Abstract

We present a new multi-illumination technique for the determination of the phase maps of unknown phase objects and wavefronts based on their diffraction patterns. A spatial light modulator is used to generate a sequence of probe-light fields that illuminate the unknown object producing different diffraction patterns. Compared with similar diffraction-pattern-based approaches, our technique benefits from a motionless multiview operation and a significantly improved deconvolution algorithm convergence speed (tens of iterations versus hundreds). Computer simulations indicate that the extra information brought by the different diffraction patterns prevents convergence of the phase retrieval algorithm to spurious local minima solutions and results in faster convergence. We describe an experimental system built based on our approach using readily available, relatively low-cost components. Successful reconstructions of test targets from experimental diffraction patterns confirm the feasibility of the technique. Major sources of error are identified, solutions to these problems suggested, and potential extensions to multiresolution analysis of unknown wavefronts are proposed.

© 2011 Optical Society of America

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2010 (1)

2009 (1)

N. Pandey, D. Kelly, T. Naughton, and B. Hennelly, “Speed up of Fresnel transforms for digital holography using pre-computed chirp and GPU processing,” Proc. SPIE 7442, 744205 (2009).
[CrossRef]

2007 (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2005 (2)

L. Seifert, H. J. Tiziani, and W. Osten, “Wavefront reconstruction with the adaptive Shack–Hartmann sensor,” Opt. Commun. 245, 255–269 (2005).
[CrossRef]

G. Pedrini, W. Osten, and Y. Zhang, “Wavefront reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

J. Primot, “Theoretical description of Shack–Hartmann sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

2002 (1)

R. Vincent, “Phase retrieval in TEM using Fresnel images,” Ultramicroscopy 90, 135–151 (2002).
[CrossRef] [PubMed]

2001 (4)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

G.-C Jin and N.-K Bao, “Surface detection and 3D profilometry for microstructure using optical metrology,” Opt. Lasers Eng. 36, 1–9 (2001).
[CrossRef]

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

2000 (1)

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1998 (1)

T. Kotzer, N. Cohen, and J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

1997 (1)

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

1996 (2)

G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

1993 (1)

1991 (1)

1990 (1)

1988 (2)

M. Suzuki and M. Kanaya, “Applications of moiré topography measurement methods in industry,” Opt. Lasers Eng. 8, 171–188 (1988).
[CrossRef]

K. Patorski, “Moiré methods in interferometry,” Opt. Lasers Eng. 8, 147–170 (1988).
[CrossRef]

1986 (1)

1983 (1)

1982 (2)

1978 (1)

P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numer. Anal. 15, 977–992 (1978).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1958 (1)

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. 29, 610–611(1958).
[CrossRef]

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Arieli, Y.

Balmer, J. E.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Bao, N.-K

G.-C Jin and N.-K Bao, “Surface detection and 3D profilometry for microstructure using optical metrology,” Opt. Lasers Eng. 36, 1–9 (2001).
[CrossRef]

Barbastathis, G.

Barton, D. R.

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Ben-Yosef, N.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Box, G. E. P.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. 29, 610–611(1958).
[CrossRef]

Cohen, N.

T. Kotzer, N. Cohen, and J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

Doleschal, W.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Fienup, J. R.

Fracasso, B.

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Gehner, A.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gill, P. E.

P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numer. Anal. 15, 977–992 (1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Grundke, W.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Gustafsson, M.

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

Heggarty, K.

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

Hennelly, B.

N. Pandey, D. Kelly, T. Naughton, and B. Hennelly, “Speed up of Fresnel transforms for digital holography using pre-computed chirp and GPU processing,” Proc. SPIE 7442, 744205 (2009).
[CrossRef]

Hornbeck, J. L.

J. L. Hornbeck, “Current status of the digital micromirror device (DMD) for projection television applications,” in International Electron Devices Meeting IEDM ’93 Technical Digest (IEEE, 1993), pp. 381–384.

Jin, G.-C

G.-C Jin and N.-K Bao, “Surface detection and 3D profilometry for microstructure using optical metrology,” Opt. Lasers Eng. 36, 1–9 (2001).
[CrossRef]

Kanaya, M.

M. Suzuki and M. Kanaya, “Applications of moiré topography measurement methods in industry,” Opt. Lasers Eng. 8, 171–188 (1988).
[CrossRef]

Kelly, D.

N. Pandey, D. Kelly, T. Naughton, and B. Hennelly, “Speed up of Fresnel transforms for digital holography using pre-computed chirp and GPU processing,” Proc. SPIE 7442, 744205 (2009).
[CrossRef]

Kotzer, T.

T. Kotzer, N. Cohen, and J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

Kuck, H.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Lalor, M. J.

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Leone, G.

Loewenthal, F.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mazine, A.

A. Mazine, “La caractérisation de front d’onde dans un système de propagation à multi-illumination gérée par un SLM,” doctoral thesis (Ecole Nationale Supérieure des Télécommunications Paris, 2006), pp. 1–126.

Melcher, R.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Muller, M. E.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. 29, 610–611(1958).
[CrossRef]

Murray, W.

P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numer. Anal. 15, 977–992 (1978).
[CrossRef]

Naughton, T.

N. Pandey, D. Kelly, T. Naughton, and B. Hennelly, “Speed up of Fresnel transforms for digital holography using pre-computed chirp and GPU processing,” Proc. SPIE 7442, 744205 (2009).
[CrossRef]

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

G. Pedrini, W. Osten, and Y. Zhang, “Wavefront reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005).
[CrossRef] [PubMed]

L. Seifert, H. J. Tiziani, and W. Osten, “Wavefront reconstruction with the adaptive Shack–Hartmann sensor,” Opt. Commun. 245, 255–269 (2005).
[CrossRef]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Pandey, N.

N. Pandey, D. Kelly, T. Naughton, and B. Hennelly, “Speed up of Fresnel transforms for digital holography using pre-computed chirp and GPU processing,” Proc. SPIE 7442, 744205 (2009).
[CrossRef]

Patorski, K.

K. Patorski, “Moiré methods in interferometry,” Opt. Lasers Eng. 8, 147–170 (1988).
[CrossRef]

K. Patorski, The Self-Imaging Phenomenon and Its Applications, Vol. 27 of Progress in Optics (Elsevier, 1989) pp. 1–110.

Paufler, J.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Pedrini, G.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

G. Pedrini, W. Osten, and Y. Zhang, “Wavefront reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30, 833–835 (2005).
[CrossRef] [PubMed]

Pierri, R.

Primot, J.

J. Primot, “Theoretical description of Shack–Hartmann sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Optical Measurements Techniques and Applications (Artech, 1997).

Roddier, C.

Roddier, F.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Seifert, L.

L. Seifert, H. J. Tiziani, and W. Osten, “Wavefront reconstruction with the adaptive Shack–Hartmann sensor,” Opt. Commun. 245, 255–269 (2005).
[CrossRef]

Seltmann, R.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Shamir, J.

T. Kotzer, N. Cohen, and J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

Siegel, C.

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

Sogno, L.

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

Soldovieri, F.

Suzuki, M.

M. Suzuki and M. Kanaya, “Applications of moiré topography measurement methods in industry,” Opt. Lasers Eng. 8, 171–188 (1988).
[CrossRef]

Teague, M. R.

Tian, L.

Tiziani, H. J.

L. Seifert, H. J. Tiziani, and W. Osten, “Wavefront reconstruction with the adaptive Shack–Hartmann sensor,” Opt. Commun. 245, 255–269 (2005).
[CrossRef]

Vanderlugt, A.

Vincent, R.

R. Vincent, “Phase retrieval in TEM using Fresnel images,” Ultramicroscopy 90, 135–151 (2002).
[CrossRef] [PubMed]

Wackerman, C. C.

Waller, L.

Wolfling, S.

Wu, F.

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, H.

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

Zhang, Y.

Zimmer, G.

H. Kuck, W. Doleschal, A. Gehner, W. Grundke, R. Melcher, J. Paufler, R. Seltmann, and G. Zimmer, “Deformable micromirror devices as phase-modulating high-resolution light valves,” Sens. Actuators A 54, 536–541 (1996).
[CrossRef]

Ann. Math. Stat. (1)

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. 29, 610–611(1958).
[CrossRef]

Appl. Opt. (4)

J. Microsc. (1)

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (7)

T. Kotzer, N. Cohen, and J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

J. Primot, “Theoretical description of Shack–Hartmann sensor,” Opt. Commun. 222, 81–92 (2003).
[CrossRef]

L. Seifert, H. J. Tiziani, and W. Osten, “Wavefront reconstruction with the adaptive Shack–Hartmann sensor,” Opt. Commun. 245, 255–269 (2005).
[CrossRef]

F. Wu, H. Zhang, M. J. Lalor, and D. R. Barton, “A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry,” Opt. Commun. 187, 347–357 (2001).
[CrossRef]

C. Siegel, F. Loewenthal, and J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[CrossRef]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Opt. Eng. (1)

J. Primot, L. Sogno, B. Fracasso, and K. Heggarty, “Wavefront sensor prototype for industrial applications based on three-level phase grating,” Opt. Eng. 36, 901–904 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (3)

G.-C Jin and N.-K Bao, “Surface detection and 3D profilometry for microstructure using optical metrology,” Opt. Lasers Eng. 36, 1–9 (2001).
[CrossRef]

M. Suzuki and M. Kanaya, “Applications of moiré topography measurement methods in industry,” Opt. Lasers Eng. 8, 171–188 (1988).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Basic beam propagator model including an object plane, ( x 0 , y 0 , 0 ) , separated from a Fresnel diffraction pattern plane, ( x 1 , y 1 , z ) , by a distance z.

Fig. 2
Fig. 2

Basic IFTA cycle adapted to the Fresnel diffraction approach. The phase retrieval starts from an initial random or uniform phase guess to produce after k iterations the estimations g k , g k , g k + 1 , G k , G k of unknown wavefront.

Fig. 3
Fig. 3

Simulated IFTA beam retrieval (intensity and phase map with the number of iterations k) from the diffraction data obtained at the distance z = 35 mm . The target to be reconstructed is a squared pure three-phase level [ 0 , 5 4 π , 7 4 π ] staircase.

Fig. 4
Fig. 4

Multiplane beam propagator using a back and forth iteration between the object plane ( x 0 , y 0 , 0 ) and the Fresnel diffraction pattern planes ( x m , y m , Δ z m ) , m = 1 , , M .

Fig. 5
Fig. 5

Two pure phase simulated targets and their reconstructed maps after k iterations by means of the multiplane technique; 256 gray levels encode the relative phase range from 0 to 2 π .

Fig. 6
Fig. 6

Schema of the multi-illumination pattern projection optical setup where the SLM is imaged onto the unknown object plane.

Fig. 7
Fig. 7

Schema of the multi-illumination pattern projection optical setup where the SLM diffracts light onto the unknown object plane.

Fig. 8
Fig. 8

(a), (b) Binary phase patterns used to modulate illuminating beams; (c), (d) corresponding simulated Fresnel diffraction patterns at z = 35 mm from the object plane for a conic test surface.

Fig. 9
Fig. 9

Phase retrieval of two pure phase targets (conic and stairlike shapes) from their simulated diffraction patterns when sets of binary square-modulated phase patterns are used as probe-light fields (top) and for phase gratings light probe patterns (bottom).

Fig. 10
Fig. 10

Simulated illuminating pattern intensities with Gaussian noise added in the target’s plane: noise mean m = 0.2 , standard deviation σ 2 (top), their Fresnel diffraction views (center), and the phase profile (bottom) of a pure phase test target reconstructed with four illuminating light probes after k = 30 iterations.

Fig. 11
Fig. 11

Phase maps recovered after k = 30 iterations in presence of an error ϵ z 2 in the propagation distance z 2 = 35 mm .

Fig. 12
Fig. 12

Architecture of a phase profilometer using active light probes.

Fig. 13
Fig. 13

View from above of the experimental setup.

Fig. 14
Fig. 14

Binary pure phase samples reconstructed by a multiplane diffraction method (200 iterations): (a), (b) experiment without light modulation, (c) experiment with SLM-modulated light.

Fig. 15
Fig. 15

CGHs and their focal plane images used to calibrate the system propagation distances optically.

Fig. 16
Fig. 16

(top) Schema of the distance calibration process. (bottom) Observed characteristic spots (left to right) in focal plane, defocused by Δ z = 0.25 mm , defocused by Δ z = 0.5 mm .

Fig. 17
Fig. 17

(top) Experimental intensity patterns of six grating-modulated illuminating waves propagated at the distance z 1 = 100 mm ; (center) corresponding phase distributions; (bottom) corresponding Fresnel diffraction patterns at z 2 = 50 mm . The “T” values are the grating periods.

Fig. 18
Fig. 18

Two binary phase maps reconstructed from experimental diffraction patterns with our multi-illumination technique ( k = 10 iterations).

Equations (10)

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U ( x 1 , y 1 , z ) = e 2 π z 1 λ λ z 1 e π 1 λ z ( x 1 2 + y 1 2 ) + + U ( x 0 , y 0 , 0 ) e 1 π λ z ( x 0 2 + y 0 2 ) e 1 2 π λ z ( x 0 x 1 + y 0 y 1 ) d x 0 d y 0 .
U ( p , q , z ) = e 2 π z 1 λ λ z 1 e π δ z 2 1 λ z ( p 2 + q 2 ) i = 0 N 1 j = 0 N 1 U ( i , j , 0 ) e π δ 0 2 1 λ z ( i 2 + j 2 ) e 2 π 1 N ( i p + j q ) , i , j , p , q = 0 , 1 , 2 , , N 1 , ( x 0 , y 0 ) = δ 0 × ( i , j ) , ( x 1 , y 1 ) = δ z × ( p , q ) ,
Z limit = δ 0 2 ( N 1 ) λ ,
MSE z , k 2 = 1 N 2 p q ( | U ( p , q , z ) | 2 | DFrT { U k ( i , j , 0 ) e ϕ k ( i , j , 0 ) 1 } | 2 ) 2 .
MSE 0 , k 2 = 1 N 2 i j ( | U k ( i , j , 0 ) | 2 DFrT 1 { | U ( p , q , z ) · e ϕ k ( p , q , z ) 1 | 2 ) 2 ,
U m = 1 , , M ( x 0 , y 0 , 0 ) = U illum ( x 0 , y 0 , 0 ) { exp [ i ϕ 1 llum ( x 0 , y 0 , 0 ) ] , exp [ i ϕ 2 illum ( x 0 , y 0 , 0 ) ] , , exp [ i ϕ M illum ( x 0 , y 0 , 0 ) ] } × e i Φ ( x 0 , y 0 , 0 ) .
U m = 1 , , M ( x 1 , y 1 , z 1 ) = { U illum ( x 0 , y 0 , 0 ) e i ϕ 1 illum ( x 0 , y 0 , 0 ) K z 1 Fresnel U illum ( x 0 , y 0 , 0 ) e i ϕ 2 illum ( x 0 , y 0 , 0 ) K z 1 Fresnel , U illum ( x 0 , y 0 , 0 ) e i ϕ M illum ( x 0 , y 0 , 0 ) K z 1 Fresnel } × e i Φ ( x 1 , y 1 , z 1 ) ,
K z 1 Fresnel = e 2 π z 1 1 λ z 1 λ 1 · exp [ π δ 0 2 1 z 1 λ · ( x 0 2 + y 0 2 ) δ 0 2 ] .
U m = 1 , , M ( x 2 , y 2 , z 2 ) = U m = 1 , , M ( x 1 , y 1 , z 1 ) K z 2 Fresnel ,
K z 2 Fresnel = e 2 π z 2 1 λ z 2 λ 1 · exp [ π δ 0 2 1 z 2 λ · ( x 1 2 + y 1 2 ) δ 0 2 ] ,

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