coherent incoherent illumination transfer function (otf...

Today

• Review of spatial filtering with coherentcoherent illumination• Derivation of the lens law using wave optics• Point-spread function of a system with incoherentincoherent

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 Optics11/10/04 wk10-b-1

Coherent imaging as a linear, shift-invariant system

MIT 2.71/2.710 Optics11/10/04 wk10-b-2

Thin transparency( )yxt ,

( )yxg ,1

( ) ),( ,),(

1

2

yxtyxgyxg

==

output amplitude

impulse response ( )),(),(

,

2

3

yxhyxgyxg∗=

=′′

convolutionillumination

Fourier transform

Fourier transform

transfer function(≡plane wave spectrum) ),(),(

),(

2

3

vuHvuGvuG

==( )vuG ,2

multiplication

transfer function H(sx ,sy): aka pupil function

The 4F system with FP aperture1f 1f 2f 2f

( )yxg ,1⎟⎠⎞

⎜⎝⎛ ′′

×⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′Rr

fy

fxG circ,

111 λλ ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛′−′−∗ y

ffx

ffhg

2

1

2

11 ,

( )vuG ,1

θx

object planeFourier plane: aperture-limited Image plane: blurred

(i.e. low-pass filtered)MIT 2.71/2.710 Optics11/10/04 wk10-b-3

Single-lens imaging condition

ss’

lensobject

image

fss111 =

′+ Imaging condition

(aka Lens Law) Derivation usingwave optics ?!?

ssm′

−=lateral Magnification

MIT 2.71/2.710 Optics11/10/04 wk10-b-4

Single-lens imaging system

ss’

lensobject

image

spatialspatial“LSI” system“LSI” system

gin(x,y) gout(x’,y’)

MIT 2.71/2.710 Optics11/10/04 wk10-b-5

Single-lens imaging systemImpulse response (PSF)

spatialspatial“LSI” system“LSI” system

gin(x,y) gout(x’,y’)

( ) ( ) ( )myymxxyxyxh −′−′=′′ δδ,;,Ideal PSF:

Diffraction--limitedPSF:

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −′′

+⎟⎠⎞

⎜⎝⎛ −′′

=′′22

jinc,;,sy

sy

sx

sxRyxyxh

λ

MIT 2.71/2.710 Optics11/10/04 wk10-b-6

Imaging with incoherent light

MIT 2.71/2.710 Optics11/10/04 wk10-b-7

Two types of incoherencetemporaltemporal

incoherencespatialspatial

incoherenceincoherence incoherence

1r

2r1rmatched

pathspointsource

d2d1

Michelson interferometer Young interferometer

poly-chromatic light(=multi-color, broadband)

mono-chromatic light(= single color, narrowband)

MIT 2.71/2.710 Optics11/10/04 wk10-b-8

Two types of incoherencetemporaltemporal

incoherencespatialspatial

incoherenceincoherence incoherence

1r

2r1rmatched

pathspointsource

d2d1

waves from unequal pathswaves from unequal pathsdo not interfere

waves with equal pathswaves with equal pathsbut from different pointsbut from different points

on the wavefronton the wavefrontdo not interfere

do not interfere

do not interfereMIT 2.71/2.710 Optics11/10/04 wk10-b-9

Coherent vs incoherent beams

MIT 2.71/2.710 Optics11/10/04 wk10-b-10

1e11φiaa =

2e22φiaa =

Mutually coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes

221

2

212121 ee

aaaI

aaaaa ii

+==

+=+= φφ

Mutually incoherent: superposition field intensityintensityis described by sum of intensities1I sum of intensities

21 III +=(the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly — statistical optics)

2I

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

simple object: two point sourcesnarrowband, mutually incoherent(input field is spatially incoherentspatially incoherent)

MIT 2.71/2.710 Optics11/10/04 wk10-b-11

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

2x0

incoherent: adding in intensity ⇒

( ) ( ) ( ) 20

20 xxhxxhxI +′+−′=′

MIT 2.71/2.710 Optics11/10/04 wk10-b-12

Imaging with spatially incoherent light

2f 2fx ′′ x′x

1f 1f

( )xI

Generalizing:thin transparency with

sp. incoherentsp. incoherent illumination

( ) ( ) ( ) xxxhxIxI d 2−′=′ ∫intensity at the outputof the imaging system

MIT 2.71/2.710 Optics11/10/04 wk10-b-13

Incoherent imaging as a linear, shift-invariant system

MIT 2.71/2.710 Optics11/10/04 wk10-b-14

Thin transparency( )yxt ,

( )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

MIT 2.71/2.710 Optics11/10/04 wk10-b-15

Thin transparency( )yxt ,

( )yxI ,1

( ) ),( ,),(

1

2

yxtyxIyxI

=

=

(≡plane wave spectrum) ( )vuI ,2̂

incoherentimpulse response ( )

22

3

),(),(

,

yxhyxI

yxI

∗=

=′′

output intensity

convolutionillumination

Fourier transform

Fourier transform

transfer function

),(~),(ˆ),(ˆ

2

3

vuHvuI

vuI

=

=

multiplication

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

MIT 2.71/2.710 Optics11/10/04 wk10-b-16

( )H~real

1

real(H)

1

umax–umax 2umax–2umax

some terminology ...

( )vuH , Amplitude transfer function(coherent)

Optical Transfer Function (OTF)(incoherent)

Modulation Transfer Function (MTF)

( )vuH ,~

( )vuH ,~

MIT 2.71/2.710 Optics11/10/04 wk10-b-17

MTF of circular aperture

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

MIT 2.71/2.710 Optics11/10/04 wk10-b-18

MTF of rectangular aperture

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

MIT 2.71/2.710 Optics11/10/04 wk10-b-19

Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/10/04 wk10-b-20

Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/10/04 wk10-b-21

Incoherent low–pass filtering

MTF Intensity @ image planef1=20cmλ=0.5µm

MIT 2.71/2.710 Optics11/10/04 wk10-b-22

Diffraction-limited vs aberrated MTF

2umax–2umax

( )H~real

1ideal thin lens,ideal thin lens,finite aperturefinite aperture

realistic lens,realistic lens,finite aperturefinite aperture& aberrations& aberrations

MIT 2.71/2.710 Optics11/10/04 wk10-b-23

Imaging with polychromatic light

Monochromatic, spatially incoherent responseat wavelength λ0:

( ) ( ) ( ) yxyyxxhyxIyxI dd ;,;,;, 2000 λλλ −′−′=′′ ∫∫

Polychromatic (temporally and spatially incoherent) response:

( ) ( )

( ) ( ) 02

00

00

d dd ;,;,

d ;,,

λλλ

λλ

∫ ∫∫∫

−′−′=

′′=′′

yxyyxxhyxI

yxIyxI

MIT 2.71/2.710 Optics11/10/04 wk10-b-24

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 phase

objects• Polychromatic imaging introduces further blurring due to

chromatic aberration (dependence of the MTF on wavelength)

MIT 2.71/2.710 Optics11/10/04 wk10-b-25

Resolution

MIT 2.71/2.710 Optics11/10/04 wk10-b-26

Connection between PSF and NA

MIT 2.71/2.710 Optics11/10/04 wk10-b-27

( ) ( ) ( )yxyxg δδ=,in

( )

λπ

λπ

rfR

rfR

′

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′

≡

1

11

2

2J2.,.jinc

object planeimpulse

Fourier planecirc-aperture

image planeobserved field

(PSF)

1f 1f

( ) ⎟⎠⎞

⎜⎝⎛ ′′

=′′′′RryxH circ,

monochromaticcoherent on-axis

illumination

ℑ Fouriertransform

x ′′ x′x1f 1f

radial coordinate@ Fourier plane

22 yxr ′′+′′=′′

22 yxr ′+′=′radial coordinate@ image plane

2R

(unit magnification)

Connection between PSF and NA

MIT 2.71/2.710 Optics11/10/04 wk10-b-28

Fourier planecirc-aperture

image plane

1f 1f

monochromaticcoherent on-axis

illumination

x ′′ x′x1f 1f

( )

( )λ

π

λπ

λπ

λπ

λλ r

r

rfR

rfR

yfRx

fR

′

⎟⎠⎞

⎜⎝⎛ ′

=′

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′

≡⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−

′−

NA2

NA2J2

2

2J22,2jinc

1

1

11

11

( )1

NAfR

≡Numerical Aperture (NA)by definition:

NA: angleof acceptancefor on–axispoint object

2R

Numerical Aperture and Speed (or F–Number)

medium ofrefr. index n

θ

θ: half-angle subtended by the imaging system from an axial object

Numerical Aperture(NA) = n sinθ

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 Optics11/10/04 wk10-b-29

Connection between PSF and NA

( )NA61.0 @ null λ

=′r

( )( )

( )λ

π

λπ

r

r

yxh ′

⎟⎠⎞

⎜⎝⎛ ′

=′′NA2

NA2J2,

1

MIT 2.71/2.710 Optics11/10/04 wk10-b-30

Connection between PSF and NA

( )( )

( )λ

π

λπ

r

r

yxh ′

⎟⎠⎞

⎜⎝⎛ ′

=′′NA2

NA2J2,

1

( )NA22.1 width lobe λ

=′∆r

MIT 2.71/2.710 Optics11/10/04 wk10-b-31

NA in unit–mag imaging systems1f 1f

monochromaticcoherent on-axis

illumination

x ′′ x′x1f 1f

2R

MIT 2.71/2.710 Optics11/10/04 wk10-b-32

12 f x ′′ x′xmonochromaticcoherent on-axis

illumination

12 f2R

( )1

NAfR

≡

( )12

NAf

R≡

( ) ( ) ( ) ⎟⎠⎞

⎜⎝⎛ ′

=′=′′=λrrhyxh NA2jinc,PSFin both cases,

MIT 2.71/2.710 Optics11/10/04 wk10-b-33

The incoherent case:

( )NA61.0 @ null λ

=′r

( ) ( ) 2,,~ yxhyxh ′′=′′

( )( )

( )

2

1

NA2

NA2J2,~

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

′

⎟⎠⎞

⎜⎝⎛ ′

=′′

λπ

λπ

r

r

yxh

The two–point resolution problem

object: two point sources,mutually incoherent

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

x′x

imagingsystem intensity

patternobserved(e.g. with

digitalcamera)

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)?

MIT 2.71/2.710 Optics11/10/04 wk10-b-34

The meaning of “resolution”

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

resolve v : 1 to break up into constituent parts: ANALYZE;2 to find an answer to : SOLVE; 3 DETERMINE, DECIDE;4 to 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 Optics11/10/04 wk10-b-35

MIT 2.71/2.710 Optics11/10/04 wk10-b-36

Resolution in optical systemsx

( ) ( )NA61.0

NA0.3 λλ

>=∆r

( )⎟⎟⎠⎞

⎜⎜⎝

⎛+′

NA5.1~ λxh

( )⎟⎟⎠⎞

⎜⎜⎝

⎛−′

NA5.1~ λxh

MIT 2.71/2.710 Optics11/10/04 wk10-b-37

Resolution in optical systemsx

( ) ( )NA61.0

NA0.3 λλ

>=∆r

( )

( )⎟⎟⎠⎞

⎜⎜⎝

⎛−′+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+′

NA5.1~

NA5.1~

λ

λ

xh

xh

MIT 2.71/2.710 Optics11/10/04 wk10-b-38

x

( ) ( )NA61.0

NA4.0 λλ

<=∆r

Resolution in optical systems

( )⎟⎟⎠⎞

⎜⎜⎝

⎛+′

NA2.0~ λxh ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−′

NA2.0~ λxh

MIT 2.71/2.710 Optics11/10/04 wk10-b-39

Resolution in optical systemsx

( ) ( )NA61.0

NA4.0 λλ

<=∆r

( ) ( )⎟⎟⎠⎞

⎜⎜⎝

⎛−′+⎟⎟

⎠

⎞⎜⎜⎝

⎛+′

NA2.0~

NA2.0~ λλ xhxh

MIT 2.71/2.710 Optics11/10/04 wk10-b-40

x

( )NA61.0 λ

=∆r

Resolution in optical systems

( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛+′

NA305.0~ λxh ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−′

NA305.0~ λxh

MIT 2.71/2.710 Optics11/10/04 wk10-b-41

Resolution in optical systemsx

( )NA61.0 λ

=∆r

( ) ( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛−′+⎟⎟

⎠

⎞⎜⎜⎝

⎛+′

NA305.0~

NA305.0~ λλ xhxh

MIT 2.71/2.710 Optics11/10/04 wk10-b-42

Resolution in noisynoisy optical systems

x

( )NA61.0 λ

=∆r

MIT 2.71/2.710 Optics11/10/04 wk10-b-43

x

( )NA22.1 λ

=∆r

“Safe” resolution in optical systems

( ) ( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛−′+⎟⎟

⎠

⎞⎜⎜⎝

⎛+′

NA61.0~

NA61.0~ λλ xhxh

Diffraction–limited resolution (safe)Two point objects are “just resolvablejust 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)

( )NA22.1 λ

=′∆r

( )NA61.0 λ

=′∆r

( )NAλ

=′∆x

( )NA5.0 λ

=′∆x

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

MIT 2.71/2.710 Optics11/10/04 wk10-b-44