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Brandoch Calef Wavelength Diversity

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2 Introduction Wavelength diversity = Imaging using simultaneous measurements at different wavelengths. Why should this help? Diversity: the PSF is different in each band Wavefront estimation at longer wavelengths is easier How could it be used? Collect simultaneously in multiple bands, postprocess all data together by coupling wavefront phases. See work of Stuart and Doug. Or: recover wavefront in one band (e.g. LWIR) and use it to partially correct other band (e.g. with a DM). Star observed in LWIR exhibits speckle

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3 Spectral coverage at AMOS 480—660 nm raw ASIS 700 — 950 nm raw ASIS 1 — 1.2 μm raw NIRVIS 4 μm — 5 μm raw LWIR 11 μm — 12 μm raw LWIR AMOS sensors can collect simultaneously from visible to LWIR.

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4 IR image limited by diffraction MFBD processing of simulated MWIR (3.5 μm) data: At longer wavelengths, high spatial frequencies are lost due to diffraction. Resulting reconstructed image lacks fine detail.

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5 Visible image limited by poor wavefront estimate MFBD processing of simulated visible (500 nm) data: At shorter wavelengths, MFBD becomes trapped in a local maximum of the cost function and fails to find true wavefront → Recovered image has artifacts.

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6 Wavelength diversity: linking spectral bands Each wavelength experiences ~same optical path difference (OPD) due to atmospheric turbulence Wavefront phase is θ λ = OPD × 2π/λ, point-spread function is |F[P exp(i θ λ )]| 2 Longer wavelength Shorter wavelength Longer wavelength: turbulence less severe, diffraction more severe Shorter wavelength: turbulence more severe, diffraction less severe OPD in telescope pupil

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7 Spectral variation of imagery OPD can be linked from band to band, but images cannot: To demonstrate insensitivity to spectral variation, use satellite defined in two bands for wavelength-diverse processing example: 800 nm4.7 μm11 μm 3.5 μm500 nm

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8 Wavelength-diverse MFBD processing of visible and MWIR data: Combination of sensors yields better reconstructed image Two reconstructions, one in each band MWIR onlyVisible onlyJoint reconstruction

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9 OPD invariance breakdown: diffraction Basic assumption in coupling phase at different wavelengths is that and that OPD is not a function of wavelength. But OPD actually does depend on wavelength to some degree. Geometrical optics: OPD is sum of delays along path. But diffraction is wavelength- dependent. Mean-square phase error between λ 1 and λ 2 due to neglected diffraction: in rad 2 at λ 1 where k i = 2π/λ i, h 0 = telescope altitude, h 1 = top of atmosphere, x = zenith angle, D = diameter (Hogge & Butts 1982).

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10 OPD invariance breakdown: diffraction OPD error due to diffraction as function of wavelength, λ 2 =10 µm, r 0 =5 cm, zenith angle=30° OPD error due to diffraction as function of wavelength, λ 2 =500 nm, r 0 =5 cm, zenith angle=30° OPD error due to diffraction as function of r 0, λ 1 =800 nm, λ 2 =10 µm, zenith angle=30° OPD error due to diffraction as function of zenith angle, λ 1 =800 nm, λ 2 =10 µm, r 0 =5 cm Wavefront error in waves rms at λ 1 (λ1)(λ1)(λ1)(λ1) 600 nm

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11 OPD invariance breakdown: path length error Geometrical approximation: Wavelength dependence of n is usually ignored, but can be significant for wavelength diversity. Assume n is separable in λ and (z, x). Tilt-removed mean-square phase error due to path length error is in rad 2 at λ 1. Should be at least partially correctible based on approximate knowledge of n(λ). n -1 Mathar, “Refractive index of humid air in the infrared,” J. Opt. A 9 (2007)

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12 OPD invariance breakdown: path length error OPD error as function of wavelength, λ 2 =10 µm, r 0 =5 cm, zenith angle=30° OPD error as function of wavelength, λ 2 =500 nm, r 0 =5 cm, zenith angle=30° OPD error as function of r 0, λ 1 =800 nm, λ 2 =10 µm, zenith angle=30° OPD error as function of zenith angle, λ 1 =800 nm, λ 2 =10 µm, r 0 =5 cm Wavefront error in waves at λ 1 (λ1)(λ1)(λ1)(λ1)

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13 top of atmosphere observatory Different colors follow different paths through atmosphere: Illustration not to scale! Actual pupil displacement at top of atmosphere ~few cm except at very low elevation. Mean-square phase error between λ 1 and λ 2 due to chromatic anisoplanatism in rad 2 at λ 1 where a(h) is air density at height h (Nakajima 2006). Projected pupils diverge → OPD depends on wavelength OPD invariance breakdown: chromatic anisoplanatism

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14 OPD invariance breakdown: chromatic anisoplanatism OPD error as function of wavelength, λ 2 =10 µm, r 0 =5 cm, zenith angle=30° OPD error as function of wavelength, λ 2 =500 nm, r 0 =5 cm, zenith angle=30° OPD error as function of r 0, λ 1 =800 nm, λ 2 =10 µm, zenith angle=30° OPD error as function of zenith angle, λ 1 =800 nm, λ 2 =10 µm, r 0 =5 cm Wavefront error in waves at λ 1 (λ1)(λ1)(λ1)(λ1) Totals assume independent error contributions.

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15 OPD invariance breakdown is small relative to turbulence OPD error as function of wavelength, λ 2 =10 µm, r 0 =5 cm, zenith angle=30° OPD error as function of wavelength, λ 2 =500 nm, r 0 =5 cm, zenith angle=30° OPD error as function of r 0, λ 1 =800 nm, λ 2 =10 µm, zenith angle=30° OPD error as function of zenith angle, λ 1 =800 nm, λ 2 =10 µm, r 0 =5 cm Wavefront error in waves at λ 1 (λ1)(λ1)(λ1)(λ1) OPD error not sensitive to elevation angle above 40 degrees If wavefront is measured at 10 µm, total error at 800 nm about ¼ wave, increases rapidly for shorter wavelengths, vs. 1.29 waves atmospheric turbulence Dominant error source is almost every case is path length error, which is partially correctible

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16 Cramér-Rao bounds on variance of wavefront estimate 800 nm989 nm1.98 µm3.5 µm4.7 µm9.9 µm11 µm Pristine image Measured image QE 0.50.150.50.4 0.5 Read noise 7 e - 50 e - 1300 e - PSNR 10072170298242004300 Renderings from SVST (TASAT), range to satellite (SEASAT) ~450 km Includes solar spectral irradiance, atmospheric extinction, thermal foreground Δλ/λ = 1/8, D=3.6 m, 1/60 sec integration time, r 0 =6 cm at 500 nm, telescope optics throughput = 30% at all wavelengths Next step: Characterize effect of radiometry/sensor noise on wavefront estimate with Cramér-Rao bounds.

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17 CRB caveats Calculating CRB from pseudoinverse of full FIM is not consistent from band to band Here only first 88 Zernikes beyond piston, tip, and tilt participate. Residual rms OPD ≈ 1830 nm! Possibly better approach would be to integrate Fisher information matrix over residual wavefront. CRB results here provide lower bounds and illustrate trends. True OPDOPD estimated in MWIR vs.

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18 CRBs: single wavelengths 3.5 µm 4.7 µm 11 µm 9.9 µm 2 µm 990 nm MWIR: low signal, high noise LWIR: high SNR, low sensitivity to wavefront NIR/SWIR: moderate SNR, high sensitivity to wavefront Aberrations very small in LWIR, so modulation corresponding to Zernike orders is evident.

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19 CRBs: NIR + second band 800 nm + second band (988 nm – 11µm)

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20 CRBs: 11 µm + second band 11µm + second band (988 nm – 9.9 µm)

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21 Summary of CRB analysis Wavelength Single-channel OPD CRB 1/2 (nm) Two-channel OPD CRB 1/2 (nm) with 11 µm Two-channel OPD CRB 1/2 (nm) with 800 nm 989 nm6.55.62.9 1.98 µm9.88.42.9 3.5 µm5501003.4 4.7 µm420953.4 9.9 µm77603.1 11 µm128–3.2 LWIR preferable to MWIR Two LWIR channels preferable to one LWIR + one MWIR SNR trumps diversity, perhaps because object is independent in each band NIR/SWIR results much better than longer wavelengths, but probably not achievable because of local minima traps.

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22 Conclusions and future steps Wavelength-diverse MFBD is a promising technique for combining data from multiple sensors to yield a higher-quality reconstructed image. “Diversity” offered by multi-wavelength imaging is less important than the fact that wavefront estimation is just easier at longer wavelengths Local minima traps at shorter wavelengths, even in joint processing with longer wavelengths Coupling between bands is not sufficiently strong unless some coupling of images is assumed (compare with phase diversity) For a reasonable range of conditions, the OPD changes ¼ wave or less (rms @ 800nm) between 800 nm and 10 µm, potentially half of this if path length error can be approximated. This is a small fraction of the total wavefront error. CRB analysis shows greater advantage in using LWIR bands than MWIR bands. Good characterization of the LWIR path is likely to be critical. Experimental studies: On 1.6 m telescope using GEMINI (visible) and ADET (1-2 μm) cameras On AEOS 3.6 m using range of sensors from visible to LWIR

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