today imaging with coherent lightweb.mit.edu/2.710/fall06/2.710-wk10-b-ho.pdfillumination x x ′′...

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MIT 2.71/2.710 Optics11/09/05 wk10-b-1

Today

• Coherent image formation– space domain description: impulse response– spatial frequency domain description: coherent transfer

function

Imaging with coherent light

MIT 2.71/2.710 Optics11/09/05 wk10-b-2

Coherent imaging with a telescope

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MIT 2.71/2.710 Optics11/09/05 wk10-b-3

The 4F system1f 1f 2f 2f

Fourier transformrelationship

Fourier transformrelationship

MIT 2.71/2.710 Optics11/09/05 wk10-b-4


( ){ }{ } ( )yxgyxg −−=ℑℑ ,,Theorem:

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MIT 2.71/2.710 Optics11/09/05 wk10-b-5


( )yxg ,1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

111 ,

fy

fxG

λλ ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′− y

ffx

ffg

2

1

2

11 ,

object planeFourier plane Image plane

MIT 2.71/2.710 Optics11/09/05 wk10-b-6


( )yxg ,1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

111 ,

fy

fxG

λλ ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′− y

ffx

ffg

2

1

2

11 ,

object planeFourier plane Image plane

( )yx ssG ,1

θx

λθλθ

yy

xx

s

s

sin

sin

=

=

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MIT 2.71/2.710 Optics11/09/05 wk10-b-7

The 4F system with FP aperture1f 1f 2f 2f

( )yxg ,1⎟⎠⎞

⎜⎝⎛ ′′

×⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′Rr

fy

fxG circ,

111 λλ ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛′−′−∗ y

ffx

ffhg

2

1

2

11 ,

object planeFourier plane: aperture-limited Image plane: blurred

(i.e. low-pass filtered)

( )vuG ,1

θx

MIT 2.71/2.710 Optics11/09/05 wk10-b-8

Impulse response & transfer function

A point source at theinput plane ...

... results not in a point imagebut in a diffraction pattern

h(x’,y’)

Point source at the origin ↔ delta function δ(x,y)h(x’,y’) is the inpulse response of the system

More commonly, h(x’,y’) is called theCoherent Point Spread Function (Coherent PSF)

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MIT 2.71/2.710 Optics11/09/05 wk10-b-9

Coherent imaging as a linear, shift-invariant system

Thin transparency( )yxt ,

( )yxg ,1

( ) ),( ,),(

1

2

yxtyxgyxg

==

(≡plane wave spectrum) ( )vuG ,2

impulse response

transfer function

( )),(),(

,

2

3

yxhyxgyxg∗=

=′′

Fourier transform

),(),(),(

2

3

vuHvuGvuG

==

output amplitude

convolution

multiplication

Fourier transform

illumination

transfer function H(u ,v): aka pupil function

MIT 2.71/2.710 Optics11/09/05 wk10-b-10

Transfer function & impulse response of circular aperture

Transfer function:circular aperture

Impulse response:Airy function

⎟⎠⎞

⎜⎝⎛ ′′Rrcirc ⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′

2

jincfRr

λ

2R Rf222.1 λ

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MIT 2.71/2.710 Optics11/09/05 wk10-b-11



( )yxg ,1

(≡plane wave spectrum)

impulse response

transfer function

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′=

2

jinc),(fRryxh

λ

( )),(),(

,

2

3

yxhyxgyxg∗=

=′′

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

RvufvuH

22

circ, λ

Fourier transform

output amplitude

convolution

multiplication

Fourier transform

illumination

( ) ),( ,),(

1

2

yxtyxgyxg

==

( )vuG ,2 ),(),(),(

2

3

vuHvuGvuG

==

Example: 4F system with circularcircular aperture @ Fourier plane

MIT 2.71/2.710 Optics11/09/05 wk10-b-12

Transfer function & impulse response of rectangular aperture

a af22 λ

⎟⎠⎞

⎜⎝⎛ ′′

⎟⎠⎞

⎜⎝⎛ ′′

by

ax rectrect ⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′⎟⎟⎠

⎞⎜⎜⎝

⎛ ′

22

sincsincfby

fax

λλ

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MIT 2.71/2.710 Optics11/09/05 wk10-b-13



( )yxg ,1

(≡plane wave spectrum)

impulse response

transfer function

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′⎟⎟⎠

⎞⎜⎜⎝

⎛ ′=

22

sincsinc),(fby

faxyxh

λλ

( )),(),(

,

2

3

yxhyxgyxg∗=

=′′

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

bfv

afuvuH λλ rectrect,

Fourier transform

output amplitude

convolution

multiplication

Fourier transform

illumination

( ) ),( ,),(

1

2

yxtyxgyxg

==

Example: 4F system with rectangularrectangular aperture @ Fourier plane

( )vuG ,2 ),(),(),(

2

3

vuHvuGvuG

==

MIT 2.71/2.710 Optics11/09/05 wk10-b-14

Aperture–limited spatial filtering

1f 1f 2f 2f

object plane:grating generates

one spatial frequencyFourier plane:

aperture unlimited

Image plane:grating is imaged

with lateralde-magnification

(all orders pass)

x ′′ x′x

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MIT 2.71/2.710 Optics11/09/05 wk10-b-15

x ′′ x′x

Aperture–limited spatial filtering

1f 1f 2f 2f

object plane:grating generates

one spatial frequencyFourier plane: aperture limited

Image plane:grating is not imaged;

only 0th order(DC component)

surviving(some orders cut off)

MIT 2.71/2.710 Optics11/09/05 wk10-b-16

Spatial frequency clipping

( ) ( )[ ] ( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ ++−+=⇒+= 00in0in 2

121

212cos1

21 uuuuuuGxuxg δδδπ

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ +′′+−′′+′′=′′− 0101f 2

121

21 vfxufxxxg λδλδδfield before filter

field after filter ( ) ( )xxg ′′=′′+ δ21

f

( ) ( ) ( )21

21

outout =′⇒= xguuG δfield at output(image plane)

field after input transparency

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MIT 2.71/2.710 Optics11/09/05 wk10-b-17

Effect of spatial filtering

Original object(sinusoidal spatial variation,

i.e. grating)

Frequency-filtered image(spatial variation blurred out,

only average survives)

Fourier plane filterwith circ-aperture

MIT 2.71/2.710 Optics11/09/05 wk10-b-18


object planetransparency

Fourier planecirc-aperture

1f 1f

monochromaticcoherent on-axis

illumination

x ′′x x′1f 1f

f1=20cmλ=0.5µm

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MIT 2.71/2.710 Optics11/09/05 wk10-b-19

Space-Fourier coordinate transformations

δx :pixel size

∆x :field sizespace

domainspatial

frequency domain

δfx :frequencyresolution

fx,max

xf

xf xx ∆

==2

1 21

max, δδ

Nyquistrelationships:

MIT 2.71/2.710 Optics11/09/05 wk10-b-20

4F coordinate transformations

δx :pixel size

∆x :field sizespace

domainFourierplane

δx”

xfx

xfx

∆=′′=′′

2

2maxλδ

δλNyquist

relationships:

x”max

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MIT 2.71/2.710 Optics11/09/05 wk10-b-21




Image planeobserved field

1f 1f


illumination

x ′′ x′x1f 1f

f1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/09/05 wk10-b-22

x ′′ x′x

Formation of the impulse response

1f 1f 2f 2f

object plane:pinhole generates

spherical waveFourier plane:

circ-aperture limited

Image plane:Fourier transform

of aperture,Airy pattern

(plane wave is clipped)

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MIT 2.71/2.710 Optics11/09/05 wk10-b-23

Low–pass filtering

( ) ( ) ( ) ( ) 1,, inin =⇒= vuGyxyxg δδ

( ) 1,f =′′′′− yxgfield before filter

field after filter ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′′+′′=′′′′+ R

yxyxg

22

f circ,

( )

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′+′∝′′

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

fyxR

yxg

RvufvuG

λ

λ

22

out

22

out

jinc,

circ,field at output(image plane)

field after input transparency

(Airy pattern)

ℑ

MIT 2.71/2.710 Optics11/09/05 wk10-b-24

Effect of spatial filtering

Original object(small pinhole ⇔ impulseimpulse,generating spherical wave

past the transparency)

Impulse reponse(aka pointpoint--spread functionspread function,original point has blurred to

an Airy pattern, or jinc)

Fourier plane filterwith circ-aperture

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MIT 2.71/2.710 Optics11/09/05 wk10-b-25

Low–pass filtering the impulse



1f 1f


illumination

x ′′x x′1f 1f

f1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/09/05 wk10-b-26





1f 1f


illumination

x ′′ x′x1f 1f

f1=20cmλ=0.5µm

note: pseudo-accentuatedsidelobes

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MIT 2.71/2.710 Optics11/09/05 wk10-b-27

Low-pass filtering with the 4F system

( )yxg ,in

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′′+′′×⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′Ryx

fy

fxG

22

11in circ,

λλ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′−∗ y

ffx

ffg

2

1

2

1in , jinc




1f 1f 2f 2f

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

11in ,

fy

fxG

λλ

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ′′+′′=′′′′

Ryx

yxH22

circ,

field arrivingat Fourier plane


illumination

field departingfrom Fourier plane

ℑℑFourier

transformFourier

transform

x ′′ x′x

MIT 2.71/2.710 Optics11/09/05 wk10-b-28

Spatial filtering with the 4F system

( )yxg ,in

( )yxHfy

fxG ′′′′×⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′′′′,,

11in λλ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′−∗ y

ffx

ffhg

2

1

2

1in ,


Fourier planetransparency


1f 1f 2f 2f

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

11in ,

fy

fxG

λλ

( )yxH ′′′′ ,

field arrivingat Fourier plane


illumination

field departingfrom Fourier plane

ℑ

ℑFourier

transformFourier

transform

x ′′ x′x

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MIT 2.71/2.710 Optics11/09/05 wk10-b-29

Examples: the amplitude MIT pattern

MIT 2.71/2.710 Optics11/09/05 wk10-b-30

Weak low–pass filtering

Fourier filter Intensity @ image planef1=20cmλ=0.5µm

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MIT 2.71/2.710 Optics11/09/05 wk10-b-31

Moderate low–pass filtering

Fourier filter Intensity @ image plane

(aka blurringblurring)

f1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/09/05 wk10-b-32

Strong low–pass filtering


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MIT 2.71/2.710 Optics11/09/05 wk10-b-33

Moderate high–pass filtering


MIT 2.71/2.710 Optics11/09/05 wk10-b-34

Strong high–pass filtering

Fourier filter Intensity @ image plane

(aka edge enhancementedge enhancement)

f1=20cmλ=0.5µm

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MIT 2.71/2.710 Optics11/09/05 wk10-b-35

1-dimensional blurring


MIT 2.71/2.710 Optics11/09/05 wk10-b-36

1-dimensional blurring


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MIT 2.71/2.710 Optics11/09/05 wk10-b-37

Phase objects

glass plate(transparent)

protruding partphase-shifts

coherent illuminationby amount φ=2π(n-1)t/λ

thicknesst

Often useful in imaging biological objects (cells, etc.)

MIT 2.71/2.710 Optics11/09/05 wk10-b-38

Viewing phase objects

Intensity(object is invisible)

Amplitude(need interferometer)

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MIT 2.71/2.710 Optics11/09/05 wk10-b-39

Zernicke phase-shift mask

π/25mm

1mm

MIT 2.71/2.710 Optics11/09/05 wk10-b-40

Imaging with Zernicke mask


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MIT 2.71/2.710 Optics11/09/05 wk10-b-41

Imaging with Zernicke mask


MIT 2.71/2.710 Optics11/09/05 wk10-b-42

Coherent imaging with a single lens

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MIT 2.71/2.710 Optics11/09/05 wk10-b-43

Single-lens imaging condition

ss’

object

image

lens

fss111 =

′+ Imaging condition

(aka Lens Law) Derivation usingwave optics ?!?

ssm′

−=lateral Magnification

MIT 2.71/2.710 Optics11/09/05 wk10-b-44

Single-lens imaging system

ss’

object

image

lens

spatialspatial“LSI” system“LSI” system

gin(x,y) gout(x’,y’)

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MIT 2.71/2.710 Optics11/09/05 wk10-b-45

Single-lens imaging systemImpulse response (PSF)

spatialspatial“LSI” system“LSI” system

gin(x,y) gout(x’,y’)

Ideal PSF: ( ) ( ) ( )myymxxyxyxh −′−′=′′ δδ,;,

Diffraction--limitedPSF:

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −′′

+⎟⎠⎞

⎜⎝⎛ −′′

=′′22

jinc,;,sy

sy

sx

sxRyxyxh

λ

today imaging with coherent lightweb.mit.edu/2.710/fall06/2.710-wk10-b-ho.pdfillumination x x ′′...

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