Koji Murakawa (ASTRON)B. Tubbs, R. Mather, R. Le Poole, J. Meisner,

E. Bakker (Leiden), F. Delplancke, K. Scale (ESO)

Conceptual Design Review for PRIMA

@Lorentz Center, Leiden on 29 Sep., 2004

PRIMA Astrometric ObservationsPolarization effectsTechnical Report

AS-TRE-AOS-15753-0011

Frosty Leo CW Leo

- OUTLINE -1. Introduction Why instrumental polarization analysis?2. Effects of phase error on astrometry Operation principle of the FSU3. Polarization properties of PRIMA optics Basic concepts of polarization model

Introduction

Why instrumental polarization analysis? changes phase and amplitude VLT telescope, StS, base line, etc (telescope pointing, separation, station…) the fringe sensor unit detects a wrong phase delay. provide an error in astrometry what kind of error? (</100?)

What we have to do?

Establish a strategy of analysis Study the operation principle of FSU Make a polarization model of VLTI opticsAnalysis Fringe detection by FSU polarization model analysis of VLTI optics

telescope, StS, base line optics time evolution (as a function of hour angle) difference between the ref. and the obj.

The Operation Principleof the Fringe Sensor Unit

achromaticλ/4

compensator light

from T1

light from T2

BC

p1 & s1

p2 & s2τp1 + ρp2 τs1 + ρs 2

|ϕp2 - ϕs2| = 90°

PBS

τp1 + ρp2

PBS

τs1 + ρs 2

ρp1 + τp2

ρs1 + τs 2

ρp1 + τp2 ρs1 + τs 2

Φ0

Φ2 = Φ

0 +

A

C

Φ1 = Φ

0 + /2

Φ

Ck

ΦΦ

3 = Φ

0 + 3/2

D

B

Alenia Co., VLT-TRE-ALS-15740-0004

The original ABCD Algorithm

Complex AmplitudeEA = -(P1-P2)EB = (S1+S2)EC = (P1+P2)ED = -(S1-S2)

Identical polarizationS1 = expi(kLopl,1)S2 = expi(kLopl,2)P1 = expi(kLopl,1) P2 = expi(kLopl,2 +/2)

k: wave number (k=2/λ)Lopl,i: optical path length at the station i

The original ABCD Algorithm

ABCD signalsIA = 2||2{1+sin(kLopd)}IB = 2||2{1+cos(kLopd)}IC = 2||2{1-sin(kLopd)}ID = 2||2{1-cos(kLopd)}

VisibilityV = 1/2(IA+IB+IC+ID)=4||2

Phase delay ϕ = kLopd

= arctan(IA-IC/IB-ID)

Lopd: optical path difference Lopd = Lopl,1 - Lopl,2

The phase delay can be measured with a simple way.

The original ABCD Algorithm

Complex AmplitudeEA = -(P1-P2)EB = (S1+S2)EC = (P1+P2)ED = -(S1-S2)

Different polarizationS1 = S1expi(kLopl,1)S2 = S1expi(kLopl,2)P1 = P1expi(kLopl,1) P2 = P1expi(kLopl,2+/2)

k: wave number (k=2/λ)Lopl,i: optical path length at the station i

The original ABCD AlgorithmABCD signalsIA = 2|P1|2{1+sin(kLopd)}IB = 2|S1|2{1+cos(kLopd)}IC = 2|P1|2{1-sin(kLopd)}ID = 2|S1|2{1-cos(kLopd)}

VisibilityV = 1/2(IA+IB+IC+ID) = 2||2(|P1|2+|S1|2)Phase delay ϕ = kLopd

= arctan(IA-IC/IA+IC * IB+ID/IB-ID)

Lopd: optical path difference Lopd = Lopl,1 - Lopl,2

The phase delay can be measured not affectedby different polarization status between S and P.

A Modified ABCD Algorithm

Complex AmplitudeEA = -(P1-P2)EB = (S1+S2)EC = (P1+P2)ED = -(S1-S2)

Different polarizationS1 = S1expi(kLopl,1)S2 = S2expi(kLopl,2)P1 = P1expi(kLopl,1+ϕS) P2 = P2expi(kLopl,2+ϕP+/2)

Different polarization between beam 1 and 2• phase ϕS = ϕS,2-ϕS,1, and ϕP = ϕP,2-ϕP,1 • amplitude S2≠S1, P2≠P1

A Problem on the ABCD Algorithm

ABCD signalsIA = ||2{P1

2+P22+2P1P2sin(kLopd+ϕP)}

IB = ||2{S12+S2

2+2S1S2cos(kLopd+ϕS)}IC = ||2{P1

2+P22-2P1P2sin(kLopd+ϕP)}

ID = ||2{S12+S2

2-2S1S2cos(kLopd+ϕS)}

The ABCD algorithm tells a wrong phase delay.

A Modified ABCD Algorithm

Get another sampling with a /2(=λ/4) stepIA0 = ||2{P1

2+P22+2P1P2sin(kLopd+ϕP)}

IA1 = ||2{P12+P2

2+2P1P2cos(kLopd+ϕP)}IC0 = ||2{P1

2+P22-2P1P2sin(kLopd+ϕP)}

IC1 = ||2{P12+P2

2-2P1P2cos(kLopd+ϕP)}

• only P-polarization is described above.• assume fixed P1 and P2

A Modified ABCD Algorithm& Polarization Effects

Phase delay ΦP = kLopd + ϕP = arctan(IA0-IC0/IA1+IC1) ΦS = kLopd + ϕS = arctan(IB0-ID0/IB1+ID1)The FSU may correct (detect) 1/2(ΦP+ΦS) = kLopd+1/2(ϕP+ϕS)

• Instrumental polarization between two beams cannot be principally corrected.• a phase delay of |ϕS-ϕP| still remains.

Impact on Astrometry- Polarization Effects on Object -

Visibility of the object V = <|ES,1+ES,2+EP,1+EP,2|2> = <|ES,1|2>+<|ES,2|2>+<|EP,1|2>+<|EP,2|2> +<ES,1ES,2

*>+<ES,1*ES,2>

+<ES,1EP,1*>+<ES,1

*EP,1> +<ES,1EP,2

*>+<ES,1*EP,2>

+<ES,2EP,1*>+<ES,2

*EP,1> +<ES,2EP,2

*>+<ES,2*EP,2>

+<EP,1EP,2*>+<EP,1

*EP,2>

ES,1 = S1expi(kLopl,1’)ES,2 = S2expi(kLopl,2’+ϕS’)EP,1 = P1expi(kLopl,1’+ϕSP’)EP,2 = P2expi(kLopl,2’+ϕSP’+ϕP’)

Impact on Astrometry- Polarization Effects on Object -

Cross correlation <ES,1ES,2

*>+<ES,1*ES,2> = 2S1S2<cos(klopd’-ϕS’)>

<ES,1EP,1*>+<ES,1

*EP,1> = 2S1P1<cos(ϕSP’)><ES,1EP,2

*>+<ES,1*EP,2> = 2S1P2<cos(klopd’-ϕSP’-ϕP’)>

<ES,2EP,1*>+<ES,2

*EP,1> = 2S2P1<cos(klopd’+ϕSP’-ϕS’)><ES,2EP,2

*>+<ES,2*EP,2> = 2S2P2<cos(ϕSP’+ϕP’-ϕS’)>

<EP,1EP,2*>+<EP,1

*EP,2> = 2P1P2<cos(klopd’-ϕP’)>

Impact on Astrometry- Polarization Effects on Object -

Visibility of the unpolarized object V = <|ES,1+ES,2+EP,1+EP,2|2> = <|ES,1|2>+<|ES,2|2>+<|EP,1|2>+<|EP,2|2> +2<S1S2cos(klopd’-ϕS’)>+2<P1P2cos(klopd’-ϕP’)>Because of <cos(ϕSP’)>=0….unpolarized lightAstrometry of the unpolarized object k(Lopd-Lopd’)+{(ϕS-ϕP)-(ϕS’-ϕP’)}= kLBLsin+{(ϕS-ϕP)-(ϕS’-ϕP’)} … : astrometry

Impact on Astrometry- Summary -

1. Operation principle of FSU Phase delay measurement not affected by polarization status of the reference. A modified ABCD algorithm to calibrate instrumental polarization

2. Impact on astrometry {(ϕS-ϕP)-(ϕS’-ϕP’)} gives error in astrometry Similar beam combiner to the FSU is

encouraged to science instrument

Polarization Model

Optics can work as a phase retarder or a polarizer So = J Si … S: Stokes parm, J: Jones matrix Sf = JNJN-1…J1 S*

Grouping Jtel(Az(h), El(h), r, , λ, St): telescope optics JStS(r, , λ): star separator optics JBL(λ, St): base line opticsModel Sf = JBL JStS Jtel S*

Future Activities

1. Telescope optics (Jtel) time evolution: |ϕS-ϕP|(h, Dec, r, )2. Star separator optics (JStS) |ϕS-ϕP|(r)3. Base line optics (JBL) |ϕS-ϕP|(St)4. Color dependence ϕopd(λ), Ix(λ)@FSU, group delay