national aeronautics and space administration 1 traub principles of coronagraphy wesley a. traub jet...

National Aeronautics and Space Administration

1 Traub

Principles of Coronagraphy

Wesley A. TraubJet Propulsion Laboratory,

California Institute of Technology

International Young Astronomer SchoolOn High Angular Resolution Techniques

CIEP and Paris Observatory, 1-5 Nov. 2010Sevres and Meudon

Copyright 2010 California Institute of Technology. Government sponsorship acknowledged.


Outline

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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.


1. Introduction

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What is a Coronagraph?

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• 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.

pupilplane

distant Sun& corona

2nd pupil plane& Lyot stop

image plane& Sun blocker 2nd image plane

& detector


Internal & external coronagraphs

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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.


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 brightnessand separation

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Specific intensity

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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.


Photon rates

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photon / (s cm2 Hz sr)

photon / (s cm2 cm sr)

photon / (s cm2 μm sr)


Star flux

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photon / (s cm2 μm )

etendue is conserved(etendue = “throughput”)


Flux & magnitudes

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erg/(s cm2 μm)


Kepler’s 3rd law

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year, AU, solar mass, & negligible planet mass

θ = (1 + e) a/d arcsec, AU, pc


Contrast vs wavelength

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Contrast (visible)

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p is geometric albedoΦ(α) is phase functionrp is planet radius

α is phase angle

Earth visible contrast

Jupiter visible contrast


Contrast vs separation

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Contrast (thermal infrared)

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T is effectivetemperature ofstar or planet

Luminosity of star

Set incident flux = radiated flux,over fraction f of planet


Zodiacal light (zodi)

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Empirical relation from SS obs & Zodipic calc.

Ωas = solid angle (arcsec2)

RAU = radius (AU)


Color and spectra

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“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.


3. Photonsas waves

& particles

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A traveling photon is a wavefront

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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.


A detected photon is a particle

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Poisson process (all values of average rate)

Gaussian (normal) process(large values of average rate)


Photons in radio and optical

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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 asingle electromagnetic mode,

per polaristion state (2 of these),from a blackbody source


Why radio and optical are different

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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 signalsinterferometrically, ie, including the phase information.

However in an array of optical telescopes, only one photon in a single modecan be detected at a time, so all telescopes must feed a single collection point. Ie, we cannot coherently combine data from several independent telescopes afterdetection; 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.


4. Coronagraph& Interferometer

Concepts

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Types of Direct Imaging

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4 coronagraph planes

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pupil plane (x1)

image plane (x2 or -θ2)

2nd image plane (x4 or θ4)

2nd pupil (Lyot) plane (x3)

distant star & planet, etc.


Classical single pupil (1 of 2)

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The phase of a tilted wavefront is:

The amplitude at the focus is:


Classical single pupil (2 of 2)

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1-D case 2-D case

Simple to calculate in 1-D; but cannot easily extrapolate to 2-D case.


FT relations

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Note:

Result: The amplitudes insuccessive planes are FourierTransforms, so it follows thatPlane 3 is an image of plane 1,& plane 4 is an image of plane 2.


Convolution picture

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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 amultiply operation and “*” is a convolution.

This means that we can write the amplitude in a plane asthe 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.


Imaging Recipe

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Practical considerations

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• Kramers-Kronig relation says that an absorption always has a related phase shift (and vice versa). So when you deposit an absorbing or reflecting metalspot 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 avariable 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).


Off-axis performance

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Because stars have finite diameters, and telescopes do not point perfectly, it is helpful if a nulling instrumentsuch 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 ifthe instrument has an intensity null that is proportional tooff-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.


Obscurations in the pupil

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Circular telescope witharea AD and central obscuration of area Ad.

Circular telescope witharea AD and spider ofwidth 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).


Pupil apodization

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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.


Pupil masking

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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.


Pupil mapping (PIAA)

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Ray-trace images of PIAA

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Lyot (hard-edge) mask coronagraph

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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.


Gaussian mask coronagraph

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where z± = πθg(±D/2-x3)/λ

is a gaussian mask, black in the center.

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.


Band-limited mask coronagraph (1/2)

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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.


Band-limited mask coronagraph (2/2)

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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.


Phase mask coronagraph

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If a phase mask in inserted into the image plane, such that halfof the star image is delayed by a half-wavelength with respectto 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.


Vector vortex mask coronagraph

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1-D example, phase mask in image plane.

2-D example (Mawet 2005)

Calc. 2nd pupil amplitude (m = 2x3/D)

Small in middle of pupil, bright near edge,zero at edge, but not a good coronagraph

Calc. 2nd pupil amplitude, get zero in middle of full pupil,bright outside edge, a perfect coronagraph


External occulter coronagraph

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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.


Nulling interferometer (1/4)

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Phase of wavefront at plane-1.

Amplitude in pupil plane is sum of amplitudes in each of the pupils, same as anintegral over sparse pupil elements, with single-pupil shape factored out.

Intensity in plane-2, where detector is located, for 3 differentconfigurations of delay.



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3 different delays,rapid chopping.

Star signal from this chop.

Exozodi signal from this chop.



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For exoplanet detection,use this delay set.

Chop rapidly between these 2 states.

Final output is planet signal.


Visible nuller coronagraph/interferometer

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This design is well-suited to segmented pupils, such as the EELT,where the image of one segment can be superposed on the imageof another, effectively making the telescope into a nulling interferometer with many parallel baselines.


4. Speckles

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Speckles from a phase step in pupil

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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.


Speckles from phase and amplitude ripples

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Standard Fourieranalysis on a finiteInterval.

Coefficients are projectionsof the wavefront to be approximated

Same as above, but in complexNotation.


Example of speckles in focal plane

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Phase and amplitude ripple in pupil

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Suppose pupil has a phase rippleand 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.


Speckles in image plane

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Let A0(θ) be the usual Airy pattern.

Then the amplitude in the pupilis a sum of 3 Airy patterns.

The intensity is the sum of 6 terms.

These are symmetrically-placed speckles.

These are pinned speckles,amplified by the original Airy pattern.


Speckles from mirror errors

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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 thearea under the PSD curve.


Contrast in a dark hole

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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:


Single-speckle nulling (1/2)

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Recall that if this ripple is inthe pupil plane, then …

… we get this amplitude in the image plane, i.e., we get a main star imageplus a speckle ghost on either side of it.


Single-speckle nulling (2/2)

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text

text

text


Multi-speckle nulling

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Suppose that we start with this (unknown) phase in thepupil. (1-D for illustration)

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 outboth phase and amplitude in a half-square dark hole.

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.


Multi-speckle energy minimization (1/3)

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Assume this is the unknown inputphase. Φ1 can be complex.

This is the amplitude in plane 2, the image plane.

Expand in a Fourier series.

This is a DM just after this pupil.

Expand in Fourier series.



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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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If we minimize with respect to the parameters of the DM, the wavefront distortion is canceled, up tothe highest frequency of the DM.

If we actually integrate over the entire focal plane, we get this total energy.

This is the remaining energy, high freqs. of distortion plus all of the absorption part of the wavefront.


Talbot effect (1/2)

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This is the condition for a line of holesto be reproduced a distance z downstream.

There will be infinitely many such planes lyingbetween z=0 and z=zTC. This is the Talbot carpet.

If the wavelength is small, this is z.


Talbot effect (2/2)

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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 periodic intensity.


6. Ground vs Space

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Ground vs space direct imaging

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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.


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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