Nonlinear waveguiding

Figure 16 shows the dependence of the SH intensity at 45° incidence of p-polarized fundamental light on the number of deposited bilayers. The SH intensity increases quadratically with the film thickness of up to 400 bilayers (2 pm), as predicted theoretically in the case of the nonlinear slab with a thickness much smaller than the coherence length. The quadratic dependence demonstrates that the highly ordered noncentrosymmetric molecular orientation, which was confirmed in the relatively thin LB film from the SHG and X-ray diffraction measurements, was preserved in the alternating LB film with the thickness enough to be applied to the nonlinear waveguide devices. 

Using the LB technique, one can control the film thickness and the molecular arrangement along the film normal at molecular level. These features are very useful for the construction of sophisticated nonlinear waveguide with high... 

The main purpose of this paper is to consider a two-dimensional non-stationary (2D-I-T) problem of a nonlinear waveguide excitation by a non-stationary light beam and to study spatiotemporal phenomena arising upon propagation of the beam in a step-index waveguide, first, in the quasi-static approximation and, second, with account of MD and SS effects. 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

We find below a set of stationary nonlinear modes for the step-index nonlinear waveguide, investigate their stability and global dynamics. The latter is simulated numerically by the FD-BPM as a solution to the Cauchy problem for waveguide junctions under consideration. 

For the analysis of a nonlinear mode stability which is important for a problem of nonlinear waveguide excitation, consider the power dispersion diagrams (Fig.5). 

As it is well known, stationary solutions to Eq.(3.2) occur at the extrema of the Hamiltonian for a given power. The solutions that correspond to global or local minimum of H for a family of solitons are stable. The representation of the output nonlinear waveguide as a nonlinear dynamical system by the Hamiltonian allows to predict, to some extent, the dynamics of the total field behind the waveguide junction. 

If a nonlinear waveguide is excited by a light beam which is not matched with nonlinear mode, the initial value of the Hamiltonian (0) is greater than. In order to estimate how far is the structure A initially from its stable state, the Hamiltonian (3.4) with a linear mode profile given by (2.11) has been calculated. In Fig.7, (0) depending on V is presented for some values of light beam power. 

Figure 9. Unsteady-state regime in nonlinear waveguide of the structure A, or=1.8 m,P = l.

Figure 10. Longitudinal variation of the normalized power (3.5) propagating within the core of the nonlinear waveguide of the structure A for some values of power P (3.3), a = 1.8jum.

The transmittance of the structure A depending on the input power P was evaluated via calculation of Tfz) (Fig.l3) and T2 z) (Fig.l4). It is seen that self-focusing of the light beam in the core of nonlinear waveguide increases with input power, but rate of the increase diminishes so that for the powers P > 7 (a = 1.8 pm) and P > 4 (a = 3.0 pm) the dependence is weak. Negative slope of the curve in this range results from the mentioned above soliton-like... 

Figure 14. Normalized power (3.5) propagating within the computational window in a nonlinear waveguide A versus the initial power P for some values of the core half-width a,...

In this section, a problem of nonlinear waveguide excitation by stationary light beam has been investigated. In the analysis, an approach traditional for nonlinear optics and based on solution to nonlinear paraxial wave equation has been used. The range of light beam powers that induce nonlinear variation of refractive index comparable with linear contrast of the step-index waveguide has been considered. 

Two kinds of nonlinear junctions considered above have different functions with respect to the power of input light beam. The transmittance of the linear/nonlinear junction decreases with input power. The efficiency of the nonlinear action of the structure is greater in narrower waveguides. The transmittance of the junction of nonlinear waveguides has extremes in dependence on the input power but grows up to unity in the limit of high-intensity light beams. 

In this section, propagation of an optical pulse in a step-index nonlinear waveguide is simulated by the numerical technique described in Section 2. As a comparative parameter, the power evaluated by (3.2) p) at... 

In the structure A, the transverse profile of the initial pulse varies behind the junction due to the self-focusing effect. In the nonlinear waveguide of the structure, a fraction of an initial pulse power is emitted from the guiding region (Fig. 19). 

Thus, in the quasi-static approximation, the length of the unsteady-state regime in the core of the nonlinear waveguide is finite, as in the case of the stationary light beam propagation (see Fig. 10, 11). 

The pulse duration at the nonlinear waveguide axis and integrated within the waveguide core decreases with the peak power of the initial pulse (Fig.23). 

Figure 24. Longitudinal dependence of the relative variation of the pulse duration (2.16) hehind the junction of nonlinear waveguides. Ui = 3.0 pm, U2 = 4.0 pm (solid line) ai = 3.0 pm, 02 = 2.0 pm (dashed line), = 3.

Figure 26. Longitudinal variation of the pulse displacement (2.13) at the waveguide axis in nonlinear waveguide of the structure A with respect to the center of moving coordinate system, ro=70fs (1), 60fs (2), 50fs (3), a=3.0pm (solid lines), a= 2.4pm (dashed line),...

Power losses resulting from the SS effect are observed at the earliest stages of the unsteady-state regime in the nonlinear waveguide of the junction (Fig.28). The losses grow significantly beginning at some distance... 

Figure 28. Longitudinal variation of power of the light beam calculated over the computational window in nonlinear waveguide of the stmcture A, To(0)=20fs (dashed line), To(0)= lOfs (dotted line), solid line corresponds to the quasi-static approximation.

Consider now how the solution obtained in the quasi-static approximation changes if the second-order group velocity dispersion is taken into account. It is seen from Fig. 29 that in a nonlinear waveguide of the structure A with the MD and the SS effects is first observed that spatial and temporal parameters of the field vary similarly to the case of the quasi-static approximation. Then, at a given power, spatiotemporal distribution varies depending on the value and sign of the dispersion coefficient 2- The pulse duration decreases in the case of anomalous GVD (k2 < 0) and increases in the case of normal GVD ( 2 > 0). 

Figure 32. Spatiotemporal distribution of optical pulse propagating in a nonlinear waveguide with positive GVD k2 > 0) (a) and negative GVD ( 2 < 0) (b), z = 4 mm, = 3, tq = 16 fs,...

In this paper, spatiotemporal dynamics of non-stationary light beam propagating through the junctions of step-index nonlinear waveguides have been investigated. 

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