coherent imaging - mitweb.mit.edu/2.710/fall06/2.710-wk11-a-sl.pdf · coherent imaging as a linear,...
TRANSCRIPT
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
MIT 2.71/2.710 Optics11/14/05 wk11-a-5
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 ,~
MIT 2.71/2.710 Optics11/14/05 wk11-a-15
MTF of circular aperture
physical aperture filter shape (MTF)f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/14/05 wk11-a-16
MTF of rectangular aperture
physical aperture filter shape (MTF)f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/14/05 wk11-a-17
Incoherent low–pass filtering
MTF Intensity @ image planef1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/14/05 wk11-a-18
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)