Diffraction spectrum. School Encyclopedia Spectral decomposition of white light diffraction grating

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, light diffraction is the rounding of light rays around the boundaries of opaque bodies, 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 (illuminated) 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.

Decision

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.

Decision

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.

A light breeze ran up, and ripples (a wave of small length and amplitude) ran across the surface of the water, meeting various obstacles on its way, above the surface of the water, plant stems, boughs of a tree. On the leeward side, behind the bough, the water is calm, there is no unrest, and the wave bends around the stems of plants.

DIFFRACTION OF WAVES (from lat. diffractus- broken) rounding waves of various obstacles. Wave diffraction is inherent in any wave motion; occurs if the dimensions of the obstacle are less than or comparable to the wavelength.

Diffraction of light is the phenomenon of deviation of light from the rectilinear direction of propagation when passing near obstacles. During diffraction, light waves bend around the boundaries of opaque bodies and can penetrate into the region of a geometric shadow.
An obstacle can be a hole, a gap, the edge of an opaque barrier.

Diffraction of light is manifested in the fact that light penetrates into the region of a geometric shadow in violation of the law of rectilinear propagation of light. For example, passing light through a small round hole, we find on the screen a bright spot of a larger size than one would expect in a rectilinear propagation.

Due to the fact that the wavelength of light is small, the angle of deviation of light from the direction of rectilinear propagation is small. Therefore, to clearly observe diffraction, you need to use very small obstacles or place the screen far from obstacles.

Diffraction is explained on the basis of the Huygens-Fresnel principle: each point of the wave front is a source of secondary waves. The diffraction pattern is the result of the interference of secondary light waves.

The waves formed at points A and B are coherent. What is observed on the screen at points O, M, N?

Diffraction is well observed only at a distance

where R are the characteristic dimensions of the obstacle. At smaller distances, the laws of geometric optics apply.

The phenomenon of diffraction imposes a limitation on the resolution of optical instruments (for example, a telescope). As a result, a complex diffraction pattern is formed in the focal plane of the telescope.

Diffraction grating - is a collection of a large number of narrow, parallel, closely spaced areas (slits) transparent to light, located in the same plane, separated by opaque gaps.

Diffraction gratings are either reflective or transmissive. The principle of their action is the same. The grating is made using a dividing machine that applies periodic parallel strokes on a glass or metal plate. A good diffraction grating contains up to 100,000 lines. Denote:

a is the width of the slits (or reflective stripes) that are transparent to light;
b- the width of opaque gaps (or areas that scatter light).
Value d = a + b is called the period (or constant) of the diffraction grating.

The diffraction pattern created by the grating is complex. It exhibits main maxima and minima, secondary maxima, and additional minima due to slit diffraction.
Of practical importance in the study of spectra using a diffraction grating are the main maxima, which are narrow bright lines in the spectrum. If white light falls on a diffraction grating, the waves of each color included in its composition form their diffraction maxima. The position of the maximum depends on the wavelength. Zero highs (k = 0 ) for all wavelengths are formed in the directions of the incident beam (β = 0 ), so there is a central bright band in the diffraction spectrum. To the left and to the right of it, colored diffraction maxima of different orders are observed. Since the angle of diffraction is proportional to the wavelength, red rays are deflected more than violet ones. Note the difference in the order of colors in the diffraction and prism spectra. Due to this, a diffraction grating is used as a spectral apparatus, along with a prism.

When passing through a diffraction grating, a light wave of length λ on the screen will give a sequence of intensity minima and maxima. The intensity maxima will be observed at the angle β:

where k is an integer, called the order of the diffraction maximum.

Basic summary:

DEFINITION

Diffraction spectrum called the intensity distribution on the screen, which is obtained as a result of diffraction.

In this case, the main part of the light energy is concentrated in the central maximum.

If we take a diffraction grating as the device under consideration, with the help of which diffraction is carried out, then from the formula:

(where d is the grating constant; is the diffraction angle; is the wavelength of light; . is an integer), it follows that the angle at which the main maxima occur is related to the wavelength of the light incident on the grating (light falls on the grating normally). This means that the intensity maxima produced by light of different wavelengths occur in different places in the observation space, which makes it possible to use a diffraction grating as a spectral instrument.

If white light falls on a diffraction grating, then all the maxima, with the exception of the central maximum, are decomposed into a spectrum. It follows from formula (1) that the position of the maximum of the th order can be determined as:

It follows from expression (2) that with increasing wavelength, the distance from the central maximum to the maximum with number m increases. It turns out that the violet part of each main maximum will be turned towards the center of the diffraction pattern, and the red part will be outward. It should be remembered that in the spectral decomposition of white light, violet rays are deflected more than red ones.

A diffraction grating is used as a simple spectral instrument that can be used to determine the wavelength. If the grating period is known, then finding the wavelength of light will be reduced to measuring the angle that corresponds to the direction to the chosen line of the order of the spectrum. Typically, spectra of the first or second order are used.

It should be noted that high-order diffraction spectra are superimposed on each other. Thus, when decomposing white light, the spectra of the second and third orders already partially overlap.

Diffraction and dispersion decomposition into a spectrum

With the help of diffraction, as well as dispersion, a light beam can be decomposed into components. However, there are fundamental differences in these physical phenomena. So, the diffraction spectrum is the result of light bending around obstacles, for example, darkened zones near a diffraction grating. This spectrum spreads evenly in all directions. The violet part of the spectrum is facing the center. A dispersion spectrum can be obtained by passing light through a prism. The spectrum is stretched in the violet direction and compressed in the red direction. The violet part of the spectrum occupies a greater width than the red part. Red rays in the spectral decomposition deviate less than violet, which means that the red part of the spectrum is closer to the center.

The maximum order of the spectrum during diffraction

Using formula (2) and taking into account that it cannot be more than one, we get that:

Examples of problem solving

EXAMPLE 1

Exercise	Light with a wavelength equal to = 600 nm falls on a diffraction grating perpendicular to its plane, the grating period is m. What is the highest order of the spectrum? What is the number of maxima in this case?
Decision	The basis for solving the problem is the formula for the maxima that are obtained by diffraction on a grating under given conditions: The maximum value of m will be obtained at Let's carry out calculations if =600 nm=m: The number of maxima (n) will be equal to:
Answer	=3;

EXAMPLE 2

Exercise	A monochromatic beam of light is incident on a diffraction grating perpendicular to its plane. A screen is located at a distance L from the grating, and a spectral diffraction pattern is formed on it using a lens. It is obtained that the first main diffraction maximum is located at a distance x from the central one (Fig. 1). What is the grating constant (d)?
Decision	Let's make a drawing.

the phenomenon of dispersion when white light is passed through a prism (Fig. 102). When leaving the prism, white light is decomposed into seven colors: red, orange, yellow, green, blue, indigo, violet. The red light is deflected the least, the violet the most. This suggests that glass has the highest refractive index for violet light and the lowest for red light. Light with different wavelengths propagates in a medium with different speeds: violet with the lowest, red with the highest, since n= c/v,

As a result of the passage of light through a transparent prism, an ordered arrangement of monochromatic electromagnetic waves of the optical range is obtained - the spectrum.

All spectra are divided into emission spectra and absorption spectra. The emission spectrum is created by luminous bodies. If a cold, non-radiating gas is placed in the path of rays incident on a prism, then dark lines appear against the background of the continuous spectrum of the source.

Light

Light is transverse waves

An electromagnetic wave is the propagation of an alternating electromagnetic field, and the strengths of the electric and magnetic fields are perpendicular to each other and to the line of wave propagation: electromagnetic waves are transverse.

polarized light

Polarized light is called light, in which the directions of oscillations of the light vector are ordered in some way.

The light falls from the environment with a large showing. Refraction into a medium with less

Methods for obtaining linear polarized light

Birefringent crystals are used to produce linearly polarized light in two ways. The first one uses crystals that do not have dichroism; prisms are made from them, composed of two triangular prisms with the same or perpendicular orientation of the optical axes. In them, either one beam deviates to the side, so that only one linearly polarized beam comes out of the prism, or both beams come out, but separated by a large angle. In the second way is used strongly dichroic crystals, in which one of the rays is absorbed, or thin films - polaroids in the form of sheets of a large area.

Brewster's Law

Brewster's law is a law of optics that expresses the relationship of the refractive index with such an angle at which the light reflected from the interface will be completely polarized in a plane perpendicular to the plane of incidence, and the refracted beam is partially polarized in the plane of incidence, and the polarization of the refracted beam reaches its greatest value. It is easy to establish that in this case the reflected and refracted rays are mutually perpendicular. The corresponding angle is called the Brewster angle.

Brewster's law: where n21 is the refractive index of the second medium relative to the first, θBr is the angle of incidence (Brewster angle)

Law of light reflection

The law of light reflection - establishes a change in the direction of the light beam as a result of a meeting with a reflective (mirror) surface: the incident and reflected rays lie in the same plane with the normal to the reflecting surface at the point of incidence, and this normal divides the angle between the rays into two equal parts. The widely used but less accurate formulation "angle of incidence equals angle of reflection" does not indicate the exact direction of reflection of the beam.

The laws of light reflection are two statements:

1. The angle of incidence is equal to the angle of reflection.

2. The incident ray, the reflected ray and the perpendicular restored at the point of incidence of the ray lie in the same plane.

Law of refraction

When light passes from one transparent medium to another, the direction of its propagation changes. This phenomenon is called refraction. The law of refraction of light determines the relative position of the incident beam, refracted and perpendicular to the interface between two media.

The law of refraction of light determines the relative position of the incident beam AB (Fig. 6), refracted by DB and the perpendicular CE to the media interface, restored at the point of incidence. Angle a is called the angle of incidence, and angle b is called the angle of refraction.

White and any complex light can be considered as a superposition of monochromatic waves with different wavelengths, which behave independently when diffracted by a grating. Accordingly, conditions (7), (8), (9) for each wavelength will be satisfied at different angles, i.e. the monochromatic components of the light incident on the grating will be spatially separated. The set of main diffraction maxima of the m-th order (m≠0) for all monochromatic components of the light incident on the grating is called the diffraction spectrum of the m-th order.

The position of the main zero-order diffraction maximum (central maximum φ=0) does not depend on the wavelength, and for white light it will look like a white stripe. The diffraction spectrum of the m-th order (m≠0) for incident white light has the form of a colored band in which all the colors of the rainbow occur, and for complex light in the form of a set of spectral lines corresponding to monochromatic components incident on the diffraction grating of complex light (Fig. 2).

A diffraction grating as a spectral device has the following main characteristics: resolution R, angular dispersion D, and dispersion region G.

The smallest difference between the wavelengths of two spectral lines δλ, at which the spectral apparatus resolves these lines, is called the spectral resolvable distance, and the value is the resolution of the apparatus.

Spectral resolution condition (Rayleigh criteria):

Spectral lines with close wavelengths λ and λ' are considered resolved if the main maximum of the diffraction pattern for one wavelength coincides in position with the first diffraction minimum in the same order for another wave.

According to the Rayleigh criterion, we get:

, (10)

where N is the number of grooves (slots) of the grating involved in diffraction, m is the order of the diffraction spectrum.

And the maximum resolution:

, (11)

where L is the total width of the diffraction grating.

Angular dispersion D is a value defined as the angular distance between directions for two spectral lines that differ in wavelength by 1

and
.

From the condition of the main diffraction maximum

(12)

Dispersion region G is the maximum width of the spectral interval Δλ, at which there is still no overlap of the diffraction spectra of neighboring orders

, (13)

where λ is the initial boundary of the spectral interval.

Description of the installation.

The problem of determining the wavelength using a diffraction grating is reduced to measuring the diffraction angles. These measurements in this work are made by a goniometer (goniometer).

The goniometer (Fig. 3) consists of the following main parts: a base with a table (I), on which the main scale is applied in degrees (limb -L); a collimator (II) rigidly fixed to the base and an optical tube (III) fixed to a ring that can rotate about an axis passing through the center of the table. Two verniers N are located opposite each other on the ring.

The collimator is a tube with a lens F 1, in the focal plane of which there is a narrow slit S, about 1 mm wide, and a movable eyepiece O with an index thread H.

Installation data:

The price of the smallest division of the main scale of the goniometer is 1 0 .

The price of division of the vernier is 5.

Grating constant
, [mm].

A mercury lamp (DRSH 250 - 3), which has a discrete emission spectrum, is used as a light source in laboratory work. In this work, the wavelengths of the brightest spectral lines are measured: blue, green, and two yellow (Fig. 2b).