Abstract

Through-focus phase retrieval methods aim to retrieve the phase of an optical field from its intensity distribution measured at different planes in the focal region. By using the concept of spatial correlation for propagating fields, for both the complex amplitude and the intensity of a field, we can infer which planes are suitable to retrieve the phase and which are not. Our analysis also reveals why all techniques based on measuring the intensity at two Fourier-conjugated planes usually lead to a good reconstruction of the phase. The findings presented in this work are important for aberration characterization of optical systems, adaptive optics and wavefront metrology.

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References

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  2. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart)35, 237–246 (1972).
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    [CrossRef] [PubMed]
  4. H. M. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett.93, 023903 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).
    [CrossRef]
  11. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998).
    [CrossRef]
  12. B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, “Phase retrieval for high-numerical-aperture optical systems,” Opt. Lett.28, 801–803 (2003).
    [CrossRef] [PubMed]
  13. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  14. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2007), 883–891.

2011 (1)

2009 (1)

2005 (1)

H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy103153–164 (2005).
[CrossRef] [PubMed]

2004 (1)

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

2003 (1)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

1998 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

1985 (1)

1982 (2)

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

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

1972 (1)

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

Agard, D.

Alieva, T.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2007), 883–891.

Brady, G. R.

Calvo, M. L.

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

Cristbal, G.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Faulkner, H. M.

H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy103153–164 (2005).
[CrossRef] [PubMed]

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

Fienup, J.

Fienup, J. R.

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 (Stuttgart)35, 237–246 (1972).

Gonsalves, R. A.

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

Guizar-Sicairos, M.

Gustafsson, M.

Hanser, B.

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Rodenburg, J. M.

H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy103153–164 (2005).
[CrossRef] [PubMed]

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

Rodrigo, J. A.

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 (Stuttgart)35, 237–246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998).
[CrossRef]

Sedat, J.

Teague, M. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2007), 883–891.

Appl. Opt. (1)

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

Nature (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature400342–344 (1999).
[CrossRef]

Opt. Eng. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Optik (Stuttgart) (1)

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

Phys. Rev. Lett. (2)

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

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ultramicroscopy (1)

H. M. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy103153–164 (2005).
[CrossRef] [PubMed]

Other (2)

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 2007), 883–891.

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Figures (6)

Fig. 1
Fig. 1

Degree of spatial correlation c(0, z) between the field in focus and the field in any other plane in the focal region, as function of the dimensionless variable z NA2/λ, with NA = 0.3. This curve corresponds to the case of a field with uniform angular spectrum A(0)(p,q) within NA.

Fig. 2
Fig. 2

Degree of spatial correlation cint (0, z) between the intensity in focus and that in any other plane in the focal region, as function of the dimensionless variable zNA2/λ, with NA = 0.3. This curve corresponds to the case of a field with uniform angular spectrum A(0)(p,q) within NA.

Fig. 3
Fig. 3

Schematic of the experimental setup for through-focus phase retrieval.

Fig. 4
Fig. 4

Panel a: measured degree of spatial correlation cint (0, z) between the intensity in focus and that in any other plane in the focal region, as function of the dimensionless variable zNA2/λ. The curve has been generated by using actual through-focus intensity measurements of a field focused by a microscope objective with nominal NA = 0.4. In panel b we plot the ideal degree of spatial correlation for the intensity (blue line) and field (red line) for an aberration-free system of same NA illuminated by a uniform field distribution. Panel c and d contain the 2D plots of the two intensity distributions chosen for the phase reconstruction.

Fig. 5
Fig. 5

Measured intensities (first row) and retrieved phases (second row) of the field U(x,y,0) in focus and the field at position z = 7.83 μm that has a minimum degree of spatial correlation with U(x,y,0).

Fig. 6
Fig. 6

Comparison between a set of intensity measurements (first row) and the simulated ones. The simulated intensities have been obtained by propagating the field in the focal plane (U(x,y,0), not shown in the figure), at different distances. The field in focus is made of a measured amplitude and a retrieved phase. The excellent agreement between the measured and simulated intensity distributions confirms that the phase of the field in focus has been successfully reconstructed. The RMS deviation between measurements and simulations is equal to 0.68% at z = 4.04 μm, 0.38% at z = 7.83 μm, 0.47% at z = 11 μm and finally 0.39% at z = 15.17μm.

Tables (1)

Tables Icon

Table 1 Positions z, intensity correlation (cint (0, z)), signal to noise ratio (SNR) and root mean square error (RMSE) in the reconstruction of the field at z = −4.62μm for different intensity measurements with known NA

Equations (8)

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c ( 0 , z ) = U ( x , y , 0 ) , U ( x , y , z ) = U ( x , y , 0 ) U ( x , y , z ) d x d y | U ( x , y , 0 ) | 2 d x d y | U ( x , y , z ) | 2 d x d y .
c ( 0 , z ) = ( p , q ) 𝔇 | A ( 0 ) ( p , q ) | 2 exp ( i 2 π m z ) d p d q ( p , q ) 𝔇 | A 0 ( p , q ) | 2 d p d q .
A ( 0 ) ( p , q ) = U ( x , y , 0 ) exp [ i 2 π ( p x + q y ) ] d x d y .
m = ( 1 λ 2 p 2 q 2 ) .
U ( x , y , 0 ) , U ( x , y , z ) = U ( x , y , z ) , U ( x , y , 0 ) .
c ( 0 , z ) = ( p , q ) 𝔇 | A ( 0 ) ( p , q ) | 2 exp ( i 2 π m z ) d p d q ( p , q ) 𝔇 | A ( 0 ) ( p , q ) | 2 d p d q = 0 2 π 0 N A / λ exp ( i 2 π 1 / λ 2 ρ 2 z ) ρ d ρ d φ 0 2 π 0 N A / λ ρ d ρ d φ = λ 2 NA 2 π [ exp ( i 2 π z / λ ) 2 π z 2 ( 1 + i 2 π z λ ) exp ( i 2 π z 1 NA 2 / λ ) 2 π z 2 ( 1 + i 2 π z 1 NA 2 / λ ) ] .
c ( 0 , z ) | A ( 0 ) ( 0 , 0 ) | 2 exp ( i 2 π z / λ ) [ exp ( i 2 π z NA 2 / λ ) 1 ] i 2 λ z ( p , q ) 𝔇 | A 0 ( p , q ) | 2 d p d q .
c int ( 0 , z ) = I ( x , y , 0 ) , I ( x , y , z ) = I ( x , y , 0 ) I ( x , y , z ) d x d y | I ( x , y , 0 ) | 2 d x d y | I ( x , y , z ) | 2 d x d y .

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