Reflection and refraction at the boundary of two ideal dielectrics. Fresnel formulas (classical electrodynamics)

Fresnel formulas determine the amplitudes and intensities of a refracted and reflected electromagnetic wave when passing through a flat interface between two media with different refractive indices. Named after Auguste Fresnel, the French physicist who developed them. The reflection of light described by Fresnel's formulas is called Fresnel reflection.

Fresnel's formulas are valid in the case when the interface between two media is smooth, the media are isotropic, the angle of reflection is equal to the angle of incidence, and the angle of refraction is determined by Snell's law. In the case of an uneven surface, especially when the characteristic dimensions of the irregularities are of the same order of magnitude as the wavelength, the diffuse reflection of light on the surface is of great importance.

When incident on a flat boundary, two polarizations of light are distinguished. s-Polarization is the polarization of light for which the electric field strength of an electromagnetic wave is perpendicular to the plane of incidence (i.e., the plane in which both the incident and reflected beams lie). p

Fresnel formulas for s-polarization and p-polarizations differ. Because light with different polarizations reflects differently from a surface, the reflected light is always partially polarized, even if the incident light is unpolarized. The angle of incidence at which the reflected beam is completely polarized is called Brewster angle; it depends on the ratio of the refractive indices of the media forming the interface.

s-Polarization

Angles of incidence and refraction for μ = 1 (\displaystyle \mu =1) related by Snell's law

sin ⁡ α sin ⁡ β = n 2 n 1 . (\displaystyle (\frac (\sin \alpha )(\sin \beta ))=(\frac (n_(2))(n_(1))).)

Attitude n 21 = n 2 n 1 (\displaystyle n_(21)=(\cfrac (n_(2))(n_(1)))) is called the relative refractive index of two media.

R s = | Q | 2 | P | 2 = sin 2 ⁡ (α − β) sin 2 ⁡ (α + β) . (\displaystyle R_(s)=(\frac (|Q|^(2))(|P|^(2)))=(\frac (\sin ^(2)(\alpha -\beta))( \sin ^(2)(\alpha +\beta))).) T s = 1 − R s . (\displaystyle T_(s)=1-R_(s).)

Please note that the transmittance is not equal | S | 2 | P | 2 (\displaystyle (\frac (|S|^(2))(|P|^(2)))), since waves of the same amplitude in different media carry different energies.

p-Polarization

p-Polarization is the polarization of light for which the electric field strength vector lies in the plane of incidence.

( S = 2 μ 1 ε 1 μ 2 ε 2 ⋅ sin ⁡ 2 α μ 1 μ 2 sin ⁡ 2 α + sin ⁡ 2 β P ⇔ 2 cos ⁡ α sin ⁡ β sin ⁡ (α + β) cos ⁡ (α − β) P , Q = μ 1 μ 2 sin ⁡ 2 α − sin ⁡ 2 β μ 1 μ 2 sin ⁡ 2 α + sin ⁡ 2 β P ⇔ t g (α − β) t g (α + β) P , ( \displaystyle \left\((\begin(matrix)S=2(\sqrt (\cfrac (\mu _(1)\varepsilon _(1))(\mu _(2)\varepsilon _(2))) )\cdot (\cfrac (\sin 2\alpha )((\cfrac (\mu _(1))(\mu _(2)))\sin 2\alpha +\sin 2\beta ))P\; \Leftrightarrow \;(\cfrac (2\cos \alpha \sin \beta )(\sin(\alpha +\beta)\cos(\alpha -\beta)))P,\\\;\\Q=( \cfrac ((\cfrac (\mu _(1))(\mu _(2)))\sin 2\alpha -\sin 2\beta )((\cfrac (\mu _(1))(\mu _(2)))\sin 2\alpha +\sin 2\beta ))P\;\Leftrightarrow \;(\cfrac (\mathrm (tg\,) (\alpha -\beta))(\mathrm (tg \,) (\alpha +\beta)))P,\end(matrix))\right.)

The notation is retained from the previous section; the expressions after the arrows again correspond to the case μ 1 = μ 2 (\displaystyle \mu _(1)=\mu _(2))

Fresnel formulas

Fresnel's formulas are valid in the case when the interface between two media is smooth, the media are isotropic, the angle of reflection is equal to the angle of incidence, and the angle of refraction is determined by Snell's law. In the case of an uneven surface, especially when the characteristic dimensions of the irregularities are of the same order of magnitude as the wavelength, diffuse scattering of light on the surface is of great importance.

When incident on a flat boundary, two polarizations of light are distinguished. s p

Fresnel formulas for s-polarization and p-polarizations differ. Because light with different polarizations reflects differently from a surface, the reflected light is always partially polarized, even if the incident light is unpolarized. The angle of incidence at which the reflected beam is completely polarized is called Brewster's angle; it depends on the ratio of the refractive indices of the media forming the interface.

s-Polarization

s-Polarization is the polarization of light for which the electric field strength of an electromagnetic wave is perpendicular to the plane of incidence (i.e., the plane in which both the incident and reflected beams lie).

where is the angle of incidence, is the angle of refraction, is the magnetic permeability of the medium from which the wave falls, is the magnetic permeability of the medium into which the wave passes, is the amplitude of the wave that falls on the interface, is the amplitude of the reflected wave, is the amplitude of the refracted wave. In the optical frequency range with good accuracy, the expressions are simplified to those indicated after the arrows.

The angles of incidence and refraction are related by Snell's law

The ratio is called the relative refractive index of the two media.

Please note that the transmittance is not equal to , since waves of the same amplitude in different media carry different energies.

p-Polarization

p-Polarization is the polarization of light for which the electric field strength vector lies in the plane of incidence.

where , and are the amplitudes of the wave that falls on the interface, the reflected wave and the refracted wave, respectively, and the expressions after the arrows again correspond to the case.

Reflection coefficient

Transmittance

Normal fall

In the important special case of normal incidence of light, the difference in reflection and transmission coefficients for p- And s- polarized waves. For normal fall

Notes

Literature

Sivukhin D.V. General physics course. - M.. - T. IV. Optics.
Born M., Wolf E. Fundamentals of optics. - “Science”, 1973.
Kolokolov A. A. Fresnel formulas and the principle of causality // UFN. - 1999. - T. 169. - P. 1025.

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FRESNEL FORMULA- determine the relationship between the amplitude, phase and state of polarization of reflected and refracted light waves that arise when light passes through the interface of two transparent dielectrics to the corresponding characteristics of the incident wave. Installed... ... Physical encyclopedia

FRESNEL FORMULA- determine the amplitudes, phases and polarizations of reflected and refracted plane waves that arise when a plane monochromatic light wave falls on a stationary plane interface between two homogeneous media. Installed O.Zh. Fresnel in 1823... Big Encyclopedic Dictionary

Fresnel formula- determine the amplitudes, phases and polarizations of reflected and refracted plane waves that arise when a plane monochromatic light wave falls on a stationary plane interface between two homogeneous media. Installed by O. J. Fresnel in 1823. * *… … encyclopedic Dictionary

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Fresnel formula- determine the relationship between the amplitude, phase and state of polarization of reflected and refracted light waves that arise when light passes through a stationary interface between two transparent dielectrics and the corresponding characteristics... ... Great Soviet Encyclopedia

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Light*- Contents: 1) Basic concepts. 2) Newton's theory. 3) Huygens ether. 4) Huygens' principle. 5) The principle of interference. 6) Huygens Fresnel principle. 7) The principle of transverse vibrations. 8) Completion of the ethereal theory of light. 9) The basis of the ether theory.… …

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Fresnel, Augustin Jean- Augustin Jean Fresnel Augustin Jean Fresnel Augustin ... Wikipedia

Fresnel formulas

Let us determine the relationship between the amplitudes of the incident, reflected and refracted waves. Consider first an incident wave with normal polarization. If the incident wave has normal polarization, then both the reflected and refracted waves will have the same polarization. The validity of this can be verified by analyzing the boundary conditions at the interface between the media.

If you have a component with parallel polarization, then the boundary conditions will not be satisfied at any point of the boundary surface.

The plane of incidence of the wave is parallel to the plane (ZoY). The directions of propagation of reflected and refracted waves will also be parallel to the plane (ZoY) and for all waves the angle between the X axis and the direction of propagation of the wave will be equal to: , and the coefficient

In accordance with the above, the vector of all waves is parallel to the X axis, and the vectors are parallel to the plane of incidence of the wave (ZoY), therefore, for all three waves, the projection of the vector on the X axis is equal to zero:

The incident wave vector is given by:

The incident wave vector has two components:

The equations for the reflected wave vectors have the form:

The equations for the refracted wave field vectors are:

To find the connection between the complex amplitudes of the incident, reflected and refracted waves, we use the boundary conditions for the tangential components of the electromagnetic field vectors at the interface:

The field in the first medium at the interface between the media in accordance with (1.27) will have the form:

The field in the second medium is determined by the field of the refracted wave:

Since the vector of all three waves is parallel to the interface, and the tangential component of the vector is a component, the boundary conditions (1.27) can be represented as:

The incident and reflected waves are homogeneous, therefore the equalities are valid for them:

where is the characteristic impedance of the first medium.

Since the fields of any of the waves under consideration are related to each other by a linear dependence, then for the refraction of waves we can write:

where is the coefficient of proportionality.

From expressions (1.29) we obtain the projections of vectors:

Substituting equalities (1.31) into equations (1.28) and taking into account equality (1.30), we obtain a new system of equations:

Reflection and refraction at the boundary of two ideal dielectrics

Ideal dielectrics have no losses and. Then the dielectric constants of the media are real values and the Fresnel coefficients will also be real values. Let us determine under what conditions the incident wave passes into the second medium without reflection. This occurs when the wave completely passes through the interface and the reflection coefficient in this case should be equal to zero:

Consider an incident wave with normal polarization.

The reflection coefficient will be equal to zero: if the numerator in formula (1.34) is equal to zero:

However, therefore, for a wave with normal polarization at any angle of incidence of the wave on the interface. This means that a wave with normal polarization is always reflected from the interface between the media.

Waves with circular and elliptical polarization, which can be represented as a superposition of two linearly polarized waves with normal and parallel polarization, will be reflected at any angle of incidence on the media interface. However, the ratio between the amplitudes of normally and parallel polarized components in the reflected and refracted waves will be different than in the incident wave. The reflected wave will be linearly polarized, and the refracted wave will be elliptically polarized.

Let us consider an incident wave with parallel polarization.

The reflection coefficient will be equal to zero: if the numerator in formula (1.35) is equal to zero:

Having solved equation (1.37), we obtain:

Thus, an incident wave with parallel polarization passes through the interface without reflection if the angle of incidence of the wave is given by expression (1.38). This angle is called Brewster's angle.

Let us determine under what conditions the complete reflection of the incident wave from the interface between two ideal dielectrics will occur. Let us consider the case when the incident wave propagates in a denser medium, i.e. .

It is known that the angle of refraction is determined from Snell’s law:

Since: , then from expression (1.38) it follows that:.

At a certain value of the angle of incidence of the wave on the interface, we obtain:

From equality (1.40) it is clear that: and the refracted wave slides along the interface between the media.

The angle of incidence of the wave on the interface, determined by equation (1.40), is called the critical angle:

If the angle of incidence of the wave on the interface is greater than critical: , then. The amplitude of the reflected wave, regardless of the type of polarization, is equal in amplitude to the incident wave, i.e. The incident wave is completely reflected.

It remains to be seen whether the electromagnetic field penetrates the second medium. Analysis of the refracted wave equation (1.26) shows that the refracted wave is a plane inhomogeneous wave propagating in a second medium along the interface. The greater the difference in the permeability of the media, the faster the field in the second medium decreases with distance from the interface. The field practically exists in a fairly thin layer at the interface between the media. Such a wave is called a surface wave.

Fresnel formulas (classical electrodynamics).

Let us consider the incidence of a plane harmonic electromagnetic wave at the interface between two homogeneous isotropic non-conducting media (Fig.). The normal to the interface is defined by the vector, the angles between the normal and the directions of propagation of the incident, reflected and refracted waves are indicated by the symbol with the subscript , or, respectively. The directions of propagation of the described plane waves are given by the unit unit vectors , and . The vector in subsequent calculations is the radius vector of the observation point, and the quantities and are the phase velocities of wave propagation in the first (incident and reflected wave) and in the second (refracted wave) medium. We believe that the plane of polarization of an electromagnetic wave is the plane of oscillations of the electric field strength vector. We represent an electromagnetic wave with an arbitrary orientation of the plane of polarization as a superposition of two waves - a wave with a plane of polarization parallel to the plane of incidence, and a wave with a plane of polarization perpendicular to the plane of incidence. Thus, we get the relation:

If the amplitudes of oscillations of the electric field strength vector of the incident wave are equal, respectively, for a particular orientation of the plane of polarization, then the following relations hold:

. (3)

These relations are valid for the selected positive directions of the vectors and shown in Fig. (the axis is perpendicular to the plane of the drawing and directed “towards us”, the vector is directed along the axis).

For the magnetic field strength vector in the incident wave, we use the results obtained earlier:

In relation (4) the vector is the wave vector ( , where is the wavelength). In accordance with result (4), we write down the coordinate representation of the magnetic field strength vector of the incident wave:

Let be the complex amplitude of the refracted wave, directed “at us” along the axis, and perpendicular to the vector and directed towards the axis. The described amplitude orientations are conventionally assumed to be positive. For the components of the electromagnetic field in the refracted wave, as well as in the incident wave, we obtain the following dependences:

, ,

, , (6)

, .

In expressions (6), the instantaneous phase of harmonic oscillations has the form:

. (7)

Let us continue the description of the interaction of a plane wave with the interface between media. Let be the complex amplitude of the reflected wave, directed “at us” along the axis, and perpendicular to the vector and directed towards the axis. The described amplitude orientations are conventionally assumed to be positive. For the components of the electromagnetic field in the reflected wave, as well as in the incident wave, we obtain the following dependences:

, ,

, , (8)

, .

For a reflected wave, the instantaneous phase of harmonic oscillations has the form:

. (9)

The above expressions for the instantaneous values of the coordinate components of the electromagnetic field are valid at any point in the plane of incidence and at any time.

In accordance with the general integral theorems of electrodynamics at the interface between two media ( - the coordinate of the radius vector of the observation point is zero), at any moment of time the conditions of continuity of the tangent components of the electric field strength vector and the tangent components of the magnetic field strength must be satisfied. The last condition is valid if there is no surface conduction current density at the interface between the media.

So, when z=0 We require the following conditions to be met:

, , (10)

, . (11)

It is possible to ensure the fulfillment of conditions (10)-(11) at an arbitrary moment of time only if we require the equality of exponential factors in the expressions for the components of the vectors and at the interface. Equating expressions and with each other z=0, we make sure that the angle of incidence is equal to the angle of reflection: . Equating expressions and with each other z=0, we are convinced that Snell’s law of sines is valid: the sine of the angle of incidence is related to the sine of the angle of refraction as the phase velocity of the incident wave is to the phase velocity of the refracted wave (or as the refractive index of the second medium is related to the refractive index of the first medium). The previously described technique was used regardless of the nature of the plane wave (section). Below we will use the established results.

Four equations (10)-(11) fall into two independent systems:

(12)

(13)

The fact that the conditions for conjugating the electromagnetic field at the interface between media are split into two independent systems of equations serves as a basis for Fresnel’s hypothesis about the possibility of considering separately the phenomena of reflection and refraction of light waves, the oscillations of which are parallel or perpendicular to the plane of incidence of the wave.

Equations (12)-(13) are written using the approximation , while , . All that remains is to solve the systems of equations (12) and (13). After simple calculations using known relationships between trigonometric functions, we obtain the following results:

(14)

(15)

For the convenience of practical calculations, we present solutions to systems of equations (12)-(13) using the concept of refractive index:

(16)

(17) Relations (14) and (15) allow us to obtain the corresponding expressions for the components of the magnetic field strength; if desired, the reader has the opportunity to do these calculations independently.

Relations (14)-(15) completely solve the problem under consideration. They were obtained using the conditions of continuity of the tangent components of the electric and magnetic field strength vectors at the interface between two media (10)-(11). But from the integral theorems of classical electrodynamics certain conditions follow that the components of the same vector fields normal to the interface must satisfy:

In condition (18), the quantity is the surface density of free electric charges. If we substitute the solutions obtained above into equation (18) and use the approximation of a vanishingly small difference in the magnetic permeability of media from unity,

then we obtain, taking into account the second of the equations of system (12), which was used above to obtain the solution that on the interface between the media there really cannot be a nonzero surface density of free electric charges. And if we substitute the solutions obtained above into equation (19), then with the same degree of accuracy we obtain the second of the equations of system (13). Thus, it can be considered proven that the normal components of the electric and magnetic field strength vectors

satisfy the conditions at the interface between two media. We once again have the opportunity to verify how strictly internally an electromagnetic wave is organized.

Experimental verification of Fresnel's formulas is based on measuring the ratio of the intensity of the reflected wave to the intensity of the incident wave. If the incident light is natural, the averaged values of the squared amplitudes of oscillations and coincide, and the following relation is valid:

, (20)

where is the intensity of natural incident light, is the intensity of reflected partially polarized light. Relation (20) has been experimentally verified many times; it describes the experimental results well. For the sake of completeness of the discussion of the problem, we note that cases of deviation from the Fresnel formulas are known in optics, but they are not related to the fundamentals of electrodynamics, but to the fact that an idealized model of the phenomenon was considered above, which simply describes the properties of the interface and, generally speaking, the dynamic properties of material media.

Comparing expressions (14) and (15) with the “Fresnel formulas”, we are convinced of their identity. But within the framework of classical electrodynamics, unlike Fresnel’s theory, there are no internally contradictory elements; however, and we should not forget about this, physicists have been working towards such a triumph for about 40 years.

Oblique incidence of a plane harmonic electromagnetic wave at the dielectric-conductor interface.

The purpose of this section is to describe the phenomenon of reflection-refraction of a plane homogeneous harmonic wave when it is obliquely incident on a flat interface between a dielectric medium and a conducting medium. The need to return to this issue after considering the Fresnel formulas for the case of oblique incidence of an electromagnetic wave at the interface between two dielectric media is due to some new specific patterns of the phenomenon that arise due to the fact that one of the media is conductive.

An alternating electromagnetic field is described by a system of Maxwell's equations in differential form; the values of dielectric and magnetic permeabilities and electrical conductivity of a hypothetical (i.e., model) medium are considered independent of time and spatial coordinates. In a non-conducting medium (dielectric), the condition is satisfied.

We represent the solution to the system of Maxwell's equations in the form of plane harmonic traveling waves:

where is the current time, is the circular frequency of the wave, is the period of oscillation of the physical quantity taking part in the wave process. Here is the electric field strength vector, - the magnetic field strength vector, - the electric displacement vector, - the magnetic induction vector, - the volumetric density of third-party electric charges. We assume, as before, that the circular frequency is a real constant scalar quantity, and the vector is the radius vector of the observation point. The wave vector below is considered as a vector with complex components:

where vectors different in magnitude and direction have real components.

Vector quantities in relation (1) we will consider constant vector quantities (amplitudes of plane harmonic waves). The results of calculating the divergence and rotor of vector quantities (1) have been described more than once in previous sections. Thus, the system of equations of an alternating harmonic electromagnetic field, written for the electric and magnetic field strength vectors, formally takes on an “algebraic” form.

FRESNEL FORMULA

They determine the ratio of the amplitude, phase and polarization of reflected and refracted light waves that arise when light passes through the interface of two transparent dielectrics to the corresponding characteristics of the incident wave. French installed physicist O. J. Fresnel in 1823 based on ideas about elastic transverse vibrations of the ether. However, the same relations - F. f. follow as a result of strict derivation from el.-magn. theory of light when solving Maxwell's equations.

Let a plane light wave fall on the interface between two media with refractive indices n1 and n2 (Fig.).

Angles j, j" and j" are respectively the angles of incidence, reflection and refraction, and always n1sinj=n2sinj" (law of refraction) and |j|=|j"| (law of reflection). Electrical amplitude vector of the incident wave A will be decomposed into a component with amplitude Ap, parallel to the plane of incidence, and a component with amplitude As, perpendicular to the plane of incidence. Similarly, let us decompose the amplitudes of the reflected wave R into components Rp and Rs, and the amplitudes of the refracted wave D into Dp and Ds (only p-components are shown in the figure). F. f. for these amplitudes have the form:

It follows from (1) that for any value of the angles j and j" the signs of Ap and Dp, as well as the signs of As and Ds, coincide. This means that the phases also coincide, i.e., in all cases, the refracted wave retains the phase of the incident wave. For components of the reflected wave (Rp and Rs) phase relationships depend on j, n1 and n2, if j=0, then at n2>n1 the phase of the reflected wave is shifted by p. i.e. the energy flux carried by it, proportional to the square of the amplitude (see Poynting VECTOR. The ratio of the average energy fluxes over a period in the reflected and refracted waves to the average energy flux in the incident wave is called the reflection coefficient r and the transmission coefficient d. From (1 ) we obtain F. f., which determine the coefficients of reflection and refraction for the s- and p-components of the incident wave, taking into account that

In the absence of light absorption, rs+ds=1 and rp+dp=1 in accordance with the law of conservation of energy. If , i.e., all directions of electrical oscillations, falls on the interface. vectors are equally probable, then the waves are equally divided between the p- and s-oscillations, the total coefficient. reflections in this case: r=1/2(rs+rp). If j + j "= 90 °, then tg (j + j") ® ?, and rp \u003d 0, that is, under these conditions, polarized so that its electric. the vector lies in the plane of incidence and is not reflected at all from the interface. When nature falls light at this angle, the reflected light will be completely polarized. The angle of incidence at which this occurs is called. the angle of full polarization or Brewster's angle (see BREWSTER'S LAW), the ratio tgjB = n2/n1 is valid for it.

At normal incidence of light on the interface between two media (j=0) F. f. for the amplitudes of reflected and refracted waves can be reduced to the form

From (4) it follows that at the interface, the greater the abs. the value of the difference n2-n1; coefficient, r and A do not depend on which side of the interface the incident light wave comes from.

The condition for applicability of F. f. is the independence of the refractive index of the medium from the amplitude of the electric vector. light wave intensity. This condition is trivial in the classical (linear) optics, is not carried out for high-power light fluxes, for example. emitted by lasers. In such cases, F. f. do not give satisfaction. descriptions of the observed phenomena and it is necessary to use the methods and concepts of nonlinear optics.

Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .

FRESNEL FORMULA

Determine the relationship between the amplitude, phase and state of polarization of reflected and refracted light waves that arise when light passes through the interface of two transparent dielectrics to the corresponding characteristics of the incident wave. Established by O. J. Fresnel in 1823 on the basis of ideas about elastic transverse vibrations of the ether. However, the same relationships - F. f. - follow as a result of a strict derivation from the electric magnetic field. theory of light when solving Maxwell's equations.

Let a plane light wave fall on the interface between two media with refractive indices P 1 . And P 2 (fig.). Angles j, j" and j" are respectively the angles of incidence, reflection and refraction, and always n 1 . sinj= n 2 sinj "(law of refraction) and |j|=|j"| (law of reflection). Amplitude of the electric vector of the incident wave A Let's decompose it into a component with amplitude A r, parallel to the plane of incidence, and a component with amplitude As, perpendicular to the plane of incidence. Let us similarly expand the amplitudes of the reflected wave R into components Rp And R s and a refracted wave D- on Dp And D s(the figure shows only R-components). F. f. for these amplitudes have the form

From (1) it follows that for any value of the angles j and j " the signs A r And Dp match up. This means that the phases also coincide, i.e. in all cases the refracted wave retains the phase of the incident one. For the components of the reflected wave ( Rp And R s)phase relationships depend on j, n 1 and n 2 ; if j=0, then when n 2 >n 1, the phase of the reflected wave shifts by p.

In experiments, they usually measure not the amplitude of a light wave, but its intensity, i.e., the energy flow it carries, proportional to the square of the amplitude (see.

Poynting vector). The ratio of the period-average energy flows in reflected and refracted waves to the average energy flow in the incident wave is called. coefficient reflections r and coefficient passing d. From (1) we obtain the functional functions that determine the coefficient. reflection and refraction for s- And R-components of the incident wave, taking into account that

In the absence of light absorption there are relationships between the coefficients in accordance with the laws of conservation of energy r s +d s=1 and r p + d p=1. If the interface falls natural light, i.e. all directions of electrical oscillations. vectors are equally probable, then the wave energy is equally divided between R- And s- fluctuations, full coefficient. reflections in this case r=(1/2)(r s +r p) If j+j "=90 o , then And r p=0 i.e., under these conditions, light is polarized so that its electric the vector lies in the plane of incidence and is not reflected at all from the interface. When nature falls light at this angle, the reflected light will be completely polarized. The angle of incidence at which this occurs is called. full polarization angle or Brewster angle (see. Brewster's law) for it the relation logj B = n 2 /n 1 .

With normal incidence of light on the interface between two media (j = 0) F.f. for the amplitudes of reflected and refracted waves can be reduced to the form

Here the difference between the components disappears s And p, because the concept of the plane of incidence loses its meaning. In this case, in particular, we obtain

From (4) it follows that light reflection at the interface, the greater the abs. magnitude of the difference n 2 - n 1 ; coefficient r And d do not depend on which side of the interface the incident light wave comes from.

The condition for applicability of F. f. is the independence of the refractive index of the medium from the amplitude of the electric vector. light wave intensity. This condition is trivial in the classical (linear) optics, is not carried out for high-power light fluxes, for example. emitted by lasers. In such cases, F. f. do not give satisfaction. descriptions of observed phenomena and it is necessary to use methods and concepts nonlinear optics.

Lit.: Born M., Wolf E., Fundamentals of Optics, trans. from English, 2nd ed., M., 1973; Kaliteevsky N.I., Volnovaya, 2nd ed., M., 1978. L. N. Kaporsky.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

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