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

Seite 1 von 1

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