Abstract

A numerical analysis of the reflection of a two dimensional Gaussian beam from the interface between a linear and a nonlinear medium is presented. The refractive index of the nonlinear medium is a function of the intensity of the radiation field, having a smaller value than the linear refractive index for zero field intensity. The Gaussian beam is incident from the linear medium and suffers total reflection at low intensity. At sufficiently high intensity nonlinear effects are observed. Above a threshold value the incident beam breaks up into a reflected wave and a surface wave. Once the beam is sufficiently strong for a surface wave to form, its interaction with the boundary becomes surprisingly independent of field intensity; but for very strong fields the reflectivity is increased at the expense of the surface wave. A very different behavior is observed when the refractive index is constrained to remain below a certain maximum value. Now the field detaches itself from the surface and penetrates into the nonlinear medium forming one or more distinct beams. The plane wave theory predicts the existence of hysteresis so that two different solutions should exist for the same physical parameters. A second solution was indeed found in one case with constrained refractive index, but its validity is somewhat uncertain at this time.

© 1980 Optical Society of America

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References

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  1. A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).
  2. A. E. Kaplan, Radiophys. Quantum Electron. 22, 229 (1979).
    [CrossRef]
  3. P. W. Smith, J.-P. Hermann, W. J. Tomlinson, P. J. Maloney, Appl. Phys. Lett. 35, 846 (1979).
    [CrossRef]
  4. W. J. Tomlinson, Opt. Lett., 5, 323 (1980).
    [CrossRef] [PubMed]
  5. D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 610.
  6. Ref. 5, Vol. 1, p. 458.
  7. Ref. 5, p. 1077.
  8. D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972), p. 234.
  9. Ref. 8, p. 232.

1980 (1)

1979 (2)

A. E. Kaplan, Radiophys. Quantum Electron. 22, 229 (1979).
[CrossRef]

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

1977 (1)

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

Gregory, R. T.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 610.

Hermann, J.-P.

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, Radiophys. Quantum Electron. 22, 229 (1979).
[CrossRef]

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

Maloney, P. J.

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

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972), p. 234.

Smith, P. W.

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

Tomlinson, W. J.

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]

Young, D. M.

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 610.

Appl. Phys. Lett. (1)

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

Opt. Lett. (1)

Radiophys. Quantum Electron. (1)

A. E. Kaplan, Radiophys. Quantum Electron. 22, 229 (1979).
[CrossRef]

Sov. Phys. JETP (1)

A. E. Kaplan, Sov. Phys. JETP 45, 896 (1977).

Other (5)

D. M. Young, R. T. Gregory, A Survey of Numerical Mathematics (Addison-Wesley, Reading, Mass., 1973), Vol. 2, p. 610.

Ref. 5, Vol. 1, p. 458.

Ref. 5, p. 1077.

D. Marcuse, Light Transmission Optics (Van Nostrand-Reinhold, New York, 1972), p. 234.

Ref. 8, p. 232.

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Figures (10)

Fig. 1
Fig. 1

Geometry of beam and interface.

Fig. 2
Fig. 2

(a) Linear case r2 = 0. Cross section of Gaussian beam at the input plane. (b) Gaussian beam at z = 0. (c) Gaussian beam at z = 75 μm. (d) The beam at its output position resumes its input shape except for a slight asymmetry relative to z = 0 due to the Goos-Haenchen shift.

Fig. 3
Fig. 3

Nonlinear case r2 = 0.01 without clipping. At the output plane z = +200 μm a surface wave and a reflected beam are present.

Fig. 4
Fig. 4

Very strong nonlinearity r2 = 0.1 without clipping. The reflected beam is stronger and the surface wave is weaker than in Fig. 3.

Fig. 5
Fig. 5

(a) Nonlinear case r2 = 0.01, n2max = 1.52, shown at z = 75 μm. The input field is identical to Fig. 2(a). (b) The field at the output plane z = 200 μm, where it has formed a surface wave.

Fig. 6
Fig. 6

(a) Nonlinear case r2 = 0.01, n2max = 1.52; attempt at finding a second solution. (b) The difference between this figure and Fig. 3(b) suggests that a second solution has been found.

Fig. 7
Fig. 7

(a) Nonlinear case r2 = 0.015, n2max = 1.52, field at z = 75 μm, where a beam has already broken through the interface. (b) Field at output plane z = 200 μm.

Fig. 8
Fig. 8

(a) Nonlinear case r2 = 0.02, n2max = 1.52 at z = 0. (b) z = 75 μm. (c) z = 200 μm. A self-guided self-curving beam and a surface wave have formed.

Fig. 9
Fig. 9

(a) Nonlinear case r2 = 0.03, n2max = 1.52 shown at z = 0, where a beam has already broken through the interface. (b) The field at z = 75 μm, where two beams have formed. (c) The two beams at output plane z = 200 μm.

Fig. 10
Fig. 10

(a) Strongly nonlinear case r2 = 0.1, n2max = 1.52 at z = 0. (b) Multiple beams at z = 75 μm. (c) Five beams have formed in the nonlinear medium and are shown here at the output plane at z = 200 μm.

Equations (21)

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n 2 2 = n 02 2 + r 2 | E | 2 .
n 2 = { n 20 + r 2 | E | 2 for n 2 < n 2 n 2 max for r 2 | E | 2 > n 2 max n 20 .
2 E x 2 + 2 E z 2 + n 2 ( x , z ) k 2 E = 0
k = 2 π / λ ,
E = ϕ ( x , z ) exp ( i n 0 k z ) .
2 ϕ x 2 2 i n 0 k ϕ z + [ n 2 ( x , z ) n 0 2 ] k 2 ϕ = 0.
ϕ = ϕ Re + i ϕ Im
ϕ Re z = 1 2 n 0 k { 2 ϕ Im x 2 + [ n 2 ( x , z ) n 0 2 ] k 2 ϕ Im } ,
ϕ Im z = 1 2 n 0 k { 2 ϕ Re x 2 + [ n 2 ( x , z ) n 0 2 ] k 2 ϕ Re } .
2 ϕ x 2 = ϕ i + 2 , j + 16 ϕ i + 1 , j 30 ϕ i , j + 16 ϕ i 1 , j ϕ i 2 , j 12 h x 2 .
ϕ ( x i , z j ) = ϕ i , j
ϕ z = F ( z ) ,
ϕ i , j + 1 ( P ) = ϕ i , j + h z 24 ( 55 F j 59 F j 1 + 37 F j 2 9 F j 3 ) .
ϕ i , j + 1 = ϕ i , j + h z 24 ( 9 F j + 1 ( P ) + 19 F j 5 F j 1 + F j 2 ) .
ϕ i , j + 1 = ϕ i + h z F j .
h z < K h x 2 ,
ϕ = exp { u 2 w 2 ( υ ) i [ k ( n 1 υ n 0 L ) + n 1 π u 2 λ R ( υ ) 1 2 arctan ( λ υ n 1 π w 0 2 ) ] } .
w ( υ ) = w 0 [ 1 + ( λ υ n 1 π w 0 2 ) 2 ] 1 / 2 ,
R ( υ ) = υ [ 1 + ( n 1 π w 0 2 λ υ ) 2 ] .
u = ( x x i ) cos θ ,
υ = L + ( x x i ) sin θ .

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