electric field has the form where is the (constant) complex amplitude vector, is the imaginary unit, is the wave vector (whose magnitude is the angular wavenumber), is the position vector, ω is the angular frequency, is time, and it is understood that the real part of the expression is the physical field. The magnetizing field has the same form with the same and ω. The value of the expression is unchanged if the position varies in a direction normal to ; hence is normal to the wavefronts. If ℓ is the component of in the direction of the field () can be written If the argument of is to be constant, ℓ must increase at the velocity known as the phase velocity. This in turn is equal to where is the phase velocity in the reference medium (taken as vacuum) and is the local refractive index w.r.t. the reference medium. Solving for gives i.e. where is the wavenumber in vacuum. From (), the electric field in the "external" medium has the form where is the wave vector for the transmitted wave (we assume isotropic media, but the transmitted wave is not yet assumed to be evanescent). In Cartesian coordinates , let the region have refractive index and let the region have refractive index . Then the plane is the interface, and the axis is normal to the interface (Fig.11). Let and (in bold roman type) be the unit vectors in the and directions, respectively. Let the plane of incidence (containing the incident wave-normal and the normal to the interface) be the plane (the plane of the page), with the angle of incidence θ measured from towards . Let the angle of refraction, measured in the same sense, be θt (t for transmitted, reserving r for reflected). From (), the transmitted wave vector has magnitude . Hence, from the geometry, where the last step uses Snell's law. Taking the dot product with the position vector, we get so that Eq.() becomes In the case of TIR, the angle θt does not exist in the usual sense. But we can still interpret () for the transmitted (evanescent) wave, by allowing to be complex. This becomes necessary when we write in terms of and thence in terms of using Snell's law: For θ greater than the critical angle, the value under the square-root symbol is negative, so that To determine which sign is