School encyclopedia. Diffraction spectrum Why does a diffraction grating split light into a spectrum?

Diffraction is an optical phenomenon that limits the sharpness of a photograph as the relative aperture of the lens decreases. Unlike other optical aberrations, diffraction is fundamentally irremovable, universal and equally characteristic of all photographic lenses without exception, regardless of their quality and cost.

Diffraction can only be seen at 100% magnification. Notice how the image becomes less and less sharp as the aperture number increases.

	f/4
	f/5.6
	f/8
	f/11
	f/16
	f/22

Nature of diffraction

As light passes through the aperture, the bulk of the light waves continue to move in a straight line. However, those waves whose path lies near the very edge of the diaphragm deviate from their original direction, trying to go around the obstacle that appears in their path. The smaller the size of the aperture opening, the greater the percentage of rays that touch its edge, and the more light is scattered. Due to the diffraction of light waves, the image of a point light source takes the form not of a point (as it would be in an ideal optical system), but of a blurred spot called Airy disk.

Despite some similarities between the Airy disk and the scattering circle that appears when a lens is defocused, the Airy disk has three very characteristic features.

Firstly, the circle of confusion is illuminated more or less evenly, while the brightness of the Airy disk rapidly decreases as it moves away from its center.

Secondly, unlike the scattering circle, which is a single round spot, the Airy disk is surrounded by a series of concentric rings. These rings arise due to the interference of light waves that have deviated from the original path with each other, as well as with waves that have retained their straight direction. Together with the Airy disk, the rings form a characteristic diffraction pattern known as the Airy pattern. 85% of the illumination comes from the Airy disk itself, and 15% from the rings surrounding it.

Thirdly, when the lens is apertured, the diameter of the scattering circle decreases, while the diameter of the Airy disk, on the contrary, increases. Accordingly, as the relative aperture decreases (i.e., as the aperture number increases), the depth of sharpness of the imaged space increases, but the overall sharpness of the photograph decreases.

Diffraction and camera resolution

According to the Rayleigh criterion, in order for two adjacent Airy disks to be visually distinguishable, their radius should not exceed the distance between the centers of the disks. Otherwise, the disks are perceived as one point. Since, at a constant light wavelength, the radius of the Airy disk depends solely on the size of the aperture, then for any distance between the disks there is a certain maximum aperture value, after which the disks increase so much that they merge together.

What does this have to do with digital photography? The most direct thing. Two theoretical points can be distinguished in an image only if the distance between them is not less than the distance between the centers of two adjacent pixels of the matrix. If the two points are Airy disks (and in reality it cannot be otherwise), then at a certain aperture value they will still cease to be distinguishable due to the effect of diffraction. Thus, the potential resolution of the system is limited on the one hand by the pixel density of the matrix, and on the other hand by the relative aperture size.

The aperture value at which the Airy disk radius is equal to the pixel size of the matrix of a particular digital camera is called the diffraction-limited aperture value or simply diffraction-limited aperture(tracing paper from English diffraction limited aperture - DLA). At aperture numbers greater than the diffraction-limited value, image degradation due to diffraction becomes visually visible.

The diffraction-limited aperture value for any digital camera can be calculated using the following formula:

, Where

K– diffraction-limited aperture;

n– matrix pixel size in micrometers (microns);

λ – wavelength of light in nanometers.

The pixel size n (see "") corresponds to the maximum radius of the Airy disk or, if you prefer, the diffraction limit of the optical system. I advise you to take 540 nm as the wavelength λ, since both the human eye and the digital photo matrix are most sensitive to green color. For blue, diffraction will be less pronounced, and for red, diffraction will be more pronounced.

To save your time, the author was not too lazy to calculate the values of the diffraction-limited aperture for matrices with various parameters and create a corresponding table. By using these or smaller apertures, you can be sure that your photographs are free from the negative effects of diffraction and that their blurriness is due either to flaws in the photographic equipment or, more likely, to your own negligence.

The values of the diffraction-limited aperture depending on the camera resolution and its crop factor.

Resolution, MP	Crop factor
Resolution, MP	1 *	1,5	1,6	2	2,7
10		f/9.4	f/8.8		f/5.2
12	f/12.9	f/8.6	f/8	f/6.4
14		f/7.9			f/4.4
16	f/11.2	f/7.4		f/5.6
18	f/10.5		f/6.6		f/3.9
20	f/10	f/6.7	f/6.2		f/3.7
22	f/9.5
24	f/9.1	f/6.1	f/5.7
28		f/5.6
36	f/7.4
42	f/6.9
50	f/6.3

* A crop factor equal to one corresponds to
full frame (36 × 24 mm).

The accuracy of the aperture values given in the table is excessive. Since aperture can usually only be set to within 1/3 of a stop, choose the actual aperture value that is closest to the theoretical aperture.

The words “loss of sharpness” or “image degradation” sound scary, but in fact, diffraction is not nearly as bad as it is made out to be. Nobody forbids you to use larger aperture values if there is an objective need for it. A very slight decrease in sharpness can be noticed with the naked eye only by setting the aperture one full stop larger than the diffraction-limited value. Sometimes sharpness can even increase (especially with inexpensive lenses) because stopping down reduces the optical aberrations that cause blurring when shooting wide open. If you stop the aperture down another stop, diffraction becomes slightly more obvious, but overall image quality remains quite acceptable. And only by moving three stops away from the diffraction-limited aperture do we get a noticeable loss of detail. But even this can be tolerated if the frame requires a particularly large depth of field. But it’s better to refrain from further reducing the relative opening.

Diffraction and lenses

A lens whose resolution is limited primarily by diffraction is called diffraction-limited. This means that for a given lens, at a given aperture, optical aberrations are eliminated so well that their contribution to image degradation does not exceed the diffraction effect. Actually, all our theoretical discussions about the diffraction limitation of the resolution of digital cameras imply the use of just such ideal lenses. In reality, very few lenses are diffraction-limited when the aperture is wide open, and then only in the center of the frame. Usually, to achieve optimal sharpness, you have to close the aperture a couple of stops, after which the lens still has a chance to become diffraction-limited, but its resolution will, of course, be lower than that of a lens that has reached its sharpness limit with a larger relative aperture.

Diffraction and focal length

There is a fairly common misconception that diffraction also depends on the focal length of the lens. After all, the aperture number is the ratio of the focal length to the diameter of the aperture hole, which means that for the same aperture value, the physical size of the hole in a long-focus lens will be larger than that of a short-focus lens, and an increase in the hole leads to a decrease in the Airy disk. This is true, but we must not forget that as the focal length of the lens increases, the distance that the rays of light must travel when they touch the edge of the aperture and deviate from the straight path also increases, as a result of which the scattering of light increases with increasing focal length. As a consequence, the positive effect of increasing the physical size of the aperture is counteracted by the negative effect of increasing the focal length. So, the size of the Airy disk really depends only on the magnitude relative holes.

The surprising thing is that, contrary to theory, when using telephoto lenses, large apertures often actually steal sharpness less blatantly than when using wide-angle lenses. Most likely, this can be explained by the fact that shooting with long-focus lenses very often involves an acute lack of depth of field, and therefore, even with a strong lens aperture, the damage caused by diffraction is compensated by an increase in depth of field, which creates the illusion of increased sharpness. At short focal lengths, however, depth of field is usually not an issue even at moderate apertures, so stopping down too much will only make the image look worse.

Thank you for your attention!

Vasily A.

Post scriptum

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A light breeze came, and ripples (a wave of small length and amplitude) ran along the surface of the water, encountering various obstacles on its way, above the surface of the water, plant stems, tree branches. On the leeward side behind the branch, the water is calm, there is no disturbance, and the wave bends around the plant stems.

WAVE DIFFRACTION (from lat. difractus– broken) waves bending around various obstacles. Wave diffraction is characteristic of any wave motion; occurs if the dimensions of the obstacle are smaller than the wavelength or comparable to it.

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 geometric shadow.
An obstacle can be a hole, a gap, or the edge of an opaque barrier.

Diffraction of light manifests itself 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 a larger bright spot on the screen than would be expected with linear propagation.

Due to the short wavelength of light, the angle of deflection of light from the direction of rectilinear propagation is small. Therefore, to clearly observe diffraction, it is necessary to use very small obstacles or place the screen far from the obstacles.

Diffraction is explained on the basis of the Huygens–Fresnel principle: each point on the wave front is a source of secondary waves. The diffraction pattern results from 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 clearly observed only at distances

where R is the characteristic dimensions of the obstacle. At shorter 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, close to each other transparent to light areas (slits) located in the same plane, separated by opaque spaces.

Diffraction gratings can be either reflective or transmitting light. The principle of their operation is the same. The grating is made using a dividing machine that makes periodic parallel strokes on a glass or metal plate. A good diffraction grating contains up to 100,000 lines. Let's denote:

a– the width of the slits (or reflective stripes) transparent to light;
b– the width of the opaque spaces (or light-scattering areas).
Magnitude 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 diffraction by the slit.
The main maxima, which are narrow bright lines in the spectrum, are of practical importance when studying spectra using a diffraction grating. If white light falls on a diffraction grating, the waves of each color included in its composition form their own 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 ), therefore there is a central bright band in the diffraction spectrum. To the left and right of it, color diffraction maxima of different orders are observed. Since the diffraction angle is proportional to the wavelength, red rays are deflected more than violet rays. Note the difference in the order of colors in the diffraction and prismatic spectra. Thanks to this, a diffraction grating is used as a spectral apparatus, along with a prism.

When passing through a diffraction grating, a light wave with a length λ the screen will give a sequence of minimums and maximums of intensity. Intensity maxima will be observed at angle β:

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

Basic summary:

From the relation d sin j = ml it is clear that the positions of the main maxima, except for the central one ( m= 0), in the diffraction pattern from the slit grating depend on the wavelength of the light used l. Therefore, if the grating is illuminated with white or other non-monochromatic light, then for different values l all diffraction maxima, except for the central one, will be spatially separated. As a result, in the diffraction pattern of a grating illuminated by white light, the central maximum will look like a white stripe, and all the rest will look like rainbow stripes, called diffraction spectra of the first ( m= ± 1), second ( m= ± 2), etc. orders of magnitude. In the spectra of each order, the red rays will be the most deviated (with a large value l, since sin j ~ 1 / l), and the least - violet (with a lower value l). The more slits there are, the clearer the spectra are (in terms of color separation) N contains a grid. This follows from the fact that the linear half-width of the maximum is inversely proportional to the number of slits N). The maximum number of observed diffraction spectra is determined by relation (3.83). Thus, the diffraction grating decomposes complex radiation into individual monochromatic components, i.e. conducts a harmonic analysis of the radiation incident on it.

The property of a diffraction grating to decompose complex radiation into harmonic components is used in spectral devices - devices used to study the spectral composition of radiation, i.e. to obtain the emission spectrum and determine the wavelengths and intensities of all its monochromatic components. The schematic diagram of the spectral apparatus is shown in Fig. 6. Light from the source under study enters the entrance slit S device located in the focal plane of the collimator lens L 1. The plane wave formed when passing through the collimator falls on the dispersing element D, which uses a diffraction grating. After spatial separation of the beams by a dispersing element, the output (chamber) lens L 2 creates a monochromatic image of the entrance slit in radiation of different wavelengths in the focal plane F. These images (spectral lines) in their totality constitute the spectrum of the radiation under study.

As a spectral device, a diffraction grating is characterized by angular and linear dispersion, free region of dispersion and resolution. As a spectral device, a diffraction grating is characterized by angular and linear dispersion, free region of dispersion and resolution.

Angular dispersion Dj characterizes the change in the angle of deflection j beam when its wavelength changes l and is defined as

Dj= dj / dl,

Where dj- angular distance between two spectral lines differing in wavelength by dl. Differentiating the ratio d sin j = ml, we get d cos j× j¢l = m, where

Dj = j¢l = m / d cos j.

Within small angles cos j@ 1, so we can put

Dj@m / d.

Linear dispersion is given by

D l = dl / dl,

Where dl– linear distance between two spectral lines differing in wavelength dl.

From Fig. 3.24 it is clear that dl = f 2 dj, Where f 2 – lens focal length L 2. Taking this into account, we obtain a relation connecting angular and linear dispersion:

D l = f 2 Dj.

Spectra of neighboring orders may overlap. Then the spectral apparatus becomes unsuitable for studying the corresponding part of the spectrum. Maximum width D l spectral interval of the radiation under study, in which the spectra of neighboring orders do not yet overlap, is called the free dispersion region or the dispersion region of the spectral apparatus. Let the wavelengths of radiation incident on the grating lie in the range from l to l+D l. Maximum D value l, at which the spectra do not overlap yet, can be determined from the condition of overlap of the right end of the spectrum m-th order for wavelength l+D l to the left end of the spectrum

(m+ 1)th order for wavelength l, i.e. from the condition

d sin j = m(l+D l) = (m + 1)l,

D l = l / m.

Resolution R of a spectral device characterizes the ability of the device to separately produce two close spectral lines and is determined by the ratio

R = l / d l,

Where d l– the minimum difference in wavelengths of two spectral lines at which these lines are perceived as separate spectral lines. Size d l is called the resolvable spectral distance. Due to diffraction at the active lens aperture L 2, each spectral line is depicted by a spectral apparatus not in the form of a line, but in the form of a diffraction pattern, the intensity distribution in which has the form of a sinc 2 function. Since spectral lines with different

If these wavelengths are not coherent, then the resulting diffraction pattern created by such lines will be a simple superposition of diffraction patterns from each slit separately; the resulting intensity will be equal to the sum of the intensities of both lines. According to the Rayleigh criterion, spectral lines with similar wavelengths l And l + d l are considered permitted if they are at this distance d l that the main diffraction maximum of one line coincides in its position with the first diffraction minimum of the other line. In this case, a dip is formed on the curve of the total intensity distribution (Fig. 3.25) (depth equal to 0.2 I 0 , where I 0 is the maximum intensity, the same for both spectral lines), which allows the eye to perceive such a picture as a double spectral line. Otherwise, two closely spaced spectral lines are perceived as one broadened line.

Position m th main diffraction maximum corresponding to the wavelength l, determined by the coordinate

x¢ m = f tg j@f sin j = ml f/ d.

Similarly we find the position m-th maximum corresponding to the wavelength l + d l:

x¢¢ m = m(l + d l) f / d.

If the Rayleigh criterion is fulfilled, the distance between these maxima will be

D x = x¢¢ m - x¢ m= md l f / d

equal to their half-width d x =l f / d(here, as above, we determine the half-width by the first intensity zero). From here we find

d l= l / (mN),

and, therefore, the resolution of the diffraction grating as a spectral device

Thus, the resolution of a diffraction grating is proportional to the number of slits N and spectrum order m. Putting

m = m max @d / l,

we get the maximum resolution:

R max = ( l /d l)max = m max N@L/ l,

Where L = Nd– width of the working part of the grille. As we can see, the maximum resolution of a slot grating is determined only by the width of the working part of the grating and the average wavelength of the radiation being studied. Knowing R max , let’s find the minimum resolvable wavelength interval:

(d l) min @l 2 / L.

1. Diffraction of light. Huygens-Fresnel principle.

2. Diffraction of light by slits in parallel rays.

3. Diffraction grating.

4. Diffraction spectrum.

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

6. X-ray structural 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 bending 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 in the behavior of light from the laws of geometric optics. (Diffraction is not limited to light.)

Diffraction is a wave phenomenon that manifests itself most clearly in the case when the dimensions of the obstacle are commensurate (of the same order) with the wavelength of light. The rather late discovery of light diffraction (16th-17th centuries) is associated with the small lengths of visible light.

21.1. Diffraction of light. Huygens-Fresnel principle

Diffraction of light is a complex of phenomena that are caused by 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 for constructing the wave front at time t + Δt if its position at time t is known.

1.According to Huygens' principle each point on 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 Huygens' principle using the following example. Let a plane wave fall on an obstacle with a hole, the front of which is parallel to the obstacle (Fig. 21.1).

Rice. 21.1. Explanation of Huygens' principle

Each point of the wave front isolated 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 secondary waves. This drawback was eliminated by Fresnel, who supplemented Huygens' principle with the idea 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 Huygens-Fresnel principle the magnitude of light vibrations at a certain point O is the result of the interference at this point of coherent secondary waves emitted everyone elements of the wave surface. The amplitude of each secondary wave is proportional to the area of the element dS, inversely proportional to the distance r to point O and decreases with increasing angle α between normal n to element dS and direction to point O (Fig. 21.2).

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

21.2. Slit diffraction in parallel beams

Calculations associated with the application of the Huygens-Fresnel principle are, in general, 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 flat monochromatic light wave fall on a narrow slit (AB) in an opaque barrier, the direction of propagation of which is perpendicular to the surface of the slit (Fig. 21.3, a). We place a collecting lens behind the slit (parallel to its plane), in focal plane which we will place the screen E. All secondary waves emitted from the surface of the slit in the direction parallel optical axis of the lens (α = 0), the lens comes into focus in the same phase. Therefore, at 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 slit 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 - light wavelength.

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

Rice. 21.3. Diffraction at 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 slit. 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 suppression of rays from all Fresnel zones occurs and at point O" a minimum of the interference pattern is observed.

Minimum intensity during diffraction by a slit is observed for the directions of rays of secondary waves satisfying the condition

The integer k is called on the order of the minimum.

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

The maximum intensity during diffraction by a slit is observed for the directions of rays of secondary waves satisfying the condition:

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

From formula (21.3) it follows 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 stripe is furthest 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 quickly decreases. Calculations show that I 0:I 1:I 2 = 1:0.047:0.017.

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

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

21.3. Diffraction grating

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

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

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

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

The path of rays in a 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 surfaces of the slots belong to the same wave surface and are sources of coherent secondary waves. Let us consider secondary waves whose direction of propagation 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 adjacent slits. When condition (21.4) is satisfied, secondary waves arrive at point O" in the same phase and a maximum interference pattern appears on the screen. Maxima that satisfy condition (21.4) are called main maxima of order k. Condition (21.4) itself is called the basic formula of a diffraction grating.

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

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

Rice. 21.5. Diffraction of light by a diffraction grating

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

Condition (21.4), which determines the positions of all main maxima, does not take into account diffraction at a separate slit. It may happen that for some direction the condition will be simultaneously satisfied maximum for the lattice (21.4) and the condition minimum for the slot (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 slits 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 intensity distribution graphs obtained from gratings with different numbers of slits (N). The periods (d) and slot widths (b) are the same for all gratings.

Rice. 21.6. Intensity distribution at different values of N

21.4. Diffraction spectrum

From the basic formula of a diffraction grating (21.4) it is clear that the diffraction angle α, at which the main maxima are formed, depends on the wavelength of the incident light. Therefore, intensity maxima corresponding to different wavelengths are obtained in different places on the screen. This allows the grating to be used as a spectral device.

Diffraction spectrum- spectrum obtained using a diffraction grating.

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

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

When writing 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 width of the grating, and N is the number of lines.

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 device

The basic formula of a diffraction grating (21.4) allows you to determine the wavelength of light by measuring the angle α corresponding to the position of the kth maximum. Thus, a diffraction grating makes it possible to obtain and analyze 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 maximum, α - the angle at which it is observed.

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

Resolution(resolving power) of a diffraction grating - a quantity characterizing its ability to produce

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

It is clear from the formula that close lines that merge in a first-order spectrum can be perceived separately in second- or third-order spectra.

21.6. X-ray diffraction analysis

The basic diffraction grating formula 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 at 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

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

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

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

crystal and the incident X-ray beam (grazing angle), λ is the wavelength of the X-ray radiation. Relationship (21.11) is called Bragg-Wolfe condition.

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

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

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

In the case when the substance is a mixture of a large number of very small crystals (as in a metal or powder), a series of rings appears (Fig. 21.8, b). Each ring corresponds to a diffraction maximum of a certain order k, and the x-ray pattern 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 using this method.

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

In conclusion, let us consider the issue of light diffraction by a round hole, which is of great practical interest. Such openings are, for example, the pupil of the eye and the lens of a microscope. Let light from a point source fall on the lens. A lens is an opening that allows only Part light wave. Due to diffraction on the screen located behind the lens, a diffraction pattern will appear as shown in Fig. 21.9, a.

As for the gap, the intensities of the side maxima are low. The central maximum in the form of a light 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 a 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).

Let us present without derivation a formula that provides a separate image of close point sources on the screen (aperture resolution):

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

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

21.8. Basic concepts and formulas

End of the table

21.9. Tasks

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

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

Solution

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

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

4. The diffraction grating contains N = 600 lines at L = 1 mm. Find the highest spectral order for light with wavelength λ = 600 nm.

5. Orange light with a wavelength of 600 nm and green light with a wavelength of 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 = α or - α z = 13.88° - 12.47° = 1.41°.

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

Solution

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

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

A one-dimensional diffraction grating is a system of a large number N equal-width and parallel to each other slits in the screen, also separated by equal-width opaque spaces (Fig. 9.6).

The diffraction pattern on a grating is determined as the result of mutual interference of waves coming from all slits, i.e. V diffraction grating carried out multipath interference coherent diffracted beams of light coming from all slits.

Let's denote: b – slot width gratings; A - distance between slots; – diffraction grating constant.

The lens collects all rays incident on it at one angle and does not introduce any additional path difference.


Rice. 9.6	Rice. 9.7

Let ray 1 fall on the lens at an angle φ ( diffraction angle ). A light wave coming at this angle from the slit creates a maximum intensity at the point. The second ray coming from the adjacent slit at the same angle φ will arrive at the same point. Both of these beams will arrive in phase and will reinforce each other if the optical path difference is equal mλ:

Conditionmaximum for a diffraction grating will look like:

(9.4.4)

Where m= ± 1, ± 2, ± 3, … .

The maxima corresponding to this condition are called main maxima . Value value m, corresponding to one or another maximum is called order of the diffraction maximum.

At the point F 0 will always be observed null or central diffraction maximum .

Since light incident on the screen passes only through slits in the diffraction grating, the condition minimum for the gap and it will be conditionmain diffraction minimum for grating:

(9.4.5)

Of course, with a large number of slits, light will enter the points of the screen corresponding to the main diffraction minima from some slits and formations will form there. side diffraction maxima and minima(Fig. 9.7). But their intensity, compared to the main maxima, is low (≈ 1/22).

Given that ,

the waves sent by each slit will be canceled out as a result of interference and additional minimums .

The number of slits determines the luminous flux through the grille. The more there are, the more energy is transferred by the wave through it. In addition, the greater the number of slits, the more additional minima are placed between adjacent maxima. Consequently, the maxima will be narrower and more intense (Fig. 9.8).

From (9.4.3) it is clear that the diffraction angle is proportional to the wavelength λ. This means that a diffraction grating decomposes white light into its components, and deflects light with a longer wavelength (red) at a larger angle (unlike a prism, where everything happens the other way around).

Diffraction spectrum- Intensity distribution on the screen resulting from diffraction (this phenomenon is shown in the lower figure). The main part of the light energy is concentrated in the central maximum. The narrowing of the gap leads to the fact that the central maximum spreads out and its brightness decreases (this, naturally, also applies to other maxima). On the contrary, the wider the slit (), the brighter the picture, but the diffraction fringes are narrower, and the number of fringes themselves is greater. When in the center, a sharp image of the light source is obtained, i.e. has a linear propagation of light. This pattern will only occur for monochromatic light. When the slit is illuminated with white light, the central maximum will be a white stripe; it is common for all wavelengths (with the path difference being zero for all).