MIT 2.71/2.710 Optics 11/24/04 wk12-b-1 Today Resolution Wavefront modulation.

MIT 2.71/2.710 Optics 11/24/04 wk12-b-1

Today

• Resolution

• Wavefront modulation

MIT 2.71/2.710 Optics 11/24/04 wk12-b-2

Resolution

MIT 2.71/2.710 Optics 11/24/04 wk12-b-3

Coherent imaging as a linear, shift-invariant system

Thin transparency

illumination(field)

impulse response

convolution

output amplitude

Fourier transform

Fourier transform

(≡plane wave spectrum)

transfer function

multiplication

transfer function of coherent system H(u,v): aka amplitude transfer function

MIT 2.71/2.710 Optics 11/24/04 wk12-b-4

Incoherent imaging as a linear, shift-invariant system

Thin transparency

illumination(field)

incoherentimpulse response

convolution

output amplitude

Fourier transform

Fourier transform

transfer function

multiplication

transfer function of incoherent system: optical transfer function (OTF)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-5

Transfer Functions of Clear Aperture

Coherent Incoherent

normalized to

1D amplitude transfer function (ATF) 1D optical transfer function (OTF)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-6

Connection between PSF and NA

monochromaticcoherent on-axis

illumination

image planeobserved field

(PSF)

(unit magnification)

Fourier planecirc-aperture

radial coordinate@ Fourier plane

radial coordinate@ image plane

object planeimpulse

MIT 2.71/2.710 Optics 11/24/04 wk12-b-7


Numerical Aperture (NA)by definition:

NA: angleof acceptance

for on–axispoint object

Fourier planecirc-aperture

image plane

MIT 2.71/2.710 Optics 11/24/04 wk12-b-8

Numerical Aperture and Speed (or F–Number)

medium ofrefr. index n

θ: 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 of the imaging system

MIT 2.71/2.710 Optics 11/24/04 wk12-b-9


MIT 2.71/2.710 Optics 11/24/04 wk12-b-10


lobe width

MIT 2.71/2.710 Optics 11/24/04 wk12-b-11

The incoherent case:

MIT 2.71/2.710 Optics 11/24/04 wk12-b-12

NA in unit–mag imaging systems

Monochromaticcoherent on-axis

illumination

Monochromaticcoherent on-axis

illumination

in both cases

MIT 2.71/2.710 Optics 11/24/04 wk12-b-13

The two–point resolution problem

Imagingsystem Intensity

patternobserved

(e.g. with digital camera)

object: two point sources,mutually incoherent

(e.g. two stars in the night sky;two fluorescent beads in a solution)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-14

The meaning of “resolution”

[from the New Merriam-Webster Dictionary, 1989 ed.]:

resolve v: 1 to break up into constituent parts: ANALYZE;2to find an answer to : SOLVE; 3DETERMINE, DECIDE;4to make or pass a formal resolution

resolution n: 1 the act or process of resolving 2 the actionof solving, also: SOLUTION; 3 the quality of being resolute :FIRMNESS, DETERMINATION; 4 a formal statementexpressing the opinion, will or, intent of a body of persons

MIT 2.71/2.710 Optics 11/24/04 wk12-b-15

Resolution in optical systems

MIT 2.71/2.710 Optics 11/24/04 wk12-b-16


MIT 2.71/2.710 Optics 11/24/04 wk12-b-17


MIT 2.71/2.710 Optics 11/24/04 wk12-b-18


MIT 2.71/2.710 Optics 11/24/04 wk12-b-19


MIT 2.71/2.710 Optics 11/24/04 wk12-b-20


MIT 2.71/2.710 Optics 11/24/04 wk12-b-21

Resolution in noisynoisy optical systems

MIT 2.71/2.710 Optics 11/24/04 wk12-b-22

“Safe” resolution in optical systems

MIT 2.71/2.710 Optics 11/24/04 wk12-b-23

Diffraction–limited resolution (safe)

Two point objects are “just resolvable just resolvable” (limited by diffraction only)if they are separated by:

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” pointsis system–or application –dependent

Two–dimensional systems(rotationally symmetric PSF)

Safe definition:(one–lobe spacing)

Pushy definition:(1/2–lobe spacing)

One–dimensional systems(e.g. slit–like aperture)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-24

Also affecting resolution: aberrations

All our calculations have assumed “geometrically perfect”systems, i.e. we calculated the wave–optics behavior of

systems which, in the paraxial geometrical optics approximationwould have imaged a point object onto a perfect point image.

The effect of aberrations (calculated with non–paraxial geometricaloptics) is to blur the “geometrically perfect ”image; including

the effects of diffraction causes additional blur.

geometrical optics description

MIT 2.71/2.710 Optics 11/24/04 wk12-b-25

Also affecting resolution: aberrations

“diffraction––limited”(aberration–free) 1D MTF 1D MTF with aberrations

diffraction––limited1D PSF(sinc2)

Somethingwider

wave optics picture

Fouriertransform

Fouriertransform

MIT 2.71/2.710 Optics 11/24/04 wk12-b-26

Typical result of optical design

shiftVariantOpticalsystem

(FoV)field of view

of the system

MTF degradestowards thefield edges

MTF is neardiffraction–limitednear the center

of the field

MIT 2.71/2.710 Optics 11/24/04 wk12-b-27

The limits of our approximations

• Real–life MTFs include aberration effects, whereas our analysis has been “diffraction–limited”• Aberration effects on the MTF are FoV (field) location– dependent: typically we get more blur near the edges of the field (narrower MTF ⇔broader PSF)• This, in addition, means that real–life optical systems are not shift invariant either!• ⇒the concept of MTF is approximate, near the region where the system is approximately shift invariant (recall: transfer functions can be defined only for shift invariant linear systems!)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-28

The utility of our approximations

• Nevertheless, within the limits of the paraxial, linear shift–invariant system approximation, the concepts of PSF/MTF provide– a useful way of thinking about the behavior of

optical systems– an upper limit on the performance of a given

optical system (diffraction–limited performance is the best we can hope for, in paraxial regions of the field; aberrations will only make worse non–paraxial portions of the field)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-29

Common misinterpretations

Attempting to resolve object features smaller than the“resolution limit” (e.g. 1.22λ/NA) is hopeless.

Image quality degradation as object features become smaller than the

resolution limit (“exceed the resolution limit”) is noise dependent noise dependent and gradual.

MIT 2.71/2.710 Optics 11/24/04 wk12-b-30


• Attempting to resolve object features smaller than the• “resolution limit” (e.g. 1.22λ/NA) is hopeless.

Besides, digital processing of the acquired images (e.g. methods such as the CLEAN

algorithm, Wiener filtering, expectation maximization, etc.) can be employed.

MIT 2.71/2.710 Optics 11/24/04 wk12-b-31


By engineering the pupil function (“apodizing”) to result in a PSF with narrower side–lobe, one can “beat” the resolution limitations imposed by the angular acceptance (NA) of the system.

Super-resolution

Pupil function design always results in(i) narrower main lobe but accentuated

side–lobes(ii) lower power transmitted through the

systemBoth effects are BAD on the image

MIT 2.71/2.710 Optics 11/24/04 wk12-b-32

ApodizationClear pupil Annular pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-33

ApodizationClear pupil Gaussian pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-34

Unapodized (clear–aperture) MTFClear pupil MTF of clear pupill

auto-correlation

MIT 2.71/2.710 Optics 11/24/04 wk12-b-35

Unapodized (clear–aperture) MTF

Clear pupil MTF of clear pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-36

Unapodized (clear–aperture) PSFClear pupil PSF of clear pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-37

Apodized (annular) MTFAnnular pupil MTF of annular pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-38

Apodized (annular) PSF

Annular pupil PSF of annular pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-39

Apodized (Gaussian) MTFGaussian pupil MTF of Gaussian pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-40

Apodized (Gaussian) PSFGaussian pupil PSF of Gaussian pupil

MIT 2.71/2.710 Optics 11/24/04 wk12-b-41

Conclusions (?)• Annular–type pupil functions typically narrow the main lobe of the PS

F at the expense of higher side lobes• Gaussian–type pupil functions typically suppress the side lobes but br

oaden the main lobe of the PSF• Compromise? →application dependent

– for point–like objects (e.g., stars) annular apodizers may be a good idea– for low–frequency objects (e.g., diffuse tissue) Gaussian apodizers may image with fewer artifacts

• Caveat: Gaussian amplitude apodizersvery difficult to fabricate and introduce energy loss binary phase apodizers (lossless ⇒

by nature) are used instead; typically designed by numerical optimization

MIT 2.71/2.710 Optics 11/24/04 wk12-b-42


By engineering the pupil function (“apodizing”) to

result in a PSF with narrower side–lobe, one can “beat” the resolution limitations imposed by the angular acce

ptance (NA) of the system.

Super-resolution

main lobe size ↓⇔ sidelobes↑and vice versa

main lobe size ↑⇔ sidelobes↓

power loss an important factor

compromise application dependent

MIT 2.71/2.710 Optics 11/24/04 wk12-b-43


“This super cool digital camera has resolutionof 5 Mega pixels (5 million pixels).”

This is the most common and worst misuse of the term “resolution.” They

are actually referring to the space–bandwidth product (SBP)

bandwidth product (SBP)of the camera

MIT 2.71/2.710 Optics 11/24/04 wk12-b-44

What can a camera resolve?Answer depends on the magnification and

PSF of the optical system attached to the camera

Pixels significantly smaller than the system PSFare somewhat underutilized (the effective SBP is reduced)

PSF of opticalsystem

MIT 2.71/2.710 Optics 11/24/04 wk12-b-45

Summary of misinterpretationsof “resolution” and their refutations

• It is pointless to attempt to resolve beyond the Rayleigh criterion (however defined)–NO: difficulty increases gradually as feature size shrinks,

and difficulty is noise dependent

• Apodization can be used to beat the resolution limit imposed by the numerical aperture–NO: watch sidelobe growth and power efficiency loss

• The resolution of my camera is N×Mpixels–NO: the maximum possible SBP of your system may be

N×M pixels but you can easily underutilize it by using a suboptimal optical system

MIT 2.71/2.710 Optics 11/24/04 wk12-b-46

So, what is resolution?

• Our ability to resolve two point objects (in general, two distinct features in a more general object) based on the image

• It is relatedto the NA but not exclusivelylimited by it• Resolution, as it relates to NA:

– Resolution improves as NA increases

• Other factors affecting resolution:– aberrations / apodization (i.e., the exact shape of the PSF)– NOISE!

•Is there an easy answer? – No …… but when in doubt quote 0.61λ/(NA) as an estimate (not as an exact li

mit).

MIT 2.71/2.710 Optics 11/24/04 wk12-b-47

Wave front modulation

• Photographic film

• Spatial light modulators

• Binary optics

MIT 2.71/2.710 Optics 11/24/04 wk12-b-48

Photographic films / plates

protective layer

emulsion with silver halide (e.g. AgBr)grains

base (glass, mylar, acetate)

Exposure:

Collection of development specks = latent image

Development(1stchemical bath): converts specks to metallic silver

Fixingthe emulsion (2ndchemical bath): removal of unexposed silver halide

development speck

MIT 2.71/2.710 Optics 11/24/04 wk12-b-49

Photographic films / plates

Exposure (energy) : energy incident per unit area on a photographic emulsion during the exposure process (units: mJ/cm2)

Exposure = incident intensity ×exposure time

Intensity transmittance : average ratio of intensity transmitted over intensity incident after development

Photographic density

local

average

Transmitted at

Incident at

MIT 2.71/2.710 Optics 11/24/04 wk12-b-50

Photographic film / platesHurter-Driffield curve Gamma curve

Shoulder

Linear region

Toe

log

(min)

γhigh/low : high/low contrast film

Gross fog

Saturationregion

development time

MIT 2.71/2.710 Optics 11/24/04 wk12-b-51

Kelley model of photographic process

Optical imaging during exposure

(linear)

H&D curve

(nonlinear)

additional blur due to chemical diffusion(linear)

Measureddensity

H&D curve

inferred logarithmicexposure

MIT 2.71/2.710 Optics 11/24/04 wk12-b-52

The Modulation Transfer Function

adjacency effect(due to chemical diffusion) Exposure:

“Effective exposure”:

MIT 2.71/2.710 Optics 11/24/04 wk12-b-53

Bleaching / phase modulation

emulsion

exposure

tanning bleach(relief grating)

non-tanning bleach(index grating)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-54

Spatial Light Modulators

• Liquid crystals• Magneto-Optic• Micro-mirror• Grating Light Valve• Multiple Quantum Well• Acousto-Optic

MIT 2.71/2.710 Optics 11/24/04 wk12-b-55

Liquid crystal modulators

• Nematic• Smectic (smectic-C* phase: ferroelectric)• Cholesteric

OFF

crossed polarizers crossed polarizers

(acts as λ/2 plate;rotates polarization by 90°)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-56

Micro-mirror technology

Images removed due to copyright concerns

Lucent (Bell Labs)

Texas Instruments DMD/DLP

Sandia

MIT 2.71/2.710 Optics 11/24/04 wk12-b-57

Micro-mirror display


http://www.howstuffworks.com

MIT 2.71/2.710 Optics 11/24/04 wk12-b-58

Micromirrors for adaptive optics


Véran, J.-P. & Durand, D. 2000, ASP Conf. Ser 216, 345 (2000).

MIT 2.71/2.710 Optics 11/24/04 wk12-b-59

Grating Light Valve (GLV) display


Silicon Light Machines, www.siliconlight.com

www.meko.co.uk

MIT 2.71/2.710 Optics 11/24/04 wk12-b-60

Binary Optics

Refractive Diffractive Binary (prism) (blazed grating)

efficiency of 1stdiffracted order fromstep-wise (binary) approximation toblazed grating with N steps over 2πrange

MIT 2.71/2.710 Optics 11/24/04 wk12-b-61

Binary Optics: binary gratingAmplitude mask

MIT 2.71/2.710 Optics 11/24/04 wk12-b-62

Binary Optics: binary grating

Phase mask

MIT 2.71/2.710 Optics 11/24/04 wk12-b-63

Fourier series for binary phase grating

MIT 2.71/2.710 Optics 11/24/04 wk12-b-64


identify physical meaning:• plane waves• orientation of nth plane wave:

•diffracted orders

MIT 2.71/2.710 Optics 11/24/04 wk12-b-65




MIT 2.71/2.710 Optics 11/24/04 wk12-b-66




odd orders only!

MIT 2.71/2.710 Optics 11/24/04 wk12-b-67


Fourier transform:

• result is

(Fourier series)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-68

Diffracted spectrum from binary phase grating

MIT 2.71/2.710 Optics 11/24/04 wk12-b-69

Fourier-plane diffraction from finite binary phase grating

(x”not to scale)

MIT 2.71/2.710 Optics 11/24/04 wk12-b-1 Today Resolution Wavefront modulation.

Documents

Transcript of MIT 2.71/2.710 Optics 11/24/04 wk12-b-1 Today Resolution Wavefront modulation.