Abstract

Two computational methods are common for simulating the evolution of three beams propagating in a birefringent medium and interacting through a second-order nonlinearity: the split-step method and solution of the coupled equations for the amplitudes of the spatial frequency components of the beams (Fourier-space method). I (i) compare the accuracy and computational cost of both methods, (ii) investigate the effect of using a first-order expansion for the refractive index as a function of propagation direction, and (iii) generalize both methods to handle arbitrary propagation directions in biaxial crystals. It turns out that the Fourier-space method with a Runge–Kutta solver gives best accuracy, but a symmetrized split-step method may be faster when low accuracy is sufficient. The first-order expansion for the refractive index gives a very small error for well-collimated beams, but the approximation is not important for computational efficiency. Modeling of parametric amplification outside the principal planes of a biaxial crystal is demonstrated, and to the author's knowledge this process has not been modeled in such detail before.

© 1997 Optical Society of America

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References

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  1. S. C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
    [CrossRef]
  6. S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
    [CrossRef]
  7. M. S. Bowers and A. V. Smith, “Optical parametric oscillator modeling with diffraction, depletion, and double refraction,” in Advanced Solid-State Lasers, Vol. 20 of 1994 OSA Proceedings Series, T. Y. Fan and B. Chai, eds. (Optical Society of America, Washington, D.C., 1994), pp. 471–474.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametric generation in KTiOPO4 crystals,” Opt. Commun. 124, 512–518 (1996).
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  15. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.
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1996 (1)

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametric generation in KTiOPO4 crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

1995 (3)

1993 (2)

1990 (1)

1989 (1)

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

1984 (1)

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

1983 (1)

1980 (1)

S. C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

1978 (1)

1972 (2)

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989), Chap. 2.4.

Alford, W. J.

Banerjee, P. P.

Bloembergen, N.

Bosenberg, W. R.

Bowers, M. S.

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2260 (1995).
[CrossRef]

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

M. S. Bowers and A. V. Smith, “Optical parametric oscillator modeling with diffraction, depletion, and double refraction,” in Advanced Solid-State Lasers, Vol. 20 of 1994 OSA Proceedings Series, T. Y. Fan and B. Chai, eds. (Optical Society of America, Washington, D.C., 1994), pp. 471–474.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), Chap. 4.

Campbell, B. F.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
[CrossRef]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.

Dreger, M. A.

Fahlen, T. S.

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

Feit, M. D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 16.

Fleck, J. A.

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.

Guthals, D. M.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
[CrossRef]

Hu, P. H.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
[CrossRef]

Jarman, R. H.

Lalor, E.

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

Lera, G.

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Ma, S. S.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
[CrossRef]

McIver, J. K.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.

Nishikawa, T.

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametric generation in KTiOPO4 crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Pliszka, P.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 16.

Raymond, T. D.

Schell, A. J.

Sheng, S. C.

S. C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Siegman, A. E.

S. C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Smith, A. V.

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995).
[CrossRef]

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2260 (1995).
[CrossRef]

M. S. Bowers and A. V. Smith, “Optical parametric oscillator modeling with diffraction, depletion, and double refraction,” in Advanced Solid-State Lasers, Vol. 20 of 1994 OSA Proceedings Series, T. Y. Fan and B. Chai, eds. (Optical Society of America, Washington, D.C., 1994), pp. 471–474.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 16.

Uesugi, N.

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametric generation in KTiOPO4 crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 16.

Yao, J. Q.

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

J. Appl. Phys. (2)

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

J. Math. Phys. (2)

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

E. Lalor, “An analytical approach to the theory of internal conical refraction,” J. Math. Phys. 13, 449–454 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletion and diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametric generation in KTiOPO4 crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

S. C. Sheng and A. E. Siegman, “Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Other (6)

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), Chap. 4.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 16.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989), Chap. 2.4.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wave mixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE1868, 135–142 (1993).
[CrossRef]

M. S. Bowers and A. V. Smith, “Optical parametric oscillator modeling with diffraction, depletion, and double refraction,” in Advanced Solid-State Lasers, Vol. 20 of 1994 OSA Proceedings Series, T. Y. Fan and B. Chai, eds. (Optical Society of America, Washington, D.C., 1994), pp. 471–474.

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Figures (4)

Fig. 1
Fig. 1

Estimated relative errors of split-step (SS) and Fourier-space (FS) algorithms as functions of (a) number of steps or (b) relative run times.

Fig. 2
Fig. 2

(a) Intensity profiles of the incident beams and the output beams. Only one profile is shown for the incident beams because beams 1 and 3 are equal and beam 2 is zero. (b) Profiles of the relative amplitude errors for the Fourier-space method with two equal steps. The relative error is the maximum amplitude error divided by the maximum amplitude of the beam. (c) Corresponding error profiles for the split-step method with 16 equal steps.

Fig. 3
Fig. 3

Intensity profiles in the xz plane of the three output beams for simulations with exact and approximate refractive indices.

Fig. 4
Fig. 4

Top, interaction geometry. Note that all the beams have walk-off and that they walk in different directions. Bottom, contours of the magnitudes of the field amplitudes of the beams before and after the parametric amplifier. Beams 1 and 3 were incident, and they had the same profile. Beam 2 was generated in the interaction. The sides of the squares containing the beams are 3.2 mm.

Tables (1)

Tables Icon

Table 1 Output Power in Beam 2 Computed with Exact and Approximate Refractive Indices and Various Modifications of the Incident Beam 1a

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

ET(x, y, 0)=dkxdkyj=1,2E¯(j, kx, ky, 0)×e(j, kx, ky)exp[i(kxx+kyy)];
E¯(j, kx, ky, z)z=-αE¯(j, kx, ky, z)+ikz(j, kx, ky)E¯(j, kx, ky, z),
kz(j, kx, ky)=[k2(j, kx, ky)-kx2-ky2]1/2,
k(j, kx, ky)=2πn(j, kx, ky, kz)/λ.
P3(x, y)=P3(2)(x, y)+P3(3)(x, y)+P3(T)(x, y).
P3(2)(x, y)=2ε0χeffE1(x, y)E2(x, y),
P3(T)(x, y)=2ε0n3Δn(x, y)E3(x, y),
P3(3)(x, y)=2ε0n3ΔnNL(x, y)E3(x, y),
ΔnNL(x, y)=η{I3(x, y)+2[I1(x, y)+I2(x, y)]},
I3(x, y)=2n3Z0-1|E3(x, y)|2,
E¯3(kx, ky, z)z=ik3,z(kx, ky)E¯3(kx, ky, z)-α3E¯3+iμ0ω3c2n3F(P3),
F(P3)=1(2π)2dxdy exp[-i(kxx+kyy)]P3(x, y).
e¯3(kx, ky, z)z=iδk3,z(kx, ky)e¯3(kx, ky, z)-αe¯3+i ω3n3c[exp(-iΔkz)χeffF(e1e2)+F(n3ΔnNLe3)+F(n3Δne3)]
δk3,z(kx, ky)=k3,z(kx, ky)-k3,z
e¯3(kx, ky, z)=a¯3(kx, ky, z)exp[iδk3,z(kx, ky, z)z],
a¯3(kx, ky, z)z=-α3a¯3+i ω3n3c×exp[-iδk3,z(kx, ky, z)z]×[exp(-iΔkz)χeffF(e1e2)+F(n3ΔnNLe3)+F(n3Δne3)].
E3(x, y, z)z=iμ0ω3c2n3P3(x, y, z).

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