**6. Experimental results**

262 Soybean – Genetics and Novel Techniques for Yield Enhancement

the absorption and scattering coefficients for polarized light in soybean oil. System errors can be kept small by careful design of the system with achromatic elements, but can never be completely eliminated. Dichroism is a more serious problem when interpreting the results as solely due to birefringence. However, Mueller matrix polarimetry measurements have shown that the error due to dichroism is relatively small. [36] The variance in the computed Stokes vectors of transmitted light (excluding effect of birefringence) is due to multiple scattering, speckles, and shot noise (i.e., optimized system). At some depth, the detected signals are limited by shot noise. At shallower depths (i.e., before the shot noise limit) variance in the Stokes parameters is primarily due to the effects of multiple scattering and speckle. Multiple scattering scramble the polarization mainly in a random manner and this offers some means to distinguish it from birefringence. Thus, birefringence induced changes are relatively slow, and the Stokes parameters change according to the Mueller matrix of a linear retarder. [37] However, an optic axis that varies with depth will give changes in the polarization state that will be difficult to distinguish from the random manner of multiple scattering. More research is necessary on this complex problem. We use coherent light source and standard optical filters to minimize these errors for our Mueller

Fig. 1. Experimental setup for measurements of transmitted Mueller matrix elements. A He-Ne laser beam with an output power of 5 mW at a wavelength of 632.5 nm is used as the light source. The laser light is focused on polarizer P1 for obtaining linearly polarized light. The circularly polarized light is generated, by inserting a quarter mica retardation plates behind the linear polarizer. The output polarized light is focus to soybean oil and again

detector/CCD camera, which is controlled and operated with Lab software. The soybean oil

pass through linear polarizer, quarter wave plate and recorded on photodiode

matrix polarimeter.

is used as scattering phantom.

In this study the soybean oil is used as tissue like phantom. We recovered optical information by selectively detecting a transmitted component of the scattered photon flux that has its initial polarization state preserved. These photons transmitted through or reemitted from a multiply scattering medium by using relatively inexpensive Mueller matrix polarimeter provides the basis for several potential applications. The measurement technique is based upon an operational principle, which involves the modulation of a polarization state. The resulting modulated light signal is collected by the detector/CCD camera and is analyzed pixel by pixel to calculate individual intensity patterns, which correspond respectively to the 16 components of the scattering Mueller matrix. In brief, the two polarizers and quarter-wave plates inserted in the probing and analyzing beam paths, are generate a periodic signal, this signal carries information about the properties of the medium which induces the transformation of the polarization state of the modulated probing light. The experimental procedure requires collecting of 16 intensity images at various orientations of the polarizing components. The described procedure provides the possibility to calculate the scattering Mueller matrix for a given sample.

The Stokes parameter I of the system in Fig.2 represents the magnitude of the intensity of the scattered light. Thus, any abrupt change in the detected signal indicates strong discontinuity in the refractive index of the specimen. Along with I, other Stokes parameters, *Q, U* and *V* can be used to detect structural changes that are not simply detected from I. Other Stokes parameter images of *U* and *V* show supplement information that there is no apparent level of stress inside the scatterer. Therefore, by analyzing the corresponding series of the polarization patterns one can trace scattering events of different order. Specifically, the data analysis of Fig. 3, suggests several interesting observations regarding the general properties of the scattering Mueller matrix. First, the magnitude of the off-diagonal components of the scattering Mueller matrix is significantly smaller than the magnitude of the diagonal components.

In the present experiments, we are able to trace the polarization patterns and to verify that the existing magnitude distributions are preserved. The magnitude and the sign of most of the spatial extent of the matrix components for Fig. 3, 4 closely resemble the form of the Mueller matrix for scattering medium. The next important observation is that the experimental results clearly display several symmetry properties of certain matrix components for homogeneous scattering medium. The seven out of sixteen are independent and other can be calculated through symmetry relation. [39] By comparing the images of Fig. 3, 4, and 5, one can identify the unique features of the Mueller matrix for scattering medium. The axial symmetry of the system provides relation between all the Mueller matrix

Polarization Sensitive Optical Imaging

intensity for all the Stokes vectors.

in the fourth row, fourth column elements.

the retarder and the other nine dependent elements are:

and Characterization of Soybean Using Stokes-Mueller Matrix Model 265

Fig. 3. the Stokes vector I, Q, U, and V are shown, I present the total irradiant power profile, Q for horizontal polarized light, U for +450 polarized incident light and V the right circular polarized light. (a) Represents the 3-D images and (b) represent the direct transmitted

Where is the rotation angle of the transmission axis of the polarizer, is the phase shift of

13 12 4

<sup>31</sup> <sup>13</sup> <sup>12</sup> <sup>4</sup> *m mm* (,) (,) (, ) ,

<sup>32</sup> <sup>23</sup> <sup>23</sup> <sup>4</sup> *m mm* (,) (, ) (, ) ,

<sup>33</sup> <sup>22</sup> <sup>4</sup> *m m* (,) (, ) ,

<sup>34</sup> <sup>24</sup> <sup>4</sup> *m m* (,) (, ) ,

(, ) () () (,) (,) *m mm*

 ,

<sup>43</sup> <sup>34</sup> <sup>24</sup> <sup>4</sup> *m mm* (,) (, ) (, ) (9)

41 41 14

Fig. 6. displays the matrix array, which represents the detector reading specific to a linear polarizer lie in the first row, first column elements, those specific to a quarter wave plate lie

42 24

*m m*

(,) (, ) (,) (, )

 ,

21 12

*m m m m*

components and reduces the number of measurements, which reduces the observation time. These symmetry relations describe that the seven independent elements are, [40]

<sup>11</sup> <sup>12</sup> 14 22 <sup>23</sup> <sup>24</sup> <sup>44</sup> *m m mm m m m* ( , ), ( , ), ( ), ( , ), ( , ), ( , ), ( ) , (8)

Fig. 2. Transmitted Mueller matrix components (3D) corresponding to a 16 images of scattering medium. The images are taken with the experimental setup in Fig.1. The scale bar is adjusted so that red represent the maximum irradiance, yellow to middle one, green for minimum irradiance and blue means "no light" or component change by an order of magnitude. All displayed images are 3x3 cm.

components and reduces the number of measurements, which reduces the observation time.

<sup>11</sup> <sup>12</sup> 14 22 <sup>23</sup> <sup>24</sup> <sup>44</sup> *m m mm m m m* ( , ), ( , ), ( ), ( , ), ( , ), ( , ), ( ) , (8)

These symmetry relations describe that the seven independent elements are, [40]

Fig. 2. Transmitted Mueller matrix components (3D) corresponding to a 16 images of scattering medium. The images are taken with the experimental setup in Fig.1. The scale bar is adjusted so that red represent the maximum irradiance, yellow to middle one, green for minimum irradiance and blue means "no light" or component change by an order of

magnitude. All displayed images are 3x3 cm.

Fig. 3. the Stokes vector I, Q, U, and V are shown, I present the total irradiant power profile, Q for horizontal polarized light, U for +450 polarized incident light and V the right circular polarized light. (a) Represents the 3-D images and (b) represent the direct transmitted intensity for all the Stokes vectors.

Where is the rotation angle of the transmission axis of the polarizer, is the phase shift of the retarder and the other nine dependent elements are:

$$m\_{13}(\Phi,\mathfrak{q}) = m\_{12}(\Phi,\mathfrak{q} + \mathfrak{z}\mathfrak{z})$$

$$m\_{21}(\Phi,\mathfrak{q}) = m\_{12}(\Phi,\mathfrak{q})$$

$$m\_{31}(\Phi,\mathfrak{q}) = -m\_{13}(\Phi,\mathfrak{q}) = m\_{12}(\Phi,\mathfrak{q} - \mathfrak{z}\mathfrak{z})$$

$$m\_{32}(\Phi,\mathfrak{q}) = -m\_{23}(\Phi,\mathfrak{q}) = m\_{23}(\Phi,\mathfrak{q} \pm \mathfrak{z}\mathfrak{z})$$

$$m\_{33}(\Phi,\mathfrak{q}) = -m\_{22}(\Phi,\mathfrak{q} - \mathfrak{z}\mathfrak{z})$$

$$m\_{44}(\Phi,\mathfrak{q}) = m\_{24}(\Phi,\mathfrak{q} - \mathfrak{z}\mathfrak{z})$$

$$m\_{41}(\Phi,\mathfrak{q}) = m\_{41}(\Phi) = m\_{14}(\Phi)$$

$$m\_{42}(\Phi,\mathfrak{q}) = m\_{24}(\Phi,\mathfrak{q})$$

$$m\_{43}(\Phi,\mathfrak{q}) = -m\_{34}(\Phi,\mathfrak{q}) = m\_{24}(\Phi,\mathfrak{q} + \mathfrak{z})\tag{9}$$

Fig. 6. displays the matrix array, which represents the detector reading specific to a linear polarizer lie in the first row, first column elements, those specific to a quarter wave plate lie in the fourth row, fourth column elements.

Polarization Sensitive Optical Imaging

technique.

and Characterization of Soybean Using Stokes-Mueller Matrix Model 267

Fig. 5. Experimental transmitted polarization images of Mueller matrix components corresponding to a scattering medium of randomly distributed particles. The images corresponds the direct scatterer irradiance, measured through polarization discrimination

Mueller matrix polarimetric pattern analysis predicts interesting information about the medium. The ballistic, snake and diffuse photons reaching the camera contributes to the formation of direct image. The diffuse photons have suffered multiple scattering before exiting the scattering medium. Ballistic photons completely preserve the polarization properties of the irradiant light after passing through scattering medium. The snake photons recorded by the detector are partially polarized but to the smaller degree then the ballistic photons, because snake photons partially preserve polarization. The diffuse photon depolarized for thick sample. The collective image of these three scattered photon provides

Fig. 4. Transmitted Mueller matrix components (2D) corresponding to a scattering. These 2- D images are derived through 49 measurements of Mueller matrix polarimeter. The central spot presents the output transmitted irradiance by scatterer along with scattering pattern.

Making the above 49 measurements of polarized light from a scatterer will produce 16 matrix element pattern. Each one is an electric field dependent intensity measurement for a particular arrangement of input-output optics. These 16 curves contain all the information that can be learned from a scattering experiment. Choosing input-output optical combinations, different than described above, will produce a set of patterns drastically different in appearance but not fundamentally different in information content. When the 16 matrix elements are measured the data is ready for analysis. For certain perfect particles like spheres, fibers, and mixtures of perfect particles, the matrix elements can be exactly predicted. So the set of 16 measurements will stand as the signature of the scatterer as described by polarized scattered light.

### Polarization Sensitive Optical Imaging and Characterization of Soybean Using Stokes-Mueller Matrix Model 267

266 Soybean – Genetics and Novel Techniques for Yield Enhancement

Fig. 4. Transmitted Mueller matrix components (2D) corresponding to a scattering. These 2- D images are derived through 49 measurements of Mueller matrix polarimeter. The central spot presents the output transmitted irradiance by scatterer along with scattering pattern.

Making the above 49 measurements of polarized light from a scatterer will produce 16 matrix element pattern. Each one is an electric field dependent intensity measurement for a particular arrangement of input-output optics. These 16 curves contain all the information that can be learned from a scattering experiment. Choosing input-output optical combinations, different than described above, will produce a set of patterns drastically different in appearance but not fundamentally different in information content. When the 16 matrix elements are measured the data is ready for analysis. For certain perfect particles like spheres, fibers, and mixtures of perfect particles, the matrix elements can be exactly predicted. So the set of 16 measurements will stand as the signature of the scatterer as

described by polarized scattered light.

Fig. 5. Experimental transmitted polarization images of Mueller matrix components corresponding to a scattering medium of randomly distributed particles. The images corresponds the direct scatterer irradiance, measured through polarization discrimination technique.

Mueller matrix polarimetric pattern analysis predicts interesting information about the medium. The ballistic, snake and diffuse photons reaching the camera contributes to the formation of direct image. The diffuse photons have suffered multiple scattering before exiting the scattering medium. Ballistic photons completely preserve the polarization properties of the irradiant light after passing through scattering medium. The snake photons recorded by the detector are partially polarized but to the smaller degree then the ballistic photons, because snake photons partially preserve polarization. The diffuse photon depolarized for thick sample. The collective image of these three scattered photon provides

Polarization Sensitive Optical Imaging

oil.

effects.

**7. Conclusion**

and Characterization of Soybean Using Stokes-Mueller Matrix Model 269

incident light is of 450 polarized. From analysis of the elements of this group the normality and abnormality of the medium can easily be defined. In Fig.7 the depolarization of linearly polarized light through scattered is represented and it increases with the depth of soybean

The elements in the middle of this group m22, m23, m32, and m33 can obtain through 450 linear polarization. If some properties of the scattered cannot be obtain through the elements of other group they can be characterize through it. The elements m23 and m32 decline for dense scattering medium and the scattering angle for these elements is very small. The reduction of these elements directly related to the variation in structure of the scattered. The last row and last column of this matrix set consist on circular polarization pattern. This group exhibits the depolarization properties of the medium. The depolarization is faster in dense as in case of circular one. The light is equally right and left-hand polarized and the effect is strongest in the center, near the laser entry point. Here the scattered light has undergone only a few scattering events and the polarization effects are strongest. With increasing distance from the point of light incident, the number of scattering events increases and eventually the polarization information is lost, the value of the m44 approaches zero. The majority of the elements of this group shows decline in the magnitude for dense medium and predict that the scattering cross-section of medium is small for this wavelength

The next important observation is that the experimental results in Mueller matrix array display several symmetry properties and relations among them. This can be seen in Fig. 3-6 and these derived symmetry relations hold, of course, if the scattering medium contains one kind of randomly distributed asymmetrical particles or optically active. Some elements of the matrix have same behavior and other one are of same shape but rotated through 900 as indicated in equation 8, 9 of this paper. All sixteen Mueller matrix components together provide a "finger print" of the scattering medium under investigation. As just shown, looking at the entire Mueller matrix often enables one to distinguish qualitatively between two media. Lot of information about particle size, refractive index, particle shape etc. has to be found in the Mueller matrix by careful analysis of the matrix elements. However, further information may be gained, for example, by measuring the diffuse backscattering and backreflectance at different incident and observation angles, or time-dependent polarization

In this study we presented a polarization discrimination scattering experiment and have taken care to establish unambiguously the coordinate systems involved, the redundancy of certain measurements and the importance of particular orientations of optical element combinations. We believe that these concepts are important for understanding and fully appreciating optical polarization and that this approach is attractive because it discusses the

We describe the Mueller matrix polarization discrimination (MMPD) technique for characterization of highly scattering media(soybean oil) through laser beam. In our experiments, the scattering regime was adjusted to be at the incipient transition between single and multiple scattering. From an experimental standpoint the scattering is most challenging and on the other hand, it is rich in information content because the low-order scattering events are responsible for non-trivial polarization features. Our results

and the circular polarization preservation of light is weaker.

inexpensive and non invasive procedures that are equally valid.

useful information and characterizes the scattering medium in term of size shape under Raleigh and Mie scattering theory.

The diagonal elements of Mueller matrix consist on linear and circular polarization pattern. The Mueller matrix m11 describe the properties of the total irradiance of light source and provides less information comparing to other elements of the matrix array, but all other elements are normalized through it. The m22 composed of linear horizontal and vertical polarization state. The liner polarization preserve through longer distance in the scattered as compared to circular light The concentration , size and shape of the particles in scattering medium can be predicted through careful analysis of this element alomgwith other linear elements of the matrix. If the value of this element is zero or below, then the medium obey Raleigh theory and the size of the particle is small as compared to the irradiated wavelength. If its value is greater than zero than it can be explain through Mie theory and particles of the scattering medium are larger in size. m33 depend on 450 linear polarization state and describes the properties almost close to m22 element. The last element m44 of the Mueller matrix compose on circular polarization. If the size of scatterer is larger, then the magnitude of this element will be in negative otherwise greater than or equal to zero. The difference between normal and malignant biological tissues can be characterized through this element. But for larger scattered concentration, it is less informatics because the data is taken through diffuse photon and the preservation of circular polarization is not dominant in this medium. If this element is measured through the ballistic and snake photons contribution then it reviles a significant role in characterization of biological tissues. In our case the experimental data shows decline in the diagonal elements from top to bottom that conforms the preservation of linear polarization in diffuse medium for longer distance compared to circular one. As m44 is greater than zero, which predicts that the size of scattered is larger than the irradiating wavelength. The size of the particle can be numerically calculated through Mie scattering theory. [41]



Fig. 6. Mueller matrix data for transmitted polarized laser beam (a) no sample (air) and (b) for scattering turbid sample.

First row and first column of the Mueller matrix except m14 and m41 describe the linear polarization pattern. Each and every element of this group is very informatics, and describes the structure of the dense diffuse scattered. From Fig. 3, 4, and 5 we see that the intensity contrast reduces from right to left and top to bottom of the Mueller intensity matrices except m13, which tells about the enriched optical activity and highly birefringence of the sample. The higher value of element m13 is due to randomization of the sample molecules, when the

useful information and characterizes the scattering medium in term of size shape under

The diagonal elements of Mueller matrix consist on linear and circular polarization pattern. The Mueller matrix m11 describe the properties of the total irradiance of light source and provides less information comparing to other elements of the matrix array, but all other elements are normalized through it. The m22 composed of linear horizontal and vertical polarization state. The liner polarization preserve through longer distance in the scattered as compared to circular light The concentration , size and shape of the particles in scattering medium can be predicted through careful analysis of this element alomgwith other linear elements of the matrix. If the value of this element is zero or below, then the medium obey Raleigh theory and the size of the particle is small as compared to the irradiated wavelength. If its value is greater than zero than it can be explain through Mie theory and particles of the scattering medium are larger in size. m33 depend on 450 linear polarization state and describes the properties almost close to m22 element. The last element m44 of the Mueller matrix compose on circular polarization. If the size of scatterer is larger, then the magnitude of this element will be in negative otherwise greater than or equal to zero. The difference between normal and malignant biological tissues can be characterized through this element. But for larger scattered concentration, it is less informatics because the data is taken through diffuse photon and the preservation of circular polarization is not dominant in this medium. If this element is measured through the ballistic and snake photons contribution then it reviles a significant role in characterization of biological tissues. In our case the experimental data shows decline in the diagonal elements from top to bottom that conforms the preservation of linear polarization in diffuse medium for longer distance compared to circular one. As m44 is greater than zero, which predicts that the size of scattered is larger than the irradiating wavelength. The size of the particle can be

> 0.986 0.007 0.004 0.0003 0.007 1.003 -0.007 0.009 0.008 -0.007 0.992 -0.003 0.003 -0.006 -0.007 0.989 a

0.910 0.657 -0.314 0.097 0.739 0.732 0.793 0.083 0.435 0.243 0.620 -0.213 0.133 0.421 0.136 0.751 b Fig. 6. Mueller matrix data for transmitted polarized laser beam (a) no sample (air) and (b)

First row and first column of the Mueller matrix except m14 and m41 describe the linear polarization pattern. Each and every element of this group is very informatics, and describes the structure of the dense diffuse scattered. From Fig. 3, 4, and 5 we see that the intensity contrast reduces from right to left and top to bottom of the Mueller intensity matrices except m13, which tells about the enriched optical activity and highly birefringence of the sample. The higher value of element m13 is due to randomization of the sample molecules, when the

Raleigh and Mie scattering theory.

numerically calculated through Mie scattering theory. [41]

for scattering turbid sample.

incident light is of 450 polarized. From analysis of the elements of this group the normality and abnormality of the medium can easily be defined. In Fig.7 the depolarization of linearly polarized light through scattered is represented and it increases with the depth of soybean oil.

The elements in the middle of this group m22, m23, m32, and m33 can obtain through 450 linear polarization. If some properties of the scattered cannot be obtain through the elements of other group they can be characterize through it. The elements m23 and m32 decline for dense scattering medium and the scattering angle for these elements is very small. The reduction of these elements directly related to the variation in structure of the scattered.

The last row and last column of this matrix set consist on circular polarization pattern. This group exhibits the depolarization properties of the medium. The depolarization is faster in dense as in case of circular one. The light is equally right and left-hand polarized and the effect is strongest in the center, near the laser entry point. Here the scattered light has undergone only a few scattering events and the polarization effects are strongest. With increasing distance from the point of light incident, the number of scattering events increases and eventually the polarization information is lost, the value of the m44 approaches zero. The majority of the elements of this group shows decline in the magnitude for dense medium and predict that the scattering cross-section of medium is small for this wavelength and the circular polarization preservation of light is weaker.

The next important observation is that the experimental results in Mueller matrix array display several symmetry properties and relations among them. This can be seen in Fig. 3-6 and these derived symmetry relations hold, of course, if the scattering medium contains one kind of randomly distributed asymmetrical particles or optically active. Some elements of the matrix have same behavior and other one are of same shape but rotated through 900 as indicated in equation 8, 9 of this paper. All sixteen Mueller matrix components together provide a "finger print" of the scattering medium under investigation. As just shown, looking at the entire Mueller matrix often enables one to distinguish qualitatively between two media. Lot of information about particle size, refractive index, particle shape etc. has to be found in the Mueller matrix by careful analysis of the matrix elements. However, further information may be gained, for example, by measuring the diffuse backscattering and backreflectance at different incident and observation angles, or time-dependent polarization effects.

In this study we presented a polarization discrimination scattering experiment and have taken care to establish unambiguously the coordinate systems involved, the redundancy of certain measurements and the importance of particular orientations of optical element combinations. We believe that these concepts are important for understanding and fully appreciating optical polarization and that this approach is attractive because it discusses the inexpensive and non invasive procedures that are equally valid.
