Reflection and refraction at the boundary of two ideal dielectrics. Reflection and refraction of light (Boundary conditions

Let us assume that the interface between the media is flat and motionless. A plane monochromatic wave is incident on it:

the reflected wave then has the form:

for the refracted wave we have:

the reflected and refracted waves will also be plane and have the same frequency: $(\omega )_(pad)=\omega_(otr)=\omega_(pr)=\omega $. The frequency equality follows from the linearity and homogeneity of the boundary conditions.

Let us decompose the electric field of each wave into two components. One, located in the plane of incidence, the other in the perpendicular plane. These components are called the main components of the waves. Then you can write:

where $((\overrightarrow(e))_x,\overrightarrow(e))_y,\ (\overrightarrow(e))_z$ are unit vectors along axes $X$,$Y$,$Z.$ $( \overrightarrow(e))_1,\ (\overrightarrow(e))"_1,(\overrightarrow(e))_2$ -- unit vectors that are in the plane of incidence and are perpendicular, respectively, to the incident, reflected, and refracted rays ( Fig. 1) That is, you can write:

Picture 1.

We scalarly multiply the expression (2.a) by the vector $(\overrightarrow(e))_x,$ and get:

In a similar way, get:

Thus, expressions (4) and (5) give $x-$, $y-$. $z-$ components of the electric field at the interface between substances (for $z=0$). If we do not take into account the magnetic properties of matter ($\overrightarrow(H)\equiv \overrightarrow(B)$), then the magnetic field components can be written as:

The corresponding expressions for the reflected wave have the form:

For a refracted wave:

To find $E_(pr\bot )$,$\ E_(pr//),\ E_(otr\bot ),\ E_(otr//)$ boundary conditions are used:

We substitute formulas (10) into expressions (11), we get:

From the system of equations (12), taking into account the equality of the angle of incidence and the angle of reflection ($(\alpha )_(pad)=\alpha_(otr)=\alpha $), we obtain:

Relations that are in the left parts of expressions (13) are called Fresnel coefficients. These expressions are Fresnel formulas.

For ordinary reflection, the Fresnel coefficients are real. This proves that reflection and refraction do not accompany a phase change, except for a phase change of the reflected wave by $180^\circ$. If the incident wave is polarized, then the reflected and refracted waves are also polarized.

When obtaining the Fresnel formulas, we assumed the light to be monochromatic, however, if the medium is not dispersive and ordinary reflection occurs, then these expressions are also valid for non-monochromatic waves. It is only necessary to understand the components ($\bot $ and //) as the corresponding components of the electric field strengths of the incident, reflected, and refracted waves at the interface.

Example 1

Exercise: Explain why the image of the setting sun under the same conditions is not inferior in brightness to the sun itself.

Solution:

To explain this phenomenon, we use the following Fresnel formula:

\[\frac(E_(otr\bot ))(E_(pad\bot ))=-\frac(sin (\alpha -(\alpha )_(pr)))(sin (\alpha +(\alpha ) _(pr)));\ \frac(E_(otr//))(E_(pad//))=\frac(tg (\alpha -(\alpha )_(pr)))(tg (\alpha +(\alpha )_(pr)))(1.1).\]

Under conditions of grazing incidence, when the angle of incidence ($\alpha $) is almost equal to $90^\circ$, we get:

\[\frac(E_(otr\bot ))(E_(pad\bot ))=\frac(E_(otr//))(E_(pad//))\to -1(1.2).\]

With grazing incidence of light, the Fresnel coefficients (in modulus) tend to unity, that is, the reflection is almost complete. This explains the bright images of the shores in the calm water of the reservoir and the brightness of the setting sun.

Example 2

Exercise: Get an expression for reflectivity ($R$), if that's what the reflection coefficient is when light is normally incident on a surface.

Solution:

To solve the problem, we use the Fresnel formulas:

\[\frac(E_(otr\bot ))(E_(pad\bot ))=\frac(n_1cos\left(\alpha \right)-n_2cos\left((\alpha )_(pr)\right)) (n_1cos\left(\alpha \right)+n_2cos\left((\alpha )_(pr)\right)),\ \frac(E_(otr//))(E_(pad//))=\frac (n_2(cos \left(\alpha \right)\ )-n_1(cos \left((\alpha )_(pr)\right)\ ))(n_2(cos \left(\alpha \right)\ )+ n_1(cos \left((\alpha )_(pr)\right)\ ))\left(2.1\right).\]

Under normal incidence of light, the formulas are simplified and turn into expressions:

\[\frac(E_(otr\bot ))(E_(pad\bot ))=-\frac(E_(otr//))(E_(pad//))=\frac(n_1-n_2)(n_1 +n_2)=\frac(n-1)(n+1)(2.2),\]

where $n=\frac(n_1)(n_2)$

The reflection coefficient is the ratio of reflected energy to incident energy. It is known that the energy is proportional to the square of the amplitude, therefore, we can assume that the desired coefficient can be found as:

Answer:$R=(\left(\frac(n-1)(n+1)\right))^2.$

FRESNEL FORMULA- determine the ratio of the amplitude, phase and state of the reflected and refracted light waves that occur when light passes through the interface between two transparent ones, to the corresponding characteristics of the incident wave. Established by O. Zh. Fresnel in 1823 on the basis of ideas about elastic transverse oscillations of the ether. However, the same ratios - F. f. - follow as a result of a rigorous derivation from the 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 P 1 and P 2 (Fig.). The angles j, j" and j"" are respectively the angles of incidence, reflection and refraction, and always n 1 sinj= n 2 sinj"" (the law of refraction) and |j|=|j"| (the law of reflection). The amplitude of the electric vector of the incident wave A expand into a component with amplitude A r, parallel to the plane of incidence, and a component with amplitude A s perpendicular to the plane of incidence. Let us similarly expand the amplitude of the reflected wave R into components Rp And Rs, and the refracted wave D- on Dp And Ds(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 wave. For the reflected wave components ( Rp And Rs) phase relations depend on j, n 1 and n 2; if j=0, then n 2 >n 1 phase of the reflected wave is shifted by p.

In experiments, it is usually not the amplitude of a light wave that is measured, but its intensity, i.e., the energy flux carried by it, which is proportional to the square of the amplitude (see Fig.

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

Fresnel formulas

Fresnel formulas determine the amplitudes and intensities of the refracted and reflected electromagnetic waves 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 the Fresnel formulas is called Fresnel reflection.

Fresnel formulas are valid 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 scattering of light on the surface is of great importance.

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

Fresnel formulas for s-polarization and p polarizations are different. Since 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

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 the reflected beam lie).

where is the angle of incidence; In the optical frequency range with good accuracy and the expressions are simplified to those indicated after the arrows.

The angles of incidence and refraction for 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, since waves of the same amplitude in different media carry different energies.

p-Polarization

p-Polarization - 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 the coefficients of reflection and transmission disappears for p- And s-polarized waves. For a normal fall

Notes

Literature

Sivukhin D.V. General course of physics. - 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. - S. 1025.

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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 a 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 on the media interface.

If we 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 the 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 wave propagation 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 are:

The equations for the field vectors of the refracted wave have the form:

To find the relationship 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 media 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 between the media, and the tangent component of the vector is a component, then 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 wave resistance of the first medium.

Since the fields of any of the waves under consideration are interconnected 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 permittivities 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 passes completely through the interface between the media, 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.

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:

Solving equation (1.37), we get:

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

Let us determine under what conditions there will be a complete reflection of the incident wave from the interface between two ideal dielectrics. 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:.

For a certain value of the angle of incidence of the wave on the interface between the media, we obtain:

Equation (1.40) shows that: and the refracted wave slides along the interface between the media.

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

If the angle of incidence of the wave on the interface between the media is greater than the critical one: , 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 find out whether the electromagnetic field penetrates into the second medium. Analysis of the refracted wave equation (1.26) shows that the refracted wave is a plane inhomogeneous wave propagating in the 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 near the interface between the media. Such a wave is called a surface wave.

1.1. Border conditions. Fresnel formulas

A classical problem for which the orientation of the vector is important E, is the passage of a light wave through the interface between two media. Due to the geometry of the problem, a difference arises in the reflection and refraction of two independent components polarized parallel and perpendicular to the plane of incidence, and, consequently, the initially unpolarized light becomes partially polarized after reflection or refraction.

The boundary conditions for the vectors of tension and induction, known from electrostatics, equalize the tangential components of the vectors at the interface E And H and normal components of the vectors D And B, in fact, expressing the absence of currents and charges along the boundary and the weakening of the external electric field by a factor of e when it enters the dielectric:

In this case, the field in the first medium consists of the fields of the incident and reflected waves, and in the second medium it is equal to the field of the refracted wave (see Fig. 2.1).

The field in any of the waves can be written as relations of the type . Since the boundary conditions (5.1) must be satisfied at any point of the interface and at any time, from them it is possible to obtain the laws of reflection and refraction:

1. The frequencies of all three waves are the same: w 0 \u003d w 1 \u003d w 2.

2. The wave vectors of all waves lie in the same plane: .

3. The angle of incidence is equal to the angle of reflection: a = a".

4. Snell's Law: . It can be shown that the product n×sin a remains constant for any law of change in the refractive index along the Z axis, not only stepwise at the interfaces, but also continuous.

Wave polarization does not affect these laws.

On the other hand, the continuity of the corresponding components of the vectors E And H leads to the so-called Fresnel formulas, allowing one to calculate the relative amplitudes and intensities of the reflected and transmitted waves for both polarizations. The expressions turn out to be significantly different for a parallel (vector E lies in the plane of incidence) and perpendicular polarization, naturally coinciding for the case of normal incidence (a = b = 0).

The field geometry for parallel polarization is shown in fig. 5.2a, for perpendicular - in fig. 5.2b. As noted in Section 4.1, in an electromagnetic wave, the vector E, H And k form a right orthogonal triple. Therefore, if the tangential components of the vectors E 0 and E 1 of the incident and reflected waves are directed in the same way, then the corresponding projections of the magnetic vectors have different signs. With this in mind, the boundary conditions take the form:

(5.2)

for parallel polarization and

(5.3)

for perpendicular polarization. In addition, in each of the waves, the strengths of the electric and magnetic fields are related by the relations . With this in mind, from the boundary conditions (5.2) and (5.3), we can obtain expressions for amplitude reflection and transmission coefficients :

(5.4)

In addition to amplitude, are of interest energy reflection coefficients R and transmission T, equal to relation energy flows corresponding waves. Since the intensity of the light wave is proportional to the square of the electric field strength, for any polarization the equality holds. In addition, the relation R+T= 1, which expresses the energy conservation law in the absence of absorption at the interface. Thus,

(5.5)

The set of formulas (5.4), (5.5) is called Fresnel formulas . Of particular interest is the limiting case of normal light incidence on the interface (a = b = 0). In this case, the difference between parallel and perpendicular polarizations disappears and

(5.6)

From (5.6) we find that with normal incidence of light from air ( n 1 = 1) on glass ( n 2 = 1.5) 4% of the energy of the light beam is reflected, and 96% passes through.

1.2. Analysis of Fresnel formulas

Consider first the energy characteristics. It can be seen from (5.5) that for a + b = p/2 the reflection coefficient of the parallel component vanishes: R|| = 0. The angle of incidence at which this effect occurs is called Brewster angle . It is easy to find from Snell's law that

, (5.7)

Where n 12 - relative refractive index. At the same time, for the perpendicular component R^ ¹ 0. Therefore, when unpolarized light is incident at the Brewster angle, the reflected wave turns out to be linearly polarized in a plane perpendicular to the plane of incidence, and the transmitted wave is partially polarized with a predominance of the parallel component (Fig. 5.3a) and the degree of polarization

For the air-glass transition, the Brewster angle is close to 56°.

In practice, obtaining linearly polarized light by reflection at the Brewster angle is rarely used due to the low reflectivity. However, it is possible to construct a transmissive polarizer using feet of Stoletov (Fig. 5.3b). Stoletov's foot consists of several plane-parallel glass plates. When light passes through it at the Brewster angle, the perpendicular component is almost completely scattered at the interfaces, and the transmitted beam is polarized in the plane of incidence. Such polarizers are used in high power laser systems where other types of polarizers can be destroyed by laser radiation. Another application of the Brewster effect is to reduce reflection losses in lasers by mounting optical elements at the Brewster angle to the optical axis of the resonator.

The second most important consequence of the Fresnel formulas is the existence total internal reflection (TIR) from an optically less dense medium at angles of incidence greater than the limiting angle determined from the relation

The effect of total internal reflection will be considered in detail in the next section; for now, we only note that it follows from formulas (5.7) and (5.8) that the Brewster angle is always less than the limiting angle.

On the graphs of Fig. 5.4a shows the dependences of the reflection coefficients for the incidence of light from air on the boundaries with media with n 2" = 1.5 (solid lines) and n 2 "" = 2.5 (dashed lines). On fig. 5.4b, the direction of passage of the interface is reversed.

Let us now turn to the analysis of the amplitude coefficients (5.4). It is easy to see that for any ratio between the refractive indices and for any angles, the transmittances t are positive. This means that the refracted wave is always in phase with the incident wave.

Reflection coefficients r, on the other hand, can be negative. Since any negative value can be written as , the negativity of the corresponding coefficient can be interpreted as a phase shift by p upon reflection. This effect is often referred to as loss of half a wave upon reflection.

It follows from (5.4) that upon reflection from an optically denser medium ( n 1 < n 2 , a > b) r ^ < 0 при всех углах падения, а r || < 0 при углах падения меньших угла Брюстера. При отражении от оптически менее плотной среды (n 1 > n 2 , a< b) отражение софазное за исключением случая падения света с параллельной поляризацией под углом большим угла Брюстера (но меньшим предельного угла). Очевидно, что при нормальном падении на оптически более плотную среду фаза отраженной волны всегда сдвинута на p.

Thus, naturally polarized light, when passing through the interface between two media, turns into partially polarized light, and when reflected at the Brewster angle, even into linearly polarized light. Linearly polarized light remains linearly polarized upon reflection and refraction, but the orientation of the plane of polarization may change due to the difference in the reflectances of the two components.