Transcript Quarterly Review ORIGINS & Astrophysics Overview

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Principles of Coronagraphy
Wesley A. Traub
Jet Propulsion Laboratory,
California Institute of Technology
International Young Astronomer School
On High Angular Resolution Techniques
CIEP and Paris Observatory, 1-5 Nov. 2010
Sevres and Meudon
Copyright 2010 California Institute of Technology. Government sponsorship acknowledged.
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Outline
1. Introduction
2. Exoplanet Brightness and Separation
3. Photons as Waves and Particles
4. Coronagraph and Interferometer Concepts
5. Speckles
6. Ground vs Space
Reference for this talk: W. Traub & B. Oppenheimer, Chapter on “Direct Imaging”,
in book “Exoplanets”, edited by Sara Seager, late 2010 publication.
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1. Introduction
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What is a Coronagraph?
• Lyot (1933) invented the coronagraph to observe the corona of the Sun.
• Lyot used a simple lens to image the Sun, an over-sized blocking disk to
Stop direct sunlight, and a photographic plate to image the faint (~10-6) corona.
• The big problems he faced were scattered light and diffracted light.
• These are our problems today too, but at the 10-10 level of contrast.
distant Sun
& corona
pupil
plane
image plane
& Sun blocker
2nd pupil plane
& Lyot stop
2nd image plane
& detector
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Internal & external coronagraphs
Internal: Sun blocker & Lyot stop come after pupil lens,
shown on chart above
External: Sun blocker comes before pupil lens,
like your hand in front of your eye to view the Sun;
also called “starshade”.
Nulling interferometers: “blocking” of star provided by interference,
not a physical stop.
Many combinations are possible.
Encyclopedic classification is not very helpful.
What is helpful is to know how to calculate the action of all these types.
This is the purpose of this lecture.
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Example 1: Fomalhaut b
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Example 2: HR 8799 b, c, d
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Example 3: Beta Pictoris
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2. Exoplanet brightness
and separation
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Specific intensity
erg/(s cm2 Hz sr)
erg/(s cm2 cm sr)
You can change from one to the other using
Bλ dλ = Bν dν
and
λν = c.
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Photon rates
photon / (s cm2 Hz sr)
photon / (s cm2 cm sr)
photon / (s cm2 μm sr)
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Star flux
photon / (s cm2 μm )
etendue is conserved
(etendue = “throughput”)
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Flux & magnitudes
erg/(s cm2 μm)
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Kepler’s 3rd law
year, AU, solar mass,
& negligible planet mass
θ = (1 + e) a/d
arcsec, AU, pc
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Contrast vs wavelength
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Contrast (visible)
p is geometric albedo
Φ(α) is phase function
rp is planet radius
α is phase angle
Earth visible contrast
Jupiter visible contrast
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Contrast vs separation
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Contrast (thermal infrared)
T is effective
temperature of
star or planet
Luminosity of star
Set incident flux = radiated flux,
over fraction f of planet
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Zodiacal light (zodi)
Empirical relation from
SS obs & Zodipic calc.
Ωas = solid angle (arcsec2)
RAU = radius (AU)
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Color and spectra
“color” means low (R ~ 5) spectral resolution,
e.g., U, B, V, R, I etc.
“spectra” are medium resolution (R ~ 100)
or high (R ~> 1000) resolution
Color and spectra are needed to characterize the
atmosphere and possibly the surface of an exoplanet in
terms of composition, temperature, motions, seasons, etc.,
and on Earthlike planets to search for signs of life.
Both topics are in chapter, but skipped in this lecture.
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3. Photons
as waves
& particles
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A traveling photon is a wavefront
A is amplitude of electric vector of wavefront
I is intensity of wavefront
Aout(x) is sum of path-delayed
individual Ain(x’) wavelets
Input (Huygens) wavelets
M(x’) is mask example:
left edge is hard (step fn),
right edge is soft (tapered)
Hard edge diffracts a lot;
Soft edge diffracts less.
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A detected photon is a particle
Poisson process
(all values of average rate)
Gaussian (normal) process
(large values of average rate)
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Photons in radio and optical
Etendue of focused beam [D×λ/D]2 = λ2
defines a single electromagnetic mode
Photon rate × area × solid angle = photons/(sec Hz)
in single electromagnetic mode from a blackbody
Uncertainty relation ΔpΔx = h for a photon
is Δ(hν/c)Δ(ct) = h
So this is the number of photons in a
single electromagnetic mode,
per polaristion state (2 of these),
from a blackbody source
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Why radio and optical are different
Radio example: T = 5000K, λ = 1 cm, get 3400 photons per single mode.
Optical example: T = 5000K, λ = 0.5 μm, get 0.002 photons per mode.
So an array of radio antennas can collect photons from a single mode,
one or more photons per antenna, and later combine these signals
interferometrically, ie, including the phase information.
However in an array of optical telescopes, only one photon in a single mode
can be detected at a time, so all telescopes must feed a single collection point.
Ie, we cannot coherently combine data from several independent telescopes after
detection; we must combine the amplitudes from all telescopes immediately.
Exception: In the Hanbury-Brown Twiss Intensity Interferometer,
each telescope collects light independently of the others,
and the cases of simultaneous arrival are noted.
If fewer such simultaneous arrivals occur than for a point source,
then the source is inferred to be extended.
So stellar diameters are measured this way.
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4. Coronagraph
& Interferometer
Concepts
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Types of Direct Imaging
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4 coronagraph planes
distant star & planet, etc.
pupil plane (x1)
image plane (x2 or -θ2)
2nd pupil (Lyot) plane (x3)
2nd image plane (x4 or θ4)
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Classical single pupil (1 of 2)
The phase of a tilted wavefront is:
The amplitude at the focus is:
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Classical single pupil (2 of 2)
1-D case
2-D case
Simple to calculate in 1-D; but cannot easily
extrapolate to 2-D case.
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FT relations
Result: The amplitudes in
successive planes are Fourier
Transforms, so it follows that
Plane 3 is an image of plane 1,
& plane 4 is an image of plane 2.
Note:
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Convolution picture
The amplitude in each image or pupil plane is the Fourier
transform of the amplitude, times the intervening mask,
in the plane just before it.
We know that FT(f×g) = FT(f)*FT(g) where “×” is a
multiply operation and “*” is a convolution.
This means that we can write the amplitude in a plane as
the Fourier Transform of the incident amplitude in the
preceding plane convolved with the Fourier Transform
of the mask in that plane.
This idea is conceptually elegant, and sometimes even useful.
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Imaging Recipe
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Practical considerations
• Kramers-Kronig relation says that an absorption always
has a related phase shift (and vice versa).
So when you deposit an absorbing or reflecting metal
spot on a mask, there will be an automatic phase shift
that occurs to the passing wavefront.
• Lab air is always convecting, so therewill always be a
variable wavefront error in a normal lab experiment.
Vacuum is needed to eliminate this error.
We find this limit to be a contrast of roughly 10-6 or so.
• Poor optical mounts can bend mirrors at the λ level.
• Beam walk will add wavefront error.
• Talbot effect will add wavefront error (later charts).
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Off-axis performance
Because stars have finite diameters, and telescopes do not
point perfectly, it is helpful if a nulling instrument
such as a coronagraph or an interferometer will still reject
most of the star-light in these practical situations.
The key to knowing how well this will work is to know if
the instrument has an intensity null that is proportional to
off-axis angle to the power n = 2, 4, 6, or even higher power.
Most of the coronagraphs in this talk are n=4 instruments,
which is adequate in practice.
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Obscurations in the pupil
Circular telescope with
area AD and central
obscuration of area Ad.
Circular telescope with
area AD and spider of
width w and area Aw.
Babinet’s principle says amplitude contribution of an opaque part
is given by the negative amplitude of the same transmitting part.
Example: Contrast of a gap across a mirror is C = (w/D)2 .
If w = 80μm (a hair), and D=1m, then C = 80×10-10 = 80×Cvis(Earth).
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Pupil apodization
We can minimize the Airy rings if we apodize (taper,
or make to have “no feet”) the edge of the pupil.
Suppose we taper the amplitude with black spray paint:
A1(x1) = exp[ -(x/x0)2 ] where x0 < D/2.
Then the image will have a Gaussian intensity pattern:
This image will reach 10-10 at θ = 2λ/Deff .
So in principle this is a powerful technique.
In practice we use a prolate spheroid function and get 4λ/D.
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Pupil masking
Here we replace continuous apodization with a discrete pattern.
The nominal shape is a prolate spheroid function (D. Spergel).
Numerical optimization gives the 6-hole pupil mask (left).
The image plane intensity (right) has dark holes on 2 sides.
Theoretical contrast is 10-10 beyond IWA of 4λ/D.
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Pupil mapping (PIAA)
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Ray-trace images of PIAA
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Lyot (hard-edge) mask coronagraph
Assume an on-axis star (A1 = 1). Get A2 = Airy pattern, as usual.
Insert top-hat mask in focal plane:
Calculate A3:
So the 2nd pupil amplitude is a copy of the input pupil minus an
oscillating function of position x3.
Note A3 has a zero at x3 = ±D/2, a common trait of some designs.
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Gaussian mask coronagraph
is a gaussian mask, black in the center.
where z± = πθg(±D/2-x3)/λ
This is an error function. A simple approximation is:
which is small inside ±D/2, zero at the edge, and large beyond,
so a Lyot mask (M3) will remove most of the star.
Refocussing onto plane 4 will give a weak star and strong planet.
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Band-limited mask coronagraph (1/2)
Focal-plane mask, c=1/2, θB = few*λ/D,
no high frequencies (--> “band limited”)
Calc. 2nd pupil amplitude.
Result is sum of 3 rect functions,
giving A3=0 over middle of pupil,
most of light at the edge,
and a zero at the exact edge.
Alternate way to write result,
zero in middle, and
+/- wiggles at edges.
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Band-limited mask coronagraph (2/2)
Another example: 1 – sinc.
2nd pupil: zero in middle,
wiggles near edges,
to be blocked by Lyot pupil mask.
Plane 1, input pupil
FT of mask at plane 2; convolve with pupil.
Result of FT operation, or integral for A3.
Lyot mask, transmits center, blocks edges.
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Phase mask coronagraph
If a phase mask in inserted into the image plane, such that half
of the star image is delayed by a half-wavelength with respect
to the other half, then in the amplitude in the following pupil
plane will be mostly (or all) diffracted to the edge of that pupil.
If a Lyot mask blocks that diffracted light, then in the following
image plane the starlight will be small (or zero).
An off-axis planet will not be so affected.
This is the principle of a large family of image-plane phase masks.
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Vector vortex mask coronagraph
1-D example, phase mask in image plane.
Calc. 2nd pupil amplitude (m = 2x3/D)
Small in middle of pupil, bright near edge,
zero at edge, but not a good coronagraph
2-D example (Mawet 2005)
Calc. 2nd pupil amplitude,
get zero in middle of full pupil,
bright outside edge,
a perfect coronagraph
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External occulter coronagraph
Mask is in plane-0, before the
telescope, toward star.
Original (hypergaussian)
version is shown.
IWA drives large size and distance of occulter.
Fresnel number ~70 says need
full Fresnel theory.
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Nulling interferometer (1/4)
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Nulling interferometer (2/4)
Phase of wavefront at plane-1.
Amplitude in pupil plane
is sum of amplitudes in each
of the pupils, same as an
integral over sparse pupil
elements, with single-pupil
shape factored out.
Intensity in plane-2,
where detector is
located, for 3 different
configurations of delay.
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Nulling interferometer (3/4)
3 different delays,
rapid chopping.
Star signal from this chop.
Exozodi signal
from this chop.
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Nulling interferometer (4/4)
For exoplanet detection,
use this delay set.
Chop rapidly between
these 2 states.
Final output is planet signal.
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Visible nuller coronagraph/interferometer
This design is well-suited to segmented pupils, such as the EELT,
where the image of one segment can be superposed on the image
of another, effectively making the telescope into a nulling
interferometer with many parallel baselines.
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4. Speckles
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Speckles from a phase step in pupil
Suppose we delay the wavefront
by φ across half the telescope pupil.
We get this image-plane amplitude
If φ = 0, recover standard Airy pattern.
If φ = π, get 2 peaks, or speckles.
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Speckles from phase and amplitude ripples
Standard Fourier
analysis on a finite
Interval.
Coefficients are projections
of the wavefront
to be approximated
Same as above, but in complex
Notation.
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Example of speckles in focal plane
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Phase and amplitude ripple in pupil
Suppose pupil has a phase ripple
and an amplitude ripple.
Let the peak to valley wavefront
ripple have height h0 (cm).
The intensity varies as exp(-2b*cos(…)).
Amplitude in image plane.
Amplitude in image plane,
assuming φ1 << 1.
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Speckles in image plane
Let A0(θ) be the usual Airy pattern.
Then the amplitude in the pupil
is a sum of 3 Airy patterns.
The intensity is the sum of 6 terms.
These are symmetricallyplaced speckles.
These are pinned speckles,
amplified by the original
Airy pattern.
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Speckles from mirror errors
Define spatial frequency k on the
surface of a mirror.
Empirical result is that mirrors have a
Power Spectral Density of this form,
where n ~ 3.
The rms error of the surface is the
area under the PSD curve.
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Contrast in a dark hole
Assume that we have an N×N element deformable
mirror, with N/2 periods in each direction.
There are then M modes of ripple on the surface.
If a mode has amplitude h0,
then it generates a speckle.
The rms amplitude is the sum of M random vectors,
so hrms = M1/2 h0.
The average contrast is:
or
Conversely, we have
Finally, the radius of the dark hole is:
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Single-speckle nulling (1/2)
Recall that if this ripple is in
the pupil plane, then …
… we get this amplitude
in the image plane, i.e.,
we get a main star image
plus a speckle ghost on
either side of it.
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Single-speckle nulling (2/2)
text
text
text
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Multi-speckle nulling
Suppose that we start with
this (unknown) phase in the
pupil. (1-D for illustration)
We turn on the DM (in a pupil
image) and add this phase to the
the original phase.
Then change the DM to a
different pattern, and add this
phase instead.
Net result is 3N data points measured, and 2N unknowns,
so we can solve for them, including the signs. Combining this
method with the previous example, we see that we can null out
both phase and amplitude in a half-square dark hole.
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Multi-speckle energy minimization (1/3)
Assume this is the unknown input
phase. Φ1 can be complex.
Expand in Fourier series.
This is a DM just after this pupil.
Expand in a Fourier series.
This is the amplitude in plane 2,
the image plane.
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Multi-speckle energy minimization (2/3)
Insert phases, keep to 1st order.
Assume a perfect coronagraph
follows, so drop the “1”, and
insert phase expressions,
Define the total energy in the dark hole.
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Multi-speckle energy minimization (3/3)
If we actually integrate over
the entire focal plane, we get
this total energy.
If we minimize with respect to the parameters of
the DM, the wavefront distortion is canceled, up to
the highest frequency of the DM.
This is the remaining energy, high
freqs. of distortion plus all of the
absorption part of the wavefront.
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Talbot effect (1/2)
This is the condition for a line of holes
to be reproduced a distance z downstream.
If the wavelength is small, this is z.
There will be infinitely many such planes lying
between z=0 and z=zTC. This is the Talbot carpet.
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Talbot effect (2/2)
Now assume an incident plane wave, A1=1, and let the
array of points be replaced by a phase ripple.
Let this propagate freely, with no lens.
Here “l” is the distance from x1 to x2.
Expand, assume amplitude is small,
and integrate. Get plane wave plus
added periodic copy.
This is the intensity, periodically repeating.
Talbot distance.
Distance from plane of
const. intensity to
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6. Ground vs Space
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Ground vs space direct imaging
See text for full explanation. Bottom line: to achieve 10-10 contrast
at a ground-based telescope would seem to require a star brighter
than any that exist, according to the above logic. Even if there is
a way to solve this problem, it will likely not be easy.
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Current, planned, & proposed projects for exoplanets imaging
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Example contrast vs separation
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Thank you!
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