Английская Википедия:Fresnel equations

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Файл:Partial transmittance.gif

Partial transmission and reflection of a pulse travelling from a low to a high refractive index medium.

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 (Шаблон: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.

Overview

When light strikes the interface between a medium with refractive index Шаблон:Math and a second medium with refractive index Шаблон: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. 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 polarizations

Файл:Plane of incidence.svg

The plane of incidence is defined by the incoming radiation's propagation vector and the normal vector of the surface.

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 Шаблон: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 Шаблон:Mvar 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 (Шаблон: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).

Configuration

Файл:Fresnel1.svg

Variables used in the Fresnel equations

In the diagram on the right, an incident plane wave in the direction of the ray Шаблон:Math strikes the interface between two media of refractive indices Шаблон:Math and Шаблон:Math at point Шаблон:Math. Part of the wave is reflected in the direction Шаблон:Math, and part refracted in the direction Шаблон:Math. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as Шаблон:Math, Шаблон:Math and Шаблон: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 coefficients

Файл:Fresnel power air-to-glass.svg

Power coefficients: air to glass

Файл:Fresnel power glass-to-air.svg

Power coefficients: glass to air (Total internal reflection starts from 42° making reflection coefficient 1)

We call the fraction of the incident power that is reflected from the interface the reflectance (or reflectivity, or power reflection coefficient) Шаблон:Math, and the fraction that is refracted into the second medium is called the transmittance (or transmissivity, or power transmission coefficient) Шаблон: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.

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 Шаблон:Math and Шаблон:Math are the wave impedances of media 1 and 2, respectively.

We assume that the media are non-magnetic (i.e., Шаблон:Math), 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 Шаблон:Math and Шаблон:Math: <math display=block>Z_i = \frac{Z_0}{n_i}\,,</math> where Шаблон:Math is the impedance of free space and Шаблон: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 Шаблон: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:Шаблон:Hsp <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 Шаблон:Math for a wave at an angle Шаблон: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 Шаблон: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 cases

Normal incidence

For the case of normal incidence, Шаблон: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 (Шаблон:Math) surrounded by air (Шаблон: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 angle

Шаблон:Main At a dielectric interface from Шаблон:Math to Шаблон:Math, there is a particular angle of incidence at which Шаблон: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 Шаблон:Math and Шаблон:Math (typical glass).

Total internal reflection

Шаблон:Main When light travelling in a denser medium strikes the surface of a less dense medium (i.e., Шаблон:Math), beyond a particular incidence angle known as the critical angle, all light is reflected and Шаблон: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 Шаблон:Math for all real Шаблон:Math). For glass with Шаблон:Math surrounded by air, the critical angle is approximately 42°.

45° incidence

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 (Шаблон:Math), it follows algebraically from the above equations that Шаблон:Math equals the square of Шаблон: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 Шаблон:Math and Шаблон: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 Шаблон:Math and Шаблон:Math at 45° can be used to estimate the reflectivity at normal incidence.Шаблон:Cn The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of Шаблон:Math and Шаблон:Math, and then averaging these two averages again arithmetically, gives a value for Шаблон:Math with an error of less than about 3% for most common optical materials.Шаблон: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 Шаблон:Math and Шаблон: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 coefficients

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 Шаблон:Math and Шаблон:Math (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, Шаблон:Math of both media to be equal to the permeability of free space Шаблон:Math as is essentially true of all dielectrics at optical frequencies.

Файл:Fresnel amplitudes air-to-glass.svg

Amplitude coefficients: air to glass

Файл:Fresnel amplitudes glass-to-air.svg

Amplitude coefficients: glass to air

In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient Шаблон: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 Шаблон: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 Шаблон:Math is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients Шаблон:Math and Шаблон: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 Шаблон: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 Шаблон:Nowrap, a wave reflected at angle Шаблон:Nowrap, and a wave transmitted at angle Шаблон:Nowrap. In the case of an interface into an absorbing material (where Шаблон: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, <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>

One can see that Шаблон:Math and Шаблон: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 Шаблон:Math is just the squared magnitude of Шаблон:Math:Шаблон:Hsp <math display=block>R = |r|^2.</math>

On the other hand, calculation of the power transmission coefficient Шаблон: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: <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 Шаблон:Math. The introduced factor of Шаблон:Math is the reciprocal of the ratio of the media's wave impedances. The Шаблон: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 Шаблон:Math.

In the case of total internal reflection where the power transmission Шаблон:Mvar is zero, Шаблон: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 Шаблон: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 Шаблон:Math and Шаблон: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 forms

In the above formula for Шаблон: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 Шаблон:Math, we obtainШаблон:Hsp <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 Шаблон:Math, the result is easily shown to be equivalent toШаблон:Hsp <math display=block>r_\text{p}=\frac{\tan(\theta_\text{i}-\theta_\text{t})}{\tan(\theta_\text{i}+\theta_\text{t})}. </math>

These formulasШаблон:Hsp 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 asШаблон:Px2 Шаблон:Math.

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, 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 methodШаблон:Hsp can be used to solve multiple-surface problems.

History

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. 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. But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write: Шаблон:Blockquote 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. 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.

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. 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 (Шаблон:Math and Шаблон: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. 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.

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 rhombШаблон:Hsp §Шаблон: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.

Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoirШаблон:Hsp in which he introduced the needed terms linear polarization, circular polarization, and elliptical polarization, 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).

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 and homogeneous. If the medium is also isotropic, the four field vectorsШаблон:Px2 Шаблон:MathШаблон:Tsp are related by <math display=block>\begin{align} \mathbf{D} &= \epsilon \mathbf{E} \\ \mathbf{B} &= \mu \mathbf{H}\,, \end{align} </math> where Шаблон:Math and Шаблон:Math are scalars, known respectively as the (electric) permittivity and the (magnetic) permeability of the medium. For a vacuum, these have the values Шаблон:Math and Шаблон:Math, respectively. Hence we define the relative permittivity (or dielectric constant) Шаблон:MathШаблон:Hsp, and the relative permeability Шаблон:Math.

In optics it is common to assume that the medium is non-magnetic, so that Шаблон:Math. For ferromagnetic materials at radio/microwave frequencies, larger values of Шаблон:Math must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials), Шаблон:Math is indeed very close to 1; that is, Шаблон:Math.

In optics, one usually knows the refractive index Шаблон:Math of the medium, which is the ratio of the speed of light in a vacuum (Шаблон: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 Шаблон:Mvar, which is the ratio of the amplitude of Шаблон:Math to the amplitude of Шаблон:Math. It is therefore desirable to express Шаблон:Math and Шаблон:Mvar in terms of Шаблон:Math and Шаблон:Math, and thence to relate Шаблон:Mvar to Шаблон:Math. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave admittance Шаблон:Mvar, which is the reciprocal of the wave impedance Шаблон: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 waves

In a uniform plane sinusoidal electromagnetic wave, the electric field Шаблон:Math has the form Шаблон:NumBlk where Шаблон:Math is the (constant) complex amplitude vector, Шаблон:Math is the imaginary unit, Шаблон:Math is the wave vector (whose magnitude Шаблон:Mvar is the angular wavenumber), Шаблон:Math is the position vector, Шаблон:Math is the angular frequency, Шаблон:Math is time, and it is understood that the real part of the expression is the physical field.  The value of the expression is unchanged if the position Шаблон:Math varies in a direction normal to Шаблон:Math; hence Шаблон:Math is normal to the wavefronts.

To advance the phase by the angle ϕ, we replace Шаблон:Math by Шаблон:MathШаблон:Tsp (that is, we replace Шаблон:Math by Шаблон:Math), with the result that the (complex) field is multiplied by Шаблон:Math. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. This becomes more obvious when the field (Шаблон:EquationNote) is factored asШаблон:Px2 Шаблон:Math, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by Шаблон:Math.Шаблон:Hsp

If ℓ is the component of Шаблон:Math in the direction of Шаблон:Math, the field (Шаблон:EquationNote) can be writtenШаблон:Px2 Шаблон:Math.  If the argument of Шаблон:Math is to be constant,  ℓ must increase at the velocityШаблон:Px2 <math>\omega/k\,,\,</math> known as the phase velocityШаблон:Hsp Шаблон:Math. This in turn is equal toШаблон:Px2 Шаблон:Nowrap Solving for Шаблон:Mvar gives Шаблон:NumBlk

As usual, we drop the time-dependent factor Шаблон: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 Шаблон:NumBlk.</math>|Шаблон:EquationRef}} For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce toШаблон:Hsp <math display=block>\begin{align}

 \omega\mathbf{B} &=  \mathbf{k}\times\mathbf{E}\\
 \omega\mathbf{D} &= -\mathbf{k}\times\mathbf{H}\,.

\end{align}</math>

PuttingШаблон:Tsp Шаблон:MathШаблон:Tsp andШаблон:Tsp Шаблон:Math,Шаблон:Px2 as above, we can eliminate Шаблон:Math and Шаблон:Math to obtain equations in only Шаблон:Math and Шаблон: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 Шаблон:Math and Шаблон:Math are real (as in a lossless dielectric), these equations show that Шаблон:MathШаблон:Tsp 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 (Шаблон:EquationNote), we obtain <math display=block>\begin{align}

 \mu cH &= nE\\
 \epsilon cE &= nH\,,

\end{align}</math> where Шаблон:Mvar and Шаблон:Mvar are the magnitudes of Шаблон:Math and Шаблон:Math. Multiplying the last two equations gives Шаблон:NumBlk Dividing (or cross-multiplying) the same two equations givesШаблон:Tsp Шаблон:Math, where Шаблон:NumBlk This is the intrinsic admittance.

From (Шаблон:EquationNote) we obtain the phase velocityШаблон:Px2 Шаблон:NowrapШаблон:Tsp For a vacuum this reduces toШаблон:Px2 Шаблон:NowrapШаблон:Tsp 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 Шаблон:Tmath. Шаблон:LargerTaking the reciprocal of (Шаблон:EquationNote), we find that the intrinsic impedance is Шаблон:Nowrap In a vacuum this takes the value <math display="inline">Z_0=\sqrt{\mu_0/\epsilon_0}\,\approx 377\,\Omega\,,</math>Шаблон:Px2 known as the impedance of free space. By division, Шаблон:Nowrap/\epsilon_{\text{rel}}}</math>.}}Шаблон:Tsp For a non-magnetic medium, this becomes <math>Z=Z_0\big/\!\sqrt{\epsilon_{\text{rel}}}=Z_0/n.</math>Шаблон:Larger

Wave vectors

Файл:Wave vectors n1 to n2.svg

Incident, reflected, and transmitted wave vectors (Шаблон:Math, and Шаблон:Math), for incidence from a medium with refractive index Шаблон:Math to a medium with refractive index Шаблон:Math. The red arrows are perpendicular to the wave vectors.

In Cartesian coordinates Шаблон:Math, let the regionШаблон:Px2 Шаблон:MathШаблон:Px2 have refractive index Шаблон:Math, intrinsic admittance Шаблон:Math, etc., and let the regionШаблон:Px2 Шаблон:MathШаблон:Px2 have refractive index Шаблон:Math, intrinsic admittance Шаблон:Math, etc. Then the Шаблон:Math plane is the interface, and the Шаблон:Math axis is normal to the interface (see diagram). Let Шаблон:Math and Шаблон:Math (in bold roman type) be the unit vectors in the Шаблон:Math and Шаблон:Math directions, respectively. Let the plane of incidence be the Шаблон:Math plane (the plane of the page), with the angle of incidence Шаблон:Math measured from Шаблон:Math towards Шаблон:Math. Let the angle of refraction, measured in the same sense, be Шаблон:Math, where the subscript Шаблон:Math stands for transmitted (reserving Шаблон:Math for reflected).

In the absence of Doppler shifts, ω does not change on reflection or refraction. Hence, by (Шаблон:EquationNote), the magnitude of the wave vector is proportional to the refractive index.

So, for a given Шаблон:Math, if we redefine Шаблон:Mvar as the magnitude of the wave vector in the reference medium (for which Шаблон:Math), then the wave vector has magnitude Шаблон:Math in the first medium (regionШаблон:Px2 Шаблон:MathШаблон:Px2 in the diagram) and magnitude Шаблон: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 (Шаблон:EquationNote) are Шаблон:NumBlk Hence: Шаблон:NumBlk

The s components

For the s polarization, the Шаблон:Math field is parallel to the Шаблон:Math axis and may therefore be described by its component in the Шаблон:Math direction. Let the reflection and transmission coefficients be Шаблон:Math and Шаблон:Math, respectively. Then, if the incident Шаблон:Math field is taken to have unit amplitude, the phasor form (Шаблон:EquationNote) of its Шаблон:Math-component is Шаблон:NumBlk,</math>|Шаблон:EquationRef}} and the reflected and transmitted fields, in the same form, are Шаблон:NumBlk\\

 E_\text{t} &= t_{s\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.

\end{align}</math>|Шаблон: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 Шаблон:Math field. If the incident, reflected, and transmitted Шаблон:Math fields (in the above equations) are in the Шаблон:Math-direction ("out of the page"), then the respective Шаблон:Math fields are in the directions of the red arrows, sinceШаблон:Tsp Шаблон:MathШаблон:Tsp form a right-handed orthogonal triad. The Шаблон:Math fields may therefore be described by their components in the directions of those arrows, denoted byШаблон:Tsp Шаблон:Math.  Then, sinceШаблон:Tsp Шаблон:Math, Шаблон: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>|Шаблон:EquationRef}}

At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the Шаблон:Math and Шаблон:Math fields must be continuous; that is, Шаблон:NumBlk When we substitute from equations (Шаблон:EquationNote) to (Шаблон:EquationNote) and then from (Шаблон:EquationNote), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations Шаблон:NumBlk which are easily solved for Шаблон:Math and Шаблон:Math, yielding Шаблон:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}</math>|Шаблон:EquationRef}} and Шаблон:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\,.</math>|Шаблон:EquationRef}} At normal incidenceШаблон:Px2 Шаблон:Math, indicated by an additional subscript 0, these results become Шаблон:NumBlk and Шаблон:NumBlk At grazing incidenceШаблон:Px2 Шаблон:Math, we haveШаблон:Tsp Шаблон:Math, henceШаблон:Tsp Шаблон:MathШаблон:Tsp andШаблон:Tsp Шаблон:Math.

The p components

For the p polarization, the incident, reflected, and transmitted Шаблон: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 beШаблон:Tsp Шаблон:Math (redefining the symbols for the new context). Let the reflection and transmission coefficients be Шаблон:Math and Шаблон:Math. Then, if the incident Шаблон:Math field is taken to have unit amplitude, we have Шаблон: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>|Шаблон:EquationRef}} If the Шаблон:Math fields are in the directions of the red arrows, then, in order forШаблон:Tsp Шаблон:MathШаблон:Tsp to form a right-handed orthogonal triad, the respective Шаблон:Math fields must be in the Шаблон: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 Шаблон:Largerthe Шаблон:Math field in the case of the p polarizationШаблон: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.

So, for the incident, reflected, and transmitted Шаблон:Math fields, let the respective components in the Шаблон:Math direction beШаблон:Tsp Шаблон:Math.  Then, sinceШаблон:Tsp Шаблон:Math, Шаблон: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>|Шаблон:EquationRef}}

At the interface, the tangential components of the Шаблон:Math and Шаблон:Math fields must be continuous; that is, Шаблон:NumBlk When we substitute from equations (Шаблон:EquationNote) and (Шаблон:EquationNote) and then from (Шаблон:EquationNote), the exponential factors again cancel out, so that the interface conditions reduce to Шаблон:NumBlk Solving for Шаблон:Math and Шаблон:Math, we find Шаблон:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}</math>|Шаблон:EquationRef}} and Шаблон:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\,.</math>|Шаблон:EquationRef}} At normal incidenceШаблон:Px2 Шаблон:Math indicated by an additional subscript 0, these results become Шаблон:NumBlk and Шаблон:NumBlk At Шаблон:Itco Шаблон:Math, we again haveШаблон:Tsp Шаблон:Math, henceШаблон:Tsp Шаблон:MathШаблон:Tsp andШаблон:Tsp Шаблон:Math.

Comparing (Шаблон:EquationNote) and (Шаблон:EquationNote) with (Шаблон:EquationNote) and (Шаблон: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)

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 isШаблон:Px2 Шаблон:Math, where Шаблон:Math and Шаблон:Math are due only to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), Шаблон:Math and Шаблон:Math are in phase, and at right angles to each other and to the wave vector Шаблон:MathШаблон:Hsp; so, for s polarization, using the Шаблон:Mvar and Шаблон:Mvar components of Шаблон:Math and Шаблон:Math respectively (or for p polarization, using the Шаблон:Mvar and Шаблон:Mvar components of Шаблон:Math and Шаблон:Math), the irradiance in the direction of Шаблон:Math is given simply by Шаблон:Math,Шаблон:Px2 which isШаблон:Px2 Шаблон:MathШаблон:Tsp in a medium of intrinsic impedance Шаблон: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 Шаблон:Mvar component (rather than the full Шаблон:Mvar component) of Шаблон:Math or Шаблон:Math or, equivalently, simply multiply Шаблон:Math by the proper geometric factor, obtaining Шаблон:Math.

From equations (Шаблон:EquationNote) and (Шаблон:EquationNote), taking squared magnitudes, we find that the reflectivity (ratio of reflected power to incident power) is Шаблон:NumBlk{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\right|^2</math>|Шаблон:EquationRef}} for the s polarization, and Шаблон:NumBlk{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\right|^2</math>|Шаблон:EquationRef}} for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same Шаблон:TspcosШаблон: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 Шаблон:Mvar direction) is to use Шаблон:MathШаблон:Px2 (conservation of energy). In this way we find Шаблон: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>|Шаблон:EquationRef}} for the s polarization, and Шаблон: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>|Шаблон: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, Шаблон:MathШаблон:Hsp).

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 indices

From equations (Шаблон:EquationNote) and (Шаблон: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 haveШаблон:Tsp Шаблон:MathШаблон:Px2 (that is, the transmitted ray is undeviated), so that the cosines in equations (Шаблон:EquationNote), (Шаблон:EquationNote), (Шаблон:EquationNote), (Шаблон:EquationNote), and (Шаблон:EquationNote) to (Шаблон: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. When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering.

Non-magnetic media

Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing (Шаблон:EquationNote) by (Шаблон:EquationNote)) yields <math display=block>Y=\frac{n}{\,c\mu\,}\,.</math> For non-magnetic media we can substitute the vacuum permeability Шаблон:Math for Шаблон:Math, so that <math display=block>Y_1=\frac{n_1}{\,c\mu_0} ~~;EducationBot (обсуждение) 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 (Шаблон:EquationNote) to (Шаблон:EquationNote) and equations (Шаблон:EquationNote) to (Шаблон:EquationNote), the factor cμ₀ cancels out. For the amplitude coefficients we obtain:

Шаблон:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}</math>|Шаблон:EquationRef}} Шаблон:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}\,</math>|Шаблон:EquationRef}}

Шаблон:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}</math>|Шаблон:EquationRef}} Шаблон:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}\,.</math>|Шаблон:EquationRef}}

For the case of normal incidence these reduce to:

The power reflection coefficients become: Шаблон:NumBlk{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}\right|^2</math>|Шаблон:EquationRef}} Шаблон:NumBlk{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}\right|^2\,.</math>|Шаблон:EquationRef}} The power transmissions can then be found from Шаблон:Math.

Brewster's angle

For equal permeabilities (e.g., non-magnetic media), if Шаблон:Math and Шаблон:Math are complementary, we can substituteШаблон:Px2 Шаблон:MathШаблон:Px2 forШаблон:Px2 Шаблон:Math, andШаблон:Px2 Шаблон:MathШаблон:Px2 forШаблон:Px2 Шаблон:Math, so that the numerator in equation (Шаблон:EquationNote) becomesШаблон:Px2 Шаблон:Math,Шаблон:Tsp which is zero (by Snell's law). HenceШаблон:Tsp Шаблон:MathШаблон:Tsp and only the s-polarized component is reflected. This is what happens at the Brewster angle. SubstitutingШаблон:Px2 Шаблон:MathШаблон:Px2 forШаблон:Px2 Шаблон:MathШаблон:Px2 in Snell's law, we readily obtain Шаблон:NumBlk 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 (Шаблон:EquationNote) and (Шаблон:EquationNote), if Шаблон:Math is fixed instead of Шаблон:Math, then Шаблон:Mvar becomes inversely proportional to Шаблон:Mvar, with the result that the subscripts 1 and 2 in equations (Шаблон:EquationNote) to (Шаблон:EquationNote) are interchanged (due to the additional step of multiplying the numerator and denominator by Шаблон:Math). Hence, in (Шаблон:EquationNote) and (Шаблон:EquationNote), the expressions for Шаблон:Math and Шаблон:Math in terms of refractive indices will be interchanged, so that Brewster's angle (Шаблон:EquationNote) will giveШаблон:Tsp Шаблон:MathШаблон:Tsp instead ofШаблон:Tsp Шаблон:Math,Шаблон:Px2 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|§Шаблон: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. Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.

See also

• 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

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, Шаблон: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, Шаблон:ISBN.
• A. Fresnel, 1866Шаблон:Hsp (ed.Шаблон:Tsp H. de Senarmont, E. Verdet, and L. Fresnel), Oeuvres complètes d'Augustin Fresnel, Paris: Imprimerie Impériale (3 vols., 1866–70), vol.Шаблон:Tsp1 (1866).
• E. Hecht, 1987, Optics, 2nd Ed., Addison Wesley, Шаблон:ISBN.
• E. Hecht, 2002, Optics, 4th Ed., Addison Wesley, Шаблон:ISBN.
• F.A. Jenkins and H.E. White, 1976, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill, Шаблон: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.Шаблон: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.Шаблон: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.

Further reading

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• 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, Шаблон:ISBN

External links

• 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 dielectricsШаблон: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.