Researching | Fast reconstruction of multiple off-axis holograms based on a combination of complex encoding and digital spatial multiplexing

Journals >Chinese Optics Letters >Volume 14 >Issue 6 >Page 060902 > Article

Chinese Optics Letters
Vol. 14, Issue 6, 060902 (2016)

Fast reconstruction of multiple off-axis holograms based on a combination of complex encoding and digital spatial multiplexing

Bei Sha^1、2, Yujie Lu¹, Yiyan Xie¹, Qingyang Yue¹, and Chengshan Guo^1、*

Author Affiliations

¹College of Physics and Electronics, Shandong Normal University, Jinan 250014, China

²College of Physics and Electronic Engineering, Qilu Normal University, Jinan 250200, China

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DOI: 10.3788/COL201614.060902 Cite this Article Set citation alerts

Bei Sha, Yujie Lu, Yiyan Xie, Qingyang Yue, Chengshan Guo. Fast reconstruction of multiple off-axis holograms based on a combination of complex encoding and digital spatial multiplexing[J]. Chinese Optics Letters, 2016, 14(6): 060902 Copy Citation Text

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Abstract

An algorithm is proposed for the fast reconstruction of off-axis digital holograms based on a combination of complex encoding (CE) and spatial multiplexing (SM). In this algorithm, every two off-axis holograms recorded in sequence are first assembled into a CE hologram using the CE method, and then four of the CE holograms are again encoded into one complex spatial multiplexing (CSM) hologram based on the SM algorithm. It is demonstrated that the eight holograms encoded into such one CSM hologram can be quickly reconstructed by performing a two-dimensional (2D) Fourier transform (FT) on the CSM hologram. Using this method, the eight 2D FTs required for the reconstruction of the eight holograms in the conventional spatial filtering methods can be simplified to a process with only one 2D FT, which can largely improve the computation efficiency with the resolution of the reconstructed images nearly unchanged.

H_{t} (x, y) = R^{2} + {| O_{t} (x, y) |}^{2} + O_{t} (x, y) R \exp (- i 2 π ξ_{0} x) + O_{t}^{*} (x, y) R \exp (i 2 π ξ_{0} x),

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{\tilde{H}}_{t} = F {H_{t}} = F {R^{2} + {| O_{t} |}^{2}} + R F {O_{t}} \otimes δ (ξ + ξ_{0}) + R F {O_{t}^{*}} \otimes δ (ξ - ξ_{0}) = C_{t} (ξ, η) + R {\tilde{O}}_{t} (ξ, η) \otimes δ (ξ + 3 ξ_{m}) + R {\tilde{O}}_{t}^{*} (ξ, η) \otimes δ (ξ - 3 ξ_{m}),

(2)

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{\begin{matrix} H_{1, 2} = H_{1} + j H_{2} \\ H_{3, 4} = H_{3} + j H_{4} \\ H_{5, 6} = H_{5} + j H_{6} \\ H_{7, 8} = H_{7} + j H_{8} \end{matrix},

(3)

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H = H_{1, 2} \exp (- j 6 π ξ_{m} y) + H_{3, 4} \exp (- j 2 π ξ_{m} y) + H_{5, 6} \exp (j 2 π ξ_{m} y) + H_{7, 8} \exp (j 6 π ξ_{m} y) .

(4)

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H = A_{1} \exp [- j 6 π ξ_{m} y] + S_{1} \exp [- j 2 π (ξ_{0} x + 3 ξ_{m} y)] + S_{2} \exp [j 2 π (ξ_{0} x - 3 ξ_{m} y)] + A_{2} \exp [- j 2 π ξ_{m} y] + S_{3} \exp [- j 2 π (ξ_{0} x + ξ_{m} y)] + S_{4} \exp [j 2 π (ξ_{0} x - ξ_{m} y)] + A_{3} \exp [j 2 π ξ_{m} y] + S_{5} \exp [- j 2 π (ξ_{0} x - ξ_{m} y)] + S_{6} \exp [j 2 π (ξ_{0} x + ξ_{m} y)] + A_{4} \exp [j 6 π ξ_{m} y] + S_{7} \exp [- j 2 π (ξ_{0} x - 3 ξ_{m} y)] + S_{8} \exp [j 2 π (ξ_{0} x + 3 ξ_{m} y)],

(5)

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{\begin{matrix} S_{1} = O_{1} + j O_{2}, & S_{2} = O_{1}^{*} + j O_{2}^{*} \\ S_{3} = O_{3} + j O_{4}, & S_{4} = O_{3}^{*} + j O_{4}^{*} \\ S_{5} = O_{5} + j O_{6}, & S_{6} = O_{5}^{*} + j O_{6}^{*} \\ S_{7} = O_{7} + j O_{8}, & S_{8} = O_{7}^{*} + j O_{8}^{*} \end{matrix} .

(6)

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\tilde{H} = F {H} = [{\tilde{A}}_{1} + R {\tilde{S}}_{1} \otimes δ (ξ + ξ_{0}) + R {\tilde{S}}_{2} \otimes δ (ξ - ξ_{0})] \otimes δ (η + 3 ξ_{m}) + [{\tilde{A}}_{2} + R {\tilde{S}}_{3} \otimes δ (ξ + ξ_{0}) + R {\tilde{S}}_{4} \otimes δ (ξ - ξ_{0})] \otimes δ (η + ξ_{m}) + [{\tilde{A}}_{3} + R {\tilde{S}}_{5} \otimes δ (ξ + ξ_{0}) + R {\tilde{S}}_{6} \otimes δ (ξ - ξ_{0})] \otimes δ (η - ξ_{m}) + [{\tilde{A}}_{4} + R {\tilde{S}}_{7} \otimes δ (ξ + ξ_{0}) + R {\tilde{S}}_{8} \otimes δ (ξ - ξ_{0})] \otimes δ (η - 3 ξ_{m}),

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{\begin{matrix} {\tilde{O}}_{1} (ξ, η) = \frac{1}{2} [{\tilde{S}}_{1} (ξ, η) + {\tilde{S}}_{2} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{2} (ξ, η) = \frac{1}{2 j} [{\tilde{S}}_{1} (ξ, η) - {\tilde{S}}_{2} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{3} (ξ, η) = \frac{1}{2} [{\tilde{S}}_{3} (ξ, η) + {\tilde{S}}_{4} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{4} (ξ, η) = \frac{1}{2 j} [{\tilde{S}}_{3} (ξ, η) - {\tilde{S}}_{4} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{5} (ξ, η) = \frac{1}{2} [{\tilde{S}}_{5} (ξ, η) + {\tilde{S}}_{6} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{6} (ξ, η) = \frac{1}{2 j} [{\tilde{S}}_{5} (ξ, η) - {\tilde{S}}_{6} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{7} (ξ, η) = \frac{1}{2} [{\tilde{S}}_{7} (ξ, η) + {\tilde{S}}_{8} {(- ξ, - η)}^{*}] \\ {\tilde{O}}_{8} (ξ, η) = \frac{1}{2 j} [{\tilde{S}}_{7} (ξ, η) - {\tilde{S}}_{8} {(- ξ, - η)}^{*}] \end{matrix},

(8)

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Bei Sha, Yujie Lu, Yiyan Xie, Qingyang Yue, Chengshan Guo. Fast reconstruction of multiple off-axis holograms based on a combination of complex encoding and digital spatial multiplexing[J]. Chinese Optics Letters, 2016, 14(6): 060902

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