Dynamic holography for optical interconnections. II. Routing holograms with predictable location and intensity of each diffraction order

Kim L. Tan; Stephen T. Warr; Ilias G. Manolis; Timothy D. Wilkinson; Maura M. Redmond; William A. Crossland; Robert J. Mears; Brian Robertson

doi:10.1364/JOSAA.18.000205

Journal of the Optical Society of America A
Vol. 18,
Issue 1,
pp. 205-215
(2001)
•https://doi.org/10.1364/JOSAA.18.000205

Dynamic holography for optical interconnections. II. Routing holograms with predictable location and intensity of each diffraction order

Kim L. Tan, Stephen T. Warr, Ilias G. Manolis, Timothy D. Wilkinson, Maura M. Redmond, William A. Crossland, Robert J. Mears, and Brian Robertson

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Kim L. Tan, Stephen T. Warr, Ilias G. Manolis, Timothy D. Wilkinson, Maura M. Redmond, William A. Crossland, Robert J. Mears, and Brian Robertson, "Dynamic holography for optical interconnections. II. Routing holograms with predictable location and intensity of each diffraction order," J. Opt. Soc. Am. A 18, 205-215 (2001)

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Abstract

An analysis of dynamic phase-only holograms, described by fractional notation and recorded onto a pixelated spatial light modulator (SLM) in a reconfigurable optical beam-steering switch, is presented. The phase quantization and arrangement of the phase states and the SLM pixelation and dead-space effects are decoupled, expressed analytically, and simulated numerically. The phase analysis with a skip–rotate rule reveals the location and intensity of each diffraction order at the digital replay stage. The optical reconstruction of the holograms recorded onto SLM’s with rectangular pixel apertures entails sinc-squared scaling, which further reduces the intensity of each diffraction order. With these two factors taken into account, the highest values of the nonuniform first-order diffraction efficiencies are expected to be 33%, 66%, and 77% for two-, four-, and and eight-level one-dimensional holograms with a 90% linear pixel fill factor. The variation of the first-order diffraction efficiency and the relative replay intensities were verified to within 1 dB by performing the optical reconstruction of binary phase-only holograms recorded onto a ferroelectric liquid crystal on a silicon SLM.

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Power Loss Component	Loss Expression and Assumptions	Power Loss Value
Hologram-dependent losses
Phase effect: number of phase levels, required routing fraction	${sinc}^{2} (π / 2) / {sinc}^{2} (π / x_{0})$ for $σ = x / x_{0},$ first-order digital diffraction efficiency for binary holograms of replay fraction σ	$σ = 1 / 2 : 0 dB$ $σ = 1 / 4 : - 3 dB$ $σ = 1 / 8 : - 3.7 dB$

Spatial effect: SLM pixelation, dead-space for isolation	$ρ^{2} {sinc}^{2} (ρ π σ),$ $ρ = fill factor at 82 %$	$σ = 1 / 2 : - 4.27 dB$ $σ = 1 / 4 : - 2.33 dB$ $σ = 1 / 8 : - 1.87 dB$
Hologram-independent losses
Retardation efficiency: non-λ/2 liquid-crystal cell configuration	${sin}^{2} (π Δ n 2 T / λ),$ $Δ n = 0.145$ ^a at $λ = 690 nm,$ and T is the liquid-crystal cell thickness at 1.95 μm	-5.4 dB
Phase efficiency of electro-optic modulation: non-π/2 FLC switching angle	${sin}^{2} (θ_{0}),$ $θ_{0} = FLC$ switch angle at 68°^b	-0.66 dB
Gaussian beam aperturing	$[\erf (γ_{x} / \sqrt{2}) \times \erf (γ_{y} / \sqrt{2})]^{2}$ at $γ_{x} = 1.26,$ $γ_{y} = 1.29$	-3.96 dB

Sundry losses in the optical system, including lens transmission and back-reflections	Three ${MgF}_{2}$ coated lenses with six reflecting surfaces each giving 1.5% loss, SLM front glass with $2 \times 4 %$ loss and 95% reflectance of Al SLM pixels	1 dB

Fiber coupling loss: losses unaccounted for	Replay beam–fiber axial tilts, strains on the output fiber glued onto a quarter disk, chromatic dispersions of source, lens aberrations, and system misalignment	∼2.3 dB
Total hologram-independent losses		13.32

Input power: The $λ = 690 - nm$ laser diode has 15-mW output at 17 °C; the optical powermeter has a -5-dB responsitivity at 750-nm calibration		6.8 dBm

Expected power	After the major loss components in the above have been taken into account	$σ = 1 / 2 : - 10.8 dBm$ $σ = 1 / 4 : - 11.9 dBm$ $σ = 1 / 8 : - 12.0 dBm$
Detected power	Actual measured first-order intensity values	$σ = 1 / 2 : - 10.8 dBm$ $σ = 1 / 4 : - 11.9 dBm$ $σ = 1 / 8 : - 13.6 dBm$

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Journal of the Optical Society of America A