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Introduction to Coronagraph Optics

Michelson Summer School on High-Contrast Imaging

Caltech, Pasadena

20-23 July 2004

Wesley A. TraubHarvard-Smithsonian Center for Astrophysics

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• 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

Outline of talk

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Solar system at 10 pc

• At visible wavelengths:

• Earth/sun = 10-10

= 25 mag

• Zodi per pixel is small

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Discovery space for coronagraphs

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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.

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Planet albedo and color

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Bernard Lyot and his coronagraph machines

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Early solar coronagraphs

1932

1963

radial angle

inte

nsi

ty

Bernard Lyot

Lyot

2004 corona

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Stellar coronagraphs

Ref: McCarthy & Zuckerman (2004); Macintosh et al (2003)

20 arcsec radius circle

K~20 magBkgd objects

7 arcsec wand

J~21 magBkgd object

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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

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Basic photon-wave-photon processWe 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…..………...photon1x

1y

1z

Ex

Ey

1 countdetected

1 photon emitted

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.

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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.

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Huygens wavelets

Wavelets align here, and make nearly flat wavefront, as expected from geometric optics.

Wavelets add with variousphases here, reducing the net amplitude, especially

at large angles.

Portion of large, spherical wavefrontfrom distant atom.

Blocking screen,with slit.

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Image-plane coronagraphs

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Huygens’ wavelets --> Fraunhofer --> Fourier transformThe phase of each wavelet on a surfaceTilted by theta = x/f and focussed by theLens 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

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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

Ref.: Sivaramakrishnan et al., ApJ, 552, p.397, 2001; Kuchner 2004.

L(u)·[M(u)*A(u)]~0L(x)*[M(x)·A(x)]~0

u

u

u

u

x

x

x

x

A(u)

A(x)

M(x)

MA

M*A

L(u)

L[M*A]

L*[MA]

aperture

imagemask

Lyotstop

detector

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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

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How to satisfy L·(M*A)(u) = 0

L(u) = 0 here L(u) = 0 here

M(u)=0 here M(u)=0 here

∫M(u)du=0 here

Lyot stop

Nominal pupil diameter

1/21/2-e/2e/2

M(u)=anything = 0 (band-limited) ≠ 0 (notch)

M(u)=anything = 0 (band-limited) ≠ 0 (notch)

u0

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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.

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ˆ M (x)

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ˆ M (x) = 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 < 0 gives M(u) = sinc

which has ∫M(u) ~ 0 inside ± e/2 and M(u) ~ 0 outside ± e/2, but not exactly.

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ˆ M (x)

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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 transmissionof on-axis star

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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.€

€

€

0

€

0

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4-Quadrant phase mask

Sub-wavelength phasemask, from silicon, for K-band region.

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X-Y phase knife

experiment theory

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X-Y phase knife: double star in lab

Binary star without coronagraph

Binary with X Phase knife

Binary with X + Yphase knives; Bright star nulled

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Band-limited masks

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ˆ M (x)

€

ˆ M (x)

= sin2(kx) (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.

= 1 - sin(kx)/kx ([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

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Band-Limited Image Mask

Example: this 1-D image mask transmits the

band-limited function (1-sin x/x)2 .

Ref.: Kuchner & Traub ApJ 570, 900, 2002

On-axis star istotally blocked In re-imaged pupil.

Off-axis planet is~fully transmitted In re-imaged pupil.

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Image-plane coronagraph simulation

Ref.: Pascal Borde 2004

1stpupil

1stimagewithAiry rings

mask, centered on starimage

2ndpupil

Lyotstop,blocks bright edges

2nd image,no star,brightplanet

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Band-limited (1 - sin x/x) mask

FT(1 - sin x/x) =rect(u) + delta(u)

Convolution

Lyot stop blocks bright edges

Zero transmissionof on-axis star

Amplitude of on-axis star = 1 ei0

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sin2x, 1-sin x/x and other band-limited masks

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Notch-filter masks

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ˆ M (x) = 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

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Null depth vs mask type

Mask Leak near axis Pointing/IWA

Tophat 0 --Disk phase mask 0 --Phase knife 2 0.00014-quad phase mask 2 0.0001All masks > 1st order 2 0.0001Notch filter 4 0.01Band-limited 4 0.01Gaussian 4 0.01 + stopsAchromatic dual zone 4 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

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Shaped-pupil mask

Kasdin, Vanderbei, Littman, & Spergel, preprint, 2004

Pupil: Spergel-Kasdin prolate-spheroidal mask

Image: dark areas < 10-10 transmission

Image: cut along the x-axis

v

u

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

x

y

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Discrete-transmission masks

Bar-code mask (many slots not visible here) Concentric ring mask

6-opening mask; (right) black < 10-10

(left) 20-star mask; (right) PSF for 150-point star mask

Kasdin, Vanderbei, Littman, & Spergel, preprint, 2004

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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.

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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

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.

Image with many aliases Densified

pupil

Clean image,narrow FOV

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Continuous-mapped pupil

Guyon, A&A 404, p.379, 2003; Traub & Vanderbei, ApJ 599, 2003

Input wavefront: uniform amplitude.

Mirror 1

Mirror 2

Output wavefront:prolate-spheroidal amplitude.

100 dB = 10-10 = 25 mag

Output image:prolate spheroid

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Achromatic nulling coronagraph

split recombine

π phase &rotate pupil

Solution with mirrors

Solution with lenses

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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 Airyring, 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.

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Nuller with π-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.

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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.

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Perturbation #1: ripples and speckles

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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/DThe amplitude across the pupil is then A(u ) = eikh(u) 1 + i[kn ancos(Knu) + bnsin(Knu)]

In the image plane at angle the amplitude will be A() = A(u) eiku du = (0) + (i/2) n [(an-ibn)(k-Kn) + [(an+ibn)(k+Kn)] The image intensity is then I() = (0) + (1/4) n (an

2+bn2) [(k-Kn) + (k+Kn)] = speckles

at = n/D

If we add a deformable mirror (DM), then anan+An and bnbn+Bn Commanding An=-an and Bn=-bn forces all speckles to zero.

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Phase + amplitude ripple and specklesSuppose the height error h(u) across the pupil is complex, where h(u) = n (an+ian')cos(Knu) + (bn+ibn')sin(Knu) = ripplei.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.

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Phase ripple and speckles

No DM:

With DM:

Phase ripples from primarymirror errors

Polishing errorson primary

Speckles generated by 3 sinusoidalcomponents of thepolishing errors

Pupil plane

Image plane

Image plane

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Full-field correction of phase and amplitude

Half-silvered mirror

DM-2corrects amplitude

DM-1corrects phase

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

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S-P phase shift

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.

S-P

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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.

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Perturbation #3:Fresnel is not Fraunhofer

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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.

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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.

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Perturbation # 4:n+ik index of refraction

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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?

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ScatteringExperience 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.

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Perturbation #6:Geometric distortions

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“Top 10” Contrast ContributionsMajor 14 Contrast Contributors

Source Cotributor

Contrast (3?/D)

Mask Leakage z3 4.34E-13

Mask Leakage z2 4.33E-13

Rigid Body Pointing Super OAP1 3.22E-13

Rigid Body Pointing Super Fold Mirror 2 1.22E-13

Rigid Body Pointing Super Fold Mirror 1 6.84E-14

Thermal Structural Deformation Aberrations Secondary (z4 for dL ) 3.48E-14

Thermal Structural Deformation Aberrations Secondary (z8 for dL ) 1.14E-14

Thermal Structural Deformation Aberrations

Secondary (z4 for ?f/f)

4.28E-14

Thermal Deformation of Optics Primary (z4) 5.78E-14

Thermal Deformation of Optics Primary (z12) 1.79E-14

Dynamics Deformation of Optics Primary (z4) 2.23E-14Dynamics Deformation of Optics Primary (z8) 2.90E-14

Dynamics Deformation of Optics Primary (z12) 1.79E-14 <Id> Total

Dynamics Structural Deformation Aberrations Secondary (z4) 7.30E-14

Contrast at 3?/D

6.75E-12 7.09E-12

For TPF-C, this table shows that deformations of the optical system are second only to mask leakage and telescope pointingas sources of speckles in the focal plane. Ref.: Shaklan and Marchen (2004).

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SummaryExtrasolar 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.