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

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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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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

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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

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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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Magnitude

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x [m]

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|γ|

R [m]

0.9

0.95

1

1.05

0 20 40 60 80 100

|γ|

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

tens

ity

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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Reconstructed ΦReal Φ

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Reconstructed ImageReal Image

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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Reconstructed Image

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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)

Real radius (mas)

PSF

0.185

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Error close-up

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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

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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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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

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Time (ns)

Channel 1Channel 2

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-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.