Microstrip circuit analog of a complex diffraction phenomenon

Microstrip circuit analog of a complex diffraction phenomenon A. Fernández-Prieto,1F. Medina,1,a兲and F. Mesa2 1Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012-Sevilla, Spain 2Departamento de Física Aplicada 1, Universidad de Sevilla, 41012-Sevilla, Spain 共Received 23 May 2009; accepted 21 June 2009; published online 14 July 2009兲 This letter presents a methodology to reproduce at microwave frequencies the physical behavior of certain diffraction structures that have been investigated in the optics community. The methodology will be exemplified with a microstrip circuit implementation of a transmission line system exhibiting an electromagnetic response that mimics the response of simple and compound diffraction gratings. The identification of such “bridge” circuit analogies could be very fruitful for the understanding and development of design strategies to devise practical components based on phenomena employed in the affine areas of optics and microwave engineering. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3176436兴 Extraordinary transmission 共ET兲of electromagnetic waves through 1D 共one-dimensional兲or two-dimensional periodic diffraction gratings has been a very popular research topic during the last ten years, since the discovery of the phenomenon1at the optical range. Even though ET was soon connected with periodicity, in a first stage, the plasmalike behavior of metals at optical frequencies 共frequency dependent complex permittivity with large and negative real part兲 was considered essential for the phenomenon. In this regard, the excitation of surface plasmon polaritons 共SPPs兲was key to explain the unexpected frequency-selective enhanced transmission through otherwise almost opaque screens. However, ET peaks can also be observed at microwave or millimeter wave frequencies.2At these frequencies, metals can no longer be regarded as plasmas, and therefore the specific constitutive relation of the metal should not play any essential role. Indeed, even perfect-conductor perforated screens exhibit enhanced transmission peaks. The SPP theory was accordingly extended to conform with these observations.3,4 The basic idea of that extension was that the structured metal surface supports certain surface-wave eigenmodes, which act as intermediary in the electromagnetic field enhancing process required for the appearance of transmission peaks 共these waves have been called spoof plasmons or SPP-Bloch waves兲. Recently some of the authors of this letter have developed an alternative equivalent theory to explain ET. Our theory does not explicitly rely on the excitation of SPP but on the perfect impedance matching condition. A comprehensive account of this point of view can be found in some recent papers that describe the nature of the model and its qualitative and quantitative predictions.5,6Although the model involves components that are familiar to electrical engineers 共transmission lines, waveguides, lumped capacitors, and inductors兲, it is not only a mere circuit analog but rather a different paradigm that can explain most of the ET systems previously studied on the basis of SPP, and also other ET systems where SPPs are not present 共for instance, total transmission through small diaphragms inside closed waveguides5,7,8兲. This equivalent-circuit approach was later extended by some of the authors9to explain the complex transmission spectra observed in 1D compound gratings. Our proposed equivalent transmission line circuit reproduces all the qualitative and quantitative details of these optical system, which were previously studied both theoretically10,11 and experimentally.12–14 Starting from this circuit analogy, the present contribution will focus on the design of a microwave passive system that behaves in a manner totally analogous to that of single/compound gratings. More specifically, what we will propose in this paper is a simple microstrip circuit that directly implements the essential details of the transmission spectra observed in an optical compound grating. This work is expected to provide a useful approach to introduce some aspects of physical optics to microwave students or engineers that are not familiar with the topic. However, more important is that it provides a “bridge” methodology to go back and forth from optics to microwave engineering, thus paving the way to design devices based on phenomena known or observed in the complementary discipline. A compound grating is a 1D infinite periodic array of groups of slits practiced on an opaque and electrically thick screen 关see an example in Fig. 1共a兲兴. A planar transverse electromagnetic wave that normally impinges on the array is a兲Electronic mail: [email protected]. x y z E H k ac1c2 h W (a) unit cell d symmetry planes AA´ z Y1 Y2 Y0Y0 C1C1 Cm Cm C2C2 port 1 port 2 A A´ h(b) z FIG. 1. 共Color online兲共a兲Two cells of a compound grating having three slits per period of length d. The structure is electrically large along xand y directions. 共b兲Equivalent transmission line circuit following Refs. 9and 15. APPLIED PHYSICS LETTERS 95, 021108 共2009兲 0003-6951/2009/95共2兲/021108/3/$25.00 © 2009 American Institute of Physics95, 021108-1 reflected and transmitted. Due to the symmetry of the problem 共geometry and excitation兲as well as its periodic nature, the study of the complete structure can be reduced to the study of a unit cell of arbitrary width Wand vertical size d, such as that depicted in Fig. 1共a兲. Following the guidelines in Refs. 9and 15, and provided that the wavelength of the impinging radiation is larger than the period of the structure, a transmission line problem can readily be identified with exactly the same scattering coefficients as those of the original diffraction problem. In particular, the transmission line circuit shown in Fig. 1共b兲is the equivalent circuit for the three slits 共per period兲system in Fig. 1共a兲. The two transmission line sections with characteristic admittances Y1and Y2 replace the slits. 共Due to the symmetry of the unit cell only two different transmission lines are required to account for the three slits兲. The free space above and below the screen is replaced by the two transmission lines with characteristic admittance Y0. In a typical situation Y1,Y2ⰇY0because c1,2Ⰶd. Note that this same model is also valid for the case of four slits per period 共using the appropriate values for Y1 and Y2兲. In general, Ntransmission lines should be used for 2Nor 2N−1 slits per period. As it is explained in Refs. 9and 15, the compoundgrating structure behaves as a Fabry–Pérot system where the transmission-line resonators are capacitively loaded. In the case of one or two slits per period, a typical Fabry–Pérot transmission/reflection spectrum is observed. For wavelengths smaller than the period d, the situation is more involved because the impinging power will split into several grating lobes. However, even within the Fabry–Pérot regime and in the frequency range where a Fabry–Pérot resonance is expected for the case of a single slit per period, both theoretical10,11 and experimental12–14 investigations show that the transmission spectrum presents profound dips when the number of slits per period is equal or larger than three. This fact was explained in10,11 using the concept of phase resonances. However, the equivalent circuit in Fig. 1共b兲can be used to deduce in a simple manner why and where the transmission dips 共zero transmission in ideal systems兲must appear. Thus, in each Fabry–Pérot order, a number of total transmission peaks equal to the number of transmission lines with different electrical lengths involved in the model is obtained. The difference between the electrical lengths of the lines in Fig. 1共b兲comes from the different loading capacitances. However this difference can also be achieved using different values of hfor each slit. A close inspection of the equations derived from the circuit model reveals that a transmission dip will always appear between each couple of total transmission peaks. For lossy structures, the transmission peaks will not reach total transmission. Additionally, and as expected from Kramers–Kronig relationships, the model also predicts sharp absorption peaks at the frequencies of the dips. Our challenge now is to find a microstrip circuit that accounts for the main features of the transmission-line network shown in Fig. 1共b兲关which is the circuit analog of the grating in Fig. 1共a兲兴. By inspection of the circuit in Fig. 1共b兲 we can see that the input and output transmission lines 共characteristic admittance Y0兲are connected to a system of much lower characteristic impedance lines 共lines with Y1and Y2兲, which would be series connected if the capacitances were ignored. Our proposal for the physical microstriplike implementation of this system is the printed circuit shown in Fig. 2. The characteristic impedance of the input and output transmission lines is 50 ⍀in order to match the measurement system. These lines are connected to a much wider microstrip 共much lower characteristic impedance兲in order to simulate the high characteristic impedance ratios involved in compound grating experiments. The number of floating conductor patches below the main resonator 共the wide microstrip section at the top surface兲should only be one to account for the circuit in Fig. 1共b兲共which is valid for three or four slits per period兲. Two floating patches below the top resonator correspond to a circuit model such as that in Fig. 1共b兲but with three intermediate transmission lines 共instead of two兲. This case mimics the behavior of compound gratings having five or six slits per period. The multilayer printed circuit has been simulated 共using the full-wave commercial software ADS MOMENTUM兲, fabricated, and measured. Since differences between the electrical lengths are essential to have transmission dips, we have chosen the lengths of the patches slightly different to make the effect more noticeable in the measured patterns. This is not required in the original compound grating because the electrical lengths of the various transmission lines involved in the problem are different enough due to differences between the edge capacitances. In Fig. 3it is shown the transmission spectra measured for two different multilayer structures. The figure shows two high transmission regions 共corresponding to first and second order Fabry–Pérot resonances兲where one or two transmission dips are clearly observed. This spectrum is qualitatively identical to the transmission spectra reported in previous theoretical and experimental works for compound gratings.11–14 In particular, when two transmission lines are involved in the microstrip circuit analog, a single transmission dip is observed. Two separate dips are obtained when three microstrip lines are present, what corresponds to the cases of five or six slits per period. This is exactly what happens in the compound grating problems and what our transmission line model predicts. From the measurement of the reflection coefficient we can obtain the power loss, which has also been plotted in Fig. 3. Note that strong absorption peaks are observed at the frequencies of the transmission dips 共as it has been verified in Ref. 12 for gratings兲. Similar results have been found by the authors for other geometries and substrates.16 In brief, this letter has shown a simple microstrip circuit whose behavior is fully equivalent to the observed transmission behavior of compound 1D gratings studied by optics substrate TOP VIEW 50 W 50 W50 W 50 W l1 l2 l3 LATERAL VIEW er er W FIG. 2. Microstrip circuit that mimics compound gratings having five or six slits per period. For three or four slits per period one of the floating patches embedded in the substrate should be removed. 021108-2 Fernández-Prieto, Medina, and Mesa Appl. Phys. Lett. 95, 021108 共2009兲 researchers. This equivalence has been achieved after finding a transmission-line circuit analog of the phenomenon, and the implementation of this equivalent circuit at microwave frequencies. The experiments demonstrate that the circuitlike approach completely captures the physics of the transmission in these systems. The advantage is that a few parameters, several of which are known in closed form, are enough to characterize the whole reflection/absorption/transmission spectrum. Moreover, the rationale presented in this work is expected to be very fruitful in the design of experiments and/or devices in both disciplines: optics and microwaves. For instance, the electrical response observed in the case here treated of compound gratings with three/four slits per period can be seen as a notch filter of poor behavior. However, using a microwave engineering standpoint, one can envisage that if, instead of a single resonant stage, something similar to a Chebyshev transformer was used to increase the bandwidth of the pass band, the resulting optical structure could behave as a good notch filter. This work has been supported by the Spanish Ministerio de Ciencia e Innovación 共Project No. TEC2007-65376兲, Consolider Ingenio 2010 “Engineering Metamaterials” 共Grant No. CSD2008-00066兲, and by Spanish Junta de Andalucía 共Project No. TIC-253兲. 1T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature 共London兲391, 667 共1998兲. 2M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín Moreno, and F. J. García-Vidal, IEEE Trans. Antennas Propag. 53, 1897 共2005兲. 3J. B. Pendry, L. Martín-Moreno, and F. J. 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Skigin, “Extraordinary transmission through arrays of slits: A circuit theory model,” IEEE Trans. Microwave Theory Tech. 共to be published兲. 16F. Medina, A. Fernández-Prieto, and F. Mesa, Proceedings of the 39th European Microwave Conference 共EuMC’09兲, Rome, Italy, 2009 共unpublished兲. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 56 Power Loss ( ma g nitude ) |S | 12 one patch 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 Power Loss (magnitude) Frequency (GHz) Two patches |S | 12 FIG. 3. Measured 兩S12兩共black lines兲of the microstrip circuit analog with one/two patches corresponding to compound 1D gratings with three共four兲/ five共six兲slits per period. Power loss is included 共dashed gray lines兲. One dip per Fabry–Pérot resonance is found in the first case and two dips appear in the second case. Absorption peaks are found at the resonances. Parameters of the structure in Fig. 2:W=30.25 mm, l1=38.43 mm, l2=42.0 mm, l3=45.57 mm, h=0.49 mm, ␧r=2.43, and tan ␦ =0.002. 021108-3 Fernández-Prieto, Medina, and Mesa Appl. Phys. Lett. 95, 021108 共2009兲