Abstract

A quadratic cost functional for reconstruction of a high-resolution wave front from a coarse wave front is presented. The functional uses as data the position and the direction of the coarse wave front that had previously been computed with a ray-tracing method. This functional uses an optical relationship between the ray information and the wave front’s shape to reconstruct a high-density wave front. The performance of the proposed functional is illustrated by reconstruction of complicated wave fronts for which this functional has an accuracy that is superior to that of conventional interpolation methods.

© 2002 Optical Society of America

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References

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  1. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 2.3.
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  7. G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems,” Appl. Opt. 36, 5303–5309 (1997).
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  8. G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. II. Three-dimensional systems with symmetry,” Appl. Opt. 36, 5106–5111 (1998).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Secs. 3.2.1 and 3.1.2.
  10. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chaps. IX and XI.
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    [CrossRef]
  14. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 10.
  15. M. Bertero, P. Boccaci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, UK, 1998), Sec. 5.6.
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.
  17. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Sec. 6.4.
  18. R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
    [CrossRef]

1999

R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
[CrossRef]

1998

G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. II. Three-dimensional systems with symmetry,” Appl. Opt. 36, 5106–5111 (1998).
[CrossRef]

1997

1995

1987

1985

1982

1902

J. Hadamard, “Sur les probléms aux dérivées partielles et leur signification physique,” Princeton U. Bull. 13, (1902).

Andersen, A. H.

Arsenin, V. Y.

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Winston and Sons, Washington, D.C., 1977).

Beliakov, G.

G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. II. Three-dimensional systems with symmetry,” Appl. Opt. 36, 5106–5111 (1998).
[CrossRef]

G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems,” Appl. Opt. 36, 5303–5309 (1997).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero, P. Boccaci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, UK, 1998), Sec. 5.6.

Boccaci, P.

M. Bertero, P. Boccaci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, UK, 1998), Sec. 5.6.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Secs. 3.2.1 and 3.1.2.

Chan, D. Y. C.

G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. II. Three-dimensional systems with symmetry,” Appl. Opt. 36, 5106–5111 (1998).
[CrossRef]

G. Beliakov, D. Y. C. Chan, “Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems,” Appl. Opt. 36, 5303–5309 (1997).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.

Forbes, G. W.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 10.

Hadamard, J.

J. Hadamard, “Sur les probléms aux dérivées partielles et leur signification physique,” Princeton U. Bull. 13, (1902).

Legarda-Sáenz, R.

R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
[CrossRef]

Marroquin, J. L.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.

Rivera, M.

R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
[CrossRef]

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

Rodriguez-Vera, R.

R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 2.3.

Sharma, A.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chaps. IX and XI.

Stone, B. D.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 2.3.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.

Thikonov, A. N.

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Winston and Sons, Washington, D.C., 1977).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 10.

Vest, C. M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.

Vizia, D.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Secs. 3.2.1 and 3.1.2.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

R. Legarda-Sáenz, R. Rodriguez-Vera, M. Rivera, “Nonparaxial method for computing the gradient field of a wavefront using moiré deflectometry,” Opt. Commun. 160, 214–218 (1999).
[CrossRef]

Princeton U. Bull.

J. Hadamard, “Sur les probléms aux dérivées partielles et leur signification physique,” Princeton U. Bull. 13, (1902).

Other

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Winston and Sons, Washington, D.C., 1977).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 2.3.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 10.

M. Bertero, P. Boccaci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, UK, 1998), Sec. 5.6.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C., 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 3.6.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Sec. 6.4.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Secs. 3.2.1 and 3.1.2.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chaps. IX and XI.

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Figures (5)

Fig. 1
Fig. 1

Wave front that results from tracing rays through an inhomogeneous medium.

Fig. 2
Fig. 2

(a) Ray information. (b) Ray information and unknown values.

Fig. 3
Fig. 3

(a) Synthetic real wave front defined by Eq. (9), (b) coarse wave front, (c) wave front computed with Eq. (8), (d) MSE of (c), (e) wave front computed with bicubic interpolation, and (f) MSE of (e).

Fig. 4
Fig. 4

(a) Synthetic real wave front defined by Eq. (10), (b) coarse wave front, (c) wave front computed with Eq. (8), and (d) MSE of (c).

Fig. 5
Fig. 5

(a) Temperature distribution defined by Eq. (11) for z = 0 and centered at (0, 0, 0). (b) Wave front computed by tracing of a fine lattice of 81 × 81 rays through the inhomogeneous medium; time employed, 15,697 s. (c) Wave front computed by tracing of a coarse lattice of 9 × 9 rays through the inhomogeneous medium and then interpolating by use of Eq. (8); time employed, 194 s.

Equations (16)

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

ddsnrdrds=nr,
d2rdξ2=nrnr,
t=drdξ=nrdrds=nrcos α, cos β, cos γ.
Wu, v=Ru, v, Tu, vT,
Ru, v=Xu, v, Yu, v, Zu, vT,Tu, v=nu, vαu, v, βu, v, Yu, vT,
Tu, vRu, vu=0, Tu, vRu, vv=0.
Ri,jq=Xi,j, Yi,j, Zi,jT, Ti,jq=n0αi,j, βi,j, Yi,jT,
Ŵk,l=Rˆk,l, Tˆk,lT,
URˆk,l=k,lΩ |Rˆk,l-Rk,l|2× Sk,l+k+1,lk,lΩk-1,lTk,lRˆk+1,l-Rˆk-1,l22×Sk,l+k,l+1k,lΩk,l-1Tk,lRˆk,l+1-Rˆk,l-122×Sk,l+ρ k+1,lk,lΩk-1,l |Rˆk+1,l-2Rˆk,l+Rˆk-1,l|2×1-Sk,l+ρ k,l+1k,lΩk,l-1 |Rˆk,l+1-2Rˆk,l+Rˆk,l-1|21-Sk,l,
URˆk,lRˆk,l=0.
Rˆk,lt+1=Rˆk,lt-τ URˆk,ltRˆk,lt,
MSE=1M×Nk,l |Rk, l-Rˆk,l|2,
R1u, v=u, v, u exp-u2-v2T,
R2u, v=u-u33+uv2, u-u33+uv2, u2-v2T,
Tx, y, z=25+1001+y-y01.52exp-1.25×x-x02+z-z021/2,
n=1+0.29201×10-31+Temperature°C×0.368184×10-2.

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