What does a diffraction grating look like? What is a diffraction grating: definition, length and operating principle

When a parallel beam of monochromatic light is incident perpendicularly (normally) onto a diffraction grating on a screen in the focal plane of a collecting lens located parallel to the diffraction grating, a non-uniform pattern of illumination distribution in different areas of the screen (diffraction pattern) is observed.

Main the maxima of this diffraction pattern satisfy the following conditions:

Where n- order of the main diffraction maximum, d - constant (period) diffraction grating, λ - wavelength of monochromatic light,φn- the angle between the normal to the diffraction grating and the direction to the main diffraction maximum n th order.

Constant (period) of the diffraction grating length l

where N - the number of slits (lines) per section of the diffraction grating with length I.

Along with the wavelengthfrequently used frequency v waves.

For electromagnetic waves(light) in a vacuum

where c = 3 * 10 8 m/s - speed propagation of light in a vacuum.

Let us select from formula (1) the most difficult mathematically determined formulas for the order of the main diffraction maxima:

where denotes the whole part numbers d*sin(φ/λ).

Underdetermined analogues of formulas (4, a, b) without the symbol [...] on the right-hand sides contain the potential danger of substituting a physically based selection operation integer part of a number operation rounding a number d*sin(φ/λ) to an integer value according to formal mathematical rules.

Subconscious tendency (false trail) to substitute the operation of isolating an integer part of a number d*sin(φ/λ) rounding operation

This number to an integer value according to mathematical rules is further enhanced when it comes to test tasks type B to determine the order of the main diffraction maxima.

In any type B test tasks, the numerical values ​​of the required physical quantities by agreementrounded to integer values. However, in the mathematical literature there are no uniform rules for rounding numbers.

In the reference book by V. A. Gusev, A. G. Mordkovich on mathematics for students and Belarusian textbook L. A. Latotina, V. Ya. Chebotarevsky in mathematics for fourth grade give essentially the same two rules for rounding numbers. They are formulated as follows: “When rounding decimal Before any digit, all digits following this digit are replaced by zeros, and if they are after the decimal point, they are discarded. If the first digit following this digit is greater than or equal to five, then the last remaining digit is increased by 1. If the first digit following this digit is less than 5, then the last remaining digit is not changed."

In M. Ya. Vygodsky’s reference book on elementary mathematics, which has gone through twenty-seven (!) editions, it is written (p. 74): “Rule 3. If the number 5 is discarded, and there are no significant figures behind it, then rounding is done to the nearest even number, i.e., the last digit stored remains unchanged if it is even, and is enhanced (increased by 1) if it is odd.”

In view of the existence of various rules for rounding numbers, the rules for rounding decimal numbers should be explicitly formulated in the “Instructions for Students” attached to the tasks of centralized testing in physics. This proposal acquires additional relevance, since not only citizens of Belarus and Russia, but also other countries, enter Belarusian universities and undergo mandatory testing, and it is certainly unknown what rules for rounding numbers they used when studying in their countries.

In all cases, we will round decimal numbers according to rules, given in , .

After a forced retreat, let us return to the discussion of the physical issues under consideration.

Taking into account zero ( n= 0) of the main maximum and the symmetrical arrangement of the remaining main maxima relative to it total observed main maxima from the diffraction grating are calculated using the formulas:

If the distance from the diffraction grating to the screen on which the diffraction pattern is observed is denoted by H, then the coordinate of the main diffraction maximum n th order when counting from the zero maximum is equal to

If then (radians) and

Problems on the topic under consideration are often offered during physics tests.

Let's start the review by looking at Russian tests used by Belarusian universities at initial stage, when testing in Belarus was optional and carried out by separate educational institutions at your own peril and risk as an alternative to the usual individual written and oral form of entrance examinations.

Test No. 7

A32. The highest spectral order that can be observed by diffraction of light with a wavelength λ on a diffraction grating with a period d=3.5λ equals

1) 4; 2) 7; 3) 2; 4) 8; 5) 3.

Solution

Monochromaticno light spectra out of the question. In the problem statement, we should talk about the main diffraction maximum of the highest order when monochromatic light is perpendicularly incident on the diffraction grating.

According to formula (4, b)

From an underdetermined condition

on the set of integers, after rounding we getn max=4.

Only due to the mismatch of the integer part of the number d/λ with its rounded integer value the correct solution is ( n max=3) differs from incorrect (n max=4) at the test level.

An amazing miniature, despite the flaws in the wording, with a delicately verified false trail across all three versions of rounding numbers!

A18. If the diffraction grating constant d= 2 µm, then for normally incident on the grating white light 400 nm<λ < 700 нм наибольший полностью наблюдаемый порядок спектра равен

1)1; 2)2; 3)3; 4)4; 5)5.

Solution

It's obvious that n sp =min(n 1max, n 2max)

According to formula (4, b)

Rounding numbers d/λ to integer values ​​according to the rules - , we get:

Due to the fact that the integer part of the number d/λ 2 differs from its rounded integer value, this task allows you to objectively distinguish the correct solution(n sp = 2) from incorrect ( n sp =3). A great problem with one false lead!

CT 2002 Test No. 3

AT 5. Find the highest spectral order for the yellow Na line (λ = 589 nm), if the diffraction grating constant is d = 2 µm.

Solution

The task is formulated scientifically incorrectly. Firstly, when illuminating the diffraction gratingmonochromaticWith light, as noted above, there can be no talk of a spectrum (spectra). The problem statement should deal with the highest order of the main diffraction maximum.

Secondly, the task conditions should indicate that light falls normally (perpendicularly) onto a diffraction grating, since only this particular case is considered in the physics course of secondary educational institutions. This limitation cannot be considered as implied by default: all restrictions must be specified in tests obviously! Test tasks must be self-sufficient, scientifically correct tasks.

The number 3.4, rounded to an integer value according to the rules of arithmetic - , also gives 3. Exactly therefore, this task should be considered simple and, by and large, unsuccessful, since at the test level it does not allow one to objectively distinguish the correct solution, determined by the integer part of the number 3.4, from the incorrect solution, determined by the rounded integer value of the number 3.4. The difference is revealed only with a detailed description of the solution process, which is done in this article.

Addendum 1. Solve the above problem by replacing in its condition d=2 µm by d= 1.6 µm. Answer: n max = 2.

CT 2002 Test 4

AT 5. Light from a gas-discharge lamp is directed onto the diffraction grating. The diffraction spectra of the lamp radiation are obtained on the screen. Line with wavelength λ 1 = 510 nm in the fourth order spectrum coincides with the wavelength line λ 2 in the third order spectrum. What is it equal to λ 2(in [nm])?

Solution

In this problem, the main interest is not the solution of the problem, but the formulation of its conditions.

When illuminated by a diffraction gratingnon-monochromatic light( λ 1 , λ 2) quite it is natural to talk (write) about diffraction spectra, which in principle do not exist when illuminating a diffraction gratingmonochromatic light.

The task conditions should indicate that the light from the gas-discharge lamp falls normally on the diffraction grating.

In addition, the philological style of the third sentence in the task condition should be changed. The turnover of the "line with wavelength" hurts the ear λ "" , it could be replaced by “a line corresponding to radiation with a wavelength λ "" or in shorter form - “a line corresponding to the wavelength λ "" .

Test formulations must be scientifically correct and literary impeccable. Tests are formulated completely differently from research and Olympiad tasks! In tests, everything should be precise, specific, unambiguous.

Taking into account the above clarification of the task conditions, we have:

Since according to the conditions of the task That

CT 2002 Test No. 5

AT 5. Find the highest order of the diffraction maximum for the yellow sodium line with a wavelength of 5.89·10 -7 m if the diffraction grating period is 5 µm.

Solution

Compared to task AT 5 from test No. 3 TsT 2002, this task is formulated more precisely, however, in the task conditions, we should talk not about the “diffraction maximum”, but about “ main diffraction maximum".

Along with main diffraction maxima there are always also secondary diffraction maxima. Without explaining this nuance in a school physics course, it is all the more necessary to strictly adhere to the established scientific terminology and talk only about the main diffraction maxima.

In addition, it should be noted that the light falls normally on the diffraction grating.

Taking into account the above clarifications

From an undefined condition

according to the rules of mathematical rounding of the number 8.49 to an integer value, we again obtain 8. Therefore, this task, like the previous one, should be considered unsuccessful.

Addendum 2. Solve the above problem by replacing in its condition d =5 µm per (1=A µm. Answer:n max=6.)

RIKZ manual 2003 Test No. 6

AT 5. If the second diffraction maximum is located at a distance of 5 cm from the center of the screen, then when the distance from the diffraction grating to the screen increases by 20%, this diffraction maximum will be located at a distance... cm.

Solution

The condition of the task is formulated unsatisfactorily: instead of “diffraction maximum” you need “main diffraction maximum”, instead of “from the center of the screen” - “from the zero main diffraction maximum”.

As can be seen from the above figure,

From here

RIKZ manual 2003 Test No. 7

AT 5. Determine the highest spectral order in a diffraction grating having 500 lines per 1 mm when illuminated with light with a wavelength of 720 nm.

Solution

The conditions of the task are formulated extremely unsuccessfully from a scientific point of view (see clarifications of tasks No. 3 and 5 from the CT 2002).

There are also complaints about the philological style of wording the assignment. Instead of the phrase “in a diffraction grating” one would have to use the phrase “from a diffraction grating”, and instead of “light with a wavelength” - “light whose wavelength”. The wavelength is not the load on the wave, but its main characteristic.

Taking into account clarifications

Using all three rules for rounding numbers above, rounding 2.78 to a whole number results in 3.

The last fact, even with all the shortcomings in the formulation of the task conditions, makes it interesting, since it allows us to distinguish the correct (n max=2) and incorrect (n max=3) solutions.

Many tasks on the topic under consideration are contained in the CT 2005.

In the conditions of all these tasks (B1), you need to add the keyword “main” before the phrase “diffraction maximum” (see comments to task B5 CT 2002 Test No. 5).

Unfortunately, in all versions of the V1 TsT 2005 tests, the numerical values d(l,N) And λ poorly chosen and always given in fractions

the number of “tenths” is less than 5, which does not allow at the test level to distinguish the operation of separating an integer part of a fraction (correct decision) from the operation of rounding a fraction to an integer value (false trace). This circumstance calls into question the advisability of using these tasks to objectively test applicants’ knowledge on the topic under consideration.

It seems that the test compilers were carried away, figuratively speaking, by preparing various “side dishes for the dish”, without thinking about improving the quality of the main component of the “dish” - the selection of numerical values d(l,N) And λ in order to increase the number of "tenths" in fractions d/ λ=l/(N* λ).

CT 2005 Option 4

IN 1. On a diffraction grating whose periodd 1=1.2 µm, a normally parallel beam of monochromatic light with a wavelength of λ =500 nm. If we replace it with a lattice whose periodd 2=2.2 µm, then the number of maxima will increase by... .

Solution

Instead of "light with wavelength λ"" you need "light wavelength λ "" . Style, style and more style!

Because

then, taking into account the fact that X is const, and d 2 >di,

According to formula (4, b)

Hence, ΔN total max =2(4-2)=4

When rounding the numbers 2.4 and 4.4 to integer values, we also get 2 and 4, respectively. For this reason, this task should be considered simple and even unsuccessful.

Addendum 3. Solve the above problem by replacing in its condition λ =500 nm at λ =433 nm (blue line in the hydrogen spectrum).

Answer: ΔN total. max=6

CT 2005 Option 6

IN 1. On a diffraction grating with a period d= A normally parallel beam of monochromatic light with a wavelength of λ =750 nm. Number of maxima that can be observed within an angle A=60°, the bisector of which is perpendicular to the plane of the lattice, is equal to... .

Solution

The phrase "light with a wavelength λ " has already been discussed above in CT 2005, option 4.

The second sentence in the conditions of this task could be simplified and written as follows: “The number of observed main maxima within the angle a = 60°” and further according to the text of the original task.

It's obvious that

According to formula (4, a)

According to formula (5, a)

This task, like the previous one, does not allow objectively determine the level of understanding of the topic being discussed by applicants.

Appendix 4. Complete the above task, replacing in its condition λ =750 nm at λ = 589 nm (yellow line in the sodium spectrum). Answer: N o6ш =3.

CT 2005 Option 7

IN 1. On a diffraction grating havingN 1- 400 strokes per l=1 mm in length, a parallel beam of monochromatic light with a wavelength of λ =400 nm. If it is replaced with a lattice havingN 2=800 strokes per l=1 mm in length, then the number of diffraction maxima will decrease by... .

Solution

We will omit the discussion of inaccuracies in the wording of the task, since they are the same as in previous tasks.

From formulas (4, b), (5, b) it follows that

The diffraction grating device is based on the property of diffraction. A diffraction grating is a collection of a very large number of narrow slits that are separated by opaque spaces.

The general view of the diffraction grating is shown in the following figure.

Lattice period and principle of its operation

The grating period is the sum of the width of one slit and one opaque gap. The letter d is used for designation. The diffraction grating period often fluctuates around 10 µm. Let's look at how a diffraction grating works and why it is needed.

A plane monochromatic wave is incident on a diffraction grating. The length of this wave is equal to λ. Secondary sources located in the grating slits create light waves that will travel in all directions. We will look for conditions under which waves coming from different slits will reinforce each other.

To do this, consider the propagation of waves in any one direction. Let these be waves propagating at an angle φ.
The difference in path between the waves will be equal to the segment AC. If an integer number of wavelengths can be placed in this segment, then the waves from all the slits will overlap each other and reinforce each other.

The length Ac can be found from the right triangle ABC.

AC = AB*sin(φ) = d*sin(φ).

We can write down the condition for the angle at which the maxima will be observed:

d*sin(φ) = ±k*λ.

Here k is any positive integer or 0. A quantity that determines the order of the spectrum.

A collecting lens is placed behind the grating. With its help, rays running parallel are focused. If the angle satisfies the maximum condition, then on the screen it determines the position of the main maxima. Since the position of the maxima will depend on the wavelength, the grating will decompose white light into a spectrum. This is shown in the following figure.

picture

picture

Between the maximum there will be intervals of minimum illumination. The greater the number of slits, the more clearly defined the maxima will be, and the greater the width of the minima.

A diffraction grating is used to accurately determine the wavelength. With a known grating period, it is very easy to determine the wavelength; you just need to measure the direction angle φ to the maximum.

DEFINITION

Diffraction grating- This is the simplest spectral device. It contains a system of slits that separate opaque spaces.

Diffraction gratings are divided into one-dimensional and multidimensional. A one-dimensional diffraction grating consists of parallel light-transparent sections of the same width, which are located in the same plane. Transparent areas are separated by opaque spaces. Using these gratings, observations are carried out in transmitted light.

There are reflective diffraction gratings. Such a grating is, for example, a polished (mirror) metal plate onto which strokes are applied using a cutter. The result is areas that reflect light and areas that scatter light. Observation using such a grating is carried out in reflected light.

The diffraction pattern on the grating is the result of mutual interference of waves that come from all the slits. Consequently, with the help of a diffraction grating, multi-beam interference of coherent beams of light that have undergone diffraction and coming from all slits is realized.

Diffraction grating period

If we denote the width of the slit in the grating as a, the width of the opaque section as b, then the sum of these two parameters is the grating period (d):

The diffraction grating period is sometimes also called the diffraction grating constant. The period of a diffraction grating can be defined as the distance through which the lines on the grating are repeated.

The diffraction grating constant can be found if the number of lines (N) that the grating has per 1 mm of its length is known:

The period of the diffraction grating is included in the formulas that describe the diffraction pattern on it. Thus, if a monochromatic wave is incident on a one-dimensional diffraction grating perpendicular to its plane, then the main intensity minima are observed in the directions determined by the condition:

where is the angle between the normal to the grating and the direction of propagation of diffracted rays.

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

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

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

The value of the sine cannot exceed one, therefore, the number of main maxima (m):

Examples of problem solving

EXAMPLE 1

One of the important optical instruments that has found its application in the analysis of emission and absorption spectra is a diffraction grating. This article provides information that allows you to understand what a diffraction grating is, what the principle of its operation is, and how you can independently calculate the position of the maxima in the diffraction pattern that it produces.

At the beginning of the 19th century, the English scientist Thomas Young, studying the behavior of a monochromatic beam of light when it was divided in half by a thin plate, obtained a diffraction pattern. It was a sequence of bright and dark stripes on the screen. Using the concept of light as a wave, Jung correctly explained the results of his experiments. The picture he observed arose due to the phenomena of diffraction and interference.

Diffraction is understood as the curvature of the rectilinear path of wave propagation when it hits an opaque obstacle. Diffraction can occur as a result of a wave bending around an obstacle (this is possible if the wavelength is much larger than the obstacle) or as a result of trajectory curvature when the size of the obstacle is comparable to the wavelength. An example for the latter case is the penetration of light into cracks and small round holes.

The phenomenon of interference consists of the superposition of some waves on others. The result of this superposition is a bending of the resulting sinusoidal waveform. Special cases of interference are either maximum amplitude enhancement, when two waves arrive in the considered zone of space in the same phase, or complete attenuation of the wave process, when both waves meet in a given zone in antiphase.

The described phenomena allow us to understand what a diffraction grating is and how it works.

Diffraction grating

The name itself says what a diffraction grating is. It is an object that consists of periodically alternating transparent and opaque stripes. This can be achieved by gradually increasing the number of slits onto which the wave front falls. This concept is generally applicable for any wave, but it has found use only for the region of visible electromagnetic radiation, that is, for light.

A diffraction grating is usually characterized by three main parameters:

• The period d is the distance between the two slits through which light passes. Since the wavelengths of light lie in the range of several tenths of a micrometer, the value of d is of the order of 1 micrometer.
• The lattice constant a is the number of transparent slits that are located along the length of 1 mm of the lattice. The lattice constant is the inverse of the period d. Its typical values ​​are 300-600 mm-1. Typically, the value of a is written on the diffraction grating.
• The total number of slits is N. This value can be easily obtained by multiplying the length of the diffraction grating by its constant. Since typical lengths are several centimeters, each grating contains about 10-20 thousand slits.

Transparent and reflective grilles

It was described above what a diffraction grating is. Now let's answer the question of what it really is. There are two types of such optical objects: transparent and reflective.

A transparent grille is a thin glass plate or a transparent plastic plate on which strokes are applied. The lines of a diffraction grating are an obstacle to light; it cannot pass through them. The width of the stroke is the aforementioned period d. The remaining transparent gaps between the strokes act as slits. When performing laboratory work, this type of grating is used.

A reflective grid is a polished metal or plastic plate on which, instead of strokes, grooves of a certain depth are applied. Period d is the distance between grooves. Reflective gratings are often used in the analysis of emission spectra because their design allows the intensity of the diffraction pattern maxima to be distributed in favor of higher order maxima. The optical CD is a prime example of this type of diffraction grating.

Operating principle of the grid

For example, consider a transparent optical device. Let us assume that light with a flat front is incident on the diffraction grating. This is a very important point, since the formulas below take into account that the wavefront is flat and parallel to the plate itself (Fraunhofer diffraction). The strokes distributed according to a periodic law introduce a disturbance into this front, as a result of which at the exit from the plate a situation is created as if many secondary coherent radiation sources are operating (the Huygens-Fresnel principle). These sources lead to diffraction.

From each source (the gap between the lines) a wave propagates, which is coherent with all other N-1 waves. Now suppose that a screen is placed at some distance from the plate (the distance should be sufficient so that the Fresnel number is much less than one). If you look at the screen along a perpendicular drawn to the center of the plate, then as a result of the interference superposition of waves from these N sources, for some angles θ, bright stripes will be observed, between which there will be a shadow.

Since the condition of the interference maxima is a function of wavelength, if the light incident on the plate was white, multi-colored bright stripes would appear on the screen.

Basic formula

As mentioned, a plane wave front incident on a diffraction grating is displayed on the screen in the form of bright stripes separated by a shadow region. Each bright band is called a maximum. If we consider the condition for the amplification of waves arriving in the region under consideration in the same phase, then we can obtain a formula for the maxima of the diffraction grating. It looks like this:

Where θ m are the angles between the perpendicular to the center of the plate and the direction to the corresponding maximum line on the screen. The quantity m is called the order of the diffraction grating. It accepts integer values ​​and zero, that is, m = 0, ±1, 2, 3 and so on.

Knowing the grating period d and the wavelength λ that falls on it, the position of all maxima can be calculated. Note that the maxima calculated using the formula above are called the main ones. In fact, between them there is a whole set of weaker maxima, which are often not observed in experiment.

You should not think that the picture on the screen does not depend on the width of each slit on the diffraction plate. The width of the slit does not affect the position of the maxima, but it does affect their intensity and width. Thus, with a decrease in the gap (with an increase in the number of lines on the plate), the intensity of each maximum decreases, and its width increases.

Diffraction grating in spectroscopy

Having dealt with the questions of what a diffraction grating is and how to find the maxima that it gives on the screen, it is interesting to analyze what will happen to white light if a plate is irradiated with it.

Let us write again the formula for the main maxima:

If we consider a specific order of diffraction (for example, m = 1), it is clear that the larger λ, the farther from the central maximum (m = 0) the corresponding bright line will be located. This means that white light is split into a series of rainbow colors that are displayed on the screen. Moreover, starting from the center, violet and blue colors will appear first, and then yellow, green will appear, and the farthest maximum of the first order will correspond to the red color.

The wavelength diffraction grating property is used in spectroscopy. When it is necessary to find out the chemical composition of a luminous object, for example, a distant star, its light is collected by mirrors and directed to a plate. By measuring the angles θ m, it is possible to determine all the wavelengths of the spectrum, and therefore the chemical elements that emit them.

Below is a video that demonstrates the ability of gratings with different N numbers to split light from a lamp.

The concept of "angular dispersion"

This value refers to changes in the angle of occurrence of the maximum on the screen. If we change the length of monochromatic light by a small amount, we get:

If the left and right sides of the equality in the formula for the main maxima are differentiated by θ m and λ, respectively, then we can obtain an expression for the dispersion. It will be equal to:

The dispersion must be known when determining the resolution of the plate.

What is resolution?

In simple terms, it is the ability of a diffraction grating to separate two waves with similar λ values ​​into two separate peaks on the screen. According to Lord Rayleigh's criterion, two lines can be distinguished if the angular distance between them is greater than half their angular width. The half-width of the line is determined by the formula:

Δθ 1/2 = λ/(N*d*cos(θ m))

The distinction between lines in accordance with the Rayleigh criterion is possible if:

Substituting the formula for dispersion and half-width, we obtain the final condition:

The resolution of the grating increases with the number of slits (lines) on it and with the increase in the diffraction order.

The solution of the problem

Let's apply the acquired knowledge to solve a simple problem. Let light fall on the diffraction grating. It is known that the wavelength is 450 nm, and the grating period is 3 μm. What is the maximum diffraction order that can be observed on a tap?

To answer the question, you need to substitute the data into the lattice equation. We get:

sin(θ m) = m*λ/d = 0.15*m

Since the sine cannot be greater than one, then we find that the maximum diffraction order for the specified conditions of the problem is 6.

What is a diffraction grating: definition, length and principle of operation - all about traveling to the site

How to find the period of a diffraction grating?

well it's a shame not to know

Apparently, it's just a number of units.
That is, it does not have any specific unit of measurement.
http://dic.academic.ru/dic.nsf/bse/84886/Diffraction
Well, at least here I read that R=mN, where m is just an integer, and N is again the number of slits, and since no units of measurement are implied by them, then one should expect some kind of unit of measurement from them works should not either.
The same follows from this formula “R=λ/dλ”: it’s like dividing time by the change in time - there will be just units, if my logic is correct.

• DIFFRACTION OF LIGHT

  in the narrow (most common) sense - the phenomenon of light rays bending around the contour of opaque bodies and, consequently, the penetration of light into the geometric region. shadows; in a broad sense - the manifestation of wave properties of light under conditions close to the conditions of applicability of the representation of geometric optics.
  In natural conditions of D. s. usually observed as a blurred, blurred boundary of the shadow of an object illuminated by a distant source. The most contrasting D. s. in spaces. areas where the ray flux density undergoes a sharp change (in the region of a caustic surface, focus, boundary of a geometric shadow, etc.). In laboratory conditions, it is possible to detect the structure of light in these areas, manifested in the alternation of light and dark (or colored) areas on the screen. Sometimes this structure is simple, as, for example, with D. s. on a diffraction grating, often very complex, e.g. in the focal area of ​​the lens. D. s. on bodies with sharp boundaries is used in instrumental optics and, in particular, determines the limit of optical capabilities. devices.
  First element. quantity theory D. s. French was developed. physicist O. Fresnel (1816), who explained it as a result of the interference of secondary waves (see HUYGENS - FRESNEL PRINCIPLE). Despite the shortcomings, the method of this theory has retained its importance, especially in calculations of an evaluative nature.
  The method consists of dividing the front of the incident wave, cut off by the edges of the screen, into Fresnel zones.
  Rice. 1. Diffraction rings when light passes: on the left - through a round hole, into which an even number of zones fits; on the right - around the round screen.
  It is believed that secondary light waves are not generated on the screen and the light field at the observation point is determined by the sum of contributions from all zones. If the hole in the screen leaves an even number of zones open (Fig. 1), then in the center of the diffraction. The picture turns out to be a dark spot, and with an odd number of zones - a light spot. In the center of the shadow from a round screen covering not too many Fresnel zones, a light spot is obtained. The magnitudes of the zone contributions to the light field at the observation point are proportional to the areas of the zones and slowly decrease with increasing zone number. Adjacent zones make contributions of opposite signs, since the phases of the waves emitted by them are opposite.
  The results of O. Fresnel's theory served as decisive proof of the wave nature of light and provided the basis for the theory of zone plates. There are two types of diffraction - Fresnel diffraction and Fraunhofer diffraction, depending on the relationship between the size of the body b, on which diffraction occurs, and the size of the Fresnel zone? (zl) (and therefore, depending from the distance z to the observation point). The Fresnel method is effective only when the size of the hole is comparable to the size of the Fresnel zone: b = ?(zl) (diffraction in converging beams). In this case, a small number of zones into which the spherical zone is divided. the wave in the hole determines the picture of the D. s. If the hole in the screen is smaller than the Fresnel zone (b<-?(zl), дифракции Фраунгофера), как, напр., при очень удалённых от экрана наблюдателя и источника света, то можно пренебречь кривизной фронта волны, считать её плоской и картину дифракции характеризовать угловым распределением интенсивности потока. При этом падающий параллельный пучок света на отверстии становится расходящимся с углом расходимости j = l/b. При освещении щели параллельным монохроматич. пучком света на экране получается ряд тёмных и светлых полос, быстро убывающих по интенсивности. Если свет падает перпендикулярно к плоскости щели, то полосы расположены симметрично относительно центр. полосы (рис. 2), а освещённость меняется вдоль экрана периодически с изменением j, обращаясь в нуль при углах j, для к-рых sinj=ml/b (m=1, 2, 3, . . .).
  Rice. 2. Fraunhofer diffraction by a slit.
  For intermediate values ​​of j, the illumination reaches a maximum. values. Ch. the maximum occurs at m=0 and sinj=0, i.e. j=0. As the slot width decreases, the center. the light stripe expands, and for a given slit width the position of the minima and maxima depends on l, i.e., the greater the l, the greater the distance between the stripes. Therefore, in the case of white light, there is a set of corresponding patterns for different colors; Ch. the maximum will be common to all l and is represented as a white stripe, turning into colored stripes with alternating colors from violet to red.
  In math. Fraunhofer diffraction is simpler than Fresnel diffraction. Fresnel's ideas were mathematically embodied by him. physicist G. Kirchhoff (1882), who developed the theory of boundary dynamic systems, used in practice. However, his theory does not take into account the vector nature of light waves and the properties of the screen material itself. Mathematically correct theory of D. s. on bodies requires solving complex boundary value problems of electric-magnetic scattering. waves that have solutions only for special cases.
  The first exact solution was obtained by him. physicist A. Sommerfeld (1894) for the diffraction of a plane wave by a perfectly conducting wedge. At distances greater than l from the tip of the wedge, Sommerfeld's result predicts a deeper penetration of light into the shadow region than follows from Kirchhoff's theory.
  Diffraction phenomena arise not only at the sharp boundaries of bodies, but also in extended systems. Such a voluminous D. s. is caused by large-scale dielectric inhomogeneities compared to l. permeability of the environment. In particular, volumetric D. s. occurs during the diffraction of light by ultrasound, in holograms in a turbulent environment and nonlinear optics. environments Often, volumetric dispersion, in contrast to boundary dispersion, is inseparable from the accompanying phenomena of reflection and refraction of light. In cases where there are no sharp boundaries in the environment and the reflection plays insignificantly. role in the nature of light propagation in the medium, for diffraction. processes apply asymptotic. methods of the theory of differential equations. Such approximate methods, which form the subject of the diffusion theory of diffraction, are characterized by a slow (at size H) change in the amplitude and phase of the light wave along the beam.
  In nonlinear optics D. s. occurs on inhomogeneities of the refractive index, which are created by the radiation itself propagating through the medium. The non-stationary nature of these phenomena further complicates the picture of the dynamic system, in which, in addition to the angular transformation of the radiation spectrum, a frequency transformation also occurs.