Calculations using the formula
Represents in general case Very difficult task. However, as Fresnel showed, in cases characterized by symmetry, the amplitude of the resulting vibration can be found by simple algebraic or geometric summation.
We'll find it in arbitrary point M amplitude of a spherical light wave propagating in homogeneous environment from a point source S.
According to the Huygens-Fresnel principle, we replace the action of the source S by the action of imaginary sources located on the auxiliary surface F, which is the surface of the wave front coming from S(surface of a sphere with center S). Fresnel broke the wave surface A into ring zones of such a size that the distances from the edges of the zone to M differed by λ/2,
A similar division of the wave front into zones can be done by drawing with the center at the point. The wave front can be divided into zones by drawing with the center at the point M spheres with radii 
Since oscillations from neighboring zones travel to the point M distances differing by λ/2, then to the point M they arrive in opposite phases and when superimposed, these oscillations will mutually weaken each other. Therefore, the amplitude of the resulting light vibration at the point M:
Where A 1, A 2, … A m− amplitudes of oscillations excited 1st, 2nd, …, m-th zones.
To estimate vibration amplitudes let's find the area Fresnel zones. Let the outer border m-th zone identifies a spherical segment of height on the wave surface h m(rice.). 
Designating the radius of this segment as r m, we find that the area m-th Fresnel zones:
Here σ m-1− area of the spherical segment allocated by the outer boundary m − 1st zones. From the figure it follows that
After elementary transformations, given that λ << a
And λ << b
, we get 
Area of a spherical segment and area m-th Fresnel zones: 
Where Δσ m square m-th Fresnel zone, which, as the last expression shows, does not depend on m. Not too big m The areas of the Fresnel zones are the same.
Thus, the construction of Fresnel zones divides the wave surface of a spherical wave into equal zones.
Let's find the radii of the Fresnel zones 
where 
According to Fresnel's assumption, the action of individual zones at a point M the smaller the larger the angle φ m between the normal to the surface of the zone and the direction to M, i.e. the effect of the zones gradually decreases from the central one (about P 0) to peripheral. In addition, the radiation intensity in the direction of the point M decreases with growth m and due to an increase in the distance from the zone to the point M. Taking both these factors into account, we can write: 
The phases of oscillations excited by neighboring zones differ by π
. Therefore, the amplitude of the resulting oscillation at the point M is determined by the expression
Let's write the last expression in the form:
Due to the monotonic decrease in the amplitudes of the Fresnel zones with increasing zone number, the amplitude of the oscillation Am from some m-th Fresnel zone is equal to the arithmetic mean of the amplitudes of the adjacent zones 
Then 
Thus, the amplitude of the resulting oscillations at an arbitrary point M determined by the action of only half of the central Fresnel zone. Consequently, the action of the entire wave surface on point M is reduced to the action of its small section, smaller than the central zone.
If an opaque screen with a hole is placed in the path of the wave, leaving only the first Fresnel zone open, the amplitude at the point M equal to A 1, and the intensity in 4 times more than in the absence of an obstacle between the points S And M.
Spread of light from S To M it is as if the light flux is propagating inside a very narrow channel along a straight line S.M., i.e. straight forward. Thus, the Huygens-Fresnel principle allows us to explain the rectilinear propagation of light in a homogeneous medium.
The validity of dividing the wave front into Fresnel zones has been confirmed experimentally. If you place a plate in the path of a light wave that would cover all even or odd Fresnel zones, then the intensity of light at a point M increases sharply. With closed even Fresnel zones, the amplitude at the point M will be equal
In experiment, a zone plate increases the intensity of light at a point many times M, acting like a converging lens.
An even greater effect can be achieved without overlapping the even (or odd) Fresnel zones, but by changing the phase of their oscillations by 180°. Such a plate is called a phase zone plate. Compared to the amplitude zone plate, the phase plate gives an additional increase in amplitude in 2 times, and the light intensity is in 4 times.
To simplify calculations when determining the wave amplitude at a given point in the production line. Z.F.'s 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 a point Q (source) to a cl. 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, the quality of which is chosen is the surface of the front spherical. 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=PO+l/2; Рb=Ra+l/2; Рс=Рb+l/2 (О - point of intersection of the wave surface with the line PQ, l - ). Educated thus. equal-sized areas of the surface S are called. Z.F. Plot Oa is spherical. surface S called the first Z.F., ab - the second, bc - the third Z.F., etc. The radius of the mth Z.F. in the case of diffraction by round holes and screens is determined by . approximate expression (with 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 by rectilinear structures (straight edge of the screen, slit), the size of the m-th Z.F. (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 Z. F. separately, taking into account that from each zone the phase of oscillations caused at point P by adjacent zones slowly decreases with increasing zone number, opposite. Therefore, waves arriving at the observation point from two adjacent zones weaken each other; the resulting amplitude at point P is less than the amplitude created by the action of one center. zones.
The method of partitioning into ZF clearly explains the rectilinear propagation of light from the point of view of waves. nature of light. It allows you to simply compile high-quality, and in some cases, fairly accurate quantities. an idea of the results of wave diffraction at decomposition. difficult conditions for their distribution. A screen consisting of a concentric system. rings corresponding to the Z.F. (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 studying the propagation of radio and radio waves. waves. . 1983 .
Physical encyclopedic dictionary. - M.: Soviet Encyclopedia
FRESNEL ZONES Cm.
Fresnel zone.. Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. 1988 .
Editor-in-chief A. M. Prokhorov
Areas into which the surface of a light (or sound) wave can be divided to calculate the results of diffraction of light (See Diffraction of Light) (or sound). This method was first used by O. Fresnel in 1815 19. The essence of the method is as follows. Let from... ...
FRESNEL- (1) diffraction (see) of a spherical light wave, when considering which the curvature of the surface of the incident and diffracted (or only diffracted) waves cannot be neglected. In the center of the diffraction pattern from a round opaque disk there is always... ... Big Polytechnic Encyclopedia
Areas into which the wave surface is divided when considering diffraction waves (Huygens Fresnel principle). Fresnel zones are selected so that the distance of each subsequent zone from the observation point is half the wavelength greater than... ...
Diffraction spherical light wave on an inhomogeneity (for example, a hole in a screen), the size of the swarm b is comparable to the diameter of the first Fresnel zone?(z?): b=?(z?) (diffraction in converging rays), where z is the distance of the observation point to the screen . Name in honor of the French... Physical encyclopedia
Areas into which the wave surface is divided when considering wave diffraction (Huygens Fresnel principle). Fresnel zones are selected so that the distance of each subsequent zone from the observation point is half the 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
Areas into which the surface of the light wave front is divided to simplify calculations when determining the wave amplitude at a given point in space. Method F. z. 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 a 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 after O. J. Fresnel... Big Encyclopedic Dictionary
Diffraction of a spherical electromagnetic wave by an inhomogeneity, for example a hole in a screen, the size of which 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 after O. J. Fresnel... encyclopedic Dictionary
Areas into which the wave surface is divided when considering wave diffraction (Huygens Fresnel principle). F. z. are chosen so that every trace is deleted. zone from the observation point was half the wavelength greater than the distance from the previous one... ... Natural history. encyclopedic Dictionary
To find the result of the interference of secondary waves, Fresnel proposed a method of dividing the wave front into zones called Fresnel zones.
Let us assume that the light source S (Fig. 17.18) is point and monochromatic, and the medium in which the light propagates is isotropic. The wave front at an arbitrary moment of time will have the shape of a sphere with radius \(~r=ct.\) Each point on this spherical surface is a secondary source of waves. Oscillations at all points of the wave surface occur with the same frequency and in the same phase. Therefore, all these secondary sources are coherent. To find the amplitude of oscillations at point M, it is necessary to add up the coherent oscillations from all secondary sources on the wave surface.
Fresnel divided the wave surface Ф into ring zones of such a size that the distances from the edges of the zone to point M differed by \(\frac(\lambda)(2),\) i.e. \(P_1M - P_0M = P_2M - P_1M = \frac(\lambda)(2).\)
Since the difference in path from two adjacent zones is equal to \(\frac(\lambda)(2),\), then the oscillations from them arrive at point M in opposite phases and, when superimposed, these oscillations will mutually weaken each other. Therefore, the amplitude of the resulting light vibration at point M will be equal to
\(A = A_1 - A_2 + A_3 - A_4 + \ldots \pm A_m,\) (17.5)
where \(A_1, A_2, \ldots , A_m,\) are the amplitudes of oscillations excited by the 1st, 2nd, .., m-th zones.
Fresnel also suggested that the action of individual zones at point M depends on the direction of propagation (on the angle \(\varphi_m\) (Fig. 17.19) between the normal \(~\vec n \) to the surface of the zone and the direction to point M). With increasing \(\varphi_m\), the effect of the zones decreases and at angles \(\varphi_m \ge 90^\circ\) the amplitude of the excited secondary waves is equal to 0. In addition, the intensity of radiation in the direction of point M decreases with increasing and due to increasing distance from zones to point M Taking into account both factors, we can write that
\(A_1 >A_2 >A_3 > \cdots\)
1. Explanation of the straightness of light propagation.
The total number of Fresnel zones that fit on a hemisphere of radius SP 0, equal to the distance from the light source S to the wave front is very large. Therefore, as a first approximation, we can assume that the amplitude of oscillations A m from a certain m-th zone equal to the arithmetic mean of the amplitudes of the adjacent zones, i.e.
\(A_m = \frac( A_(m-1) + A_(m+1) )(2).\)
Then expression (17.5) can be written in the form
\(A = \frac(A_1)(2) + \Bigr(\frac(A_1)(2) - A_2 + \frac(A_3)(2) \Bigl) + \Bigr(\frac(A_3)(2) - A_4 + \frac(A_5)(2) \Bigl) + \ldots \pm \frac(A_m)(2).\)
Since the expressions in parentheses are equal to 0, and \(\frac(A_m)(2)\) is negligible, then
\(A = \frac(A_1)(2) \pm \frac(A_m)(2) \approx \frac(A_1)(2).\) (17.6)
Thus, the amplitude of oscillations created at an arbitrary point M spherical wave surface, is equal to half the amplitude created by one central zone. From Figure 17.19, the radius r of the m-th zone of the Fresnel zone \(r_m = \sqrt(\Bigr(b + \frac(m \lambda)(2) \Bigl)^2 - (b + h_m)^2).\) Since \(~h_m \ll b\) and the wavelength of light is small, then \(r_m \approx \sqrt(\Bigr(b + \frac(m \lambda)(2) \Bigl)^2 - b^2 ) = \sqrt(mb \lambda + \frac(m^2 \lambda^2)(4)) \approx \sqrt(mb\lambda).\) So, the radius of the first Considering that \(~\lambda\) the wavelength can have values from 300 to 860 nm, we get \(~r_1 \ll b.\) Consequently, the propagation of light from S to M occurs as if the light flux propagates inside a very narrow channel along SM, the diameter of which less than radius the first Fresnel zone, i.e. straight forward.
2. Diffraction by a round hole.
A spherical wave propagating from a point source S meets on its path a screen with a round hole (Fig. 17.20). The type of diffraction pattern depends on the number of Fresnel zones that fit into the hole. According to (17.5) and (17.6) at the point B amplitude of the resulting oscillation
\(A = \frac(A_1)(2) \pm \frac(A_m)(2),\)
where the plus sign corresponds to odd m, and the minus sign to even m.
When the hole opens odd number Fresnel zones, then the amplitude of oscillations at point B will be greater than in the absence of a screen. If one Fresnel zone fits in the hole, then at point B the amplitude \(~A = A_1\) i.e. twice as much as in the absence of an opaque screen. If two Fresnel zones are placed in a hole, then their action at the point IN practically destroy each other due to interference. Thus, diffraction pattern from a round hole near a point IN will have the appearance of alternating dark and light rings with centers at the point IN(if m is even, then there is a dark ring in the center, if m is odd, there is a light ring), and the intensity of the maxima decreases with distance from the center of the picture.
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 514-517.
A spherical wave propagating from a point source S encounters a disk on its path. The diffraction pattern is observed on the screen E in the vicinity of point P, lying on the line connecting S with the center of the disk.
In this case, the section of the wave front covered by the disk must be excluded from consideration and the Fresnel zones must be constructed starting from the edges of the disk.
Let the disk cover the first m Fresnel zones. Then the amplitude of the resulting oscillation at point P is equal to
because expressions in parentheses are equal to zero. Consequently, at point P there is always an interference max corresponding to half the action of the first open Fresnel zone. Orago was the first to obtain a bright spot (Poisson's spot) experimentally. As in the case of diffraction by a circular hole, the central max is surrounded by concentric dark and light rings, and the intensity of the maxima decreases with distance from the center of the pattern.
As the disk radius increases, the first open Fresnel zone moves away from point P and, what is especially significant, the angle α between the normal to the surface of this zone and the direction to point P increases. As a result, the intensity of the central maximum decreases with increasing disk size. At large sizes disk (its radius is many times greater than radius the central Fresnel zone closed by it), an ordinary shadow is observed behind it, near the boundaries of which there is a very weak diffraction pattern. In this case, the diffraction of light can be neglected and light can be considered to propagate rectilinearly.
Diffraction by a circular hole and a disk was first considered by Fresnel using the Huygens-Fresnel method and the Fresnel zone method based on it.
Disadvantages of Fresnel theory:
1. In Fresnel’s theory, it is assumed that the opaque parts of the screens are not sources of secondary waves and also that the amplitudes and initial phases vibrations at a point on the surface Ф, not covered by opaque screens, are the same as in the absence of the latter. This is incorrect, because. border conditions on the surface of the screen depend on its material. True, this only affects small distances from the screen, on the order of λ. For holes and screens whose dimensions are significantly larger than λ, Fresnel's theory agrees well with experiment.
2. Fresnel theory gives an incorrect value for the phase of the resulting wave. For example, when graphically adding the vectors of vibration amplitudes excited at point P by all small elements open front waves, it turns out that the phase of the resulting vector A differs from the initial phase of the oscillations at point P that actually occur.
3. Based on a purely qualitative postulated assumption about the dependence of the amplitude of secondary waves on the angle α.
Fresnel's theory provides only an approximate calculation method. The mathematical justification and refinement of the Huygens-Fresnel method was made in 1882 by Kirchhoff.
§ Fraunhofer diffraction.
The phenomenon of diffraction is usually classified depending on the distances of the source and the observation point (screen) from the obstacle placed in the path of light propagation. Diffraction spherical waves, the intensity distribution pattern of which is observed at a finite distance from the obstacle that caused diffraction, is called Fresnel diffraction. If the distances from the obstacle to the source and the observation point are very large (infinitely large), they speak of Fraunhofer diffraction.
There is no fundamental difference or sharp boundary between Fresnel and Fraunhofer diffraction. One continuously transforms into the other. If, for an observation point lying on the axis of the system, in the obstacle hole, for example, a noticeable part of the first zone or several Fresnel zones fits, then diffraction is considered Fresnel. If a small part of the first Fresnel zone fits in the hole, then the diffraction will be Fraunhofer.
The Huygens-Fresnel principle explains the straightness of light propagation in a homogeneous medium free of obstacles. To show this, consider the action of a spherical light wave from a point source S 0 at an arbitrary point in space P (Fig. 4.1). The wave surface of such a wave is symmetrical relative to a straight line S 0 P . Amplitude of the desired wave at a point P depends on the result of the interference of secondary waves emitted by all sections dS surfaces S . The amplitudes and initial phases of secondary waves depend on the location of the corresponding sources dS relative to point P .
Fresnel proposed a method of dividing the wave surface into zones (Fresnel zone method). According to this method, the wave surface is divided into ring zones (Fig. 4.1), constructed so that the distances from the edges of each zone to the point P differ by l/2(l - wavelength of light). If we denote by b distance from the top of the wave surface 0 to the point P , then the distances b + k (l/2) form the boundaries of all zones where k - zone number. Vibrations coming to a point P from similar points of two adjacent zones are opposite in phase, since the path difference from these zones to the point P equal to l/2. Therefore, when superimposed, these oscillations mutually weaken each other, and the resulting amplitude will be expressed by the sum:
A = A 1 - A 2 +A 3 - A 4 + ... . (4.1)
Amplitude value A k depends on area D.S. k k th zone and angle a k between the outer normal to the surface of the zone at any point and the straight line directed from this point to point P .
It can be shown that the area D.S. k k th zone does not depend on the zone number in the conditions l<< b . Thus, in the considered approximation, the areas of all Fresnel zones are equal in size and the radiation power of all Fresnel zones - secondary sources - is the same. At the same time, with an increase k angle increases a k between the normal to the surface and the direction to the point P , which leads to a decrease in radiation intensity k th zone in a given direction, i.e. to a decrease in amplitude A k compared to the amplitudes of previous zones. Amplitude A k also decreases due to an increase in the distance from the zone to the point P with growth k . Eventually
A 1 > A 2 > A 3 > A 4 > ... > A k > ...
Due to large number zones decreasing A k is monotonic in nature and we can approximately assume that
. (4.2)
Rewriting the resulting amplitude (4.1) in the form
we find that, according to (4.2) and taking into account the small amplitude of the remote zones, all expressions in brackets are equal to zero and equation (4.1) is reduced to the form
A = A 1 / 2. (4.4)
The result obtained means that the vibrations caused at the point P spherical wave surface, have an amplitude given by half of the central Fresnel zone. Therefore, the light from the source S 0 exactly P propagates within a very narrow direct channel, i.e. straight forward. As a result of the interference phenomenon, the effect of all zones except the first is destroyed.
Fresnel diffraction from simple obstacles
Action of a light wave at a certain point P reduces to the action of half of the central Fresnel zone if the wave is unlimited, since only then the actions of the remaining zones are mutually compensated and the action of the remote zones can be neglected. For a finite section of the wave, the diffraction conditions differ significantly from those described above. However, here too, the use of the Fresnel method makes it possible to predict and explain the features of the propagation of light waves.
Let us consider several examples of Fresnel diffraction from simple obstacles.
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Diffraction by a circular hole . Let the wave from the source S 0 meets an opaque screen with a round hole on the way B.C. (Fig. 4.2). The result of diffraction is observed on the screen E , parallel to the plane holes. It is easy to determine the diffraction effect at a point P screen located opposite the center of the hole. To do this, it is enough to build waves on the open part of the front B.C. Fresnel zones corresponding to the point P . If in the hole B.C. fits k Fresnel zones, then the amplitude A resulting oscillations at a point P depends on whether the number is even or odd k , and also on how big absolute value this number. Indeed, from formula (4.1) it follows that at the point P amplitude of the total oscillation
(the first equation of the system for odd k , the second - when even) or, taking into account formula (4.2) and the fact that the amplitudes of two neighboring zones differ little in value and can be considered A k-1 approximately equal Ak, we have
where plus corresponds to an odd number of zones k , fitting on the hole, and the minus is even.
With a small number of zones k amplitude A k little different from A 1 . Then the result of diffraction at the point P depends on parity k : if odd k a diffraction maximum is observed, and a minimum is observed when the diffraction is even. Minimums and maximums will be more different from each other the closer they get A k To A 1 those. the less k . If the hole only opens central zone Fresnel, amplitude at point P will be equal A 1 , it is twice as large as that which occurs with a completely open wave front (4.4), and the intensity in this case is four times greater than in the absence of an obstacle. On the contrary, with an unlimited increase in the number of zones k , amplitude A k tends to zero (A k<< A 1 ) and expression (4.5) turns into (4.4). In this case, light actually spreads in the same way as in the absence of a screen with a hole, i.e. straight forward. This leads to the conclusion that the consequences of wave concepts and concepts of rectilinear propagation of light begin to coincide when the number of open zones is large.
Oscillations from even and odd Fresnel zones mutually weaken each other. This sometimes leads to an increase in light intensity when part of the wave front is covered by an opaque screen, as was the case with an obstacle with a round hole on which only one Fresnel zone is placed. The light intensity can be increased many times by making a complex screen - the so-called zone plate (a glass plate with an opaque coating), which covers all even (or odd) Fresnel zones. The zone plate acts like a converging lens. Indeed, if the zone plate covers all even zones, and the number of zones k = 2m , then from (4.1) it follows
A = A 1 + A 3 +...+ A 2m-1
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or with a small number of zones, when A 2m-1 approximately equal A, A = mA 1 , i.e. intensity of light at a point P at 2 m ) 2 times more than with unhindered propagation of light from the source to the point P , wherein A = A 1 / 2, and intensity accordingly / 4 .
Diffraction by a circular disk. When placed between the source S 0 and a screen of a round opaque disk NE one or several first Fresnel zones closes (Fig. 4.3). If the disk closes k Fresnel zones, then at the point P sum wave amplitude
and, since the expressions in brackets can be taken equal to zero, similarly to (4.3) we obtain
A = A k +1 / 2. (4.6)
Thus, in the case of a round opaque disk in the center of the picture (point P ) for any (both even and odd) k it turns out to be a bright spot.
If the disk covers only part of the first Fresnel zone, there is no shadow on the screen, the illumination at all points is the same as in the absence of an obstacle. As the radius of the disk increases, the first open zone moves away from the point P and the angle increases a between the normal to the surface of this zone at any point and the direction of radiation towards the point P (see Huygens-Fresnel principle). Therefore, the intensity of the central maximum weakens as the disk size increases ( A k+1 << A 1 ). If the disk covers many Fresnel zones, the light intensity in the region of the geometric shadow is almost everywhere equal to zero and only near the observation boundaries there is a weak interference pattern. In this case, we can neglect the phenomenon of diffraction and use the law of rectilinear propagation of light.
