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2.5.9 Light reflection

The reflectance R at normal incidence at the interface between two non-absorbing media of refractive index n₀ and n₁ is given by

      R = [(n₀ − n₁)/(n₀ + n₁)]²

R(%) for various values of n₁ and n₀ = 1 (air) are tabulated below

For light at an angle of incidence θ₀ and angle or refraction θ₁ in the two media, the reflectance R depends on the state of polarization of the light and is given by

         R = [(N₀ − N₁)/(N₀ + N₁)]²

where for light polarized with the electric vector parallel to the plane of incidence (p vibration, TM wave)

         N₀ = n₀/cosθ₀   N₁ = n₁/cosθ₁

while for light with the electric vector perpendicular to the plane of incidence (s vibration, TE wave)

         N₀ = n₀cosθ₀   N₁ = n₁cosθ₁

For incident unpolarized light, the reflectance is the mean of R_p and R_s. Values tabulated below are for an air/glass interface (n₀ = 1, n₁ = 1.52)

The angle at which R_p = 0 is given by tan θ₀ = n₁/n₀ and is known as the Brewster angle.

At grazing incidence (θ₀ = 90°), R_p = R_s = 100%.

The reflection from a dielectric surface may readily be modified by the application of thin film coatings. For a single dielectric film located between the two bulk media of refractive indices n₀ and n₁, the reflectance varies with the film thickness d_f and refractive index n_f. When the film optical thickness (n_fd_f) equals one quarter of the wavelength of the incident light, the reflectance is a minimum if n_f is between n₀ and n₁, or a maximum if it lies outside this range. The reflectance R at normal incidence is then given by

Typical refractive indices of common thin film materials (wavelength 550 nm) and the reflectance of quarter-wave films on glass (n₀ = l, n₁ = 1.52) are shown below. The reflectances of quarter-wave multi-layer stacks of alternately high and low refractive index films on a glass substrate are also tabulated.

For complete elimination of the reflected beam the refractive index of a single film should equal , when the reflectance will be zero for the wavelength at which the film is quarter-wave in optical thickness. With the range of thin film materials available, complete suppression of the reflected beam cannot be achieved using a single film on common optical glasses. However, with a multilayer design, high-efficiency antireflection coatings on glass can be produced with a reflectance of less than 0.5% through the visible spectrum.

High refractive index films can usefully be employed as beam splitters. At oblique angles of incidence, appropriate design of a multilayer system allows the intensity and state of polarization of the transmitted and reflected components to be varied over wide limits.

For metals and other opaque media the normal incidence reflectance R is given by

where n₀ is the refractive index of the entrance medium and n₁ and k₁ are the real and imaginary parts of the complex refractive index n₁ = n₁ − ik₁of the absorbing medium. n₁ and k₁ are usually known as the refractive index and absorption index respectively.

Typical optical constants and reflectance (normal incidence) of metals and semi-conductors in air

Values of the optical constants exhibit significant variations depending on whether the material is in bulk or thin film form and its method of preparation. The reflectance of a metal surface is altered by the build up of oxide or other surface layers, while very dramatic changes can be produced by the deposition of thicker dielectric films.

Opaque metal films, e.g. aluminium, silver and gold (in the infrared), are commonly used as high reflectance mirrors, while very thin semi-transparent films (i.e. a few nm thick) can be employed as beam splitters, although there is significant light loss due to absorption.

R.J.King / K.W.Raine

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