Abstract

We present a formulation of optical point spread function based on a scaled three-dimensional Fourier transform expression of focal field distribution and the expansion of generalized aperture function. It provides an equivalent but more flexible representation compared with the analytic expression of the extended Nijboer-Zernike approach. A phase diversity algorithm combined with an appropriate regularization strategy is derived and analyzed to demonstrate the effectiveness of the presented formulation for phase retrieval and deconvolution. Experimental results validate the performance of presented algorithm.

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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  17. J. J. M. Braat, S. van Haver, and S. F. Pereira, “Microlens quality assessment using the Extended Nijboer-Zernike (ENZ) diffraction theory,” presented at EOS Optical Microsystems, Capri, Italy, 27–30 Sept. 2009.
  18. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A20(11), 2156–2162 (2003).
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  19. S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt.41(11), 2095–2102 (2002).
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  20. C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
    [CrossRef]
  21. J. Nocedal and S. J. Wright, Numerical Optimization 2nd ed. (Springer, 2006).
  22. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).
  23. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express14(23), 11277–11291 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11277 .
    [CrossRef] [PubMed]

2011

2010

A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, paper A6 (2010).

2007

2006

2005

J. B. Sibarita, “Deconvolution Microscopy,” Adv. Biochem. Eng. Biotechnol.95, 201–243 (2005).
[PubMed]

2004

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

2003

2002

1998

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

1993

1992

1982

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982).
[CrossRef] [PubMed]

1981

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun.39(4), 205–210 (1981).
[CrossRef]

1964

Agard, D. A.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

Ariste, A. L.

A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, paper A6 (2010).

Braat, J. J. M.

Chan, T.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Chenegros, G.

Dainty, J. C.

Dirksen, P.

Fienup, J. R.

Georges, J.

Glanc, M.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).

Gustafsson, M. G. L.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

Hanser, B. M.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

Hege, E. K.

Janssen, A. J. E. M.

Jefferies, S. M.

Kou, S. S.

Lacombe, F.

Lasser, T.

Leitgeb, R. A.

Leutenegger, M.

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun.39(4), 205–210 (1981).
[CrossRef]

Lin, J.

Lloyd-Hart, M.

McCutchen, C. W.

Mugnier, L. M.

Paxman, R. G.

Plemmons, R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Ramos, A. A.

A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, paper A6 (2010).

Rao, R.

Rodríguez-Herrera, O. G.

Schulz, T. J.

Sedat, J. W.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Sibarita, J. B.

J. B. Sibarita, “Deconvolution Microscopy,” Adv. Biochem. Eng. Biotechnol.95, 201–243 (2005).
[PubMed]

Török, P.

Vogel, C. R.

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Wolf, E.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun.39(4), 205–210 (1981).
[CrossRef]

Yuan, X.-C.

Adv. Biochem. Eng. Biotechnol.

J. B. Sibarita, “Deconvolution Microscopy,” Adv. Biochem. Eng. Biotechnol.95, 201–243 (2005).
[PubMed]

Appl. Opt.

Astron. Astrophys

A. A. Ramos and A. L. Ariste, “Image reconstruction with analytical point spread functions,” Astron. Astrophys. 518, paper A6 (2010).

J. Microsc.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc.216(1), 32–48 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun.39(4), 205–210 (1981).
[CrossRef]

Opt. Eng.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).

Opt. Express

Opt. Lett.

Proc. SPIE

C. R. Vogel, T. Chan, and R. Plemmons, “Fast algorithms for phase diversity-based blind deconvolution,” Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Other

J. Nocedal and S. J. Wright, Numerical Optimization 2nd ed. (Springer, 2006).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).

S. van Haver, “The Extended Nijboer-Zernike diffraction theory and its applications,” PhD Dissertation, Delft University of Technology (2010).

J. J. M. Braat, S. van Haver, and S. F. Pereira, “Microlens quality assessment using the Extended Nijboer-Zernike (ENZ) diffraction theory,” presented at EOS Optical Microsystems, Capri, Italy, 27–30 Sept. 2009.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed., (Elsevier, 2008), 51, 349–468.

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Figures (8)

Fig. 1
Fig. 1

Geometry for computing the optical field distribution in the focal region.

Fig. 2
Fig. 2

Simulation results for the two-dimensional phase retrieval (all units are the wavelength), where (a-c) are the cases of circular aperture and (d-f) are the cases of annular aperture. (a) and (d) are the randomly generated phase aberrations with the same RMS = 0.222λ. (b) and (e) are the reconstructed phases with amplitude regularization, where the RMS of residual errors are 0.023λ and 0.032λ, respectively. (c) and (f) are the reconstructed phases without amplitude regularization, where the RMS of residual errors are 0.031λ and 0.068λ, respectively.

Fig. 3
Fig. 3

Simulation results for the three-dimensional deconvolution, where (a) is the stack of the simulated object, (b) is the stack of the focused image, and (c) is the stack of the reconstructed object. The RMS of randomly generated phase aberrations is 0.182λ, and the RMS of residual error of reconstructed phase is 0.07λ.

Fig. 4
Fig. 4

Zernike coefficients for a generalized aperture function with uniform amplitude and large phase aberration of RMS = 1.398λ, where the charts from top to bottom are the α coefficients and the corresponding imaginary and real parts of β coefficients, respectively.

Fig. 5
Fig. 5

Layout of the experimental setup.

Fig. 6
Fig. 6

Experimental results for small to large phase aberrations, where the graphs from top to bottom are the RMS of residual errors and the times of optimizing iterations, respectively.

Fig. 7
Fig. 7

Experimental results of phase retrieval and deconvolution for the phase aberration of RMS = 0.647λ. (a) and (d) are the measured and reconstructed phases respectively, where the RMS of residual error is 0.046λ. (b) and (e) are the measured and reconstructed focal PSF, respectively. (c) and (f) are the simulated focal image and reconstructed object, respectively.

Fig. 8
Fig. 8

Experimental results of phase retrieval and deconvolution for the phase aberration of RMS = 1.203λ. (a) and (d) are the measured and reconstructed phases respectively, where the RMS of residual error is 0.202λ. (b) and (e) are the measured and reconstructed focal PSF, respectively. (c) and (f) are the simulated focal image and reconstructed object, respectively.

Tables (1)

Tables Icon

Table 1 Experimental results for small to large phase aberrations

Equations (37)

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U( P )= iA λ exp( ikf ) f W U( Q ) exp( ikr ) r cosθdS ,
U F ( x,y,z )= iA λ d f 3 V U W ( u,v,w )A( u,v,w )exp( ik xu+yv+zw f )dudvdw ,
A( u,v,w )=δ( u 2 + v 2 + w 2 f ),
P( u,v,w )= U W ( u,v,w )A( u,v,w ).
z ¯ '= z ¯ / ( 1+ z ¯ λ fπ ( NA ) 2 ) , r ¯ '= r ¯ / ( 1+ z ¯ λ fπ ( NA ) 2 ) ,
z ¯ =z π ( NA ) 2 λ , r ¯ = x 2 + y 2 2π( NA ) λ .
x'= f f+z x,y'= f f+z y,z'= f f+z z.
U F ( x',y',z' )= iA λ fz' f 3 exp( ik x ' 2 +y ' 2 +2z ' 2 2( fz' ) ) × V U W ( u,v,w )A( u,v,w )exp( ik x'u+y'v+z'w f )dudvdw .
U( x,y,z )=g( x,y,z ) F 3 [ P( u,v,w ) ],
P( u,v,w )= nm β n m Z ' n m ( u,v,w ) ,
Z ' n m ( u,v,w )= Z n m ( u,v )A( u,v,w ),
U( x,y,z )= nm β n m V ' n m ( x,y,z ) ,
V ' n m ( x,y,z )=g( x,y,z ) F 3 [ Z ' n m ( u,v,w ) ].
U( r,φ,z )=h( r,φ,z ) nm 2 i m β n m V n m ( r,z )exp( imφ ) ,
V n m ( r,z )= 0 1 exp( iz ρ 2 ) R n | m | ( ρ ) J m ( 2πrρ )ρdρ ,
V ' n m ( r,φ,z )=2 i m h( r,φ,z ) V n m ( r,z )exp( imφ ).
S( x,y,z )= | nm β n m V ' n m ( x,y,z ) | 2 .
I( x,y,z )=O( x,y,z )S( x,y,z )+N( x,y,z ),
E 3D = 1 2 uvw | I ˜ ( u,v,w ) O ˜ ( u,v,w ) S ˜ ( u,v,w ) | 2 + γ 2 uvw | O ˜ ( u,v,w ) | 2 +μ uvw [ | P( u,v,w ) | 2 | P 0 ( u,v,w ) | 2 ] 2 ,
E 2D = 1 2 uv z | I z ˜ ( u,v ) O ˜ ( u,v ) S z ˜ ( u,v ) | 2 + γ 2 uv | O ˜ ( u,v ) | 2 +μ uvw [ | P( u,v,w ) | 2 | P 0 ( u,v,w ) | 2 ] 2 .
E ' 3D = uvw | I ˜ * ( u,v,w ) S ˜ ( u,v,w ) | 2 γ+ | S ˜ ( u,v,w ) | 2 + uvw | I ˜ ( u,v,w ) | 2 +μ uvw [ | P( u,v,w ) | 2 | P 0 ( u,v,w ) | 2 ] 2 ,
E ' 2D = uv z | I z ˜ * ( u,v ) S z ˜ ( u,v ) | 2 γ+ z | S z ˜ ( u,v ) | 2 + uv z | I z ˜ ( u,v ) | 2 +μ uvw [ | P( u,v,w ) | 2 | P 0 ( u,v,w ) | 2 ] 2 ,
O ˜ 3D ( u,v,w )= I ˜ ( u,v,w ) S ˜ * ( u,v,w ) γ+ | S ˜ ( u,v,w ) | 2 ,
O ˜ 2D ( u,v )= z I z ˜ ( u,v ) S z ˜ * ( u,v ) γ+ z | S z ˜ ( u,v ) | 2 .
S= kl β k β l * V ' k V ' l * = kl β k β l * W kl , W kl =V ' k V ' l * ,
S ˜ = kl β k β l * W kl ˜ , S z ˜ = kl β k β l * W kl ( z ) ˜ .
S z ˜ β k Re = l β l * W kl ( z ) ˜ + l β l W lk ( z ) ˜ , S z ˜ β k Im =i l β l * W kl ( z ) ˜ i l β l W lk ( z ) ˜ ,
2 S z ˜ β k Re β l Re = 2 S z ˜ β k Im β l Im = W lk ( z ) ˜ + W kl ( z ) ˜ , 2 S z ˜ β k Re β l Im = 2 S z ˜ β k Im β l Re =i W lk ( z ) ˜ i W kl ( z ) ˜ .
| P | 2 = kl β k β l * Z ' k Z ' l = kl β k β l * X kl , X kl =Z ' k Z ' l ,
| P | 2 β k Re =2 l β l Re X kl , | P | 2 β k Im =2 l β l Im X kl ,
2 | P | 2 β k Re β l Re = 2 S z ˜ β k Im β l Im =2 X kl , 2 S z ˜ β k Re β l Im = 2 | P | 2 β k Im β l Re =0.
E ' 2D = uv | T | 2 D + uv z | I z ˜ | 2 +μ uvw Q 2 , T= I z ˜ * S z ˜ ,D=γ+ z | S z ˜ | 2 ,Q= | P | 2 | P 0 | 2 .
| T | 2 β k Re,Im =2Re[ T * z I z ˜ * S z ˜ β k Re,Im ], 2 | T | 2 β k Re,Im β l Re,Im =2Re[ T * z I z ˜ * 2 S z ˜ β k Re,Im β l Re,Im + z I z ˜ * S z ˜ β k Re,Im z I z ˜ S z ˜ * β l Re,Im ],
D β k Re,Im =2Re[ z S z ˜ * S z ˜ β k Re,Im ], 2 D β k Re,Im β l Re,Im =2Re[ z ( S z ˜ * 2 S z ˜ β k Re,Im β l Re,Im + S z ˜ β k Re,Im S z ˜ * β l Re,Im ) ],
Q 2 β k Re,Im =2[ Q | P | 2 β k Re,Im ], 2 Q 2 β k Re,Im β l Re,Im =2[ Q 2 | P | 2 β k Re,Im β l Re,Im + | P | 2 β k Re,Im | P | 2 β l Re,Im ].
E ' 2D β k Re,Im = uv [ | T | 2 D 2 D β k Re,Im 1 D | T | 2 β k Re,Im +μ Q 2 β k Re,Im ] ,
2 E ' 2D β k Re,Im β l Re,Im = uv [ 2 | T | 2 D 3 D β k Re,Im D β l Re,Im + | T | 2 D 2 2 D β k Re,Im β l Re,Im 1 D 2 | T | 2 β k Re,Im β l Re,Im + 1 D 2 D β k Re,Im | T | 2 β l Re,Im + 1 D 2 | T | 2 β k Re,Im D β l Re,Im +μ 2 Q 2 β k Re,Im β l Re,Im ] .

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