Abstract

We show that if two waves are incident on a quadratically nonlinear crystal, with the third wave generated entirely within the crystal, a phase-velocity mismatch (Δk ≠ 0) leads to intensity-dependent phase shifts of the generated wave only if there is walk-off, linear absorption, or significant diffraction of at least one of the waves as well as significant energy exchange among the waves. The result is frequency broadening and wave-front distortion of the generated wave. Although the induced phase distortions are usually quite small, they may be significant in applications that require high spectral resolution or pointing accuracy.

© 1995 Optical Society of America

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References

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  1. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [Crossref]
  2. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Ascend, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
    [Crossref] [PubMed]
  3. R. DeSalvo, D. J. Hagar, M. Sheik-Bahae, G. Stegeman, and E. Van Stryland, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [Crossref] [PubMed]
  4. J.-M. R. Thomas and J.-P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–333 (1972).
    [Crossref]
  5. P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993);“Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
    [Crossref]
  6. R. L. Byer and R. L. Herbst, “Parametric oscillation and mixing,” in Nonlinear Infrared Generation, Y. R. Shen, ed. (Springer-Verlag, Berlin, 1977).
    [Crossref]
  7. J. A. Armstrong, N. Bloebergen, J. Ducuin, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  8. M. A. Dreger and J. K. McIver, “Second-harmonic generation in a nonlinear anisotropic medium with diffraction and depletion,” J. Opt. Soc. Am. B 7, 776–784 (1990).
    [Crossref]
  9. T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).
  10. S. Gangopadhyay, N. Melikechi, and E. E. Eyler, “Optical phase perturbations in nanosecond pulsed amplification and second harmonic generation,” J. Opt. Soc. Am. B 11, 231–241 (1994).
    [Crossref]
  11. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  12. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [Crossref]
  13. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
    [Crossref]

1994 (1)

1993 (3)

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Ascend, “Large nonlinear phase shifts in second-order nonlinear-optical processes,” Opt. Lett. 18, 13–15 (1993).
[Crossref] [PubMed]

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993);“Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[Crossref]

T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).

1992 (1)

1991 (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

1990 (1)

1972 (1)

J.-M. R. Thomas and J.-P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–333 (1972).
[Crossref]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

1962 (1)

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

Alford, W. J.

T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).

Armstrong, J. A.

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

Ascend, G.

Banerjee, P. P.

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993);“Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[Crossref]

Bloebergen, N.

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

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Byer, R. L.

R. L. Byer and R. L. Herbst, “Parametric oscillation and mixing,” in Nonlinear Infrared Generation, Y. R. Shen, ed. (Springer-Verlag, Berlin, 1977).
[Crossref]

DeSalvo, R.

Dreger, M. A.

Ducuin, J.

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

Eyler, E. E.

Gangopadhyay, S.

Hagar, D. J.

Herbst, R. L.

R. L. Byer and R. L. Herbst, “Parametric oscillation and mixing,” in Nonlinear Infrared Generation, Y. R. Shen, ed. (Springer-Verlag, Berlin, 1977).
[Crossref]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

McIver, J. K.

Melikechi, N.

Pershan, P. S.

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

Pliszka, P.

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993);“Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[Crossref]

Raymond, T. D.

T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).

Sheik-Bahae, M.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[Crossref]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Smith, A. V.

T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).

Stegeman, G.

Stegeman, G. I.

Taran, J.-P. E.

J.-M. R. Thomas and J.-P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–333 (1972).
[Crossref]

Thomas, J.-M. R.

J.-M. R. Thomas and J.-P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–333 (1972).
[Crossref]

Van Stryland, E.

Bull. Am. Phys. Soc. (1)

T. D. Raymond, W. J. Alford, and A. V. Smith, “Frequency shifts in seeded optical parametric oscillators with phase mismatch,” Bull. Am. Phys. Soc. 38, 1715 (1993).

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

J. Mod. Opt. (1)

P. Pliszka and P. P. Banerjee, “Self-phase modulation in quadratically nonlinear media,” J. Mod. Opt. 40, 1909–1916 (1993);“Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[Crossref]

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

Opt. Commun. (1)

J.-M. R. Thomas and J.-P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun. 4, 329–333 (1972).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

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

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

R. L. Byer and R. L. Herbst, “Parametric oscillation and mixing,” in Nonlinear Infrared Generation, Y. R. Shen, ed. (Springer-Verlag, Berlin, 1977).
[Crossref]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Example fluence profiles for (a) the input fundamental wave and (b) the output second-harmonic wave. The second-harmonic wave walks off in the +x direction. Δk = 0.15 mm−1.

Fig. 2
Fig. 2

Phases of the output second-harmonic optical field for Δk = 0.15 mm−1 at three positions on the calculational grid. The y positions are zero, and the x positions are as labeled.

Fig. 3
Fig. 3

Numerical results for frequency doubling a 7-ns (FWHM Gaussian), 0.25-mm-diameter (FWHM lowest-order Gaussian) pulse of 645-nm light in a 3-cm-long KDP crystal. The conditions are described for Case 1 in the text. Results for (a)–(e) Δk = 0, (f)–(j) Δk = 0.15 mm−1. In (a) and (f) the solid curve is one half of the 645-nm power and the dashed curve is the full 322.5-nm power. The tilt, curvature, and M2 characterize the wave-front distortions in the walk-off direction. The heterodyne phases of the depleted fundamental and generated second harmonic are at the exit face of the crystal.

Fig. 4
Fig. 4

Numerical results for frequency doubling a 7-ns (FWHM), 14-μm waist-diameter (FWHM) pulse of 645-nm light in a 2-mm-long KDP crystal with Δk = 0. The conditions are described for Case 2 in the text. In (a) the solid curve is one half of the 645-nm power and the dashed curve is the full 322.5-nm power. The tilt, curvature, and M2 characterize the wave–front distortions in the walk-off direction. The heterodyne phases of the depleted fundamental and generated second harmonic are at the exit face of the crystal.

Equations (31)

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d ɛ s d z = i d eff c ω s n s ɛ p ɛ i * exp ( i Δ k z ) α s ɛ s , d ɛ i d z = i d eff c ω i n i ɛ p ɛ i * exp ( i Δ k z ) α i ɛ i , d ɛ p d z = i d eff c ω p n p ɛ s ɛ i exp ( i Δ k z ) α p ɛ p ,
E ω = ½ { ɛ ω exp [ i ( ω t k z ) ] + ɛ ω * exp [ i ( ω t k z ) ] } ,
Δ k = k p k s k i ,
ω p = ω s ω i .
ɛ n = n exp ( i ϑ n ) ,
d s d z = p i d eff c ω s n s sin θ α s s , d i d z = p s d eff c ω i n i sin θ α i i , d p d z = + s i d eff c ω p n p sin θ α p p , d ϑ s d z = d eff c ω s n s p i s cos θ , d ϑ i d z = d eff c ω i n i p s i cos θ , d ϑ p d z = d eff c ω p n p s i p cos θ ,
θ = ϑ p ϑ i ϑ s + Δ k z .
cos θ = ( Γ c Δ k n p p 2 2 d eff ω p ) s i p ,
cos θ = Δ k c 2 d eff n i ω i i s p
cos θ = Δ k c 2 d eff n p ω p p s i
d ϑ i d z = + Δ k 2 .
d ϑ p d z = Δ k 2 ,
d ϑ i d z = d eff ω I c n i i 2 { i ( 0 ) s ( 0 ) p ( 0 ) cos θ ( 0 ) + Δ k c n p 2 d eff ω p [ p 2 ( 0 ) p 2 ] } .
n i ω i [ i 2 i 2 ( 0 ) ] = n p ω p [ p 2 ( 0 ) p 2 ]
d ϑ i d z = d eff ω I c n i i 2 [ s ( 0 ) i ( 0 ) p ( 0 ) cos θ ( 0 ) Δ k c n i 2 d eff ω i i 2 ( 0 ) ] + Δ k 2 .
cos θ ( 0 ) + Δ k c n i 2 d eff ω i i ( 0 ) s ( 0 ) p ( 0 ) .
ɛ j ( x , y , z , t ) z = i 2 k j [ 2 ɛ j ( x , y , z , t ) y 2 + 2 ɛ j ( x , y , z , t ) x 2 ] tan ρ ɛ j ( x , y , z , t ) x + P j ( x , y , z , t ) α j ɛ j ( x , y , z , t ) ,
P s ( x , y , z , t ) = i d eff c ω s n s ɛ p ( x , y , z , t ) × ɛ i * ( x , y , z , t ) exp ( i Δ k z ) , P i ( x , y , z , t ) = i d eff c ω i n i ɛ p ( x , y , z , t ) × ɛ s * ( x , y , z , t ) exp ( i Δ k z ) , P p ( x , y , z , t ) = i d eff c ω p n p ɛ i ( x , y , z , t ) × ɛ s ( x , y , z , t ) exp ( i Δ k z ) .
ɛ j ( x , y , z , t ) = ɛ j ( s x , s y , z , t ) × exp [ i 2 π ( s x x , s y y ) ] d s x d s y , P j ( s x , s y , z , t ) = P j ( s x , s y , z , t ) × exp [ i 2 π ( s x x , s y y ) ] d s x d s y ,
ɛ j ( s x , s y , z , t ) z = i [ 2 π 2 k j ( s x 2 + s y 2 ) + 2 π s y tan ρ ] × ɛ j ( s x , s y , z , t ) + P j ( s x , s y , z , t ) .
β x ( t ) = λ s x | ɛ ( s x , s y , t ) | 2 d s x d s y | ɛ ( s x , s y , t ) | 2 d s x d s y ,
ɛ ( s x , s y , t ) = ɛ ( x , y , t ) exp [ i 2 π ( s x x + s y y ) ] d x d y .
c x ( t ) = σ x 2 ( t ) λ R x ( t ) ,
R x ( t ) = Z x ( t ) [ 1 + σ ox 2 ( t ) λ 2 σ s x 2 ( t ) Z x 2 ( t ) ] .
Z x ( t ) = A x ( t ) + 2 β x ( t ) x ¯ ( t ) 2 λ 2 σ s x 2 ( t ) ,
σ ox 2 ( t ) = σ ox 2 ( t ) Z x 2 ( t ) λ 2 σ s x 2 ( t ) .
x ¯ ( t ) = x | ɛ ( x , y , t ) | 2 d x d y | ɛ ( x , y , t ) | 2 d x d y .
A x ( t ) = i λ 2 π s x [ ɛ ( s x , s y , t ) ɛ * ( s x , s y , t ) s x c . c . ] d s x d s y | ɛ ( s x , s y , t ) | 2 d s x d s y ;
σ x 2 ( t ) = [ x x ¯ ( t ) ] 2 | ɛ ( x , y , t ) | 2 d x d y | ɛ ( x , y , t ) | 2 d x d y ,
σ s x 2 ( t ) = [ λ s x β x ( t ) ] 2 | ɛ ( s x , s y , t ) | 2 d s x d s y λ 2 | ɛ ( s x , s y , t ) | 2 d s x d s y .
M x 2 ( t ) = 4 π σ ox ( t ) σ s x ( t ) .

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