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Graded index photonic crystals

Hamza Kurt and David S. Citrin

Open Access Open Access

Abstract

We explore two-dimensional triangular lattice photonic crystals composed of air holes in a dielectric background which are subject to a graded-index distribution along the direction transverse to the propagation. The proper choice of the parameters such as the input beam width, gradient coefficient, and the operating frequency allow the realizations of the focusing (lens) and guiding (waveguide) effects upon which more complex optical devices such as couplers can be designed. Numerical results obtained by the finite-difference time-domain and planewave expansion methods validate the application of Gaussian optics within a range of parameters where close agreement between them are observed.

©2007 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. GRIN PC structure: (a) 2D triangular lattice PCs with air holes in dielectric background and the desired refractive index variation along the transverse direction. The two different ways of the realization of such an index variation either (b) by varying the radii of holes or (c) by changing the refractive indices of holes.

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Fig. 2.
Fig. 2. GRIN PC structure: 2D triangular lattice photonic crystals with air holes in dielectric background. The right and the left plots show the refractive index variation along the dotted section.

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Fig. 3.
Fig. 3. The steady-state magnetic field variation for the (a) self-collimated and (b) converging cases for the input beam at frequencies a/λ = 0.12 and a/λ = 0.24, respectively.

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Fig. 4.
Fig. 4. Equi-frequency curves of the three lowest bands. The refractive indices of air holes are varied (εa = 1.0 and εa = 4.0) keeping the background refractive indices the same. The dotted circles and hexagonal shapes highlight the frequency curves of a/λ = 0.12 and 0.16 for the first band and a/λ = 0.24 for the second band.

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Fig. 5.
Fig. 5. The beam profiles across the y section for the (a) self-collimation at three different distances and (b) focusing cases (solid line is for the input beam and dotted line is at the focal plane).

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Fig. 6.
Fig. 6. (a)–6(c) Three different input beam widths (values are listed in Table I) and the steady-state field plot for the focusing cases at frequency below the band-gap (a/λ = 0.16) and (d–f) above the band-gap (a/λ = 0.24).

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Fig. 7.
Fig. 7. (a). Beam amplitude plots of Figs. 5 (a)–5(c) along the middle section of the GRIN PC and (b) for the Figs. 5 (d)–5(f). Solid, dashed and dotted lines correspond to different input beam width in increasing order.

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Fig. 8.
Fig. 8. The beam profiles at the input and output (focal) planes at frequency below the band-gap (a–c) and above the band-gap (e–f).

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Fig. 9.
Fig. 9. The steady-state field map for frequencies (a) inside band-gap and (b) at high frequencies.

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Fig. 10.
Fig. 10. The steady-state field map for the (a) self-collimation (a/λ = 0.24) and (b) focusing cases (a/λ = 0.16) for a long propagation distances.

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Tables (2)

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Table I. FWHM of the beam profile at the input and focal planes for two different frequencies.

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Table II. Comparison of spot-size width values obtained by FDTD and Gaussian optics for two different frequencies.

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Equations (15)

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1 q ( x ) = 1 R ( x ) − j λ 0 π ω 2 ( x ) n 0 ,
[ r 2 r 2 ′ ] = [ A B C D ] [ r 1 r 1 ′ ] ,
[ A B C D ] = [ cos ⁡ α d sin ⁡ α d α − α sin ⁡ α d cos ⁡ α d ] .
q out = cos ⁡ ( α d ) q in + sin ⁡ ( α d ) α − α sin ⁡ ( α d ) q in + cos ⁡ ( α d ) .
q in = i π n 0 w 0 2 λ 0 = i q i ′ ,
q out = i cos ⁡ ( α x ) q i ′ + sin ⁡ ( α x ) α − i α sin ⁡ ( α x ) q i ′ + cos ⁡ ( α x ) .
Re ( q out ) = Re ( A q in + B C q in + C ) = 0 .
w 0 = ( λ π n 0 α ) 1 2 .
q out ( x ) = i π n x w x 2 λ 0 ,
q i ′ cos ⁡ 2 ( α x ) + α 2 sin ⁡ 2 ( α x ) q i ′ 2 = π n x w x 2 λ 0 ,
w x = n 0 w 0 2 n x [ cos ⁡ 2 ( α x ) + α 2 sin ⁡ 2 ( α x ) π n 0 w 0 2 λ 0 ] .
H z ( x , y ) = H 0 w 0 w ( x ) exp ( − i k y 2 2 q ( x ) ) = H 0 w 0 w ( x ) exp [ − i k y 2 2 ( 1 R ( x ) − i λ 0 π w 2 ( x ) n 0 ) ]
= H 0 w 0 w ( x ) exp ( − i π n 0 y 2 λ 0 R ( x ) ) exp ( − y 2 w 2 ( x ) )
f = − 1 C = 1 n x α sin ⁡ ( α d ) .
n eff 2 = f n 1 2 + ( 1 − f ) n 2 2 + 1 3 [ a λ π f ( 1 − f ) ( n 1 2 − n 2 2 ) ] 2 ,
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