Introduction to Coronagraph Optics

1
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

2
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

3
Solar system at 10 pc

• At visible wavelengths
• Earth/sun 10-10
• 25 mag
• Zodi per pixel is small

4
Discovery space for coronagraphs
5
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.
6
Planet albedo and color
7
Bernard Lyot and his coronagraph machines
8
Early solar coronagraphs
Bernard Lyot
1932
2004 corona
Lyot
intensity
1963
radial angle
9
Stellar coronagraphs
K20 mag Bkgd objects
7 arcsec wand
J21 mag Bkgd object
20 arcsec radius circle
Ref McCarthy Zuckerman (2004) Macintosh et
al (2003)
10
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.
11
Photons and waves
12
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.
13
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
14
Fourier optics vs geometric optics
Fourier optics (or physical optics) describes
ideal diffraction- limited 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
diffraction- controlled 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.
15
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.
16
Image-plane coronagraphs
17
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
18
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.
19
Classical Coronagraph
aperture
A(u)
u
A(x)
x
image mask
M(x)
x
MA
x
MA
u
Lyot stop
L(u)
u
LMA
u
LMA
detector
x
L(u)M(u)A(u)0
L(x)M(x)A(x)0
Ref. Sivaramakrishnan et al., ApJ, 552, p.397,
2001 Kuchner 2004.
20
Final pupil L(u)M(u)(A(u)E(u))
E(u) 1 is input field across pupil A(u)
pupil transmission fn. M(u)(A(u)E(u))
pupil field L(u) Lyot pupil
transmission
A(x) FT(A(u)E(u)) image (x) M(x) mask
transmission fn
For on-axis point-like star to be zero across
exit pupil, we need
L(u)M(u)A(u) 0
21
How to satisfy L(MA)(u) 0
Nominal pupil diameter
Lyot stop
u
1/2
1/2-e/2
e/2
0
M(u)0 here
M(u)0 here
L(u) 0 here
L(u) 0 here
?M(u)du0 here
M(u)anything 0 (band-limited) ? 0 (notch)
M(u)anything 0 (band-limited) ? 0 (notch)
22
Wide-band masks
gaussian gives M(u) delta -
gaussian which has ?M(u) 0 inside e/2
and M(u) 0 outside e/2, but not
exactly.
rectangle gives M(u) delta - sinc (hard
disk mask) which has ?M(u) 0 inside e/2
and M(u) 0 outside e/2, but not
exactly.
1 if x 0 (phase mask) -1 if x gives M(u) sinc which has ?M(u) 0 inside
e/2 and M(u) 0 outside e/2,
but not exactly.
23
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
24
Wide-band (quadrant-phase) mask

• 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.
25
4-Quadrant phase mask
Sub-wavelength phase mask, from silicon, for
K-band region.
26
X-Y phase knife
experiment
theory
27
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
28
Band-limited masks
sin2(kx) (sin4(kx) transmission
mask) gives M(u) 2 delta(0) - delta(u-k)
delta(uk) which has ?M(u) 0 inside e/2
and M(u) 0 outside e/2, exactly.
1 - sin(kx)/kx (1-sinc(kx)2
transmission mask) gives M(u) delta(0) -
(p/k)?(p 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
29
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
30
Image-plane coronagraph simulation
1st image with Airy rings
mask, centered on star image
1st pupil
2nd pupil
Lyot stop, blocks bright edges
2nd image, no star, bright planet
Ref. Pascal Borde 2004
31
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 star
32
sin2x, 1-sin x/x and other band-limited masks
33
Notch-filter masks
Discrete version of continuous masks, 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
34
Null depth vs mask type
Mask Leak near axis
Pointing/IWA Tophat
? 0 -- Disk phase mask
? 0 -- Phase knife
? 2
0.0001 4-quad phase mask ? 2
0.0001 All masks 1st order ? 2
0.0001 Notch filter
? 4 0.01 Band-limited
? 4 0.01 Gaussian
? 4 0.01
stops Achromatic dual zone ? 4
0.01 stops
Ref Kuchner, a unified view preprint, 2004
35
Pupil-plane coronagraphs

• Shaped pupil mask
• Apodized pupil mask
• Discrete-transmission pupil mask
• Discrete-mapped pupil
• Continuous-mapped pupil
• Nulled pupil

36
Shaped-pupil mask
y
v
x
u
Image cut along the x-axis
Pupil Spergel-Kasdin prolate-spheroidal mask
Image dark areas 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
Kasdin, Vanderbei, Littman, Spergel, preprint,
2004
37
Discrete-transmission masks
Concentric ring mask
Bar-code mask (many slots not visible here)
(left) 20-star mask (right) PSF for 150-point
star mask
6-opening mask (right) black Kasdin, Vanderbei, Littman, Spergel, preprint,
2004
38
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.
39
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.
40
Discrete-mapped pupil (2) Densification
Clean image, narrow FOV
Image with many aliases
Densified pupil
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.
41
Continuous-mapped pupil
Input wavefront uniform amplitude.
Mirror 2
100 dB 10-10 25 mag
Output wavefront prolate-spheroidal amplitude.
Output image prolate spheroid
Mirror 1
Guyon, AA 404, p.379, 2003 Traub
Vanderbei, ApJ 599, 2003
42
Achromatic nulling coronagraph
p phase rotate pupil
split
recombine
Solution with mirrors
Solution with lenses
43
Binary stars nulled at telescopeall 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
Ref. Gay, Rivet, Rabbia, EAS Pub. 8, p.245,
2003.
44
Nuller with p-shift rotated pupil
Schematic for y-axis- symmetric pupil flip.
Sensitivity pattern on sky after x- y-axis
pupil flips.
Ref. B. Lane, pers. comm., 2003.
45
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.
46
Perturbation 1 ripples and speckles
47
Phase ripple and speckles
Suppose there is height error h(u) across the
pupil, where h( u) ?n ancos(Knu)
bnsin(Knu) ripple, K2?/DThe amplitude
across the pupil is then A(u ) eikh(u)
? 1 ik?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) (anibn)?(k?Kn)
The image intensity is then I(?) ?(0) (1/4)
?n (an2bn2) ?(k?-Kn) ?(k?Kn) speckles
at ? ?n?/D If we add a deformable mirror
(DM), then an?anAn and bn?bnBn Commanding
An-an and Bn-bn forces all speckles to zero.
48
Phase amplitude ripple and speckles
Suppose the height error h(u) across the pupil is
complex, where h(u) ?n
(anian')cos(Knu) (bnibn')sin(Knu)
ripple i.e., we have both phase and amplitude
ripples ( errors). The image intensity is then
I(?) ?(0) (1/4) ?n (anbn)2 (bn-an)2
?(k?Kn)
(an-bn)2 (bnan)2 ?(k?-Kn)
speckles If we add a deformable mirror (DM), and
command An -(an-bn) and Bn -(bnan)
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.
49
Phase ripple and speckles
Polishing errors on primary
Pupil plane
Phase ripples from primary mirror errors
Speckles generated by 3 sinusoidal components of
the polishing errors
No DM
Image plane
With DM
Image plane
50
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.
51
Perturbation 2Polarization
52
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.
53
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.
54
Perturbation 3Fresnel is not Fraunhofer
55
Approximations
Maxwells 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.
56
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
Fourier- transform 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.
57
Perturbation 4nik index of refraction
58
nik
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?
59
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.
60
Perturbation 6Geometric distortions
61
Top 10 Contrast Contributions
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).
62
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.