diffraction analysis of 2-d pupil remapping j i for high ... · -0.2 0 0.2 0.4 2nd pupil phase map...
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Diffraction Analysis of 2-D Pupil Remapping
for High-Contrast Imaging
Robert J. Vanderbei
SPIE San DiegoAug 4, 2005
Princeton Universityhttp://www.princeton.edu/∼rvdb
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Contents1 Apodization 3
2 Pupil Mapping 5
3 Notations 6
4 Huygens Wavelets 7
5 Fresnel Approximation 8
6 Fresnel Examples 9
7 Better Than Fresnel (BTF) 12
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1. Apodization
-0.5 0 0.5
-1
-0.5
0
0.5
1
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-0.5 0 0.50
0.5
1
1.5
2
2.5
3
3.5
4
-60 -40 -20 0 20 40 60-180
-160
-140
-120
-100
-80
-60
-40
-20
0
E(ξ, η) =1
λf
∫∫e2πi xξ+yη
λf A(√
x2 + y2)dydx
E(ρ) =2π
λf
∫J0
(2π
rρ
λf
)A(r)rdr.
Psf(ρ) = |E(ρ)|2
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2. Pupil Mapping
Advantages:
• 100% throughput
• Implicit magnification...effectively iwa ≈ 1λ/D.
Disadvantages:
• Diffraction effects limitachievable contrast to 10−5
for a pure pupil-mappingsystem.
−0.5 0 0.5−0.5
0
0.5
1
1.5
2
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3. Notations
R(r) = ±
√∫ r
02A2(s)sds.
∂h
∂r(r) =
r − R(r)√Q2
0 + (n2 − 1)(r − R(r))2
∂h
∂r(r) =
R(r)− r√Q2
0 + (n2 − 1)(R(r)− r)2
n is the refractive index and
Q0 = −(n− 1)z
= S(R(r), r) + n(h(r)− h(R(r)))h(r)
r
h(r)
r R(r)
R(r)
z
Z
0
Notes:
• the second expression for Q0 is in fact independent of r.
• for mirrors, put n = −1.
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4. Huygens Wavelets
Cartesian Coordinates
Eout(x, y) =
∫∫1
λQ(x, y, x, y)e2πiQ(x,y,x,y)/λdydx,
where Q denotes the optical path length:
Q(x, y, x, y) =
√(x− x)2 + (y − y)2 + (h(r)− h(r))2 + n(Z − h(r) + h(r))
Polar Coordinates
Eout(r) =
∫∫1
λQ(r, r, θ)e2πiQ(r,r,θ)/λrdθdr,
where
Q(r, r, θ) =
√r2 − 2rr cos θ + r2 + (h(r)− h(r))2 + n(Z − h(r) + h(r)).
Double integrals are hard!
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5. Fresnel Approximation
Assuming a large separation between the lenses and that the lensesare themselves thin, we get
√r2 − 2rr cos θ + r2 + (h(r)− h(r))2
≈ (h(r)− h(r)) +r2 − 2rr cos θ + r2
2z.
Fresnel uses this approximation in the exponential (and the “parax-ial” approximation in the leading factor) to get the standard Fresnelapproximation:
Eout(r) =2π
λZeπi r2
zλ+2πi (n−1)h(r)λ
∫eπi r2
zλ−2πi (n−1)h(r)λ J0(2πrr/zλ)rdr.
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6. Fresnel Examples
Flat Glass A ≡ 1
0 0.005 0.01 0.0150.4
0.6
0.8
1
1.2
1.42nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
6.1 6.2 6.3 6.4
x 10-3
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04Amplitude Map Zoomed
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.015-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.42nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-8
10-6
10-4
10-2
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
n = 1.5. D = 25mm. z = 15D. λ = 632.8nm.
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Galilean Telescope: A ≡ a
R(r) = ar R(r) = r/a
h(r) = z +
√Q2
0 + (n2 − 1)(1− 1/a)2r2 − |Q0|(n2 − 1)(1− 1/a)
h(r) =
√Q2
0 + (n2 − 1)(a− 1)2r2 − |Q0|(n2 − 1)(a− 1)
If n > 1, then both lenses are hyperbolic.
If n < 1, then both lenses are elliptical.
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Galilean Telescope A ≡ 3
0 2 4 6 8
x 10-3
1
2
3
4
5
62nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.01 0.02 0.030
0.5
1
1.5x 10-3 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 2 4 6 8
x 10-3
-10
-5
0
5
10
15
202nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-10
10-5
100
105Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
Fresnel approximation is very bad.
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7. Better Than Fresnel (BTF)
We retain the paraxial approximation for the leading factor.
We can subtract an arbitrary constant from the Q in the exponent.
Let’s choose to subtract Q(r, R(r), 0). We compute:
Q(r, r, θ)−Q(r, R(r), 0) = S(r, r, θ)− S(r, R(r), 0) + n(h(R(r))− h(r))
=S2(r, r, θ)− S2(r, R(r), 0)
S(r, r, θ) + S(r, R(r), 0)+ n(h(R(r))− h(r))
Then, we simplify the numerator:
S2(r, r, θ)− S2(r, R(r), 0) = (r −R(r))(r + R(r))− 2r(r cos θ −R(r)
)+
(h(r)− h(R(r))
) (h(r) + h(R(r))− 2h(r)
).
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Mo’ Better
So far, everything is exact (except for the paraxial approximation).The only other approximation is to replace S(r, r, θ) in the de-nominator with S(r, R(r), 0) so that the denominator becomes just2S(r, R(r), 0). Replacing the integral on θ with the appropriateBessel function, we get a new approximation:
Eout(r) ≈ 2π
λZ
∫e2πi
�r2−R(r)2+2rR(r)+(h(r)−h(R(r)))(h(r)+h(R(r))−2h(r))
2S(r,R(r),0)λ+n
λ(h(R(r))−h(r))
�
·J0
(2πrr
λS(r, R(r), 0)
)rdr.
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Example: A ≡ 3
0 2 4 6 8
x 10-3
0
1
2
3
4
52nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.01 0.02 0.030
0.5
1
1.5x 10-3 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 2 4 6 8
x 10-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.42nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-8
10-6
10-4
10-2
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
This approximation is much better than Fresnel.
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Pupil Mapping for High Contrast (PIAA)
0 0.005 0.01 0.0150
1
2
3
4
52nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.0150
1
2
3x 10-4 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.005 0.01 0.015-0.2
0
0.2
0.4
0.6
0.82nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-20
10-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
Designed for 10−10. Delivers 10−5.
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Brute Force Huygens Wavelet Integral
0 0.005 0.01 0.0150
1
2
3
4
52nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.0150
1
2
3x 10-4 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.005 0.01 0.015-0.4
-0.2
0
0.2
0.4
0.6
0.8
12nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-20
10-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
Full 2D integration: 500 r-values and 500 θ-values.Increased the wavelength by a factor 10.The degradation is fundamental physics, not numerical error.
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How about larger lenses?
0 0.5 1 1.50
1
2
3
42nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.5 1 1.50
0.005
0.01
0.015
0.02
0.025
0.03Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.5 1 1.5-0.4
-0.2
0
0.2
0.4
0.62nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-20
10-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
D = 2.5m (100 times bigger). Discretization: 500000 points.First sidelobe: 1.3× 10−7.
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Big Primary, Small Secondary
0 0.01 0.02 0.030
5
10
15
202nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.05 0.1 0.15 0.20
0.005
0.01
0.015
0.02Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.01 0.02 0.03-0.2
0
0.2
0.4
0.6
0.82nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-20
10-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
A 10 inch primary and a 2 inch secondary.
A DISASTER!
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Design for Medium Contrast
0 0.005 0.01 0.0150
0.5
1
1.5
2
2.5
32nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.0150
1
2
3x 10-4 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.005 0.01 0.015-0.2
-0.1
0
0.1
0.2
0.32nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
Designed for 10−5. Delivers 10−5.max(A) is smaller...less off-axis distortion to correct.
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Medium Contrast Pupil Mapping with Back-End
Apodizer
0 0.005 0.01 0.0150
0.5
1
1.5
2
2.5
32nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.0150
1
2
3x 10-4 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.005 0.01 0.015-0.2
-0.1
0
0.1
0.2
0.32nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
Hybrid system designed for 10−10. Delivers 10−7.5.
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Front-End Apodizer
0 0.005 0.01 0.015 0.020
0.5
1
1.5
2
2.5
3
3.52nd Pupil Amplitude Map
Radius (m)
Out
put A
mpl
itude
Rel
ativ
e to
Inpu
t
0 0.005 0.01 0.015 0.020
1
2
3
4x 10-4 Lens profiles
Radius (m)
Lens
hei
ght (
m)
0 0.005 0.01 0.015-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.62nd Pupil Phase Map
Radius (m)
Pha
se in
radi
ans
0 0.5 1 1.510-20
10-15
10-10
10-5
100Ideal and Fresnel PSFs
Image-Plane Radius (mm)
Inte
nsity
Rel
ativ
e to
Pea
k
This hybrid system is about as good as the back-end apodizer.Easier to manufacture.