Polarimeters

II Escuela de Optica Biomedica, Puebla, 2011

Polarimeters

Jessica C. Ramella-Roman, PhD


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

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥


Simple Stokes vector polarimeter

• Six intensity measurements

€

S=

IQUV

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥=

IH + IVIH −IVI 45 −I−45I R −IL

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥



• Horizontal, i.e. parallel to reference frame

€

S=

IQUV

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥=


⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

polarizer



• Vertical, i.e. perpendicular to reference frame

€

S=

IQUV

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥=


⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

polarizer



• Linear polarizer at +/-45o to reference frame

€

S=

IQUV

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥=


⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

polarizer



• Circularly polarized (left and right)Quarter-wave plate

polarizer

€

S=

IQUV

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥=


⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Polarizer and quarterwaveplate axis are at 45o to each other


A Mueller matrix polarimeter

R1 R2PqwpP qwp


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



R1 R2PqwpP qwp


Special issues in polarimetry

• Spectral stokes vector optimization

• Mueller matrix optimization


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)


Experimental Layout

LCR1- Liquid Crystal Retarder q = 0o


p polarizer

Fiber – 200µm

LED – White LED or Xenon wls

p


Experimental Layout for Mueller M

P

We observe the spectrum between550 and 750 nm



p polarizer

Fiber – 200µm

LED – White LED or Xenon wls

p

WP


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.


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


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


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


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

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SbeforeSafter[ ] =W IbeforeIafter[ ]

W = S[ ] I[ ]−1


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[ ]


Results - Incident [1 -1 0 0]

90o


Results - Incident [1 0 0 1]

45o

wp


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


More on chicken and polarization on


The same layout & calibration can be used to build a Mueller matrix polarimeter

45o

wp


Mueller matrix of air45o

wp


Conclusions

• Stokes vector polarimeter is fiber based and usable between 550-750 nm

• Point measurements of small scatterers• Miniaturizing the system


Optimization of Mueller Matrices measurements

• The classic Mueller matrix polarimeter

• Previous work on optimizing a polarimeter

• Mueller matrix polarimetry with SVD


Dual rotating retarder polarimeter

R1 R2

R1 : 5 R2

R1R2


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


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



Pq =AqT MSq =WqM

Flattened Mueller Matrix

Measurement matrix

€

Pq =AqTMSq =WqM

• The qth measurement


aqsq


M =W −1P

For 16 measurements

• W is square with a unique inverse – (if W non singular)

€

M =W −1P


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


Two main issues

• Which wave-plates are best for Mueller matrix calculation

• number of measurements to calculate the Mueller matrix


Which retarder are best for Mueller matrix calculation*


Change retardation of R1 and R2, (R1,R2 have same retardation)200 measurements to calculate WR1:R2, 1:5 ratio

Calculate cond( W)


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


Best retardance 127

Minima at 127 and 233


n of measurements*


Angular increments of source and detectorretarders are varied

Angular increments 0:60

Fixed retardance 127o

16 measurements30 measurements Calculate cond( W)


Cond(W)

• Over-determined system are better

16 measurements 30 measurements


Svd vs pseudo-inverse

1Air2 Linear P3 qwp


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


Modeled error

SVD

Smith

SVD gives low level error for broader range of retardances

SVD

pseudoinverse


Does the sample Mueller M bias results?

1Air2 Linear P3 qwp


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

∑


SVD

Smith

Error with reconstructed Mueller matrices

SVD gives low level error for broader range of retardances

SVD

pseudoinverse


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


Tomorrow

• Monte Carlo modeling basics

• Monte Carlo with Meridian planes


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


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


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

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Polarimeters

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