II Escuela de Optica Biomedica, Puebla, 2011
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
€
S =
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
ExEx* +EyEy
*
ExEx* −EyEy
*
ExEy* +EyEx
*
i ExEy* −EyEx
*( )
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥=
E0x2 +E0y
2
E0x2 −E0y
2
2E0xE0ycosδ2E0xE0 ysinδ
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Six intensity measurements
€
S=
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
IH + IVIH −IVI 45 −I−45I R −IL
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Horizontal, i.e. parallel to reference frame
€
S=
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
IH + IVIH −IVI 45 −I−45I R −IL
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Vertical, i.e. perpendicular to reference frame
€
S=
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
IH + IVIH −IVI 45 −I−45I R −IL
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Linear polarizer at +/-45o to reference frame
€
S=
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
IH + IVIH −IVI 45 −I−45I R −IL
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
polarizer
II Escuela de Optica Biomedica, Puebla, 2011
Simple Stokes vector polarimeter
• Circularly polarized (left and right)Quarter-wave plate
polarizer
€
S=
IQUV
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥=
IH + IVIH −IVI 45 −I−45I R −IL
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
Polarizer and quarterwaveplate axis are at 45o to each other
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
R1 R2PqwpP qwp
II Escuela de Optica Biomedica, Puebla, 2011
16 measurementsHH -> H source H detectorHV -> H source V detectorHP -> H source P detectorHR -> H source R detector
VH -> V source H detectorVV -> V source V detectorVP -> V source P detectorVR -> V source R detector
PH -> P source H detectorPV -> P source V detectorPP -> P source P detectorPR -> P source R detector
RH -> R source H detectorRV -> R source V detectorRP -> R source P detectorRR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
16 measurementsHH -> H source H detectorHV -> H source V detectorHP -> H source P detectorHR -> H source R detector
VH -> V source H detectorVV -> V source V detectorVP -> V source P detectorVR -> V source R detector
PH -> P source H detectorPV -> P source V detectorPP -> P source P detectorPR -> P source R detector
RH -> R source H detectorRV -> R source V detectorRP -> R source P detectorRR -> R source R detector
Handbook of opticsVol II
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
R1 R2PqwpP qwp
II Escuela de Optica Biomedica, Puebla, 2011
16 measurementsHH -> H source H detectorHV -> H source V detectorHP -> H source P detectorHR -> H source R detector
VH -> V source H detectorVV -> V source V detectorVP -> V source P detectorVR -> V source R detector
PH -> P source H detectorPV -> P source V detectorPP -> P source P detectorPR -> P source R detector
RH -> R source H detectorRV -> R source V detectorRP -> R source P detectorRR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
R1 R2PqwpP qwp
II Escuela de Optica Biomedica, Puebla, 2011
16 measurementsHH -> H source H detectorHV -> H source V detectorHP -> H source P detectorHR -> H source R detector
VH -> V source H detectorVV -> V source V detectorVP -> V source P detectorVR -> V source R detector
PH -> P source H detectorPV -> P source V detectorPP -> P source P detectorPR -> P source R detector
RH -> R source H detectorRV -> R source V detectorRP -> R source P detectorRR -> R source R detector
II Escuela de Optica Biomedica, Puebla, 2011
A Mueller matrix polarimeter
R1 R2PqwpP qwp
II Escuela de Optica Biomedica, Puebla, 2011
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
LCR1- Liquid Crystal Retarder q = 0o
LCR2- Liquid Crystal Retarder q = 45o
p polarizer
Fiber – 200µm
LED – White LED or Xenon wls
p
II Escuela de Optica Biomedica, Puebla, 2011
Experimental Layout for Mueller M
P
We observe the spectrum between550 and 750 nm
LCR1- Liquid Crystal Retarder q = 0o
LCR2- Liquid Crystal Retarder q = 45o
p polarizer
Fiber – 200µm
LED – White LED or Xenon wls
p
WP
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 measurements and the Stokes vector
*B. Boulbry, J.C. Ramella-Roman, T.A. Germer, Applied Optics, 46, pp. 8533–8541, 2007.
II Escuela de Optica Biomedica, Puebla, 2011
Theta is the orientation angle of the polarizerwith respect to the reference plane, 0 to 180o
Six spectra Ii , are acquired for each theta for differentLCR retardation
pWP achromatic ¼wave plate
Polarizer after wave plate
II Escuela de Optica Biomedica, Puebla, 2011
Polarizer before wave plateTheta is the orientation angle of the polarizerwith respect to the reference plane, 0 to 180o
Six spectra Ii , are acquired for each theta for differentLCR retardation
pWP achromatic ¼wave plate
II Escuela de Optica Biomedica, Puebla, 2011
Calibration cnt.
• The calibration polarizer and wave plate ideally create the Stokes vectors
€
Sbefore=MwavepM polarizerq( )Sunpol BEFORE
Safter=M polarizerq( )MwavepSunpol 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
• W is finally calculated using the SVD of I
€
SbeforeSafter[ ] =W IbeforeIafter[ ]
W = S[ ] I[ ]−1
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
• This is true at every wavelength.
€
S[ ] =W I[ ]
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?
Angle
Wav
elen
gth Transmitted
degree of polarization
[1 1 0 0]
P
II Escuela de Optica Biomedica, Puebla, 2011OASIS 2011
Chicken muscle ~ cylinder scattering + Rayleigh scattering
DLP
Real
Simulated
DCP
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 air45o
wp
II Escuela de Optica Biomedica, Puebla, 2011
Mueller matrix of air45o
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 R2
R1 : 5 R2
R1R2
II Escuela de Optica Biomedica, Puebla, 2011
Pq =AqT MSq
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)
Measured flux
Analyzing vector
Source vector
Sample Mueller matrix
€
Pq =AqTMSq
Calculation of Mueller Matrix
II Escuela de Optica Biomedica, Puebla, 2011
Pq measured flux for q source detector retarders combinationSq source vector (Stokes vectors of source polarizing elements)Aq detectors vector (Stokes vectors of detector analyzing elements)
= aq,0 aq,1 aq,2 aq,3⎡⎣ ⎤⎦
m 00 m 01 m 02 m 03
m10 m11 m12 m13
m 20 m 21 m 22 m 23
m 30 m 31 m 32 m 33
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
sq,0sq,1sq,2sq,3
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Pq =AqT MSq
€
Pq =AqTMSq =WqM
Calculation of Mueller Matrix
II Escuela de Optica Biomedica, Puebla, 2011
Pq =AqT MSq =WqM
Flattened Mueller Matrix
Measurement matrix
€
Pq =AqTMSq =WqM
• The qth measurement
Calculation of Mueller Matrix
aqsq
II Escuela de Optica Biomedica, Puebla, 2011
M =W −1P
For 16 measurements
• W is square with a unique inverse – (if W non singular)
€
M =W −1P
II Escuela de Optica Biomedica, Puebla, 2011
M =(W TW )−1W TPM =WP
−1P
For more than 16 measurements
• W is not square so to calculate the Mueller matrix
*M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)
€
M =W −TW( )−1W TP (*)
M =WP−1P
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*
*M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)
Change retardation of R1 and R2, (R1,R2 have same retardation)200 measurements to calculate WR1:R2, 1:5 ratio
Calculate cond( W)
II Escuela de Optica Biomedica, Puebla, 2011
Condition number
cond( A) = A A−1
A → max norm A
A =max1≤i≤n
aijj=1
n
∑
€
conδ A( )= A A−1
A → max norm A
A =max1≤i≤n
aijj=1
n∑
1 / cond( A)how close is A to a sin gular matrix1/cond(A) how close is A to a 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*
*M. Smith ‘Optimization of the dual-rotating-retarder Mueller matrix polarimeter”, Applied Optics, V. 41, No. 13 (2002)
Angular increments of source and detectorretarders are varied
Angular increments 0:60
Fixed retardance 127o
16 measurements30 measurements Calculate cond( W)
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
1Air2 Linear P3 qwp
II Escuela de Optica Biomedica, Puebla, 2011
Modeled error
error 2 = Mi
reconstructeδ −Miiδeal
( )i=1
16∑( )2
Mi
reconstructed =W TW( )−1W TP
Mi
reconstructeδ =V S−1U TP [U ,S,V ] =svδ(W )
€
Mireconstructeδ=W −TW( )
−1W TP (*)
Mireconstructeδ=VS−1U TP V ,S,U[ ] =svδW( )
error2 = Mireconstructeδ−Mi
iδeal( )
i=1
16
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
II Escuela de Optica Biomedica, Puebla, 2011
Modeled error
SVD
Smith
SVD gives low level error for broader range of retardances
SVD
pseudoinverse
II Escuela de Optica Biomedica, Puebla, 2011
Does the sample Mueller M bias results?
1Air2 Linear P3 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)
Tr( MM T ) ≤4m oo2
moo − m o12 +m o2
2 +m o32
( )2
≥ m o, j− m j,k
m o, j
m o12 +m o2
2 +m o32
k=1
3
∑⎛
⎝⎜⎜
⎞
⎠⎟⎟
j=1
3
∑
€
m oo2 ≥ m o1
2 +m o22 +m o3
2 ; m oo ≥m ij
Tr MMT( )≤4m oo
2
m oo− m o12 +m o2
2 +m o32 ⎛
⎝ ⎜ ⎞ ⎠ ⎟2
≥ m o,jj=1
3
∑ − m j,km o,j
m o12 +m o2
2 +m o32
k=1
3
∑
II Escuela de Optica Biomedica, Puebla, 2011
SVD
Smith
Error with reconstructed Mueller matrices
SVD gives low level error for broader range of retardances
SVD
pseudoinverse
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 =max1≤i≤n
aijj=1
n
∑
€
conδ A( )= A A−1
A → max norm A
A =max1≤i≤n
aijj=1
n∑
1 / cond( A)how close is A to a sin gular matrix1/cond(A) how close is A to a singular matrix
II Escuela de Optica Biomedica, Puebla, 2011
SVD
Ax =bSolution this is obtained minimizing the error
Ax −b
Fundamental property of SVD
A=USV T
€
Ax=b
Ax=b
Ax=USV T
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).
S
SS
€
S
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S
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