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Redistribution of output weighting coefficients for complex multiplexed phase-diffractive elements

Hua Liu, Zhenwu Lu, and Fengyou Li

Open Access Open Access

Abstract

The formation of multiplexed phase-only holograms with more weighted phase functions creates spurious cross terms and nonlinear scaling. We extend previously reported work [Appl. Opt. 25, 3767 (1986)] by proposing a normal method to analyze multiplexed holograms mathematically. We show that the output of holograms with any number weighted phase function can be written as a new linear combination for the original phase function with new weights. The relationship between the original weights and the new weights is developed for real-time optimization of hologram performance. We focus on the analysis of two and three multiplexed holograms to demonstrate the effectiveness of this approach.

©2004 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Input ratio versus the desired output ratio of the three multiplexed holograms.

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Fig. 2.
Fig. 2. Diffraction efficiency versus output ratio of the three multiplexed holograms.

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Fig. 3
Fig. 3 (a) Input ratio and (b) diffraction efficiency versus the desired output ratio of two multiplexed holograms.

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Fig. 4.
Fig. 4. (a) Binary representation of a trifocal lens desired for x 1 = 0.5 and x 2 = 0.6 and (b) reconstruction.

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Equations (18)

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M exp ⁡ ( ia ) = ∑ n = 1 N A n exp ⁡ ( i ϕ n )
exp ⁡ ( ia ) = ∑ n = 1 N A n M exp ⁡ ( i ϕ n )
M = [ A 1 2 + A 2 2 + ⋯ + A N 2 + 2 A 1 A 2 cos ⁡ ( ϕ 1 − ϕ 2 ) + 2 A 1 A 3 cos ⁡ ( ϕ 1 − ϕ 3 ) + ⋯ + 2 A 1 A N cos ⁡ ( ϕ 1 − ϕ N )
+ 2 A 2 A 3 cos ⁡ ( ( ϕ 1 − ϕ 3 ) − ( ϕ 1 − ϕ 2 ) ) + ⋯ + 2 A 2 A N cos ⁡ ( ( ϕ 1 − ϕ N ) − ( ϕ 1 − ϕ 2 ) )
+ ⋯⋯
+ 2 A N − 1 A N cos ⁡ ( ( ϕ 1 − ϕ N ) − ( ϕ 1 − ϕ N − 1 ) ) ] 1 2
M ′ ( β 1 β 2 ⋯ β N − 1 ) = 1 M = ∑ m 1 ⋯ ∑ m N − 1 a m 1 m 2 ⋯ m N − 1 exp ⁡ ( i m 1 β 1 + i m 2 β 2 + ⋯ + i m N − 1 β N − 1 )
β 1 = ϕ 1 − ϕ 2 ⋯ β N − 1 = ϕ 1 − ϕ N − 1
a m 1 m 2 ⋯ m N − 1 = 1 ( 2 π ) N − 1 ∫ 0 2 π ∫ 0 2 π M ′ ( − i m 1 β 1 − ⋯ − i m N − 1 β N − 1 ) d β 1 ⋯ dβ N − 1
exp ⁡ ( ia ) = ∑ m 1 ⋯ ∑ m N − 1 a m 1 ⋯ m N − 1 { A 1 exp ⁡ [ i ( m 1 + ⋯ + m N − 1 + 1 ) ϕ 1 − i m 1 ϕ 2 − ⋯ − i m N − 1 ϕ N ] + A 2 exp ⁡ [ i ( m 1 + ⋯ + m N − 1 ) ϕ 1 − i ( m 1 − 1 ) ϕ 2 − ⋯ − i m N − 1 ϕ N ] + ⋯ + A N exp ⁡ [ i ( m 1 + ⋯ + m N − 1 ) ϕ 1 − i m 1 ϕ 2 − ⋯ − i £ ‥ m N − 1 − 1 £ © ϕ N ] }
exp ⁡ ( ia ) = ⋯ + ( a 00 A 1 + a 10 A 2 + a 01 A 3 ) exp ⁡ ( i ϕ 1 ) + ( a 00 A 2 + a − 10 A 1 + a − 11 A 3 ) exp ⁡ ( i ϕ 2 )
+ ( a 00 A 3 + a 0 − 1 A 1 + a 1 − 1 A 2 ) exp ⁡ ( i ϕ 3 ) + ⋯
y 2 = ∑ i = 0 k ∑ j = 0 i m ij x 1 j x 2 i − j , y 1 = ∑ i = 0 k ∑ j = 0 i n ij x 1 j x 2 i − j
y 1 = 0.00005096072781 + 3.94783 x 1 − 0 . 00873 x 2 − 10.19480 x 1 2 + ⋯⋯ + 11.96649 x 1 2 x 2 5 +
4.82817 x 1 x 2 6 − 4.11001 x 2 7
y 2 = 0.00005096072782 − 0.00873 x 1 + 3.94783 x 2 + 0.44625 x 1 2 + ⋯⋯ + 8.09256 x 1 2 x 2 5 +
6.93143 x 1 x 2 6 + 2.49845 x 2 7
exp ⁡ ( ia ) = ∑ m 1 ∑ m 2 a m 1 m 2 { A 1 exp ⁡ { ik 2 f [ [ x + ( m 1 a − m 2 b ) ] 2 + y 2 ] } exp ⁡ [ ik 2 f ( − ( m 1 a − m 2 b ) 2 − m 1 a 2 − m 2 b 2 ) ] + A 2 exp ⁡ { ik 2 f [ [ x + ( m 1 a − m 2 b − a ) ] 2 + y 2 ] } exp ⁡ [ ik 2 f ( − ( m 1 a − m 2 b − a ) 2 − m 1 a 2 − m 2 b 2 + a 2 ) ] + A 3 exp ⁡ { ik 2 f [ [ x + ( m 1 a − m 2 b + b ) ] 2 + y 2 ] } exp ⁡ [ ik 2 f ( − ( m 1 a − m 2 b + b ) 2 − m 1 a 2 − m 2 b 2 + b 2 ) ] }
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