An optical-fiber-based multichannel polarization-sensitive Mueller optical coherence tomography (OCT) system was built to acquire the Jones or Mueller matrix of a scattering medium, such as biological tissue. For the first time to our knowledge, fiber-based polarization-sensitive OCT was dynamically calibrated to eliminate the polarization distortion caused by the single-mode optical fiber in the sample arm, thereby overcoming a key technical impediment to the application of optical fibers in this technology. The round-trip Jones matrix of the sampling fiber was acquired from the reflecting surface of the sample for each depth scan (A scan) with our OCT system. A new rigorous algorithm was then used to retrieve the calibrated polarization properties of the sample. This algorithm was validated with experimental data. The skin of a rat was imaged with this fiber-based system

Jiao, Shuliang

Stoica, George

Wang, Lihong V.

Yu, Wurong

Caltech Authors - Main

1206 OPTICS LETTERS / Vol. 28, No. 14 / July 15, 2003Optical-fiber-based Mueller optical coherence tomography
Shuliang Jiao and Wurong Yu
Optical Imaging Laboratory, Department of Biomedical Engineering, Texas A&M University,
3120 TAMU, College Station, Texas 77843-3120
George Stoica
Department of Pathobiology, Texas A&M University, College Station, Texas 77843-5547
Lihong V. Wang
Optical Imaging Laboratory, Department of Biomedical Engineering, Texas A&M University,
3120 TAMU, College Station, Texas 77843-3120Received January 8, 2003
An optical-fiber-based multichannel polarization-sensitive Mueller optical coherence tomography (OCT) system
was built to acquire the Jones or Mueller matrix of a scattering medium, such as biological tissue. For the
first time to our knowledge, fiber-based polarization-sensitive OCT was dynamically calibrated to eliminate
the polarization distortion caused by the single-mode optical fiber in the sample arm, thereby overcoming a
key technical impediment to the application of optical fibers in this technology. The round-trip Jones matrix
of the sampling fiber was acquired from the ref lecting surface of the sample for each depth scan (A scan) with
our OCT system. A new rigorous algorithm was then used to retrieve the calibrated polarization properties
of the sample. This algorithm was validated with experimental data. The skin of a rat was imaged with
this fiber-based system. © 2003 Optical Society of America
OCIS codes: 120.2130, 170.0170, 170.1870, 170.4500, 260.5430.In contrast to conventional optical coherence tomogra-
phy (OCT), polarization-sensitive optical coherence to-
mography (PS-OCT) adds the polarization properties
of the sample as a contrast mechanism.1 –6 However,
practical applications of PS-OCT have been limited by
the difficulty of its optical-fiber implementation. A
single-mode optical fiber (SMF) alters the polariza-
tion state of the guided light because of the inherent
birefringence in the f iber. The birefringence varies
with the bending and twisting of the f iber during ma-
nipulation of the imaging probes, which can result in
dynamic distortion in PS-OCT images. Therefore, a
dynamic calibration technique is required for elimina-
tion of this effect.
Based on previous studies, a Jones matrix can be ap-
plied in PS-OCT to characterize completely the polar-
ization properties of a sample.2 – 6 If the one-way Jones
or Mueller matrix of the sampling optical fiber can
be determined, the polarization distortion caused by
the sampling f iber can be eliminated from the PS-OCT
images. Multichannel Mueller OCT can measure the
Jones and Mueller matrices of a sample with a single
scan and thus offer the possibility of rigorously elimi-
nating the polarization effect of the sampling fiber.
This method allows fiber-based Mueller OCT to ac-
quire a calibrated Mueller-matrix image as rapidly as
conventional OCT acquires a regular image. In this
Letter we report a new rigorous calibration algorithm
that was validated with experimental data and was
also applied to imaging the skin of a rat.
In general, a pure retarder can be characterized by
a homogeneous Jones matrix that has two orthogo-
nal elliptical eigenvectors, each representing an eigen-
polarization. When two or more linear retarders are0146-9592/03/141206-03$15.00/0cascaded, the overall retarder is generally elliptical un-
less the axes are aligned. Because of its randomly dis-
tributed birefringence along the core, a SMF should be
treated as an elliptical retarder.
We first introduce the general properties of a
retarder. The 2 3 2 Jones matrix Jw, u, d of
an elliptical retarder can be expressed with three
independent real parameters: J1, 1  cosw2 1
i sinw2cos 2u, J1, 2  i sinw2sin 2u exp2id,
J2,1 2J1, 2, J2,2  J1, 1. The fast and
slow eigenvectors are cos u, sin u expidT and
2sin u exp2id,cos uT , respectively, where the
superscript T represents a transpose operation. The
angle u is an auxiliary angle of the fast eigenvector,
d represents the phase difference between the two
components of the fast eigenvector, and w is the phase
difference (retardation) between the two eigenvalues.
If d  0, the retarder is linear and u represents the
orientation of the fast axis.
The round-trip Jones matrix J2 of an optical
component can be calculated from its one-way Jones
matrix J1 according to the Jones reversibility
theorem: J2  J1TJ1. Subscripts 1 and 2 denote
the one-way and round-trip parameters, respectively.
Because J2 is transpose symmetric J22, 1 J21, 2,
the phase difference d2 of J2 is zero, indicating that
J2 represents a linear retarder. As a result, only
two independent real parameters are needed to de-
scribe J2.
In a fiber-based Mueller OCT system, the incident
sampling light undergoes transformation sequentially,
first through the sampling fiber and the sample in
forward propagation and then the sample and the
sampling f iber in backward propagation. Therefore,© 2003 Optical Society of America
July 15, 2003 / Vol. 28, No. 14 / OPTICS LETTERS 1207the raw round-trip Jones matrix Jsf2 can be
expressed in terms of the one-way Jones matrix of the
sampling f iber Jf1 and the round-trip Jones matrix
of the sample at a given imaging depth Js2 as
Jsf2  Jf1TJs2Jf1 . (1)
The round-trip Jones matrix of the sampling fiber
Jf2 can be calculated from the OCT signal ref lected
from the sample surface: Jf2  Jf1TJf1.
Without discriminating between the one-way and
round-trip transformations, a recently developed tech-
nique7 used an algorithm equivalent to using Jf221 di-
rectly to treat Jsf2. This algorithm yields Jsf2Jf221 
Jf1TJs2Jf1T21, which is not equal to Js2 unless Js2
and Jf1 are permutable. Since it does not ref lect the
correct physical process and it treats the sample as
a pure retarder, this algorithm is unable to correctly
recover the orientation of the birefringence and the
diattenuation.
To eliminate the distortion, the best approach is to
calculate Jf1 from Jf2 for each A scan. However, there
are three real variables in Jf1wf1,uf1, df1 but only
two in Jf2wf2,uf2. Consequently, the equation Jf2 
Jf1TJf1 provides only two independent relationships;
therefore, Jf1 can be determined only from Jf2 with a
free parameter.
For each Jf2, we can always find a unique hypotheti-
cal linear retarder Jfl1 to satisfy Jf2  Jfl1TJfl1. We
introduce the following matrix to ref lect the free
parameter:
Jfc1  Jf1Jfl121 . (2)
Removing the round-trip effect of Jfl1 from Jsf2, we
obtain a new matrix Jsc2:
Jsc2  Jfl1T 21Jsf2Jfl121 . (3)
Based on Eqs. (1)–(3), we have the following solution
representing the general calibration algorithm in a ma-
trix form:
Js2  Jfc1T 21Jsc2Jfc121 . (4)
The round-trip retardation ws2 of the sample can
be calculated with
ws2  2 cos21
(
1
2
jtr Js2 1 det Js2jdet Js2jtr Js2j
trJs2Js21 2jdet Js2j12
)
,
(5)
or, in the case of negligible diattenuation in the sample,
by ws2  2 cos21Js21, 1 1 Js22,22.
We can also prove from Eq. (2) that the elements of
Jfc1 are real numbers and that Jfc121, 11 Jfc121, 2
1. Consequently, we can introduce a new parameter g
as follows:
Jfc1 
∑
cos g sin g
2sin g cos g
∏
. (6)
Jfc1 thus represents a rotation matrix. In other
words, the Jones matrix of the sampling f iber isdecomposed into a linear retarder and a rotator
because Jf1  Jfc1Jfl1. Equation (4) is equivalent
to rotating the fast axis of Js2 along the axis of the
incident light by an angle g. This rotation does not
affect the amplitudes of either the birefringence or
diattenuation. As a result, the calibrated round-trip
retardation of the sample can be calculated exactly
from Eq. (5). From Eqs. (4) and (6), we can calculate
the calibrated orientation of birefringence as follows:
us2  usc2 2 g , (7)
where usc2 can be calculated from the fast eigenvector
of Jsc2.
The calibration in Eq. (7) has an offset g that de-
pends on the parameters of the sampling f iber only.
This offset is a constant in an image frame as long as
the parameters of the sampling f iber are kept constant
during the image acquisition of each frame, which is
the case when the fast lateral scanning of OCT does
not move the sampling f iber. Therefore, a relative dis-
tribution of the orientation of the birefringence can be
retrieved. If the parameters of the sampling f iber are
varied among the A scans, which is true when the lat-
eral scanning in OCT does move the sampling fiber,
g will differ among the A lines. In this case, if the
orientation of the birefringence of the surface layer
is constant or known a priori, or if a known thin re-
tarder is attached to the sample as the first layer, g can
be eliminated. In either case, ws2 can be calculated
exactly.
Figure 1 shows a schematic of the experimental sys-
tem. The two source beams (superluminescent diodes;
central wavelength l  850 nm, FWHM bandwidth
Dl  26 nm), one amplitude modulated at 3 kHz and
the other at 3.5 kHz, are merged by a polarizing beam
splitter (PBS1). Both the sample and the reference
beams are coupled into a 0.5-m-long single-mode fiber.
A 45± linear polarizer is used to control the polarization
state of the reference beam. The horizontally (H) and
vertically (V) polarized components of the interference
signal are detected by photodiodes PDH and PDV, re-
spectively. The data processing and the Jones matrix
calculation were described previously.4,5
We first tested the system by imaging a quarter-
wave l4 plate in combination with a mirror, for a
Fig. 1. Schematic of the f iber-based Mueller OCT system.
SLDH and SLDV, superluminescent diodes, H and V polar-
ized, respectively; PBS1 and PBS2, polarizing beam split-
ters; SF, spatial filter assembly; LP, linear polarizer; NBS,
nonpolarizing beam splitter; M, mirror; SMFs, single-mode
optical fibers; PDH and PDV, photodiodes for the H and V
polarization components, respectively.
1208 OPTICS LETTERS / Vol. 28, No. 14 / July 15, 2003Fig. 2. Round–trip phase retardation of a l4 plate cal-
culated from the measured Jones matrix before and after
cancellation of the polarization distortion caused by the
sampling optical fiber. The phase retardation of the sam-
pling fiber, which is by def inition zero after cancellation,
is shown as well.
Fig. 3. M00 image of the Mueller matrix, the retardation
images before and after cancellation of the polarization ef-
fect of the sampling f iber wsf2 and ws2 of the skin of a rat
tail measured with the f iber-based Mueller OCT system.
A histological image stained with hematoxylin and eosin is
also shown for comparison. The M00 image is on a loga-
rithmic scale, and the retardation images are on a linear
scale. The height of each image is 1 mm. EP, epidermis;
DP, dermal papilla; DJ, dermal–epidermal junction.
frame consisting of 35 A scans with a lateral span
of 1 mm. The sampling fiber was intentionally de-
formed every f ifth A scan to vary its polarization prop-
erty. In Fig. 2, one can see that the raw round-trip
retardation of the l4 plate was severely distorted
by the sampling f iber. The measured Jf2 was used
to cancel the distortion with the algorithm described
in Eqs. (2) and (3). The calibrated ws2 of the l4
plate shown in Fig. 2 accurately matches the expected
value of the half-wave, indicating the validity of our
algorithm.
We then used the fiber-based Mueller OCT sys-
tem to image a biological sample, the skin of a rat
tail (Berlin Drucrey). After the hair of the tail
was removed with hair remover lotion, the tail wasscrubbed with glycerin. Two-dimensional data of
the skin were taken by lateral movement of the
sample between A scans. The sampling f iber was
intentionally disturbed between A scans to introduce
distortions. The Jones matrix was calibrated pixel-
wise and then converted into its corresponding 4 3 4
Mueller matrix.4,5 Figure 3 shows the images of the
polarization-independent M00 element of the Mueller
matrix, the retardation before calibration wsf2,
and the retardation after calibration ws2. Some
structures, such as the dermal–epidermal junction
and the collagen-rich dermal papillae, can be clearly
seen in the M00 and ws2 images, whereas they are
blurred in the wsf2 image because of the distortion
of the sampling f iber. Also shown in Fig. 3 is the
hematoxylin and eosin histological image of the tail
skin of the same breed. The calibrated OCT images
conform well with the histological image.
In conclusion, single-mode optical f ibers have suc-
cessfully been incorporated into our Mueller OCT
system. A rigorous algorithm was invented to pre-
cisely eliminate the polarization effect of the sampling
fiber on the retardation image of a sample dynami-
cally. With this algorithm, the distribution of the
orientation of the birefringence can also be extracted
with only a constant offset in each pixel as long as
the sampling fiber is not scanned during the acqui-
sition of each image frame. Our fiber-based Mueller
OCT system was successfully applied to imaging of
biological samples.
Thanks to Geng Ku for assistance in machining
mechanical components for this study. This project
was sponsored in part by National Institutes of Health
grants R21 EB00319-02 and R01 EB000712, by Na-
tional Science Foundation grant BES-9734491, and
by Texas Higher Education Coordinating Board grant
000512-0063-2001. L. V. Wang’s e-mail address is
lwang@tamu.edu.
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Optical-fiber-based Mueller optical coherence tomography

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