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 condition
pupil
mask
arbitrary complex
input transparency
gt(x)
spatially coherent
illumination
gillum(x)
output
field
gPM(x”)
z1
z2
input field
diffraction from the
input field
diffracted field
after lens + pupil mask
wave converging
to form the image
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
Eliminating the quadratic term
pupil
mask
output
field
gPM(x”)
gt(x)
gillum(x)
z1
z2
Solution 1: shape gt(x) as a sphere of radius z1
to
lle
c
co
arbitrary complex
input transparency
gt(x)
r
pupil
mask
gPM(x”)
output
field
gillum(x)
z1
z2
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]
MIT 2.71/2.710
05/04/09 wk13-a- 3
Imaging with a single lens: PSF and ATF
pupil
mask
arbitrary complex
input transparency
gt(x)
spatially coherent
illumination
gillum(x)
output
field
gPM(x”)
z1
z2
input field
diffraction from the
input field
diffracted field
after lens + pupil mask
wave converging
to form the image
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
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
MIT 2.71/2.710
05/04/09 wk13-a- 4
Imaging with a single lens: lateral magnification
pupil
mask
arbitrary complex
input transparency
gt(x)
spatially coherent
illumination
gillum(x)
output
field
gPM(x”)
z1
z2
input field
diffraction from the
input field
diffracted field
after lens + pupil mask
wave converging
to form the image
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 So by ignoring diffraction due to the finite
lateral size and, possibly, phase-delay elements inside the clear aperture of the pupil mask gPM(x”, y”), we have essentially found that the imaging phase factor,
replica of the input field
condition and lateral magnification does not affect the
within the factor of
relationships from geometrical optics image intensity
lateral magnification
remain valid in wave optics as well for the intensity of the optical field.
MIT 2.71/2.710
05/04/09 wk13-a- 5
Block diagrams for coherent and incoherent
linear shift invariant imaging systems
spatially
thin
coherent transparency
illumination
output
field
input
field
cPSF
convolution
4F imaging system:
✳
Fourier
transform
Fourier
transform
ATF
Single lens imaging system:
multiplication
spatially
thin
incoherent transparency
illumination
output
intensity
input
intensity
iPSF
convolution
4F imaging system:
Fourier
transform
Fourier
transform
OTF
Single lens imaging system:
multiplication
MIT 2.71/2.710
05/04/09 wk13-a- 6
✳
ct
lle
pupil mask
gPM(x”,y”)
co
f1
or
e
iv
ct
je
phase object
gt(x,y)=exp{iφ(x,y)}
ob
4F
Example: Zernike phase mask
quasi-monochromatic
illumination
(sp. coherent or
incoherent)
π/2 phase
shift
quasi-monochromatic
illumination
(sp. coherent or
incoherent)
MIT 2.71/2.710
05/04/09 wk13-a- 7
π/2 phase
shift
je
c
pupil mask
gPM(x”,y”)
ob
phase object
gt(x,y)=exp{iφ(x,y)}
phase contrast
image Iout(x’,y’)
λ=1µm
f1=10cm, f2=1cm
z1=11cm, z2=1.1cm
tiv
e
SL
f2
f1
phase contrast
image Iout(x’,y’)
ATF and OTF of the Zernike phase mask
z
See also PP03, pp. 16-17
a=2cm
0.2cm
0.5µm
T2
air
glass,
n =1.5
opaque
opaque
x”
b=4cm
1cm
−0.5
0
x’’ [cm]
0.5
1
1.5
2
phase(g
PM
) [rad]
pi
pi/2
0
−pi/2
−pi
−2
−1.5
−1
−0.5
0
x’’ [cm]
0.5
1
1.5
0.5
0.25
−150
−100
−50
0
u [mm−1]
50
100
1
0.25
−150
−100
−50
0
u [mm−1]
50
100
150
200
0.5
−150
−100
−50
0
u [mm−1
50
100
150
−150
−100
−50
0
u [mm−1
50
100
150
200
0.5
0.25
0
−200
200
4F
1
0.75
SL
1
0.75
0.5
−1.5
−1
−0.5
0
x’’ [cm]
0.5
1
1.5
2
1
OTF [a.u.]
0
−2
0.75
OTF [a.u.]
0.75
0.25
OTF
Re(gPM) [a.u.]
0.5
1
0.25
0
−200
2
0.75
0
−200
200
0.75
1
Im(gPM) [a.u.]
150
Im(ATF) [a.u.]
−1
0.75
0
−200
SL
1
Re(ATF) [a.u.]
Re(ATF) [a.u.]
−1.5
ATF
0.25
Im(ATF) [a.u.]
| [a.u.]
PM
|g
0.5
0
−2
4F
1
1
0.75
0.5
0.5
0.75
0.25
0.25
0.5
0.25
MIT
2.71/2.710
0
−2
−1.5wk13-a−1
−0.5 8 0
05/04/09
x’’ [cm]
0.5
1
1.5
2
0
−200
−150
−100
−50
0
u [mm−1]
50
100
150
200
0
−200
−150
−100
−50
0
u [mm−1]
50
100
150
200
Clear circular aperture
quasi-monochromatic
illumination
(sp. coherent or
incoherent)
or
ct
pupil mask
gPM(x”,y”)=
circ(r”/R)
radius R
(NA)in=R/f1
lle
f1
co
je
ob
object
gt(x,y)
ct
iv
e
4F
(NA)out=R/f2
quasi-monochromatic
illumination
(sp. coherent or
incoherent)
MIT 2.71/2.710
05/04/09 wk13-a- 9
radius R
(NA)in=R/z1
je
c
pupil mask
gPM(x”,y”)=
circ(r”/R)
ob
object
gt(x,y)
phase contrast
image Iout(x’,y’)
λ=1µm
f1=10cm, f2=1cm
z1=cm, z2=cm
tiv
e
SL
f2
f1
(NA)out=R/z2
phase contrast
image Iout(x’,y’)
ATF, cPSF of clear circular aperture
4F
SL
Common expression
MIT 2.71/2.710
for the cPSF
05/04/09 wk13-a-10
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
1
1
Total intensity
Source 1
Source 2
0.75
I [a.u.]
I [a.u.]
0.75
0.5
0.25
0
−10
Total intensity
Source 1
Source 2
0.5
0.25
−8
−6
−4
−2
0
x [µm]
2
4
6
8
10
0
−10
−8
−6
−4
−2
0
x [µm]
2
4
6
Two point sources are well resolved if they are spaced such that:
(i) the PSF diameter
equals the point source spacing
MIT 2.71/2.710
05/04/09 wk13-a-12
(i) the PSF radius
equals the point source spacing
8
10
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2.71 / 2.710 Optics
Spring 2009
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