Spectral shadowing is a distortion of a downstream FBGspectrum that results from light having to pass twice throughthe upstream FBG. From Eq. (1), if the Bragg wavelength of theath FBG (a � i) is slightly offset from the ith FBG, it appearsthat the spectrum of the ith FBG is shifted further in thedirection of the actual Bragg wavelength. For example, if theBragg wavelength of the distal FBG in a 50-FBG serial array is1550 nm, the rest of the FBGs are aligned to the same wave-length, but have an offset from 1 pm to 300 pm. The Braggwavelength shift of the distal FBG due to this effect is shown inFigure 2. It is found that the maximum wavelength detectionerror is 18 pm when the reflectivity of the distal FBG is 1% andthe FWHM of FBGs is 0.1 nm.

Multiple reflection [3] results if the light reflected from anFBG arrives in the detection-time window allotted to another

downstream FBG, because of delays associated with multiplereflection paths. We consider only first-order crosstalk pulses,which are those that undogo a total of three reflections in thearray and are the strongest stray pulses generated. The second-order multiple reflections are reduced relative to the primarysignal by the 4th power of reflectivity, and so on for thehigher-order reflections. The graphical representation of thefirst-order multiple reflection processes shown in Figure 3.Here, the optical paths leading to primary FBG signal and allthe possible first-order crosstalk pulses for the 4th and 5th FBGin a serial FBG sensor array are shown. If we assume thereflected power from FBGs are balanced and the Bragg wave-length of FBGs (from 1 to i � 1) are aligned to the Braggwavelength of ith FBG, the wavelength detection error of the ithFBG due to the multiple reflection crosstalk is

���i�BG

� � 1

4 ln 2 �a�0

i�1

�1 � RGa�2RGi �

RG�i�1� �a�1

i�2

�1 � RGa�2�RG�i�2�RG�i�1� � 2 �

a�1

i�3

RGaRG�a�1��� �

a�3

�i/2( �even i� or �i�1�/2 �odd i�

RG�i��a�1� ��b�1

i�a

�1 � RGb�2�

� � RG�i�2�a�1��RG�i��a�1� �c�i��2�a�1��1

i���a�1��1

��1 � RGc�2

� 2 �c�1

i��2a�1� �RGcRG��a�1��c �b�1

a�2

�1 � RG��a�1��b�c��2� for i � 3. (3)

For example, if for the Bragg wavelength all FBGs are alignedto the same wavelength and the reflected power from them isbalanced, the wavelength detection error of the distal FBG dueto multiple reflection in a 50-FBG sensor array is shown inFigure 4. The wavelength detection error is 12 pm when thereflectivity of the distal FBG is 1% and the FWHM of FBGs are0.1 nm.

3. CONCLUSION

The computer simulation results of the intrinsic crosstalk of aTDM FBG serial sensor array have been reported. The mathemat-ical formula of the multiple reflections in this topology is derived.The maximum wavelength detection error is 18 pm and 12 pm forthe spectral shadowing and multiple reflection effect, respectively,when the reflectivity of the distal FBG is 1%.

ACKNOWLEDGMENT

The authors would like to acknowledge support from the HongKong CERG, research grant no. PolyU5109/99E, and the HongKong Polytechnic University, research grant no. G-YW54.

REFERENCES

1. A.D. Kersey, M.A. Davis, H.P. Patrick, M. LeBlanc, K.P. Koo, C.G.Askins, M.A. Putnam, and E.J. Friebele, Fiber grating sensors, IEEE JLightwave Technol 15 (1997), 1442–1462.

2. T. Coroy, L.M. Chappell, N.J. Guillermo, S.Y. Huang, R.M. Measures,and K.D. Chik, Peak detection demodulation of a Bragg fiber optic

sensor using a gain-coupled distributed feedback tunable laser, ProcOFS’12, 1997, pp. 210–212.

3. A.D. Kersey, K.L. Dorsey, and A. Dandridge, Cross talk in a fiber-opticFabry–Perot sensor array withring reflectors, Optics Lett 14 (1989),93–95.

© 2003 Wiley Periodicals, Inc.

PARALLEL-PLATE WAVEGUIDES WITHKRONIG–PENNEY MORPHOLOGY ASPHOTONIC BAND-GAP FILTERS

Alvaro Gomez,1,2 Akhlesh Lakhtakia,2 Miguel A. Solano,1

and Angel Vegas1

1 Departamento de Ingenierıa de ComunicacionesUniversity of CantabriaAvenida de los Castros s/n39005 Santander, Spain2 Department of Engineering Science and MechanicsPennsylvania State UniversityUniversity Park, PA 16802-6812, USA

Received 21 June 2002

ABSTRACT: Guided wave propagation in a parallel-plate waveguidewith Kronig–Penney morphology is analyzed. Modes in the photonicband-gap (PBG) structure can be classified as either transverse electricor transverse magnetic with respect to the propagation direction. Abovethe modal cut-off frequencies, band gaps exist where in propagation is

4 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003

forbidden. The locations and the widths of the band gaps depend onmodal order, waveguide height, and the permittivity contrast and rela-tive volumetric proportion of the two materials constituting the unit cellof the PBG structure. © 2003 Wiley Periodicals, Inc. Microwave OptTechnol Lett 36: 4–8, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10654

Key words: band gaps; Kronig–Penney morphology; parallel-platewaveguide; PBG filters

1. INTRODUCTION

Being periodic, the photonic band-gap (PBG) structure allowselectromagnetic field propagation through it in some frequencybands, but not in other frequency bands. This remarkable filteringproperty makes PBG devices attractive in the optical regime, aswell as in the microwave and millimeter frequency bands [1–8].PBG devices are fabricated using dielectrics, semiconductors, andeven metals. The essential feature of a PBG device is simply theperiodic arrangement of high-contrast electromagnetic properties.

Recently, Srivastava et al. [9] presented the design of PBGultraviolet filters. Their work is typical of the optical filtercommunity, which routinely employs multilayers of the. . .HLHLHL. . . configuration, with H standing for a high-permittivity dielectric material and L for a low-permittivity one[10]. Their analysis is reminiscent of the Kronig–Penney modelin the band theory of solids [11, 12].

The PBG filters of [9] employ dielectric layers of infinitetransverse extent. While this arrangement is satisfactory for anal-ysis in the optical regime because optical wavelengths are small incomparison to the transverse dimensions of typical devices, itcannot be practically scaled down to the microwave and millime-ter-wave regimes. In the latter regimes, waveguides of finite trans-verse extent must be used. Therefore, as a first step, in this paperwe analyze PBG filters with the Kronig–Penney morphology—thatis, multilayers of the . . .HLHLHL. . . type—housed in a parallel-plate waveguide. We note that this configuration is very differentfrom those employed by Chang and Hsu [1] and Kyriazidou et al.[7] for PBG structures inserted in waveguides.

The plan of this paper is as follows: section 2 contains deriva-tions of the dispersion equations for transverse electric (TEz) andtransverse magnetic (TMz) modes in a parallel-plate waveguidewith the Kronig–Penney morphology. Section 3 focuses on thenumerical results and conclusions derived therefrom. A time-dependence of exp( j�t) is implicit in both sections, with � theangular frequency.

2. THEORY

Let us consider the boundary value problem illustrated in Figure 1.The planes y � 0 and y � h are perfectly conducting, therebyconstituting a parallel-plate waveguide. Slabs of two differentmaterials are placed in the region 0�� such that the permittivity

is periodic along the z axis and has a binary distribution in eachunit cell. Mathematically,

� x, y, z � d� � x, y, z�, �x� � �, 0 � y � h, �z� � �, (1)

with the reference unit cell characterized by

� z� 0r1 if 0 � z � a0r2 if �b � z � 0, �x� � �, 0 � y � h. (2)

Here, r1,2are the relative permittivities of the two materials, and

0 is the permittivity of free space (or vacuum). Thus, d � a � bis the spatial period of the chosen structure.

For propagation along the z axis, the wave equation

� 2

y2 � 2

z2 � �2� z��0��� y, z� 0 (3)

must be obeyed inside the waveguide. Here, we have denoted� {�, �, �} � � {�}, �0 is the permeability of free space, while�{�, �} � E�{�, �} for the TEz modes and �{�, �} �H�{�, �} for the TMz modes. The two types of modes have to beanalyzed separately.

2.1. TEz ModesFor a TEz mode of propagation, the only non-zero components ofthe electromagnetic field are Ex, Hy � ( j��0)�1 Ex/ z, andHz � �( j��0)�1 Ex/ y. As the planes y � 0 and y � h areperfectly conducting, Ex must be zero identically on these planes.Therefore, the solution of Eq. (3) in the reference unit cell is

Ex� y, z� �n��

�An sin�kyn y�e�j�n1z � Bn sin�kyn y�ej�n1z,

0 � z � a, (4)

Ex� y, z� �n��

�Cn sin�kyn y�e�j�n2z � Dn sin�kyn y�ej�n2z,

�b � z � 0, (5)

where n is the modal order and � is the set of all natural numbers;kyn

� (n�/h) is the modal cut-off wavenumber; �n1,2

2 ��2�00r1,2

� kyn

2 ; while An, Bn, Cn, and Dn are the expansioncoefficients.

On ensuring the continuity of Ex and Hy across the surface z �0 inside the reference unit cell and exploiting the orthogonalities ofthe functions sin(n�y/h) over the range 0yh, the followingrelations are obtained:

An � Bn � Cn � Dn 0, � n � �, (6)

�n1An � �n1Bn � �n2Cn � �n2Dn 0, � n � �. (7)

The periodicity of the PBG structure requires the application of theBloch theorem [13], which in this case entails

Ex� y, z��z�a Ex� y, z��z��bej�d, (8)

Hy� y, z��z�a Hy� y, z��z��bej�d. (9)

The Bloch wavenumber � must be real-valued in allowed bands.Eqs. (8) and (9) lead to

Figure 1 Schematic of the periodic variation of the relative permittivityinside the parallel plate-waveguide with Kronig–Penney morphology

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003 5

Ane�j��n1���a � Bne

j��n1���a � Cnej��n2���b � Dne

�j��n2���b 0,

� n � �, (10)

�n1Ane�j��n1���a � �n1Bne

j��n1���a � �n2Cnej��n2���b

� �n2Dne�j��n2���b 0, � n � �. (11)

Equations (6), (7), (10), and (11) are written together in matrixnotation as

1 1 �1 �1

�n1 ��n1 ��n2 �n2

e�j��n1���a ej��n1���a �ej��n2���b �e�j��n2���b

�n1e�j��n1���a ��n1e

j��n1���a ��n2ej��n2���b �n2e

�j��n2���b�

� An

Bn

Cn

Dn

� 0000�, � n � �. (12)

The 4 4 matrix must be singular for Eq. (12) to have a nontrivialsolution; thus,

cos���a � b� cos��n1a�cos��n2b�

��n1

2 � �n2

2

2�n1�n2

sin��n1a�sin��n2b�, � n � �. (13)

From Eq. (13) it follows that

� 1

a � bcos�1�LTEn

z�, � n � �, (14)

where

LTEnz cos��n1a�cos��n2b� �

�n1

2 � �n2

2

2�n1�n2

sin��n1a�sin��n2b�,

� n � �. (15)

2.2. TMz ModesNext, for a TMz mode of propagation, the only non-zero compo-nents of the electromagnetic field are Hx, Ey � �( j��0)�1 Hx/ z, and Ez � ( j��0)�1 Hx/ y. Because Ez must be null-valuedon the planes y � 0 and y � h, the solution of Eq. (3) in thereference unit cell is

Hx� y, z� �n����0�

�Fn sin�kyn y�e�j�n1z � Gn sin�kyn y�ej�n1z,

0 � z � a, (16)

Hx� y, z� �n����0�

� Jn sin�kyn y�e�j�n2z � Kn sin�kyn y�ej�n2z,

�b � z � 0, (17)

where Fn, Gn, Jn, and Kn are the expansion coefficients and, inthis case, the set of all natural numbers � has to be extended with0, because the parallel-plate waveguide can support the TM0

mode, which is the TEM mode. On ensuring the continuity of Hx

and Ey across the surface z � 0 inside the waveguide, thefollowing relations are obtained:

Fn � Gn � Jn � Kn 0, � n � � � �0�, (18)

�n1

r1

�Fn � Gn� ��n2

r2

� Jn � Kn� 0, � n � � � �0�. (19)

Application of the Bloch theorem yields the following relations:

Fne�j��n1���a � Gne

j��n1���a � Jnej��n2���b � Kne

�j��n2���b 0,

� n � � � �0�, (20)

�n1

r1

�Fne�j��n1���a � Gne

j��n1���a�

��n2

r2

�Jnej��n2���b � Kne

�j��n2���b� 0, � n � � � �0�. (21)

Equations (18), (19), (20), and (21) can be written in matrixnotation in the same manner as Eq. (12). The dispersion relation

cos���a � b� cos��n1a�cos��n2b�

��n2

2 r1

2 � �n1

2 r2

2

2�n1�n2r1r2

sin��n1a�sin��n2b�, � n � � � �0� (22)

is thereby obtained. Defining

LTMnz cos��n1a�cos��n2b�

��n2

2 r1

2 � �n1

2 r2

2

2�n1�n2r1r2

sin��n1a�sin��n2b�, � n � � � �0�, (23)

we see that Eq. (22) is compactly re-expressed as follows:

� 1

a � bcos�1�LTMn

z�, � n � � � �0�. (24)

3. NUMERICAL RESULTS AND CONCLUSIONS

The functions LTEnz and LTMn

z were computed for the parallel-platewaveguide with Kronig–Penney morphology. All results presentedhere are for a � 51 mm and b � 9 mm, which correspond to the

Figure 2 Variation of LTE1z and LTM1

z with frequency � � �/2� for r1�

1, r2� 4, a � 51 mm, b � 9 mm, and h � 25 mm

6 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003

Yablonovich structure (a � 0.85 d). While r1� 1 was kept

fixed, r2� r1

was kept a variable.Figure 2 shows the computed values of LTE1

z and LTM1z when

r2� 4 and h � 25 mm. Because � must not be real-valued in a

forbidden band, satisfaction of the condition �1 � cos(�d) � 1identifies allowed bands for both TEz and TMz modes. Therefore,the lines LTE1

z �1 and LTM1z �1 are also shown in Figure 2.

Clearly, several forbidden bands exist.The dependences of the band gaps (that is, the forbidden bands)

on the permittivity contrast r2/r1

and the waveguide height h areexplored in Figures 3 and 4 for the lowest-order TEz mode.Inspecting these figures, we infer that the number of band gapsincreases for fixed values of a, b, and h, as the permittivitycontrast is increased. Naturally, the widths of both allowed andforbidden bands decrease. An increase in the waveguide heightlowers the modal cut-off wavenumbers, thereby changing thelocations of the allowed and the forbidden band gaps in thespectrum. Furthermore, the lesser of the two relative permittivitiesdetermines the lowest frequency at which a mode of order n canpropagate, because �n1,2

2 � �2�00r1,2� kyn

2 � 0 for guidedwave propagation.

Allowed and forbidden bands are identified in Figure 5 for thethree lowest-order TEz and TMz modes for waveguides withdifferent heights. The quantity �LTEn

z � � 1 in the black zones, but�1 in the white zones; similarly for �LTMn

z �. Obviously, the firstallowed band appears when the frequency is greater than the modalcut-off frequency. For higher frequencies, band gaps appear thatwould be absent if r2

were equal to r1.

A very thick waveguide would be overmoded, and the lowest-order modes would have certain characteristics not very differentfrom that of the TM0

z mode, because the TE0z mode is impossible

in a parallel-plate waveguide. Therefore, we expected that thedisplayed filtration response for the lowest-order modes would notdepend critically on the modal order and type. Indeed, this turnedout to be the case, which is evident from the profiles of the allowedand forbidden bands depicted in Figure 6 for very high values ofh. Incidentally, the TM0

z mode is insensitive to h, and its charac-

Figure 3 Variation of LTE1z with frequency � � �/2� for r1

� 1, a �51 mm, b � 9 mm, and h � 25 mm

Figure 4 Same as Figure 3, except for h � 10 mm

Figure 5 Allowed and forbidden bands for the first three TEz and TMz

modes with respect to frequency � � �/2� and the waveguide height hfor r1

� 1, r2� 2, a � 51 mm, b � 9 mm; allowed bands

(�LTEnz � � 1 or ��LTMn

z � � 1� are shown in white, forbidden bands��LTEn

z � � 1 or �LTMnz � � 1) in black

Figure 6 Allowed and forbidden bands for TE1z , TM0

z , and TM1z modes

with respect to frequency � � �/2� and waveguide height h for r1� 1,

r2� 2, a � 51 mm, b � 9 mm; allowed bands ��LTEn

z � � 1 or �LTMnz �

� 1) are shown in white, forbidden bands ��LTEnz � � 1 or �LTMn

z � � 1) inblack

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003 7

teristics are the same as those predicted by Srivastava et al. [9] forPBG structures of infinite transverse extent.

4. CONCLUSION

In this paper, we examined guided wave propagation in a parallel-plate waveguide with Kronig–Penney morphology. Modes can beclassified as either transverse electric or transverse magnetic withrespect to the propagation direction. Above the modal cut-offfrequencies, band gaps exist that do not allow propagation. Thelocations and widths of the forbidden bands depend on modalorder, waveguide height, and the permittivity contrast of the twomaterials constituting the unit cell. A dependence on the volumet-ric proportions of the two materials is also implicit. We plan toinvestigate other waveguides with the Kronig–Penney morphologynext.

ACKNOWLEDGMENT

This work and visit of A. Gomez at Pennsylvania State Universitywere partially supported by the Direccion General de Investiga-cion, MCyT, Spain, under project no. TIC2000-1612-C03-01.

REFERENCES

1. C.-Y. Chang and W.-C. Hsu, Photonic band-gap dielectric waveguidefilter, IEEE Microwave Wireless Components Lett 12 (2002), 137–139.

2. J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S.Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, and E.P. Ippen,Photonic bandgap microcavities in optical waveguides, Nature 390(1997), 143–145.

3. P.R. Villeneuve, D.S. Abrams, S. Fan, and J.D. Joannopoulos, Single-mode waveguide microcavity for fast optical switching, Opt Lett 21(1996), 2017–2019.

4. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, andE.L. Thomas, A dielectric omnidirectional reflector, Science 282(1998), 1679–1682.

5. S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, Large omnidirectionalband gaps in metallodielectric photonic crystals, Phys Rev B 54(1996), 11245–11251.

6. P.R. Villeneuve, S. Fan, J.D. Joannopoulos, K.-Y. Lim, G.S. Petrich,L.A. Kolodziejski, and R. Reif, Air-bridge microcavities, Appl PhysLett 67 (1995), 167–169.

7. C.A. Kyriazidou, H.F. Contopanagos, and N.G. Alexopoulos, Mono-lithic waveguide filters using printed photonic-bandgap materials,IEEE Trans Microwave Theory Tech 49 (2001), 297–307.

8. T. Kim and C. Seo, A novel photonic bandgap structure for low-passfilter of wide stopband, IEEE Microwave Guided Wave Lett 10 (2000),13–15.

9. S.K. Srivastava, S.P. Ojha, and K.S. Ramesh, Design of an ultravioletfilter based on photonic band-gap material, Microwave Opt Tech Lett33 (2002), 308–314. [Although Eq. (7) of this paper is correct, and infact agrees with our Eq. (22) for the TMz mode of order n � 0, Eq.(9) of this paper is incorrect. Consequently, Figures 2–4 are not correctfor the stated values of the geometric and the constitutive parameters.]

10. O.S. Heavens, Optical properties of thin solid films, Butterworths,London, 1955, Chapter 7.

11. C. Kittel, Introduction to solid state physics, Wiley, New York, 1971,695–699.

12. K. Kano, Semiconductor devices, Prentice-Hall, Upper Saddle River,NJ, 1998, 37–39.

13. C. Kittel, Introduction to solid state physics, Wiley, New York, 1971,306.

© 2003 Wiley Periodicals, Inc.

NUMERICAL ANALYSIS OF LOCALINTERPOLATION ERROR FOR2D-MLFMA

Shinichiro Ohnuki and Weng Cho ChewCenter for Computational ElectromagneticsDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL 61801-2991

Received 11 July 2002

ABSTRACT: The error control of local interpolation for a 2D MLFMAwill be discussed. The way to select proper parameters is proposed interms of both numerical accuracy and computational cost. Satisfying theconditions derived in this paper, error can be controlled at the samelevel as global interpolation, and the computational cost becomes lessexpensive than the global one. © 2003 Wiley Periodicals, Inc.Microwave Opt Technol Lett 36: 8–12, 2003; Published online in WileyInterScience (www.interscience.wiley.com). DOI 10.1002/mop.10655

Key words: multilevel fast multipole algorithm; error analysis; interpo-lation error

1. INTRODUCTION

Electromagnetic scattering problems with a large number of un-knowns can be solved by the recent development of fast algorithms[1–5]. In particular, the multilevel fast multipole algorithm(MLFMA) is a powerful one for treating very large scale problems[6, 7] and the factorization of the Green’s function is the mostimportant mathematical formula. To generalize the two-level al-gorithm to the multilevel one, interpolation is a crucial processwhich becomes a numerical error source. However, this error isfully controlled by using the global interpolation, since the treatedsignal is band-limited.

In this paper, we will discuss the numerical accuracy of localinterpolation for the 2-D MLFMA, and the way to select properparameters in terms of both error control and computational cost.Satisfying the derived conditions, the error is fully controlled to thesame level as the global interpolation and the computational costbecomes cheaper.

2. BAND-LIMITED SIGNAL

A band-limited function f(t) treated in this paper is represented by

f�t� :� exp��ikr cos t�. (1)

Figure 1 plots this signal for kr � 102 in the Fourier space. When��� � kr, it decays very fast. To control the interpolation errorprecisely, it is important to predict this decay rate for the desiredaccuracy.

The Fourier series expansion of this signal can be written as

exp��ikr cos t� �n���

�

Jn�kr�einti�n. (2)

Since Jn( x) tends to zero exponentially fast when n 3 �, theabove series has exponential convergence. To truncate this series

This work was supported by AFOSR under MURI grant no. F49620-96-1-0225, a contract from WPAFB via SAIC, and the Kajima Foundation’sassistance for research abroad.

8 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003