Analyse de la cohérence en présence de lumière partiellement polarisée

Analyse de la cohérence en présence

de lumière partiellement polarisée

François GoudailLaboratoire Charles Fabry de l’Institut d’Optique,

Palaiseau (France)

Philippe RéfrégierInstitut 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.

• Statistical relations between and are represented by a joint probability density function (PDF) :

11, trE 22 , trE

21,,, ,2211

EErr ttP

• Partially polarized light at point r and time t is represented by a random vector field :

),(),(

),(tEtE

tY

X

rr

rE

4

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) :

),(),(),(),(),(),(),(),(

**

**

tEtEtEtEtEtEtEtE

YYXY

YXXX

rrrrrrrr

P

Polarization state

),(),(),( ttt rErEr

),(),(),,,( 11222121 trtrtt EErr

)],(),([)],(),([)],(),([)],(),([

11*

2211*

22

11*

2211*

22

tEtEtEtEtEtEtEtE

YYXY

YXXX

rrrrrrrr

• The mutual coherence matrix :

5

Gaussian scalar coherent light

• The correlation coefficient depends on the intensities

Normalization to make it independent :

2222

11

11*

222121

,,

,,,,,

tEtE

tEtEtt

rr

rrrr Incoherent light = 0

1,,,0 2121 ttrr

2222

11

11*

222121

,,

,,,,,

tEtE

tEtEtt

rr

rrrr

• 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 : utEt

,, rrE

Totally coherent light = 1

Mutual coherence matrix -> correlation coefficient

Polarization matrix -> intensity 2, tE r

6

Mutual information• A standard measure of statistical dependence (information theory) is mutual information

• In the Gaussian scalar case,

21

2,1,

21,,,21,,,

2211

2211

2211

,log, EE

EEEE

EErr

rrrr dd

PPP

Ptt

ttttI

21log I

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), MI = 0

7

Measure of the degree of coherence

• For scalar light, the complex degree of coherence is a measurable quantity

),(r 11 tE

),(r 22 tE

Interferences

Contrast

arg

Its modulus is the contrast of the fringe pattern

Its phase is the phase of the fringe pattern

8

Outline




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

Wolf degree of coherence

• Approach 1 : Expression of the contrast of interference fringes between two partially polarized lights:

),(r 11 tE

),(r 22 tE

Interferences

W

Contrast

Warg

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).

),(tr),(tr

),,,(tr

2211

2121

tttt

W rrrr

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 ?

2222

11

11*

222121

,,

,,,,,

tEtE

tEtEtt

rr

rrrr

Singular value decomposition (SVD)

12

D is a diagonal matrix with real valued, positive coefficients :

N1 and N2 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.

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22 1log1log ISI

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:

21log I

It depends only on the intrinsic degrees of coherence.

can be written as

They are thus appropriate measures of statistical relations

21

2,1,

21,,,21,,,

2211

2211

2211

,log, EE

EEEE

EErr

rrrr dd

PPP

Ptt

ttttI

14

• One assumes that one can adjust the polarization with polarization modulators having Jones matrices J1 and J2

),(r 11 tE

),(r 22 tE

Interferences

1

0

J1

J2

Physical interpretation• Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ?

15

),(r 11 tE

),(r 22 tE

Interferences

1

0

J1

J2

• Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ?

Physical interpretation


16

1U

2U

),(r 11 tE

),(r 22 tE

Interferences

1

0

J1

J2

• Can the intrinsic degrees of coherence be related to interference measurements as the Wolf degree of coherence ?


• What is the maximal value of the interference fringe contrast that can be obtained by varying J1 and J2?


17

),(r 11 tE

),(r 22 tE

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) .

T1

T2

• This value is obtained when the modulator Jones matrices are :

S

Contrast


2/1 iii ΓNT

18

),(r 11 tE

),(r 22 tE

Interferences

1

0


T1

T2

• A contrast I is obtained when the modulator Jones matrices are :

I

Contrast


2/1 iii ΓNT

The maximal value of the fringe contrast is equal to the larger intrinsic degree of coherence S

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

Outline




21

An interference experiment

• Incident field : purely polarized, linear 45°.

Coherence length :

Birefringent optical fiber trE Xin ,

trE Yin ,

trE Xout ,

trE Yout ,

ex

ey

t

• Birefringent fiber with delay .

trin ,E

0 : coherence time

gr()

22


)(r, tE

)(r, tE

Contrast

• Input Eout into an interferometer with relative delay :

trin ,E

Birefringent optical fiber trE Xin ,

trE Yin ,

trE Xout ,

trE Yout ,

ex

ey

t

trin ,E gr()

23

(a)

• Wolf degree of coherence : 110 Wrg

• Intrinsic degrees :


trE YorXout ,,

trE YorXout ,,

We restrict ourselves to the case .

Comparison of Wolf and intrinsic degrees of coherence

24

• Wolf degree of coherence : 00 Wrg

(b) 0<<

• Intrinsic degrees of coherence :


trE YorXout ,,

trE YorXout ,,

25

• Wolf degree of coherence :

00 Wrg

• Intrinsic degrees of coherence :

(c)

are coherent but orthogonally polarized states : no interference !

trE Xout , trE Yout ,and

It is possible to obtain fringes with contrast 1 !


26

(c)

• No transformation

• Rotation of 90°

How to obtain fringes with contrast 1 ?


trE YorXout ,,

trE YorXout ,,

2/1 iii ΓNT

27

(c)




Rotating the polarization state makes the two states parallel. They can thus interfere. Since they are totally coherent, the fringe contrast is 1.


trE YorXout ,,

trE YorXout ,,

2/1 iii ΓNT

28

)(r, tE

)(r, tE

0Contrast


trin ,E

)(r, tE

)(r, tE ex

ex

trin ,E

1 S

Contrast

(c)

1

0

29

(c)




The complementary transformation consists in applying T1 and T2, and then polarize along ey. This gives fringe contrast equal to I, that is, 0.

How to obtain fringes with contrast 0 ?

ey


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).





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


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Conclusion• Wolf degree of coherence : contrast of interference fringes (after balancing the intensities).




• 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 S can be measured by testing all possible polarization modulators J1 and J2

-> Takes a long time !

• The mutual coherence matrix can be estimated from 4 interferometric measurements.

It is possible to estimate S and the Ti with a finite number of measurements

34

),(r 11 tE

),(r 22 tE

Measurement of

Px: (polarizer parallel to direction x)

• Mesurement of )],(),([ 11*

22 tEtEe XXi

XXXXxx rr

XX

Contrast

XX

YYYX

XYXX

35

YY),(r 11 tE

),(r 22 tE

Measurement of



22 tEtEe YYi

YYYYyy rr

YY

Contrast

/2

/2

45°

YYYX

XYXX

36

XY),(r 11 tE

),(r 22 tE

Measurement of



22 tEtEe YXi

XYXYxy rr

XY

Contrast

/2

45°

YYYX

XYXX

37

YX),(r 11 tE

),(r 22 tE

Measurement of



22 tEtEe XYi

YXYXyx rr

YX

Contrast

/2

45°

YYYX

XYXX

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Measurement

• The polarization matrices1, 2 are measured by classical Stokes polarimetry.• Ti , and can be computed from the SVD of the normalized mutual coherence matrix :

S I

211

212

M

• The largest intrinsic degree of coherence S can be measured by inspection of all possible polarization modulators U1 and U2

-> Takes a long time !

• The mutual coherence matrix can be estimated from 4 interferometric measurements.

It is possible to estimate S and the Ti with a finite number of measurements

39

),(r 11 tE

),(r 22 tE

Interferences

1

0

The maximal value of |w| is equal to the largest intrinsic degree of coherence S


T1

T2

• This value is obtained when the modulator Jones matrices are :

SW

Contrast


2/1 iii ΓNT

40

And thus if

For any J1 and J2 :

0S

),(r 11 tE

),(r 22 tE

Interferences

1

0

J1

J2

Totally incoherent light


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 :

Analyse de la cohérence en présence de lumière partiellement polarisée

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