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The radiational eigenmodes in photonic crystal slabs are classified into guided and leaky modes. The former are genuine eigenmodes with infinite lifetimes, while the latter are quasi-eigenmodes with finite lifetimes. We will show that the dispersion curves of photonic crystal slabs are well represented by the folding of those in uniform dielectric slabs into the two-dimensional Brillouin zone. This feature enables us to predict the symmetries of the eigenmodes by group theory. The optical transmission due to the leaky modes in the direction parallel to the slab surface is mainly governed by their diffraction loss, which is a particular feature of photonic crystal slabs. We will show that the diffraction loss is suppressed by symmetry mismatching between internal eigenmodes and the external radiation field [81, 82].

8.1 Eigenmodes of Uniform Slabs Before we treat photonic crystal slabs, we examine the radiational eigenmodes in uniform dielectric slabs. We assume a geometry shown in Fig. 8.1, where cb and d denote the dielectric constant and the thickness of the slab. We assume that the structure is infinite in the x andy directions and surrounded by air in the z direction. We take the x-y plane in the middle of the slab. Then the structure has mirror symmetry about the x-y plane, z. 1

a

z

Fig. 8.1. Illustration of a uniform dielectric slab 1

We denote symmetry operations by hats to distinguish them from their eigenvalues in this chapter.

K. Sakoda, Optical Properties of Photonic Crystals © Springer-Verlag Berlin Heidelberg 2001

178

8. Photonic Crystal Slabs

Guided radiational modes that are confined in a uniform slab are classified into four categories according to the symmetry for the mirror reflection az and to their polarizations. Those modes whose electric fields lie in the xy plane are referred to as transverse electric (TE) modes, whereas those modes whose magnetic fields lie in the x-y plane are referred to as transverse magnetic (TM) modes. Each mode is also characterized by the wave number k 11 in the x-y plane. Now, we derive their dispersion relation. We begin with the symmetric TE mode. We assume without the loss of generality that the mode propagates in the x direction. Hence, we have the following form for the electric field in the slab:

~ ~Y

E,

(

)

exp {i (klx-

wt)} oov (k,z)

(8.1)

The z component of the wave vector in the slab, kz, is related to k 11 and w by

(8.2) which is derived from the wave equation. From netic field is given by H

1

=

.Ely

IWf.lo

(

vXE

kz sinkzz ) 0 exp {i (kllx-

wt)}.

=

-8B I at, the mag-

(8.3)

ik11 coskzz

The electric field for z > d/2 is expressed as

(8.4) where

K

is related to k 11 and w by

"'> 0.

(8.5)

Since we are dealing with guided modes, we assumed an exponential decrease for the electric field in the air region in (8.4). The magnetic field for z > d/2 is given by

(8.6)

8.1 Eigenmodes of Uniform Slabs

179

1.0 w=ck

11

0.8

TM, az= I .--· TE, az = -1

.-· TM, az =-I TE,az=!

0.6 ~~~ 81::

0.4 0.2

1.5

0.5

2.0

Fig. 8.2. Dispe