Modal Gain and Loss in Three-Layer Optical Waveguides

A perturbation analysis based on the imaginary part of the complex permittivity rather than on the electric fields associated with stimulated emission shows that the formula for TM gain applies to any increase or decrease in the mode amplitude, whether due to boundary irregularities, material inhomogeneities, doping levels, or stimulated emissions. The resulting derived equations for the TE and TM modal gain confinement factors are superior to previously published approximate equations and provide accurate results for practical photonic waveguides. The modal gain can be larger than the gain of a plane wave traveling in a medium with the same gain if the gain is only in the middle layer (for both TE and TM modes), and in some TM cases, with gain only in the evanescent layers. In some cases, the TM modal gain or loss can be a factor of approximately 10 larger than the TE modal gain or loss. The derived analytical formulas showed excellent agreement with the results obtained from an accurate numerical approach for the gain and modal gain confinement factors for waveguides with small, medium, and large index differences. The formulae derived in this study can be useful for periodic optical waveguides, semiconductor lasers, optical amplifiers, distributed reflector lasers, optical waveguide isolators, and silicon photonic waveguides. The results of these analytical formulae can be used to verify numerical methods applied to more complex structures.