1. Analyse de la cohérence en présence de lumière partiellement polarisée François Goudail Laboratoire Charles Fabry de l’Institut d’Optique, Palaiseau (France) Philippe Réfrégier Institut Fresnel, Marseille (France)

2. 2 Outline 1. Scalar degree of coherence 2. Two ways of defining the degree of coherence for partially polarized light 3. An interference experiment

3. 3 Partially coherent light What is coherence ? Measure of the statistical dependence between the values of a light field at two points r 1 and r 2 and two times t 1 and t 2. Statistical relations between and are represented by a joint probability density function (PDF) : Partially polarized light at point r and time t is represented by a random vector field :

4. 4 Gaussian Partially Coherent light If light is Gaussian, its joint PDF is entirely defined by : The polarization matrices at (r 1,t 1 ) and (r 2,t 2 ) (pointwise properties) :   P Polarization state The mutual coherence matrix :

5. 5 Gaussian scalar coherent light The correlation coefficient depends on the intensities Normalization to make it independent : Incoherent light   = 0 The “strength” of the correlation is given be the modulus of the complex degree of coherence: Complex degree of coherence If light is scalar ( ), things are simpler : Totally coherent light   = 1 Mutual coherence matrix -> correlation coefficient Polarization matrix -> intensity

6. 6 Mutual information A standard measure of statistical dependence (information theory) is mutual information In the Gaussian scalar case, It depends only on the modulus of the complex degree of coherence  is thus an appropriate measure of coherence If the two fields are independent (incoherent light), M I = 0

7. 7 Measure of the degree of coherence For scalar light, the complex degree of coherence is a measurable quantity Interferences Its modulus is the contrast of the fringe pattern Its phase is the phase of the fringe pattern

8. 8 Outline 1. Scalar degree of coherence 2. Two ways of defining the degree of coherence for partially polarized light 3. An interference experiment

9. 9 Question How to define a degree of coherence for partially polarized Gaussian light ? Two approaches : 1.Contrast of interference fringes 2. “Normalized” measure of statistical relation.

10. 10 Wolf degree of coherence Approach 1 : Expression of the contrast of interference fringes between two partially polarized lights: Interferences This value has been proposed as the expression of the complex degree of coherence of partially polarized light. E. Wolf, Phys. Lett. A, 312, 263-267 (2003).

11. 11 Approach 2 : How “normalize” the correlation matrix with respect to the “pointwise” properties of the field ? Intrinsic degrees of coherence Normalized mutual coherence matrix : Take the modulus of M ? Singular value decomposition (SVD)

12. 12 D is a diagonal matrix with real valued, positive coefficients : N 1 and N 2 are unitary matrices equivalent to the modulus of the scalar degree of coherence With this approach, we obtain 2 parameters which are called intrinsic degrees of coherence. They are different from  w  given by Approach 1. Intrinsic degrees of coherence equivalent of the phase of the scalar degree of coherence.

13. 13 Mutual information In the Gaussian case, it can be shown that the mutual information Ph. Réfrégier, Opt. Lett. 30, 3117-3119 (2005). Analogy with the scalar case: It depends only on the intrinsic degrees of coherence. can be written as They are thus appropriate measures of statistical relations

14. 14 One assumes that one can adjust the polarization with polarization modulators having Jones matrices J 1 and J 2 Interferences 1 0 J1J1 J2J2 Physical interpretation Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ?

15. 15 Interferences 1 0 J1J1 J2J2 Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ? Physical interpretation One assumes that one can adjust the polarization with polarization modulators having Jones matrices J 1 and J 2

16. 16 Interferences 1 0 J1J1 J2J2 Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ? Physical interpretation What is the maximal value of the interference fringe contrast that can be obtained by varying J 1 and J 2 ? One assumes that one can adjust the polarization with polarization modulators having Jones matrices J 1 and J 2

17. 17 Interferences 1 0 The maximal value of the fringe contrast is equal to the larger intrinsic degree of coherence  S Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005). T1T1 T2T2 This value is obtained when the modulator Jones matrices are : Physical interpretation

18. 18 Interferences 1 0 Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005). T1T1 T2T2 A contrast  I is obtained when the modulator Jones matrices are : Physical interpretation The maximal value of the fringe contrast is equal to the larger intrinsic degree of coherence  S

19. 19 Conclusion Classical measures of the “disorder” of light (mutual information) depends only on the intrinsic degrees of coherence. Intrinsic degrees of coherence can be seen as order parameters that describe the “symmetry” of a state of light. Applications : Processing of interferometric/polarimetric SAR images Applications in optics … Wolf degree of coherence : contrast of interference fringes (after balancing the intensities). Intrinsic degrees of coherence :  S is the maximal contrast of interference fringes that can be obtained after optimizing the polarization states the two fields. “actual” fringe contrast “potential” fringe contrast Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007)

20. 20 Outline 1. Scalar degree of coherence 2. Two ways of defining the degree of coherence for partially polarized light 3. An interference experiment

21. 21 An interference experiment Incident field : purely polarized, linear 45°. Coherence length : Birefringent optical fiber  exex eyey t Birefringent fiber with delay .  0 : coherence time   gr()gr()

22. 22 An interference experiment Input E out into an interferometer with relative delay  : Birefringent optical fiber  exex eyey t   gr()gr()

23. 23 (a)   Wolf degree of coherence : Intrinsic degrees : An interference experiment We restrict ourselves to the case  . Comparison of Wolf and intrinsic degrees of coherence

24. 24 Wolf degree of coherence : (b)   0 <<    Intrinsic degrees of coherence : An interference experiment

25. 25 Wolf degree of coherence : Intrinsic degrees of coherence : (c)    are coherent but orthogonally polarized states : no interference ! and It is possible to obtain fringes with contrast 1 ! An interference experiment

26. 26 (c)    No transformation Rotation of 90° How to obtain fringes with contrast 1 ? An interference experiment

27. 27 (c)    Rotation of 90° No transformation Rotating the polarization state makes the two states parallel. They can thus interfere. Since they are totally coherent, the fringe contrast is 1. An interference experiment

28. 28 An interference experiment exex  exex (c)   1 0

29. 29 (c)    Rotation of 90° No transformation The complementary transformation consists in applying T 1 and T 2, and then polarize along e y. This gives fringe contrast equal to  I, that is, 0. How to obtain fringes with contrast 0 ? eyey An interference experiment

30. 30 Conclusion Classical measures of the “disorder” of light (mutual information) depends only on the intrinsic degrees of coherence. Intrinsic degrees of coherence can be seen as order parameters that describe the “symmetry” of a state of light. Applications : Processing of interferometric/polarimetric SAR images Applications in optics … Wolf degree of coherence : contrast of interference fringes (after balancing the intensities). Intrinsic degrees of coherence :  S is the maximal contrast of interference fringes that can be obtained after optimizing the polarization states the two fields. “actual” fringe contrast “potential” fringe contrast Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007)

31. 31 Group invariance There is still another way to consider intrinsic degrees of coherence. They can be seen as order parameters (in the sense of statistical physics) that describe changes of symmetry of the problem. 4 symmetry classes : (  S,  I )=(0,0)  I =0  S =  I  S and  I 0 Most symmetric Less symmetric Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007)

32. 32 Conclusion Wolf degree of coherence : contrast of interference fringes (after balancing the intensities). Intrinsic degrees of coherence :  S is the maximal contrast of interference fringes that can be obtained after optimizing the polarization states the two fields. “actual” fringe contrast “potential” fringe contrast Classical measure of the “disorder” of light (mutual information) depends only on the intrinsic degrees of coherence.

33. 33 Measurement The largest intrinsic degree of coherence  S can be measured by testing all possible polarization modulators J 1 and J 2 -> Takes a long time ! The mutual coherence matrix  can be estimated from 4 interferometric measurements. It is possible to estimate  S and the T i with a finite number of measurements

34. 34 Measurement of  P x : (polarizer parallel to direction x) Mesurement of

35. 35 Measurement of  P x : (polarizer parallel to direction x) Mesurement of /2 45°

36. 36 Measurement of  P x : (polarizer parallel to direction x) Mesurement of /2 45°

37. 37 Measurement of  P x : (polarizer parallel to direction x) Mesurement of /2 45°

38. 38 Measurement The polarization matrices  1,  2 are measured by classical Stokes polarimetry. T i, and can be computed from the SVD of the normalized mutual coherence matrix : The largest intrinsic degree of coherence  S can be measured by inspection of all possible polarization modulators U 1 and U 2 -> Takes a long time ! The mutual coherence matrix  can be estimated from 4 interferometric measurements. It is possible to estimate  S and the T i with a finite number of measurements

39. 39 Interferences 1 0 The maximal value of |  w | is equal to the largest intrinsic degree of coherence  S Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005). T1T1 T2T2 This value is obtained when the modulator Jones matrices are : Physical interpretation

40. 40 And thus if For any J 1 and J 2 : Interferences 1 0 J1J1 J2J2 Totally incoherent light Physical interpretation

41. 41 with the random vectors Statistical interpretation and the transformation matrices The vectors are totally depolarized. One can write and since D is a diagonal matrix :