image formation abbé theory of imaging
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Image FormationErnst Abbé and Carl Zeiss (1866)
71
Fourier Planes
Abbé theory of imaging
72-2
Diffracted orders from high spatial frequencies miss the lens
High spatial frequencies are missing from the image.
θmax defines the numerical aperture… and resolution
Limited Resolution
73
Coherent Illumination ➙ Abbé
object: u0(x0) = t(x0) uin(x0)coherent illumination: uin(x0)object transmission: t(x0)
u0(x0) uf(xf)
ui(xi)
} u
f
(xf
) =1
f
Zu0(x0)e
ix0kxf/fdx0
Fourier transforms
|ui(xi)| / |TF (uf )|
75
TF (uf ) ! ui(xi)
Wavefront Preservation
f1 f1 f2 f2
u0(x0) uf(xf)ui(xi)
sequence oftransformations
TF
(uo
) ! u
f
(xf
)
TF
(TF
(uo
)) / u
i
(xi
) =f1
f2u0(�x
i
f1/f2)
76-1
TF (uf ) ! ui(xi)
Wavefront Preservation
f1 f1 f2 f2
u0(x0) uf(xf)ui(xi)
wavefront
preserved
sequence oftransformations
TF
(uo
) ! u
f
(xf
)
TF
(TF
(uo
)) / u
i
(xi
) =f1
f2u0(�x
i
f1/f2)
76-2
Optical Point-Spread Functionor how to form an image
77
Optical Point-Spread Function
f1 f1 f2 f2
u0(x0) uf(xf)ui(xi)
?H(xf)
Imperfect, filtering function H(xf) acting on uf(xf) convolution
h(xi)
ui(xi) =f1
f2
Zu0(�x
0if1/f2)h(xi � x
0i)dx
0i
PSF
coherent illumination
TF (uf ) = ui(xi)
ui(xi) / TF [uf (xf ) ·Hf (xf )] = TF [uf ]⌦ TF [Hf ]
78
Optical Point-Spread Function
f1 f1 f2 f2
u0(x0) uf(xf)ui(xi)
H(xf)
ui(xi) =f1
f2
Zu0(�x
0if1/f2)h(xi � x
0i)dx
0i
coherent illumination
incoherent illuminationIi(xi) /
f
21
f
22
ZI0(�x
0if1/f2)|h(xi � x
0i)|2dx0
i
h(xi) = TF [Hf ](kxi/f2)
79
Notes on Fraunhofer Diffraction
f1 f1 f2 f2
u0(x0) uf(xf)ui(xi)
H(xf)
ui(xi) =f1
f2
Zu0(�x
0if1/f2)h(xi � x
0i)dx
0i
coherent illumination
incoherent illuminationIi(xi) /
f
21
f
22
ZI0(�x
0if1/f2)|h(xi � x
0i)|2dx0
i
single
sourceu0 = �(x0)
I0 = �(x0)
Ii(xi) / |h(xi)|2
80
Fourierplane
Image plane
Optical Image Processing
81-6
Optical Image Processing
82
a b
a’ b’
(a) and (b) show objects: double helix
at different angle of view
Diffraction patterns of (a) and (b) observed in
Fourier plane
Computer performs Inverse Fourier transform
To find object “shape”
X-Ray Diffraction
83-2
FourierPlane
Schlieren Photography
phase → amplitude modulation
84-2
Schlieren Photography
85
Near Field RegimeFresnel’s wave propagation
F =a2
�D⇡ 1
86
Fresnel‘s Wave Propagation
Fresnel-Kirchhoff diffraction integral
up = � i
⇤
��(⇥in, ⇥out)
u0
reikrdS
�(⇥in, ⇥out) =12(cos ⇥in + cos ⇥out)
obliquity factor
eikr �⇤ eikr0 · ei(�xx+�yy)
Fraunhofer (far field) diffraction is a special case
87
Near Field → Talbot Effect
Near-field diffractionof an optical grating
zT = 2d2/�
self-imaging at
88
Phase difference of π at edge of 1st HPZ
Fresnel‘s Theory of Wave Propagation
z
z⇢⇡,n =
p�nz
⇢2⇡2z
=�
2
89-1
Phase difference of π at edge of 1st HPZ
Fresnel‘s Theory of Wave Propagation
z
z⇢⇡,n =
p�nz
⇢2⇡2z
=�
2
1
3
57
2
4
6
89-2
First Half Period Zone
Fresnel‘s Theory of Wave Propagation
R� = 2i�u0⇥
phasor addition
z
(z + �/2)
90
n→∞ ⇒ resultant → ½ diameter of 1st HPZ
Fresnel‘s Theory of Wave Propagation
R� = i�u0⇥!= u0
91
Fresnel Zone Plate24.11.08 01:40http://upload.wikimedia.org/wikipedia/commons/9/97/Zone_plate.svg
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mask out every second HPZ
in every transparent
zone, the phase is running from
0 to π
acting asa focussing
lens
92-2
Fresnel Lens
phase jump by π from HPZ to HPZ
1. HPZ
2. HPZ
3. HPZ4. HPZ5. HPZ
sub-division into HPZ
z
(z + �/2)
93-3
Fresnel Lens
phase jump by π from HPZ to HPZ
section of a lens in every HPZ
nearly perfect focussing lens
1. HPZ
2. HPZ
3. HPZ4. HPZ5. HPZ
93-4
Poisson versus Fresnel
particles waves
94-1
Poisson versus Fresnel
particles waves
94-2
Poisson versus FresnelFrançois
Arago
Poisson Spot
94-5