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A small poster presentation made by Harsh Purwar, Student, Indian Institute of Science Education and Research, Kolkata with the contribution of others (names mentioned in the presentation) and presented in a workshop on "Trends in Optics" organized by Satendra Nath Bose National Center for Basic Sciences (SNBNCBS), Kolkata.
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Automated Spectral Mueller Matrix Polarimeter
Harsh Purwar1, Jalpa Soni1, Harshit Lakhotia1, Shubham Chandel2, Chitram Banerjee1 &
Nirmalya Ghosh1
1Department of Physical Sciences Indian Institute of Science Education and Research, Kolkata
2Cochin University of Science and Technology
I n t r o d u c t i o n
Goals to achieve:
â Develop spectral Mueller Matrix Polarimeter
â Calibrate and automate the equipment for fast and precise measurements
â Apply this approach for early stage cancer detection
Polarization: A property of the EM radiations that describes the shape and orientation of the locus of the electric field vector extremity as a function of time, at a given point in space.
⢠If the đ¸ extremity describes a stationary curve during observation, the wave is called polarized.
⢠It is called un-polarized if the extremity of vector exhibits random positions.
Stokes Vector: đ0đ1đ2đ3
=
đźđť + đźđđźđť â đźđđźđ â đźđđźđ â đźđż
S o m e B a s i c s
⢠Polarization State Generator đž : A black box that can generate different polarization states.
đ =
1 0 0 00 đđ1
2 + đ đ12 đđż đ đ1đđ1 1 â đđż âđ đ1đ đż
0 đ đ1đđ1 1 â đđż đ đ12 + đđ12 đđż đđ1đ đż
0 đ đ1đ đż âđđ1đ đż đđż
MM for QWP
Ă
1 1 0 01 1 0 00 0 0 00 0 0 0
MM for LP at H position
Ă
1000 Si
⢠Polarization State Analyzer đ¨ is dedicated to the measurement of an unknown Stokes vector. It can be described by a characteristic matrix A that links the measured intensities to the input Stokes vector.
đ´ =
1 â1 0 0â1 1 0 00 0 0 00 0 0 0MM for LP at V position
Ă
1 0 0 00 đđ1
2 + đ đ12 đđż đ đ1đđ1 1 â đđż âđ đ1đ đż
0 đ đ1đđ1 1 â đđż đ đ12 + đđ12 đđż đđ1đ đż
0 đ đ1đ đż âđđ1đ đż đđż
MM for QWP
⢠Measured MM vector, đđ = đ´đđ đ
⢠For four chosen angles of generator QWP â đ˝đ, đ˝đ, đ˝đ and đ˝đ,
đđđş =
1 1 1 1đđ12 + đ đ12 đđż đđ2
2 + đ đ22 đđż đđ3
2 + đ đ32 đđż đđ4
2 + đ đ42 đđż
đ đ1đđ1 1 â đđż đ đ2đđ2 1 â đđż đ đ3đđ3 1 â đđż đ đ4đđ4 1 â đđżđ đ1đ đż đ đ2đ đż đ đ3đ đż đ đ4đ đż
⢠Similarly, for four chosen angles of analyzer QWP â đđ, đđ, đđ and đđ,
đđđ´ =
1 â đđ12 + đ đ1
2 đđż âđđ1đ đ1 1 â đđż đ đ1đ đż
1 â đđ22 + đ đ2
2 đđż âđđ2đ đ2 1 â đđż đ đ2đ đż
1 â đđ32 + đ đ3
2 đđż âđđ3đ đ3 1 â đđż đ đ3đ đż
1 â đđ42 + đ đ4
2 đđż âđđ4đ đ4 1 â đđż đ đ4đ đż
⢠It can be shown that measured Mueller vector đđ is given by, đđ = đđđ´â đđđş
đ
đ
đđ = đ´âđđ đđ
⢠Optimal angles, đâs and đâs were computed so as to maximize the determinant of the đ matrix and are as follows,
đ1đ2đ3đ4
=
đ1đ2đ3đ4
=
đđ°đđ°đđđ°đđđ°
E x p e r i m e n t a l S e t u p
Simplified schematic of the experimental setup. Additional lenses, filters etc. may be used for focusing and collecting the incident or scattered light.
đˇđşđŽ = đˇđ +đđ
đˇđşđ¨ = đˇđ + đđ
E q u i p m e n t C a l i b r a t i o n
⢠Calibration was done using the Eigenvalue calibration method proposed by A. De. Martino et. al. in 2004.
⢠Consider, đ0 = đđ¤, đ = đđđ¤
â đ = đ0â1đ = đ¤đđ¤â1, đⲠ= đđ0
â1 = đđđâ1
⢠Mueller matrix of the sample with both diattenuation & retardance takes the form,
đ =
1 âcos 2đ 0 0â cos 2đ 1 0 00 0 sin 2đ cos Î sin 2đ sin Î0 0 sin 2đ sin Î sin 2đ cosÎ
and has four eigenvalues (2 Re and 2 Im). Matrices đ, đⲠand đ being similar have the same eigenvalues, which are đđ 1 = 2đ cos
2đ , đđ 2 = 2đ sin2đ , đđś1 = đ sin 2đ đ
âđÎ, đđś2 = đ sin 2đ đđÎ
⢠So,
đ =đđ 1 + đđ 22, đ = tanâ1
đđ 1đđ 2, Î = ln
đđś2đđś1
⢠Consider equations, đđ â đđś = 0
with a unique solution, đ = đ.
⢠4 Ă 4 matrix đ can also be written in a 16 Ă 1 basis as follows đťđđ16 = 0
where đťđ is a 16 Ă 16 matrix.
⢠Matrix đťđ is, đťđ = đ
1, đ2, đ3, ⌠, đ16
where, đđ are constructed from đşđ and đşđ is a 4 Ă 4 matrix given by, đşđ = đđđ â đđđś for đ = 1,2,3, ⌠, 16
⢠Finally the solution of the above equation is given by,
đž = đťđ1đ đťđ1 + đťđ2
đ đťđ2 +âŻ
⢠đž is a positive symmetric real matrix with a null eigenvalue, because it has a unique solution đ16 of the equation đžđ16 = 0.
⢠It has been shown that the Eigen vector of đž with zero eigenvalue gives the 16 elements of the đ (PSA) matrix.
⢠From đ, đ´ can also be obtained using, đ´ = đľ0đâ1.
M u e l l e r M a t r i x D e c o m p o s i t i o n
4 Ă 4 Mueller matrix was decomposed into three 4 Ă 4 matrices using the Polar Decomposition scheme and various polarization properties of the sample were extracted.
⢠Retardance đš is the phase shift between two orthogonal polarizations of light.
⢠Diattenuation đ is the differential attenuation of orthogonal polarizations for both linear and circular polarization states.
⢠If a completely polarized beam is incident and the emergent beam has a DOP less than unity, then the system is depolarizing.
Limitations: ⢠There should be at least two reference samples with different Mueller matrices, so that đ
and đ´ are uniquely determined.
⢠The forms of the Mueller matrices of the reference samples must be known.
Advantages: ⢠Choice of reference sample does not depend on đ or đ´.
⢠Independent of source and detector (spectrometer) polarization response.
⢠Optical elements constituting PSG and PSA need not be ideal.
⢠PSG and PSA matrices are determined using Eigenvalue calibration method for all wavelengths.
⢠System can easily be automated for fast and precise data acquisition.
L i m i t a t i o n s & A d v a n t a g e s
Mueller Matrix elements for all wavelengths measured for air as a sample after calibration (normalized with đ´đđ).
Diattenuation versus wavelength for a wide band linear polarizer.
Linear Retardance versus wavelength for a quarter wave plate.
Measured Mueller Matrix for air at 633 nm.
1 0.010 0.009 0.0020.000 0.994 0.005 0.0010.000 â0.003 0.994 â0.0010.001 â0.003 â0.007 0.999
Diattenuation and Linear Retardance plotted against wavelength for two of the
reference samples
Initial Applications on Human Cervical Tissues
⢠This approach was initially applied on the biopsy slides of human cervical cancer tissues to probe the changes in their polarization properties as compared to the normal cervical tissues.
⢠Following are some of the interesting results.
In the Backscattering Mode Geometry (scattering angle đđ°) Histopathology Report - Grade III Cancer
From Stromal Region From Epithelial Region
Retardance Plots for Grade II Cancer
Following are the retardance plots in the Transmission Mode Geometry for scattering angle eqaul to 7°, which were characterized histopathologically and were reported to have second grade cancer.
For Stromal Region For Epithelial Region
C o n c l u s i o n s
⢠A completely automated spectral Mueller matrix polarimeter has been developed.
⢠Measured Mueller matrix elements are precise up to the 2nd decimal place.
⢠Typical time taken for measurement of all 16 elements averaged over 50 spectral readings is about 3 min. for air. This may vary depending upon the nature of the sample and the signal strength.
⢠This approach helps to study polarization properties of various biological samples such as to distinguish between diseased and normal tissues.
R e f e r e n c e s
⢠General Methods for Optimized Design and Calibration of Mueller Polarimeters, A. De.
Martino et. al. 2004, Thin Solid Films, Vol. 455.
⢠General and self-consistent method for the calibration of polarization modulators,
polarimeters, and Mueller matrix ellipsometers, E. Compain, S. Poirier, B. Drevillon 1999,
Applied Optics, Vol. 38.
⢠Utilization of Mueller Matrix Formalism to Obtain Optical Targets Depolarization and
Depolarization properties. F. Le Roy â Brehonnet, B. Le Je 1997, Elsevier Science.
⢠Polarized Light: Fundamentals and Applications, E. Collette 1990, Marcel Dekker Inc., New
York.
⢠Absorption and Scattering of Light by Small Particles, C. F. Bohren, D. R. Huffman 1983,
Wiley, New York.
⢠Handbook of Optics, R. A. Chipman 2nd Edition, 1994, Vol. 2, McGraw-Hill, New York.