Mueller Matrix Measurement Techniques for various Optical Elements & Biological Tissues

Indian Institute of Science Education and Research, Kolkata 28th March – 25th April 2011

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Mueller Matrix Measurement Techniques for various Optical Elements & Biological Tissues

Optics Lab Report By

Harsh Purwar (07MS – 76) 4th Year, Int. M.S. (Physics)

Indian Institute of Science Education and Research, Kolkata

Abstract

It has been observed that in recent years, researchers have given much emphasis on exploring on various tissues through polarized light imaging technique as a potential diagnostic tool for detecting any kind of abnormality particularly chronic diseases such as Cancer, Tumor, etc. As we are aware of the fact that tissues depolarize a large fraction of incident light so that the Mueller calculus lends itself well to these applications. Cancerous tissues were well discriminated from normal tissues using Mueller imaging. Here I present a detailed analysis about the various methods that can be adopted for measuring/calculating Mueller matrix for various optical elements (such as polarizers, mirror, quarter wave plates etc.) and biological samples (cancerous human cervical tissues). The analyses presented in this report includes decomposition of the Mueller matrix constructed from the 16 polarization state measurements (images and spectra) of various samples using polarization state generators and analyzers into three optical parameters namely depolarization, di-attenuation and linear retardance.

Keywords: Mueller matrix, spectral measurement, depolarization, di-attenuation, birefringence, tissue optics, polarization imaging, retardance, Mueller decomposition, turbid polarimetry, Stokes vector, polarization, multiple scattering.

Introduction

Stokes Vector and Mueller Matrix

Stokes vector is a 4 element column matrix that describes the polarization state of electromagnetic radiations. The elements of a Stokes vector are called Stokes parameters. These were defined by George Gabriel Stokes in 1852. In terms of the components of electric field ( and ), the stokes parameters are given by (1):

⟨

⟩ ⟨

⟩

⟨

⟩ ⟨

⟩

Here are the 4 elements of Stokes vector or Stokes parameters. And the intensity of light by


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The polarization state of the scattered light in the far zone is described by the Stokes vector connected with the Stokes vector of the incident light. If is the initial incident Stokes vector and is the Stokes vector describing the scattered light then we have,

where is the normalized scattering matrix or Mueller matrix.

Polarization States

Polarization states are various orientations of a polarizer or analyzer. There can be infinitely many orientations and so infinitely many states. These all experiments are designed to use only seven such states listed below. Important point to remember is the relative differences between the states. For example horizontal polarization state (H) should be more or less from the vertical polarization state ( ). Same goes for +45 ( ) and -45 ( ).

Mueller Matrix

Out of the many ways of constructing Mueller matrix or rather measuring Mueller matrix experimentally for any given optical element one of the most standard method and adopted by most, is the one described using the concept of polarization state generators and analyzers. For a particular wavelength of light, Mueller matrix was generated from the four incident polarization states (linear polarization at angles of , , from the horizontal and right circular polarization) and recorded the intensity of the light transmitted through sample after it passed through the suitably oriented analyzers (linear polarization at angles of , , from the horizontal and right circular polarization). Mathematically we use PSG and PSA matrix respectively as,

(

) (

)

intensity measurement matrix is given as,

is the Mueller matrix of the sample, which can also be written as vector,

where is a matrix given as Kroneker product of PSA with transpose of PSG,

This kind of construction allows simpler calibration for non-ideal components since one replace the ideal stokes vector in each column of PSG or each row of PSA by measured stokes vectors that may deviate from ideal value which is not possible in direct construction of Mueller matrix from other intensity measurements. The errors arising due to the finite extinction ratios of the linear polarizer and analyzer was incorporated using the generated matrix corresponding to these polarizers. For a white incident light, the set of four incident polarization states were generated by rotating fast axis of quarter wave plate (QWP) with respect to the pass axis of first polarizer (Figure 3). For each input polarization state, intensities at four different analyzer states are measured using suitably oriented QWP and polarizer at the output. Stokes vector of light incident on sample after passing through a fixed polarizer and QWP (for a given orientation angle, ), is given as,


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( ) (

( )

( )

) (

) (

)

(

( )

)

where, , , , and is the linear retardance which for a QWP is ⁄ . Above equation represents the general Stokes vector of the incident light. Now, changing will change its polarization state. We chose four different orientation angles ( , , and ) of QWP and hence PSG comes out as,

(

( ) ( ) ( ) ( )

)

Similarly the polarization state of the detected light, analyzed by PSA optics will depend upon the orientation angle of QWP . In Mueller matrix measurements only intensity is recorded which is given by the first element of the Stokes vector. In our case this is decided by the first row of the matrix formed by the multiplication of Mueller matrix of QWP (for a given ) followed by the Mueller matrix of polarizer (kept cross with respect to which is at horizontal position). By changing orientation angles of QWP ( , , and ) four different polarizations of the scattered light can be analyzed. Hence PSA can be written as,

(

( ) ( )

( ) ( )

( ) ( )

( ) ( ) )

And then rest of the algebra remains same as mentioned above for the previous case.

Polarization Images

In one of the following experiments, polarization images are the images taken by the CCD for various polarization states like , , , etc. These polarization images may also be called as the intensity images for various states. Basically these intensity images are matrix, each element storing the value of intensity (in arbitrary units) falling at that particular location on the CCD’s chip. This intensity can have values ranging from 0 to 255. While processing the images and decomposing the matrix using a simple MATLAB script (Appendix section) the value of each parameter is calculated for all pixels of the images.


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Optical Parameters

Depolarization:

A process which couples polarized light into un-polarized light. Depolarization is intrinsically

associated with scattering and with di-attenuation and retardance which vary in space, time, and/or

wavelength. Examples of depolarizers include tissues, polystyrene microspheres, phantoms etc.

Di-attenuation:

The property of an optical element or system whereby the intensity transmittance of the exiting

beam depends on the polarization state of the incident beam. The intensity transmittance is a

maximum (Tmax) for one incident state, and a minimum (Tmin) for the orthogonal state. The di-

attenuation is defined as (Tmax - Tmin) / (Tmax + Tmin).

Any homogeneous polarization element which displays significant di-attenuation and minimal retardance is called a di-attenuator. Polarizers have a di-attenuation close to one, but nearly all optical interfaces are weak di-attenuators. Examples of di-attenuators include the following: polarizers and di-chroic materials, as well as metal and dielectric interfaces with reflection and transmission differences described by Fresnel equations; thin films (homogeneous and isotropic); and diffraction gratings.

Polarizance:

The property of an optical element or system whereby un-polarized light is transformed into

polarized light. The polarizance is described by its magnitude (equal to the degree of polarization of

light exiting the system when un-polarized light is input) and the Stokes vector of the output light.

Retardance:

A polarization-dependent phase change associated with a polarization element or system. The

phase (optical path length) of the output beam depends upon the polarization state of the input

beam. The transmitted phase is a maximum for one eigen polarization, and a minimum for the

other eigen polarization. Other states show polarization coupling and an intermediate phase.

B-irefringence:

A material property, the retardance associated with propagation through an anisotropic medium.

For each propagation direction within a bi-refringent medium there are two modes of propagation

with different refractive indices and . The bi-refringence is given by, | |.

Decomposition Scheme for Mueller matrix

An arbitrary Mueller matrix can be decomposed into three basic optical parameters namely decomposition, retardance and di-attenuation discussed in the latter half of this section. It has been shown that any Mueller matrix can be expressed as a product of three matrices called depolarizer, di-attenuator and retarder (2). So we have,

( ) The three di-attenuation components ⁄ , ⁄ and provide its complete description. The di-attenuation of a Mueller matrix described above is given by,

√

( )

Similarly di-attenuation vector is given by,


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(

)

(

) ( )

And diattenuation matrix is given by,

[

] ( )

where is given by,

√ (

) ( √ ) ( )

So, we are now left with,

( ) ( )

Let and be the eigenvalues of ( ) . So from Eq. – 7, has √ √ and √ as

eigenvalues. After calculating the eigenvalues of ( ) we evaluate the following expression,

[ ( ) (√ √ √ ) ]

[(√ √ √ ) ( ) √ ] ( )

If the determinant of is negative then minus sign is applied otherwise positive. Now to get from we first calculate polarizance vector and polarizance matrix using,

(

) ( )

( )

Now we write as a matrix in terms of and as below,

[

] ( )

The value of depolarization power or simple depolarization is also calculated and is shown as a figure (later) using,

| ( ) |

( )

Now again we pre-multiply (Eq. – 6) by

to get,

( )

The value of retardance was calculated as,

[ ( )

] ( )

Spatial Light Modulators:

A spatial light modulator (SLM) is an object that imposes some form of spatially – varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the phrase SLM is


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used, it means that the transparency can be controlled by a computer. In the 1980s, large SLMs were placed on overhead projectors to project computer monitor contents to the screen. Since then more modern projectors have been developed where the SLM is built inside the projector. These are commonly used in meetings of all kinds for presentations. Usually, an SLM modulates the intensity of the light beam. However, it is also possible to produce devices that modulate the phase of the beam or both the intensity and the phase simultaneously. SLMs are used extensively in holographic data storage setups to encode information into a laser beam in exactly the same way as a transparency does for an overhead projector. They can also be used as part of a holographic display technology. SLMs have been used as a component in optical computing. They also often find application in holographic optical tweezers.

Optically addressed spatial light modulator (OASLM):

The image on an optically addressed spatial light modulator, also known as a light valve, is created and changed by shining light encoded with an image on its front or back surface. A photo-sensor allows the OASLM to sense the brightness of each pixel and replicate the image using liquid crystals. As long as the OASLM is powered, the image is retained even after the light is extinguished. An electrical signal is used to clear the whole OASLM at once. They are often used as the second stage of a very-high-resolution display, such as one for a computer-generated holographic display. In a process called active tiling, images displayed on an EASLM are sequentially transferred to different parts on an OASLM, before the whole image on the OASLM is presented to the viewer. As EASLMs can run as fast as 2500 frames per second, it is possible to tile around 100 copies of the image on the EASLM onto an OASLM while still displaying full-motion video on the OASLM. This potentially gives images with resolutions of above 100 megapixels.

Experimental Setup

For a Fixed Wavelength:

Figure 1: Schematic Diagram of the Experimental Setup in transmission mode geometry designed for a fixed wavelength of light.

Key:

Helium (He) Neon (Ne) LASER (633 nm 12.0 mW)

Neutral Density (ND) Filter manufactured by ThorLabs was used.

Polarizers, and manufactured by ThorLabs were used.

Quarter Wave Plates, and again manufactured by ThorLabs were used.


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The rotational mounts for the polarizers and QWPs were specially designed for precise angular measurements.

Sample is an optical element whose Mueller matrix we are interested in finding out.

A collecting lens can be used for better results in case the sample is highly scattering or has high absorbance in between the sample and QWP .

Final measurement was intensity based done using a charge coupled device or a CCD camera manufactured by ThorLabs. The intensity measurement range of the CCD was from 0 – 255 in arbitrary units (0 for darkest pixel and 255 for the brightest).

Another convex lens of appropriate focal length could be used to focus the light from the polarizer to the CCD chip in case the beam diverges too much.

The CCD was connected to the computer and the intensity image (profile) as seen by the CCD chip was recorded using the software provided along with the device.

Figure 2: Experimental Setup for white light in back scattering mode.


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For White Light (Xenon Lamp):

Figure 3: Schematic Diagram of the Experimental Setup in the back-scattering mode geometry designed for white light from Xenon Lamp.

Key:

The above shown apparatus was mounted on a rail whose ends were fixed on a rotational stage. This helps in aligning the optical elements and back-scattering angle can be measured precisely.

Xenon Lamp (50 W)

Polarizers and are wide band transmission polarizers manufactured by ThorLabs.

Similarly quarter wave plates and also have a wide band transmission again manufactured by ThorLabs.

Any other optical element required in the experiment should also have a wide band transmission spectrum like focusing and collecting lenses, optical fibers etc.

The light was focused on one of the ends of an optical fiber which was then connected to a spectrometer.

The measurement was done in the form of a spectrum (Intensity versus Wavelength curve) recorded using the software supplied along with the spectrometer.

Equipment Calibration

Before starting the experiment according to the above setup, we must assign rather choose 4 states of input and output polarizations of light commonly known as Polarization State Generator (PSG) and Polarization State Analyzer (PSA). It has been shown that one can find out the Mueller matrix using any 4 arbitrary states as far as the determinant of the W-matrix is non-zero. We choose the following standard


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four states which gives the determinant of the W-matrix maximum (far away from zero) and also this arrangement suits our setup for a particular wavelength.

For a fixed Wavelength:

Remove and . Let us call to be the horizontal, state of the polarizer . The corresponding cross-state (vertical) of the polarizer can be found using one of the fundamental laws in optics known as Malus’s Law. The polarizer is rotated to obtain minimum intensity on the CCD or a photo diode. The angle for is recorded and this state is called vertical, . Similarly state for would be and corresponding state could be noted for , would be vertical ( ) for and for can be found. Now to find the of the QWP set to state and to state . Now find an angle of for which on rotating the polarizer the intensity profile remains almost constant. This angle along with state of corresponds to the right circularly polarized light abbreviated as . Same is done for the QWP . Following table shows the angles corresponding to the four states for , , and .

For Polarizer :

Horizontal ( )

Vertical ( )

+45 ( )

For Polarizer :

Horizontal ( )

Vertical ( )

+45 ( )

For QWP :

+45

For QWP :

+45

For White Light (Spectra Measurement):

The apparatus for the white light is as shown in Figure 3. The QWP was mounted on a computer controlled rotational mount. The polarizers and QWPs can easily be calibrated as mentioned above. Here we have chosen four states of QWP’s and the polarizers were fixed in a particular state. was fixed at horizontal and was fixed at vertical. The four chosen angles for the QWPs were optimized so as to get a maximum possible determinant of the W-matrix. The optimization code (written in MATLAB) for the same is attached in the Appendix at the end of this report. The four chosen angles were , , and . The corresponding determinant of the W-matrix was computed to be 18.23.


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Procedure

Among many ways of experimentally measuring Mueller matrix for an optical element the two adopted methods with followed protocol are as follows.

For a Fixed Wavelength:

After cleaning and aligning the optical equipment as mentioned above in the Experimental Setup section data images were recorded as detailed below.

Before recording the data optical alignment was checked ones again.

The optical element whose Mueller matrix is to be measured was mounted on the sample stand and it was made certain that the LASER beam passes though the approximate center of the sample.

Both the two polarizers, and were set to angles corresponding to the state (horizontal). Both the QWP’s were removed temporarily.

The image as seen through the charged coupled device (CCD) on the computer screen was saved after adjusting the integration time so that the device is not saturated. The integration time was not changed after that.

Similarly by rotating the two polarizers to proper angles known by calibrating each of them all other 8 states ( , , , , , , , ) were recorded.

The QWP’s and were then introduced and following the same procedure rest of the 7 states ( , , , , , , ) were recorded.

For White Light:

A similar method as described above was followed for recording the spectra of various input and output states. Briefly the adopted method is described below.

Before recording the spectra optical alignment was checked ones again.

The optical element whose Mueller matrix is to be measured was mounted on the sample stand and it was made certain that the LASER beam passes though the approximate center of the sample.

The two polarizers, and were set to angles corresponding to the state (horizontal) and (vertical) respectively.

The two quarter wave plates, and were rotated to the above calibrated angles and the corresponding spectrum was recorded for each state.


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Observations

Recorded Images for various Input & Output States

HH HP HR HV

PH PP PR PV

RH RP RR RV

VH VP VR VV

For Blank (No Sample)

The fringe pattern visible in almost all the intensity images above is mainly a side effect of the use of a circular pin-hole (or aperture) after beam broadening.


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Recorded Spectrum for Various Input & Output States

Following is a spectrum for one of the states with the QWP aligned at angles and for a biological sample (human tissue with cervical cancer) mounted on a microscope slide.

Figure 4: Spectrum of white light incident on a human tissue sample for the state with , , and .

Data Analysis

Analysis of data recorded in the form of images

Each image for blank (no sample) and QWP (at ) as a sample was imported in MATLAB as a matrix and was summed over all the pixels or elements so as to obtain an overall average intensity corresponding to each state.

The PSA and PSG were constructed for an ideal case and the corresponding W-matrix was calculated.

(

) (

)

And,

The Mueller column vector ( ) was then formed for blank as well as for the sample (QWP) for the measured states.

( )


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Sample Mueller vector ( ) was then computed using the relation,

This was then transformed into a matrix to obtain the Mueller matrix for blank and sample

(without blank correction).

Then the corrected Mueller matrix for the QWP sample was obtained by, ( )

Results and Conclusions

The Mueller matrix obtained for the blank (no sample) is as follows,

(

)

Ideally the Mueller matrix for blank should be identity but as can it seen it deviates from that. This is mainly due to other non-ideal optical elements in the PSA and PSG optics. Using the above matrix blank correction was done and the corrected Mueller matrix for the QWP at an angle of was calculated,

(

)

Spectral Measurements

The spectral measurement of Mueller matrix was done for a QWP at an angle of and then this matrix was decomposed to give depolarization, di-attenuation and linear retardance as a function of wavelength. Following plots (Figure 5, Figure 6 & Figure 7) show these properties for a QWP in transmission mode geometry.

Figure 5: Depolarization versus wavelength for a QWP in transmission mode geometry.


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Figure 6: Di-attenuation versus wavelength for a QWP in transmission mode geometry.

Figure 7: Linear retardance versus wavelength for a QWP in transmission mode geometry.


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The same was done for a polished mirror as well but as is obvious in the back-scattering mode. Following (Figure 8, Figure 9 & Figure 10) are the plots.

Figure 8: Depolarization versus wavelength for a polished mirror in the back-scattering mode.

Figure 9: Di-attenuation versus wavelength for a polished mirror in the back-scattering mode.


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Figure 10: Linear retardance versus wavelength for a polished mirror in the back-scattering mode.

Linear retardance for the mirror, at all wavelengths of the incident light, should be closed to zero but it seems from the above figure that it is close to . This is due to the reflection of the light from the surface of the mirror. Apart from this the reason for periodicity in the results (in all previous curves) is not very clear at this moment. It might be due to the misalignment of other optics involved in the experimental setup, or could even be due to the imperfections in the mirror itself. The exact reason could not be explained.


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Biological Samples

A very similar measurement was done for the biological tissues (human cervical cancer tissues) for three different grades of cancer. The beam was focused on the stroma region of the tissues. Obtained spectra was analyzed using the code attached at the end in the Appendix section of this report. This was done in both transmission and back-scattering mode geometries. Moreover, instead of doing it at zero degree we preferred to do at in transmission mode so as to collect most of the scattered light from the tissue. Similarly in the back-scattering mode the angle between the incoming beam and the collected light beam was . Following figure (Figure 11) shows variation of each element of Mueller matrix with wavelength measured for the human cervical cancer tissue (Slide Code: 8413) in the back-scattering mode with angular separation between the incident and the detected light being .

Figure 11: Mueller elements versus wavelength for human cervical cancerous tissue Grade 1 (Slide Code: 8413).

Each of the above Mueller elements plays an important role in the diagnostic technique for cancer. However as described above, the three more physical parameters namely depolarization, di-attenuation and linear retardance would be used as a tool here for the basic diagnostic of cancer or more precisely as parameters to distinguish between various grades/stages of cancer in these biological samples.


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Following are the plots of depolarization, di-attenuation and linear retardance for these 3 grades of cancerous tissues.

Depolarization Plots for Biological Samples

Figure 12: Depolarization versus wavelength for human cervical cancerous tissue, Grade 2 (Slide Code: 7695).



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Di-Attenuation Plots for Biological Samples

Figure 15: Di-attenuation versus wavelength for human cervical cancerous tissue, Grade 2 (Slide Code: 7695).


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Linear Retardance Plots for Biological Samples

In the following plots of linear retardance versus wavelength for the three tissue samples it seems that experiments done in two different geometries yield very different results. This is not true in fact. The value of retardance for the back-scattering geometry ( ) at each wavelength is as clear from the graph is close to which is due to the reflection of the incident light. If we subtract from these values then the results in both the cases match closely.

Figure 18: Linear retardance versus wavelength for human cervical cancerous tissue, Grade 2 (Slide Code: 7695).



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From the above figures (Figure 12 - Figure 20), one cannot precisely distinguish between various stages of cancers as there are a lot of errors associated with the measurements. Also one has to do the experiment in more idealized conditions for more number of tissue samples and with proper alignment. Also a more thorough analysis of the data is required so as to account for any associated random errors etc. But certainly this diagnostic technique could lead to a major breakthrough in the medical science.

Spatial Light Modulator (SLM)

The spectral measurement for the construction of Mueller matrix was also done with SLM as a sample so as to study the basic optical properties of it. The following gray scale (default) was set with the help of configuration software supplied along with it and the spectral measurements were done just as mentioned before in the Procedure section. Gray scale Details: Contrast: 195 (Min: 0, Max: 255) Brightness: 100 (Min: 0, Max: 255)


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Following are the decomposition plots for the spatial light modulator with above specified gray scale.

Figure 21: Depolarization versus wavelength for spatial light modulator at two angles (0° and 5°) in transmission mode.

Figure 22: Di-attenuation versus wavelength for spatial light modulator at two angles ( and ) in transmission mode.


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Figure 23: Linear retardance versus wavelength for spatial light modulator at two angles ( and ) in transmission mode.

Sources of Errors

Out of the many possible sources of errors in the above described experiments following is a description of a few major sources which could have affected the preceding results.

Major contribution to the fluctuations in all most all of the above graphs comes from the fluctuating initial (LASER/Lamp) intensity of light.

Apart from this, fluctuations in case of biological samples are due to the fact that these samples have complex structures/patterns, which even in the normal case would give a background fluctuation in almost all measured optical parameters.

Optical misalignment may lead to many spurious errors in the measurements.

All the optical elements, to be used, must be thoroughly cleaned with acetone or lens cleansing liquid/paper.

Other deviations from the standard or expected results may be due to the involvement of other non-ideal optics.

Acknowledgement

I thank Dr. Nirmalya Ghosh for his humble guidance, help and support throughout the experiment period. I would also like to thank Jalpa Soni and my group partners, Satish Kumar, Amit Anand and Irfan Raza for helping me in carrying out these experiments and analyzing the data.


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Works Cited

1. Tuchin, Valery. Optical properties of tissues with strong (multiple) scattering. Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis. Second. 1, pp. 3-108. 2. Interpretation of Mueller matrices based on polar decomposition. Lu, Shih-Yau and Chipman, Russell A. 5, Alabama : Optical Society of America, May 1996, Opt. Soc. Am. A, Vol. 13, pp. 1106-1113. 3. Wikipedia. Mueller Calculus. [Online] Wikipedia.org. http://en.wikipedia.org/wiki/Mueller_calculus. 4. Gupta, Vivek. Mueller matrix in optical imaging for cervical cancer detection. Department of Physics, Indian Institute of Technology. Kanpur : s.n., 2009. pp. 1-6. 5. Reference Page. Axometrics Web Site. [Online] http://www.axometrics.com/reference.htm. 6. Ghosh, Nirmalya. Handbook of Photonics for Biomedical Science. s.l. : Unpublished. pp. 253-282. 7. General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers. Compain, Eric, Poirier, Stephane and Drevillon, Bernard. 16, June 1, 1999, Applied Optics, Vol. 38, pp. 3490-3502. 8. Saleh, B E.A. and Teich, M. C. Fundamentals of Photonics. s.l. : John Wiley & Sons, Inc., 2007. ISBN: 9780471358329. 9. UTILIZATION OF MUELLER MATRIX FORMALISM TO OBTAIN OPTICAL TARGETS DEPOLARIZATION AND POLARIZATION PROPERTIES. Roy-Brehonnet, F. Le and Jeune, B. Le. 2, s.l. : Elsevier Science Ltd., 1997, Progress in Quantum Electronics, Vol. 21, pp. 109-151. 10. Purwar, Harsh. Mueller Imaging: An approach to Detect Abnormality in Human Brain Tissues. Department of Physical Sciences, Indian Institute of Science Education & Research. Kolkata : Unpublished, 2009. Summer Project Report.


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Appendix

Code1:

For maximizing the determinant of W-matrix by choosing appropriate states of QWPs given the two polarizers and are at and states respectively. 1 function fastest() 2 del=pi/2; 3 ang=0:2:180; 4 ang=ang/180*pi; 5 i=0; 6 [m,n]=size(ang); 7 for a=1:n 8 disp(a); 9 for b=a+1:n 10 for c=b+1:n 11 for d=c+1:n 12 PSG(:,1)=PSGG(ang(1,a),del); 13 PSG(:,2)=PSGG(ang(1,b),del); 14 PSG(:,3)=PSGG(ang(1,c),del); 15 PSG(:,4)=PSGG(ang(1,d),del); 16 PSA(1,:)=PSAG(ang(1,a),del); 17 PSA(2,:)=PSAG(ang(1,b),del); 18 PSA(3,:)=PSAG(ang(1,c),del); 19 PSA(4,:)=PSAG(ang(1,d),del); 20 W=kron(PSA,PSG'); 21 DW=det(W); 22 if(DW>23.5) 23 i=i+1; 24 detW(i,:)=[ang(1,a)/pi*180,ang(1,b)/pi*180,ang(1,c)/pi*180,ang(1,d)/pi*180,DW]; 25 end 26 clear W 27 end 28 end 29 end 30 end 31 save 'DetW.txt' detW -ascii; 32 % detW 33 end 34 35 function y1 = PSGG(t,del) 36 c2t=cos(2*t); 37 s2t=sin(2*t); 38 sl=sin(del); 39 cl=cos(del); 40 y1=[1;(c2t^2)+((s2t^2)*cl);c2t*s2t*(1-cl);s2t*sl]; 41 end 42 43 function y2 = PSAG(t,del) 44 c2t=cos(2*t); 45 s2t=sin(2*t); 46 sl=sin(del); 47 cl=cos(del); 48 y2=[1,-((c2t^2)+((s2t^2)*cl)),-c2t*s2t*(1-cl),s2t*sl]; 49 end


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For further resolving the angles (more accurate results):

1 function resolver()

2 del=pi/2;

3 i=0;

4 data=load('DetW.txt');

5 [p,q]=size(data);

6 for j=1:p

7 disp ([num2str(j),' of ',num2str(p)]);

8 for t1=(data(j,1)-1):0.2:(data(j,1)+1)

9 for t2=(data(j,2)-1):0.2:(data(j,2)+1)

10 for t3=(data(j,3)-1):0.2:(data(j,3)+1)

11 for t4=(data(j,4)-1):0.2:(data(j,4)+1)

12 PSG(:,1)=PSGG(t1/180*pi,del);




16 PSA(1,:)=PSAG(t1/180*pi,del);




20 W=kron(PSA,PSG');

21 DW=det(W);

22 i=i+1;

23 detW(i,:)=[t1,t2,t3,t4,DW];

24 clear W

25 End

26 End

27 End

28 End

29 [M,in]=max(detW(:,5));

30 detW(in,:)

31 clear detW

32 End

33 End

34

35 function y1 = PSGG(t,del)

36 c2t=cos(2*t);

37 s2t=sin(2*t);

38 sl=sin(del);

39 cl=cos(del);

40 y1=[1;(c2t^2)+((s2t^2)*cl);c2t*s2t*(1-cl);s2t*sl];

41 End

42

43 function y2 = PSAG(t,del)

44 c2t=cos(2*t);

45 s2t=sin(2*t);

46 sl=sin(del);

47 cl=cos(del);

48 y2=[1,-((c2t^2)+((s2t^2)*cl)),-c2t*s2t*(1-cl),s2t*sl];

49 End


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Code 2:

The following code is for calculating the Mueller matrix for the sample (with blank correction) whose data was recorded in the form of images for a particular wavelength of incident light. The algorithm is as described briefly above in the Analysis section of this report. 1 clear; clc; 2 cd Blank; 3 hh=sum(sum(imread('hh.jpg'))); 4 hp=sum(sum(imread('hp.jpg'))); 5 hv=sum(sum(imread('hr.jpg'))); 6 hr=sum(sum(imread('hv.jpg'))); 7 ph=sum(sum(imread('ph.jpg'))); 8 pp=sum(sum(imread('pp.jpg'))); 9 pr=sum(sum(imread('pr.jpg'))); 10 pv=sum(sum(imread('pv.jpg'))); 11 rh=sum(sum(imread('rh.jpg'))); 12 rp=sum(sum(imread('rp.jpg'))); 13 rr=sum(sum(imread('rr.jpg'))); 14 rv=sum(sum(imread('rv.jpg'))); 15 vh=sum(sum(imread('vh.jpg'))); 16 vp=sum(sum(imread('vp.jpg'))); 17 vr=sum(sum(imread('vr.jpg'))); 18 vv=sum(sum(imread('vv.jpg'))); 19 cd ..; 20 21 PSG=[1,1,1,1; 22 1,-1,0,0; 23 0,0,1,0; 24 0,0,0,1]; 25 PSA=PSG'; 26 W=kron(PSA,PSG'); 27 Winv=inv(W); 28 29 Mi_b=[hh;hv;hp;hr;vh;vv;vp;vr;ph;pv;pp;pr;rh;rv;rp;rr]; 30 Ms_b=Winv*Mi_b; 31 Ms_b=Ms_b/Ms_b(1,1); 32 Ms_b=reshape(Ms_b,4,4)' 33 34 cd QWP_45; 35 hh=sum(sum(imread('hh.jpg'))); 36 hp=sum(sum(imread('hp.jpg'))); 37 hv=sum(sum(imread('hr.jpg'))); 38 hr=sum(sum(imread('hv.jpg'))); 39 ph=sum(sum(imread('ph.jpg'))); 40 pp=sum(sum(imread('pp.jpg'))); 41 pr=sum(sum(imread('pr.jpg'))); 42 pv=sum(sum(imread('pv.jpg'))); 43 rh=sum(sum(imread('rh.jpg'))); 44 rp=sum(sum(imread('rp.jpg'))); 45 rr=sum(sum(imread('rr.jpg'))); 46 rv=sum(sum(imread('rv.jpg'))); 47 vh=sum(sum(imread('vh.jpg'))); 48 vp=sum(sum(imread('vp.jpg'))); 49 vr=sum(sum(imread('vr.jpg'))); 50 vv=sum(sum(imread('vv.jpg'))); 51 cd ..; 52 53 Mi_QWP=[hh;hv;hp;hr;vh;vv;vp;vr;ph;pv;pp;pr;rh;rv;rp;rr];


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54 Ms_QWP=Winv*Mi_QWP; 55 Ms_QWP=Ms_QWP/Ms_QWP(1,1); 56 Ms_QWP=reshape(Ms_QWP,4,4)'; 57 Ms_QWP=Ms_QWP*inv(Ms_b); 58 Ms_QWP=Ms_QWP/Ms_QWP(1,1)

Code 3:

The following code was used to analyze the data obtained in the form of spectra for white light. It calculates the Mueller matrix and then uses a decomposition scheme to decompose it into depolarization, di-attenuation and linear retardance for each wavelength of light. 1 clear; clc; 2 dummy=load('3535.txt'); 3 mvec(1,:,:)=load('3535.txt'); 4 mvec(2,:,:)=load('3570.txt'); 5 mvec(3,:,:)=load('35105.txt'); 6 mvec(4,:,:)=load('35140.txt'); 7 mvec(5,:,:)=load('7035.txt'); 8 mvec(6,:,:)=load('7070.txt'); 9 mvec(7,:,:)=load('70105.txt'); 10 mvec(8,:,:)=load('70140.txt'); 11 mvec(9,:,:)=load('10535.txt'); 12 mvec(10,:,:)=load('10570.txt'); 13 mvec(11,:,:)=load('105105.txt'); 14 mvec(12,:,:)=load('105140.txt'); 15 mvec(13,:,:)=load('14035.txt'); 16 mvec(14,:,:)=load('14070.txt'); 17 mvec(15,:,:)=load('140105.txt'); 18 mvec(16,:,:)=load('140140.txt'); 19 20 mvecb(1,:,:)=load('b3535.txt'); 21 mvecb(2,:,:)=load('b3570.txt'); 22 mvecb(3,:,:)=load('b35105.txt'); 23 mvecb(4,:,:)=load('b35140.txt'); 24 mvecb(5,:,:)=load('b7035.txt'); 25 mvecb(6,:,:)=load('b7070.txt'); 26 mvecb(7,:,:)=load('b70105.txt'); 27 mvecb(8,:,:)=load('b70140.txt'); 28 mvecb(9,:,:)=load('b10535.txt'); 29 mvecb(10,:,:)=load('b10570.txt'); 30 mvecb(11,:,:)=load('b105105.txt'); 31 mvecb(12,:,:)=load('b105140.txt'); 32 mvecb(13,:,:)=load('b14035.txt'); 33 mvecb(14,:,:)=load('b14070.txt'); 34 mvecb(15,:,:)=load('b140105.txt'); 35 mvecb(16,:,:)=load('b140140.txt'); 36 37 wavel = dummy(:,1); 38 lim = find(mvec(1,:,1)>=470 & mvec(1,:,1)<=840); 39 a = lim(1,1); 40 b = lim(1,end); 41 for m2 = a:b, 42 muela=mueller_psa(mvec(:,m2,2)'); 43 muelb=mueller_psa(mvecb(:,m2,2)'); 44 muela = muela / muela(1,1); 45 muelb = muelb / muelb(1,1);


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46 muel = muela*inv(muelb); 47 muel = muel/muel(1,1); 48 flu_mat(m2,:,:)=muel;

49 [rvec,dvec,R(m2),D(m2),depol(m2),rotation(m2),lin_reta(m2)] = polardecomposition_paperm(muel);

50 end 51 totint=(flu_mat(a:b,1,1)); 52 subplot(4,4,1) 53 plot(flu_mat(a:b,1,1)) 54 subplot(4,4,2) 55 plot(flu_mat(a:b,1,2)) 56 subplot(4,4,3) 57 plot(flu_mat(a:b,1,3)) 58 subplot(4,4,4) 59 plot(flu_mat(a:b,1,4)) 60 subplot(4,4,5) 61 plot(flu_mat(a:b,2,1)) 62 subplot(4,4,6) 63 plot(flu_mat(a:b,2,2)) 64 subplot(4,4,7) 65 plot(flu_mat(a:b,2,3)) 66 subplot(4,4,8) 67 plot(flu_mat(a:b,2,4)) 68 subplot(4,4,9) 69 plot(flu_mat(a:b,3,1)) 70 subplot(4,4,10) 71 plot(flu_mat(a:b,3,2)) 72 subplot(4,4,11) 73 plot(flu_mat(a:b,3,3)) 74 subplot(4,4,12) 75 plot(flu_mat(a:b,3,4)) 76 subplot(4,4,13) 77 plot(flu_mat(a:b,4,1)) 78 subplot(4,4,14) 79 plot(flu_mat(a:b,4,2)) 80 subplot(4,4,15) 81 plot(flu_mat(a:b,4,3)) 82 subplot(4,4,16) 83 plot(flu_mat(a:b,4,4)) 84 figure 85 plot(depol(a:b)) 86 figure 87 plot(D(a:b)) 88 figure 89 plot(lin_reta(a:b)) 90 figure 91 plot(rotation(a:b)) 92 depol = depol'; 93 D = D'; 94 lin_reta = lin_reta'; 95 rotation = rotation'; 96 wavel = wavel; 97 allpol = [wavel(a:b) depol(a:b) D(a:b) lin_reta(a:b) rotation(a:b)]; 98 save 'polpara.dat' allpol -ascii;


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Included Functions:

Mueller_psa()

1 function[Muel]=mueller_psa(m_vec) 2 psg = [1.0000 1.0000 1.0000 1.0000 3 0.1169 0.5870 0.7498 0.0302 4 0.3213 -0.4924 0.4331 -0.1713 5 0.9397 0.6427 -0.5002 -0.9848]; 6 psa = [1.0000 -0.1169 -0.3213 0.9387 7 1.0000 -0.5870 0.4924 0.6427 8 1.0000 -0.7498 -0.4331 -0.5002 9 1.0000 -0.0302 0.1713 -0.9848]; 10 m_t=kron(psa, psg'); 11 m_trans=inv(m_t); 12 M=m_trans*m_vec'; 13 Muel=[M(1:4)';M(5:8)';M(9:12)';M(13:16)']'/M(1,1);

Polardecomposition_paperm()

1 function[rvec,dvec,R,D,depol,rotation,lin_reta]=polardecomposition_paperm(muel) 2 format long 3 I=[1 0 0; 4 0 1 0; 5 0 0 1]; 6 pvec=[muel(2,1),muel(3,1),muel(4,1)]*(1/muel(1,1)); 7 dvec=[muel(1,2),muel(1,3),muel(1,4)]*(1/muel(1,1)); 8 D=((muel(1,2)^2+muel(1,3)^2+muel(1,4)^2)^0.5)*(1/muel(1,1)); 9 m=(1/muel(1,1))*[muel(2,2),muel(2,3),muel(2,4); 10 muel(3,2),muel(3,3),muel(3,4); 11 muel(4,2),muel(4,3),muel(4,4)]; 12 D1=(1-D^2)^0.5; 13 if D==0 14 muel_0=muel/muel(1,1); 15 else 16 mD=D1*I+(1-D1)*dvec'*dvec/D^2; 17 MD=muel(1,1)*[1,dvec; 18 dvec',mD] 19 diattenuation = ((MD(1,2)^2+MD(1,3)^2+MD(1,4)^2)^0.5)*(1/MD(1,1)) 20 muel_0=muel*inv(MD); 21 end 22 m_1=[muel_0(2,2) muel_0(2,3) muel_0(2,4); 23 muel_0(3,2) muel_0(3,3) muel_0(3,4); 24 muel_0(4,2) muel_0(4,3) muel_0(4,4)]; 25 l_0=eig(m_1*m_1'); 26 m_0=inv(m_1*m_1'+((l_0(1)*l_0(2))^0.5+(l_0(2)*l_0(3))^0.5+(l_0(3)*l_0(1))^0.5)*I); 27 m_00=(l_0(1)^0.5+l_0(2)^0.5+l_0(3)^0.5)*m_1*m_1'+I*(l_0(1)*l_0(2)*l_0(3))^0.5; 28 if det(m_1)>=0 29 mdelta=m_0*m_00; 30 else 31 mdelta=-m_0*m_00; 32 end 33 [v,mdeltaf] = eig(mdelta); 34 depol=1-(abs(mdelta(1,1))+abs(mdelta(2,2))+abs(mdelta(3,3)))/3 35 depol1 =1-(abs(mdeltaf(1,1))+abs(mdeltaf(2,2))+abs(mdeltaf(3,3)))/3 36 nul=(pvec'-m*dvec')/D1^2; 37 Mdelta=[1 0 0 0;


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38 nul mdelta] 39 Mdeltaf =[1 0 0 0; 40 nul mdeltaf] 41 Mdinv=inv(Mdelta); 42 MR=Mdinv*muel_0 43 trmR=(MR(2,2)+MR(3,3)+MR(4,4))/2; 44 argu=trmR-1/2; 45 if abs(argu)>1 46 if argu>0 47 R=acos(1); 48 else 49 R=acos(-1); 50 end 51 else 52 R=acos(argu); 53 end 54 cssq_10=(MR(2,2)+MR(3,3))^2+(MR(3,2)-MR(2,3))^2; 55 tan_rot=(MR(3,2)-MR(2,3))/(abs(MR(2,2))+abs(MR(3,3))); 56 de=cssq_10^0.5-1; 57 if de>0.999999999999 58 de=1; 59 end 60 if de<-0.99999999999 61 de=-1; 62 end 63 lin_reta=acos(de); 64 rotation=0.5*atan(tan_rot); 65 if tan_rot<-0.000000001 66 rotation=rotation+pi; 67 end 68 if abs(MR(3,2)-MR(2,3))<=0.000000001 & abs(MR(2,2)+MR(3,3))>0.0000000001 69 rotation=0; 70 end 71 if abs(sin(R))<=0.000000001 72 a3=((1+cos(lin_reta))/2)^0.5; 73 a1=(MR(3,4)+MR(4,3))/(4*a3); 74 a2=(MR(4,2)+MR(2,4))/(4*a3); 75 else 76 D2=1/(2*sin(R)); 77 a1=D2*(MR(3,4)-MR(4,3)); 78 a2=D2*(MR(4,2)-MR(2,4)); 79 a3=D2*(MR(2,3)-MR(3,2)); 80 end 81 rvec=[1,a1,a2,a3]' 82 if abs(cos(R))>=0.9999999999 83 C1=MR(2,2)+MR(3,3); 84 C2=MR(2,3)-MR(3,2); 85 if abs(C1)<0.0000000001 86 MR=MR*[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1]; 87 lin_reta=pi; 88 end 89 end 90 orientation = 0.5*acos(MR(3,4)/sin(lin_reta)) 91 return


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Notes / Remarks:

Documents

Mueller Matrix Measurement Techniques for various Optical Elements & Biological Tissues