Abbe Schott

Where was modern optical imaging technology born?

Jena

Zeiss

35. Diffraction and Image Formation

ff

Geometrical Optics…

…implies perfect resolution.

point images

point sources

Physical Optics…

diffracting source Imperfect image

Every lens is a diffracting aperture.

2

1

2212

2212

sin2212

2212

sinN

p

bap

bap

ksjbap

bap

ksjtkrjLP dsedsee

rE

E

a

b

b

a

b

a

b

r

Multiple Slits

Principle maxima

secondary maxima

Central maximum

Diffraction Grating

A special corner of multi-slit-space: N ~ 104, a ~ , b ~

N ~ 104: Principle maxima are very narrow! Secondary maxima are very low!

a ~ : principle maxima are highly separated!(most don’t exist)

b ~ : central maximum is very large!

typical grating specs: 900 g/mm, 1 cm grating.

N = 9,000

a = 1.11 microns

b = 1.11 microns

= 0.633 microns!

ma sin

sina

Maxima at:

... 3, 2, 1, ,0m

m = 0

m = 1“first order”

monochromatic light

grating

Abbe Theory of Image Formation

grating

m = 0

m = +1

m = -1

diffraction plane

focal plane

Abbe Theory of Image Formation

grating

m = 0

m = +1

m = -1

diffraction plane

Resulting interference pattern is the image

focal plane

m = 0

m = +1

Image formation requires a lens large enough to capture the first order diffraction.

a

f

D

ma sinGrating Equation:

To resolve a:

afD

2sin

Resolution (diffraction limited):

Df

a2

Rectangular Apertures

b krtjAP e

rdAE

dE

P(X,Y,Z)

R

r

dAeeRE

Eaperture

RYyXxjkkRtjAP

a

dA(x,y,z)

Rather than an aperture, consider an object:

dAeEeR

Eaperture

RYyXxjkFeynman

kRtjP 1

dxdyeEeR

Eaperture

RYyXxjkFeynman

kRtjP 1

Remember, the integral is over the aperture area:

Let’s rearrange that a little it (this is where the magic happens):

dxdyeEeR

Eaperture

yR

kYx

R

kXj

FeynmankRtj

P

1

THAT’S A FOURIER TRANSFORM!!

EP(X,Y,Z) = F{EFeynman}

RkX

kx R

kYk y

Where does diffraction put the spatial frequencies in EFeynman?