Propagation of light Huygens principle. Huygens–Fresnel principle. Diffuse and specular reflection
Huygens' principle
Substantiating the wave theory of light, Huygens proposed a principle that made it possible to visually solve some problems of the propagation and refraction of light. Its meaning is that: If at any moment in time the light wave front is known, then in order to determine its position after a certain time interval equal to $\\triangle t$, then each point of the front should be considered as a source of a spherical wave, construct a sphere around such a secondary wave source with a radius $c\triangle t$, where $c$ is the speed of light in vacuum. In this case, the surface that bends around the secondary spherical waves will be the front of the original wave after a given time interval $\triangle t$.
In terms of physical content, Huygens' principle expresses the view of light as a continuous process in space. Using Huygens' principle, it can be explained why light waves fall into the geometric shadow region.
The main problem with Huygens' principle is that it does not take into account the phenomenon of interference of light. This principle does not provide information about the amplitude and intensity of the waves.
The Huygens-Fresnel principle, its analytical expression
Definition 1
Fresnel developed Huygens' principle, and this position began to be formulated as follows: Any point belonging to the wave front turns into a source of secondary waves (this is from Huygens' principle), while the secondary sources are coherent with each other and the secondary waves emitted by them interfere. For a surface coinciding with the wave surface, the powers of secondary radiation of equal areas are the same. Moreover, the light propagating from each secondary source goes in the direction of the external normal.
Rayleigh summarized the above principle:
Let us surround all $S_1,S_2,S_3,\dots $ with a closed surface $(F)$ of arbitrary shape. In this case, any point on the surface $F$ can be considered a secondary source of waves that propagate in all directions. These waves are coherent, since they are excited by the same primary sources. The light field that appears as a result of their spatial interference outside the surface $F$ coincides with the field of real light sources.
Thus, real light sources can be replaced by a luminous surface that surrounds them. Moreover, coherent secondary sources of light waves seem to be continuously distributed over this entire surface. The difference with this hypothetical surface is that it is transparent to any radiation.
Let us assume that the light source is monochromatic and the medium is homogeneous and isotropic. Thus, in accordance with the corrected principle, each element of the wave surface $S$ (Fig. 1) is a source of a secondary spherical wave, which has an amplitude proportional to the size of this element ($dS$).
Picture 1.
From any section $dS$ of the wave surface, an oscillation comes to point $A$ (Fig. 1), which is located in front of the surface $S$, which can be described by the following equation:
where $\left(\omega t+(\alpha )_0\right)$ is the oscillation phase at the location of the surface $S$, $k$ is the wave number, $r$ is the distance from the surface element ($dS)$ to the point $A$, $a_0$ - amplitude of light vibration at the location of the element $dS$. $K$ is a coefficient depending on the angle $\varphi $ between the normal $\overrightarrow(n)$ to the area $dS$ and the direction from it to the point $4A$. If $\varphi =0,\ $then we have $K=K_(max)$, with $\ \ \varphi =\frac(\pi )(2)$ $K=0.$
The total oscillation at point A is found as a superposition of oscillations that are taken for the entire wave surface $S$, that is:
Formula (2) is an integral formulation of the Huygens-Fresnel principle.
Interpretation of the Huygens-Fresnel principle
Fresnel replaced Huygens's artificial assumption about the envelope of secondary waves with a clear physical position along which secondary waves, when added, interfere. In this case, light is visible at the interference maxima; where the waves cancel each other out, there is darkness. Thus, the physical meaning of the envelope is explained. The secondary waves approach the envelope in the same phases, so interference causes greater light intensity. The Huygens-Fresnel principle explains the absence of a backward wave. Secondary waves, which propagate forward from the wave front, go into space free from disturbance. At the same time, they interfere only with each other. Secondary waves that go backwards fall into space where a direct wave is already present, so the secondary waves dampen the direct wave, therefore, after the wave passes, the space on it has no disturbances.
In Rayleigh's formulation, the principle in question means that a wave, which has separated from its source, then exists autonomously, independent of the presence of sources.
The Huygens-Fresnel principle allows us to explain the phenomenon of diffraction.
Example 1
Exercise: Write down an expression for the electric field strength ($E$) in the wave, if we assume that the wave is spherical and propagates freely.
Solution:
Figure 2.
Let's consider the free propagation of a spherical wave in a homogeneous medium (Fig. 2), it can be described using the equation:
The auxiliary wave surface in our case is the surface S, which has a radius $r_0$. According to Fresnel, each element of this surface ($dS$) emits a secondary spherical wave. In this case, we find the wave field emitted by the element $dS$ at point $A$ as:
Using the Fresnel hypothesis we have:
where $K\left(\alpha \right)$ is a function depending on the wavelength and the angle between the normal to the wave front and the direction of propagation of the secondary wave (Fig. 2).
Let us represent the total wave field at point $A$ by the integral:
Let us take as the element $dS$ the area of the ring, which is cut out from the wave front by two infinitely close concentric spheres whose centers are located at point $A$ (Fig. 2). In this case, we can write that:
We take the distance $r_1.$ as the integration variable. We consider the quantities $r_0$ and $r$ constant. From triangle $DOA$ we find:
\[(r_1)^2=(r_0)^2+(\left(r_0+r\right))^2-2r_0\left(r_0+r\right)cos\beta \left(1.6\right).\ ]
Let us differentiate expression (1.6), we have:
Substituting expression (1.7) for $dS$ into formula (1.4), we obtain:
where function $K\left(\alpha \right)\ \consider\ as$ function $r_1$. In this case $r_(max)=r+2r_0.$
Answer:$E=\frac(2\pi A_0)(\left(r_0+r\right))e^(i\left(\omega t-kr_0\right))\int\limits^(r_(max))_r (K\left(r_1\right)e^(-ikr_1))dr_1.$
Example 2
Exercise: How to use the Huygens-Fresnel principle to explain the phenomenon of diffraction?
Solution:
Let us assume that a plane wave is incident on the screen perpendicular to the hole in it. According to the Huygens-Fresnel principle, each point of the section of the wave front, which is highlighted by a hole in the screen, becomes a source of secondary waves. If the medium is homogeneous and isotropic, the secondary waves are spherical. When constructing the envelope of secondary waves for a fixed moment in time, it turns out that the front of the wave enters the region of the geometric shadow, which means that the wave bends around the hole.
Diffraction of light - in a narrow, but most commonly used sense - rounding the boundaries of opaque bodies (screens) by light rays; penetration of light into an area of geometric shadow. Light diffraction manifests itself most clearly in areas of sharp changes in ray flux density: near caustics, the focus of a lens, the boundaries of a geometric shadow, etc. Wave diffraction is closely intertwined with the phenomena of wave propagation and scattering 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.
We constantly observe the bending of sound waves around obstacles (sound wave diffraction) (we hear sound around the corner of the house). To observe the diffraction of light rays, special conditions are required, this is due to the short wavelength of light waves.
There are no significant physical differences between interference and diffraction. Both phenomena involve the redistribution of light flux as a result of wave superposition.
The phenomenon of diffraction is explained using Huygens' principle , Whereby every point to which the wave reaches serves center of secondary waves, and the envelope of these waves sets the position of the wave front at the next moment in time.
Let a plane wave be normally incident on a hole in an opaque screen (Fig. 9.1). Each point in the section of the wave front isolated 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 in 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.
O. Fresnel played a decisive role in establishing the wave nature of light 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 Paris Academy Prize for his explanation of the phenomenon of diffraction and the method for its quantitative calculation.
Fresnel put a physical meaning into Huygens' principle, supplementing it with the idea of interference of secondary waves.
When considering diffraction, Fresnel proceeded from several basic principles, accepted without proof. The set of these statements is called the Huygens–Fresnel principle.
According to Huygens' principle , every front point waves can be considered as a source of secondary waves.
Fresnel significantly developed this principle.
· All secondary sources of a wave front emanating from one source are coherent between themselves.
· Equal areas of the wave surface radiate equal intensity (power) .
· Each secondary source emits light predominantly in the direction of the outer normal to the wave surface at this point. The amplitude of secondary waves in the direction making an angle α with the normal is 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 radiation from 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 could already make quantitative calculations of the diffraction pattern.
Diffraction called the deviation of light from linear propagation, the bending of waves around obstacles. Diffraction is noticeable if the size of the obstacles is comparable to the wavelength. Diffraction of light is always accompanied by interference - alternation of light and dark places for monochromatic light and colored (all colors of the rainbow) for white light. Diffraction is explained based on Huygens-Fresnel principle : each point to which the disturbance has reached becomes a source of secondary waves; secondary waves are coherent; the wave surface at any time is the result of the interference of secondary waves.
There are two special cases of diffraction. Fresnel diffraction called diffraction in converging and diverging beams. Fraunhofer diffraction observed in parallel rays. The condition of parallelism of the incident and diffracted beams can be met by placing the light source and the screen on which the diffraction pattern is observed at a great distance from the obstacle, or by using a lens that can convert the diverging beam of light into a parallel one.
In 1690, Huygens proposed a way to find the position of the wave front1 at subsequent times based on its position at a given moment.
This method is known as Huygens principle and can be formulated as follows: each point of the wave front can be considered as a source of secondary elementary spherical waves propagating into the front part of the half-space; the new position of the wave front coincides with the envelope of the elementary waves.
As a simple example of the application of Huygens' principle, consider the wave front AB in Fig. 6. Let us agree to assume that the environment isotropic, that is, the speed of the waves is the same in all directions. To find the position of the wavefront after a short period of time D t after he held the position AB, draw circles with radius . The centers of these circles lie on the original wavefront AB, and the circles themselves represent elementary Huygens waves. The envelope of these elementary waves is the line CD – determines the new position of the wave front. In Fig.6 V shows the curvature of the wave front when a wave with a flat front passes through a small hole. As a result of diffraction, the wave enters the region of the geometric shadow.
Huygens' principle allows only to qualitatively depict the diffraction pattern. Fresnel supplemented Huygens' principle with the proposition about the coherence of secondary waves. He suggested also taking into account the radiation power of secondary sources. Huygens-Fresnel principle is the set of the following statements.
1. Any real light source S 0 can be replaced by a system of fictitious secondary sources and secondary waves excited by them. As these sources, you can select small sections of the wave surface surrounding the source.
2. Secondary sources equivalent to the same source S 0 are coherent.
3. The powers of secondary sources of equal area located on the wave surface are identical.
4. Each secondary source emits light predominantly in the direction of the outer normal to the surface. The amplitudes of secondary waves in other directions are smaller, the larger the angle between the normal and a given direction, and are equal to zero at an angle equal to p/2.
To simplify the calculation of diffraction patterns, Fresnel proposed the zone method. Essence Fresnel zone method Let's consider the example of determining the amplitude of the electric field of a spherical wave excited by a point source S 0(Fig. 7). Fresnel proposed dividing the spherical wave surface into ring zones so that the distances from the edges of each zone to the observation point R differed by half the wavelength (recall that on the wave surface oscillations occur in the same phase). With such a division, for each small section of one zone there will be a corresponding section of the neighboring zone, the distances of which to the observation point will differ by l/2, and the waves from these sections will arrive at the observation point in antiphase and weaken each other. Therefore, the resulting vibrations created at the point R adjacent zones will be antiphase, i.e. differ by p. A simple calculation allows you to find expressions for the radii of the Fresnel zones depending on the wavelength l, wave surface radius A and distances b from the wave surface to the observation point (Fig. 7):
, (3)
Where m– Fresnel zone number.
A change in phase to the opposite one can be represented as a change in the sign of the amplitude to the opposite, therefore, if the amplitude of the wave arriving at R from the first Fresnel zone, denoted by E 1, then the amplitude of the wave coming from the second zone must be assigned a minus sign and denoted as – E 2. The sign of the wave amplitude from the third zone must again be changed to the opposite. Thus, the amplitude of the resulting wave at the point R can be found as the algebraic sum of the wave amplitudes from all Fresnel zones:
As calculations show, the areas of the ring zones constructed in this way are approximately the same. However, due to the increase in the angle between the normal to the zone surface areas and the direction to the observation point, the absolute values of the amplitudes monotonically decrease with increasing zone number: 
If we write the previous expression as:
, (5)
then, assuming that the expressions in brackets are equal to zero and the number of zones is large, we obtain that the resulting amplitude of the wave at the observation point is equal to half the amplitude of the wave from the first zone:
This leads to a seemingly paradoxical conclusion: if you place a screen in the path of light E with a small hole that opens only the first zone, then the amplitude of the wave at the observation point will increase by 2 times, and the intensity by four1. If there is a hole in the screen E opens two zones, then at the observation point the waves from the first and second zones will overlap in antiphase and the amplitude will be very small. Thus, when Fresnel diffraction on a round hole in the center of the geometric shadow there will be a maximum or minimum depending on the number of Fresnel zones opened by this hole (Fig. 8).
If an annular screen is placed in the path of light, which would cover the even Fresnel zones (they are shaded in Fig. 7), then the amplitude of the resulting wave at the point R will increase sharply. Indeed, in this case, the amplitudes from even zones will be equal to zero, eliminating them from formula (4), we obtain: . This screen is called zone plate
.
If you put in the path of a ray of light opaque disk
, covering a not very large integer number of Fresnel zones, then in the center of the geometric shadow there will always be a maximum - a bright spot, regardless of how many zones are closed - even or odd. Indeed, if we write down the resulting amplitude for this case in t. R(Fig. 7) in a form similar to formula (8), starting from the amplitude m-th zone, we get: 
. In Fig. Figure 9 shows the shadow of a small disk illuminated by a laser. A bright spot (Poisson's spot) is observed in the center. It can also be seen that light and dark rings are observed outside the geometric shadow. This is also the result of diffraction at different parts of the disk.
Note that the phenomena described above are observed only when certain conditions are met: the light must be monochromatic; the center of the hole (disk) must be on the straight line connecting the source to the observation point; the edges of the obstacle must be smooth (scratches must be less than the width of the nearest open area). To fulfill the last condition, a small number of Fresnel zones must be placed on the hole (disk), since the width of the annular zone decreases with increasing its number.
The zone method allowed Fresnel to explain the rectilinear propagation of light within the framework of wave theory. As follows from formula (3), the smaller the wavelength, the smaller the size of the first Fresnel zone. At a=b= 1 m and l=0.5 µm, the radius of the first zone is 0.5 mm. To ensure that placing a screen with a hole in the light path does not change the intensity at the observation point, the size of the hole must be smaller than the size of the first zone. In this case, light from the source to the observation point propagates as if within a very narrow channel, i.e. almost straight forward
 Rice. 5.7 |
Let the gap be wide b A parallel beam of light (Fig. 5.7), which has a flat wave surface, falls normally. To determine the resulting amplitude of the beam propagating behind the slit, we will divide the open part of the wave surface located in the plane of the slit into a number of parallel strips - zones. At diffraction angle j=0 waves from all zones will propagate in the same phase, therefore, when j=0 there is a maximum. At some other diffraction angle j, such that the wave surface can be divided into two
zones so that the difference in wave paths from the edges (left in Fig. 5.7) of neighboring zones D will be equal to l/2, the waves from these zones will cancel each other out and at a given diffraction angle there will be a minimum. Find the value of angle j from the triangle ABC: BC/AB= sin j or: 
. From here we obtain the condition for the first minimum: b sin j=l. At the value of the diffraction angle determined by the relation: , the wave surface can be divided into three zones of equal width, so that the path difference from the edges of adjacent zones will be equal to l/2. In this case, the waves from the two zones will completely extinguish each other, and the wave from the third zone will not be extinguished and will give a small maximum. It is not difficult to obtain the condition for this maximum: b sin j=3(l/2).
Thus, as the diffraction angle increases, the slit area can be successively divided into an even and odd number of zones. The general condition for maxima (except zero) for diffraction from a slit has the form:
, (5.3)
and the minimum condition:
J - diffraction angle, - the period of the diffraction grating (the distance between the centers of adjacent slits) is the number of slits per unit length of the grating.
A diffraction grating splits white light into a spectrum. It can be used to make very precise wavelength measurements
1 wave front is a surface separating regions of space that have not yet been reached by wave excitation from regions involved in the wave process. A wave surface is a geometric locus of points at which oscillations occur in one phase. In fact, the wave front is the very first wave surface.
1 Limiting the light beam to a small hole will darken the plane in which the so-called is located. R. An increase in amplitude occurs only when R and in a small area near it.
Gordyunin S.A. Huygens' principle //Quantum. - 1988. - No. 11. - P. 54-56.
By special agreement with the editorial board and editors of the journal "Kvant"
This principle was formulated by Christian Huygens in his Treatise on Light, published in 1690. At that time, there were no longer any great difficulties in describing the motion of particles. In free space, particles move rectilinearly and uniformly; under the influence of external influences they slow down, accelerate, change the direction of movement (are refracted or reflected) - and all this can be calculated. At the same time, the laws of wave propagation - reflection, refraction, bending around obstacles (diffraction) could not be explained. And Huygens proposed a principle on the basis of which this could be done.
Obviously, his idea was inspired by reasoning about the reasons for the propagation of wave processes. A stone thrown into water causes circular waves to run across the surface. This process continues even after the stone has fallen to the bottom, that is, when the source that generated the first waves is no longer there. It followed that the sources of waves are the wave excitations themselves. Huygens put it this way:
Each point to which the wave excitation reaches is, in turn, the center of secondary waves; the surface that bends around these secondary waves at a certain moment in time indicates the position of the front of the actually propagating wave at that moment.
It is easy to imagine, for example, how plane and spherical waves propagate (Fig. 1). Envelope of secondary waves through time Δ t is for a plane wave a plane shifted by a distance cΔ t, and for spherical - a sphere with radius R + cΔ t, Where c- speed of propagation of secondary waves, R- radius of the initial spherical wave.
In fact, Huygens' principle in this formulation is simply a geometric recipe for constructing a surface that envelops secondary waves. This surface is identified with the wave front, and thus the direction of wave propagation is determined.
Huygens initially formulated his principle for light waves and applied it to derive the laws of reflection and refraction of light at the interface between media. First of all, the very fact of the presence of reflected and refracted waves followed directly from Huygens' principle, and this was already a great success. According to Huygens, each point of the boundary of the media, as the front of the incident wave reaches it, becomes a source of secondary waves that propagate into both boundary media. The result of the superposition of these secondary waves in the first medium from which the wave falls is a reflected wave, and the result of the superposition of secondary waves in the second medium is a refracted wave.
Of course, based on Huygens’ principle, we cannot answer the question about the intensity of reflected and refracted waves, since for this we need to know at least their physical nature (which, in Huygens’ principle, is not “involved” at all). But the geometric laws of reflection and refraction are completely independent of either the physical nature of the waves or the specific mechanism of their reflection and refraction. They are the same for all waves.
Let υ - speed of a plane incident wave, α - angle of its incidence (Fig. 2). Then the front of the incident wave runs along the interface between the two media with a speed \(~\frac(\upsilon)(\sin \alpha)\). Both reflected and refracted waves are generated by the incident wave, therefore their fronts run along the boundary at the same speed, i.e.
\(~\frac(\upsilon)(\sin \alpha) = \frac(\upsilon_1)(\sin \alpha_1) = \frac(\upsilon_2)(\sin \alpha_2)\) .
Angles α 1 and α 2 determine the directions of propagation of the fronts of reflected and refracted waves. But since in a plane wave the rays are perpendicular to the wave fronts, the same relationships hold for reflected and refracted rays.
The explanation of the laws of refraction and reflection was a strong argument in favor of the validity of Huygens' principle. However, naturally, it also raised many doubts and questions. Why is there no backward wave (after all, secondary sources emit spherical waves that also propagate against the front)? Why does light pass through the hole in a straight line (after all, secondary waves should also propagate into the region of the geometric shadow)? Huygens himself believed that all this was due to the low intensity of secondary waves. But sound waves bend - we hear sound whose source is around the corner.
Answers to these and other questions were given by Augustin Fresnel at the beginning of the 19th century. He supplemented Huygens' principle with an important and natural proposition:
The resulting wave disturbance at a given point in space is a consequence of the interference of elementary secondary Huygens waves.
Secondary waves are emitted by “sources” whose amplitude and phase of oscillation are determined by the original disturbance, and therefore such sources are coherent. The combined action of these sources, i.e., the interference effect, replaces Huygens' idea of an envelope, which in Fresnel's theory acquired a clear physical meaning as a surface where the resulting wave, due to interference, has a noticeable intensity. The modified Huygens-Fresnel principle allows us to more fully explore the issue of wave propagation in an inhomogeneous medium (due to its mathematical complexity, this issue is beyond the scope of a school physics course). So, we must clearly understand both the advantages (simplicity and clarity) and the disadvantages (lack of physical content) of the first principle of the theory of wave propagation - Huygens' principle.
> Huygens principle
Explore Huygens principle– laws of light reflection and wave refraction. Read the formulation of Huygens' principle, formula, diffraction effects, wave front.
Each point on the wave front acts as a source of bursts that propagate forward at the same speed.
Learning Objective
- Express Huygens' principle.
Main points
- Diffraction is the bending of waves at the edge of an opening or obstacle.
- This principle can be used to define reflection and explain refraction and interference.
- It is transmitted in the formula: s = vt (s – distance, v – propagation speed, t – time).
Term
- Diffraction is the bending of waves around the edges of a hole or obstacle.
Review
In accordance with Huygens' principle, all points on the wave front act as sources of bursts and propagate at the same speed as the original wave. The new wave front will be straight.
The basis
Christiaan Huygens is recognized for creating a method for detecting wave propagation. In 1678, he proposed that all points encountering a light disturbance become sources of a spherical wave. The new type of wave is determined by the sum of the secondary ones.
He not only explained the linear and spherical propagation of waves, but also derived the laws of light reflection and refraction in Huygens' principle. But he failed to explain diffraction effects—the deviation from straight propagation when light hits an edge or obstacle. Augustine-Jean Fresnel already figured out this issue in 1816. Below is a presentation of Huygens' principle in the form of a diagram.
Huygens' principle can be used for the wave front. All points emit semicircular curls moving over a distances =vt
Huygens' principle
The top image shows a simple example of Huygens' principle in action. It can be expressed in the formula:
s = vt (s – distance, v – propagation speed, t – time).
The created waves form into semicircles, and the new front touches the bursts. The principle operates for all wave types and benefits reflection, refraction and interference characteristics. Visually it also clarifies reflection and is used in refraction situations.
Its principle can be applied to a straight wavefront moving into a medium where the speed is lower. The beam is deflected to the perpendicular
The principle works if the waves hit the mirror. The tangent of the bursts shows that the new wave front was reflected at an angle equal to the angle of incidence. The direction is set perpendicular (down arrows)
Examples
You often see this Huygens wave principle at work in everyday life, but you don't consciously notice it. The easiest way to explain is using sounds as an example. If someone plays a musical instrument in a room with the door tightly closed, you will not hear anything. You will have to open it and stand next to it. This is a direct effect of diffraction. When light passes through small holes, it begins to resemble sound, but on a smaller scale.
Diffraction
Diffraction is a wave bend created when it collides with the edge of an opening or an obstacle.
