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• Resolution
• Wavefront modulation
MIT 2.71/2.710 Optics11/24/04 wk12-b-1
Resolution
MIT 2.71/2.710 Optics11/24/04 wk12-b-2
Coherent imaging as a linear, shift-invariant system
MIT 2.71/2.710 Optics11/24/04 wk12-b-3
Thin transparency( )yxt ,
( )yxg ,1
( ) ),( ,),(
1
2
yxtyxgyxg
==
output amplitude
impulse response ( )),(),(
,
2
3
yxhyxgyxg∗=
=′′
convolutionillumination(field)
Fourier transform
Fourier transform
transfer function(≡plane wave spectrum) ),(),(
),(
2
3
vuHvuGvuG
==( )vuG ,2
multiplication
transfer function of coherent system H(u ,v): aka amplitude transfer function
Incoherent imaging as a linear, shift-invariant system
MIT 2.71/2.710 Optics11/24/04 wk12-b-4
Thin transparency( )yxt ,
( )yxI ,1
( ) 21
2
),( ,
),(
yxtyxI
yxI
=
=
( )vuI ,2̂
incoherentimpulse response ( )
22
3
),(),(
,
yxhyxI
yxI
∗=
=′′
output intensity
convolutionillumination
(intensity)Fourier
transformFourier
transform
transfer function
),(~),(ˆ),(ˆ
2
3
vuHvuI
vuI
=
=
multiplication
( )vuH ,~ optical transfer function (OTF)transfer function of incoherent system:
Transfer Functions of Clear Aperture
MIT 2.71/2.710 Optics11/24/04 wk12-b-5
( ) ( ){ }( ) ( )
( )∫∫∫∫
′′′′
′′−′−′′′=
ℑ≡
vuvuH
vuvvvuuHvuH
yxhvuH
dd,
dd ,,
1 tonormalized , ,~
2
*
2
umax–umax
1
2umax–2umax
real(H) ( )H~real
1
1D amplitude transfer function (ATF) 1D optical transfer function (OTF)
( ) ( ){ }yxhvuH ,, ℑ≡
CoherentCoherent IncoherentIncoherent
PSF
Connection between PSF and NA
MIT 2.71/2.710 Optics11/24/04 wk12-b-6
( ) ( ) ( )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 Optics11/24/04 wk12-b-7
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 Optics11/24/04 wk12-b-8
Connection between PSF and NA
( )NA61.0 @ null λ
=′r
( )( )
( )λ
π
λπ
r
r
yxh ′
⎟⎠⎞
⎜⎝⎛ ′
=′′NA2
NA2J2,
1
MIT 2.71/2.710 Optics11/24/04 wk12-b-9
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 Optics11/24/04 wk12-b-10
MIT 2.71/2.710 Optics11/24/04 wk12-b-11
The incoherent case:
( )NA61.0 @ null λ
=′r
( ) ( ) 2,,~ yxhyxh ′′=′′
( )( )
( )
2
1
NA2
NA2J2,~
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′
⎟⎠⎞
⎜⎝⎛ ′
=′′
λπ
λπ
r
r
yxh
NA in unit–mag imaging systems1f 1f
monochromaticcoherent on-axis
illumination
x ′′ x′x1f 1f
2R
MIT 2.71/2.710 Optics11/24/04 wk12-b-12
12 f x ′′ x′xmonochromaticcoherent on-axis
illumination
12 f2R
( )1
NAfR
≡
( )12
NAf
R≡
( ) ( ) ( ) ⎟⎠⎞
⎜⎝⎛ ′
=′=′′=λrrhyxh NA2jinc,PSFin both cases,
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 Optics11/24/04 wk12-b-13
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 Optics11/24/04 wk12-b-14
MIT 2.71/2.710 Optics11/24/04 wk12-b-15
Resolution in optical systemsx
( ) ( )NA61.0
NA0.3 λλ
>=∆r
( )⎟⎟⎠⎞
⎜⎜⎝
⎛+′
NA5.1~ λxh
( )⎟⎟⎠⎞
⎜⎜⎝
⎛−′
NA5.1~ λxh
MIT 2.71/2.710 Optics11/24/04 wk12-b-16
Resolution in optical systemsx
( ) ( )NA61.0
NA0.3 λλ
>=∆r
( )
( )⎟⎟⎠⎞
⎜⎜⎝
⎛−′+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+′
NA5.1~
NA5.1~
λ
λ
xh
xh
MIT 2.71/2.710 Optics11/24/04 wk12-b-17
x
( ) ( )NA61.0
NA4.0 λλ
<=∆r
Resolution in optical systems
( )⎟⎟⎠⎞
⎜⎜⎝
⎛+′
NA2.0~ λxh ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−′
NA2.0~ λxh
MIT 2.71/2.710 Optics11/24/04 wk12-b-18
Resolution in optical systemsx
( ) ( )NA61.0
NA4.0 λλ
<=∆r
( ) ( )⎟⎟⎠⎞
⎜⎜⎝
⎛−′+⎟⎟
⎠
⎞⎜⎜⎝
⎛+′
NA2.0~
NA2.0~ λλ xhxh
MIT 2.71/2.710 Optics11/24/04 wk12-b-19
x
( )NA61.0 λ
=∆r
Resolution in optical systems
( ) ⎟⎟⎠⎞
⎜⎜⎝
⎛+′
NA305.0~ λxh ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−′
NA305.0~ λxh
MIT 2.71/2.710 Optics11/24/04 wk12-b-20
Resolution in optical systemsx
( )NA61.0 λ
=∆r
( ) ( ) ⎟⎟⎠⎞
⎜⎜⎝
⎛−′+⎟⎟
⎠
⎞⎜⎜⎝
⎛+′
NA305.0~
NA305.0~ λλ xhxh
MIT 2.71/2.710 Optics11/24/04 wk12-b-21
Resolution in noisynoisy optical systems
x
( )NA61.0 λ
=∆r
MIT 2.71/2.710 Optics11/24/04 wk12-b-22
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 Optics11/24/04 wk12-b-23
Also affecting resolution: aberrationsAll our calculations have assumed “geometrically perfect”
systems, i.e. we calculated the wave–optics behavior ofsystems which, in the paraxial geometrical optics approximation
would 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 descriptionMIT 2.71/2.710 Optics11/24/04 wk12-b-24
Also affecting resolution: aberrations
2sx,max–2sx,max
H~
1
“diffractiondiffraction––limitedlimited”(aberration–free) 1D MTF
2sx,max–2sx,max
H~
1
1D MTF with aberrations
ℑ Fouriertransform ℑ Fourier
transformdiffraction–limited 1D PSF(sinc2)
somethingwider
wave optics picture
MIT 2.71/2.710 Optics11/24/04 wk12-b-25
MIT 2.71/2.710 Optics11/24/04 wk12-b-26
Typical result of optical design(FoV)
field of viewof the system
MTF is neardiffraction–limited
near the centerof the field
MTF degradestowards thefield edges
shiftshiftvariantvariantopticalopticalsystemsystem
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 Optics11/24/04 wk12-b-27
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 thinkingthinking 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 Optics11/24/04 wk12-b-28
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 dependentnoise dependent and gradualgradual.
MIT 2.71/2.710 Optics11/24/04 wk12-b-29
Common misinterpretations
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 Optics11/24/04 wk12-b-30
Common misinterpretationsSuper-resolution
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.
Pupil function design always results in(i) narrower main lobe but accentuated
side–lobes(ii) lower power transmitted through the
systemBoth effects are BADBAD on the image
MIT 2.71/2.710 Optics11/24/04 wk12-b-31
Apodization
MIT 2.71/2.710 Optics11/24/04 wk12-b-32
f1=20cmλ=0.5µm ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′′−⎟⎠⎞
⎜⎝⎛ ′′
=′′2
circcircRr
RrrH
Apodization
MIT 2.71/2.710 Optics11/24/04 wk12-b-33
f1=20cmλ=0.5µm ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′′−×⎟
⎠⎞
⎜⎝⎛ ′′
=′′ 20
2
2expcirc
Rr
RrrH
Unapodized (clear–aperture) MTF
MIT 2.71/2.710 Optics11/24/04 wk12-b-34
f1=20cmλ=0.5µm ( ) ⎟
⎠⎞
⎜⎝⎛ ′′
⊗⎟⎠⎞
⎜⎝⎛ ′′
=′′Rr
RrrH circcirc~
auto-correlation
Unapodized (clear–aperture) MTF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-35
Unapodized (clear–aperture) PSF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-36
Apodized (annular) MTF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-37
Apodized (annular) PSF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-38
Apodized (Gaussian) MTF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-39
Apodized (Gaussian) PSF
f1=20cmλ=0.5µm
MIT 2.71/2.710 Optics11/24/04 wk12-b-40
Conclusions (?)
• Annular–type pupil functions typically narrow the main lobe of the PSF at the expense of higher side lobes
• Gaussian–type pupil functions typically suppress the side lobes but broaden 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 apodizers very 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 Optics11/24/04 wk12-b-41
Common misinterpretationsSuper-resolution
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.
main lobe size ↓ ⇔ sidelobes ↑and vice versa
main lobe size ↑ ⇔ sidelobes ↓
power loss an important factor
compromise application dependentMIT 2.71/2.710 Optics11/24/04 wk12-b-42
Common misinterpretations
“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
spacespace––bandwidth product (SBP)bandwidth product (SBP)of the camera
MIT 2.71/2.710 Optics11/24/04 wk12-b-43
What can a camera resolve?Answer depends on the magnification and
PSF of the optical system attached to the camera
PSF of opticalsystem
pixels oncamera die
Pixels significantly smaller than the system PSFare somewhat underutilized (the effective SBP is reduced)
MIT 2.71/2.710 Optics11/24/04 wk12-b-44
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×M pixels– 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 Optics11/24/04 wk12-b-45
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 related to the NA but not exclusively limited 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 limit).
MIT 2.71/2.710 Optics11/24/04 wk12-b-46
Wavefront modulation
• Photographic film• Spatial light modulators• Binary optics
MIT 2.71/2.710 Optics11/24/04 wk12-b-47
Photographic films / platesprotective layer
emulsion with silver halide (e.g. AgBr)grains
base (glass, mylar, acetate)
Exposure: Ag+ + e– → Ag (or 2Ag+ + 2e– → Ag2 )
development speckCollection of development specks = latent image
Development (1st chemical bath): converts specks to metallic silver
Fixing the emulsion (2nd chemical bath): removal of unexposed silver halide
MIT 2.71/2.710 Optics11/24/04 wk12-b-48
Photographic film / plates• Exposure (energy) : energy incident per unit area on a photographic
emulsion during the exposure process (units: mJ/cm2)
• Intensity transmittance : average ratio of intensity transmitted over intensity incident after development
• Photographic density
Exposure = incident intensity × exposure time E=Iexpose × T
( )⎭⎬⎫
⎩⎨⎧
=yxI
yxIyx,at incident
,at ed transmittaverage
local,τ
DD −=⇔⎟⎠⎞
⎜⎝⎛= 10 1log10 ττ
MIT 2.71/2.710 Optics11/24/04 wk12-b-49
Photographic film / plates• Hurter-Driffield curve • Gamma curve
γ high/low : high/low contrast film
log E
D
Gross fog
ToeLinear region(slope γ)
Shoulder
Saturationregion
development time
γ
1
2
5 10 15 (min)
MIT 2.71/2.710 Optics11/24/04 wk12-b-50
Kelley model of photographic processOptical imagingduring exposure
(linear)
H&D curve
(nonlinear)
additional blurdue to chemicaldiffusion(linear)
MIT 2.71/2.710 Optics11/24/04 wk12-b-51
log E
DH&D curve
measureddensity
inferred logarithmicexposure
The Modulation Transfer Function
Exposure:adjacency effect
(due to chemical diffusion)MxuEEE 010 2cos π+=
MIT 2.71/2.710 Optics11/24/04 wk12-b-52
( ) xuEuMEE 0100 2cos π+=′u0
“Effective exposure”:
1.0
0.5
Bleaching / phase modulation
exposure
emulsion
MIT 2.71/2.710 Optics11/24/04 wk12-b-53
( )zyx ,,ε∆
tanning bleach(relief grating)
non-tanning bleach(index grating)
Spatial Light Modulators
• Liquid crystals• Magneto-Optic• Micro-mirror• Grating Light Valve• Multiple Quantum Well• Acousto-Optic
MIT 2.71/2.710 Optics11/24/04 wk12-b-54
Liquid crystal modulators
• Nematic• Smectic (smectic-C* phase: ferroelectric)• Cholesteric
MIT 2.71/2.710 Optics11/24/04 wk12-b-55
crossed polarizers
OFF
crossed polarizers
V(acts as λ/2 plate;rotates polarizationby 90°)
Micro-mirror technology
Images removed due to copyright concerns
Lucent (Bell Labs)
Texas Instruments DMD/DLPSandiaMIT 2.71/2.710 Optics
11/24/04 wk12-b-56
Micro-mirror display
Image removed due to copyright concerns
http://www.howstuffworks.comMIT 2.71/2.710 Optics11/24/04 wk12-b-57
Micromirrors for adaptive optics
Image removed due to copyright concerns
Véran, J.-P. & Durand, D. 2000, ASP Conf. Ser 216, 345 (2000). MIT 2.71/2.710 Optics11/24/04 wk12-b-58
Grating Light Valve (GLV) display
Images removed due to copyright concerns
www.meko.co.uk
Silicon Light Machines, www.siliconlight.com
MIT 2.71/2.710 Optics11/24/04 wk12-b-59
Binary Optics
Refractive Diffractive Binary(prism) (blazed grating)
efficiency of 1st diffracted order fromstep-wise (binary) approximation toblazed grating with N steps over 2π range
⎟⎠⎞
⎜⎝⎛= N2
1sinc21η
MIT 2.71/2.710 Optics11/24/04 wk12-b-60
Binary Optics: binary grating
0
( )xtAmplitude maskAmplitude mask1 X
x
MIT 2.71/2.710 Optics11/24/04 wk12-b-61
Binary Optics: binary grating( )xtPhase maskPhase mask
1 X
x
–1
MIT 2.71/2.710 Optics11/24/04 wk12-b-62
( ) ( )[ ]xixt φ exp=0
π( )xφ X
x
Fourier series for binary phase grating
1( )xt X
x
–1
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
MIT 2.71/2.710 Optics11/24/04 wk12-b-63
Fourier series for binary phase grating
1( )xt X
x
–1
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
identify physical meaning:• plane waves• orientation of nth plane wave:
• diffracted orders
Xnn λθ =sin
MIT 2.71/2.710 Optics11/24/04 wk12-b-64
Fourier series for binary phase grating
1( )xt X
x
–1
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
nθidentify physical meaning:• plane waves• orientation of nth plane wave:
• diffracted orders
Xnn λθ =sin
MIT 2.71/2.710 Optics11/24/04 wk12-b-65
Fourier series for binary phase grating
1( )xt X
x
–1
identify physical meaning:• plane waves• orientation of nth plane wave:
• diffracted orders
Xnn λθ =sin
nθ
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
only! orders odd 2
sinc ⇒⎟⎠⎞
⎜⎝⎛ n
MIT 2.71/2.710 Optics11/24/04 wk12-b-66
Fourier series for binary phase grating
1( )xt X
x
–1
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
Fourier transform:•
• result is
( ) ( )002exp uuxui −↔δπ
⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛∑
+∞
−∞= Xnun
nδ
2sinc
0≠n
(Fourier series)
MIT 2.71/2.710 Optics11/24/04 wk12-b-67
Diffracted spectrum from binary phase grating
1( )xt X
x
–1
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnxin
nπ2exp
2sinc
0≠n
ℑ ℑ
1( ) uT
1/X
0
⎟⎠⎞
⎜⎝⎛
2sinc n
⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛= ∑
+∞
−∞= Xnun
nδ
2sinc
0≠n
u
MIT 2.71/2.710 Optics11/24/04 wk12-b-68
Fourier-plane diffraction from finite binary phase grating
MIT 2.71/2.710 Optics11/24/04 wk12-b-69
f f
(x” not to scale)
