Abstract

By numerical simulations of the reflection of a (2-D) Gaussian beam from a nonlinear interface, we show several new features of the behavior of such an interface and resolve the difference between the results of two previous studies of this type. Newly reported features include a very large nonlinear Goos-Hänchen shift and large variations of the angle of an output beam for small changes in the input intensity. The latter phenomenon has potential applications for a light-controlled angular scanning element. The differences between prior studies of this type are shown to be an artifact of the numerical procedures used.

© 1982 Optical Society of America

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References

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  1. A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).
  2. B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
    [CrossRef]
  3. A. A. Kolokolov, A. I. Sukov, Radiophys. Quantum Electron. 21, 1013 (1978).
    [CrossRef]
  4. N. N. Rozanov, Opt. Spectrosc. (USSR) 47, 335 (1979).
  5. D. Marcuse, Appl. Opt. 19, 3130 (1980).
    [CrossRef] [PubMed]
  6. A. E. Kaplan, J. Opt. Soc. Am. 71, 1640 (1981).
    [CrossRef]
  7. P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
    [CrossRef]
  8. P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
    [CrossRef]
  9. W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
    [CrossRef] [PubMed]
  10. M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B. 25, 53 (1973).
  11. S. Kozaki, H. Sakurai, J. Opt. Soc. Am. 68, 508 (1978).
    [CrossRef]
  12. A. E. Kaplan, Sov. J. Quantum Electron. 8, 95 (1978);Radiophys. Quantum Electron. 22, 229 (1979);IEEE J. Quantum Electron. QE-17, 336 (1981).
    [CrossRef]
  13. N. L. Schryer, “A User's Guides to DODES,” Bell Laboratories Computing Science Technical Report 33 (1975);J. Phys. Chem. 81, 2335 (1977).
  14. S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
    [CrossRef]

1981

A. E. Kaplan, J. Opt. Soc. Am. 71, 1640 (1981).
[CrossRef]

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

1980

1979

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

N. N. Rozanov, Opt. Spectrosc. (USSR) 47, 335 (1979).

1978

A. A. Kolokolov, A. I. Sukov, Radiophys. Quantum Electron. 21, 1013 (1978).
[CrossRef]

S. Kozaki, H. Sakurai, J. Opt. Soc. Am. 68, 508 (1978).
[CrossRef]

A. E. Kaplan, Sov. J. Quantum Electron. 8, 95 (1978);Radiophys. Quantum Electron. 22, 229 (1979);IEEE J. Quantum Electron. QE-17, 336 (1981).
[CrossRef]

1976

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

1975

B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
[CrossRef]

1973

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B. 25, 53 (1973).

1968

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Boiko, B. B.

B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
[CrossRef]

Dzhilavdari, I. Z.

B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
[CrossRef]

Hermann, J.-P.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, J. Opt. Soc. Am. 71, 1640 (1981).
[CrossRef]

A. E. Kaplan, Sov. J. Quantum Electron. 8, 95 (1978);Radiophys. Quantum Electron. 22, 229 (1979);IEEE J. Quantum Electron. QE-17, 336 (1981).
[CrossRef]

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Kolokolov, A. A.

A. A. Kolokolov, A. I. Sukov, Radiophys. Quantum Electron. 21, 1013 (1978).
[CrossRef]

Kozaki, S.

Maloney, P. J.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

Marcuse, D.

Miyagi, M.

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B. 25, 53 (1973).

Nishida, S.

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B. 25, 53 (1973).

Petrov, N. S.

B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
[CrossRef]

Rozanov, N. N.

N. N. Rozanov, Opt. Spectrosc. (USSR) 47, 335 (1979).

Sakurai, H.

Schryer, N. L.

N. L. Schryer, “A User's Guides to DODES,” Bell Laboratories Computing Science Technical Report 33 (1975);J. Phys. Chem. 81, 2335 (1977).

Smith, P. W.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Sukov, A. I.

A. A. Kolokolov, A. I. Sukov, Radiophys. Quantum Electron. 21, 1013 (1978).
[CrossRef]

Tomlinson, W. J.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
[CrossRef] [PubMed]

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
[CrossRef]

IEEE J. Quantum Electron.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, J.-P. Hermann, IEEE J. Quantum Electron. QE-17, 340 (1981).
[CrossRef]

J. Appl. Spectrosc.

B. B. Boiko, I. Z. Dzhilavdari, N. S. Petrov, J. Appl. Spectrosc. 23, 1511 (1975).
[CrossRef]

J. Opt. Soc. Am.

JETP Lett.

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

Opt. Lett.

Opt. Spectrosc. (USSR)

N. N. Rozanov, Opt. Spectrosc. (USSR) 47, 335 (1979).

Radiophys. Quantum Electron.

A. A. Kolokolov, A. I. Sukov, Radiophys. Quantum Electron. 21, 1013 (1978).
[CrossRef]

Sci. Rep. Res. Inst. Tohoku Univ. Ser. B.

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B. 25, 53 (1973).

Sov. J. Quantum Electron.

A. E. Kaplan, Sov. J. Quantum Electron. 8, 95 (1978);Radiophys. Quantum Electron. 22, 229 (1979);IEEE J. Quantum Electron. QE-17, 336 (1981).
[CrossRef]

Sov. Phys. Usp.

S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Other

N. L. Schryer, “A User's Guides to DODES,” Bell Laboratories Computing Science Technical Report 33 (1975);J. Phys. Chem. 81, 2335 (1977).

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Figures (10)

Fig. 1
Fig. 1

Interface configuration and coordinate system.

Fig. 2
Fig. 2

Intensity distributions on various constant-z planes for a linear interface (n2 = 0): (a) z = −200; (b) z = 0; (c) z = +200; (d) comparison of distributions at z = −200 and +200.

Fig. 3
Fig. 3

Perspective plot showing intensity distributions for a linear interface (n2 = 0). This format is used to present many of the results for nonlinear interfaces.

Fig. 4
Fig. 4

Plane-wave theory predictions of the amplitude reflectivity of a nonlinear interface for ψ / ψ c = 1 / 2 as a function of normalized input intensity.

Fig. 5
Fig. 5

Perspective plots of the intensity distributions for a nonlinear interface: (a) n2 = 0.01000; (b) n2 = 0.01085; (c) n2 = 0.01100; (d) n2 = 0.01200.

Fig. 6
Fig. 6

Intensity distributions on the plane z = +200 for n2 = 0.012, for various mesh spacings (Δx) and boundary conditions. For curves (a)−(c) the calculations included a correction for the discontinuity of 2ϕ/∂x2 at the interface as described in the Appendix. For curves (d) and (e) no such correction was made.

Fig. 7
Fig. 7

Plots of the path of the peak of the self-focused channel in the nonlinear medium for various normalized intensities: A n2 = 0.01095; B n2 = 0.01100; C n2 = 0.01105; D n2 = 0.01110; E n2 = 0.01115; F n2 = 0.01120; G n2 = 0.01125; H n2 = 0.01130.

Fig. 8
Fig. 8

Reflectivity of a nonlinear interface as a function of normalized input intensity: (a) results from the present calculations for a Gaussian beam; and (b) predictions of the plane wave theory.

Fig. 9
Fig. 9

Perspective plot of the intensity distributions for a nonlinear interface with n2 = 0.020.

Fig. 10
Fig. 10

Angle of the transmitted beam(s) as a function of normalized input intensity: (a) results from the present calculations for a Gaussian beam; and (b) predictions of the plane wave theory.

Tables (1)

Tables Icon

Table I Relative Errors in Estimating the Second Derivative of the Function exp(kxx) Using the Three-Point Formula [Eq. (11)] and the Five-Point Formula [Eq. (12)]; the Constant kx was Assumed to be ( 2 π / λ ) 2 n 0 Δ, with λ = 1, n0 = 1.5, and Δ = 0.02

Equations (19)

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n ( x , y , z ) = n 0 Δ + n 2 I 1 ( x , y , z ) ,
n 2 I 0 Δ = { ½ [ 1 ( ψ / ψ c ) 2 ] 1 / 2 ψ / ψ c < 1 ( ψ c / ψ ) 2 0 < ψ / ψ c 1 / 2 .
n 2 I Δ = 1 ( 1 + r ) 2 [ 1 4 r ( 1 + r ) 2 ( ψ ψ c ) 2 ] ,
n 2 I Δ = ( ψ + ψ 1 ) 2 ( ψ 1 2 + ψ c 2 ψ 2 ) 4 ψ 2 ψ c 2 .
E ( x ) = E 0 sech ( k x x x 0 ) ,
k x = 2 π λ 2 n 0 Δ 1 ( ψ / ψ c ) 2 ,
k x = 2 π λ 2 n 0 Δ 1 + D ,
2 E + n 2 k 2 E = 0 ,
E ( x , z ) = ϕ ( x , z ) exp ( in 0 k cos ψ z ) ,
i ( 2 n 0 k cos ψ ) ϕ z = 2 ϕ z 2 + 2 ϕ x 2 + ( n 2 n 0 2 cos 2 ψ ) k 2 ϕ .
2 ϕ x 2 | i = ϕ i 1 2 ϕ i + ϕ i + 1 Δ x 2 ,
2 ϕ x 2 | i = ϕ i 2 + 16 ϕ i 1 30 ϕ i + 16 ϕ i + 1 ϕ i + 2 12 Δ x 2 ,
2 i β ϕ / z = ϕ + ( n 2 k 2 β 2 ) ϕ ,
2 i β ϕ / z = ϕ ̅ + ( n 2 ̅ k 2 β 2 ) ϕ ,
( Δ x ) 2 ( ϕ 1 + ϕ 1 2 ϕ 0 ) = ϕ ̅ + ( Δ x / 6 ) δ ϕ + O [ ( Δ x ) 2 ] ,
δ ϕ + k 2 [ ( n 1 2 n 0 2 ) ϕ + ( n 1 2 ) ϕ ] = 0 ,
ϕ ( ϕ 1 ϕ 1 ) / 2 Δ x ; ( n 1 2 ) n 0 n 2 [ | ϕ 1 | 2 | ϕ 1 | 2 ] / Δ x
ϕ ̅ = [ Δ x ] 2 ( ϕ 1 + ϕ 1 2 ϕ 0 ) + ( k 2 / 12 ) [ ( n 1 2 n 0 2 ) ( ϕ 1 ϕ 1 ) + 2 n 0 n 2 ( | ϕ 1 | 2 | ϕ 1 | 2 ) ϕ 0 ] .
2 ϕ x 2 | n = 2 ϕ x 2 | n 1 × ( ϕ n / ϕ n 1 ) ,

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