The Huygens-Fresnel principle within the framework of wave theory allows us to explain straight propagation Sveta. Let us determine the amplitude of the light wave in arbitrary point R, using Fresnel zone method. Let us first consider the case of falling plane wave(Fig. 5.2).
Let the plane wave front F, propagating from a light source located at infinity, at some point in time it is at a distance OR– r 0 from observation point R.
Rice. 5.2. Application of the Huygens-Fresnel principle to a plane wave: Fresnel zones on the surface
plane wave frontFare concentric rings
(for clarity, the image of the Fresnel zones is rotated 90°, this is how they look from point P)
All points of the wave front, according to the Huygens-Fresnel principle, emit elementary spherical waves, which propagate in all directions and after some time reach the observation point R. The resulting amplitude of oscillations at this point is determined by the vector sum of the amplitudes of all secondary waves.
Oscillations at all points of the wave front F have the same direction and occur in the same phase. On the other hand, all points of the front F are from the point R at various distances. To determine the resulting amplitude of all secondary waves at the observation point, Fresnel proposed a method of dividing the wave surface into ring zones called Fresnel zones.
Taking a point R as the center, we construct a series of concentric spheres, the radii of which begin at and increase each time by half the wavelength . When crossing a plane wave front F these spheres will produce concentric circles. Thus, ring zones (Fresnel zones) with radii, etc. will appear at the wave front.
Let us determine the radii of the Fresnel zones, bearing in mind that , 0A 2 = AR 2 – 0Р 2 , that is
Similarly we find
|
|
To estimate the oscillation amplitudes, we determine the areas of the Fresnel zones. First zone (circle):
|
|
second zone (ring):
third and subsequent zones (rings):
|
|
Thus, the areas of the Fresnel zones are approximately the same, therefore, according to the Huygens-Fresnel principle, each Fresnel zone serves as a source of secondary spherical waves, the amplitudes of which are approximately the same. In addition, the oscillations excited at the point R two adjacent zones, opposite in phase, since the difference in the path of the corresponding waves from these zones to the observation point R equal to . Therefore, when superimposed, these oscillations must mutually weaken each other, that is, the amplitude A resulting oscillation at a point R can be represented as an alternating series
Where A 1 - amplitude of oscillations at a point R excited by the action of the central (first) Fresnel zone, A 2 - amplitude of oscillations excited by the second zone, etc.
Distance from m th zone to point R slowly increases with zone number m. Angle between the normal to the zone elements and the direction to the point R also growing with m, therefore the amplitude A m vibrations excited m th zone at point R, monotonically decreases with growth m. In other words, the amplitudes of oscillations excited at a point R Fresnel zones form a monotonically decreasing sequence:
Due to the monotonous and slow decrease A t we can approximately assume that the amplitude of oscillations from the zone with number m equal to the arithmetic mean of the amplitudes of oscillations from two adjacent Fresnel zones:
|
|
In the expression for the amplitude of the resulting oscillation, all amplitudes from even zones are included with one sign, and from odd ones - with another. Let's write this expression in the following form:
The expressions in parentheses based on (5.10) will be equal to zero, so
that is, the resulting amplitude produced at the observation point R the entire surface of the wave front is equal to half the amplitude created by the central (first) Fresnel zone alone. Thus, the vibrations caused at the point R wave surface F, have the same amplitude as if only half of the first (central) zone were active. Consequently, light propagates as if in a narrow channel, the cross section of which is equal to half of the first (central) Fresnel zone - we again come to the rectilinear propagation of a plane wave.
If a diaphragm with a hole is placed in the path of the wave, leaving only the central (first) Fresnel zone open, the amplitude at the point R will be equal A 1, that is, it will be twice the amplitude created by the entire wavefront. Accordingly, the light intensity at a point R will be four times greater than in the absence of an obstacle between the light source and the point R. Amazing, isn't it? But miracles do not happen in nature: at other points on the screen the light intensity will be weakened, and the average illumination of the entire screen when using the aperture will, as one would expect, decrease.
The validity of this approach, which consists in dividing the wave front into Fresnel zones, has been confirmed experimentally. Oscillations from even and odd Fresnel zones are in antiphase and, therefore, mutually weaken each other. If you place a plate in the path of a light wave that covers all even or odd Fresnel zones, you can make sure that the light intensity at a point R will increase sharply. This plate, called zone, acts like a converging lens. Let us emphasize once again: Fresnel zones are mentally selected areas of the wave front surface, the position of which depends on the selected observation point R. At a different observation point, the location of the Fresnel zones will be different. Fresnel zone method - convenient way solving problems of wave diffraction by certain obstacles.
There are two types of diffraction. If the light source S and observation point R are far from the obstacle, the rays falling on the obstacle and going to the point R, form almost parallel beams. In this case they talk about diffraction in parallel rays , or Fraunhofer diffraction. If the diffraction pattern is considered at a finite distance from the obstacle that caused diffraction, then we talk about spherical wave diffraction, or Fresnel diffraction.
http://pymath.ru/viewtopic.php?f=77&t=757&sid=– Video lesson “Fresnel zone radius”
As a result of studying this chapter, the student should: know
- the essence of the Fresnel zone method;
- theory of diffraction by a circular hole and round disk;
- the theory of diffraction in parallel rays from one slit;
- theory diffraction grating(conditions of maxima and minima, dispersion and grating resolution);
- the theory of diffraction from volumetric gratings and the Bragg-Wulf formula; be able to
- apply the Fresnel zone method to calculate diffraction patterns;
- solve typical applications physical tasks on diffraction of light; own
- skills to use standard methods and models of mathematics in relation to the diffraction of light;
- skills to conduct physical experiment, as well as processing the results of the light diffraction experiment.
Fresnel zone method. Diffraction by a circular hole and a circular disk
Diffraction of light called the phenomenon of light deviation from the rectilinear direction of propagation when passing near obstacles. This phenomenon can be illustrated by waves on water that bend around even a fairly large obstacle, while a small (compared to the wavelength) obstacle passes as if it were not there. And light, under certain conditions, can enter the region of geometric shadow. If there is a round obstacle in the path of a parallel light beam (a round disk or a round hole in an opaque screen), then on a screen located at a sufficient distance long distance from an obstacle appears diffraction pattern - alternating light and dark rings. If the obstacle is straight (thread, crack, edge of the screen), then parallel stripes appear on the screen.
Let's consider first diffraction by a round hole - diffraction problem about the passage of a plane monochromatic wave through a small circular hole of radius R in an opaque screen (Fig. 27.1). Observation point R located on the axis of symmetry at a sufficiently large distance L from the screen, and
Where X- wavelength.
Rice. 27.1
In accordance with the Huygens-Fresnel principle, we can split wave surface plane of the hole to a set of secondary sources, the waves from which give an interference pattern at the point R. Based on the circular symmetry of the problem, Fresnel divided the wave surface of the incident wave into annular zones (Fresnel zones) so that the distances from the boundaries of neighboring zones to the point R differed by half a wavelength:
Thus, the wave surface will be divided into concentric circles (see Fig. 27.1). Let's find the radii using the Pythagorean theorem r t of these circles (Fresnel zones):
Here the condition of distance between the screen and the hole is taken into account, which is observed in experience usually with a large margin. The number of Fresnel zones placed on the hole is determined by the radius of the hole R:
Where T - not necessarily an integer. Although for a clear interference pattern, as will be seen below, T should be integer with sufficiently high accuracy. The result of interference at a point R depends on the number T participating in the interference of Fresnel zones. Let us show that all zones have the same area Sm:
Zones of the same area, emitting a wave of the same amplitude, at first glance, should make the same contribution to the illumination at the observation point. However, this is not entirely true. The larger the zone number, the larger angle and between the beam g t and normal to the radiating wave surface. In addition, the distance to the observation point increases g t. Both of these factors lead to a slight decrease in the amplitude of oscillations with increasing T at the observation point And t> provided by the zone T:
It is important that the oscillations excited by neighboring zones are in antiphase, since the distances from them to the observation point differ by X/2. Therefore, the wave from the subsequent zone almost extinguishes the wave from the previous zone. In this case, the total amplitude at the observation point is equal to final amount, the number of terms in which is limited by the value T
As a result of amplitude grouping, it is clear that the total amplitude of oscillations at the observation point is always less than the amplitude of oscillations that would be caused by the first Fresnel zone alone. If the hole were infinitely large and all Fresnel zones were open, then a wave unperturbed by the obstacle would reach the observation point with an amplitude A 0. Then, as a result of grouping the amplitudes, we have an infinite sum, which is simplified taking into account equality (27.7):
Thus, the action (amplitude) caused by the entire wave surface of the undisturbed wave is equal to only half the action of the first zone. In other words, if a hole in an opaque screen leaves one Fresnel zone open, then the amplitude of oscillations at the observation point increases by 2 times (and the intensity by 4 times) compared to the action of an unperturbed wave. If two zones are opened, the amplitude of the oscillations practically becomes zero. And if you make an opaque screen that would leave only a few odd (or only a few even) zones open, then the amplitude of the oscillations at the observation point will increase sharply. So, if the first, third, fifth and seventh zones are open, then the amplitude of the oscillations increases by 8 times, and the intensity by 64 times. It can be concluded that Such zone plates have the property of focusing light.
Let us now turn to the problem of diffraction on a circular disk, not transmitting light. Let us assume that in this case the Fresnel zones with numbers from 1 to T turn out to be closed. Then the amplitude of oscillations at the observation point, by analogy with previous reasoning, is given by an infinite sum:
Here it is taken into account that the expressions in brackets, in accordance with equality (27.7), are equal to zero. If the screen does not cover too many areas, then
and similarly to formula (27.10)
Thus, in the center of the picture, when light is diffraction on the disk, an interference maximum is observed, called Poisson's spot. This spot is surrounded by light and dark diffraction rings, and the intensity of the maxima decreases with distance from the center.
Let us now estimate the characteristic sizes of the Fresnel zones. Let, for example, the diffraction pattern be observed on a screen located at a distance L- 1m from the obstacle, and the wavelength of the light X= 0.5 µm ( green light). Then the radius of the first Fresnel zone according to formula (27.3) is equal to
p, = 4XL~ 0.71 mm, and the radius of the hundredth Fresnel zone
pwo = V100 XL~ 7.1 mm.
Diffraction phenomena appear most clearly when
obstacle, a small number of zones are laid out (27.4): t = ~gu ~ 1, or
This is the relationship between wavelength X, size of the obstacle R and the distance from the obstacle to the observation point L can be seen as applicability limit geometric optics. At long wavelengths, diffraction is significant, and at shorter wavelengths, geometric optics and the concept of geometric beam Sveta.
In Lecture 2 we looked at the phenomena of intensity redistribution luminous flux as a result of wave superposition. We called this phenomenon interference and examined the interference pattern from two sources. This lecture is a direct continuation of the previous one. There is no significant difference between interference and diffraction. physical differences. Both phenomena involve the redistribution of light flux as a result of wave superposition.
By historical reasons redistribution of intensity resulting from the superposition of waves excited finite number discrete coherent sources are usually called interference. Redistribution of intensity resulting from the superposition of waves excited coherent sources, located continuously, is usually called wave diffraction. (When there are few sources, for example two, then their result joint action usually called interference, and if there are many sources, then they often talk about diffraction.)
Diffraction is called any deviation of wave propagation near obstacles from the laws of geometric optics.
In geometric optics the concept is used light beam - a narrow beam of light propagating in a straight line. The straightness of light propagation is explained by Newton's theory and is confirmed by the presence of a shadow behind an opaque source located in the path of light from a point source. But - a contradiction with wave theory, because According to Huygens' principle, each point of the wave field can be considered as a source of secondary waves propagating in all directions, including into the region of the geometric shadow of the obstacle (the waves must bend around the obstacles). How can a shadow arise? Huygens' theory could not provide an answer. But Newton’s theory could not explain the phenomenon of interference and the violation of the law of rectilinear propagation of light when light passes through fairly narrow slits and holes, as well as when illuminating small opaque obstacles.
In these cases, on a screen installed behind holes or obstacles, instead of clearly demarcated areas of light and shadow, a system of interference maxima and minima of illumination is observed. Even for obstacles and openings that have large sizes, there is no sharp transition from shadow to light. There is always some transition region, in which weak interference maxima and minima can be detected. That is, when waves pass near the boundaries of opaque or transparent bodies, through small holes, etc., waves deviate from rectilinear propagation (laws of geometric optics), and these deviations are accompanied by their interference phenomena.
Diffraction properties:
1) Wave diffraction - characteristic feature propagation of waves regardless of their nature.
2) Waves can fall into the geometric shadow area (bending around obstacles, penetrating through small holes in screens...). For example, a sound can be clearly heard around the corner of a house - the sound wave goes around it. Diffraction of radio waves around the Earth's surface explains the reception of radio signals in the range of long and medium radio waves beyond the line of sight of the emitting antenna.
3) Diffraction of waves depends on the relationship between the wavelength and the size of the object causing diffraction. At the limit of laws wave optics deviations from the laws of geometric optics pass into the laws of geometric optics for other equal conditions turns out to be smaller, the shorter the wavelength. Therefore, it is easy to observe the diffraction of sound, seismic and radio waves, for which ~ from m to km; It is much more difficult to observe the diffraction of light without special devices. Diffraction is detected in cases where the size of obstacles around is commensurate with the wavelength.
Diffraction of light was discovered in the 17th century. by the Italian physicist and astronomer F. Grimaldi and was explained at the beginning of the 19th century. French physicist O. Fresnel, which became one of the main evidence wave nature Sveta.
Diffraction phenomenon can be explained by using Huygens-Fresnel principle.
Huygens principle: every point to which the wave reaches at the moment time, serves as the center of secondary (elementary) waves The envelope of these waves gives the position of the wave front at the next moment in time.
Assumptions:
1) the wave is flat;
2) the light falls normally on the hole;
3) the screen is opaque; The screen material is considered to be, to a first approximation, unimportant;
4) waves propagate in a homogeneous isotropic environment;
5) backward elementary waves should not be taken into account.
According to Huygens, each point of the wave front section isolated by the hole serves as a source of secondary waves (in a homogeneous isotropic medium they are spherical). Having constructed the envelope of secondary waves for a certain moment in time, we see that the wave front enters the region of the geometric shadow, i.e., the wave bends around the edges of the hole - diffraction is observed - light is a wave process.
Conclusions: Huygens principle
1) is geometric method building a wave front;
2) solves the problem of the direction of propagation of the wave front;
3) gives an explanation of wave propagation that is consistent with the laws of geometric optics;
4) simplifies the task of determining the influence of everything wave process, occurring in a certain space, on a point, reducing it to the calculation of the action on this point arbitrarily chosen wave surface.
5) But: valid provided that the wavelength is large smaller sizes wave front;
6) does not address the issue of the amplitude and intensity of waves propagating in different directions.
Huygens' principle supplemented by Fresnel
Huygens-Fresnel principle : wave disturbance at some point R can be considered as the result of the interference of coherent secondary waves emitted by each element of a certain wave surface.
Comment:
1) The result of interference of secondary elementary waves depends on the direction.
2) Secondary sources of phenomena. fictitious. They can serve as infinitesimal elements of any closed surface enclosing the source. Usually, one of the wave surfaces is chosen as the surface; all fictitious sources act in phase.
Fresnel assumptions:
1) excluded the possibility of the occurrence of reverse secondary waves;
2) assumed that if there is an opaque screen with a hole between the source and the observation point, then on the surface of the screen the amplitude of the secondary waves is zero, and in the hole it is the same as in the absence of a screen.
Conclusion: The Huygens-Fresnel principle serves as a technique for calculating the direction of propagation of waves and the distribution of their intensity (amplitude) in various directions.
1) Taking into account the amplitudes and phases of secondary waves allows in each specific case to find the amplitude (intensity) of the resulting wave at any point in space. The amplitude of the wave that has passed through the screen is determined by calculating the interference of secondary waves from secondary sources located in the screen hole at the observation point.
2) Mathematically strict solution diffraction problems based on the wave equation with boundary conditions, depending on the nature of the obstacles, presents exceptional difficulties. Approximate solution methods are used, e.g. Fresnel zone method.
3) Huygens-Fresnel principle within wave theory explained the rectilinear propagation of light.
Since ancient times, people have noticed the deflection of light rays when there is some obstacle in front of them. You can pay attention to how much light is distorted when it hits water: the beam “breaks” due to the so-called light diffraction effect. Diffraction of light is the bending or distortion of light due to various factors close up.
The operation of a similar phenomenon was described by Christian Huygens. After a certain number of experiments with light waves on water surface, he offered science a new explanation for this phenomenon and gave it the name “wave front”. Thus, Christian made it possible to understand how a ray of light would behave when it hits some other type of surface.
Its principle is as follows:
Surface points visible at a certain point in time may cause secondary elements. The area that touches all secondary waves is considered a wave sphere in subsequent periods of time.
He explained that all elements should be considered as the beginning of spherical waves, which are called secondary waves. Christian noted that the wave front is essentially a collection of these points of contact, hence his whole principle. In addition, secondary elements appear to be spherical in shape.
It's worth remembering that wave front - These are points of geometric meaning to which vibrations reach at a certain point in time.
Huygens' secondary elements are not represented as real waves, but only additional ones in the shape of a sphere, used not for calculation, but only for approximate construction. Therefore, these spheres of secondary elements inherently only have an enveloping effect, which allows a new wave front to form. This principle well explains the work of light diffraction, but it only solves the problem of the direction of the front, and does not explain where the amplitude, the intensity of the waves, the sputtering of the waves and their reverse action come from. Fresnel used Huygens' principle to eliminate these shortcomings and complement his work physical meaning. After some time, the scientist presented his work, which was fully supported by the scientific community.
Back in Newton's time, physicists had some idea about the work of light diffraction, but some points remained a mystery to them due to the small capabilities of technology and knowledge about this phenomenon. So, describe diffraction based on corpuscular theory light was impossible.
Independently, two scientists developed a qualitative explanation of this theory. French physicist Fresnel took on the task of supplementing Huygens' principle with a physical meaning, since the original theory was presented only with mathematical point vision. Thus, geometric meaning optics changed with the help of Fresnel's works.
The changes basically looked like this- Fresnel by physical methods proved that secondary waves interfere at observation points. Light can be seen in all parts of space where the force of the secondary elements is multiplied by interference: so that if a darkening is noticed, it can be assumed that the waves interact and cancel out under the influence of each other. If secondary waves fall into an area with similar types, states and phases, a strong burst of light is noticed.
Thus, it becomes clear why there is no backward wave. So, when the secondary wave returns back into space, they interact with the direct wave and, through mutual cancellation, the space turns out to be calm.
Fresnel zone method
The Huygens-Fresnel principle gives a clear idea about the possible propagation of light. The application of the methods described above became known as the Fresnel zone method, which allows the use of new and innovative ways to solve problems of finding amplitude. Thus, he replaced integration with summation, which was very positively received in the scientific community.
The Huygens-Fresnel principle gives clear answers to questions about how some important physical elements work, for example, how light diffraction works. Problem solving became possible only thanks to detailed description the work of this phenomenon.
The calculations presented by Fresnel and his method of zones are difficult work in themselves, but the formula derived by the scientist makes this process a little easier, making it possible to find exact value amplitudes. Early principle Huygens was not capable of this.
It is necessary to detect a point of oscillation in the area, which can subsequently serve as important element in the formula. The area will be presented in the form of a sphere, so using the zone method it can be divided into ring sections, which allow you to accurately determine the distances from the edges of each zone. Points passing through these zones have different vibrations, and accordingly, a difference in amplitude arises. In the case of a monotonic decrease in amplitude, several formulas can be presented:
- A res = A 1 – A 2 + A 3 – A 4 +…
- A 1 > A 2 > A 3 > A m >…> A ∞
It should be remembered that quite large number others physical elements influence the solution of a problem of this type, which also need to be looked for and taken into account.
To simplify calculations when determining the wave amplitude in given point pr-va. 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. Radius m ZF in the case of diffraction by round holes and screens is determined. 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) size m 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 create a high-quality, and in some cases sufficient exact quantities. an idea of the results of wave diffraction at decomp. 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
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" are in other dictionaries:
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 diffraction pattern from a round opaque disk always... ... Big Polytechnic Encyclopedia
Areas into which the wave surface is divided when viewed 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
Sections 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
Spherical diffraction electromagnetic wave on 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 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 science. Encyclopedic Dictionary
