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Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm

Yan Zhang, Giancarlo Pedrini, Wolfgang Osten, and Hans J. Tiziani

Open Access Open Access

Abstract

The phase retrieval algorithm has been used in this paper for whole reconstruction of the optical wave fields. The quantitative information of the phase distribution as well as the intensity distribution of the reconstruction field at different locations along the propagation direction has been achieved from double or multi in-line holograms. Numerical reconstructions of the wave fields from experimentally recorded in-line holograms are presented. This technique can be potentially applied for aberrated wave front analyzing and 3D imaging.

©2003 Optical Society of America

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Supplementary Material (6)

Media 1: AVI (1850 KB)     
Media 2: AVI (1832 KB)     
Media 3: AVI (2435 KB)     
Media 4: AVI (2474 KB)     
Media 5: AVI (2389 KB)     
Media 6: AVI (2287 KB)     

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Figures (7)
Fig. 1.
Fig. 1. Experimental setup for in-line holography.

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Fig. 2.
Fig. 2. Pure phase object used in experiment.

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Fig. 3.
Fig. 3. Holograms recorded at different distance. (a) d=7.40cm (b) d=8.15cm and (c) d=8.90cm.

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Fig. 4.
Fig. 4. Video presentation of reconstructed wave field at different distance from the hologram plane along the z axis direction, d′=8.90cm to d′=2.00cm with step of Δd′=1.0mm. (a) Intensity (1.85MB) and (b) phase (1.83MB) distributions. Constructed by using a single hologram.

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Fig. 5.
Fig. 5. Video presentation of reconstructed wave field at different distance from the hologram plane along the z axis direction, d′=8.90cm to d′=2.00cm with step of Δd′=1.0mm. (a) Intensity (2.43MB) and (b) phase (2.47MB) distributions. Constructed by using double holograms.

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Fig. 6.
Fig. 6. Schematic diagram of the iterative method of phase retrieval for whole wave field reconstruction.

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Fig. 7.
Fig. 7. Video presentation of reconstructed wave field at different distance from the hologram plane along the z axis direction, d′=8.90cm to d′=2.00cm with step of Δd′=1.0mm. (a) Intensity (2.38MB) and (b) phase (2.28MB) distributions. Constructed by using three holograms.

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Equations (11)

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U 2 ( x 2 , y 2 ) = U 1 ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , d )
h ( x 1 , y 1 , d ) = exp ⁡ ( − i 2 π d λ ) i λ d exp ⁡ [ i π λ d ( x 1 2 + y 1 2 ) ] .
I ( x 2 , y 2 ) = ∣ U 2 ( x 2 , y 2 ) ∣ 2
= ∣ 1 − a ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , d ) ∣ 2
= 1 − a * ( x 1 , y 1 ) ⊗ h * ( x 1 , y 1 , d ) − a ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , d )
+ ∣ a ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , d ) ∣ 2 .
U r ( x r , y r , d ' ) = I ( x 2 , y 2 ) ⊗ h ( x 2 , y 2 , − d ' ) ,
U r ( x r , y r , d ' ) = 1 − a ( x 1 , y 1 ) − a * ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , − 2 d )
+ ∣ a ( x 1 , y 1 ) ⊗ h ( x 1 , y 1 , d ) ∣ 2 ⊗ h ( x 1 , y 1 , − d ) ,
U r ( x r , y r , d ' ) = F − 1 { F { I ( x 2 , y 2 ) } H ( ξ , η , − d ' ) } ,
H ( ξ , η , − d ' ) = exp ⁡ ( − i 2 π d ' λ ) exp ⁡ [ i π λ d ' ( ξ 2 + η 2 ) ] .
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