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

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

Denote: b – slot width gratings; a - distance between slots; – grating constant.

The lens collects all the rays that fall on it at the same angle and does not introduce any additional path difference.

Rice. 9.6	Rice. 9.7

Let beam 1 fall on the lens at an angle φ ( diffraction angle ). A light wave traveling at this angle from the slit creates a maximum intensity at the point. The second beam coming from the neighboring slot at the same angle φ will come to the same point. Both of these beams will come in phase and will amplify each other if the optical path difference is equal to 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 major highs . The value of the quantity 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 peak .

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

.

(9.4.5)

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

Given that ,

the waves sent by each slit will be canceled out by interference and will appear additional minimums .

The number of slots determines the light flux through the grating. The more of them, the more energy is transferred by the wave through it. In addition, the greater the number of slots, the more additional minima fit between neighboring maxima. Consequently, the highs will be narrower and more intense (Figure 9.8).

From (9.4.3) it can be seen that the diffraction angle is proportional to the wavelength λ. This means that the diffraction grating decomposes white light into components, and rejects 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, obtained due to 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, of course, 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 rectilinear propagation of light. This picture will only take place for monochromatic light. When the slit is illuminated with white light, the central maximum will be a white strip, it is common for all wavelengths (when the path difference is zero for all).

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(Lesson for obtaining new knowledge, grade 11, profile level - 2 hours).

Educational objectives of the lesson:

• Introduce the concept of light diffraction
• Explain the diffraction of light using the Huygens-Fresnel principle
• Introduce the concept of Fresnel zones
• Explain the structure and principle of operation of a diffraction grating

Developmental objectives of the lesson

• Development of skills in qualitative and quantitative description of diffraction patterns

Equipment: projector, screen, presentation.

Lesson Plan

• Diffraction of light
• Fresnel diffraction
• Fraunhofer diffraction
• Diffraction grating

During the classes.

1. Organizational moment.

2. Learning new material.

Diffraction- the phenomenon of waves bending around obstacles encountered in their path, or in a broader sense - any deviation of the propagation of waves near obstacles from the laws of geometric optics. Due to diffraction, waves can fall into the region of a geometric shadow, go around obstacles, penetrate through small holes in screens, etc. For example, sound is well heard around the corner of a house, i.e., a sound wave goes around it.

If light is a wave process, as convincingly indicated by the phenomenon of interference, then light diffraction must also be observed.

Diffraction of light- the phenomenon of deflection of light rays into the region of a geometric shadow when passing by the edges of obstacles or through holes whose dimensions are comparable to the wavelength of light ( slide number 2).

The fact that light goes beyond the edges of obstacles has been known to people for a long time. The first scientific description of this phenomenon belongs to F. Grimaldi. In a narrow beam of light, Grimaldi placed various objects, in particular thin threads. In this case, the shadow on the screen turned out to be wider than it should be according to the laws of geometric optics. In addition, colored bands were found on both sides of the shadow. Passing a thin beam of light through a small hole, Grimaldi also observed a deviation from the law of rectilinear propagation of light. The bright spot opposite the hole turned out to be larger than it should be expected for rectilinear light propagation ( slide number 2).

In 1802, T. Jung, who discovered the interference of light, staged a classical experiment on diffraction ( slide number 3).

In an opaque screen, he pierced with a pin two small holes B and C at a short distance from each other. These holes were illuminated by a narrow beam of light passing through a small hole A in another screen. It was this detail, which was very difficult to think of at that time, that decided the success of the experiment. Only coherent waves interfere. The spherical wave that arose in accordance with the Huygens principle from hole A excited coherent oscillations in holes B and C. Due to diffraction from holes B and C, two light cones emerged, which partially overlapped. As a result of the interference of these two light waves, alternating light and dark stripes appeared on the screen. Closing one of the holes. Young found that the fringes disappeared. It was with the help of this experiment that Jung first measured the wavelengths corresponding to light rays of different colors, and very accurately.

Theory of diffraction

The French scientist O. Fresnel not only studied various cases of diffraction in more detail in experiment, but also built a quantitative theory of diffraction. Fresnel's theory was based on the Huygens principle, supplementing it with the idea of ​​the interference of secondary waves. The Huygens principle in its original form made it possible to find only the positions of wave fronts at subsequent moments of time, i.e., to determine the direction of wave propagation. Essentially, this was the principle of geometric optics. Fresnel replaced Huygens' hypothesis about the envelope of secondary waves with a physically clear position, according to which the secondary waves, arriving at the observation point, interfere with each other ( slide number 4).

There are two types of diffraction:

If the obstacle on which diffraction occurs is close to the light source or the screen on which the observation takes place, then the front of the incident or diffracted waves has a curved surface (for example, spherical); this case is called Fresnel diffraction.

If the dimensions of the obstacle are much smaller than the distance to the source, then the wave incident on the obstacle can be considered as a plane wave. Plane wave diffraction is often referred to as Fraunhofer diffraction ( slide number 5).

Fresnel zone method.

To explain the features of diffraction patterns on simple objects ( slide number 6), Fresnel came up with a simple and illustrative method for grouping secondary sources - the method of constructing Fresnel zones. This method makes it possible to approximate the calculation of diffraction patterns ( slide number 7).

Fresnel zones– a set of coherent sources of secondary waves, the maximum path difference between which is equal to λ/2.

If the path difference from two adjacent zones is equal to λ /2 , therefore, vibrations from them come to the observation point M in opposite phases, so that waves from any two adjacent Fresnel zones cancel each other out(slide number 8).

For example, when passing light through a small hole, both a light and a dark spot can be detected at the observation point. It turns out a paradoxical result - the light does not pass through the hole!

To explain the result of diffraction, it is necessary to look at how many Fresnel zones fit into the hole. When the hole is laid odd number of zones maximum(light spot). When the hole is laid even number of zones, then at the observation point there will be minimum(dark spot). In fact, the light, of course, passes through the hole, but the interference maxima appear at neighboring points ( slide number 9 -11).

Fresnel zone plate.

A number of remarkable, sometimes paradoxical, consequences can be obtained from Fresnel's theory. One of them is the possibility of using a zone plate as a converging lens. zone plate– a transparent screen with alternating light and dark rings. The radii of the rings are chosen so that the rings of opaque material cover all the even zones, then only the oscillations from the odd zones occurring in the same phase come to the observation point, which leads to an increase in the light intensity at the observation point ( slide number 12).

The second remarkable consequence of Fresnel's theory is the prediction of the existence of a bright spot ( poisson spots) in the area of ​​geometric shadow from an opaque screen ( slide number 13-14).

To observe a bright spot in the region of a geometric shadow, it is necessary that an opaque screen overlap a small number of Fresnel zones (one or two).

Fraunhofer diffraction.

If the dimensions of the obstacle are much smaller than the distance to the source, then the wave incident on the obstacle can be considered as a plane wave. A plane wave can also be obtained by placing a light source at the focus of a converging lens ( slide number 15).

Plane wave diffraction is often referred to as Fraunhofer diffraction after the German scientist Fraunhofer. This type of diffraction is considered especially for two reasons. Firstly, this is a simpler particular case of diffraction, and secondly, this kind of diffraction is often found in various optical devices.

Slit Diffraction

The case of light diffraction by a slit is of great practical importance. When the slit is illuminated by a parallel beam of monochromatic light, a series of dark and light bands is obtained on the screen, rapidly decreasing in intensity ( slide number 16).

If the light is incident perpendicular to the slit plane, then the fringes are arranged symmetrically with respect to the central fringe, and the illumination changes along the screen periodically, in accordance with the conditions of maximum and minimum ( slide number 17, flash animation "Diffraction of light by a slit").

Conclusion:

• a) with a decrease in the width of the slit, the central light band expands;
• b) for a given slit width, the greater the distance between the fringes, the greater the wavelength of light;
• c) therefore, in the case of white light, there is a set of corresponding patterns for different colors;
• d) in this case, the main maximum will be common for all wavelengths and will appear as a white stripe, and the side maxima are colored stripes with alternating colors from purple to red.

Diffraction at two slits.

If there are two identical parallel slits, then they give the same overlapping diffraction patterns, as a result of which the maxima are correspondingly enhanced, and, in addition, there is mutual interference of waves from the first and second slits. As a result, the minima will be in the same places, since these are the directions in which none of the slits sends light. In addition, directions are possible in which the light sent by the two slits cancels each other out. Thus, between the two main maxima there is one additional minimum, and the maxima become narrower than with one gap ( slides 18-19). The greater the number of slots, the more sharply defined the maxima and the wider the minima they are separated by. In this case, the light energy is redistributed so that most of it falls on the maxima, and an insignificant part of the energy falls into the minima ( slide number 20).

Diffraction grating.

A diffraction grating is a collection of a large number of very narrow slits separated by opaque gaps ( slide number 21). If a monochromatic wave falls on the grating, then the slots (secondary sources) create coherent waves. A converging lens is placed behind the grille, then a screen. As a result of the interference of light from different grating slits, a system of maxima and minima is observed on the screen ( slide number 22).

The position of all maxima, except for the main one, depends on the wavelength. Therefore, if white light falls on the grating, then it decomposes into a spectrum. Therefore, a diffraction grating is a spectral device that serves to decompose light into a spectrum. Using a diffraction grating, one can accurately measure the wavelength, since with a large number of slits, the regions of intensity maxima narrow, turning into thin bright bands, and the distance between the maxima (width of dark bands) increases ( slide №23-24).

Resolution of the diffraction grating.

For spectral instruments containing a diffraction grating, the ability to separately observe two spectral lines with close wavelengths is important.

The ability to separately observe two spectral lines having close wavelengths is called the grating resolution ( slide #25-26).

If we want to resolve two close spectral lines, then it is necessary to ensure that the interference maxima corresponding to each of them are as narrow as possible. For the case of a diffraction grating, this means that the total number of grooves applied to the grating should be as large as possible. So, in good diffraction gratings, having about 500 lines per millimeter, with a total length of about 100 mm, the total number of lines is 50,000.

Lattices depending on their application happen metal or glass. The best metal gratings have up to 2000 lines per millimeter of surface, while the total length of the grating is 100-150 mm. Observations on metal gratings are carried out only in reflected light, and on glass ones - most often in transmitted light.

Our eyelashes, with gaps between them, are a rough diffraction grating. If you squint at a bright light source, you can see iridescent colors. The phenomena of diffraction and interference of light help

Nature to color all living things without resorting to the use of dyes ( slide number 27).

3. Primary fixation of the material.

test questions

1. Why is the diffraction of sound everyday more obvious than the diffraction of light?
2. What are Fresnel's additions to Huygens' principle?
3. What is the principle of constructing Fresnel zones?
4. What is the principle of operation of zone plates?
5. When is Fresnel diffraction, Fraunhofer diffraction observed?
6. What is the difference between Fresnel diffraction by a round hole when it is illuminated with monochromatic and white light?
7. Why is diffraction not observed at large apertures and large disks?
8. What determines whether the number of Fresnel zones opened by a hole will be even or odd?
9. What are the characteristic features of the diffraction pattern obtained by diffraction on a small opaque disk.
10. What is the difference between the diffraction pattern on the slit when illuminated with monochromatic and white light?
11. What is the maximum width of the slit at which intensity minima will still be observed?
12. How does an increase in the wavelength and slit width affect Fraunhofer diffraction from a single slit?
13. How will the diffraction pattern change if the total number of grating lines is increased without changing the grating constant?
14. How many additional minima and maxima are produced by diffraction by six slits?
15. Why does a diffraction grating decompose white light into a spectrum?
16. How to determine the highest order of the spectrum of a diffraction grating?
17. How will the diffraction pattern change as the screen moves away from the grating?
18. Why, when using white light, is only the central high white and the side highs iridescent?
19. Why do the strokes on a diffraction grating have to be closely spaced to each other?
20. Why should there be a large number of strokes?

Examples of some key situations (primary consolidation of knowledge) (slide No. 29-49)

1. A diffraction grating with a constant of 0.004 mm is illuminated with light at a wavelength of 687 nm. At what angle to the grating should the observation be made in order to see the image of the second-order spectrum ( slide number 29).
2. Monochromatic light with a wavelength of 500 nm is incident on a diffraction grating having 500 lines per 1 mm. Light is incident on the grating perpendicularly. What is the highest order of the spectrum that can be observed? ( slide number 30).
3. The diffraction grating is located parallel to the screen at a distance of 0.7 m from it. Determine the number of lines per 1 mm for this diffraction grating if, under normal incidence of a light beam with a wavelength of 430 nm, the first diffraction maximum on the screen is at a distance of 3 cm from the central bright band. Assume that sinφ ≈ tgφ ( slide number 31).
4. A diffraction grating with a period of 0.005 mm is located parallel to the screen at a distance of 1.6 m from it and is illuminated by a beam of light with a wavelength of 0.6 μm incident along the normal to the grating. Determine the distance between the center of the diffraction pattern and the second maximum. Assume that sinφ ≈ tgφ ( slide number 32).
5. A diffraction grating with a period of 10-5 m is located parallel to the screen at a distance of 1.8 m from it. The grating is illuminated by a normally incident beam of light with a wavelength of 580 nm. The maximum illumination is observed on the screen at a distance of 20.88 cm from the center of the diffraction pattern. Determine the order of this maximum. Assume that sinφ ≈ tgφ ( slide number 33).
6. Using a diffraction grating with a period of 0.02 mm, the first diffraction image was obtained at a distance of 3.6 cm from the central one and at a distance of 1.8 m from the grating. Find the wavelength of light ( slide number 34).
7. The spectra of the second and third orders in the visible region of the diffraction grating partially overlap with each other. What wavelength in the third order spectrum corresponds to a wavelength of 700 nm in the second order spectrum? ( slide number 35).
8. A plane monochromatic wave with a frequency of 8 1014 Hz is incident along the normal onto a diffraction grating with a period of 5 μm. A converging lens with a focal length of 20 cm is placed parallel to the grating behind it. The diffraction pattern is observed on the screen in the focal plane of the lens. Find the distance between its main maxima of 1st and 2nd orders. Assume that sinφ ≈ tgφ ( slide number 36).
9. What is the width of the entire first-order spectrum (wavelengths range from 380 nm to 760 nm) obtained on a screen 3 m away from a diffraction grating with a period of 0.01 mm? ( slide number 37).
10. What should be the total length of a diffraction grating having 500 lines per 1 mm in order to resolve two spectral lines with wavelengths of 600.0 nm and 600.05 nm with its help? ( slide number 40).
11. Determine the resolution of a diffraction grating with a period of 1.5 μm and a total length of 12 mm if light with a wavelength of 530 nm falls on it ( slide number 42).
12. What is the minimum number of lines the grating should contain so that two yellow sodium lines with wavelengths of 589 nm and 589.6 nm can be resolved in the first-order spectrum. What is the length of such a grating if the grating constant is 10 µm ( slide number 44).
13. Define the number of open zones with the following parameters:
    R = 2 mm; a=2.5 m; b=1.5 m
    a) λ=0.4 µm.
    b) λ=0.76 µm ( slide number 45).
14. A 1.2 mm slit is illuminated with green light at a wavelength of 0.5 µm. The observer is located at a distance of 3 m from the slit. Will he see the diffraction pattern ( slide number 47).
15. A 0.5 mm slit is illuminated with green light from a 500 nm laser. At what distance from the slit can one clearly observe the diffraction pattern ( slide number 49).

4. Homework (slide number 50).

Textbook: § 71-72 (G.Ya. Myakishev, B.B. Bukhovtsev. Physics.11).

Collection of problems in physics No. 1606,1609,1612, 1613,1617 (G.N.Stepanova).

The propagation of a beam in an optically homogeneous medium is rectilinear, but there are a number of phenomena in nature where a deviation from this condition can be observed.

Diffraction- the phenomenon of light waves bending around the encountered obstacles. In school physics, two diffraction systems are studied (systems in which diffraction is observed during the passage of a beam):

• diffraction by a slit (rectangular hole)
• grating diffraction (a set of equally spaced slits)

- diffraction on a rectangular hole (Fig. 1).

Rice. 1. Slit diffraction

Let a plane with a slit be given, of width , on which a beam of light A falls at a right angle. Most of the light passes onto the screen, but some of the rays diffract at the edges of the slit (i.e., deviate from their original direction). Further, these rays with each other with the formation of a diffraction pattern on the screen (alternating bright and dark areas). Consideration of the laws of interference is rather complicated, so we confine ourselves to the main conclusions.

The resulting diffraction pattern on the screen consists of alternating regions with diffraction maxima (maximum light areas) and diffraction minima (maximum dark regions). This pattern is symmetrical with respect to the central light beam. The position of the maxima and minima is described by the angle relative to the vertical at which they are visible, and depends on the size of the slit and the wavelength of the incident radiation. The position of these areas can be found using a number of relationships:

• for diffraction maxima

The zero diffraction maximum is the central point on the screen under the slit (Fig. 1).

• for diffraction minima

Conclusion: according to the conditions of the problem, it is necessary to find out: the maximum or minimum of diffraction must be found and the corresponding relation (1) or (2) should be used.

Diffraction on a diffraction grating.

A diffraction grating is a system consisting of alternating slots equally spaced from each other (Fig. 2).

Rice. 2. Diffraction grating (beams)

Just as for a slit, a diffraction pattern will be observed on the screen after the diffraction grating: alternation of light and dark areas. The whole picture is the result of the interference of light rays with each other, but the picture from one slit will be affected by rays from other slits. Then the diffraction pattern should depend on the number of slits, their sizes, and proximity.

Let's introduce a new concept - grating constant:

Then the positions of the diffraction maxima and minima are:

• for the main diffraction maxima(Fig. 3)

From the relation d sin j = ml it can be seen 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 with white light, the central maximum will have the form of a white band, and all the rest will be rainbow bands, called the diffraction spectra of the first ( m= ± 1), second ( m= ± 2), etc. orders. In the spectra of each order, the most deviated will be red rays (with a large value l, since sin j ~ 1 / l), and the least purple (with a smaller value l). The spectra are clearer (in terms of color separation) the more slits N contains a grid. This follows from the fact that the linear half-width of the maximum is inversely proportional to the number of slots N). The maximum number of observed diffraction spectra is determined by relation (3.83). Thus, the diffraction grating decomposes complex radiation into separate monochromatic components, i.e. performs 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 that serve 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 hits the entrance slit S device located in the focal plane of the collimator lens L one . The plane wave formed during the passage through the collimator is incident on the dispersive element D, which is used as a diffraction grating. After the spatial separation of the beams by the dispersing element, the output (camera) objective 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 make up the spectrum of the studied radiation.

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

Angular dispersion D j characterizes the change in the deflection angle j beam when changing its wavelength l and is defined as

D j= dj / dl,

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

D j = j¢ l = m / d cos j.

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

D j @ m / d.

Linear dispersion is given by

D l = dl / dl,

where dl is the linear distance between two spectral lines that differ in wavelength dl.

From fig. 3.24 shows that dl = f 2 dj, where f 2 - lens focal length L 2. With this in mind, we obtain a relation relating the angular and linear dispersions:

D l = f 2 D j.

The spectra of adjacent orders may overlap. Then the spectral apparatus becomes unsuitable for studying the corresponding part of the spectrum. Maximum width D l of the spectral interval of the radiation under study, in which the spectra of neighboring orders do not yet overlap, is called the free region of dispersion or the dispersion region of the spectral apparatus. Let the wavelengths of the radiation incident on the grating lie in the interval from l before l+ D l. Maximum D value l, at which the overlapping of the spectra does not yet occur, can be determined from the condition of superposition 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 the 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 give separately two close spectral lines and is determined by the ratio

R = l / d l,

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

are not coherent at different wavelengths, 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 close wavelengths l and l + d l are considered permitted if they are within that 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 (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, is 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 from the first zero of the intensity). From here we find

d l= l / (mN),

and, consequently, the resolution of the diffraction grating as a spectral instrument

Thus, the resolution of the diffraction grating is proportional to the number of slots N and the order of the spectrum 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- the width of the working part of the lattice. As you can see, the maximum resolution of a slotted grating is determined only by the width of the working part of the grating and the average wavelength of the radiation under study. Knowing R max , we find the minimum resolvable wavelength interval:

(d l) min @ l 2 / L.

Topics of the USE codifier: light diffraction, diffraction grating.

If there is an obstacle in the path of the wave, then diffraction - wave deviation from rectilinear propagation. This deviation is not reduced to reflection or refraction, as well as the curvature of the path of rays due to a change in the refractive index of the medium. Diffraction consists in the fact that the wave goes around the edge of the obstacle and enters the region of the geometric shadow.

Let, for example, a plane wave be incident on a screen with a rather narrow slit (Fig. 1). A diverging wave arises at the slot exit, and this divergence increases with a decrease in the slot width.

In general, diffraction phenomena are expressed the more clearly, the smaller the obstacle. Diffraction is most significant when the size of the obstacle is less than or of the order of the wavelength. It is this condition that must be satisfied by the width of the slot in Fig. one.

Diffraction, like interference, is characteristic of all types of waves - mechanical and electromagnetic. Visible light is a special case of electromagnetic waves; It is not surprising, therefore, that one can observe
light diffraction.

So, in fig. 2 shows the diffraction pattern obtained as a result of the passage of a laser beam through a small hole with a diameter of 0.2 mm.

We see, as expected, the central bright spot; very far from the spot is a dark area - a geometric shadow. But around the central spot - instead of a clear border between light and shadow! - there are alternating light and dark rings. The farther from the center, the lighter rings become less bright; they gradually disappear into the shadow area.

Sounds like interference, doesn't it? This is what she is; these rings are interference maxima and minima. What kind of waves are interfering here? We will soon deal with this issue, and at the same time we will find out why diffraction is observed at all.

But before that, one cannot fail to mention the very first classical experiment on the interference of light - Young's experiment, in which the phenomenon of diffraction was significantly used.

Young's experience.

Every experiment with light interference contains some way of obtaining two coherent light waves. In the experiment with Fresnel mirrors, as you remember, the coherent sources were two images of the same source obtained in both mirrors.

The simplest idea that came up in the first place was the following. Let's poke two holes in a piece of cardboard and expose it to the sun's rays. These holes will be coherent secondary light sources, since there is only one primary source - the Sun. Therefore, on the screen in the area of ​​overlapping beams diverging from the holes, we should see the interference pattern.

Such an experiment was set long before Jung by the Italian scientist Francesco Grimaldi (who discovered the diffraction of light). Interference, however, was not observed. Why? This question is not very simple, and the reason is that the Sun is not a point, but an extended source of light (the angular size of the Sun is 30 arc minutes). The solar disk consists of many point sources, each of which gives its own interference pattern on the screen. Superimposed, these separate pictures "blur" each other, and as a result, a uniform illumination of the area of ​​overlapping beams is obtained on the screen.

But if the Sun is excessively "big", then it is necessary to artificially create pinpoint primary source. For this purpose, a small preliminary hole was used in Young's experiment (Fig. 3).

Rice. 3. Scheme of Jung's experiment

A plane wave is incident on the first hole, and a light cone appears behind the hole, which expands due to diffraction. It reaches the next two holes, which become the sources of two coherent light cones. Now - due to the point nature of the primary source - an interference pattern will be observed in the region of overlapping cones!

Thomas Young carried out this experiment, measured the width of the interference fringes, derived a formula, and using this formula for the first time calculated the wavelengths of visible light. That is why this experiment has become one of the most famous in the history of physics.

Huygens-Fresnel principle.

Let us recall the formulation of the Huygens principle: each point involved in the wave process is a source of secondary spherical waves; these waves propagate from a given point, as from a center, in all directions and overlap each other.

But a natural question arises: what does "superimposed" mean?

Huygens reduced his principle to a purely geometric way of constructing a new wave surface as an envelope of a family of spheres expanding from each point of the original wave surface. Secondary Huygens waves are mathematical spheres, not real waves; their total effect is manifested only on the envelope, i.e., on the new position of the wave surface.

In this form, the Huygens principle did not give an answer to the question why, in the process of wave propagation, a wave traveling in the opposite direction does not arise. Diffraction phenomena also remained unexplained.

The modification of the Huygens principle took place only 137 years later. Augustin Fresnel replaced Huygens' auxiliary geometric spheres with real waves and suggested that these waves interfere together.

Huygens-Fresnel principle. Each point of the wave surface serves as a source of secondary spherical waves. All these secondary waves are coherent due to the commonality of their origin from the primary source (and, therefore, can interfere with each other); the wave process in the surrounding space is the result of the interference of secondary waves.

Fresnel's idea filled Huygens' principle with physical meaning. Secondary waves, interfering, amplify each other on the envelope of their wave surfaces in the "forward" direction, ensuring further wave propagation. And in the "backward" direction, they interfere with the original wave, mutual damping is observed, and the reverse wave does not occur.

In particular, light propagates where the secondary waves are mutually reinforcing. And in places of weakening of the secondary waves, we will see dark areas of space.

The Huygens–Fresnel principle expresses an important physical idea: a wave, moving away from its source, subsequently "lives its own life" and no longer depends on this source. Capturing new areas of space, the wave propagates farther and farther due to the interference of secondary waves excited at different points in space as the wave passes.

How does the Huygens-Fresnel principle explain the phenomenon of diffraction? Why, for example, does diffraction occur at a hole? The fact is that only a small luminous disk cuts out the screen hole from the infinite flat wave surface of the incident wave, and the subsequent light field is obtained as a result of the interference of waves from secondary sources located no longer on the entire plane, but only on this disk. Naturally, the new wave surfaces will no longer be flat; the path of the rays is bent, and the wave begins to propagate in different directions, not coinciding with the original. The wave goes around the edges of the hole and penetrates into the region of the geometric shadow.

Secondary waves emitted by different points of the cut out light disk interfere with each other. The result of interference is determined by the phase difference of the secondary waves and depends on the deflection angle of the beams. As a result, there is an alternation of interference maxima and minima - which we saw in Fig. 2.

Fresnel not only supplemented the Huygens principle with the important idea of ​​coherence and interference of secondary waves, but also came up with his famous method for solving diffraction problems, based on the construction of the so-called Fresnel zones. The study of Fresnel zones is not included in the school curriculum - you will learn about them already in the university physics course. Here we will only mention that Fresnel, within the framework of his theory, managed to give an explanation of our very first law of geometric optics - the law of rectilinear propagation of light.

Diffraction grating.

A diffraction grating is an optical device that allows you to decompose light into spectral components and measure wavelengths. Diffraction gratings are transparent and reflective.

We will consider a transparent diffraction grating. It consists of a large number of slits of width separated by gaps of width (Fig. 4). Light only passes through cracks; gaps do not let light through. The quantity is called the lattice period.

Rice. 4. Diffraction grating

The diffraction grating is made using a so-called dividing machine, which marks the surface of glass or transparent film. In this case, the strokes turn out to be opaque gaps, and the untouched places serve as cracks. If, for example, a diffraction grating contains 100 lines per millimeter, then the period of such a grating will be: d= 0.01 mm= 10 µm.

First, we will look at how monochromatic light passes through the grating, that is, light with a strictly defined wavelength. An excellent example of monochromatic light is the beam of a laser pointer with a wavelength of about 0.65 microns).

On fig. 5 we see such a beam incident on one of the diffraction gratings of the standard set. The grating slits are arranged vertically, and periodic vertical stripes are observed behind the grating on the screen.

As you already understood, this is an interference pattern. The diffraction grating splits the incident wave into many coherent beams that propagate in all directions and interfere with each other. Therefore, on the screen we see an alternation of maxima and minima of interference - light and dark bands.

The theory of a diffraction grating is very complex and in its entirety is far beyond the scope of the school curriculum. You should know only the most elementary things related to a single formula; this formula describes the position of the screen illumination maxima behind the diffraction grating.

So, let a plane monochromatic wave fall on a diffraction grating with a period (Fig. 6). The wavelength is .

Rice. 6. Diffraction by a grating

For greater clarity of the interference pattern, you can put the lens between the grating and the screen, and place the screen in the focal plane of the lens. Then the secondary waves coming in parallel from different slits will gather at one point of the screen (side focus of the lens). If the screen is far enough away, then there is no special need for a lens - the rays coming to a given point on the screen from different slits will be almost parallel to each other anyway.

Consider secondary waves deviating by an angle . The path difference between two waves coming from adjacent slots is equal to the small leg of a right triangle with hypotenuse ; or, equivalently, this path difference is equal to the leg of the triangle. But the angle is equal to the angle, since these are acute angles with mutually perpendicular sides. Therefore, our path difference is .

Interference maxima are observed when the path difference is equal to an integer number of wavelengths:

(1)

When this condition is met, all waves arriving at a point from different slots will add up in phase and reinforce each other. In this case, the lens does not introduce an additional path difference - despite the fact that different rays pass through the lens in different ways. Why is it so? We will not go into this issue, since its discussion is beyond the scope of the USE in physics.

Formula (1) allows you to find the angles that specify the directions to the maxima:

. (2)

When we get it central maximum, or zero order maximum.The path difference of all secondary waves traveling without deviation is equal to zero, and in the central maximum they add up with a zero phase shift. The central maximum is the center of the diffraction pattern, the brightest of the maximums. The diffraction pattern on the screen is symmetrical with respect to the central maximum.

When we get the angle:

This angle sets the direction for first order maxima. There are two of them, and they are located symmetrically with respect to the central maximum. The brightness in the first-order maxima is somewhat less than in the central maximum.

Similarly, for we have the angle:

He gives directions to second order maxima. There are also two of them, and they are also located symmetrically with respect to the central maximum. The brightness in the second-order maxima is somewhat less than in the first-order maxima.

An approximate pattern of directions to the maxima of the first two orders is shown in Fig. 7.

Rice. 7. Maxima of the first two orders

In general, two symmetrical maxima k th order are determined by the angle:

. (3)

When small, the corresponding angles are usually small. For example, at µm and µm, the first-order maxima are located at an angle .The brightness of the maxima k-th order gradually decreases with increasing k. How many maximums can be seen? This question is easy to answer using formula (2). After all, the sine cannot be greater than one, therefore:

Using the same numerical data as above, we get: . Therefore, the highest possible order of the maximum for this lattice is 15.

Look again at fig. 5 . We see 11 maximums on the screen. This is the central maximum, as well as two maxima of the first, second, third, fourth and fifth orders.

A diffraction grating can be used to measure an unknown wavelength. We direct a beam of light at the grating (the period of which we know), measure the angle to the maximum of the first
order, we use formula (1) and obtain:

Diffraction grating as a spectral device.

Above, we considered the diffraction of monochromatic light, which is a laser beam. Often dealing with non-monochromatic radiation. It is a mixture of various monochromatic waves that make up range this radiation. For example, white light is a mixture of wavelengths across the entire visible range, from red to violet.

The optical device is called spectral, if it allows one to decompose light into monochromatic components and thereby investigate the spectral composition of radiation. The simplest spectral device you are well aware of is a glass prism. The diffraction grating is also among the spectral instruments.

Assume that white light is incident on a diffraction grating. Let's go back to formula (2) and think about what conclusions can be drawn from it.

The position of the central maximum () does not depend on the wavelength. In the center of the diffraction pattern will converge with zero path difference all monochromatic components of white light. Therefore, in the central maximum, we will see a bright white band.

But the positions of the maxima of the order are determined by the wavelength. The smaller the , the smaller the angle for the given . Therefore, at the maximum k th order, monochromatic waves are separated in space: the purple band will be the closest to the central maximum, and the red one will be the farthest.

Therefore, in each order, white light is decomposed by a grating into a spectrum.
The first-order maxima of all monochromatic components form a first-order spectrum; then come the spectra of the second, third, and so on orders. The spectrum of each order has the form of a colored band, in which all the colors of the rainbow are present - from purple to red.

The diffraction of white light is shown in Fig. eight . We see a white band in the central maximum, and on the sides - two spectra of the first order. As the deflection angle increases, the color of the bands changes from purple to red.

But a diffraction grating not only makes it possible to observe spectra, i.e., to conduct a qualitative analysis of the spectral composition of radiation. The most important advantage of a diffraction grating is the possibility of quantitative analysis - as mentioned above, we can use it to to measure wavelengths. In this case, the measurement procedure is very simple: in fact, it comes down to measuring the direction angle to the maximum.

Natural examples of diffraction gratings found in nature are bird feathers, butterfly wings, and the mother-of-pearl surface of a sea shell. If you squint into the sunlight, you can see the iridescence around the eyelashes. Our eyelashes act in this case like a transparent diffraction grating in fig. 6, and the optical system of the cornea and lens acts as a lens.

The spectral decomposition of white light, given by a diffraction grating, is easiest to observe by looking at an ordinary CD (Fig. 9). It turns out that the tracks on the surface of the disk form a reflective diffraction grating!