2. E. Desurvire, Erbium-doped fiber amplifiers: Principles and applica-tions, New York, Wiley, 1994.

3. M.A. Mahdi, S. Thirumeni, P. Poopalan, S. Selvakennedy, F.R. Ma-hamd Adikan, W.Y. Chan, and H. Ahmad, Effects of self-saturation inan erbium-doped fiber amplifier, Opt Fiber Technol 6 (2000), 265–274.

4. R.I. Laming, M.N. Zervas, and D.N. Payne, Erbium-doped fiber ampli-fier with 54-dB gain and 3.1 dB noise figure, IEEE Photon Technol Lett4 (1992), 1345–1347.

5. W.I. Way, T.H. Wu, A.Y. Yan, M. Andrejco, and C. Lin, Optical powerlimiting amplifier and its applications in an SONET self-healing ringnetwork, Lightwave Technol 10 (1992), 206–214.

6. C.H. Kim, C.R. Giles, and Y.C. Chung, Two-stage optical limiting fiberamplifier using a synchronized etalon filter, IEEE Photon Technol Lett10 (1998), 285–287.

7. B. Bouzid, M.B. Ali, and M.K. Abdullah, A high-gain EDFA designusing double pass amplification with a band-pass filter, IEEE PhotonTechnol Lett 15 (2003), 1195–1197.

8. B. Heens, X. Wang, L. Meuleman, F. Ravet, and M. Blondel, Improve-ment of the EDFA design for both preamplifier and high-power appli-cations, Proc MELECON’98 2 (1998), 1404–1408.

© 2004 Wiley Periodicals, Inc.

AN EFFICIENT FINITE-DIFFERENCEFREQUENCY-DOMAIN METHODINCLUDING THIN LAYERS

Carl M. Arft and Andre KnoesenDept. of Electrical and Computer Engineering2064 Kemper HallUniversity of CaliforniaDavis, CA 95616-5294

Received 11 March 2004

ABSTRACT: A method is presented to incorporate thin, lossy layersinto a finite-difference frequency-domain model while retaining a coarsegrid-spacing. This approach significantly reduces computational effort.The new method is validated by a comparison with analytical solutions.© 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 40–44,2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20369

Key words: finite-difference frequency-domain method; thin films;waveguide analysis

1. INTRODUCTION

A well-known method for the numerical solution of the propaga-tion constants and electromagnetic fields of general guided-wavestructures is the finite-difference method in the frequency domain,or the FDFD method [1–5]. Using the FDFD method, thewaveguide modes and associated propagation constants are di-rectly solved, in contrast to the finite-difference time-domain(FDTD) method, which involves multiple propagation steps andpost-processing of the fields to determine the desired modal pa-rameters. Because the solution of the large matrix eigenvalueequation inherent in the FDFD method is computationally inten-sive, finding appropriate ways to reduce the size of the matrix (thatis, reduce the number of discrete field components involved) orincrease the efficiency of its solution is critical. Initially, onlypartial-wave methods were considered [1, 2, 5]. Recently, two-dimensional full-wave FDFD (2D FDFD) algorithms were pro-posed using 6-field [3] and 4-field [4] formulations. The existingmethods are difficult to apply to structures incorporating extremelythin features, for example, active waveguide components with thin,

semi-transparent electrode or resistive layers, or optical biosensordevices incorporating adsorbed molecular layers. In this context, athin layer is defined as a layer of material that is thinner than thee�1 penetration depth of the electromagnetic field into the layer ifit is lossy, or thinner than the wavelength if the layer is a dielectric.Utilizing a globally fine computational grid in this situation leadsto a prohibitively large matrix eigenvalue equation, and evenvarious local sub-grid methods (see, for example, [6, 7]) still leadto an impractical matrix size when the layers are extremely thin.

This paper presents an FDFD method that includes the effectsof thin dielectric or lossy layers while retaining a grid spacing thatis many times larger than the thickness of the layer. This greatlyreduces the number of grid points (discrete field components)required for the numerical solution. The method is derived byincorporating the ideas of the contour-path (CP) approximation [8,9], originally developed for the FDTD method, into the 2D-FDFDmethod. The resultant method, referred to as the FDFD-CPmethod, provides a computational advantage over numerical meth-ods in which multiple grid points or elements must be placedwithin the thin layer in order to obtain accurate results.

2. THEORY

The CP approximation, as originally described in [8], is basedupon the observation that Yee’s FDTD equations [10] may beinterpreted as approximations of Maxwell’s equations in integralform, given by

�

�t ��s

�H� � ds � ��c

E� � d�, (1)

�

�t ��s

�E� � ds � �c

H� � d�, (2)

where the permittivity and permeability are diagonal tensors andthe permittivity may be complex valued to account for electricalconductivity. The effect of a sub-grid size layer is then approxi-mated by incorporating the proper field behavior into the contourand surface integrals [9]. However, it can be shown that the2D-FDFD equations, developed in [4], may also be derived fromMaxwell’s equations in integral form, which allows the CP tech-nique to be applied in the frequency domain. This is accomplishedby first approximating the surface and contour integrals in Eqs. (1)and (2), as shown in [8], assuming a uniform grid spacing accord-ing to Yee’s mesh [10]. Next, assuming time-harmonic behavior,the derivatives with respect to time are replaced by j�. Assumingthe structure is invariant in the direction of propagation z, the gridspacing in the z-direction, �z, may approach zero, essentiallyconverting the difference equations with respect to �z into partialderivatives, which in turn are replaced by �/� z � �j�, where � �(� � j�), the complex propagation constant. In this way, Eq. (1)is applied to Hx, Hy, and Hz in order to generate the 2D-FDFDequations for �Ey, �Ex, and Hz, respectively. Likewise, Eq. (2) isapplied to Ex, Ey, and Ez in order to generate the equations for�Hy, �Hx, and Ez, respectively.

To apply the CP approximation, assume that the thin layer isoriented either in the y–z or x–z plane, and is within the currentlattice cell (within one-half grid spacing of the current-field com-ponent). As an example, for a layer in the y–z plane, as shown inFigure 1, the electric field normal to the layer, Ex, is split into twoparts: Ex,in and Ex,out, which represent the fields inside and

40 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 43, No. 1, October 5 2004

outside the layer, respectively. Due to the continuity of the normalelectric-flux density, these two values are related by

Ex,in�*, 0� ��x,out

�x,inEx,out�1

2, 0� . (3)

The other fields are assumed continuous across the layer. Theinsertion of the thin layer necessitates the following modificationsto the difference equations. The difference equations for �Hx andEz (generated by applying Eq. (2) to Ey and Ez, respectively) areevaluated using the average permittivity �avg of the cell [11]. Thedifference equations for �Ex and Hz are generated by solving Eq.(1) for Hy and Hz as described previously, except that the sur-rounding contour now includes both of the split electric-fieldvalues, Ex,in and Ex,out, which are related by Eq. (3). The remain-ing difference equations are unchanged. The CP approximation islikewise applied to the case that the thin layer lies in the x–z plane,and the results are combined into four coupled difference equa-tions for the transverse-field components:

�Ex�1

2, 0� �

j

��x�Ez�1, 0� Ez�0, 0��

��y

�Hy�1

2, 0� , (4)

�Ey�0,1

2� �j

�y�Ez�0, 1� Ez�0, 0��

��x

Hx�0,

1

2� , (5)

�Hx�0,1

2� �j

�x�Hz�1

2,

1

2� Hz��1

2,1

2�� ��y,avgEy�0,

1

2�, (6)

�Hy�1

2, 0� �

j

�y�Hz�1

2,

1

2� Hz�1

2, �

1

2�� ��x,avgEx�1

2, 0�, (7)

and two equations for the longitudinal components:

Ez�0, 0� �j

��z,avg�y�Hx�0,

1

2� Hx�0, �1

2��

j

��z,avg�x�Hy�1

2, 0� Hy��

1

2, 0��, (8)

Hz�1

2,

1

2� �j

��z�x�Ey�1,

1

2� Ey�0,1

2��

j�

��z�y�Ex�1

2, 1� Ex�1

2, 0�� . (9)

The average permittivity values are defined as

�x,avg � �1 Txz

�y��x,out �Txz

�y��x,in, (10)

�y,avg � �1 Tyz

�x��y,out �Tyz

�x��y,in, (11)

�z,avg � �1 Txz

�y

Tyz

�x��z,out �Txz

�y

Tyz

�x��z,in. (12)

The constants � and are given by

� ��x Tyz

�x

Tyz

�x

�x,out

�x,in, (13)

��y Txz

�y

Txz

�y

�y,out

�y,in, (14)

where �x and �y are the grid spacings in the x and y directions andTxz and Tyz are the thicknesses of thin layers in the x–z and y–zplanes, respectively, and the subscripts in and out refer to thematerial properties inside or outside the thin layer. Txz and Tyz arenonzero only if a thin layer is present within the lattice cell (withinone-half grid spacing) of the field component being evaluated. Thisallows the equations to be valid for all points in space because,when Txz and Tyz are zero, Eqs. (4)–(9) reduce to the 2D-FDFDdifference equations given in [4]. To construct the eigenvalueequation, Eqs. (8) and (9) are substituted into Eqs. (4)–(7) and theresults of evaluating the tangential field equations at each gridlocation are arranged into a matrix, as described in [4].

3. NUMERICAL RESULTS

The application of the FDFD-CP method is next illustrated withnumerical examples. In all cases, Matlab [12] was used to solve forthe eigenvalues/vectors using the ARPACK [13] routines for theimplicitly restarted Arnoldi algorithm. While the FDFD-CPmethod may be applied to waveguides of arbitrary cross section,planar and circular structures were chosen for comparison pur-poses since there are closed-form characteristic equations that canbe solved numerically.

3.1. Parallel Plate Microwave Transmission Line with Thin,Lossy LayerThis transmission line, shown in Figure 2, we previously analyzedin [9] using the CP approximation in the time domain. It consistsof perfectly conducting, infinite parallel-plates with a lossy layer ofthickness Ts, suspended through the middle of the line. Thepermittivity, permeability, and conductivity of the layer are givenby �s, �0, and �s, respectively. The FDFD-CP method was used

Figure 1 Illustration of a thin layer, oriented in the y–z plane, insertedinto the lattice at location i* (arrows indicate the positions of the variousfield components within the lattice cell; the normal E-field is split into twocomponents due to the discontinuity of the field at the layer interface)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 43, No. 1, October 5 2004 41

to calculate the propagation and attenuation constants of the TEMand TM1 modes for a � 1 mm, �s � 5�0, and Ts � 5 �m, with�s varying from 10�4 to �max. For each mode, �max was chosento be an order of magnitude less than the conductivity at which thelayer thickness is equal to the skin depth in order to ensure thevalidity of the CP approximation. The frequency is given by � ��n�c/a, where c is the speed of light, and �n was chosen to be 0.1for the TEM mode and (2)1/2 for the TM1 mode. An infinitewaveguide in the x direction was simulated by utilizing a Neu-mann boundary condition wherein the field components outsidethe boundary were set equal to those inside the boundary.

Figure 3 compares the FDFD-CP results with the exact analyt-ical solutions given in [9]. Excellent agreement is observed overthe range of conductivities shown in the figure. Note the slighterror in the attenuation constant of the TEM mode [Fig. 3(a)] as �s

approaches �max, due to the CP approximation. In addition, it wasobserved that, as the conductivity approached �max, the thin layerbegan to act as an additional parallel electrode, and modal classi-fication became ambiguous.

3.2. Multilayer Waveguide Incorporating a Thin, Dichroic LayerThis planar dielectric waveguide, shown in Figure 4, incorporatesa thin adsorbed molecular layer. The waveguide was analyzed in[14] using the ray-optics model (ROM), a perturbational methodused to generate a closed-form expression that was shown to agreewell with electromagnetic wave theory. The indices of the sub-strate, core, and cladding are nsub � 1.46, ncore � 1.56, andnclad � 1.33, and the thickness of the core, Tcore, is 400 nm. Thethin film has a thickness TL of 3 nm and is composed of moleculeshaving a linear electronic-absorption dipole. The molecules areassumed to have a rotationally symmetric distribution in thewaveguide plane ( x–z plane), but are nonrandom with respect tothe mean molecular tilt angle , which is the angle between theaverage electric-absorption dipole and the axis perpendicular to thewaveguide surface ( y-axis). The index of refraction of the layer isgiven by the tensor nL � n � jn�, where n � 1.33, and thecomponents of the tensor n� are related to by the followingequations [14]:

n �x � n �z �3

2n�sin2� �, (15)

n �y � 3n�cos2� �, (16)

where n� is chosen to be 0.01. An infinite waveguide in the xdirection was again simulated by utilizing a Neumann boundarycondition. Figure 5 shows the calculated attenuation of the lowest-order TE and TM modes at � � 550 nm using the FDFD-CP

Figure 3 Propagation and attenuation constants (� and �) of the parallel-plate waveguide depicted in Fig. 2. The solutions obtained using theFDFD-CP method (dots) are compared with the exact analytical solutions,given in [9]: (a) TEM mode; (b) TM1 mode

Figure 4 Multilayered dielectric waveguide with thin, anisotropic layerbetween the core and cladding (parameter values are given in the text)

Figure 2 Parallel-plate transmission line with thin, lossy layer sus-pended through the center (the top and bottom plates are assumed to beperfectly conducting; the parameter values are given in the text)

42 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 43, No. 1, October 5 2004

method, as compared to the results obtained using the ROM. Theresults match very closely across the entire range of tilt angles,thus demonstrating that the FDFD-CP method properly accountsfor anisotropy in the thin, lossy layer.

3.3. Optical Fiber with Thin, Lossy Inner CladdingTo demonstrate the applicability of the FDFD-CP method tononplanar structures, it was applied to the optical fiber shown inFigure 6. The index of the cladding, no � 1.44402, is obtainedfrom the Sellmeier equation for fused silica [15] at � � 1550 nm.The core/cladding index difference �n is assumed to be 0.0045,and the core radius a is 4.15 �m. The complex index of refractionof the thin inner-cladding layer is given by no � jni. If ni � no,the weakly guiding approximation is valid and the scalar wave

equation may be used to obtain the characteristic equation for thestructure, as in [16]. The characteristic equation is then iterativelysolved for the complex propagation constants of the guided modes.For the FDFD-CP method, the thin inner-cladding layer was ap-proximated by a combination of continuously joined orthogonalsegments, as illustrated in Figure 6. In addition, electric andmagnetic symmetry walls were placed as shown in the figure inorder to reduce computational time.

Figure 7 shows the calculated attenuation of the fundamentalHE11 fiber mode for ni � 5 � 10�2, 5 � 10�3, and 5 � 10�4.The values obtained using the FDFD-CP method correlate wellwith those obtained from the characteristic equation. The effectiveindex neff is not shown, but was found to be minimally affected bythe presence of the lossy layer. For example, for ni � 5 � 10�2,the variation of neff was less than 0.01% over the range of � shownin the figure. The predicted attenuation, on the other hand, is quitelarge, reaching a peak value of approximately 270 dB/cm for ni �5 � 10�2 and � � 100 nm. This example illustrates that the CPapproximation is not restricted to planar layers, but may be accu-rately applied to waveguides incorporating arbitrarily shaped thinlayers by approximating the layer as a collection of orthogonalsegments of varying length and/or width.

4. CONCLUSION

In this paper, a FDFD method to directly solves for the modalpropagation and attenuation constants of waveguides incorporatingthin, dielectric, or lossy layers while retaining a grid size manytimes larger than the thickness of the thin layer has been described.The method was shown to be applicable to planar and nonplanarstructures. The primary advantage of this FDFD-CP method is thatit can be accurately applied to waveguides of arbitrary crosssection and is not restricted to specific geometries.

ACKNOWLEDGMENT

This work was supported by the National Science Foundationunder grant no. ECS-245716.

Figure 5 Attenuation of the lowest-order TE and TM modes of themultilayered waveguide depicted in Fig. 4. Very close correlation isobserved between the FDFD-CP method (dots) and the results obtainedusing the ray optics model (ROM), described in [14]

Figure 6 Optical fiber, with thin lossy layer of thickness � insertedbetween the core and cladding. The circular thin layer was approximatedusing orthogonal segments. The symmetry of the structure was exploited toreduce computational time by inserting electric and magnetic symmetrywalls (E.W. and M.W.) and only considering the lower left quadrant of thestructure (parameter values are given in the text)

Figure 7 Attenuation of the lowest-order (HE11) mode of the opticalfiber shown in Fig. 6. The results obtained using the FDFD-CP method(dots) are compared to the solution of the characteristic equation for theweakly guiding approximation, given in [16]. Results are shown for threedifferent values of ni, the imaginary portion of the refractive index of thethin inner cladding layer

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 43, No. 1, October 5 2004 43

REFERENCES

1. E. Schweig and W.B. Bridges, Computer analysis of dielectricwaveguides: A finite-difference method, IEEE Trans Microwave The-ory Tech 32 (1984), 531–541.

2. K. Bierworth, N. Schulz, and F. Arndt, Finite-difference analysis ofrectangular dielectric waveguide structures, IEEE Trans MicrowaveTheory Tech 34 (1986), 1104–1114.

3. M.-L. Liu and Z. Chen, A direct computation of propagation constantusing compact 2-D full-wave eigen-based finite-difference frequency-domain technique, Proc Int Comput Electromagn Conf, Beijing,China, 1999, pp. 78–81.

4. Y.-J. Zhao, K.-L. Wu, and K.-K.M. Cheng, A compact 2-D full-wavefinite-difference frequency-domain method for general guided wavestructures, IEEE Trans Microwave Theory Tech 50 (2002), 1844–1848.

5. J.A. Perede, A. Vega, and A. Prieto, An improved compact 2Dfull-wave FDFD method for general guided wave structures, Micro-wave Optical Technol Lett 38 (2003), 331–335.

6. R. Lotz, J. Ritter, and F. Arndt, 3D subgrid technique for the finitedifference method in the frequency domain, 1998 Int MicrowaveSymp Dig 3 (1998), 1739–1742.

7. L. Kulas and M. Mrozowski, Reduced order models of refined Yee’scells, IEEE Microwave Wireless Compon Lett 13 (2003), 164–166.

8. A. Taflove, K.R. Umashankar, B. Beker, F. Harfoush, and K.S. Yee,Detailed FD-TD analysis of electromagnetic fields penetrating narrowslots and lapped joints in thick conducting screens, IEEE Trans An-tennas Propagat 36 (1988), 247–257.

9. J.G. Maloney and G.S. Smith, The efficient modeling of thin materiallayers in the finite-difference time-domain (FDTD) method, IEEETrans Antennas Propagat 40 (1992), 323–330.

10. K.S. Yee, Numerical solution of initial boundary value problemsinvolving Maxwell’s equations in isotropic media, IEEE Trans Anten-nas Propagat 14 (1966), 302–307.

11. A. Taflove, Computational electrodynamics: The finite-differencetime-domain method, Artech House, Boston, 1995.

12. The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098.13. R.B. Lehoucq, D.C. Sorensen, and C. Yang, ARPACK users’ guide:

Solution of large-scale eigenvalue problems with implicitly restartedArnoldi methods, 1998, http://www.caam.rice.edu/software/ARPACK.

14. S.B. Mendes and S.S. Saavedra, Comparative analysis of absorbancecalculations for integrated optical waveguide configurations by use ofthe ray optics model and the electromagnetic wave theory, Appl Optics39 (2000), 612–621.

15. W.G. Driscoll and W. Vaughan (Eds.), OSA handbook of optics,McGraw-Hill, New York, 1978, pp. 7–85.

16. S. Kawakami and S. Nishida, Characteristics of a double clad opticalfiber with a low-index inner cladding, IEEE J Quant Electron 10(1974), 879–887.

© 2004 Wiley Periodicals, Inc.

LOW-CROSS-POLARIZED WIDEBANDV-SLOT MICROSTRIP ANTENNA

Gh. Z. Rafi and L. ShafaiDepartment of Electrical and Computer EngineeringUniversity of ManitobaWinnipeg, Manitoba, R3T 5V6

Received 9 March 2004

ABSTRACT: The performance of a V-slotted microstrip patch an-tenna, with and without additional edge slots, is investigated. It isshown that the V-slotted patch is broadband, but its cross polariza-tion increases with frequency. Placing two additional rectangularslots at the patch edges, parallel to its shorter side, eliminates theproblem without affecting the bandwidth. The cross-polarization be-

comes independent of frequency. It remains low over a bandwidth,which is wider than the antenna-impedance bandwidth. © 2004Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 44 – 47,2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20370

Key words: microstrip antennas; broadband antennas; slot antennas;low cross polarization

INTRODUCTION

Microstrip patch antennas with a folded slot is one of the mostpopular designs for wideband, single-layer applications. The slotand patch form two coupled resonators in order to broaden thebandwidth. This eliminates the need for stacking two patches,thereby simplifying the fabrication. The slot originally was bentinto a U shape to accommodate it on the patch [1], which was thenfed with a probe at the patch center. This central location of theprobe and the symmetry of the patch and slot eliminated the firstmode. The resultant antenna then operated in the second mode,with the patch currents running parallel to its shorter edge. Imped-ance bandwidths as high as 30% for a probe-fed patch wereobtained for the U-slotted patch [1]. In [2], a circular-arc slot overa rectangular patch was used to enhance its bandwidth to about30%, which is similar to that of a U-slot. Larger bandwidths werealso obtained by other slots. In [3], a bent dipole was used to feedthe same circular-arc slot, but on a circular-patch microstrip an-tenna. The bandwidth was increased to about 40%, but because ofthe bent probe shape, the configuration was more complex. Thesestudies confirmed the similarity of rectangular and circular micro-strip patches in the bandwidth performance, as well as the simi-larity of the U- and circular-arc-shaped slots on them. Recently, itwas shown that by modifying the slot shape from a U to atruncated V, the angle of the V arm can be used as an additionalparameter to further increase the impedance bandwidth to 36.5%[4]. The geometry of this antenna is shown in Figure 1. One areaof difficulty with both U and V slotted antennas is the crosspolarization. Cutting the U or V slot on the patch forces the patchcurrents to bend around the slot. Consequently, the patch currenthas an orthogonal component that contributes to cross-polar radi-ation. Because of the central location of the probe and the sym-metry of both slot and patch geometries, the generated cross-polarized current is in the same direction as the currents of thenext-higher-order mode of the patch in the horizontal W direction.Thus, their similarity aids in the excitation of this higher mode. Asa result, the cross polarization of the radiated field in the H-plane,that is, the horizontal plane in Figure 1, increases with frequency.In this paper, we investigate the influence of additional slots on theradiation properties of the slotted wideband microstrip antennas,such as gain and cross-polarization.

Figure 1 Geometry of the wideband V-slotted microstrip patch antenna:LG � WG � 60 mm, h � 5.5 mm, �r � 1.05, L � 26 mm, W � 36mm, � 10°, LV � 20.7 mm, WV � 8.8 mm, t � 2 mm, d � 2.3 mm

44 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 43, No. 1, October 5 2004