Abstract

A phase retrieval method for microscopy using multiple illumination wavelengths is proposed. A fast algorithm suitable for calculations with high numerical aperture is used for the iterative retrieval of the object wavefront. The advantages and limitations of the technique are systematically analyzed and demonstrated by both simulation and experimental results.

© 2012 Optical Society of America

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References

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    [CrossRef]
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  5. G. Popescu, L. P. Delflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
    [CrossRef]
  6. C. G. Rylander, D. Dave, T. Akkin, T. E. Milner, K. R. Diller, and A. J. Welch, “Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy,” Opt. Lett. 29, 1509–15011 (2004).
    [CrossRef]
  7. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
    [CrossRef]
  8. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  9. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
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  12. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef]
  13. B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
  14. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef]
  15. J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
    [CrossRef]
  16. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
    [CrossRef]
  17. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express 11, 3234–3241 (2003).
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  18. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
    [CrossRef]
  19. 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]
  20. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309–311 (2008).
    [CrossRef]
  21. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
    [CrossRef]
  22. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  24. Y. Takaki and H. Ohzu, “Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,” Appl. Opt. 38, 2204–2211 (1999).
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  25. H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6,411,406 B1 (25June2002).
  26. P. Ferraro, S. D. Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854–856(2004).
    [CrossRef]
  27. L. P. Yaroslavsky, “Efficient algorithm for discrete sinc interpolation,” Appl. Opt. 36, 460–463 (1997).
    [CrossRef]
  28. L. P. Yaroslavsky, “Boundary effect free and adaptive discrete signal sinc-interpolation algorithms for signal and image resampling,” Appl. Opt. 42, 4166–4175 (2003).
    [CrossRef]
  29. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20, 1755–1762 (2003).
    [CrossRef]

2008 (1)

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]

2006 (1)

2004 (5)

2003 (3)

2002 (1)

1999 (4)

1998 (2)

1997 (2)

L. P. Yaroslavsky, “Efficient algorithm for discrete sinc interpolation,” Appl. Opt. 36, 460–463 (1997).
[CrossRef]

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
[CrossRef]

1996 (1)

1981 (1)

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

1978 (1)

1973 (1)

1972 (1)

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

Adams, M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
[CrossRef]

Akkin, T.

Alfieri, D.

Almoro, P.

Badizadegan, K.

Bao, P.

Barty, A.

Bevilacqua, F.

Coppola, G.

Cuche, E.

Dasari, R. R.

Dave, D.

Delflores, L. P.

Depeursinge, C.

Diller, K. R.

Dorsch, R. G.

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Feld, M. S.

Ferraro, P.

Fienup, J. R.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef]

Finizio, A.

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gu, B.

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Iwai, H.

Jueptner, W. P. O.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
[CrossRef]

Jüptner, W.

Kao, F. J.

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer-Verlag, 2003).

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
[CrossRef]

Kreuzer, H. J.

H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6,411,406 B1 (25June2002).

Matsushima, K.

Mendlovic, D.

Milner, T. E.

Nicola, S. D.

Nugent, K. A.

Ohzu, H.

Osten, W.

Paganin, D.

Pawley, J. B.

J. B. Pawley, Handbook of Biological Confocal Microscopy(Plenum, New York, 1995).

Paxman, R. G.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Pedrini, G.

Pierattini, G.

Popescu, G.

Reiley, M. F.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Roberts, A.

Rodenburg, J. M.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Rylander, C. G.

Saxton, W. O.

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

Schimmel, H.

Seebacher, S.

Sommargren, G. E.

Takaki, Y.

Thelen, B. J.

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Thompson, B. J.

Tiziani, H.

Tiziani, H. J.

Torok, P.

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer-Verlag, 2003).

Vaughan, J. C.

Wagner, C.

Welch, A. J.

Wyrowski, F.

Yamaguchi, I.

Yang, G.

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Yaroslavsky, L. P.

Zalevsky, Z.

Zhang, F.

Zhang, T.

Zhang, Y.

Acta Opt. Sin. (1)

B. Gu and G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).

Appl. Opt. (7)

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (10)

P. Ferraro, S. D. Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854–856(2004).
[CrossRef]

C. G. Rylander, D. Dave, T. Akkin, T. E. Milner, K. R. Diller, and A. J. Welch, “Quantitative phase-contrast imaging of cells with phase-sensitive optical coherence microscopy,” Opt. Lett. 29, 1509–15011 (2004).
[CrossRef]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
[CrossRef]

G. Popescu, L. P. Delflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
[CrossRef]

P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309–311 (2008).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
[CrossRef]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Optik (1)

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

Phys. Rev. A (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]

Phys. Rev. Lett. (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef]

Proc. SPIE (2)

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224 (1997).
[CrossRef]

J. R. Fienup, R. G. Paxman, M. F. Reiley, and B. J. Thelen, “3-D imaging correlography and coherent image reconstruction,” Proc. SPIE 3815, 60–69 (1999).
[CrossRef]

Other (4)

P. Torok and F. J. Kao, Optical Imaging and Microscopy (Springer-Verlag, 2003).

J. B. Pawley, Handbook of Biological Confocal Microscopy(Plenum, New York, 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

H. J. Kreuzer, “Holographic microscope and method of hologram reconstruction,” U.S. patent 6,411,406 B1 (25June2002).

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Figures (11)

Fig. 1.
Fig. 1.

Recording geometry for lensless microscopy.

Fig. 2.
Fig. 2.

Sampling grids (a) in x, y and; (b) in vx, vy domains.

Fig. 3.
Fig. 3.

Optical setup.

Fig. 4.
Fig. 4.

Flowchart of the high NA phase retrieval: n is the iteration number, Uk are the calculated amplitudes, k is the serial number of the used illumination wavelength, a is the indicator, and Ik are the recorded intensities.

Fig. 5.
Fig. 5.

Correlation coefficient between the wavefront calculated by (a) the Rayleigh-Sommerfeld integral and the HNAA approximation or (b) the Fresnel approximation.

Fig. 6.
Fig. 6.

Comparison between the HNAA and Fresnel approximation when the NA is (a) 0.10;, (b) 0.60, and (c) 0.83. The object size is (d) 60 μm, (e) 100 μm, and (f) 290 μm.

Fig. 7.
Fig. 7.

(a) Original and (b) retrieved phase. The gray level indicates the phase value (02π).

Fig. 8.
Fig. 8.

Convergence of different algorithms when (a) NA=0.83 (z=2mm), (b) NA=0.51 (z=5mm), and (c) NA=0.20 (z=15mm).

Fig. 9.
Fig. 9.

Convergence performance versus calculation time for HNAA and interpolated AS algorithms: (a) NA=0.83; (b) NA=0.51; and (c) NA=0.20.

Fig. 10.
Fig. 10.

Convergence performance for (a) different numbers of intensity recordings M=5, 10, and 20; (b) different tuning ranges, TR=20nm and 50 nm; and (c) different signal-to-noise rate, SNR=15dB, 25 dB, and 35 dB.

Fig. 11.
Fig. 11.

Experimental investigation of a Fresnel lens. CCD captured images at (a) 750 nm and (b) 850 nm; (c) retrieved phase; (d) three-dimensional surface; (e) profile along the black line shown in (c), and (f) profile measured by a step profiler.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

U(x,y)=1iλU(x,y)K(θ)r(x,y,x,y,z)exp{2πir(x,y,x,y,z)λ}dxdy,
r(x,y,x,y,z)=(z2+(xx)2+(yy)2)1/2r(x,y,x,y,z)|x,y=0+(x[r(x,y,x,y,z)x]|x,y=0+y[r(x,y,x,y,z)y]|x,y=0)+=(z2+x2+y2)1/2xx+yy(z2+x2+y2)1/2+,
1r(x,y,x,y,z)1(z2+x2+y2)1/2,
exp{2πir(x,y,x,y,z)λ}{2πiλ[(z2+x2+y2)1/2xx+yy(z2+x2+y2)1/2]}.
vx=xλ(z2+x2+y2)1/2,vy=yλ(z2+x2+y2)1/2,
U(vx,vy)=(1λ2vx2λ2vy2)iλzexp{i2πλz(1λ2vx2λ2vy2)1/2}U(x,y)exp{i2π(vxx+vyy)}dvxdvy=(1λ2vx2λ2vy2)iλzexp{i2πλz(1λ2vx2λ2vy2)1/2}F{U(x,y)},
Δx=λ(z2+(12Mpx)2+(12Npy)2)1/2M·px,Δy=λ(z2+(12Mpx)2+(12Npy)2)1/2N·py,
Uk(vx,vy)=Ik(vx,vy)·exp{i·ϕk(vx,vy)},
Uk(x,y)=iλkz(1λk2vx2λk2vy2)·exp{i2πλz(1λk2vx2λk2vy2)1/2}·F1{Uk(vx,vy)},
Uk+a(x,y)=Uk(x,y)·exp{i·ϕk(x,y)}·exp{iϕk(x,y)·λk/λk+a},
Uk+a(vx,vy)=(1λk+a2vx2λk+a2vy2)iλk+az·exp{i2πλk+az(1λk+a2vx2λk+a2vy2)1/2}·F{Uk+a(x,y)}.
ρ=mn(AmnA¯)(BmnB¯)(mn(AmnA¯)2)(mn(BmnB¯)2),

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