Formation of all fourteen Bravais lattices of three-dimensional photonic crystal structures by a dual beam multiple-exposure holographic technique

Ashish Dwivedi; Jolly Xavier; Joby Joseph; Kehar Singh

doi:10.1364/AO.47.001973

Applied Optics
Vol. 47,
Issue 12,
pp. 1973-1980
(2008)
•https://doi.org/10.1364/AO.47.001973

Formation of all fourteen Bravais lattices of three-dimensional photonic crystal structures by a dual beam multiple-exposure holographic technique

Ashish Dwivedi, Jolly Xavier, Joby Joseph, and Kehar Singh

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Ashish Dwivedi, Jolly Xavier, Joby Joseph, and Kehar Singh, "Formation of all fourteen Bravais lattices of three-dimensional photonic crystal structures by a dual beam multiple-exposure holographic technique," Appl. Opt. 47, 1973-1980 (2008)

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Abstract

We make use of a dual beam multiple-exposure (DBME) holographic technique for the formation of all 14 Bravais lattices of three-dimensional photonic crystal microstructures. For simplicity of experimental implementation, the DBME method has been modified such that, prior to each exposure, once the proper angle between the wave vectors of the interfering beams is chosen, a single axis rotation of the recording medium gives the desired results. The parameters required for the generation of the lattice structures have been derived by appropriate modification of interference of four noncoplanar beams (IFNB) analysis for corresponding implementation in the DBME technique, and the results have been verified by computer simulations. After giving a comparative study of the results with the IFNB method, recording geometries for the DBME approach are also proposed in order to realize all 14 Bravais lattices experimentally.

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Lattice Type	Lattice Parameters $[\begin{matrix} a \\ b \\ c \end{matrix}]$ & $λ (μm)$	Wave Vectors $[\begin{array}{l} k_{1} \\ k_{2} \\ k_{3} \\ k_{4} \end{array}] = [\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \\ l_{4} & m_{4} & n_{4} \end{matrix}]$				Angles Between Wave Vectors (Degrees) $[\begin{matrix} θ_{12} & θ_{23} \\ θ_{13} & θ_{24} \\ θ_{14} & θ_{34} \end{matrix}]$
Simple cubic (P)	0.46		6.81	6.81	6.81	70.52	109.47	0
	0.46		-6.81	6.81	6.81	70.52	109.47	120
	0.46	0.532	6.81	-6.81	6.81	70.52	109.47	120
			6.81	6.81	-6.81
Face centered cubic (F)	1.38		6.81	6.81	6.81	38.94	65.95	0
	1.38		2.27	2.27	11.36	38.94	65.95	120
	1.38	0.532	11.36	2.27	2.27	38.94	65.95	120
			2.27	11.36	2.27
Body centered cubic (I)	1.38		2.27	2.27	2.27	109.47	109.47	0
	1.38		2.27	-2.27	-2.27	109.47	109.47	120
	1.38	0.532	-2.27	2.27	-2.27	109.47	109.47	120
			-2.27	-2.27	2.27
Triclinic (P)	1.0		1.03	38.35	6.33	11.12	13.29	0
	2.0		-5.26	38.63	2.15	3.27	10.38	125.65
	1.75	0.5329	1.02	35.20	7.89	1.49	4.76	177.65
			1.03	38.35	5.31
Simple monoclinic (P)	0.90		2.50	3.33	1.90	32.16	102.11	0
	0.30		0.28	3.33	3.13	93.23	73.84	107.75
	0.60	0.5321	2.50	-3.33	1.90	49.09	113.67	118.89
			2.50	3.33	-1.90
Base centered monoclinic (C)	0.5		35.25	3.49	9.88	37.19	40.36	0
	0.9		22.69	6.98	29.99	10.89	67.95	100.23
	0.6	0.5320	35.25	-3.49	9.88	31.16	33.10	91.52
			35.25	3.49	- 9.88
Simple orthorhombic (P)	0.3		10.47	3.49	1.49	140.13	164.56	0
	0.6		-10.47	3.49	1.49	36.52	143.47	155.49
	2.1	0.5323	10.47	-3.49	1.49	15.43	39.86	92.56
			10.47	3.49	-1.49
Base centered orthorhombic (C)	2.30		9.83	3.92	5.23	23.36	61.58	0
	1.60		7.10	7.85	5.23	38.83	58.32	164.09
	0.60	0.5319	9.83	-3.92	5.23	52.62	67.29	100.03
			9.83	3.92	-5.23
Body centered orthorhombic (I)	0.91		3.45	4.98	10.13	146.00	61.79	0
	0.63		3.45	-4.98	-10.13	130.04	130.04	98.18
	0.31	0.5320	-3.45	4.98	-10.13	61.79	146.00	141.10
			-3.45	-4.98	10.13
Face centered orthorhombic (F)	2.5		17.01	10.21	5.44	25.97	47.24	0
	1.5		14.50	6.02	13.30	25.97	51.25	132.4
	0.8	0.5316	19.53	6.02	-2.40	25.97	27.00	65.64
			14.50	14.40	-2.40
Simple tetragonal (P)	0.40		7.85	7.85	3.92	83.62	141.05	0
	0.40		-7.85	7.85	3.92	83.62	96.37	143.13
	0.80	0.5333	7.85	-7.85	3.92	38.94	96.37	108.43
			7.85	7.85	-3.92
Body centered tetragonal (I)	0.8		3.92	3.92	10.47	141.30	55.87	0
	0.8		3.92	-3.92	-10.47	141.30	141.30	97.08
	0.3	0.5301	-3.92	3.92	-10.47	55.87	141.30	131.45
			-3.92	-3.92	10.47
Hexagonal (P)	0.40		5.23	9.06	5.23	101.53	101.53	0
	0.40		-10.47	0	5.23	101.53	126.86	104.47
	0.60	0.5367	5.23	-9.06	5.23	53.13	126.86	127.76
			5.23	9.06	-5.23
Rhombohedral [Trigonal] (R)	0.7		24.90	10.57	6.87	27.47	49.33	0
	0.7		15.93	19.24	10.87	28.37	46.24	126.03
	0.7	0.5303	24.90	-1.90	12.44	28.37	48.602	120.00
			24.90	10.57	-6.84

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Applied Optics