16. Ch. 7, part 1

Lecture 16 - Plane Waves

We start from Maxwell’s equations with constant, real, frequency-independent ε and μ. We will then generalize to frequency-dependent, complex ε and μ, and we will also discuss briefly the case if inhomogeneous and anisotropic media.

16.1. Plane waves in a non-conducting medium

Each cartesian component of E and B (and also D, H, A, Φ) obeys the scalar wave equation

∇^{2}u − \frac{1}{v^{2}} \frac{∂^{2}u}{∂t^{2}} = 0  (16.1)

with

v = \frac{c}{\sqrt{εμ}} = \frac{c}{n}  (16.2)

The general solution of eq. (16.2) is a linear superposition of fundamental solutions of the form

u(r, t) = u_{0} exp [i(k · r − ωt)]  (16.3)

where the amplitude u_{0} is in general complex and the wave vector k must be such that

k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = \frac{ω^{2}}{v^{2}} = εμ \frac{ω^{2}}{c^{2}}  (16.4)

Solutions that are well behaved for |t| → ∞ and r → ∞ are constructed using real values of ω and k (i.e. real k_{x}, k_{y}, k_{z}). However, solutions for complex values of ω and k do exist and are useful (see section 2).

It is customary to write out separately the positive-frequency (ω > 0) and negativefrequency components of the fields, although both must be present in the physical solution, which is real. Quantum-mechanically they correspond to photon creation and annihilation operators, respectively.

Equation (16.3) describes a plane wave: the wave fronts are planes perpendicular to k. Letting ξ be the coordinate parallel to k, the fundamental plane wave is

u(ξ, t) = u_{0} exp [i(k_{ξ}ξ − ωt)]  (16.5)

with

k^{2} = k_{ξ}^{2} = εμ \frac{ω^{2}}{c^{2}} = n^{2} \frac{ω^{2}}{c^{2}}  (16.6)

In the simplest case, when ε and μ are real and positive, the index of refraction n is the positive square root of εμ, and by definition k = n|ω|/c is positive. There are then four solutions of the form (16.5): two for ω > 0 and k_{ξ} = ±k, and two for ω < 0 and k_{ξ} = ±k. It is enough to consider complex solutions with ω > 0 and pick up their real part (which automatically has ω < 0). The general real plane wave solution for a given frequency is then

u(ξ, t) = Re [Ae^{ikξ−iωt} + Be^{−ikξ−iωt}]  (16.7)

with

vk = ω,  v = c/n.  (16.8)

More explicitly, with A = |A| e^{iα} and B = |B| e^{iβ}, we have

u(ξ, t) = |A| cos(kξ − ωt + α) + |B| cos(kξ + ωt + β)  (16.9)

In (16.7) or (16.9) the first term describes a wave propagating in the direction of increasing ξ, the second term a wave propagating in the direction of decreasing ξ.

When n does not depend on ω, the general plane wave solution is of the form

u(ξ, t) = f(ξ − vt) + g(ξ + vt)  (16.10)

with f and g arbitrary functions. This is obtained by linear (Fourier) superposition of the solutions (16.7) or (16.9) with different frequencies, or by direct solution of the onedimensional wave equation.

The most general solution of the three-dimensional wave equation (16.1) is a superposition of plane waves running in different directions, or a sum (16.3) over different k values.

Dispersion

If the refractive index n = \sqrt{εμ} is a function of ω, the generalsolution is still a sum of monochromatic plane waves (16.3), but eq.(16.10) is no longer a valid expression for the general plane wave solution. Instaead, the shape of a plane wave train will change wit time, due to the ω-dependence of the phase velocity v = c/n. The medium is said to be dispersive.

Transverse polarization

While each cartesian component of the electromagnetic fields must satisfy the scalar wave equation, there are additional constraints imposed by Maxwell’s equations, in particular by ∇ · D = 0. As a result, in our case k · E = 0 and the fundamental solution for the electric field can be written

E(r, t) = (ϵ_{1} E_{1} + ϵ_{2} E_{2}) exp [i (k · r − ωt)]  (16.11)

where ϵ_{1}and ϵ_{2} are two unit vectors such that ϵ_{1}, ϵ_{2}, k define a right-handed set of orthogonal axes. The amplitudes E_{1} and E_{2} are in general complex.

Any unit vector such that k · ϵ = 0 is a possible polarization vector. It would be sufficient to consider real unit vectors ϵ_{1} and ϵ_{2}, but it is often convenient to introduce complex polarization vectors (see the treatment of cicularly polarization in section (??).

The Maxwell equation ∇ × E = −\frac{1}{c} \frac{∂B}{∂t} gives

\frac{ω}{c} B = k × E,

while ∇ × H = −\frac{1}{c} \frac{∂D}{∂t} gives

εμ \frac{ω}{c} E = k × B.

These equations are mutually compatible: both imply that B is orthogonal to E and to k, and that B = nE. Note that, for real n, B is in phase with E.

Energy and the Poynting vector

With E and B defined as above, the time-averaged energy density is

⟨U⟩ = \frac{ε}{8π} (|E_{1}|^{2} + |E_{2}|^{2})

The energy flow is in the direction of k and has magnitude given by the Poynting vector. The time-averaged energy flow is

⟨S⟩ = v ⟨U⟩ = \frac{c}{8π} \sqrt{\frac{ε}{μ}} (|E_{1}|^{2} + |E_{2}|^{2})

Both ⟨U⟩ and ⟨S⟩ are independent of the orientation of the mutually orthogonal unit vectors ϵ_{1} and ϵ_{2}.

Evanescent waves

As noted in connection with eq. (16.1), the fundamental solution still holds if some of the numbers k_{x}, k_{y}, k_{z}, ω are complex, provided that k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = εμ ω^{2}/c^{2}.

In particular, solutions with real k_{x}, k_{y}, ω and purely imaginary k_{z} occur naturally in the problem of wave propagation across a plane interface between two media at z = 0. Usually a solution of this type is of interest for z > 0 (for instance) and has k_{z} = iα with α > 0. It is then of the form

u(r, t) = u_{0} exp [i(k_{x}x + k_{y}y − ωt)] e^{−αz}

It vanishes for z → +∞ and is usually called an evanescent wave.

More generally, complex k solutions occur for wave propagation in inhomogeneous media and are called inhomogeneous plane waves by Jackson. If k = k_{R} + ik_{I}, we find that k^{2} = εμω^{2}/c^{2} gives

k_{R}^{2} − k_{I}^{2} = εμω^{2}/c^{2}, k_{R} · k_{I} = 0

In the example above, k_{R} = k_{x}x̂+k_{y}ŷ and k_{I} = αẑ

16.2. Polarization

The tip of the vector

E(r, t) = Re ((ϵ_{1} E_{1} + ϵ_{2} E_{2}) exp [i (k · r − ωt)])

decribes an ellipse (for fixed k · r as a function of t, and for fixed t as a function of k · r).

If E_{1} and E_{2} have the same phase (so that E_{2}/E_{1} is real), the ellipse degenerates into a line segment. The radiation is then linearly polarized at an angle arctan(E_{2}/E_{1}) to the direction of ϵ_{1}.

If E_{1} and E_{2} are 90^{∘} out of phase (so that E_{2}/E_{1} is pure imaginary) teh axis of the ellipse are in the directions of ϵ_{1} and ϵ_{2}, and the lengths of the semiaxes are |E_{1}| and |E_{2}| . Without loss of generality (up toan irrelevant phase factor) we can choose E_{1} to be real and ϵ_{1} to point along the major axis. Then E_{1} = a and E_{2} = ±ib, where a and b are the semiaxes.

If E_{2} = iE_{1} (or b = a and E_{2} = +ib), the tip of E describes a circle counterclockwise as a function of t (the tip E describes a circle clockwise as a function of k · r for fixetd t). In optics, light of this kind is said to be left circularly polarized. If E_{2} = iE_{1}, light is said to be right circularly polarized (in the optics literature; in particle physics the opposite convention is used). In general, for a given elliptical polarization one must still specify if the motion of the tip of E is clockwise or counterclockwise.