diffraction

In many important practical situations, beam propagation through a nonlinear media can be accurately modeled by the following nonlinear Schrödinger equation (NLS)

i Qη + ΔQ + |Q|2 Q = 0 (NLS)

with a function Q0(ρ) as an incident radially symmetric pulse Q(ρ, η=0)= Q0(ρ) (here Qη = Q / η and the Laplacian Δ is given by  Δ= ∂2/∂ρ2 + (1/ρ)(∂/∂ρ),  η=z / Ldf  is a normalized propagation distance, and ρ= (x2+y2)1/2 / R0 is the normalized distance from the center of the pulse in the transverse two-dimensional plane, Ldf  is the diffraction length and R0 is the beam’s FW1/eM radius).

However, this model is prone to exponential growth or even singularity formation of a numerical solution in the focusing regions when self-focusing is a dominating process. Another model for nonlinear propagation, which proved to better handle self-focusing regimes [6], is a nonparaxial approximation (nopaxNLS) which, compared to NLS, has an extra term of the pulse second derivative

i Qη + ē Qηη + ΔQ + |Q|2 Q = 0 (nonpaxNLS)

with the following coefficient of nonparaxiality ē = (λ0 / 2πR0)2 for the incident pulse Q(ρ, η=0)= Q0(ρ) with the radius R0 and wavelength λ0. Both equations are derived from the following nonlinear Helmholtz (NHL) equation [7,8].

Ezz(x,y,z,t) + ΔE + (k0)2(1+(2n2/n0)|E|2) E = 0 (NHL)

by assuming no backscattering happens while pulse propagate the nonlinear sample. The model for such pulse can be represented by the following SVEA form with the complex conjugate term Q*(x,y,z,t), representing the backscattered part of the pulse, left out

E(x,y,z,t) = R0 k0 (2n2/n0)1/2 Q(x,y,z,t) exp[ -i0t– k0z) ]

If λ0 << R0 , which is the case in many situations, the nonparaxiality coefficient ē is negligibly small, so that nonparaxial approximation turns to NLS – paraxial approximation. NLS can be solved numerically as an example of an evolution problem (eta acts as the time variable in this case).

Problems with a numerical solution of paraxial NLS arise when the initial power of the incident pulse N0 = (||Q0||2)2=∫ |Q0(ρ)|2 ρ dρ exceeds a threshold power Nc = 1.86 [1,2], which happen in self-focusing regimes. What may happen in such cases is that the non-paraxial term ēQηη increases substantially to become as important as other terms in nonparaxial NLS, so that disregarding this term is no longer safe. Nonparaxial term plays as a balancing term, without which the Laplacian (diffraction) and the cubic nonlinearity (Kerr factor) become misbalanced in paraxial NLS, which leads to possible numerical collapses [3,4,5].

One solution would be to relax the nonparaxiality coefficient, introducing a linear damping ε which “turns on” the nonparaxial term near self-focusing regions only [6,7,8]. An extension of split-step method based on Padé approximation operators [9] allows solving nonparaxial NLS, where the nonparaxiality coefficient ε is adapted at each propagation step based on the behavior of the current solution.

See also: Paraxial approximation, Slowly varying envelope approximation (SVEA).

References
[1] G. Fibich, A. Gaeta, Critical power for self-focusing in bulk media and in hollow waveguides, Opt. Lett. 25 (2000) 335–337.
[2] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimate, Comm. Math. Phys. 87 (1983) 567–576.
[3] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Does the nonlinear Schrödinger equation correctly describe beam propagation?, Opt. Lett. 18 (1993) 411–413.
[4] M.D. Feit, J.A. Fleck, Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams, J. Opt. Soc. Amer. B Opt. Phys. 5 (1988) 633–640.
[5] G. Fibich, Small beam nonparaxiality arrests self-focusing of optical beams, Phys. Rev. Lett. 76 (1996) 4356–4359.
[6] G. Fibich, B. Ilan, S. Tsynkov, Backscattering and nonparaxiality arrest collapse of damped nonlinear waves, SIAM J. Appl. Math. 63 (5), (2003), 1718-1736.
[7] G. Fibich A and S. Tsynkov B, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comp. Phys. 210, (2005), 183-224.
[8] G. Baruch, G. Fibich, and Semyon Tsynkov, Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams, Opt. Express 16, 13323-13329 (2008).
[9] K. Malakuti and E. Parilov, A split-step difference method for nonparaxial nonlinear Schrödinger equation at critical dimension, Appl. Num. Math. 61(7), 891-899 (2011).


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