The process of light propagation in a homogeneous transparent substance and the laws of its interaction with various obstacles (opaque objects, interfaces of transparent media, openings) throughout history have interested scientists and philosophers. A reasonable explanation and mathematical description of many phenomena in which light behaves like a wave has been obtained through the use of the Huygens-Fresnel principle. What it means and what conclusions it leads to, these and other questions are revealed in the article.
Huygens wave theory of light and its principle
The first theories about the nature of light and the laws of its distribution in media began to appear in the XVII century. One of them was the wave theory of Christian Huygens, a Dutch scientist whose contribution to the development of modern concepts of light is significant.
Huygens believed that light is a wave, like a sound. Like sound, for its existence it needs an environment. The scientist considered ether to be this medium. Using the analogy with sound, Huygens was able to understand how the movement of a light wave in the ether occurs. This movement is fully described by the so-called Huygens principle.
Suppose that there is some source of plane waves. This means that the wave surface (the surface of one phase) is a plane. Having reached an arbitrary part of space, each point of the wave front excites the local region of the ether so that it becomes the source of a new wave. The totality of the surfaces of new or secondary waves determines the resulting surface of the front during its movement.
The idea of excitation by each point of the front of the secondary waves is called the Huygens principle (see figure above).
Huygens principle and front of spherical and plane waves
We show, using the Huygens principle, how to construct wave surfaces for a front of arbitrary shape. For simplicity, we consider two types of front: flat and spherical (see the figure below).
Suppose that light moves in a homogeneous medium in all directions equally with speed c. The left figure shows the AA 'front at some zero point in time. We determine its shape at time Δt. Since each point of the front generates secondary waves, in the time Δt they will cover the distance c * Δt. If we draw circles of this radius from each of these sources (three points on the front AA '), then the tangent to the circles determines the front at time Δt (BB').
Now consider the spherical front in the right figure. We do exactly the same as in the previous case: select a series of points on a spherical front, draw circles of the same radius with a center in each of them, draw a tangent to all circles (secondary waves). As you can see, in this case a spherical front is also obtained, but with a larger radius.
Thus, the Huygens principle explains how light waves propagate in media. It is important to understand that Huygens did not use the concepts of amplitude or wavelength. Its principle is an exclusively geometric idea based on an analogy with the movement of sound.
Huygens-Fresnel principle
Many confuse it with the Huygens principle. Consider the Huygens-Fresnel principle.
After the world community of scientists adopted Newton’s corpuscular theory , Huygens’s theory of waves was forgotten. Her new revival is associated with the name of the Frenchman Augustin Fresnel.
In 1801, Thomas Jung conducted his famous experiments with two slits and received a diffraction picture from a monochromatic light source. In this case, the picture was obtained due to two phenomena: diffraction and interference. Fresnel a few years later used Jung's experimental data and Huygens' wave theory to explain the phenomena of interference and diffraction.
The formulation of the Huygens-Fresnel principle, in fact, does not differ from that for the Huygens principle, but the latter is a geometric number, Fresnel introduced physical meaning into it. Each point of the front is a source of secondary waves, the amplitude of which decreases inversely with the distance from the source. To determine the resulting oscillation at a given point, it is necessary to consider the interference of all the secondary waves of the moving front.
The formula for the resulting oscillation at an arbitrary point
From the Huygens-Fresnel principle, the formula for the amplitude of oscillations at any point that is ahead of the wave front is obtained quite easily.
Imagine some wavefront of arbitrary shape. Each of its elements with area dS generates a secondary wave. At an arbitrary point P located in front of the front, an elementary wave arrives with the following amplitude:
dA = K * a 0 * dS / r * cos (ω * tk * r + α 0 ).
Here r is the distance from the element dS to the point P, k is the wave vector, ω is the oscillation frequency, α 0 is the initial phase, a 0 is the amplitude of the secondary wave of the element dS, the coefficient K depends on the angle between the direction to the point P and the perpendicular to site dS.
If we take the integral sum over all elementary waves of the front with area S, then we get the desired oscillation at point P, that is:
A = ∫ S K (Φ) * a 0 / r * cos (ω * tk * r + α 0 ) * dS.
The coefficient K (Φ) will take its maximum value for Φ = 0 and will be equal to zero for Φ = 90 o .
Thus, when considering an arbitrary wave front, the Huygens-Fresnel principle allows one to obtain the resulting oscillation at any point.
The concept of Fresnel zones
The formula given in the previous paragraph is quite complicated from the point of view of mathematics. Even for a front of a spherical shape, the calculation of the reduced integral is a cumbersome task. However, Fresnel showed that it can be replaced by a simple sum if the wave front is divided into separate zones in the form of a circle. Moreover, the thickness of each zone will be such that the distance from its edge to the considered point (receiver) differs from the same distance for the previous zone by half the wavelength.
If the spherical front moves along a certain direction, then from the source to the receiver its shape will look like a ball stretched in one direction. The figure below shows the Fresnel zones and the wavefront between the source T and receiver R.
Using Fresnel zones to calculate wave amplitude
Using Fresnel zones is very convenient. Since each of them, according to the Huygens-Fresnel principle, emits secondary waves, then they come in antiphase from two neighboring zones to the receiver. That is, neighboring zones weaken each other. As a result of simple reasoning and calculations, it can be shown that the total amplitude of the wave that arrives at the receiver will be equal to half of the first or central Fresnel zone.
This amplitude can be increased if you close all the odd or even even zones using a special plate called the zone.
Wave diffraction
In the paragraph above, it was said that the overlapping of certain Fresnel zones allows you to increase the intensity of the light wave at the receiving point. The zone plate that is used for this is a kind of diffraction grating with gaps in the form of circles.
The Huygens-Fresnel principle allows the phenomenon of diffraction to be explained and mathematically described. The obtained formulas for the Fraunhofer and Fresnel diffraction by a slit, a lattice, a round hole, and other obstacles are the result of applying this principle.
Strict mathematical solution for the diffraction phenomenon
The Huygens-Fresnel principle was developed at the beginning of the 19th century. In the 60s of the same century, James Maxwell got his theory of electromagnetism. It is precisely its equations that must be used for the rigorous solution of the diffraction phenomenon.
As for the principle considered in the article, it uses a number of approximations that simplify mathematical calculations, and at the same time allow us to obtain results suitable for practical use.