1. 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 Observations Polarization effects Technical Report AS-TRE-AOS-15753-0011 Frosty LeoCW Leo

2. - 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

3. 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?)

4. 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.

5. The Operation Principle of the Fringe Sensor Unit Alenia Co., VLT-TRE-ALS-15740-0004

6. The original ABCD Algorithm Complex Amplitude E A = -  (P 1 -P 2 ) E B =  (S 1 +S 2 ) E C =  (P 1 +P 2 ) E D = -  (S 1 -S 2 ) Identical polarization S 1 = expi(kL opl,1 ) S 2 = expi(kL opl,2 ) P 1 = expi(kL opl,1 ) P 2 = expi(kL opl,2 +  /2) k: wave number (k=2  / ) L opl,i : optical path length at the station i

7. The original ABCD Algorithm ABCD signals I A = 2|  | 2 {1+sin(kL opd )} I B = 2|  | 2 {1+cos(kL opd )} I C = 2|  | 2 {1-sin(kL opd )} I D = 2|  | 2 {1-cos(kL opd )} Visibility V = 1/2(I A +I B +I C +I D )=4|  | 2 Phase delay  = kL opd = arctan(I A -I C /I B -I D ) L opd : optical path difference L opd = L opl,1 - L opl,2 The phase delay can be measured with a simple way.

8. The original ABCD Algorithm Complex Amplitude E A = -  (P 1 -P 2 ) E B =  (S 1 +S 2 ) E C =  (P 1 +P 2 ) E D = -  (S 1 -S 2 ) Different polarization S 1 = S 1 expi(kL opl,1 ) S 2 = S 1 expi(kL opl,2 ) P 1 = P 1 expi(kL opl,1 ) P 2 = P 1 expi(kL opl,2 +  /2) k: wave number (k=2  / ) L opl,i : optical path length at the station i

9. The original ABCD Algorithm ABCD signals I A = 2|  P 1 | 2 {1+sin(kL opd )} I B = 2|  S 1 | 2 {1+cos(kL opd )} I C = 2|  P 1 | 2 {1-sin(kL opd )} I D = 2|  S 1 | 2 {1-cos(kL opd )} Visibility V = 1/2(I A +I B +I C +I D ) = 2|  | 2 (|P 1 | 2 +|S 1 | 2 ) Phase delay  = kL opd = arctan(I A -I C /I A +I C * I B +I D /I B -I D ) L opd : optical path difference L opd = L opl,1 - L opl,2 The phase delay can be measured not affected by different polarization status between S and P.

10. A Modified ABCD Algorithm Complex Amplitude E A = -  (P 1 -P 2 ) E B =  (S 1 +S 2 ) E C =  (P 1 +P 2 ) E D = -  (S 1 -S 2 ) Different polarization S 1 = S 1 expi(kL opl,1 ) S 2 = S 2 expi(kL opl,2 ) P 1 = P 1 expi(kL opl,1 +  S ) P 2 = P 2 expi(kL opl,2 +  P +  /2) Different polarization between beam 1 and 2 phase  S =  S,2 -  S,1, and  P =  P,2 -  P,1 amplitude S 2 ≠S 1, P 2 ≠P 1

11. A Problem on the ABCD Algorithm ABCD signals I A = |  | 2 {P 1 2 +P 2 2 +2P 1 P 2 sin(kL opd +  P )} I B = |  | 2 {S 1 2 +S 2 2 +2S 1 S 2 cos(kL opd +  S )} I C = |  | 2 {P 1 2 +P 2 2 -2P 1 P 2 sin(kL opd +  P )} I D = |  | 2 {S 1 2 +S 2 2 -2S 1 S 2 cos(kL opd +  S )} The ABCD algorithm tells a wrong phase delay.

12. A Modified ABCD Algorithm Get another sampling with a  /2(= /4) step I A0 = |  | 2 {P 1 2 +P 2 2 +2P 1 P 2 sin(kL opd +  P )} I A1 = |  | 2 {P 1 2 +P 2 2 +2P 1 P 2 cos(kL opd +  P )} I C0 = |  | 2 {P 1 2 +P 2 2 -2P 1 P 2 sin(kL opd +  P )} I C1 = |  | 2 {P 1 2 +P 2 2 -2P 1 P 2 cos(kL opd +  P )} only P-polarization is described above. assume fixed P 1 and P 2

13. A Modified ABCD Algorithm & Polarization Effects Phase delay  P = kL opd +  P = arctan(I A0 -I C0 /I A1 +I C1 )  S = kL opd +  S = arctan(I B0 -I D0 /I B1 +I D1 ) The FSU may correct (detect) 1/2(  P +  S ) = kL opd +1/2(  P +  S ) Instrumental polarization between two beams cannot be principally corrected. a phase delay of |  S -  P | still remains.

14. Impact on Astrometry - Polarization Effects on Object - Visibility of the object V = = + + + + + E S,1 = S 1 expi(kL opl,1 ’) E S,2 = S 2 expi(kL opl,2 ’+  S ’) E P,1 = P 1 expi(kL opl,1 ’+  SP ’) E P,2 = P 2 expi(kL opl,2 ’+  SP ’+  P ’)

15. Impact on Astrometry - Polarization Effects on Object - Cross correlation + = 2S 1 S 2 + = 2S 1 P 1 + = 2S 1 P 2 + = 2S 2 P 1 + = 2S 2 P 2 + = 2P 1 P 2

16. Impact on Astrometry - Polarization Effects on Object - Visibility of the unpolarized object V = = + + + +2 +2 Because of =0….unpolarized light Astrometry of the unpolarized object k(L opd -L opd ’)+{(  S -  P )-(  S ’-  P ’)} = kL BL sin  +{(  S -  P )-(  S ’-  P ’)} …  : astrometry

17. 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  {(  S -  P )-(  S ’-  P ’)} gives error in astrometry  Similar beam combiner to the FSU is encouraged to science instrument

18. Polarization Model Optics can work as a phase retarder or a polarizer S o = J S i … S: Stokes parm, J: Jones matrix S f = J N J N-1 …J 1 S * Grouping J tel (Az(h), El(h), r, ,, St): telescope optics J StS (r, , ): star separator optics J BL (, St): base line optics Model S f = J BL J StS J tel S *

19. Future Activities 1. Telescope optics (J tel ) time evolution: |  S -  P |(h, Dec, r,  ) 2. Star separator optics (J StS ) |  S -  P |(r) 3. Base line optics (J BL ) |  S -  P |(St) 4. Color dependence  opd ( ), I x ( )@FSU, group delay