Pis'ma v ZhETF, vol.93, iss.8, pp.473-476
© 2011 April 25
Optical Fano resonances in photonic crystal slabs near diffraction
threshold anomalies
A.B.Akimov+1\ N.A. Gippius+*, S. G. Tikhodeev+
+ A.M. Prokhorov General Physics Institute RAS, 119991 Moscow, Russia * LASMEA, UMR 6602 CNRS, Université Blaise Pascal, 63177 Aubière, France Submitted 24 February 2011
Optical Fano resonances due to resonant eigenmodes in a layered periodically-modulated structure (photonic crystal slab) are investigated theoretically. The special attention is focused on the behavior of the resonances near a diffraction threshold. A new formulation of the resonant mode approximation for the optical scattering matrix near the diffraction threshold anomalies is proposed.
Layered periodic structures or photonic crystal slabs (PCS) [1] attract a great interest in modern nanooptics. A very convenient formalism for the numerical investigation of periodic structures is the Fourier modal method, also known as the rigorous coupled wave analysis [2], especially in the form of the optical scattering matrix (Si-matrix) approach [3-5]. It is based on a decomposition of the solutions of the Maxwell's equations into a set of the Bloch waves. S-matrix connects the hypervectors of complex amplitudes of Bloch harmonics corresponding to the incoming and outgoing waves, 11) and |O),
S\I) = IO).
(1)
The S-matrix method has been greatly improved in recent years with several techniques, such as factorization rules [6] and adaptive spatial resolution [7] (see, e.g., [8]). Due to this progress it is now possible to consider periodic structures with a complicated unit cell consisted of arbitrary materials including metals and anisotropic crystals. However, the needed computational time can still be very long. Therefore, physically clear approximations giving a qualitative, and, if possible, quantitative prediction of the system's properties are crucially important. In this paper we suggest a generalization of the resonant mode approximation (RMA) [9-11], which allows one to describe correctly the optical properties of PCS with the Fano-type resonances [12] near the Wood-Rayleigh diffraction threshold anomalies [13] (see an early analysis in [14]). The approximation is free of fitting parameters: all needed quantities are calculated from the eigenproblem for the linearized inverse S-matrix.
e-mail: toshaakimov0gmail.com
Within the scattering matrix formalism, a resonant mode corresponds to the pole of the S-matrix or zero of the inverse S-matrix,
S-1 (0r,k||) IOr) = 0.
(2)
Here fir is a complex eigenfrequency of the mode and |Or) is the corresponding output eigenvector. Equation (2) defines the dependencies of fir and |Or) on the in-plane component of the incoming light wavevector ky = (kx,ky), thus allowing to find the dispersion law of the resonant mode and its optical properties.
Far from the diffraction thresholds, if N different poles are located near the frequency range of interest, S-matrix can be represented [9-11] as a sum of slowly varying background and resonant parts Sj and Sr,
N
S — Sj + Sr — Sj + ^ ]
r=l
w — fir
(3)
Here matrices Ar depend on the corresponding input and output resonant vectors, \Ir) and |Or),
Ar = \Or)(Ir\.
(4)
The details of this resonant mode approximation can be found in [10, 11, 15]. It allows one to evaluate the S-matrix near the resonances without fitting parameters. However, RMA in form of Eqs. (3), (4) fails near the diffraction thresholds where the folded dispersion curves cross the folded light cones
fit = —= Ikii
G
(5)
where G is the two-dimensional reciprocal lattice of the PCS and e is the dielectric constant of the media to which the corresponding diffraction order is opened at this threshold photon frequency fij.
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A. B. Akímov, N. A. Gippius, S. G. Tikhodeev
On the other side, as shown in [16], S-matrix in the vicinity of the threshold can be represented as
S = S0 + Bit,
where the parameter
k =
Ve c
w*
m
(6)
(7)
is a z-component of the wavevector of the new diffracted wave. This parameter is real above the threshold (w > > fit ) where the diffracted wave is propagating, and pure imaginary below the threshold.
However, if resonances approach the diffraction threshold, Eq. (6) becomes applicable only in a very narrow frequency range around the threshold. In order to propose a better approximation for this case, let us note first of all that fit is the branching point of the S-matrix, thus, the analytical continuations of the inverse scattering matrix S_1(w) into Imw < 0 part of the complex frequency plane, needed to describe the resonances, are different depending on whether the continuation is done from the segment w > fit or w < fit of the real frequency axis.
The situation simplifies greatly if S-matrix is considered as a function of k instead of w. Then the branching point at fit is removed, and S-matrix as a two-sheet function of w is mapped into a single-sheet function of k. The "physical" segments of the real frequency axis above the threshold a; > fit and below the threshold a; < fit then correspond to the rays on the complex k plane A = (Ren > 0,1m«; = 0) and B = (Ren = 0,1m« > 0), respectively (see the straight thick lines in Fig. 1). The analytical continuation of S(«) into complex « plane cannot have poles in the quadrant Re « > 0, Im« > 0, because S(u>) is analytical when Imw > 0. Thus, all the poles of S(«,ky) are below A and/or to the left of B; the physically important resonances have to be close to these physical rays A and/or B.
In what follows we consider the simplest case when there are only two poles «i and «2 in the vicinity of the physical rays A and B, respectively. Instead of Eq. (3) we can now write the Breit-Wigner type expression for the S-matrix as a function of « as
S = Sb + Sr = Sb + ¿2 I0»-)—-—(U\- (8)
r=l,2 K Kr
We will now demonstrate that Eq. (8) is really a very convenient approximation for S-matrix in the vicinity of the diffraction threshold, on example of a one dimen-sionally periodic PCS.
Let us consider, for simplicity, the near-threshold behavior of so called quasiguided modes [5]. Such resonances exist in periodically modulated planar dielectric
2.0
1.5
1.0
0.5
m (| 0
* -0.5
F -1.0
-1.5
-2.0
-2.5
B
J
A— J* A
V 1
,200 nm 80 n ml .........A
* 300 nm * O —\/
-2
-1
Re k (|im 1)
Fig. 1. The calculated trajectories of the poles «1,2 (curves 1,2, respectively) of S'(/i)-matrix of the model structure on the complex n plane. The in-plane momentum kx is changed (see arrows) from 0 to 0.3 ¡imT1 in case of curve 2 and in the opposite direction in case of curve 1. The circles and rectangles denote the positions of poles at kx = 0.17 iimT1 and kx = 0.2 iimT1, respectively. Thick straight rays A and B correspond to the real frequency ranges above and below the threshold frequency fi*. The crossection of the model structure is shown in the insert
waveguides; they originate from guided modes of the homogenized planar waveguide but acquire a finite lifetime due to multiple scattering on the periodic inhomo-geneities of the modulated structure.
The model grating-type PCS (see the scheme in Fig. 1 inset) consists of the ZnO rectangular wires (ew = 6.25) embedded into quartz (e = 2.25) substrate. The period of the structure is d = 300 nm, the height and the width of the wires are 80 nm and 200 nm, respectively. For the sake of simplicity we restrict ourselves to the case of inclined and perpendicular to the wires (ky = 0) incidence from air of the incoming TM-polarized light wave (magnetic field parallel to the wires). We focus our attention on the first diffraction channel opening to the substrate. The corresponding folded light cone is given by
c /2?r
Ve \ d
The calculated trajectories of the poles «1,2 fe) when kx is changed between 0 and 0.3 /tm_1 are shown in Fig. 1. The calculation was done via the eigenproblem for the linearized inverse S-matrix near its zeroes as described in [10] (with the only difference that instead of the frequency plane, the linearization is done on the complex k plane). It is clear that near the threshold (k = 0) both K\ and K2 become important in Eq. (8), because they both are located at comparable distances
fit =
k3
(9)
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2
Optical Fano resonances in photonic crystal slabs
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from the physical rays A and B. However, far from the threshold only one pole remains important: «1 above the threshold and «2 below the threshold.
The poles of the S-matrix as a function of frequency are then
0
1,2
f
2„2
= A/n?
<TK-
(10)
Figure 2 shows the real and imaginary parts of Oij2, corresponding to «1,2 in Fig. 1, as functions of kx. It
>
2750
* 2700
> OJ
ES
10 0 -10
2
1
0
0.05 0.1
0.15
kx Om_1)
0.2 0.25 0.3
Fig. 2. The real and imaginary parts (upper and lowers panel, respectively) of the eigenenergies Mii,2 (curves 1,2) as functions of kx. Arrows correspond to the arrows in Fig. 1. Dashed straight line 3 in the upper panel is the folded light cone Eq. (9)
can be understood from Eq.(10) and Figs. 1,2 that fix and 02 are the poles of the analytical continuation of S-matrix as a function of energy above and below the threshold, respectively. Note that the imaginary part of the pole frequency can even become positive if it is located far from the physical segment of the real frequency axis, i.e., far into the unphysical sheet of the S-matrix as a function of frequency.
For checking the approximation Eq. (8) it is representative to consider the cases of kx = 0.17 /tm_1 and 0.2/tm-1. The results of calculations within the S-matrix method [5] (with 15 plane waves) in comparison with the modified resonant mode approximation Eq. (8) are shown in Fig. 3. As already noted above, the eigenvalues «1,2, input and outp
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