In 1669, the Danish scientist Erasmus Bartholin discovered that if you look at any object through a crystal of Icelandic spar, then at certain positions of the crystal and the object, two images of the object are visible at once. This phenomenon has been called double refraction phenomenon.
An explanation of the nature of this phenomenon was given in 1690 by Christian Huygens in his work Treatise on Light.
In the modern interpretation, the explanation of the nature of the phenomenon is as follows.
Light entering a birefringent substance is divided into two plane-polarized beams in mutually perpendicular planes.
In general, these rays propagate differently in different directions.
However, in any birefringent substance, there are one or two directions along which both beams propagate at the same speed.
These directions are called optical axes.
Depending on the number of axes, birefringent substances are divided into uniaxial and biaxial. We will consider only uniaxial birefringent materials.
It is important to note that the directions of vector oscillations E plane-polarized beams arising inside a birefringent substance are always oriented in a certain way. One of them has vector oscillations E are perpendicular to the plane in which the incident beam and the optical axis lie (this plane is commonly called main section). The second one is parallel to the main section.
The propagation velocities of these rays depend on the angle between the vector E and optical axis.
In beam with vector E, perpendicular to the main section, the angle between E and the optical axis does not depend on the angle of incidence of the beam. At any angle of incidence, the vector E perpendicular to the optical axis.
This means that at any angle of incidence, it has the same speed.
Since the speed of light in a substance is related to the refractive index of this substance, the refractive index of a birefringent substance for this beam also does not depend on the angle of incidence. In other words, this beam behaves like in an ordinary isotropic medium.
Therefore it is called ordinary. Next Vector E ordinary beam will be denoted E o.
The second beam is called extraordinary, since for it the angle between the direction of oscillations of the vector Her(hereinafter the vector E extraordinary ray will be denoted Her) and the optical axis depends on the angle of incidence (see figure). Therefore, at different angles of incidence, it propagates at different speeds and has a different refractive index, which, in general, is unusual.
Let plane-polarized light fall on a plane-parallel plate of a birefringent substance.
In this case, the plane of the main section is perpendicular to the surface of the plate.
Inside the plate, the incident beam is divided into two plane polarized beams, one of which is polarized perpendicular to the optical axis (ordinary beam), and the second is parallel (extraordinary beam).
Naturally, these beams will be in phase at the entrance to the plate.
Inside the plate, the refractive indices for these rays have different values ( n o and n e).
This means that if the ordinary and extraordinary rays pass the same distance inside the plate (for example, d- the thickness of the plate), then they will no longer be in phase. They will have a phase difference Dj equal to k o ( n o d – n e d). Here k o is the wavenumber for vacuum.
If the phase difference of the rays emerging from the plate is a multiple of 2p, the orientation of the vector oscillation plane E Will not change. The light behind the plate will be polarized in the same way as in front of it.
If the phase difference is a multiple of an odd number p, the vector oscillation plane E behind the plate will rotate 90°, but the light will still be plane polarized.
If the phase difference turns out to be equal to p / 2, then the light behind the plate will turn out to be polarized in a circle. Plates of this thickness are called quarter-wave.
Passing circularly polarized light through a second quarter-wave plate adds an additional phase difference of p/2. This will cause the transformation of circularly polarized light into plane polarized light, the plane of polarization of which is rotated by 90° compared to the light incident on the first plate*.
The wave surfaces of the ordinary and extraordinary rays have different shapes.
In an ordinary ray, this, of course, is a sphere - an ordinary ray propagates in all directions at the same speed.
In the extraordinary, the wave surface is an ellipsoid - its speed is different for different directions.
Since both ordinary and extraordinary light waves propagate at the same speed along the optical axis, their wave surfaces touch at the points of intersection with the optical axis.
Consider a natural light wave incident on the surface of a crystalline birefringent plate.
Let the optical axis of the plate be parallel to the surface of the plate.
Beam of natural light hitting the spot BUT, excites two secondary light waves - ordinary and extraordinary.
Their fronts have the form shown in the figure.
Rays of secondary waves excited between points BUT and AT, are perpendicular to the wave surfaces of the ordinary and extraordinary waves, which can be constructed by drawing from the point AT tangent to each wave surface formed by ordinary and extraordinary rays passing through the point BUT.
It can be seen from the construction shown in the figure that the ordinary and extraordinary waves propagate inside the crystal in different directions. A number of methods for obtaining polarizing devices are based on this property - by cutting off one of the rays (ordinary or extraordinary), one can obtain plane polarized light.
In conclusion, we note that crystalline substances such as quartz and Icelandic spar are birefringent.
In addition, substances with asymmetric molecules oriented in an ordered fashion along any direction can be birefringent. These can be liquids and amorphous bodies in which the orientation of molecules occurs due to external influences (mechanical stress, external electric or magnetic fields).
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Quantum mechanics
Crisis of classical physics
At the end of the nineteenth century. in physics there is an interesting situation. Scientists believed that the slender building of classical physics was close to completion. It seemed that it remained to explain some minor effects ... and the development of physics would be completed.
However, at the turn of the nineteenth and twentieth centuries. several discoveries were made that could not be explained from the standpoint of classical physics. These discoveries gave rise to the crisis of classical physics, which, in turn, revolutionized science and gave rise to quantum physics.
thermal radiation
Thermal radiation is electromagnetic radiation emitted by a substance due to its internal energy.
Thermal radiation is emitted by all bodies whose temperature is different from absolute zero.
Thermal radiation is a superposition of electromagnetic waves, the lengths of which lie in a wide range. The spectrum of thermal radiation is continuous.
The spectral composition of thermal radiation depends on temperature - the higher the temperature of the body, the greater the proportion of short-wave radiation in it.
You know perfectly well that incandescent bodies can glow. This means that the thermal radiation of such a body contains visible waves.
The color of the glow will depend on the temperature. For example, the body can be heated white hot. Cooling down, the body will change color to red, then stop glowing at all, although it will still be quite hot.
The body will stop glowing, but will radiate energy - you can feel the heat coming from it. This means that the body radiates in the infrared range.
Colder bodies mostly radiate in a range not perceived by our senses, so we do not feel it.
double refraction
To obtain polarized light, the phenomenon of birefringence is also used.
“From Iceland, an island located in the North Sea, at a latitude of 66 °,” Huygens wrote in 1678, “a stone (Icelandic spar) was brought, very remarkable in its shape and other qualities, but most of all in its strange refractive properties ".
If a piece of Icelandic spar is put on any inscription, then through it we will see a double inscription (Fig. 133).
Rice. 133. Double refraction.
The bifurcation of the image occurs due to the fact that each beam incident on the surface of the crystal corresponds to two refracted beams. On fig. 134 shows the case when the incident beam is perpendicular to the crystal surface; then the ray o, called ordinary, passes through the crystal unrefracted, and the ray O, called extraordinary, goes along the broken line shown in fig. 134.
Rice. 134. The path of rays in birefringence.
The names of the rays are clear: an ordinary ray behaves as we might expect it to on the basis of the known laws of refraction. An extraordinary ray, as it were, violates these laws: it falls along the normal to the surface, but experiences refraction. Both beams exit the crystal as plane polarized, and they are polarized in mutually perpendicular planes. This can be easily verified by a very simple experiment. Let's take some analyzer (for example, a foot) and look through it at the bifurcated picture given by the crystal. At a certain position of the foot, we will see only one of the images, the second will be canceled. When the foot rotates 90° around the line of sight, this second image will appear, but the first one will disappear. Thus, we are really convinced that both images are polarized and exactly as it was just indicated.
It is curious that in 1808 Malus quite by accident made a similar experiment and discovered the polarization of light when reflected from glass. Looking through a piece of Icelandic spar at the reflection of the setting sun in the windows of the Luxembourg Palace in Paris, he was surprised to find that the two images resulting from double refraction had different brightness. Rotating the crystal, Malus saw that the images alternately became brighter, then faded. Malus at first thought that fluctuations in sunlight in the atmosphere were affecting here, but with the onset of night he repeated the experiment with the light of a candle reflected from the surface of the water, and then glass. In both cases, however, the effect was confirmed. Malus owns the term "polarization" of light.
Let us now turn to a more detailed analysis of the phenomenon of double refraction. If we change the angle of incidence of the beam on the surface of the crystal, then a new remarkable property of the extraordinary beam will be revealed. It turns out that its refractive index is not constant, but depends on the angle of incidence. Since the direction of the refracted ray in the crystal also depends on the angle of incidence, this property can also be formulated as follows: the refractive index of an extraordinary ray depends on its direction in the crystal. Passing, finally, from the index of refraction to the velocity of propagation, we can say that the velocity of an extraordinary ray in a crystal depends on the direction of its propagation.
In this final formulation, the optical properties of the crystal coincide with its other properties: the dielectric constant, thermal conductivity, and elasticity of the crystal are also not the same in different directions. The correspondence between the anisotropy of the optical and electrical properties of a crystal becomes quite clear if we remember that the speed of light is inversely proportional to the square root of the dielectric constant of the medium. Therefore, strictly speaking, the speed of propagation of a light wave does not depend on the direction of propagation, but on the direction of the electric field of the light wave. Even if two light waves polarized in mutually perpendicular planes propagate in the same direction in a crystal, their velocities will be different (with the exception of some special cases). An example of two such waves are extraordinary and ordinary rays.
If, from a point lying on the surface of Icelandic spar, we draw radius-vectors inside the crystal, the magnitude of which is proportional to the speed of light in the corresponding directions, then their ends will lie on the surface of the ellipsoid of revolution. This is equivalent to the fact that the wave surface of light oscillations propagating from a point has an ellipsoidal shape, in contrast to the spherical shape when propagating in an amorphous body. All the time, of course, we are talking about an extraordinary ray. Ordinary rays obviously form a spherical wave surface. Thus, in a crystal we have two types of wave surfaces: ellipsoids and spheres. These ellipsoids and spheres touch at points lying on straight lines, called the optical axes of the crystal.
It is clear that the light propagates in the direction of the optical axis with a speed that is completely independent of the state of polarization. In Icelandic spar, there is only one direction of the optical axis - a uniaxial crystal.
Using a simple graphical method based on the Huygens principle, we construct a refracted wave of both ordinary and extraordinary rays. One wave will be tangent to a series of elementary spheres, the other will be tangent to a series of ellipsoids. We see that an angle is formed between these two plane waves, which corresponds to the formation of an angle between the refracted rays, i.e., birefringence.
Rice. 5. Huygens construction in a crystal.
In contrast to an isotropic medium in a crystal, the (extraordinary) ray is no longer normal to the wave surface. On fig. 5 o denotes the ordinary ray, e the extraordinary and n the normal.
However, there is also a direction in the crystal of Icelandic spar, in which both ordinary and extraordinary rays travel at the same speed, without separating. This direction is called the optical axis of the crystal. It is obvious that the points of contact of the ellipsoid with the sphere lie on the optical axis. In a plane perpendicular to the optical axis, there are directions along which the difference in velocities between the ordinary and extraordinary rays is maximum. The ordinary and extraordinary rays go in the same direction, but the extraordinary ray overtakes the ordinary.
Any plane passing through the optical axis is called the main section or the main plane of the crystal.
In addition to Icelandic spar, uniaxial crystals include, for example, quartz and tourmaline. There are crystals in which the phenomena of refraction obey even more complex laws. In particular, for them there are two directions in which both beams travel at the same speed, therefore such crystals are called biaxial (for example, gypsum). In biaxial crystals, both beams are extraordinary, i.e., the propagation velocities of both beams depend on the direction.
Tourmaline has a remarkable ability to absorb one of the rays produced by birefringence, due to which the tourmaline crystal serves as a polarizer, giving one polarized beam at once.
Back in 1850, Herapat discovered that artificially made crystals of quinine iodide sulfate have the same properties as tourmaline.
Rice. 6. The use of polaroids.
However, individual crystals were too small and quickly deteriorated in air. Only in the very last years have they learned how to manufacture on an industrial scale a celluloid film into which a large number of perfectly identically oriented crystals of quinine iodide sulfate are introduced. This film is called polaroid.
Polaroid completely polarizes light, not only passing along the normal to its surface, but retains its properties for rays that form angles up to 30 ° with the normal. Thus, a polaroid can polarize a fairly wide cone of light rays.
Polaroid has found wide application in a wide variety of areas. Let us point out the most curious application of the Polaroid in the automotive business.
Polaroid plates are fixed on the front glass of the car (Fig. 6) and on car headlights. The polaroid plate on the front glass is an analyzer, the plates on the headlights are polarizers. The planes of polarization of the plates make an angle of 45° with the horizon and are parallel to each other. The driver, looking at the road through a polaroid, sees the reflected light of his headlights, i.e., sees the road illuminated by them, since the corresponding planes of polarization are parallel, but does not see the light from the headlights of an oncoming car, which is also equipped with polaroid plates. In the latter case, as can be easily seen from Fig. 6, the polarization planes will be mutually perpendicular. Thus, the driver is protected from the blinding effect of the headlights of an oncoming vehicle.
Glasses are made from polaroids, through which the glare of light reflected from shiny surfaces becomes invisible. This is explained by the fact that the glare is usually partially or completely polarized. Polaroid glasses are very advisable to use in museums and art galleries (the surface of paintings painted with oil paints often gives glare that makes it difficult to see the paintings and distorts the shades of colors).
One of the most common polarizers is the so-called Nicol prism, or simply nicol.
Rice. 7. Section of the Nicol prism.
Nicol's prism is a crystal of Icelandic spar sawn diagonally and glued together with Canadian balsam (Fig. 7). In the Nicol prism, one of the rays resulting from double refraction is eliminated in a very ingenious way. An ordinary ray, which is refracted more strongly, falls on the border with Canadian balsam at an angle of incidence greater than that of the extraordinary ray. Since Canada balsam has a lower refractive index than Iceland spar, total internal reflection occurs and the beam hits the side face. The side face is covered with black paint and absorbs the beam falling on it. Thus, only one plane-polarized beam (extraordinary) comes out of the prism. The plane of polarization of this beam is called the principal Nicol plane.
Two nicols located one behind the other, with mutually perpendicular main planes, obviously, do not let light through at all. If the main planes are parallel, then the maximum amount of light will pass through the nicols. The question arises how much light such a combination of nicols will let through at some intermediate position, when the angle a between the main planes is greater than zero, but less than 90°.
Since each polarizer, as we have already said, can be compared with a slit that transmits only oscillations lying in its plane, the procedure for calculating the intensity of light transmitted through two nicols is clear. For this purpose, we depict the main planes of the nicols in the form of straight lines I u II (Fig. 138). Then the oscillations emerging from the first nicol coincide with I, and if we decompose them into two components (one coinciding with II and the second perpendicular to it), then the first component will pass completely, and the second, obviously, will be delayed by the nicol. The magnitude of the amplitude constituting the oscillations in the direction II, as can be seen from the drawing, is equal to A where A is the amplitude of the oscillations coming out of the first nicol. This component, as we have just said, will pass completely; consequently, this will be the amplitude of the oscillation that has passed through two nicols.
Rice. 8. To the calculation of the energy that has passed through two nicols.
The energy of a light wave, like that of any oscillation, is proportional to the square of the amplitude; therefore, finally, for the light energy that has passed through two nicols, we have the following formula - the Malus law:
where I changes from to 0 as α changes from 0 to . Thus, by rotating one of the nicols, we can attenuate the transmitted light any number of times and obtain light of any intensity.
Malus' law obviously applies to any polarizer and analyzer. In particular, the intensity of light reflected successively from two glass mirrors obeys the same law.
If the Nicol prism serves to obtain one polarized beam, then the Wollaston prism produces two beams polarized in mutually perpendicular planes and located symmetrically with respect to the incident beam. The design of the Wollaston prism is extremely ingenious and shows especially clearly how the speed of propagation of rays in a crystal depends on the direction of their plane of polarization.
Rice. 9. Wollaston prism.
The Wollaston prism consists of two pieces of Icelandic spar cut parallel to the optical axis and glued together so that the optical axis of one piece is perpendicular to the optical axis of the other piece. On fig. 9, the optical axis of the right piece is parallel to the plane of the drawing, and the optical axis of the left piece is perpendicular to it.
A beam of light incident normally on the upper boundary is divided into two beams: an ordinary beam with a plane of polarization parallel to the optical axis, and an extraordinary beam polarized in a perpendicular direction. Both beams go in the same direction, but with different velocities determined by the refractive indices and . Having reached the interface with the second piece, both beams change roles. The plane of polarization of an ordinary (in the first piece) beam already becomes perpendicular to the optical axis (in the second piece), therefore, this beam in the second piece will propagate as an extraordinary one. On the contrary, an extraordinary beam in the first piece will be already ordinary in the second piece, since its plane of polarization is parallel to the optical axis of this piece. Thus, one beam (ordinary in the first piece) passes from a medium with a refractive index to a medium with a refractive index, the other (extraordinary in the first piece) - from a medium to a medium with . Icelandic spar has more. Consequently, the first beam passes from a denser medium to a less dense one, the second - vice versa. As a result, one beam will be refracted at the boundary to the left, and the other just as much to the right, and two polarized beams will enter symmetrically from the prism.
In 1669 Danish physicist and mathematician Erasmus Bartholin published the results of experiments with crystals of Icelandic spar (calcite CaCO 3), in which “amazing and strange refraction” was discovered. The essence of this physical phenomenon, called birefringence, is illustrated in Figure 8.
A beam of natural light falling on a crystal is divided inside it into two beams: ordinary(o) obeying the law of refraction; and unusual(e), for which , and depends on the angle of incidence and on the choice of the refracting face of the crystal (Fig. 8a).
An extraordinary beam of light does not obey the usual law of refraction and can be deflected even when light is normally incident on a crystal (Fig. 8b).
Ordinary and extraordinary beams generally propagate in a crystal in different directions, with different velocities, and are linearly polarized.
Let us consider the phenomenon of birefringence from the point of view of the electromagnetic theory of light propagating in an anisotropic medium.
Anisotropic optical medium.
The optical properties of a substance (dielectric permittivity, refractive index n, wave phase velocity v=c/n and others) are determined by the properties of molecules and atoms, their mutual arrangement and the nature of interaction between themselves and with the electromagnetic field of a light wave.
If the properties of a substance do not depend on the direction of oscillation of a vector in a light wave, then the medium is optically isotropic. Amorphous substances, like ordinary glass, and cubic crystals are usually isotropic.
Wednesday is called optically anisotropic, if its properties depend on the direction of propagation and polarization of the electromagnetic wave. Optically anisotropic crystals are called "birefringent crystals".
Ordinary and extraordinary waves.
We confine ourselves to considering an anisotropic crystal, the optical properties of which have a rotational symmetry about one of the directions in the crystal, called optical axis. Such crystals are called uniaxial.
The plane in which the optical axis and the wave vector of a light wave lie is called the main section of the crystal.
The oscillations of the vector are perpendicular to the main section of the crystal.
In this case (Fig. 9a), the crystal behaves as an isotropic medium with a refractive index .
A linearly polarized wave, in which the vector oscillations occur perpendicular to the main section (), and the phase velocity is called ordinary(ordinary).
Let there be a point light source S in the crystal, which emits an ordinary wave (Fig. 9b). Oscillations of the vector , shown by dots, occur perpendicular to the main section - the ZX plane. In any direction from the source S, the phase velocity is . The situation will not change if we consider any other plane rotated around the optical axis O 1 O 2 . Putting in all directions of light propagation segments equal to the distances traveled per unit time, we obtain a spherical wave surface of an ordinary wave from a point source with radius .
Vector oscillations occur in the main section.
Let's consider three cases.
a) The vector is parallel to the optical axis (Fig. 10 a). Then
where l is the wavelength of light in vacuum.
Such a wave propagating in the direction of the optical axis has a speed .
b) The vector is perpendicular to the optical axis (Fig. 3b). In this case
The wave propagates with speed .
c) The vector is located at an angle to the optical axis (Fig. 10 c)
The vector lies in the plane of the main section due to the symmetry of rotation. But since , then the vector does not coincide in direction with the vector . The wave vector is perpendicular to the vectors and , but not perpendicular to the vector . The wave remains transverse with respect to the oscillations of the vector , that is, but (see Fig. 10 c).
Energy transfer occurs in the direction of the Poynting vector. This direction does not coincide with the direction of the vector (the direction of motion of the wave surface).
When changing the direction of propagation of a linearly polarized wave, in which oscillations occur in the plane of the main section, the phase velocity depends on the direction of propagation and varies from to (, ). Such a wave is called extraordinary (extraordinary).
In the case of a point light source S located in a crystal and emitting an extraordinary wave, setting aside the distances traveled by the wave in different directions per unit time, we obtain an ellipsoidal wave surface with semiaxes and (Fig. 11d).
Along the optical axis, ordinary and extraordinary waves propagate with the same speeds, equal (see Fig. 9b and Fig. 10d).
Positive and negative uniaxial crystals.
Instead of wave surfaces for ordinary and extraordinary waves in a uniaxial crystal (see Fig. 9 b) and 10 d)) one can construct surfaces of refractive index values . Depending on the ratios between and (or and), negative and positive crystals are distinguished (Fig. 11).
Uniaxial crystals are called:
negative, if (Fig. 11 a),
positive if (Fig. 11 b).
lecture demonstrations
Field experiment
1. Double refraction.
Video demos
2. Educational film: "Polarization of light", Fragment 3 - "Polarization at birefringence". The following people worked on the film: E. Osmolovskaya, I. Wasserman and others. A.A. Zhdanova Fragment duration: 6 min.
3. Display of computer demonstrations.
Model1. An illustration of the operation of records in half-wave, quarter-wave and wavelength.
Fig.13
Optical constructor for studying polarized light:
1 – polarization ellipses at the input to the system; 2 – parameter window; 3,5-polaroids; 4 - birefringent plate; 6 - polarization ellipses at the output of the optical system.
a) selection of one or more light waves at the entrance to the optical system; b) selection at the input of the same waves for all; c), f) polaroids (remove, put, rotate); d) birefringent plate (sub-item "Thickness" - change of the parameter , sub-item "Inside" - observation of the change in the polarization ellipse inside the plate); f) start of the computer experiment.
Model2. group speed.
Fig.14. Propagation of a wave packet in media with different dispersion laws.
1 – window for selecting the amplitudes of each of the three waves; 2 - graph of the selected dispersion law, on which the marks of the corresponding color show the frequencies of each of the three waves; 3 - a window in which the movement of each of the three waves is shown; 4 - a window in which the movement of the envelope (sum) of three waves is shown; 5 - labels showing the phase velocities of individual spectral components and the group velocity of their sum.
a) - the beginning of the demonstration; b) – change in the parameters of the spectral components; c) – choice of the dispersion law.
On the fig.14 shows a general view of the screen of a program designed to study the simultaneous propagation in a dispersive medium of a signal containing three spectral components. Such a simple group of waves allows us to illustrate the concept of group velocity and its relation to phase velocity. A linear dispersion law is used. The program allows you to change the frequencies (they are indicated on the graph in window 2) and the amplitudes of all three spectral components (window 1), as well as the constants a and d in the dispersion law (window 2). Parameters are changed using the "Parameters" menu key. After pressing the "Start" key in window 3 in dynamic mode, you can observe the movement of all three waves separately, and in window 4 - the movement of the entire group of waves as a whole, that is, their sum. For the convenience of observation, special marks of the corresponding color (5) are displayed on the screen, which show the phase velocities of the spectral components and a separate white mark showing the group velocity.
The program allows you to reproduce on the display screen a kinematic model of a group of waves propagating in a medium with normal and anomalous dispersion laws.
Educational materials
Main literature
1. Saveliev I. V. Course of general physics, book. 3. - M .: Astrel Publishing House LLC, AST Publishing House LLC, 2004, §§6.3-6.8, §§7.1-7.5.
2. I. E. Irodov, Wave Processes. Basic Laws: Textbook for High Schools. – M.: Binom. Basic Knowledge Lab, 2007, §§ 6.3-6.7, §§7.1-7.5.
additional literature
3. Sivukhin D.V. General course of physics. v. 4. M.: FIZMATLIT, 2009, §§84, 90.
4. Landsberg G.S. Optics. -M.,: FIZMATLIT, 2003, §§156, 157, 159-160, 168.
5. Losev V.V. Optical phenomena. Theory and experiment. Textbook, M., 2002, §§4.2.
Information and reference resources
6. [Electronic resource].-M.: Collection of electronic resources of MIET, 2007.- Access mode: http://orioks.miet.ru/oroks-miet/srs.shtml
7. Training program. “Open Physics 2.6. Part 2":
http://www.physics.ru/
http://www.physics.ru/courses/op25part2/design/index.htm
8. Scientific Center "PHYSICON": of the course "Wave Optics on the Computer"
http://college.ru/WaveOptics/content/chapter1/section1/paragraph1/theory.html
9. Disk or program "Physics in animations"
http://physics.nad.ru/
http://physics.nad.ru/Physics/Cyrillic/optics.htm
When light passes through some crystals, the light beam splits into two beams. This phenomenon is called double refraction. Birefringence is the bifurcation of a light beam when passing through an optically anisotropic medium, due to the dependence of the refractive index (and, consequently, the wave speed) on its polarization and the orientation of the wave vector relative to the crystallographic axes. If a narrow beam of light is directed to an Icelandic spar crystal, then two spatially separated beams will come out of the crystal, parallel to each other and to the incident beam - ordinary (o) and extraordinary (e). 
An ordinary beam satisfies the usual law of refraction and lies in the same plane as the incident beam and the normal to the interface at the point of incidence. For the extraordinary ray, the ratio 
depends on the angle of incidence. In addition, the extraordinary ray does not, as a rule, lie in the same plane as the incident ray and the normal to the interface. The experiment shows that the rays emerging from the crystal are plane polarized in mutually perpendicular directions. The phenomenon of birefringence is observed for all transparent crystals, except for crystals of the cubic system. Uniaxial crystals have a direction along which light propagates without splitting into two beams. This direction is called the optical axis of the crystal. Any plane passing through the optical axis is called main section or main plane of the crystal. The plane passing through the beam and intersecting its optical axis is called the main plane (main section) of a uniaxial crystal for this beam. The plane of oscillation of an ordinary beam is perpendicular to the main section of the crystal. The oscillations of the vector in the extraordinary ray occur in the main plane of the crystal. In addition to uniaxial, there are biaxial crystals, which have two directions along which the light is not divided into two beams. In biaxial crystals, both beams are extraordinary.
Birefringence is explained by the anisotropy of crystals. In crystals of a non-cubic system, the permittivity depends on the direction. The vector of an ordinary beam is always perpendicular to the optical axis of the crystal (perpendicular to the main section). Therefore, for any direction of propagation of an ordinary beam, the speed of a light wave will be the same, the refractive index of a crystal for an ordinary beam does not depend on the direction of the beam in the crystal and is equal to 
The vector of the extraordinary ray oscillates in the main plane of the crystal, it can make any angles with the optical axis from 0 to. Therefore, the speed of light propagation along the extraordinary ray and the refractive index of the crystal for the extraordinary ray depend on the direction of this beam with respect to the optical axis. When light propagates along the optical axis, both beams coincide, the speed of light does not depend on the direction of oscillation of the vector (in both beams, the vector is perpendicular to the optical axis), the refractive index of an extraordinary ray coincides with the refractive index of an ordinary ray: When light propagates in any other direction, its speed and the refractive index along the extraordinary ray differ from the corresponding values for the ordinary ray. The greatest difference is observed in the direction perpendicular to the optical axis. In this direction 
where is the velocity of the extraordinary ray in this direction. The refractive index of an extraordinary ray is taken as the value for the direction of propagation perpendicular to the optical axis of the crystal. There are positive and negative uniaxial crystals. For positive crystals > (< ), у отрицательных – < ( > ).
In some crystals, one of the rays is absorbed more strongly than the other. This phenomenon is called dichroism .
Using the Huygens principle, one can graphically plot the wave surfaces of the ordinary and extraordinary rays. The figure shows the wave surfaces of rays centered at the point 2
for the moment when the wavefront of the incident wave reaches the point 1
. Along the optical axis, both beams propagate at the same speed. Wave surface for an ordinary ray emanating from a point 2
, a sphere (in a plane section - a circle), for an extraordinary - an ellipsoid (in a plane section - an ellipse). Envelopes of all secondary waves whose centers are between the points 1
and 2
, are planes. The front of an ordinary wave is a tangent from a point 1
to the circle; the extraordinary wave front is a tangent from a point 1
to the ellipse. For an ordinary beam, the direction of propagation of the energy of a light wave coincides with the normal to the wave surface; an ordinary ray is perpendicular to the wave surface. For an extraordinary ray, the direction of energy propagation does not coincide with the normal to the wave surface; the extraordinary ray passes through the point where the wavefront touches the ellipse.
The phenomenon of double refraction. Properties of ordinary and extraordinary rays.
Almost all transparent dielectrics are optically anisotropic, that is, the properties of light as it passes through them depend on the direction. The physical nature of anisotropy is associated with structural features of the dielectric molecules or features of the crystal lattice, at the nodes of which there are atoms or ions.
Due to the anisotropy of crystals, when light passes through them, a phenomenon occurs called birefringence
Birefringence is caused by the unequal speed of propagation of light waves in different directions. At the point of incidence of natural light, two light waves are formed. One propagates in a crystal in all directions with the same speed - this is an ordinary beam (spherical wave front). In the other, the speed along the direction of the optical axis of the crystal is the same as the speed in the first wave, and more along the direction perpendicular to the optical axis. This is an extraordinary beam (the wave front has an ellipsoidal shape).
We will focus on the so-called uniaxial crystals. In uniaxial crystals, one of the refracted beams obeys the usual law of refraction. It is called ordinary. The other beam is called extraordinary, it does not obey the usual law of refraction. Even with a normal incidence of a light beam on the crystal surface, an extraordinary beam can deviate from the normal. As a rule, the extraordinary ray does not lie in the plane of incidence. If you look at the surrounding objects through such a crystal, then each object will split in two. When the crystal rotates around the direction of the incident beam, the ordinary beam remains motionless, and the extraordinary beam will move around it in a circle.
Uniaxial crystals include, for example, crystals of calcite or Icelandic spar (). In uniaxial crystals, there is a preferred direction along which the ordinary and extraordinary waves propagate without spatial separation and at the same speed. The direction in which no birefringence is observed is called optical axis of the crystal. It should be borne in mind that the optical axis is not a straight line passing through some point of the crystal, but a certain direction in the crystal. Any line parallel to this direction is an optical axis.
A study of the ordinary and extraordinary beams shows that both beams are completely plane polarized in mutually perpendicular directions. Oscillations of the electric field strength vector in an ordinary wave occur in the direction perpendicular to the main section of the crystal for an ordinary beam. In an extraordinary wave, the intensity vector oscillates in a plane coinciding with the main section for an extraordinary ray.
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It can be seen from the figure that in this case the planes of oscillation of the ordinary and extraordinary rays are mutually perpendicular. Note that this is observed for almost any orientation of the optical axis, since the angle between the ordinary and extraordinary rays is very small.
At the output of the crystal, both beams differ from each other only in the direction of polarization, so that the names "ordinary" and "extraordinary" make sense only inside the crystal.
As you know, the refractive index Consequently, from the anisotropy of e it follows that electromagnetic waves with different directions of vector oscillations correspond to different values of the refractive index . Therefore, the speed of light waves depends on the direction of oscillation of the light vector. In an ordinary beam, the oscillations of the light vector occur in the direction perpendicular to the main section of the crystal, therefore, for any direction of the ordinary beam, it forms a right angle with the optical axis of the crystal and the speed of the light wave will be the same, equal to .
Uniaxial crystals are characterized by the refractive index of an ordinary ray equal to , and the refractive index of an extraordinary ray perpendicular to the optical axis, equal to . The latter quantity is simply called the refractive index of the extraordinary ray. For Icelandic spar, . Note that the values of and depend on the wavelength.
The refractive index, and, consequently, the propagation velocity for an ordinary beam n o does not depend on the direction in the crystal. An ordinary beam propagates in a crystal according to the usual laws of geometric optics.
For an extraordinary ray, the refractive index varies from n o in the direction of the optical axis up to n e in a direction perpendicular to it. If a n e > n o, then the crystals are called positive, with the inverse ratio n e < n o- negative.
From the point of view of the Huygens principle, with birefringence at each point on the surface of a wave reaching the face of a crystal, not one secondary wave arises, as in ordinary media, but simultaneously two waves propagate in the crystal. The propagation speed of an ordinary wave is the same in all directions. The speed of propagation of an extraordinary wave in the direction of the optical axis coincides with the speed of an ordinary wave, but differs in other directions.
