Band Structure of Two-Dimensionally Photonic Crystal

R. García-Llamas1,*

, J. D. Valenzuela-Sau2

1Research Physics Department, University of Sonora, México

‎2Physic Doctorate, University of Sonora, México *corresponding author: ragal@cifus.uson.mx

Abstract-We calculate the photonic band structure and the electromagnetic modes of a photonic

crystal; a rectangular array of cylinders of elliptical cross section, for TE and TM polarizations. A

full band gap for both polarizations was found, and the band structure was compared with that

presented in ref. [17]. We plan to calculate the band structure and modes considering oblique

electromagnetic propagation according to the homogeneous axis of the system.

Since the first prediction of the photonic crystals (PC) [1-8], literature about this issue [9-17] has been

increased vastly. Photonic band structures represent electromagnetic modes and frequency bands for a

photonic crystal. Modes can be in-plane (stationary), and out-of-plane (propagating). Frequency bands can be

permitted or forbidden (band gaps). The plane wave method was used to obtain the band structure for two cases:

In-plane and out-of-plane. From Maxwell’s equations, we find the wave equation for the electric and magnetic

fields; in order to solve this equation, we define the fields as Bloch waves, and the inverse of the dielectric

constant as a Fourier’s series. Considering TE and TM polarizations, in-plane modes are obtained for a

rectangular array of cylinders of elliptical cross section, and for each polarization, the intensity of the fields is

presented in the primitive cell and its neighborhoods in a high symmetry point in the Brillouin zone. Considering

oblique propagation according to the homogeneous axis of the system, we plan to calculate out-of-plane modes

for the same structure.

The electric or magnetic field can be written as a Bloch wave expansion

where is the Bloch wave vector, is the z-component of the wave vector, and and are

two vectors of the reciprocal lattice. Inverse of the dielectric constant is written as a Fourier series and its coefficients

are calculated numerically.

The structure studied, is a rectangular lattice of vacuum (εc = 1) cylinders of elliptical cross section

embedded in a dielectric material (εe = 11.4).

In the figure (left hand), the photonic band structure of the studied system is presented. The continuous

green (dashed blue) curves represent the TE (TM) modes, and the continuous (dashed) black line represents the

results presented in reference [17]. A full band gap is represented for a grey area. The representation of

irreducible first Brillouin zone is shown in the left hand inset. In the right hand inset, the rectangular unit cell (uc)

is shown, where the black (white) colored zone represents the dielectric (vacuum) medium. The sides of the

rectangular uc are ax and ay= ax/0.77. The ellipse has a minor radius rx=0.38ay, and a major radius ry=0.45ay. The

filling factor is 0.696.

In the figure (right hand), the square modulus of the magnetic field is shown in the unit cell (contoured in

black lines) and its neighborhoods, in the high symmetry point M. Frequency and the speed of light in vacuum

are presented for ω and c, respectively.

ay/(2c)=0.55, q

x=/a

x, q

y=/a

y, K=3

y

x0

0.50

1.0

1.5

2.0

2.5

3.0

3.5

Results of the out-of-plane band structure of the rectangular lattice of cylinders of elliptical cross section will be

present during the meeting.

Acknowledgements, to the National Council of Science and Technology of México (CONACYT), for the

scholarship granted to one of the authors2.

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