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Self-guiding in two-dimensional photonic crystals

Dmitry N. Chigrin, Stefan Enoch, Clivia M. Sotomayor Torres, and Gérard Tayeb

Open Access Open Access

Abstract

Dielectric periodic media can possess a complex photonic band structure with allowed bands displaying strong dispersion and anisotropy. We show that for some frequencies the form of iso-frequency contours mimics the form of the first Brillouin zone of the crystal. A wide angular range of flat dispersion exists for such frequencies. The regions of iso-frequency contours with near-zero curvature cancel out diffraction of the light beam, leading to a self-guided beam.

©2003 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. Convex (red), concave (blue) and flat (black) iso-frequency contours. Iso-frequency contours on the left (right) panel correspond to the second band of a two-dimensional square (triangular) lattice photonic crystal made of dielectric rods in vacuum. Rods have the refractive index 2.9 and radius r=0.15d, where d is the lattice period. The wave vectors resulting in the Bloch eigenwaves with the group velocity pointing to the direction normal to the Brillouin zone boundary are depicted with arrows.

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Fig. 2.
Fig. 2. Photonic band structure of the square lattice photonic crystal made of dielectric rods in vacuum. Rods have the refractive index 2.9 and radius r=0.15d, where d is the lattice period. The band structure is given for TM polarization. The frequency is normalized to Ω=ωd/2πc=d/λ. c is the speed of light in the vacuum. Insets show the first Brillouin zone (left) and a part of the lattice (right).

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Fig. 3.
Fig. 3. Iso-frequency diagram for the normalized frequencies Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765 (right). The shaded regions show the wave vectors resulting in the Bloch eigenwaves with the group velocity vectors pointing to the same direction. The crystal parameters are given in Fig. 2 caption.

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Fig. 4.
Fig. 4. Radiation pattern of a point source inside the photonic crystal with parameters given in Fig. 2 caption. The normalized frequencies are Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765.

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Fig. 5.
Fig. 5. Modulus of the electric field map for a 30×30 rod photonic crystal excited by a point isotropic source. Left, Ω=d/λ=0.3567; right, Ω=d/λ=0.5765. The axis scales are in units of d and the colorscale is from 0 (blue) to 1 (red).

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Fig. 6.
Fig. 6. Self-guiding in a limit-size photonic crystal. Left: Modulus of the electric field map for a 20×40 rod photonic crystal illuminated by a Gaussian beam with W/d=2.5, Ω=d/λ=0.5765. Right: Modulus of the electric field of the incident Gaussian beam. The axis scales are in units of d and the colorscale is from 0 (blue) to 1 (red).

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Fig. 7.
Fig. 7. Modulus of the field as a function of x. The crystal is the one depicted in Fig. 6, left. The normalized frequency is Ω=d/λ=0.5765. From the top to the bottom: modulus of the incident field at y=19.5d for θ0=0°, modulus of the total field at y=-21.5d for θ0=0°, modulus of the total field at y=-21.5d for θ0=5°, modulus of the total field at y=-21.5d for θ0=10°

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Fig. 8.
Fig. 8. Crossing of two self-guided beams.Modulus of the electric field map for a 30×30 rod photonic crystal illuminated two perpendicular Gaussian beams with W/d=2.5, d/λ=0.5714 and the colorscale is from 0 (blue) to 1 (red).

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Equations (4)

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A ( r ) = − i 4 π c ω 0 V ∑ n ∫ BZ d 3 k n ( a n k * ( r 0 ) · d ) ( ω n k 2 − ω 0 2 ) a n k ( r ) e i k n ( r − r 0 ) .
A ( r , t ) ≈ ∑ ν ∑ n exp ( − i ( ω 0 t + π 4 ( sign ⁡ ( α 1 ν ) + sign ⁡ ( α 2 ν ) ) ) )
× c V ( A n k ν * ( r 0 ) · d ) A n k ν ( r ) ∣ V n k ν ∣ 8 π 3 ∣ K n k ν ∣ 1 2 ∣ r − r 0 ∣ ,
A ( α ) = W 2 π exp ⁡ ( − ( α − α 0 ) 2 W 2 4 )
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