or
ct
lle
co
je
pupil
mask
ob
input transparency
ideal point source
ct
iv
e
The Point-Spread Function (PSF) of a low-pass filter
PSF
plane
wave
illumination
ideal
(infinitely wide)
spherical wave
ideal
(infinitely wide)
plane wave
truncated
plane wave
wave converging
to form the PSF
at the output plane
Now consider the same 4F system but replace the input transparency with an ideal point source, implemented as an opaque sheet with an infinitesimally small transparent hole and illuminated with a plane wave on axis (actually, any illumination will result in a point source in this case, according to Huygens.)
In Systems terminology, we are exciting this linear system with an impulse (delta-function); therefore, the response is known as Impulse Response. In Optics terminology, we use instead the term Point-Spread Function (PSF) and we denote it as h(x’,y’).
The sequence to compute the PSF of a 4F system is:
➡ observe that the Fourier transform of the input transparency δ(x) is simply 1 everywhere at the pupil plane
➡ multiply 1 by the complex amplitude transmittance of the pupil mask
➡ Fourier transform the product and scale to the output plane coordinates x’=uλf2.
➡Therefore, the PSF is simply the Fourier transform of the pupil mask, scaled to the output coordinates x’=uλf2
MIT 2.71/2.710
04/15/09 wk10-b-22
Example: PSF of a low-pass filter
pupil mask
PSF
1
3
h(x’) [a.u.]
2
2
|gP(x’’)| [a.u.]
0.75
0.5
0.25
0
1
0
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5
x’’ [cm]
The pupil mask is
1
1.5
2
2.5
3
−1
−20
−15
−10
−5
0
x’ [µm]
5
10
15
. If the input transparency is δ(x), the field at the pupil plane to the
right of the pupil mask is
The output field, i.e. the PSF is
The scaling factor (3×) in the PSF ensures that the integral ∫|h(x’)|2dx equals the portion of the input energy
transmitted through the system
MIT 2.71/2.710
04/15/09 wk10-b-23
20
Example: PSF of a phase pupil filter
PSF
5
|h(x’)|2 [a.u.]
|gPM| [a.u.]
pupil
mask
1
0.75
0.5
0.25
0
−4
−3
−2
−1
0
x’’ [cm]
1
2
3
2
0
−20
4
−15
−10
−5
0
x’ [µm]
5
10
15
20
−15
−10
−5
0
x’ [µm]
5
10
15
20
pi
phase[h(x’)] [rad]
phase(gPM) [rad]
3
1
pi
pi/2
0
−pi/2
−pi
−4
4
−3
−2
The pupil mask is The PSF is
MIT 2.71/2.710
04/15/09 wk10-b-24
−1
0
x’’ [cm]
1
2
3
4
pi/2
0
−pi/2
−pi
−20
Comparison: low-pass filter vs phase pupil mask filter
PSF:
phase pupil mask filter
5
8
|h(x’)|2 [a.u.]
|h(x’)|2 [a.u.]
PSF:
low-pass filter
10
6
4
2
0
−20
−15
−10
−5
0
x’ [µm]
5
10
15
2
0
−20
20
−15
−10
−5
0
x’ [µm]
5
10
15
20
−15
−10
−5
0
x’ [µm]
5
10
15
20
pi
phase[h(x’)] [rad]
phase[h(x’)] [rad]
3
1
pi
pi/2
0
−pi/2
−pi
−20
4
−15
−10
MIT 2.71/2.710
04/15/09 wk10-b-25
−5
0
x’ [µm]
5
10
15
20
pi/2
0
−pi/2
−pi
−20
or
ct
lle
co
je
pupil
mask
ob
input transparency
ideal point source
ct
iv
e
Shift invariance of the 4F system
PSF
r
to
lle
c
pupil
mask
co
ob
input transparency
ideal point source
shifted
je
c
tiv
e
plane
wave
illumination
PSF, shifted
plane
wave
illumination
lateral
magnification
MIT 2.71/2.710
04/15/09 wk10-b-26
ct
lle
co
je
ob
arbitrary complex
input transparency
gt(x)
ct
iv
e
pupil
mask
gPM(x”)
or
The significance of the PSF
output
field
spatially coherent
illumination
gillum(x)
input field
diffraction from the
input field
diffracted field
after objective
diffracted field
after pupil mask
wave converging
to form the image
at the output plane
Finally, consider the same 4F system with an input transparency whose transmittance is an arbitrary complex
function gt(x) and the field gin(x) immediately to the right of the input transparency.
According to the sifting theorem for δ-functions, we may express gin(x) as
Physically, this expresses a superposition of point sources weighted by the input field, i.e. Huygens’ principle.
Using the shift invariance property, we find that the field produced at the output plane by each Huygens point
source is the PSF, shifted by −x1×f2 ⁄f1, (note: −f2 ⁄f1 is the lateral magnification) and weighted by gin(x1); i.e.,
the output field is almost a convolution (within a sign &
a scaling factor) of the input field with the PSF:
MIT 2.71/2.710
04/15/09 wk10-b-27
ob
ct
or
0
lle
je
arbitrary complex
input transparency
gt(x)
pupil
mask
gPM(x”)
co
ct
iv
e
The Amplitude Transfer Function (ATF)
output
field
spatially coherent
illumination
gillum(x)
input field
diffraction from the
input field
diffracted field
after objective
diffracted field
after pupil mask
wave converging
to form the image
at the output plane
To avoid the mathematical complications of inversion and lateral magnification at the output plane, we define
the “reduced” output coordinate
field is expressed simply as a convolution
. Inverted and re-sampled in the reduced coordinate, the output
.
Using the convolution theorem of Fourier transforms, we can re-express this input-output relationship in the
spatial frequency domain as
, where H(u) is the amplitude transfer function (ATF).
Inverting the Fourier transform relationship between gPM(x”) and the PSF, we obtain
That is, the ATF of the 4F system is obtained directly from the pupil mask, via a
coordinate scaling transformation.
MIT 2.71/2.710
04/15/09 wk10-b-28
Example: ATFs of the low-pass filter and the phase pupil mask
low pass filter
1
pupil mask
|gPM| [a.u.]
0.75
0.75
0.5
0.25
0
−4
−3
−2
−1
0
x’’ [cm]
1
2
3
4
−3
−2
−1
0
x’’ [cm]
1
2
3
4
0.5
pi
phase(gPM) [rad]
|gP(x’’)|2 [a.u.]
phase pupil filter
pupil mask
1
0.25
0
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5
x’’ [cm]
1
1.5
2
2.5
pi/2
0
−pi/2
−pi
−4
3
ATF
ATF
1
|gPM| [a.u.]
1
0.5
0.25
0
−4
−3
−2
−1
x’’ [cm]-1]
u[cm
0
1
2
3
4
×103
−3
−2
−1
0
x’’ [cm]
1
2
3
4
×103
0.5
pi
phase(gPM) [rad]
|gP(x’’)|2 [a.u.]
0.75
0.75
0.25
0
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5
x’’ [cm]
MIT 2.71/2.710
04/15/09 wk10-b-29
u[cm-1]
1
1.5
2
2.5
3
×103
pi/2
0
−pi/2
−pi
−4
u[cm-1]
Today
• Lateral and angular magnification
• The Numerical Aperture (NA) revisited
• Sampling the space and frequency domains, and
the Space-Bandwidth Product (SBP)
• Pupil engineering
next two weeks
•
•
•
•
•
Depth of focus and depth of field
The angular spectrum
“Non-diffracting” beams
Temporal and spatial coherence
Spatially incoherent imaging
MIT 2.71/2.710
04/22/09 wk11-b- 1
4F imaging as a linear shift invariant system
ob
or
ct
lle
je
arbitrary complex
input transparency
gt(x,y)
reduced
0 coordinates
co
ct
iv
e
pupil
mask
gPM(x”,y”)
output
field
spatially coherent
illumination
gillum(x,y)
input field
diffracted field
after objective
diffraction from the
input field
Illumination:
diffracted field
after pupil mask
Input transparency:
wave converging
to form the image
at the output plane
Field to the right of the input transparency (input field):
Amplitude transfer function (ATF):
Pupil mask:
∝
Point Spread Function (PSF):
∝
in actual coordinates
in reduced coordinates
in actual coordinates
Output field:
MIT 2.71/2.710
04/22/09 wk11-b- 2
∝
∝
in reduced coordinates
ct
lle
co
je
ob
arbitrary complex
input transparency
gt(x,y)
ct
iv
e
pupil
mask
gPM(x”,y”)
or
Lateral magnification
output
field
spatially coherent
illumination
gillum(x,y)
Huygens wavelet
from input field,
departing at x1
spherical
wave
plane wave
Huygens wavelet
converging to form
point image at MTx1=−x1f2/f1
clear pupil mask
An ideal imaging system with a point source as input field should form a point image at the output plane.
Therefore, the ideal PSF is a δ-function. This is of course the limit of Geometrical Optics and in practice
unachievable because of diffraction; alternatively, it implies that the spatial bandwidth is infinite, or that the
lateral extent of the pupil mask is infinite. Both conditions are non-physical; however, for the purpose of
calculating the geometrical magnification, let us indeed assume that
∝
So our calculation has reproduced the Geometrical
Optics result for a telescope with finite conjugates:
MIT 2.71/2.710
04/22/09 wk11-b- 3
ob
or
ct
lle
je
arbitrary complex
input transparency
gt(x,y)
pupil
mask
gPM(x”,y”)
co
ct
iv
e
Angular magnification
output
field
spatially coherent
illumination
gillum(x,y)
plane wave input field
(or angular spectrum
component of input field),
departing at θ1
spherical
wave
focusing
at x”=θ1f1
plane wave exiting
at angle MAθ1=−θ1f1/f2
spatial
frequency
Now let us consider a plane wave input field
Still assuming the ideal geometrical PSF, the output field is
∝
So the angular magnification is
again in agreement with Geometrical Optics.
Based on the interpretation of propagation angle as spatial frequency, the magnification results are also in
agreement with the scaling (similarity) theorem of Fourier transforms:
MIT 2.71/2.710
04/22/09 wk11-b- 4
∝
∝
Example: PM, ATF and PSF for clear apertures
Rectangular
a
b
a/λf1
b/λf1
PM
ATF
PSF
Circular
2R
PM
MIT 2.71/2.710
04/22/09 wk11-b- 5
2R/λf1
ATF
PSF
(NA)in
or
ct
lle
pupil
mask
gPM(x”,y”)
co
je
ob
arbitrary complex
input transparency
gt(x,y)
ct
iv
e
Numerical aperture (NA) and PSF size
output
field
(NA)out
spatially coherent
illumination
gillum(x,y)
on-axis point source
truncated by
pupil-plane mask
plane wave
clear pupil mask
converging spherical wave
The pupil mask is the system aperture (assuming that the lenses are sufficiently large); therefore,
Let Δx’ denote
the half-size of
the PSF main
lobe; then,
MIT 2.71/2.710
04/22/09 wk11-b- 6
Let Δr’ denote
the radius of the
PSF main lobe;
then,
PSF
PSF
A note on sampling
y
input
field
x
spatial
frequency
domain
v
pupil
mask
y”
u
x”
fx,max
δx :pixel size
Δx :field size
δu :frequency resolution
2Δu :frequency bandwidth
δx” :PM pixel size
2Δx” :pupil mask size
Nyquist
sampling
relationships
MIT 2.71/2.710
04/22/09 wk11-b- 7
SpaceBandwidth
Product
(SBP)
Pupil engineering
ct
or
reduced
0 coordinates
lle
co
je
ob
arbitrary complex
input transparency
gt(x,y)
ct
iv
e
pupil
mask
gPM(x”,y”)
output
field
spatially coherent
illumination
gillum(x,y)
input field
diffraction from the
input field
diffracted field
after objective
Amplitude transfer function (ATF):
Point Spread Function (PSF):
Output field:
∝
diffracted field
after pupil mask
wave converging
to form the image
at the output plane
∝
in actual coordinates
in actual coordinates
Pupil engineering is the design of a pupil mask gPM(x”,y”) such that the ATF, the PSF
or the output field meet requirement(s) specified by the user of the imaging system
MIT 2.71/2.710
04/22/09 wk11-b- 8
Example: spatial frequency clipping
f1=20cm
λ=0.5µm
monochromatic
coherent on-axis
illumination
object plane
transparency
MIT 2.71/2.710
04/22/09 wk11-b- 9
pupil mask
circ-aperture
Image plane
observed intensity
Spatial filtering examples: the amplitude MIT pattern
MIT 2.71/2.710
04/22/09 wk11-b- 10
Weak low-pass filtering
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 11
pupil mask
Intensity @ image plane
Moderate low-pass filtering
(moderate blurring)
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 12
pupil mask
Intensity @ image plane
Strong low-pass filtering
(strong blurring)
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 13
pupil mask
Intensity @ image plane
Moderate high-pass filtering
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 14
pupil mask
Intensity @ image plane
Strong high-pass filtering
(edge enhancement)
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 15
pupil mask
Intensity @ image plane
One-dimensional (1D) blur: vertical
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 16
pupil mask
Intensity @ image plane
One-dimensional (1D) blur: horizontal
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 17
pupil mask
Intensity @ image plane
Phase objects
thickness
t(x)
TOP
VIEW
protrusion
(transparent)
glass plate
(transparent)
protrusion
phase-shifts
coherent illumination
by amount φ(x) =2π(n-1)t(x)/λ
This model is often useful for imaging
biological objects (cells, etc.)
If we illuminate this object uniformly (e.g., with a plane wave) the resulting intensity
|gin(x)|2 is also uniform, i.e. the object is invisible.
MIT 2.71/2.710
04/22/09 wk11-b- 18
CROSS
SECTION
To visualize
phase objects in
subsequent
slides, we use the
color
representation of
phase as shown
here. Physically,
phase may be
measured with
interferometry.
The Zernike phase pupil mask
The Zernike mask is a phase pupil mask
used often to visualize input transparencies
that are themselves phase objects. The
Zernike mask imparts phase delay of π/2
near the center of the pupil plane, i.e. at
the lower spatial frequencies. The result at
the output plane is intensity contrast at the
edges of the phase object.
MIT 2.71/2.710
04/22/09 wk11-b- 19
5mm
π/2
1mm
Use of the Zernike phase mask is also
called phase contrast imaging.
ob
plane
wave
illumination
DC term
pupil plane bandwidth
or
ct
π/2 phase
shift
arbitrary
MIT 2.71/2.710
04/22/09 wk11-b- 20
lle
je
phase object
gt(x,y)=exp{iφ(x,y)}
pupil
mask
co
ct
iv
e
Phase contrast (Zernike mask) imaging
φ=π/2
phase contrast
image Iout(x’,y’)
Phase contrast imaging the MIT phase object
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 21
pupil mask
Intensity @ image plane
Phase contrast imaging the MIT phase object
f1=20cm
λ=0.5µm
MIT 2.71/2.710
04/22/09 wk11-b- 22
pupil mask
Intensity @ image plane
The transfer function of Fresnel propagation
Fresnel (free space) propagation may be expressed as a
convolution integral
MIT 2.71/2.710
04/08/09 wk9-b-26
The Talbot effect
x
x
y
MIT 2.71/2.710
04/08/09 wk9-b-27
z
y
z
Talbot effect: still shots
1000
1000
0
intensity at
z = 10, 000λ
−500
0
x(!)
500
y(!)
y(!)
−500
MIT 2.71/2.710
04/08/09 wk9-b- 28
−500
0
x(!)
500
1000
intensity at
z = 20, 000λ
500
0
−1000
−1000
0
−1000
−1000
1000
1000
500
Talbot
plane
(phase
reversal)
Talbot
subplane
−500
−500
−1000
−1000
1000
intensity at
z = 5, 000λ
500
y(!)
y(!)
intensity at
z = 0 500
Talbot
plane
(in phase)
0
−500
−500
0
x(!)
500
1000
−1000
−1000
−500
0
x(!)
500
1000
Physical explanation of the Talbot effect
one
plane
wave
three
plane
waves
!
∆r ≡ z 2 + x2 − z
er
ord
1
+
st
x
≈
x = z tan θ ≈ θ
≈
phase delay
∆r
∆φ = 2π
λ
≈
x2
2z
λ
Λ
π
λz
.
Λ2
–1 st
MIT 2.71/2.710
04/08/09 wk9-b-29
ord
er
refe
z
renc
(rad e sphere
ius z
)
0th order
0th, +1st, -1st orders are in phase
∆φ = 0, ±2π, ±4π, . . .
replicates field at grating
1
2
∆φ = ±π, ±3π, . . .
period shift wrt grating
∆φ = ± π2 , ± 32π , . . .
+1st , − 1st orders interfere
destructively
Mathematical derivation of the Talbot effect /1
•
Field immediately past grating
" x #$
1!
g(x; 0) =
1 + m cos 2π
2
Λ
"
"
1!
m
x# m
x #$
– also expressed as g(x; 0) =
1+
exp i2π
+
exp −i2π
2
2
Λ
2
Λ
•
Fourier transform (spatial spectrum) of field past grating
!
"
#
"
#$
1
m
1
m
1
G(u; 0) =
δ(0) + δ u −
+ δ u+
2
2
Λ
2
Λ
• Fourier transform (spatial spectrum) of Fresnel propagation kernel
!
$
" 2
#
H(u, v; z) = exp − iπλz u + v 2
!
"
1
λz
– value at grating’s spatial frequencies H(± , 0) = exp −iπ 2
Λ
Λ
MIT 2.71/2.710
04/08/09 wk9-b-30
Mathematical derivation of the Talbot effect /2
•
Fourier transform (spatial spectrum) of field propagated to distance z,
G(u, v; z)
=
=
=
=
•
(u, v; 0) × H(u, v; z)
G
!
"
#
"
#$
1
m
1
m
1
× H(u, v; z)
δ(0) + δ u −
+ δ u+
2
2
2
Λ
Λ
!
"
# "
#
1
m
1
1
H(0, 0; z)δ(0) + H
, 0; z δ u −
2
2
Λ
Λ
"
# "
#$
m
1
1
+ H − , 0; z δ u +
2
Λ
Λ
!
"
# "
#
1
m
λz
1
δ(0) + exp −iπ 2 δ u −
2
2
Λ
Λ
"
# "
#$
m
λz
1
+ exp −iπ 2 δ u +
2
Λ
Λ
Inverse Fourier transforming,
!
"
#
"
#
&
$
$
1
m
λz
x% m
λz
x%
g(x; z) =
1+
exp −iπ 2 exp i2π
+
exp −iπ 2 exp −i2π
2
2
Λ
Λ
2
Λ
Λ
MIT 2.71/2.710
04/08/09 wk9-b-31
Mathematical derivation of the Talbot effect /3
•
Our previous result
!
"
#
"
#
&
$
$
1
m
λz
x% m
λz
x%
g(x; z) =
1+
exp −iπ 2 exp i2π
+
exp −iπ 2 exp −i2π
2
2
Λ
Λ
2
Λ
Λ
– also expressed as
!
"
#
$ x %&
1
λz
g(x; z) =
1 + m exp −iπ 2 cos 2π
2
Λ
Λ
– intensity is
2
I(x; z) = |g(x; z)|
!
$
%
" x#
" x #&
1
λz
=
1 + m2 cos2 2π
+ 2m cos π 2 cos 2π
4
Λ
Λ
Λ
– note intensity immediately after grating
! "
# x $%&2
1
I(x; 0) =
1 + m cos 2π
2
Λ
# x$
# x $(
1'
2
2
=
1 + m cos 2π
+ 2m cos 2π
4
Λ
Λ
MIT 2.71/2.710
04/08/09 wk9-b-32
Mathematical derivation of the Talbot effect /4
•
Special cases of propagation distance
Talbot plane
(shifted) Talbot plane
Talbot subplane
MIT 2.71/2.710
04/08/09 wk9-b-33
Talbot plane
etc.
Talbot subplane
Talbot plane
f
ao
ic g
pl in
re rat
g
sh
d
rio
pe ft
i
cy
en g
qu lin
fre oub
d
2
MIT 2.71/2.710
04/08/09 wk9-b-34
1/
cy
en g
qu lin
fre oub
d
Talbot subplane
The Talbot planes
MIT OpenCourseWare
http://ocw.mit.edu
2.71 / 2.710 Optics
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.