Abstract

We consider noncollinear interaction among three amplitude-modulated plane waves in a uniaxial crystal under type I phase-matching conditions. If one among the interacting fields can be taken as undepleted during the interaction, the remaining two fields are holographic replicas of each other: the undepleted field acts as the reference field in holography. We develop a general method to calculate the holographic wave front generated by a nonplane object wave front and a plane reference wave front. We present experiments in which a holographic image of an object consisting of a pointlike light source is obtained by choosing the object- and reference-field frequencies among ω1, ω2, and ω3 (with ω3=ω1+ω2) in any of the possible combinations.

© 2003 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  2. D. M. Pepper and A. Yariv, “Optical phase conjugation using three-wave and four-wave mixing via elastic photon scattering in transparent media,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 23.
  3. D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
    [CrossRef]
  4. M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
    [CrossRef]
  5. G. Klose and A. Siahmakoun, “External mode locking with feedback from a self-pumped phase-conjugator BaTiO3 crystal,” Opt. Eng. 35, 2983–2988 (1996).
    [CrossRef]
  6. P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
    [CrossRef]
  7. P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
    [CrossRef]
  8. S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).
  9. A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).
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    [CrossRef] [PubMed]
  12. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
    [CrossRef]
  13. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
    [CrossRef]
  14. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
    [CrossRef]
  15. Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).
  16. A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
    [CrossRef]
  17. Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
    [CrossRef]
  18. Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
    [CrossRef]
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    [CrossRef]
  20. A. Andreoni, M. Bondani, and M. A. C. Potenza, “Combinational tasks performed by second harmonic generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
    [CrossRef]
  21. A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
    [CrossRef]
  22. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, New York, 1997).
  23. D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–463 (1949).
    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), Chap. 8.
  25. M. R. Fewings and A. L. Gaeta, “Compensation of pulse distortions by phase conjugation via difference-frequency generation,” J. Opt. Soc. Am. B 17, 1522–1525 (2000).
    [CrossRef]

2002 (1)

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

2001 (1)

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

2000 (7)

M. R. Fewings and A. L. Gaeta, “Compensation of pulse distortions by phase conjugation via difference-frequency generation,” J. Opt. Soc. Am. B 17, 1522–1525 (2000).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Combinational tasks performed by second harmonic generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

1999 (3)

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

1996 (3)

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

G. Klose and A. Siahmakoun, “External mode locking with feedback from a self-pumped phase-conjugator BaTiO3 crystal,” Opt. Eng. 35, 2983–2988 (1996).
[CrossRef]

L. Lefort and A. Barthelemy, “Revisiting optical phase conjugation by difference-frequency generation,” Opt. Lett. 21, 848–850 (1996).
[CrossRef] [PubMed]

1982 (1)

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

1979 (1)

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

1978 (1)

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
[CrossRef]

1977 (1)

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1949 (1)

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–463 (1949).
[CrossRef]

Ananiev, Yu. A.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Andreoni, A.

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Combinational tasks performed by second harmonic generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Avizonis, P. V.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Barthelemy, A.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bogdanov, M. P.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Bomberger, W. D.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Bondani, M.

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Combinational tasks performed by second harmonic generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Denisyuk, Yu. N.

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).

Dimakov, S. A.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fainman, S. Y.

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Fekete, D.

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
[CrossRef]

Fewings, M. R.

Gabor, D.

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–463 (1949).
[CrossRef]

Gaeta, A. L.

Gorlanov, A. M.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Gorlanov, A. V.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Goryachkin, D. A.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Grigor’ev, A. M.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Grishmanova, N. I.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Hopf, F. A.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Irtuganov, V. M.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Jacobs, S. F.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Kalinin, V. P.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Kliment’ev, S. I.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Klose, G.

G. Klose and A. Siahmakoun, “External mode locking with feedback from a self-pumped phase-conjugator BaTiO3 crystal,” Opt. Eng. 35, 2983–2988 (1996).
[CrossRef]

Kozlovskaya, I. M.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Lefort, L.

Mazurenko, Y. T.

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Oba, K.

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Orlova, I. B.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Pepper, D. M.

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Podoba, Ya. G.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Potenza, M. A. C.

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

A. Andreoni, M. Bondani, Yu. N. Denisyuk, and M. A. C. Potenza, “Holographic properties of the second-harmonic cross correlation of object and reference optical wave fields,” J. Opt. Soc. Am. B 17, 966–972 (2000).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Combinational tasks performed by second harmonic generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “Real-time holograms by second-harmonic cross correlation of object and reference optical wave fields,” Opt. Lett. 25, 890–892 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Puddu, E.

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

Sherstobitov, V. E.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Shostko, S. N.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Siahmakoun, A.

G. Klose and A. Siahmakoun, “External mode locking with feedback from a self-pumped phase-conjugator BaTiO3 crystal,” Opt. Eng. 35, 2983–2988 (1996).
[CrossRef]

Solov’yov, V. D.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Sun, P. C.

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Sventsitskaya, N. A.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Tomita, A.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Venediktov, V. Yu.

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

Villani, F.

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Volosov, B. D.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Womack, K. H.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Yariv, A.

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
[CrossRef]

Appl. Phys. Lett. (2)

D. M. Pepper, D. Fekete, and A. Yariv, “Observation of amplified phase-conjugate reflection and optical parametric oscillation by degenerate four-wave mixing in a transparent medium,” Appl. Phys. Lett. 33, 41–44 (1978).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Eur. Phys. J. D (1)

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

Kvant. Elektron. (Moscow) (1)

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Opt. Commun. (2)

M. P. Bogdanov, S. A. Dimakov, A. V. Gorlanov, D. A. Goryachkin, A. M. Grigor’ev, V. M. Irtuganov, V. P. Kalinin, S. I. Kliment’ev, I. M. Kozlovskaya, I. B. Orlova, V. E. Sherstobitov, and V. Yu. Venediktov, “Correction of segmented mirror aberrations by phase conjugation and dynamic holography,” Opt. Commun. 129, 405–413 (1996).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, ““Viewing” objects hidden in highly scattering media by cross-correlating the Fourier-transform of the image with the incident field in a second-order nonlinear crystal,” Opt. Commun. 174, 487–497 (2000).
[CrossRef]

Opt. Eng. (1)

G. Klose and A. Siahmakoun, “External mode locking with feedback from a self-pumped phase-conjugator BaTiO3 crystal,” Opt. Eng. 35, 2983–2988 (1996).
[CrossRef]

Opt. Lett. (3)

Opt. Mem. Neural Netw. (1)

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 123–137 (1999).

Opt. Spectrosc. (Russia) (1)

Yu. N. Denisyuk, A. Andreoni, M. Bondani, and M. A. C. Potenza, “The formation of the holographic image of a diffusing object in the second-harmonic light generated by a nonlinear material,” Opt. Spectrosc. (Russia) 89, 113–120 (2000).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

Pis'ma Zh. Tekh. Fiz. (1)

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Proc. IEEE (1)

P. C. Sun, K. Oba, Y. T. Mazurenko, and S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–463 (1949).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Andreoni, M. Bondani, Yu. N. Denisyuk, M. A. C. Potenza, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

Other (4)

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, New York, 1997).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), Chap. 8.

D. M. Pepper and A. Yariv, “Optical phase conjugation using three-wave and four-wave mixing via elastic photon scattering in transparent media,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 23.

B. Zel’dovich, “Wave front conjugation by three-wave mixing,” in Wave Front Conjugation (Science, Moscow, 1985), p. 194.

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Figures (9)

Fig. 1
Fig. 1

Noncollinear three-wave interaction in a type I uniaxial crystal: (x, y, z), laboratory reference frame with the (x, y) plane parallel to the crystal entrance face; α, tuning angle.

Fig. 2
Fig. 2

Wave fronts, Wi (i=1, 2, 3, dotted curves), generated by three-wave interactions in response to a plane and a spherical input wave fronts (solid curves); k^i are unit vectors parallel to the corresponding wave vectors. (a) The object, W1, and reference, W2, wave fronts are at ω and 2ω, respectively, and the SFG wave front is W3 (case 1). (b) The inputs are W3 at 3ω and W2 at 2ω, and the DFG wave front is W1 (case 2). In Fig. 2(c) the inputs are W1 at ω and W3 at 3ω, and the DFG wave front is W2 (case 3).

Fig. 3
Fig. 3

Setup for selected experiments of χ(2) interactions in a BBO crystal in which lens l1 (f1=100 mm) forms the pointlike object source O, the telescope (7.5×, see text) enlarges the reference beam, and O is the holographic image reconstructed by the interaction. The interacting-beam wavelengths are as indicated. Wavefronts W1,2,3, object and reference (solid curves), generated (dotted curve); CCD, camera. (a) Setup for the experiment in case 1 in which the virtual image O is transformed into the real image O by lens l2 (f2=460 mm). (b) Setup for the experiment in case 2a, in which the virtual image O is transformed into the real image O by lens l2 (f2=250 mm). (c) Setup for the experiment in case 3.

Fig. 4
Fig. 4

Upper panel: image O detected by the CCD camera in case 1 [see the setup in Fig. 3(a)]. Lower panel: Gaussian fit of image O.

Fig. 5
Fig. 5

Upper panel: image O detected by the CCD camera in case 2a [see the setup in Fig. 3(b)]. Lower panel: Gaussian fit of image O.

Fig. 6
Fig. 6

Upper panel: image O detected by the CCD camera in case 3 [see the setup in Fig. 3(c)]. Lower panel: Gaussian fit of image O.

Fig. 7
Fig. 7

(a) Upper panel: object O seen by the CCD through the BBO in case 2a. Lower panel: Gaussian fit of the object in case 2a. (b) Upper panel: image O detected by the CCD camera in case 2a. Lower panel: Gaussian fit of image O.

Fig. 8
Fig. 8

(a) Upper panel: focal spot formed by lens l1 in case 3. Lower panel: Gaussian fit of the spot in case 3. (b) Object O as detected by the CCD through the BBO aperture (upper panel) in case 3 and its Gaussian fit (lower panel). (c) Upper panel: image O detected by the CCD camera in case 3. Lower panel: Gaussian fit of image O.

Fig. 9
Fig. 9

Reference frames for the propagation of a Gaussian beam from the focal spot [plane (xO, yO, zO)] to the crystal entrance and exit planes [planes (xC, yC, zC) and (xf, yf, zf), respectively].

Equations (55)

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E1(r, t)=xˆ22η0ω1n1 a1(r)exp[-i(k1cos ϑ1z+k1sin ϑ1y-ω1t)]+c.c.,
E2(r, t)=xˆ22η0ω2n2 a2(r)exp[-i(k2cos ϑ2z+k2sin ϑ2y-ω2t)]+c.c.,
E3(r, t)=122η0ω3n3 [yˆa3y(r)+zˆa3z(r)]exp[-i(k3cos ϑ3z+k3sin ϑ3y-ω3t)]+c.c.,
a1(0)0,a2(z)=a2(0),a3(0)=0,
a1(0)=0,a2(z)=a2(0),a3(0)0,
a1(0)0,a2(0)=0,a3(z)=a3(0),
a3y,3z(y, z)=g+,-g a1(0)sing|a2(0)|sin(ϑ1-ϑ3) [(cos ϑ3-cos ϑ1)y-(sin ϑ3-sin ϑ1)z]×expiΛ2(0)+π2,
g+,-=d+,-2ω1ω2ω3η03n1n2n3,
d+=d22cos(α-ϑ3)+d31sin(α-ϑ3),
d-=d22sin(α-ϑ3)-d31cos(α-ϑ3),
g=g+2+g-2.
E3(y, z, t)
=122η0ω3n3yˆg++zˆ,g-g a1(0)×sing|a2(0)|cosϑ3-ϑ12  sinϑ3+ϑ12 y+cosϑ3+ϑ12 zexpiΛ2(0)+π2×exp[-i(k3cos ϑ3z+k3sin ϑ3y-ω3t)]+c.c.,
E3(y, z, t)=wˆ22η0ω3n3a3(y, z)exp[-i(k3cos ϑ3z+k3sin ϑ3y-ω3t)]+c.c.,
a3(y, z)=a1(0)sing|a2(0)|cosϑ3-ϑ12  sinϑ3+ϑ12 y+cosϑ3+ϑ12 zexpiΛ2(0)+π2.
a1(y, z)=geffg a3(0)sing|a2(0)|cosϑ3-ϑ12  sinϑ3+ϑ12 y+cosϑ3+ϑ12 zexpi-Λ2(0)+π2,
E1(y, z, t)=xˆ22η0ω1n1geffg a3(0)×sing|a2(0)|cosϑ3-ϑ12  sinϑ3+ϑ12 y+cosϑ3+ϑ12 zexpi-Λ2(0)+π2×exp[-i(k1cos ϑ1z+k1sin ϑ1y-ω1t)]+c.c..
a2(y, z)=a1*(0)sinhgeff|a3(0)|cosϑ1-ϑ22  sinϑ1+ϑ22 y+cosϑ1+ϑ22 zexpiΛ3(0)+π2,
E2(y, z, t)=xˆ22η0ω2n2 a1*(0)×sinhgeff|a3(0)|cosϑ1-ϑ22  sinϑ1+ϑ22 y+cosϑ1+ϑ22 zexpiΛ3(0)+π2×exp[-i(k2cos ϑ2z+k2sin ϑ2y-ω2gt)]+c.c..
(case1)φ3(r)=Λ1(0)+Λ2(0)+π2-k3r=φ(r)+φ2(r)+π2,
(case2)φ1(r)=-Λ2(0)+Λ3(0)+π2-k1r=φ3(r)-φ2(r)+π2,
(case3)φ2(r)=-Λ1(0)+Λ3(0)+π2-k2r=φ3(r)-φ1(r)+π2.
w0=4dO2kj21w021+4dO2kj21w04+λ2πdO2L2,
G(x, y)=A exp-(x-x0)2+(y-y0)2w2.
a1(xO, yO, zO)=A1exp-xO2+yO2w1,02,
E1(xC, yC, zC, t)
=xˆ2A1w1,0w1(zC-zO)exp-xC2+yC2w12(zC-zO)×exp-ik1z+k1xC2+yC22R1(zC-zO)-tan-1zC-zOz1,0-ω1t+c.c.=xˆ22η0ω1n1 a1(xC, yC, zC)×exp[-i(k1z-ω1t)]+c.c.,
a1(xC, yC, zC)
=A12η0ω1n1w1,0w1(zC-zO)exp-xC2+yC2w12(zC-zO)×exp-ik1xC2+yC22R1(zC-zO)-i tan-1zC-zOz1,0,
w12(zC-zO)=w1,021+zC-zOz1,02,
R1(zC-zO)=(zC-zO)1+z1,0zC-zO2.
a2(xf, yf, zf)=a1*(xC, yC, zC)×sinh[geff|a3(xC, yC, zC)|Z]×expiΛ3(0)+π2,
E2(xf, yf, zf, t)
=xˆ22η0ω2n2 a2(xf, yf, zf)×exp{-i[k2(zf-zC)-ω2t]}+c.c.xˆ2ω2n1ω1n2 A1w1,0w1(zC-zO)×exp-xf2+yf2w12(zC-zO)expik2k1k2xf2+yf22R1(zC-zO)+i π2sinh[geff|a3(xC, yC, zC)|Z]expiΛ3(0)+π2exp[-i(k2Z-ω2t)]+c.c.xˆ2A(Z)exp-xf2+yf2w22(zf)expik2xf2+yf22R2(zf)+i tan-1zfz2,0exp[-i(k2Z-ω2t)]+c.c.,
w2(zf)=w1(zC-zO)
R2(zf)=k2k1 R1(zC-zO)
w1(zC-zO)=w1,01+zC-zOz1,02=w1,01+2(zC-zO)k1w1,0222(zC-zO)k1w1,0,
w2(zf)=w2,01+zfz2,02=w2,01+2zfk2w2,0222zfk2w2,0,
R1(zC-zO)=(zC-zO)1+z1,0zC-zO2(zC-zO),
R2(zf)=zf1+z2,0zf2zf.
w2,0w1,0=1,
zfzC-zO=-k2k1.
T(xC, yC, zC)=rectxCLrectyCL.
U(xC, yC, zC)=E(xC, yC, zC)T(xC, yC, zC)
U(xC, yC, zC)
=exp(-ikzC)iλzCexp-i k2CC (xC2+yC2)×U0(xO, yO, zO)exp-i k2zC (xO2
+yO2)expi 2πλzC (xCxO+yCyO)dxOdyO,
U0(xO, yO, zO)
=iλzCexp(ikzC)expi k2zC (xO2+yO2)×-+U(xC, yC, zC)expi k2zC (xC2+yC2)exp-i 2πλzC (xCxO+yCyO)dxCdyC=iλzCA w0w(zC)expi k2zC (xO2+yO2)expi tan-1zCz0×-+exp-1w2(zC)+i k21R(zC)-1zC(xC2+yC2)T(xC, yC, zC)×exp-i 2πλzC (xCxO+yCyO)dxCdyC.
T(xC, yC, zC)exp-πL2 (xC2+yC2).
U0(xO, yO, zO)
=iλzCA w0w(zC)expi k2zC (xO2+yO2)expi tan-1zCz0×π1w2(zC)+i k21R(zC)-1zC+πL2  ×exp-π2λ2zC2(xC2+yC2)1w2(zC)+i k21R(zC)-1zC+πL2  .
U0(xO, yO, zO)
=iλzCA1w0w(ZC)π1w2(zC)+πL2  expi k2zC (xO2+yO2)expi π2exp-(xC2+yC2)w02,
w0=λ2zC2π21w2(zC)+πL2=4zC2k21w021+4zC2k21w04+λ2πzC2L2.

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