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Optics Communications 281 (2008) 2826–2833 www.elsevier.com/locate/optcom

An efficient approach toward guided mode extraction in two-dimensional photonic crystals P. Sarrafi, A. Naqavi, K. Mehrany *, S. Khorasani, B. Rashidian School of Electrical Engineering, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iran Received 12 November 2007; received in revised form 22 January 2008; accepted 22 January 2008

Abstract A rigorous, fast and efficient method is proposed for analytical extraction of guided defect modes in two-dimensional photonic crystals, where each Bloch spatial harmonic is expanded in terms of Hermite–Gauss functions. This expansion, after being substituted in Maxwell’s equations, is analytically projected in the Hilbert space spanned by the Hermite–Gauss basis functions, and then a new set of first order coupled linear ordinary differential equations with non-constant coefficients is obtained. This set of equations is solved by employing successive differential transfer matrices, whereupon defect modes, i.e. the guided modes propagating in the straight line-defect photonic crystal waveguides and coupled resonator optical waveguides, are analytically derived. In this fashion, the governing differential equations are converted into an algebraic and easy to solve matrix eigenvalue problem. Thanks to the analyticity of the proposed approach, the eigenmodes of these structures can be extracted very quickly. The validity of the obtained results is however justified by comparing them to those derived by using the standard finite-difference time-domain method. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Photonic crystal waveguide; Coupled resonator optical waveguide; Guided modes

1. Introduction The emergence of photonic crystals, artificial periodic structures with an electromagnetic bandgap, has attracted extensive research interest in the last decade [1–4]. One of the promising applications of a photonic crystal is in molding the flow of light in defect structures introduced in an otherwise perfect lattice. A waveguide can be made by introducing chains of defects, e.g. removing one row of either air columns or dielectric rods of either two-dimensional photonic crystals [5–7] or two-dimensional photonic crystal slabs [8]. These guiding structures can be analyzed by using various numerical calculations like the beam propagation method [9,10], the plane wave expansion method [11], the T-matrix method [12], the S-matrix

*

Corresponding author. Tel.: +98 21 6616 4326. E-mail address: [email protected] (K. Mehrany).

0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.01.044

method [13], the impedance matching method [14], or the finite-difference time-domain method [15–17]. However, due to the innate complexity of such structures, the computational burden of the available calculation methods is rather heavy and therefore a fast yet rigorous approach, which is capable of accurate calculation, is still needed. This is particularly true in design and synthesis, where fast and accurate methods of analysis are needed in applying optimization processes. In this manuscript, a semi-analytical method based on the polynomial expansion for electromagnetic fields is proposed for analysis of such structures. Using polynomial expansion of electromagnetic fields has been already reported for extraction of electromagnetic eigenmodes in layered structures [18–20], where algebraic and easy-to-solve dispersion equations were derived for analysis of stratified and/or graded index planar waveguides. Also, this approach has been extended to analyze longitudinally inhomogeneous diffraction gratings [21,22], wherein electromagnetic fields, in accordance with the

P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

Floquet theorem, were expanded in terms of Legendre polynomials and diffraction efficiencies were therefore semi-analytically calculated. Similarly, Mogilevtsev et. al. employed Hermite–Gaussian expansion of magnetic fields for extraction of the propagation constant in photonic crystal fibers [23]. Although their formulation is twodimensional and rather general, it cannot be employed for analysis of line defects in photonic crystal waveguides, in which the permittivity profile is longitudinally periodic and certainly variant along the axis of propagation. Here, straight single-line defects introduced in twodimensional photonic crystals are analytically analyzed by expanding the transverse electromagnetic fields in terms of Hermite–Gauss basis functions. This is in line with the standard Galerkin’s method, for which a non-Fourier basis is applied along the transverse direction. Thanks to the analytical nature of the proposed formulation and applicability of the differential transfer matrix method in longitudinal direction [24–27], extraction of the defect modes is considerably sped up as compared to brute force techniques. This article is arranged as follows: in Section 2, the main formulation of the proposed method is presented. In Section 3, numerical results are given. In Section 4, the convergence rate of the numerical results with respect to the number of retained basis functions is briefly discussed. The computational complexity of the presented approach is discussed in Section 5. Finally, conclusions are made in Section 6. 2. Formulation The two-dimensional photonic crystal to be analyzed is assumed to be invariant along the y-axis and periodic in the x–z plane. However, the rather perfect periodicity of the permittivity profile along the x-axis is assumed to be upset by introducing some defects. As for the structure still has translational symmetry in z-direction, the localized eigenmodes satisfy the Bloch condition along the z-axis and form a one-dimensional impurity band, which can be characterized by a wave number k in the z-direction. In this section, the general formulation for calculating the aforesaid impurity band is presented for both major polarizations, where the dispersion relation between the frequency x and the wave number k is given. The proposed formulation is based on the first order Maxwell’s equations rather than the second order Helmholtz equation. This way, the calculation of $ln(e) in a structure with a discontinuous permittivity profile e-as normally encountered in solving the second order wave equation of inhomogeneous structures [23], is shunned. 2.1. E polarization With reference to the aforementioned two-dimensional photonic crystal, the Maxwell’s equations for E-polarized waves read as

oEy ¼ jlxH x ; oz oEy ¼ jlxH z ; ox oH x oH z  ¼ jxeEy ; oz ox 

2827

ð1aÞ ð1bÞ ð1cÞ

in which the permittivity e(x, z) is supposed to be periodic along the z-axis and consequently eðx; z þ Lz Þ ¼ eðx; zÞ;

ð2Þ

where Lz stands for the longitudinal period of the structure. As the Hermite–Gauss functions span a complete Hilbert space, E-polarized electromagnetic fields, i.e. Ey(x, z), Hz(x, z) and Hx(x, z) can be expanded as 1 X j an ðzÞWn ðxÞ; ð3aÞ  Ey ¼ g n¼0 Hx ¼

1 X

bn ðzÞWn ðxÞ;

ð3bÞ

cn ðzÞWn ðxÞ;

ð3cÞ

n¼0

Hz ¼

1 X n¼0

where the unknown functions an(z), bn(z), and cn(z) are to be determined later and Wn(x) stands for the Hermite– Gauss functions, i.e.  2 x 1=2 Wn ðxÞ ¼ p1=4 ð2n n!Þ exp  ð4Þ H n ðxÞ; 2 in which Hn(x) denotes the ordinary Hermite polynomials. After substituting the aforementioned expansion of electromagnetic fields, i.e. Eq. (3) into Maxwell’s equations given in (1), the following equations can be obtained: 1 1 X X dan ðzÞ Wn ðxÞ ¼ k 0 bn ðzÞWn ðxÞ; dz n¼0 n¼0  X 1 1 X g an ðzÞW0n ðxÞ ¼ jlx cn ðzÞWn ðxÞ; j n¼0 n¼0 1 1 X X dbn ðzÞ Wn ðxÞ  cn ðzÞW0n ðxÞ dz n¼0 n¼0  X 1 g ¼ jxeðx; zÞ an ðzÞWn ðxÞ: j n¼0

ð5aÞ ð5bÞ

ð5cÞ

It should be noticed that in practice, the expansions given in (5) should be inevitably truncated to have a finite number of terms N, whose numerical value is to be determined by the required level of accuracy. Now, after multiplying (5) by the mth basis function Wm(x) and integrating it over the entire domain (1, +1), the x-dependence of each defect mode will be projected onto the Hilbert space spanned by the Hermite–Gauss basis functions. This projection yields the following set of linear ordinary differential equations with non-constant coefficients:

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P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

d ½ a ¼ k 0 ½  b; ð6aÞ dz ð6bÞ G½ a ¼ k 0 ½c; d  ½b  G½c ¼ k 0 D½ a; ð6cÞ dz where ½ a ¼ ½an ðzÞ, ½ b ¼ ½bn ðzÞ, ½c ¼ ½cn ðzÞ, k0 denotes the wavenumber at vacuum, and Z 1 W0n ðxÞWm ðxÞdx; ð7aÞ G ¼ ½gmn ; gmn ¼ 1 Z 1 D ¼ ½d mn ðzÞ; d mn ðzÞ ¼ Wm ðxÞWn ðxÞer ðx; zÞdx: ð7bÞ 1

It should be noticed that the calculation of the preceding coefficients gmn and functions dmn(z) calls for numerically heavy integrations. Therefore, in order to expedite the efficiency of the proposed method, these integrals are analytically calculated in the Appendix A. Now, by substituting vector c from (6b) in (6c) and using (6a), we can achieve the following set of coupled linear differential equations with non-constant coefficients: #    " 0 k0I  a d a ¼ : ð8Þ 2 1    G  k D 0 dz b b 0 k0 As it is later explained in Subsection 2.3, this set of equations can be solved by successively employing a differential transfer matrix method. 2.2. H polarization Along the same vein, the Maxwell’s equations for Hpolarized waves can be projected onto the Hilbert space spanned by the Hermite–Gauss basis functions. Yet, the discontinuity of the permittivity profile becomes rather crucial in determining the convergence rate of the formulation, where field expressions are inevitably truncated [28,29]. This issue should be therefore carefully treated by choosing the right equations in applying the Galerkin’s method [30,31]. Here, the factorization technique as proposed by Sauvan et al. [31] is followed and the Maxwell’s equations are written as 1 oH y ¼ Ex ; ð9aÞ jxe oz oH y ¼ jxeEz ; ð9bÞ ox oEz oEx  ¼ jxlH y ; ð9cÞ ox oz where the H-polarized electromagnetic fields Hy(x, z), Ez(x, z) and Ex(x, z) can be expanded as 1 X Hy ¼ an ðzÞWn ðxÞ; ð10aÞ n¼0 1 X j Ex ¼ bn ðzÞWn ðxÞ; g n¼0 1 X j Ez ¼ cn ðzÞWn ðxÞ; g n¼0

ð10bÞ ð10cÞ

By substituting (10) in Maxwell’s equations given in (9), a new set of equations can be written down: 1 1 X 1 X dan ðzÞ Wn ðxÞ ¼ k 0 bn ðzÞWn ðxÞ; er ðx; zÞ n¼0 dz n¼0 1 1 X X an ðzÞW0n ðxÞ ¼ k 0 er ðx; zÞ cn ðzÞWn ðxÞ; n¼0 1 X

ð11aÞ ð11bÞ

n¼0

cn ðzÞW0n ðxÞ 

n¼0

¼ k0

1 X

1 X dbn ðzÞ Wn ðxÞ dz n¼0

an ðzÞWn ðxÞ:

ð11cÞ

n¼0

In a similar fashion, the preceding set of equations is projected on the Hilbert space spanned by the Hermite–Gauss basis functions, where a set of linear ordinary differential equations with non-constant coefficients is obtained. This new set can be expressed in the matrix form: d ½a ¼ k 0 F1 ½b; dz D1 G½a ¼ k 0 ½c; d G½c  ½b ¼ k 0 ½a; dz

ð12aÞ ð12bÞ ð12cÞ

where G matrix and D matrix are already defined in (7a) and (7b), respectively. The newly introduced F matrix; however, is defined as Z 1 Wm ðxÞWn ðxÞ dx ð13Þ F ¼ ½fmn ðzÞ; f mn ðzÞ ¼ er ðx; zÞ 1 The analytical calculation of functions fmn(z) follows the same line of functions dmn(z). After substituting (12b) in (12c) and by using (12a) we will have #    " 0 k 0 F1 a d a ¼ ð14Þ b ; dz b  k10 GD1 G  k 0 I 0 which is quite similar to (8) and can be solved by following the same procedure explained in Subsection 2.3. As already pointed out, this formulation eliminates the need for calculation of the permittivity profile derivative, i.e. $ln(e), and alleviates the slow convergence of the truncated expansions caused by the discontinuity of the permittivity profile. 2.3. Differential transfer matrix method The governing sets of linear differential equations for Epolarized and H-polarized waves, i.e. Eqs. (8) and (14), can both be reformulated according to the following general form:      0 Qðz; xÞ a d a ¼ ; ð15Þ dz b Hðz; xÞ 0 b

P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

where ð15aÞ

Q ¼ k 0 I; 1 H ¼  G2  k 0 D; k0

ð15bÞ

for E-polarization and Q ¼ k 0 D; 1 H ¼  GD1 G  k 0 I; k0

ð15cÞ ð15dÞ

for H-polarization. An approximate solution of the preceding set of differential equations is available via the following differential transfer matrix [24,25]: Z z0 þLz         aðz0 þ Lz Þ 0 Q aðz0 Þ ð16Þ dz  : bðz0 þ Lz Þ ¼ exp z H 0 bðz0 Þ 0 Here, z0 can be any arbitrary point. Still, the overall accuracy of the aforementioned solution can be further improved by using successive differential transfer matrices, where  !   Y  Z z0 þMm Lz  M aðz0 þ Lz Þ  0 Q aðz0 Þ exp dz  :  0 þ Lz Þ ¼ bðz H 0 bðz0 Þ z0 þm1 m¼1 M Lz ð17Þ 2.4. Periodic boundary condition On the other hand, the translational periodicity along the z-direction results in localized eigenmodes whose field profiles satisfy the following Bloch condition: Eðx; z0 þ Lz Þ ¼ expðjkLz ÞEðx; z0 Þ;

ð18Þ

H ðx; z0 þ Lz Þ ¼ expðjkLz ÞH ðx; z0 Þ;

ð19Þ

where Lz stands for the period of the structure along zdirection and each of the modes is characterized by the propagation constant k. Using (18) -(19), one can write down:     aðz0 þ Lz Þ  aðz0 Þ Þ ¼ expðjkL ð20Þ z  0 þ Lz Þ  0Þ : bðz bðz Now, combining (17) and (20) results in (  !) Z z0 þMm Lz  M Y 0 Q exp expðjkLz Þ ¼ eig dz ; H 0 z0 þm1 m¼1 M Lz and consequently: ( Z z0 þMm Lz  M Y 0 j ln eig exp k¼ m1 Lz z0 þ M Lz H m¼1

Q 0

2829

 Z expðjkLz Þ ¼ eig exp 

z0 þLz



z0

Z ¼ exp eig

z0 þLz z0

0

Q

  dz

H 0    0 Q dz ; H 0

which can be further simplified to Z z0 þLz    0 Q dz =ðjLz Þ: k ¼ eig H 0 z0

ð23Þ

ð24Þ

3. Numerical example In this section the dispersion diagram of four different photonic structures – widely studied in the literature – are investigated to assess the capability of the proposed method for obtaining numerically accurate results. In justification of our obtained results, the standard finite-difference time-domain method (FDTD) is employed. As the first numerical example, consider the structure shown in the inset of Fig. 1, where the removal of a row of rods breaks the perfect periodicity of a square array of parallel, infinitely long GaAs rods of circular cross section in air. These GaAs rods, with an index of refraction of 3.4, have a radius r = 0.18a, where a denotes the distance between the centers of two neighboring rods. The dispersion diagram of the E-polarized guided mode created by removing a row of rods in the (10) direction of the crystal, i.e. normalized frequency versus normalized wavevector, is plotted in Fig. 1. In this figure, dots represent numerical results obtained by following our proposed approach, whereas circles correspond to numerical results obtained by following the standard FDTD method. It is clearly shown that those results obtained by following our proposed method are in excellent agreement with those obtained by FDTD. However, thanks to the analytic nature of our calculation, the former running on a desktop

ð21Þ

 ! )! dz : ð22Þ

In most practical cases, M = 10 yields very accurate numerical results. However, the special case of M = 1 is worthy of further simplification, where

Fig. 1. Dispersion relations for an E-polarized photonic crystal waveguide, dots: proposed approach, circles: FDTD. Field expansions are truncated by keeping N = 25 terms and M = 10 slices are employed.

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P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

PC gives the accurate numerical results in just a couple of seconds and is about a factor of 1000 faster than the latter. As another example, a new photonic crystal waveguide is made by removing three rows of rods in the (11) direction of the photonic crystal shown in the inset of Fig. 2. Again, the dispersion diagram of the E-polarized guided modes is depicted in the figure, where the observed three guided modes inside the gap – according to their intrinsic symmetry – can be classified as even and odd modes [33]. In the same fashion, the obtained results are justified by using the standard FDTD approach. Furthermore, tangential electromagnetic field profiles of the even and odd modes corresponding to different normalized frequencies, i.e. xa/2pc = 0.35 (even), xa/2pc = 0.39 (odd), and xa/ 2pc = 0.45 (even), are plotted in Fig. 3a–c, respectively, where the proposed method by keeping N = 25 terms and M = 10 slices is employed. In the third example, a triangular photonic crystal made of air holes in GaAs is considered, where the perfect peri-

odicity of the lattice is perturbed by the removal of a row of holes in the C–X direction. The radius of these air holes reads as r = 0.3a, where a denotes the lattice constant. This is shown in the inset of Fig. 4, where the dispersion diagram of the H-polarized guided modes is plotted. Once again, the presented numerical results that are obtained very quickly and without heavy numerical calculations show excellent agreement with those obtained by applying FDTD. Finally, a coupled resonator optical waveguide (CROW) [34,35], which is constructed in the C–M direction within a triangular photonic crystal made of air holes in GaAs, is analyzed. The radius of the holes is r = 0.3a (see the inset of Fig. 5). The H-polarized guided modes supported by this structure are shown in Fig. 5, where dots represent numer-

Fig. 4. Dispersion relations for an H-polarized photonic crystal waveguide, dots: proposed approach, circles: FDTD. Field expansions are truncated by keeping N = 25 terms and M = 10 slices are employed.

Fig. 2. Dispersion relations for an E-polarized photonic crystal waveguide, dots: proposed approach, circles: FDTD. Field expansions are truncated by keeping N = 25 terms and M = 10 slices are employed.

Fig. 3. Electromagnetic field profiles at different normalized frequencies: (a) xa/2pc = 0.35, (b) xa/2pc = 0.39, (c) xa/2pc = 0.45. Field expansions are truncated by keeping N = 25 terms and M = 10 slices are employed.

Fig. 5. Dispersion relations for an H-polarized coupled resonator optical waveguide, dots: proposed approach, circles: FDTD. Field expansions are truncated by keeping N = 25 terms and M = 10 slices are employed.

P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

2831

ical results obtained by following our proposed approach and circles correspond to numerical results obtained by following the standard FDTD method. As previously mentioned, our proposed method, being as accurate as FDTD, is much faster. 4. Convergence rate In following our approach, the electromagnetic fields expanded in terms of Hermite–Gauss basis functions have to be inevitably truncated by keeping N terms. In this section, the convergence of the obtained numerical results in terms of N, the number of retained basis functions used in our calculation, is numerically demonstrated. In particular, two different normalized frequencies corresponding to the E-polarized impurity bands depicted in Fig. 2, viz. xa/2pc = 0.35 and xa/2pc = 0.41, are further investigated. The former corresponds to an even impurity mode, whereas the latter corresponds to an odd impurity mode. Fig. 6 shows the convergence of the normalized propagation constant as the number of basis functions used in the calculation is increased. The propagation constant of the even mode, whose corresponding field profile varies more smoothly, converges faster than that of the odd mode, whose corresponding field profile varies more rapidly. As expected, severe variations in the field profile can be reasonably approximated at the cost of N, i.e. the total number of retained basis functions in the expansion of electromagnetic fields. However, only tens of basis functions are sufficient to yield acceptable numerical results. Similarly, two normalized frequencies corresponding to the H-polarized impurity bands depicted in Fig. 4, viz. xa/2pc = 0.265 and xa/2pc = 0.245, are further investigated. Again, the former corresponds to an even impurity mode, whereas the latter corresponds to an odd impurity mode. This time, Fig. 7 shows the convergence of the normalized propagation constant as the number of basis functions used in the calculation is increased. Predictably, the

Fig. 6. The propagation constant of even and odd modes of Fig. 2 versus the number of retained basis functions N.

Fig. 7. The propagation constant of even and odd modes of Fig. 3 versus the number of retained basis functions N.

severe variations in the field profile of the odd mode can be reasonably approximated at the cost of N, i.e. the total number of retained basis functions in the expansion of electromagnetic fields. Notwithstanding, only tens of basis functions are sufficient to yield acceptable numerical results. It should be noticed that introducing an appropriate scaling factor in the expression of the Hermite–Gauss functions can further improve the convergence rate [32]. In the same way, the thickness of the guidance layer can be tuned to best match the non-scaled Hermite–Gauss basis functions given in (4). As a rule of thumb; however, the scaling factor could be chosen to equal the thickness of the guidance layer, wherein most of the electromagnetic energy is confined. 5. Computational complexity In this section, the overall computational complexity of the presented method is numerically studied. In Fig. 8, corresponding to the structure shown in the inset of Fig. 2, the computation time for extraction of the eigenmodes at an arbitrary normalized frequency (xa/2pc) is plotted versus N, i.e. the number of retained basis functions. It should be noted that the value of the aforementioned frequency does not alter the general trend of the plotted curve. The computational complexity is demonstrated to be almost quadratic as it approximately takes one hundred times larger to extract modes with N = 50 than it does with N = 5. In this figure, a typical quadratic relation between the computation time T and N, i.e. T = 0.001N2, is also given for comparison. In this figure, all the presented results are obtained by implementing the proposed approach on a PC Pentium 4 (2.4 GHz, 512 MB RAM) platform. Now, insofar as both matrices G and D are independent of the frequency at which the calculation is made, both can be generated once the permittivity profile of the structure,

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P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833 2

10

solid: calculation time for 100 points

T, computation time in seconds

2

dotted: T = 0.01 N 1

10

0

10

solid: calculation time for one point

-1

10

2

dotted: T = 0.001 N

-2

10

1

10

N, number of retained basis functions Fig. 8. Computation time versus of retained basis funtions.

i.e. er(x, z), is known. Consequently, the governing set of differential Eq. (15) can be very easily redressed to extract defect modes corresponding to different frequencies. This interesting feature further expedites the efficiency of the proposed method. The computation time for extraction of the eigenmodes, this time for 100 different frequency points, is therefore plotted versus N. Again, the computational complexity is demonstrated to be almost quadratic; however, T = 0.01N2 instead of T = 0.001N2 befits the real time of computation better. Accordingly, a hundredfold increase in the number of frequency points at which defect modes are extracted results in an approximately tenfold increase in the overall computation time. In comparison with Fourier-based analytical techniques, e.g. plane wave expansion by using the Fourier expansion in an appropriately defined supercell, the proposed method offers a certain advantage: the Galerkin’s method is applied only along the transverse direction and the longitudinal variation of the fields is analytically incorporated into Q and H functions introduced in (14). The size of the involved matrices is therefore smaller than the size of the supercellmatrices to which two-dimensional Fourier expansion is applied. As solving eigenvalue problems for matrices with order n typically requires O(n3) operations, reduction in the size of the involved matrices usually plays the decisive role in determining the overall run time. This difference is especially pronounced for extraction of dispersion diagram when a large number of frequency points is to be considered.

polynomials. In the proposed method, first order Maxwell’s equations- instead of second order wave equations which call for the calculation of the permittivity derivativeare converted to the standard matrix eigenvalue problem via the analytical projection of the governing differential equations into the Hilbert space spanned by the Hermite– Gauss basis functions. In our calculations, both major polarizations are separately formulated and the impurity bands of the E-polarized and H-polarized modes are extracted without the evaluation of the permittivity derivative. In order to assess the accuracy of the proposed method, four different test cases are further studied and in all cases the obtained numerical results turned out to be as accurate as the reference values computed by following the standard FDTD method. The convergence rate of the proposed method with respect to N, i.e. the number of retained basis functions, is numerically studied and a good convergence is observed. Thanks to the analytical nature of this approach, all the necessary calculations can be carried out in a couple of seconds. Appendix The calculation of the gmn coefficients and dmn functions ask for numerically heavy integrations. This difficulty can be overcome by analytical calculation of the integrals given in (7a,b). Since W0n ðxÞ can be analytically projected on Wn(x)s via well-known recursive relations [36], it is straightforward to show that: rffiffiffiffiffiffiffiffiffiffiffi rffiffiffi Z 1 n nþ1 0 dm;nþ1 : gmn ¼ Wn ðxÞWm ðxÞdx ¼ dm;n1  2 2 1 ðA1Þ However, in order to calculate the dmn functions, the permittivity profile should be written as the sum of the perfectly periodic permittivity profile and the localized defect: er ¼ er;per ðx; zÞ þ er;def ðx; zÞ;

ðA2Þ

where the perfectly periodic section, i.e. er,per (x, z), can be expanded in terms of two dimensional Fourier series and the defect section, i.e. er,def(x, z), can be expanded in terms of one dimensional Fourier series and Hermite–Gauss basis functions along the z and x direction, respectively: er;per ðx; zÞ ¼

L X K X l¼0

er;def ðx; zÞ ¼

bkl expðiGkx xÞ expðiGlz zÞ;

ðA3aÞ

akl Wk ðxÞ expðiGlz zÞ;

ðA3bÞ

k¼0

L X K X l¼0

k¼0

Therefore, 6. Conclusions In this paper, straight single line defects and coupled resonator optical waveguides in two-dimensional photonic crystals have been analytically investigated by expanding the electromagnetic fields in terms of the Hermite–Gauss

er ðx; zÞ ¼

L X K X l¼0

þ

akl Wk ðxÞ expðiGlz zÞ

k¼0

L X K X l¼0

k¼0

bkl expðiGkx xÞ expðiGlz zÞ:

ðA4Þ

P. Sarrafi et al. / Optics Communications 281 (2008) 2826–2833

Now, by writing the relative permittivity in the preceding form and substituting it in (7b), the dmn functions can be analytically expressed as d mn ðzÞ ¼

L X K X l¼0

þ

akl I m;n;k expðiGlz zÞ

k¼0

L X K X l¼0

bkl P m;n;k expðiGlz zÞ;

ðA5Þ

k¼0

R1 where I m;n;k¼ 1 Wm ðxÞWn ðxÞWk ðxÞdx is analytically available [37–39], and Pm,n,k can be written down as [23] R1 P m;n;k ¼ 1 Wm ðxÞWn ðxÞ expðiGkx xÞdx ¼ 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi  

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