![Partial transmittance](https://upload.wikimedia.org/wikipedia/commons/3/30/Partial_transmittance.gif)
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
(or
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
in general) when incident on an interface between different optical
media
Media may refer to:
Communication
* Media (communication), tools used to deliver information or data
** Advertising media, various media, content, buying and placement for advertising
** Broadcast media, communications delivered over mass e ...
. They were deduced by
Augustin-Jean Fresnel
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
() who was the first to understand that light is a
transverse wave
In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time,
polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the ''s'' and ''p'' polarizations incident upon a material interface.
Overview
When light strikes the interface between a medium with
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
''n''
1 and a second medium with refractive index ''n''
2, both
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
and
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenome ...
of the light may occur. The Fresnel equations give the ratio of the ''reflected'' wave's electric field to the incident wave's electric field, and the ratio of the ''transmitted'' wave's electric field to the incident wave's electric field, for each of two components of polarization. (The ''magnetic'' fields can also be related using similar coefficients.) These ratios are generally complex, describing not only the relative amplitudes but also the
phase shift
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
s at the interface.
The equations assume the interface between the media is flat and that the media are homogeneous and
isotropic. The incident light is assumed to be a
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, ...
, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.
S and P polarizations
![Plane of incidence](https://upload.wikimedia.org/wikipedia/commons/b/b6/Plane_of_incidence.svg)
There are two sets of Fresnel coefficients for two different linear
polarization components of the incident wave. Since any
polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise,
unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.
The s polarization refers to polarization of a wave's electric field ''
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
'' to the
plane of incidence
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave opt ...
(the direction in the derivation below); then the magnetic field is ''in'' the plane of incidence. The p polarization refers to polarization of the electric field ''in'' the plane of incidence (the plane in the derivation below); then the magnetic field is ''normal'' to the plane of incidence.
Although the reflection and transmission are dependent on polarization, at normal incidence (''θ'' = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned
below in which that is true).
Power (intensity) reflection and transmission coefficients
![Fresnel power glass-to-air](https://upload.wikimedia.org/wikipedia/commons/d/d6/Fresnel_power_glass-to-air.svg)
In the diagram on the right, an incident
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, ...
in the direction of the ray IO strikes the interface between two media of refractive indices ''n''
1 and ''n''
2 at point O. Part of the wave is reflected in the direction OR, and part refracted in the direction OT. The angles that the incident, reflected and refracted rays make to the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surfac ...
:
and
Snell's law:
The behavior of light striking the interface is solved by considering the electric and magnetic fields that constitute an
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
, and the laws of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, as shown
below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine ''power'' coefficients, since power (or
irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude.
We call the fraction of the incident
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
that is reflected from the interface the ''
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
'' (or reflectivity, or power reflection coefficient) ''R'', and the fraction that is refracted into the second medium is called the ''
transmittance
Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is t ...
'' (or transmissivity, or power transmission coefficient) ''T''. Note that these are what would be measured right ''at'' each side of an interface and do not account for attenuation of a wave in an absorbing medium ''following'' transmission or reflection.
The reflectance for
s-polarized light
Polarization (American and British English spelling differences, also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation ...
is
while the reflectance for
p-polarized light is
where and are the
wave impedance The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
s of media 1 and 2, respectively.
We assume that the media are non-magnetic (i.e., ''μ''
1 = ''μ''
2 =
''μ''0), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies). Then the wave impedances are determined solely by the refractive indices ''n''
1 and ''n''
2:
where is the
impedance of free space The impedance of free space, , is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, , where is the electric field strength and is the magnetic fie ...
and =1,2. Making this substitution, we obtain equations using the refractive indices:
The second form of each equation is derived from the first by eliminating ''θ''
t using
Snell's law and
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
.
As a consequence of
conservation of energy, one can find the transmitted power (or more correctly,
irradiance: power per unit area) simply as the portion of the incident power that isn't reflected:
and
Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances ''in the direction of an incident or reflected wave'' (given by the magnitude of a wave's
Poynting vector) multiplied by cos''θ'' for a wave at an angle ''θ'' to the normal direction (or equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos''θ''
i = cos''θ''
r, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface.
Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the ''s'' and ''p'' polarizations, so that the ''effective'' reflectivity of the material is just the average of the two reflectivities:
For low-precision applications involving unpolarized light, such as
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, rather than rigorously computing the effective reflection coefficient for each angle,
Schlick's approximation In 3D computer graphics, Schlick’s approximation, named after Christophe Schlick, is a formula for approximating the contribution of the Fresnel factor in the specular reflection of light from a non-conducting interface (surface) between two medi ...
is often used.
Special cases
Normal incidence
For the case of
normal incidence The angle of incidence, in geometric optics, is the angle between a ray incident on a surface and the line perpendicular (at 90 degree angle) to the surface at the point of incidence, called the normal. The ray can be formed by any waves, such as o ...
, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to
For common glass (''n''
2 ≈ 1.5) surrounded by air (''n''
1=1), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.
Brewster's angle
At a dielectric interface from to , there is a particular angle of incidence at which goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as
Brewster's angle
Brewster's angle (also known as the polarization 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 ...
, and is around 56° for ''n''
1=1 and ''n''
2=1.5 (typical glass).
Total internal reflection
When light travelling in a denser medium strikes the surface of a less dense medium (i.e., ), beyond a particular incidence angle known as the ''critical angle'', all light is reflected and . This phenomenon, known as
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflect ...
, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin''θ'' ≤ 1 for all real ''θ''). For glass with ''n''=1.5 surrounded by air, the critical angle is approximately 42°.
Complex amplitude reflection and transmission coefficients
The above equations relating powers (which could be measured with a
photometer
A photometer is an instrument that measures the strength of electromagnetic radiation in the range from ultraviolet to infrared and including the visible spectrum. Most photometers convert light into an electric current using a photoresistor, ...
for instance) are derived from the Fresnel equations which solve the physical problem in terms of
electromagnetic field complex amplitude
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
s, i.e., considering
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
shifts in addition to their
amplitudes. Those underlying equations supply generally
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case ''r'' and ''t'' (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, of both media to be equal to the permeability of free space as is essentially true of all dielectrics at optical frequencies.
![Fresnel amplitudes glass-to-air](https://upload.wikimedia.org/wikipedia/commons/4/41/Fresnel_amplitudes_glass-to-air.svg)
In the following equations and graphs, we adopt the following conventions. For ''s'' polarization, the reflection coefficient is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for ''p'' polarization is the ratio of the waves complex ''magnetic'' field amplitudes (or equivalently, the ''negative'' of the ratio of their electric field amplitudes). The transmission coefficient is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients and are generally different between the ''s'' and ''p'' polarizations, and even at normal incidence (where the designations ''s'' and ''p'' do not even apply!) the sign of is reversed depending on whether the wave is considered to be ''s'' or ''p'' polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence).
The equations consider a plane wave incident on a plane interface at
angle of incidence a wave reflected at angle and a wave transmitted at angle In the case of an interface into an absorbing material (where ''n'' is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the
inhomogeneous waves launched into the second medium cannot be described using a single propagation angle.
Using this convention,
[Lecture notes by Bo Sernelius]
main site
, see especiall
Lecture 12
.[Born & Wolf, 1970, p.40, eqs.(20),(21).]
One can see that and . One can write very similar equations applying to the ratio of the waves' magnetic fields, but comparison of the electric fields is more conventional.
Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient R is just the squared magnitude of ''r'':
On the other hand, calculation of the power transmission coefficient is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power (
irradiance) is given by the square of the electric field amplitude ''divided by'' the
characteristic impedance of the medium (or by the square of the magnetic field ''multiplied by'' the characteristic impedance). This results in:
using the above definition of . The introduced factor of is the reciprocal of the ratio of the media's wave impedances. The factors adjust the waves' powers so they are reckoned ''in the direction'' normal to the interface, for both the incident and transmitted waves, so that full power transmission corresponds to ''T''=1.
In the case of
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflect ...
where the power transmission is zero, nevertheless describes the electric field (including its phase) just beyond the interface. This is an
evanescent field
In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillati ...
which does not propagate as a wave (thus =0) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the
phase angles of and (whose magnitudes are unity in this case). These phase shifts are different for ''s'' and ''p'' waves, which is the well-known principle by which total internal reflection is used to effect
polarization transformations.
Alternative forms
In the above formula for , if we put
(Snell's law) and multiply the numerator and denominator by , we obtain
If we do likewise with the formula for , the result is easily shown to be equivalent to
These formulas are known respectively as ''Fresnel's sine law'' and ''Fresnel's tangent law''. Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
as .
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
In physics, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves dif ...
, which for ordinary white light is few micrometers; it can be much larger for light from a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
.
An example of interference between reflections is the
iridescent colours seen in a
soap bubble or in thin oil films on water. Applications include
Fabry–Pérot interferometers,
antireflection coating
An antireflective, antiglare or anti-reflection (AR) coating is a type of optical coating applied to the surface of lenses, other optical elements, and photovoltaic cells to reduce reflection. In typical imaging systems, this improves the effici ...
s, and
optical filter
An optical filter is a device that selectively transmits light of different wavelengths, usually implemented as a glass plane or plastic device in the optical path, which are either dyed in the bulk or have interference coatings. The optical ...
s. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.
The
transfer-matrix method
In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier. In many one dimensional lattice m ...
, or the recursive Rouard method can be used to solve multiple-surface problems.
History
In 1808,
Étienne-Louis Malus
Étienne-Louis Malus (; ; 23 July 1775 – 23 February 1812) was a French officer, engineer, physicist, and mathematician.
Malus was born in Paris, France. He participated in Napoleon's expedition into Egypt (1798 to 1801) and was a member o ...
discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like ''one'' of the two rays emerging from a
doubly-refractive calcite crystal. He later coined the term ''polarization'' to describe this behavior. In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by
David Brewster.
[D. Brewster]
"On the laws which regulate the polarisation of light by reflexion from transparent bodies"
''Philosophical Transactions of the Royal Society'', vol.105, pp.125–59, read 16 March 1815. But the ''reason'' for that dependence was such a deep mystery that in late 1817,
Thomas Young was moved to write:
In 1821, however,
Augustin-Jean Fresnel
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
derived results equivalent to his sine and tangent laws (above), by modeling light waves as
transverse elastic waves with vibrations perpendicular to what had previously been called the
plane of polarization
The term ''plane of polarization'' refers to the direction of polarization of '' linearly-polarized'' light or other electromagnetic radiation. Unfortunately the term is used with two contradictory meanings. As originally defined by Étienne-Lou ...
. Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle. The experimental confirmation was reported in a "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were ''purely'' transverse.
[A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq., ''Annales de Chimie et de Physique'', vol.17, pp.102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted in Fresnel, 1866, pp.609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, &postscript", / , 2021.]
Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the
French Academy of Sciences in January 1823.
[A. Fresnel, "Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée" ("Memoir on the law of the modifications that reflection impresses on polarized light"), read 7 January 1823; reprinted in Fresnel, 1866, pp.767–99 (full text, published 1831), pp.753–62 (extract, published 1823). See especially pp.773 (sine law), 757 (tangent law), 760–61 and 792–6 (angles of total internal reflection for given phase differences).] That derivation combined conservation of energy with continuity of the ''tangential'' vibration at the interface, but failed to allow for any condition on the ''normal'' component of vibration. The first derivation from ''electromagnetic'' principles was given by
Hendrik Lorentz in 1875.
In the same memoir of January 1823,
Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients ( and ) gave complex values with unit magnitudes. Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the
argument represented the phase shift, and verified the hypothesis experimentally. The verification involved
* calculating the angle of incidence that would introduce a total phase difference of 90° between the s and p components, for various numbers of total internal reflections at that angle (generally there were two solutions),
* subjecting light to that number of total internal reflections at that angle of incidence, with an initial linear polarization at 45° to the plane of incidence, and
* checking that the final polarization was
circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circular ...
.
Thus he finally had a quantitative theory for what we now call the ''Fresnel rhomb'' — a device that he had been using in experiments, in one form or another, since 1817 (see ''
Fresnel rhomb §History'').
The success of the complex reflection coefficient inspired
James MacCullagh
James MacCullagh (1809 – 24 October 1847) was an Irish mathematician.
Early Life
MacCullagh was born in Landahaussy, near Plumbridge, County Tyrone, Ireland, but the family moved to Curly Hill, Strabane when James was about 10. He was the e ...
and
Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a
complex refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
.
Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir
[A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe" ("Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis"), read 9 December 1822; printed in Fresnel, 1866, pp.731–51 (full text), pp.719–29 (''extrait'', first published in ''Bulletin de la Société philomathique'' for 1822, pp. 191–8).] in which he introduced the needed terms ''
linear polarization
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
'', ''
circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to ...
'', and ''
elliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An el ...
'', and in which he explained
optical rotation as a species of
birefringence: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance.
Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see ''
Augustin-Jean Fresnel
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
'').
Derivation
Here we systematically derive the above relations from electromagnetic premises.
Material parameters
In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately)
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
homogeneous. If the medium is also
isotropic, the four field vectors are
related by
where ''ϵ'' and ''μ'' are scalars, known respectively as the (electric) ''
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
'' and the (magnetic) ''
permeability'' of the medium. For a vacuum, these have the values ''ϵ''
0 and ''μ''
0, respectively. Hence we define the ''relative'' permittivity (or
dielectric constant) , and the ''relative'' permeability .
In optics it is common to assume that the medium is non-magnetic, so that ''μ''
rel=1. For
ferromagnetic materials at radio/microwave frequencies, larger values of ''μ''
rel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible
metamaterial
A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...
s), ''μ''
rel is indeed very close to 1; that is, ''μ''≈''μ''
0.
In optics, one usually knows the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
''n'' of the medium, which is the ratio of the speed of light in a vacuum () to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic
wave impedance The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
, which is the ratio of the amplitude of to the amplitude of . It is therefore desirable to express ''n'' and in terms of ''ϵ'' and ''μ'', and thence to relate to ''n''. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave ''admittance'' , which is the reciprocal of the wave impedance .
In the case of ''uniform
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
sinusoidal'' waves, the wave impedance or admittance is known as the ''intrinsic'' impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived.
Electromagnetic plane waves
In a uniform plane sinusoidal
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
, the
electric field has the form
where is the (constant) complex amplitude vector, is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, is the
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
(whose magnitude is the angular
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
), is the
position vector, ''ω'' is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
, is time, and it is understood that the ''real part'' of the expression is the physical field.
[The above form () is typically used by physicists. ]Electrical engineers
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the ...
typically prefer the form that is, they not only use instead of for the imaginary unit, but also change the sign of the exponent, with the result that the whole expression is replaced by its complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, leaving the real part unchanged . The electrical engineers' form and the formulas derived therefrom may be converted to the physicists' convention by substituting for . The value of the expression is unchanged if the position varies in a direction normal to ; hence ''is normal to the wavefronts''.
To advance the
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
by the angle ''ϕ'', we replace by (that is, we replace by ), with the result that the (complex) field is multiplied by . So a phase ''advance'' is equivalent to multiplication by a complex constant with a ''negative''
argument. This becomes more obvious when the field () is factored as where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by .
[In the electrical engineering convention, the time-dependent factor is so that a phase advance corresponds to multiplication by a complex constant with a ''positive'' argument, and differentiation w.r.t. time corresponds to multiplication by . This article, however, uses the physics convention, whose time-dependent factor is . Although the imaginary unit does not appear explicitly in the results given here, the time-dependent factor affects the interpretation of any results that turn out to be complex.]
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 Solving for gives
As usual, we drop the time-dependent factor which is understood to multiply every complex field quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent ''
phasor''
For fields of that form,
Faraday's law and the
Maxwell-Ampère law respectively reduce to
[Compare M.V. Berry and M.R. Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", in E. Wolf (ed.), ''Progress in Optics'', vol.50, Amsterdam: Elsevier, 2007, pp.13–50, , at p.18, eq.(2.2).]
Putting and as above, we can eliminate and to obtain equations in only and :
If the material parameters ''ϵ'' and ''μ'' are real (as in a lossless dielectric), these equations show that form a ''right-handed orthogonal triad'', so that the same equations apply to the magnitudes of the respective vectors. Taking the magnitude equations and substituting from (), we obtain
where and are the magnitudes of and . Multiplying the last two equations gives
Dividing (or cross-multiplying) the same two equations gives where
This is the ''intrinsic admittance''.
From () we obtain the phase velocity For a vacuum this reduces to Dividing the second result by the first gives
For a ''non-magnetic'' medium (the usual case), this becomes
Taking the reciprocal of (), we find that the intrinsic ''impedance'' is In a vacuum this takes the value
known as the
impedance of free space The impedance of free space, , is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, , where is the electric field strength and is the magnetic fie ...
. By division, For a ''non-magnetic'' medium, this becomes
The wave vectors
![Wave vectors n1 to n2](https://upload.wikimedia.org/wikipedia/commons/0/03/Wave_vectors_n1_to_n2.svg)
In Cartesian coordinates , let the region have refractive index intrinsic admittance etc., and let the region have refractive index intrinsic admittance etc. Then the plane is the interface, and the axis is normal to the interface (see diagram). Let and (in bold
roman type) be the unit vectors in the and directions, respectively. Let the plane of incidence be the plane (the plane of the page), with the angle of incidence measured from towards . Let the angle of refraction, measured in the same sense, be where the subscript stands for ''transmitted'' (reserving for ''reflected'').
In the absence of
Doppler shifts, ''ω'' does not change on reflection or refraction. Hence, by (), the magnitude of the wave vector is proportional to the refractive index.
So, for a given ''ω'', if we ''redefine'' as the magnitude of the wave vector in the ''reference'' medium (for which ), then the wave vector has magnitude in the first medium (region in the diagram) and magnitude in the second medium. From the magnitudes and the geometry, we find that the wave vectors are
where the last step uses Snell's law. The corresponding dot products in the phasor form () are
Hence:
The ''s'' components
For the ''s'' polarization, the field is parallel to the axis and may therefore be described by its component in the direction. Let the reflection and transmission coefficients be and respectively. Then, if the incident field is taken to have unit amplitude, the phasor form () of its component is
and the reflected and transmitted fields, in the same form, are
Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the ''transverse'' field, meaning (in this context) the field normal to the plane of incidence. For the ''s'' polarization, that means the field. If the incident, reflected, and transmitted fields (in the above equations) are in the direction ("out of the page"), then the respective fields are in the directions of the red arrows, since form a right-handed orthogonal triad. The fields may therefore be described by their components in the directions of those arrows, denoted by . Then, since
At the interface, by the usual
interface conditions for electromagnetic fields, the tangential components of the and fields must be continuous; that is,
}
When we substitute from equations () to () and then from (), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations
which are easily solved for and yielding
and
At ''normal incidence'' indicated by an additional subscript 0, these results become
and
At ''grazing incidence'' , we have hence and .
The ''p'' components
For the ''p'' polarization, the incident, reflected, and transmitted fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be (redefining the symbols for the new context). Let the reflection and transmission coefficients be and . Then, if the incident field is taken to have unit amplitude, we have
If the fields are in the directions of the red arrows, then, in order for to form a right-handed orthogonal triad, the respective fields must be in the direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field the field in the case of the ''p'' polarization. The agreement of the ''other'' field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission.
So, for the incident, reflected, and transmitted fields, let the respective components in the direction be . Then, since
At the interface, the tangential components of the and fields must be continuous; that is,
}
When we substitute from equations () and () and then from (), the exponential factors again cancel out, so that the interface conditions reduce to
Solving for and we find
and
At ''normal incidence'' indicated by an additional subscript 0, these results become
and
At ''grazing incidence'' , we again have hence and .
Comparing () and () with () and (), we see that at ''normal'' incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at ''grazing'' incidence.
Power ratios (reflectivity and transmissivity)
The ''
Poynting vector'' for a wave is a vector whose component in any direction is the ''
irradiance'' (power per unit area) of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is where and are due ''only'' to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), and are in phase, and at right angles to each other and to the wave vector ; so, for s polarization, using the and components of and respectively (or for p polarization, using the and components of and ), the
irradiance in the direction of is given simply by which is in a medium of intrinsic impedance . To compute the irradiance in the direction normal to the interface, as we shall require in the definition of the power transmission coefficient, we could use only the component (rather than the full component) of or or, equivalently, simply multiply by the proper geometric factor, obtaining .
From equations () and (), taking squared magnitudes, we find that the ''
reflectivity
The reflectance of the surface of a material is its effectiveness in Reflection (physics), reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the respon ...
'' (ratio of reflected power to incident power) is
for the s polarization, and
for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cos''θ'', the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power ''transmission'' (below), these factors must be taken into account.
The simplest way to obtain the power transmission coefficient (''
transmissivity'', the ratio of transmitted power to incident power ''in the direction normal to the interface'', i.e. the direction) is to use (conservation of energy). In this way we find
for the s polarization, and
for the p polarization.
In the case of an interface between two lossless media (for which ϵ and μ are ''real'' and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations () and (). But, for given amplitude (as noted above), the component of the Poynting vector in the direction is proportional to the geometric factor and inversely proportional to the wave impedance . Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:
for the s polarization, and
for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, ).
For unpolarized light:
Where
.
Equal refractive indices
From equations () and (), we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have (that is, the transmitted ray is undeviated), so that the cosines in equations (), (), (), (), and () to () cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence. When extended to spherical reflection or scattering, this results in the Kerker effect for
Mie scattering
The Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the ...
.
Non-magnetic media
Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing () by ()) yields
For non-magnetic media we can substitute the
vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
''μ''
0 for ''μ'', so that
that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations () to () and equations () to (), the factor ''cμ''
0 cancels out. For the amplitude coefficients we obtain:
For the case of normal incidence these reduce to:
The power reflection coefficients become:
The power transmissions can then be found from .
Brewster's angle
For equal permeabilities (e.g., non-magnetic media), if and are ''
complementary'', we can substitute for and for so that the numerator in equation () becomes which is zero (by Snell's law). Hence and only the s-polarized component is reflected. This is what happens at the
Brewster angle
Brewster's angle (also known as the polarization 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 ...
. Substituting for in Snell's law, we readily obtain
for Brewster's angle.
Equal permittivities
Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations () and (), if ''ϵ'' is fixed instead of ''μ'', then becomes ''inversely'' proportional to , with the result that the subscripts 1 and 2 in equations () to () are interchanged (due to the additional step of multiplying the numerator and denominator by ). Hence, in () and (), the expressions for and in terms of refractive indices will be interchanged, so that Brewster's angle () will give instead of and any beam reflected at that angle will be p-polarized instead of s-polarized. Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization.
This switch of polarizations has an analog in the old mechanical theory of light waves (see ''
§History'', above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different ''densities'' and that the vibrations were ''normal'' to what was then called the
plane of polarization
The term ''plane of polarization'' refers to the direction of polarization of '' linearly-polarized'' light or other electromagnetic radiation. Unfortunately the term is used with two contradictory meanings. As originally defined by Étienne-Lou ...
, or by supposing (like
MacCullagh and
Neumann) that different refractive indices were due to different ''elasticities'' and that the vibrations were ''parallel'' to that plane.
[Whittaker, 1910, pp.133,148–9; Darrigol, 2012, pp.212,229–31.] Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.
See also
*
Jones calculus
In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by ''Jones matrices''. When light crosses an op ...
*
Polarization mixing
*
Index-matching material
*
Field and power quantities
*
Fresnel rhomb
A Fresnel rhomb is an optical prism that introduces a 90° phase difference between two perpendicular components of polarization, by means of two total internal reflections. If the incident beam is linearly polarized at 45° to the plane of inc ...
, Fresnel's apparatus to produce circularly polarised light
*
Specular reflection
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
*
Schlick's approximation In 3D computer graphics, Schlick’s approximation, named after Christophe Schlick, is a formula for approximating the contribution of the Fresnel factor in the specular reflection of light from a non-conducting interface (surface) between two medi ...
*
Snell's window
Snell's window (also called Snell's circle or optical man-hole) is a phenomenon by which an underwater viewer sees everything above the surface through a cone of light of width of about 96 degrees. This phenomenon is caused by refraction of light ...
*
X-ray reflectivity X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in chemistry, physics, and materials science to characterize surfaces, thin films and multilayers.J. ...
*
Plane of incidence
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave opt ...
*
Reflections of signals on conducting lines
A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impedance of the line, or if the far end ...
Notes
References
Sources
* M. Born and E. Wolf, 1970, ''
Principles of Optics'', 4th Ed., Oxford: Pergamon Press.
* J.Z. Buchwald, 1989, ''The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century'', University of Chicago Press, .
* R.E. Collin, 1966, ''Foundations for Microwave Engineering'', Tokyo: McGraw-Hill.
* O. Darrigol, 2012, ''A History of Optics: From Greek Antiquity to the Nineteenth Century'', Oxford, .
* A. Fresnel, 1866 (ed. H. de Senarmont, E. Verdet, and L. Fresnel), ''Oeuvres complètes d'Augustin Fresnel'', Paris: Imprimerie Impériale (3 vols., 1866–70)
vol.1 (1866)
* E. Hecht, 1987, ''Optics'', 2nd Ed., Addison Wesley, .
* E. Hecht, 2002, ''Optics'', 4th Ed., Addison Wesley, .
* F.A. Jenkins and H.E. White, 1976, ''Fundamentals of Optics'', 4th Ed., New York: McGraw-Hill, .
* H. Lloyd, 1834
"Report on the progress and present state of physical optics" ''Report of the Fourth Meeting of the British Association for the Advancement of Science'' (held at Edinburgh in 1834), London: J. Murray, 1835, pp.295–413.
* W. Whewell, 1857, ''History of the Inductive Sciences: From the Earliest to the Present Time'', 3rd Ed., London: J.W. Parker & Son
vol.2
*
E. T. Whittaker
Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
, 1910,
''A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century'', London: Longmans, Green, & Co.
Further reading
*
*
*
*
*''Encyclopaedia of Physics (2nd Edition)'',
R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
*''McGraw Hill Encyclopaedia of Physics (2nd Edition)'', C.B. Parker, 1994,
External links
Fresnel Equations– Wolfram.
Fresnel equations calculatorFreeSnell– 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 strengthsReflection and transmittance for two dielectrics– Mathematica interactive webpage that shows the relations between index of refraction and reflection.
A self-contained first-principles derivationof the transmission and reflection probabilities from a multilayer with complex indices of refraction.
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