Today
• Spatially coherent and incoherent imaging with a single lens – re-derivation of the single-lens imaging condition – ATF/OTF/PSF and the Numerical Aperture – resolution in optical systems – pupil engineering revisited
next week • Two more applications of the MTF
– defocus – diffractive optics and holography
• Multi-pass interferometers: Fabry-Perot – optical resonators and Lasers
• Beyond scalar optics: polarization and resolution
MIT 2.71/2.710 05/04/09 wk13-a- 1
Imaging with a single lens: imaging conditionpupil mask
diffracted field after lens + pupil mask
output field
arbitrary complex input transparency
gt(x)
diffraction from the input field
wave converging to form the image
input field
spatially coherent illumination
gillum(x)
gPM(x”)
z1 z2
at the output plane The field at the output plane of this imaging system is (derivation in the supplement to this lecture)
We also need to eliminate this quadratic term
To eliminate the quadratic term we satisfy the Lens Law of geometrical optics
After eliminating the quadratic, the remaining integral is the Fourier transform of the pupil mask gPM(x”, y”).
MIT 2.71/2.710 05/04/09 wk13-a- 2
MIT 2.71/2.710 05/04/09 wk13-a-
Eliminating the quadratic term
3
pupil mask
colle
ctor
output field
arbitrary complex input transparency
gt(x) gPM(x”)
z1 z2
gillum(x)
Solution 2: attach a lens of focal length z1 to gt(x) Solution 3: limit gt(x) to ¼ the lateral size of gPM(x”) [see Goodman 5.3.2 and Ref. 303]
pupil mask
output field
gillum(x)
gPM(x”)
z1 z2
Solution 1: shape gt(x) as a sphere of radius z1
gt(x)
Imaging with a single lens: PSF and ATFpupil mask
diffracted field after lens + pupil mask
output field
arbitrary complex input transparency
gt(x)
diffraction from the input field
wave converging to form the image
input field
spatially coherent illumination
gillum(x)
gPM(x”)
z1 z2
at the output plane Assuming one of the three conditions is satisfied and the last remaining quadratic term can be eliminated, the output field is
MIT 2.71/2.710 05/04/09 wk13-a- 4
where is the PSF,
i.e. the Fourier transform of the pupil mask scaled so that (x”,y”)=(λz1u,λz1v). As in the 4F system,
the scaled complex transmissivity of the pupil mask is the ATF
Imaging with a single lens: lateral magnificationpupil mask
diffracted field after lens + pupil mask
output field
arbitrary complex input transparency
gt(x)
diffraction from the input field
wave converging to form the image
input field
spatially coherent illumination
gillum(x)
gPM(x”)
z1 z2
at the output plane If the pupil mask gPM(x”, y”) is infinitely large and clear, its Fourier transform is approximated as a δ-function. Therefore, the optical field at the output plane is
replica of the input fieldwithin the factor of
lateral magnification
So by ignoring diffraction due to the finitelateral size and, possibly, phase-delay
elements inside the clear aperture of the pupil mask gPM(x”, y”), we have
essentially found that the imaging condition and lateral magnification
relationships from geometrical optics remain valid in wave optics as well for the intensity of the optical field.
phase factor, does not affect the
image intensity
MIT 2.71/2.710 05/04/09 wk13-a- 5
Block diagrams for coherent and incoherentlinear shift invariant imaging systems
thin transparency
cPSF
ATF
Fourier transform
output field
convolution
multiplication
Fourier transform
spatially coherent
illumination
input field
✳
4F imaging system:
Single lens imaging system:
spatially incoherent
thin input intensityoutput
transparencyillumination
iPSF intensity
✳ convolution
Fourier Fourier transform transform
OTF
multiplication
MIT 2.71/2.710 05/04/09 wk13-a- 6
4F imaging system:
Single lens imaging system:
Example: Zernike phase mask4F
pupil mask gPM(x”,y”) phase contrastphase object co
llecto
r
objec
tive
π/2 phase shift λ=1µm
f1=10cm, f2=1cm z1=11cm, z2=1.1cm
f1 f2
quasi-monochromatic illumination
(sp. coherent or incoherent)
gt(x,y)=exp{iφ(x,y)} image Iout(x’,y’)
MIT 2.71/2.710 05/04/09 wk13-a- 7
objec
tive
π/2 phase shift
phase object gt(x,y)=exp{iφ(x,y)}
phase contrast image Iout(x’,y’)
pupil mask gPM(x”,y”)
f1
quasi-monochromatic illumination
(sp. coherent or incoherent)
SL
MIT 2.71/2.71005/04/09 wk13-a- 8
ATF and OTF of the Zernike phase mask
a=2cm
b=4cm
0.5µm
glass,
n =1.5
x”
air
z
T2
opaque opaque
0.2cm
1cm
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.25
0.5
0.75
1
x’’ [cm]
Re(
g PM) [
a.u.
]
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.25
0.5
0.75
1
x’’ [cm]
Im(g
PM) [
a.u.
]
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.25
0.5
0.75
1
x’’ [cm]
|gPM
| [a.
u.]
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −pi
−pi/2
0
pi/2
pi
x’’ [cm]
phas
e(g PM
) [ra
d]
−200 −150 −100 −50 0 50 100 150 2000
0.25
0.5
0.75
1
u [mm−1]R
e(AT
F) [a
.u.]
−200 −150 −100 −50 0 50 100 150 2000
0.25
0.5
0.75
1
u [mm−1
Im(A
TF) [
a.u.
]−200 −150 −100 −50 0 50 100 150 200
0
0.25
0.5
0.75
1
u [mm−1]
Re(
ATF)
[a.u
.]
−200 −150 −100 −50 0 50 100 150 2000
0.25
0.5
0.75
1
u [mm−1
Im(A
TF) [
a.u.
]
−200 −150 −100 −50 0 50 100 150 2000
0.25
0.5
0.75
1
u [mm−1]
OTF
[a.u
.]
−200 −150 −100 −50 0 50 100 150 2000
0.25
0.5
0.75
1
u [mm−1]
OTF
[a.u
.]
ATF
OTF
4F SL
4F SL
See also PP03, pp. 16-17
Clear circular aperture
pupil mask gPM(x”,y”)= circ(r”/R)
colle
ctor
objec
tive
radius R (NA)in=R/f1
object gt(x,y)
phase contrast image Iout(x’,y’)
λ=1µm f1=10cm, f2=1cm
z1=cm, z2=cm
f1 f2
4F
quasi-monochromatic illumination
(sp. coherent or incoherent)
(NA)out=R/f2
objec
tive
object gt(x,y)
phase contrast image Iout(x’,y’)
f1 SL
quasi-monochromatic illumination
(sp. coherent or incoherent)
radius R (NA)in=R/z1 (NA)out=R/z2
pupil mask gPM(x”,y”)= circ(r”/R)
MIT 2.71/2.710 05/04/09 wk13-a- 9
ATF, cPSF of clear circular aperture
-a-10MIT 2.71/2.710 05/04/09 wk13
Common expression for the cPSF
4F
SL
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 action of solving, also : SOLUTION; 3 the quality of being resolute: FIRMNESS, DETERMINATION; 4 a formal statement expressing the opinion, will or, intent of a body of persons
MIT 2.71/2.710 05/04/09 wk13-a- 11
Rayleigh resolution limit
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.25
0.5
0.75
1
x [µm]
I [a.
u.]
Total intensitySource 1Source 2
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.25
0.5
0.75
1
x [µm]
I [a.
u.]
Total intensitySource 1Source 2
Two point sources are well resolved if they are spaced such that: (i) the PSF diameter (i) the PSF radius equals the point source spacing equals the point source spacing
MIT 2.71/2.710 05/04/09 wk13-a-12
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2.71 / 2.710 Optics Spring 2009
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