The diffraction of light is manifested in the light waves bending around small obstacles, while deviations from the laws of geometric optics are observed. This also applies to light waves passing through a hole, for example, in a camera lens or through the pupil of an eye. There is a Fresnel and Fraunhofer diffraction. The differences consist in the magnitude 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 the general series of optical phenomena

The passage of light (and in general electromagnetic) waves through various inhomogeneous media is accompanied by the phenomena of their reflection, diffraction and refraction. When a wave reaches the boundary of two media, it is divided into a reflected one, which remains in the original medium, but with a change in the direction of propagation, and refracted, which passes through the boundary of the media, but also with a change in direction. Fresnel diffraction is the process of changing in the direction of a light wave when it encounters not the boundaries of two media, but some opaque obstacle with a hole (or without it, but small dimensions) in the same medium. The degree of diffraction increases with increasing length of the light wave.

## Discovery phenomenon

Probably the first to observe the 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. He devoted the second half of his life to the study of astronomy and optics.

Grimaldi was famous for his work entitled “The Physical Science of Light, Flowers 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 point of view 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 spread in straight lines. After all, an object that is between, for example, a candle flame and a wall, casts a shadow with a sharp border, as if the direct rays of light break off on an opaque barrier.

However, the results of the experiments of Grimaldi contradicted these ideas established over thousands of years. It turns out that if you illuminate different objects through an obstacle with a small hole, then the shadows from them will not be the same as in the absence of an obstacle. It turned out that the light can change the direction of propagation and go around small obstacles.

## How Fresnel diffraction was detected on a round hole

Grimaldi, passing 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 needles and hair on the screen is much larger (as seen in the photo below) than it would be if the rays of light passed along straight lines.

He also noted that the circle of light formed on the screen by rays passing through a very small hole in the lead plate was clearly larger than it would be if these rays fell on the screen in a straight line. Grimaldi concluded that they change their direction when passing near the edges of the hole.

In his experiments, conducted in the same room, the light into which came through the holes in the shutters, the distance between the obstacle for light waves (a plate with a round hole) and the screen was small. The Fresnel diffraction also corresponds to these conditions. Analyzing it, one cannot neglect the curvature of the front of both the initial wave incident on the obstacle and the secondary waves. They give the screen a diffraction image of an obstacle with a hole, as shown in the photo below.

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

Grimaldi also discovered that the shadow of a small body (of irregular shape) was surrounded by three colored stripes or ribbons, which became narrower as they moved away from the center of the shadow. If the original light flow was stronger, it reproduced similar color bands in the area 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 a phenomenon that was later called “Fresnel diffraction,” as a result of which a diffraction picture of an obstacle was obtained on the screen. Analytically calculate it is very difficult. However, there are methods that allow in some particular cases to significantly simplify this calculation.

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

## First observation of interference

Passing the sunbeams into the room through several small round holes, Grimaldi received traces of overlapping cones of light on the screen. As was to be expected, in those areas where rays from two holes fell, the screen was lit more than it would have been from a single 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 an illuminated body can become darker if you add light to the one it already receives. Now we know that the reason for this is interference, i.e., mutual amplification or attenuation of light waves. It also manifests itself in the phenomenon of Fresnel diffraction. Many modern scientists even do not fundamentally distinguish between these concepts, reducing all diffractive manifestations to interference manifestations, as did, for example, R. Feynman 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. Guk. He suggested that light is the wave oscillations of the world ether, which was then understood as the all-pervading substance that fills all space. Hooke’s idea has 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 made by Christian Huyges, who formulated his famous principle in 1690. According to him, visible light is a combination 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 ether particles excited directly by a light source (for example, a candle flame), but also any other of its particles at points in space that the light passes during propagation. The resulting visible wave is at any time as an envelope of all secondary waves. The latter may well spread beyond the boundaries of the obstacles in the path of light, which is well superimposed on the patterns of their shadows observed during diffraction. Therefore, according to this theory, there is simply no obstacle around the light - from new (secondary) sources it extends beyond the obstacles.

However, according to Huygens' principle, narrow light rays are impossible at all - their edges should immediately spread in all directions. Nevertheless, they can be seen with the naked eye, as it was in the experiments of Grimaldi. There was a contradiction between theory and practice.

## Light waves return

In 1880, the English physicist T. Jung proposed to return to the wave theory of light, supplemented by the concept of the interference of light waves. It means that when coherent (with identical frequencies) waves overlap each other, the time intensity of the light intensity at some points in the field and attenuation at others depending on the phase ratio of the added light waves is stable.

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

## Passing the light through the round hole

When a light wave propagates through a hole, the ratio between its diameter and the wavelength of the incident beam determines the behavior of the light. As shown on the left side of the figure below, when the wavelength is significantly smaller than the diameter of the hole, it simply passes forward in a straight line, as if there are no obstacles at all.

On the right side of the figure, however, a different situation is shown. In this case, the wavelength of light transmitted from a point source exceeds the diameter of the opening, and Fresnel diffraction occurs on the aperture. 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 the very same secondary spherical waves, which have already been mentioned above. They propagate in the direction of the screen and reach different points with different phases, interfering with each other, i.e., increasing or weakening at each such point. Since the whole system has axial symmetry, the incident cylindrical beam of light turns into a conical, and the axisymmetric diffraction pattern of alternating bright and dark rings, also called maxima and minima of illumination, is also observed on the screen, respectively. At point P, located on the hole axis, there will be a bright spot — the main maximum, and the first of the secondary maxima of illumination will appear at point Q. The intensity of the secondary maxima decreases as their distance from the center of the diffraction pattern increases. The ratio 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 light wavelength.

The figure below shows how the intensity of the screen illumination 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 ∏.

Analytical calculation of the picture of such a phenomenon as Fresnel diffraction on a hole and a disk is significantly simplified due to axial symmetry, which will be discussed further below.

## Round disc in the path of a beam of light

If you follow the Fresnel theory, then when a round opaque disk is placed on a beam of light, all points on its edges become sources of coherent secondary spherical waves. The distances between these points and the point of intersection of the disk axis with an 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, i.e. they must be folded and significantly strengthen each other. It turns out that in the center of a circular shadow from the disk a bright illuminated spot should be observed, as in the figure below.  This circumstance was first noticed by the French physicist S. Poisson, who was an opponent of the Fresnel theory. He considered that the circumstance noted by him proves its inconsistency. What was his surprise when Fresnel together with Arago did the corresponding experience and got such a spot in the center of the shadow from the disk! The figure below schematically shows this experience.

This is how Fresnel diffraction manifests itself on the disk. The bright spot in the center of its shadow received the name of the Poisson spot. If the disk is small, then the intensity of light in the center of its diffraction image is almost the same as with 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 difficult. But in axisymmetric cases, it can be simplified, so that the whole picture of such a phenomenon as diffraction becomes simple. The method of Fresnel zones allows you to visually geometric way to break the front of a spherical wave into circular sections.

The amplitudes and relative phases of all zones are taken into account to calculate the intensity distribution. Thus, rather complex mathematical processing is used to determine the diffraction pattern. But when analyzing such a phenomenon as Fresnel diffraction on a round hole and disk, it is significantly simplified.

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 point P is determined by dividing it into annular zones. The distances from the edges of two successive zones to point P differ by λ / 2. Ring zones with this property are called Fresnel zones. The distance from the zero zone to the point P is equal to 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.

The successive edges of two adjacent zones are located at similar points. If secondary spherical waves are excited in them, then 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 round hole and disk - a picture with axial symmetry. Therefore, the use of this method can significantly simplify the construction of a diffraction pattern when light passes through such obstacles.

## How do Fresnel ring zones work on a round hole?

Consider again the case when the diffraction of light occurs on a round hole. Fresnel zones into which a wave front can be broken, laying in a hole of a given diameter at a certain wavelength λ and distance from the front to the screen b0. may be in the amount expressed by 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 for the center of the diffraction pattern the number of Fresnel zones that fit in the hole is odd (2k + 1), then the amplitude of the illumination in the center of the picture will be the sum of the remainder of the first (central) zone and nothing compensated (2k + 1) - th zones that will reinforce each other. The diffraction pattern for this case is shown in the figure below.

If the number of Fresnel zones stacked in the hole is even, then the influence of all zones in the center of the picture will be mutually compensated in pairs, and a dark spot will appear in it.