Systematic Design of Antireflection Coating for One-dimensional Photonic Crystals Using Bloch Wave Expansion

We present a systematic method for designing a perfect antireflection coating (ARC) of a one-dimensional (1D) photonic crystal (PC) with an arbitrary unit cell. We use Bloch wave expansion and time reversal symmetry, which leads exactly to analytic formulas of structural parameters for the ARC and renormalized Fresnel coefficients of the PC. Surface immittance (admittance and impedance) matching plays essential role for designing the ARC of 1D PC’s, which is shown together with a practical example. Typeset using REVTEX 1 Photonic crystals (PC’s) have unique energy dispersion due to coupling between periodic materials and electromagnetic (EM) waves. In particular, strong energy dispersion of propagation modes of PC’s has attracted much interest because of their applicability to add/drop multiplexers, dispersion compensators, polarization filters, and image processors. These transmission-type applications of PC’s require achieving negligibly small reflection-loss at the interface of the PC’s. Therefore, it is important to apply effective antireflection interface-structures, or antireflection coatings (ARC’s) to the input and output interfaces of the PC’s. There have been several reports on ARC’s designed for the PC’s. The structural ideas of those ARC’s were developed from the concept of adiabatic interconnection or weve-vector matching (k-matching) for two-dimensional (2D) PC’s. Reflectance at the interface of 1D PC’s has been calculated by plane-wave expansion and multiplication of a transfer matrix for a unit cell. However, these approaches do not tell directly the optimal parameters of the ARC to be applied, so iteration of numerical calculation is required until those parameters have been optimized. In this Letter, we derive analytic formulas of structural parameters of an ARC that is applied to a 1D PC. We deal with 1D systems for simplicity, but the formulas to be derived would be applicable to multi-dimensional PC’s under appropriate approximation. Structural parameters of a conventional ARC between two homogeneous media are readily calculated if the refractive indices of the two media are known. In Fig. 1 (a), the ARC (region 2) with refractive index n2 and thickness d is placed between two semi-infinite homogeneous media with refractive indices n1 (region 1) and n3 (region 3). These three regions are divided by two boundaries at z = 0 and −d, where the z-axis is defined as perpendicular to the surface of the ARC. Each region consists of linear and lossless dielectrics. The reflection coefficient at z = −d is then given by r = r1,2 + r2,3exp[2ik2d] 1 + r1,2r2,3exp[2ik2d] , (1) where k2 is the normal component of the wave vector in the ARC, and ri,j is the reflection coefficient of light propagating from region i to j. The reflection coefficient in Eq. (1) 2 equals zero for normal incidence light when d = λ0/4n2 and n2 = √ n1n3, where λ0 is the wavelength in vacuum. As in Fig. 1 (b), we replace the semi-infinite homogeneous medium (region 3) in Fig. 2 (a) by a semi-infinite 1D PC with lattice constant Λ and periodic refractive index function n(z). The function n(z) has an arbitrary spatial modulation in the unit cell. To determine the reflection coefficient for this case, we expand the EM waves in the PC using all the eigenmodes in the PC, namely propagating and decaying Bloch waves. Those Bloch waves in the 1D PC can be obtained as the eigenvectors of a transfer matrix under a given frequency ω, a parallel component of a wave vector k‖, and a polarization of light σ. The reflection coefficient r of the electric field at the interface (z = −d) is determined by considering the boundary condition. The expression of the derived reflection coefficient is formally the same as Eq. (1), except for r2,3. The reflection coefficient r2,3 for normal incidence light is modified into r2,3(ω) = n2 −N(ω) n2 +N(ω) , (2) where N(ω) equals to ±Yk,k(ω), surface admittance of the Bloch waves. The signs differ for two orthogonal light polarizations in Cartesian coordinate system. The surface admittance Yk,k is defined as Yk,k = Hk/Ek , (3) where Ek(Hk) stands for the tangential component of the electric(magnetic) field of the Bloch waves at the interface (z = +0), and k is the Bloch wave vector. In the derivation of Eqs. (2) and (3) we used the fact that time reversal symmetry inhibits the simultaneous appearance of both the propagating and decaying modes in the same direction for a given set {ω, k‖ and σ} in a 1D PC. Note that the function N(ω) in Eq. (2) is complex in general due to the phase difference between Ek and Hk of the Bloch waves at the interface, but that the imaginary part of N(ω) is zero if the surface of the PC’s is a mirror plane in a infinite form of the PC’s. Note also that multiple reflections in the 1D PC’s, which are represented 3 as plane waves, are renormalized into a single Fresnel coefficient in Eq. (2) by using the Bloch wave expansion. The pair of the refractive index n2 and thickness d of the ARC for 1D PC’s to achieve zero reflectance is found as follows. The extremal condition of the reflectance R(= |r|2) with respect to d, i.e., ∂R/∂d = 0, is written as D ≡ d λ0/4n2 = 1 2πi ln [ r 2,3 r2,3 ] +m , m = 0, 1, · · · . (4) We call D the “normalized thickness”. Substituting d derived from Eq. (4) into Eq. (1), we obtain the refractive index n2 that makes the reflectance zero: n2 = √