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Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis

Robert C. Gauthier and Khaled Mnaymneh

Open Access Open Access

Abstract

The TM propagation properties of planar 12-fold photonic quasi-crystal patterns are theoretically examined using FDTD. The patterns examined can be produced using a dual beam multiple exposure technique. Simulated transmission plots are shown for various fill factors, dielectric contrast and propagation direction. It is shown that low index waveguides can be produced using the quasi-crystal photonic crystal pattern.

©2005 Optical Society of America

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Figures (14)
Fig. 1.
Fig. 1. 12-fold quasi-crystal patterns produced using different threshold levels for exposure. Dark regions correspond to unexposed resist and thus the high dielectric. (a) Threshold of 12 out of a maximum of 24 produces 50 % fill. (b) 9/24 giving 80% fill. (c) 15/24 giving 20 % fill. (d) 13.5/24 giving 33.3% fill.

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Fig. 2.
Fig. 2. Curve A plots the high dielectric fill factor versus exposure threshold level used in the production of 12-fold quasi-crystal patterns. Curve B shows the change in the fill factor versus a threshold change of 0.5. Curve C shows the reverse sensitivity regions and is obtained by inverting the polarity of the photoresist.

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Fig. 3.
Fig. 3. Photonic quasi-crystal (6 by 10 µm) located in the (6 by 13 µm) discretization grid. Black regions indicate high dielectric. Solid line across point 1 indicates location of the plane wave soft source. Solid colored circles indicate points where time domain data is retained and FFT transmission plots computed.

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Fig. 4.
Fig. 4. TE polarization (Hz component) transmission spectrum for a 10 µm length 12-fold quasi-crystal of 50% fill factor. A band gap is present between the wavelengths of 1.05 µm and 1.28 µm and contains two defect states within.

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Fig. 5.
Fig. 5. TM polarization (Ez component) transmission spectrum for a 10 µm length 12-fold quasi-crystal of 50% fill factor. Several band gap regions are present over the wavelength range displayed. Due to the rich nature of the band gaps for the TM polarization we explore in detail the optical properties of the 12-fold quasi-crystal for this polarization state of light.

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Fig. 6.
Fig. 6. Line plot of the transmission spectrum for the Ez component of the TM polarization plotted versus wavelength and dielectric fill factor. The larger of the band gaps is located in the flat zone about the 1.5 µm wavelength value. A second large band gap is located in the 1.0 µm wavelength range.

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Fig. 7.
Fig. 7. Line plot of the transmission spectrum for the Ez component of the TM polarization plotted versus wavelength and 0° to 15° propagation angle in 2.5° increments. Through rotational symmetry, other propagation angle outside 15o range can be rotated into the 0-15° range displayed. The quasi-crystal displays a uniform transmission spectrum over propagation angle.

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Fig. 8.
Fig. 8. High band gap region plotted versus dielectric fill factor. This large band gap exists down to a dielectric contrast of 2:1 but is considerably narrower than for high dielectric contrasts. The information displayed in Fig. 6–Fig. 8 indicates that the design of a band gap in the 1.55 um range can be achieved through a large selection in the fill factor, dielectric contrast and propagation angle.

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Fig. 9.
Fig. 9. Plot of the Poynting vector magnitude versus wavelength and Fourier transform point location in the center of a 1.50 µm wide waveguide centered on the quasi-crystal. The top trace indicates that the low dielectric waveguide transmits strongly in the band gap range of the quasi-crystal and leaks light in those wavelength ranges which correspond to the transmission bands of the quasi-crystal.

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Fig. 10.
Fig. 10. Plot of the TM Ez component versus wavelength and Fourier transform point location in the waveguide for a 1.50 µm wide waveguide centered on the quasi-crystal. The top trace indicates that the low dielectric waveguide transmits strongly in the band gap range of the quasi-crystal and leaks light in those wavelength ranges which correspond to the transmission bands of the quasi-crystal.

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Fig, 11.
Fig, 11. Transmission profile of the 1.5 µm wavelength through a 1.50 µm low index waveguide centered on the photonic quasi-crystal. The dimensions of the quasi-crystal is 6 µm across and 17 µm long. High transmission is observed for this wavelength.

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Fig. 12.
Fig. 12. Transmission profile (b) of the 1.55 µm wavelength through a 1.50 µm low index waveguide centered on the photonic quasi-crystal (a). The dimensions of the quasi-crystal structure is 6 µm across and 17 µm long. The waveguide transmits an attenuated light for this particular wavelength.

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Fig. 13.
Fig. 13. Transmission profile (b) of the 1.65 µm wavelength through a 1.50 µm low index waveguide centered on the photonic quasi-crystal (a). The dimensions of the quasi-crystal is 6 µm across and 17 µm long. The waveguide displays a resonant back coupling of the light with only a small transmitted contribution.

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Fig. 14.
Fig. 14. Transmission profile of the 1.50 µm wavelength through a Y splitter fabricated from low index 1.50 µm waveguides positioned in the photonic quasi-crystal. Quasi-crystal 6 µm across and 17 µm long. The splitter divides the power and excites the two output ports evenly. The Y junction also generates a small backwards propagating component which exits the input waveguide. No attempts were made to optimize the Y branches splitting properties.

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Equations (1)

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I j = E j 1 2 + E j 2 2 + 2 E j 1 E j 2 cos ( θ j 12 ) cos ( [ k → j 1 − k → j 2 ] ∙ r → + φ oj 1 + φ oj 2 )
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