1. Diffraction of light. Huygens-Fresnel principle.

2. Diffraction of light by a slit in parallel beams.

3. Diffraction grating.

4. Diffraction spectrum.

5. Characteristics of a diffraction grating as a spectral device.

6. X-ray diffraction analysis.

7. Diffraction of light by a round hole. aperture resolution.

8. Basic concepts and formulas.

9. Tasks.

In a narrow, but most commonly used sense, the diffraction of light is the rounding of the borders of opaque bodies by the rays of light, the penetration of light into the region of a geometric shadow. In phenomena associated with diffraction, there is a significant deviation of the behavior of light from the laws of geometric optics. (Diffraction does not only show up for light.)

Diffraction is a wave phenomenon that is most clearly manifested when the dimensions of the obstacle are commensurate (of the same order) with the wavelength of light. The relatively late discovery of light diffraction (16th-17th centuries) is connected with the smallness of the lengths of visible light.

21.1. Diffraction of light. Huygens-Fresnel principle

Diffraction of light called a complex of phenomena that are due to its wave nature and are observed during the propagation of light in a medium with sharp inhomogeneities.

A qualitative explanation of diffraction is given by Huygens principle, which establishes the method of constructing the wave front at time t + Δt if its position at time t is known.

1. According to Huygens principle, each point of the wave front is the center of coherent secondary waves. The envelope of these waves gives the position of the wave front at the next moment in time.

Let us explain the application of the Huygens principle by the following example. Let a plane wave fall on a barrier with a hole, the front of which is parallel to the barrier (Fig. 21.1).

Rice. 21.1. Explanation of Huygens' principle

Each point of the wave front emitted by the hole serves as the center of secondary spherical waves. The figure shows that the envelope of these waves penetrates into the region of the geometric shadow, the boundaries of which are marked with a dashed line.

Huygens' principle says nothing about the intensity of the secondary waves. This drawback was eliminated by Fresnel, who supplemented the Huygens principle with the concept of the interference of secondary waves and their amplitudes. The Huygens principle supplemented in this way is called the Huygens-Fresnel principle.

2. According to the Huygens-Fresnel principle the magnitude of light oscillations at some point O is the result of interference at this point of coherent secondary waves emitted everyone wave surface elements. The amplitude of each secondary wave is proportional to the area of ​​the element dS, inversely proportional to the distance r to the point O, and decreases with increasing angle α between normal n to the element dS and direction to the point O (Fig. 21.2).

Rice. 21.2. Emission of secondary waves by wave surface elements

21.2. Slit Diffraction in Parallel Beams

Calculations related to the application of the Huygens-Fresnel principle, in the general case, are a complex mathematical problem. However, in a number of cases with a high degree of symmetry, the amplitude of the resulting oscillations can be found by algebraic or geometric summation. Let us demonstrate this by calculating the diffraction of light by a slit.

Let a plane monochromatic light wave fall on a narrow slot (AB) in an opaque barrier, the direction of propagation of which is perpendicular to the surface of the slot (Fig. 21.3, a). Behind the slit (parallel to its plane) we place a converging lens, in focal plane which we place the screen E. All secondary waves emitted from the surface of the slot in the direction parallel optical axis of the lens (α = 0), come into focus of the lens in the same phase. Therefore, in the center of the screen (O) there is maximum interference for waves of any length. It's called the maximum zero order.

In order to find out the nature of the interference of secondary waves emitted in other directions, we divide the slot surface into n identical zones (they are called Fresnel zones) and consider the direction for which the condition is satisfied:

where b is the slot width, and λ - the length of the light wave.

Rays of secondary light waves traveling in this direction will intersect at point O.

Rice. 21.3. Diffraction by one slit: a - ray path; b - distribution of light intensity (f - focal length of the lens)

The product bsina is equal to the path difference (δ) between the rays coming from the edges of the slot. Then the difference in the path of the rays coming from neighboring Fresnel zones is equal to λ/2 (see formula 21.1). Such rays cancel each other out during interference, since they have the same amplitudes and opposite phases. Let's consider two cases.

1) n = 2k is an even number. In this case, pairwise extinction of rays from all Fresnel zones occurs, and at the point O" a minimum of the interference pattern is observed.

Minimum intensity during slit diffraction is observed for the directions of rays of secondary waves that satisfy the condition

An integer k is called minimum order.

2) n = 2k - 1 is an odd number. In this case, the radiation of one Fresnel zone will remain unquenched, and at the point O" the maximum of the interference pattern will be observed.

The intensity maximum during slit diffraction is observed for the directions of rays of secondary waves that satisfy the condition:

An integer k is called maximum order. Recall that for the direction α = 0 we have maximum zero order.

It follows from formula (21.3) that as the light wavelength increases, the angle at which a maximum of order k > 0 is observed increases. This means that for the same k, the purple stripe is closest to the center of the screen, and the red one is farthest away.

In figure 21.3, b shows the distribution of light intensity on the screen depending on the distance to its center. The main part of the light energy is concentrated in the central maximum. As the order of the maximum increases, its intensity rapidly decreases. Calculations show that I 0:I 1:I 2 = 1:0.047:0.017.

If the slit is illuminated with white light, then the central maximum will be white on the screen (it is common for all wavelengths). Side maxima will consist of colored bands.

A phenomenon similar to slit diffraction can be observed on a razor blade.

21.3. Diffraction grating

In the case of slit diffraction, the intensities of the maxima of the order k > 0 are so insignificant that they cannot be used to solve practical problems. Therefore, as a spectral instrument is used diffraction grating, which is a system of parallel equidistant slots. A diffraction grating can be obtained by applying opaque strokes (scratches) to a plane-parallel glass plate (Fig. 21.4). The space between the strokes (slits) transmits light.

Strokes are applied to the surface of the grating with a diamond cutter. Their density reaches 2000 strokes per millimeter. In this case, the width of the grating can be up to 300 mm. The total number of lattice slots is denoted N.

The distance d between the centers or edges of adjacent slots is called constant (period) diffraction grating.

The diffraction pattern on the grating is defined as the result of mutual interference of waves coming from all slits.

The path of the rays in the diffraction grating is shown in Fig. 21.5.

Let a plane monochromatic light wave fall on the grating, the direction of propagation of which is perpendicular to the plane of the grating. Then the slot surfaces belong to the same wave surface and are sources of coherent secondary waves. Consider secondary waves whose propagation direction satisfies the condition

After passing through the lens, the rays of these waves will intersect at point O.

The product dsina is equal to the path difference (δ) between the rays coming from the edges of neighboring slots. When condition (21.4) is satisfied, the secondary waves arrive at the point O" in the same phase and a maximum of the interference pattern appears on the screen. The maxima satisfying condition (21.4) are called principal maxima of the order k. The condition (21.4) itself is called the basic formula of a diffraction grating.

Major Highs during grating diffraction are observed for the directions of rays of secondary waves that satisfy the condition: dsinα = ± κ λ; k = 0,1,2,...

Rice. 21.4. Cross section of the diffraction grating (a) and its symbol (b)

Rice. 21.5. Diffraction of light on a diffraction grating

For a number of reasons that are not considered here, there are (N - 2) additional maxima between the main maxima. With a large number of slits, their intensity is negligible, and the entire space between the main maxima looks dark.

Condition (21.4), which determines the positions of all the main maxima, does not take into account diffraction by a single slit. It may happen that for some direction the condition maximum for the lattice (21.4) and the condition minimum for the gap (21.2). In this case, the corresponding main maximum does not arise (formally, it exists, but its intensity is zero).

The greater the number of slots in the diffraction grating (N), the more light energy passes through the grating, the more intense and sharper the maxima will be. Figure 21.6 shows the intensity distribution graphs obtained from gratings with different numbers of slots (N). Periods (d) and slot widths (b) are the same for all gratings.

Rice. 21.6. Intensity distribution for different values ​​of N

21.4. Diffraction spectrum

It can be seen from the basic formula of the diffraction grating (21.4) that the diffraction angle α, at which the main maxima are formed, depends on the wavelength of the incident light. Therefore, the intensity maxima corresponding to different wavelengths are obtained in different places on the screen. This makes it possible to use the grating as a spectral instrument.

Diffraction spectrum- spectrum obtained using a diffraction grating.

When white light falls on a diffraction grating, all maxima, except for the central one, decompose into a spectrum. The position of the maximum of order k for light with wavelength λ is given by:

The longer the wavelength (λ), the farther from the center is the kth maximum. Therefore, the purple region of each main maximum will be facing the center of the diffraction pattern, and the red region will be outward. Note that when white light is decomposed by a prism, violet rays are more strongly deflected.

Writing down the basic lattice formula (21.4), we indicated that k is an integer. How big can it be? The answer to this question is given by the inequality |sinα|< 1. Из формулы (21.5) найдем

where L is the lattice width and N is the number of strokes.

For example, for a grating with a density of 500 lines per mm, d = 1/500 mm = 2x10 -6 m. For green light with λ = 520 nm = 520x10 -9 m, we get k< 2х10 -6 /(520 х10 -9) < 3,8. Таким образом, для такой решетки (весьма средней) порядок наблюдаемого максимума не превышает 3.

21.5. Characteristics of a diffraction grating as a spectral instrument

The basic formula of a diffraction grating (21.4) makes it possible to determine the wavelength of light by measuring the angle α corresponding to the position of the k-th maximum. Thus, the diffraction grating makes it possible to obtain and analyze the spectra of complex light.

Spectral characteristics of the grating

Angular dispersion - a value equal to the ratio of the change in the angle at which the diffraction maximum is observed to the change in wavelength:

where k is the order of the maximum, α - the angle at which it is observed.

The angular dispersion is the higher, the greater the order k of the spectrum and the smaller the grating period (d).

Resolution(resolving power) of a diffraction grating - a value that characterizes its ability to give

where k is the order of maximum and N is the number of lattice lines.

It can be seen from the formula that close lines that merge in the spectrum of the first order can be perceived separately in the spectra of the second or third orders.

21.6. X-ray diffraction analysis

The basic formula of a diffraction grating can be used not only to determine the wavelength, but also to solve the inverse problem - finding the diffraction grating constant from a known wavelength.

The structural lattice of a crystal can be taken as a diffraction grating. If a stream of X-rays is directed to a simple crystal lattice at a certain angle θ (Fig. 21.7), then they will diffract, since the distance between the scattering centers (atoms) in the crystal corresponds to

wavelength of x-rays. If a photographic plate is placed at some distance from the crystal, it will register the interference of the reflected rays.

where d is the interplanar distance in the crystal, θ is the angle between the plane

Rice. 21.7. X-ray diffraction on a simple crystal lattice; dots indicate the arrangement of atoms

crystal and the incident x-ray beam (glancing angle), λ is the wavelength of x-ray radiation. Relation (21.11) is called the Bragg-Wulf condition.

If the X-ray wavelength is known and the angle θ corresponding to condition (21.11) is measured, then the interplanar (interatomic) distance d can be determined. This is based on X-ray diffraction analysis.

X-ray diffraction analysis - a method for determining the structure of a substance by studying the patterns of X-ray diffraction on the samples under study.

X-ray diffraction patterns are very complex because a crystal is a three-dimensional object and X-rays can diffract on different planes at different angles. If the substance is a single crystal, then the diffraction pattern is an alternation of dark (exposed) and light (unexposed) spots (Fig. 21.8, a).

In the case when the substance is a mixture of a large number of very small crystals (as in a metal or powder), a series of rings appears (Fig. 21.8, b). Each ring corresponds to a diffraction maximum of a certain order k, while the radiograph is formed in the form of circles (Fig. 21.8, b).

Rice. 21.8. X-ray pattern for a single crystal (a), X-ray pattern for a polycrystal (b)

X-ray diffraction analysis is also used to study the structures of biological systems. For example, the structure of DNA was established by this method.

21.7. Diffraction of light by a circular hole. Aperture resolution

In conclusion, let us consider the question of the diffraction of light by a round hole, which is of great practical interest. Such holes are, for example, the pupil of the eye and the lens of the microscope. Let light from a point source fall on the lens. The lens is a hole that only lets through part light wave. Due to diffraction on the screen located behind the lens, a diffraction pattern will appear, shown in Fig. 21.9, a.

As for the gap, the intensities of side maxima are small. The central maximum in the form of a bright circle (diffraction spot) is the image of a luminous point.

The diameter of the diffraction spot is determined by the formula:

where f is the focal length of the lens and d is its diameter.

If light from two point sources falls on the hole (diaphragm), then depending on the angular distance between them (β) their diffraction spots can be perceived separately (Fig. 21.9, b) or merge (Fig. 21.9, c).

We present without derivation a formula that provides a separate image of nearby point sources on the screen (diaphragm resolution):

where λ is the wavelength of the incident light, d is the aperture (diaphragm) diameter, β is the angular distance between the sources.

Rice. 21.9. Diffraction by a circular hole from two point sources

21.8. Basic concepts and formulas

End of table

21.9. Tasks

1. The wavelength of light incident on the slit perpendicular to its plane fits into the width of the slit 6 times. At what angle will the 3rd diffraction minimum be seen?

2. Determine the period of a grating with a width L = 2.5 cm and N = 12500 lines. Write your answer in micrometers.

Solution

d = L/N = 25,000 µm/12,500 = 2 µm. Answer: d = 2 µm.

3. What is the diffraction grating constant if the red line (700 nm) in the 2nd order spectrum is visible at an angle of 30°?

4. The diffraction grating contains N = 600 lines per L = 1 mm. Find the largest order of the spectrum for light with a wavelength λ = 600 nm.

5. Orange light at 600 nm and green light at 540 nm pass through a diffraction grating having 4000 lines per centimeter. What is the angular distance between the orange and green maxima: a) first order; b) third order?

Δα \u003d α op - α z \u003d 13.88 ° - 12.47 ° \u003d 1.41 °.

6. Find the highest order of the spectrum for the yellow sodium line λ = 589 nm if the lattice constant is d = 2 μm.

Solution

Let's bring d and λ to the same units: d = 2 µm = 2000 nm. By formula (21.6) we find k< d/λ = 2000/ 589 = 3,4. Answer: k = 3.

7. A diffraction grating with N = 10,000 slots is used to study the light spectrum in the 600 nm region. Find the minimum wavelength difference that can be detected by such a grating when observing second-order maxima.

The grille on the side looks like this.

Also find application reflective grilles, which are obtained by applying thin strokes to a polished metal surface with a diamond cutter. Prints on gelatin or plastic after such an engraving are called replicas, but such diffraction gratings are usually of poor quality, so their use is limited. Good reflective gratings are considered to be those with a total length of about 150 mm, with a total number of strokes of 600 pieces / mm.

The main characteristics of a diffraction grating are total number of strokes N, hatch density n (number of strokes per 1 mm) and period(constant) of the lattice d, which can be found as d = 1/n.

The grating is illuminated by one wave front and its N transparent strokes are usually considered as N coherent sources.

If we remember the phenomenon interference from many identical light sources, then light intensity is expressed according to the pattern:

where i 0 is the intensity of the light wave that passed through one slit

Based on the concept maximum wave intensity obtained from the condition:

β = mπ for m = 0, 1, 2… etc.

.

Let's move on from auxiliary cornerβ to the spatial viewing angle Θ, and then:

(π d sinΘ)/ λ = m π,

The main maxima appear under the condition:

sinΘ m = m λ/ d, at m = 0, 1, 2… etc.

light intensity in major highs can be found according to the formula:

I m \u003d N 2 i 0.

Therefore, it is necessary to produce gratings with a small period d, then it is possible to obtain large beam scattering angles and a wide diffraction pattern.

For example:

Continuing the previous example Let us consider the case when in the first maximum the red rays (λ cr = 760 nm) deviate by an angle Θ k = 27 °, and the violet ones (λ f = 400 nm) deviate by an angle Θ f = 14 °.

It can be seen that with the help of a diffraction grating it is possible to measure wavelength one color or another. To do this, you just need to know the period of the grating and measure the angle, but which the beam deviated, corresponding to the required light.

Diffraction grating

DiffractionAny deviation of the propagation of light from a straight line is called, not associated with reflection and refraction. A qualitative method for calculating the diffraction pattern was proposed by Fresnel. The main idea of ​​the method is Huygens-Fresnel principle:

Each point to which the wave reaches serves as a source of coherent secondary waves, and the further propagation of the wave is determined by the interference of the secondary waves.

The locus of points for which the oscillations have the same phases is called wave surface . The wave front is also a wave surface.

Diffraction gratingis a collection of a large number of parallel slots or mirrors of the same width and spaced from each other at the same distance. The lattice period ( d) called the distance between the midpoints of adjacent slots, or what is the same, the sum of the width of the slot (a) and the opaque gap (b) between them (d = a + b).

Consider the principle of operation of a diffraction grating. Let a parallel beam of white light rays fall on the grating normally to its surface (Fig. 1). On the grating slits, the width of which is commensurate with the wavelength of light, diffraction occurs.

As a result, behind the diffraction grating, according to the Huygens-Fresnel principle, from each point of the slit, light rays will propagate in all possible directions, which can be associated with deflection angles φ light rays ( diffraction angles) from the original direction. Beams parallel to each other (diffracting at the same angle) φ ) can be focused by placing a converging lens behind the grating. Each beam of parallel rays will converge in the rear focal plane of the lens at a certain point A. Parallel rays corresponding to different diffraction angles will converge in other points of the focal plane of the lens. At these points, interference of light waves emanating from different slots of the grating will be observed. If the optical path difference between the corresponding rays of monochromatic light is equal to an integer number of wavelengths, κ = 0, ±1, ±2, …, then at the point where the beams overlap, the maximum light intensity for a given wavelength will be observed. Figure 1 shows that the optical path difference Δ between two parallel beams emerging from the corresponding points of neighboring slots is equal to

where φ is the angle of deflection of the beam by the grating.

Therefore, the condition for the occurrence main interference maxima gratings or grating equation

, (2)

where λ is the light wavelength.

In the focal plane of the lens for rays that have not experienced diffraction, a central zero-order white maximum is observed ( φ = 0, κ = 0), to the right and left of which there are colored maxima (spectral lines) of the first, second, and subsequent orders (Fig. 1). The intensity of the maxima decreases as their order increases; with increasing diffraction angle.

One of the main characteristics of a diffraction grating is its angular dispersion. Angular dispersion lattice determines the angular distance dφ between directions for two spectral lines that differ in wavelength by 1 nm ( = 1 nm), and characterizes the degree of spectrum stretching near a given wavelength:

The formula for calculating the angular dispersion of the lattice can be obtained by differentiating equation (2) . Then

. (5)

It follows from formula (5) that the angular dispersion of the grating is the greater, the greater the order of the spectrum.

For gratings with different periods, the width of the spectrum is greater for a grating characterized by a smaller period. Usually, within one order of magnitude, it varies insignificantly (especially for gratings with a small number of lines per millimeter), so the dispersion remains almost unchanged within one order of magnitude. The spectrum obtained with constant dispersion is stretched uniformly over the entire wavelength range, which favorably distinguishes the grating spectrum from the spectrum given by a prism.

Angular dispersion is related to linear dispersion. Linear dispersion can also be calculated using the formula

, (6) where is the linear distance on the screen or photographic plate between the spectral lines, f is the focal length of the lens.

The diffraction grating is also characterized resolution. This value characterizes the ability of a diffraction grating to give a separate image of two close spectral lines

R = , (7)

where l is the average wavelength of the resolved spectral lines; dl is the difference between the wavelengths of two neighboring spectral lines.

Dependence of resolution on the number of slits of a diffraction grating N is determined by the formula

R = = kN, (8)

where k is the order of the spectrum.

From the equation for the diffraction grating (1), we can draw the following conclusions:

1. A diffraction grating will give noticeable diffraction (significant diffraction angles) only if the grating period is commensurate with the wavelength of the light, that is d»l» 10 –4 cm. Gratings with a period less than the wavelength do not give diffraction maxima.

2. The position of the main maxima of the diffraction pattern depends on the wavelength. The spectral components of the radiation of a non-monochromatic beam are deflected by the grating at different angles ( diffraction spectrum). This makes it possible to use the diffraction grating as a spectral instrument.

3. The maximum order of the spectrum, with normal incidence of light on a diffraction grating, is determined by the relation:

k max £ d¤l.

Diffraction gratings used in different regions of the spectrum differ in size, shape, surface material, profile and frequency of lines, which makes it possible to cover the region of the spectrum from its ultraviolet part (l » 100 nm) to the infrared part (l » 1 μm). Engraved gratings (replicas) are widely used in spectral instruments, which are imprints of gratings on special plastics, followed by the application of a metal reflective layer.

DEFINITION

grating called a spectral device, which is a system of a certain number of slits separated by opaque gaps.

Very often, in practice, a one-dimensional diffraction grating is used, consisting of parallel slots of the same width, located in the same plane, which are separated by opaque gaps of equal width. Such a grating is made using a special dividing machine, which applies parallel strokes to a glass plate. The number of such strokes can be more than a thousand per millimeter.

Reflective diffraction gratings are considered the best. This is a collection of areas that reflect light with areas that reflect light. Such gratings are a polished metal plate on which light-scattering strokes are applied with a cutter.

The grating diffraction pattern is the result of the mutual interference of waves that come from all the slits. Therefore, with the help of a diffraction grating, multipath interference of coherent light beams that have undergone diffraction and that come from all slits is realized.

Let us assume that on the diffraction grating the width of the slit will be a, the width of the opaque section will be b, then the value:

is called the period of the (constant) diffraction grating.

Diffraction pattern on a one-dimensional diffraction grating

Let us imagine that a monochromatic wave is incident normal to the plane of the diffraction grating. Due to the fact that the slots are located at equal distances from each other, the path differences () that come from a pair of adjacent slots for the chosen direction will be the same for the entire given diffraction grating:

The main intensity minima are observed in the directions determined by the condition:

In addition to the main minima, as a result of mutual interference of light rays sent by a pair of slits, they cancel each other out in some directions, which means that additional minima appear. They arise in directions where the difference in the path of the rays is an odd number of half-waves. The additional minima condition is written as:

where N is the number of slits of the diffraction grating; k' takes any integer value except 0, . If the lattice has N slots, then between the two main maxima there is an additional minimum that separates the secondary maxima.

The condition for the main maxima for the diffraction grating is the expression:

Since the value of the sine cannot be greater than one, then the number of main maxima:

If white light is passed through the grating, then all the maxima (except the central m=0) will be decomposed into a spectrum. In this case, the violet region of this spectrum will be directed to the center of the diffraction pattern. This property of a diffraction grating is used to study the composition of the light spectrum. If the grating period is known, then the calculation of the wavelength of light can be reduced to finding the angle , which corresponds to the direction to the maximum.

Examples of problem solving

EXAMPLE 1

Exercise	What is the maximum order of the spectrum that can be obtained using a diffraction grating with a constant m, if a monochromatic beam of light with a wavelength m is incident on it perpendicular to the surface?
Solution	As a basis for solving the problem, we use the formula, which is the condition for observing the main maxima for the diffraction pattern obtained when light passes through a diffraction grating: The maximum value is one, so: From (1.2) we express , we get: Let's do the calculations:
Answer

EXAMPLE 2

Exercise

Monochromatic light with a wavelength is passed through a diffraction grating. A screen is placed at a distance L from the grating. A diffraction pattern is projected onto it using a lens located near the grating. In this case, the first diffraction maximum is located at a distance l from the central one. What is the number of lines per unit length of the diffraction grating (N) if the light falls on it normally?

Solution

Let's make a drawing.

Diffraction grating - an optical device, which is a collection of a large number of parallel, usually equidistant from each other, slots.

A diffraction grating can be obtained by applying opaque scratches (strokes) to a glass plate. Unscratched places - cracks - will let light through; strokes corresponding to the gap between the slits scatter and do not transmit light. The cross section of such a diffraction grating ( a) and its symbol (b) shown in fig. 19.12. The total slot width a and interval b between the cracks is called permanent or grating period:

c = a + b.(19.28)

If a beam of coherent waves falls on the grating, then secondary waves traveling in all possible directions will interfere, forming a diffraction pattern.

Let a plane-parallel beam of coherent waves fall normally on the grating (Fig. 19.13). Let us choose some direction of the secondary waves at an angle a with respect to the normal to the grating. The rays coming from the extreme points of two adjacent slots have a path difference d = A"B". The same path difference will be for secondary waves coming from respectively located pairs of points of adjacent slots. If this path difference is a multiple of an integer number of wavelengths, then interference will cause main highs, for which the condition ÷ A "B¢÷ = ± k l , or

With sin a = ± k l , (19.29)

where k = 0,1,2,... — order of principal maxima. They are symmetrical about the central (k= 0, a = 0). Equality (19.29) is the basic formula of a diffraction grating.

Between the main maxima minima (additional) are formed, the number of which depends on the number of all lattice slots. Let us derive a condition for additional minima. Let the path difference of secondary waves traveling at an angle a from the corresponding points of neighboring slots be equal to l /N, i.e.

d= With sin a=l /N,(19.30)

where N is the number of slits in the diffraction grating. This path difference is 5 [see (19.9)] corresponds to the phase difference Dj= 2 p /N.

If we assume that the secondary wave from the first slot has a zero phase at the moment of addition with other waves, then the phase of the wave from the second slot is equal to 2 p /N, from the third 4 p /N, from the fourth - 6p /N etc. The result of adding these waves, taking into account the phase difference, is conveniently obtained using a vector diagram: the sum N identical electric field strength vectors, the angle (phase difference) between any neighboring of which is 2 p /N, equals zero. This means that condition (19.30) corresponds to the minimum. With the path difference of the secondary waves from neighboring slots d = 2( l /N) or phase difference Dj = 2(2p/n) a minimum of interference of secondary waves coming from all slots will also be obtained, etc.

As an illustration, in fig. 19.14 shows a vector diagram corresponding to a diffraction grating consisting of six slits: etc. - vectors of intensity of the electric component of electromagnetic waves from the first, second, etc. slits. Five additional minima arising during interference (the sum of vectors is equal to zero) are observed at a phase difference of waves coming from neighboring slots of 60° ( a), 120° (b), 180° (in), 240° (G) and 300° (e).

Rice. 19.14

Thus, one can make sure that between the central and each first main maxima there is N-1 additional lows satisfying the condition

With sin a = ± l /N; 2l /N, ..., ±(N- 1)l /N.(19.31)

Between the first and second main maxima are also located N- 1 additional minima satisfying the condition

With sin a = ± ( N+ 1)l /N, ±(N+ 2)l /N, ...,(2N- 1)l /N,(19.32)

etc. Thus, between any two adjacent main maxima, there is N - 1 additional minimums.

With a large number of slits, individual additional minima hardly differ, and the entire space between the main maxima looks dark. The greater the number of slits in the diffraction grating, the sharper the main maxima. On fig. 19.15 are photographs of the diffraction pattern obtained from gratings with different numbers N slots (the constant of the diffraction grating is the same), and in Fig. 19.16 - intensity distribution graph.

Let us especially note the role of minima from one slit. In the direction corresponding to condition (19.27), each slot gives a minimum, so the minimum from one slot will be preserved for the entire lattice. If for some direction the minimum conditions for the gap (19.27) and the main maximum of the lattice (19.29) are simultaneously satisfied, then the corresponding main maximum will not arise. Usually they try to use the main maxima, which are located between the first minima from one slot, i.e., in the interval

arcsin(l /a) > a > - arcsin(l /a) (19.33)

When white or other non-monochromatic light falls on a diffraction grating, each main maximum, except for the central one, will be decomposed into a spectrum [see Fig. (19.29)]. In this case k indicates spectrum order.

Thus, the grating is a spectral device, therefore, characteristics are essential for it, which make it possible to evaluate the possibility of distinguishing (resolving) spectral lines.

One of these characteristics is angular dispersion determines the angular width of the spectrum. It is numerically equal to the angular distance da between two spectral lines whose wavelengths differ by one (dl. = 1):

D= da/dl.

Differentiating (19.29) and using only positive values ​​of quantities, we obtain

With cos a da = .. k dl.

From the last two equalities we have

D = ..k /(c cos a). (19.34)

Since small diffraction angles are usually used, cos a » 1. Angular dispersion D the higher the higher the order k spectrum and the smaller the constant With diffraction grating.

The ability to distinguish close spectral lines depends not only on the width of the spectrum, or angular dispersion, but also on the width of the spectral lines, which can be superimposed on each other.

It is generally accepted that if between two diffraction maxima of the same intensity there is a region where the total intensity is 80% of the maximum, then the spectral lines to which these maxima correspond are already resolved.

In this case, according to JW Rayleigh, the maximum of one line coincides with the nearest minimum of the other, which is considered the criterion for resolution. On fig. 19.17 intensity dependences are shown I individual lines on the wavelength (solid curve) and their total intensity (dashed curve). It is easy to see from the figures that the two lines are unresolved ( a) and limiting resolution ( b), when the maximum of one line coincides with the nearest minimum of the other.

Spectral line resolution is quantified resolution, equal to the ratio of the wavelength to the smallest interval of wavelengths that can still be resolved:

R= l./Dl.. (19.35)

So, if there are two close lines with wavelengths l 1 ³ l 2, Dl = l 1 - l 2 , then (19.35) can be approximately written as

R= l 1 /(l 1 - l 2), or R= l 2 (l 1 - l 2) (19.36)

The condition of the main maximum for the first wave

With sin a = k l 1 .

It coincides with the nearest minimum for the second wave, the condition of which is

With sin a = k l 2 + l 2 /N.

Equating the right-hand sides of the last two equalities, we have

k l 1 = k l 2 + l 2 /N,k(l 1 - l 2) = l 2 /N,

whence [taking into account (19.36)]

R =k N .

So, the resolving power of the diffraction grating is the greater, the greater the order k spectrum and number N strokes.

Consider an example. In the spectrum obtained from a diffraction grating with the number of slots N= 10 000, there are two lines near the wavelength l = 600 nm. At what is the smallest wavelength difference Dl these lines differ in the spectrum of the third order (k = 3)?

To answer this question, we equate (19.35) and (19.37), l/Dl = kN, whence Dl = l/( kN). Substituting numerical values ​​into this formula, we find Dl = 600 nm / (3.10,000) = 0.02 nm.

So, for example, lines with wavelengths of 600.00 and 600.02 nm are distinguishable in the spectrum, and lines with wavelengths of 600.00 and 600.01 nm are indistinguishable

We derive the formula for the diffraction grating for the oblique incidence of coherent rays (Fig. 19.18, b is the angle of incidence). The conditions for the formation of the diffraction pattern (lens, screen in the focal plane) are the same as for normal incidence.

Let's draw perpendiculars A "B falling rays and AB" to secondary waves propagating at an angle a to the perpendicular raised to the grating plane. From fig. 19.18 it is clear that to the position A¢B rays have the same phase, from AB" and then the phase difference of the beams is preserved. Therefore, the path difference is

d \u003d BB "-AA".(19.38)

From D AA"B we have AA¢= AB sin b = With sinb. From D BB"A find BB" = AB sin a = With sin a. Substituting expressions for AA¢ and BB" in (19.38) and taking into account the condition for the main maxima, we have

With(sin a - sin b) = ± kl. (19.39)

The central main maximum corresponds to the direction of the incident rays (a=b).

Along with transparent diffraction gratings, reflective gratings are used, in which strokes are applied to a metal surface. The observation is carried out in reflected light. Reflective diffraction gratings made on a concave surface are capable of forming a diffraction pattern without a lens.

In modern diffraction gratings, the maximum number of lines is more than 2000 per 1 mm, and the grating length is more than 300 mm, which gives the value N about a million.