Polarimeters

1. Polarimeters Jessica C. Ramella-Roman, PhD

2. Stokes vector formalism • Four measurable quantities (intensities) • Characterize the polarization state of light • E0x, E0y, Cartesian electric field component • d=dx-dy phase difference

3. Simple Stokes vector polarimeter • Six intensity measurements

4. Simple Stokes vector polarimeter • Horizontal, i.e. parallel to reference frame polarizer

5. Simple Stokes vector polarimeter • Vertical, i.e. perpendicular to reference frame polarizer

6. Simple Stokes vector polarimeter • Linear polarizer at +/-45o to reference frame polarizer

7. Simple Stokes vector polarimeter • Circularly polarized (left and right) Quarter-wave plate polarizer Polarizer and quarterwave plate axis are at 45o to each other

8. A Mueller matrix polarimeter P qwp qwp P R1 R2

9. 16 measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

10. 16 measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector Handbook of optics Vol II

11. A Mueller matrix polarimeter P qwp qwp P R1 R2

12. 16 measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

13. A Mueller matrix polarimeter P qwp qwp P R1 R2

14. 16 measurements HH -> H source H detector HV -> H source V detector HP -> H source P detector HR -> H source R detector VH -> V source H detector VV -> V source V detector VP -> V source P detector VR -> V source R detector PH -> P source H detector PV -> P source V detector PP -> P source P detector PR -> P source R detector RH -> R source H detector RV -> R source V detector RP -> R source P detector RR -> R source R detector

15. A Mueller matrix polarimeter P qwp qwp P R1 R2

16. Special issues in polarimetry • Spectral stokes vector optimization • Mueller matrix optimization

17. Motivations • Stokes vector polarimeter can be used for • rough surface measurements • characterization of particle size (partial Stokes vectors, co cross polarization) • Multi-spectral Stokes vector polarimeters are costly, often we need to sacrifice spectral performance (single wavelengths)

18. Experimental Layout p LCR1- Liquid Crystal Retarder q= 0o LCR2- Liquid Crystal Retarder q= 45o p polarizer Fiber – 200µm LED – White LED or Xenon wls

19. Experimental Layout for Mueller M P WP p LCR1- Liquid Crystal Retarder q= 0o LCR2- Liquid Crystal Retarder q= 45o p polarizer Fiber – 200µm LED – White LED or Xenon wls We observe the spectrum between 550 and 750 nm

20. Calibration • Method was originally proposed by Boulbry et al.* for an imaging system and 3 wavelengths. • Calibration does not require ANY knowledge of LCR retardation or orientation • There is a linear transformation between a set of measurements and the Stokes vector *B. Boulbry, J.C. Ramella-Roman, T.A. Germer, Applied Optics, 46, pp. 8533–8541, 2007.

21. Polarizer after wave plate Theta is the orientation angle of the polarizer with respect to the reference plane, 0 to 180o Six spectra Ii , are acquired for each theta for different LCR retardation p WP achromatic ¼ wave plate

22. Polarizer before wave plate Theta is the orientation angle of the polarizer with respect to the reference plane, 0 to 180o Six spectra Ii ,are acquired for each theta for different LCR retardation p WP achromatic ¼ wave plate

23. Calibration cnt. • The calibration polarizer and wave plate ideally create the Stokes vectors M Mueller matrices S Stokes vectors

24. Calibration cnt. • The Stokes vectors are related to the measured values Ii through the data reduction matrix W for which • W is finally calculated using the SVD of I

25. Calibration cnt. • Once W is know only 6 I measurements are necessary to build the full Stokes vector • This is true at every wavelength.

26. Results - Incident [1 -1 0 0] 90o

27. Results - Incident [1 0 0 1] 45o wp

28. Is chicken a perfect wave-plate? [1 1 0 0] P Transmitted degree of polarization Wavelength Angle

29. Chicken muscle ~ cylinder scattering + Rayleigh scattering DLP DCP Real Simulated OASIS 2011

30. More on chicken and polarization on

31. The same layout & calibration can be used to build a Mueller matrix polarimeter 45o wp

32. Mueller matrix of air 45o wp

33. Mueller matrix of air 45o wp

34. Conclusions • Stokes vector polarimeter is fiber based and usable between 550-750 nm • Point measurements of small scatterers • Miniaturizing the system

35. Optimization of Mueller Matrices measurements • The classic Mueller matrix polarimeter • Previous work on optimizing a polarimeter • Mueller matrix polarimetry with SVD

36. Dual rotating retarder polarimeter R1 : 5 R2 R1 R2 R1 R2

37. Calculation of Mueller Matrix Analyzing vector Measured flux Source vector Sample Mueller matrix • D. B. Chenault, J.L. Pezzaniti, R.A. Chipman, “Mueller matrix algorithms,” in D. Goldstein and R. Chipman (eds.) , “Polarization analysis and measurement,” in Proc. Soc. Photo-Opt. Instrum. Eng. V. 1746, pp. 231-246 (1992)

38. Calculation of Mueller Matrix Pq measured flux for q source detector retarders combination Sq source vector (Stokes vectors of source polarizing elements) Aq detectors vector (Stokes vectors of detector analyzing elements)

39. Calculation of Mueller Matrix • The qth measurement Measurement matrix aqsq Flattened Mueller Matrix

40. For 16 measurements • W is square with a unique inverse • (if W non singular)

41. For more than 16 measurements • W is not square so to calculate the Mueller matrix • *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)

42. Two main issues • Which wave-plates are best for Mueller matrix calculation • number of measurements to calculate the Mueller matrix

43. Which retarder are best for Mueller matrix calculation* • Change retardation of R1 and R2, (R1,R2 have same retardation) • 200 measurements to calculate W • R1:R2, 1:5 ratio Calculate cond( W) • *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)

44. Condition number 1/cond(A) how close is A to a singular matrix

45. Best retardance127 • Minima at 127 and 233

46. n of measurements* • Angular increments of source and detector retarders are varied • Angular increments 0:60 • Fixed retardance 127o • 16 measurements • 30 measurements Calculate cond( W) • *M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)

47. Cond(W) • Over-determined system are better 16 measurements 30 measurements

48. Svdvs pseudo-inverse 1Air 2 Linear P 3 qwp

49. Modeled error

50. Modeled error SVD gives low level error for broader range of retardances Smith pseudoinverse SVD SVD