coherent imaging - mitweb.mit.edu/2.710/fall06/2.710-wk11-a-sl.pdf · coherent imaging as a linear,...

MIT 2.71/2.710 Optics11/14/05 wk11-a-1

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(sx ,sy): aka pupil function

MIT 2.71/2.710 Optics11/14/05 wk11-a-2

Spatial filtering with the 4F system

( )yxg ,in

( )yxHf

yf

xG ′′′′×⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′,,

11in λλ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′−∗ y

ffx

ffhg

2

1

2

1in ,

object planetransparency

Fourier planetransparency

Image planeobserved field

1f 1f 2f 2f

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

11in ,

fy

fxG

λλ

( )yxH ′′′′ ,

field arrivingat Fourier plane

monochromaticcoherent on-axis

illumination

field departingfrom Fourier plane

ℑ

ℑFourier

transformFourier

transform

x ′′ x′x

MIT 2.71/2.710 Optics11/14/05 wk11-a-3

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

MIT 2.71/2.710 Optics11/14/05 wk11-a-4

Imaging with incoherent light

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Two types of incoherence

d1 d2

Michelson interferometer Young interferometer

1r1r

2r

spatialspatialincoherenceincoherence

temporaltemporalincoherenceincoherence

pointsource

matchedpaths

poly-chromatic light(=multi-color, broadband)

mono-chromatic light(= single color, narrowband)

MIT 2.71/2.710 Optics11/14/05 wk11-a-6

Two types of incoherence

d1 d2

1r1r

2r

spatialspatialincoherenceincoherence

temporaltemporalincoherenceincoherence

pointsource

matchedpaths

waves from unequal pathswaves from unequal pathsdo not interferedo not interfere

waves with equal pathswaves with equal pathsbut from different pointsbut from different points

on the wavefronton the wavefrontdo not interferedo not interfere

MIT 2.71/2.710 Optics11/14/05 wk11-a-7

Coherent vs incoherent beamsMutually coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes

1e11φiaa =

2e22φiaa =

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Mutually incoherent: superposition field intensityintensityis described by sum of intensitiessum of intensities1I

21 III +=

2I (the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly — statistical optics)

MIT 2.71/2.710 Optics11/14/05 wk11-a-8

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

simple object: two point sourcesnarrowband, mutually incoherent(input field is spatially incoherentspatially incoherent)

MIT 2.71/2.710 Optics11/14/05 wk11-a-9

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

incoherent: adding in intensity ⇒

( ) ( ) ( ) 20

20 xxhxxhxI +′+−′=′

2x0

MIT 2.71/2.710 Optics11/14/05 wk11-a-10

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

Generalizing:thin transparency with

sp. incoherentsp. incoherent illumination

( ) ( ) ( ) xxxhxIxI d 2−′=′ ∫

( )xI

intensity at the outputof the imaging system

MIT 2.71/2.710 Optics11/14/05 wk11-a-11

Incoherent imaging as a linear, shift-invariant system

Thin transparency( )yxt ,

( )yxI ,1

( ) ),( ,),(

1

2

yxtyxIyxI

=

=

incoherentimpulse response ( )

22

3

),(),(

,

yxhyxI

yxI

∗=

=′′

output intensity

convolutionillumination

Incoherent imaging is linear in intensitywith incoherent impulse response (iPSF)

where h(x,y) is the coherent impulse response (cPSF)

( ) 2),(,~ yxhyxh =

MIT 2.71/2.710 Optics11/14/05 wk11-a-12

Incoherent imaging as a linear, shift-invariant system

Thin transparency( )yxt ,

( )yxI ,1

( ) ),( ,),(

1

2

yxtyxIyxI

=

=

(≡plane wave spectrum) ( )vuI ,2̂

incoherentimpulse response

transfer function

( )2

2

3

),(),(

,

yxhyxI

yxI

∗=

=′′

Fourier transform

),(~),(ˆ),(ˆ

2

3

vuHvuI

vuI

=

=

output intensity

convolution

multiplication

Fourier transform

illumination

transfer function of incoherent system: ( )yx ssH ,~optical transfer function (OTF)

MIT 2.71/2.710 Optics11/14/05 wk11-a-13

The Optical Transfer Function

( ) ( ){ }( ) ( )

( )∫∫∫∫

′′′′

′′−′−′′′=

ℑ≡

vuvuH

vuvvuuHvuH

yxhvuH

dd,

dd ,,

1 tonormalized , ,~

2

*

2

umax–umax

1

2umax–2umax

real(H) ( )H~real

1

MIT 2.71/2.710 Optics11/14/05 wk11-a-14

some terminology ...

Amplitude transfer function(coherent)

Optical Transfer Function (OTF)(incoherent)

Modulation Transfer Function (MTF)

( )vuH ,

( )vuH ,~

( )vuH ,~

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MTF of circular aperture

physical aperture filter shape (MTF)f1=20cmλ=0.5µm

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MTF of rectangular aperture

physical aperture filter shape (MTF)f1=20cmλ=0.5µm

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Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

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Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/14/05 wk11-a-19

Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/14/05 wk11-a-20

Diffraction-limited vs aberrated MTF

2umax–2umax

( )H~real

1ideal thin lens,ideal thin lens,finite aperturefinite aperture

realistic lens,realistic lens,finite aperturefinite aperture& aberrations& aberrations

MIT 2.71/2.710 Optics11/14/05 wk11-a-21

Imaging with polychromatic light

( ) ( ) ( ) yxyyxxhyxIyxI dd ;,;,;, 2000 λλλ −′−′=′′ ∫∫

Monochromatic, spatially incoherent responseat wavelength λ0:

Polychromatic (temporally and spatially incoherent) response:

( ) ( )

( ) ( ) 02

00

00

d dd ;,;,

d ;,,

λλλ

λλ

∫ ∫∫∫

−′−′=

′′=′′

yxyyxxhyxI

yxIyxI

MIT 2.71/2.710 Optics11/14/05 wk11-a-22

Comments on coherent vs incoherent

• Incoherent generally gives better image quality:– no ringing artifacts– no speckle– higher bandwidth (even though higher frequencies are

attenuated because of the MTF roll-off)• However, incoherent imaging is insensitive to phase

objects• Polychromatic imaging introduces further blurring due to

chromatic aberration (dependence of the MTF on wavelength)