Abstract

We have determined the far-field beam profiles of an initially Gaussian laser beam that traverses a thin slab of nonlinear-optical material in which the spatial distribution of the refractive nonlinearity is anisotropic by an expansion in terms of elliptical Gaussian functions. In marked contrast to the annular structures observed with isotropic media, it is shown that the transmitted beam breaks into two beams in the far field. This nonlinear beam-splitting behavior has been confirmed experimentally.

© 1990 Optical Society of America

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References

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  1. D. Weaire, B. S. Wherrett, D. A. B. Miller, S. D. Smith, Opt. Lett. 4, 331 (1979).
    [CrossRef] [PubMed]
  2. S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
    [CrossRef]
  3. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  4. E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).
  5. E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, J. Opt. Soc. Am. B 5, 1980 (1988).
    [CrossRef]
  6. J. A. Herman, J. Opt. Soc. Am. B 1, 729 (1984).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986), p. 428.
  8. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  9. A. Yariv, Optical Electronics, 3rd ed. (Wiley, New York, 1985), p. 47.
  10. D. Bloor, F. H. Preston, Phys. Status Solidi A 37, 427 (1976).
    [CrossRef]
  11. D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
    [CrossRef] [PubMed]

1988 (1)

1986 (1)

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

1985 (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

1984 (1)

1979 (1)

1976 (1)

D. Bloor, F. H. Preston, Phys. Status Solidi A 37, 427 (1976).
[CrossRef]

1968 (1)

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

1966 (1)

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

Bloor, D.

D. Bloor, F. H. Preston, Phys. Status Solidi A 37, 427 (1976).
[CrossRef]

Boggess, T. F.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986), p. 428.

Guha, S.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Hagan, D. J.

Heremans, J.

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

Herman, J. A.

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

Kogelnik, H.

Krindach, D. D.

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

Li, T.

Mansour, K.

Migulin, A. V.

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

Miller, D. A. B.

Morelli, D. T.

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

Preston, F. H.

D. Bloor, F. H. Preston, Phys. Status Solidi A 37, 427 (1976).
[CrossRef]

Sakumoto, M.

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

Smirl, A. L.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Smith, S. D.

Soileau, M. J.

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Sukhorukov, A. P.

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

Uher, C.

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

Van Stryland, E. W.

E. W. Van Stryland, Y. Y. Wu, D. J. Hagan, M. J. Soileau, K. Mansour, J. Opt. Soc. Am. B 5, 1980 (1988).
[CrossRef]

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Vanherzeele, H.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Weaire, D.

Wherrett, B. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986), p. 428.

Woodall, M. A.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Wu, Y. Y.

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Wiley, New York, 1985), p. 47.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

S. A. Akhmanov, D. D. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, IEEE J. Quantum Electron. QE-4, 568 (1968).
[CrossRef]

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

Opt. Eng. (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, T. F. Boggess, Opt. Eng. 24, 613 (1985).

Opt. Lett. (1)

Phys. Rev. Lett. (2)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

D. T. Morelli, J. Heremans, M. Sakumoto, C. Uher, Phys. Rev. Lett. 57, 869 (1986).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

D. Bloor, F. H. Preston, Phys. Status Solidi A 37, 427 (1976).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986), p. 428.

A. Yariv, Optical Electronics, 3rd ed. (Wiley, New York, 1985), p. 47.

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Figures (3)

Fig. 1
Fig. 1

(a) Far-field intensity distribution as calculated from Eq. (3) for the parameters rx = r0, ry = 2r0, X = 4, and z = 5 diffraction lengths, (b) As in (a) except for the isotropic case, in which rx = ry = r0.

Fig. 2
Fig. 2

Peak separation as a function of distance behind the sample for an increasing anisotropy of the nonlinear refractive-index distribution. rx = r0; ry = 1.5r0 (triangles), 1.75r0 (crosses), 2.0r0 (diamonds). The incident irradiance is held constant (X = 4). Beam splitting, which is enhanced with increasing anisotropy, is shown.

Fig. 3
Fig. 3

Cross sections through the transmitted beam profile in the far field (z = 5 diffraction lengths) for an increasing incident irradiance (proportional to X): curve (a), X = 0.01; curve (b), X = 2; curve (c), X = 4. The anisotropy is held constant (ry/rx = 2). The cross sections are taken parallel to the minor axis of the refractive nonlinearity.

Equations (6)

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E ( x , y , 0 ) = E ( 0 , 0 , 0 ) exp { ( x 2 + y 2 r 0 2 ) + i X exp [ ( 2 x 2 r x 2 + 2 y 2 r y 2 ) ] } ,
X = ω c 0 D n 2 I 0 ( z ) d z ,
E ( x , y , 0 ) = E ( 0 , 0 , 0 ) m = 0 ( i X ) m × exp [ ( x 2 W m x 2 ( 0 ) + y 2 W m y 2 ( 0 ) ) ] ,
W m j 2 = r j 2 r 0 2 ( 2 m r 0 + r j ) .
E ( x , y , z ) = E ( 0 , 0 , 0 ) m = 0 ( i X ) m m ! [ 1 + ( z 2 / d m x 2 ) ] 1 / 4 × [ 1 + ( z 2 / d m y 2 ) ] 1 / 4 × exp { i P m ( z ) x 2 [ 1 W m x 2 ( z ) + i k 2 R m x ( z ) ] y 2 [ 1 W m y 2 ( z ) + i k 2 R m y ( z ) ] } ;
k = ω / c , W m j 2 ( z ) = W m j 2 ( 0 ) [ 1 + ( z / d m j ) 2 ] , d m j = k W m j 2 ( 0 ) 2 , R m j ( z ) = z [ 1 + ( d m j / z ) 2 ] , P m ( z ) = 1 / 2 tan 1 ( z / d m x ) 1 / 2 tan 1 ( z / d m y ) .

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