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amjad-sh

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- Thread starter amjad-sh
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In summary, shining a source of light on a wall with an infinitesimally small opening will result in the diffraction of an intense beam into an infinite number of light rays with the same intensity. However, this violates the law of conservation of energy and it is incorrect to assume that this is what happens. The Kirchhoff Integral and the Bessel Function Multiplication Theorem provide a more accurate explanation, showing that intensity decreases with distance and small holes diffract even less power than large holes due to destructive interference.

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amjad-sh

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From your imagination.amjad-sh said: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.

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Dale

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

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PhDeezNutz

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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)

Diffraction of light is the phenomenon where light waves bend and spread out as they pass through a narrow opening or around an obstacle. This results in the formation of a pattern of bright and dark areas, known as a diffraction pattern.

Diffraction of light occurs due to the wave nature of light. When light passes through a narrow opening or around an obstacle, it interacts with the edges and bends, resulting in the diffraction pattern.

The conservation of energy is a fundamental law of physics that states that energy cannot be created or destroyed, but can only be transformed from one form to another. This means that the total energy in a closed system remains constant over time.

The conservation of energy is related to diffraction of light as the energy of the light waves remains constant as they pass through the narrow opening or around an obstacle, even though the direction and intensity of the light may change.

Diffraction of light and conservation of energy have numerous real-life applications, such as in the design of optical devices like microscopes and telescopes, in the production of holograms, and in the study of crystal structures. They are also important concepts in the fields of optics, physics, and engineering.

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