Waveguiding in Semiconductor Lasers
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In this chapter, we discuss the basic electromagnetic theory needed to understand waveguiding in semiconductor lasers. Although this subject can become quite complicated, a few simple approximations are sufficient for device analysis. We begin in Section 4.1 by introducing the material parameters that describe wave refraction and absorption in semiconductors, ending up with the frequency-resolved macroscopic Maxwell equations. The solutions to these equations take a special, simple form in rectangular waveguide geometries, as we explain in Section 4.2. We are most interested in dielectric waveguides, and we analyze the three-layer slab waveguide in Section 4.3. Although this is only a “textbook” problem, rather than a practical laser structure, we use the solutions to motivate approximations for treating more complicated structures. In particular, we argue (hope?) that the two-dimensional scalar wave equation is a sufficiently good approximation for in-plane semiconductor lasers, and in Section 4.4 we present an elegant method for obtaining the best separable approximation to this equation. The resulting equations are simple, and we present details for solving them in arbitrary piecewise-constant geometries. The two most important parameters we can extract from our waveguide solutions are the optical confinement factor Г—which represents the overlap of the waveguide mode with the active gain region—and the free-carrier absorption coefficient αfc. Once we know the spatial profile of the waveguide modes, it is simple to calculate Г. The free-carrier losses can be approximated quite easily through first-order perturbation theory, as we show in Section 4.5.
KeywordsSemiconductor Laser Waveguide Mode Eigenvalue Equation Transverse Field Dielectric Waveguide
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- 1.This description sweeps an enormous number of approximations under the rug. Even the distinction between “free” and bound charges and currents becomes muddled, especially for time-varying sources. There is an entire field of study devoted to calculating and measuring the electromagnetic responses of materials. Fortunately, the simplifications entailed in (4.5)-(4.8) and (4.11)-(4.13) are sufficient for our purposes.Google Scholar
- 2.For a single plane wave like (4.22), the phase of E is irrelevant. But in assembling wave packets through (4.9), it becomes important.Google Scholar
- 3.This standard result is shown in almost every electromagnetics text, including J. D. Jackson, Classical Electromagnetics. Second Edition (Wiley, New York, 1975).Google Scholar
- 4.More accurately, we could always choose our axes to eliminate any x-dependence.Google Scholar
- 5.This method was introduced by P.C. Kendall, M. J. Adaras, S. Ritchie, and M. J. Robertson, “Theory for calculating approximate values for the propagation constants of an optical rib waveguide by weighting the refractive indices,” IEE Proc. A, vol. 134, pp. 699–702, Sept. 1987, and M. J. Robertson, P. C. Kendall, S. Ritchie, P. W. A. Mcllroy, and M. J. Adams, “The weighted index method: a new technique for analyzing planar optical waveguides,” IEEE J. Lightwave Tech., vol. 7, pp. 2105-2111, 1989.Google Scholar
- 6.Here “best” means that F and G minimize the function-space error where Φexact(x,y) is the exact solution to (4.74).Google Scholar
- 7.In Chapter 1, we assumed the zeroth-order wavefunctions and eigenvalues were exact eigen-states of the zeroth-order Hamiltonian. Although this is certainly desirable, it is not absolutely necessary. We can also apply perturbation techniques using approximate zeroth-order solutions, provided we are willing to sacrifice some accuracy.Google Scholar