Optics Notes#2: The wave equation and the velocity of light

This note explains how to derive Maxwell’s wave equations.

The wave equation

The Maxwell equations describe the electric and magnetic fields as a coupled system. The purpose of this note is to combine them into a single equation for one field. Once this is done, the propagation of light appears directly as a wave phenomenon.

We first record two vector identities that will be used below. For a vector field, the double curl can be written in terms of its divergence and Laplacian: \[ \begin{aligned} \\ [\nabla \times (\nabla \times \mathbf{u})]_i &= \epsilon_{ijk} \partial_j (\epsilon_{klm} \partial_l u_m) \\ &= \epsilon_{kij} \epsilon_{klm} \partial_j \partial_l u_m \\ &= (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) \partial_j \partial_l u_m \\ &= \delta_{il} \delta_{jm} \partial_j \partial_l u_m - \delta_{im} \delta_{jl} \partial_j \partial_l u_m \\ &= \delta_{il} \partial_l \partial_j u_j - \delta_{im} \partial_j \partial_j u_m \\ &= [\nabla (\nabla \cdot \mathbf{u})]_i - [\nabla^2 \mathbf{u}]_i \end{aligned} \] \[ \nabla \times (\nabla \times \mathbf{u}) = \nabla (\nabla \cdot \mathbf{u}) - \nabla^2 \mathbf{u} \tag{1}\]

The second identity is the product rule for the curl. Here \(u\) is a scalar field and \(\mathbf{v}\) is a vector field: \[ \begin{aligned} \\ [\nabla \times (u \mathbf{v})]_i &= \epsilon_{ijk} \partial_j u v_k \\ &= \epsilon_{ijk} (\partial_j u) v_k + \epsilon_{ijk} u (\partial_j v_k) \\ &= [(\nabla u) \times \mathbf{v}]_i + [u \nabla \times \mathbf{v}]_i \end{aligned} \] \[ \nabla \times (u \mathbf{v}) = (\nabla u) \times \mathbf{v} + u \nabla \times \mathbf{v} \tag{2}\]

The first identity handles the geometry of the vector field. The second becomes important when the material parameters vary from point to point, because \(\epsilon\) and \(\mu\) can no longer be moved through a spatial derivative.

Maxwell equations in a source-free medium

We consider a region containing no free current or free charge. Thus \(\mathbf{j}=0\) and \(\rho=0\). This does not mean that the electromagnetic field vanishes; it only means that the field is propagating through the region without local sources. In this case Maxwell’s equations are \[ \begin{aligned} \nabla \times \mathbf{H} - \frac{1}{c}\frac{\partial \mathbf{D}}{\partial t} &= \frac{4\pi}{c}\mathbf{j},\\ \nabla \times \mathbf{E} + \frac{1}{c}\frac{\partial \mathbf{B}}{\partial t} &= \mathbf{0}, \end{aligned} \tag{3}\] \[ \begin{aligned} \nabla \cdot \mathbf{D} &= 4\pi\rho,\\ \nabla \cdot \mathbf{B} &= 0. \end{aligned} \tag{4}\] \[ \begin{aligned} \mathbf{j} &= \sigma \mathbf{E}, \\ \mathbf{D} &= \epsilon \mathbf{E}, \\ \mathbf{B} &= \mu \mathbf{H}. \end{aligned} \tag{5}\] The first curl equation in Equation 3, after setting \(\mathbf{j}=\mathbf{0}\), differentiating with respect to time, and using \(\mathbf{D}=\epsilon\mathbf{E}\) from Equation 5, gives \[ \begin{aligned} \nabla \times \frac{\partial \mathbf{H}}{\partial t} - \frac{1}{c}\frac{\partial^2 \mathbf{D}}{\partial t^2} &= 0,\\ \nabla \times \frac{\partial \mathbf{H}}{\partial t} - \frac{\epsilon}{c}\frac{\partial^2 \mathbf{E}}{\partial t^2} &= 0,\\ \end{aligned} \tag{6}\] This relation connects the time variation of the magnetic vector to the second time derivative of the electric vector. The other curl equation in Equation 3 connects the two fields in the opposite direction: \[ \begin{aligned} \nabla \times \mathbf{E} + \frac{1}{c}\frac{\partial \mathbf{B}}{\partial t} = 0 \implies \nabla \times (\frac{1}{\mu} \nabla \times \mathbf{E}) + \frac{1}{c} \nabla \times \frac{\partial \mathbf{H}}{\partial t} &= 0 \\ \implies \nabla \times (\frac{1}{\mu} \nabla \times \mathbf{E}) + \frac{\epsilon}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \end{aligned} \tag{7}\] Using \(\mathbf{B}=\mu\mathbf{H}\) from Equation 5, divide Faraday’s law in Equation 3 by \(\mu\) and take its curl. The result contains the same term \(\nabla\times\partial_t\mathbf{H}\) as Equation 6. Substituting Equation 6 eliminates \(\mathbf{H}\) and produces Equation 7, an equation involving only \(\mathbf{E}\):

The coefficient \(1/\mu\) is inside the curl because the permeability may depend on position. Applying the product rule in Equation 2 and the double-curl identity in Equation 1 gives

\[ \begin{aligned} \nabla \times (\frac{1}{\mu} \nabla \times \mathbf{E}) + \frac{\epsilon}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \\ (\nabla \frac{1}{\mu}) \times (\nabla \times \mathbf{E}) + \frac{1}{\mu} \nabla \times (\nabla \times \mathbf{E}) + \frac{\epsilon}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} &= 0 \\ - \mu \frac{1}{\mu^2}(\nabla \mu) \times (\nabla \times \mathbf{E}) + \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} + \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} &= 0 \\ \nabla^2 \mathbf{E} - \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} + (\nabla \ln{\mu}) \times (\nabla \times \mathbf{E}) - \nabla (\nabla \cdot \mathbf{E}) &= 0 \end{aligned} \tag{8}\] The last line of Equation 8 is the wave equation for \(\mathbf{E}\) before using the divergence condition. Its terms have a simple origin: the Laplacian describes the spatial structure of the field, the second time derivative describes its temporal evolution, and the terms containing \(\nabla\mu\) and \(\nabla\cdot\mathbf{E}\) account for the material and vector-field constraints.

To remove \(\nabla\cdot\mathbf{E}\), use the electric part of Gauss’s law in Equation 4. Since \(\rho=0\) and \(\mathbf{D}=\epsilon\mathbf{E}\) from Equation 5, \[ \begin{aligned} \nabla \cdot (u \mathbf{v}) &= \partial_i (u v_i) \\ &= (\partial_i u) v_i + u (\partial_i v_i) \\ &= \mathbf{v} \cdot (\nabla u) + u \nabla \cdot \mathbf{v} \end{aligned} \tag{9}\] Therefore, applying Equation 9 to \(\mathbf{D}=\epsilon\mathbf{E}\) from Equation 5, \[ \begin{aligned} \nabla \cdot (\epsilon \mathbf{E}) &= \partial_i (\epsilon E_i) \\ &= (\partial_i \epsilon) E_i + \epsilon (\partial_i E_i) \\ &= \mathbf{E} \cdot (\nabla \epsilon) + \epsilon \nabla \cdot \mathbf{E} \end{aligned} \tag{10}\] where \(\rho=0\) has been used from Equation 4. Hence Equation 4 and Equation 10 give \[ \mathbf{E} \cdot (\nabla \epsilon) + \epsilon \nabla \cdot \mathbf{E} = 0 \tag{11}\] \[ \begin{aligned} \mathbf{E} \cdot (\nabla \epsilon) &+ \epsilon \nabla \cdot \mathbf{E} = 0 \\ \nabla \cdot \mathbf{E} &= - \frac{1}{\epsilon} \mathbf{E} \cdot (\nabla \epsilon)\\ \nabla \cdot \mathbf{E} &= - \mathbf{E} \cdot \nabla \ln{\epsilon} \end{aligned} \tag{12}\]

Substituting Equation 12 into Equation 8 gives the electric-field wave equation in an inhomogeneous isotropic medium: \[ \nabla^2 \mathbf{E} - \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} + (\nabla \ln{\mu}) \times (\nabla \times \mathbf{E}) + \nabla (\mathbf{E} \cdot \nabla \ln{\epsilon}) = 0 \tag{13}\] The two additional terms describe the effect of spatially varying material properties. The term containing \(\nabla\ln\mu\) comes from the magnetic response of the medium, while the term containing \(\nabla\ln\epsilon\) comes from the electric response. They vanish when the corresponding material parameter is constant.

By interchanging the roles of the electric and magnetic fields, and simultaneously interchanging \(\epsilon\) and \(\mu\), we obtain the corresponding equation for \(\mathbf{H}\): \[ \nabla^2 \mathbf{H} - \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{H}}{\partial t^2} + (\nabla \ln{\epsilon}) \times (\nabla \times \mathbf{H}) + \nabla (\mathbf{H} \cdot \nabla \ln{\mu}) = 0 \tag{14}\] The electric and magnetic wave equations Equation 13 and Equation 14 are therefore not independent assumptions. They are two forms of the same Maxwell structure, related by the substitutions \(\mathbf{E}\leftrightarrow\mathbf{H}\) and \(\epsilon\leftrightarrow\mu\).

Homogeneous medium

If the medium is homogeneous, \(\epsilon\) and \(\mu\) are constants, so \(\nabla \ln{\epsilon} = \nabla \ln{\mu} = 0\). The equations reduce to

\[ \nabla^2 \mathbf{E} - \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \qquad \nabla^2 \mathbf{H} - \frac{\epsilon \mu}{c^2} \frac{\partial^2 \mathbf{H}}{\partial t^2} = 0 \tag{15}\] Both fields are governed by the same wave operator in Equation 15. Maxwell’s curl equations in Equation 3 still determine how their directions and amplitudes are related; the two equations should not be interpreted as two unrelated waves.

The velocity of light

The standard wave equation for a field \(\mathbf{F}\) travelling with speed \(v\) is \[ \nabla^2 \mathbf{F} - \frac{1}{v^2}\frac{\partial^2 \mathbf{F}}{\partial t^2}=0. \tag{16}\] Comparing Equation 16 with the homogeneous-medium equations in Equation 15 gives the propagation speed \[ v = \frac{c}{\sqrt{\epsilon\mu}}. \tag{17}\] It is convenient to define the refractive index \[ n = \sqrt{\epsilon\mu}, \tag{18}\] so that, using Equation 17, \[ v=\frac{c}{n}. \tag{19}\]

In vacuum, \(\epsilon=\mu=1\) in these units and the speed is \(c\). In a material, the electric and magnetic responses change the coefficient of the time-dependent term in Maxwell’s equations, and light consequently travels more slowly. This expression describes the wave speed in a homogeneous, isotropic, nondispersive medium; when \(\epsilon\) or \(\mu\) depends on frequency, the phase and group velocities require a separate treatment.