Brandoch Calef Wavelength Diversity
Feb 01, 2016
Brandoch Calef
Wavelength Diversity
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
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.
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.
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.
6
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 |F[P exp(i θλ)]|2
Longer
wavelength
Shorterwavelength
Longer wavelength: turbulence less severe,diffraction more severe
Shorter wavelength: turbulence more
severe,diffraction less severe
OPD in telescope pupil
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 nm 4.7 μm 11 μm
3.5 μm500 nm
8
Wavelength-diverse MFBD processing of visible and MWIR data:
Combination of sensors yields better reconstructed image
Two reconstructions, one in each band
MWIR only Visible only Joint reconstruction
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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 λ1 and λ2 due to neglected diffraction:
in rad2 at λ1 where ki = 2π/λi, h0 = telescope altitude, h1 = top of atmosphere, = zenith angle, D = diameter (Hogge & Butts 1982).
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OPD invariance breakdown: diffraction
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Wavelength (um)
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Wavelength (um)
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r0 at zenith, 500 nm (cm)
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Zenith angle (deg)
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OPD error due to diffraction as function of wavelength,
λ2=10 µm, r0=5 cm,zenith angle=30°
OPD error due to diffraction as function of wavelength,
λ2=500 nm, r0=5 cm,zenith angle=30°
OPD error due to diffraction as function of r0,
λ1=800 nm, λ2=10 µm,zenith angle=30°
OPD error due to diffraction as function of zenith angle,
λ1=800 nm, λ2=10 µm,r0=5 cm
(λ1) (λ1)
600 nm
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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 rad2 at λ1. Should be at least partially correctible based on approximate knowledge of n(λ).
0 2 4 6 8 102.72
2.74
2.76
2.78
2.8x 10
-4
Wavelength (um)
n -
1
nMathar, “Refractive index of humid air in the infrared,” J. Opt. A 9 (2007)
120 10 20 30 40 50 60 70 80
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Zenith angle (deg)
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Chromatic path length error
Diffraction
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r0 at zenith, 500 nm (cm)
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aves
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Chromatic path length error
Diffraction
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0.025
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Correction wavelength (um)
Wav
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aves
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Chromatic path length error
Diffraction
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Correction wavelength (um)
Wav
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aves
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Chromatic path length error
Diffraction
OPD invariance breakdown:path length error
OPD error as function of wavelength, λ2=10 µm,
r0=5 cm, zenith angle=30°
OPD error as function of wavelength, λ2=500 nm,
r0=5 cm, zenith angle=30°
OPD error as function of r0,λ1=800 nm, λ2=10 µm,
zenith angle=30°
OPD error as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm
(λ1) (λ1)
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 λ1 and λ2 due to chromatic anisoplanatism
in rad2 at λ1 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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Zenith angle (deg)
Wav
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aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
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r0 at zenith, 500 nm (cm)
Wav
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aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
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0.015
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Correction wavelength (um)
Wav
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rror
(w
aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
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0.4
0.6
0.8
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Correction wavelength (um)
Wav
efro
nt e
rror
(w
aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
OPD invariance breakdown:chromatic anisoplanatism
OPD error as function of wavelength, λ2=10 µm,
r0=5 cm, zenith angle=30°
OPD error as function of wavelength, λ2=500 nm,
r0=5 cm, zenith angle=30°
OPD error as function of r0,λ1=800 nm, λ2=10 µm,
zenith angle=30°
OPD error as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm
(λ1) (λ1)
Totals assum
e independent error contributions.T
otals assume independent error contributions.
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0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
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Wavelength (um)
Wav
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aves
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Path-length error
TotalTilt-removed wavefront
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r0 at zenith, 500 nm (cm)
Wav
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aves
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Path-length error
TotalTilt-removed wavefront
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Zenith angle (deg)
Wav
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aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
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0.005
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0.015
0.02
0.025
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0.035
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Correction wavelength (um)
Wav
efro
nt e
rror
(w
aves
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Chromatic path length error
Chromatic anisoplanatismDiffraction
Total
OPD invariance breakdown is
small relative to turbulence
OPD error as function of wavelength, λ2=10 µm,
r0=5 cm, zenith angle=30°
OPD error as function of wavelength, λ2=500 nm,
r0=5 cm, zenith angle=30°
OPD error as function of r0,λ1=800 nm, λ2=10 µm,
zenith angle=30°
OPD error as function of zenith angle, λ1=800 nm, λ2=10 µm, r0=5 cm
(λ1) (λ1)
OPD error not sensitive to elevation angle above 40 degrees
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
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
Dominant error source is almost every case is path length error, which is partially correctible
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Cramér-Rao bounds on variance of wavefront
estimate
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
Next step: Characterize effect of radiometry/sensor noise on wavefront estimate with Cramér-Rao bounds.
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 wavefront (nm)
-1500
-1000
-500
0
500
1000
1500
Estimated wavefront (nm)
-1500
-1000
-500
0
500
1000
1500
True OPD OPD estimated in MWIR
vs.
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CRBs: single wavelengths
0 10 20 30 40 50 60 70 80 90 10010
-10
10-9
10-8
10-7
Zernike index
CR
B1/
2 (m
)
3.5 µm
4.7 µm
11 µm
9.9 µm
2 µm
990 nm
MWIR: low signal, high noise
MWIR: low signal, high noise
LWIR: high SNR, low sensitivity to
wavefront
LWIR: high SNR, low sensitivity to
wavefront
NIR/SWIR: moderate SNR,
high sensitivity to wavefront
NIR/SWIR: moderate SNR,
high sensitivity to wavefront
Aberrations very small in LWIR, somodulation corresponding to Zernikeorders is evident.
Aberrations very small in LWIR, somodulation corresponding to Zernikeorders is evident.
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CRBs: NIR + second band
0 10 20 30 40 50 60 70 80 90 10010
-10
10-9
10-8
10-7
Zernike index
CR
B1/
2 (m
)
800 nm +second band(988 nm – 11µm)
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CRBs: 11 µm + second band
0 10 20 30 40 50 60 70 80 90 10010
-10
10-9
10-8
10-7
Zernike index
CR
B1/
2 (m
)
11µm +second band(988 nm – 9.9 µm)
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Summary of CRB analysis
WavelengthSingle-channelOPD 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
• 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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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