Abstract

We study a three-dimensional model of interaction of fundamental-frequency and second-harmonic beams in a quadratically nonlinear medium. Numerical simulations of the three-dimensional propagation problem in the presence of diffraction and anisotropy are performed under the paraxial approximation. The role of the transverse effects in various regimes is investigated. We demonstrate the effect of phase modulation and an induced nonlinear focusing during the interaction of the fundamental frequency with the generated second harmonic.

© 1993 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
    [CrossRef]
  2. S. A. Akhmanov, R. V. Khoklov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972).
  3. R. L. Byer, R. L. Herbst, Nonlinear Infrared Generation (Springer-Verlag, Berlin, 1978), p. 81.
  4. G. D. Boyd, D. A. Kleinman, J. Appl. Phys. 39, 3597 (1967).
    [CrossRef]
  5. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, H. Vanherzeele, E. W. Van Stryland, Opt. Lett. 17, 28 (1991).
    [CrossRef]
  6. H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
    [CrossRef] [PubMed]
  7. U. Österberg, W. Margulis, Opt. Lett. 12, 5 (1987).
    [CrossRef]
  8. P. Pliszka, P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” Opt. Acta (to be published).
  9. V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  10. N. C. Kothari, X. Carlotti, J. Opt. Soc. Am. B 5, 756 (1988).
    [CrossRef]
  11. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  12. M. D. Feit, J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]
  13. R. C. Eckardt, J. Reinties, IEEE J. Quantum Electron. QE-20, 1178 (1984).
    [CrossRef]
  14. F. P. Mattar, M. C. Newstein, IEEE J. Quantum Electron. QE-13, 507 (1977).
    [CrossRef]
  15. S. K. Turitsin, Sov. Phys. JETP 64, 979 (1986).

1991 (1)

1990 (1)

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

U. Österberg, W. Margulis, Opt. Lett. 12, 5 (1987).
[CrossRef]

1986 (1)

S. K. Turitsin, Sov. Phys. JETP 64, 979 (1986).

1984 (1)

R. C. Eckardt, J. Reinties, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

1977 (1)

F. P. Mattar, M. C. Newstein, IEEE J. Quantum Electron. QE-13, 507 (1977).
[CrossRef]

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1967 (1)

G. D. Boyd, D. A. Kleinman, J. Appl. Phys. 39, 3597 (1967).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khoklov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Bakker, H. J.

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Banerjee, P. P.

P. Pliszka, P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” Opt. Acta (to be published).

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Boyd, G. D.

G. D. Boyd, D. A. Kleinman, J. Appl. Phys. 39, 3597 (1967).
[CrossRef]

Byer, R. L.

R. L. Byer, R. L. Herbst, Nonlinear Infrared Generation (Springer-Verlag, Berlin, 1978), p. 81.

Carlotti, X.

DeSalvo, R.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt, J. Reinties, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

Feit, M. D.

M. D. Feit, J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

M. D. Feit, J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Hagan, D. J.

Herbst, R. L.

R. L. Byer, R. L. Herbst, Nonlinear Infrared Generation (Springer-Verlag, Berlin, 1978), p. 81.

Khoklov, R. V.

S. A. Akhmanov, R. V. Khoklov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972).

Kleinman, D. A.

G. D. Boyd, D. A. Kleinman, J. Appl. Phys. 39, 3597 (1967).
[CrossRef]

Kothari, N. C.

Kuipers, L.

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Lagendijk, A.

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Margulis, W.

U. Österberg, W. Margulis, Opt. Lett. 12, 5 (1987).
[CrossRef]

Mattar, F. P.

F. P. Mattar, M. C. Newstein, IEEE J. Quantum Electron. QE-13, 507 (1977).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Newstein, M. C.

F. P. Mattar, M. C. Newstein, IEEE J. Quantum Electron. QE-13, 507 (1977).
[CrossRef]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Österberg, U.

U. Österberg, W. Margulis, Opt. Lett. 12, 5 (1987).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Planken, P. C. M.

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Pliszka, P.

P. Pliszka, P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” Opt. Acta (to be published).

Reinties, J.

R. C. Eckardt, J. Reinties, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

Sheik-Bahae, M.

Stegeman, G.

Turitsin, S. K.

S. K. Turitsin, Sov. Phys. JETP 64, 979 (1986).

Van Stryland, E. W.

Vanherzeele, H.

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. C. Eckardt, J. Reinties, IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

F. P. Mattar, M. C. Newstein, IEEE J. Quantum Electron. QE-13, 507 (1977).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd, D. A. Kleinman, J. Appl. Phys. 39, 3597 (1967).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Phys. Rev. A (1)

H. J. Bakker, P. C. M. Planken, L. Kuipers, A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

S. K. Turitsin, Sov. Phys. JETP 64, 979 (1986).

Other (4)

S. A. Akhmanov, R. V. Khoklov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972).

R. L. Byer, R. L. Herbst, Nonlinear Infrared Generation (Springer-Verlag, Berlin, 1978), p. 81.

P. Pliszka, P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” Opt. Acta (to be published).

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (9)

Fig. 1
Fig. 1

Wave-vector surfaces for the wave numbers k1 and k2/2. Θ0 is the angle corresponding to synchronous wave vectors, and Θ is the angle corresponding to a particular component of the angular spectrum of E1; ρ is the walk-off angle between k2 and the Poynting vector S2.

Fig. 2
Fig. 2

(a) Transverse profile of amplitude A1 for the plane-wave limit (κ < 10−3) and ΔL = 0 for different propagation distances z/Inl. (b) On-axis intensity and power of the FF for κ = 0 and ΔL = 0.

Fig. 3
Fig. 3

(a) Transverse profile of amplitudes A1 and A2 for κ = 0 and ΔLnl = 4 for different propagation distances z/Lnl. (b) Powers P1 of the FF beam and P2 of the SH beam versus z for κ = 0 and ΔLnl = 0.5.

Fig. 4
Fig. 4

(a) Transverse profile of amplitude A1 for κ = 1/40 and ΔL = 0 for different propagation distances z/Lnl. (b) On-axis intensities I1 (crosses) and I2 (squares) and powers P1 and P2 of FF and SH radiation (in atomic units) for κ ≈ 1/50 and ΔL = 0.

Fig. 5
Fig. 5

On-axis phases of the FF for different values of the parameters κ and Δ. The curves illustrate the following: a, phase instability in the absence of diffraction (κ = 0, Δ = 0); b, weak self-defocusing in the plane-wave limit (κ = 0, Δ = 10); c, initial diffraction followed by phase instability for κ = 1/20 and Δ = 0; d, same as c but for κ = 1/40 and Δ = 0; e, self-defocusing and diffraction in the weak nonlinearity regime (κ = 1/20, Δ = 10). The distance of propagation is given in units of a typical diffraction length Ld, since the specific values of Lnl and Ld are different for each curve. For curve d the value of Lnl is half the value of that of curve c.

Fig. 6
Fig. 6

(a) Total power and (b) on-axis intensity of the SH beam as a function of the propagation distance for different widths of the input beam (given in wavelengths). The total power of the input FF is kept fixed (at 2500 in the units given). For the w = 80λ case the distance z = 1 corresponds to Lnl and κ = 1, while, for w = 10λ, κ ≈ 8 holds since κ ~ 1/w.

Fig. 7
Fig. 7

Transverse profiles of amplitudes A1 and A2 for κ = 1/20 and ΔL = 10 for different propagation distances z/Lnl.

Fig. 8
Fig. 8

Transverse profiles (cross section along the x axis) of the FF and the SH in the presence of anisotropy in the x direction.

Fig. 9
Fig. 9

Gray-scale intensity distribution map for the FF and the SHG waves for the case of focused input beam and anisotropy (low-intensity regime): (a) the case ρ ≠ 0 and Δ = 0, (b) beams at the exit plane in the case ρ ≠ 0 and Δ ≠ 0. Note the coincidence of the focal point for the FF with the maximum intensity of the SH. The vertical and horizontal scales in the zx cross section differ by a factor of the order of Lnl/w, and ρ is of the order of w/Ld.

Equations (39)

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E 1 z = i σ exp ( i Δ z ) E 1 * E 2 i 2 k 1 2 E 1 ,
( z ρ x ) E 2 = i σ exp ( i Δ z ) E 1 2 1 2 k 2 2 E 2 ,
P 1 + P 2 = const ., P i = d x d y | E i | 2 ,
E 1 = 2 E 1 , E 2 = exp ( i Δ z ) E 2 ,
E j z = i δ H δ E j * .
H = d x d y σ E 1 2 2 E 2 * + c . c . + 1 2 k 1 | E 1 | 2 + 1 2 k 2 | E 2 | 2 | E 2 | 2 Δ i ρ ( E 2 * E 2 x c . c . )
E 2 ( x , y , z ) = c ( z ) E 1 2 ( x , y , z ) .
[ L ˆ ( z , λ ) ψ ] 2 L ˆ ( z , λ / 2 ) ψ 2 ,
I 2 ( z ) = | E 2 | 2 = I 1 ( 0 ) υ 2 sn 2 [ ( z / l nl ) υ , υ 4 ] ,
υ 1 = Δ l nl / 4 + [ 1 + ( Δ l nl / 4 ) 2 ] 1 / 2 , l nl = l nl ( x , y ) = k 1 / σ | E 1 | ( x , y , 0 ) , l nl ( 0 , 0 ) = L nl , I 1 ( 0 ) = | E 1 | 2 ( x , y , 0 ) ,
Λ = [ Δ 4 + ( 1 l nl 2 + Δ 2 16 ) 1 / 2 ] 1 .
ϕ 1 z = σ A 2 cos θ ( ϕ 1 ) 2 2 k 1 + 2 A 1 2 k 1 A 1 ,
ϕ 2 z = σ A 1 2 A 2 cos θ ( ϕ 2 ) 2 2 k 2 + 2 A 2 2 k 2 A 2 ( sin ρ ) ϕ 2 x ,
E j ( x , y , z ) = exp [ i z ( 2 / 2 k j + N ˆ j ) ] E j ( x , y , 0 ) , j = 1 , 2 ,
exp [ i z ( 2 / 2 k j + N ˆ j ) ] exp [ i z ( 2 / 2 k j ) ] exp ( i z N ˆ j ) .
E ˜ 2 ( k 2 ) : : d 3 k a d 3 k b E ˜ 1 ( k a ) E ˜ 1 ( k b ) × δ ( k 2 k a k b ) δ [ k 2 2 4 ω 2 c 2 n e 2 2 ( k 2 ) ] × δ [ k a 2 ω 2 c 2 n o 1 2 ( k a ) ] δ [ k b 2 ω 2 c 2 n o 1 2 ( k b ) ] ,
E ˜ 1 ( k 1 ) : : d 3 k a d 3 k b E ˜ 1 ( k a ) E ˜ 2 ( k b ) × δ ( k 1 + k a k b ) δ [ k 1 2 ω 2 c 2 n e 2 2 ( k 2 ) ] × δ [ k a 2 ω 2 c 2 n o 1 2 ( k a ) ] δ [ k b 2 4 ω 2 c 2 n o 1 2 ( k b ) ] .
E ˜ 1 ( k ) : : exp [ i z Δ ( k ) ] [ E ˜ 1 * ( k ) * E ˜ 2 ] ( k ) ,
E ˜ 2 ( k ) : : exp [ i z Δ ( k ) ] ( E ˜ 1 * E ˜ 1 ) ( k ) ,
Δ ( k ) = Δ ( 0 ) + k 1 cos ( Θ Θ 0 ) [ 1 ( n e 2 n o 1 ) 2 ] ( 1 cos 2 Θ cos 2 Θ ) Δ ( 0 ) + α 1 k x + α 2 k x 2 + β 2 k y 2 ,
σ exp ( i Δ z ) [ ( E 2 + ρ z E 2 x ) E 1 * + E 2 ρ z E 1 * x ] ,
σ exp ( i Δ z ) E 1 ( E 1 2 ρ z E 1 x ) ,
E 1 z = i 2 k 1 2 E 1 ,
( z ρ x ) E 2 = i σ exp ( i Δ z ) E 1 2 i 2 k 2 2 E 2 .
E 1 2 ( r , z ) = E 1 ( 0 , z ) E 1 ( 2 r , z ) ,
E ˜ j ( k , z ) = 1 2 π exp ( i k r ) E j ( r , z ) d 2 r ,
E ˜ 2 z = ( i k 2 2 k 2 i ρ k x ) E ˜ 2 i σ 2 exp ( i Δ z ) E 1 ( 0 , z ) E ˜ 1 ( k 2 ) .
E ˜ 2 ( k , z ) = i σ 2 0 z exp [ i ( k 2 2 k 2 ρ k x ) ( z z ) i Δ z ] × E 1 ( 0 , z ) E ˜ 1 ( k 2 , z ) d z
= i σ 2 exp ( ρ k x z ) E ˜ 1 ( k 2 , z ) × 0 z exp [ i ( Δ k 2 4 k 2 k 1 Δ + ρ k x ) z ] E 1 ( 0 , z ) d z .
E ˜ 1 ( k 2 , z ) = exp [ i k 2 4 k 1 ( z z ) ] E ˜ 1 ( k 2 , z ) .
E 2 ( r , z ) = [ i σ E 1 ( 0 , z ) 0 z E 1 ( 0 , z ) exp ( i Δ z ) d z ] E 1 2 ( r , z ) = c ( z ) E 1 2 ( r , z ) ,
E 1 z = i 1 2 k 1 2 E 1 i χ ( z ) E 1 | E 1 | 2 ,
E j ( r , z ) = E j , 0 ( z ) + δ j ( r , z ) .
d δ ˜ 1 ( k ) d z = i k 2 2 k 1 δ ˜ 1 ( k ) i σ [ E 1 , 0 * δ ˜ 2 ( k ) + δ ˜ 1 * ( k ) ] ,
d δ ˜ 2 ( k ) d z = ( i k 2 2 k 2 + i Δ ) δ ˜ 2 ( k ) 2 i σ E 1 , 0 δ ˜ 1 ( k ) .
d d z δ ˜ = M ˆ δ ˜ , δ ˜ = ( x ˜ 1 , y ˜ 1 , x ˜ 2 , y ˜ 2 ) T ,
M ˆ = [ b 2 a 2 + χ 1 b 1 a 1 a 2 χ 1 b 2 a 1 b 1 2 b 1 2 a 1 0 χ 2 2 a 1 2 b 1 χ 2 0 ] , χ 1 = k 2 / 2 k 1 , χ 2 = Δ + k 2 / 2 k 2 .
λ = ± ( 1 / 2 ) { D ± [ D 2 4 ( 4 a 1 4 4 a 1 2 χ 1 χ 2 a 2 2 χ 2 2 + χ 1 2 χ 2 2 ) ] 1 / 2 } 1 / 2 , D = 4 a 1 2 + a 2 2 χ 1 2 χ 2 2 .
λ = ± [ | σ E 2 | 2 ( k 2 / 2 k 1 ) 2 ] 1 / 2 , ± i k 2 / 2 k 2 .

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