Simulation of Second-Order Dispersion, Third-Order Dispersion, and SPM in Nonlinear Fiber Optics

In nonlinear fiber optics research, simulating second-order dispersion, third-order dispersion, and self-phase modulation (SPM) is essential for understanding optical pulse propagation behavior in optical fibers. These effects collectively influence pulse broadening, compression, and spectral changes, representing core issues in optical communications and ultrafast laser research.

Second-order dispersion (Group Velocity Dispersion, GVD): This describes how different frequency components propagate at different speeds in optical fibers, leading to temporal pulse broadening or compression. The sign depends on the fiber's dispersion characteristics (normal or anomalous dispersion regime). In numerical implementations, GVD is typically handled in the frequency domain using dispersion operators that apply phase shifts proportional to β₂ (second-order dispersion coefficient) and angular frequency deviation.

Third-order dispersion (TOD): In ultrashort pulse propagation, TOD effects become significant and cannot be neglected. TOD causes asymmetric pulse shaping and may generate oscillatory tails. Simulation codes must incorporate correction terms using β₃ (third-order dispersion coefficient) in the dispersion operator, which adds a cubic frequency dependence to the phase response.

Self-phase modulation (SPM): Originating from the nonlinear refractive index change in fibers, SPM causes pulse phase variations proportional to intensity, leading to spectral broadening. This effect is particularly critical in applications like supercontinuum generation. Code implementation typically uses a nonlinear operator that applies an intensity-dependent phase shift exp(iγP(z,t)L), where γ is the nonlinear coefficient and P(z,t) is the instantaneous power.

For researchers entering this field, simulation programs provide intuitive visualization of pulse evolution under different parameters, helping understand theoretical models. A typical simulation approach employs the Split-Step Fourier Method (SSFM), which alternately handles linear dispersion in the frequency domain and nonlinear effects in the time domain, balancing computational efficiency with accuracy. By adjusting fiber length, input power, and dispersion parameters, researchers can investigate pulse dynamics under various conditions, providing theoretical support for experimental design. The SSFM algorithm structure generally follows: while propagating through small fiber segments Δz, apply dispersion operator using FFT, then apply nonlinear operator, repeating until completing the full fiber length.