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Conceptual Design Review for PRIMA
Frosty Leo
PRIMA Astrometric Observations
Polarization effects
Technical Report
AS-TRE-AOS-15753-0011
CW Leo
Koji Murakawa (ASTRON)
B. Tubbs, R. Mather, R. Le Poole, J. Meisner,
E. Bakker (Leiden), F. Delplancke, K. Scale (ESO)
@Lorentz Center, Leiden on 29 Sep., 2004
- OUTLINE 1. Introduction
Why instrumental polarization analysis?
2. Effects of phase error on astrometry
Operation principle of the FSU
3. 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? (<p/100?)
What we have to do?
Establish a strategy of analysis
 Study the operation principle of FSU
 Make a polarization model of VLTI optics
Analysis
 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 Principle
of the Fringe Sensor Unit
B
achromatic

light
from T2
p2 & s2
 p
s1 + s2
p2 - s2| = 90°
p1 + p2
s1 + s2
p1 + p2
A
BC
p1 + p2
s1 + s2
PBS
PBS

p1 + p2
compensator
light
from T1
C
 p
p1 & s1
s1 + s2
D
Alenia Co., VLT-TRE-ALS-15740-0004
Ck

 p

The original ABCD Algorithm
Complex Amplitude
EA = -b(P1-P2)
EB = b(S1+S2)
EC = b(P1+P2)
ED = -b(S1-S2)
Identical polarization
S1 = expi(kLopl,1)
S2 = expi(kLopl,2)
P1 = expi(kLopl,1)
P2 = expi(kLopl,2 +p/2)
k: wave number (k=2p/)
Lopl,i: optical path length at the station i
The original ABCD Algorithm
ABCD signals
IA = 2|b|2{1+sin(kLopd)}
IB = 2|b|2{1+cos(kLopd)}
IC = 2|b|2{1-sin(kLopd)}
ID = 2|b|2{1-cos(kLopd)}
Visibility
V = 1/2(IA+IB+IC+ID)=4|b|2
Phase delay
f = 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 Amplitude
EA = -b(P1-P2)
EB = b(S1+S2)
EC = b(P1+P2)
ED = -b(S1-S2)
Different polarization
S1 = S1expi(kLopl,1)
S2 = S1expi(kLopl,2)
P1 = P1expi(kLopl,1)
P2 = P1expi(kLopl,2+p/2)
k: wave number (k=2p/)
Lopl,i: optical path length at the station i
The original ABCD Algorithm
ABCD signals
IA = 2|bP1|2{1+sin(kLopd)}
IB = 2|bS1|2{1+cos(kLopd)}
IC = 2|bP1|2{1-sin(kLopd)}
ID = 2|bS1|2{1-cos(kLopd)}
Visibility
V = 1/2(IA+IB+IC+ID)
= 2|b|2(|P1|2+|S1|2)
Phase delay
f = 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 affected
by different polarization status between S and P.
A Modified ABCD Algorithm
Complex Amplitude
EA = -b(P1-P2)
EB = b(S1+S2)
EC = b(P1+P2)
ED = -b(S1-S2)
Different polarization
S1 = S1expi(kLopl,1)
S2 = S2expi(kLopl,2)
P1 = P1expi(kLopl,1+fS)
P2 = P2expi(kLopl,2+fP+p/2)
Different polarization between beam 1 and 2
• phase fS = fS,2-fS,1, and fP = fP,2-fP,1
• amplitude S2≠S1, P2≠P1
A Problem on the ABCD Algorithm
ABCD signals
IA = |b|2{P12+P22+2P1P2sin(kLopd+fP)}
IB = |b|2{S12+S22+2S1S2cos(kLopd+fS)}
IC = |b|2{P12+P22-2P1P2sin(kLopd+fP)}
ID = |b|2{S12+S22-2S1S2cos(kLopd+fS)}
The ABCD algorithm tells a wrong phase delay.
A Modified ABCD Algorithm
Get another sampling with a p/2(=/4) step
IA0 = |b|2{P12+P22+2P1P2sin(kLopd+fP)}
IA1 = |b|2{P12+P22+2P1P2cos(kLopd+fP)}
IC0 = |b|2{P12+P22-2P1P2sin(kLopd+fP)}
IC1 = |b|2{P12+P22-2P1P2cos(kLopd+fP)}
• only P-polarization is described above.
• assume fixed P1 and P2
A Modified ABCD Algorithm
& Polarization Effects
Phase delay
P = kLopd + fP = arctan(IA0-IC0/IA1+IC1)
S = kLopd + fS = arctan(IB0-ID0/IB1+ID1)
The FSU may correct (detect)
1/2(P+S) = kLopd+1/2(fP+fS)
• Instrumental polarization between two beams
cannot be principally corrected.
• a phase delay of |fS-fP| 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,1 = S1expi(kLopl,1’)
+<ES,2EP,1*>+<ES,2*EP,1> ES,2 = S2expi(kLopl,2’+fS’)
+<ES,2EP,2*>+<ES,2*EP,2> EP,1 = P1expi(kLopl,1’+fSP’)
+<EP,1EP,2*>+<EP,1*EP,2> EP,2 = P2expi(kLopl,2’+fSP’+fP’)
Impact on Astrometry
- Polarization Effects on Object Cross correlation
<ES,1ES,2*>+<ES,1*ES,2> = 2S1S2<cos(klopd’-fS’)>
<ES,1EP,1*>+<ES,1*EP,1> = 2S1P1<cos(fSP’)>
<ES,1EP,2*>+<ES,1*EP,2> = 2S1P2<cos(klopd’-fSP’-fP’)>
<ES,2EP,1*>+<ES,2*EP,1> = 2S2P1<cos(klopd’+fSP’-fS’)>
<ES,2EP,2*>+<ES,2*EP,2> = 2S2P2<cos(fSP’+fP’-fS’)>
<EP,1EP,2*>+<EP,1*EP,2> = 2P1P2<cos(klopd’-fP’)>
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’-fS’)>+2<P1P2cos(klopd’-fP’)>
Because of <cos(fSP’)>=0….unpolarized light
Astrometry of the unpolarized object
k(Lopd-Lopd’)+{(fS-fP)-(fS’-fP’)}
= kLBLsinq+{(fS-fP)-(fS’-fP’)} … q: 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
 {(fS-fP)-(fS’-fP’)} 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, q, , St): telescope optics
JStS(r, q, ): star separator optics
JBL(, St): base line optics
Model
Sf = JBL JStS Jtel S*
Future Activities
1. Telescope optics (Jtel)
time evolution: |fS-fP|(h, Dec, r, q)
2. Star separator optics (JStS)
|fS-fP|(r)
3. Base line optics (JBL)
|fS-fP|(St)
4. Color dependence
fopd(), Ix()@FSU, group delay