towards optical intensity interferometry for high angular ...pnunez/defense.pdf · introduction...
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Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Towards optical intensity interferometry for highangular resolution stellar astrophysics
Paul D. NunezDept. of Physics & Astronomy
University of Utah
June 4 2012
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Talk outline
Motivations.
Background.
Cherenkov telescope arrays.
The imaging problem: Phase recovery.
Simulations of future intensity interferometers.
Experimental efforts.
Conclusions.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Motivations for stellar intensity interferometry
0.1mas resolution
λ = 400nm
(Simulation by B.Freitag)
∼ 1km baseline
Disk-like stars: Diameters atdifferent wavelengths.
Rotating stars: Oblatenessand pole brightening.
Interacting binaries.
Mass loss and mass transfer.
Altair (CHARA)
α − Arae (VLTI)
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Complex Degree of Coherence γ
γ(ri , ti ; rj , tj) ≡〈E ∗(ri , ti ) · E (rj , tj)〉√〈|E ∗(ri , ti )|2〉〈|E ∗(rj , tj)|2〉
Amplitude Interferometry Intensity Interferometry (I.I.)γ(x) = F [O(θ)] |γ(x)|2 = |F [O(θ)]|2
Star
E i
E j
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Complex Degree of Coherence γ
γ(ri , ti ; rj , tj) ≡〈E ∗(ri , ti ) · E (rj , tj)〉√〈|E ∗(ri , ti )|2〉〈|E ∗(rj , tj)|2〉
Amplitude Interferometry Intensity Interferometry (I.I.)γ(x) = F [O(θ)] |γ(x)|2 = |F [O(θ)]|2
Star
Inte
nsity
Phase
E i
E j
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Complex Degree of Coherence γ
γ(ri , ti ; rj , tj) ≡〈E ∗(ri , ti ) · E (rj , tj)〉√〈|E ∗(ri , ti )|2〉〈|E ∗(rj , tj)|2〉
Amplitude Interferometry Intensity Interferometry (I.I.)γ(x) = F [O(θ)] |γ(x)|2 = |F [O(θ)]|2
Star
Inte
nsity
Phase
E i
E j
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Complex Degree of Coherence γ
γ(ri , ti ; rj , tj) ≡〈E ∗(ri , ti ) · E (rj , tj)〉√〈|E ∗(ri , ti )|2〉〈|E ∗(rj , tj)|2〉
Amplitude Interferometry Intensity Interferometry (I.I.)γ(x) = F [O(θ)] |γ(x)|2 = |F [O(θ)]|2
Star
Correlator
Analysis
Cij =<∆Ii ∆Ij><Ii><Ij>
|Fourier Trans|2
I iI j
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Intensity Interferometry
Cij =< ∆Ii ∆Ij >
< Ii >< Ij >
Hanbury Brown & Twiss.1950’s
Correlation of Intensities.
No need for sub-wavelengthprecision.
Insensitive to turbulence.
Narrabri 1963
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Intensity Interferometry
Cij =< ∆Ii ∆Ij >
< Ii >< Ij >
Hanbury Brown & Twiss.1950’s
Correlation of Intensities.
No need for sub-wavelengthprecision.
Insensitive to turbulence.
Narrabri 1963
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Intensity Interferometry
Cij =< ∆Ii ∆Ij >
< Ii >< Ij >
Hanbury Brown & Twiss.1950’s
Correlation of Intensities.
No need for sub-wavelengthprecision.
Insensitive to turbulence.
Narrabri 1963
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Gamma-ray astronomy
Primary γ ray initiates electromagneticcascade, which in turn emitsCherenkov radiation.
γ
e +
e −
VERITAS array in southern arizona
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
-0.2
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0
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rat
e (1
/min
)
Phase
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
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0
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Rat
e (1
/min
)
Phase
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
-0.2
-0.1
0
0.1
0.2
0.3
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0.6
0.7
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rat
e (1
/min
)
Phase
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rat
e (1
/min
)
Phase
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rat
e (1
/min
)
Phase
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
X-ray binary LSI+ 61◦303
Be star and compactcompanion.
TeV emission is maximum atapastron.
TeV attenuation due tointeractions with stellarphotons and circumstellarhydrogen.
Compact object is ablack hole.Nunez, P.D., LeBohec, S., &
Vincent, S. 2011, ApJ, 731,
105-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.2 0.4 0.6 0.8 1 1.2 1.4
Rat
e (1
/min
)
Phase
i=64°, nH=5x1019m-3
i=22°, nH=1x1020m-3
i=22°, nH=2x1020m-3
2006-2007 Veritas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Air Cherenkov telescope arrays
The Cherenkov Telescope Array (CTA)
ACTs are ideal SII receivers.
Easily adapted for optical SII (λ ∼ 400 nm)
Future arrays will provide thousands of simultanous baselines.
SII working group in CTA.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Array design and (u, v) coverage
λ = 400nm∆θmin ∼ λ
∆xmax∼ 0.01mas
∆θmax ∼ λ∆xmin
∼ 1mas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Array design and (u, v) coverage
1.4
km
λ = 400nm∆θmin ∼ λ
∆xmax∼ 0.01mas
∆θmax ∼ λ∆xmin
∼ 1mas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Array design and (u, v) coverage
35m
λ = 400nm∆θmin ∼ λ
∆xmax∼ 0.01mas
∆θmax ∼ λ∆xmin
∼ 1mas
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Data simulation
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x [m]
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R [m]
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R [m]
0.1mas, mv = 3, 10 hrs (Nunez et al. 2012)
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
The imaging problem
γ ∼ Fourier transform of the radiance distribution of the star.
γ(x) ∼∫O(θ)e ikxθdθ
In SII we have access to the squared modulus of the complexdegree of coherence |γ|2
When |γ|2 is measured, the phase of the Fourier transform islost.
How can we recover images?
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Some alternatives?
Model fitting.Maximum likelihood approach. (Thiebaut 2006).
Phase recovery
Three point correlations. 〈Ii Ij Ik〉 (Ribak 2006)
Analytic approach. (Holmes 2004).
The Fourier transform of an object of finite size (e.g. a star)is an analytic function.Relate magnitude to phase through Cauchy-Riemannequations.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Some alternatives?
Model fitting.Maximum likelihood approach. (Thiebaut 2006).
Phase recovery
Three point correlations. 〈Ii Ij Ik〉 (Ribak 2006)
Analytic approach. (Holmes 2004).
The Fourier transform of an object of finite size (e.g. a star)is an analytic function.Relate magnitude to phase through Cauchy-Riemannequations.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Some alternatives?
Model fitting.Maximum likelihood approach. (Thiebaut 2006).
Phase recovery
Three point correlations. 〈Ii Ij Ik〉 (Ribak 2006)
Analytic approach. (Holmes 2004).
The Fourier transform of an object of finite size (e.g. a star)is an analytic function.Relate magnitude to phase through Cauchy-Riemannequations.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Some alternatives?
Model fitting.Maximum likelihood approach. (Thiebaut 2006).
Phase recovery
Three point correlations. 〈Ii Ij Ik〉 (Ribak 2006)
Analytic approach. (Holmes 2004).
The Fourier transform of an object of finite size (e.g. a star)is an analytic function.Relate magnitude to phase through Cauchy-Riemannequations.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Cauchy-Riemann phase recovery in 1-D (Holmes 2004)
γ(z) =∑j
I (j∆θ)z j ; z ≡ exp(i mk∆x∆θ) ≡ e iφ
z ≡ ξ + iψ ; γ ≡ Re iΦ
∂Φ
∂ψ=
∂ lnR
∂ξ≡ ∂s
∂ξ
∂Φ
∂ξ= −∂ lnR
∂ψ≡ − ∂s
∂ψ
∆s|| =∂s
∂ξ∆ξ +
∂s
∂ψ∆ψ =
∂Φ
∂ψ∆ξ − ∂Φ
∂ξ∆ψ ≡ ∆Φ⊥
Log-Magnitude differences ⇔ Phase differences.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Cauchy-Riemann phase recovery in 1-D (Holmes 2004)
γ(z) =∑j
I (j∆θ)z j ; z ≡ exp(i mk∆x∆θ) ≡ e iφ
z ≡ ξ + iψ ; γ ≡ Re iΦ
∂Φ
∂ψ=
∂ lnR
∂ξ≡ ∂s
∂ξ
∂Φ
∂ξ= −∂ lnR
∂ψ≡ − ∂s
∂ψ
∆s|| =∂s
∂ξ∆ξ +
∂s
∂ψ∆ψ =
∂Φ
∂ψ∆ξ − ∂Φ
∂ξ∆ψ ≡ ∆Φ⊥
Log-Magnitude differences ⇔ Phase differences.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Cauchy-Riemann phase recovery in 1-D (Holmes 2004)
γ(z) =∑j
I (j∆θ)z j ; z ≡ exp(i mk∆x∆θ) ≡ e iφ
z ≡ ξ + iψ ; γ ≡ Re iΦ
∂Φ
∂ψ=
∂ lnR
∂ξ≡ ∂s
∂ξ
∂Φ
∂ξ= −∂ lnR
∂ψ≡ − ∂s
∂ψ
∆s|| =∂s
∂ξ∆ξ +
∂s
∂ψ∆ψ =
∂Φ
∂ψ∆ξ − ∂Φ
∂ξ∆ψ ≡ ∆Φ⊥
Log-Magnitude differences ⇔ Phase differences.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
1D example
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∝ In
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θ(k ∆x)-1
Reconstructed ImageReal Image
Image
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
1D examples
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∝ In
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Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Simulation/Analysis overview
Pristine
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R [m]
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|γ|
R [m]
Data Hermite fit
Cauchy-RiemannGerchberg-SaxtonMiRA
∂Φ∂ψ = ∂ lnR
∂ξ∂Φ∂ξ = −∂ lnR
∂ψ
arg min{χ2}
Thiebaut 2006
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Simulation/Analysis overview
Pristine
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R [m]
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|γ|
R [m]
Data Hermite fit
Cauchy-RiemannGerchberg-SaxtonMiRA
∂Φ∂ψ = ∂ lnR
∂ξ∂Φ∂ξ = −∂ lnR
∂ψ
arg min{χ2}
Thiebaut 2006
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
0
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ius
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Real vs. Reconstructed radii
ErrorReal=Rec
Rec radius
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idua
l (m
as)
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PSF
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Error close-up
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as]
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1 1.2 1.4 1.6 1.8 2R
ec a
/bReal a/b
Real vs. Reconstructed a/b
Nunez, et. al. 2012,
MNRAS, 419, 172
0
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Reconstructed Image
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as]
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Re
c s
ep
ara
tio
n (
ma
s)
Real separation (mas)
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Post-processing example: Star spots
mv = 3, 10 hrs of observation
∆T = 500◦K
Nunez et al. 2012, MNRAS
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Post-processing example: Star spots
Effect of finite telescope size?
electronic noise, stray light?
Rou et al. in prep.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator
Artificial star
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator
Artificial star
Miniature telescopes
.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator
Artificial star
Miniature telescopes
Analog or digital
David Kieda
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator
Artificial star
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Attempts to measure |γ|2 in the lab
Light source .
Pinhole
.
BS PMT1
PMT2
..
Correlator
Artificial star
θ ≈ 1×10−4 m3m = 10−4rad
θ = 1.22λ/d
⇒ d = 6.5mm
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Results with the pseudo-thermal light source
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Baseline (mm)
0.2 mm0.3 mm0.5 mm
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Results with the pseudo-thermal light source
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|γ|2
Baseline (mm)
0.2 mm0.3 mm0.5 mm
.
d = 1.22λ/θ
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
Artificial star
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
Speckle image
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
Speckle image AutocorrelationAutocorrelation
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
Speckle image AutocorrelationAutocorrelation Reconstruction
Analysis
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Imaging from autocorrelation data
Laser Beam expander .
Mask
.
.
Rough surface
.
CCD
.
.
.
CPU
In collaboration with Ryan Price and Erik Johnson. .
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
StarBase
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
StarBase camera
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Starting observations of Spica
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Starting observations of Spica
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ital c
ount
s
Time (ns)
Channel 1Channel 2
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0 50 100 150 200 250 300 350 400
Cor
rela
tion
Time lag (ns)
Analyzed 1 s of data.
Need several hours of data to detect a correlation.
Expect to see modulation of correlation as a function of time.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Conclusions
Fast electronics and IACT arrays will allow to measure |γ|2
SII will provide high angular resolution images of stars.
Advantages when compared to amplitude interferometry.
Phase can be recovered via several techniques.
Images can be used for stellar astrophysics.
Experimental efforts allow to understand the advantages anddifficulties of modern SII.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Thanks to:
Stephan LeBohec, Dave Kieda and Janvida Rou.
Lina Peralta.
Physics dept: Jackie Hadley, Heidi Frank.
Michelle Hui, Gary Finnegan, Nick Borys ...
...
THE END
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Thanks to:
Stephan LeBohec, Dave Kieda and Janvida Rou.
Lina Peralta.
Physics dept: Jackie Hadley, Heidi Frank.
Michelle Hui, Gary Finnegan, Nick Borys ...
...
THE END
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
JUST IN CASE
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Two component Fourier model (Hanbury Brown 1974)Detected currents at A and B
iA = KA(E1 sin(ω1t + φ1) + E2 sin(ω2t + φ2))2
iB = KB(E1 sin(ω1(t + d1/c) + φ1) + E2 sin(ω2(t + d2/c) + φ2))2
Band pass of electronics ∆f ≈ 100MHz
iA = KAE1E2 cos((ω1 − ω2)t + (φ1 − φ2))
iB = KBE1E2 cos((ω1 − ω2)t + (φ1 − φ2) + ω1d1/c − ω2d2/c)
iA and iB are correlated.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Two component Fourier model (Hanbury Brown 1974)Detected currents at A and B
iA = KA(E1 sin(ω1t + φ1) + E2 sin(ω2t + φ2))2
iB = KB(E1 sin(ω1(t + d1/c) + φ1) + E2 sin(ω2(t + d2/c) + φ2))2
Band pass of electronics ∆f ≈ 100MHz
iA = KAE1E2 cos((ω1 − ω2)t + (φ1 − φ2))
iB = KBE1E2 cos((ω1 − ω2)t + (φ1 − φ2) + ω1d1/c − ω2d2/c)
iA and iB are correlated.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Intensity fluctuations and correlations⟨∆n2
⟩=〈n〉 + 〈n〉2 τc
T
Shot noise Wave noise
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20 25 30 35 40
P(N
)
(N) Counts per electronic time resolution
PoissonSuper Poisson
Fit
Wave noise is small but correlatedbetween neighboring detectors.
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Cauchy-Riemann phase recovery (1D)
γ(m∆x) =∑j
I (j∆θ)exp(i mk∆x j∆θ)
γ [z(m∆x)] =∑j
I (j∆θ)z j ; z ≡ exp(i mk∆x∆θ) ≡ e iφ
z ≡ ξ + iψ ; γ ≡ Re iΦ (1)
∂Φ
∂ψ=
∂ lnR
∂ξ≡ ∂s
∂ξ
∂Φ
∂ξ= −∂ lnR
∂ψ≡ − ∂s
∂ψ
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Cauchy-Riemann phase recovery
γ(m∆x) =∑j
I (j∆θ)exp(i mk∆x j∆θ) (2)
γ [z(m∆x)] =∑j
I (j∆θ)z j ; z ≡ exp(i mk∆x∆θ) ≡ e iφ (3)
z ≡ ξ + iψ ; γ ≡ Re iΦ (4)
∂Φ
∂ψ=
∂ lnR
∂ξ≡ ∂s
∂ξ(5)
∂Φ
∂ξ= −∂ lnR
∂ψ≡ − ∂s
∂ψ(6)
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
∆sξ =∂s
∂ξ∆ξ =
∂Φ
∂ψ∆ξ (7)
∆sψ =∂s
∂ψ∆ψ = −∂Φ
∂ξ∆ψ (8)
(9)
Since |z | = 1 ⇒ we can only access to the log-magnitude s alongthe unit circle in the ξ − ψ plane.
∆s|| =∂s
∂ξ∆ξ +
∂s
∂ψ∆ψ =
∂Φ
∂ξ(−∆ψ) +
∂Φ
∂ψ∆ξ ≡ ∆Φ⊥
Introduction ACTs and γ-ray astronomy SII with IACTS The imaging problem Experimental efforts Closing
Φ is in general a solution to the Laplace equation in the ξ − ψplane.
Φ(φ) =∑j
ajcos(jφ) + bjsin(jφ) (10)
∆Φ⊥(φ) =∑j
{(1− 2sin(∆φ/2))j − 1
}(ajcos(jφ) + bjsin(jφ))
(11)
There is a relation between the log-magnitude differences andthe phase.
There will be ambiguities in the phase were the magnitude iszero.
Reconstructed phase will have an unknown gross piston andtilt.