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