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Wavelength Diversity

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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? - PowerPoint PPT Presentation
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Brandoch Calef Wavelength Diversity
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Page 1: Wavelength Diversity

Brandoch Calef

Wavelength Diversity

Page 2: 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

Page 3: Wavelength Diversity

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.

Page 4: Wavelength Diversity

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.

Page 5: Wavelength Diversity

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.

Page 6: Wavelength Diversity

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

Page 7: Wavelength Diversity

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

Page 8: Wavelength Diversity

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

Page 9: Wavelength Diversity

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 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).

Page 10: Wavelength Diversity

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OPD invariance breakdown: diffraction

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

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Wavelength (um)

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aves

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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

Page 11: Wavelength Diversity

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 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)

Page 12: Wavelength Diversity

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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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Chromatic path length error

Diffraction

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Correction wavelength (um)

Wav

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Chromatic path length error

Diffraction

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

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Correction wavelength (um)

Wav

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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)

Page 13: Wavelength Diversity

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

Page 14: Wavelength Diversity

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Chromatic anisoplanatismDiffraction

Total

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r0 at zenith, 500 nm (cm)

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aves

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Chromatic path length error

Chromatic anisoplanatismDiffraction

Total

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Correction wavelength (um)

Wav

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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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1

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Correction wavelength (um)

Wav

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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.

Page 15: Wavelength Diversity

15

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

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Wavelength (um)

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Path-length error

TotalTilt-removed wavefront

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TotalTilt-removed wavefront

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Chromatic anisoplanatismDiffraction

Total

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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

Page 16: Wavelength Diversity

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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.

Page 17: Wavelength Diversity

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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.

Page 18: Wavelength Diversity

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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.

Page 19: Wavelength Diversity

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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)

Page 20: Wavelength Diversity

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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)

Page 21: Wavelength Diversity

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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.

Page 22: Wavelength Diversity

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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


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