Off-axis pulse propagation in a gradient index medium with Kerr nonlinearity - a variational approach

A gradient index medium is a medium with a parabolic refractive index distribution (equation presented) Here G is the index gradient constant and the axis is in the z-direction. A recent analysis'¹' by one of the authors (J. Wang) and his student has shown that continuously adjustable negative group-velocity dispersion up to hundreds of square femtoseconds can be produced by propagating the optical beam off the axis of a gradient index medium (see Figure 1). The idea is that the dispersion caused by the frequency dependence of G(ω) (the index gradient dispersion) is of a different sign compared to the dispersion caused by the frequency dependence of n_o(ω) (the material dispersion, usually positive). The net effect of index gradient dispersion is proportional to the square of the offset distance from the beam center to the axis, as can be guessed from the form of equation (1). Therefore, by off-axis propagating the light with a big enough offset, the net dispersion can be made negative. In a subsequent paper^[2] they further show that optical solitons in a broad spectral range can be produced by such off-axis propagation after taking into account the Kerr nonlinearity. In the present paper we develop a variational approach to study the general off-axis pulse propagation problem in a gradient index medium with Kerr nonlinearity. Compared to the method used in Ref[l,2], our variational approach has the advantages of being more general and more physically clear. Simple expressions that describe the existence criteria and properties of this new type of solitons are obtained. Our approach is based on the standard Langrangian variational formulation^[3]. The pulse evolution equation is derived under the paraxial approximation and its corresponding Langrangian is determined. We then introduce the following solution ansatz: (equation presented) Here U is the pulse envelope, which is assumed to be separable in the x, y, t dimensions. Function F determines the beam shape in space and pulse shape in time. In our analysis, F is assumed to be either Gaussian or hyperbolic secant. A is the amplitude, 9 is the phase, x_o and y_o are the transverse coordinates of the beam center, k_x and k_y are the transverse components of the beam propagation wavevector, p_x and p_y are two variables that describe the beam curvatures in x and y dimensions, w_x and w_y represent the transverse beam widths, t_o is the time-position of the pulse center, p_t is the variable that describes the chirp of the pulse, and w_t represents the pulse duration. All these variables are functions of propagation distance z and their evolution equations are determined using the usual Ritz optimization procedure. Some of these equations with F being assumed to be Gaussian are listed below: (equation presented) The equations for parameters in the y dimension is of the same form as in the x dimension. Here (equation presented) is the wavenumber inside the medium at the carrier frequency (equation presented) represents the effect of the gradient index, (equation presented) represents the effect of material dispersion, (equation presented) represents the effect of the index gradient dispersion and (equation presented) represents the effect of Kerr nonlinearity. From the above equations, it is not difficult to show that a class of pure soliton solutions exist. These solitons propagate in a helix way with the beam offset r_o = x²_o + y²_o remaining the same. The curvatures and chirp of these solitons are zero, and their widths and pulse duration remain unchanged during propagation. For a circular beam, to a very good approximation, the widths in transverse dimensions is given by (equation presented) and the amplitude-pulsewisth product is a constant: (equation presented) Equation (10) has very good physical meanings. The first two terms on the righthand side are the negative dispersion caused by index gradient dispersion and off-axis propagation while the third term is the positive material dispersion. Usually the beam width w_o is much smaller than the position offset r_o from the axis and thus the second term can be ignored. To have soliton solutions, the position offset r_o has to be large enough to overcome the material positive dispersion. The square of amplitude-pulsewidth product is proportional to the ratio between the net negative dispersion and the Kerr nonlinearity. This is analogous to the solitons described by nonlinear Schrodinger equation. Actually an equivalent nonlinear Schrodinger equation can be derive if we leave the solution form in the t-dimension unspecified. As an numerical example, for typical GRIN lens at 0.63 (equation presented) If the distance offset is 3 mm, then the net negative dispersion is (equation presented) and the beam width w_o is 14 μm. To generate a w_t=l ps soliton, the required peak power would be at the order of 1000 W. To generate the pure solitons described above, the propagation direction and the beam widths of the incident light has to be set correctly. If this is not the case, then in general all the amplitude, beam widths, curvatures, pulse duration and chirp will oscillate periodically in z. The negative dispersion produced by off-axis propagation is proportional to the square of the position offset and therefore oscillates periodically in z. The nonlinear effect is proportional to the peak intensity and therefore also oscillates periodically in z. The combined effects of such oscillating dispersion and nonlinearity will produce interesting soliton phenomena analogous to the "guiding-center solitons" in optical fibers[4]. A typical numerical example from our analysis is shown in Figure 2. It should be noted that the change of pulsewidth is actually quite small (0.00001 ps !). In off-axis beam propagation problems, since the propagation direction of the light is not necessarily along the axis, it is therefore necessarily to check the accuracy of the paraxial approximation. We will also address this issue.