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 \mathbf{\hat{d}}_\mathrm{i} from the light source to the surface and the surface normal direction \mathbf{\hat{d}}_\mathrm{n}, the specularly reflected direction \mathbf{\hat{d}}_\mathrm{s} (all
unit vectors) is: : \mathbf{\hat{d}}_\mathrm{s} = \mathbf{\hat{d}}_\mathrm{i} - 2 \mathbf{\hat{d}}_\mathrm{n} \left(\mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{d}}_\mathrm{i}\right), where \mathbf{\hat{d}}_\mathrm{n} \cdot \mathbf{\hat{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:{{cite book | last1 = Farin | first1 = Gerald | last2 = Hansford | first2 = Dianne | author2-link = Dianne Hansford | title = Practical linear algebra: a geometry toolbox | url=http://www.farinhansford.com/books/pla/ : \mathbf{\hat{d}}_\mathrm{s} = \mathbf{R} \; \mathbf{\hat{d}}_\mathrm{i}, where \mathbf{R} is the so-called
Householder transformation matrix, defined as: : \mathbf{R} = \mathbf{I} - 2 \mathbf{\hat{d}}_\mathrm{n} \mathbf{\hat{d}}_\mathrm{n}^\mathrm{T}; in terms of the
identity matrix \mathbf{I} and twice the
outer product of \mathbf{\hat{d}}_\mathrm{n}. == Reflectivity ==