IT 2.71/2.710 Optics 1/10/04 wk10-b-1 Today • Review of spatial filtering with coherent cohe rent illumination • Derivation of the lens law using wave optics • Point-spread function of a system with incohe rent incoherent illumination • The Modulation Transfer Function (MTF) and O ptical Transfer Function (OTF) • Comparison of coherent and incoherent imagin g • 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-1 Today Review of spatial filtering with coherent coherent illumination Derivation of the lens law using wave optics.
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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
MIT 2.71/2.710 Optics 11/10/04 wk10-b-2
Coherent imaging as a linear, shift-invariant system
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
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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Imaging with spatially incoherent light
simple object: two point sourcesnarrowband, mutually incoherent(input field is spatially incoherentspatially incoherent)
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Imaging with spatially incoherent light
incoherent: adding in intensity ⇒
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Imaging with spatially incoherent light
Generalizing:thin transparency with
sp. incoherentsp. incoherent illumination intensity at the outputof the imaging system
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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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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)
Polychromatic (temporally and spatially incoherent) response:
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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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Resolution
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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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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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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
Connection between PSF and NA
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MIT 2.71/2.710 Optics 11/10/04 wk10-b-31
Connection between PSF and NA
lobe width
NA in unit–mag imaging systems
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Monochromaticcoherent on-axis
illumination
Monochromaticcoherent on-axis
illumination
in both cases,
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The incoherent case:
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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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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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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in noisy optical systems
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“Safe” resolution in optical system
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Diffraction–limited resolution (safe)
Two point objects are “just resolvable” (limited by diffraction only)if they are separated by: