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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. 10 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. 12 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. 14 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. 18 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 19 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 23 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. 24 4-Quadrant phase mask Sub-wavelength phase mask, from silicon, for K-band region. 25 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 30 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 33 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 34 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. 39 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. 40 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[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 (an2+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. 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. 50 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. 62