Analyse de la cohérence en présence de lumière partiellement polarisée

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

3. Partially coherent light • What is coherence ? • Measure of the statistical dependence between the values of a light field at two points r1 and r2 and two times t1 and t2. • Partially polarized light at point r and time t is represented by a random vector field : • Statistical relations between and are represented by a joint probability density function (PDF) :

4. P 2e 2a Gaussian Partially Coherent light If light is Gaussian, its joint PDF is entirely defined by : • The polarization matrices at (r1,t1) and (r2,t2) (pointwise properties) : Polarization state • The mutual coherence matrix :

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

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

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

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

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

12. Intrinsic degrees of coherence D is a diagonal matrix with real valued, positive coefficients : equivalent to the modulus of the scalar degree of coherence N1 and N2 are unitary matrices equivalent of the phase of the scalar degree of coherence. With this approach, we obtain 2 parameters which are called intrinsic degrees of coherence. They are different from |mw| given by Approach 1.

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

14. Physical interpretation J1 1 0 J2 Interferences • Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ? • One assumes that one can adjust the polarization with polarization modulators having Jones matricesJ1 and J2

15. Physical interpretation J1 1 0 J2 Interferences • Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ? • One assumes that one can adjust the polarization with polarization modulators having Jones matricesJ1 and J2

16. Physical interpretation • Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ? J1 1 0 J2 Interferences • One assumes that one can adjust the polarization with polarization modulators having Jones matricesJ1 and J2 • What is the maximal value of the interference fringe contrast that can be obtained by varying J1 and J2?

17. Physical interpretation The maximal value of the fringe contrast is equal to the larger intrinsic degree of coherence mS Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005) . • This value is obtained when the modulator Jones matrices are : T1 1 0 T2 Interferences

18. Physical interpretation The maximal value of the fringe contrast is equal to the larger intrinsic degree of coherence mS Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005) . • A contrast mI is obtained when the modulator Jones matrices are : T1 1 0 T2 Interferences

19. Conclusion • Wolf degree of coherence : contrast of interference fringes (after balancing the intensities). “actual” fringe contrast • Intrinsic degrees of coherence :mS is the maximal contrast of interference fringes that can be obtained after optimizing the polarization states the two fields. “potential” fringe contrast • 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. Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007) • Applications : • Processing of interferometric/polarimetric SAR images • Applications in optics …

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. h gr(t) t0 t0 ex t ey An interference experiment Birefringent optical fiber • Incident field : purely polarized, linear 45°. t0 : coherence time Coherence length : • Birefringent fiber with delay h.

22. h gr(t) t0 t0 ex t ey An interference experiment Birefringent optical fiber • Input Eout into an interferometer with relative delay t :

23. An interference experiment Comparison of Wolf and intrinsic degrees of coherence We restrict ourselves to the case h>>t0. h (a) t = 0 • Wolf degree of coherence : • Intrinsic degrees :

24. An interference experiment h (b) t >> t0<<h t • Wolf degree of coherence : • Intrinsic degrees of coherence :

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

26. An interference experiment How to obtain fringes with contrast 1 ? (c) t = h t • No transformation • Rotation of 90°

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

28. An interference experiment (c) t = h l/2 ex 1 0 ex

29. An interference experiment How to obtain fringes with contrast 0 ? The complementary transformation consists in applying T1 and T2, and then polarize along ey. This gives fringe contrast equal to mI, that is, 0. (c) t = h t • Rotation of 90° ey • No transformation • Rotation of 90°

30. Conclusion • Wolf degree of coherence : contrast of interference fringes (after balancing the intensities). “actual” fringe contrast • Intrinsic degrees of coherence :mS is the maximal contrast of interference fringes that can be obtained after optimizing the polarization states the two fields. “potential” fringe contrast • 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. Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007) • Applications : • Processing of interferometric/polarimetric SAR images • Applications in optics …

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 : • (mS,mI)=(0,0) Most symmetric • mI=0 • mS=mI Less symmetric • mS and mI 0 Ph. Réfrégier, J. Math. Phys., vol. 48, 3303 (2007)

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

33. Measurement • The largest intrinsic degree of coherence mS can be measured by testing all possible polarization modulators J1 and J2 -> Takes a long time ! It is possible to estimate mS and the Ti with a finite number of measurements • The mutual coherence matrix W can be estimated from 4 interferometric measurements.

34. Measurement of W • Mesurement of Px: (polarizer parallel to direction x)

35. Measurement of W • Mesurement of Px: (polarizer parallel to direction x) 45° l/2 l/2

36. Measurement of W • Mesurement of Px: (polarizer parallel to direction x) 45° l/2

37. Measurement of W • Mesurement of Px: (polarizer parallel to direction x) l/2 45°

38. Measurement • The largest intrinsic degree of coherence mS can be measured by inspection of all possible polarization modulators U1 and U2 -> Takes a long time ! It is possible to estimate mS and the Ti with a finite number of measurements • The mutual coherence matrix W can be estimated from 4 interferometric measurements. • The polarization matrices G1, G2 are measured by classical Stokes polarimetry. • Ti , and can be computed from the SVD of the normalized mutual coherence matrix :

39. Physical interpretation The maximal value of |mw| is equal to the largest intrinsic degree of coherence mS Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005) . • This value is obtained when the modulator Jones matrices are : T1 1 0 T2 Interferences

40. Physical interpretation And thus if For any J1 and J2 : Totally incoherent light J1 1 0 J2 Interferences

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