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Hence, by , the magnitude of the wave vector is proportional to the refractive index. So, for a given ω, if we redefine k as the magnitude of the wave vector in the reference medium (for which n = 1), then the wave vector has magnitude n 1 k in the first medium (region y < 0 in the diagram) and magnitude n 2 k in the second medium.
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
Reflection from a stratified interface. The Abeles matrix method [3] [4] [5] is a computationally fast and easy way to calculate the specular reflectivity from a stratified interface, as a function of the perpendicular momentum transfer, Q z: = =
Each optical element (surface, interface, mirror, or beam travel) is described by a 2 × 2 ray transfer matrix which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system.
The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the reflectivity of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the Fresnel equations , which for normal incidence reduces to [ 42 ] : 44
In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane. Some mathematicians use "flip" as a synonym for "reflection". [2 ...
The reflection hyperplane can be defined by its normal vector, a unit vector (a vector with length ) that is orthogonal to the hyperplane. The reflection of a point about this hyperplane is the linear transformation:
Given a normalized light vector (pointing from the light source toward the surface) and a normalized plane normal vector , one can work out the normalized reflected and refracted rays, via the cosines of the angle of incidence and angle of refraction , without explicitly using the sine values or any trigonometric functions or angles: [22]