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The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by French engineer and physicist Augustin-Jean Fresnel (Template:IPAc-en) who was the first to understand that light is a transverse wave, when no one realized that the waves 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.
OverviewEdit
When light strikes the interface between a medium with refractive index Template:Math and a second medium with refractive index Template:Math, both reflection and refraction 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 shifts at the interface.
The equations assume the interface between the media is flat and that the media are homogeneous and isotropic.<ref>Born & Wolf, 1970, p. 38.</ref> The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.
S and P polarizationsEdit
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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 to the plane of incidence (the Template:Mvar 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 Template:Mvar plane in the derivation below); then the magnetic field is normal to the plane of incidence. The names "s" and "p" for the polarization components refer to German "senkrecht" (perpendicular or normal) and "parallel" (parallel to the plane of incidence).
Although the reflection and transmission are dependent on polarization, at normal incidence (Template:Math) 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).
ConfigurationEdit
In the diagram on the right, an incident plane wave in the direction of the ray Template:Math strikes the interface between two media of refractive indices Template:Math and Template:Math at point Template:Math. Part of the wave is reflected in the direction Template:Math, and part refracted in the direction Template:Math. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as Template:Math, Template:Math and Template:Math, respectively. The relationship between these angles is given by the law of reflection: <math display=block>\theta_\mathrm{i} = \theta_\mathrm{r},</math> and Snell's law: <math display=block>n_1 \sin \theta_\mathrm{i} = n_2 \sin \theta_\mathrm{t}.</math>
The behavior of light striking the interface is explained by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, 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.
Power (intensity) reflection and transmission coefficientsEdit
We call the fraction of the incident power that is reflected from the interface the reflectance (or reflectivity, or power reflection coefficient) Template:Math, and the fraction that is refracted into the second medium is called the transmittance (or transmissivity, or power transmission coefficient) Template:Math. 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.<ref>Hecht, 1987, p. 100.</ref>
The reflectance for s-polarized light is <math display=block>
R_\mathrm{s} = \left|\frac{Z_2 \cos \theta_\mathrm{i} - Z_1 \cos \theta_\mathrm{t}}{Z_2 \cos \theta_\mathrm{i} + Z_1 \cos \theta_\mathrm{t}}\right|^2,
</math> while the reflectance for p-polarized light is <math display=block>
R_\mathrm{p} = \left|\frac{Z_2 \cos \theta_\mathrm{t} - Z_1 \cos \theta_\mathrm{i}}{Z_2 \cos \theta_\mathrm{t} + Z_1 \cos \theta_\mathrm{i}}\right|^2,
</math> where Template:Math and Template:Math are the wave impedances of media 1 and 2, respectively.
We assume that the media are non-magnetic (i.e., Template:Math), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies).<ref>Template:Cite book</ref> Then the wave impedances are determined solely by the refractive indices Template:Math and Template:Math: <math display=block>Z_i = \frac{Z_0}{n_i}\,,</math> where Template:Math is the impedance of free space and Template:Math. Making this substitution, we obtain equations using the refractive indices: <math display=block>
R_\mathrm{s} = \left|\frac{n_1 \cos \theta_\mathrm{i} - n_2 \cos \theta_\mathrm{t}}{n_1 \cos \theta_\mathrm{i} + n_2 \cos \theta_\mathrm{t}}\right|^2
= \left|\frac
{n_1 \cos \theta_{\mathrm{i}} - n_2 \sqrt{1 - \left(\frac{n_1}{n_2} \sin \theta_{\mathrm{i}}\right)^2}}
{n_1 \cos \theta_{\mathrm{i}} + n_2 \sqrt{1 - \left(\frac{n_1}{n_2} \sin \theta_{\mathrm{i}}\right)^2}}
\right|^2\!,
</math> <math display=block>
R_\mathrm{p} = \left|\frac{n_1 \cos \theta_\mathrm{t} - n_2 \cos \theta_\mathrm{i}}{n_1 \cos \theta_\mathrm{t} + n_2 \cos \theta_\mathrm{i}}\right|^2
= \left|\frac
{n_1 \sqrt{1 - \left(\frac{n_1}{n_2} \sin \theta_\mathrm{i}\right)^2} - n_2 \cos \theta_\mathrm{i}}
{n_1 \sqrt{1 - \left(\frac{n_1}{n_2} \sin \theta_\mathrm{i}\right)^2} + n_2 \cos \theta_\mathrm{i}}
\right|^2\!.
</math>
The second form of each equation is derived from the first by eliminating Template:Math using Snell's law and trigonometric identities.
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:Template:Hsp<ref>Hecht, 1987, p.Template:Tsp102.</ref> <math display=block>T_\mathrm{s} = 1 - R_\mathrm{s}</math> and <math display=block>T_\mathrm{p} = 1 - R_\mathrm{p}</math>
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 Template:Math for a wave at an angle Template:Math 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 Template:Math, 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: <math display=block>R_\mathrm{eff} = \frac{1}{2}\left(R_\mathrm{s} + R_\mathrm{p}\right).</math>
For low-precision applications involving unpolarized light, such as computer graphics, rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used.
Special casesEdit
Normal incidenceEdit
For the case of normal incidence, Template:Math, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to <math display=block> R_0 = \left|\frac{n_1 - n_2 }{n_1 + n_2 }\right|^2\,. </math>
For common glass (Template:Math) surrounded by air (Template:Math), 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 angleEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} At a dielectric interface from Template:Math to Template:Math, there is a particular angle of incidence at which Template:Math 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, and is around 56° for Template:Math and Template:Math (typical glass).
Total internal reflectionEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} When light travelling in a denser medium strikes the surface of a less dense medium (i.e., Template:Math), beyond a particular incidence angle known as the critical angle, all light is reflected and Template:Math. This phenomenon, known as total internal reflection, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact Template:Math for all real Template:Math). For glass with Template:Math surrounded by air, the critical angle is approximately 42°.
45° incidenceEdit
Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence (Template:Math), it follows algebraically from the above equations that Template:Math equals the square of Template:Math: <math display=block> R_\text{p} = R_\text{s}^2 </math>
This can be used to either verify the consistency of the measurements of Template:Math and Template:Math, or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required.
Measurements of Template:Math and Template:Math at 45° can be used to estimate the reflectivity at normal incidence.Template:Cn The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of Template:Math and Template:Math, and then averaging these two averages again arithmetically, gives a value for Template:Math with an error of less than about 3% for most common optical materials.Template:Cn This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since the dependence of Template:Math and Template:Math on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam.
Complex amplitude reflection and transmission coefficientsEdit
The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase shifts in addition to their amplitudes. Those underlying equations supply generally complex-valued 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 Template:Math and Template:Math (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, Template:Math of both media to be equal to the permeability of free space Template:Math as is essentially true of all dielectrics at optical frequencies.
In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient Template:Math is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for p polarization Template:Math 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 Template:Math is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients Template:Math and Template:Math 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 Template:Math 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 Template:Nowrap, a wave reflected at angle Template:Nowrap, and a wave transmitted at angle Template:Nowrap. In the case of an interface into an absorbing material (where Template:Math 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,<ref name=Sernelius>Lecture notes by Bo Sernelius, main site Template:Webarchive, see especially Lecture 12 .</ref><ref name="Born 1970">Born & Wolf, 1970, p.Template:Hsp40, eqs.Template:Tsp(20),Template:Hsp(21).</ref> <math display="block">\begin{align}
r_\text{s} &= \frac{ n_1 \cos \theta_\text{i} - n_2 \cos \theta_\text{t}}{n_1 \cos \theta_\text{i} + n_2 \cos \theta_\text{t}}, \\[3pt]
t_\text{s} &= \frac{2 n_1 \cos \theta_\text{i}} {n_1 \cos \theta_\text{i} + n_2 \cos \theta_\text{t}}, \\[3pt]
r_\text{p} &= \frac{ n_2 \cos \theta_\text{i} - n_1 \cos \theta_\text{t}}{n_2 \cos \theta_\text{i} + n_1 \cos \theta_\text{t}}, \\[3pt]
t_\text{p} &= \frac{2 n_1 \cos \theta_\text{i}} {n_2 \cos \theta_\text{i} + n_1 \cos \theta_\text{t}}.
\end{align}</math>
For the case where the magnetic permeabilities are non-negligible, the equations change such that every appearance of <math> n_i </math> is replaced by <math> n_i/\mu_i </math> (for both <math> i=1, 2 </math>).Template:Cn
One can see that Template:Math<ref>Hecht, 2002, p.Template:Hsp116, eqs.Template:Tsp(4.49),Template:Hsp(4.50).</ref> and Template:Math. 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 Template:Math is just the squared magnitude of Template:Math:Template:Hsp<ref>Hecht, 2002, p.Template:Hsp120, eq.Template:Hsp(4.56).</ref> <math display=block>R = |r|^2.</math>
On the other hand, calculation of the power transmission coefficient Template:Mvar 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:<ref>Hecht, 2002, p.Template:Hsp120, eq.Template:Hsp(4.57).</ref> <math display=block>T = \frac{n_2 \cos \theta_\text{t}}{n_1 \cos \theta_\text{i}} |t|^2</math> using the above definition of Template:Math. The introduced factor of Template:Math is the reciprocal of the ratio of the media's wave impedances. The Template:Math 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 Template:Math.
In the case of total internal reflection where the power transmission Template:Mvar is zero, Template:Mvar nevertheless describes the electric field (including its phase) just beyond the interface. This is an evanescent field which does not propagate as a wave (thus Template:Math) 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 Template:Math and Template:Math (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 formsEdit
In the above formula for Template:Math, if we put <math>n_2=n_1\sin\theta_\text{i}/\sin\theta_\text{t}</math> (Snell's law) and multiply the numerator and denominator by Template:Math, we obtainTemplate:Hsp<ref>Fresnel, 1866, p.Template:Hsp773.</ref><ref>Hecht, 2002, p.Template:Hsp115, eq.Template:Hsp(4.42).</ref> <math display=block>r_\text{s}=-\frac{\sin(\theta_\text{i}-\theta_\text{t})}{\sin(\theta_\text{i}+\theta_\text{t})}.</math>
If we do likewise with the formula for Template:Math, the result is easily shown to be equivalent toTemplate:Hsp<ref>Fresnel, 1866, p.Template:Hsp757.</ref><ref>Hecht, 2002, p.Template:Hsp115, eq.Template:Hsp(4.43).</ref> <math display=block>r_\text{p}=\frac{\tan(\theta_\text{i}-\theta_\text{t})}{\tan(\theta_\text{i}+\theta_\text{t})}. </math>
These formulasTemplate:Hsp<ref>E. Verdet, in Fresnel, 1866, p.Template:Hsp789n.</ref><ref>Born & Wolf, 1970, p.Template:Hsp40, eqs.Template:Hsp(21a).</ref><ref>Jenkins & White, 1976, p.Template:Hsp524, eqs.Template:Hsp(25a).</ref> are known respectively as Fresnel's sine law and Fresnel's tangent law.<ref>Whittaker, 1910, p.Template:Hsp134; Darrigol, 2012, p.Template:Tsp213.</ref> Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as Template:Math.
Multiple surfacesEdit
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, which for ordinary white light is few micrometers; it can be much larger for light from a laser.
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 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, or the recursive Rouard methodTemplate:Hsp<ref>Template:Cite book chapt. 4.</ref> can be used to solve multiple-surface problems.
HistoryEdit
In 1808, Étienne-Louis Malus 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.<ref>Darrigol, 2012, pp.Template:Tsp191–2.</ref> 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.<ref name=brewster-1815b>D. Brewster, "On the laws which regulate the polarisation of light by reflexion from transparent bodies", Philosophical Transactions of the Royal Society, vol.Template:Tsp105, pp.Template:Tsp125–59, read 16 March 1815.</ref> But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write:
<templatestyles src="Template:Blockquote/styles.css" />
Template:Brackethe great difficulty of all, which is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory.<ref>T. Young, "Chromatics" (written Sep–Oct 1817), Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica, vol.Template:Tsp3 (first half, issued February 1818), pp.Template:Tsp141–63, concluding sentence.</ref>{{#if:|{{#if:|}}
— {{#if:|, in }}Template:Comma separated entries}}
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In 1821, however, Augustin-Jean Fresnel 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. 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.<ref>Buchwald, 1989, pp.Template:Tsp390–91; Fresnel, 1866, pp.Template:Tsp646–8.</ref> 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.<ref name=fresnel-1821a>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.Template:Nbsp17, pp.Template:Nbsp102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821); reprinted in Fresnel, 1866, pp.Template:Nbsp609–48; translated as "On the calculation of the tints that polarization develops in crystalline plates, &Template:Nbsppostscript", Template:Zenodo / {{#invoke:doi|main}}, 2021.</ref>
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.<ref name=fresnel-1823a>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.Template:Tsp767–99 (full text, published 1831), pp.Template:Tsp753–62 (extract, published 1823). See especially pp.Template:Tsp773 (sine law), 757 (tangent law), 760–61 and 792–6 (angles of total internal reflection for given phase differences).</ref> 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.<ref>Buchwald, 1989, pp.Template:Tsp391–3; Whittaker, 1910, pp.Template:Tsp133–5.</ref> The first derivation from electromagnetic principles was given by Hendrik Lorentz in 1875.<ref>Buchwald, 1989, p.Template:Hsp392.</ref>
In the same memoir of January 1823,<ref name=fresnel-1823a /> Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients (Template:Math and Template:Math) 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.<ref>Lloyd, 1834, pp.Template:Tsp369–70; Buchwald, 1989, pp.Template:Tsp393–4,Template:Tsp453; Fresnel, 1866, pp.Template:Tsp781–96.</ref> 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.<ref>Fresnel, 1866, pp.Template:Tsp760–61,Template:Tsp792–6; Whewell, 1857, p.Template:Tsp359.</ref>
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|Fresnel rhomb §Template:TspHistory]]).
The success of the complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index.<ref>Whittaker, 1910, pp.Template:Tsp177–179.</ref>
Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoirTemplate:Hsp<ref name=fresnel-1822z>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.Template:Tsp731–751 (full text), pp.Template:Tsp719–729 (extrait, first published in Bulletin de la Société philomathique for 1822, pp. 191–8).</ref> in which he introduced the needed terms linear polarization, circular polarization, and elliptical polarization,<ref>Buchwald, 1989, pp.Template:Tsp230–231; Fresnel, 1866, p.Template:Hsp744.</ref> 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.<ref>Buchwald, 1989, p.Template:Hsp442; Fresnel, 1866, pp.Template:Tsp737–739,Template:Tsp749. Cf. Whewell, 1857, pp.Template:Tsp356–358; Jenkins & White, 1976, pp.Template:Tsp589–590.</ref>
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).
DerivationEdit
Here we systematically derive the above relations from electromagnetic premises.
Material parametersEdit
In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous. If the medium is also isotropic, the four field vectors Template:MathTemplate:Tsp are related by <math display=block>\begin{align} \mathbf{D} &= \epsilon \mathbf{E} \\ \mathbf{B} &= \mu \mathbf{H}\,, \end{align} </math> where Template:Math and Template:Math are scalars, known respectively as the (electric) permittivity and the (magnetic) permeability of the medium. For vacuum, these have the values Template:Math and Template:Math, respectively. Hence we define the relative permittivity (or dielectric constant) Template:Math, and the relative permeability Template:Math.
In optics it is common to assume that the medium is non-magnetic, so that Template:Math. For ferromagnetic materials at radio/microwave frequencies, larger values of Template:Math must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials), Template:Math is indeed very close to 1; that is, Template:Math.
In optics, one usually knows the refractive index Template:Math of the medium, which is the ratio of the speed of light in vacuum (Template:Mvar) 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 Template:Mvar, which is the ratio of the amplitude of Template:Math to the amplitude of Template:Math. It is therefore desirable to express Template:Math and Template:Mvar in terms of Template:Math and Template:Math, and thence to relate Template:Mvar to Template:Math. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave admittance Template:Mvar, which is the reciprocal of the wave impedance Template:Mvar.
In the case of uniform plane 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 wavesEdit
In a uniform plane sinusoidal electromagnetic wave, the electric field Template:Math has the form Template:NumBlk where Template:Math is the (constant) complex amplitude vector, Template:Math is the imaginary unit, Template:Math is the wave vector (whose magnitude Template:Mvar is the angular wavenumber), Template:Math is the position vector, Template:Math is the angular frequency, Template:Math is time, and it is understood that the real part of the expression is the physical field.<ref group=Note>The above form (Template:EquationNote) is typically used by physicists. Electrical engineers typically prefer the form Template:Math that is, they not only use Template:Math instead of Template:Math 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, leaving the real part unchanged Template:Bracket. The electrical engineers' form and the formulas derived therefrom may be converted to the physicists' convention by substituting Template:Math for Template:Math.</ref> The value of the expression is unchanged if the position Template:Math varies in a direction normal to Template:Math; hence Template:Math is normal to the wavefronts.
To advance the phase by the angle ϕ, we replace Template:Math by Template:Math (that is, we replace Template:Math by Template:Math), with the result that the (complex) field is multiplied by Template:Math. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. This becomes more obvious when the field (Template:EquationNote) is factored as Template:Math, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by Template:Math.Template:Hsp<ref group=Note>In the electrical engineering convention, the time-dependent factor is Template:Math, 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 Template:Math. This article, however, uses the physics convention, whose time-dependent factor is Template:Math. 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.</ref>
If ℓ is the component of Template:Math in the direction of Template:Math, the field (Template:EquationNote) can be written Template:Math. If the argument of Template:Math is to be constant, ℓ must increase at the velocity <math>\omega/k\,,\,</math> known as the phase velocity Template:Math. This in turn is equal to Template:Nowrap Solving for Template:Mvar gives Template:NumBlk
As usual, we drop the time-dependent factor Template:Math, 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 Template:NumBlk.</math>|Template:EquationRef}} For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce toTemplate:Hsp<ref name=berry-jeffrey-2007>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.Template:Tsp50, Amsterdam: Elsevier, 2007, pp.Template:Tsp13–50, {{#invoke:doi|main}}, at p.Template:Hsp18, eq.Template:Tsp(2.2).</ref> <math display=block>\begin{align}
\omega\mathbf{B} &= \mathbf{k}\times\mathbf{E}\\
\omega\mathbf{D} &= -\mathbf{k}\times\mathbf{H}\,.
\end{align}</math>
Putting Template:Math and Template:Math, as above, we can eliminate Template:Math and Template:Math to obtain equations in only Template:Math and Template:Math: <math display=block>\begin{align}
\omega\mu\mathbf{H} &= \mathbf{k}\times\mathbf{E}\\
\omega\epsilon\mathbf{E} &= -\mathbf{k}\times\mathbf{H}\,.
\end{align}</math> If the material parameters Template:Math and Template:Math are real (as in a lossless dielectric), these equations show that Template:Math 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 (Template:EquationNote), we obtain <math display=block>\begin{align}
\mu cH &= nE\\ \epsilon cE &= nH\,,
\end{align}</math> where Template:Mvar and Template:Mvar are the magnitudes of Template:Math and Template:Math. Multiplying the last two equations gives Template:NumBlk Dividing (or cross-multiplying) the same two equations gives Template:Math, where Template:NumBlk This is the intrinsic admittance.
From (Template:EquationNote) we obtain the phase velocity Template:Nowrap For vacuum this reduces to Template:Nowrap Dividing the second result by the first gives <math display=block>n=\sqrt{\mu_{\text{rel}}\epsilon_{\text{rel}}}\,.</math> For a non-magnetic medium (the usual case), this becomes Template:Tmath. Template:LargerTaking the reciprocal of (Template:EquationNote), we find that the intrinsic impedance is Template:Nowrap In vacuum this takes the value <math display="inline">Z_0=\sqrt{\mu_0/\epsilon_0}\,\approx 377\,\Omega\,,</math> known as the impedance of free space. By division, Template:Nowrap/\epsilon_{\text{rel}}}</math>.}} For a non-magnetic medium, this becomes <math>Z=Z_0\big/\!\sqrt{\epsilon_{\text{rel}}}=Z_0/n.</math>Template:Larger
Wave vectorsEdit
In Cartesian coordinates Template:Math, let the region Template:Math have refractive index Template:Math, intrinsic admittance Template:Math, etc., and let the region Template:Math have refractive index Template:Math, intrinsic admittance Template:Math, etc. Then the Template:Math plane is the interface, and the Template:Math axis is normal to the interface (see diagram). Let Template:Math and Template:Math (in bold roman type) be the unit vectors in the Template:Math and Template:Math directions, respectively. Let the plane of incidence be the Template:Math plane (the plane of the page), with the angle of incidence Template:Math measured from Template:Math towards Template:Math. Let the angle of refraction, measured in the same sense, be Template:Math, where the subscript Template:Math stands for transmitted (reserving Template:Math for reflected).
In the absence of Doppler shifts, ω does not change on reflection or refraction. Hence, by (Template:EquationNote), the magnitude of the wave vector is proportional to the refractive index.
So, for a given Template:Math, if we redefine Template:Mvar as the magnitude of the wave vector in the reference medium (for which Template:Math), then the wave vector has magnitude Template:Math in the first medium (region Template:Math in the diagram) and magnitude Template:Math in the second medium. From the magnitudes and the geometry, we find that the wave vectors are <math display=block>\begin{align}
\mathbf{k}_\text{i} &= n_1 k(\mathbf{i}\sin\theta_\text{i} + \mathbf{j}\cos\theta_\text{i})\\[.5ex]
\mathbf{k}_\text{r} &= n_1 k(\mathbf{i}\sin\theta_\text{i} - \mathbf{j}\cos\theta_\text{i})\\[.5ex]
\mathbf{k}_\text{t} &= n_2 k(\mathbf{i}\sin\theta_\text{t} + \mathbf{j}\cos\theta_\text{t})\\
&= k(\mathbf{i}\,n_1\sin\theta_\text{i} + \mathbf{j}\,n_2\cos\theta_\text{t})\,,
\end{align}</math> where the last step uses Snell's law. The corresponding dot products in the phasor form (Template:EquationNote) are Template:NumBlk Hence: Template:NumBlk
s componentsEdit
For the s polarization, the Template:Math field is parallel to the Template:Math axis and may therefore be described by its component in the Template:Math direction. Let the reflection and transmission coefficients be Template:Math and Template:Math, respectively. Then, if the incident Template:Math field is taken to have unit amplitude, the phasor form (Template:EquationNote) of its Template:Math-component is Template:NumBlk,</math>|Template:EquationRef}} and the reflected and transmitted fields, in the same form, are Template:NumBlk\\
E_\text{t} &= t_{s\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|Template:EquationRef}}
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 Template:Math field. If the incident, reflected, and transmitted Template:Math fields (in the above equations) are in the Template:Math-direction ("out of the page"), then the respective Template:Math fields are in the directions of the red arrows, since Template:Math form a right-handed orthogonal triad. The Template:Math fields may therefore be described by their components in the directions of those arrows, denoted by Template:Math. Then, since Template:Math, Template:NumBlk\\
H_\text{r} &=\, Y_1 r_{s\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
H_\text{t} &=\, Y_2 t_{s\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|Template:EquationRef}}
At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the Template:Math and Template:Math fields must be continuous; that is, Template:NumBlk When we substitute from equations (Template:EquationNote) to (Template:EquationNote) and then from (Template:EquationNote), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations Template:NumBlk which are easily solved for Template:Math and Template:Math, yielding Template:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}</math>|Template:EquationRef}} and Template:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\,.</math>|Template:EquationRef}} At normal incidence Template:Math, indicated by an additional subscript 0, these results become Template:NumBlk and Template:NumBlk At grazing incidence Template:Math, we have Template:Math, hence Template:Math and Template:Math.
p componentsEdit
For the p polarization, the incident, reflected, and transmitted Template:Math 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 Template:Math (redefining the symbols for the new context). Let the reflection and transmission coefficients be Template:Math and Template:Math. Then, if the incident Template:Math field is taken to have unit amplitude, we have Template:NumBlk\\
E_\text{r} &= r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
E_\text{t} &= t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|Template:EquationRef}} If the Template:Math fields are in the directions of the red arrows, then, in order for Template:Math to form a right-handed orthogonal triad, the respective Template:Math fields must be in the Template:Math-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 Template:Largerthe Template:Math field in the case of the p polarizationTemplate:Larger. 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.<ref>This agrees with Born & Wolf, 1970, p.Template:Hsp38, Fig.Template:Tsp1.10.</ref>
So, for the incident, reflected, and transmitted Template:Math fields, let the respective components in the Template:Math-direction be Template:Math. Then, since Template:Math, Template:NumBlk\\
H_\text{r} &=\, Y_1 r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
H_\text{t} &=\, Y_2 t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|Template:EquationRef}}
At the interface, the tangential components of the Template:Math and Template:Math fields must be continuous; that is, Template:NumBlk When we substitute from equations (Template:EquationNote) and (Template:EquationNote) and then from (Template:EquationNote), the exponential factors again cancel out, so that the interface conditions reduce to Template:NumBlk Solving for Template:Math and Template:Math, we find Template:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}</math>|Template:EquationRef}} and Template:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\,.</math>|Template:EquationRef}} At normal incidence Template:Math indicated by an additional subscript 0, these results become Template:NumBlk and Template:NumBlk At Template:Itco Template:Math, we again have Template:Math, hence Template:Math and Template:Math.
Comparing (Template:EquationNote) and (Template:EquationNote) with (Template:EquationNote) and (Template:EquationNote), 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)Edit
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 Template:Math, where Template:Math and Template:Math are due only to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), Template:Math and Template:Math are in phase, and at right angles to each other and to the wave vector Template:Math; so, for s polarization, using the Template:Mvar and Template:Mvar components of Template:Math and Template:Math respectively (or for p polarization, using the Template:Mvar and Template:Math components of Template:Math and Template:Math), the irradiance in the direction of Template:Math is given simply by Template:Math, which is Template:Math in a medium of intrinsic impedance Template:Math. 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 Template:Mvar component (rather than the full Template:Mvar component) of Template:Math or Template:Math or, equivalently, simply multiply Template:Math by the proper geometric factor, obtaining Template:Math.
From equations (Template:EquationNote) and (Template:EquationNote), taking squared magnitudes, we find that the reflectivity (ratio of reflected power to incident power) is Template:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\right|^2</math>|Template:EquationRef}} for the s polarization, and Template:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\right|^2</math>|Template:EquationRef}} for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cosTemplate:Tspθ, 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 Template:Mvar direction) is to use Template:Math (conservation of energy). In this way we find Template:NumBlk{\cos\theta_\text{i}} =\frac{4Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}}{\left(Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}\right)^2}</math>|Template:EquationRef}} for the s polarization, and Template:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\right)^2\frac{\,Y_2\,}{Y_1}\,\frac{\cos\theta_\text{t}}{\cos\theta_\text{i}} =\frac{4Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}}{\left(Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}\right)^2}</math>|Template:EquationRef}} 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, Template:Math).
For unpolarized light: <math display="block">T={1 \over 2}(T_s+T_p)</math> <math display="block">R={1 \over 2}(R_s+R_p)</math> where <math>R+T=1</math>.
Equal refractive indicesEdit
From equations (Template:EquationNote) and (Template:EquationNote), 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 Template:Math (that is, the transmitted ray is undeviated), so that the cosines in equations (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), (Template:EquationNote), and (Template:EquationNote) to (Template:EquationNote) 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.<ref>Template:Cite journal</ref> When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering.
Non-magnetic mediaEdit
Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing (Template:EquationNote) by (Template:EquationNote)) yields <math display=block>Y=\frac{n}{\,c\mu\,}\,.</math> For non-magnetic media we can substitute the vacuum permeability Template:Math for Template:Math, so that <math display=block>Y_1=\frac{n_1}{\,c\mu_0} ~~;~~~ Y_2=\frac{n_2}{\,c\mu_0}\,;</math> that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations (Template:EquationNote) to (Template:EquationNote) and equations (Template:EquationNote) to (Template:EquationNote), the factor cμ0 cancels out. For the amplitude coefficients we obtain:<ref name=Sernelius /><ref name="Born 1970" />
Template:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}</math>|Template:EquationRef}} Template:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}\,</math>|Template:EquationRef}}
Template:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}</math>|Template:EquationRef}} Template:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}\,.</math>|Template:EquationRef}}
For the case of normal incidence these reduce to:
Template:NumBlk Template:NumBlk
Template:NumBlk Template:NumBlk
The power reflection coefficients become: Template:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}\right|^2</math>|Template:EquationRef}} Template:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}\right|^2\,.</math>|Template:EquationRef}} The power transmissions can then be found from Template:Math.
Brewster's angleEdit
For equal permeabilities (e.g., non-magnetic media), if Template:Math and Template:Math are complementary, we can substitute Template:Math for Template:Math, and Template:Math for Template:Math, so that the numerator in equation (Template:EquationNote) becomes Template:Math, which is zero (by Snell's law). Hence Template:MathTemplate:Tsp and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting Template:Math for Template:Math in Snell's law, we readily obtain Template:NumBlk for Brewster's angle.
Equal permittivitiesEdit
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 (Template:EquationNote) and (Template:EquationNote), if Template:Math is fixed instead of Template:Math, then Template:Mvar becomes inversely proportional to Template:Mvar, with the result that the subscripts 1 and 2 in equations (Template:EquationNote) to (Template:EquationNote) are interchanged (due to the additional step of multiplying the numerator and denominator by Template:Math). Hence, in (Template:EquationNote) and (Template:EquationNote), the expressions for Template:Math and Template:Math in terms of refractive indices will be interchanged, so that Brewster's angle (Template:EquationNote) will give Template:Math instead of Template:Math, and any beam reflected at that angle will be p-polarized instead of s-polarized.<ref>More general Brewster angles, for which the angles of incidence and refraction are not necessarily complementary, are discussed in C.L. Giles and W.J. Wild, "Brewster angles for magnetic media", International Journal of Infrared and Millimeter Waves, vol.Template:Tsp6, no.Template:Tsp3 (March 1985), pp.Template:Tsp187–97.</ref> 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|§Template:NnbspHistory]], 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, 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.<ref>Whittaker, 1910, pp. 133, 148–149; Darrigol, 2012, pp. 212, 229–231.</ref> Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.
See alsoEdit
- Jones calculus
- Polarization mixing
- Index-matching material
- Field and power quantities
- Fresnel rhomb, Fresnel's apparatus to produce circularly polarised light
- Reflection loss
- Specular reflection
- Schlick's approximation
- Snell's window
- X-ray reflectivity
- Plane of incidence
- Reflections of signals on conducting lines
NotesEdit
ReferencesEdit
SourcesEdit
- 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, Template:ISBN.
- 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, Template:ISBN.
- 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.Template:Tsp1 (1866).
- Template:Cite book
- E. Hecht, 1987, Optics, 2nd Ed., Addison Wesley, Template:ISBN.
- E. Hecht, 2002, Optics, 4th Ed., Addison Wesley, Template:ISBN.
- F.A. Jenkins and H.E. White, 1976, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill, Template:ISBN.
- 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.Template:Tsp295–413.
- W. Whewell, 1857, History of the Inductive Sciences: From the Earliest to the Present Time, 3rd Ed., London: J.W. Parker & Son, vol.Template:Tsp2.
- E. T. Whittaker, 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.
External linksEdit
- Fresnel Equations – Wolfram.
- Fresnel equations calculator
- 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 dielectricsTemplate:Dead link – Mathematica interactive webpage that shows the relations between index of refraction and reflection.
- A self-contained first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.