Optical Field Propagation
Like all electromagnetic phenomena, the propagation of optical fields in a linear/nonlinear, dispersive/nondispersive, homogeneous/inhomogeneous or isotropic/anisotropic media in a guided/unguided structure is governed by Maxwell’s equations. It includes a wide spectrum. For each specific case, a basic equation is obtained such as the nonlinear Schrodinger equation (NSL) for pulse propagation in a waveguide, or the paraxial Helmholtz Equation for beam propagation in free space. The equations usually are nonlinear partial differential equations that do not have analytical solutions except for some specific cases. A numerical approach is often necessary to understand the nonlinear, dispersion and other effects. Taking the pulse propagation in a waveguide as an example, one widely used method to solve NSL equation is the split-step Fourier method. The split-step Fourier method assumes that the dispersion and nonlinear effects can be pretended to act independently over a small distance dz. More specifically, pulse propagation form z to z+dz is carried out in two steps. The nonlinearity acts alone in the first step while the dispersive effect act alone in the second step indicated in figure 1 flow diagram.
Example 1: Third-order Dispersion Effect
Example 2: Cross-phase Modulation
Figure 3 shows both pump and probe pulses’ evolution and assumes both pulses have same polarization. The probe pulse experiences a much bigger broadening in both time and frequency domains induced by cross-phase modulation.
 Govind P. Agrawal, “Nonlinear Fiber Optics”, Academic Press (1995)
 Hohn M. Jarem and Partha P. Banerjee, “Computational Methods for Electromagnetic and Optical System”, Marcel Dekker Inc. (2000)
 Bahaa E.A. Saleh and Malvin Carl Teich, “Fundamentals of Photonics”, John Wiley & Sons Inc. (1991)
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