Abstract

We present a two-wave mixing configuration associated with a coherent optical processor (4-f system) to characterize the third-order susceptibility coefficient. An accurate measurement of the spatial structure of the intensity becomes possible with a CCD camera, which simplifies and improves the accuracy of third-order susceptibility measurements. To verify the validity of the method, measurements of reference materials illuminated by linearly polarized light are carried out. Good agreement with other measurements by various authors is obtained.

© 1996 Optical Society of America

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References

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  1. G. Bruhat, Optique (Masson, Paris, France, 1992), Chap. XIV, p. 326.
  2. L. Pichon and J.-P. Huignard, “Dynamic joint-Fourier-transform correlator by Bragg diffraction in photorefractive Bi12SiO20 crystals,” Opt. Commun. 36, 277–287 (1981).
    [CrossRef]
  3. N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–89.
  5. Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984), Chap. 3, pp. 42–50.
  6. R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188–5195 (1970).
    [CrossRef]
  7. W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
    [CrossRef]
  8. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  9. Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).
  10. N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
    [CrossRef]
  11. M. Paillette, “Recherche expérimentales sur les effets Kerr induits par une onde lumineuse,” Ann. Phys. 4, 617–712 (1969).
  12. P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979).
    [CrossRef]
  13. D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2,” Chem. Phys. 98, 9279–9283 (1993).
  14. A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).
  15. B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
    [CrossRef]
  16. G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

1995 (2)

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

1993 (1)

D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2,” Chem. Phys. 98, 9279–9283 (1993).

1990 (2)

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1984 (2)

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

1981 (1)

L. Pichon and J.-P. Huignard, “Dynamic joint-Fourier-transform correlator by Bragg diffraction in photorefractive Bi12SiO20 crystals,” Opt. Commun. 36, 277–287 (1981).
[CrossRef]

1979 (1)

P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979).
[CrossRef]

1971 (1)

Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).

1970 (1)

R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188–5195 (1970).
[CrossRef]

1969 (1)

M. Paillette, “Recherche expérimentales sur les effets Kerr induits par une onde lumineuse,” Ann. Phys. 4, 617–712 (1969).

Alfano, R. R.

P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979).
[CrossRef]

Boudebs, G.

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

Bourdin, J. P.

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

Brekhovskhikh, G. L.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

Bruhat, G.

G. Bruhat, Optique (Masson, Paris, France, 1992), Chap. XIV, p. 326.

Chevalier, R.

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

Fahmi, A.

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

Ferrier, J. L.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

Gazengel, J.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

Gazengrl, J.

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–89.

Hagan, D.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Ho, P. P.

P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979).
[CrossRef]

Huignard, J.-P.

L. Pichon and J.-P. Huignard, “Dynamic joint-Fourier-transform correlator by Bragg diffraction in photorefractive Bi12SiO20 crystals,” Opt. Commun. 36, 277–287 (1981).
[CrossRef]

Kudriavtseva, A. D.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

LaMacchia, J. T.

R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188–5195 (1970).
[CrossRef]

Lecoq, J. P.

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

Nalbandov, L. U.

Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).

Paillette, M.

M. Paillette, “Recherche expérimentales sur les effets Kerr induits par une onde lumineuse,” Ann. Phys. 4, 617–712 (1969).

Phu, X. N.

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

Phu Xuan, N.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

Pichon, L.

L. Pichon and J.-P. Huignard, “Dynamic joint-Fourier-transform correlator by Bragg diffraction in photorefractive Bi12SiO20 crystals,” Opt. Commun. 36, 277–287 (1981).
[CrossRef]

Rivoire, G.

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2,” Chem. Phys. 98, 9279–9283 (1993).

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

Sahraoui, B.

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Shen, Y. R.

Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984), Chap. 3, pp. 42–50.

Soileau, M. J.

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Sokolovskaia, A. I.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

Styland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sylla, M.

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

Tcherniega, N. V.

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

Townsend, R. L.

R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188–5195 (1970).
[CrossRef]

Van Stryland, E. W.

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Volkova, Y. A.

Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).

Wang, D.

D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2,” Chem. Phys. 98, 9279–9283 (1993).

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Williams, W. E.

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Xuan, N. P.

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

Zamkoff, V. A.

Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).

Zaremba, J.

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

Ann. Phys. (1)

M. Paillette, “Recherche expérimentales sur les effets Kerr induits par une onde lumineuse,” Ann. Phys. 4, 617–712 (1969).

Chem. Phys. (1)

D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2,” Chem. Phys. 98, 9279–9283 (1993).

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

J. Appl. Phys. (1)

R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188–5195 (1970).
[CrossRef]

J. Mod. Opt. (1)

B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095–2107 (1995).
[CrossRef]

Nonlin. Opt. (1)

A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165–178 (1995).

Opt. Commun. (4)

N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433–437 (1984).
[CrossRef]

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

L. Pichon and J.-P. Huignard, “Dynamic joint-Fourier-transform correlator by Bragg diffraction in photorefractive Bi12SiO20 crystals,” Opt. Commun. 36, 277–287 (1981).
[CrossRef]

N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244–250 (1990).
[CrossRef]

Opt. Spectroscp. (USSR) (1)

Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300–311 (1971).

Phys. Rev. A (1)

P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979).
[CrossRef]

Other (4)

G. Bruhat, Optique (Masson, Paris, France, 1992), Chap. XIV, p. 326.

G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non linéaire auto-induit dans Fe:NbLiO3,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 77–89.

Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984), Chap. 3, pp. 42–50.

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Figures (6)

Fig. 1
Fig. 1

4-f System used as a coherent optical processor. A nonlinear material (NL) is placed in the Fourier-transform plane of this setup. L’s, lenses.

Fig. 2
Fig. 2

Two-slit object. The circle represents the incident light beam.

Fig. 3
Fig. 3

Simulation of the central line in the image of a two-slit object filtered by a nonlinear medium: evolution of the intensities of the first diffracted orders with the exciting intensity I(0, 0) in the spectrum.

Fig. 4
Fig. 4

Schematic of the experimental setup: NL, nonlinear material; L1–L3, lenses; P1, P2, prismatic plates; M1, M2, mirrors; te, to, sf, tf, filters defined in text.

Fig. 5
Fig. 5

Experimental images of a two-slit object filtered by a CS2 cell obtained in one laser shot. The reference input image (between 100 and 250 on the abscissa) and the first lateral order (m = −1, between 0 and 50 on the abscissa) are obtained with different values of filters tf and to.

Fig. 6
Fig. 6

Refractive-index change Δn(0, 0) in the center of the Fourier spectrum in a 1-mm-long cell of CS2, versus corresponding input intensity I(0, 0).

Tables (2)

Tables Icon

Table 1 Measured Value of the Nonlinear Refractive-Index Constant of CS2 in Linear Polarization

Tables Icon

Table 2 Absolute and Relative Values of n2 for Different Materials

Equations (24)

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

S ( u , v ) = 1 λ f FT [ O ( x , y ) ] = 1 λ f O ( x , y ) exp [ 2 π j ( u x + v y ) ] d x d y ,
S ( u , v ) = S ( u , v ) T ( u , v ) ;
U ( x , y ) = λ f FT 1 [ S ( u , v ) T ( u , v ) ] ,
S z = [ α S + ( β j γ ) | S | 2 S ] ,
T ( u , v ) = exp ( α L ) [ 1 + β 1 exp ( 2 α L ) α | S ( u , v ) | 2 ] 1 / 2 × exp { j γ 2 β ln [ 1 + β 1 exp ( 2 α L ) α | S ( u , v ) | 2 ] } ,
T ( u , v ) = exp ( α L ) exp [ j γ 1 exp ( 2 α L ) 2 α | S ( u , v ) | 2 ] .
Δ n ( u , v ) = λ 2 π L γ 1 exp ( 2 α L ) 2 α | S ( u , v ) | 2 .
Δ n ( u , v ) = λ 2 π γ | S ( u , v ) | 2 .
I ( u , v ) = 2 0 n c | S ( u , v ) | 2 ,
Δ n ( u , v ) = n 2 I ( u , v ) ,
t ( x , y ) = t ( x ) = rect ( x + x 0 2 d ) + rect ( x x 0 2 d ) ,
O ( x , y ) = E ( x , y ) t ( x , y ) .
E ( x , y ) = E ( y ) = E 00 exp ( y 2 2 σ 2 ) ,
S ( u , v ) = 2 π σ λ f E 00 4 d cos ( 2 π x 0 u ) × sinc ( 2 d u ) exp ( 2 π 2 σ 2 v 2 ) ,
I im ( x , y ) α | FT 1 { S ( u , v ) exp [ j φ ( u , v ) ] } | 2 ,
φ ( u , v ) = γ L | S ( u , v ) | 2 .
O ( x , y ) E ( x 0 , y ) rect ( x + x 0 2 d ) + E ( x 0 , y ) rect ( x x 0 2 d ) .
I obj ( ± x 0 , y ) 2 0 c = E 2 ( ± x 0 , y ) ,
I im ( ± x 0 , y ) 2 0 c = E 2 ( ± x 0 , y ) ,
I im ( ± 3 x 0 , y ) 2 0 c = 9 16 ( 2 d λ f ) 4 γ 2 L 2 g ± ± 2 ( y ) ,
g ρ σ τ ( y ) = E ( ρ x 0 , y ) E ( σ x 0 , y ) E ( τ x 0 , y ) ,
I im ( ± x 0 , y ) 2 0 c = exp ( 2 α L ) { E 2 ( ± x 0 , y ) 3 2 ( 2 d λ f ) 2 × β L eff E ( ± x 0 , y ) [ g ± ± ± ( y ) + 2 g ± ( y ) ] } ,
I im ( ± 3 x 0 , y ) 2 0 c = 9 16 ( 2 d λ f ) 4 exp ( 2 α L ) × ( β 2 + γ 2 ) L eff 2 g ± ± 2 ( y ) ,
L eff = 1 exp ( 2 α L ) 2 α .

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