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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). Outline. 1. Scalar degree of coherence . - PowerPoint PPT Presentation
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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
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 :
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.
13
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
• One assumes that one can adjust the polarization with polarization modulators having Jones matrices J1 and J2
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 ?
Physical interpretation
• What is the maximal value of the interference fringe contrast that can be obtained by varying J1 and J2?
• One assumes that one can adjust the polarization with polarization modulators having Jones matrices 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
Physical interpretation
2/1 iii ΓNT
18
),(r 11 tE
),(r 22 tE
Interferences
1
0
Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005) .
T1
T2
• A contrast I is obtained when the modulator Jones matrices are :
I
Contrast
Physical interpretation
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
1. Scalar degree of coherence
2. Two ways of defining the degree of coherence for partially polarized light
3. An interference experiment
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
An interference experiment
)(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 :
An interference experiment
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 :
An interference experiment
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 !
An interference experiment
26
(c)
• No transformation
• Rotation of 90°
How to obtain fringes with contrast 1 ?
An interference experiment
trE YorXout ,,
trE YorXout ,,
2/1 iii ΓNT
27
(c)
• Rotation of 90°
• 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
trE YorXout ,,
trE YorXout ,,
2/1 iii ΓNT
28
)(r, tE
)(r, tE
0Contrast
An interference experiment
trin ,E
)(r, tE
)(r, tE ex
ex
trin ,E
1 S
Contrast
(c)
1
0
29
(c)
• Rotation of 90°
• Rotation of 90°
• No transformation
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
An interference experiment
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
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
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
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
Px: (polarizer parallel to direction x)
• Mesurement of )],(),([ 11*
22 tEtEe YYi
YYYYyy rr
YY
Contrast
/2
/2
45°
YYYX
XYXX
36
XY),(r 11 tE
),(r 22 tE
Measurement of
Px: (polarizer parallel to direction x)
• Mesurement of )],(),([ 11*
22 tEtEe YXi
XYXYxy rr
XY
Contrast
/2
45°
YYYX
XYXX
37
YX),(r 11 tE
),(r 22 tE
Measurement of
Px: (polarizer parallel to direction x)
• Mesurement of )],(),([ 11*
22 tEtEe XYi
YXYXyx rr
YX
Contrast
/2
45°
YYYX
XYXX
38
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
Ph. Réfrégier and F. Goudail, Optics Express, vol. 15, 6051, (2005) .
T1
T2
• This value is obtained when the modulator Jones matrices are :
SW
Contrast
Physical interpretation
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
Physical interpretation
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 :