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Mueller matrix approach for determination of
optical rotation in chiral turbid media in
backscattering geometry
S. Manhas 1, M. K. Swami 2, P. Buddhiwant 1, N. Ghosh 1*, P. K. Gupta 1 and K. Singh 2
1 Biomedical Applications Section,
Center for Advanced Technology, Indore, India –452013.
2 Department of Physics, Indian Institute of Technology, Delhi, India
neem@cat.ernet.in
Abstract: For in vivo determination of optically active (chiral) substances
in turbid media, like for example glucose in human tissue, the
backscattering geometry is particularly convenient. However, recent
polarimetric measurements performed in the backscattering geometry have
shown that, in this geometry, the relatively small rotation of the polarization
vector arising due to the optical activity of the medium is totally swamped
by the much larger changes in the orientation angle of the polarization
vector due to scattering. We show that the change in the orientation angle of
the polarization vector arises due to the combined effect of linear
diattenuation and linear retardance of light scattered at large angles and can
be decoupled from the pure optical rotation component using polar
decomposition of Mueller matrix. For this purpose, the method developed
earlier for polar decomposition of Mueller matrix was extended to
incorporate optical rotation in the medium. The validity of this approach for
accurate determination of the degree of optical rotation using the Mueller
matrix measured from the medium in both forward and backscattering
geometry was tested by conducting studies on chiral turbid samples
prepared using known concentration of scatterers and glucose molecules.
© 2006 Optical Society of America
OCIS Codes: (170.0170) Medical optics and biotechnology, (170.4580) Optical diagnosis for
medicine, (120.5410) Polarimetry, (290. 4210) Multiple scattering, (110.7050) Turbid media
References
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1. Introduction
Studies on polarization properties of scattered light from a turbid medium like human tissue
have received considerable attention because the depolarization of scattered light can be used
as an effective tool to discriminate against multiply scattered light and thus can facilitate
imaging through tissue [1-3]. Further, measurement of polarization properties of scattered
light, such as depolarization, retardance and diattenuation, can facilitate quantification of
useful physiological and morphological parameters of tissue [4,5]. For example, since the rate
of depolarization of incident linearly and circularly polarized light depends on the
morphological parameters like the density, size, distribution, shape and refractive index of
scatterers present in the medium [6-11], this information may be used for quantitative tissue
diagnosis. Similarly, it is well known that the presence of asymmetric chiral molecules like
glucose leads to rotation of the plane of linearly polarized light about the axis of propagation.
Motivated by the fact that measurement of optical rotation might turn out to be an attractive
approach for non-invasive monitoring of glucose level in human tissue, several studies have
addressed measurement of polarization properties of light scattered from chiral turbid medium
[12 -18]. A major problem in determining the concentration of chiral substances in a turbid
medium, is the fact that unlike dilute solutions, the incident polarized light gets strongly
depolarized due to multiple scattering and only a small fraction of the light coming out from
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the medium retains its initial state of polarization [15,17]. Techniques such as polarization
modulation and synchronous detection methods have therefore been developed to extract the
surviving polarization fraction of polarized light scattered from chiral turbid medium and
analyze this for quantification of the degree of optical rotation [19 -21]. However, recent
studies [22] based on this approach have shown that in the backscattering geometry, the
relatively small rotation of the polarization vector arising due to the optical activity of the
medium is totally swamped by the much larger changes in the orientation angle of the
polarization vector due to scattering. In contrast, this effect was observed to be minimal in the
forward scattering direction. Based on this observation, it was proposed that in order to avoid
errors in concentration determination of chiral molecules in turbid samples, the scattered light
in the forward direction should be used [22].
In this communication, we show that the change in the orientation angle of the polarization
vector of light propagating through a chiral turbid medium arises not only due to the circular
retardance property of the medium (introduced by the presence of chiral molecules) but also is
contributed by linear diattenuation (different attenuation of two orthogonal linear polarization)
and linear retardance (dephasing of two orthogonal linear polarization) of light scattered at
large angles. We present a method based on polar-decomposition of Mueller matrix that can
decouple the contribution of linear diattenuation and linear retardance and extract the
contribution of optical rotation purely due to the circular retardance property of the medium
using the Mueller matrix measured from the medium in both forward and backscattering
geometry. This method is an extension of the previously developed method for polar
decomposition of Mueller matrix [23] and incorporates the optical rotation matrix in the
decomposition process. The validity of this approach was tested by conducting studies on
chiral turbid samples prepared using known concentration of scatterers and glucose
molecules.
2. Theory
2.1 Polar decomposition process for separating out linear retardance and circular
retardance:
The process for Polar decomposition of experimentally measured Mueller matrix into Mueller
matrices of a diattenuator MD (component that causes different amplitude changes for its
orthogonal eigen states), a retarder MR (component that causes dephasing of two eigen states)
and a depolarizer MΔ(component that causes depolarization) has been described in details by
Lu and Chipman [23]. Here, we discuss this in brief.
Mueller matrix of any sample can be decomposed in the form
RD
M=M M M
(1)
The diattenuation matrix MD is defined as
T
D
1
D
=Dm
D
M (2)
where
22T
D
ˆ
ˆ
m
= 1-D I+(1- 1-D )DD (3)
I is 3Χ3 identity matrix,
D
is diattenuation vector and
ˆ
D
is its unit vector and are defined as
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[]
T
01 02 03
00
1
D= m m m
m ,
D
ˆ
D=
D
(4)
ij
m
is matrix element of ith row and jth column of Mueller matrix M.
The product of the retardance and the depolarizing matrices can then be obtained as,
′
-1
RD
M M =M =MM
(5)
The matrices MΔ, MR and M/ have the following form
10
Pm
=M
R
1
0
0m
=
R
M (6)
'
1
0
P
m
′=
M
Here,
2
P
-m
D
P=
1
-D , Polarizance vector
[]
T
10 20 30
00
1
P= m m m
m, [
m
is the sub-
matrix of
M
].
m
′is the sub matrix of M/ and can be written as
R
m
=m m
′ (7)
The sub matrix
m
can be computed using
m
′ as
T-1T
12 23 31 1 2 3 123
m=±[m'(m')+( + + )I] ×[( + + )m'(m') +
I]
(8)
Where
12
,
and
3
are eigen values of
()
T
mm
′′
. The sign “+” or “-” in the right side of
Eq. (8) is determined by the sign of determinant of
m
′.
Using Eqs. (7) and (8), the expression for the sub matrix
R
m
of the retardance matrix
MR can be obtained as
-1
R
m=mm
′ (9)
Using Eqs. (6) and (9), the total retardance matrix MR can be computed.
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The decomposed retardance matrix MR can further be used to determine the value of optical
rotation. The total retardance (R) and the elements of the retardance vector
[
]
123
R=1, r, r, r
can be written as
3
i ijk R jk
j,k=1
1
r= (m )
2sinR (10)
here
ijk
is Levi-civita permutation symbol.
-1 tr( )
R
=cos -1
2R
M (11)
The retardance matrix MR can be written as a combination of linear retardance matrix and
optical rotation matrix
R
M
=
10 0 0
22
0cos2
+sin 2 cos sin 2 cos 2 (1- cos )-sin 2 sin
22
0sin 2
cos 2 (1 -cos )sin2 + co s 2 cos cos 2 sin
0sin 2 sin -cos 2 sin cos
×
10 00
0cos2
sin2 0
0-sin2
cos2 0
00 01
(12)
Here δ, θ and ψ are linear retardance, orientation of fast axis of the linear retarder and optical
rotation respectively.
Using Eqs. (11) and (12), the total retardance (R) can be expressed as
-1 2 2
R = cos 2cos ( )cos ( )-1
2 (13)
Similarly, the square of the fourth element of the retardance vector (r32) can be expressed as
()
22
2
322
sin cos 2
r=
1-cos cos
2
(14)
It can be seen from Eqs. (13) and (14) that R and r32 are function of linear retardance (δ) and
optical rotation (ψ) and are independent of the orientation of the fast axis of the linear retarder
(θ). Therefore, using Eqs. (13) and (14), linear retardance and optical rotation (ψ) can be the
expressed as
()
{
}
-1 2 2 2
3
= 2cos r 1- cos (R/2) +cos (R/2
)
(15)
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-1
c
os(R/2)
=cos cos( /2) (16)
The orientation of the fast axis with respect to the horizontal axis (θ) can also be determined
using the relationship [23]
-1
3
2
r
1
=tan
2
r (17)
It thus, follows that following the process described above, one can obtain values for δ and ψ
separately using the total retardance matrix MR. The total polar decomposition process is
illustrated in the flow chart shown in Fig. 1.
Experimental Mueller
matrix M
Diattenuation
vector and
matrix MD
M’=MMD-1
Polarizance
vector and
depolarization
matrix MΔ
MR=MΔ-1M’
Total retardance
vector and total
retardance R
Linear retardance and
circular retardance
Experimental Mueller
matrix M
Diattenuation
vector and
matrix MD
M’=MMD-1
Polarizance
vector and
depolarization
matrix MΔ
MR=MΔ-1M’
Total retardance
vector and total
retardance R
Linear retardance and
circular retardance
It should be mentioned here that the decomposition of Mueller matrix also depends upon the
order in which the diattenuator, depolarizer and retarder matrices are multiplied. Based on the
order of these matrices, six possible decompositions can be performed [24]. These six possible
decomposition processes can again be classified in to two groups depending upon the order in
which the diattenuator and the depolarizer matrices are multiplied. The group in which the
diattenuator matrix comes ahead of the depolarization matrix always leads to a physically
realizable Mueller matrix [24]. The decomposition process discussed in this section is
therefore based on this convention.
2.2 Decomposition of single scattering Mueller matrix:
In order to demonstrate the applicability of the procedure outlined in Section 2.1, for
separation of contribution of linear retardance from circular retardance from the total
Fig. 1. Flow chart for polar decomposition of an experimentally obtained Mueller Matrix
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retardance matrix (MR), we first consider the case of scattering from spherical scatterers with
known size and refractive index.
The scattering matrix S (Θ) for a spherical scatterer with diameter 2.0 μm at a wavelength
λ= 632.8 nm was computed using Mie theory [25]. The refractive index of the scatterer (n)
and that of the surrounding medium (n medium) was taken to be n = 1.59 and n medium = 1.33
respectively. The variation in the value for the orientation angle of the polarization vector (γ)
as a function of scattering angle Θ of an incident beam polarized at an angle γ0 = 450 after
single scattering from the scatterer is shown in Fig. 2(a). Here, γ is defined as
γ = 0.5 × tan –1 (U / Q) (18)
Q and U are the second and the third element of the Stoke’s vector [25].
In agreement with the results of Cote et al [22], γ is seen to vary considerably as a function
of Θ, with its value being significantly larger in the backscattering direction (Θ >1.57 radian)
as compared to that in the forward scattering direction (Θ < 0.523 radian). The values for the
linear diattenuation (d) linear retardance (δ) and optical rotation (ψ) were calculated following
the process described in Section 2.1. The variations of d, δ and ψ as a function of Θ are
displayed in Fig. 2(b). The values for d and δ are observed to vary significantly as a function
Θ. However, as can be seen, there is no optical rotation component (ψ = 0) in single scattering
from this achiral spherical scatterer. The observed linear diattenuation and linear retardance
originate from the difference in amplitude and phase between the scattered light polarized
parallel [0.5 × {S11 (Θ)+S12 (Θ)}] and perpendicular [0.5 × {S11 (Θ)-S12 (Θ)}] to the scattering
plane. The change in orientation angle of linear polarization vector of light scattered from the
achiral scatterer is therefore a combined effect of linear diattenuation and linear retardance
and there is no contribution of circular retardance to it. In order to unambiguously detect
optical rotation introduced by the presence of chiral substances in a turbid medium (i.e., due
to circular retardance property of the medium), it would be necessary to filter out the
contribution of the additional rotation of polarization vector arising due to the combined effect
of linear diattenuation and linear retardance. To investigate the efficacy of polar
decomposition of Mueller matrix to accomplish this objective, we constructed Mueller matrix
Fig. 2. (a) The variation of the orientation angle (
γ
) as a function of scattering angle Θ. (b)
The values for linear retardance δ (solid line), diattenuation d (dotted line) and optical
rotation ψ (dash dotted line) obtained from polar decomposition of single scattering Mueller
matrix (
δ
and
ψ
are in radian).
0 1 2 3
0
1
2
3
4
Θ
γ
00.5 11.5 22.5 3
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Θ
δ
ψ
d
(a)
(b)
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for a chiral medium having scatterer with the same scattering parameters. The rotatory power
of the chiral medium was taken to be 0.01745 radian / cm.
The depolarization (represented by the variation in degree of polarization P), diattenuation (d),
linear retardance (δ) and optical rotation (ψ) maps obtained using the polar decomposition
process are shown in Fig. 3. As expected, the pure optical rotation (ψ) corresponding to the
chirality of the scatterer can be seen to be recovered by removing the contribution of the
scattering induced linear diattenuation and linear retardance. The value for ψ is however,
observed to be zero in the exact backscattering direction and increases gradually as one goes
away from the point of incidence. This is expected because due to exact backscattering, the
light traverses the same path twice but in opposite direction that leads to cancellation of the
net optical rotation.
The theoretical results presented in this section show that the change in the orientation
angle of the polarization vector of light propagating through a chiral turbid medium arises not
(a) (b)
(c) (d)
Fig. 3. (a) Depolarization (represented by degree of polarization P) (b) diattenuation (d), (c)
linear retardance (δ) and (d) Optical rotation (ψ) map of the chiral spherical scatterer obtained
from polar decomposition of M
ueller matrix (X and Y are in cm,
δ
and
ψ
are in radian).
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only due to the circular retardance property of the medium but also due to linear diattenuation
and linear retardance of light scattered at large angles. Polar-decomposition of Mueller matrix
can be used to decouple the contribution of linear diattenuation and linear retardance and
extract the contribution of optical rotation purely due to the circular retardance property of the
medium In order to test the experimental validity of this approach, Mueller matrix
measurements were performed on chiral turbid samples prepared using known concentration
of scatterers and glucose molecules. The details of these studies are described in the following
section.
3. Experimental methods
The 632.8 nm output from a He-Ne laser (Suresh Indu Lasers, India) was used as excitation
source. A set of linear polarizer (P1) and a quarter wave plate (Q1) was used to illuminate the
sample with desired polarization states. Another set of linear polarizer (P2) and quarter wave
plate (Q2) was used to analyze the polarization states of the light scattered from the sample.
The scattered light emerging from the sample was collected with an f/3 lens and after passing
through subsequent polarizing optics was imaged onto a CCD detector (ST6, SBIG, USA).
For the measurements in the backscattering geometry, the analyzers and the collection optics
were kept at an angle of ~ 30o away from the exact backscattering direction. While performing
measurements in the forward scattering direction, the analyzers and the collection optics were
kept at an angle of 0o with respect to the direction of the ballistic beam. The collection angle
was ~ 20o.
To construct the 4 × 4-intensity measurement matrix (Mi) we generated the required four
incident polarization states (linear polarization at angles of 0o, 45o, 90o 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 0o, 45o,
90o from the horizontal and right circular polarization) [26,27,28]. From this matrix the
Muller matrix for the sample (Ms) was constructed using the relation [26]
Mi = PSA MS PSG
Where, PSA and PSG are the polarization state analyzer matrix and the polarization state
generator matrix respectively [26].
This kind of construction allows calibration for non-ideal components since one can
replace the ideal stokes vector in each column of PSG or each row of PSA by the measured
Stoke’s vectors that may deviate from ideal value. Using this approach, calibration for errors
related with diattenuation and retardance of polarizer and quarter wave plate were
incorporated.
The experimental set-up was calibrated by measuring Mueller matrix from known optical
components such as mirror, linear polarizer and quarter wave plate. After calibration of the
set-up, Mueller matrix measurements were performed on both achiral and chiral turbid
samples prepared using known concentration of scatterers alone or scatterers with glucose
molecules. The samples were kept inside a quartz cuvette of path length of 10 mm while
taking measurements. The achiral turbid samples were prepared using aqueous suspension of
polystyrene microspheres (Bangs Lab., USA) with mean diameters of 2.0 µm (anisotropy
parameter g = 0.91 at 632.8 nm). The values of scattering coefficient (μs) of the samples with
known concentration of microspheres were calculated using Mie theory. In order to prepare
the chiral turbid samples, known concentration of D (dextro-rotatory) glucose solution was
added to the microspheres suspension. The values for μs of the individual chiral turbid
samples were measured separately in a spectrophotometer (GBC, Cintra 20, Australia).
4. Results and discussion
In order to test the experimental validity of the polar decomposition based approach to
decouple other polarization properties (depolarization, diattenuation and linear retardance)
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from the circular retardance property for accurate determination of optical rotation,
experiments were performed on samples with known retardance and optical rotation values.
For this purpose, Mueller matrices were recorded in transmission mode, separately from a
linear retarder (linear retardance δ at 632.8nm ∼ 1.27) and from glucose solution of known
concentration (5 M). Muller matrix was also recorded from the linear retarder with the glucose
solution placed in front of it. Table 1(a) shows the recorded Mueller matrix (M) and the
decomposed diattenuation (MD) and retardance (MR) matrices for the linear retarder. The
value for δ obtained using the decomposed retardance matrix MR was found to be 1.23.
The optical rotation from the pure glucose solution kept in a 10 mm cuvette was determined to
be ψ = 0.064 radian (Mueller matrix not shown here). In Table 1b, we show the recorded
Mueller matrix and the decomposed matrices MD and MR for the combination of the linear
retarder and glucose solution. Using the matrix MR and following the procedure described in
section 2.1, the values for the linear retardance and optical rotation were obtained to be δ =
1.16 and ψ = 0.068 radian respectively. These values are reasonably close to the
corresponding values for δ and ψ of the linear retarder and pure glucose solution obtained
from separate measurements (δ = 1.23, ψ = 0.064 radian).
M
1 -0.06 -0.031 0
-
0.06 0.856 0.284 -0.43
-
0.031 0.266 0.475 0.837
-
0.001 0.442 -0.831 0.331
MR
10 0 0
0 0.857 0.283 -0.431
0 0.265 0.475 0.839
0 0.443 -0.833 0.332
MD
1 -0.06 -0.031 0
-0.06 1 0.001 0
-0.031 0.001 1 0
0000.998
Table 1. (a) Measured Mueller matrix and the decomposed components for the linear
retarder.
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Experiments were conducted on achiral turbid samples prepared using aqueous suspension of
2.0 μm diameter polystyrene microspheres (μs = 1.5 mm-1, g = 0.91) in both forward and
backscattering geometry. The polar decomposition based approach could successfully
decouple the scattering induced linear diattenuation and linear retardance from the measured
Mueller matrix for the achiral turbid sample and yield a value of ψ = 0.003 radian (data not
shown here). The measured Mueller matrix (M) and the decomposed diattenuation (MD),
depolarization (MΔ) and retardance (MR) matrices of a chiral turbid sample (prepared using
aqueous suspension of 2.0 μm diameter polystyrene microspheres and known concentration of
glucose) for forward and backscattering geometry are displayed in Table 2(a) and 2(b)
respectively. The concentration of glucose in the sample was 5M. The value for μs of this
chiral turbid sample was determined to be μs = 0.6 mm-1. The values for the parameters, linear
diattenuation (d), linear retardance (δ), optical rotation (ψ) and degree of linear polarization
[defined here as PL = (Q2 + U2) 1/2 / I)] for this sample are summarized in Table 3.
M
1
0.026 0.044 -0.039
0
.029 0.962 -0.144 -0.047
0
.002 0.126 0.975 0.026
-
0.039 0.019 0.115 0.936
MΔ
1000
0
.008 0.976 -0.01 -0.021
-
0.023 -0.01 0.982 0.073
-
0.009 -0.022 0.073 0.941
MR
10 0 0
0
0.99 -0.138 -0.027
0
0.136 0.99 -0.048
0
0.033 0.044 0.998
MD
1
0.026 0.044 -0.039
0
.026 0.998 0.001 -0.001
0
.044 0.001 0.999 -0.001
-
0.039 -0.001 -0.001 0.999
M
1
0.09 -0.093 -0.2
0
.155 0.874 0.119 -0.435
-
0.179 0.303 0.487 0.804
0
.029 0.310 -0.837 0.383
MR
10 0 0
0
0.892 0.130 -0.432
0
0.32 0.492 0.809
0
0.318 -0.861 0.398
MD
1
0.09 -0.09 -0.2
0
.09 0.975 -0.004 -0.009
-
0.093 -0.004 0.976 0.009
-
0.2 -0.009 0.009 0.992
Table 1. (b) Measured Mueller matrix and the decomposed components for the combination
of the linear retarder and glucose (5M) solution.
Table 2. (a) Measured Mueller matrix and the decomposed components for the chiral turbid sample (μs = 0.6
mm-1, glucose 5M) in forward scattering geometry
#9466 - $15.00 USD Received 10 November 2005; revised 22 December 2005; accepted 3 January 2006
(C) 2006 OSA 9 January 2006 / Vol. 14, No. 1 / OPTICS EXPRESS 200
Parameters Forward scattering Backscattering
d 0.064 0.135
δ 0.055 2.82
ψ 0.069 0.073
PL 0.976 0.636
As expected, the values for d and δ are observed to be larger in the backscattering
geometry as compared to that obtained in the forward scattering geometry. The values for
optical rotation ψ determined using the polar decomposition of Mueller matrix for both
forward and backscattering geometry are slightly larger than the corresponding value for clear
solution of glucose (ψ = 0.064 radian) having the same concentration. This arises due to the
increased path length of light in the medium due to multiple scattering [14,18,22]. It is
interesting to note that the value for ψ is marginally higher in the backscattering geometry as
compared to that obtained for forward scattering geometry. A plausible reason for this is the
fact that for this forward scattering sample (g = 0.91), the incident photons have to suffer
significantly larger number of forward scattering events and hence have to traverse a longer
path in the medium to come out from the medium through the backward direction as
compared to that required for emerging through the forward direction. This would be the case
if there is significant contribution of multiply scattered photons to the detected light in the
backscattering geometry. Unlike the case for exact backscattering (scattering angle Θ = 3.14
radian), in this situation, the net optical rotation is not expected to cancel out, would rather
add up to yield a larger value of ψ in the backscattering geometry. The fact that the value for
PL obtained from the pure depolarizing matrix is considerably lower in the backscattering
direction as compared to that obtained for the forward scattering direction, suggests that this
M
1
-0.115 -0.066 0.023
-
0.111 0.759 -0.061 -0.001
-
0.018 0.151 -0.435 -0.139
-
0.046 0.006 0.128 -0.334
MΔ
1000
0
.028 0.756 -0.072 0.021
-
0.062 -0.072 0.488 -0.014
-
0.03 0.021 -0.014 0.358
−
MR
1000
0 0.985 -0.184 0
0
-0.175 -0.924 -0.312
0
0.057 0.306 -0.95
MD
1
-0.115 -0.066 0.023
-
0.115 0.998 0.004 -0.001
-
0.066 0.004 0.993 -0.001
0
.023 -0.001 -0.001 0.991
Table 2. (b) Measured Mueller matrix and the decomposed components for the chiral turbid sample (
μ
s =
0.6 mm-1, glucose 5M) in backscattering geometry.
Table 3. Comparison between the different polarization parameters for the chiral turbid sample (
μ
s = 0.6
mm-1, glucose 5M) in forward and backscattering geometry.
#9466 - $15.00 USD Received 10 November 2005; revised 22 December 2005; accepted 3 January 2006
(C) 2006 OSA 9 January 2006 / Vol. 14, No. 1 / OPTICS EXPRESS 201
might be the case here. This aspect is being investigated in more details by carrying out
systematic studies on chiral scattering samples having different values of μs and g.
In Table 4, we present the results for another chiral turbid sample with relatively larger
value of μs (= 5 mm-1). The value for ψ recovered after polar decomposition of Mueller matrix
detected in forward geometry was found to be ψ = 0.076 radian. This value of ψ is about 10 %
higher than the corresponding value obtained from the turbid sample with μs = 0.6 mm-1 and is
about 18 % higher than that obtained from the clear solution of glucose with the same
concentration (5M). As noted earlier, this gradual increase of the value of ψ with increasing
value of μs of the chiral scattering sample arises due to the increase in path length of photons
due to increased degree of multiple scattering of light in the medium. For the same reason, the
value for PL for this sample was also found to be lower (PL = 0.249) as compared to that
obtained for the sample with μs = 0.6mm-1.
M
1
-0.009 -0.021 -0.041
-
0.002 0.256 -0.029 -0.003
0
.024 0.045 0.235 -0.032
0
.041 0.024 0.017 0.538
MΔ
1000
0
.001 0.258 0.01 0.009
0
.028 0.01 0.241 -0.015
0
.064 0.009 -0.015 0.541
−
MR
10 0 0
0
0.988 -0.152 -0.022
0
0.151 0.986 -0.067
0
0.032 0.063 0.998
MD
1
-0.009 -0.021 -0.041
-0.009 0.999 0 0
-0.021 0 0.999 0
-0.041 0 0 1
5. Conclusion
To conclude, the results of our studies show that the ambiguity in the cause of the change in
the orientation angle of the polarization vector of light propagating through a chiral turbid
medium can be lifted with full Mueller matrix measurements. Polar decomposition of Mueller
matrix can then be used to extract the component of optical rotation arising purely due to the
circular retardance property (introduced by the presence of chiral molecules) of the medium
from the other causes (linear diattenuation and linear retardance) of optical rotation. The
validity of this approach was tested by conducting studies on chiral turbid samples prepared
using known concentration of scatterers and glucose molecules. This approach may therefore
facilitate determination of the concentration of chiral substances present in a medium using
the measured Mueller matrix from the medium in both forward and backscattering geometry.
Table 4. Measured Mueller matrix and the decomposed components for the chiral turbid sample
(μs = 5 mm-1, glucose 5M) in forward scattering geometry.
#9466 - $15.00 USD Received 10 November 2005; revised 22 December 2005; accepted 3 January 2006
(C) 2006 OSA 9 January 2006 / Vol. 14, No. 1 / OPTICS EXPRESS 202
