Abstract

We present autofocusing in optical scanning holography (OSH) with experimental results. We first record the complex hologram of an object using OSH and then create the Fresnel zone plate (FZP) that codes the object constant within the depth range of the object using Gaussian low-pass filtering. We subsequently synthesize a real-only spectrum hologram in which its phase term contains information about a distance parameter. Finally, we extract the distance parameter from the real-only spectrum hologram using fringe-adjusted filtering and the Wigner distribution. Using the extracted distance parameter, we reconstruct a three-dimensional image of the object from the complex hologram using digital convolution, which bypasses the conventional blind convolution to reconstruct a hologram. To the best of our knowledge, this is the first report with experimental results that autofocusing in OSH is possible without any searching algorithm or tracking process.

© 2009 Optical Society of America

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References

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  1. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 521-527 (1985).
    [CrossRef]
  2. T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
    [CrossRef]
  3. B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
    [CrossRef]
  4. G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
    [CrossRef]
  5. G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A 23, 1699-1707 (2006).
    [CrossRef]
  6. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46, 993-1000 (2007).
    [CrossRef] [PubMed]
  7. T. -C. Poon, Digital Holography and Three-Dimensional Display (Springer, 2006).
    [CrossRef]
  8. P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28, 1257-1259 (2003).
    [CrossRef] [PubMed]
  9. M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424-2430 (2004).
    [CrossRef]
  10. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47, D176 (2008).
    [CrossRef] [PubMed]
  11. T. Kim, Y. S. Kim, W. S. Kim, and T.-C. Poon, “Algorithm for converting full-parallax holograms to horizontal-parallax-only holograms,” Opt. Lett. 34, 1231-1233 (2009).
    [CrossRef] [PubMed]
  12. T. Kim and T.-C. Poon, “Extraction of 3-D location of matched 3-D object using power fringe-adjusted filtering and Wigner analysis,” Opt. Eng. 38, 2176-2183 (1999).
    [CrossRef]
  13. T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
    [CrossRef]
  14. T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
    [CrossRef]
  15. T.-C. Poon, “Three-dimensional television using optical scanning holography,” J. Inf. Disp. 3, 12-16 (2002).
    [CrossRef]
  16. M. S. Alam, X.-W. Chen, and M. A. Karim, “Distortion-invariant fringe-adjusted joint transform correlation,” Appl. Opt. 36, 7422-7427 (1997).
    [CrossRef]
  17. G. Indebetouw and P. Klysubun, “Space-time digital holography: a three-dimensional microscopic imaging scheme with an arbitrary degree of spatial coherence,” Appl. Phys. Lett. 75, 2017-2019 (1999).
    [CrossRef]
  18. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photon. 2, 190-195 (2008).
    [CrossRef]
  19. T.-C. Poon, “Holography: scan-free three-dimensional imaging,” Nat. Photon. 2, 131-132 (2008).
    [CrossRef]
  20. Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001).
    [CrossRef]
  21. N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
    [CrossRef] [PubMed]

2009 (1)

2008 (3)

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47, D176 (2008).
[CrossRef] [PubMed]

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photon. 2, 190-195 (2008).
[CrossRef]

T.-C. Poon, “Holography: scan-free three-dimensional imaging,” Nat. Photon. 2, 131-132 (2008).
[CrossRef]

2007 (2)

2006 (1)

2004 (1)

2003 (1)

2002 (2)

T.-C. Poon, “Three-dimensional television using optical scanning holography,” J. Inf. Disp. 3, 12-16 (2002).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

2001 (1)

2000 (2)

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

1999 (2)

T. Kim and T.-C. Poon, “Extraction of 3-D location of matched 3-D object using power fringe-adjusted filtering and Wigner analysis,” Opt. Eng. 38, 2176-2183 (1999).
[CrossRef]

G. Indebetouw and P. Klysubun, “Space-time digital holography: a three-dimensional microscopic imaging scheme with an arbitrary degree of spatial coherence,” Appl. Phys. Lett. 75, 2017-2019 (1999).
[CrossRef]

1997 (2)

1995 (1)

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

1985 (1)

Abookasis, D.

Alam, M. S.

Brooker, G.

Chen, X.-W.

Coppola, G.

De Nicola, S.

Dirksen, D.

Doh, K. B.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

Ferraro, P.

Finizio, A.

Indebetouw, G.

G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46, 993-1000 (2007).
[CrossRef] [PubMed]

G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A 23, 1699-1707 (2006).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

G. Indebetouw and P. Klysubun, “Space-time digital holography: a three-dimensional microscopic imaging scheme with an arbitrary degree of spatial coherence,” Appl. Phys. Lett. 75, 2017-2019 (1999).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

Karim, M. A.

Kemper, B.

Kim, T.

T. Kim, Y. S. Kim, W. S. Kim, and T.-C. Poon, “Algorithm for converting full-parallax holograms to horizontal-parallax-only holograms,” Opt. Lett. 34, 1231-1233 (2009).
[CrossRef] [PubMed]

T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

T. Kim and T.-C. Poon, “Extraction of 3-D location of matched 3-D object using power fringe-adjusted filtering and Wigner analysis,” Opt. Eng. 38, 2176-2183 (1999).
[CrossRef]

Kim, W. S.

Kim, Y. S.

Klysubun, P.

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

G. Indebetouw and P. Klysubun, “Space-time digital holography: a three-dimensional microscopic imaging scheme with an arbitrary degree of spatial coherence,” Appl. Phys. Lett. 75, 2017-2019 (1999).
[CrossRef]

Langehanenberg, P.

Li, Y.

Liebling, M.

Pierattini, G.

Poon, T.-C.

T. Kim, Y. S. Kim, W. S. Kim, and T.-C. Poon, “Algorithm for converting full-parallax holograms to horizontal-parallax-only holograms,” Opt. Lett. 34, 1231-1233 (2009).
[CrossRef] [PubMed]

T.-C. Poon, “Holography: scan-free three-dimensional imaging,” Nat. Photon. 2, 131-132 (2008).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

T.-C. Poon, “Three-dimensional television using optical scanning holography,” J. Inf. Disp. 3, 12-16 (2002).
[CrossRef]

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

G. Indebetouw, P. Klysubun, T. Kim, and T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380-390 (2000).
[CrossRef]

T. Kim and T.-C. Poon, “Extraction of 3-D location of matched 3-D object using power fringe-adjusted filtering and Wigner analysis,” Opt. Eng. 38, 2176-2183 (1999).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 521-527 (1985).
[CrossRef]

T. -C. Poon, Digital Holography and Three-Dimensional Display (Springer, 2006).
[CrossRef]

Rosen, J.

Schilling, B. W.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

Shaked, N. T.

Shinoda, K.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

Shinoda, K. K.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

Stern, A.

Storrie, B.

Suzuki, Y.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

Tada, Y.

Unser, M.

von Bally, G.

Wu, M. H.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, and M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506-1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

Zhong, W.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

G. Indebetouw and P. Klysubun, “Space-time digital holography: a three-dimensional microscopic imaging scheme with an arbitrary degree of spatial coherence,” Appl. Phys. Lett. 75, 2017-2019 (1999).
[CrossRef]

J. Inf. Disp. (1)

T.-C. Poon, “Three-dimensional television using optical scanning holography,” J. Inf. Disp. 3, 12-16 (2002).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nat. Photon. (2)

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photon. 2, 190-195 (2008).
[CrossRef]

T.-C. Poon, “Holography: scan-free three-dimensional imaging,” Nat. Photon. 2, 131-132 (2008).
[CrossRef]

Opt Lett. (1)

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Twin-image elimination experiments for three-dimensional Images in optical scanning holography,” Opt Lett. 25, 215-217 (2000).
[CrossRef]

Opt. Eng. (3)

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338-1344(1995).
[CrossRef]

T. Kim and T.-C. Poon, “Extraction of 3-D location of matched 3-D object using power fringe-adjusted filtering and Wigner analysis,” Opt. Eng. 38, 2176-2183 (1999).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, “Depth detection and image recovery in remote sensing by optical scanning holography,” Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (1)

T. -C. Poon, Digital Holography and Three-Dimensional Display (Springer, 2006).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

OSH: M, mirrors; AOFS1, AOFS12, acousto-optic frequency shifters; BS1, BS2, beam splitters; BE1, BE2, beam expanders; L1, focusing lens; L2, collecting lens; ⊗, electronic multiplier; LPF, low-pass filter.

Fig. 2
Fig. 2

Flow chart for extracting distance parameter z 0 from the hologram recorded by OSH.

Fig. 3
Fig. 3

Object with L x = L y = 1.5 cm and L z = 15 cm .

Fig. 4
Fig. 4

(a) Amplitude and (b) phase of the complex hologram.

Fig. 5
Fig. 5

Line impulse on the frequency-space map.

Fig. 6
Fig. 6

Reconstructed image using distance parameter z 0 .

Fig. 7
Fig. 7

Sequential reconstruction of the complex hologram shown in Fig. 4.

Equations (22)

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I s ( x , y , z , t ) = A s ( x , y ; z ) λ z sin [ π λ z ( x 2 + y 2 ) Δ Ω t ] ,
A s ( x , y ; z ) = exp [ π a ( z ) 2 ( x 2 + y 2 ) ] .
H ( x , y ) = i I ( x , y , z ) j i Q ( x , y , z ) = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) j A s ( x , y ; z ) λ z exp [ j π λ z ( x 2 + y 2 ) ] d z ,
g 1 ( x , y ) g 2 ( x , y ) = g 1 ( x , y ) g 2 ( x x , y y ) d x d y .
F { u ( x , y ) } k x , k y = u ( x , y ) exp [ j ( k x x + k y y ) ] d x d y = u ( k x , k y ) ,
H ( k x , k y ) = F { H ( x , y ) } = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( k x , k y , z ) exp { [ 1 4 π ( λ NA ) 2 + j λ z 4 π ] ( k x 2 + k y 2 ) } d z .
A g ( k x , k y ) = exp { π [ λ 2 π NA g ] 2 ( k x 2 + k y 2 ) } ,
H l p ( k x , k y ) = H ( k x , k y ) × A g ( k x , k y ) = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( k x , k y , z ) exp [ { 1 4 π ( λ NA l p ) 2 + j λ z 4 π } ( k x 2 + k y 2 ) ] d z ,
H l p ( x , y ) = F 1 { H l p ( k x , k y ) } = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) j A l p ( x , y ; z ) λ z exp [ j π λ z ( x 2 + y 2 ) ] d z ,
A l p ( x , y ; z ) = exp [ π a l p ( z ) 2 ( x 2 + y 2 ) ] ,
Δ z = 2 λ / π NA l p 2 = 2 λ / π ( NA 2 + NA g 2 ) / ( NA g NA ) 2 .
h z ( x , y ) h z 0 ( x , y ) = j A l p ( x , y ; z 0 ) / ( λ z 0 ) exp [ j π λ z 0 ( x 2 + y 2 ) ]
H l p ( x , y ) = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) d z j A l p ( x , y ; z 0 ) λ z 0 exp [ j π λ z 0 ( x 2 + y 2 ) ] .
H r - only ( k x , k y ) = Re [ F { Re [ H l p ( x , y ) ] } ] + j Re [ F { Im [ H l p ( x , y ) ] } ] ,
H r - only ( k x , k y ) = Re [ F { z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) d z } ] exp [ { 1 4 π ( λ NA l p ) 2 + j λ z 0 4 π } ( k x 2 + k y 2 ) ] ,
H 0 ( k x ) = 1 | H r - only ( k x , k y ) d k y | 2 + ε [ H r - only ( k x , k y ) d k y ] 2 exp ( j λ z 0 2 π k x 2 ) ,
W H o ( k x , x ) = H o ( k x + k x 2 ) H o * ( k x k x 2 ) exp ( j x k x ) d k x δ ( x λ z o π k x ) .
I r ( x , y , z r ) = H ( x , y ) h z r * ( x , y ) = I 0 ( x , y , z r ) + z 0 ( 1 / 2 ) δz z z r z 0 + ( 1 / 2 ) δz I 0 ( x , y , z ) j A s ( x , y ; z ) λ ( z z r ) exp [ j π λ ( z z r ) ( x 2 + y 2 ) ] d z .
Δ z = 2 λ / π × ( NA 2 + NA g 2 ) / ( NA g NA ) 2 30 cm ,
Re [ H l p ( x , y ) ] = Re [ z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) d z j A l p ( x , y ; z 0 ) λ z 0 exp [ j π λ z 0 ( x 2 + y 2 ) ] ] = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) d z Re [ j A l p ( x , y ; z 0 ) λ z 0 exp [ j π λ z 0 ( x 2 + y 2 ) ] ] = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z I 0 ( x , y , z ) d z A l p ( x , y ; z 0 ) λ z 0 sin [ π λ z 0 ( x 2 + y 2 ) ] ,
O ( x , y , z ) = | O ( x , y , z ) | exp ( j O ) = Re [ O ] + j Im [ O ] ,
Re [ H l p ( x , y ) ] = Re [ z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z O ( x , y , z ) d z j A l p ( x , y ; z 0 ) λ z 0 exp [ j π λ z 0 ( x 2 + y 2 ) ] ] = z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z Re [ O ( x , y , z ) ] d z A l p ( x , y ; z 0 ) λ z 0 sin [ π λ z 0 ( x 2 + y 2 ) ] z 0 ( 1 / 2 ) δ z z 0 + ( 1 / 2 ) δ z Im [ O ( x , y , z ) ] d z A l p ( x , y ; z 0 ) λ z 0 cos [ π λ z 0 ( x 2 + y 2 ) ] .

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