Abstract

In this paper, we present a method to recover the complex amplitude of speckle fields from measurements performed by a shear interferometer. It is based on the optimization of an objective function using the steepest descent gradient technique in combination with a heuristic initial guess. In contrast to already existing methods, the algorithm finds a local minimum least-squares solution even in the presence of Poissonian and Gaussian noise.

© 2013 Optical Society of America

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References

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  1. W. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–950-2 (1947).
    [CrossRef]
  2. C. Falldorf, E. Kolenovic, and W. Osten, “Speckle shearography using a multiband light source,” Opt. Lasers Eng. 40, 543–552 (2003).
    [CrossRef]
  3. D. L. Fried, “Least-squares fitting a wavefront distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
    [CrossRef]
  4. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
    [CrossRef]
  5. M. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
    [CrossRef]
  6. D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
    [CrossRef]
  7. C. Falldorf, “Measuring the complex amplitude of wave fields by means of shear interferometry,” J. Opt. Soc. Am. A 28, 1636–1647 (2011).
    [CrossRef]
  8. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  9. J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  10. C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39, 5353–5359 (2000).
    [CrossRef]
  11. C. Falldorf, S. Osten, C. V. Kopylow, and W. Jüptner, “Shearing interferometer based on the birefringent properties of a spatial light modulator,” Opt. Lett. 34, 2727–2729 (2009).
    [CrossRef]
  12. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
    [CrossRef]
  13. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef]
  14. J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2011 (1)

2009 (1)

2003 (1)

C. Falldorf, E. Kolenovic, and W. Osten, “Speckle shearography using a multiband light source,” Opt. Lasers Eng. 40, 543–552 (2003).
[CrossRef]

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

2000 (1)

1997 (1)

1996 (1)

1994 (1)

1983 (1)

1977 (2)

D. L. Fried, “Least-squares fitting a wavefront distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

1947 (1)

W. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–950-2 (1947).
[CrossRef]

Baranova, N. B.

Bates, W.

W. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–950-2 (1947).
[CrossRef]

Elster, C.

Falldorf, C.

Fried, D. L.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

D. L. Fried, “Least-squares fitting a wavefront distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

Jüptner, W.

Kolenovic, E.

C. Falldorf, E. Kolenovic, and W. Osten, “Speckle shearography using a multiband light source,” Opt. Lasers Eng. 40, 543–552 (2003).
[CrossRef]

Kopylow, C. V.

Malacara, D.

Mamaev, A. V.

Marroquin, J. L.

Nocedal, J.

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

Osten, S.

Osten, W.

C. Falldorf, E. Kolenovic, and W. Osten, “Speckle shearography using a multiband light source,” Opt. Lasers Eng. 40, 543–552 (2003).
[CrossRef]

Pilipetsky, N. F.

Schnars, U.

Servin, M.

Shkunov, V. V.

Wright, S.

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

Yamaguchi, I.

Zel’dovich, B. Y.

Zhang, T.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

C. Falldorf, “Measuring the complex amplitude of wave fields by means of shear interferometry,” J. Opt. Soc. Am. A 28, 1636–1647 (2011).
[CrossRef]

D. L. Fried, “Least-squares fitting a wavefront distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. A 67, 370–375 (1977).
[CrossRef]

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 375–378 (1977).
[CrossRef]

U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
[CrossRef]

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Opt. Lasers Eng. (1)

C. Falldorf, E. Kolenovic, and W. Osten, “Speckle shearography using a multiband light source,” Opt. Lasers Eng. 40, 543–552 (2003).
[CrossRef]

Opt. Lett. (2)

Proc. Phys. Soc. London (1)

W. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. London 59, 940–950-2 (1947).
[CrossRef]

Other (2)

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

J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (11)

Fig. 1.
Fig. 1.

Space of solutions [9] for two sets C1 and C2, which in this case correspond to two measurements with varying shear. The global minimum is found at position G. However, any gradient-based approach can become trapped at a local minimum, such as the one indicated by point L, because the sets are nonconvex.

Fig. 2.
Fig. 2.

Simulated speckle wave field: (a) amplitude and (b) phase distribution showing a large number of vortices. The average speckel size is four sampling points, whereas the total size of the distributions is 256×256 sampling points.

Fig. 3.
Fig. 3.

Calculated observation M1: (a) amplitude and (b) phase distribution. For the simulation, an 8-bit camera sensor with full well capacity of 500 electrons and a four-frame phase-shifting algorithm with 90° phase shift between the frames were assumed.

Fig. 4.
Fig. 4.

Reconstructed phase distribution of the speckle field obtained after (a) m=100, (b) m=1000, and (c) m=10,000 iterations of Eq. (7). The bottom row shows the respective difference between the estimated phase distribution and the true phase. A uniform random distribution was used as initial guess.

Fig. 5.
Fig. 5.

Reconstructed phase distribution of the speckle field obtained after (a) m=10, (b) m=30, and (c) m=150 iterations along with the difference from the ideal phase. A Kronecker delta function with the peak x⃗0 located in the center was used as the initial guess, and the regularization parameter was set to γ=4. The standard deviation of the phase estimation yields σϕ=0.13rad.

Fig. 6.
Fig. 6.

(a) Reconstructed amplitude distribution of the speckle field obtained after m=150 iterations along with (b) the difference from the ideal amplitude in normalized units. The standard deviation of the difference yields σa=0.016. The largest deviations are found close to the corners because the observations only provide limited information about the wave field in these regions.

Fig. 7.
Fig. 7.

Shear interferometer used for the experiments. It is based on a 4f configuration with a spatial light modulator in the Fourier domain as the shearing element. The focal length of the lenses was 150 mm, and the distance between the object and the input plane of the interferometer was 130 mm. The wave field scattered by the object is measured across the input plane by means of CoSI. For comparison, the beam splitter is inserted to provide a reference wave and the same wave field can be investigated using standard phase-shifting DH. Once the wave field is determined across the input plane, the object can be focused by numerically propagating the light in a plane close to the object. The light source (not shown) was a Nd:YAG laser with a wavelength of 532 nm, and the pixel pitch of the camera was 3.45 μm. The object was the die shown in the inset. Its size is 8 mm in any direction. Further details are found in the text.

Fig. 8.
Fig. 8.

Measured observation M1: (a) amplitude and (b) phase distribution. The shear was set to 17 sensor pixels in the horizontal direction. The inset shows the detail of the phase. The size of the distributions is 680×680 pixels with a pixel pitch of 3.45 μm.

Fig. 9.
Fig. 9.

Phase distribution determined using CoSI: (a) phase after 150 iterations, γ=2, and a Kronecker delta function as the initial guess and (b) comparison with the results obtained by standard interferometry. All vortices have been correctly identified. The standard deviation of the difference is σϕ=0.68rad. The size of the distributions is 680×680 pixels.

Fig. 10.
Fig. 10.

Comparison of the amplitude distributions obtained from (a) CoSI and (b) standard interferometry. The drop of the amplitude toward the lower right corner of (b) is an artifact of the reference wave. The size of the distributions is 680×680 pixels.

Fig. 11.
Fig. 11.

Demonstration of numerical focusing based on the wave fields determined by (a) CoSI and (b) standard interferometry. We used plane wave decomposition to calculate the intensity of the respective wave field across a plane close to the object’s surface. Across the areas bounded by the rectangles, the brightness is changed in order to visualize the structure of the noise. The size of the distributions is 680×680 pixels.

Equations (18)

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u=a·exp(iϕiωt),
|u+uR|2=|u|2+|uR|2+2aaRcos(ϕϕR).
IS(x⃗)=|u(x⃗)|2+|u(x⃗+s⃗)|2+2R{u*(x⃗)u(x⃗+s⃗)}.
Mn=u*(x⃗)·u(x⃗+s⃗n)
L=nMn(x⃗)f*(x⃗)f(x⃗+s⃗n)2,
f˜(x⃗)=argminf{L}.
f(m+1)(x⃗)=f(m)(x⃗)α(m)·L(m)(x⃗),
L(m)(x⃗)=Lf(m)(x⃗).
L(m)(x⃗)=2nf(m)(x⃗+s⃗n)·ψn(m)*(x⃗)+f(m)(x⃗s⃗n)·ψn(m)(x⃗s⃗n),
f˜(x⃗)=nf˜(x⃗+s⃗n)M*(x⃗)+f˜(x⃗s⃗n)M(x⃗s⃗n)nI˜(x⃗+s⃗n)+I˜(x⃗s⃗n).
g*(x⃗)g(x⃗+s⃗)=M(x⃗)[α2+(1α)2+α(1α)·(fϕ(x⃗+s⃗)+fϕ*(x⃗))],
LS=nMn(x⃗)f*(x⃗)f(x⃗+s⃗n)2+γΔDf(x⃗)2.
LS(m)(x⃗)=LSf(m)(x⃗)=L(m)(x⃗)+2γΔD{ΔDf(m)(x⃗)},
f(0)(x⃗;x⃗0)={1forx⃗=x⃗00elsewhere.
L(x⃗)=Lf(x⃗)=Lr(x⃗)+iLq(x⃗).
L(x⃗0)=(r(x⃗0)+iq(x⃗0))nx⃗R[Mn(x⃗)f*(x⃗)f(x⃗+s⃗n)][Mn(x⃗)f*(x⃗)f(x⃗+s⃗n)]*.
L(x⃗0)=(r(x⃗0)+iq(x⃗0))n[ψn(x⃗0)ψn*(x⃗0)+ψn(x⃗0s⃗n)ψn*(x⃗0s⃗n)],
L(x⃗0)=n[2ψn*(x⃗0)f(x⃗0+s⃗n)2ψn(x⃗0s⃗n)f(x⃗0s⃗n)].

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