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[1] A rigorous approach for modal analysis of two-dimensional photonic crystal waveguides consisting of layered arrays of circular cylinders is presented. The mode propagation constants and the mode field profiles can be accurately obtained by a simpler matrix calculus, using the one-dimensional lattice sums, the T matrix of an isolated circular cylinder, and the generalized reflection matrices for a multilayered system. Numerical examples of the dispersion characteristics and field distributions are presented for lowest even and odd transverse electric modes of a coupled two-parallel photonic crystal waveguide with a square lattice of dielectric circular cylinders in a background free space.

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[2] Photonic crystals are periodic dielectric structures in which any electromagnetic wave propagation is forbidden within a fairly large frequency range. A photonic crystal waveguide is formed by either introducing a defect layer in a photonic crystal or bounding a dielectric space by a photonic crystal. The guided fields are strongly confined because any electromagnetic energy cannot escape through the surrounding medium. Such waveguides have received growing attention in view of their promising applications to new integrated optical devices [Scherer et al., 1999]. The mode propagation in two-dimensional photonic crystal waveguides consisting of a lattice of circular cylinders has been extensively investigated, using various computational approaches such as the plane wave expansion method [Benisty, 1996], the finite difference time domain technique [Naka and Ikuno, 2000], and the beam propagation method [Koshiba et al., 2000]. Although these methods are powerful and can be universally applied to various configurations of photonic crystal waveguides, they yield approximate solutions because the electromagnetic boundary conditions between the circular cylinders and the background medium are not fully satisfied. A more rigorous treatment that takes into account the boundary conditions is an important issue to have precise understanding the guided wave phenomena in photonic crystals.

[3] In this paper, we shall present a rigorous semianalytical approach to modal analysis of a two-dimensional waveguide bounded by photonic crystals consisting of layered periodic arrays of circular cylinders. The method is an extension of the lattice sums technique combined with the T matrix approach which has been recently developed [Yasumoto et al., 2004] to analyze the electromagnetic scattering from the layered periodic arrays of circular cylinders. The dispersion equations for transverse electric (TE) and transverse magnetic (TM) guided modes are obtained in compact form in terms of the generalized reflection matrices for the layered periodic arrays. Numerical examples of the dispersion characteristics and field distributions are presented for the lowest TE modes in a coupled two-parallel photonic crystal waveguide with a square lattice of dielectric circular cylinders in a background free space. The results are compared with those [Sharkawy et al., 2002] obtained by the finite difference time domain (FDTD) method.

2. Formulation

[4] The side view of a two-dimensional waveguide is shown in Figure 1. The guiding region (−t < x < t) is bounded by two photonic crystals. The upper region 1 and the lower region 2 consist of N_{1}-layered and N_{2}-layered arrays of circular cylinders, respectively, which are infinitely long in the y direction and periodically spaced with a common distance h in the z direction. The cylindrical elements should be the same along each layer of the arrays but those in difference layers need not be necessarily same in material properties and dimensions. Figure 1 shows a typical configuration in which the identical arrays of circular cylinders with the same radius r_{1} (r_{2}) and permittivity ɛ_{1} (ɛ_{2}) are layered with an equal spacing d_{1} (d_{2}) along the x direction in the upper (lower) region. The parameter w indicates the displacement of the elements along the z direction. The background medium is a homogeneous dielectric with permittivity ɛ_{s} and permeability μ_{0}. The guided waves are assumed to be uniform in the y direction and vary in the form e^{iβz} in the z direction where β is a real propagation constant.

[5] The scattering from each layer of the arrays is characterized by the reflection and transmission matrices for the space harmonics with the z dependence as e^{iβl} where β_{l} = β + 2lπ/h and l is an integer. Let us consider an isolated jth array as shown in Figure 2 and employ the array plane x = x_{j} as the reference plane for the phase of the scattered space harmonics. Then the reflection matrix R_{j} and the transmission matrix F_{j} are derived as follows [Kushta and Yasumoto, 2000]:

with

where l, m, n = 0, ±1, ±2, ⋯, and k_{s} = ω is the wave number of the background medium. In (4), T_{j} is the T matrix of the circular cylinder in isolation which is obtained in closed form [Chew, 1990] for TE and TM waves, I is the unit matrix, and L is a square matrix whose elements are given by L_{mn} = S_{m−n}(k_{s}h, cos ϕ_{0}) where S_{m−n}(k_{s}h, cos ϕ_{0}) is the (m−n)th-order lattice sum [Nicorovici and McPhedran, 1994; Yasumoto and Yoshitomi, 1999]. Since the circular cylinders are up-down symmetric with respect to the array plane, the reflection and transmissions matrices are the same for both the downgoing and upgoing space harmonic waves.

[6] The generalized reflection and transmission matrices of an N-layered array as shown in Figure 1 can be obtained by successively concatenating the reflection and transmission matrices R_{j} and F_{j} in the x direction. Let and () be the generalized reflection and transmission matrices for the entire system of j-layered arrays, where the superscripts ± are used for the downgoing and upgoing incident waves, respectively. When the (j + 1)th array is stacked at x = x_{j+1} above the jth array located at x = x_{j}, the generalized reflection and transmission matrices and for this (j +1)-layered system are calculated using the following recursive relations [Yasumoto et al., 2004]:

with

[7] The generalized reflection and transmission matrices and for the N-layered arrays are calculated from (9)–(12) through the (N − 1) times recursion process starting with = R_{1} and = F_{1}. If the arrays consist of periodically layered identical arrays with an equal separation as shown in Figure 1, we may calculate and by using the concept of Floquet modes in the x direction [Yasumoto et al., 2004]. This approach is efficient when the number of arrays is increased.

[8] For the waveguide shown in Figure 1, the upper and lower boundaries viewed from the guiding region are characterized by the generalized reflection matrices R at x = t − 0 and R at x = −t + 0. Using the method mentioned above, these matrices are calculated as functions of β. Denoting the amplitude vectors of space harmonics incoming and outgoing for the plane x = t − 0 by a and b and those for the plane x = −t + 0 by a and b, respectively, the following relations are obtained:

where

Using (16)–(19), the following relation is derived:

[9]Equation (21) has nontrivial solutions only for discrete values β_{ν} of β which satisfy the dispersion equation

The value of β_{ν} gives the propagation constant of the νth guided mode propagating along the z direction. The result is substituted into (21) to determine the amplitude vectors a, a, b, and b for the νth mode. The mode field distribution in the plane transverse to the z axis can be calculated using a recursion formula starting from a and a (see Appendix A). The mode field pattern varies within a unit cell along the z axis but the same pattern is repeated with the period h.

3. Numerical Examples

[10] Although a substantial number of numerical examples could be generated, we shall discuss here only the results for the lowest TE modes in a coupled two-waveguide system. When two identical photonic crystal waveguides are brought in close proximity, they form a directional coupler which can be used for a variety of applications in integrated optics such as power division, switching, and wavelength or polarization selection. The characteristics of coupling are determined by the propagation constants and field distributions of two eigenmodes, an even mode and an odd mode, supported by the two-waveguide system. Their precise analysis is significantly important to design the coupler.

[11] The schematic diagram of the coupled photonic crystal waveguides is shown in Figure 3. Two parallel identical waveguides, formed by removing one row from a square lattice of dielectric circular cylinders in free space background, are separated by two rows of the lattice. The dielectric constant and radius of the cylinders are chosen to be ɛ/ɛ_{0} = 11.56 and r = 0.2h, respectively, where h is the lattice constant. Testing the convergence of solutions, the numerical results were obtained by taking account of the lowest seven space harmonics and truncating the cylindrical wave expansion at m = ±10 to calculate the T matrix of isolated circular cylinder.

[12]Figure 4 shows the dispersion curves and field distributions of the even and odd TE modes with E_{y} field in the coupled photonic crystal waveguides with N = 32. The rhombic marks on the dispersion curves indicate the mode cutoffs. The field distribution is plotted for a wavelength h/λ_{0} = 0.34 as functions of x on the plane z/h = 0.5 crossing the homogeneous background free space. The mode field patterns are rather complicated when compared with those of the conventional dielectric waveguides. The fields of both even and odd modes extend over five rows of the lattice outside of the guiding region and exhibit an oscillatory behavior as a consequence of confinement due to the Bragg reflection by the periodic lattice. The coupling length of two waveguides is described in terms of the difference β_{even} − β_{odd} between the propagation constants of the even and odd modes. Since the mode field pattern changes as a function of z with period h as a nature of a periodic system, an effective power transfer from one waveguide to the other is obtained when the coupling length takes an integral multiple of h. Let the coupling length be l_{c} = M_{1}h where M_{1} is a positive integer. Then the phase-matching condition for a coupler is given by

where M_{2} is a nonnegative integer. A very accurate calculation of β_{even} and β_{odd} as functions of the wavelength λ_{0} is required to realize the phase-matching condition (23). The rate of power transfer also depends on the field profiles of the even and odd modes. The transferred power takes a maximum when the sum and difference of field profiles in the even and odd modes coincide with that of the fundamental mode in each of two waveguides in isolation. We can see that the mode field distributions depicted in Figure 4b well satisfy such a requirement.

[13] The same configuration has been analyzed using the FDTD method, and the numerical data of the propagation constants (β_{even}, β_{odd}) and a typical coupling length L_{c} = 2π/(β_{even} − β_{odd}) have been reported as follows [Sharkawy et al., 2002]: β_{even} = 2.3487 × 10^{6} m^{−1}, β_{odd} = 1.9672 × 10^{6} m^{−1}, and L_{c} = 16.5 μm = 10.63 λ_{0} for λ_{0} = 1550 nm, h = 527 nm, and a = 106 nm. We shall compare these data with those calculated by the proposed semianalytical approach. For the same parameters, we have obtained that β_{even} = 2.342199 × 10^{6} m^{−1}, β_{odd} = 2.055885 × 10^{6} m^{−1}, and L_{c} = 21.93 μm = 14.15 λ_{0}. It is worth noting that the coupling length L_{c} predicted by the FDTD analysis is about 25% shorter than our result. This difference is serious in designing a photonic crystal waveguide directional coupler.

[14] Before concluding, we discuss the validity and limitation of the proposed method. The method uses the T matrix of an isolated circular cylinder, the lattice sums for a one-dimensional periodic system, and the matrix equations (9)–(15) for calculating the generalized reflection and transmission matrices of a layered system. First, the T matrix T_{j} of a circular cylinder is obtained in closed form and well defined if the dimension of the cylinder is moderate compared with the wavelength. Secondly, the lattice sums matrix L can be accurately calculated if the integral form [Yasumoto and Yoshitomi, 1999] of the lattice sums is used. It has been shown that the calculated values of the lattice sums enable one to obtain the free space periodic Green's function with the accuracy to fourteen decimal places [Yasumoto and Yoshitomi, 1999]. Finally, the matrix equations (9)–(14) are always stable because the transfer matrix V_{j} in (15) is given in terms of only propagating or decaying space harmonic waves. The roots searching procedure for (22) is similar to that applied to the transcendental equation for mode propagation constant in optical fibers. These facts validate that the modal solutions obtained by the present analysis are almost rigorous. The method can yield very accurate results of the propagation constants and modal field distributions even in close vicinity to cutoffs so long as the propagation constants take real values.

[15] However, the use of the lattice sums limits the applicability of the proposed method. The lattice sums technique assumes an infinite periodic system along the wave propagation. The lattice sums diverge when the propagation constants take complex values. For this reason, the method can be applied only to the straight photonic crystal waveguides and the pure guided modes with real propagation constants.

4. Conclusion

[16] We have presented a semianalytical approach to analyze the eigenmode fields supported by two-dimensional optical waveguides bounded by photonic crystals consisting of a lattice of circular cylinders. The method is rigorous since the electromagnetic boundary conditions on the circular cylinders are fully satisfied and can be applied to photonic crystal waveguides with a large difference of refractive indices between the cylinders and the background medium. If the lattice element contains two or more circular cylinders in unit cell, the T matrix in (3) is replaced by the aggregate T matrix for the composite cylindrical system. This extension is straightforward.

Appendix A: Calculation of Modal Fields

[17] A schematic of succeeding two layers of the arrays is shown in Figure A1. As in (16)–(19), the amplitudes of the space harmonics incoming on the jth array are denoted by a and those outgoing from the same array are denoted by b. Then the guided mode field within a homogeneous strip region x_{j} + r_{j} < x < x_{j+1} − r_{j+1} between two layers of arrays are expressed using a superposition of the space harmonic fields as follows:

where

[18] Taking the ray tracing for the space harmonics across the two layered arrays, on the other hand, we have the following relations:

where denotes the generalized reflection matrix of the j +1-layered arrays viewed from the downgoing incident space harmonics with a. Solving (A4) and (A5), a and b are related to a as follows:

From (A8) and (A10), b and b appeared in (A1) are related to a. Thus the mode field in the homogeneous strip region x_{j} + r_{j} < x < x_{j+1} − r_{j+1} can be calculated when a is specified. However, this field expression based on the space harmonic expansion does not converge in the inhomogeneous grating region ∣x − x_{j}∣ < r_{j} which contains the periodic arrays of circular cylinders of radius r_{j}. For the grating region, we must turn to the original expression of the scattered field using the cylindrical wave expansion. Let us consider the grating region within ∣x − x_{j+1}∣ < r_{j+1}. The incident waves on the (j + 1)th array are the downgoing and upgoing space harmonics with amplitudes a. Then the scattered field outside the zeroth cylinder within the unit cell is expressed as follows:

where

Applying the boundary condition on the cylindrical surface ρ_{0} = r_{j+1}, from (A11) the scattered field inside the zeroth cylinder are obtained as follows:

where

Thus the scattered fields Ψ_{>}(ρ_{0}, θ_{0}) and Ψ_{<}(ρ_{0}, θ_{0}) within a unit cell including the zeroth cylinder can be calculated using the amplitudes a of incoming space harmonics, where a is related to a through (32). Equations (A4) to (A10) are recursively used to obtain a for all layers of arrays by starting from a and a which are obtained as the solutions to (21). This recursion process is performed by a standard matrix calculus.

Acknowledgments

[19] This work was supported by a 2003 research grant from Hoso-Bunka Foundation.