mit 2.71/2.710 optics 11/10/04 wk10-b-1 today review of spatial filtering with coherent coherent...
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-1
Today
• Review of spatial filtering with coherent coherent illumination• Derivation of the lens law using wave optics• Point-spread function of a system with incoherent incoherent illumination• The Modulation Transfer Function (MTF) and Optical Transfer Function (OTF)• Comparison of coherent and incoherent imaging• Resolution and image quality – The meaning of resolution – Rayleigh criterion and image quality
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-2
Coherent imaging as a linear, shift-invariant system
Thin transparency
illumination
impulse response
convolution
output amplitude
Fourier transform
Fourier transform
(≡plane wave spectrum)
transfer function
multiplication
transfer function aka pupil function
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-3
The 4F system with FP aperture
object plane Fourier plane: aperture-limited Image plane: blurred(i.e. low-pass filtered)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-4
Single-lens imaging condition
object lens
image
lateral
Imaging condition(akaLens Law)
Magnification
Derivation usingwave optics ?!?
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-5
Single-lens imaging system
object lens
image
spatial“LSI” system“
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-6
Single-lens imaging systemImpulse response (PSF)
spatial“LSI” system“
Ideal PSF:
Diffraction--LimitedPSF:
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-7
Imaging with incoherent light
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-8
Two types of incoherence
temporal temporal incoherenceincoherence
spatial spatial incoherenceincoherence
matched pathspoint
source
Michelson interferometer
poly-chromaticlight(=multi-color, broadband)
Young interferometer
mono-chromaticlight(= single color, narrowband)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-9
Two types of incoherence
temporal temporal incoherenceincoherence
spatial spatial incoherenceincoherence
matched pathspoint
source
waves from unequal paths waves from unequal paths do not interferedo not interfere
waves with equal pathswaves with equal pathsbut from different pointsbut from different points
on the wavefronton the wavefrontdo not interfdo not interf
ere ere
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-10
Coherent vs incoherent beams
Mutually coherent: superposition field amplitude is described by sum of complex amplitudes
Mutually incoherent: superposition field intensityis described by sum of intensities
(the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly —statistical optics)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-11
Imaging with spatially incoherent light
simple object: two point sourcesnarrowband, mutually incoherent(input field is spatially incoherentspatially incoherent)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-12
Imaging with spatially incoherent light
incoherent: adding in intensity ⇒
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-13
Imaging with spatially incoherent light
Generalizing:thin transparency with
sp. incoherentsp. incoherent illumination intensity at the outputof the imaging system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-14
Incoherent imaging as a linear, shift-invariant system
Thin transparency
illumination
incoherentimpulse response
convolution
output intensity
Incoherent imaging is linear in intensitywith incoherent impulse response (iPSF)
where h(x,y) is the coherent impulse response (cPSF)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-15
Incoherent imaging as a linear, shift-invariant system
Thin transparency
illumination
incoherentimpulse response
convolution
output intensity
Fourier transform
Fourier transform
(≡plane wave spectrum)
transfer function
multiplication
transfer function of incoherent system: optical transfer function (OTF)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-16
The Optical Transfer Function
normalized to 1
real real
max maxmaxmax
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-17
some terminology ...
Amplitude transfer function(coherent)
Optical Transfer Function (OTF)(incoherent)
Modulation Transfer Function (MTF)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-18
MTF of circular aperture
physical aperture filter shape (MTF)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-19
MTF of rectangular aperture
physical aperture filter shape (MTF)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-20
Incoherent low–pass filtering
MTF Intensity @ image plane
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-21
Incoherent low–pass filtering
MTF Intensity @ image plane
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-22
Incoherent low–pass filtering
MTF Intensity @ image plane
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-23
Diffraction-limited vs aberrated MTF
real
max max
ideal thin lens,ideal thin lens,finite aperturezfinite aperturez
realistic lensrealistic lensfinite aperture finite aperture & aberrations & aberrations
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-24
Imaging with polychromatic light
Monochromatic, spatially incoherent responseat wavelength λ0:
Polychromatic (temporally and spatially incoherent) response:
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-25
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 phas objects• Polychromatic imaging introduces further blurring due to chromatic aberration (dependence of the MTF on wavelength)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-26
Resolution
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-27
Connection between PSF and NA
Monochromaticcoherent on-axis
illumination
object planeimpulse
Fourier planecirc-aperture
image planeobserved field
(PSF)Fourier
transform
radial coordinate@ Fourier plane
radial coordinate@ image plane
(unit magnification)
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-28
Connection between PSF and NA
Monochromaticcoherent on-axis
illumination
Fourier planecirc-aperture
image planeNA: angle
of acceptancefor on–axispoint object
Numerical Aperture (NA)by definition:
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-29
Numerical Aperture and Speed (or F–Number)
medium ofrefr. index n
half-angle subtended by the imaging system from an axial object
Numerical Aperture
Speed(f/#)=1/2(NA)pronounced f-number, e.g.f/8 means (f/#)=8.
Aperture stopthe physical element whichlimits the angle of acceptance ofthe imaging system
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Connection between PSF and NA
MIT 2.71/2.710 Optics 11/10/04 wk10-b-30
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-31
Connection between PSF and NA
lobe width
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NA in unit–mag imaging systems
MIT 2.71/2.710 Optics 11/10/04 wk10-b-32
Monochromaticcoherent on-axis
illumination
Monochromaticcoherent on-axis
illumination
in both cases,
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-33
The incoherent case:
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-34
The two–point resolution problem
Imagingsystem
intensitypattern
observed
(e.g. withdigital
camera)object: two point sources,mutually incoherent
(e.g. two stars in the night sky;two fluorescent beads in a solution)
The resolution question [Rayleigh, 1879]: when do we ceaseto be able to resolve the two point sources (i.e., tell them apart)
due to the blurring introduced in the image by the finite (NA)?
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-35
The meaning of “resolution”
[from the New Merriam-Webster Dictionary, 1989 ed.]:
resolve v: 1to break up into constituent parts: ANALYZE;2to find an answer to : SOLVE; 3DETERMINE, DECIDE;4to make or pass a formal resolution
resolution n: 1the act or process of resolving 2the actionof solving, also: SOLUTION; 3the quality of being resolute : FIRMNESS, DETERMINATION; 4a formal statementexpressing the opinion, will or, intent of a body of persons
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-36
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-37
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-38
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-39
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-40
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-41
Resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-42
Resolution in noisy optical systems
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-43
“Safe” resolution in optical system
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-44
Diffraction–limited resolution (safe)
Two point objects are “just resolvable” (limited by diffraction only)if they are separated by:
Two–dimensional systems(rotationally symmetric PSF)
One–dimensional systems(e.g. slit–like aperture)
Safe definition:(one–lobe spacing)
Pushy definition:(1/2–lobe spacing)
You will see different authors giving different definitions.Rayleigh in his original paper (1879) noted the issue of noise
and warned that the definition of “just–resolvable” points
is system–or application –dependent