diffraction diffraction is the bending and spreading of waves from apertures or obstructions,...
TRANSCRIPT
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