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

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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The 4F system with FP aperture

object plane Fourier plane: aperture-limited Image plane: blurred(i.e. low-pass filtered)

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Single-lens imaging condition

object lens

image

lateral

Imaging condition(akaLens Law)

Magnification

Derivation usingwave optics ?!?

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Single-lens imaging system

object lens

image

spatial“LSI” system“

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Single-lens imaging systemImpulse response (PSF)

spatial“LSI” system“

Ideal PSF:

Diffraction--LimitedPSF:

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Imaging with incoherent light

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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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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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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)

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The Optical Transfer Function

normalized to 1

real real

max maxmaxmax

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some terminology ...

Amplitude transfer function(coherent)

Optical Transfer Function (OTF)(incoherent)

Modulation Transfer Function (MTF)

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MTF of circular aperture

physical aperture filter shape (MTF)

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MTF of rectangular aperture

physical aperture filter shape (MTF)

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Incoherent low–pass filtering

MTF Intensity @ image plane

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Incoherent low–pass filtering

MTF Intensity @ image plane

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Incoherent low–pass filtering

MTF Intensity @ image plane

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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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Imaging with polychromatic light

Monochromatic, spatially incoherent responseat wavelength λ0:

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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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:

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