electromagnetic propagation in unbounded inhomogeneous chiral media using the coupled mode method

especially for the highest one. In addition, the measured datademonstrated that the proposed antenna maintains SWB charac-teristics, possessing a ratio bandwidth of 19.7:1 from 0.595 to8.95GHz for VSWR � 2. The radiation patterns of the antennawere simulated and the results are very close to those of a con-ventional planar monopole antenna. The simulated peak gain isplotted in Figure 6, showing a low radiation level in the notchedband.

5. CONCLUSION

In this paper, a rectangular monopole antenna having a band-notchfunction is presented for SWB applications. The study on the bandnotch function has been introduced in detail. Experimental resultshave shown an obvious rejected band in the WLAN frequencies.The bandwidth of the antenna covers 0.595–8.95GHz (15:1) forVSWR � 2, while the band rejection performance in the frequencyband of 5.07–5.85 GHz is obtained.

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foun-dation of China under grant no. 60571053 and the ShanghaiLeading Academic Discipline Project under grant no. T0102.

REFERENCES

1. S.Y. Suh, W.L. Stutzman, and W.A Davis, A new ultra-widebandprinted monopole antenna: The planar inverted cone antenna (PICA),IEEE Trans Antennas Propagat 52 (2004), 1361–1365.

2. Xian-Ling Liang, Shun-Shi Zhong, and Wei Wang, Tapered CPW-fedprinted monopole antenna, Microwave Opt Technol Lett 48 (2006),1411–1413.

3. Wooyoung Choi, Jihak Jung, Kyungho Chung, and Jaehoon Choi,Compact microstrip-fed antenna with band-stop characteristic for ultra-wideband applications, Microwave Opt Technol Lett 47 (2005), 89–92.

4. Xinan Qu, Shun-Shi Zhong, and Wei Wang, Study of the band-notchfunction for a UWB circular disc monopole antenna, Microwave OptTechnol Lett 48 (2006), 1667–1670.

5. Wen-Chung Liu and Ping-Chi Kao, CPW-fed triangular antenna with afrequency-band notch function for ultra-wideband application, Micro-wave Opt Technol Lett 48 (2006), 1032–1035.

6. CST-Microwave Studio5.0, User’s Manual, 2004.

© 2007 Wiley Periodicals, Inc.

ELECTROMAGNETIC PROPAGATION INUNBOUNDED INHOMOGENEOUSCHIRAL MEDIA USING THE COUPLEDMODE METHOD

Alvaro Gomez,1 Ismael Barba,1 Ana C. L. Cabeceira,1

Jose Represa,1 Angel Vegas,2 and Miguel Angel Solano2

1 Departamento de Electricidad y Electronica, Universidad deValladolid, Paseo Prado de la Magdalena s/n, 47011 Valladolid,Spain; Corresponding author: [email protected] Departamento de Ingenierıa de Comunicaciones, Universidad deCantabria, Avenida de los Castros s/n, 39005 Santander, Spain

Received 11 April 2007

ABSTRACT: The coupled mode method (CMM) is a seminumericalmethod for studying electromagnetic propagation, originally formulatedfor closed structures. We show how this method can be used for obtain-ing the propagation constants and electromagnetic field in unbounded

Figure 4 Simulated VSWR versus frequency of different slot width ws

(d � 1 mm, ls � 14 mm)

Figure 5 Measured and simulated VSWR of proposed antenna (Param-eters: d � 1 mm, Ls � 14 mm, ws � 0.4 mm, Hs � 3 mm)

Figure 6 Simulated antenna gain versus frequency

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 2771

isotropic chiral media. We also characterize single and periodic cas-cade discontinuities made of isotropic chiral slabs by means of the modematching method combined with the CMM. The results are tested with arobust FDTD technique, modified to model bi-isotropic media and withanalytical solutions. In all cases good agreement is found. © 2007Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2771–2779,2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22863

Key words: coupled mode method; unbounded chiral media; disconti-nuities; chiral slabs

1. INTRODUCTION

The coupled mode method (CMM) is a seminumerical tech-nique, first proposed by Schelkunoff [1], useful for studyingelectromagnetic wave propagation in any kind of nonconduct-ing material bounded by a perfect electric conductor (PEC).Typical examples are parallel-plate, rectangular, or cylindricalwaveguides partially filled with isotropic [2], anisotropic [3–7],and isotropic chiral [8 –11] or bianisotropic media [12–14].Basically, the CMM is a method of moments which consists inexpanding the electromagnetic field components inside thewaveguide containing the medium in terms of a set of basefunctions, and calculating the coefficients of the expansions [1,2]. The base functions are the components of the electromag-netic field in an empty waveguide with the same internaldimensions as the partially filled waveguide. For this reason, wewill use the term base modes throughout the article to refer tothese base functions. The CMM has two main advantages. First,all the modes in the structure under analysis (which we callcharacteristic modes) are obtained in a single calculation. Thisavoids using mathematical techniques to search for the zeroesof complex characteristic equations, and also avoids the loss ofsolutions. The second important advantage is that CMM, whencombined with a traditional mode matching method (MMM),gives a hybrid technique that allows the characterization ofdiscontinuities, without performing any kind of numerical in-tegration in the computation of the generalized scattering ma-trix [6, 7].

Now consider the following two questions. Could a closedwaveguide with an electrically large cross-section model (con-sider, for instance, an ICM inside a rectangular waveguide [seeFig. 1] whose width and height are very large when comparedwith the wavelength) an unbounded medium? And could this bedone with a technique like the CMM? The answers to thesequestions are addressed in this article. Using a formulation ofthe CMM for closed waveguides, we simulate the electromag-netic behavior of an unbounded isotropic chiral media (ICM).The strategy requires enclosing the ICM by a rectangularwaveguide (although we use a rectangular waveguide, in fact asquare waveguide, it is not essential to use such geometry.Cylindrical waveguides are equally valid) and moving away thePEC’s (see Fig. 1). We compare our results with the analyticalones that can be obtained by solving the wave equation of anICM [15, 16].

After studying the wave propagation in an unbounded ICM, weapply the MMM [6, 7] to characterize discontinuities in the powerpropagation direction of a structure composed of a number of ICMslabs. The results obtained with the hybrid CMM-MMM techniqueare compared with analytical ones and also with those obtainedwith a modified FDTD technique [17, 18] suitable for generalbi-isotropic media.

2. THEORY

2.1. Analytical Solution for an Unbounded ICMThe basic constitutive relations for an ICM can be written as [19]

D� � �r�0 E� � j� ��0�0 H� , (1a)

B� � �r�0 H� � j��0�0 E� , (1b)

where �r is the relative permittivity, �0 the vacuum permittivity, �r

the relative permeability, �0 the vacuum permeability, and � thechirality or Pasteur parameter. The solution of the wave equationfor an unbounded ICM can be split into two wavefield componentsE� �, H� � and E� �, H� � [15, 16]. The field with subindex “�” repre-sents a right-handed circular polarized (RHCP) wave and the fieldwith subindex “�” represents a left-handed circular polarized(LHCP) wave. The corresponding wavenumbers are given by[17, 18]

k� � ��1 �

��r�r�. (2)

In a homogeneous chiral medium, these two wavefields are notcoupled. The relations between the electric and magnetic fields foreach polarization are

E� � � j��H� �, (3)

where �� and �� are, respectively, the wave impedance of eachwavefield. For an ICM, these wave impedances reduce to the waveimpedance of an isotropic medium, i.e., �� /�.

2.2. Numerical Solution Using the CMMDepending on which electromagnetic fields (characteristic modes)are expanded in terms of the fields of the empty waveguide (basemodes), two types of formulations of the CMM can be seen in thebibliography [1–3, 5, 12]. The formulations can be classified asEH-type if we expand the electric E� and magnetic H� fields, and asEB-type if the H� field is replaced by the B� field [14, 20]. TheEB-formulation gives a correct description of the magnetic field incontact with the PEC’s whereas the EH-formulations do not [20].In turn, there are two types of EH-field formulations: the EHI-formulation, which requires the inversion of some matrices tocompute the coefficients of the expansions, and the EHD-formu-

Figure 1 Cross-section of a general rectangular waveguide fully loadedwith an isotropic chiral medium (ICM)

2772 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 DOI 10.1002/mop

lation, which does not require such inversions. In this work, weuse an EHI-type formulation [11] for two reasons. On the onehand, we have shown that for isotropic or anisotropic (ferrite)materials [5, 6] and for ICM [11] our EHI-formulation givesmore accurate results than an EHD-type formulation, especiallyfor high values of the constitutive parameters. On the otherhand, the most appreciable differences in the results of theelectromagnetic field provided by the EB- and the EH-formu-lations [10, 14] are observed in the proximity of the PEC’sbounding the ICM, with these differences decreasing near thecenter of the waveguide. In this article, we are going to simulatean unbounded ICM, with the PEC’s (see Fig. 1) far from thecenter of the waveguide. Consequently, we will focus ourattention only on the proximity of its center and, therefore, it isreasonable to assume that the results for the electromagneticfield provided by the EHI-formulation will be good enough forour aims. Furthermore, from the mathematical point of view,any EH-formulation is much simpler than an EB-formulation.

Therefore, we will only present results for the EHI-formu-lation. In any case, the mathematical development for bothEHD- and EHI-formulations of the CMM when applied to ICMcan be seen in detail in [11]. Here, we are only interested in theresults for the particular case of electrically large cross-sectiondimensions. As will be seen in the Results section, the numer-ical and analytical results for the unbounded ICM show verygood agreement.

2.3. Numerical Solution with FDTDTo validate the results obtained, we have also considered thesolutions provided by an extension of the FDTD technique [17,18], suitable for the modeling of bi-isotropic media.

Our FDTD algorithm basically consists of two steps: first, wecompute the values of D� and B� by means of the discretizedMaxwell’s curl equations and then the new values of H� and E� areupdated using the constitutive relationships.

In Yee’s algorithm, the electric and magnetic fields are com-puted in different space and time positions [21]. However, thepeculiar constitutive equations (1) of chiral media relate the elec-tric and magnetic fields in the same point and at the same time

instant. As a result, the unit cell defined in our FDTD schemeincludes four quantities, namely D� , B� , E� , and H� , related by theconstitutive equations as explained in [17]. The constitutive rela-tionships (1) are expressed in the frequency domain. To be incor-porated into our FDTD formulation these equations must be trans-lated to the time domain. This can be done either by means ofconvolution techniques [17] or by obtaining the differential equa-tion associated with such behavior [18]. In this case, we havechosen the second option.

3. RESULTS

The aim of this section is to check the results for unbounded ICMobtained with the CMM. These results will be compared withanalytical solutions and with results provided by our FDTD tech-nique. Several structures will be analyzed: first, ICM’s that arehomogeneous in the propagation direction, and second, inhomo-geneous structures, i.e., structures with discontinuities in the prop-agation direction. For homogeneous structures the important as-pects to be analyzed are:a. Convergence of the values of the wavenumbers k� and k�

obtained with the CMM versus the number of base modesintroduced in the expansions, and also versus the transversedimensions (width a and height b in Fig. 1) of the rectangularwaveguide PEC’s. These comparisons will establish the appro-priate values of the dimensions a and b, and the number of basemodes which have to be taken into account for a good accuracyof the numerical results.

b. The existence of an RHCP plane wave and a LHCP plane wavein the proximities of the center of the cross-section of therectangular waveguide. These plane waves correspond to thebasic wavefields E� �, H� � and E� �, H� �, respectively [16].

c. The numerical value of the wave impedance obtained with theCMM.For inhomogeneous structures, we use the hybrid technique

CMM-MMM, for characterizing the structures with discontinuitiesin the propagation direction. Two structures with a straightforwardanalytical solution are analyzed first, one is a simple vacuum-ICMdiscontinuity [see Fig. 2(a)], and the other is a double discontinu-

Figure 2 Structures under analysis. (a) Simple discontinuity, vacuum-ICM; (b) double discontinuity, vacuum-ICM slab-vacuum; (c) two slabs of ICMinserted in vacuum; (d) 10 slabs of ICM inserted in vacuum.


ity, vacuum-ICM slab-vacuum [see Fig. 2(b)]. Finally, we analyzemore complex structures composed of multiple vacuum-ICM slabdiscontinuities [see Fig. 2(c)], including periodic structures [seeFig. 2(d)]. For these last structures we make comparisons withresults provided by the FDTD technique.

3.1. Homogeneous ICM

3.1.1. Convergence of the Wavefield Wavenumbers. We havecomputed the solution of the propagation constants � � � j� for thecharacteristic modes in a square waveguide totally filled with alossless ICM characterized by �r � 9, �r � 1, � � 1.5, being the wavefrequency f � 9 GHz. As the size of the cross-section of thewaveguide is increased, the phase constants of the first half of thecharacteristic modes provided by the CMM approaches to k� (pro-vided the size of the cross-section is large enough) and the phaseconstants of the second half approaches to k�. For the authors’knowledge, this behavior has not been reported before in the literature.

It can be seen how the relative error decreases as the numberof base modes and the dimensions of the waveguide are in-creased. If the dimensions of the waveguide are not electricallylarge (a � �0), the relative error is high even though a largenumber of base modes are used. If the dimensions are electri-cally large (a �� 0) the error approaches zero as the number ofbase modes is increased. Here, it is important to note that thebase modes are introduced in the expansions by increasingcutoff frequency. But, as we are interested only in the zone nearthe center of the waveguide, the TEnm and TMnm base modescorresponding to the empty waveguide, with n � m even, arenot included, because they have no influence on the computa-tion of wavenumber either k� or k�. From Table 1, we can inferthat with a waveguide of a � b � 10 m, we obtain a very goodapproximation for k� (the same conclusion is obtained for themodes approaching to k�). These values for the width (orheight) of cross-section of the waveguide, are at f � 9 GHz (�0

� 3.3 cm) around 300 times the value of �0. This fact can begeneralized, i.e., it can be guaranteed that if we want to repro-duce the wavenumber k� (or k�) for an unbounded ICM usingthe CMM for bounded ICM we need to use a square waveguidewith transverse dimensions a � b � 300�0, and, of course, alarge enough number of base modes.

Figure 3 shows the variation of k� (� of the fundamentalmode) and k� (� of highest order mode) normalized to thevacuum wavenumber k0 versus the number of base modesintroduced in the expansions of the electromagnetic field com-ponents (convergence diagrams). On the basis of Figure 3 wecan guarantee that very good accuracy for k� and k� isachieved by using 20 –30 base modes, although with only 10base modes the relative error is less than 0.1%.

3.1.2. Electromagnetic Field. The electromagnetic field providedby the CMM converges more slowly than the wavenumbers [2, 6].Therefore, to ensure good accuracy in the electromagnetic field,the following results are obtained by using 50 base modes.

Figure 4 shows the top view of the real and imaginary partsof electric field components Ex, Ey, and Ez normalized to themodulus of the maximum value of the electric field versus thecoordinates of the cross-section of the waveguide correspond-ing to the wavefield E� �, H� �. The characteristics of the structureare displayed in the figure caption. Two important observationscan be pointed out. Firstly, as expected, the field represents aTEM mode, because the Ez-component is practically zero (thesame occurs for Hz) for the entire cross-section. Secondly, if wefocus our attention on the vicinity of the center of thewaveguide, i.e., far enough from the PEC walls to avoid edgeeffects, the field pattern is practically constant with the trans-verse coordinates. We are only interested in the behavior of theelectromagnetic field in this area. The size of this arbitrary areadepends on the frequency and on the contour of the waveguide.From a number of simulations, we have found that the electro-magnetic field of an unbounded ICM, using the results of theCMM for an ICM enclosed by a rectangular (or square)waveguide, is well-reproduced on a hypothetical surface cen-tered on the cross-section of the waveguide whose area is �100times smaller than the cross-section area of the waveguide.

Figure 5 shows the electric field components normalized tothe modulus of the maximum value of the electric field, corre-sponding to the wavefield E� �, H� �, in the vicinity of the centerof the waveguide cross-section, versus the transverse coordi-nates. The same is shown in Figure 6 but for the magnetic fieldcomponents. In this particular case, we have chosen to plot a1 � 1 m2 surface located in the center of the waveguide.

Inspection of Figures 5 and 6 shows that in the central part of thewaveguide the amplitudes of the transversal electromagnetic fieldcomponents are constant, whereas the longitudinal ones are null(infact these components are below the numerical accuracy of themethod, but they are not erroneously different from zero). It can also

TABLE 1 Relative Error (in %) of � With Respect to theExact Value k� Versus the Side (a) of the Cross-Section of aSquare Waveguide and the Number of Base Modes (N)

N

a (m)

0.033 0.1 1 10

5 3.44 2.42 2.27 2.2710 1.37 0.23 0.07 0.0720 1.25 0.17 �0.01 �0.0130 1.21 0.15 �0.01 �050 1.21 0.14 �0.01 �0

The exact value at f0 � 9 GHz is k� � 4.5k0.

Figure 3 Convergence diagrams of k� and k� normalized to thefree-space wavenumber versus the number of base modes for thestructure of Figure 1. Data: (a) and (b) �r � 4, �r� 1, and � � 0.5, sok�/k0 � 2.5 and k�/k0 � 1.5; (c) and (d) �r � 9, �r � 1, and �r � 1.5,so k�/k0 � 4.5 and k�/k0 � 1.5; a � b � 10 m and f0 � 9 GHz. [Colorfigure can be viewed in the online issue, which is available at www.in-terscience.wiley.com]


be seen that for the E��, H� � wavefield, the electric field has the formE0� ja�x � a�y and the magnetic field H0��a�x � ja�y, where E0�

and H0� are the amplitudes and a�x, a�y are the unitary vectors. There-fore, we can conclude that in the central part of the waveguide the

electromagnetic wavefield E��, H� � behaves like a RHCP plane wave.Analogous results would be obtained if we plotted the electromag-netic field of the wavefield E��, H� �, with the only difference being thatin this case the result would be a LHCP wave.

Figure 4 Electric field components (real and imaginary parts), normalized to the modulus of the maximum value of the magnetic field, as a function ofthe x- and y-coordinates over the whole cross-section of the waveguide for the structure of Figure 1. The number of base modes is 50. Data: �r � 9, �r �1, and �r � 1.5; a � b � 10 m and f0 � 9 GHz

Figure 5 Electric field components (real and imaginary parts), normalized to the modulus of the maximum value of the magnetic field, as a function ofthe x- and y-coordinates over the whole cross-section of the waveguide for the structure of Figure 1. The number of base modes is 50. Data: �r � 9, �r �1, and �r � 1.5; a � b � 10 m and f0 � 9 GHz


3.1.3. Numerical Value Impedance. The last magnitude that weanalyze for the unbounded ICM is the numerical value of thewave impedance provided by the CMM. The analytical value is� � ��/� [16]. Figure 7 shows the numerical value of the

wave impedance (real and imaginary parts) obtained by usingthe CMM for the structure of Figure 1 with a � b � 10 m andf � 9 GHz; �r � 9, �r � 1, and � � 1.5. The exact value of thewave impedance for this case is � � 120�/3 � � 125.66 �.

Figure 6 Electric field components (real and imaginary parts), normalized to the modulus of the maximum value of the magnetic field, as a function ofthe x- and y-coordinates over the central cross-section of the waveguide for the structure of Figure 1. The number of base modes is 50. Data: �r � 9, �r �1, and �r � 1.5; a � b � 10 m and f0 � 9 GHz

Figure 7 Wave impedance (real and imaginary parts), for the RCP wave (��) and for the LCP wave (��) as a function of the x- and y-coordinates over the centralcross-section of the waveguide for the structure of Figure 1. The number of base modes is 50. Data: �r � 9, �r � 1, and �r � 1.5; a � b � 10 m and f0 � 9 GHz


This value agrees perfectly with the numerical data obtainedfrom the CMM with a relative error around 0.01%.

3.2. Inhomogeneous StructuresOnce we have confirmed that the unbounded ICM can be analyzedby using a formulation of the CMM that is valid for boundedstructures, we are going to characterize some inhomogeneousstructures composed of different materials (vacuum and ICM)following the propagation direction. In particular, we are going toanalyze the variation of the modulus of the reflection coefficientversus frequency for a linear polarized plane wave incident on astructure with one or more discontinuities in the propagationdirection. Firstly, we simulate the simple discontinuity shown inFigure 2(a), and secondly, the double discontinuity of Figure 2(b).The results are compared with analytical ones obtained from [16].

To describe the constitutive parameters of the ICM’s we use adispersive model. For permittivity and permeability parameters weuse the Lorentz resonant model [22]

�r� � � ��s � � �0e

2

�0e2 � �2 � 2j�0e�e�

, (4)

�r � � � ��s � � �0m

2

�0m2 � �2 � 2j�0m�m�

, (5)

where �s and �s are, respectively, the effective relative permittivityand permeability parameters for low frequencies; � and � arethe relative permittivity and permeability parameters for highfrequencies; �0e and �0m are the resonance frequencies; and �e and�m are the electric and magnetic damping factors. For the chiralityparameter, we use the Condon model [23],

�� 0

2�

�02 � �2 � 2j�0��

, (6)

where � is the time constant, �0 is the resonance frequency, and �is the chiral damping factor.

Figure 8 shows the variation of the modulus of the reflection

coefficient versus frequency, for normal incidence from the vac-uum on a semi-infinite unbounded ICM [see Fig. 2(a)]. The con-stitutive parameter values of the ICM are �s � 6, � � 4, �s � 1.5,� � 1, � � 10 ps; �0e � �0m � �0 � 2�f0, where f0 � 5 GHz;and �e � �m � � � 0.01. For the numerical solution provided bythe hybrid CMM-MMM technique, we have chosen a squarewaveguide with a � b � 10 m and 50 base modes. In FDTD, wehave chosen a one-dimensional domain of �x � 0.033 mm and�t � 0.1 ps. The time domain results (after 100,000 time steps) aretranslated to the frequency domain using a discrete Fourier Trans-form.

It can be seen that analytical and numerical (both CMM-MMMand FDTD) results fit very well in all the frequency range. Con-sequently, the approximation of the unbounded ICM using a rect-angular waveguide with a very large cross-section is correct.

In Figure 9 the reflection coefficient modulus in normal incidenceon a slab of ICM inserted in free space is displayed versus frequency[Fig. 2(b)]. The simulation data are the same as in Figure 8 and thewidth of the slab is d � 10 mm. The agreement between the numericaland the analytical results is again very good.

Figures 10 and 11 show similar results to the two previousfigures but for the structures of Figures 2(c) and 2(d), respectively.In both cases the gap between two chiral slabs is e � 10 mm.Although it is possible to obtain analytical results for these struc-tures, the mathematical process is so bothersome, that we decidedto compare the results of the CMM-MMM with the FDTD. Theresults obtained by the two numerical techniques are very similar.

4. CONCLUSIONS

In this article, we have shown that an unbounded ICM can besimulated by using a waveguide with a sufficiently large cross-section filled with the ICM (in our analysis we use a rectangularwaveguide). We have shown that the phase constants obtainedwith the CMM are practically identical to the analytical ones. Therelative error when using about 40 base modes is less than 0.01%.This difference may be due to the approximation related to thefinite dimensions of the rectangular waveguide. We have seen thatin the vicinity of the center of the cross-section of a rectangular

Figure 8 Reflection coefficient module versus frequency for a semi-unbounded bi-isotropic chiral medium and free-space discontinuity [Fig.2(a)]. �s � 6, � � 4, �s � 1.5, � � 1, � � 10 ps, f0 � 5 GHz, and � �0.01. For the CMM solution a � b � 10 m with 50 base modes. In FDTD�x � 0.033 mm and �t � 0.1 ps. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com]

Figure 9 Reflection coefficient module versus frequency for a bi-isotro-pic chiral slab (d � 10 mm) inserted in free-space [Fig. 2(b)]. �s � 6, � �4, �s � 1.5, �s � 1, � � 10 ps, f0 � 5 GHz, and � � 0.01. For the CMMsolution a � b � 10 m with 50 base modes. In FDTD �x � 0.033 mm and�t � 0.1 ps. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]


waveguide, the CMM gives the behavior of the electromagneticfield in an ICM, i.e., two plane waves, with the wavefield E� �, H� �

being a RHCP wave and the wavefield E� �, H� � a LHCP wave. Wehave also computed numerically the values of the wave impedanceof both wavefields, and we have shown that these values agreevery well with the analytical ones. Finally, using two differentnumerical techniques, one in the frequency domain (CMM-MMM)and the other in the time domain (FDTD), we have computed themodulus of the reflection coefficient, as a function of frequency,for some different structures containing discontinuities in the prop-agation direction. In all cases, the results obtained by the two

techniques are practically identical, and they agree very well withanalytical results when these are available.

ACKNOWLEDGMENTS

This work was supported by the Direccion General de Investiga-cion of the Spanish Ministerio de Educacion y Ciencia, under theprojects/grants TEC2006-13268-C03-01 and TEC2006-13268-C03-03.

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Figure 10 Reflection coefficient module versus frequency for a twobi-isotropic chiral slabs (d � 10 mm) inserted in free-space [Fig. 2(c)].Distance between the chiral slabs e � 10 mm. �s � 6, � � 4, �s � 1.5,�s � 1, � � 10 ps, f0 � 5 GHz, and � � 0.01. For the CMM solution a �b � 10 m with 50 base modes. In FDTD �x � 0.033 mm and �t � 0.1 ps.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

Figure 11 Reflection coefficient module versus frequency for 10 bi-isotropic chiral slabs (d � 10 mm) inserted in free-space [Fig. 2(d)].Distance between chiral slabs e � 10 mm. �s � 6, � � 4, �s � 1.5, �s �1, � � 10 ps, f0 � 5 GHz, and � � 0.01. For the CMM solution a � b �10 m with 50 base modes. In FDTD �x � 0.033 mm and �t � 0.1 ps.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]


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© 2007 Wiley Periodicals, Inc.

EXPERIMENTAL INVESTIGATION OFOPTICAL THYRISTOR OPERATING ASOPTICAL HARD-LIMITER

Tae-Gu Kang,1 Dong-Muk Choi,1 Woon-Kyung Choi,2

Young-Wan Choi,2 Seong-Joo Kim,2 and Woo-Kyung Choi21 School of Electrical and Electronic Engineering and ComputerScience, Kyungpook National University, 1370 Sangyeok-dong, Buk-gu, Daegu 702–701, South Korea; Corresponding author:[email protected] School of Electrical and Electronic Engineering, Chung-AngUniversity, 221 Heuksuk-Dong, Dongjak-ku, Seoul 156–756, SouthKorea

Received 16 April 2007

ABSTRACT: We demonstrate the optical thyristor operating as opticalhard-limiter (OHL) in the optical channel and AND gate logic element(AGLE). The oxidized PnpN vertical-cavity laser (VCL)–depleted opticalthyristor (DOT) clearly shows the nonlinear s-shaped current-voltageand lasing characteristics. A switching voltage of 5.24 V, a holding volt-age of 1.50 V, and a threshold current of 0.65 mA is measured, makingthese devices attractive for optical processing applications. The VCL–DOT operating as OHL would enhance the system performance becauseit would exclude some combinations of interference patterns from caus-ing errors as in all previous works. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 2779–2780, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22850

Key words: AND gate logic element; optical hard-limiter; vertical-cav-ity laser—depleted optical thyristor

1. INTRODUCTION AND CONCEPT

Recently, great interest has been given to the design and analysisof optical communication networks. A fiber-optic code division

multiple access (FO-CDMA) system provides a natural mean tobuild all optical signal processors capable of implementing high-speed all optical encoders and decoders, thus avoiding the expen-sive electrooptic and optoelectronic conversions and foreseeableelectronic bottleneck in future multigigabit networks [1, 2]. Inparticular, we discuss a means of reducing the effective multiple-access interference (MAI) signal by placing an optical thyristoroperating as OHL, as shown in Figure 1.

In conventional FO-CDMA decoder without an optical thyris-tor operating as OHL, the received optical signal pattern[1004000100000] has a maximum value of “6” at the output of itsmatched filter (correlator). If the threshold level of the receiver isset at “5,” the error would be caused. But if the interferencepatterns were hard-limited by an optical thyristor operating asOHL, then it would reduce to [1001000100000]. This patternwould not cause an error at the output of decision circuit when thethreshold is at “5.” On the other hand, the interference signal bydecoding process can be eliminated when an optical thyristoroperating as OHL constituted in the AGLE operates the ON statefor the same or bigger optical intensity than specified level andOFF state for the smaller signal than specified level [3]. In thisarticle, we demonstrate the oxidized PnpN VCL–DOT operating asOHL in the optical channel and AGLE for FO-CDMA system.This oxidized laser also has a high sensitivity to optical input light,because it is transparent to the light, which is absorbed in the activeregion [4]. The oxidized device exhibits a low threshold currentand is simultaneously sensitive to the optical input light.

2. DEVICE DESIGN AND FABRICATION

Figure 2 shows top view of the VCL–DOT, which was grown onn-GaAs substrates by metal organic chemical vapor deposition(MOCVD). These devices have a PnpN triple junction structurewith two distributed Bragg reflector (DBR) mirrors consisting ofAl0.9Ga0.1As/Al0.16Ga0.84As layers with linearly graded transitionlayers. The alloy composition grading allows reduction in theseries resistance of the devices. The cavity space between the topand the bottom mirrors is 4�, where � is the emission wavelengthin the semiconductor medium. This device structure allows eitherelectrical or optical switching from the OFF state to the ON state.During the ON state the VCL–DOT has low impedance and emitslaser light. To increase the efficiency of optical emission, anundoped multiple quantum well layer has been incorporated as theactive layer of the VCL–DOT. The active region also acts as anabsorption region for optically switching the VCL–DOT. Althoughthe active region is thin, enhancement of absorption is expectedbecause of multi-reflection by the two DBR mirrors. Design pa-rameters such as doping concentration and layer thickness of the

Figure 1 Optical hard-limiters (VCL–DOT) placed in the channel and AGLE


electromagnetic propagation in unbounded inhomogeneous chiral media using the coupled mode method

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