Abstract

Phase diversity (PD) is a powerful technique for estimating wavefront aberrations from two-dimensional images of extended scenes. PD can work with extended incoherent images and, in an adaptive optics system, does not need extra hardware in addition to the deformable mirror. For these reasons, PD should be well suited to aberration measurement in microscopy applications. But, in biological widefield microscopy, the objects being imaged are frequently three-dimensional, and the images contain out-of-focus light. In this paper, we introduce multiplane PD and show that PD can be extended to widefield imaging of three-dimensional objects. This should be particularly useful in the field of biological fluorescence microscopy where the objects are very light sensitive and the aberrations cannot easily be determined.

© 2013 Optical Society of America

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References

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2011 (2)

2010 (1)

P. Kner, L. Winoto, D. A. Agard, and J. W. Sedat, “Closed loop adaptive optics for microscopy without a wavefront sensor,” Proc. SPIE 7570, 757006 (2010).
[CrossRef]

2008 (1)

2007 (5)

2006 (1)

2005 (1)

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

2000 (1)

1993 (1)

1992 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

1978 (1)

1972 (1)

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

1969 (1)

Agard, D.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef]

Agard, D. A.

Albert, O.

Azucena, O.

Booth, M. J.

Burns, D.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Campbell, H. I.

Campbell, M.

Chen, D. C.

Chenegros, G.

Donnelly, W.

Fernandez, B.

Fienup, J. R.

Fu, M.

Garcia, D.

Gerchberg, R. W.

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

Girkin, J. M.

S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. 47, 731–736 (2008).
[CrossRef]

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Glanc, M.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Greenaway, A. H.

Gustafsson, M. G.

Haase, S.

Hanser, B. M.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series in Optical and Imaging Sciences; 16 (Oxford University, 1998).

Hebert, T.

Hom, E. F.

Janssen, A. J. E. M.

Kam, Z.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef]

Kner, P.

P. Kner, L. Winoto, D. A. Agard, and J. W. Sedat, “Closed loop adaptive optics for microscopy without a wavefront sensor,” Proc. SPIE 7570, 757006 (2010).
[CrossRef]

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef]

Kubby, J.

Kubby, J. A.

J. A. Kubby, Adaptive Optics for Biological Imaging (CRC press, 2013).

Lacombe, F.

Lee, T. K.

Marchis, F.

Marron, J. C.

Mourou, G.

Mugnier, L. M.

Norris, T. B.

Patterson, B. A.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Paxman, R. G.

Poland, S. P.

S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. 47, 731–736 (2008).
[CrossRef]

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Queener, H.

Restaino, S.

Romero-Borja, F.

Roorda, A.

Saxton, W. O.

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

Schulz, T. J.

Sedat, J. W.

P. Kner, L. Winoto, D. A. Agard, and J. W. Sedat, “Closed loop adaptive optics for microscopy without a wavefront sensor,” Proc. SPIE 7570, 757006 (2010).
[CrossRef]

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef]

E. F. Hom, F. Marchis, T. K. Lee, S. Haase, D. A. Agard, and J. W. Sedat, “AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data,” J. Opt. Soc. Am. A 24, 1580–1600 (2007).
[CrossRef]

B. M. Hanser, M. G. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett. 28, 801–803 (2003).
[CrossRef]

Seldin, J. H.

Sherman, L.

Stokseth, P. A.

Tao, X.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997).

Valentine, G. J.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Vdovin, G.

Winoto, L.

P. Kner, L. Winoto, D. A. Agard, and J. W. Sedat, “Closed loop adaptive optics for microscopy without a wavefront sensor,” Proc. SPIE 7570, 757006 (2010).
[CrossRef]

Wright, A. J.

S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. 47, 731–736 (2008).
[CrossRef]

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Zhang, S.

Zuo, Y.

Appl. Opt. (2)

J. Microsc. (1)

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226, 33–42 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Microsc. Res. Tech. (1)

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Optik (1)

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

Phil. Trans. R. Soc. A (1)

M. J. Booth, “Adaptive optics in microscopy,” Phil. Trans. R. Soc. A 365, 2829–2843 (2007).

Proc. SPIE (1)

P. Kner, L. Winoto, D. A. Agard, and J. W. Sedat, “Closed loop adaptive optics for microscopy without a wavefront sensor,” Proc. SPIE 7570, 757006 (2010).
[CrossRef]

Other (3)

J. A. Kubby, Adaptive Optics for Biological Imaging (CRC press, 2013).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series in Optical and Imaging Sciences; 16 (Oxford University, 1998).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

Example of two-dimensional PD on a fluorescence image. (a) In focus image of GFP-labelled cell wall in Arabidopsis thaliana. (b) Image taken 1 μm out of focus. (c) Object estimated from PD calculation on images (a) and (b). (d) Wavefront estimated from PD calculation. (e) Axial cross section of the sample showing spherical aberration in agreement with the PD calculation and the wavefront in (d). Scale bar is 5 μm. These images were taken with a 40× air objective.

Fig. 2.
Fig. 2.

Simulation using the imaging model from Eq. (15). (a) Three planes from simulated object composed of lateral lines. The planes are 0.25 μm apart, and the total image is 32 planes. (b) Image of the central plane. (c) Image of the central plane including all out-of-focus planes using Eq. (15). The out-of-focus OTF was generated using the Stokseth approximation [24]. Scale bar is 1 μm.

Fig. 3.
Fig. 3.

Three-dimensional PD. (a) and (b) PD images of the simulated sample. These are the central planes from a 32×128×128 stack with dz=0.5μm and dx=50nm. The aberration is ϕ(u)=0.5Z31(u). (a) Image with θ1(u)=0. (b) Image with PD aberration θ2(u)=0.5Z22(u). (c) Central plane of the recovered image from the PD calculation with μ=0.001. (d) The error metric, L(α¯), as the amplitude of aberration is varied. Blue solid curve: aberration is Z31 and goes to 0 at the value of the actual aberration. Black dashed curve: varying another aberration, Z31, does not bring the error metric, L, to 0. Scale bar is 1 μm.

Fig. 4.
Fig. 4.

Comparison of two-dimensional PD with three-plane PD on one plane of a three-dimensional stack. (a) PD image with θ1(u)=0. (b) PD image with θ2(u)=0.5Z22(u). The wavefront aberration is ϕ(u)=0.5Z31(u). (c) Optimal PD image from two-dimensional PD equation. (d) Optimal PD image from three-plane PD. (e) Error metric, L(α¯), versus Zernike amplitude for two-dimensional PD. Black dashed curve: error metric versus Z31 blue solid curve: error metric versus Z31 (f) error metric for three-plane PD. (g) Cross section across images (c) (black, dashed) and (d) (blue, solid). The three-plane PD results in lower background. μ=0.1. Simulated object has 32 planes spaced 0.5 μm apart. Scale bar is 1 μm.

Fig. 5.
Fig. 5.

Comparison of three-plane and five-plane PD. The aberration is ϕ(u)=0.5Z31(u). (a) Plot of the error function versus Zernike amplitude. The black curves with circles are the three-plane error curves and the blue curves with squares are the five-plane error curves. The solid curves show the error versus the amplitude of Z31(u). The dashed curves show the error versus the amplitude of Z31(u). (b) PD image with no induced aberration. (c) Simulated object. (d) Result of three-plane PD calculation. (e) Result of five-plane PD calculation. (1) First column is 0.5 μm below focus. (2) Second column is focal plane. (3) Third column 0.5 μm above focus. μ=0.1. Simulated object has 32 planes spaced 0.5 μm apart. Scale bar is 1 μm.

Fig. 6.
Fig. 6.

Comparison of three-plane and five-plane PD for a simulated object with spacing of dz=0.2μm. The aberration is ϕ(u)=0.5Z31(u). (a) Plot of the error function versus Zernike amplitude. The black curves with circles are the three-plane PD error curves with dz=0.4μm, and the blue curves with squares are the five-plane PD error curves with dz=0.2μm. The solid curves show the error versus the amplitude of Z31(u). The dashed curves show the error versus the amplitude of Z31(u). (b) PD image with no induced aberration. (c) Simulated object. (d) Result of three-plane PD calculation. (e) Result of five-plane PD calculation. (1) The first column is the plane below focus. (2) The second column is the focal plane. And (3) the third column is the plane above focus. μ=0.1. Scale bar is 1 μm.

Fig. 7.
Fig. 7.

Comparison of two-dimensional PD and three-plane PD with noise. Poisson noise is added to the image assuming the object emits 1000 photons per pixel. (a) PD image with θ1(u)=0. (b) PD image with θ2(u)=0.5Z22(u). The wavefront aberration is ϕ(u)=0.5Z31(u). (c) Optimal PD image from two-dimensional PD equation. (d) Optimal PD image from three-plane PD. (e) Error metric, L(α¯) versus Zernike amplitude for two-dimensional PD. Black dashed curve: error metric versus Z31. Blue solid curve: error metric versus Z31. (f) Error metric for three-plane PD. (g) Intensity map of the 2D error metric. Black shows the minimum of the error metric, and 2D PD does not accurately find the correct amplitude. (h) Intensity map of the three-plane error metric. The minimum remains near 0.5Z31. Scale bar is 1 μm.

Fig. 8.
Fig. 8.

Comparison of Wiener deconvolution and three-plane deconvolution. (a) Two-dimensional image from the center of a 32 plane stack with 0.5 μm spacing. (b) Three-plane deconvolution. (c) Wiener deconvolution. (d) Wiener deconvolution with an aberrated image (θ(u)=0.5Z22(u)). For (b)–(d), μ=0.01. Scale bar is 1 μm.

Equations (21)

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Hk(u)=|P(u)|exp(iϕ(u)+iθk(u)).
hk(x)=F1{Hk(u)},
sk(x)=|hk(x)|2.
L(f,α¯)=k=1Kx[dk(x)f(x)sk(x)]2.
ϕ(u)=j=1Jαjϕj(u)
L(F,α¯)=k=1Ku|Dk(u)F(u)Sk(u)|2
LFre=0andLFim=0.
F(u)=D1(u)S1*(u;α¯)+D2(u)S2*(u;α¯)|S1(u;α¯)|2+|S2(u;α¯)|2
L(α¯)=u|D1(u)S2(u;α¯)D2(u)S1(u;α¯)|2|S1(u;α¯)|2+|S2(u;α¯)|2.
d(x,y,z)=f(x,y,z)s(x,y,z).
sk(x,y,z)=|hk(x,y,z)|2,
hk(x,y,z)=F1{Hk(u)exp(izγ(u))},
γ(u)=2πnλ1(λun)2.
sk(x⃗)=s(x⃗dz^)dk(x⃗)=sk(x⃗)f(x⃗)=F1{ej2πudS(u⃗)F(u⃗)}=d(x⃗dz^).
dk(x,y)=i=1Pf(x,y,zi)sk(x,y;zizd).
L(f,α¯)=k=1Kx,y(dk(x,y)i=1Pf(x,y,zi)sk(x,y;zi))2,
L(F,α¯)=k=1Ku|Dki=1PFz(u)Sk(u;zi)|2,
Mz,z[Fz]=[kDkSk,z*],
Mz,z=kMz,zk=[kSk,z*Sk,z].
F(u)=D1(u)S1*(u)+D2(u)S2*(u)|S1(u)|2+|S2(u)|2+μ2
([kSk,z*Sk,z]+μ2I)[Fz]=[kDkSk,z*],

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