Diffraction of light is manifested in the light waves around small obstacles, while deviations from the laws of geometric optics are observed. This also applies to light waves passing through the hole, for example, in the camera lens or through the pupil of the eye. There is a diffraction of Fresnel and Fraunhofer. Differences consist of the distance between the light source, the obstacle and the screen on which the picture of this phenomenon is observed.

## The place of diffraction in a general series of optical phenomena

The passage of light (and generally electromagnetic) waves through various inhomogeneous media is accompanied by phenomena of their reflection, diffraction and refraction. When a wave reaches the boundary of two media, it divides into a reflected, remaining in the initial medium, but with a change in the direction of propagation, and a refracted one that passes through the media boundary, but also with a change in direction. Fresnel diffraction is a process of change in the direction of a light wave when it encounters, on its way, not the boundary of two media, but some opaque obstacle with orifice (or without it, but of small dimensions) in the same medium. The degree of diffraction increases with the length of the light wave.

## The discovery of phenomenon

Probably the first who observed diffraction was Francesco Maria Grimaldi (April 2, 1618 - December 28, 1663), an Italian Jesuit priest and at the same time a mathematician and physicist who taught at a Jesuit college in Bologna. The second half of his life he devoted to the study of astronomy and optics.

Grimaldi made his famous work "Physical Science of Light, Colors and the Rainbow", which was published in Bologna in 1665. Most of it consists of a tedious discussion of the nature of light from a theological standpoint and today is of no interest. However, in addition, the book contains accounts of numerous experiments related to the diffraction of light rays.

Based on everyday experience, people in ancient times concluded that the rays of light propagate along straight lines. After all, an object between the flame of a candle and a wall, for example, casts a shadow with a sharp boundary, as if the direct rays of light were breaking off on an opaque barrier.

However, the results of Grimaldi's experiments contradicted these ideas that had been established over millennia. It turns out that if you illuminate different objects through a barrier with a small hole, then the shadows from them will not be the same as in the absence of a barrier. It turned out that the light is able to change the direction of propagation and to sketch small obstacles.

## How Fresnel diffraction on a circular aperture was detected

Grimaldi, having passed the light of the sun into a dark room through a small hole (aperture), noticed that the width of the shadow of thin objects like a needle and hair on the screen is much larger (as seen in the photo below) than it would be if the rays of light passed through straight lines.

He also noted that the circle of light formed on the screen by the rays passing through a very small hole in the lead plate was clearly larger than it would have been if these rays had been incident on the screen rectilinearly. Grimaldi came to the conclusion that they change their direction when passing near the edges of the hole.

In his experiments, which were conducted inside one room, the light that entered through the holes in the shutters, the distance between the obstacle for light waves (a plate with a circular hole) and the screen was small. These conditions also correspond to the phenomenon of Fresnel diffraction. Analyzing it, we can not neglect the curvature of the front as the initial wave incident on the obstacle, or the secondary waves. They give on the screen a diffraction image of an obstacle with a hole, as shown in the photo below.

## What happens if light falls on a small opaque obstacle

Grimaldi also discovered that the shadow of a small body (irregular shape) was surrounded by three colored stripes or bands that became narrower as they moved away from the center of the shadow. If the original light flux was stronger, it reproduced similar color bands and in the zone of the shadow itself: there were two or more such bands, and their number increased in proportion to the distance between the shadow and the illuminated body.

And in this case Grimaldi managed to observe the phenomenon, which was later called "Fresnel diffraction", as a result of which a diffraction picture of the obstacle was obtained on the screen. It is very difficult to calculate analytically. However, there are methods that make it possible to simplify this calculation in some particular cases.

And one more remark about Grimaldi's experiments. If he used as a screen a round disk (for example, a saucer), then perhaps he would have managed a century and a half earlier than it actually happened to observe such a phenomenon as Fresnel diffraction on a circular disk. But history in general and the history of science in particular do not know the subjunctive mood. Therefore, this experience was carried out only in the early 19th century. (see below).

## First observation of interference

Passing the sun's rays into the room through several small round holes, Grimaldi received traces of overlapping cones of light on the screen. As might be expected, in those areas on which the rays from the two holes fell, the screen was illuminated more strongly than it would be from one cone of light; but the researcher was surprised to find that those parts of the total shadow in which the cones of light were superimposed on each other turned out to be darker than the corresponding parts in which there was no overlap (figure below).Thus, for the first time it was recorded that the illuminated body could become darker if you add light to the one it already receives. Now we know that the reason for this is interference, that is, mutual amplification or weakening of light waves. It also manifests itself in such a phenomenon as the diffraction of Fresnel. Many modern scientists do not even fundamentally distinguish between these concepts, reducing all diffraction to interference phenomena, as, for example, R. Feynman did in the 3rd volume of his "Feynman lectures on physics".

## From Grimaldi to Huygens and Newton

The first attempt to explain the reason for the deviation of light from rectilinear propagation was made by the famous English scientist R. Hooke. He suggested that light is the wave oscillations of the world ether, by which then the all-pervading substance filling the entire space was understood. Hooke's idea already laid the foundation for a future correct explanation of what Fresnel diffraction and all optical phenomena are. However, he was unable to create an appropriate quantitative theory.

The next step was taken by Christian Guyges, who formulated his famous principle in 1690. According to him, visible light is the totality of spherical waves propagating from the source in all directions in the ether. In this case, the source of these waves can be not only the particles of the ether excited directly by the light source (for example, the flame of a candle), but also any of its other particles at points of space that pass the light during propagation. The resulting visible wave is at any time as enveloping all secondary waves. The latter may well extend beyond the boundaries of obstacles in the path of light, which is well superimposed on the patterns of their shadows observed during the diffraction. Therefore, according to this theory, there is simply no obeying of light obstacles - from new (secondary) sources it spreads beyond the obstacles.

However, according to the Huygens principle, narrow light beams are generally impossible - their edges should immediately creep in all directions. Nevertheless, they can be seen with the naked eye, as was the case with Grimaldi. There was a contradiction between theory and practice.

## Return of light waves

In 1880 the English physicist T. Jung proposed to return to the wave theory of light, supplemented by the concept of interference of light waves. It means that when superimposed coherent waves (with identical frequencies) are coherent (with the same frequencies), a time-stable increase in the intensity of light at certain points of the field and a weakening in others are possible, depending on the ratio of the phases of the folded light waves.

The concept of interference was used by the French physicist O. Fresnel to supplement the Huygens principle. According to his version, all secondary spherical waves are coherent and interfere upon application. What is the physical mechanism of the Huygens-Fresnel diffraction?

## We pass light through a circular hole

When a light wave propagates through an aperture, the relationship between its diameter and the wavelength of the incident beam determines the behavior of light. As shown on the left side of the figure below, when the wavelength is much smaller than the diameter of the hole, it simply runs forward along a straight line, as if there were no obstacles at all.

On the right side of the figure, however, another situation is shown. In this case, the wavelength of the light transmitted from the point source exceeds the opening diameter, and Fresnel diffraction occurs at the hole. When analyzing this phenomenon, the hole is considered to be absent, and instead of it a set of fictitious secondary light sources is placed, which excite those very secondary spherical waves, which have already been mentioned above. They propagate in the direction of the screen and reach its various points with different phases, interfering with each other, i.e. amplifying or weakening at each such point. Since the entire system has axial symmetry, the incident cylindrical beam of light turns into a conical, and on the screen there is also an axisymmetric diffraction pattern from alternating bright and dark rings, also called light maxima and minima, respectively. At point P, located on the axis of the hole, there will be a bright spot - the main maximum, and the first of the secondary highlights of illumination will appear at point Q. The intensity of secondary maxima decreases with increasing distance from the center of the diffraction pattern. The relationship between the size of the hole and the degree of diffraction is determined by the following equation:

sinθ = λ / d, where

• θ is the angle between the direction to the center of the diffraction pattern and the direction to its first minimum,
• λ is the wavelength of the light wave.

The figure below shows how the intensity of the illumination of the screen varies depending on the angular distance from the center. Note that the minima between the secondary maxima are located at points that are multiples of Π.

An analytical calculation of the picture of such a phenomenon as Fresnel diffraction on the aperture and disk is substantially simplified due to the axial symmetry, which will be further discussed below.

## Round disk on the path of a beam of light

If we follow the Fresnel theory, then, when a circular opaque disk is placed on the path of the light beam, all the points on its edges become sources of coherent secondary spherical waves. The distances between these points and the point of intersection of the axis of the disk with the opaque screen perpendicular to it are the same. Therefore, the waves from all points on the edge of the disk must intersect at the same time and in the same phase, that is, they must form and significantly strengthen each other. It turns out that in the center of the circular shadow from the disk there should be a bright illuminated spot, as in the figure below.  This circumstance was first noticed by the French physicist S. Poisson, who was an opponent of the theory of Fresnel. He believed that the circumstance noted by him proves her failure. Imagine his surprise when Frenel together with Arago made the corresponding experience and received just such a spot in the center of the disc shadow! The diagram below shows this experience schematically.

Thus, the Fresnel diffraction on the disk manifests itself. A bright spot in the center of his shadow was called the Poisson spot. If the disc is small, then the intensity of light at the center of its diffraction image is almost the same as for its (disk) absence.

## How to calculate diffraction patterns

In the general case, the calculation of the interference of secondary waves to obtain a diffraction pattern is complex. But in axisymmetric cases it can be simplified, so that the whole picture of such a phenomenon as diffraction becomes uncomplicated. The Fresnel zone method allows us to visually geometrically divide the front of a spherical wave into annular sections.

The amplitudes and relative phases of all zones are taken into account for calculating the intensity distribution. Thus, a rather complicated mathematical treatment is used to determine the diffraction pattern. But when analyzing a phenomenon such as Fresnel diffraction on a circular aperture and a disk, it becomes much simpler.

In the figure below, S is a point source of light.S emits a spherical light wave of length λ in the direction from left to right. Let the radius of its front at time t be equal to R. The effect of this wave front at the point P is determined by dividing it into annular zones. The distances from the edges of the two successive zones to the point P differ by λ / 2. The ring zones possessing this property are called the Fresnel zones. The distance from the zero zone to the point P is b0 .

The first zone is at a distance b1  = b0  + λ / 2; second: b2  = b0  + 2λ / 2; third: b3  = b0  + 3λ / 2; i-th zone: bi  = b0  + iλ / 2.

Sequentially located edges of two adjacent zones are at similar points. If secondary spherical waves are excited in them, they come to the observation point P with a phase difference of 180 ° and mutually weaken each other when superimposed (but not destroyed).

Fresnel diffraction on a circular hole and a disk is a picture with axial symmetry. Therefore, the application of this method makes it possible to simplify considerably the construction of the diffraction pattern when light passes such obstacles.

## How do the Fresnel ring zones work on a circular hole

Let us again consider the case when diffraction of light occurs on a circular hole. Fresnel zones, on which the wave front can be divided, placed in a hole of a given diameter for a certain wavelength λ and distance from the front to the screen b0. may be in an amount expressed as an odd or even number. As noted above, the secondary waves from two adjacent zones at each point of the screen weaken, although they do not destroy each other. Therefore, if the number of Fresnel zones in the hole is odd (2k + 1) for the center of the diffraction pattern, then the amplitude of the illumination at the center of the picture will be the sum of the remainder of the action of the first (central) zone and the uncompensated (2k + 1) - th zone, which will strengthen each other. The diffraction pattern for this case is shown in the figure below.

If, however, the number of Fresnel zones that fit into the hole is even, the influence of all zones in the center of the picture will be mutually compensated, and a dark spot will appear in it.