automated spectral mueller matrix polarimeter
DESCRIPTION
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.TRANSCRIPT
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:
โ Develop spectral Mueller Matrix Polarimeter
โ Calibrate and automate the equipment for fast and precise measurements
โ 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.
โข If the ๐ธ extremity describes a stationary curve during observation, the wave is called polarized.
โข It is called un-polarized if the extremity of vector exhibits random positions.
Stokes Vector: ๐0๐1๐2๐3
=
๐ผ๐ป + ๐ผ๐๐ผ๐ป โ ๐ผ๐๐ผ๐ โ ๐ผ๐๐ผ๐ โ ๐ผ๐ฟ
S o m e B a s i c s
โข Polarization State Generator ๐พ : A black box that can generate different polarization states.
๐ =
1 0 0 00 ๐๐1
2 + ๐ ๐12 ๐๐ฟ ๐ ๐1๐๐1 1 โ ๐๐ฟ โ๐ ๐1๐ ๐ฟ
0 ๐ ๐1๐๐1 1 โ ๐๐ฟ ๐ ๐12 + ๐๐12 ๐๐ฟ ๐๐1๐ ๐ฟ
0 ๐ ๐1๐ ๐ฟ โ๐๐1๐ ๐ฟ ๐๐ฟ
MM for QWP
ร
1 1 0 01 1 0 00 0 0 00 0 0 0
MM for LP at H position
ร
1000 Si
โข Polarization State Analyzer ๐จ 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.
๐ด =
1 โ1 0 0โ1 1 0 00 0 0 00 0 0 0MM for LP at V position
ร
1 0 0 00 ๐๐1
2 + ๐ ๐12 ๐๐ฟ ๐ ๐1๐๐1 1 โ ๐๐ฟ โ๐ ๐1๐ ๐ฟ
0 ๐ ๐1๐๐1 1 โ ๐๐ฟ ๐ ๐12 + ๐๐12 ๐๐ฟ ๐๐1๐ ๐ฟ
0 ๐ ๐1๐ ๐ฟ โ๐๐1๐ ๐ฟ ๐๐ฟ
MM for QWP
โข Measured MM vector, ๐๐ = ๐ด๐๐ ๐
โข For four chosen angles of generator QWP โ ๐ฝ๐, ๐ฝ๐, ๐ฝ๐ and ๐ฝ๐,
๐๐๐บ =
1 1 1 1๐๐12 + ๐ ๐12 ๐๐ฟ ๐๐2
2 + ๐ ๐22 ๐๐ฟ ๐๐3
2 + ๐ ๐32 ๐๐ฟ ๐๐4
2 + ๐ ๐42 ๐๐ฟ
๐ ๐1๐๐1 1 โ ๐๐ฟ ๐ ๐2๐๐2 1 โ ๐๐ฟ ๐ ๐3๐๐3 1 โ ๐๐ฟ ๐ ๐4๐๐4 1 โ ๐๐ฟ๐ ๐1๐ ๐ฟ ๐ ๐2๐ ๐ฟ ๐ ๐3๐ ๐ฟ ๐ ๐4๐ ๐ฟ
โข Similarly, for four chosen angles of analyzer QWP โ ๐๐, ๐๐, ๐๐ and ๐๐,
๐๐๐ด =
1 โ ๐๐12 + ๐ ๐1
2 ๐๐ฟ โ๐๐1๐ ๐1 1 โ ๐๐ฟ ๐ ๐1๐ ๐ฟ
1 โ ๐๐22 + ๐ ๐2
2 ๐๐ฟ โ๐๐2๐ ๐2 1 โ ๐๐ฟ ๐ ๐2๐ ๐ฟ
1 โ ๐๐32 + ๐ ๐3
2 ๐๐ฟ โ๐๐3๐ ๐3 1 โ ๐๐ฟ ๐ ๐3๐ ๐ฟ
1 โ ๐๐42 + ๐ ๐4
2 ๐๐ฟ โ๐๐4๐ ๐4 1 โ ๐๐ฟ ๐ ๐4๐ ๐ฟ
โข It can be shown that measured Mueller vector ๐๐ is given by, ๐๐ = ๐๐๐ดโ ๐๐๐บ
๐
๐
๐๐ = ๐ดโ๐๐ ๐๐
โข Optimal angles, ๐โs and ๐โs were computed so as to maximize the determinant of the ๐ matrix and are as follows,
๐1๐2๐3๐4
=
๐1๐2๐3๐4
=
๐๐ยฐ๐๐ยฐ๐๐๐ยฐ๐๐๐ยฐ
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.
๐ท๐บ๐ฎ = ๐ท๐ +๐๐
๐ท๐บ๐จ = ๐ท๐ + ๐๐
E q u i p m e n t C a l i b r a t i o n
โข Calibration was done using the Eigenvalue calibration method proposed by A. De. Martino et. al. in 2004.
โข Consider, ๐0 = ๐๐ค, ๐ = ๐๐๐ค
โ ๐ = ๐0โ1๐ = ๐ค๐๐คโ1, ๐โฒ = ๐๐0
โ1 = ๐๐๐โ1
โข Mueller matrix of the sample with both diattenuation & retardance takes the form,
๐ =
1 โcos 2๐ 0 0โ cos 2๐ 1 0 00 0 sin 2๐ cos ฮ sin 2๐ sin ฮ0 0 sin 2๐ sin ฮ sin 2๐ cosฮ
and has four eigenvalues (2 Re and 2 Im). Matrices ๐, ๐โฒ and ๐ being similar have the same eigenvalues, which are ๐๐ 1 = 2๐ cos
2๐ , ๐๐ 2 = 2๐ sin2๐ , ๐๐ถ1 = ๐ sin 2๐ ๐
โ๐ฮ, ๐๐ถ2 = ๐ sin 2๐ ๐๐ฮ
โข So,
๐ =๐๐ 1 + ๐๐ 22, ๐ = tanโ1
๐๐ 1๐๐ 2, ฮ = ln
๐๐ถ2๐๐ถ1
โข Consider equations, ๐๐ โ ๐๐ถ = 0
with a unique solution, ๐ = ๐.
โข 4 ร 4 matrix ๐ can also be written in a 16 ร 1 basis as follows ๐ป๐๐16 = 0
where ๐ป๐ is a 16 ร 16 matrix.
โข Matrix ๐ป๐ is, ๐ป๐ = ๐
1, ๐2, ๐3, โฆ , ๐16
where, ๐๐ are constructed from ๐บ๐ and ๐บ๐ is a 4 ร 4 matrix given by, ๐บ๐ = ๐๐๐ โ ๐๐๐ถ for ๐ = 1,2,3, โฆ , 16
โข Finally the solution of the above equation is given by,
๐พ = ๐ป๐1๐ ๐ป๐1 + ๐ป๐2
๐ ๐ป๐2 +โฏ
โข ๐พ is a positive symmetric real matrix with a null eigenvalue, because it has a unique solution ๐16 of the equation ๐พ๐16 = 0.
โข It has been shown that the Eigen vector of ๐พ with zero eigenvalue gives the 16 elements of the ๐ (PSA) matrix.
โข From ๐, ๐ด can also be obtained using, ๐ด = ๐ต0๐โ1.
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 ร 4 Mueller matrix was decomposed into three 4 ร 4 matrices using the Polar Decomposition scheme and various polarization properties of the sample were extracted.
โข Retardance ๐น is the phase shift between two orthogonal polarizations of light.
โข Diattenuation ๐ is the differential attenuation of orthogonal polarizations for both linear and circular polarization states.
โข If a completely polarized beam is incident and the emergent beam has a DOP less than unity, then the system is depolarizing.
Limitations: โข There should be at least two reference samples with different Mueller matrices, so that ๐
and ๐ด are uniquely determined.
โข The forms of the Mueller matrices of the reference samples must be known.
Advantages: โข Choice of reference sample does not depend on ๐ or ๐ด.
โข Independent of source and detector (spectrometer) polarization response.
โข Optical elements constituting PSG and PSA need not be ideal.
โข PSG and PSA matrices are determined using Eigenvalue calibration method for all wavelengths.
โข 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 ๐ด๐๐).
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 โ0.003 0.994 โ0.0010.001 โ0.003 โ0.007 0.999
Diattenuation and Linear Retardance plotted against wavelength for two of the
reference samples
Initial Applications on Human Cervical Tissues
โข 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.
โข Following are some of the interesting results.
In the Backscattering Mode Geometry (scattering angle ๐๐ยฐ) 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 7ยฐ, 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
โข A completely automated spectral Mueller matrix polarimeter has been developed.
โข Measured Mueller matrix elements are precise up to the 2nd decimal place.
โข 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.
โข 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
โข General Methods for Optimized Design and Calibration of Mueller Polarimeters, A. De.
Martino et. al. 2004, Thin Solid Films, Vol. 455.
โข 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.
โข Utilization of Mueller Matrix Formalism to Obtain Optical Targets Depolarization and
Depolarization properties. F. Le Roy โ Brehonnet, B. Le Je 1997, Elsevier Science.
โข Polarized Light: Fundamentals and Applications, E. Collette 1990, Marcel Dekker Inc., New
York.
โข Absorption and Scattering of Light by Small Particles, C. F. Bohren, D. R. Huffman 1983,
Wiley, New York.
โข Handbook of Optics, R. A. Chipman 2nd Edition, 1994, Vol. 2, McGraw-Hill, New York.