# Diffraction of light and conservation of energy.

Suppose that we shined a source of light on a wall with infinitismal small opening. As the opening is infinitismly small, only one ray of light will pass through the opening ( suppose it has an intensity ##I_0##) and this ray of light will diffract into an infinite number of light rays with the same intensity ##I_0##. What we will see on the screen is a bright fringe with intensity ##I_0## and has an infinite width. But doesn't this violate the law of conservation of energy? From where all the other infinite number of light rays come from?

PhDeezNutz
From where all the other infinite number of light rays come from?

An intense beam diffracts into a wave-front whose average intensity reduces with distance from the source of the diffraction: just like the intensity of any light reduces with distance from the source.

But doesn't this violate the law of conservation of energy?
Yes, so clearly your supposition that this is what happens is incorrect.

Are you familiar with the Kirchhoff Integral?

If you carry out the Vectorial Kirchhoff Integral for a large circular hole (compared to wavelength) you get something like the following for the Poynting Vector

##\vec{S} \left(r , \theta, \phi \right) = \frac{1}{2} \left( \frac{E_0}{4 Z_0}\right) \left(k^4 a^4 \right)\left( \frac{1}{k^2 r^2}\right) \left[ \frac{2 J_1 \left(k a \sin \theta \right)}{ka \sin \theta}\right]^2 \left( \sin^2 \phi + \cos^2 \phi \cos^2 \theta \right) \hat{r}##

Edit: This is time averaged poynting vector hence the lack of imaginary numbers ##\langle \vec{S} \rangle = \frac{1}{2}\text{Re} \left( \vec{E} \times \vec{H}^*\right)##

As @PeroK pointed out intensity decreases with distance.

The Kirchhoff Integral only works for large ##ka## for small ##ka## as you specified we need a different approach. One of the most important works done on this subject was done in 1944 by none other than Hans Bethe

http://www.physics.miami.edu/~curtright/Diffraction/Bethe1944.pdf

He argues (that after invoking a small argument for ##ka## and applying it to the Kirchhoff Integral) that the actual total radiation for small holes is substantially less than that predicted by the Kirchhoff Integral (after smallness parameter ##ka## is invoked). You can use the so-called "Bessel Function Multiplication Theorem to confirm that the above expression is on the order of ##k^2 a^4##.

He argues (See section comparison with Kirchhoff Integral) that the total radiation of small holes is reduced by a factor of ##k^2 a^2## where again ##ka## is small.

##(\text{Bethe}) \sim k^2 a^2 (\text{Small Kirchhoff})##

Which would make ##\left( Bethe \right) \sim k^4 a^6##, notice that this is the scattering cross section for Rayleigh Scattering (long wavelength short obstacle).

In Summary; Intensity drops off with inverse distance squared and small holes (compared to wavelength) diffract even less power than large holes (compared to wavelength)

## What is diffraction of light?

Diffraction of light is the phenomenon in which a beam of light bends or spreads out when it passes through an opening or around an obstacle. This is due to the wave nature of light, where the light waves interfere with each other as they pass through a narrow opening.

## How does diffraction of light relate to conservation of energy?

According to the law of conservation of energy, energy cannot be created or destroyed, only transformed from one form to another. In the case of diffraction of light, the energy of the light beam is conserved, but it is transformed from a concentrated beam to a spread out beam as it passes through an opening or around an obstacle.

## What factors affect the amount of diffraction in light?

The amount of diffraction that occurs in light is affected by the wavelength of the light, the size of the opening or obstacle, and the distance between the light source and the opening/obstacle. Longer wavelengths, smaller openings, and shorter distances result in more diffraction.

## Can diffraction of light be observed in everyday life?

Yes, diffraction of light can be observed in everyday life. Some examples include the colorful patterns seen when light passes through a CD or DVD, the blurring of light around the edges of a doorway, and the rainbow effect seen through a prism.

## What practical applications does diffraction of light have?

Diffraction of light has many practical applications, including in the fields of optics, photography, and telecommunications. It is used to create diffraction gratings, which are used to split light into its component wavelengths. It is also utilized in laser technology, where diffraction of light is used to create precise patterns and holograms. In telecommunications, diffraction of light is used to bend and focus light signals in fiber optic cables.