Specular reflection

The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal. The incident and reflected rays lie in a plane known as the plane of incidence. The angles of the two rays to the normal are known as the angle of incidence and angle of reflection.

The earliest known description of this behavior was recorded by Hero of Alexandria (AD c. 10–70). Later, Alhazen gave a complete statement of the law of reflection. He was first to state that the incident ray, the reflected ray, and the normal to the surface all lie in a same plane perpendicular to reflecting plane.

Specular reflection may be contrasted with diffuse reflection, in which light is scattered away from the surface in a range of directions.

Law of reflection

When light encounters a boundary of a material, it is affected by the optical and electronic response functions of the material to electromagnetic waves. Optical processes, which comprise reflection and refraction, are expressed by the difference of the refractive index on both sides of the boundary, whereas reflectance and absorption are the real and imaginary parts of the response due to the electronic structure of the material. The degree of participation of each of these processes in the transmission is a function of the frequency, or wavelength, of the light, its polarization, and its angle of incidence. In general, reflection increases with increasing angle of incidence, and with increasing absorptivity at the boundary. The Fresnel equations describe the physics at the optical boundary.

Reflection may occur as specular, or mirror-like, reflection and diffuse reflection. Specular reflection reflects all light which arrives from a given direction at the same angle, whereas diffuse reflection reflects light in a broad range of directions. The distinction may be illustrated with surfaces coated with glossy paint and matte paint. Matte paints exhibit essentially complete diffuse reflection, while glossy paints show a larger component of specular behavior. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

Light propagates in space as a wave front of electromagnetic fields. A ray of light is characterized by the direction normal to the wave front (wave normal). When a ray encounters a surface, the angle that the wave normal makes with respect to the surface normal is called the angle of incidence and the plane defined by both directions is the plane of incidence. Reflection of the incident ray also occurs in the plane of incidence.

The law of reflection states that the angle of reflection of a ray equals the angle of incidence, and that the incident direction, the surface normal, and the reflected direction are coplanar.

When the light is incident perpendicularly to the surface, it is reflected straight back in the source direction.

The phenomenon of reflection arises from the diffraction of a plane wave on a flat boundary. When the boundary size is much larger than the wavelength, then the electromagnetic fields at the boundary are oscillating exactly in phase only for the specular direction.

Vector formulation

The law of reflection can also be equivalently expressed using linear algebra. The direction of a reflected ray is determined by the vector of incidence and the surface normal vector. Given an incident direction ${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{i}}}$ from the light source to the surface and the surface normal direction ${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{n}},}$ the specularly reflected direction ${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{s}}}$ (all unit vectors) is:

${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{s}}={\hat{\mathbf{d}}}_{\mathrm{i}}-2{\hat{\mathbf{d}}}_{\mathrm{n}}\left({\hat{\mathbf{d}}}_{\mathrm{n}}\cdot {\hat{\mathbf{d}}}_{\mathrm{i}}\right),}$

where ${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{n}}\cdot {\hat{\mathbf{d}}}_{\mathrm{i}}}$ is a scalar obtained with the dot product. Different authors may define the incident and reflection directions with different signs. Assuming these Euclidean vectors are represented in column form, the equation can be equivalently expressed as a matrix-vector multiplication:

${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{s}}=\mathbf{R}{\hat{\mathbf{d}}}_{\mathrm{i}},}$

where ${\displaystyle \mathbf{R}}$ is the so-called Householder transformation matrix, defined as:

${\displaystyle \mathbf{R}=\mathbf{I}-2{\hat{\mathbf{d}}}_{\mathrm{n}}{\hat{\mathbf{d}}}_{\mathrm{n}}^{\mathrm{T}};}$

in terms of the identity matrix ${\displaystyle \mathbf{I}}$ and twice the outer product of ${\displaystyle {\hat{\mathbf{d}}}_{\mathrm{n}}}$.