How to find the refractive index. Absolute refractive index and its relationship with relative refractive index
Optics is one of the old branches of physics. Since the times of ancient Greece, many philosophers have been interested in the laws of the movement and propagation of light in various transparent materials, such as water, glass, diamond and air. This article discusses the phenomenon of light refraction, focusing on the refractive index of air.
Light beam refraction effect
Everyone in their life has encountered hundreds of times the manifestation of this effect when they looked at the bottom of a reservoir or at a glass of water with some object placed in it. At the same time, the pond did not seem as deep as it actually was, and the objects in the glass of water looked deformed or broken.
The phenomenon of refraction consists of a break in its rectilinear trajectory when it intersects the interface of two transparent materials. Summarizing a large amount of experimental data, at the beginning of the 17th century, the Dutchman Willebrord Snell obtained a mathematical expression that accurately described this phenomenon. This expression is usually written in the following form:
n 1 *sin(θ 1) = n 2 *sin(θ 2) = const.
Here n 1, n 2 are the absolute refractive indices of light in the corresponding material, θ 1 and θ 2 are the angles between the incident and refracted rays and the perpendicular to the interface plane, which is drawn through the intersection point of the ray and this plane.
This formula is called Snell's or Snell-Descartes' law (it was the Frenchman who wrote it down in the form presented, while the Dutchman used units of length rather than sines).
In addition to this formula, the phenomenon of refraction is described by another law, which is geometric in nature. It consists in the fact that the marked perpendicular to the plane and two rays (refracted and incident) lie in the same plane.
Absolute refractive index
This quantity is included in the Snell formula, and its value plays an important role. Mathematically, the refractive index n corresponds to the formula:
The symbol c is the speed of electromagnetic waves in a vacuum. It is approximately 3*10 8 m/s. The value v is the speed of light moving through the medium. Thus, the refractive index reflects the amount of retardation of light in a medium relative to airless space.
Two important conclusions follow from the formula above:
- the value of n is always greater than 1 (for vacuum it is equal to unity);
- it is a dimensionless quantity.
For example, the refractive index of air is 1.00029, while for water it is 1.33.
The refractive index is not a constant value for a particular medium. It depends on the temperature. Moreover, for each frequency of an electromagnetic wave it has its own meaning. Thus, the above figures correspond to a temperature of 20 o C and the yellow part of the visible spectrum (wavelength - about 580-590 nm).
The dependence of n on the frequency of light is manifested in the decomposition of white light by a prism into a number of colors, as well as in the formation of a rainbow in the sky during heavy rain.
Refractive index of light in air
Its value has already been given above (1.00029). Since the refractive index of air differs only in the fourth decimal place from zero, for solving practical problems it can be considered equal to one. A slight difference between n for air and unity indicates that light is practically not slowed down by air molecules, which is due to its relatively low density. Thus, the average air density is 1.225 kg/m 3, that is, it is more than 800 times lighter than fresh water.
Air is an optically weak medium. The process of slowing down the speed of light in a material is of a quantum nature and is associated with the acts of absorption and emission of photons by atoms of the substance.
Changes in the composition of air (for example, an increase in the content of water vapor in it) and changes in temperature lead to significant changes in the refractive index. A striking example is the mirage effect in the desert, which occurs due to differences in the refractive indices of air layers with different temperatures.
Glass-air interface
Glass is a much denser medium than air. Its absolute refractive index ranges from 1.5 to 1.66, depending on the type of glass. If we take the average value of 1.55, then the refraction of the beam at the air-glass interface can be calculated using the formula:
sin(θ 1)/sin(θ 2) = n 2 /n 1 = n 21 = 1.55.
The value n 21 is called the relative refractive index of air - glass. If the beam comes out of the glass into the air, then the following formula should be used:
sin(θ 1)/sin(θ 2) = n 2 /n 1 = n 21 = 1/1.55 = 0.645.
If the angle of the refracted ray in the latter case is equal to 90 o, then the corresponding one is called critical. For the glass-air boundary it is equal to:
θ 1 = arcsin(0.645) = 40.17 o.
If the beam falls on the glass-air boundary with larger angles than 40.17 o, then it will be reflected completely back into the glass. This phenomenon is called “total internal reflection”.
The critical angle exists only when the beam moves from a dense medium (from glass to air, but not vice versa).
In your 8th grade physics course, you learned about the phenomenon of light refraction. Now you know that light is electromagnetic waves of a certain frequency range. Based on knowledge about the nature of light, you can understand the physical cause of refraction and explain many other light phenomena associated with it.
Rice. 141. Passing from one medium to another, the ray is refracted, i.e. changes the direction of propagation
According to the law of light refraction (Fig. 141):
- the incident, refracted and perpendicular rays drawn to the interface between two media at the point of incidence of the ray lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for these two media
where n 21 is the relative refractive index of the second medium relative to the first.
If the beam passes into any medium from vacuum, then
where n is the absolute refractive index (or simply refractive index) of the second medium. In this case, the first “medium” is vacuum, the absolute value of which is taken as unity.
The law of light refraction was experimentally discovered by the Dutch scientist Willebord Snellius in 1621. The law was formulated in a treatise on optics, which was found in the scientist’s papers after his death.
After Snell's discovery, several scientists hypothesized that the refraction of light is due to a change in its speed when passing through the boundary of two media. The validity of this hypothesis was confirmed by theoretical proofs carried out independently by the French mathematician Pierre Fermat (in 1662) and the Dutch physicist Christiaan Huygens (in 1690). They came to the same result in different ways, proving that
- the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for these two media, equal to the ratio of the speeds of light in these media:
(3)
From equation (3) it follows that if the angle of refraction β is less than the angle of incidence a, then light of a given frequency in the second medium propagates more slowly than in the first, i.e. V 2 The relationship between the quantities included in equation (3) served as a compelling reason for the emergence of another formulation for the definition of the relative refractive index: n 21 = v 1 / v 2 (4) Let a beam of light pass from a vacuum into some medium. Replacing v1 in equation (4) with the speed of light in a vacuum c, and v 2 with the speed of light in a medium v, we obtain equation (5), which is the definition of the absolute refractive index: According to equations (4) and (5), n 21 shows how many times the speed of light changes when it passes from one medium to another, and n - when passing from vacuum to medium. This is the physical meaning of refractive indices. The value of the absolute refractive index n of any substance is greater than one (this is confirmed by the data contained in the tables of physical reference books). Then, according to equation (5), c/v > 1 and c > v, i.e., the speed of light in any substance is less than the speed of light in vacuum. Without giving strict justifications (they are complex and cumbersome), we note that the reason for the decrease in the speed of light during its transition from vacuum to matter is the interaction of the light wave with atoms and molecules of matter. The greater the optical density of a substance, the stronger this interaction, the lower the speed of light and the higher the refractive index. Thus, the speed of light in a medium and the absolute refractive index are determined by the properties of this medium. Based on the numerical values of the refractive indices of substances, their optical densities can be compared. For example, the refractive index of different types of glass ranges from 1.470 to 2.040, and the refractive index of water is 1.333. This means that glass is a medium optically denser than water. Let us turn to Figure 142, with the help of which we can explain why at the boundary of two media, with a change in speed, the direction of propagation of the light wave also changes. Rice. 142. When light waves pass from air to water, the speed of light decreases, the front of the wave, and with it its speed, changes direction The figure shows a light wave passing from air into water and incident on the interface between these media at an angle a. In air, light travels at a speed v 1, and in water at a lower speed v 2. Point A of the wave reaches the boundary first. Over a period of time Δt, point B, moving in the air with the same speed v 1, will reach point B." During the same time, point A, moving in water with a lower speed v 2, will travel a shorter distance, reaching only point A." In this case, the so-called front of the AB wave in the water will be rotated at a certain angle relative to the front of the AB wave in the air. And the velocity vector (which is always perpendicular to the front of the wave and coincides with the direction of its propagation) rotates, approaching the straight line OO", perpendicular to the interface between the media. In this case, the angle of refraction β turns out to be less than the angle of incidence α. This is how the refraction of light occurs. It is also clear from the figure that when moving to another medium and rotating the wave front, the wavelength also changes: when moving to an optically denser medium, the speed decreases, the wavelength also decreases (λ 2< λ 1). Это согласуется и с известной вам формулой λ = V/v, из которой следует, что при неизменной частоте v (которая не зависит от плотности среды и поэтому не меняется при переходе луча из одной среды в другую) уменьшение скорости распространения волны сопровождается пропорциональным уменьшением длины волны. Laboratory work Light refraction. Measuring the refractive index of a liquid using a refractometer Goal of the work: deepening understanding of the phenomenon of light refraction; study of methods for measuring the refractive index of liquid media; studying the principle of working with a refractometer. Equipment: refractometer, sodium chloride solutions, pipette, soft cloth for wiping optical parts of instruments. Theory Laws of reflection and refraction of light. Refractive index. At the interface between the media, light changes the direction of its propagation. Part of the light energy returns to the first medium, i.e. light is reflected. If the second medium is transparent, then part of the light, under certain conditions, passes through the interface between the media, usually changing the direction of propagation. This phenomenon is called refraction of light
(Fig. 1). Rice. 1. Reflection and refraction of light at a flat interface between two media. The direction of reflected and refracted rays when light passes through a flat interface between two transparent media is determined by the laws of reflection and refraction of light. Law of light reflection. The reflected ray lies in the same plane as the incident ray and the normal restored to the plane of separation of the media at the point of incidence. Angle of incidence The law of light refraction. The refracted ray lies in the same plane as the incident ray and the normal restored to the plane of separation of the media at the point of incidence. Angle of incidence sine ratio α
to the sine of the angle of refraction β
there is a constant value for these two media, called the relative refractive index of the second medium in relation to the first: Relative refractive index
two media is equal to the ratio of the speed of light in the first medium v 1 to the speed of light in the second medium v 2: If light comes from a vacuum into a medium, then the refractive index of the medium relative to the vacuum is called the absolute refractive index of this medium and is equal to the ratio of the speed of light in vacuum With to the speed of light in a given medium: Absolute refractive indices are always greater than unity; for air n taken as one. The relative refractive index of two media can be expressed in terms of their absolute indices n 1
And n 2
: Determination of the refractive index of a liquid To quickly and conveniently determine the refractive index of liquids, there are special optical instruments - refractometers, the main part of which are two prisms (Fig. 2): auxiliary Etc. 1 and measuring Pr.2. The liquid to be tested is poured into the gap between the prisms. When measuring indicators, two methods can be used: the grazing beam method (for transparent liquids) and the total internal reflection method (for dark, turbid and colored solutions). In this work, the first of them is used. In the grazing beam method, light from an external source passes through the face AB prisms Project 1, dissipates on its matte surface AC and then penetrates through the layer of the liquid under study into the prism Pr.2. The matte surface becomes a source of rays in all directions, so it can be observed through the edge EF
prisms Pr.2. However, the edge AC can be seen through EF only at an angle greater than a certain minimum angle i. The magnitude of this angle is uniquely related to the refractive index of the liquid located between the prisms, which is the main idea behind the design of the refractometer. Consider the passage of light through the face EF lower measuring prism Pr.2. As can be seen from Fig. 2, applying the law of light refraction twice, we can obtain two relationships: Solving this system of equations, it is easy to come to the conclusion that the refractive index of the liquid depends on four quantities: Q,
r,
r 1
And i. However, not all of them are independent. For example, r+
s=
R
,
(4) Where R
-
refractive angle of prism Project 2. In addition, by setting the angle Q the maximum value is 90°, from equation (1) we obtain: But the maximum angle value r
,
as can be seen from Fig. 2 and relations (3) and (4), the minimum angle values correspond i
And r 1 ,
those. i min
And r min .
Thus, the refractive index of a liquid for the case of “grazing” rays is associated only with the angle i. In this case, there is a minimum angle value i,
when the edge AC is still visible, that is, in the field of view it appears mirror-white. For smaller viewing angles, the edge is not visible, and in the field of view this place appears black. Since the telescope of the device captures a relatively wide angular zone, light and black areas are simultaneously observed in the field of view, the boundary between which corresponds to the minimum observation angle and is uniquely related to the refractive index of the liquid. Using the final calculation formula: (its conclusion is omitted) and a number of liquids with known refractive indices, you can calibrate the device, i.e., establish a unique correspondence between the refractive indices of liquids and angles i min .
All formulas given are derived for rays of one particular wavelength. Light of different wavelengths will be refracted taking into account the dispersion of the prism. Thus, when the prism is illuminated with white light, the interface will be blurred and colored in different colors due to dispersion. Therefore, every refractometer has a compensator that eliminates the result of dispersion. It may consist of one or two direct vision prisms - Amici prisms. Each Amici prism consists of three glass prisms with different refractive indices and different dispersion, for example, the outer prisms are made of crown glass, and the middle one is made of flint glass (crown glass and flint glass are types of glass). By rotating the compensator prism using a special device, a sharp, colorless image of the interface is achieved, the position of which corresponds to the refractive index value for the yellow sodium line λ
=5893 Å (the prisms are designed so that rays with a wavelength of 5893 Å do not experience deflection). The rays passing through the compensator enter the lens of the telescope, then pass through the reversing prism through the eyepiece of the telescope into the eye of the observer. The schematic path of the rays is shown in Fig. 3. The refractometer scale is calibrated in the values of the refractive index and the concentration of the sucrose solution in water and is located in the focal plane of the eyepiece. experimental part Task 1. Checking the refractometer. Direct the light using a mirror onto the refractometer's auxiliary prism. With the auxiliary prism raised, pipette a few drops of distilled water onto the measuring prism. By lowering the auxiliary prism, achieve the best illumination of the field of view and set the eyepiece so that the crosshair and refractive index scale are clearly visible. By rotating the camera of the measuring prism, you get the boundary of light and shadow in the field of view. Rotate the compensator head until the color of the border between light and shadow is eliminated. Align the light and shadow boundary with the crosshair point and measure the refractive index of water n change .
If the refractometer is working properly, then for distilled water the value should be n 0
=
1.333, if the readings differ from this value, an amendment must be determined Δn=
n change -
1.333, which should then be taken into account when further working with the refractometer. Please make corrections to Table 1. Table 1. n 0
n change Δ
n N 2
ABOUT Task 2. Determination of the refractive index of a liquid. Determine the refractive indices of solutions of known concentrations, taking into account the found correction. Table 2. C, vol. % n change n ist Plot a graph of the dependence of the refractive index of table salt solutions on the concentration based on the results obtained. Draw a conclusion about the dependence of n on C; draw conclusions about the accuracy of measurements using a refractometer. Take a salt solution of unknown concentration WITH x ,
determine its refractive index and use the graph to find the concentration of the solution. Clean the work area and carefully wipe the refractometer prisms with a damp, clean cloth. Control questions Reflection and refraction of light. Absolute and relative refractive indices of the medium. The principle of operation of a refractometer. Sliding beam method. Schematic path of rays in a prism. Why are compensator prisms needed? Propagation, reflection and refraction of light The nature of light is electromagnetic. One proof of this is the coincidence of the speeds of electromagnetic waves and light in a vacuum. In a homogeneous medium, light travels in a straight line. This statement is called the law of rectilinear propagation of light. An experimental proof of this law is the sharp shadows produced by point light sources. The geometric line indicating the direction of propagation of light is called a light ray. In an isotropic medium, light rays are directed perpendicular to the wave front. The geometric location of points in the medium oscillating in the same phase is called the wave surface, and the set of points to which the oscillation has reached at a given point in time is called the wave front. Depending on the type of wave front, plane and spherical waves are distinguished. To explain the process of light propagation, the general principle of wave theory about the movement of a wave front in space, proposed by the Dutch physicist H. Huygens, is used. According to Huygens' principle, each point in the medium to which light excitation reaches is the center of spherical secondary waves, which also propagate at the speed of light. The surface surrounding the fronts of these secondary waves gives the position of the front of the actually propagating wave at that moment in time. It is necessary to distinguish between light beams and light rays. A light beam is a part of a light wave that carries light energy in a given direction. When replacing a light beam with a light beam describing it, the latter must be taken to coincide with the axis of a sufficiently narrow, but at the same time having a finite width (the cross-sectional dimensions are much larger than the wavelength) light beam. There are divergent, converging and quasi-parallel light beams. The terms beam of light rays or simply light rays are often used, meaning a set of light rays that describe a real light beam. The speed of light in vacuum c = 3 108 m/s is a universal constant and does not depend on frequency. For the first time, the speed of light was experimentally determined by the astronomical method by the Danish scientist O. Roemer. More accurately, the speed of light was measured by A. Michelson. In matter the speed of light is less than in vacuum. The ratio of the speed of light in a vacuum to its speed in a given medium is called the absolute refractive index of the medium: where c is the speed of light in a vacuum, v is the speed of light in a given medium. The absolute refractive indices of all substances are greater than unity. When light propagates through a medium, it is absorbed and scattered, and at the interface between the media it is reflected and refracted. The law of light reflection: the incident beam, the reflected beam and the perpendicular to the interface between two media, restored at the point of incidence of the beam, lie in the same plane; the angle of reflection g is equal to the angle of incidence a (Fig. 1). The law of light refraction: the incident ray, the refracted ray and the perpendicular to the interface between two media, restored at the point of incidence of the ray, lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction for a given frequency of light is a constant value called the relative refractive index of the second medium relative to the first: The experimentally established law of light refraction is explained on the basis of Huygens' principle. According to wave concepts, refraction is a consequence of changes in the speed of wave propagation when passing from one medium to another, and the physical meaning of the relative refractive index is the ratio of the speed of propagation of waves in the first medium v1 to the speed of their propagation in the second medium For media with absolute refractive indices n1 and n2, the relative refractive index of the second medium relative to the first is equal to the ratio of the absolute refractive index of the second medium to the absolute refractive index of the first medium: The medium that has a higher refractive index is called optically denser; the speed of light propagation in it is lower. If light passes from an optically denser medium to an optically less dense one, then at a certain angle of incidence a0 the angle of refraction should become equal to p/2. The intensity of the refracted beam in this case becomes equal to zero. Light falling on the interface between two media is completely reflected from it. The angle of incidence a0 at which total internal reflection of light occurs is called the limiting angle of total internal reflection. At all angles of incidence equal to and greater than a0, total reflection of light occurs. The value of the limiting angle is found from the relation 2 The refractive index of a substance is a value equal to the ratio of the phase speeds of light (electromagnetic waves) in a vacuum and in a given medium. They also talk about the refractive index for any other waves, for example, sound The refractive index depends on the properties of the substance and the wavelength of the radiation; for some substances, the refractive index changes quite strongly when the frequency of electromagnetic waves changes from low frequencies to optical and beyond, and can also change even more sharply in certain areas of the frequency scale. The default usually refers to the optical range or the range determined by the context. There are optically anisotropic substances in which the refractive index depends on the direction and polarization of light. Such substances are quite common, in particular, they are all crystals with a fairly low symmetry of the crystal lattice, as well as substances subjected to mechanical deformation. The refractive index can be expressed as the root of the product of the magnetic and dielectric constants of the medium (it should be taken into account that the values of magnetic permeability and absolute dielectric constant for the frequency range of interest - for example, optical - can differ very much from the static value of these values). To measure the refractive index, manual and automatic refractometers are used. When a refractometer is used to determine the concentration of sugar in an aqueous solution, the device is called a saccharimeter. The ratio of the sine of the angle of incidence () of the beam to the sine of the angle of refraction () when the beam passes from medium A to medium B is called the relative refractive index for this pair of media. The quantity n is the relative refractive index of medium B in relation to medium A, аn" = 1/n is the relative refractive index of medium A in relation to medium B. This value, other things being equal, is usually less than unity when a beam passes from a more dense medium to a less dense medium, and more than unity when a beam passes from a less dense medium to a denser medium (for example, from a gas or from a vacuum to a liquid or solid ). There are exceptions to this rule, and therefore it is customary to call a medium optically more or less dense than another (not to be confused with optical density as a measure of the opacity of a medium). A ray falling from airless space onto the surface of some medium B is refracted more strongly than when falling on it from another medium A; The refractive index of a ray incident on a medium from airless space is called its absolute refractive index or simply the refractive index of a given medium; this is the refractive index, the definition of which is given at the beginning of the article. The refractive index of any gas, including air, under normal conditions is much less than the refractive index of liquids or solids, therefore, approximately (and with relatively good accuracy) the absolute refractive index can be judged by the refractive index relative to air. Rice. 3. Operating principle of an interference refractometer. The light beam is divided so that its two parts pass through cuvettes of length l filled with substances with different refractive indices. At the exit from the cuvettes, the rays acquire a certain path difference and, being brought together, give on the screen a picture of interference maxima and minima with k orders (shown schematically on the right). Refractive index difference Dn=n2 –n1 =kl/2, where l is the wavelength of light. Refractometers are instruments used to measure the refractive index of substances. The operating principle of a refractometer is based on the phenomenon of total reflection. If a scattered beam of light falls on the interface between two media with refractive indices and, from a more optically dense medium, then, starting from a certain angle of incidence, the rays do not enter the second medium, but are completely reflected from the interface in the first medium. This angle is called the limiting angle of total reflection. Figure 1 shows the behavior of rays when falling into a certain current of this surface. The beam comes at an extreme angle. From the law of refraction we can determine: , (since). The magnitude of the limiting angle depends on the relative refractive index of the two media. If the rays reflected from the surface are directed to a collecting lens, then in the focal plane of the lens you can see the boundary of light and penumbra, and the position of this boundary depends on the value of the limiting angle, and therefore on the refractive index. A change in the refractive index of one of the media entails a change in the position of the interface. The interface between light and shadow can serve as an indicator when determining the refractive index, which is used in refractometers. This method of determining the refractive index is called the total reflection method In addition to the total reflection method, refractometers use the grazing beam method. In this method, a scattered beam of light hits the boundary from a less optically dense medium at all possible angles (Fig. 2). The ray sliding along the surface () corresponds to the limiting angle of refraction (the ray in Fig. 2). If we place a lens in the path of the rays () refracted on the surface, then in the focal plane of the lens we will also see a sharp boundary between light and shadow. Rice. 2 Since the conditions determining the value of the limiting angle are the same in both methods, the position of the interface is the same. Both methods are equivalent, but the total reflection method allows you to measure the refractive index of opaque substances Path of rays in a triangular prism Figure 9 shows a cross section of a glass prism with a plane perpendicular to its side edges. The beam in the prism is deflected towards the base, refracting at the edges OA and 0B. The angle j between these faces is called the refractive angle of the prism. The angle of deflection of the beam depends on the refractive angle of the prismj, the refractive index n of the prism material and the angle of incidencea. It can be calculated using the law of refraction (1.4). The refractometer uses a white light source 3. Due to dispersion, when light passes through prisms 1 and 2, the boundary of light and shadow turns out to be colored. To avoid this, a compensator 4 is placed in front of the telescope lens. It consists of two identical prisms, each of which is glued together from three prisms with different refractive indexes. Prisms are selected so that a monochromatic beam with a wavelength
= 589.3 µm. (sodium yellow line wavelength) was not tested after passing the deflection compensator. Rays with other wavelengths are deflected by prisms in different directions. By moving the compensator prisms using a special handle, we ensure that the boundary between light and darkness becomes as clear as possible. The light rays, having passed the compensator, enter the lens 6 of the telescope. The image of the light-shadow interface is viewed through the eyepiece 7 of the telescope. At the same time, scale 8 is viewed through the eyepiece. Since the limiting angle of refraction and the limiting angle of total reflection depend on the refractive index of the liquid, the values of this refractive index are immediately marked on the refractometer scale. The optical system of the refractometer also contains a rotating prism 5. It allows you to position the axis of the telescope perpendicular to prisms 1 and 2, which makes observation more convenient. Ticket 75. Law of Light Reflection: the incident and reflected rays, as well as the perpendicular to the interface between the two media, reconstructed at the point of incidence of the ray, lie in the same plane (plane of incidence). The angle of reflection γ is equal to the angle of incidence α. Law of light refraction: the incident and refracted rays, as well as the perpendicular to the interface between the two media, reconstructed at the point of incidence of the ray, lie in the same plane. The ratio of the sine of the angle of incidence α to the sine of the angle of refraction β is a constant value for two given media: The laws of reflection and refraction are explained in wave physics. According to wave concepts, refraction is a consequence of changes in the speed of propagation of waves when passing from one medium to another. Physical meaning of the refractive index is the ratio of the speed of propagation of waves in the first medium υ 1 to the speed of their propagation in the second medium υ 2: Figure 3.1.1 illustrates the laws of reflection and refraction of light. A medium with a lower absolute refractive index is called optically less dense. When light passes from an optically denser medium to an optically less dense medium n 2< n 1
(например, из стекла в воздух) можно
наблюдать total reflection phenomenon, that is, the disappearance of the refracted ray. This phenomenon is observed at angles of incidence exceeding a certain critical angle α pr, which is called limiting angle of total internal reflection(see Fig. 3.1.2). For the angle of incidence α = α pr sin β = 1; value sin α pr = n 2 / n 1< 1. If the second medium is air (n 2 ≈ 1), then it is convenient to rewrite the formula in the form The phenomenon of total internal reflection is used in many optical devices. The most interesting and practically important application is the creation of optical fibers, which are thin (from several micrometers to millimeters) arbitrarily curved threads made of optically transparent material (glass, quartz). Light incident on the end of the light guide can travel along it over long distances due to total internal reflection from the side surfaces (Figure 3.1.3). The scientific and technical direction involved in the development and application of optical light guides is called fiber optics. Dispersion of light (decomposition of light)- this is a phenomenon caused by the dependence of the absolute refractive index of a substance on the frequency (or wavelength) of light (frequency dispersion), or, the same thing, the dependence of the phase speed of light in a substance on the wavelength (or frequency). It was discovered experimentally by Newton around 1672, although theoretically quite well explained much later. Spatial dispersion is called the dependence of the dielectric constant tensor of the medium on the wave vector. This dependence causes a number of phenomena called spatial polarization effects. One of the most clear examples of dispersion - white light decomposition when passing through a prism (Newton's experiment). The essence of the dispersion phenomenon is the difference in the speed of propagation of light rays of different wavelengths in a transparent substance - an optical medium (while in a vacuum the speed of light is always the same, regardless of wavelength and therefore color). Typically, the higher the frequency of a light wave, the higher the refractive index of the medium for it and the lower the speed of the wave in the medium: Newton's experiments Experiment on the decomposition of white light into a spectrum: Conclusions:- a prism decomposes light - white light is complex (composite) - violet rays are refracted more strongly than red ones. The color of a light beam is determined by its vibration frequency. When moving from one medium to another, the speed of light and wavelength change, but the frequency that determines the color remains constant. The boundaries of the ranges of white light and its components are usually characterized by their wavelengths in vacuum. White light is a collection of waves with lengths from 380 to 760 nm. Ticket 77. Absorption of light. Bouguer's law The absorption of light in a substance is associated with the conversion of the energy of the electromagnetic field of the wave into the thermal energy of the substance (or into the energy of secondary photoluminescent radiation). The law of light absorption (Bouguer's law) has the form: I=I 0
exp(-
x),(1) Where I 0
,
I-light intensity at the input (x=0) and leaving the layer of medium thickness X,
-
absorption coefficient, it depends on .
For dielectrics =10
-1
10
-5
m -1
, for metals =10
5
10
7
m -1
,
Therefore, metals are opaque to light. Dependency (
)
explains the color of absorbing bodies. For example, glass that absorbs red light poorly will appear red when illuminated with white light. Scattering of light. Rayleigh's law Diffraction of light can occur in an optically inhomogeneous medium, for example in a turbid environment (smoke, fog, dusty air, etc.). By diffracting on inhomogeneities of the medium, light waves create a diffraction pattern characterized by a fairly uniform distribution of intensity in all directions. This diffraction by small inhomogeneities is called scattering of light. This phenomenon is observed when a narrow beam of sunlight passes through dusty air, scatters on dust particles and becomes visible. If the sizes of inhomogeneities are small compared to the wavelength (no more than 0,1
), then the intensity of the scattered light turns out to be inversely proportional to the fourth power of the wavelength, i.e. I diss ~
1/
4
,
(2) this dependence is called Rayleigh's law. Light scattering is also observed in clean media that do not contain foreign particles. For example, it can occur on fluctuations (random deviations) of density, anisotropy or concentration. This type of scattering is called molecular scattering. It explains, for example, the blue color of the sky. Indeed, according to (2), blue and blue rays are scattered more strongly than red and yellow ones, because have a shorter wavelength, thereby causing the blue color of the sky. Ticket 78. Polarization of light- a set of wave optics phenomena in which the transverse nature of electromagnetic light waves is manifested. Transverse wave- particles of the medium oscillate in directions perpendicular to the direction of propagation of the wave ( Fig.1). Fig.1
Transverse wave Electromagnetic light wave plane polarized(linear polarization), if the directions of oscillation of vectors E and B are strictly fixed and lie in certain planes ( Fig.1). A plane polarized light wave is called plane polarized(linearly polarized) light. Unpolarized(natural) wave - an electromagnetic light wave in which the directions of oscillation of the vectors E and B in this wave can lie in any planes perpendicular to the velocity vector v. Unpolarized light- light waves in which the directions of oscillations of the vectors E and B change chaotically so that all directions of oscillations in planes perpendicular to the ray of wave propagation are equally probable ( Fig.2). Fig.2
Unpolarized light Polarized waves- in which the directions of the vectors E and B remain unchanged in space or change according to a certain law. Radiation in which the direction of vector E changes chaotically - unpolarized. An example of such radiation is thermal radiation (chaotically distributed atoms and electrons). Plane of polarization- this is a plane perpendicular to the direction of oscillations of the vector E. The main mechanism for the occurrence of polarized radiation is the scattering of radiation by electrons, atoms, molecules, and dust particles. 1.2. Types of polarization There are three types of polarization. Let's give them definitions. 1. Linear
Occurs if the electric vector E maintains its position in space. It seems to highlight the plane in which vector E oscillates. 2. Circular
This is polarization that occurs when the electric vector E rotates around the direction of wave propagation with an angular velocity equal to the angular frequency of the wave, while maintaining its absolute value. This polarization characterizes the direction of rotation of the vector E in a plane perpendicular to the line of sight. An example is cyclotron radiation (a system of electrons rotating in a magnetic field). 3. Elliptical
It occurs when the magnitude of the electric vector E changes so that it describes an ellipse (rotation of the vector E). Elliptical and circular polarization can be right-handed (vector E rotates clockwise when looking towards the propagating wave) and left-handed (vector E rotates counter-clockwise when looking towards the propagating wave). In reality, it occurs most often partial polarization
(partially polarized electromagnetic waves). Quantitatively, it is characterized by a certain quantity called degree of polarization
R, which is defined as: P = (Imax - Imin) / (Imax + Imin) Where Imax,Immin- the highest and lowest density of electromagnetic energy flux through the analyzer (Polaroid, Nicolas prism...). In practice, radiation polarization is often described by Stokes parameters (they determine radiation fluxes with a given polarization direction). Ticket 79. If natural light falls on the interface between two dielectrics (for example, air and glass), then part of it is reflected, and part of it is refracted and spreads in the second medium. By installing an analyzer (for example, tourmaline) in the path of the reflected and refracted rays, we make sure that the reflected and refracted rays are partially polarized: when the analyzer is rotated around the rays, the light intensity periodically increases and weakens (complete quenching is not observed!). Further studies showed that in the reflected beam, vibrations perpendicular to the plane of incidence predominate (they are indicated by dots in Fig. 275), while in the refracted beam, vibrations parallel to the plane of incidence (depicted by arrows) predominate. The degree of polarization (the degree to which light waves are separated with a certain orientation of the electric (and magnetic) vector) depends on the angle of incidence of the rays and the refractive index. Scottish physicist D. Brewster(1781-1868) installed law, according to which at the angle of incidence i B (Brewster angle), determined by the relation (n 21 - refractive index of the second medium relative to the first), the reflected beam is plane polarized(contains only vibrations perpendicular to the plane of incidence) (Fig. 276). The refracted ray at the angle of incidencei B polarized to the maximum, but not completely. If light strikes an interface at the Brewster angle, then the reflected and refracted rays mutually perpendicular(tg i B = sin i B/cos i B, n 21 =
sin i B /
sin i 2
(i 2
-
angle of refraction), whence cos i B=sin i 2). Hence, i B +
i 2
=
/2, but i’
B= i B (law of reflection), therefore i’
B+ i 2
=
/2. The degree of polarization of reflected and refracted light at different angles of incidence can be calculated from Maxwell’s equations, if we take into account the boundary conditions for the electromagnetic field at the interface between two isotropic dielectrics (the so-called Fresnel formulas). The degree of polarization of refracted light can be significantly increased (by multiple refraction, provided that the light is incident each time on the interface at the Brewster angle). If, for example, for glass ( n= 1.53) the degree of polarization of the refracted beam is 15%, then after refraction into 8-10 glass plates superimposed on each other, the light emerging from such a system will be almost completely polarized. Such a collection of plates is called foot. The foot can be used to analyze polarized light both during its reflection and during its refraction. Ticket 79 (for Spur) As experience shows, during the refraction and reflection of light, the refracted and reflected light turns out to be polarized, and the reflection. light can be completely polarized at a certain angle of incidence, but incidentally. light is always partially polarized. Based on Frinell's formulas, it can be shown that reflection. Light is polarized in a plane perpendicular to the plane of incidence and refracted. the light is polarized in a plane parallel to the plane of incidence. The angle of incidence at which the reflection the light is completely polarized is called the Brewster angle. The Brewster angle is determined from Brewster's law: - Brewster's law. In this case, the angle between the reflections. and refraction. rays will be equal. For an air-glass system, the Brewster angle is equal. To obtain good polarization, i.e. , when refracting light, many edible surfaces are used, which are called Stoletov’s Stop. Ticket 80. Experience shows that when light interacts with matter, the main effect (physiological, photochemical, photoelectric, etc.) is caused by oscillations of the vector, which in this regard is sometimes called the light vector. Therefore, to describe the patterns of light polarization, the behavior of the vector is monitored. The plane formed by the vectors and is called the plane of polarization. If vector oscillations occur in one fixed plane, then such light (ray) is called linearly polarized. It is conventionally designated as follows. If the beam is polarized in a perpendicular plane (in the plane xoz, see fig. 2 in the second lecture), then it is designated. Natural light (from ordinary sources, the sun) consists of waves that have different, chaotically distributed planes of polarization (see Fig. 3). If, as the wave propagates, the vector rotates and the end of the vector describes a circle, then such light is called circularly polarized, and the polarization is called circular or circular (right or left). There is also elliptical polarization. There are optical devices (films, plates, etc.) - polarizers, which extract linearly polarized light or partially polarized light from natural light. The polarizer (or analyzer) plane is the plane of polarization of light transmitted by the polarizer (or analyzer). Let linearly polarized light with amplitude fall on a polarizer (or analyzer) E 0 . The amplitude of the transmitted light will be equal to E=E 0 cos j, and intensity I=I 0 cos 2 j. This formula expresses Malus's law: The intensity of linearly polarized light passing through the analyzer is proportional to the square of the cosine of the angle j between the plane of oscillation of the incident light and the plane of the analyzer. Ticket 80 (for spur) Polarizers are devices that make it possible to obtain polarized light. Analyzers are devices that can be used to analyze whether light is polarized or not. Structurally, a polarizer and an analyzer are one and the same. Zn Malus. Let intensity light fall on the polarizer, if the light is natural -th then all directions of vector E are equally probable. Each vector can be decomposed into two mutually perpendicular components: one of which is parallel to the plane of polarization of the polarizer, and the other is perpendicular to it. Obviously, the intensity of the light emerging from the polarizer will be equal. Let us denote the intensity of the light emerging from the polarizer by (). If an analyzer is placed on the path of the polarized light, the main plane of which makes an angle with the main plane of the polarizer, then the intensity of the light emerging from the analyzer is determined by the law. Ticket 81. While studying the glow of a solution of uranium salts under the influence of radium rays, the Soviet physicist P. A. Cherenkov drew attention to the fact that the water itself also glows, in which there are no uranium salts. It turned out that when rays (see Gamma radiation) are passed through pure liquids, they all begin to glow. S. I. Vavilov, under whose leadership P. A. Cherenkov worked, hypothesized that the glow was associated with the movement of electrons knocked out of atoms by radium quanta. Indeed, the glow strongly depended on the direction of the magnetic field in the liquid (this suggested that it was caused by the movement of electrons). But why do electrons moving in a liquid emit light? The correct answer to this question was given in 1937 by Soviet physicists I.E. Tamm and I.M. Frank. An electron, moving in a substance, interacts with the atoms surrounding it. Under the influence of its electric field, atomic electrons and nuclei are displaced in opposite directions - the medium is polarized. Polarized and then returning to their original state, the atoms of the medium located along the electron trajectory emit electromagnetic light waves. If the speed of the electron v is less than the speed of light propagation in the medium (the refractive index), then the electromagnetic field will overtake the electron, and the substance will have time to polarize in space ahead of the electron. The polarization of the medium in front of and behind the electron is opposite in direction, and the radiations of oppositely polarized atoms, “added”, “quench” each other. When atoms that have not yet been reached by an electron do not have time to polarize, and radiation appears directed along a narrow conical layer with an apex coinciding with the moving electron and an angle at the apex c. The appearance of the light "cone" and the radiation condition can be obtained from the general principles of wave propagation. Rice. 1. Mechanism of wavefront formation Let the electron move along the axis OE (see Fig. 1) of a very narrow empty channel in a homogeneous transparent substance with a refractive index (the empty channel is needed so that collisions of the electron with atoms are not taken into account in the theoretical consideration). Any point on the OE line successively occupied by an electron will be the center of light emission. Waves emanating from successive points O, D, E interfere with each other and are amplified if the phase difference between them is zero (see Interference). This condition is satisfied for a direction that makes an angle of 0 with the trajectory of the electron. Angle 0 is determined by the relation:. Indeed, let us consider two waves emitted in a direction at an angle of 0 to the electron velocity from two points of the trajectory - point O and point D, separated by a distance . At point B, lying on line BE, perpendicular to OB, the first wave at - after time To point F, lying on line BE, a wave emitted from the point will arrive at the moment of time after the wave is emitted from point O. These two waves will be in phase, i.e. the straight line will be a wave front if these times are equal:. That gives the condition of equality of times. In all directions for which, the light will be extinguished due to the interference of waves emitted from sections of the trajectory separated by a distance D. The value of D is determined by the obvious equation, where T is the period of light oscillations. This equation always has a solution if. If , then the direction in which the emitted waves, when interfering, are amplified, does not exist and cannot be greater than 1. Rice. 2. Distribution of sound waves and the formation of a shock wave during body movement Radiation is observed only if . Experimentally, electrons fly in a finite solid angle, with some spread in speed, and as a result, radiation propagates in a conical layer near the main direction determined by the angle. In our consideration, we neglected the electron slowdown. This is quite acceptable, since the losses due to Vavilov-Cerenkov radiation are small and, to a first approximation, we can assume that the energy lost by the electron does not affect its speed and it moves uniformly. This is the fundamental difference and unusualness of the Vavilov-Cherenkov radiation. Typically, charges emit while experiencing significant acceleration. An electron outpacing its light is similar to an airplane flying at a speed greater than the speed of sound. In this case, a conical shock sound wave also propagates in front of the aircraft (see Fig. 2). This article reveals the essence of such an optics concept as refractive index. Formulas for obtaining this quantity are given, and a brief overview of the application of the phenomenon of electromagnetic wave refraction is given. At the dawn of civilization, people asked the question: how does the eye see? It has been suggested that a person emits rays that feel surrounding objects, or, conversely, all things emit such rays. The answer to this question was given in the seventeenth century. It is found in optics and is related to what refractive index is. Reflecting from various opaque surfaces and refracting at the border with transparent ones, light gives a person the opportunity to see. Our planet is shrouded in the light of the Sun. And it is precisely with the wave nature of photons that such a concept as the absolute refractive index is associated. Propagating in a vacuum, a photon encounters no obstacles. On the planet, light encounters many different denser environments: the atmosphere (a mixture of gases), water, crystals. Being an electromagnetic wave, photons of light have one phase speed in a vacuum (denoted c), and in the environment - another (denoted v). The ratio of the first and second is what is called the absolute refractive index. The formula looks like this: n = c / v. It is worth defining the phase velocity of the electromagnetic medium. Otherwise, understand what the refractive index is n, it is forbidden. A photon of light is a wave. This means that it can be represented as a packet of energy that oscillates (imagine a segment of a sine wave). The phase is the segment of the sinusoid that the wave travels at a given moment in time (remember that this is important for understanding such a quantity as the refractive index). For example, the phase may be the maximum of a sinusoid or some segment of its slope. The phase speed of a wave is the speed at which that particular phase moves. As the definition of the refractive index explains, these values differ for a vacuum and for a medium. Moreover, each environment has its own value of this quantity. Any transparent compound, whatever its composition, has a refractive index that is different from all other substances. It was already shown above that the absolute value is measured relative to the vacuum. However, this is difficult on our planet: light more often hits the boundary of air and water or quartz and spinel. For each of these media, as mentioned above, the refractive index is different. In air, a photon of light travels along one direction and has one phase speed (v 1), but when it gets into water, it changes the direction of propagation and phase speed (v 2). However, both of these directions lie in the same plane. This is very important for understanding how the image of the surrounding world is formed on the retina of the eye or on the matrix of the camera. The ratio of the two absolute values gives the relative refractive index. The formula looks like this: n 12 = v 1 / v 2. But what if light, on the contrary, comes out of the water and enters the air? Then this value will be determined by the formula n 21 = v 2 / v 1. When multiplying the relative refractive indices, we obtain n 21 * n 12 = (v 2 * v 1) / (v 1 * v 2) = 1. This relationship is valid for any pair of media. The relative refractive index can be found from the sines of the angles of incidence and refraction n 12 = sin Ɵ 1 / sin Ɵ 2. Do not forget that angles are measured from the normal to the surface. A normal is a line perpendicular to the surface. That is, if the problem is given an angle α
fall relative to the surface itself, then we must calculate the sine of (90 - α). On a calm sunny day, reflections play on the bottom of the lake. Dark blue ice covers the rock. A diamond scatters thousands of sparks on a woman’s hand. These phenomena are a consequence of the fact that all boundaries of transparent media have a relative refractive index. In addition to aesthetic pleasure, this phenomenon can also be used for practical applications. Here are examples: However, the Sun supplies us with photons not only in the visible spectrum. Infrared, ultraviolet, and x-ray ranges are not perceived by human vision, but they affect our lives. IR rays warm us, UV photons ionize the upper layers of the atmosphere and enable plants to produce oxygen through photosynthesis. And what the refractive index is equal to depends not only on the substances between which the boundary lies, but also on the wavelength of the incident radiation. What exact value we are talking about is usually clear from the context. That is, if the book examines x-rays and its effect on humans, then n there it is defined specifically for this range. But usually the visible spectrum of electromagnetic waves is meant unless something else is specified. As it became clear from what was written above, we are talking about transparent environments. We gave air, water, and diamond as examples. But what about wood, granite, plastic? Is there such a thing as a refractive index for them? The answer is complex, but in general - yes. First of all, we should consider what kind of light we are dealing with. Those media that are opaque to visible photons are cut through by X-ray or gamma radiation. That is, if we were all supermen, then the whole world around us would be transparent to us, but to varying degrees. For example, concrete walls would be no denser than jelly, and metal fittings would look like pieces of denser fruit. For other elementary particles, muons, our planet is generally transparent through and through. At one time, scientists had a lot of trouble proving the very fact of their existence. Millions of muons pierce us every second, but the probability of a single particle colliding with matter is very small, and it is very difficult to detect this. By the way, Baikal will soon become a place for “catching” muons. Its deep and clear water is ideal for this - especially in winter. The main thing is that the sensors do not freeze. So the refractive index of concrete, for example, for x-ray photons makes sense. Moreover, irradiating a substance with x-rays is one of the most accurate and important ways to study the structure of crystals. It is also worth remembering that in a mathematical sense, substances that are opaque for a given range have an imaginary refractive index. Finally, we must understand that the temperature of a substance can also affect its transparency.
Questions
Exercise
equal to the angle of reflection 
.
(1)
(2)
(3)
(5)
This law coincides with the law of reflection for waves of any nature and can be obtained as a consequence of Huygens' principle.
If n2 = 1 (vacuum), then 
Newton directed a beam of sunlight through a small hole onto a glass prism. When hitting the prism, the beam was refracted and on the opposite wall gave an elongated image with a rainbow alternation of colors - a spectrum. Experiment on the passage of monochromatic light through a prism:
Newton placed red glass in the path of the sun's ray, behind which he received monochromatic light (red), then a prism and observed on the screen only the red spot from the light ray. Experience in the synthesis (production) of white light: First, Newton directed a ray of sunlight onto a prism. Then, having collected the colored rays emerging from the prism using a collecting lens, Newton received a white image of a hole on a white wall instead of a colored stripe. Newton's conclusions:- a prism does not change light, but only decomposes it into its components - light rays that differ in color differ in the degree of refraction; Violet rays refract most strongly, red ones less strongly - red light, which refracts less, has the highest speed, and violet has the least, which is why the prism decomposes the light. The dependence of the refractive index of light on its color is called dispersion.
Natural light is sometimes conventionally designated as such. It is also called non-polarized.
Polarizers used to analyze the polarization of light are called analyzers.
Vision and refractive index
Light and refractive index
Phase speed
Absolute and relative refractive index
The beauty of refractive index and its applications
Wavelength
Refractive index and reflection
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