1. 1 Introduction to Coronagraph Optics Michelson Summer School on High-Contrast Imaging Caltech, Pasadena 20-23 July 2004 Wesley A. Traub Harvard-Smithsonian Center for Astrophysics

2. 2 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

3. 3 Solar system at 10 pc At visible wavelengths: Earth/sun = 10 -10 = 25 mag Zodi per pixel is small

4. 4 Discovery space for coronagraphs

5. 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. 6 Planet albedo and color

7. 7 Bernard Lyot and his coronagraph machines

8. 8 Early solar coronagraphs 1932 1963 radial angle intensity Bernard Lyot Lyot 2004 corona

9. 9 Stellar coronagraphs Ref: McCarthy & Zuckerman (2004); Macintosh et al (2003) 20 arcsec radius circle K~20 mag Bkgd objects 7 arcsec wand J~21 mag Bkgd object

10. 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 ~ SNR 2 * (RMS wavefront) 2 / D 4, so 30 m on ground is equivalent to ~ 2 m in space. Ref.: Stapelfeldt et al., SPIE 2002; Dekaney etal 2004.

11. 11 Photons and waves

12. 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 = amplitude 2 to detect the photon.

13. 13 Photon………..wave…..………...photon 1x1x 1y1y 1z1z ExEx EyEy 1 count detected 1 photon emitted E(x,y,z) = 1 x E x sin(kz-wt-p x ) + 1 y E y sin(kz-wt-p y ) where the electric field amplitude in the x direction is sin(kz-wt-p x ) = Im{ e i(kz-wt-px) } and likewise for the y-amplitude. At detector, add the waves from all possible paths.

14. 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. 15 Huygens wavelets Wavelets align here, and make nearly flat wavefront, as expected from geometric optics. Wavelets add with various phases here, reducing the net amplitude, especially at large angles. Portion of large, spherical wavefront from distant atom. Blocking screen, with slit.

16. 16 Image-plane coronagraphs

17. 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. 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. 19 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 image mask Lyot stop detector

20. 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. 21 How to satisfy L·(M*A)(u) = 0 L(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) u 0

22. 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 < 0 gives M(u) = sinc which has ∫ M(u) ~ 0 inside ± e/2 and M(u) ~ 0 outside ± e/2, but not exactly.

23. 23 Wide-band (gaussian) mask Amplitude of on-axis star = 1 e i0 FT( gauss(x) ) = delta(u) - gauss(u) Convolution Lyot stop blocks bright edges Leakage transmission of on-axis star

24. 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. 25 4-Quadrant phase mask Sub-wavelength phase mask, from silicon, for K-band region.

26. 26 X-Y phase knife experimenttheory

27. 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. 28 Band-limited masks = sin 2 (kx) (sin 4 (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

29. 29 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 is totally blocked In re-imaged pupil. Off-axis planet is ~fully transmitted In re-imaged pupil.

30. 30 Image-plane coronagraph simulation Ref.: Pascal Borde 2004 1st pupil 1st image with Airy rings mask, centered on star image 2nd pupil Lyot stop, blocks bright edges 2nd image, no star, bright planet

31. 31 Band-limited (1 - sin x/x) mask FT(1 - sin x/x) = rect(u) + delta(u) Convolution Lyot stop blocks bright edges Zero transmission of on-axis star Amplitude of on-axis star = 1 e i0

32. 32 sin 2 x, 1-sin x/x and other band-limited masks

33. 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. 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. 35 Pupil-plane coronagraphs Shaped pupil mask Apodized pupil mask Discrete-transmission pupil mask Discrete-mapped pupil Continuous-mapped pupil Nulled pupil

36. 36 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(-u 2 ). Then star at (x,y)=(0,0) gives A(x,0) =   inf du  v=  g(u) e ikxu dv =  exp(-u 2 + 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) =   inf du  v=  g(u) e ikyv dv =  [exp(iky exp(-u 2 )) - exp(-iky exp(-u 2 ) )]du = periodic & messy x y

37. 37 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

38. 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. 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. 40 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

41. 41 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

42. 42 Achromatic nulling coronagraph splitrecombine π phase & rotate pupil Solution with mirrors Solution with lenses

43. 43 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-axisMain 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. 44 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.

45. 45 Nulling-shearing coronagraph The central star is nulled by 180 o 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. 46 Perturbation #1: ripples and speckles

47. 47 Phase ripple and speckles Suppose there is height error h(u) across the pupil, where h( u) =  n a n cos(Knu) + b n sin(Knu) = ripple, K=2  /D The amplitude across the pupil is then A(u ) = e ikh(u)  1 + i[k  n a n cos(Knu) + b n sin(Knu)] In the image plane at angle  the amplitude will be A(  ) =  A(u) e ik  u du =  (0) + (i/2)  n [(a n -ib n )  (k  -Kn) + [(a n +ib n )  (k  +Kn)] The image intensity is then I(  ) =  (0) + (1/4)  n (a n 2 +b n 2 ) [  (k  -Kn) +  (k  +Kn)] = speckles at  =  n /D If we add a deformable mirror (DM), then a n  a n +A n and b n  b n +B n Commanding A n =-a n and B n =-b n forces all speckles to zero.

48. 48 Phase + amplitude ripple and speckles Suppose the height error h(u) across the pupil is complex, where h(u) =  n (a n +ia n ')cos(Knu) + (b n +ib n ')sin(Knu) = ripple i.e., we have both phase and amplitude ripples (= errors). The image intensity is then I(  ) =  (0) + (1/4)  n [(a n +b n ’) 2 + (b n -a n ’) 2 ]  (k  +Kn) + [(a n -b n ’) 2 + (b n +a n ’) 2 ]  (k  -Kn)] = speckles If we add a deformable mirror (DM), and command A n = -(a n -b n ’) and B n = -(b n +a n ’) Then we get I(  ) =  (0) +  n [(b n ’) 2 + (a n ’) 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. 49 Phase ripple and speckles No DM: With DM: Phase ripples from primary mirror errors Polishing errors on primary Speckles generated by 3 sinusoidal components of the polishing errors Pupil plane Image plane

50. 50 Full-field correction of phase and amplitude Half-silvered mirror DM-2 corrects amplitude DM-1 corrects phase With two DMs we can correct both phase and amplitude errors across the pupil. This is a conceptual diagram.

51. 51 Perturbation #2: Polarization

52. 52 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

53. 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. 54 Perturbation #3: Fresnel is not Fraunhofer

55. 55 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.

56. 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. 57 Perturbation # 4: n+ik index of refraction

58. 58 n+ik Image mask: For the case of the image mask (e.g., transmission =(1-sinc 2 ) 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. 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. 60 Perturbation #6: Geometric distortions

61. 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. 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.