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.
Total number Fresnel zones located on a hemisphere with a radius SP 0 equal to the distance from the light source S to the wave front are 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 by a 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 is less than the radius of the first zone Fresnel, i.e. straight forward.
2. Diffraction by 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 an odd number of Fresnel zones, 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, the diffraction pattern from a circular hole near the 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. benefits 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.
Wave Diffraction- the phenomenon of waves bending around obstacles and penetrating into the geometric shadow area. The phenomenon of diffraction can be qualitatively explained by applying the Huygens principle to the propagation of waves in a medium in the presence of obstacles.
Let us consider a flat obstacle ab (Fig. 69). The figure shows wave surfaces constructed according to Huygens' principle behind the obstacle. It can be seen that the waves act
bend tightly into the shadow area. But Huygens' principle says nothing about the amplitude of oscillations in a wave behind an obstacle. It can be found by considering the interference of waves arriving in the region of the geometric shadow. The distribution of vibration amplitudes behind an obstacle is called diffraction pattern. Full view diffraction pattern behind the obstacle depends on the relationship between the wavelength A, the size of the obstacle d and the distance L from the obstacle to the observation point. If the wavelength A is greater than the size of the obstacle d, then the wave almost does not notice it. If the wavelength A is of the same order as the size of the obstacle d, then diffraction occurs even at a very small distance L, and the waves behind the obstacle are only slightly weaker than in the free wave field on both sides. If, finally, there are many wavelengths smaller sizes obstacles, then the diffraction pattern can only be observed at a large distance from the obstacle, the magnitude of which depends on A and d.
The Huygens-Fresnel principle is a development of the principle that Christiaan Huygens introduced in 1678: each point on the front (the surface reached by the wave) is a secondary (i.e. new) source of spherical waves. The envelope of the wave fronts of all secondary sources becomes the wave front at the next moment in time.
Huygens' principle explains the propagation of waves, consistent with the laws of geometric optics, but cannot explain the phenomena of diffraction. Augustin Jean Fresnel in 1815 supplemented the Huygens principle by introducing the concepts of coherence and interference of elementary waves, which made it possible to consider diffraction phenomena on the basis of the Huygens-Fresnel principle.
The Huygens-Fresnel principle is formulated as follows:
Gustav Kirchhoff gave Huygens' principle a strict mathematical form, showing that it can be considered an approximate form of the theorem called Kirchhoff's integral theorem.
The wave front of a point source in a homogeneous isotropic space is a sphere. The amplitude of the disturbance at all points of the spherical front of a wave propagating from a point source is the same.
A further generalization and development of Huygens' principle is its formulation through path integrals, which serves as the basis of modern quantum mechanics.
Fresnel zone method Fresnel proposed a method of dividing the wave front into annular zones, which was later called Fresnel zone method.
Let a monochromatic spherical wave propagate from a light source S, P is the observation point. A spherical wave surface passes through point O. It is symmetrical with respect to straight line SP.
Let us divide this surface into annular zones I, II, III, etc. so that the distances from the edges of the zone to point P differ by l/2 - half the light wavelength. This partition was proposed by O. Fresnel and the zones are called Fresnel zones.
Let's take an arbitrary point 1 in the first Fresnel zone. In zone II there is, by virtue of the rule for constructing zones, a point corresponding to it such that the difference in the paths of the rays going to point P from points 1 and 2 will be equal to l/2. As a result, the oscillations from points 1 and 2 cancel each other at point P.
From geometric considerations it follows that if the numbers of zones are not very large, their areas are approximately the same. This means that for each point in the first zone there is a corresponding point in the second, the oscillations of which cancel each other out. The amplitude of the resulting oscillation arriving at point P from zone number m decreases with increasing m, i.e.
Fresnel proposed an original method for dividing the wave surface S into zones, which made it possible to greatly simplify 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 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 should mutually weaken each other:
| , | (9.2.2) |
Where A– amplitude of the resulting oscillation, – amplitude of oscillations excited 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 when i is not too large, the areas of neighboring zones are the same.
At the same time, with an increase in the zone number, the angle increases and, consequently, the intensity of the zone radiation in the direction of the point decreases 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 travels 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 amplitude of the oscillation from a certain 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 in the form
| . | (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 is .
Radiation intensity.
Thus, the resulting amplitude created at some point M by the entire spherical surface , equal to half the amplitude created by the central zone alone, and 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 (since). The light intensity increases if all even-numbered zones are covered.
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. For this purpose, 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 illumination at a point M, like a converging lens.
Diffraction of light (from lat. diffractus- broken, refracted) - a 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 region of a geometric shadow, and the bending of light around small obstacles. Diffraction occurs when light propagates through a medium with pronounced inhomogeneities. Diffraction of light - manifestation wave properties light under extreme conditions of transition from wave optics to geometric. The phenomenon of light diffraction can be explained on the basis of Huygens' principle.
Huygens' principle is the principle according to which each point on the wave front at a given moment in time is the center of secondary elementary waves, the envelope of which gives the position of the wave front at the next moment in time. Huygens' principle makes it possible to explain the laws of reflection and refraction of light, but it is insufficient to explain diffraction phenomena. Fresnel, who supplemented 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 idea of coherence and interference of secondary elementary waves. According to the Huygens-Fresnel principle, a wave disturbance 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 area is a source of a secondary spherical wave, the amplitude of which is proportional to the area of the element. A vibration comes to the observation point 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; - phase of oscillation 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 vibration for cases differing in symmetry, Fresnel proposed a method called the Fresnel zone method. There are two types of diffraction: Fraunhofer diffraction and Fresnel diffraction.
Fraunhofer diffraction (in parallel rays) - diffraction of plane waves on an obstacle (the light source is at an infinitely large distance 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 occurs in cases where 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 ring sections into which the spherical surface of the light wave front is divided when considering problems of wave diffraction in accordance with the Huygens-Fresnel principle to simplify calculations when determining the wave amplitude in given point space. Let a monochromatic wave propagate from point to observation point. 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 front of a spherical wave, which is divided into annular zones so that the distances from the edges of adjacent 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 Fresnel zone
(6.37.23)
area of the th zone
(6.37.24)
when the areas of the Fresnel zones are not too large, they are the same.
Since oscillations from neighboring zones travel to a distance point, they differ in that they arrive at the point 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 as the zone number increases, the amplitude of oscillations arriving at the point ,
monotonically decreasing: A 1 > A 2 > A 3 > A 4 > …. It can be assumed that the amplitude of the oscillation A m equal to the arithmetic mean of the amplitudes of the adjacent zones: 
Therefore, the amplitude of the resulting light vibration coming from the entire wave front to the point will be equal to:
A = A 1 - A 2 + A 3 - A 4 + …….. A k.
This expression can be represented as follows:
since the expressions in brackets are equal to zero, and the amplitude from the last Fresnel zone is infinitesimal. Therefore, the amplitude created at a point by the entire spherical wavefront is equal to half the amplitude created by the central Fresnel zone. If 1m, 0.5 µm, then the radius of the first Fresnel zone is 0.5 mm. Consequently, light from the source to the observation point propagates as if within a narrow direct channel, i.e. almost straight forward.
Oscillations from even and odd Fresnel zones are in antiphase and mutually weaken each other. If any obstacle blocks part of the spherical wave front, then when calculating the amplitude of the resulting oscillation at the observation point by the Fresnel zone method, only open Fresnel zones are taken into account. If you place a plate in the path of the light wave that covers all even or odd Fresnel zones, then the amplitude of the oscillation at the observation point increases sharply. This record is called zone. The zone plate increases the light intensity at the point many times over, acting like a converging lens.
Huygens-Fresnel principle. Fresnel zone method.
Lecture 3. Diffraction of light
Lecture outline
3.1. Huygens-Fresnel principle. Fresnel zone method.
3.2. Fresnel diffraction by a circular hole and a disk.
3.3. Fraunhofer diffraction by slit and grating.
3.4. X-ray diffraction methods for studying building materials.
Huygens-Fresnel principle. Fresnel zone method.
Diffraction is the deviation of waves from rectilinear propagation when the waves, bending around obstacles, enter the region of a geometric shadow.
Diffraction of light - special case wave diffraction. It manifests itself in alternating max and min intensities when the light wave front is partially shielded.
As experiments and calculations show, the condition for obtaining diffraction of light with wavelength λ from an obstacle (or hole) of size b located at a distance l from the source, are the ratios:
Therefore, they distinguish two types light diffraction:
1) Fresnel diffraction – diffraction in converging light beams, when the diffraction pattern is observed at a finite distance from the obstacle, i.e. When b 2 ~ lλ;
2) Fraunhofer diffraction 1 – diffraction in parallel rays, when the light source and the screen are located far from each other, i.e. When b 2 << l λ.
The direction of propagation of the wave front can be explained by Huygens' principle 2, which sets the method for constructing the wave front at the moment of time t + Dt according to the known position of the front at the moment of time t(see Fig. 3.1) .
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Rice. 3.1
I. Fraunhofer (1787 - 1826), German physicist.
2 H. Huygens (1629 – 1695), Dutch scientist
Huygens' principle states: every point the wave reaches (at time t) , serves as the center of secondary waves, the envelope of which gives the position of the wave front at the next moment in time (t + Δt) .
However, Huygens' principle, being a purely geometric method for constructing wave surfaces, does not essentially address the issue of amplitude, and, consequently, the intensity of waves propagating in different directions.
Fresnel introduced Huygens' principle physical meaning , supplementing it with the interference of secondary waves.
Huygens-Fresnel principle reads : a light wave excited by some source S can be represented as a result of superposition(interference)secondary coherent light waves “emitted” by fictitious sources. Such sources can be physically infinitesimal elements of any closed surface enclosing the source S. Usually one of the wave surfaces is chosen as such a surface, so all fictitious sources act in phase.
Fresnel excluded the possibility of the occurrence of backward secondary waves and suggested that if there is an opaque screen with a hole between the source and the screen, 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.
Taking into account the amplitudes and phases of secondary waves makes it possible in each specific case to find the amplitude (intensity) of the resulting wave at any point in space.
Using Huygens-Fresnel principle for secondary waves, the resulting amplitude of the light wave can be calculated, taking into account the phases of the interfering waves.
However, it is easier to do this by Fresnel zone method (see Fig. 3.2). Let us find at an arbitrary point P of the screen the amplitude of a light wave propagating in a homogeneous medium from a point source S. According to the Huygens-Fresnel principle, we replace the action of the source S with the action of imaginary sources located on an imaginary surface Ф, which is the surface of the wave front coming from S.
 |
Rice. 3.2
According to the Fresnel zone method, on the wave surface Ф (radius a) ring zones differing in radius are drawn from point P of the screen r by the amount.
In this case, the areas of each zone will be approximately the same:
and, consequently, the amplitudes of light vibrations at the screen point P will be practically equal, i.e.
.
Because oscillations from neighboring zones travel distances to the screen that differ by λ/2, then they arrive at the observation point P in antiphase. This means that the amplitude of the resulting light vibration at point P will be:
Where i- number of zones (number of the last zone).
The number of zones on the hemisphere will be
At A = r= 10 cm and λ = 0.5 µm: , i.e. N very large.
Therefore, for an open front, i.e.