Waves

Light in darkness

Fresnel zones plate and diffraction

Let us consider the simplest case - the propagation of a plane wave in a vacuum (without obstacles). Applying the Huygens-Fresnel principle, let's look into the details...

A plane wave propagates from left to right. Consider two vertical planes (Р₁ and P₂) parallel to the wave front and located far enough from each other (the distance is many times greater than the wavelength). According to the Huygens-Fresnel principle, each point of the front in the first (Р₁) plane "emits" a spherical wave. How do these - secondary - waves add up at some point of the second plane (say, at point N)?

Let's discuss. Distances from different points of the first plane to point N are different. So you can always find a pair of points such that the path difference from them to N will be half the wavelength. Moreover, for any point of the first plane one can find such a "pair". Hmm! Does this mean that in N all secondary waves will cancel each other? But this is contrary to reality!

Let's not rush, but think. Let's start with the shortest distance from the first plane to N:

L₀ = ОN.

Now let's find the points, the distance from which to N is half the wavelength greater:

Fig.1. Selection of points in the plane P1 at a given distance from the observation point N.

Obviously, these points form a circle on the first plane with the center at the point O. The radius of the circle (according to the Pythagorean theorem):

Since the distance L₀ is many times greater than the wavelength, the term

can be neglected. Then we get:

Note that waves from points inside the circle do not cancel each other at the point N (because the path difference is less than half the wavelength). Let's denote their total amplitude as А₁.

Now we choose points, the distance from which to N is equal to

It will be a circle of radius r₂ > r₁. The points inside this circle (with the exception of the points of the first circle) form an annular region on the P1 plane (centered at O). The distances from these points to N differ from each other by less than half wavelength. This means that the waves from the points of this ring again do not cancel each other at N. Let us denote the total amplitude of the waves from the ring as А₂.

Fig.2. Construction of ring regions with increasing distance to the observation point.

However, sum of waves from the first region (circle disc) and sum of waves from the ring have the path difference (to point N) around half the wavelength.

That is, the contributions of the waves from the first and second regions at point N have different signs.

Next, one can build a third, fourth, etc. ring regions, the points of which are from N at a distance less than

Lo + the corresponding number of half wavelength.

The contributions of the waves from the regions will alternately change sign (due to the difference in distances by half the wavelength). So the sum of all contributions from all (secondary) waves from the plane P₁ at N is:

А₁ - А₂+А₃ - А₄+А₅ - А₆+ ...

The "power" radiated by each region depends on its area. The area of the first region (circle) is

Ring area is

Here r_n is the outer radius of area number n.

According to the Pythagorean theorem (see Fig.1, 2)

Here

Since the distance L₀ is many times greater than the wavelength, the term

can be neglected.

Thus,

Then the area of the circle (of the first region) is:

The area of ring number n is:

And it turns out that the areas of all regions are (approximately) equal to each other.

At the same time, the distances from each subsequent region to the observation point N increase.
The angles at which the waves come to N from different areas are also changes. This leads to a decrease in the impact of (spherical) waves as the areas move away from N.

Thus, in the (infinite) sum А₁ - А₂+А₃ - А₄+А₅ - А₆+ … each subsequent term is less than the previous one in absolute value. And this means that the sum is not equal to zero.
Already not bad!

If we take a closer look at the infinite sum and think...
What happens if we cover all areas, starting with the second one, with a screen and leave only a round hole corresponding to the first region?...
Then only the first term remains from the infinite sum. And nothing is subtracted from it. This means that the intensity of light at point N will increase! Most of the light flux was blocked, but we got more light at N?!?!
Error? No!

This is true. At point N, the light intensity will increase. However, at other - distant from N - points of the second plane will become darker. The law of conservation of energy has not gone away. But for one point N, the result is amazing! And it is impossible to predict it without using the Huygens-Fresnel principle.

Even greater effect can be obtained by covering all even areas. Then only terms with one sign will remain in the infinite sum and the light intensity at N will increase even more! The system of open ring zones will work... like a lens!

Рис.3. Схематическое изображение 10 первых зон Френеля.

Finally, let's note that the "miracles" and "magic" areas described above were invented / discovered by Augustin Jean Fresnel, who developed the theory of diffraction. Since then, these areas are called Fresnel zones, and a transparent plate with opaque even (or odd) areas is called Fresnel zone plate.

Recall that here we were talking about a single wavelength. For different wavelengths (which make up white light), the Fresnel zones are different.

Diffraction

Let's take the intensity of light in a completely open space as 1. What happens if a screen with a small round hole is placed in the path of a plane wave? According to Fresnel's diffraction theory, the intensity of light at a point below the center of the hole will depend on the distance from the screen to the point of observation (on how many Fresnel zones fit in the hole). If the hole corresponds to one zone, then the intensity at the observation point (under the center) is greater than 1. If an even number of zones are placed in the hole, then the intensity will be less than 1.
When the observation point is shifted from the center, the distance to the edges of the hole becomes different. As a result, alternating light and dark rings are formed.

Diffraction on a circular hole for a different number of Fresnel zones.

BTW, not everyone accepted Fresnel's ideas and theory of diffraction straightaway. A funny incident happened with the criticism of the theory of diffraction. Siméon Denis Poisson (critically) showed that, according to Fresnel's theory of diffraction, under certain conditions, there should be a bright point in the very center of the geometric shadow of a circular screen. And since this statement is sounds as absurd, the theory is wrong! But... François Jean Dominique Argo set up a corresponding experiment and... proved that a bright spot appears in the center of the shadow (!) and, thus, Poisson's conclusions only confirm Fresnel's theory.

Poisson's spot in the center of the geometric shadow.

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