Mathematical analysis of the lasing eigenvalue problem for the optical modes in a layered dielectric cavity with a quantum well and distributed Bragg reflectors

Optical modes and associated linear threshold values of material gain bringing them to lasing are investigated for a VCSEL-type cavity with a quantum well, sandwiched between two distributed Bragg reflectors. They are found as solutions to a specific novel eigenvalue problem with the “active” imaginary part of the quantum well refractive index. For the calculation of the Bragg mirror reflection coefficients, well-established method of the transfer matrices is used. The presented results accurately quantify intuitively predictable lowering of the modal thresholds for the modes whose lasing frequencies lay inside the reflectors rejection bands. Besides, they demonstrate that this approach automatically incorporates the account of overlapping between the active region and the modal E-field patterns and its effect on the thresholds.