Generation and evaluation of d-dimensional (d ≥ 3) optical shuffle patterns

Josef Giglmayr

doi:10.1364/AO.31.004662

Applied Optics
Vol. 31,
Issue 23,
pp. 4662-4675
(1992)
•https://doi.org/10.1364/AO.31.004662

Generation and evaluation of d-dimensional (d ≥ 3) optical shuffle patterns

Josef Giglmayr

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Josef Giglmayr, "Generation and evaluation of d-dimensional (d ≥ 3) optical shuffle patterns," Appl. Opt. 31, 4662-4675 (1992)

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Abstract

We use d-dimensional shuffle patterns to describe optical frequency division multiplexing. A complexity measure is derived from the geometry of common shuffle patterns and is extended to evaluate d-dimensional shuffle patterns. The least complexity of the generated patterns is determined with respect to the size and the dimension of the data sets. Trade-offs between the dimension of the data sets and the frequency-conversion-based processing of the pattern generation are discussed. A comparison of multiplexing on vectors and arrays is presented. The relation of the complexity analysis to the design of optical frequency division multiplexing systems is briefly discussed.

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	Data Set

	16 × 16 (16 Frequency Channels)		32 × 32 (64 Frequency Channels)		64 × 64 (256 Frequency Channels)

	3-D (2-D)	4-D (3-D)	3-D (2-D)	5-D (4-D)	3-D (2-D)	6-D (5-D)
Tuning range d_max^f	7	5	31	21	127	85
Crossed channels ∑_i d_i^f	56	36	992	584	16256	9360
Switch size 2^d × 2^d	8 × 8 (4 × 4)	16 × 16 (8 × 8)	8 × 8 (4 × 4)	32 × 32 (16 × 16)	8 × 8(4 × 4)	64 × 64 (32 × 32)

2-D			3-D		4-D

Pattern (S, F) and (S²)	d_max^c	d_max^s	Pattern (S², F)	d_max^c	Pattern (S², F²)	d_max^c
S_2–D^SGPS (4, 4; 4, 4)	18	12.72	S_3–D^NSGPS (2, 2; 2, 2; 4, 4)	10.41	S_4–D^SGPS (2, 2; 2, 2; 2, 2; 2, 2)	6.41
S_2–D^NSGPS (2, 8; 4, 4)	16	11.40	S_3–D^NSGPS (2, 2; 2, 2; 2, 8)	8.41
S_2–D^SGPS (2, 8; 2, 8)	14	9.89	S_3–D^NSGPS (2, 4; 2, 4; 2, 2)	5.24

Partition S × S × F	d_max^c
Start: 2048 × 2048	1446.74
1024 × 1024 × 4	723.66
1024 × 512 × 8	574.09
512 × 512 × 16	367.62
512 × 256 × 32	299.87
256 × 256 × 64	210.60
256 × 32 × 256	254.88
256 × 128 × 128	204.76

Partition S × F^d⁻¹	d_max^c
Start: 2048 × 2048	2046
2048 × 64 × 32	2014
2048 × 128 × 16	1982
2048 × 256 × 8	1918
2048 × 512 × 4	1790
2048 × 128 × 4 × 4	1726
2048 × 32 × 4 × 4 × 4	1710
2048 × 8 × 4 × 4 × 4 × 4	1706

	3-D		4-D		5-D		6-D

	S^a	F^b	S	F	S	F	S	F

Case			1st Realization				2nd Realization
5	128 × 128	64	128 × 128	8 × 8	128 × 128	4 × 4 × 4	64 × 64	4 × 4 × 4 × 4
6	64 × 64	64	64 × 64	16 × 4	64 × 64	4 × 4 × 4	32 × 32	4 × 4 × 4 × 4
	1st Realization				2nd Realization		3rd Realization

7	64 × 64	16	64 × 64	4 × 4	32 × 32	4 × 4 × 4	16 × 16	4 × 4 × 4 × 4
8	32 × 32	16	32 × 32	4 × 4	16 × 16	4 × 4 × 4	8 × 8	4 × 4 × 4 × 4
9	16 × 16	16	16 × 16	4 × 4	8 × 8	4 × 4 × 4	4 × 4	4 × 4 × 4 × 4

Partition S × S × F^d⁻²	d_max^c
Start: 256 × 256 × 64	210.60
128 × 128 × 64 × 4	184.09
128 × 128 × 16 × 4 × 4	176.09
128 × 128 × 4 × 4 × 4 × 4	174.09
64 × 64 × 4 × 4 × 4 × 4 × 4	384.84
Start: 256 × 128 × 128	204.76
256 × 128 × 32 × 4	188.76

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Applied Optics