Contrast mechanisms in polarization-sensitive
Mueller-matrix optical coherence tomography and
application in burn imaging
Shuliang Jiao, Wurong Yu, George Stoica, and Lihong V. Wang
We investigate the various contrast mechanisms provided by polarization-sensitive 共PS兲 Mueller-matrix
optical coherence tomography 共OCT兲. Our PS multichannel Mueller-matrix OCT is the first, to our
knowledge, to offer simultaneously comprehensive polarization-contrast mechanisms, including the am-
plitude of birefringence, the orientation of birefringence, and the diattenuation in addition to the
polarization-independent intensity contrast, all of which can be extracted from the measured Jones or the
equivalent Mueller matrix. Theoretical analysis shows that when diattenuation is negligible, the round-
trip Jones matrix represents a linear retarder, which is the foundation of conventional PS-OCT, and can
be calculated with a single incident polarization state, although the one-way Jones matrix generally
represents an elliptical retarder; otherwise, two incident polarization states are needed. The experi-
mental results obtained from rat skin samples, which conform well with the histology, show that Mueller
OCT provides complementary structural and functional information on biological samples and reveal that
polarization contrast is more sensitive to thermal degeneration of biological tissue than amplitude-based
contrast. Thus, Mueller OCT has significant potential for application in the noninvasive assessment of
burn depth. © 2003 Optical Society of America
OCIS codes: 120.2130; 170.4500; 260.5430; 170.1870.
1. Introduction
Since it was first reported approximately a decade
ago as a high-resolution noninvasive medical imaging
modality, optical coherence tomography 共OCT兲 has
received tremendous attention. Several branches of
OCT have been developed based on various contrast
mechanisms, such as polarization
1–6
and Doppler
shift,
7,8
in addition to the amplitude-based contrast in
conventional OCT. OCT has found applications in
imaging of the retina, cornea, gastrointestinal tract,
artery, tooth, bladder, blood flow, and brain cortex.
9
Another potential application of OCT is the evalua-
tion of burns in biological tissue.
10
The contrast of
an OCT image is provided by the optical properties of
a sample that modify the parameters of the light
field, including the amplitude and the polarization
state. The parameters characterizing the structur-
ally isotropic or averaged optical properties
11
of a
sample include the absorption coefficient 共
a
兲, scat-
tering coefficient 共
s
兲, scattering anisotropy 共g兲, and
refractive index 共n兲; and the parameters characteriz-
ing the polarization properties of a sample include
birefringence 共amplitude ␦n, orientation, and elliptic-
ity兲 and diattenuation 共amplitude D, orientation, and
ellipticity兲, which provide polarization-based contrast
in polarization-sensitive OCT 共PS-OCT兲.
Because of the interference-based heterodyne de-
tection scheme used in OCT, a scattering sample acts
as a nondepolarizing medium.
4
The polarization
properties of a nondepolarizing sample can be com-
pletely characterized by either a Mueller matrix or a
Jones matrix, and the two matrices are equivalent.
12
Therefore, to provide comprehensive information
about polarization of a sample, the most general PS-
OCT should measure the Jones or the Mueller ma-
trix. Upon acquisition of the Jones or the Mueller
matrix, any polarization parameters can be ex-
tracted. We define Mueller-matrix OCT as PS-OCT
that can measure the Mueller or the Jones matrix of
a sample. Therefore, Mueller-matrix OCT is the
most general form of PS-OCT.
The authors are with Texas A&M University, 3120 TAMU, Col-
lege Station, Texas 77843-3120. S. Jiao, W. Yu, and L. V. Wang
共LWang@tamu.edu兲 are with the Optical Imaging Laboratory, De-
partment of Biomedical Engineering. G. Stoica is with the De-
partment of Pathobiology.
Received 18 January 2003; revised manuscript received 28 May
2003.
0003-6935兾03兾255191-07$15.00兾0
© 2003 Optical Society of America
1 September 2003 兾 Vol. 42, No. 25 兾 APPLIED OPTICS 5191
In this paper, we investigate the various contrast
mechanisms provided by Mueller-matrix OCT. The
properties of the round-trip Jones matrix are ana-
lyzed for conditions with and without diattenuation
in a sample. The analyses indicate that when diat-
tenuation is negligible, one incident polarization
state is adequate for the acquisition of the Jones ma-
trix. When diattenuation cannot be neglected, two
incident polarization states are necessary, and the
transpose symmetric property of the round-trip Jones
matrix 共first discovered by our group
5
兲 offers a critical
condition for the calculation of the Jones matrix cor-
rectly. Experimental results with biological sam-
ples are presented.
2. Polarization-Based Contrast
Diattenuation is a description of the dependence of
transmittance on the incident polarization states and
is defined as
D ⫽ 共P
q
2
⫺ P
r
2
兲兾共P
q
2
⫹ P
r
2
兲 , (1)
where P
q
and P
r
represent the amplitude transmit-
tances for the two orthogonal eigenpolarizations of a
polarization element. Therefore, diattenuation pro-
vides anisotropic amplitude-based contrast, as it in-
curs no phase retardation. Birefringence is a
description of the anisotropic dependence of the
phase velocity of light in a sample on the incident
polarization states. The phase retardation of a light
field, induced by the local birefringence between the
two orthogonal eigenpolarizations, can be expressed
as
d ⫽ k
␦n共L
s
⬘兲dL
s
⬘ ,
where k
is the wave vector corresponding to the cen-
tral wavelength of the incident light in vacuum; L
s
⬘ is
the physical path length that the light travels in the
birefringent medium; ␦n共L
s
⬘兲 is the local birefrin-
gence; and dL
s
⬘ is the local physical path length.
The phase retardation provides a unique phase-based
polarization-contrast mechanism reflecting the am-
plitude of birefringence, which exists in various bio-
logical components such as collagen, keratin, myelin,
and elastic fibers. Because highly birefringent col-
lagen is a predominant structural component in most
biological tissues, this intrinsic contrast mechanism
is prevalent in the biomedical applications of Mueller
OCT. In addition, many degenerative processes of
biological tissues alter birefringence and should,
thus, be detectable by Mueller-matrix OCT.
In a PS-OCT system, the detected variation of the
polarization state of the scattered light in reference to
the incident light is affected by the round-trip polar-
ization effect of a sample, which can be characterized
with a round-trip Jones matrix 共J
2
兲. We will use sub-
scripts 1 and 2 to describe the one-way and round-trip
parameters, respectively. After acquisition of the
round-trip Jones matrix, the round-trip retardation
共
2
兲 and diattenuation 共D
2
兲 for each pixel of the OCT
image can be calculated with the following formulas,
13
2
⫽ 2cos
⫺1
再
1
2
兩trJ
2
⫹ 关det J
2
兾兩det J
2
兩兴trJ
2
*兩
关tr共J
2
*J
2
兲 ⫹ 2兩det J
2
兩兴
1兾2
冎
, (2)
D
2
⫽
再
1 ⫺
4兩det J
2
兩
2
关tr共J
2
*J
2
兲兴
2
冎
1兾2
, (3)
respectively, where *, tr, and det represent the Her-
mitian 共transpose conjugate兲, trace, and determinant
of the matrix, respectively. The fast eigenvector of
J
2
at each pixel of the OCT image,
E
2
⫽
冋
E
2h
E
2v
册
,
can be calculated through standard algorithms. The
orientation of the fast axis can thus be calculated as
2
⫽ arc tan
冉
E
2v
E
2h
冊
. (4)
3. Calculation of the Round-Trip Jones Matrix
The round-trip Jones matrix J
2
can be expressed with
the one-way Jones matrix 共J
1
兲, according to the Jones
reversibility theorem, as
J
2
⫽ J
1
T
J
1
, (5)
where the superscript T represents the transpose op-
eration. A polarization element is called homoge-
neous when the two eigenvectors of its Jones matrix
are orthogonal. A retarder is called elliptical when
its eigenpolarizations are elliptical polarization
states. A linear retarder is a special case in which
the eigenpolarizations are linear, and a Faraday ro-
tator is another special case in which the eigenpolar-
izations are circular. We can prove that when two or
more linear retarders are cascaded, the overall re-
tarder is generally elliptical unless the axes of the
retarders are aligned. Except in some special sam-
ples, the orientations of the birefringent fibers in bi-
ological samples, take skin, for example, are not
collinear, and as a result, J
1
generally represents a
homogeneous elliptical retarder if diattenuation is
negligible in the sample.
When diattenuation is negligible in a sample, J
1
can be expressed as
J
1
共
1
,
1
, ␦
1
兲 ⫽
冋
cos共
1
兾2兲 ⫹ i sin共
1
兾2兲cos 2
1
i sin共
1
兾2兲sin 2
1
exp共⫺i␦
1
兲
i sin共
1
兾2兲sin 2
1
exp共i␦
1
兲 cos共
1
兾2兲 ⫺ i sin共
1
兾2兲cos 2
1
册
⫽
冋
J
1
共1,1兲 J
1
共1,2兲
⫺ J
1
共1,2兲* J
1
共1,1兲*
册
. (6)
5192 APPLIED OPTICS 兾 Vol. 42, No. 25 兾 1 September 2003
The fast and slow eigenvectors are
冋
cos
1
sin
1
exp共i␦
1
兲
册
and
冋
⫺ sin
1
exp共⫺i␦
1
兲
cos
1
册
,
respectively, where
1
is an auxiliary angle and ␦
1
represents the phase difference between the two com-
ponents of the fast eigenvector.
1
is the phase dif-
ference between the two eigenvalues 共the
retardation兲. The azimuth 共␣
1
兲 of the major axis of
its fast eigenpolarization can be expressed as
tan共2␣
1
兲⫽tan共2
1
兲cos ␦
1
.If␦
1
⫽ 0, J
1
is transpose
symmetric, representing a linear retarder, and
1
represents the orientation of the fast axis.
From Eq. 共5兲, we have J
2
⫽ J
2
T
, i.e., J
2
is transpose
symmetric. As a result, J
2
represents a linear re-
tarder, and we can thus conclude that the round-trip
transformation effect of an elliptical retarder is
equivalent to the one-way transformation of a linear
retarder. This conclusion is the foundation of con-
ventional PS-OCT, in which a sample is treated as a
linear retarder. Since only two parameters are
needed to characterize a linear retarder, the number
of parameters needed to characterize the round-trip
polarization properties of a sample is reduced to two.
This conclusion allows the acquisition of this type of
round-trip Jones matrix with only one incident polar-
ization state. For an incident polarization state
E
i
⫽
冋
E
ih
E
iv
册
,
the output polarization state
E
o
⫽
冋
E
oh
E
ov
册
detected by PS-OCT can be expressed as
冋
E
oh
E
ov
册
⫽ J
2
冋
E
ih
E
iv
册
. (7)
Because of the orthonormal transformation property
of J
2
, the inherent property of a retarder, we also
have
冋
E
ov
*
⫺E
oh
*
册
⫽ J
2
冋
E
iv
*
⫺E
ih
*
册
. (8)
The round-trip Jones matrix can thus be calculated
as
J
2
⫽
冋
E
oh
E
ov
*
E
ov
⫺E
oh
*
册冋
E
ih
E
iv
*
E
iv
⫺E
ih
*
册
⫺1
. (9)
When diattenuation cannot be neglected in a sam-
ple, one incident polarization state is not sufficient to
acquire its round-trip Jones matrix because five real
parameters 关
2
,
2
, amplitude transmittances 共P
q2
and P
r2
兲, and the orientation of diattenuation 共
d2
兲兴
are needed to characterize such a system. There-
fore, at least two incident polarization states, either
applied at the same time or applied sequentially, are
required. The transpose symmetry in the round-
trip Jones matrix is critical for eliminating the arbi-
trary phase difference between the two measured
Jones vectors corresponding to the two incident po-
larization states to yield the correct Jones matrix.
This arbitrary phase difference can be caused either
by the nonidentity of the power spectra when two
light sources are used or by the imperfection of the
longitudinal scanning mechanism when the two in-
cident polarization states are applied sequentially.
Because it ignores the diattenuation effect com-
pletely, conventional PS-OCT is not valid for biolog-
ical samples possessing diattenuation and cannot
provide diattenuation contrast.
4. Experiment
We have built a novel multichannel Mueller OCT,
5,6
which can acquire the Jones matrix of a sample with
a single scan for each one-dimensional depth image
共A line image兲. The Jones matrix can be further
transformed into an equivalent Mueller matrix.
The Mueller matrix is preferred because its first el-
ement, M
00
, represents the intensity transformation
property of a sample and is free of both the effects of
the sample polarization and the polarization state of
the incident light. Therefore, a Mueller matrix re-
veals the real morphologic structure as well as the
polarization-based features of a sample.
The tail of a rat was imaged in situ with Mueller
OCT after the skin was shaved and scrubbed with
glycerin. The OCT and polarization-histologic im-
ages are shown in Figs. 1共a兲–1共f 兲. There are no sig-
nificant differences between the M
00
image 关Fig. 1共b兲兴
and the conventional OCT image for this particular
sample 关Fig. 1共a兲兴, both of which are amplitude based.
The effect of polarization on a conventional OCT im-
age depends on several parameters, for example, the
incident polarization state, the value and orientation
of the birefringence, and the accumulated phase re-
tardation. When fringes are present in the conven-
tional OCT image, the difference between these two
images is dramatic.
5
The intensity and retardation
images reveal different characteristics of the sample.
The intensity images clearly reveal the boundaries of
the structures in the epidermis and only the shallow
dermal region. In contrast, the retardation image
关Fig. 1共c兲兴 reveals the distribution of birefringent
components deeper into the dermis. The absolute
value of the retardation difference between each
pixel and its previous pixel in the same A line is
calculated to obtain a differential retardation image
关Fig. 1共d兲兴. The birefringent regions 共correspond-
ing to the superficial keratin layer and collagen-rich
dermal papillae兲 and nonbirefringent regions 共cor-
responding to fat and the living epidermis兲 are
shown more clearly in the differential retardation
image than in the raw retardation image. The im-
1 September 2003 兾 Vol. 42, No. 25 兾 APPLIED OPTICS 5193
age of the orientation of the fast axis 关Fig. 1共e兲兴
revealed structures that we believe to be related to
the distribution of the orientation of the birefrin-
gent fibers 共collagen and keratin兲. In the figure,
we can see that the orientation of the fast axis
varies from region to region as also observed in the
polarization histology. Although the amplitude-
and phase-based polarization signals should have
comparable signal-to-noise ratios because they are
computed from the same measurements, the
contrast-to-noise ratio can be different depending
on the availability of the two contrasts in the sam-
ple; therefore, the two contrast mechanisms can
provide information into different depths.
To evaluate the sensitivity of the phase-based po-
larization contrast in burn-depth determination, we
imaged an ex vivo skin sample—from a rat belly—
containing a burn lesion. The burn lesion was made
by touching the skin with a heated 共approximately
100 °C兲 electric iron for less than 1 s. The calculated
intensity image, the retardation image, the diattenu-
ation image and the histological image are shown in
Figs. 2共a兲–2共d兲. The burn region cannot be identified
in the intensity image; but it can be clearly seen with
marked contrast in the retardation and diattenuation
images as verified by the polarization histological im-
age.
Figure 3 shows the depth profiles of retardation of
the burn and normal regions, respectively. Each
curve is an average of 10 profiles in the central area
of the burn region and in the normal region to the
right side of the burn region. The loss of birefrin-
gence in the burn region compared with the normal
tissue can be seen clearly. This figure further dem-
onstrates that phase-based polarization contrast pro-
vides a sensitive mechanism for evaluating thermal
degeneration of biological tissue. Because birefrin-
gence and diattenuation are related to the function of
Fig. 1. 共a兲 Conventional OCT image 共in logarithmic scale兲, 共b兲 intensity image 共M
00
, in logarithmic scale兲, 共c兲 retardation image, 共d兲
differential retardation image, 共e兲 image of the orientation of the fast axis, 共f 兲 polarization histologic image of an in situ rat tail. The height
of each image is 750 m. The gray scales are for the orientation 共
2
兲 and the retardation 共
2
兲 images. The conventional OCT image was
obtained with vertical linear polarization states for both the incident and the reference beams. F, fat; K, keratin; DP, dermal papilla.
Fig. 2. 共a兲 Intensity image 共M
00
, in logarithmic scale兲, 共b兲 retardation image, 共c兲 diattenuation image, 共d兲 polarization histologic image of
a piece of ex vivo rat skin with a burn lesion. The height of each image is 750 m. The gray scales are for the retardation 共
2
兲 and the
diattenuation 共D
2
兲 images. B, burn region.
5194 APPLIED OPTICS 兾 Vol. 42, No. 25 兾 1 September 2003
several kinds of biological component such as colla-
gen, Mueller OCT is a type of functional imaging.
5. Discussion
The differences between conventional OCT and
Mueller OCT in their sensitivities to different optical
properties of a sample result from their different con-
trast mechanisms. Conventional OCT is an
amplitude-based detection system, which detects the
local relative variations of path-length-resolved re-
flectance from tissues. By modifying an existing
theoretical model of OCT
14
to include the effect of
polarization, we can express the signal in conven-
tional OCT as
I
˜
d
共L
r
兲 ⫽ 2共I
s
I
r
兲
1兾2
兰
⫺⬁
⬁
关R共L
s
兲兴
1兾2
cos关共L
s
兲兴exp
⫻ 关⫺4共⌬L兾L
c
兲
2
兴 cos共k
⌬L兲dL
s
, (10)
where L
s
and L
r
are the round-trip optical path
lengths of the sample and reference arms, respec-
tively; ⌬L ⫽ L
s
⫺ L
r
is the round-trip optical path-
length difference; L
c
is the coherence length of the
light source; I
r
is the intensity of the reference beam;
I
s
is the reflected intensity of the sample arm; R共L
s
兲⫽
关dI
s
共L
s
兲兾dL
s
兴兾I
s
is the path-length-resolved reflec-
tance of the sample; and 共L
s
兲 is an equivalent angle
between the polarization states of the reference and
the backscattered sample beams, defined as
cos关共L
s
兲兴 ⫽ 具E
s
共L
s
兲 䡠 E
r
典兾共兩E
s
共L
s
兲储E
r
兩兲 ,
where E
s
共L
s
兲 and E
r
are the electric vectors of the
sample and the reference beams, respectively, and
the angle brackets denote a time average. The in-
tegrand is nonzero mainly in the interval 兩⌬L兩 ⱕ L
c
.
The integration produces a significant value only
when R共L
s
兲 varies sharply across a dimension of L
c
;
otherwise, the integral tends to be zero because of the
cosine term in the integrand. A sharp variation of
R共L
s
兲 is caused by interfaces between regions of dif-
ferent optical properties. Conventional OCT is, in
principle, very sensitive to discontinuity of the refrac-
tive index 共⌬n兲 as a result of specular reflection. As
studied by Pan et al.,
14,15
conventional OCT is also
sensitive to variations of the anisotropy 共⌬g兲 and the
scattering coefficient 共⌬
s
兲, but it is insensitive to
variation of the absorption coefficient 共⌬
a
兲. We can
see in Eq. 共10兲 that the polarization effect of a sample
contributes to the recorded conventional OCT signal
as an amplitude modulation and is superimposed on
the backreflection effect; consequently, conventional
OCT has difficulty in separating the polarization ef-
fect from the real morphologic effect of the sample.
To account for the meanings of the measured re-
tardation image, we can divide each depth scan into
a number of homogenous segments, each of which has
a length less than the axial resolution; each segment
can be characterized by a Jones matrix J
1
共i兲共i ⫽ 1, 2,
...兲, which is a function of the equivalent local bire-
fringence 关␦n共i兲兴, orientation of the fast axis 关
1
共i兲兴,
amplitude transmittances 关P
q1
共i兲 and P
r1
共i兲兴, and ori-
entation of the diattenuation 关
d1
共i兲兴. For single
backscattering and even multiple small-angle scat-
tering, the equivalent round-trip Jones matrix of con-
tiguous m segments of the sample from the surface to
the mth segment can be expressed as
J
2
共m兲
⫽
兿
i⫽1
m
J
1
T
共i兲
兿
i⫽m
1
J
1
共i兲 .
The equivalent round-trip parameters for the m seg-
ments, such as the retardation
2
共m兲
, orientation of the
fast axis
2
共m兲
, and diattenuation, can be calculated
from J
2
共m兲
. When
1
共1兲⫽
d1
共1兲⫽
1
共2兲⫽
d1
共2兲⫽
...⫽
1
共m兲⫽
d1
共m兲,if
2
共m兲
ⱕ ,
2
共m兲
in the retar-
dation image increases with depth, whereas
2
共m兲
keeps constant; if
2
共m兲
covers a range greater than ,
it causes fringes in both the retardation and the ori-
entation images because a retarder J共
2
共m兲
⫹,
2
共m兲
兲
is equivalent to a retarder J共 ⫺
2
共m兲
,
2
共m兲
⫾兾2兲,
共
2
共m兲
,
2
共m兲
⑀关0,兴兲, a phenomenon observed in the
retardation and orientation images of samples like
porcine tendon.
5
In this case, the differential retar-
dation image reflects a map of the local birefringence.
Otherwise,
2
共m兲
and
2
共m兲
are also functions of both
1
共i兲 and
d1
共i兲 in the optical path, making the retar-
dation image complex to interpret rigorously unless
the local polarization properties can be calculated,
which is possible only with Mueller OCT.
The Jones matrix of the first pixel of each A line
represents the round-trip Jones matrix of the first
segment, i.e., J
2
共1兲
⫽ J
1
T
共1兲J
1
共1兲.IfJ
1
共1兲 can be cal-
culated from J
2
共1兲 by developing some effective algo-
rithms, the first segment can be peeled off to yield the
round-trip Jones matrix of the second segment, J
1
T
共2兲J
1
共2兲⫽关J
1
T
共1兲兴
⫺1
J
2
共2兲
J
1
⫺1
共1兲. By use of this
strategy layer by layer, the one-way Jones matrix of
each segment can thus be extracted, and the images
of the local polarization parameters can be calcu-
lated, which should be free of fringes because the
retardation of each segment should be much less than
. This algorithm is important in fiber-based PS-
OCT system for eliminating the polarization dis-
Fig. 3. Average of 10 depth profiles of the retardation around the
center of the burn area and the normal region to the right of the
burn area.
1 September 2003 兾 Vol. 42, No. 25 兾 APPLIED OPTICS 5195
