2.717J/MAS.857J Optical Engineering - MITweb.mit.edu/2.717/www/2.717-wk1-b.pdf · Example: 4F...
Transcript of 2.717J/MAS.857J Optical Engineering - MITweb.mit.edu/2.717/www/2.717-wk1-b.pdf · Example: 4F...
MIT 2.717Jwk1-b p-1
2.717J/MAS.857JOptical Engineering
Welcome to ...
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This class is about• Statistical Optics
– models of random optical fields, their propagation and statistical properties (i.e. coherence)
– imaging methods based on statistical properties of light: coherence imaging, coherence tomography
• Inverse Problems– to what degree can a light source be determined by measurements
of the light fields that the source generates?– how much information is “transmitted” through an imaging
system? (related issues: what does _resolution_ really mean? what is the space-bandwidth product?)
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The van Cittert-Zernike theorem
Galaxy, ~100 millionlight-years away
radiowaves
Very Large Array (VLA)
Cross-Correlation+
Fouriertransform
image
( )optical image
Image credits:hubble.nasa.gov
www.nrao.edu
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Optical coherence tomography
Coronary artery
Intestinal polyps
Esophagus
Image credits:www.lightlabimaging.com
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Inverse Radon transform(aka Filtered Backprojection)
Magnetic Resonance Imaging (MRI)
The principle
The hardware
The image
Image credits:www.cis.rit.edu/htbooks/mri/www.ge.com
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You can take this class if• You took one of the following classes at MIT
– 2.996/2.997 during the academic years 97-98 and 99-00– 2.717 during fall ’00– 2.710 at anytimeOR
• You have taken a class elsewhere that covered Geometrical Optics, Diffraction, and Fourier Optics
• Some background in probability & statistics is helpful but not necessary
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Syllabus (summary)• Review of Fourier Optics, probability & statistics 4 weeks• Light statistics and theory of coherence 2 weeks• The van Cittert-Zernicke theorem and applications of statistical optics
to imaging 3 weeks• Basic concepts of inverse problems (ill-posedness, regularization) and
examples (Radon transform and its inversion) 2 weeks• Likelihood and information methods for imaging channel inversion 2
weeks
Textbooks:• J. W. Goodman, Statistical Optics, Wiley.• M. Bertero and P. Boccacci, Introduction to Inverse Problems in
Imaging, IoP publishing.• Richard E. Blahut, Theory of Remote Image Formation, Cambridge
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What you have to do• 4 homeworks (1/week for the first 4 weeks)• 3 Projects:
– Project 1: a simple calculation of intensity statistics from a model in Goodman (~2 weeks, 1-page report)
– Project 2: study one out of several topics in the application ofcoherence theory and the van Cittert-Zernicke theorem from Goodman (~4 weeks, lecture-style presentation)
– Project 3: a more elaborate calculation of information capacity of imaging channels based on prior work by Barbastathis & Neifeld (~4 weeks, conference-style presentation)
• Alternative projects ok (please propose early)• No quizzes or final exam
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Administrative• Website http://web.mit.edu/2.717/www• Broadcast list will be setup soon• Instructor’s coordinates
George Barbastathis 3-461c [email protected]• Please do not phone-call• Office hours TBA• Admin. Assistant
Nikki Hanafin 3-461 [email protected] 4-0449 & 3-5592• Class meets in 1-242
– Mondays 1-3pm (main coverage of the material)– Wednesdays 2-3pm (examples and discussion)– presentations only: Wednesdays 7pm-??, pizza served
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The 4F system1f 1f 2f 2f
( )yxg ,1 ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′
111 ,
fy
fxG
λλ ⎟⎟⎠
⎞⎜⎜⎝
⎛′−′− y
ffx
ffg
2
1
2
11 ,
object planeFourier plane Image plane
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The 4F system1f 1f 2f 2f
( )yxg ,1 ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′
111 ,
fy
fxG
λλ ⎟⎟⎠
⎞⎜⎜⎝
⎛′−′− y
ffx
ffg
2
1
2
11 ,
object planeFourier plane Image plane
( )vuG ,1
θx
λθλθ
y
x
v
u
sin
sin
=
=
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Low-pass filtering with the 4F system
( )yxg ,in
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′′+′′×⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′R
yxf
yf
xG22
11in circ,
λλ
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛′−′−∗ y
ffx
ffg
2
1
2
1in , jinc
object planetransparency
Fourier planecirc-aperture
Image planeobserved field
1f 1f 2f 2f
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′
11in ,
fy
fxG
λλ
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′′+′′=′′′′
Ryx
yxH22
circ,
field arrivingat Fourier plane
monochromaticcoherent on-axis
illumination
field departingfrom Fourier plane
ℑℑFourier
transformFourier
transform
x ′′ x′x
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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
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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
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Coherent imaging as a linear, shift-invariant system
Thin transparency( )yxt ,
( )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
==
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Coherent imaging as a linear, shift-invariant system
Thin transparency( )yxt ,
( )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
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Examples: the amplitude MIT pattern
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Weak low–pass filtering
Fourier filter Intensity @ image planef1=20cmλ=0.5µm
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Strong low–pass filtering
Fourier filter Intensity @ image planef1=20cmλ=0.5µm
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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.)
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Imaging with Zernicke mask
Fourier filter Intensity @ image planef1=20cmλ=0.5µm
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Coherent vs incoherent imaging
Coherentopticalsystem
field in field out
Incoherentopticalsystem
intensity in intensity out
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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
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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 =
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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)
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The Optical Transfer Function
( ) ( ){ }( ) ( )
( )∫∫∫∫
′′′′
′′−′−′′′=
ℑ≡
vuvuH
vuvvuuHvuH
yxhvuH
dd,
dd ,,
1 tonormalized , ,~
2
*
2
umax–umax
1
2umax–2umax
real(H) ( )H~real
1
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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
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Coherent vs incoherent imaging
( )yxh ,
( ) ( ) 2,,~ yxhyxh =
( ) ( ){ }yxhvuH ,FT, =
( ) ( ){ }( ) ( )vuHvuH
yxhvuH,,
,~FT,~
⊗==
Coherent impulse response(field in ⇒ field out)
Coherent transfer function(FT of field in ⇒ FT of field out)
Incoherent impulse response(intensity in ⇒ intensity out)
Incoherent transfer function(FT of intensity in ⇒ FT of intensity out)
( ) (MTF)Function Transfer Modulation :,~ vuH
( ) (OTF)Function Transfer Optical :,~ vuH
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Coherent vs incoherent imaging 1f 1f 2f 2f
2a
c2u−
( )uH~
1
c2uu
1c f
auλ
=cu−
1
u
Coherent illumination Incoherent illumination
( )uH
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Aberrations: geometricalParaxial
(Gaussian)image point
Non-paraxial rays“overfocus”
Spherical aberration
• Origin of aberrations: nonlinearity of Snell’s law (n sinθ=const., whereas linear relationship would have been nθ=const.)• Aberrations cause practical systems to perform worse than diffraction-limited• Aberrations are best dealt with using optical design software (Code V, Oslo, Zemax); optimized systems usually resolve ~3-5λ (~1.5-2.5µm in the visible)
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Aberrations: wave
( )yxh ,limited
ndiffractio
( ) ( ) ( )yxiyxhyxh ,
limitedndiffractioaberrated
aberratione,, ϕ=
Aberration-free impulse response
Aberrations introduce additional phase delay to the impulse response
c2u−
( )uH~
1
c2uu
unaberrated(diffraction
limited)
aberrated
Effect of aberrationson the MTF