When calculating the total action of the wave front at some point in space, we must take into account that the light oscillations coming from individual points of the front come to the "observation point" with different phases. In this case, all points of the wave front itself are in the same phase. To simplify the calculation of the total action of the entire wave front, we will assume that the light source is very far away and, therefore, the wave can be considered plane. Let the distance of observation point A from the wave front be (Fig. 86). All points of the wave front oscillate in the same phase. At the same time, all points of the front 5 are located at different distances, as a result of which the total action of the entire front will be determined by the phase difference of the interfering oscillations coming in from the individual elements of the wave front

Rice. 86. Fresnel zones

To consider the corresponding interference pattern, we make the following construction. From observation point A, we draw a series of spheres with radii:

On the surface of the wavefront, these spheres will carve out a series of rings called Fresnel zones (Figures 86 and 87). Each subsequent zone is located half a wave further from point A than the previous one. On fig. 87 the aspect ratios are, of course, distorted, since the wavelength of the light is too short to be depicted in the figure. Consequently, oscillations arrive at point A from two neighboring Fresnel zones in the opposite phase and, when added, partially destroy each other.

Rice. 87. Formation of Fresnel zones

There is no complete elimination of oscillations under the joint action of two adjacent Fresnel zones. This is evident from the following considerations. Calculate the area of ​​the Fresnel zone:

Considering that the value of k is very small compared to the distance, we can neglect the second term in brackets and consider the areas of all Fresnel zones to be approximately the same, equal to

At the same time, the angle between the line connecting the zone with point A and the normal to the wave front for each subsequent zone is greater than for the previous one, as a result of which the amplitude of oscillations coming to gradually decreases with increasing zone number. After all,

as was indicated in the previous paragraph, the radiation of individual points of the wave front has the greatest intensity in the direction of the normal. This weakening is further enhanced by an increase in the distance from the Fresnel zone to A with an increase in the zone number. This circumstance causes the incomplete mutual annihilation of oscillations of two adjacent Fresnel zones. Without making any special assumptions about the law of decreasing amplitude of elementary oscillations with distance, we can still assert that, with sufficient approximation, the amplitude at point A of a wave from some zone is the arithmetic mean of the amplitudes of waves from two adjacent zones. On fig. 88 shows the area between the two shaded halves of two adjacent areas. By virtue of the property indicated above, the action of this entire part of the wave front at point a (Fig. 87) is equal to zero. The same can be said about each zone: half of the central zone (zero) together with half of the second will destroy the first, half of the second and fourth will destroy the third, etc. We get that only half of the central Fresnel zone remains uncompensated. Thus, the oscillations caused at point A by a large section of the wave surface have the same amplitude as if only half of the central zone were acting.

Rice. 88. Compensation for the action of neighboring Fresnel zones.

As a result, we can talk about a rectilinear propagation of light from one point to another. The light going to this point is, as it were, concentrated in a channel, the cross section of which in any place is equal to half of the central Fresnel zone.

The action of a light wave on some point is reduced to the action of half of the central Fresnel zone only if the wave is infinite; only in this case the actions of the remaining zones are mutually compensated, and the action of the remote zones can be neglected. If we are dealing with the final section of the wave, then the conditions become significantly different.

Characteristic diffraction phenomena can be observed when light passes through a small aperture or near a screen.

1. Small round hole. On fig. 89 shows a segment of an opaque screen with a round hole, the dimensions of which are shown here magnified several thousand times; a parallel beam of light falls on the hole from below, the center of the hole, two arbitrary points on a straight line perpendicular to and passing through O. From the center

we describe concentric spheres, of which the inner one with radius a passes through O, and each next one has a radius greater than the previous one. Thus,

A row of the same concentric spheres with radii gradually increasing by y will be described from a point. Both rows of spheres will be cut out in the hole of the Fresnel zone. On fig. 89 spheres described around cut out three zones, and those described around - four zones.

Rice. 89. Explanation of diffraction by a round hole (the upper part of the figure is a section, the lower part is a plan).

For a much greater than the radius of the hole, the angles formed by the straight lines with the normal are very small, and therefore we can assume that the amplitudes of the waves emanating from the points of the small hole and reaching the point are equal to each other (the same is true for the amplitudes of the waves emanating from and reaching

Since the zones have practically the same area, the action of two adjacent zones at a point cancels each other out. It follows that the bright points will be those that are located from the center of the hole O at such a distance that an odd number of Fresnel zones will fit in the hole. In this case, the action of the entire hole will be equal to the action of one uncompensated Fresnel zone. Conversely, points such as those for which the number of zones that fit in the hole is even should be dark, since in this case the action of one half of the zones compensates for the action of the other half.

Thus, if we put a white screen behind the hole, which we will bring closer to the hole or move away from it, then the center of the screen will become either dark or light as we move. From the law of conservation of energy, one can further

conclude that the lateral points (located away from the axis) should be alternately either light or dark: the central spot will be surrounded by a number of light and dark rings.

2. Small round screen. On fig. 90 shows a small round screen with edges. Parallel rays fall on the screen. If the rays propagate quite rectilinearly, then a shadow cylindrical space would form behind the screen with the axis being a perpendicular drawn from the center of the screen. However, the wave theory leads to a different conclusion.

Let the front of a plane wave extend infinitely in all directions from the screen. We again draw spherical surfaces, the center of which is a point lying on the axis. The radius of the first sphere the radii of the following spheres will be:

These spheres cut out Fresnel zone waves on the plane, the areas of which are equal to each other. We can apply to these zones the same considerations that we used for the case of an infinite plane wave.

Rice. 90. Explanation of diffraction on a round screen (the upper part of the figure is a section, the lower part is a plan).

In the case of normal incidence of a parallel beam on a small round screen, the axial Point of space behind the screen is illuminated as if only half of the first Fresnel zone directly adjacent to the edges of the screen were active.

Thus, the light propagates beyond the screen.

In accordance with this, experience shows that a bright point is obtained in the center of the screen shadow (Fig. II at the end of the book). This phenomenon can be observed, however, only with screens close in size to the central Fresnel zone, since the intensity of the light spot is very low for much larger objects.

Note a curious historical fact. The famous mathematician Poisson, who was one of the sharpest opponents of the wave theory of light, presented as the most convincing argument against the theory, in his opinion, that according to it, light should always be obtained in the center of the shadow from the screen. This seemed to him quite improbable, and he was greatly embarrassed when

a simple experiment carried out by Fresnel confirmed this conclusion from the wave theory made by its ardent opponent.

It is possible to make a screen (the so-called zone plate) that will cover all even or odd Fresnel zones. Thus, the interference conditions, which we took into account above when calculating the effect of the wave surface, will be artificially violated. In this case, only zones will remain that send oscillations in one phase to point A. As a result, we obtain in A an image of the light source (Fig. 91), formed by oscillations coming in one phase from the entire area of ​​the zone plate. The action of the plate will be like the action of a lens; this fact is one of the clearest examples of non-rectilinear propagation of light.

Rice. 91. Section of the zone plate

A large screen at a sufficiently large distance from the observation point gives a noticeable diffraction pattern. Some phenomena observed during solar eclipses, when the screen is the Moon - a body with a diameter, can be explained using diffraction. At the same time, a small screen located close to the observation point does not give a diffraction pattern. It is often pointed out as a necessary condition for the observation of diffraction - the comparability of the size of the screen or hole with the wavelength. It can be seen from the above that this is not the case. In experience, most often to obtain a diffraction pattern, objects are used that are hundreds of times longer than the wavelength of light.

We get a noticeable diffraction pattern in the form of bands or rings, which account for a significant proportion of the transmitted light energy, if the screen or hole, placed at a certain distance from the observation point, has dimensions comparable to the dimensions of the central Fresnel zone. In this case, the independence of the course of individual beams is violated. If the objects are very large compared to the central Fresnel zone, the diffraction pattern is obtained only in the form of an insignificant detail at the edge of the geometric shadow, which accounts for a negligible fraction of the radiant energy involved in the formation of the entire image.

In the first case, we have a significant deviation from the rectilinear propagation of light, in the second case, the laws of ray optics will practically be valid.

To simplify calculations when determining the amplitude of the wave at a given point in the pr-va. The ZF method is used when considering problems of wave diffraction in accordance with the Huygens-Fresnel principle. Let us consider the propagation of a monochromatic light wave from the point Q(source) to the C.L. observation point P (Fig.).

According to the Huygens-Fresnel principle, the source Q is replaced by the action of imaginary sources located on the auxiliary. surface S, as a swarm choose the surface of the front spherical. a wave coming from Q. Next, the surface S is divided into annular zones so that the distances from the edges of the zone to the observation point P differ by l / 2: Pa \u003d PO + l / 2; Pb=Pa+l/2; Рс=Рb+l/2 (О - point of intersection of the wave surface with the line PQ, l - ). Educated so. equal parts of the surface S called. ZF Plot Oa spherical. surface S called. the first Z. F., ab - the second, bc - the third Z. F., etc. The radius of the m-th Z. F. in the case of diffraction on round holes and screens is determined. approximate expression (for ml

where R is the distance from the source to the hole, r0 is the distance from the hole (or screen) to the observation point. In the case of diffraction on rectilinear structures (rectilinear edge of the screen, slit), the size of the mth ZF (the distance of the outer edge of the zone from the line connecting the source and the observation point) is approximately equal to O(mr0l).

Waves. the process at point P can be considered as the result of the interference of waves arriving at the observation point from each ZF separately, taking into account that it slowly decreases from each zone with increasing zone number, and the phases of oscillations caused at point P by adjacent zones, are opposite. Therefore, waves arriving at the observation point from two adjacent zones weaken each other; the resulting amplitude at the point P is less than the amplitude created by the action of one center. zones.

The method of partitioning into ZFs clearly explains the rectilinear propagation of light from the point of view of waves. the nature of the world. It allows you to simply compile high-quality, and in some cases fairly accurate quantities. representation of the results of wave diffraction at dec. difficult conditions for their distribution. Screen consisting of a concentric system. rings corresponding to ZF (see ZONE PLATE), can give, like , an increase in illumination on the axis or even create an image. Z. F.'s method is applicable not only in optics, but also in the study of the propagation of radio and. waves.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .

FRESNEL ZONES

Cm. Fresnel zone.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .

See what "FRESNEL ZONES" is in other dictionaries:

Areas into which the surface of a light (or sound) wave can be subdivided to calculate the results of light diffraction (See Light diffraction) (or sound). This method was first used by O. Fresnel in 181519. The essence of the method is as follows. Let from ... ...

FRESNEL- (1) diffraction (see) of a spherical light wave, when considering which one cannot neglect the curvature of the surface of the incident and diffracted (or only diffracted) waves. In the center of the diffraction pattern from a round opaque disk is always ... ... Great Polytechnic Encyclopedia

Sections into which the wave surface is divided when considering diffraction waves (Huygens Fresnel principle). Fresnel zones are chosen so that the distance of each next zone from the observation point is half a wavelength greater than ... ...

Spherical diffraction. of a light wave on an inhomogeneity (for example, a hole in the screen), the swarm size b is comparable with the diameter of the first Fresnel zone? (z?): b =? . Name in honor of the French... Physical Encyclopedia

Sections into which the wave surface is divided when considering the diffraction of waves (Huygens Fresnel principle). The Fresnel zones are chosen so that the distance of each next zone from the observation point is half a wavelength greater than the distance ... encyclopedic Dictionary

Diffraction of a spherical light wave by an inhomogeneity (for example, a hole), the size of which is comparable to the diameter of one of the Fresnel zones (See Fresnel zones). The name is given in honor of O. J. Fresnel, who studied this type of diffraction (See Fresnel). ... ... Great Soviet Encyclopedia

Sections into which the surface of the front of a light wave is divided to simplify calculations when determining the amplitude of the wave at a given point in space. Method F. h. used when considering problems of wave diffraction in accordance with Huygens ... ... Physical Encyclopedia

Diffraction of a spherical electromagnetic wave by an inhomogeneity, for example, a hole in the screen, the size of which b is comparable to the size of the Fresnel zone, i.e., where z is the distance of the observation point from the screen, ?? wavelength. Named for O. J. Fresnel ... Big Encyclopedic Dictionary

Diffraction of a spherical electromagnetic wave by an inhomogeneity, such as a hole in a screen, whose size b is comparable to the size of the Fresnel zone, that is, where z is the distance of the observation point from the screen, λ is the wavelength. Named for O. J. Fresnel ... encyclopedic Dictionary

Sections into which the wave surface is divided when considering the diffraction of waves (Huygens Fresnel principle). F. h. are chosen so that the removal of each trace. the zone from the observation point was half the wavelength longer than the removal of the previous one ... ... Natural science. encyclopedic Dictionary

Fresnel proposed an original method for splitting the wave surface S into zones, which greatly simplifies the solution of problems ( fresnel zone method ).

The boundary of the first (central) zone is the surface points S, located at a distance from the point M(Fig. 9.2). sphere points S, located at distances , , etc. from the point M, form 2, 3, etc. Fresnel zones.

Oscillations excited at a point M between two adjacent zones are opposite in phase, since the path difference from these zones to the point M .

Therefore, when adding these oscillations, they must mutually weaken each other:

,

(9.2.2)

where A is the amplitude of the resulting oscillation, is the amplitude of the oscillations excited by i th Fresnel zone.

The value depends on the area of ​​the zone and the angle between the normal to the surface and the straight line directed to the point M.

Area of ​​one zone

This shows that the area of ​​the Fresnel zone does not depend on the zone number i. It means that for i not too large, the areas of neighboring zones are the same.

At the same time, as the zone number increases, the angle increases and, consequently, the radiation intensity of the zone decreases in the direction of the point M, i.e. amplitude decreases. It also decreases due to an increase in the distance to the point M:

The total number of Fresnel zones that fit on the part of the sphere facing the point M, is very large: at , , the number of zones is , and the radius of the first zone is .

It follows that the angles between the normal to the zone and the direction to the point M neighboring zones are approximately equal, i.e. what amplitudes of waves arriving at a point M from neighboring areas ,approximately equal.

A light wave propagates in a straight line. The phases of oscillations excited by neighboring zones differ by π. Therefore, as an acceptable approximation, we can assume that the oscillation amplitude from some m-th zone is equal to the arithmetic mean of the amplitudes of the zones adjacent to it, i.e.

.

Then expression (9.2.1) can be written as

.

(9.2.2)

Since the areas of neighboring zones are the same, the expressions in brackets are equal to zero, which means the resulting amplitude .

Radiation intensity .

Thus, the resulting amplitude generated at some point M by the entire spherical surface , is equal to half the amplitude created by the central zone alone, and the intensity .

Since the radius of the central zone is small (), therefore, we can assume that the light from the point P to the point M propagates in a straight line .

If an opaque screen with a hole is placed in the path of the wave, leaving only the central Fresnel zone open, then the amplitude at the point M will be equal to . Accordingly, the intensity at the point M will be 4 times more than in the absence of a screen (because ). The light intensity increases if all even zones are closed.

Thus, the Huygens–Fresnel principle makes it possible to explain the rectilinear propagation of light in a homogeneous medium.

The legitimacy of the division of the wave front into Fresnel zones has been confirmed experimentally. For this, zone plates are used - a system of alternating transparent and opaque rings.

Experience confirms that with the help of zone plates it is possible to increase the illumination at a point M like a converging lens.

Diffraction of light - in a narrow but most commonly used sense - rounding rays of light borders of opaque bodies (screens); penetration of light into the region of geometric shadow. The diffraction of light manifests itself most prominently in areas of sharp changes in the flux density of rays: near the caustics, the focus of the lens, the boundaries of the geometric shadow, etc. Wave diffraction is closely intertwined with the phenomena of propagation and scattering of waves in inhomogeneous media.

Diffraction called set of phenomena,observed during the propagation of light in a medium with sharp inhomogeneities, the dimensions of which are comparable to the wavelength, and associated with deviations from the laws of geometric optics.

The rounding of obstacles by sound waves (diffraction of sound waves) is constantly observed by us (we hear the sound around the corner of the house). To observe the diffraction of light rays, special conditions are needed, this is due to the short wavelength of light waves.

There are no significant physical differences between interference and diffraction. Both phenomena consist in the redistribution of the light flux as a result of the superposition of waves.

The phenomenon of diffraction is explained using Huygens principle , Whereby each point that the wave reaches serves as center of secondary waves, and the envelope of these waves sets the position of the wave front at the next moment of time.

Let a plane wave normally fall on a hole in an opaque screen (Fig. 9.1). Each point of the section of the wave front highlighted by the hole serves as a source of secondary waves (in a homogeneous isotopic medium they are spherical).

Having constructed the envelope of secondary waves for a certain moment of time, we see that the wave front enters the region of the geometric shadow, i.e. the wave goes around the edges of the hole.

Huygens' principle solves only the problem of the direction of propagation of the wave front, but does not address the issue of the amplitude and intensity of waves propagating in different directions.

A decisive role in establishing the wave nature of light was played by O. Fresnel at the beginning of the 19th century. He explained the phenomenon of diffraction and gave a method for its quantitative calculation. In 1818 he received the Prize of the Paris Academy for his explanation of the phenomenon of diffraction and his method of quantifying it.

Fresnel put physical meaning into Huygens' principle, supplementing it with the idea of ​​interference of secondary waves.

When considering diffraction, Fresnel proceeded from several basic assumptions accepted without proof. The totality of these statements is called the Huygens–Fresnel principle.

According to Huygens principle , each front point waves can be considered as a source of secondary waves.

Fresnel significantly developed this principle.

· All secondary sources of the wave front emanating from one source, coherent between themselves.

· Sections of the wave surface equal in area radiate equal intensities (power) .

· Each secondary source emits light predominantly in the direction of the outer normal to the wave surface at that point. The amplitude of the secondary waves in the direction making the angle α with the normal is the smaller, the larger the angle α, and is equal to zero at .

· For secondary sources, the principle of superposition is valid: radiation of some sections of the wave surfaces does not affect to the radiation of others(if part of the wave surface is covered with an opaque screen, secondary waves will be emitted by open areas as if there were no screen).

Using these provisions, Fresnel was already able to make quantitative calculations of the diffraction pattern.

Diffraction of light (from lat. diffractus- broken, refracted) - deviation in the propagation of light from the laws of geometric optics, expressed in the bending of light rays around the boundaries of opaque bodies, the penetration of light into the area of ​​\u200b\u200bthe geometric shadow, the light bending around small obstacles. Diffraction is observed when light propagates in a medium with pronounced inhomogeneities. Diffraction of light is a manifestation of the wave properties of light under the limiting conditions of the transition from wave optics to geometric. The phenomenon of light diffraction can be explained on the basis of Huygens' principle.

Huygens' principle - the principle according to which each point of the wave front at a given moment of time is the center of secondary elementary waves, the envelope of which gives the position of the wave front at the next moment of time. The Huygens principle makes it possible to explain the laws of reflection and refraction of light, but it is not sufficient to explain diffraction phenomena by Fresnel, who supplemented the Huygens principle with the idea of ​​the interference of secondary waves.

The Huygens-Fresnel principle is a further development of the H. Huygens principle by O. Fresnel, who introduced the concept of coherence and interference of secondary elementary waves. According to the Huygens-Fresnel principle, a wave perturbation at a certain point can be represented as a result of the interference of coherent secondary elementary waves emitted by each element of a certain wave surface (wave front). The Huygens-Fresnel principle also makes it possible to explain diffraction phenomena. Each element of the wave surface with an area is a source of a secondary spherical wave, the amplitude of which is proportional to the area of ​​the element. An oscillation comes to the point of observation from this element

(6.37.21)

where is a coefficient depending on the angle between the normal to the surface and the direction to the observation point; - distance from the surface element to the observation point; - oscillation phase at the location of the element .

The resulting oscillation at the observation point is a superposition of coherent oscillations from all elements of the wave surface that arrived at the observation point. To calculate the amplitude of the resulting oscillation for cases that differ in symmetry, Fresnel proposed a method called the method of Fresnel zones. There are two types of diffraction: Fraunhofer diffraction and Fresnel diffraction.

Fraunhofer diffraction (in parallel beams) is the diffraction of plane waves by an obstacle (the light source is infinitely far away from the obstacle).

Fresnel diffraction is the diffraction of a spherical light wave by an inhomogeneity (for example, a hole in a screen). Fresnel diffraction is carried out in those cases when the light source and the screen used to observe the diffraction pattern are at finite distances from the obstacle that caused the diffraction.

Fresnel zone method.

Fresnel zones are annular sections into which the spherical surface of the front of a light wave is divided when considering problems of wave diffraction in accordance with the Huygens-Fresnel principle to simplify calculations when determining the wave amplitude at a given point in space. Let a monochromatic wave propagate from point to point of observation. The position of the wave front at a certain point in time is indicated in the figure. According to the Huygens-Fresnel principle, the action of the source is replaced by the action of secondary (imaginary) sources located on the surface of the spherical wave front, which is divided into annular zones so that the distances from the edges of neighboring zones to the observation point differ by where is the wavelength. (In the figure - the point of intersection of the wave front with the line , distance = , = ). Then the distance from the edge of the th zone to the observation point is

(6.37.22)

Outer radius of the th Fresnel zone

(6.37.23)

area of ​​-th zone

(6.37.24)

for not too large areas, the Fresnel zones are the same.

Since oscillations from neighboring zones pass to the distance point, which differ in that point, they arrive in antiphase. When calculating the amplitude of the resulting oscillation at a point using the Fresnel zone method, it is also necessary to take into account that with an increase in the number of the zone, the amplitude of oscillations arriving at the point , decrease monotonically: A 1 > A 2 > A 3 > A 4 > .... It can be assumed that the oscillation amplitude A m is equal to the arithmetic mean of the amplitudes of the zones adjacent to it: Therefore, the amplitude of the resulting light vibration coming from the entire wave front to a point will be equal to:

A \u003d A 1 - A 2 + A 3 - A 4 + ... ... .. A to.

This expression can be represented in the following form:

since the expressions in brackets are equal to zero, and the amplitude from the last Fresnel zone is infinitesimal. Therefore, the amplitude generated at a point by the entire spherical wavefront is equal to half the amplitude generated by the central Fresnel zone. If 1m, 0.5 µm, then the radius of the first Fresnel zone is 0.5 mm. Consequently, the light from the source to the point of observation propagates, as it were, within the limits of a narrow straight channel, i.e. almost straight.

Oscillations from even and odd Fresnel zones are in antiphase and mutually weaken each other. If any obstacle overlaps a part of the spherical wave front, then only open Fresnel zones are taken into account when calculating the amplitude of the resulting oscillation at the observation point using the Fresnel zone method. If a plate is placed in the path of the light wave, which would cover all even or odd Fresnel zones, then the amplitude of the oscillation at the point of observation increases sharply. This plate is called zone. The zone plate multiplies the intensity of the light at point , acting like a converging lens.