What does the refractive index of a substance depend on? How is the refractive index calculated? The refractive index of a medium does not depend on

Let us turn to a more detailed consideration of the refractive index, which we introduced in §81 when formulating the law of refraction.

The refractive index depends on the optical properties of both the medium from which the beam falls and the medium into which it penetrates. The refractive index obtained when light from a vacuum falls on any medium is called the absolute refractive index of that medium.

Rice. 184. Relative refractive index of two media:

Let the absolute refractive index of the first medium be and that of the second medium - . Considering refraction at the boundary of the first and second media, we make sure that the refractive index during the transition from the first medium to the second, the so-called relative refractive index, is equal to the ratio of the absolute refractive indices of the second and first media:

(Fig. 184). On the contrary, when passing from the second medium to the first, we have a relative refractive index

The established connection between the relative refractive index of two media and their absolute refractive indices could be derived theoretically, without new experiments, just as this can be done for the law of reversibility (§82),

A medium with a higher refractive index is called optically denser. The refractive index of various media relative to air is usually measured. The absolute refractive index of air is . Thus, the absolute refractive index of any medium is related to its refractive index relative to air by the formula

Table 6. Refractive index of various substances relative to air

The refractive index depends on the wavelength of light, i.e. on its color. Different colors correspond to different refractive indices. This phenomenon, called dispersion, plays an important role in optics. We will deal with this phenomenon repeatedly in subsequent chapters. The data given in table. 6, refer to yellow light.

It is interesting to note that the law of reflection can be formally written in the same form as the law of refraction. Let us remember that we agreed to always measure angles from the perpendicular to the corresponding ray. Therefore, we must consider the angle of incidence and the angle of reflection to have opposite signs, i.e. the law of reflection can be written as

Comparing (83.4) with the law of refraction, we see that the law of reflection can be considered as a special case of the law of refraction at . This formal similarity of the laws of reflection and refraction is of great benefit in solving practical problems.

In the previous presentation, the refractive index had the meaning of a constant of the medium, independent of the intensity of light passing through it. This interpretation of the refractive index is quite natural, but in the case of high radiation intensities, achievable using modern lasers, it is not justified. The properties of the medium through which strong light radiation passes depend in this case on its intensity. As they say, the environment becomes nonlinear. The nonlinearity of the medium manifests itself, in particular, in the fact that a high-intensity light wave changes the refractive index. The dependence of the refractive index on the radiation intensity has the form

Here is the usual refractive index, and is the nonlinear refractive index, and is the proportionality factor. The additional term in this formula can be either positive or negative.

The relative changes in the refractive index are relatively small. At nonlinear refractive index. However, even such small changes in the refractive index are noticeable: they manifest themselves in a peculiar phenomenon of self-focusing of light.

Let us consider a medium with a positive nonlinear refractive index. In this case, areas of increased light intensity are simultaneously areas of increased refractive index. Typically, in real laser radiation, the intensity distribution over the cross section of a beam of rays is nonuniform: the intensity is maximum along the axis and smoothly decreases towards the edges of the beam, as shown in Fig. 185 solid curves. A similar distribution also describes the change in the refractive index across the cross section of a cell with a nonlinear medium along the axis of which the laser beam propagates. The refractive index, which is greatest along the axis of the cuvette, smoothly decreases towards its walls (dashed curves in Fig. 185).

A beam of rays leaving the laser parallel to the axis, entering a medium with a variable refractive index, is deflected in the direction where it is larger. Therefore, the increased intensity near the cuvette leads to a concentration of light rays in this area, shown schematically in cross-sections and in Fig. 185, and this leads to a further increase. Ultimately, the effective cross section of a light beam passing through a nonlinear medium is significantly reduced. Light passes through a narrow channel with a high refractive index. Thus, the laser beam of rays is narrowed, and the nonlinear medium, under the influence of intense radiation, acts as a collecting lens. This phenomenon is called self-focusing. It can be observed, for example, in liquid nitrobenzene.

Rice. 185. Distribution of radiation intensity and refractive index over the cross section of a laser beam of rays at the entrance to the cuvette (a), near the input end (), in the middle (), near the output end of the cuvette ()

The laws of physics play a very important role when carrying out calculations to plan a specific strategy for the production of any product or when drawing up a project for the construction of structures for various purposes. Many quantities are calculated, so measurements and calculations are made before planning work begins. For example, the refractive index of glass is equal to the ratio of the sine of the angle of incidence to the sine of the angle of refraction.

So first there is the process of measuring the angles, then their sine is calculated, and only then can the desired value be obtained. Despite the availability of tabular data, it is worth carrying out additional calculations each time, since reference books often use ideal conditions, which are almost impossible to achieve in real life. Therefore, in reality, the indicator will necessarily differ from the table, and in some situations this is of fundamental importance.

Absolute indicator

The absolute refractive index depends on the brand of glass, since in practice there are a huge number of options that differ in composition and degree of transparency. On average it is 1.5 and fluctuates around this value by 0.2 in one direction or another. In rare cases, there may be deviations from this figure.

Again, if an accurate indicator is important, then additional measurements cannot be avoided. But they also do not give a 100% reliable result, since the final value will be influenced by the position of the sun in the sky and cloudiness on the day of measurement. Fortunately, in 99.99% of cases it is enough to simply know that the refractive index of a material such as glass is greater than one and less than two, and all other tenths and hundredths do not matter.

On forums that help solve physics problems, the question often comes up: what is the refractive index of glass and diamond? Many people think that since these two substances are similar in appearance, then their properties should be approximately the same. But this is a misconception.

The maximum refraction of glass will be around 1.7, while for diamond this indicator reaches 2.42. This gemstone is one of the few materials on Earth whose refractive index exceeds 2. This is due to its crystalline structure and the high level of scattering of light rays. The cut plays a minimal role in changes in the table value.

Relative indicator

The relative indicator for some environments can be characterized as follows:

• - the refractive index of glass relative to water is approximately 1.18;
• - the refractive index of the same material relative to air is equal to 1.5;
• - refractive index relative to alcohol - 1.1.

Measurements of the indicator and calculations of the relative value are carried out according to a well-known algorithm. To find a relative parameter, you need to divide one table value by another. Or make experimental calculations for two environments, and then divide the data obtained. Such operations are often carried out in laboratory physics classes.

Determination of refractive index

Determining the refractive index of glass in practice is quite difficult, because high-precision instruments are required to measure the initial data. Any error will increase, since the calculation uses complex formulas that require the absence of errors.

In general, this coefficient shows how many times the speed of propagation of light rays slows down when passing through a certain obstacle. Therefore, it is typical only for transparent materials. The refractive index of gases is taken as the reference value, that is, as a unit. This was done so that it was possible to start from some value when making calculations.

If a sunbeam falls on the surface of glass with a refractive index that is equal to the table value, then it can be changed in several ways:

• 1. Glue a film on top whose refractive index will be higher than that of glass. This principle is used in car window tinting to improve passenger comfort and allow the driver to have a clearer view of traffic conditions. The film will also inhibit ultraviolet radiation.
• 2. Paint the glass with paint. Manufacturers of cheap sunglasses do this, but it is worth considering that this can be harmful to vision. In good models, the glass is immediately produced colored using a special technology.
• 3. Immerse the glass in some liquid. This is only useful for experiments.

If a ray of light passes from glass, then the refractive index on the next material is calculated using a relative coefficient, which can be obtained by comparing table values. These calculations are very important in the design of optical systems that carry practical or experimental loads. Errors here are unacceptable, because they will lead to incorrect operation of the entire device, and then any data obtained with its help will be useless.

To determine the speed of light in glass with a refractive index, you need to divide the absolute value of the speed in a vacuum by the refractive index. Vacuum is used as a reference medium because refraction does not operate there due to the absence of any substances that could interfere with the smooth movement of light rays along a given path.

In any calculated indicators, the speed will be less than in the reference medium, since the refractive index is always greater than unity.

If a light wave is incident on a flat boundary separating two dielectrics having different relative dielectric constants, then this wave is reflected from the interface and refracted, passing from one dielectric to the other. The refractive power of a transparent medium is characterized by its refractive index, which is more often called the refractive index.

Absolute refractive index

DEFINITION

Absolute refractive index name a physical quantity equal to the ratio of the speed of propagation of light in a vacuum () to the phase speed of light in the medium (). This refractive index is designated by the letter . Mathematically, we write this definition of the refractive index as:

For any substance (with the exception of vacuum), the value of the refractive index depends on the frequency of light and the parameters of the substance (temperature, density, etc.). For rarefied gases, the refractive index is taken equal to .

If the substance is anisotropic, then n depends on the direction in which the light travels and how the light wave is polarized.

Based on definition (1), the absolute refractive index can be found as:

where is the dielectric constant of the medium, and is the magnetic permeability of the medium.

The refractive index can be a complex quantity in absorbing media. In the optical wavelength range at =1, the dielectric constant is written as:

then the refractive index:

where the real part of the refractive index is equal to:

reflects refraction, the imaginary part:

is responsible for absorption.

Relative refractive index

DEFINITION

Relative refractive index() of the second medium relative to the first is called the ratio of the phase speeds of light in the first substance to the phase speed in the second substance:

where is the absolute refractive index of the second medium, is the absolute refractive index of the first substance. In the event that title="Rendered by QuickLaTeX.com" height="16" width="60" style="vertical-align: -4px;">, то вторая среда считается оптически более плотной, чем первая.!}

For monochromatic waves, the lengths of which are much greater than the distance between molecules in a substance, Snell’s law is satisfied:

where is the angle of incidence, is the angle of refraction, is the relative refractive index of the substance in which the refracted light propagates, relative to the medium in which the incident wave of light propagated.

Units

The refractive index is a dimensionless quantity.

Examples of problem solving

EXAMPLE 1

Exercise	What will be the limiting angle of total internal reflection () if a ray of light passes from glass into air. The refractive index of glass is considered to be n=1.52.
Solution	With total internal reflection, the angle of refraction () is greater than or equal to ). For an angle, the law of refraction is transformed to the form: since the angle of incidence of the beam is equal to the angle of reflection, we can write that: According to the conditions of the problem, the beam passes from the flow into the air, this means that Let's carry out the calculations:
Answer

EXAMPLE 2

or, which is the same thing, the dependence of the phase speed of light waves on frequency:

Dispersion - the dependence of the refractive index of a substance on the wave frequency - manifests itself especially clearly and beautifully together with the effect of birefringence (see Video 6.6 in the previous paragraph), observed when light passes through anisotropic substances. The fact is that the refractive indices of ordinary and extraordinary waves depend differently on the frequency of the wave. As a result, the color (frequency) of light passing through an anisotropic substance placed between two polarizers depends both on the thickness of the layer of this substance and on the angle between the planes of transmission of the polarizers.

For all transparent, colorless substances in the visible part of the spectrum, as the wavelength decreases, the refractive index increases, that is, the dispersion of the substance is negative: . (Fig. 6.7, areas 1-2, 3-4)

If a substance absorbs light in a certain range of wavelengths (frequencies), then in the absorption region the dispersion

turns out to be positive and is called abnormal (Fig. 6.7, area 2–3).

Rice. 6.7. Dependence of the square of the refractive index (solid curve) and the light absorption coefficient of the substance
(dashed curve) versus wavelengthlnear one of the absorption bands()

Newton studied normal dispersion. The decomposition of white light into a spectrum when passing through a prism is a consequence of light dispersion. When a beam of white light passes through a glass prism, a multi-colored spectrum (Fig. 6.8).

Rice. 6.8. The passage of white light through a prism: due to the difference in the refractive index of glass for different
wavelengths, the beam is decomposed into monochromatic components - a spectrum appears on the screen

Red light has the longest wavelength and the smallest refractive index, so red rays are deflected less than others by the prism. Next to them will be rays of orange, then yellow, green, blue, indigo and finally violet light. The complex white light incident on the prism is decomposed into monochromatic components (spectrum).

A prime example of dispersion is a rainbow. A rainbow is observed if the sun is behind the observer. Red and violet rays are refracted by spherical water droplets and reflected from their inner surface. Red rays are refracted less and enter the observer's eye from droplets located at a higher altitude. Therefore, the top stripe of the rainbow always turns out to be red (Fig. 26.8).

Rice. 6.9. The emergence of a rainbow

Using the laws of reflection and refraction of light, it is possible to calculate the path of light rays with total reflection and dispersion in raindrops. It turns out that the rays are scattered with the greatest intensity in a direction forming an angle of about 42° with the direction of the sun's rays (Fig. 6.10).

Rice. 6.10. Rainbow location

The geometric locus of such points is a circle with center at the point 0. Part of it is hidden from the observer R below the horizon, the arc above the horizon is the visible rainbow. Double reflection of rays in raindrops is also possible, leading to a second-order rainbow, the brightness of which, naturally, is less than the brightness of the main rainbow. For her, the theory gives an angle 51 °, that is, the second-order rainbow lies outside the main one. In it, the order of colors is reversed: the outer arc is colored purple, and the lower one is painted red. Rainbows of the third and higher orders are rarely observed.

Elementary theory of dispersion. The dependence of the refractive index of a substance on the length of the electromagnetic wave (frequency) is explained on the basis of the theory of forced oscillations. Strictly speaking, the movement of electrons in an atom (molecule) obeys the laws of quantum mechanics. However, for a qualitative understanding of optical phenomena, we can limit ourselves to the idea of ​​electrons bound in an atom (molecule) by an elastic force. When deviating from the equilibrium position, such electrons begin to oscillate, gradually losing energy to emit electromagnetic waves or transferring their energy to lattice nodes and heating the substance. As a result, the oscillations will be damped.

When passing through a substance, an electromagnetic wave acts on each electron with the Lorentz force:

Where v- speed of an oscillating electron. In an electromagnetic wave, the ratio of the magnetic and electric field strengths is equal to

Therefore, it is not difficult to estimate the ratio of the electric and magnetic forces acting on the electron:

Electrons in matter move at speeds much lower than the speed of light in a vacuum:

Where - amplitude of the electric field strength in a light wave, - phase of the wave, determined by the position of the electron in question. To simplify calculations, we neglect damping and write the electron motion equation in the form

where, is the natural frequency of vibrations of an electron in an atom. We have already considered the solution of such a differential inhomogeneous equation earlier and obtained

Consequently, the displacement of the electron from the equilibrium position is proportional to the electric field strength. Displacements of nuclei from the equilibrium position can be neglected, since the masses of the nuclei are very large compared to the mass of the electron.

An atom with a displaced electron acquires a dipole moment

(for simplicity, let us assume for now that there is only one “optical” electron in the atom, the displacement of which makes a decisive contribution to the polarization). If a unit volume contains N atoms, then the polarization of the medium (dipole moment per unit volume) can be written in the form

In real media, different types of oscillations of charges (groups of electrons or ions) are possible, contributing to polarization. These types of oscillations can have different amounts of charge e i and masses t i, as well as various natural frequencies (we will denote them by the index k), in this case, the number of atoms per unit volume with a given type of vibration Nk proportional to the concentration of atoms N:

Dimensionless proportionality coefficient fk characterizes the effective contribution of each type of oscillation to the total polarization of the medium:

On the other hand, as is known,

where is the dielectric susceptibility of the substance, which is related to the dielectric constant e ratio

As a result, we obtain the expression for the square of the refractive index of a substance:

Near each of the natural frequencies, the function defined by formula (6.24) suffers a discontinuity. This behavior of the refractive index is due to the fact that we neglected attenuation. Similarly, as we saw earlier, neglecting damping leads to an infinite increase in the amplitude of forced oscillations at resonance. Taking into account attenuation saves us from infinities, and the function has the form shown in Fig. 6.11.

Rice. 6.11. Dependence of the dielectric constant of the mediumon the frequency of the electromagnetic wave

Considering the relationship between frequency and electromagnetic wavelength in vacuum

it is possible to obtain the dependence of the refractive index of a substance P on the wavelength in the region of normal dispersion (sections 1–2 And 3–4 in Fig. 6.7):

The wavelengths corresponding to the natural frequencies of oscillations are constant coefficients.

In the region of anomalous dispersion (), the frequency of the external electromagnetic field is close to one of the natural frequencies of oscillations of molecular dipoles, that is, resonance occurs. It is in these areas (for example, area 2–3 in Fig. 6.7) that significant absorption of electromagnetic waves is observed; the light absorption coefficient of the substance is shown by the dashed line in Fig. 6.7.

The concept of group velocity. The concept of group velocity is closely related to the phenomenon of dispersion. When real electromagnetic pulses propagate in a medium with dispersion, for example, wave trains known to us, emitted by individual atomic emitters, they “spread out” - an expansion of extent in space and duration in time. This is due to the fact that such pulses are not a monochromatic sine wave, but a so-called wave packet, or a group of waves - a set of harmonic components with different frequencies and different amplitudes, each of which propagates in the medium with its own phase velocity (6.13).

If a wave packet were propagating in a vacuum, then its shape and spatio-temporal extent would remain unchanged, and the speed of propagation of such a wave train would be the phase speed of light in vacuum

Due to the presence of dispersion, the dependence of the frequency of an electromagnetic wave on the wave number k becomes nonlinear, and the speed of propagation of the wave train in the medium, that is, the speed of energy transfer, is determined by the derivative

where is the wave number for the “central” wave in the train (having the greatest amplitude).

We will not derive this formula in general form, but we will use a particular example to explain its physical meaning. As a model of a wave packet, we will take a signal consisting of two plane waves propagating in the same direction with identical amplitudes and initial phases, but differing frequencies, shifted relative to the “central” frequency by a small amount. The corresponding wave numbers are shifted relative to the “central” wave number by a small amount . These waves are described by expressions.

Topics of the Unified State Examination codifier: the law of light refraction, total internal reflection.

At the interface between two transparent media, along with the reflection of light, it is observed refraction- light, moving to another medium, changes the direction of its propagation.

The refraction of a light ray occurs when it inclined falling on the interface (though not always - read on about total internal reflection). If the ray falls perpendicular to the surface, then there will be no refraction - in the second medium the ray will retain its direction and will also go perpendicular to the surface.

Law of refraction (special case).

We will start with the special case when one of the media is air. This is exactly the situation that occurs in the vast majority of problems. We will discuss the corresponding special case of the law of refraction, and only then we will give its most general formulation.

Suppose that a ray of light traveling in air falls obliquely onto the surface of glass, water or some other transparent medium. When passing into the medium, the beam is refracted, and its further path is shown in Fig. 1 .

At the point of impact, a perpendicular is drawn (or, as they also say, normal) to the surface of the medium. The beam, as before, is called incident ray, and the angle between the incident ray and the normal is angle of incidence. Ray is refracted ray; The angle between the refracted ray and the normal to the surface is called refraction angle.

Any transparent medium is characterized by a quantity called refractive index this environment. The refractive indices of various media can be found in tables. For example, for glass, and for water. In general, in any environment; The refractive index is equal to unity only in a vacuum. In air, therefore, for air we can assume with sufficient accuracy in problems (in optics, air is not very different from vacuum).

Law of refraction (air-medium transition) .

1) The incident ray, the refracted ray and the normal to the surface drawn at the point of incidence lie in the same plane.
2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the refractive index of the medium:

. (1)

Since from relation (1) it follows that , that is, the angle of refraction is less than the angle of incidence. Remember: passing from air to the medium, the ray, after refraction, goes closer to the normal.

The refractive index is directly related to the speed of light propagation in a given medium. This speed is always less than the speed of light in vacuum: . And it turns out that

. (2)

We will understand why this happens when we study wave optics. For now, let's combine the formulas. (1) and (2) :

. (3)

Since the refractive index of air is very close to unity, we can assume that the speed of light in air is approximately equal to the speed of light in a vacuum. Taking this into account and looking at the formula. (3) , we conclude: the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in air to the speed of light in the medium.

Reversibility of light rays.

Now let's consider the reverse path of the beam: its refraction when passing from the medium to the air. The following useful principle will help us here.

The principle of reversibility of light rays. The beam path does not depend on whether the beam is propagating in the forward or backward direction. Moving in the opposite direction, the beam will follow exactly the same path as in the forward direction.

According to the principle of reversibility, when transitioning from a medium to air, the beam will follow the same trajectory as during the corresponding transition from air to medium (Fig. 2). The only difference in Fig. 2 from fig. 1 is that the direction of the beam has changed to the opposite.

Since the geometric picture has not changed, formula (1) will remain the same: the ratio of the sine of the angle to the sine of the angle is still equal to the refractive index of the medium. True, now the angles have changed roles: the angle has become the angle of incidence, and the angle has become the angle of refraction.

In any case, no matter how the beam travels - from air to medium or from medium to air - the following simple rule applies. We take two angles - the angle of incidence and the angle of refraction; the ratio of the sine of the larger angle to the sine of the smaller angle is equal to the refractive index of the medium.

We are now fully prepared to discuss the law of refraction in the most general case.

Law of refraction (general case).

Let light pass from medium 1 with a refractive index to medium 2 with a refractive index. A medium with a high refractive index is called optically more dense; accordingly, a medium with a lower refractive index is called optically less dense.

Moving from an optically less dense medium to an optically more dense one, the light beam, after refraction, goes closer to the normal (Fig. 3). In this case, the angle of incidence is greater than the angle of refraction: .

Rice. 3.

On the contrary, moving from an optically denser medium to an optically less dense one, the beam deviates further from the normal (Fig. 4). Here the angle of incidence is less than the angle of refraction:

Rice. 4.

It turns out that both of these cases are covered by one formula - the general law of refraction, valid for any two transparent media.

Law of refraction.
1) The incident ray, the refracted ray and the normal to the interface between the media, drawn at the point of incidence, lie in the same plane.
2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive index of the second medium to the refractive index of the first medium:

. (4)

It is easy to see that the previously formulated law of refraction for the air-medium transition is a special case of this law. In fact, putting in formula (4) we arrive at formula (1).

Let us now remember that the refractive index is the ratio of the speed of light in a vacuum to the speed of light in a given medium: . Substituting this into (4), we get:

. (5)

Formula (5) naturally generalizes formula (3). The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium.

Total internal reflection.

When light rays pass from an optically denser medium to an optically less dense medium, an interesting phenomenon is observed - complete internal reflection. Let's figure out what it is.

For definiteness, we assume that light comes from water into air. Let us assume that in the depths of the reservoir there is a point source of light emitting rays in all directions. We will look at some of these rays (Fig. 5).

The beam hits the water surface at the smallest angle. This ray is partially refracted (ray) and partially reflected back into the water (ray). Thus, part of the energy of the incident beam is transferred to the refracted beam, and the remaining part of the energy is transferred to the reflected beam.

The angle of incidence of the beam is greater. This beam is also divided into two beams - refracted and reflected. But the energy of the original beam is distributed between them differently: the refracted beam will be dimmer than the beam (that is, it will receive a smaller share of energy), and the reflected beam will be correspondingly brighter than the beam (it will receive a larger share of energy).

As the angle of incidence increases, the same pattern is observed: an increasingly larger share of the energy of the incident beam goes to the reflected beam, and an increasingly smaller share to the refracted beam. The refracted beam becomes dimmer and dimmer, and at some point disappears completely!

This disappearance occurs when the angle of incidence corresponding to the angle of refraction is reached. In this situation, the refracted beam would have to go parallel to the surface of the water, but there is nothing left to go - all the energy of the incident beam went entirely to the reflected beam.

With a further increase in the angle of incidence, the refracted beam will even be absent.

The described phenomenon is complete internal reflection. Water does not release rays with angles of incidence equal to or exceeding a certain value - all such rays are completely reflected back into the water. The angle is called limiting angle of total reflection.

The value is easy to find from the law of refraction. We have:

But, therefore

So, for water the limiting angle of total reflection is equal to:

You can easily observe the phenomenon of total internal reflection at home. Pour water into a glass, lift it and look at the surface of the water just below through the wall of the glass. You will see a silvery sheen on the surface - due to total internal reflection, it behaves like a mirror.

The most important technical application of total internal reflection is fiber optics. Light rays launched into a fiber optic cable ( light guide) almost parallel to its axis, fall onto the surface at large angles and are completely reflected back into the cable without loss of energy. Repeatedly reflected, the rays travel further and further, transferring energy over a considerable distance. Fiber optic communications are used, for example, in cable television networks and high-speed Internet access.