Introduction to Coronagraph Optics
Wesley A. Traub
Harvard-Smithsonian Center for Astrophysics
Michelson Summer School on High-Contrast Imaging
Caltech, Pasadena
20-23 July 2004
1
Outline of talk
•
•
•
•
•
Extrasolar planet science goals
Bernard Lyot and his coronagraph machines
Photons and waves
Current coronagraphs
Prototype coronographs:
1. Image plane
2. Pupil plane
3. Pupil mapping
4. Nulling coronagraph
• Perturbations:
1. Speckles
2. Polarization
3. Fraunhofer vs Fresnel
4. Refractive index of real materials
5. Internal scattering
6. Geometrical stability
2
Solar system at 10 pc
• At visible
wavelengths:
• Earth/sun = 10-10
= 25 mag
• Zodi per pixel is small
3
Discovery space for coronagraphs
4
Key coronagraph parameters
Contrast C: The ratio dark/bright parts of image. Specifically,
the average background brightness in the search area,
divided by the central star brightness. Speckle/star.
Example: C = 10-10 driven by Earth/Sun = 2x10-10.
Inner working angle IWA: Smallest angle at which a planet
can be detected. Inner boundary of high-contrast search area.
Example: IWA = 3 /D driven by 1 AU/10pc = 0.100 arcsec.
Outer working angle OWA: Largest angle at which a planet
can be detected. Outer boundary of high-contrast search area.
Example: OWA = 48 /D driven by N = 96 actuator DM.
5
Planet albedo and color
6
Bernard Lyot and his
coronagraph machines
7
Early solar coronagraphs
Bernard Lyot
1932
2004 corona
Lyot
1963
radial angle
8
Stellar coronagraphs
K~20 mag
Bkgd objects
7 arcsec wand
J~21 mag
Bkgd object
20 arcsec radius
circle
Ref: McCarthy & Zuckerman (2004); Macintosh et al (2003)
9
Extrasolar coronagraphs on the ground
• Jupiters: need 30-m telescope,
with essentially perfect adaptive optics,
and will still have very large background.
• Earths: need 100-m telescope,
with essentially perfect adaptive optics.
• Note: T ~ SNR2 * (RMS wavefront)2 / D4 ,
so 30 m on ground is equivalent to ~ 2 m in space.
Ref.: Stapelfeldt et al., SPIE 2002; Dekaney etal 2004.
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Photons and waves
11
Basic photon-wave-photon process
We see individual photons. Here is the life history of each one:
Each photon is emitted by a single atom somewhere on the star.
After emission, the photon acts like a wave.
This wave expands as a sphere, over 4 steradians (Huygens).
A portion of the wavefront enters our telescope pupil(s).
The wave follows all possible paths through our telescope
(Huygens again).
Enroute, its polarization on each path may be changed.
Enroute, its amplitude on each path may be changed,.
Enroute, its phase on each path may be changed.
At each possible detector, the wave “senses” that it has followed
these multiple paths.
At each detector, the electric fields from all possible paths
are added, with their polarizations, amplitudes, and phases.
Each detector has probability = amplitude2 to detect the photon.
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Photon………..wave…..………...photon
1x
Ex
1 photon
emitted
Ey
1z
1y
E(x,y,z) = 1xExsin(kz-wt-px) + 1yEysin(kz-wt-py)
where the electric field amplitude in the x direction is
sin(kz-wt-px) = Im{ ei(kz-wt-px) }
and likewise for the y-amplitude.
At detector, add the waves from all possible paths.
1 count
detected
13
Fourier optics vs geometric optics
Fourier optics (or physical optics) describes ideal diffractionlimited optical situations (coronagraphs, interferometers,
gratings, lenses, prisms, radio telescopes, eyes, etc.):
If the all photons start from the same atom, and follow the same
many-fold path to the detectors, with the same amplitudes &
phase shifts & polarizations, then we will see a diffractioncontrolled interference pattern at the detectors.
In other words, waves are needed to describe what you see.
Geometric optics describes the same situations but in the limit
of zero wavelength, so no diffraction phenomena are seen.
In other words, rays are all you need to describe what you see.
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Huygens wavelets
Portion of large,
spherical wavefront
from distant atom.
Blocking screen,
with slit.
Wavelets add with various
phases here, reducing the
net amplitude, especially
at large angles.
Wavelets align here, and make nearly flat
wavefront, as expected from geometric optics.
15
Image-plane coronagraphs
16
Huygens’ wavelets --> Fraunhofer --> Fourier transform
The phase of each wavelet on a surface
Tilted by theta = x/f and focussed by the
Lens at position x in the focal plane is
The sum of the wavelets across the
potential wavefront at angle theta is
All waves add in phase here
The net amplitude is zero here
The net amplitude mostly
cancels, but not exactly,
here
17
Fourier relations: pupil and image
We see that an ideal lens (or focussing mirror) acts on the
amplitude in the pupil plane, with a Fourier-transform
operation, to generate the amplitude in the image plane.
A second lens, after the image plane, would convert the
image-plane amplitude, with a second Fourier-transform,
to the plane where the initial pupil is re-imaged.
A third lens after the re-imaged pupil would create a
re-imaged image plane, via a third FT.
At each stage we can modify the amplitude with masks, stops,
polarization shifts, and phase changes. These all go into the
net transmitted amplitude, before the next FT operation.
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Classical Coronagraph
aperture
image
mask
A(u) u
A(x)
x
M(x)
MA
x
x
M*A u
Lyot
stop
detector
L*[MA]
L(u) u
L[M*A]
u
L(x)*[M(x)·A(x)]~0
x
L(u)·[M(u)*A(u)]~0
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Ref.: Sivaramakrishnan et al., ApJ, 552, p.397, 2001; Kuchner 2004.
Final pupil = L(u)·[M(u)*(A(u)·E(u))]
E(u) = 1 is input
field across pupil
A(x)
= FT(A(u)E(u))
= image (x)
A(u) = pupil
transmission fn.
M(x) = mask
transmission fn
M(u)*(A(u)E(u))
= pupil field
L(u) = Lyot pupil
transmission
For on-axis point-like star to be zero across exit pupil, we need
L(u)·[M(u)*A(u)] = 0
20
How to satisfy L·(M*A)(u) = 0
Nominal pupil diameter
Lyot stop
0
M(u)=0 here
L(u) = 0 here
M(u)=anything
= 0 (band-limited)
≠ 0 (notch)
e/2
1/2-e/2
1/2
u
M(u)=0 here
∫M(u)du=0
here
L(u) = 0 here
M(u)=anything
= 0 (band-limited)
≠ 0 (notch)
21
Wide-band masks
ˆ (x) = gaussian gives M(u) = delta - gaussian
M
which has ∫M(u) ~ 0 inside ± e/2
and M(u) ~ 0 outside ± e/2, but not exactly.
ˆ (x) = rectangle gives M(u) = delta - sinc (hard disk mask)
M
which has ∫M(u) ~ 0 inside ± e/2
and M(u) ~ 0 outside ± e/2, but not exactly.
ˆ (x) = 1 if x > 0
M
(phase mask)
-1 if x < 0 gives M(u) = sinc
which has ∫M(u) ~ 0 inside ± e/2
and M(u) ~ 0 outside ± e/2, but not exactly.
22
Wide-band (gaussian) mask
Amplitude of
on-axis star = 1 ei0
FT( gauss(x) ) =
delta(u) - gauss(u)
Convolution
Lyot stop blocks
bright edges
Leakage transmission
of on-axis star
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Wide-band (quadrant-phase) mask
•


0
0


Star image is centered on mask which transmits half of
image shifted by 1/2 wavelength, and 1/2 unshifted, so
symmetric parts cancel.

Ref.: Riaud et al., PASP 113 1145 2001.
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4-Quadrant phase mask
Sub-wavelength phase
mask, from silicon,
for K-band region.
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X-Y phase knife
experiment
theory
26
X-Y phase knife: double star in lab
Binary star
without coronagraph
Binary with
X Phase knife
Binary with X + Y
phase knives;
Bright star nulled
27
Band-limited masks
ˆ (x) = sin2(kx)
M
(sin4(kx) transmission mask)
gives M(u) = 2 delta(0) - delta(u-k) + delta(u+k)
which has ∫M(u) = 0 inside ±e/2
and M(u) = 0 outside ±e/2, exactly.
ˆ (x) = 1 - sin(kx)/kx
M
([1-sinc(kx)]2 transmission mask)
gives M(u) = delta(0) - (π/k)·∏(π u/k)
which has ∫M(u) = 0 inside ±e/2
and M(u) = 0 outside ±e/2, exactly.
Kuchner and Traub, ApJ 570, 900-908, 2002
28
Band-Limited Image Mask
On-axis star is
totally blocked
In re-imaged pupil.
Off-axis planet is
~fully transmitted
In re-imaged pupil.
Example: this 1-D image mask transmits the
band-limited function (1-sin x/x)2 .
Ref.: Kuchner & Traub ApJ 570, 900, 2002
29
Image-plane coronagraph simulation
1st
pupil
1st
image
with
Airy
rings
Ref.: Pascal Borde 2004
mask,
centered
on star
image
2nd
pupil
Lyot
stop,
blocks
bright
edges
2nd
image,
no star,
bright
planet
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Band-limited (1 - sin x/x) mask
Amplitude of
on-axis star = 1 ei0
FT(1 - sin x/x) =
rect(u) + delta(u)
Convolution
Lyot stop blocks
bright edges
Zero transmission
of on-axis star31
sin2x, 1-sin x/x and other band-limited masks
32
Notch-filter masks
ˆ (x) = Discrete version of continuous masks,
M
i.e., discrete grey levels or opaque/transmitting,
will have sharp edges,
and therefore high-frequency components,
but if these all lie outside the ±(1/2 - e/2) range,
then they will be blocked by the Lyot stop.
Kuchner & Spergel, ApJ 594, 617-626, 2003
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Null depth vs mask type
Mask
Leak near axis
Tophat
Disk phase mask
Phase knife
4-quad phase mask
All masks > 1st order
Notch filter
Band-limited
Gaussian
Achromatic dual zone
0
0
2
2
2
4
4
4
4
Pointing/IWA
--0.0001
0.0001
0.0001
0.01
0.01
0.01 + stops
0.01 + stops
Ref: Kuchner, “a unified view…” preprint, 2004
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Pupil-plane coronagraphs
•
•
•
•
•
•
Shaped pupil mask
Apodized pupil mask
Discrete-transmission pupil mask
Discrete-mapped pupil
Continuous-mapped pupil
Nulled pupil
35
Shaped-pupil mask
y
v
x
u
Pupil: Spergel-Kasdin
prolate-spheroidal mask
Image: dark areas
< 10-10 transmission
Image: cut along
the x-axis
Let pupil shape be g(u) = exp(-u2). Then star at (x,y)=(0,0) gives
A(x,0) = infdu v=g(u) eikxu dv =  exp(-u2 + ikxu)du = exp(-(x/)2)
So I(x) = exp( -2(x/)2 ) gives the very dark area along ± x axis.
Along the ± y axis the integral is:
A(0,y) = infdu v=g(u) eikyv dv
=  [exp(iky exp(-u2)) - exp(-iky exp(-u2) )]du = periodic & messy
36
Kasdin, Vanderbei, Littman, & Spergel, preprint, 2004
Discrete-transmission masks
Concentric ring mask
Bar-code mask (many slots not visible here)
6-opening mask; (right) black < 10-10
(left) 20-star mask;
(right) PSF for 150-point star mask
Kasdin, Vanderbei, Littman, & Spergel, preprint, 2004
37
Apodized pupil mask
• Telescope pupil is fully transmitting in center,
tapering to dark at edges.
• Image ringing due to hard pupil edge is eliminated,
and Airy rings are dramatically suppressed.
Ref.: Nisenson and Papaliolios, ApJ 548, p.L201, 2001.
38
Discrete-mapped pupil (1): quadrant shifts
Contiguous output pupil permits
coronagraphic supression of on-axis star,
but “Golden Rule” of pupil mapping is violated, therefore FOV is small.
Refs: Aime, Soummer, & Gori, EAS Pub. 8, p.281, 2003;
Traub, AO 25, p.528, 1986.
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Discrete-mapped pupil (2): Densification
Image with
many aliases
Densified
pupil
Clean image,
narrow FOV
Entrance pupil,
sparsely filled
“Golden rule” is violated, therefore FOV is small.
Refs: Traub, AO 25, p.528, 1986; Labeyrie, EAS Pub. 8, p.327, 2003.
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Continuous-mapped pupil
Input wavefront:
uniform amplitude.
Mirror 2
Output wavefront:
prolate-spheroidal
amplitude.
Mirror 1
100 dB
= 10-10
= 25 mag
Output image:
prolate spheroid
Guyon, A&A 404, p.379, 2003; Traub & Vanderbei, ApJ 599, 2003
41
Achromatic nulling coronagraph
π phase &
rotate pupil
split
recombine
Solution with lenses
Solution with mirrors
42
Binary stars nulled at telescope
all images are reflection-symmetric
OHP 1.5m, AO,
K-band, 72 Peg,
Separation 0.53 asec
circle is 1st dark Airy
ring, at 0.35 asec
Main star off-axis
Main star on axis
Nulled binary HIP 97339,
separation 0.13 asec,
Main star on axis
43
Ref.: Gay, Rivet, & Rabbia, EAS Pub. 8, p.245, 2003.
Nuller with π-shift & rotated pupil
Schematic for y-axissymmetric pupil flip.
Sensitivity pattern on sky
after x- & y-axis pupil flips.
44
Ref.: B. Lane, pers. comm., 2003.
Nulling-shearing coronagraph
• The central star is nulled by 180o delays of sub-pupil pairs.
• The wavefront is cleaned up with single-mode fibers.
• The wavefront is flattened with 2 deformable mirrors.
Ref.: Mennesson, Shao, et al., SPIE, 2002.
45
Perturbation #1:
ripples and speckles
46
Phase ripple and speckles
Suppose there is height error h(u) across the pupil, where
h( u) = n ancos(Knu) + bnsin(Knu) = ripple, K=2/D
The amplitude across the pupil is then
A(u ) = eikh(u)  1 + i[kn ancos(Knu) + bnsin(Knu)]
In the image plane at angle  the amplitude will be
A() =  A(u) eiku du
= (0) + (i/2) n [(an-ibn)(k-Kn) + [(an+ibn)(k+Kn)]
The image intensity is then
I() = (0) + (1/4) n (an2+bn2) [(k-Kn) + (k+Kn)] = speckles
at  = n/D
If we add a deformable mirror (DM), then anan+An and bnbn+Bn
Commanding An=-an and Bn=-bn forces all speckles to zero.
47
Phase + amplitude ripple and speckles
Suppose the height error h(u) across the pupil is complex, where
h(u) = n (an+ian')cos(Knu) + (bn+ibn')sin(Knu) = ripple
i.e., we have both phase and amplitude ripples (= errors).
The image intensity is then
I() = (0) + (1/4) n [(an+bn’)2 + (bn-an’)2 ] (k+Kn)
+ [(an-bn’)2 + (bn+an’)2 ] (k-Kn)] = speckles
If we add a deformable mirror (DM), and command
An = -(an-bn’) and Bn = -(bn+an’)
Then we get
I() = (0) + n [(bn’)2 + (an’)2 ] (k+Kn)  bigger speckles
+ [ 0 + 0 ] (k-Kn)]  smaller (zero) speckles
So in half the field of view we get no speckles,
but in the other half we get stronger speckles.
48
Phase ripple and speckles
Polishing errors
on primary
Phase ripples
from primary
mirror errors
No DM:
With DM:
Pupil plane
Speckles generated
by 3 sinusoidal
components of the
polishing errors
Image plane
Image plane
49
Full-field correction of phase and amplitude
DM-1
corrects
phase
DM-2
corrects
amplitude
Half-silvered
mirror
With two DMs we can correct both phase and amplitude
errors across the pupil. This is a conceptual diagram.
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Perturbation #2:
Polarization
51
S-P phase shift
S-P
S and P refer to the electric vector components
perpendicular and parallel to the plane of incidence.
For a curved mirror, these axes vary from point to point.
52
S-P shift consequences
A stellar wavefront will have the same amplitude and phase
at all points in the plane perpendicular to the line of sight to
the star, i.e., across the pupil plane for an on-axis star.
After reflecting from the curved primary and secondary mirrors,
the wavefront will no longer have the same electric field
amplitude or phase from point to point.
Therefore it will not interfere with itself in the focal plane in
the same way that a perfect wavefront would interfere.
The amplitude and phase will vary across the wavefront, and
therefore there will be ripple components, and speckles will
be formed unless they are corrected by reversing these effects.
53
Perturbation #3:
Fresnel is not Fraunhofer
54
Approximations
Maxwell’s equations are exact, and predict wave propagation in
Free space, where there are no electric charges or currents.
However at a real boundary, electric currents are induced by an
Incident wave, and the free-space equations are no longer exact.
Furthermore the actual wave is a vector, but is usually approximated
As a scalar wave.
For scalar propagation, the integral theorem of Helmholtz and
Kirchoff applies, and is the basic idea of Huygens’ wavelets.
55
More approximations
When the light source, diffracting aperture, and detectors are all
Infinitely far apart (or coupled by an ideal lens), then the
Fresnel-Kirchoff integral equation becomes linear in the coordinates
of the aperture, and we get Fraunhofer diffraction, with Fouriertransform relations between the pupil and image planes.
If we fail to have an ideal lens, or fail to have a perfect conductor
for an aperture, then Fraunhofer fails too.
The resulting equations can only be solved numerically, at great
effort.
56
Perturbation # 4:
n+ik index of refraction
57
n+ik
Image mask:
For the case of the image mask (e.g., transmission =(1-sinc2 )2 ),
the photo-resist materials used to form the mask have a measurable
phase shift that is a function of the density of the mask.
Also the materials cannot be made infinitely opaque, nor do they
have the same opacity at all wavelengths.
Pupil mask:
No mask material is perfectly conducting, as required by the theory.
Question: will non-metal masks perform like metal ones?
58
Scattering
Experience shows that rough edges on pupil masks will cause
high levels of scattered light to enter the detectors.
Any dust or inhomogeneity in the pupil lens or mirror will also
cause much scattered light to enter the detectors.
Photos by B. Lyot of the main lens in his coronagraph, showing
scattering by dust, glass inhomogeneities, scratches, and diffraction
around the edge. Solutions were a better lens, less dust, oil, and an
external occulter to prevent direct sunlight on the lens.
59
Perturbation #6:
Geometric distortions
60
“Top 10” Contrast Contributions
Major 14 Contrast Contributors
Source
Mask Leakage
Mask Leakage
Rigid Body Pointing
Rigid Body Pointing
Rigid Body Pointing
Thermal Structural Deformation Aberrations
Thermal Structural Deformation Aberrations
Thermal Structural Deformation Aberrations
Thermal Deformation of Optics
Thermal Deformation of Optics
Dynamics Deformation of Optics
Dynamics Deformation of Optics
Dynamics Deformation of Optics
Dynamics Structural Deformation Aberrations
Contrast (3λ/D)
z3
4.34E-13
z2
4.33E-13
Super OAP1
3.22E-13
Super Fold Mirror 2
1.22E-13
Super Fold Mirror 1
6.84E-14
Secondary (z4 for dL )
3.48E-14
Secondary (z8 for dL )
1.14E-14
Secondary (z4 for Δf/f)
4.28E-14
Primary (z4)
5.78E-14
Primary (z12)
1.79E-14
Primary (z4)
2.23E-14
Primary (z8)
2.90E-14
Primary (z12)
1.79E-14
Secondary (z4)
7.30E-14
Cotributor
6.75E-12
<Id> Total
Contrast at 3λ/D
7.09E-12
For TPF-C, this table shows that deformations of the optical
system are second only to mask leakage and telescope pointing
as sources of speckles in the focal plane. Ref.: Shaklan and Marchen (2004).
61
Summary
Extrasolar planets can be detected and characterized
in visible light with a coronagraph.
One of the key challenges to overcome is to eliminate
even the smallest optical imperfections in the system,
because each imperfection can be decomposed into
constituent ripples, and each ripple generates a pair
of speckles, and each speckle looks just like a planet.
Infrared interferometers can also detect and
characterize extrasolar planets, and they will be subject
to all of the same caveats about optical imperfections,
though sometimes arising from different mechanisms.
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