Fresnel equations - AbsoluteAstronomy.com

The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean Fresnel

Augustin-Jean Fresnel

Augustin-Jean Fresnel , was a French engineer who contributed significantly to the establishment of the theory of wave optics. Fresnel studied the behaviour of light both theoretically and experimentally....

(icon), describe the behaviour of light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

when moving between media

Medium (optics)

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it...

of differing refractive indices

Refractive index

In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

. The reflection

Reflection (physics)

Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

of light that the equations predict is known as Fresnel reflection.

Explanation

When light moves from a medium of a given refractive index

Refractive index

n₁ into a second medium with refractive index n₂, both reflection

Reflection (physics)

and refraction

Refraction

Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

of the light may occur.

In the diagram on the right, an incident light ray

Ray (optics)

In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

PO strikes at point O the interface between two media of refractive indices n₁ and n₂. Part of the ray is reflected as ray OQ and part refracted as ray OS. The angles that the incident, reflected and refracted rays make to the normal

Surface normal

A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

of the interface are given as θ_i, θ_r and θ_t, respectively.
The relationship between these angles is given by the law of reflection: θ_i = θ_r; and Snell's law

Snell's law

In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...

: sin(θ_i)/sin(θ_t) = n₂/n₁.

The fraction of the incident power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance

Transmittance

In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, or absorption factor, which is the fraction of radiation absorbed by a sample at a specified wavelength...

T. The media are assumed to be non-magnetic.

The calculations of R and T depend on polarisation of the incident ray. If the light is polarised with the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

of the light perpendicular to the plane of the diagram above (s-polarised), the reflection coefficient

Reflection coefficient

The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...

is given by

,

where the second form is derived from the first by eliminating θ_t using Snell's law

Snell's law

and trigonometric identities.

If the incident light is polarised in the plane of the diagram (p-polarised), the R is given by

.

As a consequence of the conservation of energy, the transmission coefficient

Transmission coefficient

The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered...

in each case is given by T_s = 1 − R_s and T_p = 1 − R_p.

If the incident light is unpolarised (containing an equal mix of s- and p-polarisations), the reflection coefficient is R = (R_s + R_p)/2.

Equations for coefficients corresponding to ratios of the electric field

Electric field

amplitude

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

s of the waves can also be derived, and these are also called "Fresnel equations". These take several different forms, depending on the choice of formalism and sign convention

Sign convention

In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...

used. The amplitude coefficients are usually represented by lower case r and t. In some formalisms they satisfy:

At one particular angle for a given n₁ and n₂, the value of R_p goes to zero and a p-polarised incident ray is purely refracted. This angle is known as Brewster's angle

Brewster's angle

Brewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly...

, and is around 56° for a glass medium in air or vacuum. Note that this statement is only true when the refractive indices of both materials are real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, as is the case for materials like air and glass. For materials that absorb light, like metal

Metal

A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

s and semiconductor

Semiconductor

A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

s, n is complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, and R_p does not generally go to zero.

When moving from a denser medium into a less dense one (i.e., n₁ > n₂), above an incidence angle known as the critical angle, all light is reflected and R_s = R_p = 1. This phenomenon is known as total internal reflection

Total internal reflection

Total internal reflection is an optical phenomenon that happens when a ray of light strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is...

. The critical angle is approximately 41° for glass in air.
When the light is at near-normal incidence to the interface (θ_i ≈ θ_t ≈ 0), the reflection and transmission coefficient are given by:

For common glass, the reflection coefficient is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 + R), when interference can be neglected (see below).

The discussion given here assumes that the permeability

Permeability (electromagnetism)

In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

μ is equal to the vacuum permeability μ₀ in both media. This is approximately true for most dielectric

Dielectric

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

materials, but not for some other types of material. The completely general Fresnel equations are more complicated.

Effect from multiple surfaces

When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length

Coherence length

In physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. The significance is that interference will be strong within a coherence length of the source, but not beyond it...

, which for ordinary white light is few micrometers; it can be much larger for light from a laser

Laser

A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

. An example of this effect is the iridescent

Iridescence

Iridescence is generally known as the property of certain surfaces which appear to change color as the angle of view or the angle of illumination changes...

colours seen in a soap bubble

Soap bubble

A soap bubble is a thin film of soapy water enclosing air, that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also...

or in thin oil films on water. Applications include Fabry–Pérot interferometers, antireflection coatings, and optical filters. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

The transfer-matrix method

Transfer-matrix method (optics)

The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified medium. This is for example relevant for the design of anti-reflective coatings and dielectric mirrors.The reflection of light from a single...

, or the recursive Rouard method can be used to solve multiple-surface problems.

External links

Fresnel Equations – Wolfram
FreeSnell – Free software computes the optical properties of multilayer materials
Thinfilm – Web interface for calculating optical properties of thin films and multilayer materials. (Reflection & transmission coefficients, ellipsometric parameters Psi & Delta)
Simple web interface for calculating single-interface reflection and refraction angles and strengths.
Reflection and transmittance for two dielectrics – Mathematica interactive webpage that shows the relations between index of refraction and reflection.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Explanation

Effect from multiple surfaces

See also

External links