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