RAPID COMMUNICATIONS
Applicabili y o he.Lo en zian peak me hod o
analyze leaky and lossy op ical wa eguides
LIuis To ne , Josep Fe e , and Fe nando Canal
Poly echnic Uni e si y o Ca alonia, Depa men o Sig-
nal Theo y
&
Communica ions,
Apdo.
30002.08080 Ba ce-
lona, Spain.
Recei ed 29 Janua y 1991.
Sponso ed by Elsa M. Ga mi e, Uni e si y o Sou he n
Cali o nia.
0003-6935/91/182418-03$05.00/0.
© 1991 Op ical Socie y o Ame ica.
Se e al conclusions conce ning he applicabili y o he
Lo en zian peak me hod o compu e he complex p opaga-
ion cons an o leaky and lossy wa eguide modes
a e
es ab-
lished.
Usually he calcula ion o he loss coe icien o leaky and
lossy modes appea ing in common in eg a ed op ical ech-
nology cons i u es a e y cumbe some and long p ocess.
This di icul y is a esul o such modes being complex guid-
ed
ones,
so
ha hei p opaga ion cons an exhibi s bo h eal
and imagina y pa s. To ind he e ec i e index o hese
modes (N = N + jNi, complica ed anscenden al equa-
ions ha e o be sol ed in mos cases. This poin equi es a
high numbe o i e a ions in he complex plane, and as a
consequence a g ea olume o calcula ions a e gene a ed.
Recen ly, an app oxima e p ocedu e, which is e e ed o as
he Lo en zian peak me hod (LPM), has been epo ed1 o
analyze his p oblem. This me hod is based on he ac ha
o he cases o p ac ical in e es he imagina y pa o he
e ec i e index amoun s o e y small alues. Typically N ~
1 and Ni ~ -4,-5. Un o una ely, he e
is
no clea c i e ion
o es ablish he applicabili y o he me hod o a pa icula
s uc u e. The main aim o his wo k is o add ess his
ques ion by means o a sligh ly di e en app oach han he
o iginal p ocedu e.
The eigen alue equa ion o a gene al wa eguiding sys em
can be w i en as D(N) = 0, D(N) being a anscenden al
complex unc ion. Acco dingly, he e ec i e indices o he
modes suppo ed by he wa eguiding s uc u e a e ob ained
as he eal and complex ze os o ha unc ion. So, i Nm =
N m +
jNim
is he e ec i e index o a gi en mode, one has
D(Nm)
= 0,
and in he neighbo hood o Nm he unc ion D(N)
can be app oxima ed as D(N) ≃ A(N - Nm), A being a
complex cons an . Then, in oducing he unc ion (N ) ≡
|D(N ,Ni
= 0)|2, which is only de ined along he eal N-axis,
one ob ains (N ) ≃ |A|2[(N , -
N m)2
+
N2im.
In a i s app oach (LPM), he unc ion
1/ (N )
can be
plo ed o ob ain Lo en zian beha io . Using he ac ha
Nim « 1, i can be assumed ha app oxima e alues o N m
and Nim can be ob ained by cu e- i ing he esul ing plo .1
Howe e , usually he alues ound o N m and Nim using his
p ocedu e depend on he sample ange used in he cu e-
i ing
p ocess.
This di icul y can be a oided by aking in o
accoun ha (N ) can be w i en also as
2418 APPLIED OPTICS / Vol. 30, No. 18 / 20 June 1991
Fig.
1.
Loss
coe icien
as a
unc ion o he op ical
axis
o ien a ion
o he s ep index wa eguide ( =
2
μm).
We
ha e no included he
s ongly inaccu a e LPM esul s a
θ
≃
θc.
Now, he N m and (N m) a e known, his equa ion de ines a
unipa ame ic eal unc ion, and Nim can
be
s aigh o wa d-
ly calcula ed by a simple linea eg ession p ocedu e. Also,
i allows us o ob ain app oxima e alues o N m and
(N m),
since (N ) exhibi s a ela i e minimum a N = N m. This is
a single and objec i e alue, i.e., i does no depend on he
used sample ange, and hus ep oducible esul s a e ob-
ained. The amoun o calcula ion equi ed is simila o
wha
is
needed in he usual oo - inding schemes o he pu e
guided modes (Nim
=
0),
since he i e a ion p ocedu e can be
pe o med by a as con e ging minimizing algo i hm.
To check he applicabili y o he abo e p ocedu e
we
ha e
conside ed he o -axis p opaga ion in a ypical X,Y-cu
LiNbO3-based wa eguide, in which he c ys al op ical axis
lies in he wa eguide plane making an angle θ wi h he
wa eguide
axis.
The co esponding eigen alue equa ion has
been de i ed om he s anda d ans e -ma ix o malism.2
Fi s , we conside a s ep index wa eguide o hickness in
which he co e is in ai . The wa eguide pa ame e s a e no
= 2.2946, ne = 2.2108, nos = 2.2866, nes = 2.2028, = 2 μm,
and λ - 633 nm. He e nos,nes a e, espec i ely, he o dina y
and he ex ao dina y e ac i e index o he subs a e,
whe eas no ,ne co espond o he ilm. When θ = 0°, his
wa eguide suppo s he TE0 and TM0 modes. The TE0
mode a θ = 0° emains guided o all alues o
θ;
meanwhile
he mode which
is
he TM0 a θ = 0° becomes leaky beyond
θc
≃
14°.
In Fig.
1
we plo ed he loss coe icien o his mode
as a unc ion o
θ.
The con inuous line co esponds o he
exac solu ion ob ained by di ec ly sol ing he eigen alue
equa ion by means o a nume ical zoom oo - inding algo-
i hm. The dashed line ollows om he modi ied LPM.
Table I. Inhomogeneous Wa eguide
The ag eemen be ween bo h se s o alues is excellen , e en
in he egion o highes losses.
Also,
he e o in he calcula ion o N m amoun s o a
negligible alue in mos cases, so ha i is only no iceable in
he egion whe e Nim has g ea alues. This is because,
gene ally speaking, o he poin s belonging o he eal N-
axis,
he linea app oxima ion o D(N) becomes less accu-
a e when he ac ual oo mo es away om his eal axis.
The e o
is
educed o a negligible alue by making a second
calcula ion by means o he new unc ion ƒ1(N ) ≡
|D(N ,N0im)
N0im
being he alue ob ained a he i s s ep.
Also,
i is wo h no icing ha he LPM does no wo k in he
egion
θ
~
θc,
since his is a cu o poin and he ac ual oo s
s em om a sha p s uc u e o D(N) in he complex plane.
On he o he hand, e y good ag eemen has also been ob-
se ed o a mul imode e sion ( = 3 μm) o he abo e
example. This ac has o be emphasized, since in his case
he loss coe icien shows a complica ed beha io as a unc-
ion o
θ
wi h a ious maxima and minima.3
Now, we a e going o examine an inhomogeneous wa e-
guide wi h a Gaussian p o ile in bo h no and ne , which
suppo s also a pu e guided mode and a leaky guided mode.
The wa eguide pa ame e s a e iden ical as in he o me
single-mode example. Again, he accu acy ob ained in his
case using he modi ied LPM is e y good when Nim o he
leaky mode has small alues. Ne e heless, he esul s in
he egions wi h mode a ely high losses s ongly disag ee
wi h he exac alues.2 Table I shows some calcula ed al-
ues.
This impo an disag eemen occu s in a ange o ~25°
beyond θc ≃ 11° and comes om he pa icula o m o he
unc ion D(N). In Fig. 2 he shape o he unc ion (N ) in
he neighbo hood o N m is shown o θ = 80°. The plo has
wo minima. The i s one, which occu s o he smalle
alue o N , is a local minimum and does no co espond o
any oo o he equa ion D(N) = 0. The emaining mini-
mum, which occu s a a highe alue o N (on he igh o he
s eep maximum),
is
he one associa ed wi h he ac ual ze o
o
he unc ion D(N) and leads o he esul s gi en in Table I.
The shape o unc ion (N ) shown in Fig. 2 is he same,
wha e e he alue o
θ.
Ne e heless in he ange
θc<θ
≤
37°,
he ze o o D(N) p oduces a e y sha p minimum in he
complex plane o such an ex en ha , al hough i occu s o
ypical alues o Nim ~ lO-4, i s exis ence is no e ealed in
he axis Ni
=
0. In hese condi ions, in he plo equi alen o
he one shown in Fig. 2 o
(N ),
he co ec minimum does
no appea , leading o he ailu e o he basic assump ion o
Fig.
2.
Decimal
loga i hm o
ƒ
as
a unc ion o N o
he
inhomoge-
neous wa eguide wi h
θ =
80°. N m = 2.20805 co esponds o he
ac ual oo o he unc ion
D{N)
o he leaky mode; o he pu e
guided
mode
N m =Nm
=
2.28942.
Fig.
3.
Decimal loga i hm o
ƒ1
as
a unc ion o
N
o
he
inhomoge-
neous
wa eguide wi h
θ
=
20°:
a,
N0im
= 0;
b,
N0im„
=
10-4;
c,
N0im
=
1.3
X 10-4.
he LPM. This beha io can be clea ly seen in
Fig.
3
o θ
=
20°.
Howe e , in he abo e men ioned ha m ul ange o
θ
alues, he unc ion (N ) s ill shows one local minimum
simila o he one appea ing in Fig. 2, which, as al eady
poin ed ou , does no co espond o any ze o o D
(N).
This
local minimum misleads he minimiza ion algo i hm and
leads o he e oneous alues o N m shown in Table I. The
ob ained alues o Nim in hese condi ions make no sense, so
hey a e no included in he able.
In conclusion, ou esul s show ha he LPM p o ides
e y accu a e esul s in a ious cases wi h small compu e
20 June 1991 / Vol. 30, No. 18 / APPLIED OPTICS 2419
2420 APPLIED OPTICS / Vol. 30. No. 18 / 20 June 1991
imes.
Un o una ely, i does no wo k in o he cases, when
he eigen alue equa ion shows a sha ply peaked o oscilla-
o y beha io in he complex
plane.
These esul s ha e deep
implica ions because hey show ha when he LPM is ap-
plied o a gi en leaky o lossy s uc u e, jus o ob ain a good
ag eemen in a pa ial check o he app oxima e esul s
is
no
enough o gua an ee he iabili y o he p ocedu e when he
me hod is applied o a di e en sys em. In ac , i is no
ob ious how o
a
p io i es ablish he condi ions a which he
unc ion D(N) exhibi s he ha m ul beha io leading o he
ailu e o he p ocedu e. Howe e , i seems ha he p ob-
lema ic cases ollow om special si ua ions o
be
iden i ied, a
ques ion ha can only be answe ed a e u he in es iga-
ion. Finally, we ha e shown also ha when his ha m ul
beha io does no occu , he LPM cons i u es a e y powe -
ul compu a ional ool.
This wo k was sponso ed by a g an o he CIRIT o he
Ca alonia Go e nmen . The au ho s a e g a e ul o Elsa
Ga mi e o he Uni e si y o Sou he n Cali o nia o alu-
able sugges ions.
Re e ences
1.
M. R. Ramadas, E. M. Ga mi e, A. K. Gha ak, K. Thyaga ajan,
and M. R. Shenoy, "Analysis o Abso bing and Leaky Plana
Wa eguides: a No el Me hod," Op . Le .
14,
376-378 (1989).
2.
L. To ne , F. Canal, and J. He nandez-Ma co, "Leaky Modes in
Mul ilaye Uniaxial Op ical Wa eguides," Appl. Op . 29, 2805-
2814 (1990).
3.
J. C y oky and
M.
Cada, "Guided and Semileaky Modes in
Aniso-
opic Wa eguides o he LiNbO3 Type," Op . Commun.
27,
353-
357 (1978).