Wave optics

Wave optics is a major sub-field of optics, where light is modelled as a wave. We use this over geometric optics when light no longer travels as a straight line, as in complicated phenomena like interference or diffraction.

Basics

The basis of the theory of wave optics are Maxwell’s equations.¹ We make a few simplifying assumptions for our analysis:

• $\vec{J}=\vec{0}$. Generally materials won’t carry a current.
• $\rho =0$. Generally we deal with charge distribution-free materials.
• We have a linear medium, i.e., $\vec{D}\propto \vec{E}$ and $\vec{B}\propto \vec{H}$.
• We also have an isotropic medium, i.e., the permittivity $ϵ$ and permeability $\mu$ are scalars, not tensors.

Thus, Maxwell’s equations reduce down to:

$$\nabla \cdot \vec{E}=0$$ $$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \cdot \vec{B}=0$$ $$\nabla \times \vec{B}=\mu ϵ\frac{\partial \vec{E}}{\partial t}$$

From here, we can make a few manipulations to Maxwell’s equations (we curl both sides of equation 2/4, use an identity, and substitute equation 1/3). This nets us wave equations for both the electric and magnetic field:

$${\nabla }^2\vec{E}=\mu ϵ\frac{{\partial }^2\vec{E}}{\partial t^2}$$ $${\nabla }^2\vec{H}=\mu ϵ\frac{{\partial }^2\vec{H}}{\partial t^2}$$

i.e., electromagnetic fields propagate as waves with speed:

$$c=\frac{1}{\sqrt{\mu ϵ}}$$

where $c$ is the speed of light, i.e., light is an electromagnetic wave.

$\vec{E},\vec{H}$, and the direction of propagation $\vec{k}$, are all mutually perpendicular, related by:

$$\vec{k}\times \vec{E}=\mu \omega \vec{H}$$

We re-define the refractive index as:

$$n=\sqrt{{ϵ}_r{\mu }_r}\approx \sqrt{{ϵ}_r}$$

Sub-pages

• Wave polarisation
  • Jones vector
  • Fresnel coefficient
• Total internal reflection
• Interference
  • Coherence (temporal, spatial)
• Diffraction

Footnotes

1. “Optics […] is historically the most important application of Maxwell’s theory.” - David Griffiths, in Introduction to Electrodynamics ↩