Geometric optics. The phenomenon of total internal reflection. Limiting angle of total reflection. The course of the rays. Fiber optics. Refraction of light. Total internal reflection

Total internal reflection

Internal reflection- the phenomenon of reflection of electromagnetic waves from the interface between two transparent media, provided that the wave is incident from a medium with a higher refractive index.

Incomplete internal reflection- internal reflection, provided that the angle of incidence is less than the critical angle. In this case, the beam splits into refracted and reflected.

Total internal reflection- internal reflection, provided that the angle of incidence exceeds a certain critical angle. In this case, the incident wave is completely reflected, and the value of the reflection coefficient exceeds its highest values for polished surfaces. In addition, the reflectance of total internal reflection is independent of wavelength.

This optical phenomenon is observed for a wide range of electromagnetic radiation including the X-ray range.

Within the framework of geometric optics, the explanation of the phenomenon is trivial: based on Snell’s law and taking into account that the angle of refraction cannot exceed 90°, we obtain that at an angle of incidence whose sine is greater than the ratio of the smaller refractive index to the larger coefficient, the electromagnetic wave must be completely reflected into the first medium .

In accordance with the wave theory of the phenomenon, the electromagnetic wave still penetrates into the second medium - the so-called “non-uniform wave” propagates there, which decays exponentially and does not carry energy with it. The characteristic depth of penetration of an inhomogeneous wave into the second medium is of the order of the wavelength.

Total internal reflection of light

Let us consider internal reflection using the example of two monochromatic rays incident on the interface between two media. The rays fall from a zone of a denser medium (indicated by a darker blue) with a refractive index to the boundary with a less dense medium (indicated in light blue) with a refractive index.

The red beam falls at an angle , that is, at the boundary of the media it bifurcates - it is partially refracted and partially reflected. Part of the beam is refracted at an angle.

The green beam falls and is completely reflected src="/pictures/wiki/files/100/d833a2d69df321055f1e0bf120a53eff.png" border="0">.

Total internal reflection in nature and technology

X-ray reflection

The refraction of X-rays at grazing incidence was first formulated by M. A. Kumakhov, who developed the X-ray mirror, and theoretically substantiated by Arthur Compton in 1923.

Other wave phenomena

Demonstration of refraction, and therefore the effect of total internal reflection, is possible, for example, for sound waves on the surface and in the thickness of the liquid during the transition between zones of different viscosity or density.

Phenomena similar to the effect of total internal reflection of electromagnetic radiation are observed for beams of slow neutrons.

If a vertically polarized wave is incident on the interface at the Brewster angle, then the effect of complete refraction will be observed - there will be no reflected wave.

Notes

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TOTAL INTERNAL REFLECTION- reflection el. mag. radiation (in particular, light) when it falls on the interface between two transparent media from a medium with a high refractive index. P.v. O. occurs when the angle of incidence i exceeds a certain limiting (critical) angle... Physical encyclopedia

Total internal reflection- Total internal reflection. When light passes from a medium with n1 > n2, total internal reflection occurs if the angle of incidence a2 > apr; at angle of incidence a1 Illustrated Encyclopedic Dictionary

Total internal reflection- reflection of optical radiation (See Optical radiation) (light) or electromagnetic radiation of another range (for example, radio waves) when it falls on the interface of two transparent media from a medium with a high refractive index... ... Great Soviet Encyclopedia

TOTAL INTERNAL REFLECTION- electromagnetic waves, occurs when they pass from a medium with a large refractive index n1 to a medium with a lower refractive index n2 at an angle of incidence a exceeding the limiting angle apr, determined by the ratio sinapr=n2/n1. Full... ... Modern encyclopedia

TOTAL INTERNAL REFLECTION- COMPLETE INTERNAL REFLECTION, REFLECTION without REFRACTION of light at the boundary. When light passes from a denser medium (for example, glass) to a less dense medium (water or air), there is a zone of refraction angles in which the light does not pass through the boundary... Scientific and technical encyclopedic dictionary

total internal reflection- Reflection of light from a medium that is optically less dense with complete return to the medium from which it falls. [Collection of recommended terms. Issue 79. Physical optics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1970] Topics… … Technical Translator's Guide

TOTAL INTERNAL REFLECTION- electromagnetic waves occur when they are obliquely incident on the interface between 2 media, when radiation passes from a medium with a large refractive index n1 to a medium with a lower refractive index n2, and the angle of incidence i exceeds the limiting angle... ... Big Encyclopedic Dictionary

total internal reflection- electromagnetic waves, occurs with oblique incidence on the interface between 2 media, when radiation passes from a medium with a large refractive index n1 to a medium with a lower refractive index n2, and the angle of incidence i exceeds the limiting angle ipr ... Encyclopedic Dictionary

Physical meaning of the refractive index. Light is refracted due to changes in the speed of its propagation when passing from one medium to another. The refractive index of the second medium relative to the first is numerically equal to the ratio of the speed of light in the first medium to the speed of light in the second medium:

Thus, the refractive index shows how many times the speed of light in the medium from which the beam exits is greater (smaller) than the speed of light in the medium into which it enters.

Since the speed of propagation of electromagnetic waves in a vacuum is constant, it is advisable to determine the refractive indices of various media relative to vacuum. Speed ratio With propagation of light in a vacuum to the speed of its propagation in a given medium is called absolute refractive index of a given substance () and is the main characteristic of its optical properties,

those. the refractive index of the second medium relative to the first is equal to the ratio of the absolute indices of these media.

Typically, the optical properties of a substance are characterized by its refractive index n relative to air, which differs little from the absolute refractive index. At the same time, the environment in which absolute indicator more is called optically denser.

Limit angle of refraction. If light passes from a medium with a lower refractive index to a medium with a higher refractive index ( n 1< n 2 ), then the angle of refraction is less than the angle of incidence

r< i (Fig. 3).

Rice. 3. Refraction of light during transition

from an optically less dense medium to a medium

optically denser.

When the angle of incidence increases to i m = 90° (beam 3, Fig. 2) light in the second medium will propagate only within the angle r pr called limiting angle of refraction. In the region of the second medium within an angle additional to the limiting angle of refraction (90° - i pr ), light does not penetrate (in Fig. 3 this area is shaded).

Limit angle of refraction r pr

But sin i m = 1, therefore .

The phenomenon of total internal reflection. When light travels from a medium with a high refractive index n 1 > n 2 (Fig. 4), then the angle of refraction is greater than the angle of incidence. Light is refracted (passes into a second medium) only within the angle of incidence i pr , which corresponds to the angle of refraction r m = 90°.

Rice. 4. Refraction of light when passing from an optically denser medium to a medium

optically less dense.

Light incident at a large angle is completely reflected from the boundary of the media (Fig. 4, ray 3). This phenomenon is called total internal reflection, and the angle of incidence i pr – limiting angle of total internal reflection.

Limiting angle of total internal reflection i pr determined according to the condition:

, then sin r m =1, therefore, .

If light comes from any medium into a vacuum or air, then

Due to the reversibility of the ray path for two given media, the limiting angle of refraction during the transition from the first medium to the second is equal to the limiting angle of total internal reflection when the ray passes from the second medium to the first.

The limiting angle of total internal reflection for glass is less than 42°. Therefore, rays traveling through glass and falling on its surface at an angle of 45° are completely reflected. This property of glass is used in rotating (Fig. 5a) and reversible (Fig. 4b) prisms, often used in optical instruments.

Rice. 5: a – rotary prism; b – reversible prism.

Fiber optics. Total internal reflection is used in the construction of flexible light guides. Light, entering a transparent fiber surrounded by a substance with a lower refractive index, is reflected many times and propagates along this fiber (Fig. 6).

Fig.6. Passage of light inside a transparent fiber surrounded by a substance

with a lower refractive index.

To transmit large light fluxes and maintain the flexibility of the light-conducting system, individual fibers are collected into bundles - light guides. The branch of optics that deals with the transmission of light and images through optical fibers is called fiber optics. The same term is used to refer to the fiber optic parts and devices themselves. In medicine, light guides are used to illuminate internal cavities with cold light and transmit images.

Practical part

Devices for determining the refractive index of substances are called refractometers(Fig. 7).

Fig.7. Optical diagram of the refractometer.

1 – mirror, 2 – measuring head, 3 – prism system to eliminate dispersion, 4 – lens, 5 – rotating prism (beam rotation by 90 0), 6 – scale (in some refractometers

There are two scales: the refractive index scale and the solution concentration scale),

7 – eyepiece.

The main part of the refractometer is the measuring head, which consists of two prisms: the lighting one, which is located in the folding part of the head, and the measuring one.

At the exit of the lighting prism, its matte surface creates a scattered beam of light, which passes through the liquid under study (2-3 drops) between the prisms. The rays fall onto the surface of the measuring prism at different angles, including at an angle of 90 0 . In the measuring prism, the rays are collected in the region of the limiting angle of refraction, which explains the formation of the light-shadow boundary on the device screen.

Fig.8. Beam path in the measuring head:

1 – lighting prism, 2 – test liquid,

3 – measuring prism, 4 – screen.

When waves propagate in a medium, including electromagnetic ones, to find a new wave front at any time, use Huygens' principle.

Each point on the wave front is a source of secondary waves.

In a homogeneous isotropic medium, the wave surfaces of secondary waves have the form of spheres of radius v×Dt, where v is the speed of wave propagation in the medium. By drawing the envelope of the wave fronts of the secondary waves, we obtain a new wave front in at the moment time (Fig. 7.1, a, b).

Law of Reflection

Using Huygens' principle, it is possible to prove the law of reflection of electromagnetic waves at the interface between two dielectrics.

The angle of incidence is equal to the angle of reflection. The incident and reflected rays, together with the perpendicular to the interface between the two dielectrics, lie in the same plane.Ð a = Ð b. (7.1)

Let a plane light wave (rays 1 and 2, Fig. 7.2) fall on a flat LED interface between two media. The angle a between the beam and the perpendicular to the LED is called the angle of incidence. If at a given moment in time the front of the incident OB wave reaches point O, then according to Huygens’ principle this point

Rice. 7.2

begins to emit a secondary wave. During the time Dt = VO 1 /v, the incident beam 2 reaches point O 1. During the same time, the front of the secondary wave, after reflection in point O, propagating in the same medium, reaches points of the hemisphere with radius OA = v Dt = BO 1. The new wave front is depicted by the plane AO 1, and the direction of propagation by the ray OA. Angle b is called the angle of reflection. From the equality of triangles OAO 1 and OBO 1, the law of reflection follows: the angle of incidence is equal to the angle of reflection.

Law of refraction

An optically homogeneous medium 1 is characterized by , (7.2)

Ratio n 2 / n 1 = n 21 (7.4)

called

(7.5)

For vacuum n = 1.

Due to dispersion (light frequency n » 10 14 Hz), for example, for water n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. If the speed of light propagation in the first medium is v 1, and in the second - v 2,

Rice. 7.3

then during the time Dt the incident plane wave travels the distance AO 1 in the first medium AO 1 = v 1 Dt. The front of the secondary wave, excited in the second medium (in accordance with Huygens' principle), reaches points of the hemisphere, the radius of which is OB = v 2 Dt. The new front of the wave propagating in the second medium is represented by the BO 1 plane (Fig. 7.3), and the direction of its propagation by the rays OB and O 1 C (perpendicular to the wave front). Angle b between the ray OB and the normal to the interface between two dielectrics at point O called the angle of refraction. From the triangles OAO 1 and OBO 1 it follows that AO 1 = OO 1 sin a, OB = OO 1 sin b.

Their attitude expresses law of refraction(law Snell):

. (7.6)

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the relative refractive index of the two media.

Total internal reflection

Rice. 7.4

According to the law of refraction, at the interface between two media one can observe total internal reflection, if n 1 > n 2, i.e. Ðb > Ða (Fig. 7.4). Consequently, there is a limiting angle of incidence Ða pr when Ðb = 90 0 . Then the law of refraction (7.6) takes the following form:

sin a pr = , (sin 90 0 =1) (7.7)

With a further increase in the angle of incidence Ða > Ða pr, the light is completely reflected from the interface between the two media.

This phenomenon is called total internal reflection and are widely used in optics, for example, to change the direction of light rays (Fig. 7.5, a, b).

It is used in telescopes, binoculars, fiber optics and other optical instruments.

In classical wave processes, such as the phenomenon of total internal reflection of electromagnetic waves, phenomena similar to the tunnel effect in quantum mechanics are observed, which is associated with the wave-corpuscular properties of particles.

Indeed, when light passes from one medium to another, refraction of light is observed, associated with a change in the speed of its propagation in different media. At the interface between two media, a light beam is divided into two: refracted and reflected.

A ray of light falls perpendicularly onto face 1 of a rectangular isosceles glass prism and, without refraction, falls on face 2, total internal reflection is observed, since the angle of incidence (Ða = 45 0) of the beam on face 2 is greater than the limiting angle of total internal reflection (for glass n 2 = 1.5; Ða pr = 42 0).

If the same prism is placed at a certain distance H ~ l/2 from face 2, then the beam the light will pass through face 2 * and will exit the prism through face 1 * parallel to the ray incident on face 1. The intensity J of the transmitted light flux decreases exponentially with increasing gap h between the prisms according to the law:

where w is a certain probability of the beam passing into the second medium; d is the coefficient depending on the refractive index of the substance; l is the wavelength of the incident light

Therefore, the penetration of light into the “forbidden” region is an optical analogy of the quantum tunneling effect.

The phenomenon of total internal reflection is truly complete, since in this case all the energy of the incident light is reflected at the interface between two media than when reflected, for example, from the surface of metal mirrors. Using this phenomenon, one can trace another analogy between the refraction and reflection of light, on the one hand, and Vavilov-Cherenkov radiation, on the other hand.

WAVE INTERFERENCE

7.2.1. The role of vectors and

In practice, several waves can propagate simultaneously in real media. As a result of the addition of waves, a number of interesting phenomena are observed: interference, diffraction, reflection and refraction of waves etc.

These wave phenomena are characteristic not only of mechanical waves, but also electrical, magnetic, light, etc. All elementary particles also exhibit wave properties, which has been proven by quantum mechanics.

One of the most interesting wave phenomena, which is observed when two or more waves propagate in a medium, is called interference. An optically homogeneous medium 1 is characterized by absolute refractive index , (7.8)

where c is the speed of light in vacuum; v 1 - speed of light in the first medium.

Medium 2 is characterized by the absolute refractive index

where v 2 is the speed of light in the second medium.

Attitude (7.10)

called the relative refractive index of the second medium relative to the first. For transparent dielectrics in which m = 1, using Maxwell's theory, or

where e 1, e 2 are the dielectric constants of the first and second media.

For vacuum n = 1. Due to dispersion (light frequency n » 10 14 Hz), for example, for water n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. Light is electromagnetic waves. Therefore, the electromagnetic field is determined by the vectors and , which characterize the strengths of the electric and magnetic fields, respectively. However, in many processes of interaction of light with matter, for example, such as the effect of light on the organs of vision, photocells and other devices, the decisive role belongs to the vector, which in optics is called the light vector.

We pointed out in § 81 that when light falls on the interface between two media, the light energy is divided into two parts: one part is reflected, the other part penetrates through the interface into the second medium. Using the example of the transition of light from air to glass, i.e. from a medium that is optically less dense to a medium that is optically denser, we saw that the proportion of reflected energy depends on the angle of incidence. In this case, the fraction of reflected energy increases greatly as the angle of incidence increases; however, even at very large angles of incidence, close to , when the light beam almost slides along the interface, some of the light energy still passes into the second medium (see §81, tables 4 and 5).

A new interesting phenomenon arises if light propagating in any medium falls on the interface between this medium and a medium that is optically less dense, that is, having a lower absolute refractive index. Here, too, the fraction of reflected energy increases with increasing angle of incidence, but the increase follows a different law: starting from a certain angle of incidence, all light energy is reflected from the interface. This phenomenon is called total internal reflection.

Let us consider again, as in §81, the incidence of light at the interface between glass and air. Let a light beam fall from the glass onto the interface at different angles of incidence (Fig. 186). If we measure the fraction of reflected light energy and the fraction of light energy passing through the interface, we obtain the values given in Table. 7 (glass, like in Table 4, had a refractive index).

Rice. 186. Total internal reflection: the thickness of the rays corresponds to the fraction of light energy charged or passed through the interface

The angle of incidence from which all light energy is reflected from the interface is called the limiting angle of total internal reflection. For the glass for which the table was compiled. 7 (), the limiting angle is approximately .

Table 7. Fractions of reflected energy for various angles of incidence when light passes from glass to air

Angle of incidence

Angle of refraction

Reflected energy percentage (%)

Let us note that when light is incident on the interface at a limiting angle, the angle of refraction is equal to , i.e., in the formula expressing the law of refraction for this case,

when we have to put or . From here we find

At angles of incidence greater than that, there is no refracted ray. Formally, this follows from the fact that at angles of incidence large from the law of refraction for, values larger than unity are obtained, which is obviously impossible.

In table Table 8 shows the limiting angles of total internal reflection for some substances, the refractive indices of which are given in table. 6. It is easy to verify the validity of relation (84.1).

Table 8. Limiting angle of total internal reflection at the boundary with air

Substance

Carbon disulfide

Glass (heavy flint)

Glycerol

Total internal reflection can be observed at the boundary of air bubbles in water. They shine because what falls on them sunlight is completely reflected without passing into the bubbles. This is especially noticeable in those air bubbles that are always present on the stems and leaves of underwater plants and which in the sun appear to be made of silver, that is, from a material that reflects light very well.

Total internal reflection finds application in the design of glass rotating and turning prisms, the action of which is clear from Fig. 187. The limiting angle for a prism is depending on the refractive index of a given type of glass; Therefore, the use of such prisms does not encounter any difficulties with regard to the selection of the angles of entry and exit of light rays. Rotating prisms successfully perform the functions of mirrors and are advantageous in that their reflective properties remain unchanged, whereas metal mirrors fade over time due to oxidation of the metal. It should be noted that the wrapping prism is simpler in design than the equivalent rotating system of mirrors. Rotating prisms are used, in particular, in periscopes.

Rice. 187. Path of rays in a glass rotating prism (a), a wrapping prism (b) and in a curved plastic tube - light guide (c)

At a certain angle of incidence of light $(\alpha )_(pad)=(\alpha )_(pred)$, which is called limit angle, the angle of refraction is equal to $\frac(\pi )(2),\ $in this case the refracted ray slides along the interface between the media, therefore, there is no refracted ray. Then from the law of refraction we can write that:

Figure 1.

In the case of total reflection, the equation is:

has no solution in the region of real values of the refraction angle ($(\alpha )_(pr)$). In this case, $cos((\alpha )_(pr))$ is a purely imaginary quantity. If we turn to the Fresnel Formulas, it is convenient to present them in the form:

where the angle of incidence is denoted by $\alpha $ (for brevity), $n$ is the refractive index of the medium where the light propagates.

Note 1

It should be noted that the inhomogeneous wave does not disappear in the second medium. So, if $\alpha =(\alpha )_0=(arcsin \left(n\right),\ then\ )$ $E_(pr\bot )=2E_(pr\bot ).$ Violations of the law of conservation of energy in a given case no. Since Fresnel's formulas are valid for a monochromatic field, that is, for a steady-state process. In this case, the law of conservation of energy requires that the average change in energy over the period in the second medium be equal to zero. The wave and the corresponding fraction of energy penetrates through the interface into the second medium to a small depth of the order of the wavelength and moves in it parallel to the interface with a phase velocity that is less than the phase velocity of the wave in the second medium. It returns to the first medium at a point that is offset relative to the entry point.

The penetration of the wave into the second medium can be observed experimentally. The intensity of the light wave in the second medium is noticeable only at distances shorter than the wavelength. Near the interface on which the light wave falls and undergoes total reflection, the glow of a thin layer can be seen on the side of the second medium if there is a fluorescent substance in the second medium.

Total reflection causes mirages to occur when the earth's surface is hot. Thus, the complete reflection of light that comes from clouds leads to the impression that there are puddles on the surface of heated asphalt.

Under ordinary reflection, the relations $\frac(E_(otr\bot ))(E_(pad\bot ))$ and $\frac(E_(otr//))(E_(pad//))$ are always real. At full reflection they are complex. This means that in this case the phase of the wave undergoes a jump, while it is different from zero or $\pi $. If the wave is polarized perpendicular to the plane of incidence, then we can write:

where $(\delta )_(\bot )$ is the desired phase jump. Let us equate the real and imaginary parts, we have:

From expressions (5) we obtain:

Accordingly, for a wave that is polarized in the plane of incidence, one can obtain:

The phase jumps $(\delta )_(//)$ and $(\delta )_(\bot )$ are not the same. The reflected wave will be elliptically polarized.

Applying Total Reflection

Let us assume that two identical media are separated by a thin air gap. A light wave falls on it at an angle that is greater than the limiting one. It may happen that it penetrates the air gap as a non-uniform wave. If the thickness of the gap is small, then this wave will reach the second boundary of the substance and will not be very weakened. Having passed from the air gap into the substance, the wave will turn back into a homogeneous one. Such an experiment was carried out by Newton. The scientist pressed another prism, which was ground spherically, to the hypotenuse face of the rectangular prism. In this case, the light passed into the second prism not only where they touch, but also in a small ring around the contact, in a place where the thickness of the gap is comparable to the wavelength. If observations were carried out in white light, then the edge of the ring had a reddish color. This is as it should be, since the penetration depth is proportional to the wavelength (for red rays it is greater than for blue ones). By changing the thickness of the gap, you can change the intensity of the transmitted light. This phenomenon formed the basis of the light telephone, which was patented by Zeiss. In this device, one of the media is a transparent membrane, which vibrates under the influence of sound falling on it. Light that passes through an air gap changes intensity in time with changes in sound intensity. When it hits a photocell, it generates alternating current, which changes in accordance with changes in sound intensity. The resulting current is amplified and used further.

The phenomena of wave penetration through thin gaps are not specific to optics. This is possible for a wave of any nature if the phase velocity in the gap is higher than the phase velocity in environment. This phenomenon is of great importance in nuclear and atomic physics.

The phenomenon of total internal reflection is used to change the direction of light propagation. Prisms are used for this purpose.

Example 1

Exercise: Give an example of the phenomenon of total reflection, which occurs frequently.

Solution:

We can give the following example. If the highway is very hot, then the air temperature is maximum near the asphalt surface and decreases with increasing distance from the road. This means that the refractive index of air is minimal at the surface and increases with increasing distance. As a result of this, rays that have a small angle relative to the highway surface are completely reflected. If you concentrate your attention, while driving in a car, on a suitable section of the highway surface, you can see a car driving quite far ahead upside down.

Example 2

Exercise: What is the Brewster angle for a beam of light that falls on the surface of a crystal if the limiting angle of total reflection for a given beam at the air-crystal interface is 400?

Solution:

\[(tg(\alpha )_b)=\frac(n)(n_v)=n\left(2.2\right).\]

From expression (2.1) we have:

Let's substitute the right side of expression (2.3) into formula (2.2) and express the desired angle:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left((\alpha )_(pred)\right)\ ))\right).\]

Let's carry out the calculations:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left(40()^\circ \right)\ ))\right)\approx 57()^\circ .\]

Answer:$(\alpha )_b=57()^\circ .$