One of the important optical devices that have found their 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 gives.

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 was due to the phenomena of diffraction and interference.

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

The phenomenon of interference is the superposition of one wave on another. The result of this overlay is a curvature of the sinusoidal form of the resulting wave. Particular cases of interference are either the maximum amplification of the amplitude, when two waves arrive in the considered zone of space in one phase, or the complete attenuation of the wave process, when both waves meet in the 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. It can be obtained by gradually increasing the number of slots on which the wave front falls. This concept is generally applicable to any wave, however, 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:

• Period d is the distance between two slits through which light passes. Since the wavelengths of light are in the range of a few tenths of a micrometer, the value of d is of the order of 1 μm.
• The grating constant a is the number of transparent slots that are located on a length of 1 mm of the grating. The lattice constant is the reciprocal of the period d. Its typical values ​​are 300-600 mm-1. As a rule, the value of a is written on the diffraction grating.
• The total number of slots is N. This value is 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 slots.

Transparent and reflective grilles

It has been 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 grating is a glass thin plate or a transparent plastic plate on which strokes are applied. The grooves of the diffraction grating are an obstacle for light, it cannot pass through them. The stroke width is the aforementioned period d. The transparent gaps remaining between the strokes play the role of slits. When performing laboratory work, this type of lattice is used.

A reflective grating is a polished metal or plastic plate, on which grooves of a certain depth are applied instead of strokes. The period d is the distance between the grooves. Reflective gratings are often used in the analysis of radiation spectra, since their design allows the distribution of the intensity of the diffraction pattern maxima in favor of higher-order maxima. The CD optical disc is a prime example of this kind of grating.

The principle of operation of the lattice

For example, consider a transparent optical device. Let us assume that light having a flat front is incident on a 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). Strokes distributed according to the periodic law introduce a perturbation into this front, as a result of which a situation is created at the exit from the plate, as if many secondary coherent radiation sources are operating (the Huygens-Fresnel principle). These sources lead to the appearance of diffraction.

From each source (the gap between the strokes) a wave propagates that is coherent to all other N-1 waves. Now suppose that a screen is placed at some distance from the plate (the distance must be sufficient for the Fresnel number to be 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 interference maxima is a function of the wavelength, if the light falling on the plate was white, multi-colored bright stripes would appear on the screen.

Basic Formula

As mentioned, the incident flat wave front on the diffraction grating is displayed on the screen in the form of bright bands separated by a shadow region. Each bright band is called a maximum. If we consider the amplification condition for waves arriving in the region under consideration in the same phase, then we can obtain the 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 value m is called the order of the diffraction grating. It takes 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, we can calculate the position of all the maxima. Note that the maxima calculated by the formula above are called principal. In fact, between them there is a whole set of weaker maxima, which are often not observed in the 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 slit width 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 strokes on the plate), the intensity of each maximum decreases, and its width increases.

Diffraction grating in spectroscopy

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

We write again the formula for the main maxima:

If we consider a specific order of diffraction (for example, m = 1), then it is clear that the larger λ, the farther from the central maximum (m = 0) the corresponding bright line will be. This means that white light is split into a range 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 go, and the farthest maximum of the first order will correspond to red.

A property of the wavelength diffraction grating is used in spectroscopy. When it is necessary to know 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, one can determine all the wavelengths of the spectrum, and hence the chemical elements that emit them.

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

The concept of "angular dispersion"

This value is understood as the change 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 parts of the equality in the formula for the main maxima are differentiated with respect to θ m and λ, respectively, then an expression for the dispersion can be obtained. It will be equal to:

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

What is resolution?

In simple terms, this is the ability of a diffraction grating to separate two waves with close λ values ​​into two separate maxima 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(θm))

The difference between the lines according to the Rayleigh criterion is possible if:

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

The resolution of the grating increases with an increase in the number of slots (strokes) on it and with an increase in the order of diffraction.

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 order of diffraction that can be observed on a crane?

To answer the question, you should 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 obtain that the maximum order of diffraction 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

Continuing the reasoning for five, six slots, etc., we can establish the following rule: if there are slots between two adjacent maxima, minima are formed; the difference in the path of the rays from two adjacent slits for the maxima should be equal to an integer X, and for the minima - The diffraction spectrum from the slits has the form shown in Fig. Additional maxima located between two adjacent minima create a very weak illumination (background) on the screen.

The main part of the energy of the light wave passed through the diffraction grating is redistributed between the main maxima, formed in the directions where 3, is called the "order" of the maximum.

Obviously, the greater the number of slits, the greater the amount of light energy that passes through the grating, the more minima are formed between adjacent main maxima, the more intense and sharper the maxima will be.

If the light incident on the diffraction grating consists of two monochromatic radiations with wavelengths and their main maxima are located in different places on the screen. For wavelengths very close to each other (single-color radiation), the maxima on the screen can turn out so close to each other that they merge into one common bright band (Fig. IV.27, b). If the top of one maximum coincides with or is located further (a) than the nearest minimum of the second wave, then the presence of two waves can be confidently established by the distribution of illumination on the screen (or, as they say, "resolve" these waves).

Let us derive the condition for the solvability of two waves: the maximum (i.e., the maximum order) of the wave will turn out, according to formula (1.21), at an angle that satisfies the condition.

the minimum of the wave closest to its maximum (Fig. IV.27, c). According to the above, in order to obtain the nearest minimum, an additional addition should be added to the path difference. Thus, the condition for the coincidence of the angles at which the maximum and minimum are obtained leads to the relation

If greater than the product of the number of slots by the order of the spectrum, then the maxima will not be resolved. Obviously, if two maxima are not resolved in the order spectrum, then they can be resolved in the spectrum of higher orders. According to expression (1.22), the greater the number of beams interfering with each other and the greater the path difference A between them, the closer waves can be resolved.

In a diffraction grating, i.e., the number of slots is large, but the order of the spectrum that can be used for measuring purposes is small; in the Michelson interferometer, on the contrary, the number of interfering beams is two, but the path difference between them, which depends on the distances to the mirrors (see Fig. IV. 14), is large, so the order of the observed spectrum is measured by very large numbers.

The angular distance between two neighboring maxima of two nearby waves depends on the order of the spectrum and the grating period

The grating period can be replaced by the number of slots per unit length of the grating:

It was assumed above that the rays incident on the diffraction grating are perpendicular to its plane. With oblique incidence of the rays (see Fig. IV.22, b), the zero maximum will be shifted and will turn out in the direction.

are close to each other in size, so

where is the angular deviation of the maximum from zero. Let us compare this formula with expression (1.21), which we write in the form since the angular deviation with oblique incidence is greater than with perpendicular incidence of rays. This corresponds to a decrease in the grating period by a factor. Consequently, at large angles of incidence a, it is possible to obtain diffraction spectra from short-wavelength (for example, X-ray) radiation and measure their wavelengths.

If a plane light wave does not pass through slits, but through round holes of small diameter (Fig. IV.28), then the diffraction spectrum (on a flat screen located in the focal plane of the lens) is a system of alternating dark and light rings. The first dark ring is obtained at an angle satisfying the condition

At the second dark ring The share of the central light circle, called the Airy spot, accounts for about 85% of the total radiation power that has passed through the hole and lens; the remaining 15% is distributed between the light rings surrounding this spot. The size of the Airy spot depends on the focal length of the lens.

The diffraction gratings discussed above consisted of alternating "slits" that completely transmit the light wave, and "opaque strips" that completely absorb or reflect the radiation incident on them. We can say that in such gratings the transmittance of a light wave has only two values: over the gap it is equal to unity, and over an opaque strip it is zero. Therefore, at the interface between the slot and the strip, the transmittance changes abruptly from unity to zero.

However, diffraction gratings can also be made with a different transmission coefficient distribution. For example, if an absorbing layer with a periodically changing thickness is applied to a transparent plate (or film), then instead of alternating completely

transparent slits and completely opaque stripes, it is possible to obtain a diffraction grating with a smooth change in the transmittance (in the direction perpendicular to the slits or stripes). Of particular interest are gratings in which the transmittance varies according to a sinusoidal law. The diffraction spectrum of such gratings does not consist of many maxima (as shown for ordinary gratings in Fig. IV.26), but only of a central maximum and two symmetrically located first-order maxima

For a spherical wave, it is possible to make diffraction gratings consisting of a plurality of concentric annular slots separated by opaque rings. It is possible, for example, to ink concentric rings on a glass plate (or on a transparent film); while the central circle, covering the center of these rings, can be either transparent or shaded. Such diffraction gratings are called "zone plates" or gratings. For diffraction gratings consisting of rectilinear slits and stripes, in order to obtain a distinct interference pattern, it was necessary that the slit width and grating period be constant; for zone plates, the necessary radii and thicknesses of the rings must be calculated for this purpose. Zone gratings can also be made with a smooth, for example sinusoidal, change in the transmittance along the radius.

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 constant 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.

The first experiments and active research into the nature of light began as far back as the 17th century, when the Italian scientist Francesco Grimaldi first discovered such an interesting physical phenomenon as light diffraction. What is light diffraction? This is the deviation of light from rectilinear propagation due to certain obstacles in its path. A more scientific explanation of the causes of light diffraction was given at the beginning of the 19th century by the English scientist Thomas Young, according to whom light diffraction is possible due to the fact that light is a wave coming from its source and naturally bending when it hits certain obstacles. He also invented the first diffraction grating, which is an optical device that works on the basis of light diffraction, that is, it specifically bends a light wave.

Diffraction and interference of light

Studying the behavior of a monochromatic beam of light, Thomas Young, dividing it in half, obtained a diffraction pattern, which was a successive alternation of bright and dark stripes on the screen. The wave theory of the nature of light, formed by Jung, perfectly explained this phenomenon. Being a wave, a beam of light, when it hits an opaque obstacle, bends and changes the trajectory of its movement. This is how light diffraction appears, in which light can either completely go around obstacles (if the wavelength of light is greater than the dimensions of the obstacle) or bend its trajectory (when the dimensions of the obstacles are comparable to the wavelength of the light). An example here would be light entering through narrow slits or small holes, as in the photo below.

A beam of light in a cave, a clear illustration of the diffraction of light in nature.

And here the picture shows a more schematic representation of diffraction.

The physical phenomenon of light diffraction complements another important property of a light wave - light interference. The essence of light interference is the superposition of one light wave on another. As a result, a curvature of the sinusoidal shape of the resulting wave may occur.

This is what interference looks like.

At the same time, the waves that are superimposed can both increase the power of the total light wave (if the amplitudes coincide), and vice versa, extinguish it.

As we wrote above, a diffraction grating is a simple optical device that bends a light wave.

This is how she looks.

Or even a slightly smaller copy.

Also, the diffraction grating can be characterized by three parameters:

• Period d. It is the distance between two slits through which light passes. Since the wavelength of light is usually in the range of a few tenths of a micrometer, the value of d is usually 1 micrometer.
• Permanent lattice a. This is the number of transparent slots on a length of 1 mm of the grating surface. This value is inversely proportional to the diffraction grating period d. Usually has 300-600 mm -1
• The total number of slots N. Calculated by multiplying the length of the diffraction grating by its constant a. Typically, the length of the grating is several centimeters, and the number of slots in this case is 10-20 thousand.

Types of diffraction gratings

In fact, there are two types of diffraction gratings: transparent and reflective.

A transparent grill is a transparent thin plate of glass or transparent plastic, on which strokes are applied. These strokes are precisely the obstacles for the light wave, it cannot pass through them. The stroke width is, in fact, the period of the diffraction grating d. And the transparent gaps remaining between the strokes are the gaps. Such gratings are most often used in laboratory work.

A reflective diffraction grating is either a plastic and polished plate. Instead of strokes, grooves of a certain depth are applied to it. The period d is, respectively, the distance between these grooves. A simple example of a reflective diffraction grating would be an optical CD.

Such gratings are often used in the analysis of radiation spectra, since their design makes it possible to conveniently distribute the intensity of the diffraction pattern maxima in favor of higher-order maxima.

The principle of operation of a diffraction grating

Let's imagine that light having a flat front falls on our grating. This is an important point, since the classical formula will be correct provided that the wavefront is flat and parallel to the plate itself. The grating strokes will introduce a perturbation into this light front and, as a result, a situation will be created at the output of the grating, as if a lot of coherent (synchronous) radiation sources are working. These sources are the cause of diffraction.

From each source (essentially a gap between the grating strokes) light waves will propagate, which will be coherent (synchronous) to each other. If a screen is placed at some distance from the grate, then we can see bright stripes on it, between which there will be a shadow.

Grating Formula

The bright bands that we see on the screen can also be called lattice maxima. If we consider the conditions for the amplification of light waves, then we can derive the formula for the maximum of the diffraction grating, here it is.

sin(θ m) = m*λ/d

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 value m is called the order of the diffraction grating. It takes integer values ​​and zero, that is, m = 0, ±1, 2, 3, and so on. λ is the light wavelength and d is the grating period.

Resolution of the diffraction grating

Resolution refers to the ability of a grating to separate two waves with similar wavelengths λ into two separate maxima on the screen.

Application of a diffraction grating

What is the practical application of a diffraction grating, what is its specific use? A diffraction grating is an important and indispensable tool in spectroscopy, as it can be used to find out, for example, the chemical composition of a distant star. The light coming from this star is collected by mirrors and directed to the grate. By measuring the values ​​of θ m, you can find out all the wavelengths of the spectrum, and hence the chemical elements that emit them.

Diffraction of light and diffraction grating, video

And in conclusion, an interesting educational video on the topic of our article from the honored teacher of Ukraine - Pavel Viktor, in our opinion, his video lectures on YouTube on physics can be very useful for everyone who studies this subject.

When writing the article, I tried to make it as interesting, useful and of high quality as possible. I would be grateful for any feedback and constructive criticism in the form of comments on the article. You can also write your wish / question / suggestion to my mail [email protected] or on Facebook, with respect, the author.

One of the well-known effects that confirm the wave nature of light are diffraction and interference. Their main field of application is spectroscopy, in which diffraction gratings are used to analyze the spectral composition of electromagnetic radiation. The formula that describes the position of the main maxima given by this lattice is discussed in this article.

Before considering the derivation of the formula for a diffraction grating, one should become familiar with the phenomena due to which this grating is useful, that is, with diffraction and interference.

Diffraction is the process of changing the motion of a wave front when it encounters an opaque obstacle on its way, the dimensions of which are comparable to the wavelength. For example, if sunlight is passed through a small hole, then on the wall one can observe not a small luminous point (which should happen if the light propagated in a straight line), but a luminous spot of some size. This fact testifies to the wave nature of light.

Interference is another phenomenon that is unique to waves. Its essence lies in the imposition of waves on each other. If the waveforms from multiple sources are matched (coherent), then a stable pattern of alternating bright and dark areas on the screen can be observed. The minima in such a picture are explained by the arrival of waves at a given point in antiphase (pi and -pi), and the maxima are the result of waves hitting the point under consideration in one phase (pi and pi).

Both of these phenomena were first explained by the Englishman Thomas Young when he investigated the diffraction of monochromatic light by two thin slits in 1801.

The Huygens-Fresnel principle and far and near field approximations

The mathematical description of the phenomena of diffraction and interference is a non-trivial task. Finding its exact solution requires performing complex calculations involving the Maxwellian theory of electromagnetic waves. Nevertheless, in the 1920s, the Frenchman Augustin Fresnel showed that, using Huygens' ideas about secondary sources of waves, one can successfully describe these phenomena. This idea led to the formulation of the Huygens-Fresnel principle, which currently underlies the derivation of all formulas for diffraction by obstacles of arbitrary shape.

Nevertheless, even with the help of the Huygens-Fresnel principle, it is not possible to solve the problem of diffraction in a general form, therefore, when obtaining formulas, some approximations are resorted to. The main one is a flat wave front. It is this waveform that must fall on the obstacle so that a number of mathematical calculations can be simplified.

The next approximation is the position of the screen where the diffraction pattern is projected relative to the obstacle. This position is described by the Fresnel number. It is calculated like this:

Where a is the geometric dimensions of the obstacle (for example, a slot or a round hole), λ is the wavelength, D is the distance between the screen and the obstacle. If for a particular experiment F<<1 (<0,001), тогда говорят о приближении дальнего поля. Соответствующая ему дифракция носит фамилию Фраунгофера. Если же F>1, then near field approximation or Fresnel diffraction takes place.

The difference between Fraunhofer and Fresnel diffraction lies in the different conditions for the phenomenon of interference at small and large distances from the obstacle.

The derivation of the formula for the main maxima of the diffraction grating, which will be given later in the article, involves the consideration of Fraunhofer diffraction.

Diffraction grating and its types

This grating is a plate of glass or transparent plastic a few centimeters in size, on which opaque strokes of the same thickness are applied. The strokes are located at a constant distance d from each other. This distance is called the lattice period. Two other important characteristics of the device are the lattice constant a and the number of transparent slits N. The value of a determines the number of slits per 1 mm of length, so it is inversely proportional to the period d.

There are two types of diffraction gratings:

• Transparent, as described above. The diffraction pattern from such a grating results from the passage of a wave front through it.
• Reflective. It is made by applying small grooves to a smooth surface. Diffraction and interference from such a plate arise due to the reflection of light from the tops of each groove.

Whatever the type of grating, the idea of ​​its effect on the wave front is to create a periodic perturbation in it. This leads to the formation of a large number of coherent sources, the result of the interference of which is a diffraction pattern on the screen.

The basic formula of a diffraction grating

The derivation of this formula involves considering the dependence of the radiation intensity on the angle of its incidence on the screen. In the far-field approximation, the following formula for the intensity I(θ) is obtained:

I(θ) = I 0 *(sin(β)/β)2*2, where

α = pi*d/λ*(sin(θ) - sin(θ 0));

β = pi*a/λ*(sin(θ) - sin(θ 0)).

In the formula, the width of the slit of the diffraction grating is denoted by the symbol a. Therefore, the factor in parentheses is responsible for the diffraction by one slit. The value of d is the period of the diffraction grating. The formula shows that the factor in square brackets where this period appears describes the interference from the set of grating slots.

Using the above formula, you can calculate the intensity value for any angle of incidence of light.

If we find the value of the intensity maxima I(θ), then we can conclude that they appear under the condition that α = m*pi, where m is any integer. For the maximum condition, we get:

m*pi = pi*d/λ*(sin(θ m) - sin(θ 0)) =>

sin (θ m) - sin (θ 0) \u003d m * λ / d.

The resulting expression is called the formula for the maxima of the diffraction grating. The numbers m are the order of diffraction.

Other ways to write the basic formula for the lattice

Note that the formula given in the previous paragraph contains the term sin(θ 0). Here, the angle θ 0 reflects the direction of incidence of the front of the light wave relative to the grating plane. When the front falls parallel to this plane, then θ 0 = 0o. Then we get the expression for the maxima:

Since the grating constant a (not to be confused with the slit width) is inversely proportional to the value of d, the formula above can be rewritten in terms of the diffraction grating constant as:

To avoid errors when substituting specific numbers λ, a and d into these formulas, you should always use the appropriate SI units.

The concept of the angular dispersion of the grating

We will denote this value by the letter D. According to the mathematical definition, it is written by the following equality:

The physical meaning of the angular dispersion D is that it shows by what angle dθ m the maximum will shift for the diffraction order m if the incident wavelength is changed by dλ.

If we apply this expression to the lattice equation, then we get the formula:

The dispersion of the angular diffraction grating is determined by the formula above. It can be seen that the value of D depends on the order m and the period d.

The greater the dispersion D, the higher the resolution of a given grating.

Grating resolution

Resolution is understood as a physical quantity that shows by what minimum value two wavelengths can differ so that their maxima appear separately in the diffraction pattern.

The resolution is determined by the Rayleigh criterion. It says: two maxima can be separated in a diffraction pattern if the distance between them is greater than the half-width of each of them. The angular half-width of the maximum for the grating is determined by the formula:

Δθ 1/2 = λ/(N*d*cos(θ m)).

The resolution of the grating in accordance with the Rayleigh criterion is:

Δθ m >Δθ 1/2 or D*Δλ>Δθ 1/2 .

Substituting the values ​​of D and Δθ 1/2 , we get:

Δλ*m/(d*cos(θ m))>λ/(N*d*cos(θ m) =>

Δλ > λ/(m*N).

This is the formula for the resolution of a diffraction grating. The greater the number of strokes N on the plate and the higher the order of diffraction, the greater the resolution for a given wavelength λ.

Diffraction grating in spectroscopy

Let us write out once again the basic equation of maxima for the lattice:

It can be seen here that the more the wavelength falls on the plate with strokes, the greater the values ​​of the angles will appear on the screen maxima. In other words, if non-monochromatic light (for example, white) is passed through the plate, then the appearance of color maxima can be seen on the screen. Starting from the central white maximum (zero-order diffraction), maxima will appear further for shorter waves (violet, blue), and then for longer ones (orange, red).

Another important conclusion from this formula is the dependence of the angle θ m on the order of diffraction. The larger m, the larger the value of θ m . This means that the colored lines will be more separated from each other at the maxima for a high diffraction order. This fact was already consecrated when the grating resolution was considered (see the previous paragraph).

The described capabilities of the diffraction grating make it possible to use it to analyze the emission spectra of various luminous objects, including distant stars and galaxies.

Problem solution example

Let's show how to use the diffraction grating formula. The wavelength of light that falls on the grating is 550 nm. It is necessary to determine the angle at which first-order diffraction appears if the period d is 4 µm.

Convert all data to SI units and substitute into this equality:

θ 1 \u003d arcsin (550 * 10-9 / (4 * 10-6)) \u003d 7.9o.

If the screen is at a distance of 1 meter from the grating, then from the middle of the central maximum, the line of the first order of diffraction for a wave of 550 nm will appear at a distance of 13.8 cm, which corresponds to an angle of 7.9o.