Abbe
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Transcript of Abbe
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?