DIFFRACTION THEORY

Background:

Diagram of Diffraction Problem :

How does light from source point Q get through an aperture to reach point P?

Time-Independent Electromagnetic Wave Equations:

U = Ae-i (k r - fo)	(Time-independent equation)
	(Helmholz equation)
	(Green's function)
	(Fresnel-Kirchhoff integral)

Diffraction Theory

Historical Development:

• Huygen - Each point on wavefront treated as a point source
• Fresnel - Put Huygen's work in mathematical form, but no explanation of obliquity effect
• Kirchhoff - Treated Fresnel's work as a boundary value problem, introduced obliquity factor
• Sommerfeld - Corrected some errors in Kirchhoff's work pertaining to boundary conditions
  
  

Treating Aperture Effect As a Transmission Function

• Fresnel Integral- Corrects for obliquity, makes simplying assumptions.
• Integral Transforms - Fresnel's equation can be structured as an integral equation with a kernal to handle coordinate transformations
• Fourier Integral Transform - Adds a few more assumptions (especially the assumption stated in the following equation (where w is the width of the aperture):

Using the changes and adaptions listed above, the Fraunhofer approximation to the Fresnel-Kirchhoff integral equation

can be expressed as a Fourier transform

with f = x₂/(lz₂) as the spatial frequency variable, C as an amplitude scaling factor, and the exponential as a phase factor.