Abstract

We show through numerical simulations and experimental data that a fast and simple iterative loop known as the Fienup algorithm can be used to process the measured Maker-fringe curve of a nonlinear sample to retrieve the sample’s nonlinearity profile. This algorithm is extremely accurate for any profile that exhibits one or two dominant peaks, which covers a wide range of practical profiles, including any nonlinear film of crystalline or organic material (rectangular profiles) and poled silica, for which an excellent experimental demonstration is provided. This algorithm can also be applied to improve the accuracy of the nonlinearity profile obtained by an inverse Fourier transform technique.

© 2004 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [Crossref]
  2. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
    [Crossref]
  3. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
    [Crossref]
  4. C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
    [Crossref]
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  6. T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 29, 1187–1193 (1981).
    [Crossref]
  7. M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
    [Crossref]
  8. V. Oppenheim and R. W. Schafer, Digital Signal Processing, (Prentice Hall, 2002), Chap. 7.
  9. R. A. Myers, N. Mukerjkee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16, 1732–1734 (1991).
    [Crossref] [PubMed]
  10. Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
    [Crossref]
  11. A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
    [Crossref]
  12. A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”
  13. M. Mukherjee, R. A. Myers, and S. R. J. Brueck, “Dynamics of second-harmonic generation in fused silica,” J. Opt. Soc. Am. B 11, 665–669 (1994).
    [Crossref]
  14. T. G. Alley, S. R. J. Brueck, and R. A. Myers, “Space charge dynamics in thermally poled fused silica,” J. Non-Cryst. Solids. 242, 165–176 (1998).
    [Crossref]
  15. D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
    [Crossref]

2004 (1)

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
[Crossref]

2003 (4)

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
[Crossref]

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
[Crossref]

2001 (1)

D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
[Crossref]

1998 (1)

T. G. Alley, S. R. J. Brueck, and R. A. Myers, “Space charge dynamics in thermally poled fused silica,” J. Non-Cryst. Solids. 242, 165–176 (1998).
[Crossref]

1994 (1)

1991 (1)

1981 (1)

T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 29, 1187–1193 (1981).
[Crossref]

1980 (1)

M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
[Crossref]

1978 (1)

1962 (1)

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Alley, T. G.

T. G. Alley, S. R. J. Brueck, and R. A. Myers, “Space charge dynamics in thermally poled fused silica,” J. Non-Cryst. Solids. 242, 165–176 (1998).
[Crossref]

Brueck, S. R. J.

Corbari, C.

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

Deparis, O.

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

Digonnet, M. J. F.

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”

Douay, M.

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

Faccio, D.

D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
[Crossref]

Fienup, J. R.

Ginzton, Edward L.

A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”

Hayes, M.

M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
[Crossref]

Kazansky, P. G.

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
[Crossref]

Kino, G. S.

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”

Klappauf, B. G.

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

Lim, J. S.

M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
[Crossref]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Mukerjkee, N.

Mukherjee, M.

Myers, R. A.

Niay, P.

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

Nisenhoff, M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Oppenheim, A. V.

T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 29, 1187–1193 (1981).
[Crossref]

M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
[Crossref]

Oppenheim, V.

V. Oppenheim and R. W. Schafer, Digital Signal Processing, (Prentice Hall, 2002), Chap. 7.

Ozcan, A.

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”

Poumellec, B.

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

Pruneri, V.

D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
[Crossref]

Quatieri, T. F.

T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 29, 1187–1193 (1981).
[Crossref]

Quiquempois, Y.

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Schafer, R. W.

V. Oppenheim and R. W. Schafer, Digital Signal Processing, (Prentice Hall, 2002), Chap. 7.

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Appl. Phys. Lett. (3)

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Inverse Fourier transform technique to determine second-order optical nonlinearity spatial profiles,” Appl. Phys. Lett. 82, 1362–1364 (2003).
[Crossref]

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Improved technique to determine second-order optical nonlinearity profiles using two different samples,” Appl. Phys. Lett. 84, 681–683 (2004).
[Crossref]

D. Faccio, V. Pruneri, and P. G. Kazansky, “Dynamics of the second-order nonlinearity in thermally poled silica glass,” Appl. Phys. Lett. 79, 2687–2689 (2001)
[Crossref]

Current Opinion in Solid State & Materials Science (1)

Y. Quiquempois, P. Niay, M. Douay, and B. Poumellec, “Advances in poling and permanently induced phenomena in silica-based glasses,” Current Opinion in Solid State & Materials Science 7, 89–95 (2003)
[Crossref]

Electron. Lett. (2)

A. Ozcan, M. J. F. Digonnet, and G. S. Kino, “Cylinder-assisted Maker-fringe technique,” Electron. Lett. 39, 1834–1836 (2003).
[Crossref]

C. Corbari, O. Deparis, B. G. Klappauf, and P. G. Kazansky, “Practical technique for measurement of second-order nonlinearity for poled glass,” Electron. Lett. 39, 197–198 (2003).
[Crossref]

IEEE Trans. Acoust., Speech, Signal Processing (2)

T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 29, 1187–1193 (1981).
[Crossref]

M. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Processing 28, 672–680 (1980).
[Crossref]

J. Non-Cryst. Solids. (1)

T. G. Alley, S. R. J. Brueck, and R. A. Myers, “Space charge dynamics in thermally poled fused silica,” J. Non-Cryst. Solids. 242, 165–176 (1998).
[Crossref]

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[Crossref]

Other (2)

V. Oppenheim and R. W. Schafer, Digital Signal Processing, (Prentice Hall, 2002), Chap. 7.

A. Ozcan, M. J. F. Digonnet, G. S. Kino, and Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, are preparing a manuscript to be called “Detailed analysis of inverse Fourier transform techniques to uniquely infer second-order nonlinearity profile of thin films.”

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Figures (7)

Fig. 1.
Fig. 1.

Flow chart of the iterative Fienup algorithm.

Fig. 2.
Fig. 2.

Examples of possible physical nonlinearity profiles that can be recovered from their FT magnitudes only by using the Fienup algorithm assuming zero initial FT phase.

Fig. 3.
Fig. 3.

The same buried-Gaussian profile as in Fig. 2 with uniform noise added (blue trace). The result of the recovery with 100 iterations is shown in red. The green curve is the difference between the original noisy profile and the recovered profile.

Fig. 4.
Fig. 4.

An arbitrarily three-peaked profile (solid blue curve) and the profile recovered (black dashed curve) with the Fienup algorithm after 100 iterations. The red curve shows the recovered profile when d(0) is increased to d(0)=10·max{d(z)}.

Fig. 5.
Fig. 5.

Measured MF curve of the poled sample (circles) compared with the theoretically computed MF curve of the processed nonlinearity profile.

Fig. 6.
Fig. 6.

Recovered FT phases with the two-sample technique (blue) and with the Fienup algorithm (green) assuming zero initial FT phase.

Fig. 7.
Fig. 7.

Nonlinearity profiles recovered with the two-sample technique and with the Fienup algorithm.

Equations (1)

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n = 0 m 1 d min ( n ) 2 n = 0 m 1 d ( n ) 2

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