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Abbe Schott
Where was modern optical imaging technology born?
Jena
Zeiss
35. Diffraction and Image Formation
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
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!
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?