Diffraction of light and conservation of energy.

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SUMMARY

The discussion centers on the diffraction of light through an infinitesimally small opening and its implications for the conservation of energy. It is established that while a single ray of light with intensity ##I_0## can diffract into multiple rays, the total intensity remains consistent due to the inverse square law governing light intensity. The application of the Kirchhoff Integral, particularly for large circular holes, and the work of Hans Bethe in 1944 clarifies that the total radiation from small holes is significantly less than predicted by the Kirchhoff Integral, specifically reduced by a factor of ##k^2 a^2##. This indicates that intensity decreases with distance and that small holes diffract even less power than larger ones.

PREREQUISITES
  • Understanding of light diffraction principles
  • Familiarity with the Kirchhoff Integral
  • Knowledge of Poynting Vector and its applications
  • Basic concepts of Bessel Functions and their multiplication theorem
NEXT STEPS
  • Study the Vectorial Kirchhoff Integral in detail
  • Explore Hans Bethe's 1944 paper on diffraction and its implications
  • Learn about the Bessel Function Multiplication Theorem
  • Investigate Rayleigh Scattering and its relation to diffraction
USEFUL FOR

Physicists, optical engineers, and students studying wave optics and light behavior in various mediums will benefit from this discussion.

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