Diffraction
Diffraction is the bending and spreading of waves from apertures or obstructions, including interference of the waves.
( , , ) ( ( ', ',0)E x y z f E x y
Diffraction for increasing screen distance
Aperture 200x100 /2l p
z screen20 /2l p
z screen100 /2l p
z screen 500 /2l p
z screen 2500 /2l p
Looks like the aperture with fringes! (Fresnel)
“Far field” looks like |FT|2 of aperture!(Fraunhoffer diffraction)
Diffraction
How could we solve with no approximations?
( , , ) ( ( ', ',0)E x y z f E x y
22
20o o
EE
t
plus boundary conditions.
…but there are easier approximations!
Huygens’ principle 1678
Every point on a wavefront acts like a “forward spherical” scalar source.
Conceptual tool: gave Snell’s law, finds diffraction maxes, mins
1 2i kr te
r zr
( )
cos , ˆ
Fresnel’s update --- make it formal:
Obeys a scalar wave equation
Helmholtz equation
i kr te
r
( )
2 2 0E r k E r
22
20o o
EE
t
vs
Works when: essentially single frequency E doesn’t change significantly over a distance of l Forget about polarization
Hard to solve
(if we further required small l, we’d get the Eikenol equation…then no diffraction)
Fresnel-Kirchoff diffraction formula
0 1 2i kr
aperture
eE x y z C E x y r z dx dy
R x y z
( )
, , , , cos , ' 'ˆ( ', ', ')
Kirchhoff found the factor:
Put on firm math foundation with Green’s theorem and Helmholtz equation
Fresnel’s diffraction model: add these Huygen waves…it works pretty well!
iC
i meaning?
Fresnel approximation
Becomes: (know how to do this step with small angle/binomial approx’s)
2 2
2 2220
ki x y k kikz z i x y i xx yy
z z
aperture
ie eE x y z E x y e e dx dy
z
, , , ,
0 1 2i kr t
aperture
i eE x y z E x y r z dx dy
r
( )
, , , , cos , ' 'ˆ
restrictions:a (size of aperture) > l [scalar wave approx]z of screen > a (but if get far enough, becomes simpler Fraunhofer)
x,y of screen <<z, so angles on screen are small
Aperture 200x100 /2l p
z screen20 /2l p
z screen100 /2l p
z screen 500 /2l p
z screen 2500 /2l p
Looks like the aperture with fringes! (Fresnel diffraction)
“Far field” looks like |FT|2 of aperture!(Fraunhoffer diffraction)
Diffraction for increasing z, using Fresnel equations
Fresnel diffraction for slit, increasing z
Babinet’s principle for all diffraction patterns
Complimentarity principle
The diffraction pattern for an aperture is similar (but not identical) to the pattern for a block of the same shape
The principle describes the fields, not intensities
Circular hole diffractiona = 1 to 4 mm, screen 1 meter away, HeNe light
Center alternates bright/dark
Complimentarity principle
Center is always bright…similar but not identical
Poisson’s spot in shadow of ball bearing
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