Abstract

We demonstrate the use of transverse translation-diverse phase retrieval as a method for the measurement of wavefronts in situations where the detected intensity patterns would be otherwise undersampled. This technique involves using a smaller moving subaperture to produce a number of adequately sampled intensity patterns. The wavefront is then retrieved using an optimization jointly constrained by them. Expressions for the gradient of an error metric with respect to the optimization parameters are given. An experimental arrangement used to measure the transmitted wavefront of a plano-convex singlet using this technique is described. The results of these measurements were repeatable to within approximately λ/100 RMS.

© 2009 Optical Society of America

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  1. G. R. Brady and J. R. Fienup, "Range of Phase Retrieval in Optical Metrology," in Frontiers in Optics2005 / Laser Science XXI (Optical Society of America, Washington DC, 2005), paper FTuS3.
  2. J. R. Fienup, "Phase Retrieval for Undersampled Broadband Images," J. Opt. Soc. Am. A 16, 1831-1839 (1999).
    [CrossRef]
  3. P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).
  4. M. Bray, "Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics]," Proc. SPIE 3047, 911-18 (1997).
  5. M. Guizar-Sicairos and J. R. Fienup, "Phase retrieval with transverse translation diversity: a nonlinear optimization approach," Opt. Express 16, 7264-78 (2008).
    [CrossRef] [PubMed]
  6. 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] [PubMed]
  7. J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
    [CrossRef]
  8. J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
    [CrossRef]
  9. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
    [CrossRef] [PubMed]
  10. H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
    [CrossRef] [PubMed]
  11. M. Guizar-Sicairos and J. R. Fienup, "Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity," submitted toOpt. Express.
  12. J. R. Fienup, "Phase Retrieval Algorithms: A Comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  13. J. R. Fienup, "Phase-Retrieval Algorithms for a Complicated Optical System," Appl. Opt. 32, 1737-1746 (1993).
    [CrossRef] [PubMed]
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986), Chap. 10.
  15. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, "Efficient Subpixel Image Registration Algorithms," Opt. Lett. 33, 156-158 (2008).
    [CrossRef] [PubMed]

2008 (2)

2007 (2)

J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

2005 (1)

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

2004 (2)

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] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

2003 (1)

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

1999 (1)

1997 (1)

M. Bray, "Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics]," Proc. SPIE 3047, 911-18 (1997).

1993 (1)

1982 (1)

Bray, M.

M. Bray, "Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics]," Proc. SPIE 3047, 911-18 (1997).

Bunk, O.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

David, C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Dumas, P.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

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] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

Fienup, J. R.

Fleig, J.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

Forbes, G.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

Guizar-Sicairos, M.

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Murphy, P. E.

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

Pfeiffer, F.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

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] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

Thurman, S. T.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004).
[CrossRef]

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

Opt. Express (1)

Opt. Express. (1)

M. Guizar-Sicairos and J. R. Fienup, "Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity," submitted toOpt. Express.

Opt. Lett. (1)

Phys. Rev. Lett. (2)

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] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007).
[CrossRef] [PubMed]

Proc. SPIE (2)

P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).

M. Bray, "Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics]," Proc. SPIE 3047, 911-18 (1997).

Ultramicroscopy (2)

H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005).
[CrossRef] [PubMed]

J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007).
[CrossRef]

Other (2)

G. R. Brady and J. R. Fienup, "Range of Phase Retrieval in Optical Metrology," in Frontiers in Optics2005 / Laser Science XXI (Optical Society of America, Washington DC, 2005), paper FTuS3.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986), Chap. 10.

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Figures (11)

Fig. 1.
Fig. 1.

Experimental arrangement for phase retrieval with subaberture tranverse translation diversity. (a) Perspective view of the arrangement. (b) Without tranverse translation diversity, if the numerical aperture of the lens is too large the intensity pattern at the CCD will be undersampled. (c) With translation diversity, the numerical aperture of the beam is reduced by truncating it with a smaller subaperture. A larger area is mapped out by making multiple subaperture measurements, translating the subaperture between each.

Fig. 2.
Fig. 2.

Diagram of the experimental arrangement used to collect subaperture intensity pattern data (not to scale). The CCD camera is placed near the focus. The subaperture and CCD are mounted on computer-controlled translation stages.

Fig. 3.
Fig. 3.

A portion of the experimental setup including the lens that forms the wavefront of interest, the moving subaperture, the CCD camera, and the associated motion control equipment. The illumination point source is out of the frame to the left.

Fig. 4.
Fig. 4.

An image of the subaperture from a flatbed scanner. The diameter is 9.85 mm.

Fig. 5.
Fig. 5.

Double pinhole interference pattern that was analyzed to determine the distance between the pinhole (subaperture) plane and the CCD camera plane.

Fig. 6.
Fig. 6.

A 9.85 mm subaperture is moved to 43 positions to measure a composite aperture with a diameter of 33.32 mm. The aperture postions are shown above, with the grey level indicating how many times a given point in the composite aperture was sampled by a subaperture.

Fig. 7.
Fig. 7.

The left column indicates the subaperture position of the measured intensity patterns (center column) and retrieved intensity patterns (right column). Intensities are shown raised to the 0.4 power.

Fig. 8.
Fig. 8.

Amplitude (left) and phase (right) retrieved over the composite aperture using the intensity patterns measured at the 43 subaperture positions shown in Fig. 6. The phase shown has tip, tilt and focus removed.

Fig. 9.
Fig. 9.

Pattern of subaperture positions for a smaller data set taken to check the phase retrieval results. The color scale shows the number of times a particular point is sampled by a subaperture. The outer white line indicates the edge of the larger data set.

Fig. 10.
Fig. 10.

(a) Wavefront of Fig. 8 cropped to the size of the small composite aperture pattern. Note that some artifacts are visible at the edge of the subapertures on the scale of this figure as compared to that of Fig. 8. (b) Wavefont retrieved using independent data collected over the small composite aperture shown in Fig. 9. (c) Difference between (a) and (b). Tip, tilt and piston have been removed from each, and all are on the same color scale.

Fig. 11.
Fig. 11.

Data of Fig. 10 fit to 36 Zernike polynomials. (a) Cropped and Zernike fit version of the wavefront in Fig. 8. (b) Zernike fit version of the phase shown in Fig. 10 (b). (c) Difference between (a) and (b). Tip, tilt and piston have been removed from each, and all are on the same color scale.

Equations (50)

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

f/#Qduλ
NAλ2Qdu,
g˜n(x,y)=h(x,y)a(xxn,yyn),
gn(x,y)=g˜n(x+xn,y+yn)=h(x+xn,y+yn)a(x,y),
Gn(u,v)=P [gn(x,y)],
Gnf(u,v)=exp{λz[(vdu)2+(vdv)2]}DFT[gn(x,y)],
Gn (u,v)=IDFT [DFT[Gnf(u,v)]exp{i2πΔz[1λ2(rNdu)2(sNdv)2]12}] ,
du,v=λzNdx,y .
E=n=1q u,v wn (u,v)[Fn(u,v)Gn(u,v)]2,
L=s2{tan[arcsin(λν2)]}
Eα=n=1q u,v Gnw* (u,v)Gn(u,v)α+c.c.
Gnw(u,v)=Wn (u,v)[Fn(u,v)Gn(u,v)Gn(u,v)Gn(u,v)],
Eα=n=1q u,v Gnw* (u,v)P [α{h(x+xn,y+yn)a(x,y)}]+c.c.
Eα=n=1q x,y {P[Gnw(u,v)]}* α [h(x+xn,y+yn)a(x,y)]+c.c.,
gnw(x,y)=P[Gnw(u,v)]
Eα=n=1q x,y gnw* (x,y) α [h(x+xn,y+yn)a(x,y)]+c.c.
Eα=n=1q x,y gnw* (xxn,yyn) α [h(x,y)a(xxn,yyn)]+c.c.
h(x,y)=h(x,y)exp[iθh(x,y)].
θh(x,y)[h(x,y)a(xxn,yyn)]=ia(xxn,yyn)h(x,y)δ(x,x;y,y).
Eθh(x,y)=in=1qgnw*(xxn,yyn) h (x,y) a xxn,yyn)+c.c.
=2Im[n=1qgnw*(xxn,yyn)h(x,y)a(xxn,yyn) ] .
h(x,y)=h(x,y)exp[ikckhZk(x,y)],
ckh[h(x,y)a(xxn,yyn)]=ia(xxn,yyn)h(x,y)Zk(x,y).
ckh=2Im[n=1qx,ygnw*(xxn,yyn)a(xxn,yyn)h(x,y)Zk(x,y)].
h(x,y)[h(x,y)a(xxn,yyn)]=a(xxn,yyn)exp[iθh(x,y)]δ(x,x;y,y)
h(x,y)=2Re[n=1qgnw*(xxn,yyn)a(xxn,yyn)exp[iθh(x,y)]]
h(x,y)=hR(x,y)+ihI (x,y) .
hR(x,y)[h(x,y)a(xxn,yyn)]=a(xxn,yyn)δ(x,x;y,y),
EhR(x,y)=n=1q gnw* (xxn,yyn) a (xxn,yyn)+c.c.
EhI(x,y)=in=1q gnw* (xxn,yyn) a (xxn,yyn)+c.c.
EhR(x,y)+iEhI(x,y)=2n=1q gnw (xxn,yyn) a* (xxn,yyn).
a(x,y)=a(x,y)exp[iθa(x,y)]
Eθa(x,y)=2Im[n=1qgnw*(x,y)h(x+xn,y+yn)a(x,y)].
an(x,y)=a(x,y)exp[ik(cka+ck,na)Zk(x,y)].
cka[h(x+xn,y+yn)a(x,y)]=ih (x+xn,y+yn) an (x,y)Zk (x,y).
∂Ecka =2Im[n=1qx,ygnw*(x,y)h(x+xn,y+yn)an(x,y)Zk(x,y)].
ck',na[h(x+xn,y+yn)a(x,y)]=ih(x+xn,y+yn)an(x,y)Zk'(x,y)δ(n,n).
∂Eck,na =2Im[x,ygnw*(x,y)h(x+xn',y+yn)an'(x,y)Zk(x,y)].
a(x,y)[h(x+xn,yyn)a(x,y)]=h(x+xn,y+yn)exp[iθa(x,y)]δ(x,x;y,y)
a(x,y)=2Re[n=1qgnw*(x,y)h(x+xn,y+yn)exp[iθa(x',y')]]
a(x,y)=aR(x,y)+iaI(x,y).
aR(x,y)[h(x+xn,y+yn)a(x,y)]=h(x+xn,y+yn)δ(x,x;y,y),
EaR(x',y')=n=1q gnw* (x,y)h(x+xn,y+yn) +c.c.
EaI(x',y')=in=1q gnw* (x,y)h(x+xn,y+yn) +c.c.
EaR(x,y)+iEaI(x,y)=2n=1qgnw (x',y')h*(x+xn,y+yn).
xn[h(x+xn,y+yn)a(x,y)]=ia(x,y)xnh(x+xn,y+yn),
h(x+xn,y+yn)=IDFT [exp[i2π(xnuM+ynvM)]DFT[h(x,y)]] .
xn′h(x+xn,y+yn)=IDFT[i2πuMexp[i2π(xnuM+ynvM)]DFT[h(x,y)]]δ(n,n).
Exn=4πMIm{x,ygnw*(x,y)a(x,y)IDFT[uexp[i2π(xn′uM+yn′vM)]DFT[h(x,y)]]} .
Eyn=4πMIm{x,ygnw*(x,y)a(x,y)IDFT[vexp[i2π(xn′uM+yn′vM)]DFT[h(x,y)]]} .

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