Understanding Speckle Noise and Dynamic Range in Coronagraphic Images

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2 Understanding Speckle Noise and Dynamic Range in Coronagraphic Images Remi Soummer American Museum of Natural History New York

3 Collaborators University of Nice: Claude Aime, Andre Ferrari AMNH: B. Oppenheimer, A. Sivaramakrishnan, A. Digby, S. Hinkley Cornell J. Lloyd STScI: R. Makidon Berkeley J. Graham, M. Perrin, M. Fitzgerald HIA L. Jolissaint (PAOLA package)

4 55 Cnc Lyot Project Coronagraph first light

5 planet is a faint point-like source over a noisy background Several sources of noise: speckle photon detector etc. The stability and statistical properties of these noise source will define the actual dynamic range

6 Lyot project dynamic range Vega dynamic range, 5 Σ, 3s $m Long exposure 4 sec 35 # "7. 3! "7. 5! "8.! "9. 5! ".! ". 5! ".! " Contrast levels (dynamic range) for a sec exposure Short Exposure ( sec) Short exposure sec 35 Long exposure 4 sec (tip-tilt off) Hinkley etal in prep

7 Lyot project dynamic range Long exposure 4 sec Short Exposure ( sec) Short exposure sec Contrast levels (dynamic range) for a sec exposure Main limitation in this case: Quasi-Static Speckles Long exposure 4 sec (tip-tilt off) Hinkley etal in prep

8 A few questions... What is the statistics of the speckle noise in direct high Strehl images? What is the effect of a coronagraph on the statistics of the noise? What is the interaction between coronagraph, static, quasi-static and residual atmospheric speckles? Need for a statistical model Start with direct images Study the effect of a coronagraph Study the effect of static or quasi-static aberrations

9 Speckle and photon noise in direct images

10 imaging in the presence of wavefront errors Speckle pinning (Bloemof etal, Sivaramakrishnan etal ) Palomar AO juin 4 K band 5. CORRECTOR PROOF-OF-CONCEPT DEMONSTRATION The technical feasibility of correcting the mid-spatial-frequency surface errors in large optical components (primary & secondary mirrors) using a small, microfigured mirror was recently demonstrated by ASML Inc. (previously the precision lithographic optics division of Tinsley SVG). They were contracted to produce a small (7 mm) mirror that would correct a provided wavefront error map to a specified level. They were given a map (Figure 6) based on the surface errors of the Hubble Space Telescope (HST) that was derived from phase retrieval measurements from on-orbit data. This map was chosen because it was readily available and represents a worst-case scenario for a future optical Accurate and stable pointing 4) is critical to maintaining alignment of the beam on the corrector mirror and keeping the system like JPF; the mirrors in a new optical system would be much smoother (Figure and would lackthethe record star centered on the occulting spot. Thermally induced variations in the telescope structure can also alter the optical grove polishing errors seen in the HST maps. alignment. The ISS alt-az mounting from Lockheed used by JPF has a pointing and tracking accuracy of., and finer Simulated AO PSF origin: wavefront errors control is maintained using the FSM. The optical bench is kept at a constant temperature using heating elements and thermal isolators. Pointingsof are time. avoided that ASML was able to create a corrector within spec and within the allotted amount In could the introduce criticalsignificant 4-5 thermal variations (for example, the telescope never points at the Earth, the major source of thermal instability in the Hubble Space Telescope). cycles/aperture range, the best-achieved corrected wavefront error (x effective surface error) was.5 nm RMS (Figure To suppress the diffraction below thefrom expected of scatter 7). The corrector would reduce scattered light by a factor of 5x to x, depending onpattern the radius thelevel star. Withlight, we investigated the performance of not only conventional Lyot coronagraphs but also shaped and apodized apertures, such as the Gaussian-like shaped or apodizedadditional time or a smoother wavefront map, they would be able to produce even better results. square apertures favored by other teams. In the end, the apodized-spot Lyot coronagraph was selected because it atmospheric residuals and/or quasi static aberrations (space & ground) provided the best combination of field-of-view, resolution, and throughput. Apodized spots permit more efficient suppression of the diffracted light while making the system slightly less sensitive to jitter. Such spots, however, are difficult to make accurately. For JPF, we have Gaussian-apodized spots with half-widths-at-half-maximum of.4 and.8. Because they are tapered in transmission, imaging is possible within the spot. Target -Corrector Residual Figure 4. This is the surface error map of a. meter diameter mirror produced by ASML (formerly Tinsley) for UV photolithography. It has an RMS midfrequency surface (/ wavefront) error of.5 nm. It demonstrates the current state-ofthe-art in optical surface figuring. A version of this map was used as the assumed residual wavefront error pattern in JPF optical simulations. Compare the smoother surface of this mirror with that of the HST primary (of which a modified form is shown in Figure 4). HST (Krist) TINSLEY (Krist) 7 mm 3. MICROFIGURED WAVEFRONT Figure 6. (Left) Modified surface amplitude map derived from HST phase retrieval measurements that was provided to ASML as the CORRECTOR AO simulation using PAOLA

11 Speckles Two complementary approaches: Wavefront propagation using Tayor expansion (Sivaramakrishnan etal., Perrin etal. 3). Statistical optics approach (Aime & Soummer 4, Soummer & Aime 4)

12 Statistical model: pupil plane Wave amplitude at the pupil plane: Ψ (x, y) = [A + a(x, y)] P (x, y) Plane wave uncorrected part of the wavefront Phase simulation Amplitude (scintillation)

13 Statistical model: pupil plane Wave amplitude at the pupil plane: Ψ (x, y) = [A + a(x, y)] P (x, y) Plane wave uncorrected part of the wavefront Im pupille avec phase seule This decomposition is also valid in space Complex PDF AO simulation phase + amplitude Re Phase simulation Amplitude (scintillation)

14 Statistical model: focal plane Ψ (x, y) = [A + a(x, y)] P (x, y) Ψ (x, y) = A F[P (x, y)] + F[a(x, y) P (x, y)] C (r) S(r) Constant deterministic term Speckle random term P(ξ, η) = exp π < S(r) >! (ξ Re[C (r)]) + (η Im[C (r)]) < S(r) > decentered gaussian statistics (complex amplitude in the focal plane) same solution as for holographic speckle (Goodman 975) "

15 Statistical model: focal plane P(ξ, η) = exp π < S(r) >! (ξ Re[C (r)]) + (η Im[C (r)]) < S(r) > " decentered gaussian statistics (complex amplitude in the focal plane) same solution as for holographic speckle (Goodman 975)

16 Statistical model: focal plane!! # (ξ I + IcRe[C (r)]) I +Ic PI (I) = exp exp I P(ξ, η) = Is Is Is π < S(r) > < S(r) " Statistics of the intensity in the focal plane: modified Rice distribution! " I + Ic PI (I) = exp Is Is. Ic and Is r "arcsec# # $ I Ic I Is. Ic and Is r "arcsec# ".5 ".5 "5. - "5. - "7.5 "7.5 "Log# -4 ". -6 "5. ".5 "#m# "Log# -4 ". -6 "5. ".5 "#m# "7.5 "7.5 ". -8 ". -8 ".5 ".5 Λ!D r "Λ!D# "5. Λ!D Ic = C (r). Λ!D r "Λ!D# "5. Λ!D Is =< S(r) >

17 Statistical model: focal plane Modified Rice distribution! I + Ic PI (I) = exp Is Is " # $ I Ic I Is

18 Photon counting statistics Poisson Mandel Transformation! P(n) = n! PI (I)I n exp( I)dI Photon counting statistics:! n Ic P(n) = ( + ) exp Is + Is Is P(n) " F! Ic n + ; ; Is + Is P(n).35 Ic= Is=..3.5 Ic=4 Is= n n "

19 Mean and Variance Mean intensity: diffraction pattern + halo < Ψ (x, y) >= C(x, y) + < S(x, y) >= Ic + Is Variance speckle: σi = Is + Is Ic

20 Mean and Variance Mean intensity: diffraction pattern + halo < Ψ (x, y) >= C(x, y) + < S(x, y) >= Ic + Is Variance speckle: σi = Is + Is Ic Variance speckle + photon noise: σ = Is + Is Ic + Ic + Is Same variance expression used in the TPF error budget by Shaklan 4

21 Explanation of speckle pinning P(I) P(I).35.5 σi = Is + Is Ic.3 P(I).5.35 P(I)..3 High continuous value amplification of the fluctuation C(r) I P(I) A C(r) 4 4 B C r(λ/d) r(λ/d). 8 6 r(λ/d).5.3 zero of the PSF exponential statistics P(I) ! I PI (I) = exp Is Is I.5 r(λ/d) I I..8 P(I) P(I).5.6 A B C P(I).8.4 low value of the continuous level I. P(I) I C(r) P(I) C(r). I..5. P(I) I I. I "

22 Lick ao data Lick AO First verification on sky of speckles rician statistics: Fitzgerald & Graham (5)

23 Palomar ao data Palomar AO single point statistics through the data cube Exposure time ms in K band, need for a model of integrated speckles $ " # # $!!" M c) I + I c M (i+i I I M i i i M c c PI (I) = exp Iis IM e Iiss ic Is Is is Non central chi square distribution of order M Soummer etal in prep

24 Palomar ao data Maximum Likelihood: Non central chi square vs. data!ic!" 59.58, Is!" 6.635".5 Kolmogorov Smirnoff test i.e. lifetime of 3ms consistent with K band M= 3 4 M=4 3 4 M=6 3 4 dark rings pixels fail the KS test need to include detector noise in the model... work in progress Soummer etal in prep

25 Effect of a coronagraph on the speckle and photon noise

26 ExAOC CODD Figure 8 Left: Gemini pupil without spiders. Right: An optimized shaped pupil that allows for geometry, and provides a -7 contrast floor in a wedge with 5!/D `waist ! Figure 8 Left: perfect PSF of the pupil in Figure 8, with no residual WFE. The green line c contour of -6 relative to the bright central peak. The waist of suppression correspondi peak s PSF is approximately 5!/D across horizontally. Right: The same pupil, with a 96 SFWFS AO-corrected wave front, produces a PSF which is brighter than -6 everywhere w area of the AO system and deformable mirror. The -6 contour lies completely outside the p!/d on a side !"#$%&'()'*+,--"."/,"'.'/%#%,34"/'//&3-5' Phase Masks (PM) 4.4. Systematic Classification of Approaches OPD (x ) OPD (x ) Simple Phase Mask (PM) coronagraphs modify the wave front at the image plane b regions of " radians of phase difference between different parts of the image to crea interference of on-axis light. Recent advances in PM coronagraphy (Soummer et a more sophisticated phase masks to create achromatic (i.e. exact at two wavelengths apochromatic cancellation over wide bandpasses. While recent work on the PM co has indeed shown progress, we have rejected this design, citing three major concern manufacturability, chromatic effects with wide bandwidths, and residual image mo requirements. $!%&'()*'+)(!,(*)-.!)(/(*%,!*!-3-(4',!&!-3)335)*4.6!$!,.3)'!7(,-)&4'&3!*7!38 (-%*'+)(!&,!,.39!&!:*;%(!"<!9&'.!)(=()(-(,!'3!&')37+-'3)!*)'&-%(,!*7!*!*''(4'!'3! 73-+('!'.(!*/*&%*;%(!7&,-+,,&3!3=!'.(!-3-(4'!&!'.(!%&'()*'+)(6!$!4)(%&&*)!-%*,,&=&8 -*'&3!3=!'.(!=*&%&(,!3=!-3)335)*4.,!'.*'!.*/(!7(/(%34(7!&,!4)(,('(7!&!>&5+)(!?6!:.&,! ')((8;*,(7! -%*,,&=&-*'&3! -*4'+)(,!,(/()*%!,*%&('! 7&==()(-(,! ;('9((! -3)335)*4.! 7( D - February 5, 5 D ICD.9.x Page 7

27 Effect of a coronagraph The direct focal plane amplitude is: Ψ (r) = Cd (r) + S(r) Static direct response Speckle term

28 Effect of a coronagraph The direct focal plane amplitude is: Ψ (r) = Cd (r) + S(r) Static direct response Speckle term The coronagraphic focal plane amplitude is: Ψ4 (r) = Cc (r) + S(r) Static coronagraph response Speckle term A coronagraph has a no effect on the speckle term outside the mask area

29 Suppression of speckle amplification Speckle term: (not affected by a coronagraph)..4 Ic and Is r "arcsec# Ic and Is r arcsec " " "7.5 "Log# ".5 "#m# " ". -4 Log "7.5 " ".5 Λ!D Ic without coronagraph Λ!D r "Λ!D# "5. 3 Λ!D.4 Λ D Λ D r Λ D 3 Λ D 5. Ic with coronagraph Suppression of the speckle amplification due to the Ic term Dm

30 Suppression of speckle amplification A coronagraph can suppress the speckle noise coherent amplification (speckle pinning) Direct coronagraphic gain where Ic>Is σ = (Is Ic + Ic ) + Can be removed by a coronagraph (Is + Is ) = σc Unaffected by a coronagraph + σs

31 Semi-analytical method Ic and Is from simulations Variance from analytical expressions..4.6 r(arcsec) (Log) -7 Dynamic Range (Δm) Photons Speckle r (λ/d) 3 4

32 Static aberrations The static coronagraph response includes the static aberrations Ψ4 (r) = Cc (r) + S(r) Residual pinning amplification by static aberrations through the coronagraph! space observations like TPF

33 Static + Quasi-static Quasi-static speckles can also be included Decomposition into: perfect, static, quasi static and atmospheric terms: Ψ = A + As (x) + a (x) + a (x) Ψ4 = C + S (x) + S (x)! " σ = N τ Is + kis + I c (Is + kis ) + Is Is

34 Conclusions (I) The speckle statistics of direct and coronagraphic images is given by a modified Rice distribution Ground based: (static + ao) Space: (static + quasi static) Generalization possible for the ground (static + ao + quasi static) Model consistent with real data

35 Conclusions (II) Semi-analytical method to study/predict dynamic range Speckle calibration/cancellation (slow) necessary to reach the atmospheric variance level Performance of the speckle reduction can be derived from the analysis of the limiting noise contribution

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