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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    [CrossRef]
  5. M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletionand 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 wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 135–142 (1993).
    [CrossRef]
  7. M. S. Bowers and A. V. Smith, “Optical parametric oscillatormodeling with diffraction, depletion, and double refraction,” in Advanced Solid-State Lasers, Vol. 20 of 1994 OSA ProceedingsSeries, 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]
  12. T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametricgeneration in KTiOPO4crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
    [CrossRef]
  13. T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametricgeneration in KTiOPO4crystals,” Opt. Commun. 124, 512–518 (1996).
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    [CrossRef]

1996 (1)

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametricgeneration in KTiOPO4crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

1995 (3)

1993 (3)

P. Pliszka and P. P. Banerjee, “Nonlinear transverse effects in second-harmonic generation,” J. Opt. Soc. Am. B 10, 1810–1819 (1993).
[CrossRef]

W. R. Bosenberg and R. H. Jarman, “Type-II phase matched KNbO3optical parametric oscillator,” Opt. Lett. 18, 1323–1325 (1993).
[CrossRef]

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 135–142 (1993).
[CrossRef]

1990 (1)

1989 (1)

M. Nieto-Vesperinas and G. Lera, “Solution to non-linear optical frequency mixing equations with depletionand 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-harmonicgeneration 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]

Alford, W. J.

Banerjee, P. P.

Bloembergen, N.

Bosenberg, W. R.

Bowers, M. S.

Campbell, B. F.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 135–142 (1993).
[CrossRef]

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.

Fleck, J. A.

Guthals, D. M.

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 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 wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 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 depletionand 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 wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 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 depletionand diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

Nishikawa, T.

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametricgeneration in KTiOPO4crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametricgeneration in KTiOPO4crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Pliszka, P.

Raymond, T. D.

Schell, A. J.

Sheng, S. C.

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

Smith, A. V.

Uesugi, N.

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametricgeneration in KTiOPO4crystals,” Opt. Commun. 124, 512–518 (1996).
[CrossRef]

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametricgeneration in KTiOPO4crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

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 parametricgeneration in KTiOPO4crystals,” 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 depletionand diffraction: difference frequency generation,” Opt. Commun. 69, 329–333 (1989).
[CrossRef]

T. Nishikawa and N. Uesugi, “Transverse beam profile characteristics of traveling-wave parametricgeneration in KTiOPO4crystals,” 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-harmonicgeneration with depletion and diffraction,” Phys. Rev. A 21, 599–606 (1980).
[CrossRef]

Proc. SPIE (1)

S. S. Ma, D. M. Guthals, B. F. Campbell, and P. H. Hu, “Three-dimensional anisotropic physical optics modeling of three wavemixing,” in Laser Resonators and Coherent Optics: Modeling, Technology, and Applications, A. Bhowmik, ed., Proc. SPIE 1868, 135–142 (1993).
[CrossRef]

Other (5)

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

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. (CambridgeU. Press, New York, 1992), Chap. 16.

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

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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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