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Mathematics for Physics

81 Reflection

A phenomenon in which an electromagnetic wave bounces at a boundary between media that is used to continue the wave in the original medium with a changed direction.

The law of reflection. The angle of incidence equals the angle of reflection. This principle is used to aim a reflected ray from a known incoming ray.

The law of reflection is

\[ \theta_{i} = \theta_{r} \]

where

  • \(\theta_{i}\) is the angle of incidence measured from the normal.
  • \(\theta_{r}\) is the angle of reflection measured from the normal.

Coplanarity of the rays. The incident, reflected, and transmitted wave vectors lie in one plane with the surface normal. This principle is used to set the geometry of the boundary-value problem.

The Fresnel reflection coefficients. The amplitude of the reflected wave is fixed by the Fresnel coefficients. A Fresnel coefficient is the ratio of a reflected or transmitted field amplitude to the incident amplitude. This principle is used to compute how much of the wave is reflected at a dielectric interface.

The Fresnel coefficient for \(\mathbf{E}\) perpendicular to the plane of incidence is

\[ \tilde{r}_{\perp} = \dfrac{n_{1}\cos\theta_{i} - n_{2}\cos\theta_{t}}{n_{1}\cos\theta_{i} + n_{2}\cos\theta_{t}} \]

where

  • \(\tilde{r}_{\perp}\) is the reflected amplitude ratio.
  • \(n_{1}\) and \(n_{2}\) are the indices of refraction of the two media.
  • \(\theta_{i}\) is the angle of incidence.
  • \(\theta_{t}\) is the angle of transmission.