# Maxwell's Equations for Semiconductor Lasers

This is a continuation from the previous tutorial - operating principles of semiconductor lasers.

Since the mathematical description of all optical phenomena is based on Maxwell's equations, it is appropriate to start our discussion of semiconductor lasers by considering these equations in some detail.

In the MKS system of units, the field equations take the following form:

$\tag{2-2-1}\pmb{\nabla}\times\pmb{\mathscr{E}}=-\frac{\partial\pmb{\mathscr{B}}}{\partial{t}}$

$\tag{2-2-2}\pmb{\nabla}\times\pmb{\mathscr{H}}=\pmb{\mathscr{J}}+\frac{\partial\pmb{\mathscr{D}}}{\partial{t}}$

$\tag{2-2-3}\pmb{\nabla}\cdot\pmb{\mathscr{D}}=\rho_f$

$\tag{2-2-4}\pmb{\nabla}\cdot\pmb{\mathscr{B}}=0$

where $$\pmb{\mathscr{E}}$$ and $$\pmb{\mathscr{H}}$$ are the electric and magnetic field vectors, respectively, and $$\pmb{\mathscr{D}}$$ and $$\pmb{\mathscr{B}}$$ are the corresponding electric and magnetic flux densities. The current density vector $$\pmb{\mathscr{J}}$$ and the free charge density $$\rho_f$$ represent the sources for the electromagnetic field.

The flux densities $$\pmb{\mathscr{D}}$$ and $$\pmb{\mathscr{B}}$$ arise in response to the electric and magnetic fields $$\pmb{\mathscr{E}}$$ and $$\pmb{\mathscr{H}}$$ propagating inside the medium. In general, their relationship depends on the details of the matter-radiation interaction. For a nonmagnetic dielectric medium the relationship can be expressed in terms of the constitutive relations given by

$\tag{2-2-5}\pmb{\mathscr{D}}=\epsilon_0\pmb{\mathscr{E}}+\pmb{\mathscr{P}}$

$\tag{2-2-6}\pmb{\mathscr{B}}=\mu_0\pmb{\mathscr{H}}$

$\tag{2-2-7}\pmb{\mathscr{J}}=\sigma\pmb{\mathscr{E}}$

where $$\epsilon_0$$ is the vacuum permittivity, $$\mu_0$$ is the vacuum permeability, and $$\sigma$$ is the conductivity of the medium.

The induced electric polarization $$\pmb{\mathscr{P}}$$ is calculated quantum mechanically; for a semiconductor material its evaluation requires the knowledge of the Bloch wave functions and the density of states for the conduction and valence bands.

Maxwell's equations can be used to obtain the wave equations that describes the propagation of an optical field inside the medium. We take the curl of Equation (2-2-1) to obtain

$\tag{2-2-8}\pmb{\nabla}\times\pmb{\nabla}\times\pmb{\mathscr{E}}=-\mu_0\frac{\partial}{\partial{t}}(\pmb{\nabla}\times\pmb{\mathscr{H}})$

where Equation (2-2-6) has been used.

With the help of Equations (2-2-2), (2-2-5) and (2-2-7), we can eliminate $$\pmb{\mathscr{H}}$$, $$\pmb{\mathscr{J}}$$, and $$\pmb{\mathscr{D}}$$ in favor of $$\pmb{\mathscr{E}}$$ and $$\pmb{\mathscr{P}}$$ to obtain

$\tag{2-2-9}\pmb{\nabla}\times\pmb{\nabla}\times\pmb{\mathscr{E}}=-\mu_0\sigma\frac{\partial{\pmb{\mathscr{E}}}}{\partial{t}}-\mu_0\epsilon_0\frac{\partial^2\pmb{\mathscr{E}}}{\partial{t^2}}-\mu_0\frac{\partial^2\pmb{\mathscr{P}}}{\partial{t^2}}$

The left-hand side of Equation (2-2-9) can be simplified by using the vector identity

$\tag{2-2-10}\pmb{\nabla}\times\pmb{\nabla}\times\pmb{\mathscr{E}}=\pmb{\nabla}(\pmb{\nabla}\cdot\pmb{\mathscr{E}})-\pmb{\nabla}^2\pmb{\mathscr{E}}$

In the absence of free charges, $$\rho_f=0$$, and from Equations (2-2-3) and (2-2-5) we obtain

$\tag{2-2-11}\pmb{\nabla}\cdot\pmb{\mathscr{D}}=\epsilon_0\pmb{\nabla}\cdot\pmb{\mathscr{E}}+\pmb{\nabla}\cdot\pmb{\mathscr{P}}=0$

The term $$\pmb{\nabla}\cdot\pmb{\mathscr{P}}$$ is negligible in most cases of practical interest and, consequently, to a good degree of approximation, $$\pmb{\nabla}\cdot\pmb{\mathscr{E}}=0$$ in Equation (2-2-10). Equation (2-2-9) then becomes

$\tag{2-2-12}\pmb{\nabla}^2\pmb{\mathscr{E}}-\frac{\sigma}{\epsilon_0{c^2}}\frac{\partial\pmb{\mathscr{E}}}{\partial{t}}-\frac{1}{c^2}\frac{\partial^2\pmb{\mathscr{E}}}{\partial{t^2}}=\frac{1}{\epsilon_0c^2}\frac{\partial^2\pmb{\mathscr{P}}}{\partial{t^2}}$

where we have used the familiar relation

$\tag{2-2-13}\mu_0\epsilon_0=\frac{1}{c^2}$

and $$c$$ is the speed of light in vacuum.

The wave equation in (2-2-12) is valid for arbitrary time-varying fields.

Of particular interest are the optical fields with harmonic time variations since any field can be decomposed into its sinusoidal Fourier components. Using the complex notation, we write

$\tag{2-2-14}\pmb{\mathscr{E}}(x,y,z,t)=\text{Re}[\mathbf{E}(x,y,z)\exp(-\text{i}\omega{t})]$

$\tag{2-2-15}\pmb{\mathscr{P}}(x,y,z,t)=\text{Re}[\mathbf{P}(x,y,z)\exp(-\text{i}\omega{t})]$

where $$\omega=2\pi\nu$$ is the angular frequency and $$\nu=c/\lambda$$ is the oscillation frequency of the optical field at the vacuum wavelength $$\lambda$$. The notation $$\text{Re}$$ stands for the real part of the bracketed expression.

Note that $$\mathbf{E}$$ and $$\mathbf{P}$$ are generally complex since they contain the phase information.

Using Equations (2-2-14) and (2-2-15) in Equation (2-2-12), we obtain

$\tag{2-2-16}\pmb{\nabla}^2\mathbf{E}+k_0^2[1+\text{i}\sigma/(\epsilon_0\omega)]\mathbf{E}=-(k_0^2/\epsilon_0)\mathbf{P}$

where $$k_0=\omega/c=2\pi/\lambda$$ is the vacuum wave number.

Under steady-state conditions the response of the medium to the electric field is governed by the susceptibility $$\boldsymbol{\chi}$$ defined by

$\tag{2-2-17}\mathbf{P}=\epsilon_0\boldsymbol{\chi}(\omega)\mathbf{E}$

where the frequency dependence of $$\boldsymbol{\chi}$$ has been explicitly shown to emphasize the dispersive nature of the medium response.

In general, $$\boldsymbol{\chi}$$ is a second-rank tensor. For an isotropic medium $$\boldsymbol{\chi}$$ is a scalar.

It is useful to decompose $$\boldsymbol{\chi}$$ into two parts

$\tag{2-2-18}\boldsymbol{\chi}=\boldsymbol{\chi}_0+\boldsymbol{\chi}_\text{p}$

where $$\boldsymbol{\chi}_0$$ is the medium susceptibility in the absence of external pumping and $$\boldsymbol{\chi}_\text{p}$$ is the additional contribution to the susceptibility related to the strength of pumping.

In the case of semiconductor lasers, current injection is the source of pumping and $$\boldsymbol{\chi}_\text{p}$$ depends on the concentration of charge carriers (electrons and holes) in the active layer.

For the moment we leave $$\boldsymbol{\chi}_\text{p}$$ unspecified except for noting that both $$\boldsymbol{\chi}_0$$ and $$\boldsymbol{\chi}_\text{p}$$ are generally complex and frequency-dependent.

If we use (2-2-17) to eliminate the electric polarization $$\mathbf{P}$$ in Equation (2-2-16), we obtain the time-independent wave equation

$\tag{2-2-19}\boldsymbol{\nabla}^2\mathbf{E}+\boldsymbol{\epsilon}k_0^2\mathbf{E}=0$

where we have introduced the complex dielectric constant

\tag{2-2-20}\begin{align}\boldsymbol{\epsilon}&=\boldsymbol{\epsilon}'+\text{i}\boldsymbol{\epsilon}^"\\&=\epsilon_\text{b}+\text{i}\text{Im}(\boldsymbol{\chi}_0)+\boldsymbol{\chi}_\text{p}+\text{i}\sigma/(\epsilon_0\omega)\end{align}

and $$\epsilon_\text{b}=1+\text{Re}(\boldsymbol{\chi}_0)$$ is the background dielectric constant of the unpumped material and is real, as defined.

The notations $$\text{Re}$$ and $$\text{Im}$$ stand for the real and imaginary parts, respectively.

The wave equation (2-2-19) can be used to obtain the spatial mode structure of the optical field. However, considerable insight can be gained by considering plane-wave solutions of Equation (2-2-19) even though these are not the spatial modes of the semiconductor laser.

In place of using the complex dielectric constant given by Equation (2-2-20), the propagation characteristics of a plane wave in a medium are conveniently described in terms of two optical constants, the index of refraction $$\mu$$ and the absorption coefficient $$\alpha$$.

Consider a plane wave propagating in the positive $$z$$ direction such that

$\tag{2-2-21}\mathbf{E}=\hat{\mathbf{x}}E_0\exp(\text{i}\tilde{\beta}z)$

where $$\hat{\mathbf{x}}$$ is the polarization unit vector and $$E_0$$ is the constant amplitude.

The complex propagation constant $$\tilde{\beta}$$ is determined by substituting Equation (2-2-21) in Equation (2-2-19) and is given by

$\tag{2-2-22}\tilde{\beta}=k_0\sqrt{\boldsymbol{\epsilon}}=k_0\tilde{\mu}$

where $$\tilde{\mu}$$ is the complex index of refraction. It can be written as

$\tag{2-2-23}\tilde{\mu}=\mu+\text{i}(\alpha/2k_0)$

where $$\mu$$ is the refractive index of the medium, $$\alpha$$ is the power-absorption coefficient, and usually $$\alpha\ll\mu{k_0}$$.

Using $$\boldsymbol{\epsilon}=\tilde{\mu}^2$$ and equating the real and imaginary parts, we find

$\tag{2-2-24}\mu=(\boldsymbol{\epsilon}')^{1/2}=[\epsilon_\text{b}+\text{Re}(\boldsymbol{\chi}_\text{p})]^{1/2}$

$\tag{2-2-25}\alpha=\frac{k_0\boldsymbol{\epsilon}^"}{\mu}=\frac{k_0}{\mu}[\text{Im}(\boldsymbol{\chi}_0+\boldsymbol{\chi}_\text{p})+\sigma/(\epsilon_0\omega)]$

where Equation (2-2-20) has been used.

Equations (2-2-24) and (2-2-25) show explicitly how the refractive index and the net absorption coefficient are affected by external pumping of the semiconductor material.

The next tutorial talks about the threshold condition and longitudinal modes of semiconductor lasers