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 ...

MIT 2.717Jwk1-b p-2

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?)

MIT 2.717Jwk1-b p-3

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

MIT 2.717Jwk1-b p-4

Optical coherence tomography

Coronary artery

Intestinal polyps

Esophagus

Image credits:www.lightlabimaging.com

MIT 2.717Jwk1-b p-5

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

MIT 2.717Jwk1-b p-6

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

MIT 2.717Jwk1-b p-7

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

MIT 2.717Jwk1-b p-8

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

MIT 2.717Jwk1-b p-9

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

MIT 2.717Jwk1-b p-10

The 4F system1f 1f 2f 2f

( )yxg ,1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

111 ,

fy

fxG

λλ ⎟⎟⎠

⎞⎜⎜⎝

⎛′−′− y

ffx

ffg

2

1

2

11 ,

object planeFourier plane Image plane


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

=

=


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


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


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




( )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

==




( )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


Examples: the amplitude MIT pattern


Weak low–pass filtering

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


Strong low–pass filtering



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.)


Imaging with Zernicke mask



Coherent vs incoherent imaging

Coherentopticalsystem

field in field out

Incoherentopticalsystem

intensity in intensity out


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


Incoherent imaging as a linear, shift-invariant system


( )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 =


Incoherent imaging as a linear, shift-invariant system


( )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)


The Optical Transfer Function

( ) ( ){ }( ) ( )

( )∫∫∫∫

′′′′

′′−′−′′′=

ℑ≡

vuvuH

vuvvuuHvuH

yxhvuH

dd,

dd ,,

1 tonormalized , ,~

2

*

2

umax–umax

1

2umax–2umax

real(H) ( )H~real

1


some terminology ...

Amplitude transfer function(coherent)

Optical Transfer Function (OTF)(incoherent)

Modulation Transfer Function (MTF)

( )vuH ,

( )vuH ,~

( )vuH ,~


MTF of circular aperture

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


MTF of rectangular aperture

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


Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm


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


Coherent vs incoherent imaging 1f 1f 2f 2f

2a

c2u−

( )uH~

1

c2uu

1c f

auλ

=cu−

1

u

Coherent illumination Incoherent illumination

( )uH


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)


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

2.717J/MAS.857J Optical Engineering - MITweb.mit.edu/2.717/www/2.717-wk1-b.pdf · Example: 4F...

Documents

Transcript of 2.717J/MAS.857J Optical Engineering - MITweb.mit.edu/2.717/www/2.717-wk1-b.pdf · Example: 4F...