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:

– Develop spectral Mueller Matrix Polarimeter

– Calibrate and automate the equipment for fast and precise measurements

– 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 𝐸 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: 𝑆0𝑆1𝑆2𝑆3

=

𝐼𝐻 + 𝐼𝑉𝐼𝐻 − 𝐼𝑉𝐼𝑃 − 𝐼𝑀𝐼𝑅 − 𝐼𝐿

S o m e B a s i c s

• Polarization State Generator 𝑾 : A black box that can generate different polarization states.

𝑊 =

1 0 0 00 𝑐𝜃1

2 + 𝑠𝜃12 𝑐𝛿 𝑠𝜃1𝑐𝜃1 1 − 𝑐𝛿 −𝑠𝜃1𝑠𝛿

0 𝑠𝜃1𝑐𝜃1 1 − 𝑐𝛿 𝑠𝜃12 + 𝑐𝜃12 𝑐𝛿 𝑐𝜃1𝑠𝛿

0 𝑠𝜃1𝑠𝛿 −𝑐𝜃1𝑠𝛿 𝑐𝛿

MM for QWP

×

1 1 0 01 1 0 00 0 0 00 0 0 0

MM for LP at H position

×

1000 Si

• Polarization State Analyzer 𝑨 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.

𝐴 =

1 −1 0 0−1 1 0 00 0 0 00 0 0 0MM for LP at V position

×

1 0 0 00 𝑐𝜃1

2 + 𝑠𝜃12 𝑐𝛿 𝑠𝜃1𝑐𝜃1 1 − 𝑐𝛿 −𝑠𝜃1𝑠𝛿

0 𝑠𝜃1𝑐𝜃1 1 − 𝑐𝛿 𝑠𝜃12 + 𝑐𝜃12 𝑐𝛿 𝑐𝜃1𝑠𝛿

0 𝑠𝜃1𝑠𝛿 −𝑐𝜃1𝑠𝛿 𝑐𝛿

MM for QWP

• Measured MM vector, 𝑀𝑖 = 𝐴𝑀𝑠𝑊

• For four chosen angles of generator QWP – 𝜽𝟏, 𝜽𝟐, 𝜽𝟑 and 𝜽𝟒,

𝑃𝑆𝐺 =

1 1 1 1𝑐𝜃12 + 𝑠𝜃12 𝑐𝛿 𝑐𝜃2

2 + 𝑠𝜃22 𝑐𝛿 𝑐𝜃3

2 + 𝑠𝜃32 𝑐𝛿 𝑐𝜃4

2 + 𝑠𝜃42 𝑐𝛿

𝑠𝜃1𝑐𝜃1 1 − 𝑐𝛿 𝑠𝜃2𝑐𝜃2 1 − 𝑐𝛿 𝑠𝜃3𝑐𝜃3 1 − 𝑐𝛿 𝑠𝜃4𝑐𝜃4 1 − 𝑐𝛿𝑠𝜃1𝑠𝛿 𝑠𝜃2𝑠𝛿 𝑠𝜃3𝑠𝛿 𝑠𝜃4𝑠𝛿

• Similarly, for four chosen angles of analyzer QWP – 𝝓𝟏, 𝝓𝟐, 𝝓𝟑 and 𝝓𝟒,

𝑃𝑆𝐴 =

1 − 𝑐𝜙12 + 𝑠𝜙1

2 𝑐𝛿 −𝑐𝜙1𝑠𝜙1 1 − 𝑐𝛿 𝑠𝜙1𝑠𝛿

1 − 𝑐𝜙22 + 𝑠𝜙2

2 𝑐𝛿 −𝑐𝜙2𝑠𝜙2 1 − 𝑐𝛿 𝑠𝜙2𝑠𝛿

1 − 𝑐𝜙32 + 𝑠𝜙3

2 𝑐𝛿 −𝑐𝜙3𝑠𝜙3 1 − 𝑐𝛿 𝑠𝜙3𝑠𝛿

1 − 𝑐𝜙42 + 𝑠𝜙4

2 𝑐𝛿 −𝑐𝜙4𝑠𝜙4 1 − 𝑐𝛿 𝑠𝜙4𝑠𝛿

• It can be shown that measured Mueller vector 𝑀𝑖 is given by, 𝑀𝑖 = 𝑃𝑆𝐴⊗ 𝑃𝑆𝐺

𝑇

𝑄

𝑀𝑠 = 𝐴⊗𝑊𝑇 𝑀𝑠

• Optimal angles, 𝜃’s and 𝜙’s were computed so as to maximize the determinant of the 𝑄 matrix and are as follows,

𝜃1𝜃2𝜃3𝜃4

=

𝜙1𝜙2𝜙3𝜙4

=

𝟑𝟓°𝟕𝟎°𝟏𝟎𝟓°𝟏𝟒𝟎°

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.

𝑷𝑺𝑮 = 𝑷𝟏 +𝐐𝟏

𝑷𝑺𝑨 = 𝑷𝟐 + 𝐐𝟐

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, 𝑏0 = 𝑎𝑤, 𝑏 = 𝑎𝑚𝑤

⇒ 𝑐 = 𝑏0−1𝑏 = 𝑤𝑚𝑤−1, 𝑐′ = 𝑏𝑏0

−1 = 𝑎𝑚𝑎−1

• Mueller matrix of the sample with both diattenuation & retardance takes the form,

𝑀 =

1 −cos 2𝜓 0 0− cos 2𝜓 1 0 00 0 sin 2𝜓 cos Δ sin 2𝜓 sin Δ0 0 sin 2𝜓 sin Δ sin 2𝜓 cosΔ

and has four eigenvalues (2 Re and 2 Im). Matrices 𝑐, 𝑐′ and 𝑚 being similar have the same eigenvalues, which are 𝜆𝑅1 = 2𝜏 cos

2𝜓 , 𝜆𝑅2 = 2𝜏 sin2𝜓 , 𝜆𝐶1 = 𝜏 sin 2𝜓 𝑒

−𝑖Δ, 𝜆𝐶2 = 𝜏 sin 2𝜓 𝑒𝑖Δ

• So,

𝜏 =𝜆𝑅1 + 𝜆𝑅22, 𝜓 = tan−1

𝜆𝑅1𝜆𝑅2, Δ = ln

𝜆𝐶2𝜆𝐶1

• Consider equations, 𝑀𝑋 − 𝑋𝐶 = 0

with a unique solution, 𝑋 = 𝑊.

• 4 × 4 matrix 𝑋 can also be written in a 16 × 1 basis as follows 𝐻𝑀𝑋16 = 0

where 𝐻𝑀 is a 16 × 16 matrix.

• Matrix 𝐻𝑀 is, 𝐻𝑀 = 𝑔

1, 𝑔2, 𝑔3, … , 𝑔16

where, 𝑔𝑖 are constructed from 𝐺𝑖 and 𝐺𝑖 is a 4 × 4 matrix given by, 𝐺𝑖 = 𝑀𝑈𝑖 − 𝑈𝑖𝐶 for 𝑖 = 1,2,3, … , 16

• Finally the solution of the above equation is given by,

𝐾 = 𝐻𝑀1𝑇 𝐻𝑀1 + 𝐻𝑀2

𝑇 𝐻𝑀2 +⋯

• 𝐾 is a positive symmetric real matrix with a null eigenvalue, because it has a unique solution 𝑊16 of the equation 𝐾𝑋16 = 0.

• It has been shown that the Eigen vector of 𝐾 with zero eigenvalue gives the 16 elements of the 𝑊 (PSA) matrix.

• From 𝑊, 𝐴 can also be obtained using, 𝐴 = 𝐵0𝑊−1.

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 × 4 Mueller matrix was decomposed into three 4 × 4 matrices using the Polar Decomposition scheme and various polarization properties of the sample were extracted.

• Retardance 𝜹 is the phase shift between two orthogonal polarizations of light.

• Diattenuation 𝒅 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 𝑊

and 𝐴 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 𝑊 or 𝐴.

• 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 𝑴𝟏𝟏).

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 −0.003 0.994 −0.0010.001 −0.003 −0.007 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 𝟕𝟑°) 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 – 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.