Development of a Three-Dimensional Camera Based on
Subsampled Optical Coherence Tomography (OCT)
by
Meena Siddiqui
M.S. Electrical Engineering and Computer Science, MIT (2013)
B.S., Bioengineering, UC-San Diego (2009)
Submitted to the Harvard-MIT Division of Health Sciences & Technology
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Medical Engineering and Medical Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
@
I
Massachusetts Institute of Technology 2015. All rights reserved.
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HarvarV-MIT Dvision of Health Sciences & Technology
August 28, 2015
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Benjamin J. Vakoc, PhD
Associate Professor of Dermatology and Health Sciences & Technology
Thesis Supervisor
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Emery N. Brown MD, PhD
Director, arvard-MIT Program in Health Sciences & Technology
Professor of Computational Neuroscience and Health Sciences & Technology
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MITLibraries
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2
Development of a Three-Dimensional Camera Based on
Subsampled Optical Coherence Tomography (OCT)
by
Meena Siddiqui
Submitted to the Harvard-MIT Division of Health Sciences & Technology
on August 28, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Medical Engineering and Medical Physics
Abstract
Optical coherence tomography (OCT) allows label-free, three-dimensional imaging of tissue structure. Current implementations of OCT can either image over long depth ranges
at slow imaging speeds, or over limited depth ranges at high speeds. Here, we describe a
new OCT paradigm that supports simultaneous high speed and long depth range imaging
through subsampling bandwidth compression. We show that this requires replacing the
conventional wavelength-swept OCT laser source with a wavelength-stepped laser. First
we validated this concept by modifying a slow, conventional wavelength-swept source with
an intra-cavity Fabry-Perot etalon to provide a wavelength-stepped output. Using this
source in an existing OCT system, we show that we can passively compress signals across
a large depth range into a limited RF bandwidth. Next, to demonstrate high-speed optical domain subsampled imaging, we developed a novel wavelength-stepped laser source
based on intra-cavity pulse compression/stretching; this source provided an A-line rate
of ~19 MHz. We then built a polarization-based quadrature interferometer to remove
imaging artifacts induced by subsampling and comple-conjugate ambiguity. A calibration
and error compensation method was developed to fully remove residual artifacts in the
image. We combined the high speed laser and the interferometer to demonstrate the
first OCT camera-like imaging across several centimeters of depth range. The optically
subsampled OCT technology developed in this work may offer a new three-dimensional
camera platform for endoscopic and intraoperative imaging applications.
Thesis Supervisor: Benjamin J. Vakoc, PhD
Title: Associate Professor of Dermatology and Health Sciences & Technology
3
4
Acknowledgments
This thesis has been possible due to immeasurable support from my supervisor Dr. Benjamin J. VAKOC. I am truly grateful for his mentorship and for the opportunity to work
with him and learn from him. I also appreciate his patience with me as I explored a new
field. I would like to thank Dr. Elfar ADALSTEINSSON for serving as the chair of my
committee. His guidance and support was extremely valuable in navigating the final years
of my PhD. I am inspired by his immense knowledge and I will remember his valuable
advice for the rest of my career. My thanks also goes to Dr. Guillermo J. TEARNEY for
serving as my committee member and thesis reader. His vast insight on OCT technology
and his clinical perspective were invaluable during the final stages of this work. And
finally, a sincere thanks to Dr. Anantha CHANDRAKASAN who oversaw the production
of my MS thesis in the EECS Department.
I am grateful for the company and help from my fellow lab mates and colleagues over
the years. I would like to acknowledge Dr. Serhat TOZBURUN for his contributions to
the high-speed dispersion-based laser. A special thank you to my colleague and friend Dr.
Norman LIPPOK for his mentorship in the last year of my PhD, and for being a willing
volunteer for imaging. Thanks to Ahhyun Stephanie NAM and Dr. Ellen Ziyi ZHANG
for enduring the full duration of the thesis with me, and for many stimulating discussions.
I have also had many insightful conversations and lunches with: Dr. Nishant MOHAN,
Dr. Isabel CHICO-CALERO, Hongying TANG, and Jonathan WELT, Petronella BODO,
and Ashley FLIBOTTE.
This journey would not have been the same without my friends and the many hikes,
climbing excursions, ski trips, picnics, get-togethers, gym days, and endless adventures
we've had. They were an integral part of my PhD life and my happiness/well-being. A
special thank you to Dr. Alexander J. NICHOLS for his support and kindness, and for
introducing me to the east coast winters.
Above all I would like to thank my family for their unconditional love and support.
I am grateful to have my siblings, Hadia SIDDIQUI, Harris A. SIDDIQUI, Edrees M.
SIDDIQUI, and Faria M. SIDDIQUI. I have shared so many happy moments with them,
and with my little niece, Sophia BADIHI. Thanks to my aunt, Mina M. Sara, for always thinking of me and sending care packages. And finally, with my deepest love and
gratitude, I thank my parents Hafizullah K. SIDDIQUI and Soraya SIDDIQUI for always
believing in me.
I would like to acknowledge the following organizations for their generous support:
National Science Foundation - Graduate Research Fellowships Program (NSF-GRFP)
Harvard-MIT Health Sciences & Technology and NIH - SHBT Training Grant
Wellman Center for Photomedicine - Graduate Scholarship
Thanassis and Marina Martinos - Medical Imaging Scholarship
5
6
Contents
Introduction
1.1 Applications of OCT . . . . . . . . . . .
1.2 Practical limitations of OCT . . . . . . .
1.2.1 Long-Range Imaging . . . . . . .
1.2.2 High-Speed Imaging . . . . . . .
1.2.3 Acquisition Bandwidth Limitation
1.3 Thesis Organization . . . . . . . . . . . .
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2
Fundamental Concepts in OCT
2.1 Monochromatic interference ......
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2.2 Time-domain OCT (TD-OCT) . . . . . . . . . . . . . .
2.2.1 Correlation functions . . . . . . . . . . . . . . .
2.2.2 Gaussian sources . . . . . . . . . . . . . . . . .
2.2.3 Low-coherence interferometry . . . . . . . . . .
2.3 Fourier-Domain OCT . . . . . . . . . . . . . . . . . . .
2.3.1 Swept-source OCT (SS-OCT) . . . . . . . . . .
2.3.2 Sensitivity advantage over TD-OCT . . . . . . .
2.3.3 Balanced Detection . . . . . . . . . . . . . . . .
2.4 Practical considerations and limitations of FD-OCT . .
2.4.1 Coherence Length . . . . . . . . . . . . . . . . .
2.4.2 Discrete sampling with acquisition card . . . . .
2.4.3 Complex-conjugate ambiguity . . . . . . . . . .
2.5 Transverse scanning and microscopy . . . . . . . . . . .
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OCT.
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3 Theory of Discrete Optical Sampling
3.1 Sparsity in Extended Depth Range OCT . .
3.2 Bandpass Sampling . . . . . . . . . . . . . .
3.2.1 Electrical-Domain Subsampling . . .
3.2.2 Optical-Domain Subsampling . . . .
3.3 Subsampled OCT imaging parameters . . .
3.4 Complex-conjugate ambiguity in subsampled
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Experimental Validation of Subsampling in a Slow-Speed System
4.1 Relevant Work . . .. .. .. . .. ....
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4.1.1 Polygon-based wavelength-swept laser ................
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4.2 Fabry-Perot comb filter .........................
4.3 Laser construction and performance . . . . . . . . . . . . . . . . . .
4.3.1 Coherence length . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Chirp and nonlinear tuning . . . . . . . . . . . . . . . . . .
4.4 Interferometer and acquisition . . . . . . . . . . . . . . . . . . . . .
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4.5 Experimental validation of circular wrapping . . . . . . . ......
4.5.1 Signal loss due to higher order harmonics . . . . . . . . . . .
4.6 Experimental validation of imaging . . . . . . . . . . . . . . . . . .
4.6.1 Im age Processing . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Finger and phantom imaging . . . . . . . . . . . . . . . . .
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High-Extinction Complex-Conjugate Ambiguity Removal
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Experimental system design . . . . . . . . . . . . . . . . . .
6.2.1 Polarization-based demodulation circuit . . . . . . .
6.2.2 OCT system . . . . . . . . . . . . . . . . . . . . . . .
6.3 Mathematical framework describing errors and error-correction
optical quadrature demodulation circuits . . . . . . . . . . .
6.4 Calibrating the optical demodulation circuit . . . . . . . . .
6.4.1 Coherent fringe averaging . . . . . . . . . . . . . . .
6.4.2 Correcting only spectral errors . . . . . . . . . . . . .
6.4.3 Correcting both spectral and RF errors . . . . . . . .
6.5 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Im aging . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6
Novel High-Speed Subsampled Laser
5.1 Relevant work . . . . . . . . . . . . . . . . . . .
5.2 Laser operating principle . . . . . . . . . . . . .
5.2.1 Dispersion compensation . . . . . . . . .
5.3 Practical considerations . . . . . . . . . . . . .
5.3.1 Intensity modulator pulse synchronization
5.3.2 Polarization-mode dispersion . . . . . . .
5.4 Laser Perform ance . . . . . . . . . . . . . . . .
5.4.1 Subsampled operation . . . . . . . . . .
5.4.2 Continuously-swept operation . . . . . .
5.4.3 Coherence length measurements . . . . .
5.4.4 Future laser modifications . . . . . . . .
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7 3D Camera Imaging
7.1 Hardware system integration . . . . . . . .
7.1.1 Acquisition configuration . . . . . .
7.1.2 Microscope . . . . . . . . . . . . .
7.2 Performance characterization . . . . . . .
7.2.1 Coherent averaging . . . . . . . . .
7.3 Image processing . . . . . . . . . . . . . .
7.3.1 Complex conjugate demodulation in
7.3.2 Dispersion removal . . . . . . . . .
7.4 Imaging . . . . . . . . . . . . . . . . . . .
7.5 Future Work . . . . . . . . . . . . . . . . .
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10
Chapter 1
Introduction
1.1
Applications of OCT
Optical coherence tomography (OCT) is a high-resolution, three-dimensional imaging
modality that uses infrared light to probe depths within tissues [15, 441. For many applications, OCT is advantageous because it offers a resolution and penetration depth that
is not achievable by other modalities. Figure 1-1 shows a comparison of various imaging techniques. Confocal microscopy provides the highest resolution, however it is used
primarily as a research tool because of its limited penetration into tissue and because of
the challenges associated with implementing it clinically. On the other hand, technologies
like ultrasound, CT, and MRI have low spatial resolutions in standard clinical practice
and cannot visualize the microstructure of tissues. With a resolution on the order of a
few pm, and a penetration depth on the order of 1-2 mm in highly scattering tissues (i.e.
skin, GI tissue), OCT offers a new medical diagnostic and disease monitoring technique.
The contrast in OCT is provided by intrinsic variations in tissue scattering based on inhomogeneous optical index of refraction and therefore does not require exogenous contrast
agents. This enables non-invasive three-dimensional visualization of tissue morphology as
well as depth-resolved functional imaging. OCT is often compared to histology because
it is on a similar size-scale and in fact one of the original goals of this technology was to
perform optical biopsy
1451.
11
1o 00-Q-0p
1 cm
0
c
0-
-
10 cm
mm
Research
Imaging
ex vivo
U
150 PM
300 pm
Entire
body
b
resolution
Medical
imaging
in vivo
1mm
Figure 1-1: A schematic comparison of OCT and other imaging modalities based on
resolution, penetration depth, and clinical utility.
The earliest time-domain OCT systems (TD-OCT) focused on applications in ophthalmology
[15]. In 1993, the first in vivo tomograms of the human optic disc and macula
were demonstrated [411. With extensive research effort over the following decade, longer
wavelengths and higher power lasers gave rise to imaging of optically scattering and nontransparent tissues
[4].
Ex vivo investigation of OCT was conducted in a variety of organ
systems including cartilage, gastrointestinal tissues, upper respiratory tract, and and uro-
logic tissues [4,19,36,37,44,45]. All of these studies showed that OCT has strong clinical
potential for a wide range of diseases and organ systems.
In epithelial cancers, for in-
stance, disruption of cellular organization beneath the surface of the tissue can provide
indicators of dysplasia.
Figure 1-2A is an OCT image of a segment of normal human
esophagus, with Figure 1-2B showing the corresponding histology. In this health tissue,
the organized structure of the tissue layers is apparent in the OCT image, and this organization is disrupted in the case of sub-squamous Barrett's epithelium as shown in the
12
Figure 1-2: Representative OCT images of human esophagus and corresponding histology.
A & B) Normal esophagus with squamous epithelium (SE), lamina propria (LP), and muscularis mucosa (MM) are clearly visible on the OCT image. C & D) Barrett's esophagus
with disrupted architecture and multiple subsquamous Barrett's epithelial (SBE) glands
beneath the SE are visible on both histology and OCT image [7].
OCT and histology images of panel C and D
17].
Evidently, this technology can have a
huge impact on the standard-of-care for disease screening, monitoring, and treatment.
In recent years, advancements in functional OCT have further broadened the potential
for OCT to make a clinical impact. Angiographic OCT, for example, has been used to
visualize flow in many systems including retina vasculature, skin vasculature, and tumor
models [24,51,62]. There have been several algorithms used to perform angiographic OCT
including Doppler methods and speckle decorrelation methods 127]. Figure 1-3A shows
an image of a tumor micro-vasculature in a mouse brain imaged with a Doppler OCT algorithm. Polarization-sensitive OCT (PS-OCT) is another functional OCT method that
adds contrast for tissue composition. It detects the depth-dependent changes in the po13
Figure 1-3: A) OCT angiography projection of vasculature in a mouse brain with a
glioblastoma tumor (Scale: 500 pm) 1511. B) OCT generated birefringence map of chicken
muscle (ROI(m)) and tendon (ROI(t)). Colorbar: 0-2 deg/pm [611.
larization state of light through a sample; Figure 1-3B is an example of a PS-OCT image
showing the local birefringence of tendon/muscle junction [61].
1.2
Practical limitations of OCT
Since OCT is based on fiber optics, it can be incorporated into many existing in vivo
imaging modules, i.e. endoscopes. Initially, however, clinical imaging was only performed
on external organ systems that were easy to access such as the skin, oral cavity, and
eye. With the introduction of catheter-based fiber optic probes around 1997, imaging
of internal organs first became possible
[451.
These initial systems were limited in their
utility due to a combination of small imaging fields, motion artifacts, and difficulty meeting
geometric constraints in the organs. The potential of in vivo imaging drove research efforts
in following years to focus on: 1) longer imaging ranges that reduce sensitivity roll-off due
to organ geometry
[33],
and 2) higher speed laser sources that enable real-time image
rendering and minimize motion artifacts [17,28, 421. Although some improvements have
been made in these areas individually, it is the combination of high-speed and long-range
imaging that enables wide-field, camera-like imaging with OCT.
14
Sd&VbW
Famaimod SW
P
Balm
W
ale
ca
C
Figure 1-4: Top: A balloon catheter centration mechanism that allows for circumferential
imaging of the esophagus. Botton: Physical implementation of the balloon probe [521.
1.2.1
Long-Range Imaging
The first generation TD-OCT systems relied on a translating reference arm for depth
scanning. However TD-OCT was impractical for clinical imaging because of slow imaging
speeds and low sensitivity. Consequently, motion artifacts were severe and imaging could
only reliably be performed on small volumes [15, 19, 361. The introduction of Fourierdomain OCT (FD-OCT) obviated the need for a translating reference arm and instead
relied on the laser source for imaging speed. In these systems, however, the coherence
length, or length over which light in a sample arm is well correlated with light in a reference
arm became a relevant parameter in determining the imaging range [46, 59]. Until some
recent advances in swept laser technology, the limited coherence length of OCT lasers
required tight control of the distance between the imaging probe and the tissue surface.
If the tissue was located more than a few millimeters from its ideal location, the OCT
system rapidly lost its sensitivity and the tissue could not be imaged. For this reason,
existing clinical applications of OCT use careful engineering to meet this low depth range
criteria. In endoscopic OCT, for example, the smooth and tubular nature of the organ
15
(a) esophagus
(b) duodenum
Figure 1-5: (a) Endoscopic OCT image of esophagus obtained with a balloon catheter.
G = gland, MM = muscularis mucosa; (b) Analogous image of the duodenum where
comprehensive imaging is difficult because of villi and uneven surfaces [401.
allows imaging through a balloon-centration catheter, which arranges the tissue within
the coherence length with millimeter-level accuracy (Figure 1-4) [52].
Organs with more complex geometries, or clinical applications that require wide-field
imaging cannot be accommodated. For instance, because tissue along the intestines are
have irregular crypts and varying diameters at different sections, balloon catheterization
is not as effective in centering the imaging probe. Figure 1-5b shows a cross-section of the
duodenum that is imaged with the same balloon catheter as in the esophagus. The left
side of the duodenum has fallen beyond the imaging range of the system and these regions are potentially missed during screening of disease. If the imaging probe is moved to
visualize the left region of the duodenum, the right edge is out of the field-of-view (FOV).
This example demonstrates the difficulty of achieving comprehensive imaging without
longer imaging ranges. Until now, the limited capability to acquire long-range signals has
slowed efforts to explore long range laser sources, however some recent lasers have demonstrated multi-cm scale coherence lengths [33,391. As more sources are demonstrated with
these capabilities, new clinical and industrial applications of OCT based on simultaneous
high-speed and multi-cm depth ranges can be envisioned.
16
B
Figure 1-6: A) A longitudinal cross-sectional image of a tissue with arrows pointing to
location of motion artifacts. B) A three-dimensional rendering of the esophagus image
showing severe motion artificts in the left section 152].
1.2.2
High-Speed Imaging
High-speed imaging is important for minimizing motion artifacts during the imaging session, as well as for real-time image rendering. Most tissues in the body do not remain
stable for prolonged periods of time, and are subject to various motions, whether from
breathing, heart beating, peristalsis, or other biological functions. Significant motion of
the tissue relative to the imaging probe within this time induces artifacts in the image
that are difficult to remove in the processing stage Figure 1-6a). To minimize these artifacts, tissue must be immobilized for the duration of the imaging session; while this
is straightforward in external tissues, it poses a larger challenge in internal organs. In
the esophagus, the aforementioned balloon catheter provides limits motion during the
imaging procedure, however, some motion is unavoidable even in these applications (Figure 1-6b) [52].
Furthermore, stabilization via a balloon catheter cannot be applied to
many other internal organs, and as such motion artifacts are prohibitively large.
In OCT, the A-line rate (given in Hz) is the number of axial scans that can be completed in one second. The faster the A-line rate, the faster one depth scan is acquired
17
Table 1.1: Approximate OCT Imaging Times for Various Volumes
A-line Rate
1 kHz
100 kHz
1 MHz
10 MHz
20 MHz
1cm x 1cm x 2mm
1.1 hours
6 mins
4s
0.4s
0.2 s
5cm x 5cm x 2mm
27 hours
2.7 hours
1.7 mins
10 s
5 s
and thus less motion can occur during that interval. Table 1.1 provides some exemplary
values for the time it takes to acquire a 1cm x 1cm x 2mm or a 5cm x 5cm x 2mm
volume with different A-line rates. Realistically, in vivo imaging should be performed
in a fraction of a second since cardiac motion occurs on the order of once per second.
In 2004, swept-wavelength OCT imaging was demonstrated at 10 kHz A-line rates [57].
Since then, multiple new swept-wavelength technologies have been developed and laser
speeds have increased to the order of MHz [17,31]. With this increase in speed, in vivo
imaging is becoming easier and more informative.
1.2.3
Acquisition Bandwidth Limitation
In standard Fourier-domain OCT (FD-OCT), the frequency of the signal you must acquire scales linearly with how far away you tissue is from your imaging probe. It also
scales linearly with how fast you image, thus when the requirements of high-speed are
combined with those of extended depth range imaging, current acquisition electronics are
unable to accommodate the resulting signal bandwidth. Figure 1-7 is a schematic of the
physical depth space (top) and the corresponding RF space that maps the OCT signal
(bottom). In this example, assuming a 20 MHz Aline rate (high-speed laser), a tissue approximately spanning the physical space of 3.7 mm - 4.3 mm results in an RF bandwidth
approximately spanning 9.75 GHz - 10.25 GHz. This RF signal is well beyond what we
are able to acquire with modern digitizers and because of this we are currently limited by
the acquisition bandwidth this limits the depth range over which we can image our tissue.
18
Air/saline
OCT
Tissue
Attenuated tissue
b~
mm
4mm
2mm
mm
8MM
depth
co
(D
0
I
0 GHz
5 GHz
10 GHz
15 GHz
20 GHz
RF Bandwidth
Figure 1-7: Top: A schematic of tissue placed at a 4 mm depth away from the imaging
probe. Bottom: The corresponding tissue signal in the RF bandwidth based on simulated
tissue signal spanning a 1 GHz range. The grey shaded area is not acquirable by current
acquisition electronics.
19
In this work, we demonstrate a method to dramatically reduce the acquisition bandwidth required for extended depth range imaging, thereby enabling high-speed and long
depth range OCT with current acquisition electronics. Notice in Figure 1-7 that although
the tissue signal occupies a high RF frequency space, the bandwidth of the tissue signal
occupies a small region of the total RF bandwidth. This is because the penetration depth
of light into tissue (referring to how far into the tissue light travels) is much smaller than
the depth range of imaging. This penetration depth depends on the wavelength of the
light, the output power of the light source, the sensitivity of the imaging system, and the
scattering of the tissue. Typically for highly scattering tissues such as skin or esophagus,
penetration depths are ~2-3 mm. In this work we take advantage of the sparsity in the
RF space to compress the tissue signal into a lower baseband frequency. Our approach is
based on modifying the optical sampling approach in OCT so that wavelengths are discretely instead of continuously sampled. With this subsampling method, the acquisition
bandwidth is no longer limiting the depth range, and we can acquire tissues along the
entire coherence length of the source.
1.3
Thesis Organization
The goal of this work is to create a new platform technology that enables three-dimensional
camera-like imaging with OCT. This requires fundamental changes to the laser, interferometer, and signal processing, so that long-range, high-speed, and wide-field imaging can
be performed with minimized acquisition bandwidths.
Chapter 2 describes the funda-
mental background of OCT and section 2.4 focuses salient concepts that were utilized
extensively in this work. Chapter 3 introduces the theory behind incorporating optical
subsampling into OCT, and makes the connection between subsampling parameters and
standard imaging parameters. Chapter 4 describes our proof-of-concept set-up and our
experience with the first subsampled imaging system. Chapter 5 describes the design
and performance of our novel high-speed dispersion-based subsampled laser. Chapter 6
describes how we removed complex-conjugate artifacts in our images by developing a
20
new method of high-extinction quadrature interferometry. In Chapter 7, we integrate the
subsampling concept, our novel dispersion-based laser, and our novel quadrature interferometer to acquire unprecedented wide-field images with our camera-like OCT system.
21
22
Chapter 2
Fundamental Concepts in OCT
The fundamental structure of OCT systems consists of three major components: a light
source, an interferometer, and a data acquisition/processing unit. At the core of OCT
theory is the concept of light interferometry.
This chapter begins by introducing the
concept of interferometry in the context of time-domain OCT (TD-OCT). The evolution
of OCT into the Fourier-domain is then described as well as some prevailing concepts that
this work builds upon.
2.1
Monochromatic interference
Although monochromatic plane waves are never found in nature, they provides a good
model for studying phenomenon like light interference. In OCT, the Michelson interferometer is employed as an essential tool to indirectly measure backscattered light from
different depths within a sample. The light is otherwise traveling too fast for photodetectors to acquire. A common schematic of this interferometer is shown in Figure 2-1.
Light that is generated from a laser source enters the beam splitter (BS) and is divided
into the reference arm and the sample arm. The light that is backreflected from each
arm recombines at the beam splitter and the interference of these two beams is received
,
by a photodetector. Assuming that the monochromatic source has a wavenumber k0
and that both the reference and sample arms have a single reflector located at
23
ZR
and
I
reference
E
Z
ER ZR
AZ=Z -Zs
BS
laser D ----- -------
E= ER+ E
PetorI
Figure 2-1: A schematic of a simple Michelson interferometer where the single-pass reference arm distance is ZR and the sample arm is zs.
zs respectively, the complex electric field amplitude of the light in the reference arm is
and in the sample arm Es(z) = Esejkozs. When the light recombines
at the beam splitter the total electric field is the superposition of these electric fields
R(z)
EReikozR
(following the linearity of the Helmholtz equation) [13]:
ET(z)
=
ERe 2 jkZR
+ Es.2kozs
(2.1)
The factor of 2 results from the double-pass travel in each arm of the interferometer. For
simplicity, the amplitude changes and phase delays induced by the components in the
optical beam path are ignored. Because photodetectors detect the irradiance (energy per
unit area per unit time) rather the the electric field, the interference is expressed in terms
of average intensity [131,
I
=(_E
= IR
z2
+ Is + 2
(2.2)
IKIs cos(2k, A z))
24
where ()T denotes time average over time T, which is chosen to be much longer than an
optical period. AZz = z-
zS refers to the optical path difference between the sample and
the reference arm as shown in Figure 2-1. Thus the intensity of the total electric field
at the detector is a sum of time-independent and depth-independent DC terms and an
interference term that sinusoidally varies with path length differences. The latter term
forms the basis of backscattered light detection in OCT.
This equation makes sense intuitively because varying the optical path difference Az
causes the sample and reference arm waves to alternatively constructively and destructively interfere with each other. The average intensity does not vary with time, and this is
also true for stationary polychromatic waves as we will see later [5,131. Note that in this
case of monochromatic light, the wavenumber can equivalently be expressed as ko = o
and the time it takes for the light to travel a distance
ZR
is given by tR = ZRa and simi-
larly for the sample arm, ts = zsg. When considering low-coherence light, the properties
of the material through which light propagates becomes important in determining the
dispersion relation. Hence,
Az =
where r
tR
-
TC
(2.3)
ts represents the time difference of travel between the two reflectors. Thus
the intensity can also be expressed as
I
IR+
[5],
Is + 2IR
1
S cos(2w0 T))
(2.4)
This provides a more convenient representation when describing polychromatic or low
coherence waves in following sections. Interestingly the DC terms also represent an interference, however because it is the interference of reference and sample arm light with
itself, it is always constructively interfering because Az = r = 0.
If the reference mirror were scanned back and forth in time, Az(t), then the interference oscillations can be detected with a photodetector, which converts the irradiance to
25
an analog current based on the following:
idet(T(t))=
where p =
q
14,51,
p[ PR + PS + 2
PRPS cos(2wr (t))]
(2.5)
is the responsivity of the detector (units Amperes/Watt), r is the quan-
tum efficiency of the detector, q is the quantum electric charge (1.6 x 10- 19C), and hV
is the photon energy. PR and Ps are the powers detected by each reflector in the sample/reference arm and are proportional to
IR
and Is multiplied by the receiving area of the
photodetector. Notice that the amplitude of the signal is proportional to the product of
the magnitude of the reference and sample electric fields, implying that a weak backscattered field from the sample can be amplified by mixing with a strong reference field. In
this hypothetical monochromatic case, the interference oscillations will be observed for
infinitely wide path differences (Figure 2-2a), which has limited utility in OCT since we
are interested in measuring intensity at a particular location in the sample field. This is
why broadband light sources that produce low-coherent light are used.
2.2
2.2.1
Time-domain OCT (TD-OCT)
Correlation functions
Low-coherence or broadband polychromatic light cannot be assumed to be a time-independent
deterministic complex function; a randomness is introduced, which gives the wave function
a dependence on time and position and requires statistical methods to describe. First, we
can think of polychromatic light as a superposition of monochromatic waves. Since the
wave equations are homogeneous linear partial differential equations, any linear combination of a solution is also a solution. The complex wave equation can thus be expressed as
a Fourier integral,
E(z, t)
j
Eo(z, w)eil3 zeiwtdw
26
(2.6)
(b)
(a)
--
+
+--
-M
AIFWHM
At
Finite coherence length
Infinite coherence length
Figure 2-2: (a) Interference fringe resulting from a source with infinitely long coherence
length. (b) Interference fringe resulting from a source with a short coherence length.
#
where -w and +w are the lower and upper limits of the spectral bandwidth, Aw, and
is the propagation constant defined as:
(w) = n(w)-
=k(w)
(2.7)
In the case of monochromatic light, there was only one frequency, wO so that O(w) =
nr(wo)w 0/c = k,. This more general representation accounts for propagation through dispersive materials where the index of refraction is frequency-dependent.
As before, the intensity of light is given by the absolute square of the complex wave
2
function, however, for broadband light E(z, t) is a function of time as well, (E(z, t) ).
Its instantaneous intensity is random and the average intensity must be taken
-
( E(z, t)I 2
)
I(z)
[13,34]:
(2.8)
where (-) represents the ensemble average over many realization of the random function.
Assuming that the partially coherent waves in OCT are stationary, meaning that the
average intensity is constant over time, this intensity becomes a function of position only,
1(z). Referring again to the Michelsoni interferometer of Figure 2-1, if a reference arm has
27
depth of
ZR
and a sample arm depth of zS, the mutual coherence or correlation function
is given by:
g(zR, zs, 7)
where r =
at a fixed
(ZR - ZS)/C
ZR
T =
(2.9)
is the double-pass time delay. Since we are evaluating this function
and zS the simplified correlation function is written as:
gRS(T)
where gRR(T)
(_E*(zR, t)E(zs, t + T))
=
=
(ER(t)*ER(t +
T))
=
(E
(t)Es(t + T))
(2.10)
is the case of autocorrelation, which in the case of
0 reduces to intensity IR and similarly Is for the sample arm. After normalizing
gRs(T) we arrive at the complex degree of coherence or normalized correlation function:
9RSRT gRs(T)
gss()
V"I
~RS(T) =
V gRR (0)
(2.11)
Recall from Eqn. 2.8 that we have assumed a stationary wave, and since we are evaluating
at a particular depth, we have dropped the z-dependence for convenience. Notice that
Jgj(-r)j is the normalized shape of the correlation function and does not account for the
magnitude of intensity of light in the system [34]. Instead the magnitude of intensity
coming from the source, and the fraction of it that reflects back from the reference and
sample arms are combined in the terms IR and Is.
In the case of autocorrelation of deterministic and monochromatic light, jj(T) = exp(iwor)
so that gi (r)j, the degree of correlation = 1 for all T. However if light is not monochromatic, gii(T) I drops to a value of 1/e at a delay, Tc or the coherence time. The coherence
time is related to the coherence length through the relation:
Ale = -c
n
(2.12)
In OCT, this length is a very important characteristic as it directly relates to the axial
resolution of the imaging system [10, 34]. In the next section we will establish that the
28
coherence length relates to the spectral bandwidth and shape of the light source.
2.2.2
Gaussian sources
We established the intuition that monochromatic light is always perfectly correlated, thus
r, = oc. Hence we can imagine that the more polychromatic our light becomes, the
faster it becomes uncorrelated and r, is smaller (portrayed in Figure 2-2b). The WeinerKhinchin theorem formally establishes this relationship, saying that the autocorrelation
of a stationary random process is related to the power spectral density, S(w) through a
Fourier transform relationship:
S(W)
g(T)e-wdr
=
(2.13)
S(w) is the average power per unit area per frequency (W/cm 2-Hz), or average intensity
per frequency. This implies that the wider the source bandwidth (most commonly defined
by its full-width-half-maximum, FWHM, value), the narrower the autocorrelation. And
the shape of g(r) is determined by the shape of the power spectral density. In OCT, ideal
broadband sources have Gaussian shaped power spectrum,
S(
- wo)xexp[or
1
(2.14)
(P - WO) 2
2o.2
27
where IS(w - w o ) is the normalized shape of the spectrum. The source FWHM bandwidth
is given by
w = 2a/
2 1n2. This yields a correlation function that is also has a Gaussian
shape:
()
1
exp(-
o7, V2 -7
r
2a;
)
(2.15)
where I (T)j is the shape of the correlation function and a, is the standard deviation of
Gaussian envelope. The FWHM of
|(T) I as
a function of coherence time of the light
source is given by m, = 2ou/2 1n2. Using that oxax
=
between coherence time and spectral width Tc
for Gaussian sources. The axial
= 2 "2
1 we arrive at a relationship
resolution, 6z, of double-pass light path through the sample is !Al,, leading to the axial
29
resolution expression:
41n2 c
n Aw
6z
A22.6
_21n2
0
irn AA
(2.16)
where A 0 is the center wavelength, AA is the spectral bandwidth and n is the refractive
index of the sample [4]. Therefore, it is clear that in order to have high axial resolutions,
OCT sources must have high spectral bandwidths. It is noteworthy that sometimes the
power spectral density function has more of a rectangular shape, and in this case the axial
resolution can be calculated by 6z = A2/(2nAA)
2.2.3
f5,10,341.
Low-coherence interferometry
Interference with single sample reflector
As in the monochromatic case, we derive the intensity of the interference of two partially
coherence waves:
I(T) =(ER(t
IR
where
p(T) = arg{Rs(T)
) + Es(t + T)1 2 )
+ Is + 2VIRIS
-- aRS(T) -
(2.17)
gRS(T)j Cos(pRS(r))
wr and where again w, is the center frequency of
the spectrum and aRS(T) is a phase shift that is independent of frequency. Notice that the
intensity is directly related to the correlation function,
g(T),
and the normalized correla-
tion function, g(T), and the interferogram is a harmonic function with a frequency that is
proportional to the center frequency of the optical broadband spectrum. If
qRs(T)I =
1
for all T we have the case of monochromatic light and we arrive at Eqn. 2.4. If
gSRs(T)I =
0
we have completely incoherent light, and if 0 < J Rs() 1< 1 the light is partially coherent
and IgRs(T) itself represents the degree ofaoherence. As stated before, the r which causes
IRs(-r) ; 1/e is the coherence time,
T,
and is proportional to the OCT axial resolution
Interference with multiple sample reflectors
The interference equation in Eqn. 2.17 was derived with the assumption that there was one
reflector in the sample arm, however in tissue there are multiple backscattering positions
30
zi with time delays ri = (zi
-
ZR)/c.
Each reflector results in a cross-correlation with the
reference arm light, as well as a self-correlation from different depths within the sample. In
this multiple backscatter case, we can rewrite the intensity as I(T) = (|ER(t) +f_+o E (t+
Tj)
dril2), where Ej is the wave back-reflected from the sample position zi. The sample
arm light is represented as a continuous sum of backreflected field of light arising from
different depths within the sample arm, hence the integral ranging from
oc. Following
Eqn. 2.17 we can write the intensity of the interference as a function of time delay:
I(T) =
(ER(t) E t
+ f(E(t
+ f
+
E (t + -)Ej *(t + T2)) dcr
+Ti)Ej(t + -j)*) + (Ej(t + T)*Ej(t +Tj)) drdT,
(ER(t)Ei(t + T )*)
(2.18)
(ER(t)*Ei(t + Ti)) dr
As in the case of a single reflector, the first term accounts for the interference of reference
arm with itself and contributes a DC term. The second accounts for the sum of all selfinterferences within the the sample reflectors, which arise from the same delay,
T
within
the sample and also only contributes a DC term. The third term refers to the interference
of sample reflectors arising from different depths within the sample, only for the case where
Tj
-Z
mj.
And the last term, of greatest interest in OCT, represents the interference of
the sample reflectors with the reference arm light. The meaning of this equation becomes
more clear when it is cast in terms of the normalized correlation function, g r):
I(r) = gRR(T = 0)
+ j
ii(r
+
R{gj(w
=
+ 2
iR{gi(T =
i)
ri
=
-
0) dT,
Tj)} dridrj
(2.19)
d-ri
Note that T = Tr - Tj = (zi - zj)/c and represents the time delay between two sample
reflectors while ri
= (zi - ZR)/c
and r = (zj - zR)/c represent delays between sample and
reference reflectors. This expression says that the intensity of light after interference is the
31
sum of sample and reference autocorrelations, intra-sample cross-correlations, and samplereference cross-correlations. As in Eqn. 2.17 we will write the final intensity function in
terms of the normalized correlation function:
I(T) = IR -
Ii di
+ 11
+2
123
i - 7j) I cos[aij - wO(T - T )] drid-rj
(2.20)
VIi IR |47i
r COSI [~iR - wori ] dri
Here again we see that the cross-correlation functions are modulated by a carrier wave
that has a frequency proportional to the center frequency of the light source, w0 , and a
frequency-independent phase term that is a function of optical path delay, ai
(Tr
- r) and
aiR(Ti). Recall that the amplitude of the normalized correlation represents the degree of
coherence and 4(ri)I > 1/e only when T < r. This means that an interferogram is only
present at a small depth location equivalent to Al, as per Eqn. 2.12. The rate at which
(-ri)|Idrops to zero is determined by the shape of the spectral density function S(w) in
Eqn. 2.13. In TD-OCT, it is the amplitude of the last term that is proportional to the
sample reflectivity at a certain depth location within the sample; lock-in detection at the
carrier wave modulation frequency are frequently used to obtain the reflectivity envelope
of the last term. The reference mirror is translated in order to detect different depths
within the sample and create a backscatter map at each point in the sample. Notice that
the intra-sample cross-correlation term did not exist as long as we were only considering
a single reflector within the sample. However, because of the low backscattering intensity
from the sample their interferometric gain,
hJj, is often negligible compared to the
heterodyne gain, /i=R, of the sample-reference interferogram. Additionally, in TD-OCT
there is only a small range where
(Tr) I = 0 so the sum of intra-sample cross-correlation
contributions to the third term are smaller, hence this term is often ignored. Assuming a
Gaussian source, and assuming that the reference mirror is moved at
32
Tr(t),
the detector
current for TD-OCT can be represented as:
(2.21)
idet[ri(t)] oc 2 p - 1g(Fi(t)) V/PRPs cos(2wTi(t))
where again p is the responsively of the detector (Ampere/Watt) and PR and Ps are optical
powers reflected from the reference and sample (respectively) onto the photodetector.
2.3
Fourier-Domain OCT
The transition from time-domain OCT to Fourier domain (FD-OCT) followed closely from
the development of optical frequency domain reflectometry (OFDR). This major technological advancement for OCT imaging gave way to improved detection sensitivity, as well
as increased imaging speeds as a translating reference mirror was no longer required [57].
In the frequency domain, full sample depth structure is encoded as a depth-dependent
modulation of the broadband light. This follows again from the Weiner-Khinchin Theorem (Eqn. 2.13) because it says that the depth profile can be obtained from the Fourier
transform of the power spectral density function without the need of changing the optical
path length in the reference arm. If we start with the Fourier transform pair:
(2.22)
SiR(w)eir dw
giR(T) =
and apply it to Eqn. 2.19, express the cross-spectraldensity as SiR(w)
wr)], with aiR(w)
quency.
WT
=
SiR(w)I exp[i(aiR(w)-
arg {9iR(W)}, we arrive at the intensity as a function of fre-
Although this function is intrinsically complex, we measure only the real part
and thus express it as:
I(w) =ISRR(w)l +]
+ 2
I
+00
f+00
S (w)I di +
0SiR(w) cos
[(aiR - wi)
+ O
[
ISi (w) Icos (ai -
w(Ti - T))J dTi dTj
dTi
(2.23)
33
......
...
where again, ri = (zi - zR)/c.
Unlike in Eqn. 2.20, where the interferogram had a
modulation that was proportional to the center frequency of the light source, wO, now
the interferogram modulation is a function of optical path delay, Tr. Information from
all depths are contained within this intensity function, hence the reference arm does not
need to be translated. This serves an improvement in sensitivity as we will see later in
section 2.3.2. The depth information in the time-domain can be obtained by taking the
inverse Fourier transform of Eqn. 2.23,
I(T)
F 1 {I(w)}
=gR(0)
+
+ 2
+
gii(0)
f{[Tgij[T+
{YiR(T
di
(Ti
-
Ti)]
+gij[T
-
(TZ -
T3i)]
dTidT
(2.24)
dri
- Ti) + giR(T - Ti)
This expression is analogous to Eqn. 2.20 in TD-OCT. The first two terms correspond
to the unmodulated DC intensities resulting from the reference arm reflection and the
sum of back-reflected intensity from all scattering sites within the sample.
The third
term is the undesired intra-sample cross-correlation contribution; note that in TD-OCT
this only resulted from sample scatterers within the coherence time, Tc, however now this
term can contain intra-sample cross-correlations from the entire sample depth. Only the
sample-reference cross-correlation term in the fourth term contains information about the
backscattering coefficient and the sample structure. The correlation function is centered
about the variable r and has a value g(TTry)l ; 1/e only when r tril
<
. In TD-OCT
the correlation function was always assumed to be centered about zero delay
(T =
0) and
the reference arm was moved to scan the sample depth. Now the intensity function inherently contains information about all depths (within a certain sensitivity roll-off as we
will see in section 2.4.1). Also, the intensity signal is symmetric about the zero delay,
hence two scatterers located at zi
ZR -
Az and zi =
ZR
+ Az will result in the same
frequency modulation. To avoid this, the sample can be placed such that the surface and
entire penetration depth lies on one side of the zero delay. Alternative ways to avoid this
34
.Wh.
I
I
1
I'll _
_YAL
I
I
overlap are discussed in Section 2.4.3, and is a major consideration in subsampled imaging.
For convenience, the sample-reference cross-correlation term is often written as a convolution,
IiR(
(T) I 0iR
IiR (7) -
- SR(W)-
())
(T)'
(2.25)
where IS(w)I is the shape of the source spectrum, I'(T)I is the shape of the correlation
function (this does not have a dependence on the reflectivities). We have defined
SiR(w)'
as
the interference modulation of the spectrum scaled by the sample and reference reflected
intensities (IR and Is). And similarly
iR(T)'
is the Fourier transform of SiR(W)' and
represents the location and intensity of the reflector at each depth within the sample.
I+00
SiR(w)'
giR(T)
VfIRIi - cos(wTi) dTj -2
-
j
(2.26)
6
'RIi [ iR(T
IR and Is are reference/sample reflectivities,
6
iR
+
Ti)
+
6
iR(T
-
Ti)] dT
is the Dirac delta function, and 0 denotes
the convolution. Eqn. 2.25 highlights that the in the time-domain, the signal of a single
sample reflector is a delta function with a shift proportional to optical path delay, Ti, an
amplitude proportional to the the sample/reference reflectivities, and an axial resolution
proportional to the envelope of the correlation function, similar to TD-OCT as was shown
in Section 2.2.2.
The signal described by Eqn. 2.23 can be obtained either by spectral-domain OCT (SDOCT) or by swept-source OCT (SS-OCT), as depicted in Figure 2-3. The first technique
is to use a continuous wave (cw) broadband light source and detect the spectral components of the power spectral density function by separating the optical frequencies with
a spectrometer at the interferometer output. This technique is termed spectral-domain
OCT (SD-OCT). Another technique is to use a tunable laser with a narrowband linewidth,
where the center wavelength is swept with time over a broadband range (i.e. spanning
35
I-
o
Spectrometer
Broadband
Light Source
Itreoee
A
Interferometer
I-C.).
o
Broadband
BrLight Source
TnbePoo
Filter
DAQ
detecr
Interferometer
Figure 2-3: Schematic of swept-source (SS-OCT) and spectral-domin (SD-OCT) implementations of FD-OCT.
100nm centered at 1550 nm). This is termed swept-source OCT (SS-OCT) and the advantage of it is that the interferogram can be obtained with a single photodetector. The
subsampling concept we have introduced in this thesis applies to both SS-OCT and SDOCT setups, however, we focused primarily on SS-OCT.
2.3.1
Swept-source OCT (SS-OCT)
In SS-OCT the spectral interferogram is captured sequentially by recording the signal
with a single detector while tuning the frequency of a narrowband laser. The detector
current can be written as,
idet[W(t)] = P
(W(t)) PR, + P I
+ p 5(w(t))
+ 2p I (w(t))
W(t)) IJ+0Pi dri
j/fFP
3
- cos[w(t)2(Tr
-
Tj))] dTidTr
(2.27)
/ PR-i - cos[w(t)2r] dc1T
where again p is the responsivity of the detector and the factor of 2 in the modulation
results from the double-path travel of light in the Michelson interferometer. In the simplest
case of linear tuning, where w(t) is varied linearly in time with a constant slope a'
-
dw
(units Hz/s) then w(t) = w0 + a't where w is the lowest frequency in the spectral profile.
36
Again, the last term is the sample-reference cross-correlation.
P P -cos[2wri + 2Tia't] dTj
idet[w(t)] oc 2pjS(w(t))
(2.28)
The frequency of the detector current is directly proportional to the tuning slope and the
optical path delay,
fdet
=
2Tri'
(2.29)
27r
The tuning rate is often expressed in terms of the "A-line rate" (expressed as fA) and
a' =
(2.30)
Duty cycle
where /w is the FWHM spectral bandwidth of the source as we referenced in section 2.2.2
for Gaussian sources. Sometimes the duty cycle of swept-sources are not 100% so the Aline
rate must be divided by the duty cycle to achieve an "effective A-line rate". The frequency
can be rewritten in terms of this rate:
2TrAw -(fA)
27 (Duty cycle)
2AzAw - (fA)
27c (Duty cycle)
2AzAA - (fA)
A2(Duty cycle)
where A A is the FWHM spectral bandwidth expressed in wavelength and AO is the center
wavelength. Thus it is formally shown that the interference fringe frequency is directly
proportional to the A-line rate of the swept source and the depth range that you are
imaging over [10.
Since modern digitizers are limited in bandwidth, simultaneous long range and high speed
imaging cannot be performed without compression (Figure 1-7 has already suggested this
in section 1.2.3). It is easily shown that for SS-OCT, the sampling interval is related to
the fringe frequency,
fdet
and the digitization frequency,
Aw fdet
fdig
37
fdig,
(2.32)
where 6w, is the electronic sampling interval induced by digitizing the interference fringe.
We will see in section 2.4.2 that if this interval is not small enough, i.e. because the
digitization rate is not fast enough, it can limit the depth range of the OCT system.
Nonlinear tuning
We have established that the optical properties of the light source have an impact on
imaging parameters like axial resolution. In SS-OCT, it is not only the optical properties
of the source that effect the imaging parameters, but also the manner of sweeping. For
instance, we assumed above that the laser sweep was performed linearly in time, however
it is often the case that the optical frequencies are swept nonlinearly in time:
w(t) = wea't + a"t 2 + . . + a&t"
(2.33)
If we limit ourselves to the case of linear chirp, then the detector current becomes
idet[w(t)] oc 2p|S(w)jI
PRPi - cos[2Ti(w, - alt + a"t2
dTj
(2.34)
Figure 2-4A shows a simulated swept interferogram with no chirp (black) and one with
linear chirp (blue) wherein the phase of the interferogram changes linearly with time in
the latter case. In Figure 2-4B we take the Fourier transform of those interferograms and
we see that linear chirp causes broadening and shift of the point spread function (PSF).
When the same linear chirp is applied to an interferogram with a higher frequency (red),
there is a delay-dependent broadening of the axial resolution. This causes broadening of
the correlation function with depth [4,101.
2.3.2
Sensitivity advantage over TD-OCT
The sensitivity of an OCT system refers to the minimum reflectivity in the sample arm
that provides a detectable signal; this term encompasses the entire system ranging from
the laser, interferometer, detector, and acquisition card. In contrast, the dynamic range
38
A
-
Do
0.6
-
04
-
0.2
0
-0.2
-0.4
-
-0.
-0:
100
0
300
200
400
S00
S0
0
time
FFT
S4 I
IB
20
0-
1
9518
8
9
9
2
4
T1
Figure 2-4: A) The non-chirped interferogram (black) has a constant phase whereas the
chirped interferogram (blue) has a time dependent phase. B) The Fourier transform of
the two interferograms shows broadening and shift in the delay space (Ti). The same
linear chirp applied to an interferogram with 2x the frequency shows a delay-dependent
broadening of axial resolution (red).
39
and signal-to-noise ratio (SNR) relate only to the detector (i.e. photodiode, or pixel in
CCD array). In order to take advantage of the full system sensitivity, the range between
the smallest and largest measurable reflection must not be greater than the dynamic range
of the detector. In SS-OCT, this range can be adjusted by selecting an appropriate detector gain p (Ampere/Watt).
We have thus far expressed the detector current only as a function of the interferogram,
however, the true detector signal contains both signal and noise components such that
idet(t) -
is (t) - i-i(t). There are three dominant sources of noise in OCT systems, receiver
noise (i ), relative intensity noise
("2IN),
and shot noise (i2). Receiver noise, containing
thermal noise and dark noise, arises from the amplification and filtering process and is independent of the incident light. Shot noise is the consequence of the the quantized nature
of light and charge and is proportional to the quantum electric charge, q. It is dependent
on the incident light as the square root of the power returning from the sample/reference
(VPR +
Ps), where Ps is the total power returning from all sample reflectors. Relative
intensity noise (RIN) arises from the stochastic fluctuations in the instantaneous source
intensity, and is directly proportional to the power returning from the sample /reference.
The well known expression for the noise power (i2(t)) is
(i2 t))
=
14]:
[i2 + 2p2 q(PR + Ps)+ p 2RIN(PR
s) 2] - NEB
(2.35)
where NEB is the detection bandwidth, and again q is the quantum electric charge
(1.6 x 10-19C), and p is the responsivity of the detector (A/W). A primary goal in OCT
is to have a shot-noise-limited system. While the other sources of noise can be minimized
by high-gain electrical amplification, selecting appropriate reference arm power, and/or
using dual-balanced detection as discussed in the next section 2.3.3, shot noise is fundamental to the detection of the optical interference fringes.
In this shot noise limit, FD-OCT has a significant advantage over TD-OCT. In OCT,
sensitivity is defined as the minimum reflectivity that produces signal power equal to the
40
PTT '
-
,
- ---
_
,
-.-
,
_ , --
ty .....
'.W.,
., -
_ _.' - MVWMOMMM
'M_'
--
'
, --
, " -
-,
"
.' __
. TP1?MTM"MT
I'l I I'll''
11-1,
A-
.........
-
1~-
-1.1-
uUtU.-_
noise power, or when the signal-to-noise ratio (SNR) is equal to one,
SNR - (ij(t)(t))
where brackets
()
(2.36)
1
denote time average. For a shot-noise-limited TD-OCT system, the
signal-to-noise ratio has been shown to be [4j,
SNRTD
7PS
2hv(.NEB)
-
(2.37)
where again q is the quantum efficiency of the photodetector and Ps is the power backreflected from the sample arm. Notice that the NEB is equivalent to the maximum
the detection bandwidth (fdct) because low-pass-filtering can be applied to remove excess
noise. As we saw in section 2.3.1, fdct is inversely proportional to the A-line rate of the
laser (fA) and spectral bandwidth, hence there is a tradeoff between SNR, imaging speed
and axial resolution.
In FD-OCT, there is no tradeoff, offering a significant advantage in imaging speed. Recall
that in FD-OCT, the depth information in the time-domain can be obtained by taking an
inverse Fourier transform. Assuming that N, spectral samples are obtained by digitizing
with an acquisition card with a Dirac comb,
+o
6(w - m - 6w,)
p(w) =
(2.38)
M=-00
then the digitized fringe,
idig(w(t))
is equal to the product of the sampling comb with
the detector current idet(w(t)) and the discrete Fourier transform (DFT) of this yields
[6,23, 571:
Ns-1
idet(W)e-j27r
idig(T) =
(.9
/NS
i=O
Parseval's theorem says that E idi,(T)
Fourier domain is given by
(i (w) 2 )
=
NS E idet (U) 2 , the noise power level in the
= Ns(in(7,) 2 ) while the signal power
41
is(T)2
is zero
everywhere except at
7j.
Thus at each of the peaks, the power is [57],
NT2
1
is(Ti) 2
E
Ns
2
is()
-2)
(2.40)
Therefore the SNR of SS-OCT becomes,
2
SNRFD
Ii (T,)1
-
s
(in(7-)2 )
=
N
-SNRTD
2
(2.41)
This is the SNR of TD-OCT scaled by N,/2. Thus while the noise power is distributed
across all frequencies, the signal power is concentrated at two peaks with frequencies
corresponding to a specific depth in the sample (+Ti) [6]. Note that this relies on noise
currents that are mutually uncorrelated and thus relies on white noise powers adding
incoherently.
2.3.3
Balanced Detection
Balanced detection takes advantage of the z phase shift between the two output ports of
the beam splitter in the interferometer. In the previous, simplified derivation of irradiance,
equation 2.4, amplitudc of phase variations of the beam splitter and mirrors were not
considered. The propagation of light through a beamsplitter induces an amplitude and
phase modulation of the light traveling through it and this modulation can be described
by the matrix [5]:
[li]
(2.42)
[2
Similarly, light reflecting from mirrors or samples result in w phase shifts. Figure 2-5 is
a modified schematic of the Michelson interferometer with two separate detection arms
and input source light of Eso. For simplicity we assume a monochromatic source here.
The the irradiance in detector 1 is:
I1 = ET1E*1 = (ER + ES)(ER + Es)* = I
42
+ Is
+ 2f Inscos(2kAz)
(2.43)
reference
ER
ZR
ET=iER-iEs
zs
ES~
Eso
ET= ER+ Es
Figure 2-5: A schematic of a balanced detection set-up wherein two detectors, deti and
det2, record interference fringes that are 7 out-of-phase. ESO = input source light, ER
= electric field returning from reference arm, Es = electric field returning from sample
arm, E1 = total electric field at DI, ET2 = total electric field at det2.
whereas the irradiance in detector 2 is:
I2= ET2E* 2 =(iER - iEs)(iER - iEs)*= IR
IS --
2
IRIScOS(2kAZ)
To achieve balanced detection, equation 2.44 is subtracted from equation 2.43.
(2.44)
This
detection scheme removes the DC component and reduces the source random intensity
noise (RIN), i.e. noise resulting from mode hopping and/or mode competition in the laser
source. Furthermore, balanced detection can suppresses self-interference noise resulting
from back-reflections of components within the laser, and it can improve fixed pattern
noise by reducing strong background signal from the reference arm [23].
2.4
Practical considerations and limitations of FD-OCT
Although FD-OCT has clear advantages over TD-OCT, there are practical considerations
that must be addressed for adapting it to clinical imaging. Until now we have assumed
43
Time-domain
Frequency-domain
Axial resolution
Sensitivity roll-off
Source spectrum
Linewidth
Depth structure
Modulation
Complex conjugate ambiguity
Sampling rate
Max detectable depth
F-1
-. 4.
Y(q
to)
T
d
-
'U.
Aw = N 6o,
(
=s(0)R
=iR
Az 6 dH
'iR
Y(01} (w)
(r)
=
[19(0 -Y(] ® L R)
1
Figure 2-6: Schematic of the relationship between FD-OCT parameters in the optical
frequency domain (left) and the delay/depth space (right).
that the limits of sample depth are
+00,
however the sensitivity of the system rolls-off over
a finite optical path delay (Ti). The maximum sensitivity roll-off is fundamentally limited
by the coherence length of the laser source, which is related to the instantaneous linewidth
(6w) of the light. Furthermore, as we described earlier in section 2.3.1, the spectrum is
swept in time and the interferogram has a modulation frequency that is proportional to the
tuning rate
(fdet)
of the laser and the delay, T,. In continuously swept OCT, the frequency
spacing of the digitization samples, 6w, determines the maximum resolvable depth in the
sample space. Furthermore, as we discussed previously, the acquisition of the real part of
the cross-spectral density function (Eqn 2.23) induces an ambiguity over whether a sample
reflector is located at a positive delay (+Ti) or a negative one (-Ti). These parameters are
summarized in Figure 2-6. This figure connects the optical frequency domain parameters
with the imaging parameters of the delay/depth domain.
44
2.4.1
Coherence Length
Now that the reference mirror is stationary, there is a finite depth over which there is a
-6 dB loss in sensitivity in the system. In SS-OCT this "ranging depth" depends on the
instantaneous linewidth, 6w, of the swept laser source, and defines the coherence length of
the laser, or the maximum optical path difference where fringe visibility is good. In SDOCT, 6w is determined by the spectral resolution of the spectrometer. We can represent
the sensitivity roll-off due to the finite instantaneous linewidth by convolving the ideal
interferogram of Eqn. 2.25 with a Gaussian linewidth function, YG(w), having a FWHM
of 6w.
YG(w) = exp(-
41n(2)w2
)
(2.45)
By the convolution property, the delay-domain profile is multiplied by the Fourier transform of the Gaussian sensitivity roll-off function, yG,
IiR(WM jSI(W) 10 YG~w M S iRw) P IiR (-I ( )I yG (T) 09 giR (T)'
(2.46)
where,
eXp(
(2.47)
6W 2
41n( 2
2
)
G(T)
This means that the sensitivity of the FD-OCT signal drops with depth in a Gaussian
manner, as highlighted in red in Figure 2-6. For simplicity, only a single reflector in the
sample arm is considered, resulting in a point spread function (PSF) peak at
-Tj
and +Tj.
The depth range that results in a 6 dB drop in sensitivity is characterized as [4,57J,
zwd
21n(2)c
6W
ln(2) A2
0
7r 6A
(2.48)
where again AO is the center wavelength, and n is the index of refraction of the sample
arm. To truly appreciate the effect of coherence length, a simulation of multiple reflectors
placed at equally spaced locations in the sample space shows the visibility of the PSF as a
function of delay (or depth). The reflectors were assumed to be perfect, hence the power
drop with depth is caused solely by roll-off pertaining to the instantaneous linewidth.
45
6w
10 GHz
0
0
.8
.4
2
a.
6w
100
0
2
4
6
20
40
so
8
1e
Depth, mm
1 GHz
-60
-80
-40
-20
0
80
100
Depth, mm
Figure 2-7: Simulated coherence lengths as a function of depth for a 1 GHz (top) and 10
GHz (bottom) Gaussian instantaneous linewidth source.
The simulation was performed for 6w ~10 GHz and for 6w ~1 GHz. Notice that both
sensitivity roll-off envelopes have Gaussian shapes, however the system with the smaller
instantaneous linewidth has proportionally larger sensitivity over longer depth ranges in
the sample arm. Hence in imaging, this roll-off results in fading of the image with depth.
In most conventional OCT systems, this range is 5mm - 10mm and limits the flexibility
of where the tissue can be placed in the sample arm.
2.4.2
Discrete sampling with acquisition card
Another major consideration is the detection rate of the interference fringe. Now that
the frequency of the interferogram modulation represents the sample depth, the rate of
digitization is important in determining the maximum path length difference that can be
acquired. If the acquisition card is sampling at an effective sampling interval of, 6W5, this
means that the fringe is being sampled by N,
46
-
Aw/6w, samples. Hence the sampling
interval in the time domain becomes 27r/6w and by the Nyquist criterion,
127rc
AZ6dB
-
4 6ws
1 A2
= -=
4 6As
(2.49)
This corresponds to a maximum one-sided detectable delay of,
Ti6dB =--
1 27r
1 A2
4 6w8
4c 6A,
-
(2.50)
In Figure 2-6, this is represented by the black dotted line in the time-domain. In SS-OCT,
the maximum detectable sample reflector can be limited by the digitization electronics
that are, available; we saw in section 2.3.1 that the interference fringe frequency is proportional to the swept source speed and the optical path delay. Hence for high-speed imaging
systems, the maximum detectable depth can be much lower than the coherence length.
For the simulated 6w = 10 GHz Gaussian sensitivity roll-off above, we now plot the electronic bandwidth associated with an 18.9 MHz laser. We assume a spectral bandwidth
Aw = 5.6 x 1013 GHz based on a source that ranges from A = 1505 nm and A2 = 1575
GHz. This continuous spectrum, convolved with the Gaussian linewidth function YG(w)
is shown in Figure 2-8A. The optical interference fringe AR(w) resulting from multiplying
the source spectrum with an interference modulation resulting from a reflector near the
edge of the sensitivity roll-off region shows a high frequency fringe (Figure 2-8)B that is
beyond the resolution of electronic sampling interval 6w,. The inset of this figure shows a
region of the optical fringe being sampled by the digitizer; we can see that electric sampling with a digitizer Dirac comb (Eqn. 2.38) yields an undersampled fringe. Hence in
Figure 2-8C we show the Fourier transformed delay-space of the same multiple reflectors
as in section 2.4.1, this time as a function of the RF bandwidth. Modern digitizers can acquire -1 GHz of RF bandwidth so we have shaded the areas that are beyond our digitizer
bandwidth (grey shaded area). Notice that even if the sensitivity roll-off were increased
to accommodate long range imaging, we are still greatly limited by the acquisition of the
signals. If we compare Figure 2-8C to Figure 2-7A, we see that within our parameters,
we are limited to a range of
600 pm in the sample arm. If we wanted to increase this
47
()= i ())P(OJ)
Optical Spectrum, I S(w) I
Opt"lInterlrence Fringe, 6(.)a 19 a >I -
Y 0 (C)
0
Iw
i
"
&ws
B
I
-
3.7
0 60
4-
___L_
ONINGOess4i
-
I
C 0.
'E
.
e 0 30. 2-
0.
1.51
-1.6
1.2
1.56
1.54
153
Wavelength, (m)
-1
1.6
1.57
x1e
-0.5
4
Thne m ed on 18A MIz Aline RFte@, (a)
0
0.6
Fringe Frequency [Based on 18.9 MHz Aline Rate, (ft)
1
1J
1.6
x0
Figure 2-8: Simulated electronic bandwidth resulting from an 18.9 MHz A-line rate, and
10 GHz Gaussian instantaneous linewidth source. A) The optical source spectrum. B)
The optical interference fringe of a reflector placed at 1.8 mm. The inset figure shows
a part of the optical fringe that is discretely sampled with a digitizer (black dot). C)
The Fourier transform of the fringe with multiple reflectors. The shaded region is beyond
what we can acquire with modern digitizers.
48
LPF
-t
Optical
Processing interference
fringe
Ii(W)
-
Photoreceiver
d(
let
)
i
-21r
28,
-
HLPT)
()
h ()
d'
Digital Signal
(W)
+2x
2&o,
Figure 2-9: Schematic of acquiring and low-pass filtering an interference fringe signal.
range to match the sensitivity roll-off of -6mm, we would need to slow down the Aline
rate 10-fold (fA
1.89 MHz). And if we were to try to match the ~6cm sensitivity-roll
of shown in Figure 2-7, we would need to further slow down the laser to
fA
= 189 kHz.
Hence the limitations of the sampling rate of the digitizer, which we have represented as
+00
(w -m -6w), induces a trade-off between imaging speed and imaging range.
p(w) = E
'M-00
One important point that is that low-pass filtering should always be applied in order
to prevent aliasing of noise (i.e. optical interference noise, or receiver noise) into the
electric bandwidth region that we can acquire. For instance, if we digitize with an electric sampling interval of 6w, we should apply a low-pass filter with a frequency response
function,
HLP
ideally with cutoff frequencies at
2'-. Figure 2-9 shows a flow schematic
of acquiring fringe signals in a conventional SS-OCT system. The optical interference
fringe is detected with a photo receiver, low-pass filtered, and electrically sampled with a
digitizer.
2.4.3
Complex-conjugate ambiguity
Although the correlation function g(T) is inherently real, as we saw in section 2.17, in
FD-OCT we measure the real part of the cross-spectral density function and thus we
are left with a complex-conjugate ambiguity. In Figure 2-6 this is represented as a peak
in both
-Ti
and +Ti, with ambiguity regarding whether the sample reflector was placed
before or after the zero optical path delay. This can cause severe artifacts in FD-OCT
images because if a sample is placed across the zero OPD, it will cause overlapping of the
49
zZ
0
-
+2.9 -
(mm)
0
-2.9
z (mm)
-2.9
0
- +2.9
Figure 2-10: (A) Image segments that fall in both +z and -z regions overlap without
complex conjugate removal. (B) Image segments can be removed with conjugate demodulation 1601.
image signals. Hence, initial FD-OCT interferometers required the sample to be placed
on either the +z or -z region to avoid overlap such as in Figure 2-10A. In the simulations
above, we have assumed that complex conjugate ambiguity has been removed, hence we
had reflectors placed on both sides of the zero optical path delay. If we were not to assume
ambiguity removal, we would notice overlapping of our PSFs. For a better demonstration
of this principle, a simulated a 550 pm tissue signal spanning from +150 pm to +700 pm in
the delay-space is shown in Figure 2-11A. And the same tissue signal spanning from -200
pm to +250 pm is shown in Figure 2-11B. The true tissue signal is plotted in blue while the
complex conjugate artifact is plotted in red. Notice that in case (A), overlap is avoided
because the sample is placed entirely in one half of the delay space. In (B) however,
overlap artifacts could not be avoided because the tissue signal spans the zero OPD.
In conventional OCT, approach (A) is commonly taken to avoid this artifact, however
it is not ideal as it halves the usable depth range. Typical polygon-based swept lasers
used in SS-OCT have instantaneous linewidths 6u~ 13 GHz, and this half depth space
would correspond to -5 mm depth range whereas full-range imaging would enable -1 cm
of imaging range. Full-range imaging and removal of the complex conjugate ambiguity
was extensively explored in FD-OCT because of the already limited imaging ranges in
OCT [11, 25, 35, 55, 60]. We will see that in optical-domain subsampling, the removal
of complex conjugate ambiguity is unavoidable and this motivates the development of a
50
A
.5
1
0.
0
Depth, mm
0.6
1
0S
I
1.5
B
-4.
_0A
1 6
Depth, mm
Figure 2-11: Demonstration of complex conjugate ambiguity artifact in a simulation. (A)
When the tissue is not crossing the zero OPD (spanning from +150 pin to +700 Pill),
there is no overlap. (B) When zero OPD is crossed, the tissue signal (spanning from -200
ptm to +250 pin) overlaps with its conjugate signal.
low-bandwidth solution to ambiguity removal (discussed in Chapter 6).
2.5
Transverse scanning and microscopy
Three-dimensional imaging in OCT is obtained by scanning the tissue in all three spatial
directions. In the case of depth scan (z direction in Figure 2-12) the A-line is an intensity
map of the sample along depth.
A galvanometer mirror rotates to provide scanning
of a single frame or B-scan, and another galvanometer mirror scans in the orthogonal
direction (C-scan) to provide a three-dimensional image. As in conventional microscopy,
an -objective lens is used to focus the collimated light coming from the galvanometer
mirrors onto the tissue. Unlike the axial resolution, which was determined by the spectral
bandwidth of the source, transverse resolution is determined by the focusing optics in
the microscope. We make the general assumption of a Gaussian beam profile (the field
amplitude in the transverse direction has a Gaussian distribution) and thus we derive our
51
Galvo
mirror
Sample arm probe
Objectiv
Backscattered
Oeci>
Intensity
l.
CLD
'0
CD
Figure 2-12: Galvanometer mirrors scan the tissue in the transverse B-scan and C-scan
directions to render a three-dimensional image.
microscope parameters with this assumption. The minimum spot size
()
width) can be
calculated by [4],
6X = 6y =
where
f
7 D
(2.51)
is the focal length of the lens, A is the wavelength of the beam, and D is the
diameter of the beam that enters the focusing lens. Thus the larger the numerical aperture
(NA a
D)
of the beam, the the higher the transverse resolution at the minimum waist
spot (Figure ??). However, because the penetration of OCT light into tissue is typically
on the order of 2 mm, low numerical apertures are needed in order to increase the confocal
parameter so that the whole depth of the tissue being imaged is in relatively good focus.
The confocal parameter, b, is twice the Raleigh length and denotes the range over which
the beam waist expands to v/2ro. The confocal parameter for a Gaussian beam is given
by the following relation 14],
b =
2A
2A
(2.52)
In some applications higher NA can be used to provide ultra-high transverse resolutions at
the expense of depth-of-field. Some microscopes with adjustable focus have been explored
52
as alternative ways to have very high transverse resolution while maintaining large field
depths, however as long as the field depth remains small, the transverse resolution generally suffices and many elect to avoid the complexity of adjustable focus microscopes [141.
53
54
kki
WAWrAk._-_'- - ,
Chapter 3
Theory of Discrete Optical Sampling
In the previous chapter, many principles of continuously-swept OCT systems were reviewed; now concepts that are specific to subsampling will be introduced. Prior to this
work, optical domain subsampling did not exist in OCT, hence we provide a rigorous
description of how it induces acquisition bandwidth compression, and how subsampling
parameters are related to OCT imaging parameters.
3.1
Sparsity in Extended Depth Range OCT
We saw in section 2.4.2 that the acquisition bandwidth of modern digitizers limits the
electronic sampling interval 6w, and limits the maximum acquirable depth within the
sample arm. We try to address this problem by compressing signals over a large sample
range into the small segment of the RF bandwidth that we can acquire
(t 1 GHz). In
most tissues, the penetration depth of light in the near infrared (NIR) region into the
tissue is limited 1-2 mm because of tissue scattering, regardless of the imaging range. Let
us assume hypothetically that we have both the sensitivity and RF bandwidth available to
acquire a 1 mm tissue signal spanning from 1.6 cm to 1.7 cm along an extended range of 3
cm. This means that we will be able to detect tissue anywhere along a 3 cm region. And
lets assume (quite accurately) that the tissue signal is attenuated significantly beyond
1.7 cm and that the system sensitivity cannot detect signals beyond that region (signal
55
- 11-11 111 -1 -
.
,
, , ''
attenuated
tissue
ascattering
0
C.
0
10
5
20
i5
25
J0
Depth, mm
--+I1
14- B
Electric Frequency, Hz
fL
c
U
max
Figure 3-1: Schematic of an R.F spectrum of an extended depth range OCT image where
the tissue lies in a region 17.5 mm away from the zero OPD.
power drops to zero). In this extended depth-range OCT scenario, a tissue signal falls
between two large regions of negligible signal both superficial to the tissue surface, and
beyond the tissue where the signal drops (Figure 3-1). We have represented notable RF
frequencies corresponding to the lower, center, and upper edge of the tissue signal (fL,
fc, fu), the bandwidth corresponding to the total width of the tissue signal, B, and the
frequency corresponding to the maximum expected signal, fm.
It is exceedingly clear
that acquiring this full A-line is data inefficient because a large fraction of the acquisition
bandwidth is dedicated to the signal-absent (ascattering or attenuated) regions. Figure 33 shows a fringe with frequency corresponding to fu and the electric sampling interval 6W,
corresponding to 2
1
. One approach to reducing the acquisition bandwidth is to mix
the fringe signal out of the receiver with a local oscillator and downconvert the signal to
the baseband location as shown in Figure 3-2. We would low-pass filter at B and digitize
our signal at 2B instead of
2
fu. However, because the location of the tissue signal is
not known a priori, a limited and targeted acquisition of the depth range containing the
tissue is practically challenging. Hence, conventional FD-OCT continues to operate in an
oversampled or Nyquist regime. The acquisition bandwidth required for extended range
56
2B
Low-pass filter
id,.do)
idet(W
receiver
&
-
Photo-
)
HUT)
-B
+B
local oscillator
e-j(2rf,,e)co
Figure 3-2: Schematic of downconversion of fringe signal after the receiver idet(w) with a
local oscillator.
imaging can be reduced by finding a way to eliminate this inefficiency while preserving
the ability to image over extended depth ranges.
3.2
Bandpass Sampling
The theory of bandpass sampling has existed in communications and signal processing for
many years [1,8,53]. We begin our discussion of optical domain subsampling by reviewing
the basics of this concept.
Consider a bandwidth limited signal located at fc with a
bandwidth B (Figure 3-4a,b). Nyquist sampling at
2 fu
captures this signal fully, but is
data inefficient because a large fraction of the detected bandwidth does not contain signal
(Figure 3-4b). Alternatively, because the signal is bandwidth limited, sampling the signal
directly at twice its bandwidth, i.e., 2B, can capture its information content (Figure 34c). This approach is termed subsampling because it samples the signal at rates below
twice the highest frequency content of the signal,
2
fu. Higher frequencies appear in the
baseband window through aliasing (Figure 3-4d,e). Implementing bandpass sampling into
OCT requires discretely sampling the continuous optical interference fringe and this can
be done either electrically by reducing the digitizer clock rate, or optically by modifying
the laser.
57
7
freq
=fu
Figure 3-3: Interference fringe corresponding to fu with the inset showing electronic
sampling interval corresponding to 6a, = 2.-
3.2.1
(black dots).
Electrical-Domain Subsampling
The most straightforward implementation of subsampling is in RF-domain, i.e., to maintain full RF bandwidth on all receivers but operate the digitization clock at a reduced
rate. This is the identical acquisition set-up as in conventional OCT in Figure 2-9, now
however we electrically sample at a rate that corresponds to twice the tissue equivalent
bandwidth, 6w, = 2. 1 instead of twice the maximum frequency. For simplicity we have
reproduced a flow schematic for the case of electric-domain subsampling in Figure 3-5.
Notice that although digitization occurs at 2B, our low-pass filter has a cut-off frequency
at fax. Recall from section 2.3.2 that in FD-OCT we achieve a sensitivity advantage
over TD-OCT because we have assumed that the noise-equivalent bandwidth (NEB)
is equivalent to the maximum detection bandwidth, fdet, and that we can use low-pass
filtering to remove noise beyond that bandwidth. In this case of electric-domain subsampling, fdct -fmax,
however fdig = B and the noise integrated along the large detection
bandwidth aliases into the electric baseband window. For some applications requiring
58
(c)
(a)
E
cc
(D
4*004*0
II
'I
M
Wtime
ime
11
(d)
B
B120. Bl3r\
(Cr
EI~9
L
(e)
B
(b)
fc fu
B
B/2 B/2
B12 B/2
0
CL
0
CL
z
fL
Ifc fu
-
aliased
rea
freq
Figure 3-4: Subsampling of bandwidth limited signals. (a,b) A bandwidth limited signal
sampled at twice its highest frequency content (2fu) yields the full frequency content.
However this is data inefficient because non-aliased sampling frequencies increase with
signal frequency. (c-e) Direct subsampling of the signal at twice its bandwidth (2B)
captures its information content by repeated aliasing of the original frequency space to
the baseband window 1391.
&, =2
M=-
HLPT)
Optical
interference
fringe
IM(C)
PhotoSreceiver
'(O
t
-max
T
(
)
LPF
id
Digital Signal
Processing
+fm.
Figure 3-5: Schematic of acquisition in electrical-domain subsampling.
59
a modest decrease in acquisition bandwidth, the associated noise increase might be an
acceptable penalty to achieve a corresponding reduction in digital acquisition bandwidth.
For more aggressive applications of subsampling, this noise penalty would be prohibitive.
If we go back to our example of the 18.9 MHz swept-source laser with the Gaussian
instantaneous linewidth of 6w = 10 GHz, we can simulate electric-domain subsampling
by acquiring our above tissue signal (spanning from 1.6cm to 1.655cm this time) with a
maximum digitization bandwidth of 1 GHz. In electric-domain subsampling, our light
source does not change so we again have a Gaussian source spectrum spanning AA =
70 nm. The digitized interference fringe with the integrated detector noise over a total
sample region of 3 cm is displayed in Figure 3-6A. For comparison Figure 3-6B shows
an equivalent fringe digitized with no receiver noise. In Figure 3-6C we show the delay space as a function of RF frequency assuming we have complex conjugate ambiguity
removal. Note that the conventional electric frequency corresponding to the edges of a
tissue placed at 1.6 cm and 1.655 cm with 18.9 MHz laser speed, 100% duty cycle, and
spectral bandwidth AA = 70 nm is fL= 18.04 GHz and fu = 18.87 GHz. We can see
that the signal has been down-converted to a
1 GHz region, however, at the expense of
severe distortions caused by receiver noise. If we could reduce the detection bandwidth
to
fdct
= B we would avoid such noise penalties. This was the motivation for exploring
optical-domain subsampling as a form of bandwidth compression.
3.2.2
Optical-Domain Subsampling
Detecting a compressed optical interference fringe signal requires subsampling in the
optical-domain.
As in electric-domain subsampling where we used a sampling comb,
p(w), in optical-domain subsampling we define an ideal Dirac optical comb as FFSR(W)
+oO
Z
6(W -
m - 6 WFSR) with an optical sampling interval
6WFSR.
Multiplying the Dirac
m=-00
comb with our source spectrum yields a discretely sampled optical spectrum, as shown in
Figure 3-7. The interference fringe resulting from this ideal discrete source will also be
discretely sampled and we can represent the modification interferogram intensity function
60
Digitized Interference Fringe, l
(w) = ldet()p(W)
B
A
c
U
C
o 1
Time [Based on 18.A MHz Aline
s
0
Rate],
(a)
6x
x 10"3
0
4
3
2
Timo Wased on 18.9 MHz Aline Rate], (a)
1
5
x 10
C
a.
1
-0.8
-0.6
O.4
02
0
-0.2
-0.4
Fringe Frequency [Based on 18.9 MHz Aline Rate], (Hz)
0.6
U.0
x 1010
Figure 3-6: Simulated electric-domain subsampling resulting from an 18.9 MHz swept
source laser. A) Fringe signal after digitization. B) Equivalent fringe signal without
noise. C) RF spectrum showing noisy tissue signal after electric subsampling.
61
FFSR(W~~
X
(-m
M=-WS
OFSR)
6FSR
I
-
-
Sk
II
I
I
I
w
F
Optical Spectrum
1
0.9
0.8
0.7
6
WFSR
I0.6
5
0.4
S0.3
0.2
0.1
1..54
d
1.3
1.11.Ii
1
1.6
1.62
1.61
1.65
Wavelength,
(in)
1.66
11.1
54.
1.57
1.57
x 106
Figure 3-7: An ideal Dirac frequency comb multiplied by the source spectrum yields a
discretely sampled source.
62
IM(W) 1S( m)
I- F
(w) -
SIR(m)'
U
C
C
4
3
2
1
Time EBfsed on 18.9 MHz Alnne Rate], (e)
0
6
x 10~
Figure 3-8: Optical interference fringe of a subsampled source.
as,
IiR (U)
=
5(w)j
- FFSR(W)
iR(W)
(3.1)
-
Using again our simulated 18.9 MHz laser example and a tissue placed between 1.6 cm
1.655 cm, we display an optically subsampled fringe corresponding to an optical sampling
interval of 200 GHz
( 6 WFSR=
27r. 200 GHz) in Figure 3-8. Because we are now sampling
the interferogram with a finite number of optical samples, we force high-frequency fringes
3
to compress into a small bandwidth (baseband window). In the case of WFSR = 27-'200
GHz, we have an optical baseband window of
300pm. Now the tissue can be placed
anywhere along the depth range and its RF signal will alias into this region (corresponding
to an electrical bandwidth of
431 MHz in our example). This can be appreciated in the
delay space by taking the Fourier transform of the subsampled optical interference fringe
(Figure 3-9). Note that this figure represents a hypothetical continuous delay-space prior
to digitization with an acquisition card. The grey shaded region represents our optical
baseband window, hence we can efficiently sample only this electrical region with our
digitizer. Although the tissue signal falls midway in the baseband window, the entire
signal is still preserved. Figure 3-10A shows an acquisition flow schematic for an optically
subsampled imaging system. Now that the optical interference fringe has a compressed
63
(m
,) 10
fFSR
R
)
i
0
ILL
4
.0.8
0.6
0.2
.2
0
-. 4
Fringe Frequency [Base A on 1 9 MHz Aline Rate],
-2;r
+21ir
2&OFR
2&oFsR
0.4
0.6
0.8
1
X10
Figure 3-9: The optical intensity of a tissue signal spanning from 1.6 cm - 1.655 cm
representing aliasing of signals from an extended depth range into the optical baseband
window (grey shaded).
frequency, we can low-pass filter at
j3WFSR.
Ideally
6
WFSR
B
so that fdet = B but in
most cases it will be slightly larger; as long as the digitizer frequency is greater or equal
to fdet, we will completely avoid the noise aliasing that results from electric subsampling.
In Figure 3-10B we display the tissue signal in delay-space after digitization at 1 GHz;
the electric baseband window is twice the size of the optical baseband window so two
optical aliases are visible. Hence, optical-domain subsampling achieves the compression
of subsampling without a proportional noise increase.
3.3
Subsampled OCT imaging parameters
In subsampled OCT, the acquisition bandwidth is decoupled from the maximum detectable depth as we saw was the case in Eqn. 2.49 for the continuously sampled OCT
system.
We can see from Figure 3-9 that the edge of the baseband window falls at a
frequencies equal to i2'.
Hence the total double pass optical baseband range can be
described by,
1
AZmaxFSR
1
27rc
-
-
4n6WFSR
64
F
c
2nFSR
(3.2)
A
LPF
p(w)=
HLPT)
Optical
interference
fringe
IR(W)
M
Photo-
d,,
idet
X
(w-m.&O,)
0
Digital Signal
receiver
Processing
(
=i dt(%) H LPU) 0PG)
B
0
n.
-1
-0.8
-0.6
0.4
0.2
0
-0.2
-0.4
Fringe Frequency [lased on 18.9 MHz Aline Rate], (Hz)
0,6
0.8
1
x 101 0
Figure 3-10: A) Schematic of electronic acquisition of a subsampled interference fringe.
B) Plot of tissue signal in the delay-space as a function of electric frequency. Electrig
sampling is twice the optical sampling in this schematic.
65
I
IRg=1( W) 1 0 AR)
0 FR(w3)
I
~
I
-1
-0.8
-0.6
0
-0.2
.0.4
0.2
0.4
0.6
Fringe Frequency Wooed on 18. MHz Aline Rate], (Hz)
0.8
1
X 10
Figure 3-11: If the baseband window (grey) is too small, tissue signals can overlap and
cause distortion.
where c is the speed of light, n is the index of refraction of the sample medium we have
which is the free-spectral range of the optical comb. This is
defined FSR - 61FS,
always assuming a constant FSR in optical frequency, otherwise distorting effects caused
by nonlinear-in-k will ensue, as discussed later in section 4-5. Now the maximum detector
frequency, fdct is defined by
fdet
6
2wfA-
WFSR - DutyCycle
(3.3)
where again Aw is the spectral bandwidth and fA is the Aline rate with a certain duty cycle. Notice that the detection bandwidth has no dependence on the location of the tissue
in the sample depth space and instead is inversely proportional to the optical sampling
interval. This means that no matter where in the sample range you place your tissue,
your total acquisition bandwidth will remain the same, assuming your tuning speed does
not change. The FSR should be chosen carefully depending on the tissue being imaged;
optical baseband window should be large enough to avoid overlapping of neighboring optical aliases. In the example of the 18.9 MHz laser, if the FSR was chosen to be 300 GHz
instead of 200 GHz, there would be too few optical samples, the baseband window would
be smaller, and the tissue signals would overlap as shown in Figure 3-11.
66
MMMMMMR
Time-domain
Axial resolution
Sensitivity roll-off
Depth structure
Complex conjugate ambiguity
Baseband window size
Frequency-domain
Source spectrum
Linewidth
Modulation
Optical sampling rate
1(o),
AZ
1(T)
*
*
I
I
I
*
I
*
P
I
*I
i
to)
OFSR
Aw = N
[
=(
LL~
I
I
a
I
-
Ti
#
I
* *
I
g
*
II
0
I
*
*
I
2rc
4n&WFSR
.........
ti
6WFSR
FF
Fw)
Y(0]-IR ().
1R (0 =
[f9MT
(& fFSR (r)
(&
k R(T)
Figure 3-12: A schematic of the subsampled imaging parameters in the optical frequency
domain (left) and time/delay domain (right).
Assuming that the digitizer electronic bandwidth is large enough to acquire the optical baseband window, now the only limitation on the imaging range in the sample arm
is the sensitivity roll-off. In Figure 3-9 the source was assumed to be an ideal Dirac frequency comb and we did not observe a sensitivity roll-off with increasing delay. However
as was the case in continuous OCT sources, real subsampled sources have finite instantaneous linewidths 6w. The width and shape of this linewidth will similarly determine the
range of the sensitivity roll-off; Figure 3-12 summarizes the relationship between subsampling parameters and imaging parameters. We previously defined a Gaussian linewidth
function, YG(w), we will also introduce Lorentzian linewidth functions taking the form
7
wr(w2 + y 2
(3.4)
)
YL (P
The variable 'y is defined as ln(1/R), where R is the parameter determining the width
of the Lorentzian function. The coherence length can be found by taking the Fourier
67
transform of the Lorentzian impulse response,
YL(W; -Y)
yL(T; -Y) =
(3.5)
This means that if the instantaneous linewidth is Lorentzian, the sensitivity roll-off will be
exponential instead of Gaussian, as depicted in Figure 3-12. Like in continuously sampled
OCT, the axial resolution of subsampled OCT is determined by the spectral bandwidth
(Aw) and shape of the light source.
3.4
Complex-conjugate ambiguity in subsampled OCT
To be effective in the context of imaging, subsampling must ensure that signals from each
depth within the penetration depth of tissue, can be measured independently from those
at other depths, i.e., overlap artifacts must not compromise the resulting image. In the
examples above, we always assumed that the fringes were complex and the ambiguity
between the positive and negative delay space does not exist. Unlike in the continuous
case, where the sample can be placed on one side of the delay to prevent overlapping, in
subsampled OCT both the real and conjugate tissue signals are aliasing and it is impossible
to control where the aliases might overlap. This is independent of the size of the baseband
window; the tissue signal can overlap even if the optical baseband window is bigger than
twice the size of the tissue signal as demonstrated in Figure 3-13. In the two dimensional
image, complex conjugate ambiguity results in reflective wrapping instead of circular
wrapping. To illustrate this better, we generated a numerical OCT phantom structure,
derived associated fringe signals from this phantom, and explored the effect of subsampling
on the imaging (Figure 3-14). First, we placed the phantom at varying locations in depth
(Figure 3-14A). Next, we derived the associated fringes assuming continuous wavelength
sampling across this depth range. We then subsampled the real-valued fringe data by a
ratio of 1:4, and presented the compressed image (Figure 3-14B, showing only the positive
frequencies). Note that the non-circular mapping of signal frequency to aliased frequency
results in image overlap for most locations of the image. We then repeated this analysis
68
IiR(T) = I g(
I0
f)
W
A\
n.
.0.8
-1
0.6
0.4
0.2
0
-0.2
-0.4
Fringe Frequency (Based on 18.9 MHz Aline Rate], (Hz)
.0.6
I
1
0.8
x 10
= g( CD) I & gRT)' 0F8 SR(W)
B
/,
r
-~
-~
I
J-
-
-0.5
1
L
r~ U
-0.6
.1~~~
0.4
0.2
0
.0.2
.0.4
Fringe Frequency [Based on 18.9 MHz Aline Rate], (Hz)
0.6
0.8
Figure 3-13: A) Optically subsampled tissue signal (blue) with overlapping complex conjugate subsamples (red). B) Overlapping can occur even if the baseband window is more
than 2x tissue signal (in this case FSR = 50 GHz).
69
1
x 1010
E
IOI
Figure 3-14: Real-valued and complex-valued signals are mapped differently into the
aliased frequency space. For real-valued signals, signals located at varying locations (A)
can induce distortions due to non-circular wrapping in the aliased image (B). For complex
signals, wrapping is circular and overlap is avoided as long as the baseband window is
large enough to contain the depth extent of the signal (C) [39].
but subsampled the complex fringe signal at a ratio of 1:8 (to give the same baseband
window depth, the ratio was decreased by a factor of two because the signals are complex
valued). In Figure 3-14C, the complex subsampled images are presented. Note that the
image is wrapped circularly, and signals never overlaps onto itself for any depth location.
70
Chapter 4
Experimental Validation of
Subsampling in a Slow-Speed System
In this chapter we describe a slow-speed proof-of-concept subsampled laser using a polygonbased swept-source laser and a Fabry-Perot etalon. We characterize the sensitivity and
coherence length of this laser, and discuss losses due to higher order harmonics that are
inherent to the sweeping of subsampled lasers. We validate that subsampling results in
bandwidth compression and perform imaging with this first-ever subsampled OCT system.
4.1
Relevant Work
As discussion in section 1.2.1, increasing the coherence length of the OCT laser increases
the ranging depth improves image quality as well as clinical versatility. Recall that the coherence length/sensitivity roll-off function is associated with the instantaneous linewidth,
6w of the light source
14].
In most modern OCT systems, the inability to reduce this
linewidth severely limits the ranging depth. For instance, FDML lasers (discussed further
in Chapter 5) suffer a -5 dB sensitivity drop over a 3 mm depth range [16,181. Similarly,
polygon-based OCT lasers have ranging depths on the order of ~5mm [57]. One method
of reducing this instantaneous linewidth is to introduce a narrow passband spectral filter into the laser cavity. Various groups explored using a Fabry-Perot (FP) etalon as
71
the filtering component in order to achieve better sensitivity in their principle imaging
range 12, 3, 20, 481.
When swept sources were first employed in OCT, it was believed
that they needed to be continuously tunable, however, in 2005 Amano et al. provided a
first demonstration that mode hopping in the laser source does not affect image data [2].
To experimentally demonstrate that discontinuously tuned lasers can be stably built for
OCT imaging purposes, they designed a superstructure-grating distributed Bragg reflector (SSG DBR)-based imaging system [2]. Their experimental set-up was not suitable
for imaging, however, because of the low spectral bandwidth ( 40 nm) and very low scan
speeds (250 Hz A-line rate).
This work was followed up by Tsai et al. in 2009, who
adapted a frequency comb swept laser into an FDML imaging system. This cavity had an
instantaneous bandwidth 6f ~ 2.5 GHz and an FSR ~ 25 GHz. This resulted in a discretely sampled OCT laser; however, the focus of the study was to reduce the sensitivity
roll (from -5 dB to -1.2 dB) over their 3 mm principle imaging range rather than to minimize the acquisition bandwidth of the imaging system. Interestingly, they reported that
interferometric signals that have frequencies higher than !2 of the optical sampling rate,
'Ns, were aliased into their principle range of 3 mm. This same aliasing phenomenon
was also reported by Jung et al. who constructed an external frequency comb filter for
SD-OCT [20].
In this work, we take advantage of this aliasing phenomenon to drive down the number of samples Ns needed to image tissue over a wide ranging depth. We provide the first
demonstration that a swept frequency comb laser can be used to simultaneously increase
ranging depth (from ~5mm to -10cm)
while minimizing acquisition bandwidth. Our
approach to constructing the subsampled laser was to insert a Fabry-Perot (FP) etalon
into the cavity of a polygon-based laser.
4.1.1
Polygon-based wavelength-swept laser
Polygon-based swept lasers were one of the first robust wavelength-swept lasers designed
for SS-OCT imaging and is still widely used today [58]. In this implementation of swept72
(a)
F,
(b)
F:
F,
F,
Al
I
a'
P
Figure 4-1: (a) A polygon mirror based spectral filter [30]. (b) A schematic of a diffraction
grating where o = incident angle and = diffraction angle.
source laser, light that is amplified by an SOA is filtered by combination of a grating
and a polygon mirror that spins to reflect one wavelength at a time. A schematic of this
polygon scanning filter is shown in Figure 4-1
[301.
This filter is comprised of a diffraction
grating that angularly disperses the light, a telescope of two lenses with focal lengths F
and F2 , and a polygonal spinning mirror (typically 72 facets). A collimated Gaussian
beam that comes from the cavity is incident upon the grating at an angle, a, and diffracts
as a function of wavelength, A, with an angle 3. According to the grating equation, the
filter's tuning wavelength is [30,31,581,
A = p(sin a + sin f)
73
(4.1)
where p is the grating pitch as shown in Figure 4-1b. The center wavelength, A 0 , of the
spectral bandwidth is the wavelength for which /0 is the angle between the optical axis
of the telescope and the grating normal. The instantaneous linewidth of light from the
filter output is given by [30],
6
AFWHM
(4.2)
AoA(p/m) cos a/W
where A = V41n2/7r, m is the diffraction order and W is the 1/e 2 width of the Gaussian
beam at the collimator. Given that the facet-to-facet polar angle of the polygon, 0 =
27r/N ~ L/R, where N is the number of facets, L is the facet width, and R is the radius
of the polygon, then the free spectral range (FSR) is shown to be
AAFSR = p Cos N0
F2
F1
[30],
( 4.3)
This denotes the spectral spacing of the two wavelengths that are retroflected from different facets of the polygon, and ultimately this determines the spectral bandwidth, AA,
of the laser source. In alignment of this laser, this spectral bandwidth is maximized so
that good axial resolution can be achieved [58].
4.2
Fabry-Perot comb filter
Fabry-Perot (FP) etalons are frequency comb filters that, through a series of constructive
and destructive interferences, pass a narrow and discrete sets of wavelengths while suppressing others. Figure 4-2 shows a sample schematic of a FP that transmits frequencies
equally spaced by 6 WFSR with FWHM bandwidth, 6w. The spectral distance between
these highly transmitted wavelengths defines the free spectral range (FSR) of the etalon
and is defined as FSR =
WFSR.
This parameter is directly related to the width of the
etalon through the relation,
FSR= c
2nL
(4.4)
where c is the speed of light in vacuum and L is the width of etalon. The transmission
74
AL
'6'FSR
0.
1
U.
0
E
(01
0
K)
K4
J
finesse = 100
Angular frequency, w
Figure 4-2: A schematic of the transmission spectrum of a Fabry-Perot etalon. 6W
instantaneous angular frequency,
6WFSR
= free spectral range in angular frequency units.
function of the FP filter as displayed in Figure 4-2 and is given by the function
T(w) =
2
(1- R)
1 - 2R cos (2kL) + R 2
[48]:
(4.5)
where T is the transmission efficiency, R is the reflection coefficient, L is the cavity length
in the FP resonator, and k is the wavenumber of the incident light. The FP finesse is
defined according to:
FSR
finesse = F
where 6f =
(4.6)
-. If the instantaneous linewidth of the polygon filter, 6AokIogtn, is smaller
than the FSR of the FP, then ideally one wavelength would exist in the cavity at a time.
We can also represent the transmission function of the FP etalon as a convolution of the
Dirac frequency comb, FFSR(w) and a Lorentzian function YL(w; 7).
2wrT 2
TFP) =212 R2
I -
L
(W; ) 0 FFSR(w)
(4.7)
R is related to the reflectivity of FP mirrors; the more reflective the surfaces of the mirrors,
the higher the
Q factor of the cavity which
makes sense because the higher the reflectivity,
the higher the finesse, and the narrower the instantaneous linewidth.
75
PMF
G
SOA
BBS
,
a E
)--5-
/
-W2
FR
A F
M
FR
SA
PBSBSPM
/,
FP
Figure 4-3: Sample figure of a free space wavelength swept laser with a Fabry-Perot (FP)
etalon inserted in the cavity. PMF -= polarization maintaining fiber; FR.= Faraday rotator; BBS = broadband beam splitter; FP = Fabry-Perot; G = grating; PBS = polarization
beam splitter; A/2 = half wave plate [39].
4.3
Laser construction and performance
To generate the optical-domain wavelength stepped laser source, a continuously wavelengthswept laser based on a free space polygon mirror sliding filter was modified (Figure 4-3).
A free space fused silica Fabry-Perot (FP) etalon (LightMachinery) was inserted into the
laser cavity to select discrete and linear-in-k wavelengths. From section 3.3 we saw that
the FSR of the FP is related to the principle imaging depth (a.k.a baseband window
depth), thus we selected a FP with an FSR of 80 GHz (6A =0.45 im), providing a baseband window depth of 1.4 mm (assuming a tissue index of n = 1.38). The finesse was
greater than 80 over a 100 nm spectral bandwidth and centered at 1300 nm. The laser
was implemented using a free-space optical circulator design that allowed smaller cavity
lengths and shorter build-up times for the laser than fiber-based cavities. The linewidth
of the polygon-mirror based filter was designed to be approximately 0.21 nm (or 6f
37
GHz), providing sufficient extinction of the neighboring FP modes while retaining a high
laser duty cycle. A booster amplifier was placed outside the laser cavity to compensate
for power losses induced by the insertion of the FP filter. Output power was measured
at ~46 mW. The laser was operated at 27 kHz for imaging experiments and 5.4 kHz for
mirror translation experiments as presented in later sections. The transmission function
76
of the polygon filter reflected the continuous reflection of dispersed light from the grating
and its spectral bandwidth was given in equation 4.3. It is clear that the laser operates
at frequencies /wavelengths given by the product of the transmission of two filters, the
polygon filter and the FP etalon:
poly *TFP
TFP,poly =
4.3.1
(4.8)
Coherence length
Since the finesse of the FP we used was 80, the instantaneous linewidth, 6W
1 GHz we
would expected a coherence length of approximately 6 cm assuming a Gaussian profile.
To measure the laser coherence length, we used the imaging interferometer described in
section 4.4. Fringe data were recorded from a fixed sample arm mirror while translating
the reference arm over a 12.5 cm range (25 cm optical path variation). Fringe visibility
calculated as the standard deviation of the fringe showed a single-pass coherence length
of approximately 7.4 cm (Figure 4-4). We noted that without the intra-cavity FP filter, the laser coherence length was limited to several millimeters, demonstrating that
inclusion of the fixed FP etalon can both force optical-domain subsampling and also contribute to significant extension of the laser coherence length. Note that because the FP
etalon transmission causes a Lorentzian linewidth shape (Eqn. 4.7), the coherence function theoretically should have more of an exponential decay rather than a Gaussian decay.
Although the linewidth function can also be significantly affected by the gain medium,
which we will discuss more in section 5.4.4.
4.3.2
Chirp and nonlinear tuning
In grating-based swept sources, tuning is often not performed linearly in k-space because
the diffraction grating induces a linear-A space tuning. This tuning chirp results in degradation of axial resolution (see section 2.3.1) and can be avoided by chirped acquisition
clocks (k-clocking) or interpolation of the fringes after digitization. In optical-domain
subsampling, however, two separate sources of laser chirp can exist; the source can be
77
1.0-
E0.50
5
10
15
20
Optical Path Length (cm)
25
Figure 4-4: The measured single-pass coherence length of the optical-domain subsampled
laser incorporating an intra-cavity FP etalon exceeded 7 cm single pas [39].
nonlinear-in-k or nonlinear-in-time (or both) as illustrated in Figure 4-5. Theoretically,
optical-domain subsampling should be compatible with nonlinear-in-time chirping through
previously established methods. However, interpolation and/or clocking cannot be used
to address sources that are chirped in k, i.e., which feature a varying FSR. In this case,
the underlying fringes do not repeat periodically with depth, and a depth-dependent distortion is directly induced. The applications of advanced approaches for spectral analysis
of non-uniformly sampled signals may be applicable in such cases, but are beyond the
scope of this work.
Thus, for conventional optical-domain subsampling of bandwidth
limited signals, it is important that the source have a constant FSR, but is not critical
that each wavelength step occur at a fixed rate in time. Recall that the FSR is related
to the the FP cavity length L and the index of refraction n. Thus the material index n
must not be dispersive over the spectral bandwidth AA otherwise a non-linear-in-k sweep
will result. We selected our FP to have a constant FSR, so no nonliearities in k were
expected. However, because tuning of wavelengths in time was provided by the polygon
laser design, non-linear-in-time chirping was expected.
78
(A)
(B)
linear k/nonlinear time
01)
nonlinear k/linear time
k
k
uniform
time
time
&
chirped A
Figure 4-5: Laser chirping in optical-domain subsampled sources. A subsampled source
can be chirped either in time (A) or k-space (B). In time, interpolation and/or k-clocking
can be used to correct the nonlinearity. A nonlinear-in-k chirp, however, distorts the
optical aliasing properties, and cannot trivially be corrected citeSiddiqui2012.
4.4
Interferometer and acquisition
We discussed in section 3.4 that complex-conjugate ambiguity removal is necessary to
avoid image overlap in subsampled systems. One existing technique used to remove ambiguity is to add an acousto-optic frequency shifter (AOFS) in the reference arm of the
interferometer [601. In this method, the reference arm light is combined with a local oscillator providing a shift of Afshift. In the frequency domain the detector frequency is
given by [4,60],
fdet
7r
r i + Afshiftj
(4.9)
where again a' is the tuning slope (Eqn. 2.30) and Ti is the optical path delay. While
the Fourier transform of the detector current Idet is still Hermitian symmetric, the shift
places signals from +ri depths to the right of Afshift and -i
depths to the left of Afshift.
This separates image segments from either side of the path matched delay and results in
a continuous image with no overlap (Figure 2-10B). Typically Afshift is selected to be the
full bandwidth of the tissue signal and thus forces an undesirable signal doubling. Because
of its availability and ease of implementation, however, we used the AOFS method in our
first slow version of a subsampled imaging system. AOFS for high detection frequencies
79
Depth/OPD
Depth/OPD
2
-T
+Ti
-Ti
a
+T
b
C
U-
I
0
0
RF Frequency
RF Frequency
I
Afshift
Figure 4-6: (a) Without the frequency shifter, only the positive frequency region can be
used. (b) With the frequency shifter both positive and negative frequency regions can be
used but entire spectrum is moved to Afshift.
are difficult to find and have a lot of loss associated with them, therefore we avoided their
use in our high speed imaging system by developing a quadrature interferometry method
as later described in Chapter 6.
The interferometer used for data collection in following experiments is summarized in
Figure 4-7. An AOFS at Af = 25 MHz was used in the reference arm to provide complex
fringe demodulation. Trigger signals were generated from a fiber Bragg grating (FBG)
with a center wavelength chosen with sufficient overlap to one of the FP transmission
peaks. The FBG bandwidth of 42 GHz was small relative to the FP FSR to ensure consistent trigger pulse generation from a single FP transmission peak. Fringes were detected
using balanced 80 MHz balanced receivers (New Focus, 1817-FC). In this dual balanced
interferometer, a beam splitter splits the light 50/50 into two different ports. Each port
enters a polarization beam splitter, which further splits light into an x-polarized arm and
a y-polarized arm. Both x-polarized arms are then received by a balanced receiver (BR1);
the y-polarized arms are received in BR2. This dual detection scheme ensures that both
x-polarized light and y-polarized light coming from the sample can be interfered and received. As discussed later in the processing, the two channels can then be averaged to
further reduce random noise and increase contrast in the image. Electronic low-pass filtering at 50 MHz was implemented prior to digitization at 100 MS/sec to prevent electronic
80
reference
mirror
AOFS
20/80
20/80
subsampled
laser
microscope
4~
FBG
tissue
sample
arm
BB1S
PBS
L
reference
arm
lSchmitt
trigger
circuit
LP @ 50 MHz- --
PBSBR
trigger
CH 1
D
A
PBR2
Q
|
.f.
LP @ 50 MHz---
CH 2
Figure 4-7: The dual channel balanced interferometer and data acquisition scheme used in
subsampled imaging experiments; PBS = polarization beam splitter; BBS = broadband
beam splitter; BR = balanced receiver; AOFS = acousto-optic frequency shifter; FBG
Fiber-Bragg grating.
81
aliasing.
4.5
Experimental validation of circular wrapping
We performed a mirror-translation experiment to validate the circular wrapping and bandwidth compression characteristics of subsampling. To examine the data compression provided by optical-domain subsampling, point spread functions (PSF) were acquired over
a 15 mm (single-pass) optical path difference (Az). The interference fringes between a
sample mirror and reference mirror were recorded continuously as the reference mirror
was translated with a motorized stage over 7.5 mm (double pass) at a constant speed.
Hence, each A-line was measured at successively larger Az values. Recall from equation 2.29 that the frequency of the fringe signal increases as Az increases, thus a linear
increase in frequency with depth was expected. Illustrated in Figure 4-8 is the detection
receiver along with the frequency content of the A-lines over a subset of this 7.5 mm
translation. This data was Fourier transformed in MATLAB and displayed as frequency
vs. depth using the imagesc function; thus areas of the high intensity (white) correspond
to the PSF and the black background indicates no/low signal. The image is divided into
three regions; the baseband window region, the higher order harmonics, and the complex
conjugate signals. Because of the 25 MHz AOFS, the fundamental frequencies (baseband
window) was located in the center of the dataset. The AOFS described above was used
to provide complex signals and induce circular wrapping. The circular nature of wrapping is confirmed in the baseband window; when the fringe signal reaches the edge of the
window, it circularly wraps back to the beginning (opposite side) of the window at the
next incremental depth. This pattern continues over the extent of the 7.5 mm translation
range, and confirms that even as depth increases the frequency of the signal does not
exceed the baseband window bandwidth (approximately 6 MHz). Thus signals over an
extended depth range are confined to a 6 MHz acquisition bandwidth. The signals were
also repeated in the complex conjugate domain due to Hermitian symmetry of the real
signals detected with the optical receiver.
82
+50 MHz
complex
conjugate
(redundant)
g
higher orders
U-
0
r
(redundant)
baseband
A-line
higher orders
(redundant)
-50 MHz---
A-line/Depth
interference
signal
Optical
Receiver
LP @ 50 MHz
100 MS/s
Figure 4-8: Experimental demonstration of optical-domain subsampled OCT. Interference
fringes were acquired of a fixed sample arm mirror while translating the reference arm
mirror. The frequency content of the interference fringes demonstrates the wrapping of
the mirror signal in the baseband window and presence of higher order powers [391.
83
1n
-0.5
-
0.2U.
0
5
time (msec)
10
0
5
10
time (msec)
Figure 4-9: Interference fringe signals at two depths demonstrate optical-domain generation of baseband signals. Left: lower frequency fringe corresponding to a smaller OPD;
Right: higher frequency fringe corresponding to a larger OPD [391.
4.5.1
Signal loss due to higher order harmonics
The subsampled interferogram presented in section 3.2.2 was a Dirac idealization of the
wavelength sweeping in subsampled OCT. Realistically optical wavelengths in a subsampled source can transition directly from one wavelength to the next in a zero-order-hold
fashion, or the laser power can drop to zero between wavelength transitions. The manner
of sweeping determines the extend of interpolation associated with acquiring the interferogram. For instance, as evidenced by the measured interference fringes of the polygon-based
subsampled laser (Figure 4-9), this laser sweeping had a zero-order hold characteristic.
These interference fringes were acquired using a simple Michelson interferometer. The
A-line rate was reduced to 5.4 kHz, and the interferometer output was detected using
a 200 MHz receiver (Thorlabs, BDB460C) without balanced detection. The continuoustime sampling provided by using a receiver with sufficient bandwidth allows us to see the
wavelength-stepped nature of the laser (Figure 4-9).
The zero-order hold function results in a non-ideal sinc low-pass filter and there is
a depth-dependent loss in sensitivity over the width of the baseband window. This loss
comes from power contained in the higher-order harmonics of the interference fringe, and
the information content of these harmonics is redundant to the baseband window as shown
in the previous experiment. To explore this further, we analyzed the frequency content
84
(A) Path mismatch = (m)Ds
-2nd order
-1st order
Baseband
+1st order
+2nd order
CG
time
>Q
=(m+0.125)Ds
(B) Path mismatch
frequency
-0.3
-vim
frequency
time
C
(C) Path mismatch = (m+0.25)Ds
-0.9 d
frequency
time
(D) Path mismatch = (m+0.375)Ds
U)I
-2.1dB
frequency
time
LM
(E) Path mismatch = (m+0.5)Ds
0)
-3.8dB
pf
frequency
time
Figure 4-10: Optical-domain subsampling induces a small periodic loss in baseband signal
strength due to its stepwise nature and the resulting placement of signal power into higher
orders. The signal variation is limited to 3.8 dB over the aliased baseband depth window.
m is an integer and Ds is the baseband window depth [39].
85
of a simulated step-wise interference fringe as a function of location (Figure 4-10). The
depth-dependence of harmonic generation can be explained by the varying magnitude of
the step changes in associated interference fringes; greater magnitude of step changes occur
at fringes that are closer to being critically sampled by the discrete set of wavelengths
(Figure 4-10e). This corresponds to signals located at (m + 0.5) * Ds where Ds is the
baseband window depth and m is an integer; hence there is a maximum loss of -3.8 dB
in baseband signal power here due to higher-order harmonics. Note that these results
assume detection by a conventional low-pass filtered digitizer and would be avoided if
specialized detector circuits employing integrate and hold amplifier circuitry were to be
used [32].
4.6
Experimental validation of imaging
We validated imaging with a subsampled system for the first time with this slow-speed
system. Because of our increased coherence length and our ability to acquire over this
extended range, we had the opportunity to image wider fields with a microscope that accommodates a larger field-of-view (FOV). The design of focusing optics of the microscope
is not affected by the subsampled source and as such have the same transverse resolution
qualities as described for the conventional system in section 2.5. In this work, however,
we used a modified microscope wherein a long focal length lens (f = 10 cm) was placed
before the 2-axis galvometer mirrors instead of after it (Figure 4-11). The loose focusing
of the optical beam increased the depth-of-focus so that imaging could be performed over
an extended depth range at a reduced transverse resolution of 6x =
6y
=96 ftm. The
system also allows for an increased FOV with dependence on how far away the sample
was placed from the galvometer mirrors. This kind of imaging scheme is common in other
in vivo imaging modalities such as white-light endoscopy, and can be very advantageous
for screening applications where large volumes of tissue need to be scanned. Microscopy
like this has not been possible in OCT before because imaging ranges have not been long
enough to allow it.
86
CM d 01
f =10m
Lens
galvo
---
FOV2
FOV3
tissue
FOV1
Figure 4-11: A schematic of the wide-field microscope we use in our subsampled OCT
system. Focal length of lens =10 cm; 0 -= 300 FOV; d = distance between galvo and
sample
4.6.1
Image Processing
The processing protocol for imaging with the subsampled laser is outlined in Figure 4-12.
A trigger pulse, which was created by sending a part of the laser of the laser output
through a Fiber-Bragg grating (see Figure 4-7), initiated the recording of each A-line. As
suggested in Figure 2-12, two galvanometer mirrors scan the tissue in a raster scan pattern
in order to get traverse scans in both the x and y-directions (a.k.a. B-scan and C-scans).
In many of our experiments there were 2400 points/A-line, 1024 B-scan points, and 1024
C-scan points. The 2400-point DFT of each A-line was calculated and windowed with a
hanning function was used to prevent ringing in the frequency domain
[12].
The frequency
shift of Af that was induced by the AOFS in the interferometer (refer to Figure 4-6) was
then numerically removed. The dataset was zero padded to the next power of 2 to make
the interpolation and FFT step more efficient in MATLAB. If non-linearities in sampling
were present, the next two steps (taking the inverse Fourier Transform and re-interpolating
the dataset) linearized sampling in time to remove the effects of chirp in the laser. The
mapping function was obtained by placing a mirror in the sample arm and recording
fringes at two separate locations (one near path matched and one near the edge of the
baseband depth) and deriving chirp from non-linearities in the phase of the FFT signal.
In the final processing stage, the 4096 x 1024 x 1024 matrix corresponding to each channel
87
CH1
trigger
CH2
DAQ
Computer
CH2 A-lines
CH1 A-lines
DFT
FFT shift (-Af)
hanning window
I
zero-pad up to 4096
iFFT
I
reinterpolation
FFT
I
average CH1 + CH2
crop/tile frame'
Figure 4-12: Experimental processing flowchart for subsampled images.
88
-
..
was averaged and cropped to remove the redundant higher order harmonic signals. The
cropped images were tiled to produce a continuous image as shown in section 4.6.2.
4.6.2
Finger and phantom imaging
To validate that subsampling is applicable to OCT imaging, images of a finger and a
rubber phantom were acquired with the same detection receiver outlined in Figure 4-7.
The A-line rate was 27 kHz for these experiments. Figure 4-13A shows the baseband
window of one longitudinal cross-section of a finger. The sensitivity of the system was
sufficient to see the stratum corneum layer beneath the skin. The higher order harmonic
signals were cropped since they contained redundant information and were a consequence
of using 100 MS/s digitizer in lieu of a slower digitizer. Since there was curvature in the
finger, aliasing from the subsampling caused a circular wrap of the tissue signal upon
reaching the edge of the baseband window, which made the image appear discontinuous
(arrows: location of aliasing of the surface of the sample). Interestingly, tiling identical
copies of this baseband window lets us appreciate the continuity of the sample (Figure 413B). The fixed frequency noise coming from the intensity modulation of the laser can be
seen in this image at the edge of the baseband window of one of the cropped cross-sections
(yellow arrow). In Figure 4-13C, a depth cross section reveals how these tiled baseband
windows compile to form an en face image of the finger (arrow: junction between skin
and finger nail). The wrapping caused by subsampling resulted in numerous depth slices
being visualized in one en face cross-section. Theoretically, a surface finding algorithm
can be employed to eliminate this effect, however, this redundant depth signals does not
interfere with the interpretation of the image, and can help visualize the contour of the
sample. Figure 4-13D shows an average intensity image wherein the average value of the
entire en face stack is averaged and displayed in one image. Note that because the source
FSR gives a ~1.358 mm baseband depth, the signal has fully dropped below the system
noise floor before signal from the surface reappears. Because the new microscope scheme
(outlined in section 4.6) was used, the B-scan and C-scan had an area of approximately
1.5 cm x 1.5 cm. The ability of subsampled OCT to support imaging over extended
89
(A)
(B)
(C)
(D)
Figure 4-13: Cross-sectional images of a finger resting on a small breadboard, imaged with
the subsampled OCT set-up. (A) Baseband window cross-section of the skin. Curvature
of the sample causes wrapping of the surface at the location of the arrows. Scale bar:
500 Jijm. (B) Tiling the baseband window (outlined in yellow) allows for continuous
visualization of the sample. Arrow: fixed frequency noise resulting from laser (C) Left:
En face view of the cropped tiled image. Right: Average intensity image of the enface
stack. Bar: location of longitudinal cross section in (A). Arrow: junction between the
finger and the nail 139].
90
(A)
(B
(B)
(C)
Figure 4-14: (A) Rubber phantom resting against a metal post on a small optical breadboard. The tilted rubber phantom spans 2 cm in depth. (B) An en face cross section of
the rubber, post, and breadboard. Aliases of the tilted rubber phantom from different
depth planes into this one make it possible to visualize numerous surface reflections. (C)
An en face cross-section from another depth results in displacement of the high intensity
surface reflections (white) [39]. (D) Average intensity projection of the en-face stack.
91
depth ranges is better demonstrated with a rubber phantom that is placed at a tilt so
that the depth of the sample spans a range of 2 cm (Figure 4-14A). An en face image
of this set-up shows the numerous aliased surface reflections of the rubber phantom, the
metal post, and the small optical breadboard that the objects are resting on (Figure 414B). Since these objects were mostly opaque, the high intensity (white) signals reveal
mostly the surface of the object, which have high surface reflections. Because of the tiled
baseband windows, the full set-up can be interpreted from one en face view; this imaging
paradigm has never been shown in OCT before. Notice that because of the steepness of
the angle of the rubber phantom, the baseband windows appear to be closer together on
the rubber, illuminating the three-dimensional contour of the set-up. Also, because the
set-up was placed over 2 cm from the imaging probe a larger B-scan and C-scan area was
imaged (approximately 3.5 cm x 3.5 cm). Figure 4-14C provides another en face image
at a further depth cross-section. Although the overall set-up looks the same, the location
of the surface reflections have moved slightly, meaning that in each baseband window, a
different depth is being probed. Thus, subsampled imaging not only increases the ranging
depth of imaging while maintaining a small acquisition bandwidth, but it can also provide
important additional information about the overall geometry of the structures that are
imaged. The entire en face field of the rubber phantom, post, and optical table can been
appreciated in the average intensity image in Figure 4-14D.
92
Chapter 5
Novel High-Speed Subsampled Laser
The proof-of-concept work in Chapter 4 provided intuition on how we can construct and
image with subsampled lasers. In this chapter we describe the design and construction of
a novel laser with unprecedented speeds of -19 MHz Aline rates. This laser can operate
in both subsampled and continuously-swept regimes [47].
5.1
Relevant work
Since the advent of Fourie-domain OCT (FD-OCT), there has been much effort dedicated
to building high-speed swept source lasers. The polygon scanning laser we talked about
in section 4.1.1 was one of the first wavelength-swept lasers utilizing a polygonal scannerbased wavelength filter with a sweep repetition rate of 16 kHz
[581.
Increasing the speed
of the polygon-based laser was explored in subsequent years, and A-line rates exceeding
400 kHz were achieved by using multiple copy and paste delay lines [31J. Later, Fourierdomain mode-locking lasers with a tunable Fabry-Perot provided A-line rates ranging up
to 5 MHz, however with limited stability at these speeds [17, 21, 54]. Tuning based on
dispersion or dispersion mode-locking have more recently been demonstrated that reach
A-line rates up to 250 kHz [9,29,42,56]. The VCSEL-based laser has been demonstrated
at 1 MHz and also allows for long imaging ranges because of very narrow instantaneous
93
linewidths
133].
However, these and other high-speed sources have not been able to take
advantage of long range imaging because of the acquisition bandwidth problem that is
central to this work.
In this segment of our work, we demonstrated a novel high-speed laser specially designed to provide a wavelength-stepped output so that we can take advantage of opticaldomain subsampling.
We achieved wavelength tuning by incorporating both positive
and negative dispersion in our laser cavity and using an intensity modulator to provide
amplitude modulation. This tuning scheme is similar in operating principle to a laser
previously demonstrated for time-division multiplexed addressing of fiber Bragg grating
(FBG) sensors [22]. However, in our laser, wavelength selection was performed by a fixed
Fabry-Perot etalon rather than FBGs, which ensures wavelength-steps are equally spaced
in optical frequency.
5.2
Laser operating principle
The design of the laser cavity is shown in Figure 5-1. First, a high-extension lithium niobate intensity modulator (MXER-LN, Photline Technologies, >30 dB extinction at 1550
nm) creates 320 ps FWHM broadband pulses. The loss from the intensity modulator was
measured to be 4.3 dB at maximum transmission. The light then undergoes a positive
dispersion (+D) that is generated by 19.929 km of SMF-28e+ fiber spool. We placed a
circulator and Faraday rotator mirror at the end of the +D fiber to create a double-pass
through it and generated a total dispersion of +655.7 ps/nm. This produced a continuously swept laser with a stretched pulse of -50 ns. Because of the large loss incurred by
the long length of dispersion fiber (single-pass loss of 4.13 dB), the light was the boosted
with an optically isolated broadband semiconductor optical amplifier or SOA (Covega,
BOA1004S). We again used a free-space FP etalon in the cavity to create discrete optical
frequencies for subsampling. This time, however, we used a FP that transmits in the
1550 nm wavelength region and with an FSR of 200 GHz (1.6 nm) so that we can have
94
f=
+-
N(I/r,,)
320 ps
time
WN
Intensity
P
P
modulator
1
WI
/A
-D+D
D
FR
0time
IIT
80/20FPC
C.)
I
I
JI
> time
PC
OA1
WN
e
time
Figure 5-1: The laser design is composed of positive (+D) and negative (-D) chromatically
dispersive elements, a fast intensity modulator and a Fabry Perot (FP) etalon to generate
a rapid wavelength- stepped laser output. SOA = semiconductor optical amplifier, PC =
polarization controller, FR = faraday rotator mirror, BOA1 = booster optical amplifier.
95
-1.12 ns delay between adjacent wavelengths. We selected this FSR so that there was full
temporal separation between the optical pulses at the SOA to avoid non-linear mixing.
With a finesse of ~100 (custom manufactured from LightMachinery), we could achieve a
theoretical instantaneous linewidth of 2 GHz (0.016nm). The loss in the free-space path
was 5.27 dB at each transmission peak due to fiber-to-fiber coupling loss and etalon loss.
The output coupler was placed directly after the FP etalon so that discrete wavelength
sweeping was achieved. This location for the laser output also provided the narrowest
instantaneous linewidths. The light that continues in the cavity is boosted with another
SOA and travels through a negative dispersion fiber (-D) that was designed to be dispersion slope matched to the +D. For the -D, we used 5.26 km of dispersion compensating
fiber with a measured loss of 2.96 dB. The light that is output with a 20% tap coupler is
further amplified outside the laser to increase power and reduce intensity noise through
gain saturation effects.
By matching cavity dispersion, the round-trip time was approximately the same for all of
the wavelengths within the lasing bandwidth. This was important because the intensity
modulator was driven at a repetition rate equal to a harmonic of this round trip time,
f = N(
1
Tcavity
)
(5.1)
and discrete wavelengths that do not arrive within the 320 ps transmission window will
not accumulate enough resonant power. A pattern generator (Sympuls, PAT 3000) provided these short pulses to the intensity modulator via a high-speed amplifier, and the
pattern generator was externally clocked by an RF signal generator (Stanford Research
System, SG384).
In this laser there is a trade-off between pulse width, the free spectral range (FSR) of
the etalon, and the required dispersion. As the pulse width increases, larger dispersions
are required to separate pulses. If smaller dispersions are used, the FSR of the FP etalon
must be increased so that the wavelengths are further spectrally separated and are ade96
quately separated in time by the dispersion fiber. From these relationships, one can note
that this design is more appropriate for rapid sources using short pulses (and therefore
requiring less dispersion) than for slow sources. The sweep rate of the laser is fundamentally limited by the product of the lasing bandwidth and the dispersion provided by
each dispersive element, and is independent of the overall cavity round trip time (when
operated in a harmonic resonant mode).
5.2.1
Dispersion compensation
The core concept of the high-speed laser is based on chromatic dispersion because we rely
on it to stretch our broadband pulse so that we have sweeping of our wavelengths in time.
Dispersion is described as the dependency of the propagation constant,
# on the frequency
of light. Since we utilize broadband sources in OCT, we cannot assume a constant phase
velocity for all of the optical frequencies within our spectrum [4,5,131. The propagation
constant can be expressed as a Taylor series expansion around the center frequency, wo,
O(w) = n(w) c =
W
1
+3(P - wO) + -2(w
o(wO) +
2
-
_1O2
wo)2
+ I-/2(W
6
-_W)
wo)3
+
...
(5.2)
The first order term (01 with units [s/m]) caues a difference from the vacuum speed of
light. The second order dispersion (/2 with units [fs 2 /mM]) is the group velocity dispersion (GVD) and causes different wavelengths to travel through the medium at different
velocities. In imaging, this parameter generally causes axial resolution broadening with
depth. However in our laser, it is essential for separating our wavelengths in time. The
GVD is related to the dispersion coefficient, DA with units [ps/nm/km], by the following
expression: DA
-X
/2.
The total dispersion (+D) is the product of the dispersion
coefficient and the length of dispersive medium that light propagates through.
The spectral bandwidth of this laser relies on carefully matching the +D and -D fiber
total dispersion slopes (ps/nm). If the broadband pulse is not completely recompressed
by the end of one circulation in the laser cavity, then the pulse will continue to expand
97
A q
p hoto detecto r
RC+
Vector
network
analyzer-D
A AA/2
Tunable laser
(1500 - 1630 nm)
Figure 5-2: Schematic of the phase-shift method for measuring dispersion in combined
positive dispersion (+D) and negative dispersion (-D) fibers.
with each circulation and eventually trailing wavelengths will be attenuated by the intensity modulator's "off" sequence and fail to acquire enough gain to lase. This results
in reduced overall spectral bandwidth, or in the cases of very poorly matched dispersion,
failure to lase altogether. Thus characterizing and carefully matching the total dispersion
in the laser cavity is key to building a well balanced laser.
To achieve this matching, we measured the group delay for the combination of the
+D
and -D fibers using the phase-shift method. A schematic of the set-up is shown in Figure 5-2. A narrowband tunable laser source (TSL-510, Santec) produced optical signals
at wavelengths tunable from 1500 nm and 1630 nm and this light was intensity modulated
with a vector network analyzer (MS2036C, Anritsu) at a driver frequency of f m z= 97 MHz.
The light was transmitted though the dispersion fibers and the transmitted signal was
detected with a photodetector. The phase, q, of its modulation was measured relative to
the electrical modulation source at A
+ AA/2. This phase measurement was repeated at
another wavelength interval, A - AA/2 and from the measurements at any two adjacent
wavelengths, the change in group delay, AT corresponding to the wavelength interval AA
is calculated as,
AT =
360fm
2
x 1012
(5.3)
where A is the center wavelength of the interval AA.
98
I
237,10428
-
237,104.26
-
237,104.24
-
237,104.22
237,104.18
237,104.AG -
1500
.
.
-
237,104.20
1610
1520
1530
..
1640
I
1560
1550
Wavelength (rm)
1570
1580
1590
1600
1610
Figure 5-3: The absolute group delay of the combined positive and negative dispersion
fibers as a function of wavelength. The group delay averaged 237.1 [is across the lasing
bandwidth with variations of 60 ps [47].
70
G60
0%
coo
01520
1540
1560
Wavelength (rim)
1580
1600
1620
Figure 5-4: A plot of chromatic dispersion across the lasing bandwidth
t47].
Figure 5-3 presents the total group delay measurements of the combined dispersive fibers
(not including the SOA, output coupler, and FP) over the spectral bandwidth. There
were variations on the order of
60 ps in group delay across the wavelengths, which met
the criterial for variations that are less than the pulse width (320 ps). The blue curve
provides a 3rd-order polynomial fitting to this data. Reducing group delay variation by
higher order dispersion engineering may be a path to further improve laser performance.
The chromatic dispersion was calculated as the derivative of the group delay as shown in
Figure 5-4. We achieved a matched total dispersion (D =0) across the lasing bandwidth.
99
5.3
5.3.1
Practical considerations
Intensity modulator pulse synchronization
The driver for the intensity modulator must have timing accuracy at least within the 320
ps pulse window over the relevant cavity roundtrip time of 237.1 ps. Therefore we used a
low-jitter bit pattern generator (Sympuls, PAT 3000) with fast rise times to provide the
pulses to the lithium niobate intensity modulator. This pattern generator was externally
clocked with a high-speed RF signal generator (Stanford Research System, SG384). We
also used a low-jitter amplifier (Photline DR-PL-10-MO-LR) after the pattern generator to
amplify the electric signal into the intensity modulator. The jitter across these electronics
were measured to be less than 50 ps; further reducing jitter and rise times will lead to
more stable lasers in the future.
5.3.2
Polarization-mode dispersion
There was significant polarization mode dispersion (PMD) induced by the long positive
dispersion fiber, which resulted in an unstable source in our first iterations of this laser. We
eliminated this effect by replacing our initial single-pass SMF-28e+ spool with a doublepass design (of half the length) and using a Faraday rotator mirror to reflect and cancel
the effect of PMD through non-reciprocal polarization changes. Using the wavelengthscanning method centered at 1550 nm, we measured the PMD in 40 km of SMF-28e+ to
be 0.46 ps and in the dispersion compensating fiber to be 0.16 ps.
5.4
5.4.1
Laser Performance
Subsampled operation
If the laser were operated at it's fundamental frequency, it would produce an -18.9 MHz
A-line rate laser with a very low duty cycle. In order to achieve almost 100% duty cycle
we operated this at the 4336th-order harmonic of the laser cavity roundtrip time. The
100
Subsampled Laser + BOAI
01
SI
-30
1554.5
1554.0
1555.0
1556.0
1555.5
Waveo4.gth (nm)
1556.5
zz
Subsampled Laser + BOAl
-5-
-10-
-150
.3
-20-
UUUUUUU
-25
1500
1510
1520
1530
1540
1550
1560
L Lk4
1570
1580
1590
1600
Wavelength (nm)
Figure 5-5: A typical optical spectrum of the wavelength-stepped laser source. The
total tuning range is approximately 70 nm. The inset shows the shape and width of the
instantaneous linewidth and a -1.6 nm FSR spacing.
101
0.250.20
0.15
-
0.10
0.05-
aCh
0
-
-0.05
-0.10
I
I
120
140
-0.15-0.20
0
-
|
20
|
|
40
60
100
80
160
180
200
Time (na)
Figure 5-6: A time trace of the laser output showing a repetition rate of over 18.9 MHz.
Discrete wavelength sweeping is evident in this laser.
roundtrip time of the cavity is related to c = 3 x10 8 is the speed of light, L is the total
cavity length, and n = 1.4682 is the index of refraction of SMF 28e+.
Tcavity =N
n
c
(5.4)
The optical spectrum after the external booster amplifier, shown in Figure 5-5 reveals
the expected frequency comb structure with an expected FSR of 1.6 nm (200 GHz). As
expected from the chromatic dispersion plot, the lasing bandwidth spanned from approximately ~1520 nm to ~1590 nm, yielding a total spectral bandwidth of AA = 70 nm.
It was determined that this bandwidth was limited by a combination of high cavity loss,
variations in cavity group delay (see Figure 5-3), and to a lesser degree the PMD. We determined that the bipolar transmission peaks were caused by the frequency shift induced
by nonlinear four-wave mixing in the SOA gain medium.
This increased the FWHM
instantaneous linewidth to -0.06 nm from the expected 0.016 nm expected by the 100
finesse of the FP etalon.
In the time domain, the laser trace (shown in Figure 5-6) was detected with an amplified
102
............................
photodetector (New Focus 1617-AC, 800 MHz), filtered with a 550 MHz low-pass filter,
and digitized with a high-speed oscilloscope (Tektronix MSO 5204, 2 GHz, 10 GS/s). In
this laser, discrete wavelengths can be distinguished in the sweep; recall in section 4.5.1
that the polygon-based laser had a zero-order hold sweeping mode. We also expect losses
to higher order harmonics in this sweep configuration; in fact more power is expected
to be found in the harmonics because of the high frequency components in the sweeping
function. Unlike in the zero-order hold sweep, however, there is expected to be less of
a sensitivity drop over different depths within the baseband window. The experimental
validation of this is shown in section 7.2 and we determine that the sensitivity variation
across the depths in a given baseband window is only 1-2 dB, compared to the 3.8 dB
loss of the zero-order hold sweep (discussed in section 4.5.1). The average output power
of the laser was 10.41 dBm. The source RIN was measured to be approximately white
(aside from structures attributed to A-line and pulse repetitions) across a 1 GHz spectral
range, with a value of -126.18 dB/Hz at 499.8 MHz.
5.4.2
Continuously-swept operation
While the initial motivation of this laser was to create a subsampled laser source, removing
the intra-cavity FP etalon provided a rapid continuously-swept laser source. The operation of the continuously-swept laser provided a good comparison for changes in i.e. SNR
or coherence length due to the addition of the FP etalon. First, because of the absence
of the FP etalon, the loss in the cavity in the continuously swept operation was slightly
lower. The average output power was ~14.7 dBm, however the power in the FP transmission peaks was only 1 dB lower than the analogous wavelengths in the continuously-swept
mode (reflective of the loss through the transmission peaks of the FP). The spectral bandwidth of the continuously-swept source is shown in Figure 5-7, and has approximately the
same bandwidth of ~70 nm. The time-domain laser output was measured similarly to the
subsampled laser trace and shows the expected continuous shape (Figure 5-8). Notably,
the amplitude of the continuously-swept laser was -150 mV, the same as the amplitude
range of the subsampled sweep shown in Figure 5-6. The source RIN was measured to be
103
Continuous Laser + BOAt
0]
-5-10
'
-20-25
1510
1500
1520
1530
1540
1550
Wavelength
1570
1560
1600
1590
1580
(nm)
Figure 5-7: The optical spectrum of the dispersion-based laser operating in the
continuously-swept operation. This laser also has a -70 nm optical bandwidth.
0.25-
Continuous +BOA1 Laser Time Trace (200 ns)
--
0.20
0.15
E
0.10-
0.05
0
0
-0.05-0.10-0.15
.-
'
-0.200
20
40
60
80
100
120
140
160
180
200
Time (ns)
Figure 5-8: A trace of the time sweep of the continuously-swept laser.
114.1 dB/Hz at 498.9 MHz.
5.4.3
Coherence length measurements
The coherence length of the laser in both the subsampled and continuously-swept operation were measured with a variable delay Mach-Zehnder interferometer. The fringe visibility was measured with an amplified balanced receiver (ThorLabs PDB480C, 1.6 GHz)
as a function of the interferometer delay. While it is difficult to confirm long coherence
lengths with rapid wavelength-swept sources due to electronic and receiver bandwidth
limitations 133], subsampled lasers have limited bandwidths 1391, making these measure104
(A)
Conventional Coherence Length
0
200-
0
100
-
4c
-
E
0
0
0
0.4
0.6
1.4
1.2
1.0
0.8
1.6
1.8
0
0
0
2.6
2.4
2.2
2.0
40
OPD (mm)
100-
(B)
Subsampled-BOA1
S0''
000
E
*T
I*
50
U.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
OPD (mm)
Figure 5-9: (A) The measured double-pass coherence length of the continuously-swept
laser was approximately 1.2 mm. (B) The double-pass coherence length of the subsanpled
laser was approximately 14 mm.
105
Intensity
+DPC
modulator
-Do
+D
FIR
-+
0
>
FPPC
P
80/20
PC
FP2
OA2--
0A1
Figure 5-10: The design of a modified laser with an external cavity Fabry-Perot (FP)
etalon and two booster optical amplifiers (BOA). This laser has increased output power.
ments more straightforward. Fringe amplitudes were measured to the oscilloscope limit
of 2 GHz.
The detected fringe amplitude is plotted as function of double-pass mirror
distance for both the continuously-swept source (Figure 5-9a), and the subsampled source
(Figure 5-9b).
For the continuously-swept source, the fringe amplitude decay showed a
drop to approximately 50% of its path-matched value at 1.2 mm, suggesting a doublepass coherence length on the order of -1.2 mm. Note that this coherence length was on
a single side of the depth space and after complex conjugate removal, the total imaging
range is twice this length, or ~2.4 mm. The coherence length in the continuously-swept
configuration was greatly dependent on the modulator pulse width; smaller pulses provide greater wavelength discrimination and increase coherence length, but narrow lasing
bandwidth and reduce stability (likely due in part to the effects of timing jitter).
For
the subsampled laser, the fringe visibility dropped to 50% of its path-matched value at
a double-pass mirror displacement of approximately 14 mm (total imaging range of ~2.8
cm). This confirms the ability of the FP etalon to force the laser to operate with relatively
long instantaneous linewidth.
106
Fgue :Las
aim
20
30
Trace w External Cavity P + oA2 (100 nas)
apt .Tsm
0.20-0.15
0.10
-0.0
-0.15-0.20
0
10
50
Tite (its)
40
60
70
80
90
100
Figure 5-11: Laser time trace of amplified laser showing increased amplitude. This amplified laser also displayed discrete sweeping characteristics, and also appeared more stable.
5.4.4
Future laser modifications
Future works on the subsampled laser will be focused on improving the stability of the
laser and reducing the effect of nonlinearities in the cavity. We hypothesize that because
of the frequency shift induced by the SOA, the fixed transmission comb of the FP etalon
filters out much of the shifted power in the cavity during each round-trip. This shift is
proportional to the instantaneous linewidth of the light propagating through the cavity
and the power of the wavelength modes. Hence the power in the cavity cannot be increased
without causing instability in the laser. Although the imaging in subsequent chapters
were performed with the subsampled configuration of this laser shown in Figure 5-1, we
also explored ways to increase the output power of the laser in order to achiever higher
sensitivity. In one configuration, we introduced an external-cavity FP etalon and a second
external-cavity BOA as shown in Figure 5-10.
The external cavity FP was included
to filter the ASE light and to reduce the instantaneous linewidth of the wavelengths
prior to amplification with BOA2. The output power of this laser was 17.07 dBm (51
mW), however with slightly increased instantaneous linewidth (0.09 nm). Because of the
external-cavity filtering, this configuration of the laser was more stable. The time trace of
the output of this laser (measured in the same way as before) indeed shows an amplified
output with -335 mV amplitude, in comparison with -150 mV shown in Figure 5-6.
107
Sensitivity measurements with this amplified laser revealed more than 3 dB of sensitivity
gain as compared to the non-amplified laser. In future designs of this laser, gain medium
with suppressed nonlinear mixing can be explored (i.e. erbium-doped fibers or Raman
amplifiers) to increase output power, reduce noise, and further increase coherence lengths.
108
Chapter 6
High-Extinction Complex-Conjugate
Ambiguity Removal
In section 3.4 we established that complex conjugate ambiguity removal is required for
subsampled imaging. In our slow-speed system, we used an AOFS to achieve ambiguity
removal, but at the cost of RF frequency doubling. Quadrature demodulation allows for
detection of both the real and imaginary components of an interference fringe, however
in practice these interferometers have not been able to achieve good removal of conjugate
artifacts in OCT signals. In this chapter, we provide a framework for compensating errors
in quadrature interferometers in the post-processing stage to achieve >60 dB extinction.
We demonstrate that this error correction algorithm is stable and easy to calibrate [38].
6.1
Introduction
Without complex demodulation, Fourier domain optical coherence tomography (FDOCT) cannot differentiate between signals located at an equivalent distance above and
below the zero-delay location.
This leads to a depth ambiguity (alternatively termed
complex conjugate ambiguity or depth degeneracy). If the sample is physically located
entirely on one side of the zero delay position, this ambiguity does not induce imaging
artifacts. However, for many applications, it is preferable to use both the positive and
109
negative depth space to achieve a larger imaging range. In these applications, complex
demodulation is used to discriminate between positive and negative depth space signals.
There are many implementations of complex demodulation in OCT. Phase-shifting methods were the first attempt to achieve full-range OCT. However these methods did not
provide high extinction, and required multiple A-lines per measurement location [25, 43,
55, 63].
Later, simultaneous detection of real and imaginary fringe components were
achieved by interferometer modifications. A common technique is to use an active phase
modulating element in the sample or reference arm. These include configurations based
on phase modulators [11] and acousto-optic frequency shifters (AOFS) [57]. While the
AOFS method is straightforward to implement, it has several limitations. First, AOFS
devices operate over a finite optical bandwidth, limiting their utility in wide-band OCT
systems. Second, the AOFS drive frequency is optimal at 1/4 the digitizer acquisition
clock. This necessitates a change in AOFS hardware if the acquisition clock is changed.
Finally, the AOFS method doubles the required digitizer bandwidth per channel. For
some applications, this doubling of the digitizer bandwidth while keeping the same number of channels is advantageous.
While in others (i.e.
high speed systems operating
near the digitizer maximum speed) it may instead be advantageous to keep the same RF
bandwidth and add a quadrature signal on a second digitizer channel. Passive optical
demodulation approaches exist for acquiring this quadrature information [26,35,50].
Generating accurate quadrature signals has been a challenge for passive optical quadrature
demodulation circuits. With imperfect quadrature demodulation, a mirror will generate
signal power on both sides of the zero delay position. To match the 60 dB extinction
provided by AOFS and required in many applications, quadrature errors below one part
per thousand (0.1%) are required (errors scale with the -/extinction).
It was demon-
strated that a polarization-based demodulation circuit can create quadrature relationships
by manipulating the light polarization within a demodulation circuit
[50].
Because the
birefringent properties of the demodulation circuit vary as a function of wavenumber, a
110
quadrature correction algorithm was implemented in post-processing to reduce the signal
power on one side of zero delay, and increase the power on the other. As a result, an
extinction ratio greater than 50 dB was achieved for a low-speed (10 kHz A-line rate)
swept-source system.
Here, we use the polarization-based demodulation circuit as a template for quadrature
hardware that induces errors that can be compensated in the processing stage. First we
identify a second source of error that is dependent on the fringe RF frequency rather
than wavenumber. The magnitude of this error scales with the RF bandwidth, and will
be limiting for high-speed imaging systems operating in the 100 to 1000 MHz RF signal
range (the prior system operated with 10 MHz bandwidth, and the effect of this RF error
was small). Existing algorithms that address these types of errors do not exist, so in
this work we also describe the origin of these RF errors and present a combined quadrature correction framework that addresses both wavenumber-dependent (spectral) and RF
errors. We demonstrate that RF errors are created by path length mismatches in the
demodulation circuit and non-flat electronic RF filtering, and that these errors are stable
in time (and may only need to be derived once for a given system). In contrast, the spectral errors vary with system temperature and remain valid for several to tens of minutes.
Therefore, two methods to derive correction parameters empirically from mirror signals
are presented. The performance of these methods including their stability are presented
for a polarization-based demodulation circuit. However, these methods are generalizable
to other passive quadrature demodulation schemes used in swept-source OCT.
6.2
6.2.1
Experimental system design
Polarization-based demodulation circuit
The polarization-based demodulation circuit is presented in (Fig. 6-1) [50] . The quadrature demodulation is independent of changes in the light amplitude, phase, or polarization that occur with the reference or sample arm. The device is constructed entirely
111
Optical Demodulation Circuit
P~x PBS
50/501
B R
DAQ
2 Channel X
S
a:PBS
PCQX
PayM
PB
2
PBC
Ref
+B.R.
MQ
BR
1DAQ
2 Channel Y
50/50
PCQY
Figure 6-1: Schematic of a polarization-based quadrature demodulation circuit with polarization diversity and balanced detection. Components contained within the blue dotted
box are part of the optical demodulation circuit. PC, = in-phase polarization controller,
PCQ = quadrature polarization controller, M = measured in-phase signal, MQ = measured quadrature signal, PBS/PBC = polarization beam splitter /combiner 1381.
by fiber-based components including a 50/50 fused coupler (Gould Fiber Optics), five
polarization-beam splitters /combiners (PBS/PBC) (Oz Optics 1x2 and 2x2), and polarization controllers (PCs)(General Photonics).
No polarization-maintaining fibers were
used within the circuit or interferometer. In addition to providing quadrature signals,
this circuit supports both balanced-detection (to reduce laser amplitude noise and autocorrelation noise) and polarization diverse detection. As in other polarization diverse
circuits, the sample arm light is returned in both the X and Y polarization states and
is directed to separate detector subcircuits for independent detection. Here, sample arm
light returning in the X polarization is directed to the channel X subcircuit (after the
2x2 PBC), and sample light in the orthogonal Y polarization is directed to the channel Y
subcircuit. Note that the sample arm light and the reference arm light are orthogonally
polarized in each of the 2x2 PBC output fibers thus there is no interference at this 2x2
PBC. Once these optical signals pass through the second PBS, interference signals are
generated and the amplitude and phase of these interference signals is determined by
the specific orientation of the polarizations relative to the PBS axis. By controlling the
birefringence of the fiber between the 2x2 PBC and 1x2 PBSs with the PCs, it is possible
to tune the interference amplitude and phase relationship of one interference fringe (MI)
112
relative to the other (MQ).
Amplitude noise is removed by balanced detection of the complementary output from
each 1x2 PBS. We note that the demodulation properties of the circuit are based on the
optical properties within the circuit, and are independent of the interferometer sample
arm. Thus, the demodulation of Channel X is independent of changes to the sample arm
light (or the reference arm light). This decouples the demodulation from the varying
properties of the interferometer and sample. The fibers connecting the 1x2 PBSs and the
detectors only transmit power and are not phase or birefringence sensitive. Nominally, all
the PCs are configured such that the reference arm power is equally distributed to each
output of the 1x2 PBS, and the interference signals are approximately equal in amplitude
and phase shifted by 90 degrees. To achieve this, the following procedure is used. First, a
reference arm power is equally split between the X and Y subcircuits using the PC within
the reference arm (before the 2x2 PBC). Next, a sample arm mirror signal is provided
to generate a single depth fringe. The PCIx of the X subcircuit is then configured to
achieve equal reference arm power at the output of its 1x2 PBS, which will also maximize
the interference fringe amplitude. Then, the PCQx of the X subcircuit is configured to
provide equal reference arm power on each of its 1x2 PBS outputs (which again maximizes fringe amplitude). Now, the controllers PCIx and PCQx are perturbed to induce
an approximate 90 degree phase shift relative to each other while maintaining high fringe
amplitude. Once obtained, this sets the physical state of the X subcircuit. The same procedure can be applied independently to the Y subcircuit. It is not essential that the phase
shift be accurately set to 90 degrees as these errors will be removed in post-processing.
Therefore, care is taken to ensure that fringe amplitudes remain high and reference arm
powers balanced while setting the phase relationship approximately to 90 degrees.
6.2.2
OCT system
To validate this method, we used two imaging systems. The first used a polygon-based
swept-source operating at 46.8 kHz and two balanced receivers (ThorLabs PDB460C).
113
We acquired our fringes at 100 MS/s with a two-channel data acquisition card (Signatec
PX14400A), allowing detection of complex fringes from the X or Y subcircuit.
Four
channels would be needed for simultaneous polarization-diverse detection. We used an
-
AC coupled DAQ card, which provided a highly non-linear frequency response from 0
3.5 MHz (corresponded approximately to delays between 0 - 0.6 mm double-pass). This
allowed us to test the ability of our technique to remove large RF errors.
A simple
scanning microscope was constructed using a 2-axis galvo (ThorLabs GVS011) and a
40 mm objective lens (Edmund Optics) for imaging. The second system used a novel
dispersion-based swept source operating at an 18 MHz A-line rate [471 and two balanced
receivers (New Focus 1617-AC-FC). We used a two-channel 1.8 GS/s data acquisition
card (Alazar ATS9360) and measured delays between 0 - 240pm (corresponding to an RF
span of 0 - 350 MHz). This system allowed for confirmation of our error compensation
method for high-speed systems.
6.3
Mathematical framework describing errors and errorcorrection in passive optical quadrature demodulation circuits
An ideal optical quadrature demodulation circuit provides two quadrature signals for
each polarization-diverse channel, S1 (k) and SQ(k). In any implemented circuit, we are
provided two measured fringes Mr(k) and MQ(k) which are not in perfect quadrature relation. Here, we provide a generalized framework that describes how the measured fringes
(M1 (k) and MQ(k)) are related to the ideal fringes (S1 (k) and SQ(k)) as a function of
spectral and RF errors.
As a starting point, we note that quadrature refers only to a relative relationship between the I and
Q signals.
Thus, without loss of generality, we can initially define our
114
_'__
,
. ,
_'h_ -
-_
- -
- -
_-AA
,,, --
, -
,
I...
_.. -.
--
- - II
-
I .. -
-
_
_
I
1.1-11 -
ideal I signal, SI(k), as the measured fringe M1 (k),
S1 (k) = M1 (k)
(6.1)
Now we seek expressions relating the ideal quadrature signal SQ (k) to the measured fringes
M1 (k) and MQ(k). We first consider a polarization-based demodulation circuit wherein
the birefringence of the
Q path is
not accurately set to achieve a 90 degrees relationship
between signals. In this case, we hypothesize that the ideal
Q signal can then
be generally
written as
SQ(k)
=
Re[a(k)Mj(k) + /(k)MQ(k)]
(6.2)
for all conditions except that in which M1 (k) and MQ(k) are equal (i.e., degenerate setting
of the I and
Q paths).
Here, a(k) and O(k) are complex wavenumber-dependent correction
vectors that are related to the birefringent state of the demodulation circuit. They include
both amplitude and phase errors. Importantly, the ideal signal, Sq(k), does not dependent
on the electrical frequency of the signal, only the wavenumber, k. Thus, the parameters
oz(k) and /(k) are independent of the RF signal (i.e., the fringe frequency or signal depth).
Next we consider a demodulation circuit that is perfectly optimized with respect to
wavenumber, i.e., a(k) = 0 and /(k) = 1 and SQ(k) = MQ(k).
If in this circuit, the
transmission length (e.g., fiber length within the demodulation circuit or electronic cable
length to the digitizer) of the
Q channel
is longer than that of the I channel, it induces
a constant propagation delay of T on the measured signal MQ(k) relative to M1 (k). The
effect of this delay is a phase shift in the
Q channel
relative to the I, but this phase shift
is proportional to the fringe frequency. The framework of Eq. 6.2 cannot account for this
error since it does not include a frequency-dependent correction. Instead, we can write
the ideal
Q signal
as the convolution of a delta function at delay T,
SQ(t ) = I6(T
-
T)[MQ (t - T)]dT
115
( 6 (--
T)),
(6.3)
-_
For a more general delay function (i.e. a varying electronic frequency responses from a
low-pass filter), we can use a correction kernel h(t) that describes the
Q
signal impulse
response relative to the I signal. In this case we can write the ideal quadrature signal
SQ(t) as
SQ(t)
h()[MQ(t - T)]dT
(6.4)
We note that in the same way that the error described by h(t) cannot be described by
a(k) and /(k), the error described by a(k) and /(k) cannot be described by the parameter
h(t). Thus, both the spectral and RF errors must be corrected with Eq. 6.2 and Eq. 6.4
respectively to remove errors in quadrature demodulators.
Assuming both spectral and RF-frequency errors are present, we write a combined correction framework as
SQ(kt) =
/(k)
FT-1I[FT{MQ(k, t)}I
H(A)
a(k) M1 (k, t)
/(k)
(6.5)
where we have moved the RF correction (Eq. 6.4) to the Fourier domain via the Fourier
convolution theorem. Here H(A) is the Fourier transform of the convolution kernel h(t).
We now show that correction of the measured quadrature signals via Eq. 6.5 sufficiently
removes demodulation circuit error.
6.4
Calibrating the optical demodulation circuit
In this work, we implemented an empirical approach to solving for the correction parameters (a(k), /(k),
and H(ZA)) from measured mirror signal data. We use numerical
optimization routines to solve for these parameters by maximizing extinction (maximum
difference in peak power between positive and negative depth space). We first increase the
SNR of our measured signals by coherently averaging a set of fringes, as we will describe
in the next section. Then we perform one of two calibration procedures. The first solves
for all parameters (a(k), /(k),
and H(A)) and requires a series of mirror signals at depths
spanning the positive and negative delay space. The second procedure assumes that H(ZA)
116
M, + iMa
M, , Ma
raw
frne
Hanning
window
FT
Sum of residual
error
Update coefficientsA - T
coherent
average
Spectral error
correction
rsidual peak
finder
i T
RF errOr
correction
a(k), P(k)
spectral
Figure 6-2: Calibration algorithm flowchart outlining the steps in computing
A
(a(k) and 0(k)) and RF (H(A)) errors from an input of measured 1 and MQ.
minimizing
MATLAB minimization function is used to update coefficients A - T based on
can
the residual error. If parameters M through T from H(A) are known, a(k) and 0(k)
be easily updated with inputs from a single depth [381.
is known and updates the correction vectors a(k) arid 0(k) using a mirror signal at a single
depth location. The general calibration process is summarized in Fig. 6-2.
6.4.1
Coherent fringe averaging
the
If using a single A-line, or if using multiple A-lines that are incoherently averaged,
greater than
optimization can achieve extinction ratios up to the signal SNR; extinctions
the
this will drive the power on one side of the zero delay below the noise floor. Since
extinction ratio target was 60-65 dB, it was necessary to optimize correction parameters
for
with signals of SNR, >65 dB. To address the challenge of obtaining such high SNR
the
calibration, we coherently combined multiple sequentially acquired A-lines to improve
SNR. Because the averaging is performed coherently, the signal amplitude is preserved
has
but the noise floor is reduced. In swept-source OCT implementations, each A-line
jitter by
phase jitter that mnust be compensated before averaging. We removed this phase
finding the phase at the signal peak for each A-line, deriving a phase compensating array
that is linear across depth and equal to the measured phase at the depth of the signal
array
peak, and multiplying each A-line by this (complex) linear-in-depth compensating
or the MQ signal can be used
(see Ref. [491 for a more detailed description). Either the M,
to both
to generate this phase ramp, but it is necessary to apply the same phase ramp
117
120-
(A) NON-COHERENT AVERAGING
1008060-
2
4020-
0
Depth (mm)
-
20
C
-
60
-4
-3
-2
-1
0
Depth (mm)
1
2
3
Figure 6-3: (A) Incoherent averaging of 600 A-lines. (B) Coherent averaging of the same
A-lines. A 28 dB improvement in SNR was achieved. The signal SNR in (B) was 75 dB,
sufficient to optimize parameters to extinctions of 60-65 dB 1381.
the I and Q channels (in order to preserve their relative phase). The benefit of coherent
averaging is illustrated in Fig. 6-3. In this example, the M, signal was coherently averaged
across 600 A-lines and there was an approximately 28 dB improvement (Fig. 6-3B) over a
non-coherent average across the same 600 A-lines (Fig. 6-3A). Note that because only the
I channel signal is plotted, peaks appear on both positive and negative depth spaces. The
side-peaks outside of the point-spread function (PSF) region do not affect our optimization
because the optimization procedure uses only the peak region to compute the correction
parameters.
6.4.2
Correcting only spectral errors
To solve for the spectral correction parameters (a(k) and 0(k)), we used a single coherently averaged mirror signal. The parameters are are modeled as complex and varying
quadratically with k,
a(k) = Ak 2 + Bk +C +i[Dk 2 + Ek + F]
118
(6.6)
140
,
11 An
(A)
OPD = -1
Optimized with:
mm
120.
120-
100-
100-
80.
80-
4-
-
CD 60
6040-
40.
20-
20-3
-2
-1
0
1
2
-3
-4
3
-2
I
(C) OPD = -2.5 mm
2
4
3
Optimized with: -2.5 mm
100-
1008060-
-
80-
.2
60-
40-
40-
20-
20-
-4
1
120-
120-
--
6
(D) OPD = -2.5 mm
Optimized with: -1 mm
-3
-2
-1-Depth (mm)
2
-4
3
-
14n
-1
Depth (mm)
Depth (mm)
-2
--- 0
1
------2
-
0-4
a
Optimized with: -2.5 mm
(B) OPD = -1 mm
-1 mm
Depth (mm)
Figure 6-4: Quadrature signals at two depths (-1 mm and -2.5 mm) after coherent averag(A)
ing. Black = prior to spectral error calibration. Red = after spectral error calibration.
-1 mm PSF self-calibrated. (B) -1 mn PSF corrected with -2.5 mm error calibration. (C)
-2.5 mm PSF corrected with -1 mm error calibration. (D) -2.5 mm PSF self-calibrated [38].
119
3(k) = Gk2 + Hk + I + i[Jk2 + Kk + L]
(6.7)
where the coefficients A through L are real-valued and k is a normalized k-vector of the
fringe that spans a range from -1 to 1 (from the first point of the fringe to the last
point). To test performance, we acquired 600 A-lines from each I and Q channel at -1
mm and -2.5 mm depths. After windowing we coherently averaged to yield a pair of
higher SNR Mr and MQ fringes. The two channels were added in quadrature (MI+iMQ)
and a simple peak finding algorithm was used to locate the the conjugate residual peak.
For each depth we separately solved for coefficients A through L by setting the residual
peaks as the function to be minimized through a Nelder-Mead optimization. We applied
each of the two solutions to each other and show the resulting PSF prior to error removal
(black) and after error removal (red) in Fig. 6-4. The correction parameters achieved
excellent extinction (>55 dB) at the depth at which they were derived (Fig. 6-4A,D).
However, when these solutions were applied to the other depth, poor extinction was
achieved (Fig. 6-4B,C). Note that the PSF at -2.5mm is broader because this quadrature
correction and optimization are done prior to fringe dechirping and dispersion correction.
This demonstrates that errors can be corrected to yield accurate quadrature demodulation,
but that correction based solely on the a(k) and /3(k) parameters within Eq. 6.5 is limited
to a single depth location. This can be explained by the additional presence of RF errors
(described by the parameter H(A) in Eq. 6.5), as we confirm in the next section.
6.4.3
Correcting both spectral and RF errors
To correct for both RF and spectral errors, we include the H(A) parameter modeled as
H(A)
= MA 3
+ NA 2 + OA + P + i[QA3 + RA2 + SA + T]
(6.8)
where again A is a normalized depth vector ranging from -1 to +1 (from the first point of
the depth spectrum to the last point). The degree of the polynomial was somewhat arbitrarily chosen, but in general should be high enough to properly model the error function
(i.e. more non-linearities require higher degrees). Because H(A) is a delay-dependent
120
120
(A)
100
80-
40
20
0
-1.5
-1.0
-0.5
0
Depth (mm)
-1.0
-0.5
-0.5
0
Depth
(mm)
Depth (mm)
0.5
1.0
1.5
1.0
1.0
1.5
1.5
120
10 (B)
0)
801
60-
C
0540
20--
0J
-1.5
-1.5
0.5
Figure 6-5: PSFs without (black) and with (red) quadrature correction. Multiple depths
are concatenated on the same plot for convenience, although the PSFs were recorded
and
separately. Dechirping and dispersion correction was applied to limit PSF overlap
imp~rove clarity. (A) Only spectral error, ca(k), 0(k), correction. (B) Only RF error,
H(A), correction. (C) Both spectral and RF error correction [381.
error, it was necessary to run the optimization on PSFs from multiple depths simultaneously. We acquired 600 A-lines from each of 14 mirror signals spaced in equal increments
between 0.25mm
-
1.25mmn (data taken sequentially). Measurements from both positive
and negative depth spaces were necessary to adequately constrain our algorithm. Like
before, we windowed and coherently averaged the A-lines at each depth then added them
in quadrature. This time, we used the summed power in the residual peaks at each depth
as the function to be minimized through a Nelder-Mead optimization across coefficients
A through T. This provided a single solution for a(k), 03(k), and H(A). Greater than 60
dB extinction is achieved at all depths after chirp and dispersion correction (performed as
a separate calibration), as presented in Fig. 6-5C. The 7 signals from the negative depth
121
OPD
=
OPD = +80um
+40um
100
-
100 -
80
60-
60
-
-
so-
D
40D-
t40h (h
20 -20-"
-300
-200
-20
300
-
-100
OPD+200um
(
(PD
0
100
200
300
200
30
Depth (um)
Depth (ur)
= +240um
10oo-
SO -
80
-
100 -
-
Q40-
4020-
20-
-300
-3-
-2 00
6
-100
100
200
300
-300
-3-
-200
-1.00
0
100
Depth (um)
Depth (um)
Figure 6-6: The algorithm successfully removes residual errors at various depths at 18
MHz Aline rates. Black = prior to spectral error calibration. Red = after spectral error
calibration. (A) +40pim (B) +80pm (C) +120pm (D) +240pim [38].
space are shown Fig. 6-5, however the solution was confirmed to be valid in both positive
and negative depth spaces.
To confirm again that it is necessary to correct with both spectral and RF parameters, we
attempted to optimize on multiple depth measurements using only spectral corrections
ce(k) and /3(k) (Fig. 6-5A), or RF correction H(A) (Fig. 6-5B). As expected, when only
spectral corrections are applied, good extinction is achieved at one depth and as the signal
moves to different depths, the RF error changes, and the correction is not longer accurate.
When only RF correction is applied, poor extinction (<40 dB) is achieved at all depths
because wavenumber-dependent errors are not being addressed for any depth.
To confirm that this algorithm is successful in high-speed systems, we performed a
similar experiment using the high-speed laser at 18 MHz Aline rates. We measured six
depths equally spaced between 40pm and 240pm with an SNR of ~45 dB after coherent
122
averaging. The first and last two depths are displayed in Fig. 6-6. The signals spanned a 0
- 350 MHz RF bandwidth, a significantly larger range than our slow-speed system (0 - 15
MHz). As predicted, even small mismatches in the transmission length caused significant
RF errors at these high speeds. The extinction between the primary and residual peak
prior to error removal (black) visibly varies with depth as the relative phase between I
and
Q changes
due to these mismatches. For this reason, an additional polynomial degree
was added to H(A) to accommodate the highly varying RF error. With our algorithm
we are able to removal the residual peak down to the noise floor with the same set of
correction parameters (red).
6.5
Stability analysis
We evaluated the stability of the correction parameters obtained from the previously described calibration procedures. We acquired fringe data at 7 depth locations over 3 hours
and then daily for two weeks. The system was not modified during this time. In Fig. 6-7,
datasets from the first time point was used to calculate a complete set of correction parameters (oz(k),
#(k),
and H(A) and these parameters were applied to each subsequent
time point. This figure shows that the achieved extinction at each depth dropped measurably within 20 minutes and continued to degrade to 50 dB in 2 hours. Over 13 days,
the extinction fluctuated between 50 dB and 60 dB.
To investigate the nature of the instability, we used the same longitudinal data but updated the spectral correction parameters at each time point (while re-using the original
RF correction parameter from the first time point). A single coherently averaged A-line
(we used 1.75 mm) was used to update the spectral correction as per Section 6.4.2. We
note that because this required a single depth mirror signal, it was performed quickly by
opening a shutter and acquiring a single frame before an imaging session. This updated
spectral correction resulted in stable extinctions >60 dB at all time points (Fig. 6-8).
This confirmed our prior assertion that RF errors are stable and that spectral errors need
to be updated before each imaging session.
123
on
* 1.00mm
* 1.2s mm
S1.50 mm
1.75mm
0
-
02.00mm
,
a5
S2.25
V
,I
mm
0 2.somm
C
Ii
0
eoi I
i
0
20
40
60
80
100
120
140
160
i
180
1
I
2
7
8
9
10
13
I
I
lime (mins)
Time (days)
Figure 6-7: Plot of the extinction for seven depths ranging from 1 mm to 2.5 mm over a
period of 13 days. The depths from Time 0 mins were used to compute a(k), O(k), and
H(A) and these were used to calibrate all subsequent timepoints [381.
0
5
C
a
C
8m
e1.0
n
* 1.25 mm
* 2.00mm
0
2.25 mm
o 2.SOmm
0
20
40
60
80
100
120
140
160
180
1
2
7
8
9
10
13
I
lime (days)
lime (mins)
Figure 6-8: Plot of the extinction for seven depths ranging from 1 mm to 2.5 mm over a
period of 13 days. The depths from Time 0 mins were used to initially compute a(k), 0(k),
and H(A) and a(k), 3(k) was renewed with depth 1.75 mm from each time point [38].
124
6.6
Imaging
-- 2.5
- Omm
-- 1.5
- 2.5
-- 2.5
-- 1.5
-0mm
-- 1.5
-
2.5
-- 2.5
-- 1.5
-O mm
-- 1.5
- 2.5
Figure 6-9: Image of an IR card on a tilt so that it spans both positive and negative
depth spaces; the vertical axis is depth. (A) No optical demodulation used. (B) Optical
demodulation with a polarization-demodulation circuit alone. (C) Optical demodulation
with a polarization-demodulation circuit and our error removal algorithm [381.
Once the calibration is performed, a(k), 0(k) and H(A) values are saved and applied to
images the post-processing stage in accordance with Eq.6.5. An image of a tilted IR card
was taken with a standard OCT microscope (Fig. 6-9). With no optical demodulation and
only real-valued fringes, the conjugate ambiguity results in overlapping images (Fig. 69A). With complex fringe data obtained using the polarization demodulation circuit but
without error correction, the overlap artifact is reduced but still present (Fig. 6-9B). With
125
the demodulation circuit and our error correction algorithm, the image is free of overlap
(Fig. 6-9C).
6.7
Discussion
Two broad comments are relevant for the practical use of quadrature demodulation methods and error correction in OCT imaging. First, as we described in Section 6.4.3, it is
critical that when computing both spectral and RF error parameters, fringe signals from
both sides of the zero delay are used. We observed that optimization on multiple depth
signals that are all located on one side of the zero delay location could not achieve high extinction for signals located on the other side of the zero delay. For only spectral correction
parameters, a single depth location signal on either the positive or negative space is sufficient. Second, the form of Eq. 6.5 implies an apparent computational cost to performing
spectral and frequency correction (i.e., applying the correction once the parameters are
known). However, we note that in systems that utilize computational chirp correction,
these FFT steps are already performed as part of the interpolation procedure [57]; our
quadrature error correction can be embedded within this procedure. In this scenario, the
additional computational burden associated with applying this correction is small relative
to that burden associated with dechirping and dispersion correction. For systems utilizing hardware resampling (k-clocking) the additional computational burden of corrected
quadrature demodulation is more significant.
In this work, we demonstrated that it is possible to computationally eliminate complex
conjugate ambiguity from imperfect quadrature signals. The presented method solves for
correction parameters based on either one mirror signal (for spectral parameters alone), or
a set of mirror positions (for both RF and spectral parameters together). The computation time for each of these procedures is a few minutes and once the correction parameters
are found they can be applied to images in the post-processing stage. We investigated
the stability of the correction parameters and showed that RF and spectral errors are
126
Av.
" , _-_.
., . __
..- , .
,
'h-
x6mia
-111--
stable over the duration of an imaging experiment (minutes), and that RF errors are
stable across time periods exceeding 13 days. The correction algorithm developed in this
work enables high-extinction complex demodulation that is compatible with balanced
detection, polarization-diverse detection, high-speed imaging, and broadband imaging.
127
128
k
, __
. " -
.
I I
-
1-
. -
---
___-
_-
1.
__".
- ., --
QdQQ1iAa".,_
6".".
Chapter 7
3D Camera Imaging
In this chapter, we describe the process of integrating the high-speed dispersion-based laser
from Chapter 5, and the polarization-based interferometer from Chapter 6 to achieve a
complete high-speed subsampled system. We present the first three-dimensional cameralike imaging results with this subsampled OCT system.
7.1
Hardware system integration
A schematic of the hardware of the complete system is shown in Figure 7-1. Because of
the drift in laser harmonic over time, we used a 1% tap coupler at the output of the laser
to monitor and compensate for the drift over time. The relatively low output power of the
laser warranted using a 50/50 coupler at the input of the interferometer so that we have
sufficient reference arm power amplifying our interference signal. The sample arm was
split with a 99%/1% coupler into a calibration mirror arm and the scanning microscope
arm. The calibration mirror was used between imaging experiments to re-calibrate the
k-dependent errors in the interferometer. We used 1.6 GHz balanced receivers (ThorLabs
PBD48OC-AC) to detect the interference fringe signals. With a FP with FSR = 200 GHz
the maximum expected signal for the subsampled system was idet(w)= 500 MHz and two
550 MHz low-pass filters (Mini-Circuits BLP-550+) were used after the receivers. A 1.8
GS/s two-channel AlazarTech acquisition card was used to digitize the fringe signals. A
129
- -
-6,
11
1 1
1
Calibration
Scanning
mirror
microscope
1%
99%
External
High-speed
1%
s
50
Subsampledrzi sDemodulation M
Optical h
LPF1 D
Circuit
th
A
99%
Laser
Clock
PC
Laser
Monitor
50
Q
PC
Reference
arm
2
LPF
Trigger
EMS
Trigger
-dDelay
-
Figure 7-1: High-speed integrated system featuring the dispersion-based laser and the
polarization-based interferometer.
function generator (Stanford Research Systems SG384) was used to provide an external
clock to the acquisition card, such that an integer number of samples are used per A-line;
this clock speed was computed from the electronics driving the intensity modulator of the
dispersion-based laser so it was perfectly synchronized with the laser frequency.
7.1.1
Acquisition configuration
The high laser speed requires transferring large datasets at rates that exceeded the transfer
rate of our acquisition card. The AlazarTech 9360 board provides a simple data streamning operation (DSO) that provides high speed streaming across the PCIe x8 bus at 3.5
GBytes/s. Because this interface bypasses the Windows buffer and we could achieve an
experimental transfer rate of 3.45 GBytes/s by streaming to the RAM. Recording with
the DSO required breaking down the data into "records" that occur once per trigger, and
on-board "buffers" that accumulate "records" until the buffer is full and a transfer to
the PC is made. The on-board buffer size was
82 MBytes, while the maximum record
size was 8.2 kBytes. Figure 7-2 outlines our acquisition approach that satisfies recording
4 million samples from two separate channels within a 48 GByte RAM constraint. For
imaging that requires more samples, additional RAM can be installed in the future.
130
Buffer
1
2
3
4000
22
2 2
2 2
2 2
0 o
1
4 44
8 81
2
0 0
0 o
4
8
00
CD
4 4g
8
4
88
22
2
0 0
1000 4
8
2
0 0
4 4g
8
a 8
0 0
4
8 :68
4
4 4
8
00
8
4
8
I~
2 2
0 0
4
8 8
] CHI
H
CH2
2 2
0 0
0 0
4 4
8 8
4 4
8 6
32.8 GBytes / Acquisition
Figure 7-2: The AlazarTech DSO streaming approach records with 1000 records/ buffer,
4000 buffers acquisition.
Although the DSO provides a simple way to stream data to memory or the disk at
high speeds, its visual interface capabilities only provided real-time display of the interferogram signal. Prior to each imaging session we required visualization of PSFs so
that we can optimize reference arm power to achieve shot-limited SNR, and to adjust
our sample within the Rayleigh range of the focusing optics. Furthermore, we required
visualization of our complex conjugate terms so that we could adjust the PCI and PCQ
polarization controllers in the quadrature interferometer (refer to Figure 6-1) to achieve
the best possible quadrature relationship prior to our post-processing calibration.
We
built a LabVIEW visual interface to visualize our B-scans in real time prior to recording
with the DSO. LabVIEW utilizes the Windows buffer, which had a measured transfer rate
of 0.4 GBytes/s, hence for the real-time visualization we use a delay generator (Stanford
Research Systems DG645) to reduce our trigger frequency.
With our 3.5 GBytes/s transfer constraint, we could riot acquire with both I and
131
Q
channels of our interferometer simultaneously. Since our slow axis galvo has a maximum
speed of 65 Hz, we used a delay generator to artificially slow the A-line triggers such that
we acquire exactly 2000 A-lines in a B-scan. Each recorded Aline trigger recorded 2048
point, corresponding to 22 consecutive Alines, which we use to coherently average in order
to get a better SNR, as we discuss later in section 7-4. The electro-optic laser operates
at an A-line rate that is a harmonic of the cavity length, hence we use an external clock
to digitize at 94 S/Aline. Although we imaged with slow triggers in this first demonstration of our camera-like OCT system, we still operated our laser at its maximum A-line
rate of ~19 MHz and acquired interferograms at these high speeds, therefore it is a good
demonstration of the advantage of subsampled imaging at high speeds.
7.1.2
Microscope
The microscope we used for imaging with our high speeds system was similar in construction to the one in our slow-speed system shown in Figure 4-11. For these first imaging
experiments, we elected to use slow galvos (ThorLabs GVS012) with a maximum full
scan bandwidth of 65 Hz and 10 mm beam diameter. Because we were aiming for longrange imaging, we designed our optics so that we may optimize the trade-off between the
Rayleigh range, the transverse resolution, and the field-of-view. We used a lens with a
focal length of 250 mm (ThorLabs AC254-250) and a collimator with diameter D = 6.7
mm (ThorLabs F810APC-1310) giving an axial resolution of 6x = 70 Pm and a Rayleigh
range of -2 cm. Our sample was placed 17.8 cm from the scanning galvo mirror; we could
adjusted our scan angle 0 according to the FOV we wanted to achieve (see Figure 4-11).
Because we used a collimator centered at 1310 nm instead of 1550 nm, we experienced a
total beam divergence of 0.06 degrees, which allowed us only 330 resolvable points. Hence
we oversampled our B-scans and C-scans by acquiring over 1000 Alines in each acquisition. In future imaging experiments, we will use a 1550 centered collimator (ThorLabs
F810APC-1550) with diameter D = 7 mm, which has a total beam divergence of 0.009
degrees, providing 1100 resolvable Alines per B/C-scan.
132
7.2
Performance characterization
We characterize the performance of our system by measuring the sensitivity of our total
system and we compare the sensitivity of the system operating in both subsampled mode
and continuous mode to assess losses that are inherent to subsampling. The 2x2 PBC of
our polarization-demodulation interferometer has the capability to split light evenly between an x-polarized channel and a y-polarized channel. For these imaging experiments
we measured only x-polarized light and adjusted both sample and reference polarization
controllers to maximize light to that channel. The total power loss in our optical demodulation circuit was measured to be -2.42 dB. The measured attenuation in the sample
calibration arm (due to the 1/99 coupler) was -20.07 dB single pass, hence a double pass
loss of -40.14 dB.
We measured the sensitivity of our system by measuring point spread functions (PSFs)
with the calibration mirror in both the continuously-swept laser configuration and the
subsampled laser configuration. Note that for the continuous system, a 850 MHz LPF
was used instead of a 550 MHz LPF, otherwise the systems were identical for these measurements. Because the average output power of the continuously running system was
32.7 mW compared to 11.61 mW when the laser was running in the subsampled configuration, the reference arm power was attenuated to achieve 223 pm in each input of the
balanced receiver. The subsampled system was attenuated (less attenuation was needed)
to a similar power into the receiver. Figure 7-3 shows plotted fringes and PSFs at two
different delays for the continuous system and the subsampled system. The blue traces
represent the signal prior to chirp removal, and red is after chirp removal. As we saw in
section 2.3.1, chirp broadens the PSF and hence the axial resolution, and the SNR gain
from its removal is more pronounced at larger delays. After chirp removal the sensitivity,
(calculated by adding the SNR with the sample arm attenuation) for continuously-swept
laser is ~85 dB for depth 1, and ~81 dB for depth 2. The loss in sensitivity at the two
different depths could be explained by imperfect chirp removal, or by the drift in the laser
harmonic during the depth 2 measurement causing a reduction in laser output power and
133
Id
((f))
- - -- -
-
-- - - - - -
--
8000
-
110
100-
-
-- -
--.-.-
-
-
- ---
---
90
-
2000 -
/-
-. -.-.--.---
-
4000
0
0
a.
- -
-4000
--
70
-.--.-.-.--.-.
-
-600
400
-
-
60
- -
-
000--
so
0
1
2
3
4
401
5
x 10-1
time (a)
4300
'
4000-
Interference Fringe, Continuous Depth 2
-200
0
200
Electric Frequency (MH
FFT Fringe ,(I(tf))
400
600
800
110
6000
100
4000
90
2000
.- -
80
- - -- -
-
-
0
a.
-2000
70
80
4000
80
4000
4O00
1
2
3
4
800
5
x1'
time (9)
-40
-400
Interference Fringe, Subsampled Depth 1
600
-20
0
200
400
Electric Frequency (MHZ
FFT Fringe . (1(f))
800
1 10.
6000
100
4000
....
....
.....
...
90
2000
0
V
I.
.4000
4-
70
.-
-.-.-.-.-.-
.
..-.
.--..
.-.
-..
.
460 0 0 -
-- -
-
-
-
1
0
- -
4
3
2
--
-
- -- -
-
80
60
4004001
5
-400
x 10
time (a)
Interference Fringe, Subsampted Depth 2
-20
0
200
400
Electric Frequency (MHz
FFT Fringe I (TfO)
600
800
110
8000
6000
100
--------
--
90
.. . .... . . .. ..
.
-- - -
--
-
K
-
2000
80
-2000 --
a.
...
- ---
70
..
. ..
.
4000-
......
.....................
80
.4000-
46000 0
--
-
-
1
80
-
-
.-
FFT Fringe
Interference Fringe, Continuous Depth 1
3
2
time (S)
6
4
x
104
-800
40
40
-20
0
20
400
Electric Frequency (MWZ
600
600
80
ew
Figure 7-3: PSFs at two different OPDs for both the continuous system and the subsampled system. Blue = prior to chirp removal. Red = after chirp removal.
134
reduced sensitivity. The continuously-swept laser was more prone to harmonic frequency
drift than the subsampled laser. For the subsampled laser, the sensitivity was -78.54 dB
for depth 1 and ~77.36 dB for depth 2. The sensitivity drop in the subsampled system
was more than the expected -4.3 dB drop in laser power in the continuously-swept system.
This again suggests that there was loss in the subsampled system due to higher order harmonic generation in the interferogram signal. There was no significant sensitivity drop
between the two depths of the subsampled signal (after chirp removal), confirming that
the discrete sweeping nature of this laser resulted in equally distributed harmonic loss
across all depths in the baseband window. Notice also that in depth 2 of the subsampled
PSF, we begin to see a second alias move into the baseband window. This is a result
of the electric sampling rate being higher than the optical sampling rate; optical aliases
from adjacent baseband windows are apparent in the RF spectrum.
7.2.1
Coherent averaging
Although the sensitivity of this initial system is low, we can gain as much as 10 dB in
sensitivity by coherently averaging A-lines.
The phase stability of the laser makes it
possible to add A-lines in phase to improve SNR by reducing white noise. We determined
that 20 A-lines results in a 5-7 dB reduction in the noise floor and in order to achieve the
maximum 10 dB reduction in noise floor, we would need to induce a 50 A-line average.
Figure 7-4 shows a cross section of an IR card comparing no coherent averaging and a
20 A-line average. In the coherently averaged frame, there is a visible drop in the noise
floor, corresponding to the predicted value of -7 dB. Because of the 18.9 MHz rate of the
laser, the 20 Alines were recorded in 1.06 ps, which is a small fraction of a displacement
considering the galvo's fast axis scan was 16.5925 Hz. If coherent averaging is used in
future fast scans, averaging can be performed over repeated frames instead of adjacent
A-lines. Ideally, we can increase the seiisitivity of our hardware system to avoid the need
for coherent averaging in future systems.
135
no coherent average
-375pm
dB
110
0
105
100
95
+375pm
-90
coherent average
85
80
75
70
0
85
80
-
+375pm
Figure 7-4: An IR card image with no coherent averaging (top) and with a 20 A-line
average (bottom). Colorbar: signal power in dB.
7.3
Image processing
The image processing of the high-speed imaging featured some challenges that were not
apparent in the slow-speed imaging system. Our first challenge was to confirm that we can
apply the quadrature error removal method to subsampled images. Although we would
not expect the quadrature relationship between I and Q channels to change in different
subsampled windows, it was not clear whether performing a calibration with the baseband
window would apply to all aliases.
7.3.1
Complex conjugate demodulation in subsampled imaging
In Chapter 6 we showed that we can use a quadrature interferometer with error removal
in the post-processing to achieve high-extinction complex conjugate demodulation. We
also showed in Section 4.4 that complete conjugate demodulation is required for circular
wrapping in subsampled images.
We now confirm that our demodulation with error
removal was a valid method for subsampled imaging. Mainly, we want to verify that we
can derive a single error solution that applies to all aliases resulting from subsampling.
Figure 7-5 shows mirror signals from different locations in the depth space. As before,
136
A)
OPD a -250 um
B) OPD= 150 um
100-
80
80
-
100
*1
rC, so-
60-
40-
40-
20
20
-0.3
1
-,
-0.2
.
C) OPD = -1.675
-0.1
0
0.1
Depth (mm)
0.2
120
mm
D) OPD
-0.1
-0.2
-0.3
0.3
=.
0.1
0
Depth (mm)
0.2
0.3
-1.5
-1.4
Depth (mm)
-1.3
-1.2
1.375 mm
100-
100-
80-
80-
CO
as.
60-
60-
40-
404PI
-1.8
-1.7
-1.6
-1.5
-1.4
Depth (mm)
-1.3
-1.8
-1.2
-1.7
-1.6
19fl
F) OPD = 3.626 mm
E) OPD = 3.1 mm
100-
100
80
80a1
WS
~360-
80
40
40-
I
'
3.0
3.1
3.2
3.4
3.3
Depth (mm)
3.5
3.6
, I
3.0
3.7
'
3.1
I
1
3.2
i
I I
3.3
3.4
Depth (mm)
'
1
3.5
3.6
3.7
Figure 7-5: Complex conjugate error removal at six depths spanning the extended depth
window. Black = prior to error removal. Red = after error removal. The error correction
parameters were derived from A and B and applied to C-F.
137
black signifies the complex signal prior to error removal, and red signifies after error
removal. For this experiment, the errors correction parameters were derived from the first
two datasets (A & B), which are within the true baseband window (depth 0 is the true
path matched and the baseband window spans from -375 um to +375 pm). And the
parameters are applied to signals from -1.575 mm and -1.375 mm (these signals undergo
five aliases from the negative depth region to reach the baseband window) and to signals
from +3.1 mm and +3.525 mm (these undergo 10 aliases from the positive depth region to
reach the baseband window). As we can see, the error solution that was derived using PSFs
in the true baseband can be applied successfully to remove quadrature errors in optical
aliases. Note that because we are operating at high speeds, our electric bandwidth for
the baseband window ranged from
440 MHz as opposed to the
3.5 MHz from the slow
speed OCT system in Section 6.2.2. Recall that mismatches in the interferometer cause
RF errors that are a function of RF frequency. Hence over a significantly larger electric
bandwidth, we increased the degree of the polynomial to 4 in order to accommodate larger
fluctuations in the RF error:
H(A)
=
MA 4 + NA 3 + OA 2 + PA + Q + i[RA 4 + SA 3 + TA 2 + UA + V]
(7.1)
Here again A is a normalized depth vector ranging from -1 to +1 and the error correction
algorithm solves for the coefficients M - V.
Figure 7-6C shows a cross-section of a tilted IR card imaged with the high-speed subsampled laser. Here the surface of the IR card (yellow arrow) circularly wraps in the
baseband window, resembling the result we had in Figure 4-8, which used a frequency
shifter to achieve circular wrapping with a mirror. For comparison in Figure 7-6A there is
no demodulation therefore the image is mirrored across the zero path delay and overlapping is evident. In Figure 7-6B optical demodulation was used without the error removal
algorithm and a residual artifact remains.
138
+375 um
dB
110
Gum
-105
-375 um
100
+375 um
95
90
-375 um
85
Figure 7-6: A) No demodulation. B) Optical demodulation only. C) Optical demodulation
with error removal.
7.3.2
Dispersion removal
the
Applying chirp to quadrature signals is different than non-quadrature signals because
Si + iS9 . Dispersion compensation
dispersion vector operates on the complex fringe, S
is carried out by applying a complex vector that subtracts the effect of dispersion from
be
the dataset. Typically the 02 value (higher order dispersion terms are ignored) can
found through several methods. Empirically by measuring two PSFs at different depths,
subtracting the chirp, and fitting the phase of the interferogram to a linear fit (this
method was used in the polygon-based subsampled imaging). In this work we found /2
7-7
by minimizing the FWHM of a PSF from the negative half of the delay space. Figure
shows the intensity signal over 1000 B-scans from a stationary mirror placed at +200
jim, taken with the 1% calibration arm described above. Note that we purposely induced
the
poor complex conjugate removal with our algorithm so that we can visualize both
real and conjugate peaks. First we applied our complex dispersion to both delay spaces
by
(Figure 7-7A) and saw that the width of the conjugate peak was actually broadened
the dispersion vector. The reverse case was true when the conjugate of the dispersion was
the
applied in Figure 7-7B. We adapted our processing to apply the dispersion vector to
half
positive half of the depth space, and the conjugate dispersion vector to the negative
of the depth space in Figure 7-7C, with good compensation to the entire range.
139
A) dispersion
-375pm
0
dB
120
+375pm
0
100
200
300
400
500
600
700
800
900
1000
B) conj (dispersion)
-375pm
115
110
105
100
0
95
90
+375pm
0
100
200
300
400
500
600
700
800
900
1000
80
dispersion, conj(dispersion)
-375pm
85
75
0
+375pm
0
100
200
300
400
500
600
700
800
900
1000
Frame
Figure 7-7: An intensity image of a stationary sample mirror over 1000 A-lines, spanning
a depth space of -375pm to +375pm. The removal of dispersion is shown by multiplying
the interferogram in each A-line by A) A dispersion vector, B) the conjugate of dispersion vector, and C) top half multiplied by the conjugate of dispersion and bottom half
multiplied by the dispersion vector. Colorbar: Signal power in dB.
140
Ul)
0
CL
C
Figure 7-8: One en face cross-section of fingers in A) the continuously- swept OCT configuration and (C) the subsampled configuration. The maximum intensity image of (C)
the continuously-swept stack, and (D) the subsampled stack. FOV = 3.1 cm x 3.1 cm.
7.4
Imaging
We described in section 5.2 that we used a FP etalon with 200 GHz FSR and finesse of
100. This gives us ain expected baseband window of t375 [ti
in air and t 270 pmti
in
tissue assuming an index of n =- 1.388. For all of these imaging experiments we measured
a spectral bandwidth AA = 70 nmi, giving us ail axial resolution of ~15 /,mi. In the first
set of subsamnpled imaging experiments we wanted to compare the conventional continuous version of our system (recall that our dispersion-based laser call operate in both
continuous and subsamnpled regimies). We imaged fingers placed on an optical table with
a galvo total scan angle of 0 = 10' optical angle in both transverse directions (yielding a
FOV of 3.1 cmn x 3.1 cmn), a fast axis framne rate of 16.5925 Hz, and a 33.3 kHz effective
A-line rate (laser trigger delay produced by the digital delay generator).
Figure 7-8A shows one en
face
slice of the continuous system.
141
Notice the absence of
Figure 7-9: (A) Cross-sectional image of finger taken with the high-speed subsampled
system; yellow arrows point to the surface of the finger. (B) The same image without
complex conjugate ambiguity removal shows significant overlap of the tissue image.
optical aliases that are present in the same image taken with the subsampled system
(Figure 7-8C). As in the case of the polygon-based subsampled images, the separation
between the optical aliases in this subsampled en face projection represents the size of
the optical baseband window (540 pm in this case). Figure 7-8B shows the maximum intensity projection of the entire stack (consisting of 2000 B-scans) to give an appreciation
for the sensitivity-roll off of the system. In the conventional image, the imaging range
was approximately 3 mm, which was consistent with the low coherence length measured
in section 5.4.3 for this laser. The white represents region of high intensity within the
coherence length, whereas black is little or no signal. Although the sensitivity rolls of
quickly in the finger tissue, the optical table is still slightly visible because of its high
back-reflection. From Figure 7-8B alone we cannot visualize the overall structure of the
sample we are imaging and we might not even know that we are imaging fingers. Figure 7-8D shows the maximum intensity projection of the subsampled image where the
entire finger and the underlying breadboard fall within the sensitivity roll-off region. This
camera-like image allows us to appreciate the fine details of the structure like fingerprints
that were not visible in the continoulsy-swept system.
With the subsampled system we can also look at. cross-sectional B-scans to visualize beneath the surface of the tissue. Figure 7-9A shows a cross-section of the fingers with the
142
(A:
(C)
(B)
Figure 7-10: (A) Photograph of the sample field of a hand resting on an optical table. (B)
An en face cross-section of the sample field showing sub-surface features. (C) An average
intensity image of the en face stack showing and extended depth-range capability and a
total FOV = 6.6 cm x 6.6 cm.
complex conjugate ambiguity removed using k-dependent error update method. Although
the sensitivity of the system was quite low (~86 (lB with coherent averaging), we can still
make out the highly scattering stratum corneum layer (highlighted with blue arrow in the
topmost alias). Notice that the curvature of the finger results in multiple aliases stacked
oii top of one another (yellow arrow) and the steeper the grade of the curvature, the closer
the aliases are to each other. For comparison, a cross-section without complex conjugate
ambiguity removal is displayed in Figure 7-9B. It is clear that the conjugate image cre-
ates mirror artifacts that conceals the stratum corneum layer that was apparent in the
ambiguity-free image.
As another demonstration of our increased depth-of-field, we imaged a hand resting
143
on top of an optical breadboard. A photograph of the sample field is shown in Figure 710A, one en face cross-section is shown in Figure 7-10B and the average intensity image
is shown in Figure 7-10C. This image was acquired with a 94 Samples/A-line with 2000
A-lines in a B-scan, and 2000 B-scans in a C-scan (the image utilized a 20 adjacent A-line
average). With a scan angle of 6 = 30', the FOV was 6.6 cm x 6.6 cm. As described in section 5.4.3, the double-pass coherence length of the subsampled laser was 14 mm, resulting
in a total coherence length of 28 mm after ambiguity removal. The extended depth range
imaging enabled us to visualize the entire field of the hand in three-dimensions. This
demonstrations shows that if we are not constrained by the limitation of coherence length
and acquisition electronics, we can image large fields and extract important sub-surface
information about tissues with ease.
The wide-field capabilities of our system is demonstrated in Figure 7-11. We imaged
a face with a galvo total scan angle of 0 = 360 optical angle (FOV of 8 cm x 8 cm) and
again a fast-axis frame rate of 16.5925 Hz with a 33.3 kHz effective A-line rate. Figure 7-11A is a single en face cross-section with apparent subsampled aliases; notice that
the aliases serve as a topologic map of depth with each continuous high-surface reflection
serving as iso-depth lines. Interestingly, this gives a visual intuition of the contours on
the face. Interestingly, it would be possible to extract quantitative values of the gross and
fine topology by measuring inter-alias distances and intra-alias depths respectively. The
average intensity image highlights the the surface of the face, with a detailed recreation
of features such as wrinkles and pores. This image also demonstrates the full coherence
range of the imaging system as we can see the sensitivity roll-off both distally (at the
outer edges of the face) and proximally (around the nose). Recall that the Rayleigh range
was slightly less than our measured coherence length we had some degradation of our
transverse resolution in our total image range.
144
--
--
Figure 7-11: Left: One en face cross-section of a wide-field subsampled image of a face.
Subsampled aliases are evident and help visualize the contours of the face. Right: The
average intensity image over the entire depth stack shows high reflections from the surface
of the skin, giving a clear visualization of surface topology. FOV = 8 cm x 8 cm.
7.5
Future Work
Individually, the demonstration of a high-speed system, a long-range system, and a datacompressed system can have a large impact on OCT. The combination of these technological advances has made three-dimensional camera-like imaging with OCT possible for
the first time. The research presented here is a platform for many future developments
and applications. The first steps beyond this would be focused on optimizing the subsystems for better sensitivity. The laser performance would benefit from higher output
power, possibly by reducing the loss in the cavity, or by using linear Raman/EDFA amplification. We can also improve the performance of the interferometer by reducing the
losses in the demodulation circuit (i.e. splicing fibers). Although we have the ability to
image at high-speeds, in this initial demonstration of the system we have been limited to
slow-scanning speeds because of the 65 Hz galvo scanner. In future imaging experiments,
we can replace these with high-speed resonant scanners that will enable us to perform
145
-
~-.--
~1
video-rate imaging. With an 18.9 MHz Aline rate, we can acquire 10 frames of size 2000
x 1000 Alines per second. These image rates have never been possible in OCT before and
will allow us to image fast moving samples (i.e., motion of the tempanic membrane of the
ear). In future long-range subsampled implementations, microscope design will be key to
acquiring images with good transverse resolution throughout the extended depth range.
Beyond improving the performance of the system, we could introduce functional imaging
with subsampling. With polarization-sensitive capabilities in our subsampled system, for
instance, we can envision a wide-field camera providing en face polarization maps that can
be used for various applications (i.e., surgical screening of arthritic lesions in cartilage).
Because of the high-speed imaging capabilities, our system would also be amenable to
imaging high-velocity blood vessels in angiographic implementations. This could be used
to image vasculature in a single cardiac cycle, or to observe blood flow velocity in high
flow vessels. With subsampled OCT enabling three-dimensional camera imaging, there is
significant potential for impact in a wide range of clinical applications.
146
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