Purpose of the work: familiarization with diffraction patterns of various types; determination of the width of a rectangular slit when studying the phenomenon of diffraction in monochromatic light; determination of wavelengths of red and violet light.

Instruments and accessories: diffraction grating, screen with a slit, ruler with divisions, illuminator, tripod; installation of RMS 3.

Theoretical information

The phenomenon of diffraction consists in the deviation of light from rectilinear propagation in a medium with sharp inhomogeneities in the form of edges of opaque and transparent bodies, narrow holes, protrusions, etc., as a result of which light penetrates into the region of a geometric shadow, and an interference redistribution of light intensity occurs. Diffraction should be understood as any deviation from the rectilinear propagation of rays, unless it is a consequence of the usual laws of geometric optics - reflection and refraction. The phenomenon of diffraction is explained by the wave properties of light using the Huygens-Fresnel principle.

The main provisions of this principle:

Each element of the wave surface that the light wave has reached at a given moment serves as a source of secondary waves, the amplitude of which is proportional to the area of ​​the element.

Secondary waves created by elements of the same surface are coherent and can interfere when superimposed.

The radiation is maximum in the direction of the outer normal to the surface element. The amplitude of a spherical wave decreases with distance from the source. Only open areas of the wave surface emit.

This principle makes it possible to approve deviations from straight-line propagation in the event of any obstacle. Let us consider the case of a plane wave (a parallel beam of light) falling on an obstacle in the form of a hole MN in an opaque plate (Fig. 2.1).

elementary waves at time t 2 determines the wave front with the surface P 2 .

From Fig. 2.1 it is clear that the light rays, being perpendicular to the wave front, deviate from their original direction and fall into the region of the geometric shadow.

Solving the problem of light diffraction means exploring questions related to the intensity of the resulting light wave in different directions. The main issue in this research is the study of light interference, in which overlapping waves can not only be amplified, but also weakened. One important case of diffraction is diffraction in parallel rays. It is used when considering the operation of optical instruments (diffraction gratings, optical instruments, etc.). In the simplest case, a diffraction grating is a transparent glass plate on which strokes of equal width are applied at the same distance from each other. Such a grating can be used in a conventional spectral installation instead of a prism as a dispersive system. To make it easier to understand the rather complex physical phenomenon of interference of diffracted light beams at N slits of a grating, let us first consider diffraction at one, then at two slits, and finally write down an expression for N slits. To simplify the calculation, we use the Fresnel zone method.

Single slit diffraction. Let us consider diffraction in parallel rays at one slit. The type of diffraction that examines the diffraction pattern formed by parallel rays is called parallel ray diffraction, or Fraunhofer diffraction. A slit is a rectangular hole in an opaque plate, with one side much larger than the other. The smaller side is called the slot width A. Such a slit is an obstacle to light waves, and diffraction can be observed through it. Under laboratory conditions, slit diffraction is clearly observed if the slit width A comparable in magnitude to the wavelength of light. Let a monochromatic light wave be incident normally to the plane of a slit of width a(distance AB). Behind the slit there is a collecting lens and a screen placed in the focal plane of the lens. The diagram is shown in Fig. 2.2.

According to Huygens' principle, each point of the wave front that reaches the slit is a new source of oscillations, and the phases of these waves are the same, since with normal incidence of light the plane of the slit coincides with the plane of the wave front. Let us consider rays of monochromatic light from points lying on the front AB, the direction of propagation of which makes an angle with normal. Let us lower the perpendicular AC from point A to the direction of the ray propagating from point B. Then, propagating further from AC, the rays will not change the path difference. The difference in the path of the rays is the segment BC. To calculate the interference of these rays, we use the Fresnel zone method.

Let us divide the segment BC into segments of length . The aircraft will accommodate the following statements:

Drawing lines parallel to AC from the ends of these segments until they meet AB, we divide the wave front in the slot into a number of strips of equal width, the number of which is equal to z. They are Fresnel zones, since the corresponding points of these strips are sources of waves that reach the observation point M in a given direction with a mutual path difference. The amplitudes of the waves from the strips will be the same, because the front is flat and their areas are equal. According to the theory of Fresnel zones, rays from two adjacent zones cancel each other because their phases are opposite. Then, with an even number of Fresnel zones (z=2m, where m is an integer, m=1,2,3...) fitting into the slits, there will be a diffraction minimum at point M, and with an odd number (z=(2m+1) ) – maximum. We will then write Equation (1) as follows:

The intensity distribution in the diffraction pattern from one slit is shown in Fig. 2.3. The abscissa axis shows the distance from the zero maximum along the screen on which the spectral pattern is located.

Double-slit diffraction. To increase intensity and clearer color separation, they use not one slit, but a diffraction grating, which is a series of parallel slits of the same width a, separated by opaque intervals of width b. Sum a+ b= d is called the period or constant of the diffraction grating.

In order to find the distribution of illumination on the screen in the case of a grating, it is necessary to take into account not only the interference of waves emerging from each individual slit, but also the mutual interference of waves arriving at a given point on the screen from neighboring slits. Let's assume that there are only two slits. A monochromatic wave is incident normally to the plane of the slits. When an even number of Fresnel zones fit into the slots, the minimum condition for the slot is satisfied. Since the minimum condition is satisfied for each slit, the same holds true for the entire lattice. Consequently, the minimum condition for the lattice coincides with the minimum condition for the slit; it is called the main minimum condition, and has the form:

.

Let us consider the case when an odd number of Fresnel zones fit into the slots. In this case, in each slit there will remain one uncompensated Fresnel zone, in which all light sources oscillate in the same phase. These uncompensated rays passing through one of the slits will interfere with uncompensated rays passing through the other slit. Let us select two arbitrarily directed rays (Fig. 2.4), emanating from the corresponding points of adjacent slits and falling at one point on the screen. Their interference is determined by the path difference BC= d sin . If BC = , then at point M the light is intensified. The equation

determines the main maxima. If, , then at point M the light is weakened. The equation

is the condition for additional minima that appear due to the presence of a second gap.

If b a, then the width of the main part of the diffraction pattern from the two slits remains the same. Most of the energy is concentrated within the central maximum. The dotted line shows the intensity distribution for one slit. If b a the diffraction pattern will be somewhat narrowed. At b=0, peaks are obtained that are 2 times narrower, since there are not two slits wide a, and one slot is 2 wide a.

Diffraction byNcracks. Calculating the diffraction pattern on a diffraction grating is quite complex from a mathematical point of view, but in principle it is no different from considering diffraction on two slits. It should be taken into account that in the case of diffraction by two slits, a certain number of additional maxima and minima appear. If there is a third slit, their number increases, since it is necessary to take into account the contribution to the diffraction pattern from each slit. As the number of slits on the diffraction grating increases, the number of additional maxima and minima increases. The condition for the main maxima and minima for the diffraction grating remains the same as for two slits:

,m=0,1,2… (main maxima), (2.2)

,m=1,2,3… (main minima), (2.3)

and additional minima are determined by the condition:

,m=0,1,2… (2.4)

If the diffraction grating consists of N slits, then the condition for the main maxima is condition (2.2), and the condition for the main minima is condition (2.3).

Condition of additional minimums:

where N is the total number of grid slits (m=1, 2,…,N-1,N+1,…, 2N-1, 2N+1,…). In formula (2.5) m takes all integer values ​​except 0, N, 2N, i.e., except those under which condition (2.5) turns into (2.2).

Comparing formulas (2.2) and (2.5), we see that the number of main maxima is N times less than the total number of additional minima. Indeed, the number (or order) of additional minima corresponding to the angle , is obtained from formula (2.2) as follows:

and the total number of additional minima, as can be seen from formula (2.5),

whence follows .

Thus, between the two main maxima there are (N-1) additional minima, separated by secondary maxima. The contribution of these side maxima to the overall diffraction pattern is small, since their intensity is low and quickly decreases with distance from the main maximum of a given order. Because as the number of grating lines increases, an increasing amount of light energy passes through it and at the same time the number of additional maxima and minima increases. This means that the main maxima become narrower and their brightness increases, that is, the resolution of the grating increases.

If light containing a number of spectral components falls on the grating, then, in accordance with formula (2.2), the main maxima for different components are formed at different angles. Thus, the grating splits the light into a spectrum.

The characteristics of a grating as a spectral device are angular dispersion and resolution.

Angular dispersion is the quantity
, Where
- angular distance between two spectral lines differing in wavelength by
. Differentiating formula (2), we obtain:

Resolution is the quantity
, Where
- the smallest difference in wavelengths of two spectral lines that are visible separately in the spectrum.

According to the Rayleigh criterion, two close lines are considered resolved (visible separately) if the intensity in the interval between them is no more than 80% of the intensity of the maximum, i.e. I=0.8I 0 , where I 0 is the intensity of the main maximum, I is the intensity of the gap between two adjacent maxima (Fig. 2.6).

From the Rayleigh condition it follows:

those. The resolution of the grating increases with the number of slits N and depends on the order of the spectrum.

TASK 1. Determining the wavelengths of red and violet light.

The experimental setup consists of a tripod on which is mounted a horizontal ruler with divisions, a diffraction grating, a screen with a slit (to obtain a narrow beam of light) and an illuminator. The diffraction grating used in this work has 100 lines per 1 mm, i.e. lattice period d=0.01 mm. A beam of light, passing through a narrow slit and then a diffraction grating, hits the lens of the eye, which plays the role of a biconvex lens. With further propagation, the image of the spectra and scale with divisions on a screen with a slit reaches the retina. Thus we see the image of the spectra on the scale.

From the condition of the mth order maximum for a diffraction grating, the wavelength is expressed:

Where d is the period of the diffraction grating, sin φ is the sine of the angle at which a given line is observed in the spectrum, m is the order of the spectrum in which the line is observed.

The angles φ m at which the lines are observed in the spectra are small, therefore sin φ m ≈ tan φ m . Using this condition, we get:

Formula (2.6) is working for determining the wavelength of the observed line in the mth order spectrum.

Work order

Turn on the light.

Install the screen with the slit at a distance L from the diffraction grating.

Bring your eye closer to the grating at a convenient distance (diffraction spectra should be visible on both sides of the slit on the black background of the scale). In this case, the eye should be at a close distance from the grid (Fig. 2.7).

Using the screen scale, determine the position of the red and violet lines S in the spectra of the 1st and 2nd order, located to the right and left of the slit for various distances L (L = 15 cm, 20 cm, 25 cm). Enter the measurement results in the table. 1.

Table 1

Calculate tgφ using the formula:

.

Using formula (2.6), calculate the wavelengths of red and violet light for spectra of different orders and for different distances L.

Calculate the arithmetic average of the wavelengths for red and violet light using the formula:

,

where n is the number of measurements.

.

,

where t α (n) – Student’s coefficient, α=0.95, t 0.95 (6)=2.6.

λ= ±Δλ, nm; α=0.95.

TASK 2. Determination of the wavelength of radiation during diffraction by a slit.

Description of the laboratory setup

The MOL-1 object is a thin glass disk with an opaque coating and transparent structures located in three rows: row A - double slits, row B - round holes, row C - single slits. The total number of slits in row C is 16. Radiation from the laser is directed to the desired structure on the surface of the MOL-1 object. In this case, a corresponding diffraction pattern is observed on the screen.

From the condition of the mth order minimum for the gap, the radiation wavelength is expressed:

Where A is the slit width, sin φ is the sine of the angle at which the minimum is observed, m is the order of the minimum.

The angles φ m at which the minima are observed are small, therefore sin φ m ≈ tan φ m . Using this condition, we get:

Formula (2.7) is working for determining the wavelength of laser radiation.

Work order

According to table. 2 select slots to study in row C - at least three (as directed by the teacher).

table 2

Turn on the laser. Set the slot at a distance L to the screen. By adjusting the adjustment screws, achieve the desired direction of radiation to the slit under study in row C on the test object MOL-1. Obtain a clear diffraction pattern.

Place a blank sheet of paper on the screen. Mark on it the distances S from the middle of the central maximum to the middle of the minima of the first, second and third orders to the right and left of the central maximum (i.e. for orders m=±1, ±2, ±3). Measure distance L.

After removing the sheet, carefully measure the marked distances S with a ruler. Enter the measurement results in the table. 3.

Table 3

.

Calculate tgφ using the formula:

Calculate the arithmetic mean value of the wavelength using the formula:

,

where n is the number of measurements.

Calculate the estimate of the mean square error using the formula:

.

Calculate the random error limit using the formula:

,

where t α (n) – Student’s coefficient, α=0.95, t 0.95 (9)=2.31.

Write the final result as:

λ= ±Δλ, nm; α=0.95.

Control questions

What waves are called coherent?

What are the phenomena of interference and diffraction of light?

What is called a wave front, wave surface?

What is the Fresnel zone method?

Formulate the Huygens–Fresnel principle.

Draw and explain the diffraction patterns obtained from a single slit and from a diffraction grating when illuminated with monochromatic and white light.

Explain the appearance of the main maximum, main minimum and additional minimum during diffraction by a grating. Write down their formulas.

How will the appearance of the diffraction pattern from the grating change if the light source is replaced by a monochromatic one?

Explain the applications of diffraction in science and technology.

LABORATORY WORK No. 3

Definition 1

Diffraction of light is the phenomenon of deviation of light from the rectilinear direction of propagation when passing near obstacles.

In classical physics, the phenomenon of diffraction is described as wave interference in accordance with the Huygens-Fresnel principle. These characteristic patterns of behavior occur when a wave encounters an obstacle or gap that is comparable in size to its wavelength. Similar effects occur when a light wave passes through a medium with a changing refractive index, or when a sound wave passes through a medium with a change in acoustic impedance. Diffraction occurs with all types of waves, including sound waves, wind waves, and electromagnetic waves, as well as visible light, X-rays, and radio waves.

Since physical objects have wave properties (at the atomic level), diffraction also occurs with substances and can be studied according to the principles of quantum mechanics.

Examples

Diffraction effects are common in everyday life. The most striking examples of diffraction are those associated with light; for example, closely spaced tracks on CDs or DVDs act as a diffraction grating. Diffraction from small particles in the atmosphere can result in a bright ring that is visible near a bright light source such as the sun or moon. Speckle, which occurs when a laser beam hits an optically uneven surface, is also diffraction. All of these effects are a consequence of the fact that light travels as a wave.

Note 1

Diffraction can occur with any type of wave.

Ocean waves dissipate around jetties and other obstacles. Sound waves can bend around objects, so you can hear someone calling even when they're hiding behind a tree.

Story

The effects of light diffraction were well known in the time of Grimaldi by Francesco Maria, who also coined the term diffraction. The results obtained by Grimaldi were published posthumously in $1665. Thomas Young performed a famous experiment in 1803 demonstrating interference from two closely spaced slits. Explaining his results in terms of the interference of waves emanating from two different slits, he concluded that light must travel in the form of waves. Fresnel made more accurate studies and calculations of diffraction, which were published in $1815. Fresnel based his theory on the definition of light developed by Christian Huygens, supplementing it with the idea of ​​the interference of secondary waves. Experimental confirmation of Fresnel's theory became one of the main proofs of the wave nature of light. This theory is now known as the Huygens-Fresnel principle.

Diffraction of light

Slit diffraction

A long slit of infinitesimal width, which is illuminated by light, refracts the light into a series of circular waves and into a wavefront that emerges from the slit and is a cylindrical wave of uniform intensity. A slit that is wider than the wavelength produces interference effects in the space exiting the slit. They can be explained by the fact that the slit behaves as if it had a large number of point sources that are distributed evenly across the entire width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase.

Diffraction grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into multiple beams traveling in different directions.

The light diffracted by the grating is determined by summing the light diffracted from each element, and is essentially the convolution of the diffraction and interference patterns.

Definition 1

Diffraction of light is the phenomenon of deviation of light from the rectilinear direction of propagation when passing near obstacles.

In classical physics, the phenomenon of diffraction is described as wave interference in accordance with the Huygens-Fresnel principle. These characteristic patterns of behavior occur when a wave encounters an obstacle or gap that is comparable in size to its wavelength. Similar effects occur when a light wave passes through a medium with a changing refractive index, or when a sound wave passes through a medium with a change in acoustic impedance. Diffraction occurs with all types of waves, including sound waves, wind waves, and electromagnetic waves, as well as visible light, X-rays, and radio waves.

Since physical objects have wave properties (at the atomic level), diffraction also occurs with substances and can be studied according to the principles of quantum mechanics.

Examples

Diffraction effects are common in everyday life. The most striking examples of diffraction are those associated with light; for example, closely spaced tracks on CDs or DVDs act as a diffraction grating. Diffraction from small particles in the atmosphere can result in a bright ring that is visible near a bright light source such as the sun or moon. Speckle, which occurs when a laser beam hits an optically uneven surface, is also diffraction. All of these effects are a consequence of the fact that light travels as a wave.

Note 1

Diffraction can occur with any type of wave.

Ocean waves dissipate around jetties and other obstacles. Sound waves can bend around objects, so you can hear someone calling even when they're hiding behind a tree.

Story

The effects of light diffraction were well known in the time of Grimaldi by Francesco Maria, who also coined the term diffraction. The results obtained by Grimaldi were published posthumously in $1665. Thomas Young performed a famous experiment in 1803 demonstrating interference from two closely spaced slits. Explaining his results in terms of the interference of waves emanating from two different slits, he concluded that light must travel in the form of waves. Fresnel made more accurate studies and calculations of diffraction, which were published in $1815. Fresnel based his theory on the definition of light developed by Christian Huygens, supplementing it with the idea of ​​the interference of secondary waves. Experimental confirmation of Fresnel's theory became one of the main proofs of the wave nature of light. This theory is now known as the Huygens-Fresnel principle.

Diffraction of light

Slit diffraction

A long slit of infinitesimal width, which is illuminated by light, refracts the light into a series of circular waves and into a wavefront that emerges from the slit and is a cylindrical wave of uniform intensity. A slit that is wider than the wavelength produces interference effects in the space exiting the slit. They can be explained by the fact that the slit behaves as if it had a large number of point sources that are distributed evenly across the entire width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase.

Diffraction grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into multiple beams traveling in different directions.

The light diffracted by the grating is determined by summing the light diffracted from each element, and is essentially the convolution of the diffraction and interference patterns.

Double-slit diffraction

Diffraction- a phenomenon that occurs when waves propagate (for example, light and sound waves). The essence of this phenomenon is that the wave is able to bend around obstacles. This results in the wave motion being observed in an area behind the obstacle where the wave cannot reach directly. The phenomenon is explained by the interference of waves at the edges of opaque objects or inhomogeneities between different media along the path of wave propagation. An example would be the appearance of colored light stripes in the shadow area from the edge of an opaque screen.

Diffraction manifests itself well when the size of the obstacle in the path of the wave is comparable to its length or less.

Acoustic diffraction- deviation from straight-line propagation of sound waves.

1. Slit diffraction

Scheme of the formation of regions of light and shadow during diffraction by a slit

In the case when a wave falls on a screen with a slit, it penetrates due to diffraction, but a deviation from the rectilinear propagation of the rays is observed. The interference of waves behind the screen leads to the appearance of dark and light areas, the location of which depends on the direction in which the observation is being made, the distance from the screen, etc.

2. Diffraction in nature and technology

Diffraction of sound waves is often observed in everyday life as we hear sounds that reach us from behind obstacles. It is easy to observe the waves on the water going around small obstacles.

The scientific and technical uses of the diffraction phenomenon are varied. Diffraction gratings are used to split light into a spectrum and to create mirrors (for example, for semiconductor lasers). X-ray, electron, and neutron diffraction is used to study the structure of crystalline solids.

Diffraction time imposes limitations on the resolution of optical instruments, such as microscopes. Objects whose dimensions are smaller than the wavelength of visible light (400-760 nm) cannot be viewed with an optical microscope. A similar limitation exists in the lithography method, which is widely used in the semiconductor industry for the production of integrated circuits. Therefore, it is necessary to use light sources in the ultraviolet region of the spectrum.

3. Diffraction of light

The phenomenon of light diffraction clearly confirms the theory of the corpuscular-wave nature of light.

It is difficult to observe the diffraction of light, since the waves deviate from the interference at noticeable angles only under the condition that the size of the obstacles is approximately equal to the wavelength of the light, and it is very small.

For the first time, having discovered interference, Young performed an experiment on the diffraction of light, with the help of which the wavelengths corresponding to light rays of different colors were studied. The study of diffraction was completed in the works of O. Fresnel, who constructed the theory of diffraction, which in principle allows one to calculate the diffraction pattern that arises as a result of light bending around any obstacles. Fresnel achieved such success by combining Huygens' principle with the idea of ​​interference of secondary waves. The Huygens-Fresnel principle is formulated as follows: diffraction occurs due to the interference of secondary waves.

Topics of the Unified State Examination codifier: diffraction of light, diffraction grating.

If an obstacle appears in the path of the wave, then diffraction - deviation of the wave from rectilinear propagation. This deviation cannot be 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 of the fact that the wave bends around the edge of the obstacle and enters the region of the geometric shadow.

Let, for example, a plane wave fall on a screen with a fairly narrow slit (Fig. 1). A diverging wave appears at the exit from the slit, and this divergence increases as the slit width decreases.

In general, diffraction phenomena are expressed more clearly the smaller the obstacle. Diffraction is most significant in cases where the size of the obstacle is smaller or on the order of the wavelength. It is precisely this condition that the slot width in Fig. should satisfy. 1.

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
diffraction of light.

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

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

Reminds me of interference, doesn't it? This is what she is; these rings are interference maxima and minima. What waves are interfering here? Soon we will deal with this issue, and at the same time we will find out why diffraction is observed in the first place.

But first, 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.

Jung's experience.

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

The simplest idea that came to mind first was this. 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. Consequently, on the screen in the area of ​​overlap of the beams diverging from the holes, we should see an interference pattern.

Such an experiment was carried out long before Jung by the Italian scientist Francesco Grimaldi (who discovered the diffraction of light). However, no interference was 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 produces its own interference pattern on the screen. Overlapping, these individual patterns “smear” each other, and as a result, the screen produces uniform illumination of the area where the beams overlap.

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

Rice. 3. Jung's experience diagram

A plane wave falls on the first hole, and a light cone appears behind the hole, expanding due to diffraction. It reaches the next two holes, which become the sources of two coherent light cones. Now - thanks to the point nature of the primary source - an interference pattern will be observed in the area where the cones overlap!

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 is one of the most famous in the history of physics.

Huygens–Fresnel principle.

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

But a natural question arises: what does “overlap” mean?

Huygens reduced his principle to a purely geometric method of constructing a new wave surface as the 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 manifests itself only on the envelope, i.e., on the new position of the wave surface.

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

The modification of 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 their common 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, reinforce each other on the envelope of their wave surfaces in the “forward” direction, ensuring further propagation of the wave. And in the “backward” direction, they interfere with the original wave, mutual cancellation is observed, and a backward wave does not arise.

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

The Huygens–Fresnel principle expresses an important physical idea: a wave, having moved away from its source, subsequently “lives its own life” and no longer depends on this source. Capturing new areas of space, the wave propagates further and further 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 from the infinite flat wave surface of the incident wave, the screen hole cuts out only a small luminous disk, and the subsequent light field is obtained as a result of the interference of waves from secondary sources located not 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 that do not coincide with the original one. The wave goes around the edges of the hole and penetrates into the geometric shadow area.

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 angle of deflection of the rays. As a result, an alternation of interference maxima and minima occurs - which is what we saw in Fig. 2.

Fresnel not only supplemented 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 so-called Fresnel zones. The study of Fresnel zones is not included in the school curriculum - you will learn about them in a university physics course. Here we will only mention that Fresnel, within the framework of his theory, managed to provide 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 slots of width , separated by intervals of width (Fig. 4). Light only passes through slits; the gaps do not allow light to pass through. The quantity is called the lattice period.

Rice. 4. Diffraction grating

The diffraction grating is made using a so-called dividing machine, which applies streaks to the surface of glass or transparent film. In this case, the strokes turn out to be opaque spaces, 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 equal to: d = 0.01 mm = 10 microns.

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

In Fig. In Fig. 5 we see such a beam falling on one of the standard set of diffraction gratings. The grating slits are located vertically, and periodically located vertical stripes are observed on the screen behind the grating.

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

The theory of diffraction gratings is very complex and in its entirety is far beyond the scope of the school curriculum. You should know only the most basic things related to one single formula; this formula describes the positions of the maximum illumination of the screen 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 grating

To make the interference pattern clearer, you can place a lens between the grating and the screen, and place the screen in the focal plane of the lens. Then the secondary waves, traveling in parallel from different slits, will converge at one point on the screen (the side focus of the lens). If the screen is located far enough away, then there is no special need for a lens - the rays arriving at a given point on the screen from various slits will already be almost parallel to each other.

Let's consider secondary waves deviating by an angle. The path difference between two waves coming from adjacent slits is equal to the small leg of a right triangle with the hypotenuse; or, which is the same thing, 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 equal to .

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

(1)

If this condition is met, all waves arriving at a point from different slits 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 along different paths. Why does this happen? We will not go into this issue, since its discussion goes beyond the scope of the Unified State Exam 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 difference in the path of all secondary waves traveling without deviation is equal to zero, and at 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 relative to the central maximum.

When we get the angle:

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

Similarly, at we have the angle:

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

An approximate picture of the 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-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. Brightness of the maxima k-order gradually decreases with growth k. How many maxima can you see? This question is easy to answer using formula (2). After all, sine cannot be greater than one, therefore:

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

Look again at Fig. 5 . On the screen we can see 11 maxima. This is the central maximum, as well as two maxima of the first, second, third, fourth and fifth orders.

Using a diffraction grating, you can measure an unknown wavelength. We direct a beam of light onto the grating (the period of which we know), measure the angle at the maximum of the first
order, we use formula (1) and get:

Diffraction grating as a spectral device.

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

The optical device is called spectral, if it allows you to decompose light into monochromatic components and thereby study the spectral composition of the radiation. The simplest spectral device is well known to you - it is a glass prism. Spectral devices also include a diffraction grating.

Let us assume that white light is incident on a diffraction grating. Let's return 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. At the center of the diffraction pattern they will converge with zero path difference All monochromatic components of white light. Therefore, at the central maximum we will see a bright white stripe.

But the positions of the order maxima are determined by the wavelength. The smaller the , the smaller the angle for a given . Therefore, to the maximum k The th-order monochromatic waves are separated in space: the violet stripe will be closest to the central maximum, the red stripe will be the farthest.

Consequently, in each order, white light is laid out by a lattice into a spectrum.
The first-order maxima of all monochromatic components form a first-order spectrum; then there are spectra of the second, third, and so on orders. The spectrum of each order has the form of a color band, in which all the colors of the rainbow are present - from violet to red.

Diffraction of white light is shown in Fig. 8 . We see a white stripe in the central maximum, and on the sides there are two first-order spectra. As the deflection angle increases, the color of the stripes changes from purple to red.

But a diffraction grating not only allows one to observe spectra, that is, to carry out 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, with its help we can to measure wavelengths. In this case, the measuring 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 and look at the sunlight, you can see a rainbow color around the eyelashes. Our eyelashes act in this case like a transparent diffraction grating in Fig. 6, and the lens is the optical system of the cornea and lens.

The spectral decomposition of white light, given by a diffraction grating, is most easily observed by looking at an ordinary compact disc (Fig. 9). It turns out that the tracks on the surface of the disk form a reflective diffraction grating!