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