Abstract

The simulation of nonlinear wave propagation in the ray regime, i.e., in the limit of geometrical optics, is discussed. The medium involved is nonlinear, which means that the field amplitudes affect the constitutive parameters (e.g., dielectric constant) involved in the propagation formalism. Conventionally, linear ray propagation is computed by the use of Hamilton’s ray equations whose terms are derived from the appropriate dispersion equation. The formalism used to solve such a set of equations is the Runge–Kutta algorithm in one of its variants. In the present case of nonlinear propagation, a proper dispersion equation must first be established from which the rays can be computed. Linear ray tracing with Hamilton’s ray theory allows for the computation of ray trajectories and wave fronts. The convergence or divergence of rays suggests heuristic methods for computing the variation of amplitudes. Here, terms appearing in the Hamiltonian ray equations involve field amplitudes, which themselves are determined by the convergence (or divergence) of the rays. This dictates the simultaneous computation of a beam comprising many rays, so it is necessary to modify the original Runge–Kutta scheme by building into it some iteration mechanism such that the process converges to the values that take into account the amplitude effect. This research attempts to modify the existing propagation formalism and apply the new algorithm to simple problems of nonlinear ray propagation. The results display self-focusing effects characteristic of nonlinear optics problems. The influence of weak losses on the beam propagation and its self focusing is also discussed. Some displayed results obtained by simulating the modified formalism seem to be physically plausible and are in excellent agreement with experimental results reported in the literature.

© 1998 Optical Society of America

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References

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  1. D. Censor, “Ray tracing in weakly nonlinear moving medium,” J. Plasma Phys. 16, 415–426 (1976).
    [CrossRef]
  2. D. Censor, “Ray theoretic analysis of spatial and temporal self-focusing in general weakly nonlinear medium,” Phys. Rev. A 16, 1673–1677 (1977).
    [CrossRef]
  3. D. Censor, “Ray propagation and self-focusing in nonlinear absorbing medium,” Phys. Rev. A 18, 2614–2617 (1978).
    [CrossRef]
  4. D. Censor, “Scattering by weakly nonlinear objects,” SIAM J. Appl. Math. 43, 1400–1417 (1983).
    [CrossRef]
  5. D. Censor, “Waveguide and cavity oscillations in the presence of nonlinear medium,” IEEE Trans. Microwave Theory Tech. 33, 296–301 (1985).
    [CrossRef]
  6. I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
    [CrossRef]
  7. I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).
  8. I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
    [CrossRef]
  9. M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
    [CrossRef]
  10. M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
    [CrossRef]
  11. D. Censor, I. Gurwich, and M. Sonnenschein, “Volterra’s functionals series and wave propagation in weakly nonlinear medium: the problematics of first-principles physical modeling,” in Volterra Equations and Applications, Proceedings of The Volterra Centennial Symposium, C. Corduneanu and I. W. Sandberg, eds. (Marcel Dekker, New York, 1998).
  12. D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
    [CrossRef]
  13. D. Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ. 15, 215–223 (1978).
  14. J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
    [CrossRef]
  15. B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
    [CrossRef]
  16. J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).
    [CrossRef]

1997

D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
[CrossRef]

1996

I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
[CrossRef]

1995

I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).

1994

I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
[CrossRef]

1991

M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
[CrossRef]

1990

M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
[CrossRef]

1989

B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
[CrossRef]

1986

J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
[CrossRef]

1985

D. Censor, “Waveguide and cavity oscillations in the presence of nonlinear medium,” IEEE Trans. Microwave Theory Tech. 33, 296–301 (1985).
[CrossRef]

1983

D. Censor, “Scattering by weakly nonlinear objects,” SIAM J. Appl. Math. 43, 1400–1417 (1983).
[CrossRef]

1978

D. Censor, “Ray propagation and self-focusing in nonlinear absorbing medium,” Phys. Rev. A 18, 2614–2617 (1978).
[CrossRef]

D. Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ. 15, 215–223 (1978).

1977

D. Censor, “Ray theoretic analysis of spatial and temporal self-focusing in general weakly nonlinear medium,” Phys. Rev. A 16, 1673–1677 (1977).
[CrossRef]

1976

D. Censor, “Ray tracing in weakly nonlinear moving medium,” J. Plasma Phys. 16, 415–426 (1976).
[CrossRef]

1972

J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).
[CrossRef]

Ben-Shimol, Y.

D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
[CrossRef]

Carman, R. L.

J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).
[CrossRef]

Censor, D.

D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
[CrossRef]

I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
[CrossRef]

I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).

I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
[CrossRef]

J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
[CrossRef]

D. Censor, “Waveguide and cavity oscillations in the presence of nonlinear medium,” IEEE Trans. Microwave Theory Tech. 33, 296–301 (1985).
[CrossRef]

D. Censor, “Scattering by weakly nonlinear objects,” SIAM J. Appl. Math. 43, 1400–1417 (1983).
[CrossRef]

D. Censor, “Ray propagation and self-focusing in nonlinear absorbing medium,” Phys. Rev. A 18, 2614–2617 (1978).
[CrossRef]

D. Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ. 15, 215–223 (1978).

D. Censor, “Ray theoretic analysis of spatial and temporal self-focusing in general weakly nonlinear medium,” Phys. Rev. A 16, 1673–1677 (1977).
[CrossRef]

D. Censor, “Ray tracing in weakly nonlinear moving medium,” J. Plasma Phys. 16, 415–426 (1976).
[CrossRef]

Gurwich, I.

I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
[CrossRef]

I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).

I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
[CrossRef]

Hasan, M. A.

M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
[CrossRef]

M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
[CrossRef]

Meier, B.

B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
[CrossRef]

Molcho, J.

J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
[CrossRef]

Penzkofer, A.

B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
[CrossRef]

Reintjes, J.

J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).
[CrossRef]

Uslenghi, P. L. E.

M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
[CrossRef]

M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
[CrossRef]

Am. J. Phys.

J. Molcho and D. Censor, “A simple derivation and a classroom example for Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 (1986).
[CrossRef]

Appl. Phys. B

B. Meier and A. Penzkofer, “Determination of nonlinear refractive indices by external self-focusing,” Appl. Phys. B 49, 513–519 (1989).
[CrossRef]

Electromagnetics

M. A. Hasan and P. L. E. Uslenghi, “Higher-order harmonics in electromagnetic scattering from a nonlinear anisotropic cylinder,” Electromagnetics 11, 377–391 (1991).
[CrossRef]

IEEE Trans. Antennas Propag.

M. A. Hasan and P. L. E. Uslenghi, “Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency,” IEEE Trans. Antennas Propag. 38, 523–533 (1990).
[CrossRef]

IEEE Trans. Magn.

I. Gurwich and D. Censor, “Steady state electromagnetic wave propagation in weakly nonlinear medium,” IEEE Trans. Magn. 30, 3192–3195 (1994).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

D. Censor, “Waveguide and cavity oscillations in the presence of nonlinear medium,” IEEE Trans. Microwave Theory Tech. 33, 296–301 (1985).
[CrossRef]

Int. J. Electr. Eng. Educ.

D. Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ. 15, 215–223 (1978).

J. Electromagn. Waves Appl.

D. Censor and Y. Ben-Shimol, “Wave propagation in weakly nonlinear bi-anisotropic and bi-isotropic medium,” J. Electromagn. Waves Appl. 11, 1763–1779 (1997).
[CrossRef]

I. Gurwich and D. Censor, “Existence problems in steady state theory for electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 9, 1115–1139 (1995).

I. Gurwich and D. Censor, “On the propagation of multi-band spectrum electromagnetic waves in weakly nonlinear medium,” J. Electromagn. Waves Appl. 10, 889–907 (1996).
[CrossRef]

J. Plasma Phys.

D. Censor, “Ray tracing in weakly nonlinear moving medium,” J. Plasma Phys. 16, 415–426 (1976).
[CrossRef]

Phys. Rev. A

D. Censor, “Ray theoretic analysis of spatial and temporal self-focusing in general weakly nonlinear medium,” Phys. Rev. A 16, 1673–1677 (1977).
[CrossRef]

D. Censor, “Ray propagation and self-focusing in nonlinear absorbing medium,” Phys. Rev. A 18, 2614–2617 (1978).
[CrossRef]

Phys. Rev. Lett.

J. Reintjes and R. L. Carman, “Direct observation of the orientational Kerr effect in the self-focusing of picosecond pulses,” Phys. Rev. Lett. 28, 1697–1700 (1972).
[CrossRef]

SIAM J. Appl. Math.

D. Censor, “Scattering by weakly nonlinear objects,” SIAM J. Appl. Math. 43, 1400–1417 (1983).
[CrossRef]

Other

D. Censor, I. Gurwich, and M. Sonnenschein, “Volterra’s functionals series and wave propagation in weakly nonlinear medium: the problematics of first-principles physical modeling,” in Volterra Equations and Applications, Proceedings of The Volterra Centennial Symposium, C. Corduneanu and I. W. Sandberg, eds. (Marcel Dekker, New York, 1998).

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Figures (15)

Fig. 1
Fig. 1

Geometry of the system; in the present model, only a simple case of two Cartesian coordinates x, z is discussed, with the field E polarized along the y axis. A beam of rays enters the sample from the left, propagating to the right with z as the optical axis.

Fig. 2
Fig. 2

Schematic description of the energy-flux propagation between two rays. The rays propagate from left to right, starting from point 1 and propagating to point 2. Δl1, Δl2 denote the distances between the rays measured by lines normal to the rays. To maintain energy conservation, the power flux through Δl1 must be identical to the flow through Δl2.

Fig. 3
Fig. 3

Schematic description of external self-focusing measurement setup. A beam of rays enters the sample (S). The propagation through the nonlinear sample’s refractive index Δn decreases the incident angle θi by the amount θsf, and thus the ray leaves the sample with an angle θ0; here ri is the height of the ray in the beam profile and ls is the sample’s length. The rays cross each other in the region l>lsf,min.

Fig. 4
Fig. 4

Initial spatial Gaussian beam used for almost all the simulations. The 1/e beam radius is ai=1.25 mm, and the peak Gaussian intensity is IL=1.1×1013 W/m2.

Fig. 5
Fig. 5

Propagation of 40 rays with an initial Gaussian profile. The rays propagate from left to right. The more heavily shaded region to the left is the 5-cm-wide nonlinear sample region. In the focal region, owing to the convergence, the beam is narrower. ε¯(3)=4.4946×10-21 m2V-2, γΔz=0, and other parameters of the beam and the sample are given in Table 1.

Fig. 6
Fig. 6

Intensity profile of the propagating beam from Fig. 5. The x’s are the initial intensity distribution. During propagation the rays’ intensity profiles become narrower, and the intensity at the peak increases rapidly. The profile loses its initial shape. The initial Gaussian shape might develop humps. This is an example of filamentation created by numerical errors without a physically significant effect.

Fig. 7
Fig. 7

Simulation of an arbitrary medium with parameters similar to those of benzene. In this case the nonlinearity was changed to ε¯(3)=9.9421×10-21 m2V-2, as opposed to ε¯(3)=4.4946×10-21 m2V-2 in the original parameter for benzene. As expected, the results display a stronger convergence of the beam.

Fig. 8
Fig. 8

Simulation for a medium with weaker nonlinearity, ε¯(3)=1.9884×10-21 m2V-2, as compared with Figs. 5 and 6. As expected, the results display a smaller convergence of the beam.

Fig. 9
Fig. 9

Evolution of the beam profile of the propagating beam in Fig. 8. The initial profile is chosen to be Gaussian. During propagation, the rays converge and generate high intensities close to the peak, but the convergence is smaller than in Fig. 6.

Fig. 10
Fig. 10

Gaussian-beam propagation through a nonlinear medium with losses. The loss parameter γ is chosen arbitrarily to demonstrate the effect on the simulation of propagation in a nonlinear medium in the presence of losses. The results display reduced convergence of the beam as compared with the simulation of propagation in a lossless medium (Fig. 5). The simulation parameters are nonlinear dielectric ε¯(3)=4.4946×10-21 m2V-2 and losses γΔz=0.02 Nepers.

Fig. 11
Fig. 11

Initial and evolved propagation profiles of the beam propagating in a medium with losses. In this case, where losses are included, the intensity profiles are broader, compared with the case of the lossless medium (see Fig. 6). The global intensity of the profile decreases. The profile shape is retained for some distance of propagation, but the nonlinear self-focusing effect causes convergence of the rays and changes the initial profile.

Fig. 12
Fig. 12

Whole-beam narrowing for temporal Gaussian-pulse propagation through a nonlinear sample. The solid curve is the temporal pulse. At the leading and trailing edges of the pulse the amplitude is low, and consequently nonlinear phenomena are negligible and cannot be observed. Near the peak, the pulse amplitude increases, causing a stronger nonlinear interaction of the medium and the propagating pulse, hence stronger self-focusing and convergence of the rays.

Fig. 13
Fig. 13

Time-resolved beam narrowing; the solid curve shows the temporal pulse profile. The input-pulse maximum intensity is 1.1×1013 W/m2 and pulse duration is 35 ns. The circles are taken from the Meier–Penzkofer experiment and include the relaxation time effect. The dashed curve is interpolated from the dotted curve, which is the trajectory of the rays that match the HWHM of the beam.

Fig. 14
Fig. 14

Result of interpolation, with an order-four polynomial-fitting algorithm, of the trajectory of the rays that match the HWHM of the beam. Best fitting to the experimental result is achieved by using ε¯(3)=4.5733×10-21 m2V-2 corresponding to n¯2=3.05×10-21 m2V-2, which is close to n¯2=3.00×10-21m2V-2, as indicated in the Meier–Penzkofer experiment.

Fig. 15
Fig. 15

Time-integrated beam narrowing versus input-pump pulse. The circles are data from the Meier–Penzkofer experiment. The dashed curve is interpolated from the dotted curve, which is the simulation results. Best fitting to the experimental result is achieved with ε¯(3)=4.4938×10-21 m2V-2, corresponding to n¯2=3.0×10-21 m2V-2.

Tables (1)

Tables Icon

Table 1 Main Parameters for the Self-Focusing Simulation

Equations (36)

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f(X)=f(r, t)=I(X)exp[iθ(X)],
θ(r, t)=(r0, t0)(r, t)[k(r, t)dr-ω(r, t)dt],
θt-ω,θ=θrk,
F(K, X)=F(k, ω, r, t)=0,
drdt=-F/kF/ω=vg,
dkdt=F/rF/ω,
dωdt=-F/tF/ω,
Di(K)=εij(K)Ej(K),i, j=1, 2, 3,
Di(X)=(d4X1)εij(X1)Ej(X-X1).
D(X)=n=1Dn(X)=n=1Pn{X, E},
Di(n)(X)=(d4X1)(d4Xn)εijk(n)(X1, , Xn)
Ej(X-X1)Ek(X-Xn),
Di(n)(K)=(2π)1-n(d4K1)(d4Kn-1)εijk(n)
(K1Kn)Ej(K1)Ek(Kn),
k=k1++kn,ω=ω1++ωn,
E(X, K)=m=-Em(K)exp(imKX)=mEm(K)exp(imθ),
D(n)(X, K)=pDp(n)(K)exp(ipKX),
Dp;i(n)(K)=m1, , mnεijk(n)(m1K, , mnK)
Em1;j(m1K)Emn;k(mnK),
Dp,i(n)(pK)=ε¯p,i, jk(n)(pK)Ep, j(pK)Ep,k(pK),
F(K, X; E)=0,
Di=ε¯i,j(1)Ej+ε¯i,jkl(3)EjEkEl.
ε˜=ε¯(1)+ε¯(3)E2,
F=F(k, ω, x)=kx2+ky2+kz2-ω2μ0ε=0,
F=F(k, ω, x, E)=kx2+kz2-ω2μ0ε˜=0,
dxdt=kxωμ0ε˜,
dzdt=kzωμ0ε˜,
dkxdt=ωε˜/x2ε˜,
dkzdt=ωε˜/z2ε˜.
P=E×H.
f=E12ε˜1μ Δl1=E22ε˜2μ Δl2.
E2=E1Δl1/Δl2.
E2=E1Δl1/Δl2(1-γΔz).
Δn=n2,stnLε0c0 IL,
ε¯(3)=n2,stnL.
Δε=2ε˜(3)nLε0c0 IL

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