Diffraction of light and conservation of energy.

[amjad-sh]

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?

 

[PeroK]

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

From your imagination.

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.

 

[Dale]

But doesn't this violate the law of conservation of energy?

Yes, so clearly your supposition that this is what happens is incorrect.

 

[PhDeezNutz]

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)