the Appendix C,
(∂
∂z +no
c
∂
∂t)D(z, t) = κ∗(z)U(z, t)
(∂
∂z −no
c
∂
∂t)U(z, t) = κ(z)D(z, t)
,(3.12)
where Dis the electric field matrix of the forward (Down), and Uis the electric field matrix
ofthe backward(Up)propagatingwaves that governs the propagationofanelectromagnetic
waves in time tand 1-D space coordinate zfor such a medium. κ(z)in (3.12) is the
coupling coefficient between the forward and backward propagating waves. When the
system parameter such as refractive index do not vary with 1-D space coordinate z, then
κ(z)=0and the above two first-order coupled differential equations decouples. The
physical meaning of the decoupling of Dand Uis that the two waves do not interact.
When the refractive index changes either continuously or discontinuously, then the Dand
Uwaves do interact. These waves had the property that they do not interact unless there
is a change in the value of at least one of the material parameters of the medium. It is this
interaction that complicates the problem of transmission and reflection of electromagnetic
wavesfromthemediumwithperiodicallyvarying refractiveindex change suchastheBragg
gratings in fibers. Therefore we need to analyze the waves in two situations:
(1) as they travel along the region where material parameter is constant;
(2) as they cross the region where material parameter is discontinuous.
Inverse problem assumes that the systems impulse response U0, t =h(t)is known
in addition to the given Dirac impulse D0, t = 1δ(t). It is assumed that the output, U0, t
(impulse response) is caused by the input, D0, t (delta function) by the fundamental concept
of casualty. Principle of casualty states that the cause must precede the effect. Therefore,
it is possible to find the whole casual solutions Dµ, ν,Uµ, ν, and in addition, be able to
3.2 Longitudinal Structure Characterization of FBG 43
find κ(z). In order to calculate κ(z), we discretize (3.12) using a time step ∆tand a
corresponding position step ∆z= (c/no)∆t. A pair of discrete equations
Dµ, ν =τ−1
µ−1(k∗
µ−1Uµ−1, ν+1 +Dµ−1, ν−1)
Uµ, ν =τ−1
µ−1(Uµ−1, ν+1 +kµ−1Dµ−1, ν−1)(3.13)
are derived from the piecewise solutions of (3.12) under the assumption of locally constant
k=κ∆zas derived in Appendix D.
Figure 3.3: Wave propagation and solution sequence for inverse scattering algorithm
Figure 3.3 shows wave propagation and solution sequence given by dashed arrow in a
grid of time and space. Solution starts from the knowledge of incident unity matrix D0,0
(the only nonzero component of a discretized D0, t = 1δ(t)) and the measured impulse
response matrix U0, ν (discretized U0, t =h(t)). Waves Dµ, ν,Uµ, ν and as far as yet
unknown matrix κ(z)are calculated in a sequence given by the dashed arrow.
Figure 3.4: Calculation of Dµ, ν and Uµ, ν for one position step ∆Z
Mathematically speaking for calculation of km, all kµwith 0<µ<mmust be
known. This is automatically the case if one starts with m= 1. Each solution steps
then requires (3.13) to be solved successively for µ= 1 ···m,ν= 2m.
44 Chapter 3 Longitudinal Structure Characterization of FBG
Figure 3.4 shows the quantities required to calculate Dµ, ν and Uµ, ν for one position
step ∆Z. The transmission coefficient τµand the coupling coefficient kµin (3.13) is given
by
τµ=q1−|kµ|2
kµ=−Uµ, µ
Dµ, µ
.(3.14)
The refractive index modulation ∆n(z)which is directly proportional to the complex
coupling coefficient κ(z)of the FBG under test is
∆n(z) = λB
πκ(z) where κ(z) = kµ
∆z.(3.15)
Therefore peak to peak amplitude and phase of the refractive index modulation equals
∆npp(z) = 2 ·∆n(z)
arg[∆npp(z)] = arg[kµ]−π
2
.(3.16)
3.2.4 Scalar Measurements
Fiber Bragg grating of reflectivity >95% and ∆λ≈0.2nm at λB= 1548.25 nm, is used
as an object for the longitudinal structure characterization, in forward as well as backward
direction. Temperature of the fiber Bragg grating is kept constant during experiment by
mounting it on to an aluminium plate whose temperature is sensed using a thermistor and
kept constant using a set of Peltier elements and a simple proportional integral temperature
controller. For scalar measurement, the manual polarization controllers (in Figure 2.1)
are adjusted for the maximum interference contrast assuming that only co-polarized waves
interferes in the output 3x3 fiber coupler of the measurement interferometer. As it has been
mentioned in section 3.2.1, the optical frequency is symmetrically scanned to measure
the complex reflection coefficient ρ(f)at 2048 points. The inverse Fourier transform and
inverse scattering algorithm mentioned in section 3.2.2 and 3.2.3 are used to determine the
longitudinal grating structure. Results for the forward as well as backward measurement
are summarized below.
Forward Measurement
Figure 3.5(a) shows the magnitude of the measured frequency-dependent complex reflec-
tion coefficient |ρ(f)|while Figure 3.5(b) shows the magnitude of the scalar transmission
coefficient |τ(f)|of the FBG in forward direction.
Figure 3.6(a) shows the impulse response h(t)while Figure 3.6(b) shows the longitu-
dinal grating structure ∆n(z)with a corresponding grating burst for this FBG in forward
direction.
3.2 Longitudinal Structure Characterization of FBG 45
193.75 193.76 193.77 193.78 193.79
0.0
0.2
0.4
0.6
0.8
1.0
|
ρ
(f)|
Frequency [ THz ]
(a)
193.75 193.76 193.77 193.78 193.79
0.2
0.4
0.6
0.8
1.0
|
τ
(f)|
Frequency [ THz ]
(b)
Figure 3.5: (a) Complex reflection coefficient |ρ(f)|and (b) scalar transmission coefficient |τ(f)|of this
FBG in forward direction
0 100 200 300 400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
h(t) [ps]-1
Time [ps]
(a)
0246810
0.0
0.5
1.0
1.5
2.0
2.5
∆
npp [a.u.] (x10 -3)
z [cm]
(b)
Figure 3.6: (a) Impulse response h(t)and (b) longitudinal grating structure ∆n(z)of this FBG in forward
direction
46 Chapter 3 Longitudinal Structure Characterization of FBG
193.75 193.76 193.77 193.78 193.79
0.0
0.2
0.4
0.6
0.8
1.0
|
ρ
(f)|
Frequency [ THz ]
(a)
193.75 193.76 193.77 193.78 193.79
0.2
0.4
0.6
0.8
1.0
|
τ
(f)|
Frequency [ THz ]
(b)
Figure 3.7: (a) Complex reflection coefficient |ρ(f)|and (b) scalar transmission coefficient |τ(f)|of this
FBG in backward direction
0 100 200 300 400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
h(t) [ps]-1
Time [ps]
(a)
0246810
0.0
0.5
1.0
1.5
2.0
2.5
∆
npp [a.u.] (x10 -3)
z [cm]
(b)
Figure 3.8: (a) Impulse response h(t)and (b) longitudinal grating structure ∆n(z)of this FBG in backward
direction
3.3 Vectorial Structure Characterization of FBG 47
Backward Measurement
Figure 3.7(a) shows the magnitude of the measured frequency-dependent complex reflec-
tion coefficient |ρ(f)|while Figure 3.7(b) shows the magnitude of the scalar transmission
coefficient |τ(f)|of the FBG in backward direction.
Figure 3.8(a) shows the impulse response h(t)while Figure 3.8(b) shows the longitu-
dinal grating structure ∆n(z)with a corresponding grating burst for this FBG in backward
direction.
Conclusion
This validates the measurement principle and grating structure since “forward” and “back-
ward” measurement results agree very well. This grating is a very strong one and has a
single burst in its longitudinal grating structure.
3.3 Vectorial Structure Characterization of FBG
Grating behavior is dependent on polarization, at least when the period of a grating and
wavelength of the incident light are of the same order. Consequently the best scalar theories
are often unable to predict the efficiencies of modern gratings since they have not taken
the vectorial character of light into account. During the last decade, new techniques have
been derived from Maxwell’s equations [58]. One of the method that is used here is briefly
outlined, then studied in grater depth and applied for the vectorial structure characterization
of fiber sensor grating.
3.3.1 Concept of Birefringence and Dichroism
The propagation of monochromatic plane electromagnetic waves in an infinite isotropic
dielectric media is characterized by their refractive index n. For a monochromatic plane
wave with an angular frequency ω, the wave vector of the plane wave is k= (nω/c)b
u,
where b
uis a unit vector. The plane wave is related to a notion of light ray whose direction
is given by the direction of the Poynting vector S=E×H. In isotropic media the vectors
kand Sare parallel no matter what the direction of propagation is. In general, this does
not hold true for anisotropic media.
Generally, optical properties of an anisotropic dielectric medium are determined by its
refractive index, as seen by a plane electromagnetic wave passing through it and depend on
its direction of propagation b
u. In fact, it can be shown, for a given direction of propagation,
that two refractive indices may co-exist. The latter are associated with electromagnetic
waves having states of polarization that can propagate without any alteration. These partic-
ular states of polarization are called eigenstates for the considered direction of propagation.
Usually, two different kinds of optical anisotropy can be considered:
48 Chapter 3 Longitudinal Structure Characterization of FBG
–The linear anisotropy, whose eigenstates of propagation are the linear states
of polarization.
–The circular anisotropy, whose eigenstates of propagation are the circular
states of polarization.
–These two kinds of anisotropy can coexist in the media and in such a case, the
eigenstates of polarization are in general elliptical.
Whenever an electromagnetic wave passes through a material medium, it induces an
electric polarization P1which is added to the vacuum polarization P0, defined by: P0=
ε0E. The total polarization P=P0+P1is represented by a vectorial function P(E),
where Eis the exciting field of the electromagnetic wave. Thus, in the approximation of
low electric field intensities, the induced polarization
P1=ε0[χ]E(3.17)
is a linear function of Ewhere [χ]is the first-order electric susceptibility tensor. Since the
electric displacement vector Dis such that D=ε0[ε]E+P1we get:
D=ε0[ε]E(3.18)
where [ε]is the relative permittivity tensor and is related to the electric susceptibility tensor
[χ]through the relation: [ε] = [I]+[χ], where Iis the identity matrix.
As the electric permittivity of materials is closely related to the notion of the refractive
index (ε=n2), the properties of the dielectric tensor [ε]are immediately translated to
the optical properties of the materials. With the exception of some peculiar cases, the
considered media are non-magnetic, i.e. their magnetic permeability constant µis equal to
µ0. The relative permittivity tensor [ε]is symmetrical for non-absorbent materials, i.e.
εij =εji.(3.19)
It can also be shown that, in the case where the terms of the tensor [ε]are complex numbers,
without the medium being absorbent, the previous relation would become:
εij =ε∗
ji (3.20)
which more generally shows that the permittivity tensor is Hermitian. Only the media
possessing circular anisotropy exhibit this special property.
The case of absorbent media is non-negligible complication because they exhibit not
only the linear anisotropy but also an absorption related anisotropy. The latter phenomenon
is known as dichroism.
The electric field associated with a monochromatic plane electromagnetic waves with
an angular frequency ω= 2πν and a wavevector k, propagating through a non-absorbent
medium exhibiting a linear anisotropy are characterized by the dielectric tensor [ε]which
can be expressed using the complex notation:
E=E0exp[−i(ωt −kr)] (3.21)
3.3 Vectorial Structure Characterization of FBG 49
The other components of the electromagnetic field, D,B, and H, naturally exhibit the
same spatio-temporal dependency. Without any source term, the solution of Maxwell’s
equations in terms of field Dby eliminating the field Hand by introducing a unit vector b
u
in the direction of propagation of the wave phase, i.e. k=kb
u, we obtain:
D=k2
ω2µ0
[E−(b
u.E)b
u](3.22)
It is useful to note, in more general case, that the vector Dis not parallel to vector Eany
more. Therefore, the scalar product b
u.Eis not null and the electric field is not transverse
as was the case with an isotropic medium. Only the fields D,H(or B) are transverse, i.e.
perpendicular to the wave vector k. The electric field Eusually is not.
The solutions to Maxwell’s equations as plane waves of wave vector k=kb
ufor which
the components of the vector b
uare [α, β, γ]are given by the Fresnel’s equation:
n2
xα2
n2−n2
x
+n2
yβ2
n2−n2
y
+n2
zγ2
n2−n2
z
= 0 (3.23)
Fresnel’s equation leads to concept of birefringence and to following conclusions.
–For a given direction of propagation b
u, there are two solutions for n2, lets say ±n1and
±n2. The ±sign corresponds to the two possible directions of propagation along the unit
vector b
u. The positive values n1and n2are the solutions of Fresnel’s equation and existence
of these two values and therefore, their difference (∆n=n2−n1) gives the birefringence.
–Each of the two values of refractive index is related to a plane wave characterized by
its electric displacement vector Dand its phase velocity v. Hence, for a given direction
propagation b
u, there are two plane waves characterized by D1and D2which are orthog-
onal to vector b
u. Moreover, they travel with phase velocities v1=c/n1and v2=c/n2,
respectively.
3.3.2 Birefringence and Dichroism in Optical FBG
As it has been mentioned in the previous section that any birefringent medium can be
characterized by the dielectric tensor εr. The electric displacement vector Dis not parallel
to the electric field vector Eand this relation, in general, is written as
D=ε0εrE=ε0
εr,xx εr,xy εr,xz
εr,yx εr,yy εr,yz
εr,zx εr,zy εr,zz
E.(3.24)
The vector electromagnetic wave equation now has the dielectric tensor εrinstead of rela-
tive permittivity εrand is written as
∇E=µ0ε0εr∂
∂t2
E.(3.25)
50 Chapter 3 Longitudinal Structure Characterization of FBG
Sincethequantitiesεt=εr(x, y)andhencethetransversefieldcomponentsEt= (Ex,Ey)
are independent of z, one can write
∂
∂z2
E=µ0ε0εr∂
∂t2
E.(3.26)
For source free medium ∇ · D= 0 and moreover, there is no Dzcomponent. It can be
shown that it is indeed zero as follows:
Dz=ε0εr,zx εr,zy εr,zz
Ex
Ey
Ez
= 0 ⇔Ez=−εr,zxEx+εr,zyEy
εr,zz
.(3.27)
Therefore, the three dimensional vector wave equation reduces to two dimensional one with
only transverse components Et:
∂
∂z2Ex
Ey=µ0ε0εr,xx εr,xy εr,xz
εr,yx εr,yy εr,yz ∂
∂t2
Ex
Ey
Ez
(3.28-a)
with
Ex
Ey
Ez
=
1 0
0 1
−εr,zx
εr,zz −εr,zy
εr,zz
Ex
Ey(3.28-b)
Inserting 3.28-b into 3.28-a, we get
∂
∂z2
Ex
Ey=µ0ε0
εr,xx εr,xy εr,xz
εr,yx εr,yy εr,yz
1 0
0 1
−εr,zx
εr,zz −εr,zy
εr,zz
∂
∂t2
Ex
Ey.(3.29)
The matrix multiplication in (3.29) gives
∂
∂z2
Ex
Ey=µ0ε0
"εr,xx −εr,zx
εr,zz εr,xz εr,xy −εr,zy
εr,zz εr,xy
εr,yx −εr,zx
εr,zz εr,yz εr,yy −εr,zy
εr,zz εr,yy #∂
∂t2
Ex
Ey
=µ0ε0
εxx εxy
εyx εyy∂
∂t2
Ex
Ey.(3.30)
The refractive index matrix nis directly related to the dielectric tensor ε(n=√ε). This
refractive index matrix nwill have to eigen values, say ±n1and ±n2. If one considers only
positive eigen values, say n1and n2along the positive x- and y- direction, respectively, then
their difference with respect to the average refractive index ngives the birefringence
∆n=n2−n1
nand n=n1+n2
2.(3.31)
3.3 Vectorial Structure Characterization of FBG 51
Due to birefringence, Bragg grating with the given refractive index modulation (∆npp)
and the grating period Λwill always reflect two discrete wavelengths say, λB 1 and λB 2,
corresponding to these two eigenvalues, and are equal to
λB 1,2= 2n1,2Λ.(3.32)
Therefore, there will be two reflectivities ρ1and ρ2and two eigenvectors, say E1and E2,
corresponding to these two eigenvalues of the refractive index matrix n. The resultant
electric field of the output wave reflected by the grating depends on the polarization or the
electric field of the input wave that is parallel to one of the eigenvectors. This resultant
electric field could then be written as
Eout =ρ1Ein,1E1+ρ2Ein,2E2.(3.33)
The reflection coefficient is no more a scalar quantity. It represents a 2x2 Jones reflectance
matrix:
Eout =ρEin =ρ11 ρ12
ρ21 ρ22 Ein.(3.34)
3.3.3 Polarization Mode Coupling
For the medium with no birefringence, the dielectric tensor obeys the relation ε=ε†. With
eigenvalue decomposition the dielectric tensor εis decomposed into
ε=Eεε10
0ε2E+
ε(3.35)
with ε1,2∈Rand orthogonal eigenvectors Eε= [E1,E2]with scalar product equal to the
Kronecker delta (Ei·E∗
j=δij).
Thedielectrictensorforanisotropicopticalgratingwithaperiodicmodulationofrefractive
index is written as
ε(z) = ε0εb+εgekgz+ε+
ge−kgz.(3.36)
The first term εbcomes from the static birefringence while the second and third term comes
from the grating structure that couples the forward and the backward propagating modes.
Since ∆nis directly proportional to ∆ε; the above equation directly transforms into
∆n(z) = ∆nb+ ∆nkekgz+ ∆n+
ke−kgz,(3.37)
where the birefringence and refractive index modulation matrices are
∆nb=∆n11 ∆n12
∆n∗
12 −∆n11
∆nκ=1
4eφk∆npp11 ∆npp12
∆npp21 ∆npp22.(3.38)
52 Chapter 3 Longitudinal Structure Characterization of FBG
The electric field of the forward and backward propagating waves is given by
Ef(z, t) = D(z)eωt−kz
Eb(z, t) = U(z)eωt+kz with k=ω
c0
n(z),(3.39)
while there superposition gives the total field as
E(z, t) = D(z)eωt−kz +U(z)eωt+kz.(3.40)
The electromagnetic wave equation with this electric field and refractive index modulation
is written as ∂
∂z2
E(z, t) = n(z)
c02∂
∂t2
E(z, t).(3.41)
Simplification of this equation similar to the one derived in Appendix C yields
∂
∂zD(z) = −β(z)D(z) + κ∗(z)e(2k−kg)zU(z)
∂
∂zU(z) = β(z)U(z) + κ(z)e−(2k−kg)zD(z)
,(3.42)
with
β(z) = k
n0
∆nb(z)
κ(z) = k
n0
∆nk(z)
.(3.43)
3.3.4 Measurement of Complex Reflectance Jones Matrix
Anybirefringentordichroic optical device likeFBGhasthe frequency-dependentreflection
coefficient ρ(f)which depends on the input polarization and the reference polarization
which is analyzed at its output. An interferometric measurement determines a complex
reflection coefficient
ρ(f) = ρEr,Em(f)(3.44)
where Eris the Jones vectors in the reference branch while Emis the Jones vector in the
measurement branch where the device under test is inserted. The interference contrast is
at its maximum when the polarizations in the reference and measurement branch of the
interferometer are identical.
In order to determine the complete 2x2 Jones reflectance matrix ρ, the four combina-
tions of ρr,mhave to be measured and these will be the elements of ρ. The above scalar
equation for the complex reflection coefficient is modified to
ρ(f) = E+
rρr,m(f)Em(3.45)
3.3 Vectorial Structure Characterization of FBG 53
where Erand Emare the two pairs of Jones vectors. Even if Jones vector Erand Emare
unknown the knowledge of elements of the transformed ρmatrix is generally sufficient to
reveal the birefringence and dichroism because the transformed ρmatrix is no more than
the original ρmatrix which is pre- and post- multiplied by the unitary matrices [Er|| ,Er⊥]+
and [Em|| ,Em⊥], respectively. It is indeed difficult to generate such orthogonal pairs of
polarization by using the calibrated polarization transformers based on LiNbO3due to DC
drift. The polarization orthogonalization scheme derived by Sandel [16] which makes use
of Poincar´
e’s sphere based tetrahedrons is used here to measure the complete reflectance
Jones matrix of the device under test which is complex.
To use orthogonalization scheme, it is necessary to generate two sets of four arbitrary
polarizations which form a tetrahedron with nonzero volume on the Poincar´
e sphere: one
for reference- and another for measurement- branch. These polarizations are generated
using the calibrated polarization transformers that are based on X-cut, Z-propagation
LiNbO3(see Figure 2.1).
The transformed elements of ρr,mcorresponding to orthogonal pairs of Erand Emcan
be deduced from 16 reflectances
ρri,mj(f) = ρEri,Emj=E+
riρi, j(f)Emji, j ∈ {0···3}(3.46)
measured with four arbitrary Eriand four arbitrary Emj. These two sets of four arbitrary
polarizations enable us to derive two sets of orthogonal polarization pairs:
Er|| ,Er⊥with Er|| ·Er⊥= 0
Em|| ,Em⊥with Em|| ·Em⊥= 0.(3.47)
With this orthogonal pairs of polarization, the measured 2x2 Jones reflectance matrix now
becomes
"ρEr|| ,Em|| (f)ρEr|| ,Em⊥(f)
ρEr⊥,Em|| (f)ρEr⊥,Em⊥(f)#=Er||
Er⊥·ρ·Em|| Em⊥=RL·ρ·RR=e
ρ,(3.48)
where the subscripts || and ⊥refers to orthogonal pairs of polarizations, both at the input
of the device under test which is inserted in the measurement branch and in the reference
branch. Polarization transformers which generates these waves must operate reproducibly
but calibration is not needed. The rotation matrices RLand RRare in fact unitary. This
frequency-dependent transformed ρmatrix enables us to calculate the impulse response
and hence vectorial grating structure.
3.3.5 Polarization Orthogonalization
For some special cases which are excluded here, every polarization Eriand Emjin section
3.3.4 can be expressed by a linear combination of two others (see Figure 3.9), for example,
Em0=ϑ1Em1+ϑ2Em2=ξ1Em1+ξ3Em3,(3.49)
54 Chapter 3 Longitudinal Structure Characterization of FBG
where ϑ1, ϑ2, ξ1, ξ3are scalers. For fixed i, the reflectances ρi, j can also be expressed
using the same scalers, for example,
ρi, 0=ϑ1ρi, 1+ϑ2ρi, 2=ξ1ρi, 1+ξ3ρi, 3.(3.50)
This is obvious because the same polarizations are used during the measurement of ρi, j.
Figure 3.9: Schematic of a Poincar´
e sphere-based tetrahedron
One possibility to determined the coefficients ϑ1, ϑ2, ξ1, and ξ3from the measured
frequency-dependent reflection coefficients ρi, j(f)is to effectively use (3.50). In fact, this
can be easily done by integrating the matrix product of ρ+
i, j(f)and ρi, j(f)where ’+’ sign
indicates the Hermitian conjugate over the whole frequency range, i.e. νgoing from 0 to
N.
H=
h00 h01 h02 h03
h10 h11 h12 h13
h20 h21 h22 h23
h30 h31 h32 h33
=ZN
ν=0
ρ+
i, j(fν)·ρi, j(fν)df (3.51)
where each element of His given by
hi, j =
3
X
k=0
N
X
ν=0
ρ∗
i,k(fν)ρj,k(fν)df. (3.52)
For any Hermitian matrix, say H, the eigenvalues of Hare real, His diagonalizable,
eigenvectors corresponding to distinct and nonzero eigenvalues are orthogonal, and the
matrix Hpossesses a complete orthonormal set of eigenvectors. Eigenvalue decomposition
of this Hmatrix yields four eigenvalues λj(where j= 0 ···3) and the corresponding four
eigenvectors χj(where j= 0 ···3). The two out of four eigenvalues should come out
to be identically zero because two out of four polarizations which are used during the
3.3 Vectorial Structure Characterization of FBG 55
measurement of ρi, j are linearly dependent on two other polarizations and hence two out
of four columns and rows of ρi, j are also linearly dependent on two other columns and
rows of ρi, j. In practice, the two out of four eigenvalues are not identically zero but are
rather close to zero due to measurement errors and polarization drift. It is indeed possible
to select two eigenvectors corresponding to two eigenvalues which are close to zero, say
for example, λ1and λ2, because these two eigenvectors are linearly dependent on each
other. One linear combination of these two eigenvectors yields ϑ1and ϑ2when first and
last element is set to -1 and 0, respectively, while their second linear combination yields ξ1
and ξ3when first and third element is set to -1 and 0, respectively. Therefore, one can write
−1
ϑ1
ϑ2
0
=α1χλ1+α2χλ2and
−1
ξ1
0
ξ3
=β1χλ1+β3χλ3,(3.53)
where α1,α2,β1, and β3are two set of constants. Every Emjcan be further expressed by
the desired orthogonal pair Em|| and Em⊥as
Emj=CjEm|| +DjEm⊥(|Cj|2+|Dj|2= 1, j = 1 ···3).(3.54)
For j= 0, let Em0=Em|| by definition. There are nine degrees of freedom in the choice
of Cjand Dj. The eight real equations in (3.49) balance them except for the phase of one
of the component of Em⊥, which can be freely chosen. For example, D1can be set to a
real positive number. For Em|| and Em⊥polarization components, the insertion of (3.54)
into (3.49) results in
1 = ϑ1C1+ϑ2C21 = ξ1C1+ξ3C3(3.55)
and
0 = ϑ1D1+ϑ2D20 = ξ1D1+ξ3D3(3.56)
respectively. This allows C2,C3,D2, and D3to be expressed in terms of C1and D1as
C2=1−ϑ1
ϑ2
C1C3=1−ξ1
ξ3
C1,(3.57)
D2=−ϑ1
ϑ2
D1D3=−ξ1
ξ3
D1.(3.58)
The expressions (|Cj|2+|Dj|2= 1, j = 1 ···3) with (3.57) and (3.58) inserted for j= 2
and 3lead to a set of two equations linear in Re(C1)and Im(C1)whose solution is
Re(C1)
Im(C1)=1
2Re(ϑ1)−Im(ϑ1)
Re(ξ1)−Im(ξ1)−11−|ϑ2|2+|ϑ1|2
1−|ξ3|2+|ξ1|2.(3.59)
The positive real scalar D1can be derived from the complex C1as
D1=p1−|C1|2.(3.60)
56 Chapter 3 Longitudinal Structure Characterization of FBG
In principle, the scalers C1and D1are sufficient to calculate Em⊥from Em0and Em1and
corresponding reflection factors:
Em⊥=Emj−C1Em0
D1
, ρri,m⊥=ρri,mj−C1ρri,m0
D1
(i= 0 ···3, j = 1).(3.61)
In practice, there are always some random errors in every measured ρri,mj. This lead errors
in ρri,m⊥which are enhanced by the factor 1/D1. Since C2,C3,D2, and D3can be easily
derived from C1and D1using (3.57) and (3.58). The index j= 1 in (3.61), may be replaced
by kgoing from 2to 3or an average over all j= 1 ···3is taken which is in fact a better
choice. The weighted averages of |Dj|2results in
Em⊥=P3
j=1 |Dj|2Emj−C1Em0
D1
P3
j=1 |Dj|2
ρri,m⊥=P3
j=1 |Dj|2ρri,mj−C1ρri,m0
D1
P3
j=1 |Dj|2
(3.62)
with minimum errors for a given set of Emjfor j= 0 ···3. An analogous process
would orthogonalize the reference polarization and a (transformed) 2x2 Jones reflectance
matrix would result (3.34). This polarization orthogonalization scheme was proposed and
successfully used by D. Sandel [16].
3.3.6 Calculation of Impulse Response Matrix
For vectorial grating structure characterization, the matrix impulse response
h(t) = h11(t)h12(t)
h21(t)h22(t)⇔ρ(f) = ρ11(f)ρ12(f)
ρ21(f)ρ22(f)(3.63)
is calculated by inverse Fourier transform of each of the reflectances
hi, j(t) = F−1(ρi, j(f)).(3.64)
3.3.7 Calculation of Vectorial Grating Structure
Thedifference equations which are discretized using the position step ∆zand corresponding
time step ∆t= (n0/c)∆zare derived in
Dµ, ν =τ−1
µ−1(−b−1
µ−1k+
µ−1Uµ−1, ν+1 +bµ−1Dµ−1, ν−1)
Uµ, ν =τ−1
µ−1(b−1
µ−1Uµ−1, ν+1 +bµ−1kµ−1Dµ−1, ν−1)(3.65)
where Dis the electric field matrix of the forward- and Uthe electric field matrix of the
backward-propagating waves from piecewise solutions of
(∂
∂z +no
c
∂
∂t)D(z, t) = −β(z)D(z, t) + κ∗(z)U(z, t)
(∂
∂z −no
c
∂
∂t)U(z, t) = β(z)U(z, t) + κ(z)D(z, t)
(3.66)
3.3 Vectorial Structure Characterization of FBG 57
under the assumptions of locally constant κand βmatrices which are given here in the
diagonalized form
k=κ∆z=Ekk10
0k2E+
k
b=exp(−β∆z) = Ebb0
0b∗E+
b
,(3.67)
represents the grating coupling coefficient and birefringence, respectively, for one position
step ∆z. Transmission matrix must satisfy the lossless relation
ττ ++kk+=I.(3.68)
This matrix is Hermitian and has same eigenvectors as that of k. In the calculation of
τ=√I−kk+=Ekt10
0t2E+
k(t1, t2>0) (3.69)
positive signs must be chosen for its real eigenvalues t1,t2.
Figure 3.3 shows the wave propagation and solution sequence in a grid of time, t, and
one-dimensional space coordinate z. Solution starts from the knowledge of incident the
unity matrix D0,0(the only nonzero component of a discretized D=Iδt) and the measured
impulse response matrix U0,ν (discretized U0,t=h(t)). Waves Dµ,ν and Uµ,ν, and as far as
yet unknowns, kand bare calculated in a sequence given by dashed arrow. For calculation
of kmand bm, all kµand bµwith 0<µ<mmust be known. This is automatically the
case if one starts with m= 1. Each solution step requires the difference equations to be
successively solved for µ= 1 ···mand ν= 2m−µ.Dm,m and Um,m are obtained as an
intermediate result and the matrix quotient
Q=−Um,mD−1
m,m =bmkmbm(3.70)
has to be decomposed in such way that the first product
Q+Q= (bkb)+bkb =b+EkΛ+
kΛkE+
kb(3.71)
delivers |k1|,|k2|as the square root of its eigenvalues. The missing argument is
arg(k1) = arg(k2) = 1
2arg(det(k)) or 1
2arg(det(k)) + π. (3.72)
Second eigenvector matrix (b+Ek)of (3.71) allows for the calculation of matrix
(bEk) = Q(b+Ek)Λ−1
k.(3.73)
Third, now, we can obtain
b=q(bEk)(E+
kb)(3.74)
58 Chapter 3 Longitudinal Structure Characterization of FBG
which successively leads to
Ek=b+(bEk)
k=EkΛkE+
k
.(3.75)
Expecting a small retardation due to bwithin ∆z, the ambiguities of (3.75) and (??) can
be settled so that the argument of eigenvalues of bare closest to zero. Finally, locally
varying κ=k/∆zand β=ln(b)/∆zare calculated to yield the gratings peak-to-peak
refractive index modulation amplitudes , grating phase, and refractive index differences
(birefringence).
3.3.8 Vectorial Measurements
Fiber Bragg grating of reflectivity >95% and ∆λ≈0.2nm at λB= 1548.25 nm is used
as an object for the vectorial structure characterization in both forward as well as back-
ward direction. During the measurements, the temperature of the fiber Bragg grating’s is
kept constant by mounting it on to a aluminium plate whose temperature is sensed using a
thermistor and kept constant using a Peltier element and a simple proportional integral tem-
perature controller. Initially, for vectorial measurement, manual polarization controllers (in
Figure 2.1) are adjusted for the maximum interference contrast assuming that only copo-
larized waves interfere in the output 3x3 fiber coupler of the measurement interferometer.
For vectorial structure characterization, the broadband polarization transformers which are
on the lithium niobate-based integrated optical network analyzer circuit are initially used to
generate the two sets of four arbitrary polarizations for the implementation of orthogonal-
ization scheme. Number of trials has been made to implement this scheme with available
DFB laser. Main problem in implementing this scheme is that DFB laser was never sta-
ble during the scan and from scan to scan. Therefore, these polarization transformers are
calibrated using the commercial rotating quarter waveplate polarimeter to generate the pair
of orthogonal polarizations for vectorial measurement. This scheme has worked positively.
The results of vectorial structure characterization are summarized below. Scattering pa-
rameters, impulse response matrix elements and derived grating structures were essentially
identical when like measurements were performed from both sides of the same grating.
Forward Measurement
Figure 3.10(a) shows the magnitude of the elements of the measured frequency-dependent
complex reflectance matrix |ρ(f)|while Figure 3.10(b) shows the magnitude of the scalar
transmission coefficient |τ(f)|of this FBG in forward direction.
Figure 3.11(a) shows the impulse response matrix elements h(t)while Figure 3.11(b)
shows the longitudinal grating structure ∆n(z)with the corresponding grating burst for
this FBG in forward direction.
3.3 Vectorial Structure Characterization of FBG 59
193.75 193.76 193.77 193.78 193.79
0.0
0.2
0.4
0.6
0.8
1.0
ρ
11
ρ
12
ρ
21
ρ
22
|
ρ
(f)|
Frequency [ THz ]
(a)
193.75 193.76 193.77 193.78 193.79
0.2
0.4
0.6
0.8
1.0
|
τ
(f)|
Frequency [ THz ]
(b)
Figure 3.10: (a) 2x2 Jones reflectance matrix elements |ρ(f)|and (b) scalar transmission coefficient |τ(f)|
of FBG in forward direction
0 100 200 300 400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
h11(t)
h12(t)
h21(t)
h22(t)
h(t) [ps] -1
Time [ps]
(a)
0246810
0.0
0.5
1.0
1.5
2.0
2.5
J11
J12
J21
J22
∆
npp [a.u.] (x10 -3)
z [cm]
(b)
Figure 3.11: (a) Impulse response matrix elements h(t)and (b) vectorial grating structure ∆n(z)of FBG in
forward direction
60 Chapter 3 Longitudinal Structure Characterization of FBG
193.75 193.76 193.77 193.78 193.79
0.0
0.2
0.4
0.6
0.8
1.0
ρ
11
ρ
12
ρ
21
ρ
22
|
ρ
(f)|
Frequency [THz]
(a)
193.75 193.76 193.77 193.78 193.79
0.2
0.4
0.6
0.8
1.0
|
τ
(f)|
Frequency [ THz ]
(b)
Figure 3.12: (a) 2x2 Jones reflectance matrix elements |ρ(f)|and (b) scalar transmission coefficient |τ(f)|
of FBG in backward direction
0 100 200 300 400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
h11(t)
h12(t)
h21(t)
h22(t)
h(t) [ps] -1
Time [ps]
(a)
0246810
0.0
0.5
1.0
1.5
2.0
2.5
J11
J12
J21
J22
∆
npp [a.u.] (x10 -3)
z [cm]
(b)
Figure 3.13: (a) Impulse response matrix elements h(t)and (b) vectorial grating structure ∆n(z)of FBG in
backward direction
3.4 Conclusion 61
Backward Measurement
Figure 3.12(a) shows the magnitude of the elements of the measured frequency-dependent
complex reflectance matrix |ρ(f)|while Figure 3.12(b) shows magnitude of the scalar
transmission coefficient |τ(f)|of the FBG in backward direction.
Figure 3.13(a) shows the impulse response matrix elements h(t)while Figure 3.13(b)
shows the longitudinal grating structure ∆n(z)with the corresponding grating burst for
this FBG in backward direction.
Conclusion
This validates the measurement principle and the vectorial grating structure since “for-
ward” and “backward” results agree very well. This grating is a very strong one and the
calculated birefringence (not shown) of this commercial fiber Bragg grating was negligi-
ble. This commercial Bragg grating was also free from dichroism. The vectorial structure
characterization results are summarized in [59].
3.4 Conclusion
The given FBG is evaluated for both scalar and vectorial structure characterization using
the in-house developed integrated optical network analyzer. This would demonstrate the
functionality and versatility of integrated optical circuits in lithium niobate and their direct
application to optical instrumentation and communication.
62 Chapter 3 Longitudinal Structure Characterization of FBG
Chapter 4
Integrated Optical PMD Compensator
4.1 Design Issues for Integrated Optical PMDC
Integrated Optical PMD compensator in X-cut, Y-propagation lithium niobate is designed,
and characterized for different differential group delay profiles. It is based on cascaded
TE↔TM mode converters with endlessly adjustable coupling phases. The natural birefrin-
gence (0.22 ps/mm) of this birefringent crystal cut can be used to compensate the DGD at
the same time. Operational principle of this type of PMD compensator is described in the
next section.
4.1.1 Operational Principle
The operational principle is based on the spatially weighted coupling between two waves
with different propagation constants [20]. The phase difference between one mode and
the coupled mode therefore depends on the position where coupling occurs and is periodic
with the beat length Λ = λ/∆n. The TE–TM refractive index difference in a Ti-indiffused
waveguide in X-cut, Y-propagation LiNbO3is ∆n= 0.0679 at a free space wavelength
λ= 1550 nm, thereby giving Λ = 22 µm. The interdigital electrodes are needed for
phase matching with a period equal to one optical beat length, Λ. The widths and gaps
are equal to 1/4of the optical beat length, and subsequent electrode pairs are additionally
spaced by 3/4of the optical beat length which allows mode coupling to be adjusted in
both quadratures endlessly via the electrooptic coefficient r51. The coupling coefficient κ
is given by
κ∼
=ˆ
Γ(π/2)n3r51(V/G)λ−1,(4.1)
where ˆ
Γis a weighted field overlap integral factor as defined later and n= 2.1785 is the
average refraction index of the waveguide. r51 = 28·10−12 m/V is the relevant electrooptic
coefficient and Vis the interelectrode voltage.
Figure 4.1 shows the schematic of such PMD compensator in the X-cut, Y-propagation
LiNbO3. Voltage V1nacts on one set of comb electrodes and performs mode conversion in
phase. Voltage V2nacts on another set of comb electrodes which are translated by 3/4of the
63
64 Chapter 4 Integrated Optical PMD Compensator
Figure 4.1: Schematic of PMD compensator on X-cut, Y-propagation LiNbO3
optical beat length with respect to the first and performs mode conversion in quadrature.
The resulting complex coupling coefficient is proportional to V1n+V2nfor nin-phase
and quadrature electrode pairs. The need for 2quadratures at least doubles the necessary
chip length. The longitudinal electrode cross-section as well as local field overlap integral
factors Γ(y)for one quadrature of 2–phase TE–TM mode converter electrodes that are used
in [22] are shown in Figure 4.2.
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
Γ
y/
Λ
Figure 4.2: 2–phase electrodes with corresponding voltages and local field overlap integral factors vs nor-
malized longitudinal coordinates
4.1.2 Two vs. Three Phases
As has been mentioned in [20], this 2–phase implementation is not the only possible choice.
If isolated electrode crossings are available, 3–phase electrodes can be used with electrode
widths and gaps equal to Λ/6. In-phase and quadrature mode conversion can be produced
by the linear combinations of the “cosine” and “sine” cases as shown in Figure 4.3 and
4.1 Design Issues for Integrated Optical PMDC 65
Figure 4.4. Even and odd voltage distributions are applied by choosing electrode voltages
V0f(ˆy)where V0serves as a reference voltage and ˆyis the longitudinal position of the
center of an electrode. Here f(y) = cos(2πy/Λ) and f(y) = sin(2πy/Λ) are structure
functions needed for cosine and sine cases, respectively.
The Point Matching Method [40] has been chosen to calculate the electrostatic fields
of these periodic electrode structures. The transversal optical field Eo(x,z) is assumed to
be Gaussian and Hermite-Gaussian along width and depth of the single-mode Ti-indiffused
waveguide in LiNbO3with mode field diameters matched to our experimental values. The
position-dependent overlap integral Γ(y)[39] must be multiplied by f(y), integrated over
one beat length and normalized to obtain the weighted overlap integral factor
ˆ
Γ = 2
ΛZΛ
0
Γ(y)f(y)dy with
Γ(y) = G
VRR|Eo(x,z)|2Ex(x,y,z)dxdz
RR|Eo(x,z)|2dxdz .
(4.2)
0.0 0.2 0.4 0.6 0.8 1.0
-0.1
0.0
0.1
y/
Λ
Γ
Figure 4.3: 3–phase cosine electrodes with corresponding voltages and local field overlap integral factors vs
normalized longitudinal coordinates
For cosine and sine cases, ˆ
Γis the real or imaginary part of the spatial Fourier coefficient
of Γ(y), respectively. Ex(x,y,z) is the vertical component of the electrostatic field in the
crystal. In Γ(y)we have multiplied by the applicable gap Gand divided by the maximum
interelectrode voltage Vas defined in (4.2). Figure 4.2 shows the local overlap factors
Γ(y)for one quadrature of the 2–phase TE-TM mode converter. For the two cases of
the 3–phase TE-TM mode converters Γ(y)is shown in Figure 4.3 and Figure 4.4. The
resulting weighted overlap factors are ˆ
Γ=0.198,0.11 and 0.096, respectively. The
2–phase ˆ
Γis resized to an effective value of ∼0.086 ... 0.098, if one takes into account
66 Chapter 4 Integrated Optical PMD Compensator
the fact that two quadratures of the 2–phase design need at least twice the length of the
3–phase design. If the maximum permissible field strength limits the design, the factor
V/G in κas defined in (4.1) is replaced by a constant. The 3–phase design performs in
its worst case 0.096 roughly equal or slightly better than the 2–phase design. If the output
range of the voltage sources is the limiting factor, κas defined in (4.1) is obtained through
a multiplication by the same V/G (8/Λ,9/Λ,6√3/Λ) by which we have divided in
calculating Γ(y)as defined in (4.2). This yields equal κvalues for both 3–phase cases, and
these are 1.26 ... 1.44 as high as κin the 2–phase case. The above simulation results are
published in [60].
0.0 0.2 0.4 0.6 0.8 1.0
-0.1
0.0
0.1
y/
Λ
Γ
Figure 4.4: 3–phase sine electrodes with corresponding voltages and local field overlap integral factors vs
normalized longitudinal coordinates
Figure 4.5 shows the photograph of a portion of the chip used in [22]. This device
was fabricated by the Applied Physics group of Prof. W. Sohler, here at the University of
Paderborn. This device was fiber pigtailed and packaged with slanted endfaces to improve
the input/output optical return loss. The insertion loss was 4 dB and PDL 1.2 dB. Operating
voltages were <50 V. Thermal tuning is possible with 100 GHz/K.
4.1.3 New Proposals for High-Bit Rate PMD Compensators
This type of lithium niobate-based integrated optical PMD compensator (PMDC) should
work up to at least 40 Gbit/s. At 160 Gbit/s a poor performance is to be expected because
the experimentally needed length for one full mode conversion is on the order of 5 mm.
This means that the corresponding DGD of about 1.2 ps is only partly orientable. How-
ever, PMD compensation at 160 Gbit/s or beyond seems to be mandatory to maximize
dispersion-shifted fiber capacity, for example, in particular in all Japan. To reach higher
4.1 Design Issues for Integrated Optical PMDC 67
Figure 4.5: Photograph of in-phase and quadrature TE↔TM mode converter electrode pairs (in dark) on
Ti:LiNbO3PMD compensator
and higher bit rates one needs to tailor the birefringence of lithium niobate. This is possi-
ble in principle in two ways: one possibility is to use tilted waveguide in Y Z-plane which
will reduce the birefringence and hence the differential group delay. The other possibil-
ity is to use the mixture of lithium niobate (LN) and lithium tantalate (LT) also known as
lithium–niobate–tantalate (LNT).
Tilted Waveguide
We have also investigated the case of a larger Λ. For large Λthe electrical field reaches
deeper into the waveguide and are also more uniform, thereby increasing the overlap in-
tegral as shown in Figure 4.6. Similar characteristics are found for all three cases. The
weighted field overlap integral factor ˆ
Γgrows almost ∝Λin the range considered.
20 25 30 35 40 45
0.0
0.1
0.2
0.3
0.4
Γ
ˆ
2-Phase Cosine
3-Phase Cosine
3-Phase Sine
Λ µm
Figure 4.6: Weighted field overlap integral factor ˆ
Γas a function of beat length Λin µm
A high Λ∝cos−2ϑcan be achieved in LiNbO3if the waveguide is tilted by an
angle ϑin the YZ plane. The coupling factor then becomes κ(ϑ) =ˆ
Γ(π/2)n3(r51cosϑ −
68 Chapter 4 Integrated Optical PMD Compensator
r22sinϑ)(V/G)λ−1with n=no+ (1/2)(ne−no)cos2ϑand r22 = 6.8·10−12 m/V. To
give an example, for ϑ=−π/4the waveguide runs halfway between +Y and +Z axes. Λ
doubles, ˆ
Γmore than doubles, and κ(−π/4) ∼
=1.75·κ(0) in the field strength limited case.
Furthermore, since the DGD per length is halved the DGD spent to implement a full mode
conversion is ∼3.5times smaller than for Y–axis propagation. This means a 3.5times
more accurate PMD compensation becomes possible. However, twice as high a voltage V
is required to keep V/G constant. Furthermore, a PMD compensator with a given length
can compensate only half as much DGD. A large Λis particularly advantageous if neither
available driving voltages nor total chip length are limiting factors. This is the case at high
bit-rates, say ≥40Gbit/s, where truly bad fibers have to be ruled out anyway.
It is indeed necessary to consider the substrate radiation modes which may cause high
propagation losses in tilted waveguides. They may become significant when the waveguide
is tilted in practice because the propagation constants of the guided mode and the substrate
radiation mode may become identical. However no attempt has been made to verify this
possibility independently as it is clear from [61,62].
Lithium–Niobate–Tantalate Crystals
The mixed ferroelectrics have been the focus of intensive fundamental and applied research
for many years. Interest in the study of these materials arises from the fact that the physical
properties of crystalline materials are governed to a large extent by the composition of the
crystals. Therefore, the physical properties can be tuned by varying the composition. One
of the simplest ferroelectric mixed crystal systems is lithium–niobate–tantalate, as both end
members exhibit the same crystal structure (space group R3c) with only slight differences
in the lattice and positional parameters. The physical properties can be very easily tuned
by varying the parameter yin the composition of LNT crystals. To a certain degree, the
mixed system yields a simple crystal modelling that may lead to functional materials and
devices, which has direct implication for PMD compensation in optical communication.
Lithium niobate is a slightly nonstoichiometric, typically Li-deficient, preferably grown at
the congruently melting composition with 48.5 Mol% of Li2O. A large variety of dopants
ranging from +1 valent state H+ to the +3 valent state such as rare earth cations can be
introduced into the crystal structure frame of lithium niobate. Most are known to occupy
Li-sites. In contrastto theseLi-site dopants, tantalum isisomorphic toniobium andreplaces
niobium when introduced into the crystal structure frame of LN. Tantalum can substitute
niobium up to 100%. Any changes in the crystal composition will finally affect all physical
properties of the crystal such as the linear dielectric response, i.e. refractive index, electro-
optic coefficients and so on. It has been shown in [63] that refractive index and electro-optic
coefficients depend linearly on the Ta content yin LNT crystals. Therefore one can tailor
the birefringence of this mixed crystal especially for PMD compensation at higher bit rates.
The ordinary refractive index noand the relevant electro-optic coefficient r51 depend
linearly on the Ta content yin LNT crystals: no= 2.2125 −0.07y, and r51 = 28 −8y.
Figure4.7shows the calculated optimum weighted fieldoverlap integralfactor ˆ
Γ, as defined
in [60], for 2-phase as well as two representative cases of 3-phase TE-TM mode converters
4.1 Design Issues for Integrated Optical PMDC 69
0.0 0.2 0.4 0.6 0.8 1.0
0.1
0.2
0.3
0.4
0.5
0.6 2-phase cos
3-phase cos
3phase sin
Γ
y in LiNb1-yTayO3
Figure 4.7: Weighted field overlap integral factor ˆ
Γas a function of Ta content yin lithium–niobate–tantalate
crystals
with interdigital electrodes. The numbers can be directly compared if the 2-phase ˆ
Γhas
been halved due to the fact that the 2-phase design need at least twice the length of the
3-phase design. Using this ˆ
Γand assuming 2-phase electrodes, the achievable number of
full mode conversions per DGD at electric field strength near breakdown (10 V/mm), and
the required length per DGD have been calculated as a function of the Ta content yin the
LNT crystals (Figure 4.8). Pure LN allows for ≈8full mode conversions / ps in theory if
ˆ
Γ=1, and ≈0.8/ps experimentally, in agreement with theory (ˆ
Γ≈0.1). Pure LT allows
for 20 time more mode conversions/DGD. The length/DGD is ≈4.2mm/ps in LN, and
≈42 mm/ps in LT. But this should not be problematic since less PMD may be expected
in links for highest bit rates. LT alone should work in principle up to at least 640 Gbit/s.
Appreciable advantages over pure LN with the potential of reaching 160 Gbit/s can also
be expected for low y which may be accessible either by incorporating Ta into LN during
crystal growth or later by thermal in-diffusion. An interesting situation occurs near y= 0.9
where the sign reversal of ∆npromises Tbit/s PMD compensation. A major problem for
large y in LNT and pure LT are the large beat lengths, which scale proportional to the
length/DGD. High voltages are required to reach fields near breakdown, even for 3-phase
electrodes where the gaps are smaller (≈370 V in LT).
4.1.4 Conclusion
We found that a 3–phase TE-TM mode converter can (but need not in all cases) outperforms
a2–phase one. Tilting the waveguide in the YZ plane can drastically increase the efficiency
70 Chapter 4 Integrated Optical PMD Compensator
00.2 0.4 0.6 0.8 1
100
101
102
y
LiNbO3LiTaO3
mode conversion
efficiency / DGD
length needed / DGD
[a. u.]
Figure 4.8: Mode Conversion/DGD and length needed/DGD for a TE↔TM in LNT crystals as function of
Ta content y
but the leaky modes losses are the main issues involved in its realization. The other possi-
bility is to use mixed ferroelectric crystals such as the lithium–niobate–tantalate to realize
high bit rate PMD compensators where just a little DGD needs to be compensated. The
birefringence ∆nand electrooptic coefficient r51 decreases linearly with increasing the Ta
content yin LNT crystals. A Ta content yof up to 0.5 is good to realize a PMD compensator
at about 160 Gbit/s. These two possibilities are explored and the simulation results are sum-
marized in [24]. These solutions combines optimum performance and high-speed with a
high degree of integration and hence low cost potential.
4.2 DGD Profile Characterization
Different kinds of PMD compensators (PMDCs) based on all optical as well as electrical
filters have been demonstrated or proposed. Most of these were designed for first-order
PMD compensation. PMDCs suitable for higher order compensation like the one described
above is proposed by R. No´
e [19]. Experimental validation of such a filter is not have been
reported so far nor a PMD medium with complicated structure have been analyzed. B. L.
Heffner first demonstrated the use of Jones matrix eigenanalysis technique to accurately
measure both the PSP and DGD as a function of optical frequency. This has allows to
verify the structure with only 3known DGS section which were precharacterized [64,65].
On the other hand, L. M¨
oller from Bell Labs has synthesized a 2x2 Jones matrix filter
4.2 DGD Profile Characterization 71
for broadband PMD compensation [66] generalizing the basic work done by S. E. Harris
and his coworkers [67]. Later on A. Eyal and A. Yariv also simulated this concept for
minimization of the maximum differential group delay within a given frequency band for
broadband PMD compensation filters [68]. In M¨
oller’s paper, a synthesis algorithm and
a design concept that descends directly from Ozeki [69] is described. Filter types which
were proposed by R. No`
e can not only be represented but also analyzed by means of this
algorithm. Therefore, we proposed a nondestructive method based on this algorithm for
determining the DGD profiles. This method would be of particular help for identification,
emulation, and compensation of higher-order PMD effects that persists after compensation
of first-order PMD.
4.2.1 Measurement Setup
A measurement setup consist of external cavity tuneable laser source, an integrated optical
network analyzer circuit having on chip broadband electrooptic polarization transformers
which are based on X-Cut, Z-propagation LiNbO3and a set of two polarimeters. External
cavity tunable laser source is connected to the input of the fiber pigtailed and packaged
integrated optical network analyzer circuit. This integrated optical circuit is used to split
the laser signal between the reference and measurement branch. Electrooptic polarization
transformer which is on the measurement branch is used to generate 8different polarization
states that are equally distributed onto the Poincar sphere. These output polarization states
are given as input to the device under test. The output of the reference branch is given to
reference polarimeter through ≈25 ps DGD section and it works as a frequency meter for
frequency correction. A measurement polarimeter is connected to the output of the device
under test. The laser is swept in the steps of 10 GHz between the 1525 nm and 1545 nm
where it had no mode hops. Figure 4.9 shows the measurement setup.
Figure 4.9: Measurement setup for DGD profile characterization
72 Chapter 4 Integrated Optical PMD Compensator
The frequency-dependent 3x3 rotation matrix Rof the device, i.e. rows and columns 2
to 4of its M¨
uller matrix are thereby measured. Each of the n(= 8in our case) normalized
input (in) Stokes vectors results in corresponding output (out) Stokes vectors. Both vector
groups are arranged in form of a matrix
Sout =RSin.(4.3)
The rotation matrix Ris then obtained by
R=SoutST
in(SinST
in)−1.(4.4)
Compared to a more compact method [70] with two launched polarizations this gives a
better immunity against polarization measurement errors. Any existing nonorthogonality
of R is removed by singular value decomposition of R according to
R=usv†,(4.5)
where u, and vare orthogonal (or more generally: unitary) matrices and sis a diagonal
matrix with the singular values. Ris then redefined as an orthogonal matrix
R=uv†.(4.6)
From this frequency-dependent rotation matrix, R, the frequency-dependent Jones matrix,
J, is obtained in the form
J=A B
−B∗A∗,|A|2+|B|2= 1.(4.7)
The frequency domain results are multiplied by a cos2window centered at 1535 nm to
suppress errors introduced by the discontinuous borders. The inverse Fourier transform
yields the time-dependent Jones matrix with impulse response as elements. Its first column
is the finite impulse response to a horizontally polarized pulse while last column is the finite
impulse response to an orthogonally polarized pulse (vertical). It is sampled with a 785 fs
period. The structure is analyzed on the basis of sections having DGDs equal to this value.
The following section gives the core of the inverse scattering algorithm.
4.2.2 Inverse Scattering Algorithm
Analysis by means of an impulse response is a concept that is familiar to the electrical
engineers. If an impulse i.e. a Dirac delta function is applied to a linear time independent
network, the Fourier transform of the impulse response of the network is the frequency
domain transfer function of the network.
Wefirst consider the impulse response of the single birefringent crystal (e. g. polarization
maintaining fiber) of Figure 4.10. The crystal is cut with its optic axis perpendicular to its
4.2 DGD Profile Characterization 73
Figure 4.10: Impulse response of a single birefringent crystal
length and with end faces flat and parallel. A linearly polarized impulse of the optical elec-
tric field is assumed to be normally incident on the crystal. Since the incoming signal is
normally incident, double refraction will not occur. The impulse will divide into two or-
thogonally polarized impulses whose amplitudes depend on the polarization of the incident
impulse with respect to the principle axes of the crystal. These two impulses travel with
different velocities and therefore, emerge at different times. The difference in time at which
they emerge from the crystal is given by
tF−tS=L∆n
v,(4.8)
where ∆nis the birefringence of the crystal of length L, and vis velocity of the light in the
crystal.
Next, we would like to consider the impulse response of nbirefringent crystals of
arbitrary birefringence, lengths, and orientations. In general, we can draw some conclu-
sions: Impulse response of nbirefringent crystals is a set of 2nimpulses of finite duration.
The magnitude and polarization of these impulses are determined by the crystal orientations
while their relative times of emergence is determined by birefringence and lengths of the
crystals used. The most important conclusion is that the impulse response of series of bire-
fringent crystals is a train of impulses of finite duration. In contrast, the impulse response
of Fabry-Perot and multilayer dielectric-film filters consist of infinite train of impulses.
At the outset, two points should be stressed. First, it is assumed that the birefringent
crystals within the network are lossless. This means that at all points between first and last
crystal energy must be conserved. Energy conservation principle puts certain restrictions
on the impulse which have travel along fast (Fn
n) and slow (Sn
n) axes respectively. This
condition is give by
Fn
0Fn
n−1+Sn
1Sn
n= 0.(4.9)
74 Chapter 4 Integrated Optical PMD Compensator
Second, it should be noted that
Fn
n=Sn
0= 0.(4.10)
This is just the statement of the fact that the first and last impulses out of nth crystal must
have propagated along its fast and slow axes, respectively.
To do analysis of such a chain of crystals of arbitrary birefringence, orientation, and
lengths, one can write an expression relating the input and output of each crystals. Angle
(Θ) represents the input coupling ratio and is known as orientation angle while angle (Ψ)
represents the retardation or the differential phase. Therefore, one can write for the output
of the first (4.11-a), second (4.11-b), and third (4.11-c) crystals, respectively.
F1
0
S1
1=−sin Θ1e+Ψ1
cos Θ1e−Ψ1I0
0(4.11-a)
F2
0
F2
1
S2
1
S2
2
=
cos Θ2e+Ψ20
0−sin Θ2e+Ψ2
sin Θ2e−Ψ20
0 cos Θ2e−Ψ2
F1
0
S1
1(4.11-b)
F3
0
F3
1
F3
2
S3
1
S3
2
S3
3
=
cos Θ3e+Ψ30 0 0
0 cos Θ3e+Ψ3−sin Θ3e+Ψ30
0 0 0 −sin Θ3e+Ψ3
sin Θ3e−Ψ30 0 0
0 sin Θ3e−Ψ3cos Θ3e−Ψ30
0 0 0 cos Θ3e−Ψ3
F2
0
F2
1
S2
1
S2
2
(4.11-c)
Since the pattern has be established, one can write an expression for the output from
the nth crystal (4.11-d).
Fn
0
Fn
1
.
.
.
Fn
n−1
Sn
1
Sn
2
.
.
.
Sn
n
=
cos Θne+Ψn0. . . 0 0
0 cos Θne+Ψn. . . 0.
.
.
.
.
.0. . . −sin Θne+Ψn0
0.
.
.. . . 0−sin Θne+Ψn
sin Θne−+Ψn0. . . .
.
.0
0 sin Θne−+Ψn
. . . 0.
.
.
.
.
.0. . . cos Θne−+Ψn0
0 0 . . . 0 cos Θne−Ψn
Fn−1
0.
.
.
Fn−1
n−2
Sn−1
1.
.
.
Sn−1
n−1
(4.11-d)
This is a set of non-homogeneous equations and has a solution if and only if the rank
of the matrix of the coefficients is equal to the rank of the augmented matrix. Several
possibility exists for determining the rank of the matrix. Applying one of these criteria, we
4.2 DGD Profile Characterization 75
get an expression
tan Θne2Ψn=−Fn
n−1
Sn
n
(4.12)
that relates orientation and retardation angles with the impulse response elements. The
calculation is an easy one, involving for any stage no more than the solution of only two
simultaneous equations.
Our procedure is to start with the output from the nth crystal. From these two impulse
response elements Fn
n−1and Sn
n, we can calculate the crystal angles Θnand Ψnor the mode
conversion effect and the input to this crystal (Fn−1
nand Sn−1
n). Since the input to the nth
crystal is nothing but the output of the (n−1)th crystal. Thus we work our way back
through the entire network alternately finding crystal angles and crystal inputs. Successive
repetition of this procedure yields the input vector which is two element smaller than the
output vector. Finally, when all crystal angles are found out, one is left with a two element
vector with one zero element that describes the input signal.
We have used a matrix of an elliptical retarder to describe the mode conversion effect
instead of the one used by S. E. Harris [67] or L. M¨
oller [66] because transmission fiber
can have not only linear birefringence (due to core ellipticity, micro-bending, transverse
stress) but also circular birefringence (due to fiber twist) and may vary along the fiber [18].
Single-mode fiber must, therefore, be represented by an elliptical retarder matrix at a given
optical frequency.
Our inverse scattering algorithm can display a full length DGD profile with 40 sections.
Experimentally inverse scattering range was chosen to be ≈31.5ps.
Figure 4.11: DGD profile for back-to-back measurement
4.2.3 DGD Profiles in Fibers
At first a back-to-back measurement without DUT was performed. In this case, all DGD
sections (= rods) should be cancelled by oppositely directed adjacent ones. This is indeed
76 Chapter 4 Integrated Optical PMD Compensator
the case in the above Figure 4.11. The DGD profile travels 20 sections forth, then another
20 sections back on the same path, as if returning from a dead end. As a consequence, one
has the impression to see only 20 sections, which in reality hide the other 20 sections. Input
arrow tip and output arrow back end coincide within <100 fs which is a measurement error
since the true DGD was ∼0fs. So the simplified back-to-back DGD profile is a frequency-
independent polarization transformation specified by the arrows where input and output
arrows indicates the local principle states of polarization of the relevant DGD-sections.
Next DUT was a ∼11 m long polarization maintaining fiber (PMF) with ≈25 ps of
DGD. It yielded a straight, 25.12 ps long line (32 sections) followed by a short dead end of
2×4sections (Figure 4.12).
Figure 4.12: DGD profile of one ≈25 ps DGD section
Figure 4.13 shows a DGD profile when two pieces of PMF each with ∼22 ps and
∼6ps of DGD are concatenated with 50 % mode conversion (45◦rotation) in between.
The long section and the short section which also contains a short dead end of 2×1section
are clearly identified. Angle between the sections was ≈90◦as expected.
Figure 4.13: DGD profile of two (≈22 ps and ≈6ps) DGD sections with 50 %mode conversion in between
4.2 DGD Profile Characterization 77
4.2.4 DGD Profiles of Distributed PMD Compensator in LiNbO3
Fiber-pigtailed and packaged X-cut, Y-propagation LiNbO3-based PMD compensator was
fabricated in-house, similar to that in [22]. About 70 in-phase and quadrature TE-TM mode
converters were distributed over 95 mm long waveguide. An average DGD was about
25 ps. Without any applied voltages the DGD profile was similar to that of Figure 4.10.
Figure 4.14 shows the DGD profile bent into a full circle when two full mode conversions
were distributed over the whole chip occur in one quadrature. A dead end with approx-
imately 2×4sections is also seen because the total DGD value of the chip is less than
inverse scattering range. Pigtail-shaped profile of Figure 4.15 results when only one and
half mode conversions occur in one quadrature, but they were concentrated at about 3/4of
the total DGD. This demonstrates the versatility of the distributed PMD compensator with
respect to emulation and compensation of higher-order PMD effects which persists after
compensation of first-order PMD.
Figure 4.14: DGD profile of LiNbO3PMDC with 2 full mode conversions distributed over whole chip length
in one quadrature
4.2.5 Conclusion
The frequency-dependent reduced M¨
uller matrix measurement enables us to calculate the
corresponding Jones matrix and hence the impulse response of the devices with polarization
78 Chapter 4 Integrated Optical PMD Compensator
Figure 4.15: DGD profile of LiNbO3PMDC with 1 and 1/2 mode conversions distributed over whole chip
length in one quadrature
mode dispersion. From the latter, the differential group delay profiles are determined by
inverse scattering. These results are summarized in [71]. These allow to identity, emulate
and compensate the effects of higher-order PMD that persists after compensation of first-
order PMD.
4.3 Conclusion
An integrated optical PMD compensator is designed and optimized for maximum electro-
optic efficiency. PMD compensator for high bit rate applications are not only proposed but
also evaluated for their performance. This includes the use of tilted waveguides or mixed
ferroelectric materials such as lithium–niobate–tantalate crystals where birefringence and
hence DGD can be tailored. This is particulary advisable at data rates of 40 Gbit/s and
beyond where small DGD needs to be compensated. An inverse scattering algorithm is
implemented for characterizing the devices with PMD. Some fiber and integrated optical
devices with PMD are characterized for different DGD profiles using in-house developed
integrated optical network analyzer. The versatility of such a PMDC is demonstrated and
is found to be suitable for generation and compensation of higher-order PMD effects.
Chapter 5
Result Discussion and Future Scope
The integrated optical network analyzer based on X-cut, Z-propagation lithium niobate
for the vectorial structure characterization of optical fiber Bragg gratings is reported. The
frequency-dependent complex reflectance Jones matrix is measured by interferometry and
transformed into time-domain. From the impulse response matrix, the vectorial grating
structure is determined by inverse scattering. Optical network analyzer was also used to
measure frequency-dependent M¨
uller matrix of the optimized polarization mode dispersion
compensator in X-cut, Y-propagation lithium niobate and its different differential group
delay profiles are determined by inverse scattering using time domain impulse response.
5.1 Characterization of LiNbO3-based IOC
On this integrated optical circuit several optical components such as polarizers, phase
shifters, and broadband TE↔TM mode converters are integrated onto each branch of the 3
dB optical power splitter. This IOC together with the 3x3 fiber coupler forms a measure-
ment interferometer. Initially, all of the above mentioned components are integrated onto a
test substrate. This test circuit was used to evaluate the performance of each of these inte-
grated optical components. Indeed this has complicated the whole work. Several problems
were encountered during the entire development of this integrated optical circuit and are
listed below.
(1) Higher propagation losses for TE-polarized modes
(2) Poor power splitting uniformity of the 3 dB optical power splitter
(3) Poor polarization extinction ratio of on-chip TE-pass polarizers
(4) Static TE–TM mode conversion
(5) Nonideal behavior of on-chip TE–TM mode converters
Step by step the characterization data was analyzed. Latter on, the device fabrication
steps were also reviewed. It was found from the device fabrication history that this test
integrated optical circuit was not realized using the principal-axis (Z-axis) propagation but
was realized to use near-Z-axis propagation. Specifically, the substrate was cut an angle
of 2◦degrees with respect to the principal Z-axis. This off-axis propagation was used to
79
80 Chapter 5 Result Discussion and Future Scope
compensate the modal birefringence and to make the on-chip TE–TM mode converters
more efficient.
Ti-indiffused channel waveguides in LiNbO3themselves were having low propaga-
tion losses. But the developed device shows very high insertion loss due to the fact that
1.5 mm long TE-pass polarizes were realized directly on top of the waveguide using an
aluminum as a cladding metal. This not only gave excess propagation losses (first issue)
but also the differential propagation losses for TE-polarized modes. This resulted into poor
power splitting ratios for the TE-polarized modes (second issue), due to nonuniform metal
cladding. It also resulted in poor polarization extinction ratios for the TM-polarized modes
(third issue).
This off-axis propagation results in the undesired static mode conversion. This not
only makes the polarizers but also mode converters nonideal. This is because the TM-
polarized mode is regenerated in the waveguide after TE-pass polarizers. This regenerated
TM-polarized mode degrades the polarization extinction ratio of the on-chip polarizers
(third issue). Whether one physically rotates the direction of propagation in the Y Z-
plane globally or rotate the index ellipsoid electro-optically in the Y Z-plane locally, the
net modal birefringence compensation effect is the same. In this compensation process,
one always ends up with the undesired imaginary coupling coefficient between two, other-
wise, orthogonally polarized modes and this mode coupling results in the undesired static
mode conversion (fourth issue). If one compensates for this modal birefringence globally,
then the phenomenon is length-dependent and beyond any physical control due to inher-
ent lithium niobate anisotropy. If one compensates for it locally, then it is field-dependent
(fifth-issue). In principle, it is possible to compensate for this undesired polarization mode
coupling electro-optically.
The only option to reduce this undesired static mode coupling is to fabricate optical
waveguides along the principal axis. It is to be ensured thatthe waveguide is exactly parallel
to the Z–axis during photolithography. Otherwise, the LiNbO3-anisotropy will play the
undesired role. If the waveguide is exactly parallel to the Z-axis then the voltages required
for compensation are ideally symmetric; otherwise, they are asymmetric in practice. Of
course a relatively high bias voltages will result.
Therefore, in the next generation of these integrated optical circuits, principal-axis
propagation was used. Each component is fabricated separately on a Ti in-diffused optical
channel waveguide in lithium niobate and characterized rigorously to give feedback to
device fabrication steps and for the design of new versions of the mask plates. Three design
iterations were done including the mask design in order to achieve the target specifications
set for this integrated optical circuit. The issues which were resolved and gave significant
improvement are summarized below.
The only remedy to improve the performance of the polarizers was to use principal-axis
propagation for device development. The other possibility was to introduce a buffer layer
of right dielectric constant and right thickness in between the waveguide and aluminum
metal cladding so as to improve the phase matching condition between the surface plasmon
mode and the TM-polarized mode. Both options were implemented. They have drastically
5.2 Longitudinal Structure Characterization 81
improved the polarization extinction ratio of the TE-pass polarizers and minimized the
excess propagation losses for the TE-polarized modes. Several materials have been tried
out. It was found out experimentally that 27 nm thick yttrium oxide based polarizers gave
an extinction ratio of ≈30 dB for 3 mm long polarizers with 500 nm thick aluminum metal
cladding. To some extent, this solution has also improved uniformity of power splitting for
the TE-polarized modes.
Several 15 mm and 20 mm long phase shifters were realized on straight optical channel
waveguides. They were characterized using the Fabry-Perot technique. Voltage x length
product for these phase shifters was found out to be 60 V-cm.
The mode converters with 30 mm long electrodes with 6 and 8 µmgaps were charac-
terized. The voltage required for full TE-to-TM mode conversion and vice-versa was 10 V.
Needed bias voltages were ±38 V.
5.2 Longitudinal Structure Characterization
Optical frequency domain interferometry was used in the longitudinal structure charac-
terization of optical fiber Bragg gratings. Once the frequency-dependent complex reflec-
tion coefficient for reflective devices or complex transmission coefficient for transmissive
devices under test are measured correctly in the frequency domain then the task of deriving
the longitudinal structure is straight forward. This type of optical characterization method
needs high-speed, single-frequency, broadband tunable laser. The tuning range limits the
spatial resolution but large grating lengths may be investigated due to high coherence of a
single-mode laser. Such type of lasers are very difficult to find in the shops. External cavity
single-frequency tunable lasers with sweeping option are now commercially available in
the shops but are very expensive.
The integrated optic on-chip phase modulators are used for digital phase shifting and
performs AC rather than DC measurement in order to increase sensitivity. In fact, these
phase modulators on X-cut, Z-propagation lithium niobate are also not ideal if the polarity
of the applied voltages is such that it reduces the modal birefringence then the net effect is
that this applied voltages creates an electric field Eywhich will cause both stretching of the
index ellipsoid along the principal-axes as well as the rotation of the index ellipsoid cross-
section in the yz-plane. This indeed gives undesired static mode conversion. There are
polarizers after the phase shifters which are used to suppress negative frequencies generated
due to serrodyne modulation; in fact, these polarizers do convert these polarization changes
into undesirable intensity changes. One solution was to found out the correct polarity
of the applied voltages so that only phase is modulated and not the polarization. These
measures were taken into account while configuring the hardware of the network analyzer.
This concept of digital phase shifting to increase sensitivity was successfully demonstrated
but finally was not used for measurement because it simply took more recording time.
Especially, for such interferometric measurements, the measurement time has to be shortest
in order to minimize the DC thermal drift. There is always a trade off between sensitivity
and DC thermal drift.
82 Chapter 5 Result Discussion and Future Scope
Initially, tunable twin-guide laser was employed as tunable laser sources for measure-
ments. Unfortunately, this laser died due to malfunction in the laser control unit. In fact,
the temperature controller failed to work. Latter on, narrow linewidth DFB lasers from
two well known companies were tried out during this work. The Furukawa laser was hav-
ing very strange behavior and could not be used even for scalar measurement. On the other
hand, JDS-uniphase laser worked satisfactorily for scalar measurements. But JDS-uniphase
laser was never stable within the scan and from scan-to-scan. Moreover, the TE–TM mode
converters also had severe DC drift. This gave lot of problems in finding the common
phase for implementing the orthogonalization scheme devised by D. Sandel. Therefore,
instead of using this scheme, the on-chip TE–TM mode converters were calibrated with
respect to the commercial rotating waveplate polarimeter to generate orthogonal pairs for
the vectorial measurements.
One uniform fiber Bragg grating at 1548.25 nm of >95% reflectivity and 0.2 nm
bandwidth is characterized for longitudinal structure in order to demonstrate the utility of
this integrated optical network analyzer. The results are summarized in chapter 3. The
longitudinal structure of this commercial fiber Bragg grating showed a single burst. This
was also free from birefringence and dichroism. Shortcoming of this method is that it needs
single-frequency, broadband, and high-speed, tunable laser source. Future scope would be
either to realize it somehow or to buy such laser with sweeping option and integrate it with
the integrated optical network analyzer to complete this work.
5.3 Integrated Optical PMD Compensator
The integrated optic TE–TM mode converters are optimized to have highest electro-optic
efficiency by simulation studies. Two-phase verses three-phase TE–TM mode converters
are compared and latter outperform the former one but need not in all cases. Frequency-
dependent reduced M¨
uller matrix of the fabricated, fiber pigtailed, and packaged PMD
compensator is measured using the integrated optical network analyzer from which the
Jones matrix is calculated and hence the impulse response of the devices with polarization
mode dispersion. Differential group delay profiles of this device are determined from the
time domain impulse response by inverse scattering. This allows to identify, emulate, and
compensate the effects of higher-order PMD that persists after compensation of first-order
PMD.
Distributed PMD compensator performance can be pushed toward highest bit rates if
they are implemented in mixed ferroelectric crystals like lithium–niobate–tantalate. A Ta
content yin LiNb1−yTayO3of up to 0.5 is good to realize a PMD compensator for about
160 Gbit/s. Future scope would be to try this option.
5.4 Conclusion 83
5.4 Conclusion
This chapter has summarized these two application of the lithium niobate-based integrated
optical circuits to optical communication. In this chapter, some of the fundamental and
engineering problems encountered during this work, are described, in details. Moreover,
the attempt has been made to systematically analyze and solve most of these problems.
Future scope of this work is also presented.
84 Chapter 5 Result Discussion and Future Scope
Appendix A
Point Matching Method
Point Matching Method proposed by Marcuse [40], in 1989, is an effective method for
solving the Laplace equation for the electric fields in inhomogeneous medium by employ-
ing a series expansion of the potential in terms of functions that are themselves the solution
of Laplace equation for a homogeneous medium. The solution for stratified dielectric in-
homogeneous medium is found by matching the solutions in the different homogenous
layers with the help of boundary (continuity) conditions at a finite set of discrete points
(Point Matching). This yields solution with a sufficient accuracy with series expansion
method that does not involve any iteration.
A.1 Formulation
Figure A.1 shows the geometry of the problem. It consists of a x-cut, z-propagation
LiNbO3crystal (εx= 43 and εy= 28) that extends from x=−Dto x=−d. From
x=−dto x= 0 there exists a buffer layer with ε2=εb= 3.8(typically SiO2), and at
x= 0, plane metal electrodes are deposited which are assumed to be infinitely thin and per-
fectly conducting. A medium with dielectric constant ε1=εair = 1 (usually air), infinitely
extends in +xdirection from x= 0 plane. The number of contacts are arbitrary, but they
are assumed to be parallel and to extend infinitely in positive and negative z-direction.
The electrostatic field vector E, generated by potentials applied to the electrodes, can
be expressed in terms of potential function ψas
E=−∇ψ. (A.1)
In each dielectric region, the potential must be a solution of Laplace equation, which in
anisotropic medium assumes the form,
εx
εy
∂2ψ
∂x2+∂2ψ
∂y2= 0.(A.2)
In the isotropic regions 1 (air) and 2 (silicon dioxide buffer layer), we have εx=εy=εj
with j= 1 or 2.
85
86 Appendix A Point Matching Method
Figure A.1: Schematic of a typical electrode structure (CPW) on top of the layered dielectric substrate like
LiNbO3
The electric potentials applied to the metal electrodes imposes a definite ydependence
on the potential function in the plane x= 0. This field variation could be described by
the Fourier integral, but for numerical calculations the discrete series are easier to handle
than integrals. Therefore, potential function is expanded in terms of cosine functions. The
functions cos(ν(π/L)y)forms a complete orthogonal set over the domain 0< y < L for
integral values of ν. The functions
φ1ν= exp(±νκ(π
L)x) cos(ν(π
L)y)(A.3)
φ2ν= cosh(νκ(π
L)x) cos(ν(π
L)y)
φ3ν= sinh(νκ(π
L)x) cos(ν(π
L)y)
φ4ν=a+bx with
κ=rεy
εx
are the solutions of Laplace equation (A.2). Thus, one may express a potential function ψ
in region 1 and 2 as follows:
ψ1=a0+∞
X
ν=1
aνe−ν(π
L)xcos(ν(π
L)y) for x≥0(A.4)
ψ2=b0+c0x+∞
X
ν=1
[bνe−ν(π
L)x+cνeν(π
L)x]cos(ν(π
L)y) for 0 ≥x≥ −d(A.5)
In region 3, the form of potential function depends on whether the region is infinitely
extended or it is terminated by a ground plane. In Case of an infinitely extended medium,
A.1 Formulation 87
0.0 0.2 0.4 0.6 0.8 1.0
-0.050
-0.025
0.000
0.025
0.050
Ex
x = -0.5 µm
Normalized y [µm]
(c)
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.25
0.50
0.75
1.00
1.25
x = -0.5 µm
ψ [V]
Normalized y [µm]
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.25
0.50
0.75
1.00
1.25
x = 0.0 µm
ψ [V]
Normalized y [µm]
(a)
0.0 0.2 0.4 0.6 0.8 1.0
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
x = -0.5 µm
Ey
Normalized y [µm]
(b)
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
Ey
x = 0.0 µm
Normalized y [µm]
(b)
0.0 0.2 0.4 0.6 0.8 1.0
-0.50
-0.25
0.00
0.25
0.50
Ex
x = 0.0 µm
Normalized y [µm]
(c)
Figure A.2: Potential and electric fields of a two-electrode (CS) phase shifter computed using PMM in the
plane of electrodes. WPS = 50 µm,GPS = 15 µm,0.3µmthick buffer with εb= 3.8. (a) ψ(b) Ey(c) Ex
potential function is expressed as
ψ3=d0+∞
X
ν=1
dνeνκ(π
L)(x+d)cos(ν(π
L)y) for x≤ −d. (A.6)
The different forms of the series expansions in (A.4)–(A.6) are dictated by the boundary
conditions. Since potential can not become infinite as |x| → ∞, the linear term in x
is absent from (A.4) and (A.6). For the same reason, only the exponential function whose
value decreases with increasing value of |x|is permitted in the series expansion of (A.4) and
(A.6). Since the function cos(ν(π/L)y)is an even function, the potential is automatically
continued as an even function in the domain −L < y < 0. Moreover, it is periodic with a
period 2Land implies that the solution behaves as though the structure were continued as
a mirror images, imaged on the plane y= 0 and as if this extended system is periodically
repeated with period 2L.
88 Appendix A Point Matching Method
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
1.5
Ex
x = 0.0 µm
Normalized y [µm]
(c)
0.0 0.2 0.4 0.6 0.8 1.0
-0.05
0.00
0.05
0.10
x = -0.5 µm
Ex
Normalized y [µm]
(c)
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
x = -0.5 µm
Ey
Normalized y [µm]
(b)
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
1.0
Ey
x = 0.0 µm
Normalized y [µm]
(b)
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.25
0.50
0.75
1.00
1.25
x = -0.5 µm
ψ [V]
Normalized y [µm]
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.25
0.50
0.75
1.00
1.25
x = 0.0 µm
ψ [V]
Normalized y [µm]
(a)
Figure A.3: Potential and electric fields of a coplanar-electrode (CPW) mode converter computed using
PMM in the plane of electrodes. WMC = 10 µm,GMC = 6 µm,0.3µmthick buffer with εb= 3.8. (a) ψ
(b) Ey(c) Ex
A.2 Satisfying Boundary Conditions
At dielectric interfaces, the electric field must satisfy two boundary conditions, i.e. Eyand
εxExmust pass continuously through the interface. The continuity of Ey=−∂ψ/∂y ev-
erywhere in the plane x= constant is assured if ψis continuous at the interface. Therefore,
we may replace the condition for continuity of Eyby the requirement that ψis continuous
at the dielectric interfaces. Continuity of ψi.e. (ψ2=ψ3at x=−d) and εx∂ψ/∂x i.e.
(ε2∂ψ2/∂x =εx∂ψ3/∂x at x=−d) for all values of yrequires that the coefficients of
corresponding terms cos(ν(π/L)y)of the expansions (A.5) and (A.6) are identical at plane
x=−d. This leads to two simultaneous equations expressing bνand cνcoefficients in
terms of dνcoefficients and leads to the following expressions for a structure without a
A.2 Satisfying Boundary Conditions 89
ground plane:
bν=1
2(1 −κεx
εb
)e−ν(π
L)ddν(A.7)
cν=1
2(1 + κεx
εb
)eν(π
L)ddν(A.8)
b0=d0and (A.9)
c0= 0.
The boundary condition in the plane of the (vanishingly thin) electrodes at x= 0 requires
that xassumes same value on either side of the interface. This requirement (∂ψ1/∂y =
∂ψ2/∂y at x=0) immediately leads to the conditions
aν=bν+cνfor ν= 1,2,· · · (A.10)
a0=b0.(A.11)
On the electrode, we require instead that the potential function ψshould be equal to the
applied voltages V(y)(ψ1=V(y)at x=0). We write the electrode potential as a function of
yto indicate that its value is different on different electrodes even though V(y)is constant
on each individual electrodes. Therefore, one obtains (A.12) directly with the help of
(A.10) and (A.11). But the continuity of εx∂ψ/∂x now holds only in the gaps between
the electrodes and not on the electrodes themselves i.e. to say (ε1∂ψ1/∂x =ε2∂ψ2/∂x at
x= 0) and therefore one obtains (A.13)
b0+
N−1
X
ν=1
(bν+cν)cos(ν(π
L)y) = V(yj) on the electrodes (A.12)
L
πc0+
N−1
X
ν=1
ν[(ε1
ε2−1)bν+ (ε1
ε2
+ 1)cν]cos(ν(π
L)y) = 0 in the gaps (A.13)
The label jis attached to ycoordinate to indicate that the one satisfies the boundary
conditions in the plane x= 0 only at a finite set of discrete points y=yj. If there are N
points in plane x= 0, there will be Nterms in the series expansion in order to provide as
many equations as there are unknown coefficients dν. It is necessary to calculate only dν
coefficients since aν,bν, and cνall depend on dν. There are only Nequations since (A.12)
is used only on electrodes and (A.13) is used in the gaps. Therefore, one needs only to
compute the elements of matrix Ajνin the system of equations given by (A.14).
N−1
X
ν=0
Ajνdν=V(yj) on the electrodes
0 in the gaps (A.14)
90 Appendix A Point Matching Method
Matrix inversion of Ajνleads to the determination of the unknown coefficients dνvia
(A.15).
dν=
N−1
X
ν=0
(A−1)νj V(yj) on the electrodes
0 in the gaps (A.15)
The matrix inversion can, of course, be done numerically with the help of computer.
A.3 Computation
Depending upon the device geometry, the electrode structure is defined and a volt array
V(yj)is assigned with appropriate voltages and Ajνmatrix is generated with the help of
(A.12 and A.13). The Ax =Btype of matrix equation is solved numerically with the
help of computer by using lower upper (LU) triangularization and lower upper (LU) back
substitution subroutines. The solution array replaces the volt array V(yj). Once the dν
coefficients are known, then aν,bν, and cνcan be calculated. After this the potentials ψ1,
ψ2, and ψ3are generated with the help of (A.4)–(A.6). The electric field components Ex
and Eyare calculated by analytically taking the first derivatives of ψ1,ψ2, and ψ3with
respect to xand y. This completes the computation of electrostatic fields in air, buffer and
lithium niobate regions of a layered dielectric structure.
Figure A.2 above shows the potential, electrostatic field EyV/µm, and electrostatic
fieldExV/µmcalculatedfortherealizedphaseshifter(Vπ= 40 V) onX-cut, Z-propagation
LiNbO3, where y-directed field Eyis used for the phase modulation.
Figure A.3 above shows the potential, electrostatic field EyV/µm, and electrostatic
field ExV/µmcalculated for the realized TE-TM mode converter (VMC = 10 V) on X-cut,
Z-propagation LiNbO3, where x-directed field Exis used for the mode conversion.
Appendix B
Generation of Optical Fields
Optical fieldswithin thetitanium (Ti)in-diffused optical waveguidesin X-cut, Z-propagation
LiNbO3are well approximated by Gaussian and Hermite-Gaussian functions in width and
depth directions, respectively [72]. Electric field Eo(x, y)becomes
Eo(x, y) = ( x
wx
)e
−x2
w2
xe
−y2
w2
y(B.1)
where wyand wxare the (1/e)widths of the electric field profiles in width and depth
directions, respectively. In the computation of field overlap integral, Γ, as defined in [39],
one needs to use the normalized intensity E2
o(x, y). It is possible to measure only the Full
Width Half Maximum (FWHM) of the intensity profiles wyfwhm and wxfwhm along the width
and depth of the Ti in-diffused waveguides in LiNbO3experimentally. One needs to fit wx
and wyto experimentally measured values of wxfwhm and wyfwhm of the measured intensity
profiles.
For Gaussian approximation, one can write E2
oy=e−2(y2
1/2/w2
y)= 1/2for the half
of the maximum intensity and y1/2=wypln 2/2. But wyfwhm = 2y1/2and therefore,
wy=wyfwhm /√2 ln 2.
For Hermite-Gaussian approximation, the situation is little bit different, one can write
E2
ox= (x2/w2
x)e−2(x2/w2
x)for the intensity as before. This function will have a maximum
value at x= (1/2)e−1. For half of the maximum value, E2
ox= (x2
1/2/w2
x)e−2(x2
1/2/w2
x)=
(1/4)e−1. There are two solutions: x1= 1.1572wxand x2= 0.34056wx, if one solves this
for x1/2. For wxfwhm =x1−x2and wx=wxfwhm /0.81667.
It is necessary to adjust the peak of the optical intensity at right position along the
width and depth of the Ti in-diffused optical waveguide. Therefore, one can introduce the
normalization constants Cxand Cyas well as the p and q variables for the optical intensity
as
E2
o(x, y) = C2
x(x−q
wx
)2e−2( x−q
wx)2C2
ye−2( y−p
wy)2.(B.2)
The p value shifts the peak position of E2
o(x, y)along width of the waveguide while the q
value shifts the peak position of E2
o(x, y)along the depth of the waveguide.
91
92 Appendix B Generation of Optical Fields
It is essential to take into account the boundary conditions imposed on the Eoxat the
interface between the lithium niobate and buffer layer, which is usually, a silicon dioxide
layer. The reason because optical index of lithium niobate is 2.2 and that of silicon dioxide
is 1.46. Due to this large discontinuity in the optical refractive indices of these two ma-
terials, the optical field intensity decays exponentially in the buffer layer regardless of the
thickness of the buffer layer. In the buffer layer, the optical field Eoxobeys the equation
∂Eox
∂x =γEox(B.3)
where γ=kpN2
eff −N2
band k= 2π/λ.
Boundary conditions imposed on Eox, for the TE-polarized optical field are
Eoxb=EoxLN (B.4-a)
∂Eoxb
∂x =∂EoxLN
∂x (B.4-b)
where Eoxband EoxLN are the optical fields in the buffer layer and in the LiNbO3, respec-
tively. If one solves (B.3) then Eoxb=e−γx for the fields in the buffer region. Since,
one needs to use the normalized intensity, a normalization constant Cbis introduced in the
solution of (B.3) and E2
oxb=C2
be−2γx. One has to apply the second boundary condition
(B.4-b) at y= 0, to get the relation between the two normalization constants Cxand Cbas
Cb=−Cx
γwx
(B.5)
To find the value of q, one has to apply the boundary condition (B.4-a) at x= 0 to get the
equation
qe
−q2
w2
x−1
γ= 0.(B.6)
Solving for q, one gets
q=1
γe(1
2Lambert[w(−2
γ2w2
x)])
Lambert w(x) =x−x2+3
2x3−8
3x4+125
24 x5and x= ( −2
γ2w2
x
)
(B.7)
Similarly the boundary conditions imposed on Eoxfor the TM-polarized optical mode
are
εbEoxb=εLNEoxLN (B.8-a)
εb
∂Eoxb
∂y =εLN
∂EoxLN
∂x (B.8-b)
93
where εband εxare the dielectric constants of the buffer and the X-cut lithium niobate
respectively. These conditions modifies the equation (B.7) to
εx
εb
qe
−q2
w2
x−1
γ= 0 (B.9)
where εb=N2
band εx=N2
o. Solving for q, one gets
q=N2
b
N2
oγe(1
2Lambert[w(−2N4
b
N4
oγ2w2
x)])
Lambert w(x)=x−x2+3
2x3−8
3x4+125
24 x5and x=( −2N4
b
N4
oγ2w2
x
).
(B.10)
For normalization of the Gaussian profile, one uses the condition
C2
yZ∞
−∞ |Eoy|2dy = 1
Cy=s1
wyr2
π
(B.11)
For normalization of the Hermite-Gaussian profile, one uses the condition
C2
xZ0
−∞ |Eox|2dx +C2
bZ∞
0|Eox|2dx = 1.
Cx=1
q1
2γ3w2
x+1
16 wx√2π
Cb=−Cx
γwx
(B.12)
ThesenormalizationconstantsCx,Cb, and Cyareusedtogeneratethenormalizedpolarization
dependent optical intensity profiles in the buffer region and lithium niobate region, respec-
tively.
E2
o(x, y) =C2
be−2γ(d+x)C2
ye−2( y−p
wy)2buffer region
E2
o(x, y) =C2
x(d+x−q
wx
)2e−2( d+x−q
wx)2C2
ye−2( y−p
wy)2LiNbO3region (B.13)
94 Appendix B Generation of Optical Fields
Appendix C
Derivation of Coupled Differential
Equations for FBG
C.1 Maxwell Equations
Lightisanelectromagneticwavephenomenon. Itselectric and magnetic field is represented
by the four electromagnetic field vectors that are functions of position r[m] and time
t[s]. The four vectors — electric field E[V/m], magnetic field H[A/m], electric flux
density D[C/m2], and magnetic flux density B[Wb/m2]governs the well known Maxwell
equations:
∇×E=−∂B
∂t
∇×H=J+∂D
∂t
(C.1)
Here J[A/m2]is the current density and ∇= (∂/∂x, ∂/∂y, ∂/∂z)is the del operator.
The following continuity equation governs the current density Jand the charge density
ρ[C/m3]:
∇·J=−∂ρ
∂t .(C.2)
Use of (C:1) and (C:2) and the vector identity ∇·∇×a= 0 one gets
∇·B= 0
∇·D=ρ.(C.3)
Fields are assume to have a periodic time dependence and are written as
E(r, t) = Re[E(r)e(ωt−kz)],etc.(C.4)
95
96 Appendix C Derivation of Coupled Differential Equations for FBG
C.2 Constitutive Relations
If the polarization generated in the light transmitting medium is P[C/m2]and the magneti-
zation is M[A/m], the electric flux density Dand magnetic flux density Bare, respectively,
D=ε0E+P
B=µ0(H+M).(C.5)
Here ε0[F/m] and µ0[H/m] are the permittivity of vacuum and the permeability of free
space, respectively and their values are as follows.
ε0= 8.8541878 ×10−12 [F/m]
µ0= 4π×10−7[H/m] (C.6)
If the medium is isotropic, linear, and non-dispersive, the polarization Pand the magneti-
zation Mare obtained as follows:
P=ε0χeE
M=µ0χmH.(C.7)
Here χeand χmare the electric susceptibility and the magnetic susceptibility respectively.
If equations (C:7) are substituted into equations (C:5), respectively, then we get the relative
permittivity εrand the relative permeability µras follows.
εr= 1 + χe
µr= 1 + χm
.(C.8)
Therefore the permittivity εand permeability µare
ε=ε0εr
µ=µ0µr.(C.9)
The constitutive relations are as follows:
D=εEand
B=µH.(C.10)
The refractive index ncan be obtained from
n=√εrµr.(C.11)
But for most optical materials, (say glass), µ=µ0and µr= 1, the refractive index n
simply becomes
n=√εr.(C.12)
C.3 Wave Equation 97
C.3 Wave Equation
For deriving the vector wave equation for the electric field E, for an isotropic lossless
medium with no wave sources (J= 0 and ρ= 0) with uniform permeability µ0, one has to
take the curl of the above Maxwell equation (C:1) and make use of (C:10) to get
∇×∇×E=∇×−∂B
∂t =−∂
∂t∇×B=−µ0
∂
∂t∇×H
=−µ0
∂2D
∂t2=−µ0ε∂2E
∂t2
(C.13)
The equation (C:3) for the source free medium becomes (∇ · D= 0) and making use of
the following vector identities
∇·(ab) = a(∇·b) + b·(∇a)
∇×∇×a=−∇2a+∇(∇·a)(C.14)
one gets the wave equation
∇2E+∇(E·∇ln ε) = µ0ε∂2E
∂t2and ε=ε0εr.(C.15)
C.4 Coupled Differential Equations for FBG
One can note that the quantities εt=εr(x, y)and hence the transverse components Et=
(Ex,Ey)are independent of z.
εt=εr(x, y) = εr, core for x2+y2< R2
εr, clad for x2+y2≥R2,(C.16)
where Ris the core radius of the single mode fiber. Therefore, for the fiber Bragg grating,
the above vector wave equation (C:15) with E(z, t)and εr=εr(z)simplifies to
∂2
∂z2E(z, t) = µ0ε∂2
∂t2E(z, t)(C.17)
For Bragg grating εr(z) = n2(z)from where
n(z) = n0+1
2∆npp(z) cos(2π
Λz+φk(z))
=n0+ Re[1
2∆npp(z)e(kgz+φk(z))] and kg=2π
Λ
=n0+ Re[∆nk(z)ekgz] and ∆nk(z) = 1
2∆npp(z)eφk(z)
.(C.18)
Peak to peak refractive index modulation amplitude satisfies the condition
∆npp << neff.(C.19)
98 Appendix C Derivation of Coupled Differential Equations for FBG
Using the nomenclature [54], electric field of the forward, Ef(z, t)and backward, Eb(z, t)
propagating waves in the Bragg grating are written as
Ef(z, t) = Re[D(z)e(ωt−kz)] = Re[d(z)e(ωt−kz)]
Eb(z, t) = Re[U(z)e(ωt+kz)] = Re[u(z)e(ωt+kz)]and k=n0
c0
ω, (C.20)
where n0is the effective refractive index of the grating and c0is the free space velocity of
light. Taking the second derivatives of (C:20) with respect to time t, one gets
∂2
∂t2Ef(z, t) = Re[−ω2d(z)e(ωt−kz)]
∂2
∂t2Eb(z, t) = Re[−ω2u(z)e(ωt+kz)]
(C.21)
Similarly by taking the second derivative of (C:20) with respect to one dimensional space
coordinate z, one gets
∂2
∂z2Ef(z, t) = Re[(−k2d(z)−2k∂
∂zd(z) + ∂2
∂z2d(z))e(ωt−kz)]
∂2
∂z2Eb(z, t) = Re[(−k2u(z) + 2k∂
∂zu(z) + ∂2
∂z2u(z))e(ωt+kz)]
.(C.22)
Term with second order derivative of fields with respect to one dimensional space coordi-
nate zare so small that these terms can be neglected to give
∂2
∂z2Ef(z, t) = Re[(−k2d(z)−2k∂
∂zd(z)))e(ωt−kz)]
∂2
∂z2Eb(z, t) = Re[(−k2u(z) + 2k∂
∂zu(z))e(ωt+kz)]
.(C.23)
Total field Eis the superposition of the forward Ef(z, t)and backward Eb(z, t)propagat-
ing fields in the wave equation (C:17).
∂2
∂z2(Ef(z, t) + Eb(z, t)) = n(z)2
c2
0
∂2
∂t2(Ef(z, t) + Eb(z, t)) (C.24)
Substituting for the second derivatives of Ef(z, t)and Eb(z, t)with respect to tand zfrom
(C:21) and (C:22) into (C:24) yields
Re[(−k2d(z)−2k∂
∂zd(z)))e(ωt−kz)] + Re[(−k2u(z)+2k∂
∂zu(z))e(ωt+kz)] =
1
c2
0
(n2
0+ 2n0Re[∆nk(z)e(kgz)])Re[−ω2d(z)e(ωt−kz)]+Re[−ω2u(z)e(ωt+kz)].
(C.25)
Simplification of (C:25) using (C:20) gives
−2k∂
∂zd(z)e−kz +2k∂
∂zu(z)ekz =
−ω2
c2
0
2n0Re[∆nk(z)e(kgz)](d(z)e−kz +u(z)ekz).
(C.26)
C.4 Coupled Differential Equations for FBG 99
Grating constant kgas defined in (C:27) is approximately equal to twice the propagation
constant kof the wave (k= 2π/λ).
kg≈2k(C.27)
Equation (C:26) contains to first order differential equations:
−2k∂
∂ze−kzd(z) = −ω2
c2
0
n0ekz[∆n∗
k(z)e−kgz]u(z)
2k∂
∂zekzu(z) = −ω2
c2
0
n0e−kz[∆nk(z)ekgz]d(z)
(C.28)
Simplification of (C:28) delivers
∂
∂zd(z) = −ω
2c0
[∆n∗
k(z)e(2k−kg)z]u(z)
∂
∂zu(z) = ω
2c0
[∆nk(z)e−(2k−kg)z]d(z)
.(C.29)
The coupling coefficient
κ(z) = ω
2c0
∆nk(z)(C.30)
modifies the simplified expressions for the coupled differential equations for the fiber Bragg
grating as
∂
∂zd(z) = κ∗(z)e(2k−kg)zu(z)
∂
∂zu(z) = κ(z)e−(2k−kg)zd(z)
.(C.31)
The forward and backward propagating waves are also the functions of time t. At the Bragg
wavelength, ω=ω0and the propagation constant k=k0,
Ef(z, t) = Re[e
d(z, t)e(ω0t−k0z)]
Eb(z, t) = Re[e
u(z, t)e(ω0t+k0z)](C.32)
Subtracting (C:20) from (C:32) respectively,
e
d(z, t)e[(ω0−ω)t−(k0−k)z]=e
d(z, t)e[(ω0−ω)(t−n0
c0z)] =d(z, t)
e
u(z, t)e[(ω0−ω)t+(k0−k)z]=e
u(z, t)e[(ω0−ω)(t+n0
c0z)] =u(z, t)(C.33)
Differentiating (C:33) with respect to 1-D space coordinate z,
∂
∂z e
d(z, t)e[(ω0−ω)(t−n0
c0z)]=∂
∂zd(z)
∂
∂z e
u(z, t)e[(ω0−ω)(t+n0
c0z)]=∂
∂zu(z)
(C.34)
100 Appendix C Derivation of Coupled Differential Equations for FBG
Simplification of (C:34) gives
e[(ω0−ω)(t−n0
c0z)] ∂
∂z e
d(z, t) + n0
c0
(ω−ω0)e
d(z, t)=∂
∂zd(z)
e[(ω0−ω)(t+n0
c0z)] ∂
∂z e
u(z, t)−n0
c0
(ω−ω0)e
u(z, t)=∂
∂zu(z)
(C.35)
The terms containing (ω−ω0)e
d(z, t)and (ω−ω0)e
u(z, t)are obtained if one differen-
tiates (C:33) with respect to time t.
(ω−ω0)e
d(z, t) = ∂
∂te
d(z, t)
(ω−ω0)e
u(z, t) = ∂
∂te
u(z, t)
(C.36)
Therefore (C:35) becomes
e[(ω0−ω)(t−n0
c0z)][∂
∂z e
d(z, t) + n0
c0
∂
∂te
d(z, t)] = ∂
∂zd(z)
e[(ω0−ω)(t+n0
c0z)][∂
∂z e
u(z, t)−n0
c0
∂
∂te
u(z, t)] = ∂
∂zu(z)
.(C.37)
The equations (C:35) and (C:37) into (C:31) gives
e[(ω0−ω)(t−n0
c0z)][∂
∂z +n0
c0
∂
∂t ]e
d(z, t) =
κ(z)∗e(2k−kg)ze
u(z, t)e[(ω0−ω)(t+n0
c0z)]
e[(ω0−ω)(t+n0
c0z)][∂
∂z −n0
c0
∂
∂t ]e
u(z, t) =
κ(z)e−(2k−kg)ze
d(z, t)e[(ω0−ω)(t−n0
c0z)]
.(C.38)
Simplification of (C:38) yields the equations (3.12) used in chapter 3.
(∂
∂z +n0
c0
∂
∂t)e
d(z, t) = κ∗(z)e
u(z, t)
(∂
∂z −n0
c0
∂
∂t)e
u(z, t) = κ(z)e
d(z, t)
.(C.39)
Appendix D
Derivation of Discrete Equations for
FBG
D.1 S-Matrix Characterization of Optical Components
At network level, a component is considered as a “black box”, isolated from rest of the
world except for a few designated ports that are accessible for external connection. For
network analysis purpose, a component is completely characterized by the relations be-
tween the signals or fields available at these ports. This type of component characterization
is called port characterization. In the frequency domain, the relations between the various
fields at the ports of a linear, time-independent optical component such as a FBG are es-
sentially a set of linear equations. These linear equations are algebraically represented by
a matrix. For the characterization of active component, an extra vector describing a signal
generator may be needed.
The S-matrix characterization is well suited as a port characterization of optical com-
ponents. However, due to the fact that in optical waveguides there are two guided modes,
each physical port of an optical component is actually equivalent to two strictly single mode
ports. An optical component could be treated in exactly same fashion as a microwave com-
ponent by separating logically two virtual single mode ports that corresponds to each one
of the physical ports. This logical separation is a problematic and confusing and therefore,
it is better to regard each port both physically and logically as a single entity. This approach
is more promising, but there is a price to it: the scattering parameters become 2x2 matrices
and the S-matrix becomes a “super matrix”, i.e. a matrix whose elements are 2x2 matrices
instead of regular numbers. This is most fundamental difference between the optical S-
matrix as will be presented here and the corresponding S-matrix used in microwave theory.
If the optical component is linear and time-independent, then in the frequency domain,
the input and output Jones vectors A0and Aare related by a set of linear equations
A0=SA +C,(D.1)
where Sis a complex 2x2 matrix and Cis a complex 2 element vector, both independent
101
102 Appendix D Derivation of Discrete Equations for FBG
of A. The matrix Sis called the Scattering-matrix (S-matrix) of the component. Smay be
regarded as a Jones matrix of the component. The Jones vector Cmay represent optical
field generators that may be present in the component. It has a significant value only in the
active components. In passive components such as FBG, Cis normally neglected, since at
the frequencies that are currently used in the optical network applications, the blackbody
radiation at room temperature is negligible.
Some important properties of Smatrices are derived from the energy conservation con-
siderations. The input and output powers Pin and Pout are give by
Pin =|A|2P0and Pout =|A0|2P0=|SA +C|2P0(D.2)
where P0is a unit power. For passive component, C= 0, and energy conservation implies
that Pin≥Pout, so that
A†(I−|S|2)A≥0(D.3)
for any A. For passive component, the matrix I− |S|2is semipositive. Component for
which the power is conserved (Pin =Pout), S-matrix is unitary:
|S|2=I.(D.4)
Power loss can occur as result of absorption,scattering, and coupling to radiation modes.
Under normal conditions, optical components may be considered as a reciprocal and are
characterized by the symmetric S-matrix:
S=ST.(D.5)
D.2 Calculation of S-matrix for FBG
In FBG, the forward, d(z), and backward, u(z), propagating waves obeys the coupled
differential equations (as derived in Appendix C):
∂
∂zd(z) = κ(z)∗e(2k−kg)zu(z)
∂
∂zu(z) = κ(z)e−(2k−kg)zd(z)
(D.6)
Fiber Bragg Grating is a reciprocal (S11 =S22 and S12 =S21) and lossless optical network,
and therefore characterized by the scattering matrix:
S=S11 S12
S21 S22 .(D.7)
For the Bragg grating of length L, i.e. 0≤z≤Lone can write:
u(0)
d(L)=Sd(0)
u(L)=S11 S12
S21 S22 d(0)
u(L)(D.8)
D.2 Calculation of S-matrix for FBG 103
Now one needs to solve the above coupled differential equations given in (D:6) with appro-
priate boundary conditions in order to calculate individual elements of the scattering matrix
S. The exponential term in above coupled differential equations
2k−kg= 0,(D.9)
if the forward, d, and backward, u, propagating waves are phase matched. This condition
is called the longitudinal phase matching. This condition is a spatial analogue of conser-
vation of energy in time-dependent perturbation theory and therefore, may be called as
conservation of momentum. This is a special case of resonant coupling. For significant
mode coupling to takes place between the forward and backward propagating modes, two
conditions must be satisfied. The first is kinematical condition expressed by (D:9). Second,
the coupling coefficient κ(z)in (D:6) must not vanish. The later is also called as dynamical
condition, since it depends upon the characteristics of the waves such as polarization and
mode profiles, etc. The term (k−kg/2=∆β) and it represents the detuning factor which
is proportional to difference between the incident frequency and resonant frequency. The-
oretically ∆βis defined in same units as that of κ. The coupled mode equations becomes
∂
∂zd(z) = κ(z)∗e2(∆β)zu(z)
∂
∂zu(z) = κ(z)e−2(∆β)zd(z).
(D.10)
Since the coupling coefficient κis complex, the sign of the right hand side of both the cou-
pled differential equations is different and is very important. This sign will determine the
behavior of the coupling. These signs, of course, depends on the direction of propagation
of the coupled modes. The coupling is therefore divided into two categories: codirectional
and contradirectional. In FBG, as the coupled modes travels in opposite direction, we are
dealing with the contradirectional coupling. Boundary conditions that are used to solve the
contradirectional coupled mode equations are d(0) = 1 at z= 0 and u(L) = 0 at z=L.
The net power flow in +zdirection for this case is |d|2−|u|2. The coupled mode equations
are again consistent with the conservation of energy, which requires that
∂
∂z(|d|2−|u|2) = 0.(D.11)
The field amplitudes are say constants, D for the forward and U for the backward propa-
gating waves and therefore, one can write:
d(z) = D·e(s+∆β)z
u(z) = U·e(s−∆β)z.(D.12)
Differentiating above with respect to 1-D space coordinate z, we get
∂
∂zd(z)= ∂
∂zD·e(s+∆β)z=(s+∆β)·e(s+∆β)z=κ∗U·e(s−∆β)ze2(∆β)z
∂
∂zu(z)= ∂
∂zU·e(s−∆β)z=(s−∆β)U·e(s−∆β)z=κD·e(s+∆β)ze−2(∆β)z
(D.13)
104 Appendix D Derivation of Discrete Equations for FBG
Multiplication of these two equations in (D:13) and its simplification gives:
s2=|κ|2−(∆β)2
s=±p|κ|2−(∆β)2.(D.14)
Both the equations in (D:13) together yields:
D=κ∗
(s+∆β)U⇔U=κ
(s−∆β)D. (D.15)
Therefore, we can write d(z)and u(z)in terms of a single constant Das
d(z) = D·e(s+∆β)z
u(z) = κ
(s−∆β)D·e(s−∆β)z.(D.16)
The common solution of these two first order coupled differential equation contains two
constants corresponding to sand −s, therefore, we can write,
d(z) = D1·e(s+∆β)z+D2·e(−s+∆β)z
u(z) = κ
(s−∆β)D1·e(s−∆β)z+κ
(−s−∆β)D2·e(−s−∆β)z.(D.17)
Once the constants D1and D2are calculated, one can immediately calculate the elements
of the scattering matrix Sfor the FBG. As the grating physically exists between z= 0 and
z=L, we can apply these boundary conditions to (D:17) to get
d(0) = D1+D2
u(0) = κ
(s−∆β)D1+κ
(−s−∆β)D2
d(L) = D1·e(s+∆β)L+D2·e(−s+∆β)L
u(L) = κ
(s−∆β)D1·e(s−∆β)L+κ
(−s−∆β)D2·e(−s−∆β)L.
(D.18)
Expression for d(0) and u(L)forms the simultaneous equations. These are solved simul-
taneously to obtain the unknown constants:
D1=[κ
(−s−∆β)e(−s−∆β)L]d(0) −u(L)
κ
(−s−∆β)e(−s−∆β)L−κ
(s−∆β)e(s−∆β)L
D2=−[κ
(s−∆β)e(s−∆β)L]d(0) + u(L)
κ
(−s−∆β)e(−s−∆β)L−κ
(s−∆β)e(s−∆β)L.
(D.19)
These recently calculated constants D1and D2in (D:19) are substituted into the expres-
sions for u(0) and d(L). After simplification, we directly get the S-matrix (D:20) for the
D.2 Calculation of S-matrix for FBG 105
FBG.
u(0)
d(L)=S
d(0)
u(L)=−κsinh(sL)e−(∆β)Ls
s κ∗sinh(sL)e(∆β)L
(scosh(sL) + ∆βsinh(sL)e−(∆β)Ld(0)
u(L)(D.20)
If the term ∆β= 0 then s=±κ=|κ|and therefore we can write
u(0)
d(L)="−κ
|κ|tanh(|κ|L)1
cosh(|κ|L)
1
cosh(|κ|L)
κ∗
|κ|tanh(|κ|L)#d(0)
u(L).(D.21)
We experimentally measured the local reflectivity and hence locally constant κ’s at discrete
points, n, as a function of 1-D space coordinate zover the whole grating length L(=µ∆z)
and µ=n. As the increment ∆zin zapproaches 0, the limit of above scattering matrix S
as ∆z→0reduces it to
Sµ= lim
∆z→0Swith κ(µ∆z)∆z=κµ= constant,and (D.22)
Sµ="−κµ
|κµ|tanh(|κµ|)1
cosh(|κµ|)
1
cosh(|κµ|)
κ∗
µ
|κµ|tanh(|κµ|)#.(D.23)
The local reflectivity ρµequals
ρµ=κµ
|κµ|tanh(|κµ|)≃ |κµ|,(D.24)
and the local transmission coefficient τµequals
τµ=1
cosh(|κµ|)=q1−tanh2(|κµ|) = q1−|κµ|2.(D.25)
Therefore, now we can write the locally constant scattering matrix Sµin a simplified form
as
Sµ=−κµτµ
τµκ∗
µ.(D.26)
From this we can see that the power exchange between the forward and backward propagat-
ing modes in the region between the z= 0 and z=Lwhere the Bragg grating physically
exists is given by
|κ|2sinh2(sL)
s2cosh2(sL)+(∆β)2sinh2(sL).(D.27)
We notice that the fractional power exchange decreases as ∆βincreases. A complete power
exchange for contradirectional coupling, however, only occurs when the phase matching
condition is satisfied (∆β= 0) and Lis infinite. This situation is different from that
of codirectional coupling, where the complete power is periodically exchanged (back and
forth) between the coupled modes as a function of 1-D space coordinate zprovided ∆β=
0. Fiber Bragg grating is a typical example of contradirectional coupling.
106 Appendix D Derivation of Discrete Equations for FBG
D.3 Down–Up Difference Schemes
Instead of constructing difference schemes in terms of input Dirac impulse unity matrix
and the output impulse response matrix, we will use the down and up combinations of
propagating waves.
For example, the medium in which the wave travels is specified by a single system
parameter, the impedance, Z, as function of 1-D space coordinate z. These waves has a
property that they do not interact unless Zchanges it’s value. D(z, t)and U(z, t)waves
are only discontinuous at z=µ∆zwhere there exists a discontinuity in Z, otherwise they
are continuous.
Suppose that Z(z)is piecewise constant, with possible discontinuities only at ∆z,2∆z,
···µ∆zetc. Thus, we have,
Z(z) = Zµz∈[µ∆,(µ+ 1)∆].(D.28)
Therefore, we need to analyze the waves into two situations:
1. as they travel along a part where Z= constant;
2. as they cross the discontinuity in Z.
Figure D.1: Calculation of Dµ, ν and Uµ, ν for one position step ∆Z
Figure D.1 shows the quantities required to calculate the Dand Uwaves for one posi-
tion step ∆z. Suppose z∈[µ∆,(µ+ 1)∆], then Z=constant and the waves are separated
into down-wave (right moving) and up-wave (left moving) and are defined as
D(z, t) = f(z−t),
U(z, t) = g(z−t).(D.29)
Figure D.2 shows the down and up waves, Dand U, in a piecewise constant portion of the
medium.
Above equations states that down-wave just to the left of the discontinuity at (µ+1)∆z
at any given time tmust have departed from µ∆zat the instant t−∆. Since the wave
travels without any change of shape, we can write
D[(µ+ 1)∆z−, t] = D[µ∆z+, t −∆].(D.30)
D.3 Down–Up Difference Schemes 107
Figure D.2: Dand Uwaves in a piecewise constant portion of the medium
This wave on the right hand side of above equation is obtained from the left hand side wave
by just operating on it by a delay operator ∆and this ∆has a property such that
∆f(t) = f(t−∆),(D.31)
so that we can write equation (D:30) as
D[(µ+ 1)∆z−, t] = ∆D[µ∆z+, t].(D.32)
We can also say that the up-wave at µ∆z+and time tmust be that which departed from
(µ+ 1)∆z−at time t−∆. Thus we can write
U[µ∆z+, t] = U[(µ+ 1)∆z−, t −∆] = ∆U[(µ+ 1)∆z−, t],(D.33)
which we may invert it to give,
U[(µ+ 1)∆z−, t] = ∆−1U[µ∆z+, t].(D.34)
The last equation must be used with care since it expresses U[(µ+ 1)∆z−, t]in terms
of U[µ∆z+, t + ∆], i.e. in terms of an event in the future. It is a non-casual relation and
therefore we will work with the previous version of it. We can combine these two equations
into one matrix equation
D[(µ+ 1)∆z−, t]
U[(µ+ 1)∆z−, t]=∆
∆−1D[µ∆z+, t]
U[µ∆z+, t].(D.35)
This is the first of two equations that well describes the evolution of waves. It expresses the
fact that D,Uwave travel without any change of shape in the parts of the medium where
Zis constant. The second evolution equation for D,Uto accompany (D:35), describes
how these waves interact as they pass through the discontinuity between Zµand Zµ+1 at
(µ+ 1)∆z.D[(µ+ 1)∆z+, t]
U[(µ+ 1)∆z+, t]=ΘµD[(µ+ 1)∆z−, t]
U[(µ+ 1)∆z−, t].(D.36)
108 Appendix D Derivation of Discrete Equations for FBG
where Θµis given
Θµ=Zµ+Zµ+1
2pZµpZµ+1 "1−(Zµ−Zµ+1)
(Zµ+Zµ+1)
−(Zµ−Zµ+1)
(Zµ+Zµ+1)1#.(D.37)
If we define µth local reflection and transmission coefficient as
κµ=(Zµ−Zµ+1)
(Zµ+Zµ+1)
τµ=Zµ+Zµ+1
2pZµpZµ+1
,
(D.38)
then the matrix Θµbecomes
Θµ=1
τµ1−κµ
−κµ1.(D.39)
The matrix Θµ, which describes how the D,Uwaves interact as they cross the discon-
tinuity, is called a scattering matrix. The pair of equations (D:35) and (D:36) completely
describes the evolution of the waves Dand U. They may be combined into one equation
D[(µ+ 1)∆z+, t]
U[(µ+ 1)∆z+, t]=Θµ∆
∆−1D[µ∆z+, t]
U[µ∆z+, t].(D.40)
Equation (D:40) is easy to use; it expresses the quantities at one discontinuity in terms
of those at the previous one (in space). This means that we can obtain quantities at (µ+
1)∆(z+) in terms of quantities at 0+ simply by multiplying the appropriate operators:
D[(µ+ 1)∆z+, t]
U[(µ+ 1)∆z+, t]=Θµ∆
∆−1. . . Θ1∆
∆−1D[0+, t]
U[0+, t].(D.41)
We can call such a composition rule as a natural cascade rule. Second line of (D:40) is a
non-casual relation because it expresses U[µ∆z−, t]in terms of U[(µ−1)∆z+, t + ∆],
i.e. in terms of a quantity in future. In order to obtain a casual relationship, we need to
rearrange it as follows. We use t=ν∆in (D:40) and drop the suffixes ∆z+and ∆tand
rewrite (D:40) as Dµ+1, ν
Uµ+1, ν =Θµ∆
∆−1Dµ, ν
Uµ, ν .(D.42)
In full this equation is
Dµ+1, ν =τ−1
µ(∆Dµ, ν −∆−1κµUµ, ν)
Uµ+1, ν =τ−1
µ(−∆κµDµ, ν + ∆−1Uµ, ν)(D.43)
Eliminating ∆−1Uµ, ν from (D:40), we get one casual relation as
Dµ+1, ν = ∆τµDµ, ν −κµUµ+1, ν.(D.44)
D.3 Down–Up Difference Schemes 109
Similarly solving (D:40) simultaneously for ∆−1Uµ, ν, we get second casual relations as
Uµ, ν = ∆2κµDµ, ν + ∆τµUµ+1, ν.(D.45)
Equations (D:44) and (D:45) can be written into the matrix form as
Dµ+1, ν
Uµ, ν =∆τµ−κµ
∆2κµ∆τµ Dµ, ν
Uµ+1, ν .(D.46)
The matrix operator in (D:46) may be factorized in the form
∆τµ−κµ
∆2κµ∆τµ=1
∆τµ−κµ
κµτµ∆
1(D.47)
Therefore we can rewrite (D:46) as
Dµ+1, ν
Uµ, ν =1
∆τµ−κµ
κµτµ∆
1 Dµ, ν
Uµ+1, ν .(D.48)
This equation (D:48) fits our intuition. The Dwave moves down so that it will reach
(µ+ 1)∆z+after µ∆z+; the Uwaves moving up so that it reaches (µ+ 1)∆z+before
µ∆z+. Multiply (D:48) on left by the matrix operator
∆
1−1
=∆−1
1(D.49)
to obtain
∆−1
1Dµ+1, ν
Uµ, ν =1
∆τµ−κµ
κµτµ Dµ, ν
Uµ+1, ν .(D.50)
Carrying out the matrix operator multiplication in (D:50) and is rearranged to get
Uµ, ν
Dµ+1, ν+1 =−κµτµ
τµκµ Dµ, ν
Uµ+1, ν−1=SµDµ, ν
Uµ+1, ν−1.(D.51)
Sµis a scattering matrix for FBG as derived in (D:20). In full, the final expression for the
inverse problem is
Dµ+1, ν+1 =τµDµ, ν +κµUµ+1, ν−1
Uµ, ν =−κµDµ, ν +τµUµ+1, ν−1.(D.52)
Simplification of (D:52) gives Dµ, ν and Uµ, ν in terms of Dµ+1, ν+1 and Uµ+1, ν−1as
Uµ, ν =τ−1
µ(−κµDµ+1, ν+1 +Uµ+1, ν−1)
Dµ, ν =τ−1
µ(Dµ+1, ν+1 −κµUµ+1, ν−1).(D.53)
110 Appendix D Derivation of Discrete Equations for FBG
(D:54) can be written in matrix form as
Uµ, ν
Dµ, ν =1
τµ1−κµ
−κµ1Dµ+1, ν+1
Uµ+1, ν−1.(D.54)
Therefore, we can write for Dµ+1, ν+1 and Uµ+1, ν−1in terms of Dµ, ν and Uµ, ν as
Dµ+1, ν+1
Uµ+1, ν−1=1
τµ1−κµ
−κµ1−1Uµ, ν
Dµ, ν .(D.55)
Simplification of (D:55) with µand νreplaced by µ−1by ν−1in the first row, and µand
νreplaced by µ−1and by ν+ 1 in the second row of the matrix equation (D:48) gives the
output waves Dµ, ν and Uµ, ν (D:49) in terms of input waves at Dµ−1, ν−1and Uµ−1, ν+1.
Uµ, ν
Dµ, ν =1
τµ−1κµ−11
1κµ−1Dµ−1, ν−1
Uµ−1, ν+1 .(D.56)
Hence we can write the discrete equations from (D:56) and are used as it is to construct the
downward continuation down-up algorithm for the solution of inverse scattering problem
(3.13) in chapter 3, which enables us to calculate the coupling matrix κas function of 1-D
space coordinate zfor FBG under test and as well as the whole casual solutions Dµ, ν and
Uµ, ν, where it is assumed that the measured impulse response matrix of FBG under test is
as an output if the given input is Dirac impulse unity matrix.
Uµ, ν =τ−1
µ−1(κµ−1Dµ−1, ν−1+Uµ−1, ν+1)
Dµ, ν =τ−1
µ−1(Dµ−1, ν−1+κ∗
µ−1Uµ−1, ν+1).(D.57)
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Acknowledgements 117
Acknowledgements
First, I would like to sincerely thank my Prof. Reinhold No`
e for offering me a golden oppor-
tunity to work in his internationally recognized group in an area of optical communication and
high-frequency engineering. He constantly gave encouragement during this work. He also gave a
freedom to access and use very expensive tools in the optical communication laboratory. He was
very much keen on the technical discussions and gave lot of ideas through out the entire course.
Without him, it would never have been possible for me to cross this river.
Second, I would like to thank Prof. Wolfgang Sohler for providing me an opportunity to work
in his internationally recognized group in an area of lithium niobate-based integrated optics for
characterizing, pigtailing and packaging of the integrated optical circuits in lithium niobate and for
helpful technical discussions. He also gave me constant encouragement throughout this work.
Third, I would like to thank Dr. Hubertus Suche and Mr. Raimund Ricken who fabricated the
integrated optical circuits in lithium niobate and for useful technical discussions.
Fourth, I would like to thank Dr. David Sandel who generously helped me during the entire
course and for very useful mathematical and technical discussions. Here, I would like to thank Mr.
Schiff, Head, mechanical workshop of electrical engineering for fabricating the required housing
and mechanical jigs for pigtailing and packaging of integrated optical circuits in lithium niobate.
Fifth, I would like to thank Mr. Sebastian Hoffman, Dr. Mrs. Mihoko Yoshida–Dierolf, Mr.
Vitali Mirvoda, Dr. Frank W¨
ust, Mr. Stephan Hinz, Mr. S. K. Ibrahim, Mr. A. Hidayat, Mr. A.
Fauzi, Mrs. B. Milivojeic, Mr. Gerhard Wieseler , and Mr. Bernd Bartsch from whom I received
enormous help as group members.
Sixth, I would like to sincerely thank Dr. A. D. Shaligram, my M.Phil. supervisor and Prof. K.
Thyagarajan, IIT Delhi for their constant encouragement and moral support. I would like to thank
Dr. Y.G.K. Patro, Program Director, Optoelectronics Division and Dr. K. Chalapathi, Head, Opto-
electronics Division of SAMEER, Mumbai for their constant encouragement and moral support. I
also would like to thank Dr. B. K. Das who arranged my initial stay and latter as a family friend
during his entire stay in Deutschland.
Last but not the least, I would like to appreciate and thank my dear wife Mrs. Kranti and children
Mrunmayi and Saumitra for continuous moral support, in spite of the long periods of my physical
and mental absence they had to put with. I also would like to thank my parents and my sister Dr.
Sunita Bhandare for giving constant moral support throughout the entire stay in Deutschland.
Paderbron, Germany Suhas Bhandare
June 10, 2003
