#### Transcript Wavelength diversity

Wavelength Diversity Brandoch Calef 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 2 Spectral coverage at AMOS raw ASIS raw NIRVIS 1—1.2 μm 480—660 nm raw LWIR raw ASIS 4 μm—5 μm raw LWIR AMOS sensors can collect simultaneously from visible to LWIR. 700—950 nm 11 μm—12 μm 3 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. 4 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. 5 Wavelength diversity: linking spectral bands • Each wavelength experiences ~same optical path difference (OPD) due to atmospheric turbulence • Wavefront phase is θλ = OPD × 2π/λ, point-spread function is Longer wavelength: |F[P exp(i θλ)]|2 turbulence less severe, diffraction more severe Shorter wavelength: turbulence more severe, diffraction less severe OPD in telescope pupil 6 Spectral variation of imagery OPD can be linked from band to band, but images cannot: 800 nm 4.7 μm 11 μm To demonstrate insensitivity to spectral variation, use satellite defined in two bands for wavelength-diverse processing example: 500 nm 3.5 μm 7 Combination of sensors yields better reconstructed image Wavelength-diverse MFBD processing of visible and MWIR data: MWIR only Visible only Joint reconstruction Two reconstructions, one in each band 8 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 wavelengthdependent. Mean-square phase error between λ1 and λ2 due to neglected diffraction: in rad2 at λ1 where ki = 2π/λi, h0 = telescope altitude, h1 = top of atmosphere, x = zenith angle, D = diameter (Hogge & Butts 1982). 9 OPD invariance breakdown: diffraction 0.03 OPD error due to diffraction as function of wavelength, λ2=10 µm, r0=5 cm, zenith angle=30° 0.15 Wavefront error (waves) 0.2 0.1 0.05 OPD error due to diffraction as function of wavelength, λ2=500 nm, r0=5 cm, zenith angle=30° 0.02 0.015 0.01 0.005 0 0.5 0.6 0.7 0.8 0.9 1 Wavelength (um) (λ1) 1.1 1.2 0 1.3 0.4 0 1 2 3 4 5 6 Wavelength (um) 7 8 (λ1) 9 10 0.45 OPD error due to diffraction as function of r0, λ1=800 nm, λ2=10 µm, zenith angle=30° 0.35 0.3 0.25 0.35 0.15 0.3 0.25 0.2 0.1 0.15 0.05 0.1 0 2 4 6 8 10 r at zenith, 500 nm (cm) 12 14 OPD error due to diffraction as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm 0.4 0.2 0 600 nm 0.025 Wavefront error (waves) Wavefront error (waves) Wavefront error (waves) Wavefront error in waves rms at λ1 0.25 16 0.05 0 10 20 30 40 50 Zenith angle (deg) 60 70 80 10 OPD invariance breakdown: path length error Geometrical approximation: Wavelength dependence of n is usually ignored, but can be significant for wavelength diversity. -4 2.8 x 10 2.78 n -1 n-1 • 2.76 Mathar, “Refractive index of humid air in the infrared,” J. Opt. A 9 (2007) 2.74 2.72 0 2 4 6 Wavelength (um) 8 10 Assume n is separable in λ and (z, x). Tilt-removed mean-square phase error due to path length error is in rad2 at λ1. Should be at least partially correctible based on approximate knowledge of n(λ). 11 OPD invariance breakdown: path length error 1.4 0.03 Chromatic path length error Diffraction 0.025 0.8 Wavefront error (waves) OPD error as function of wavelength, λ2=10 µm, r0=5 cm, zenith angle=30° 1 0.6 0.4 0 0.5 0.6 0.7 0.8 0.9 1 Correction wavelength (um) 1.1 (λ1) 1.2 0.02 0.015 0.01 0 1.3 Chromatic path length error Diffraction 0.6 0.4 0.25 0.1 0.1 0.05 6 8 10 r at zenith, 500 nm (cm) 12 4 5 6 7 Correction wavelength (um) 14 16 8 9 (λ1) 10 0.2 0.15 4 3 OPD error as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm 0.3 0.2 2 2 Chromatic path length error Diffraction 0.35 0.3 0 1 0.4 OPD error as function of r0, λ1=800 nm, λ2=10 µm, zenith angle=30° 0.5 0 0.45 0.7 0 OPD error as function of wavelength, λ2=500 nm, r0=5 cm, zenith angle=30° 0.005 0.2 Wavefront error (waves) Wavefront error (waves) Wavefront error in waves at λ1 Wavefront error (waves) 1.2 Chromatic path length error Diffraction 0 10 20 30 40 50 Zenith angle (deg) 60 70 80 12 OPD invariance breakdown: chromatic anisoplanatism • Different colors follow different paths through atmosphere: top of atmosphere observatory Projected pupils diverge → OPD depends on wavelength • Illustration not to scale! Actual pupil displacement at top of atmosphere ~few cm except at very low elevation. • Mean-square phase error between λ1 and λ2 due to chromatic anisoplanatism in rad2 at λ1 where a(h) is air density at height h (Nakajima 2006). 13 OPD invariance breakdown: chromatic anisoplanatism 1.4 0.045 0.035 Wavefront error (waves) 1 OPD error as function of wavelength, λ2=10 µm, r0=5 cm, zenith angle=30° 0.8 0.6 0.4 OPD error as function of wavelength, λ2=500 nm, r0=5 cm, zenith angle=30° 0.03 0.025 0.02 0.015 0.01 0.2 0.005 0 0.5 0.6 0.7 0.8 0.9 1 Correction wavelength (um) 0.8 1.1 (λ1) 1.2 0.6 0.4 1 2 3 4 5 6 7 Correction wavelength (um) 0.3 0.2 8 9 (λ1) 10 Chromatic path length error Chromatic anisoplanatism Diffraction Total 1.2 OPD error as function of r0, λ1=800 nm, λ2=10 µm, zenith angle=30° 0.5 0 1.4 Chromatic path length error Chromatic anisoplanatism Diffraction Total 0.7 1 OPD error as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm 0.8 0.6 0.4 0.2 0.1 0 0 1.3 Wavefront error (waves) Wavefront error (waves) Wavefront error in waves at λ1 Wavefront error (waves) 1.2 Chromatic path length error Chromatic anisoplanatism Diffraction Total 0.04 0 2 4 6 8 10 r0 at zenith, 500 nm (cm) 12 14 16 0 0 10 20 30 40 50 Zenith angle (deg) 60 70 Totals assume independent error contributions. Chromatic path length error Chromatic anisoplanatism Diffraction Total 80 14 OPD invariance breakdown is small relative to turbulence 0.045 Path-length error Total Tilt-removed wavefront Dominant error source OPD error is as almost function of wavelength, λ2=500 nm, every case is path r0=5 cm, zenith angle=30° length error, which is partially correctible 0.035 If wavefront is measured 0.03 OPD error as function at of 10 µm, total error at 800 wavelength, λ2=10 µm, 0.025 nm about ¼ wave, r0=5 cm, zenith angle=30° increases rapidly 0.02 for 0.015 shorter wavelengths, vs. 0.01 1.29 waves atmospheric 0.005 turbulence Wavefront error (waves) 1 0.8 0.6 0.4 0.2 0 0.5 0.6 0.7 0.8 0.9 1 Wavelength (um) 2 1.1 (λ1) 1.2 0 1.3 1.8 1.6 1.4 1 1 2 3 4 5 6 7 Correction wavelength (um) 0.8 0.6 8 9 (λ1) 10 Chromatic path length error Chromatic anisoplanatism Diffraction Total 1.2 OPD error as function of r0, λ1=800 nm, λ2=10 µm, zenith angle=30° 1.2 0 1.4 Path-length error Total Tilt-removed wavefront Wavefront error (waves) Wavefront error (waves) Wavefront error in waves at λ1 Wavefront error (waves) 1.2 Chromatic path length error Chromatic anisoplanatism Diffraction Total 0.04 OPD error not sensitive OPD errortoaselevation function ofangle above zenith angle, λ1=800 nm, 40 degrees 1 0.8 λ2=10 µm, r0=5 cm 0.6 0.4 0.4 0.2 0.2 0 0 2 4 6 8 10 r0 at zenith, 500 nm (cm) 12 14 16 0 0 10 20 30 40 50 Zenith angle (deg) 60 70 80 15 Cramér-Rao bounds on variance of wavefront estimate Next step: Characterize effect of radiometry/sensor noise on wavefront estimate with Cramér-Rao bounds. 800 nm 989 nm 1.98 µm 3.5 µm 4.7 µm 9.9 µm 11 µm Pristine image Measured image QE 0.5 0.15 0.5 0.4 0.4 0.5 0.5 Read noise 7 e- 7 e- 50 e- 1300 e- 1300 e- 1300 e- 1300 e- PSNR 100 72 170 29 82 4200 4300 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, r0=6 cm at 500 nm, telescope optics throughput = 30% at all wavelengths 16 CRB caveats True OPD OPD estimated in MWIR True wavefront (nm) Estimated wavefront (nm) 1500 1500 1000 1000 500 500 0 0 -500 -500 -1000 -1000 -1500 -1500 • Calculating CRB from pseudoinverse of full FIM is not consistent from band to band vs. • 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. 17 CRBs: single wavelengths -7 10 3.5 µm 4.7 µm MWIR: low signal, high noise 11 µm Aberrations very 9.9small µmin LWIR, so -8 10 LWIR: high SNR, low sensitivity to wavefront CRB1/2 (m) modulation corresponding to Zernike orders is evident. NIR/SWIR: -9 10 moderate SNR, high sensitivity to wavefront 2 µm 990 nm -10 10 0 10 20 30 40 50 60 Zernike index 70 80 90 100 18 CRBs: NIR + second band -7 10 -8 CRB1/2 (m) 10 -9 10 800 nm + second band (988 nm – 11µm) -10 10 0 10 20 30 40 50 60 Zernike index 70 80 90 100 19 CRBs: 11 µm + second band -7 10 -8 CRB1/2 (m) 10 11µm + second band (988 nm – 9.9 µm) -9 10 -10 10 0 10 20 30 40 50 60 Zernike index 70 80 90 100 20 Summary of CRB analysis • • • 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. Single-channel OPD CRB1/2 (nm) Two-channel OPD CRB1/2 (nm) with 11 µm Two-channel OPD CRB1/2 (nm) with 800 nm 989 nm 6.5 5.6 2.9 1.98 µm 9.8 8.4 2.9 3.5 µm 550 100 3.4 4.7 µm 420 95 3.4 9.9 µm 77 60 3.1 11 µm 128 – 3.2 Wavelength 21 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 22