Angle of incidence

The angle of incidence $\theta_i$ is the angle between a ray arriving at a surface and the surface normal at the point where it lands. A beam striking a mirror head-on has $\theta_i = 0°$, called normal incidence; a beam skimming along the surface approaches 90°, called grazing incidence. The incident ray and the normal together define the plane of incidence, which is the reference for the s and p polarizations. Optical components are specified at a stated angle of incidence, abbreviated AOI: most laser mirrors and filters at 0°, most fold mirrors, beam splitters and dichroics at 45°.

Reflection and refraction

The law of reflection states that the reflected ray leaves at an angle equal to the angle of incidence, on the other side of the normal and in the same plane. The transmitted ray obeys Snell's law,

$$n_1 \sin\theta_i = n_2 \sin\theta_t ,$$

where $\theta_t$ is the angle of refraction, also measured from the normal. A ray in air meeting glass of index 1.5 at 30° continues inside the glass at 19.5°.

Going the other way, from glass into air, $\sin\theta_t$ reaches 1 when $\sin\theta_i = n_2/n_1$. This is the critical angle, 41.8° for $n = 1.5$; above it all the light is reflected, which is total internal reflection, the mechanism behind right-angle prisms and optical fiber.

Reflectance versus angle

How much light reflects at a bare interface is given by the Fresnel equations, and it depends on both the angle and the polarization. For air to glass of index 1.5:

At normal incidence the two polarizations are indistinguishable and $R = ((n-1)/(n+1))^2 = 0.04$. As the angle grows, s-polarized reflectance rises steadily while p-polarized reflectance falls to zero at the Brewster angle, $\theta_B = \arctan n = 56.3°$, before both climb toward 100% at grazing incidence. The s- and p-polarization entry explains why the two differ and why a component used at 45° almost always behaves differently for the two.

Coatings and filters

Interference coatings are designed for one angle. Tilting a multilayer optical filter shortens the optical path difference between reflections inside each layer, so its passband or edge moves to shorter wavelengths:

$$\lambda(\theta) = \lambda_0 \sqrt{1 - \left(\frac{\sin\theta}{n_\text{eff}}\right)^{2}} ,$$

where $n_\text{eff}$ is an effective index of the layer stack. For a 1550 nm bandpass filter with $n_\text{eff} = 2.0$, a 10° tilt moves the center to 1544.1 nm, a shift of 5.9 nm; a 1° tilt moves it by only 0.06 nm, because the shift grows roughly as $\theta^2$ near normal incidence. Narrowband filters are tuned this way on purpose, and a filter in a converging beam sees a range of angles that broadens and shifts its passband. A dichroic mirror designed for 45° has its edge in the wrong place if mounted at 40° or 50°, and its s and p edges are separated by several nanometers even at the design angle.

Setting and measuring the angle in the lab

Normal incidence is found by retroreflection: the optic is adjusted until the reflected beam returns through an iris placed in the input beam. Rotating the optic from there by $\theta$ on a calibrated rotation stage swings the reflected beam by $2\theta$, which gives a check that is twice as sensitive as reading the stage. A 45° AOI can also be set by aligning the reflected beam to a path at 90° to the incoming one, using a hole pattern on the optical table as the reference.

Tilting a component also elongates the beam footprint on it by $1/\cos\theta$, a factor of 1.41 at 45°, so a 25 mm mirror at 45° presents an effective aperture of 17.7 mm in the plane of incidence.

Pitfalls

Some fields measure the angle from the surface instead of the normal: X-ray diffraction uses the glancing angle, and grazing-incidence optics quote angles of a degree or less from the surface. A coating specified at 0° used at a few degrees off normal to avoid back-reflections is usually fine, but a stack specified at 45° is not interchangeable with one specified at 0°.

Common questions

Is the angle of incidence measured from the surface or the normal?

From the normal, in optics. Normal incidence is 0° and grazing incidence approaches 90°. Fields that quote the angle from the surface usually call it the glancing or grazing angle.

Does the angle of incidence equal the angle of reflection?

Yes, for a flat specular surface: both are measured from the same normal and lie in the plane of incidence. For a rough surface the light scatters into many angles, and only the specular component obeys the law.

Why are many optics specified at 45°?

A 45° mirror or splitter turns a beam through 90°, which fits the rectangular layout of an optical table; the coating is then designed for that angle.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010).