Translation of interference pattern by phase shift for diamond photonic crystals

Jun Hyuk Moon; Shu Yang; David J. Pine; Seung-Man Yang

doi:10.1364/OPEX.13.009841

Optics Express
Vol. 13,
Issue 24,
pp. 9841-9846
(2005)
•https://doi.org/10.1364/OPEX.13.009841

Translation of interference pattern by phase shift for diamond photonic crystals

Jun Hyuk Moon, Shu Yang, David J. Pine, and Seung-Man Yang

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Jun Hyuk Moon, Shu Yang, David J. Pine, and Seung-Man Yang, "Translation of interference pattern by phase shift for diamond photonic crystals," Opt. Express 13, 9841-9846 (2005)

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Abstract

We demonstrate the construction of diamond photonic crystal structures by the translation of a multi-beam interference pattern. Using phase shift of each beam, the double-exposed interference patterns can be aligned in the [111] direction for a face-centered cubic (FCC) and [210] direction for a body-centered cubic (BCC), respectively, producing diamond D from FCC and BCC-diamond like structure from BCC. The present result shows that the complete bandgap has been retained with slight deviation from ideal diamond symmetry.

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Fig. 1. Translation of (a) FCC interference pattern in the [111]direction and (b) BCC interference in the [210] direction.

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Fig. 2. The relative position of the FCC interference pattern (red) and the shifted pattern (blue) in the [111] direction by (a) 0.20d(1,1,1), (b) 0.25d(1,1,1), and (c) 0.29d(1,1,1). The level surface of each double-exposed FCC interference pattern with a filling fraction of 0.22 is shown in (d -f) for their respective cases.

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Fig. 3. Intensity profile of 8-term and 4-term interference patterns along [111] direction

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Fig. 4. The relative position of the BCC interference pattern (red) and the shifted pattern (blue) in the [210] direction by (a) 0.20d (1,1,1), (b) 0.25d (1,1,1), and (c) 0.29d (1,1,1). The level surface of each double-exposed BCC interference pattern with a filling fraction of 0.20 is shown in (d-f) for their respective cases.

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Fig. 5. Complete PBG of double-exposed (a) FCC and (b) BCC interference patterns as a function of the normalized pattern shift in [111] and [210], respectively.

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Equations (17)

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I = \sum_{n = 1}^{4} E_{n}^{2} + \sum_{m < n}^{4} E_{n} E_{m} ∣ ε_{n} \cdot ε_{m}^{*} ∣ \cos [(k_{n} - k_{m}) \cdot r + (ϕ_{n 0} - ϕ_{m 0})] .

I_{nm} = E_{n} E_{m} ∣ ε_{n} \cdot ε_{m}^{*} ∣ \cos [(k_{n} - k_{m}) \cdot r + (ϕ_{n 0} - ϕ_{m 0}) + (ϕ_{n} - ϕ_{m})] .

I_{nm} = E_{n} E_{m} ∣ ε_{n} \cdot ε_{m}^{*} ∣ \cos [(k_{n} - k_{m}) \cdot (r + r') + (ϕ_{n 0} - ϕ_{m 0})] .

(ϕ_{n} - ϕ_{m}) = (k_{n} - k_{m}) \cdot r' .

I_{nm} ~ \cos [\frac{2 π}{d} (x - y + z) + (ϕ_{1} - ϕ_{3})] + \cos [\frac{2 π}{d} (x + y + z) + (ϕ_{1} - ϕ_{4})]

+ \cos [\frac{2 π}{d} (- x - y + z) + (ϕ_{2} - ϕ_{3})] + \cos [\frac{2 π}{d} (- x + y + z) + (ϕ_{2} - ϕ_{4})]

I_{nm} ~ \cos [\frac{4 π}{d} (y + z) + (ϕ_{1} - ϕ_{2})] + \cos [\frac{4 π}{d} (x + y) + (ϕ_{1} - ϕ_{3})]

+ \cos [\frac{4 π}{d} (x + z) + (ϕ_{1} - ϕ_{4})] + \cos [\frac{4 π}{d} (x - z) + (ϕ_{2} - ϕ_{3})]

+ \cos [\frac{4 π}{d} (x - y) + (ϕ_{2} - ϕ_{4})] + \cos [\frac{4 π}{d} (- y + z) + (ϕ_{3} - ϕ_{4})]

I_{nm} ~ \cos [\frac{2 π}{d} (x - y + z)] + \cos [\frac{2 π}{d} (x + y + z)]

+ \cos [\frac{2 π}{d} (- x - y + z)] + \cos [\frac{2 π}{d} (- x + y + z)]

+ \sin [\frac{2 π}{d} (x - y + z)] - \sin [\frac{2 π}{d} (x + y + z)]

- \sin [\frac{2 π}{d} (- x - y + z)] + \sin [\frac{2 π}{d} (- x + y + z)]

I_{nm} ~ \cos [\frac{4 π}{d} (x + y)] + \cos [\frac{4 π}{d} (y + z)]

+ \cos [\frac{4 π}{d} (- y + z)] + \cos [\frac{4 π}{d} (x - y)]

+ \sin [\frac{4 π}{d} (x + y)] - \sin [\frac{4 π}{d} (y + z)]

- \sin [\frac{4 π}{d} (- y + z)] - \sin [\frac{4 π}{d} (x - y)]

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Equations (17)

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