### Class 2 - Polarimeters

```Polarimeters
Jessica C. Ramella-Roman, PhD
II Escuela de Optica Biomedica, Puebla, 2011
Stokes vector formalism
•
•
•
•
Four measurable quantities (intensities)
Characterize the polarization state of light
E0x, E0y, Cartesian electric field component
d=dx-dy phase difference
 E E*  E E*
I   x x
y y
   E E*  E E*
x x
y y
Q 

S
 
U   Ex E*y  E y E*x
  
V  i Ex E*y  E y E*x


  2
2

E

E
  0x
0 y 
  E 2  E 2 
0y
  0x

 2E0x E0 y cos  
 2E E sin 

  0x 0 y

II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Six intensity measurements
I   I H  IV 

  
Q   I H  IV 

S

U  I 45  I 45 

  
V   I R  I L 
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Horizontal, i.e. parallel to reference frame
I   I H  IV 

  
Q   I H  IV 

S

U  I 45  I 45 

  
V   I R  I L 
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Vertical, i.e. perpendicular to reference frame
I   I H  IV 

  
Q   I H  IV 

S

U  I 45  I 45 

  
V   I R  I L 
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Linear polarizer at +/-45o to reference frame
I   I H  IV 

  
Q   I H  IV 

S

U  I 45  I 45 

  
V   I R  I L 
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Circularly polarized (left and right)
Quarter-wave plate
I   I H  IV 

  
Q   I H  IV 

S

U  I 45  I 45 

  
V   I R  I L 
polarizer
Polarizer and quarterwave
plate axis are at 45o
to each other
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
P
qwp
R1
qwp
R2
II Escuela de Optica Biomedica, Puebla, 2011
P
16 measurements
HH -> H source H detector
HV -> H source V detector
HP -> H source P detector
HR -> H source R detector
PH -> P source H detector
PV -> P source V detector
PP -> P source P detector
PR -> P source R detector
VH -> V source H detector
VV -> V source V detector
VP -> V source P detector
VR -> V source R detector
RH -> R source H detector
RV -> R source V detector
RP -> R source P detector
RR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
16 measurements
HH -> H source H detector
HV -> H source V detector
HP -> H source P detector
HR -> H source R detector
PH -> P source H detector
PV -> P source V detector
PP -> P source P detector
PR -> P source R detector
VH -> V source H detector
VV -> V source V detector
VP -> V source P detector
VR -> V 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
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
P
qwp
R1
qwp
R2
II Escuela de Optica Biomedica, Puebla, 2011
P
16 measurements
HH -> H source H detector
HV -> H source V detector
HP -> H source P detector
HR -> H source R detector
PH -> P source H detector
PV -> P source V detector
PP -> P source P detector
PR -> P source R detector
VH -> V source H detector
VV -> V source V detector
VP -> V source P detector
VR -> V source R detector
RH -> R source H detector
RV -> R source V detector
RP -> R source P detector
RR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
P
qwp
R1
qwp
R2
II Escuela de Optica Biomedica, Puebla, 2011
P
16 measurements
HH -> H source H detector
HV -> H source V detector
HP -> H source P detector
HR -> H source R detector
PH -> P source H detector
PV -> P source V detector
PP -> P source P detector
PR -> P source R detector
VH -> V source H detector
VV -> V source V detector
VP -> V source P detector
VR -> V source R detector
RH -> R source H detector
RV -> R source V detector
RP -> R source P detector
RR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
P
qwp
R1
qwp
R2
II Escuela de Optica Biomedica, Puebla, 2011
P
Special issues in polarimetry
• Spectral stokes vector optimization
• Mueller matrix optimization
II Escuela de Optica Biomedica, Puebla, 2011
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)
II Escuela de Optica Biomedica, Puebla, 2011
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
II Escuela de Optica Biomedica, Puebla, 2011
Experimental Layout for Mueller M
P
WP
p
LCR1- Liquid Crystal Retarder q = 0o
LCR2- Liquid Crystal Retarder q = 45o
p polarizer
We observe the spectrum between
550 and 750 nm
Fiber – 200µm
LED – White LED or Xenon wls
II Escuela de Optica Biomedica, Puebla, 2011
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 J.C.
measurements
andApplied
theOptics,
Stokes
*B. Boulbry,
Ramella-Roman, T.A. Germer,
46, pp.vector
8533–8541, 2007.
II Escuela de Optica Biomedica, Puebla, 2011
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
II Escuela de Optica Biomedica, Puebla, 2011
WP achromatic ¼
wave plate
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
II Escuela de Optica Biomedica, Puebla, 2011
WP achromatic ¼
wave plate
Calibration cnt.
• The calibration polarizer and wave plate
ideally create the Stokes vectors

Sbefore  M wavep M polarizer q Sunpol
BEFORE
Safter  M polarizer q M wavepSunpol
AFTER
M Mueller matrices
S Stokes vectors
II Escuela de Optica Biomedica, Puebla, 2011
Calibration cnt.
• The Stokes vectors are related to the
measured values Ii through the data reduction
matrix W for which
S
S
before after
 W I
I
before after

W  S I 
1
• W is finally calculated using the SVD of I

II Escuela de Optica Biomedica, Puebla, 2011
Calibration cnt.
• Once W is know only 6 I measurements are
necessary to build the full Stokes vector
S W I 
• This is true at every wavelength.

II Escuela de Optica Biomedica, Puebla, 2011
Results - Incident [1 -1 0 0]
90o
II Escuela de Optica Biomedica, Puebla, 2011
Results - Incident [1 0 0 1]
45o
wp
II Escuela de Optica Biomedica, Puebla, 2011
Is chicken a perfect wave-plate?
[1 1 0 0]
Wavelength
P
Transmitted
degree of polarization
Angle
II Escuela de Optica Biomedica, Puebla, 2011
Chicken muscle ~ cylinder scattering +
Rayleigh scattering
DLP
DCP
Real
Simulated
OASIS 2011
II Escuela de Optica Biomedica, Puebla, 2011
More on chicken and polarization
on
II Escuela de Optica Biomedica, Puebla, 2011
The same layout & calibration can be
used to build a Mueller matrix
polarimeter
45o
wp
II Escuela de Optica Biomedica, Puebla, 2011
Mueller matrix of air
45o
wp
II Escuela de Optica Biomedica, Puebla, 2011
Mueller matrix of air
45o
wp
II Escuela de Optica Biomedica, Puebla, 2011
Conclusions
• Stokes vector polarimeter is fiber based and
usable between 550-750 nm
• Point measurements of small scatterers
• Miniaturizing the system
II Escuela de Optica Biomedica, Puebla, 2011
Optimization of Mueller Matrices
measurements
• The classic Mueller matrix polarimeter
• Previous work on optimizing a polarimeter
• Mueller matrix polarimetry with SVD
II Escuela de Optica Biomedica, Puebla, 2011
Dual rotating retarder polarimeter
R1 : 5 R2
R1
R2
R2
R1
II Escuela de Optica Biomedica, Puebla, 2011
Calculation of Mueller Matrix
Analyzing vector
Measured flux
Pq  A MS
MS
qq
q
TT
qq
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)

II Escuela de Optica Biomedica, Puebla, 2011
Calculation of Mueller Matrix
Pq  A MSq
T
q
  aq,0
Pq 
aq,1
aq,2
 m00
T 
Aq MS
m10q

aq,3  
m20

 m30
m01
m02
11
12
m21
m22
m31
m32
m WqmM
m03   sq,0 
 
m13   sq,1 
 

m23  sq,2 

m33   sq,3 
 
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)

II Escuela de Optica Biomedica, Puebla, 2011

Calculation of Mueller Matrix
• The qth measurement
Pqq  AA MSq  Wq M
TT
qq
Measurement matrix
aq s q
Flattened Mueller Matrix
II Escuela de Optica Biomedica, Puebla, 2011
For 16 measurements
• W is square with a unique inverse
– (if W non singular)
11
 WPP
MM
W

II Escuela de Optica Biomedica, Puebla, 2011
For more than 16 measurements
• W is not square so to calculate the Mueller
matrix


M W
T
1
T
1
M  (W WT) W P
T
M W W
M
1
 WP P
W
P
1
P
PÊ
*M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix
polarimeter”, Applied Optics, V. 41, No. 13 (2002)
II Escuela de Optica Biomedica, Puebla, 2011
( *)
Two main issues
• Which wave-plates are best for Mueller matrix
calculation
• number of measurements to calculate the
Mueller matrix
II Escuela de Optica Biomedica, Puebla, 2011
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)
II Escuela de Optica Biomedica, Puebla, 2011

Condition number

cond A  A
A
1
A  max norm A
A  max
1in

n
cond( A)  A A1
j1
A  maxÊnormÊA
aij
n
A  max  aij
1in
j1
1 / 1/cond(A)
cond( A)how
howÊ
closeÊ
toÊaÊsin
gularÊmatrix
close
is A isÊ
to aAÊ
singular
matrix
II Escuela de Optica Biomedica, Puebla, 2011
Best retardance 127
Minima at 127 and 233
II Escuela de Optica Biomedica, Puebla, 2011
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)
II Escuela de Optica Biomedica, Puebla, 2011
Cond(W)
• Over-determined system are better
16 measurements
30 measurements
II Escuela de Optica Biomedica, Puebla, 2011
Svd vs pseudo-inverse
1Air
2 Linear P
3 qwp
II Escuela de Optica Biomedica, Puebla, 2011
Modeled error
  
1
1
reconstructed
T
T
reconstructed
M

W
W
W
P
M ii
 W
W P
T
T
( *)


W 
1 T
reconstructed
reconstructed
1PÊ
T
M
V

U
Ê
]  svd(W
)
M ii
 V U ÊÊÊ
PÊÊÊÊ[U ,,V
V ,,U
 svd
error 
2

16
M
reconstructed
i
M

2
ideal
i
 16
2
reconstructed
ideal

error 
Mi
 Mi

i1

i1
II Escuela de Optica Biomedica, Puebla, 2011





2
Modeled error
SVD gives low level error for
retardances
Smith
pseudoinverse
SVD
SVD
II Escuela de Optica Biomedica, Puebla, 2011
Does the sample Mueller M bias results?
1Air
2 Linear P
3 qwp
II Escuela de Optica Biomedica, Puebla, 2011
Generating a sample Mueller matrix
• Generate 4 different Mueller matrices with
2,4,6,and 8 degrees of freedom (100 MM
total)
• Check for physical plausibility of Mueller
matrix (Handbook of Optics Vol II)
2MM T )2 4m
22
2
Tr(
moo  mo1  mo2oo mo3
; moo  mij



Tr MM2 
T
2
4m
2 oo
2
moo  mo1  mo2  mo3

3
  mo, j   m j,k

2j1  3
k 1 3

2

2
3

2
2
 mo2
 mo3  
moo  mo1


m
o,j

m


2
2
2 
mo1
 mo2

m
mo , j o3 
mo, j
j ,k
j1
k 1
II Escuela de Optica Biomedica, Puebla, 2011
2
2
2
mo1
 mo2
 mo3
Error with reconstructed Mueller
matrices
SVD gives low level error for
retardances
SVD
pseudoinverse
Smith
SVD
II Escuela de Optica Biomedica, Puebla, 2011
Conclusions
• Using SVD a broader range of retarders may
be used in DRR polarimetry
• Several numerical programs (such as Matlab)
use SVD in their pseudo-inverse algorithms
• In most cases the error due to use of SVD is
minimal compared to instrumental errors
II Escuela de Optica Biomedica, Puebla, 2011
Tomorrow
• Monte Carlo modeling basics
• Monte Carlo with Meridian planes
II Escuela de Optica Biomedica, Puebla, 2011

Condition number

cond A  A
A
1
A  max norm A
A  max
1in

n
cond( A)  A A1
j1
A  maxÊnormÊA
aij
n
A  max  aij
1in
j1
1 / 1/cond(A)
cond( A)how
howÊ
closeÊ
toÊaÊsin
gularÊmatrix
close
is A isÊ
to aAÊ
singular
matrix
II Escuela de Optica Biomedica, Puebla, 2011
SVD
Ax
Axbb
Solution this is obtained minimizing the error
Ax
Axb b
Fundamental property of SVD
T
T
AAx
 UV
UV
II Escuela de Optica Biomedica, Puebla, 2011
SVD
U and V are respectively mxm and nxn unitary matrices
 and is a diagonal whose elements are the singular value of the matrix A.

 is a mxn matrix, the singular values of A are written along the main diagonal
often in descending order.

The columns of U are eigenvectors (left singular vectors) of AAT and the columns
of V are eigenvectors of ATA (right singular vectors).
II Escuela de Optica Biomedica, Puebla, 2011
```