Abstract

This paper presents a technique for reconstructing two interfering wavefronts by analyzing their 3D interference field pattern. The method is based on the numerical inverse problem and will present a robust algorithm for reconstructing of wavefronts. Several simulations are done to validate the proposed method.

© 2011 OSA

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  2. D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).
  3. R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
  4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
  5. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).
  6. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
    [CrossRef]
  7. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
    [CrossRef]
  8. J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
    [CrossRef]
  9. 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).
  10. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996).
    [CrossRef] [PubMed]
  11. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001).
    [CrossRef] [PubMed]
  12. R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
    [CrossRef]
  13. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993).
    [CrossRef] [PubMed]
  14. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006).
    [CrossRef] [PubMed]
  15. R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express 15(7), 3876–3887 (2007).
    [CrossRef] [PubMed]
  16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000).
    [CrossRef]
  17. A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001).
    [CrossRef]
  18. A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).
  19. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).
  20. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006).
    [CrossRef] [PubMed]
  21. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
    [CrossRef]
  22. M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.
  23. http://www.zemax.com

2007

2006

2001

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001).
[CrossRef] [PubMed]

A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001).
[CrossRef]

2000

M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000).
[CrossRef]

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

1996

1993

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

1983

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[CrossRef]

1972

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).

1971

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Ai, C.

Baade, T.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Darudi, A.

A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001).
[CrossRef]

M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000).
[CrossRef]

de la Fuente, R.

Dorsch, R. G.

Esselbach, M.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Freischlad, K. R.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[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).

Griesmann, U.

Kiessling, A.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Kowarschik, R.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

López Lago, E.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Mendlovic, D.

Notni, G.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Platt, B. C.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Ragazzoni, J.

J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
[CrossRef]

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).

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Shen, F.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

Tavassoly, M. T.

A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001).
[CrossRef]

M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000).
[CrossRef]

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[CrossRef]

Uhlendorf, K.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Wang, A.

Wenke, L.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

Wyant, J. C.

Zalevsky, Z.

Appl. Opt.

Appl. Phys. B

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999).
[CrossRef]

J. Mod. Opt.

J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996).
[CrossRef]

J. Opt. Soc. Am.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. Opt. Soc. Am. A

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[CrossRef]

Opt. Commun.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[CrossRef]

M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001).
[CrossRef]

Opt. Lett.

Optik

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

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).

J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.

http://www.zemax.com

A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).

M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

Supplementary Material (2)

» Media 1: MOV (1271 KB)     
» Media 2: MOV (857 KB)     

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of the two interfering waves at planes z = z 1 and z = z 2 .

Fig. 2
Fig. 2

(a) Phase distribution of a wavefront that is generated by ZEMAX. (b) Propagated wavefront of Fig. 2(a) by 15cm distance. (c) The difference in phase distribution of the propagated wavefront by our algorithm and by ZEMAX.

Fig. 3
Fig. 3

(a) phase distribution of a −1m-radius spherical wave. (b) Phase distribution of a 4m-radius cylindrical wave. (c) Interference patterns of spherical and cylindrical waves at first observation plane (Media 1). (d) Difference of estimated and simulated phase distribution.

Fig. 4
Fig. 4

(a) phase distribution of a −2m-radius cylindrical wave. (b) Phase distribution of a −1m-radius cylindrical wave. (c) Interference patterns of two cylindrical waves at first observation plane (Media 2). (d) Difference of estimated and simulated phase distribution.

Equations (7)

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U j , z ( x , y ) = A j , z ( x , y ) e i ϕ j , z ( x , y ) j = 1 , 2
I z j ( x , y ) = A 1 z j 2 ( x , y ) + A 2 z j 2 ( x , y ) + 2 A 1 z j ( x , y ) A 2 z j ( x , y ) × cos ( Δ ϕ z j ) , j = 1 , 2
Δ ϕ z j = ϕ 2 , z j ( x , y ) ϕ 1 , z j ( x , y ) , j = 1 , 2 .
ϕ 2 z 1 = ϕ 1 z 1 + Δ ϕ z 1 ( exp ) .
Δ ϕ z 2 ( cal ) = ϕ 2 z 2 ϕ 1 z 2 .
ϕ z 2 ( x , y ) = ϕ z 1 ( x , y ) + Δ ϕ ( x x , y y , z 2 z 1 )     ,
E = i , j | [ e ] i , j | N 2     , [ e ] i , j = [ Δ ϕ z 2 ( cal ) ] i , j [ Δ ϕ z 2 ( exp ) ] i , j , i and j = 1... N .

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