Abstract

A new zonal wavefront reconstruction method for lateral shearing interferometry was presented. The proposed algorithm allows shear amounts equal to arbitrary integral multiple of the sample intervals. High spatial resolution reconstruction is achieved with only two difference wavefronts measured in orthogonal shear directions. The presented algorithm was generalized to be applicable for general aperture shape by using zero padding and Gerchberg-type iterative methods. The capability of the presented algorithm was demonstrated by some numerical examples. Also, the reconstruction error was analyzed theoretically and numerically.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).
  2. J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).
  3. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef]
  4. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  5. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  6. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. J. Herrmann, “Least-square wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  9. D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef]
  10. X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
    [CrossRef]
  11. H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  12. H. Takajo and T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  13. W. Zou and Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
    [CrossRef]
  14. W. Zou and J. P. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with generalshaped pupils,” J. Opt. Soc. Am. A 22, 938–951 (2005).
    [CrossRef]
  15. C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
    [CrossRef]
  16. C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16, 2281–2285 (1999).
    [CrossRef]
  17. C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39, 5353–5359 (2000).
    [CrossRef]
  18. Z. Yin, “Exact wavefront recovery with tilt from lateral shear interferograms,” Appl. Opt. 48, 2760–2766 (2009).
    [CrossRef]
  19. T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
    [CrossRef]
  20. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  21. W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. A 23, 2629–2638 (2006).
    [CrossRef]
  22. G.-m. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
  23. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  24. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673(1987).
    [CrossRef]
  25. F. Roddier and C. Roddier, “Wave-front reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327(1991).
    [CrossRef]
  26. J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI, Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

2009 (2)

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
[CrossRef]

Z. Yin, “Exact wavefront recovery with tilt from lateral shear interferograms,” Appl. Opt. 48, 2760–2766 (2009).
[CrossRef]

2006 (1)

2005 (1)

2002 (1)

2000 (2)

1999 (2)

1991 (1)

1989 (1)

1988 (2)

1987 (1)

1980 (2)

1979 (1)

1978 (1)

1977 (2)

1974 (1)

1972 (1)

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

1961 (1)

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).

Chang, M.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
[CrossRef]

Creath, K.

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI, Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Dai, G.-m.

G.-m. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).

Elster, C.

Fried, D. L.

Gao, Y.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
[CrossRef]

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 35, 237–246 (1972).

Ghiglia, D. C.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Kamiya, K.

Liu, X.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).

Noll, R. J.

Nomura, T.

Okuda, S.

Rimmer, M. P.

Roddier, C.

Roddier, F.

Rolland, J. P.

Romero, L. A.

Saunders, J. B.

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).

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 35, 237–246 (1972).

Southwell, W. H.

Takahashi, T.

Takajo, H.

Tashiro, H.

Weingärtner, I.

Wyant, J. C.

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI, Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Yin, Z.

Yoshikawa, K.

Zhang, Z.

Zou, W.

Appl. Opt. (8)

J. Opt. A Pure Appl. Opt. (1)

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A Pure Appl. Opt. 11, 045702 (2009).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (5)

J. Res. Natl. Bur. Stand. Sect. B (1)

J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).

Opt. Lett. (1)

Optik (1)

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

Other (3)

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI, Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).

G.-m. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Schematic of (a) original square wavefront, (b) shearing in x direction, (c) shearing in y direction, (d) difference wavefront in x direction, and (e) difference wavefront in y direction.

Fig. 2.
Fig. 2.

Schematic of the subgrid at which the values of the test wavefront are estimated first and are served as initial conditions.

Fig. 3.
Fig. 3.

Schematic of (a) extension of the test circular wavefront, (b) shearing in x direction, (c) shearing in y direction, (d) extension of the difference wavefront in x direction, and (e) extension of the difference wavefront in y direction.

Fig. 4.
Fig. 4.

Flow chart of Gerchberg-type iterative least-squares wavefront estimation from two orthogonal difference wavefronts.

Fig. 5.
Fig. 5.

Simulation conditions: (a) Zernike coefficients of the test wavefront and (b) the test wavefront.

Fig. 6.
Fig. 6.

Reconstructed wavefront under the condition that the initial values on the subgrid are the same as the test wavefront at the pairs (a) Sx=1, Sy=1; (b) Sx=4, Sy=6; (c) Sx=15, Sy=13, and (d) Sx=24, Sy=25; (e) to (f) their corresponding reconstruction errors.

Fig. 7.
Fig. 7.

Reconstructed wavefront under the condition that the initial values on the subgrid are determined by linear interpolation at the pairs (a) Sx=1, Sy=1; (b) Sx=4, Sy=6; (c) Sx=15, Sy=13, and (d) Sx=24, Sy=25. Their corresponding relative RMS reconstruction errors (e) r=0; (f) r=0.57%; (g) r=2.73%, and (h) r=5.38%.

Fig. 8.
Fig. 8.

Relative RMS reconstruction error as a function of the iterations at some pairs of the two shear amounts.

Tables (1)

Tables Icon

Table 1. Error Propagation Coefficients η for Some Pairs of Shear Amounts under the Sample Sizes of N=64, N=128, and N=256

Equations (47)

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

Wi=[Wi1Wi2WiN].
ΔWix=[ΔWi1xΔWi2xΔWi,NSxx].
ΔWiy=[ΔWi1yΔWi2yΔWiNy].
W=[W1W2WN].
ΔWx=[ΔW1xΔW2xΔWNx].
ΔWy=[ΔW1yΔW2yΔWNSyx].
ΔWi1x=Wi,1+SxWi1ΔWi2x=Wi,2+SxWi2ΔWi,NSxx=WiNWi,NSx}.
ΔWix=MxWi,
Mx[m,n]={1,m=n1,m=nSx0,otherwise,
[ΔW1xΔW2xΔWNx]=[Mx0Mx0Mx][W1W2WN].
MxW=ΔWx,
Mx=[Mx0Mx0Mx].
ΔWi1y=Wi+Sy,1Wi,1ΔWi2y=Wi+Sy,2Wi,2ΔWi,Ny=Wi+Sy,NWi,N}.
ΔWiy=I×Wi+SyI×Wy,
ΔW1y=I×W1+SyI×W1ΔW2y=I×W2+SyI×W2ΔWNSyy=I×WNI×WNSy}.
MyW=ΔWy,
My[m,n]={1,m=n1,m=nNSy0,otherwise,
MW=ΔW,
ΔW=[ΔWxΔWy]andM=[MxMy].
MTMW=MTΔW,
rank(MTM)=rank(M)=N2SxSy.
HW=W˜s,
Hij={1,j=[r+(k2)]×N+c+(l1)0,otherwise,
MeW=ΔWe,
Me=[MH]andΔWe=[ΔWW˜s].
r=c={N/2,Nis even(N+1)/2,Nis odd.
ΔWrcx=Wr,c+SxWrcΔWrcy=Wr+Sy,cWrc}.
W˜r+i,c+j=ΔWrcySy×i+ΔWrcxSx×j,
W˜is=[W˜r+i,c+1W˜r+i,c+2W˜r+i,c+Sx],
W˜s=[W˜1sW˜2sW˜Sys].
W^=(MeTMe)1MeTΔWe.
ΔW^x=MxW^ΔW^y=MyW^},
r=RMSof(WrWΔP)/RMSofW×100%,
δWr+i,c+j=Wr+i,c+jW˜r+i,c+j,
W+ε=(MeTMe)1[MTHT][εWWs+e],
ε=(MeTMe)1HTe.
σW2=j=1N2εj2/N2=tr(εεT)/N2,
εεT=(MeTMe)1HTeeTH[(MeTMe)1]T.
eeT=σ2I,
η=σW2/σ2=tr{(MeTMe)1HTH[(MeTMe)1]T}/N2,
Mx=[100100001001000101](NSx)×N.
My=[IIII](NSy)×N,
M=[MxMx(1+Sy)thcolumnMx(1+Sy)throwMxII(N+1)throwII].
M[MxMx(1+Sy)thcolumnMxMx(1+Sy)throwMxMx0I(N+1)throw0I].
M[MxMx(1+Sy)thcolumn0Mx(1+Sy)throw0Mx0I(N+1)throw0I].
M[MxMx(1+Sy)thcolumn00(1+Sy)throw000I(N+1)throw0I].
rank(M)=Sy×rank(Mx)+(NSy)×N=Sy×(NSx)+(NSy)×N=N2SxSy.

Metrics