SYSTEM AND METHOD FOR LOW COHERENCE BROADBAND QUADRATURE INTERFEROMETRY
GOVERNMENT RIGHTS
This invention was made with United States Government support under Federal
Grant No. R24 EB000243 awarded by the National Institutes of Health. The United
States Government may have certain rights in this invention.
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of U.S. Provisional Patent Application Serial
No. 60/474,222, filed on May 30, 2003. The contents of this provisional application is
incorporated herein by reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention.
The invention relates to broadband interferometry and, more particularly, to an
interferometry system and method in which all components of a complex interferometric
signal can be simultaneously acquired.
2. Background of the Related Art.
Interferometry with broadband light sources has become a widely used technique
for imaging in biologic samples using time-domain optical coherence tomography (OCT),
optical coherence microscopy (OCM), spectral domain OCT (which encompasses
spectrometer based Fourier domain OCT and swept source OCT), color Doppler OCT,
and phase-referenced interferometry. In all of these interferometry techniques, light
traveling a reference path is mixed with light returning from or traversing a sample on the
surface of a single or multiple detectors.
In homodyne interferometry, the optical frequency of the sample and reference
light is the same, and mixing of the fields on the detector results in sum and difference
frequency terms corresponding to a second harmonic frequency component and a DC
frequency component. The second harmonic frequency component is at twice the optical
frequency, and is therefore not resolved by conventional square-law electronic detectors.
In heterodyne interferometry, either the reference or sample arm light is
purposefully modulated at a carrier frequency, which results in the difference frequency
component residing on a carrier frequency which is electronically detectable. The
complete interferometric signal consists of DC components arising from non-mixing
light from each of the arms, and interferometric components arising from mixed light. In
heterodyne interferometry it is straightforward to separate the DC from interferometric
components, since the latter are distinguished by their carrier frequency. In homodyne
interferometry, it is impossible to separate the interferometric and non-interferometric
components based on their frequency content alone.
In both homodyne and heterodyne interferometers, the interferometric
component of the detector signal depends sinusoidally on both the optical path length
difference between the arms of the interferometer, and also on an additional phase term
which specifies the phase delay between the reference and sample arm fields when the
path length difference is zero. When this phase term is zero, the interferometric signal
varies as a cosine of the optical path length difference between the arms, and when the
phase term is 90 degrees, the interferometric signal varies as a sine of the path length
difference. Although a single detector can only detect one of these phase components at
a time, it is convenient to refer to the zero and 90 degree phase delayed versions of the
interferometric signal as the real and imaginary components (or zero and 90 degree
quadrature components) of a complex interferometric signal.
In OCT and many of its variations discussed above, it is often useful or necessary
to have access to the entire complex interferometric signal in order to extract amplitude
and phase information encoding, scatterer locations and/or motions. For example, in
Doppler OCT, multiple phase measurements are required to extract the magnitude and
direction of sample motions. In spectral domain OCT, acquiring only one quadrature
component of the interferometric signal results in a complex phase ambiguity which does
not allow for separation of image information resulting from positive and negative spatial
frequencies of the detected data. This ambiguity results in double images, image
contamination with undesirable autocorrelation terms, and is wasteful of detector pixels
(in Fourier domain OCT) and data collection time (in swept source OCT). Unfortunately,
the square-law detector output which is available in previously disclosed OCT systems
and their variations obtains only the real part of the complex signal (in the case of single
receiver systems), or the real part and its inverse (in the case of differential receiver
systems).
Several methods have been reported which allow for instantaneous or sequential
retrieval of both quadrature components of the complex interferometric signal. These
include 1) polarization quadrature encoding, where orthogonal polarization states encode
the real and imaginary components; 2) phase stepping, where the reference reflector is
serially displaced, thus time encoding the real and imaginary components; and 3)
synchronous detection, where the photodetector output is mixed with an electronic local
oscillator at the heterodyne frequency. Synchronous detection methods include lock-in
detection and phase-locked loops. Polarization quadrature encoding and phase stepping
can be genetically called quadrature interferometry since the complex signal is optically
generated. As such, they are useful in both homodyne and heterodyne systems.
Each of these techniques suffers from shortcomings. Polarization quadrature
encoding is instantaneous, but it requires a complicated setup, and suffers from
polarization fading. Phase shifting requires a stable and carefully calibrated reference arm
step, is not instantaneous, and is sensitive to interferometer drift between phase-shifted
acquisitions. Synchronous detection is not instantaneous, and depends on the presence of
an electronic carrier frequency. Systems based on synchronous detection are thus not
useful in an important class of homodyne systems, such as en-face imaging schemes,
those which take advantage of array detection (e.g. Fourier domain OCT), and in swept
source OCT.
There is thus a clear need for a system and method for instantaneous and
simultaneous acquisition of both quadrature components of the complex interferometric
signal in OCT and related systems.
SUMMARY OF THE INVENTION
An object of the invention is to solve at least the above problems and/or
disadvantages and to provide at least the advantages described hereinafter.
Another object of the invention is to provide a quadrature broadband
interferometry system and method that can obtain the complete complex interferometric
signal instantaneously in both homodyne and heterodyne systems in a simple, compact,
and inexpensive setup.
To achieve these and other objects and advantages, the present invention provides
an improved system and method for quadrature broadband interferometry which exploit
the inherent phase shifts of NxN (N>2) fiber optic couplers to obtain the complete
complex interferometric signal instantaneously in both homodyne and heterodyne
systems.
In accordance with one embodiment, the present invention provides a signal
processing method which includes separating interferometric components from non-
interferometric components in each of at least two detector signals of an interferometer
having a number of NxN couplers, scaling the interferometric components, and
generating real and imaginary parts of a complex interferometric signal from the scaled
interferometric components. The detector signals preferably derive from a broadband
light source coupled to the interferometer. Also, the number of NxN couplers may be
one or more, with N ≥ 3.
Separating the interferometric components may be performed by high-pass or
band-pass filtering signals output from the detectors. This separation may also be
accomplished by extracting AC-coupled signals from signals output from the detectors.
Such an extraction may be performed by removing at least one of DC components,
source noise components, and auto-correlation terms from the detector outputs prior to
re-scaling.
Scaling the interferometric components may be performed by phase shifting the
inteϊferometric components of the two detector signals to have a predetermined phase
difference. The interferometric components are preferably shifted to have orthogonal
phases, although other phase relationships are possible. Scaling may also be performed to
reduce optical losses which correspond, for example, to at least one of optical path losses
and variations in gain between different ones of the detectors.
Generating the imaginary part of the complex interferometric signal from the real
part may be performed based on a method derived from the cosine sum rule. Once the
real and imaginary parts are obtained, a Doppler shift value may be calculated based on a
time derivative of the calculated phase.
In accordance with another embodiment, the present invention provides an
analyzer which includesan interferometer including at least one fiber-optic coupler, a
number of detectors that detect light from said at least one coupler, and a processor
which acquires interferometric components from signals output from the detectors,
scales the interferometric components, and generates real and imaginary parts of a
complex interferometric signal from the scaled interferometric components.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be described in detail with reference to the following drawings
in which like reference numerals refer to like elements wherein:
Fig. IA is a schematic diagram of a differential low-coherence Michelson
interferometer based on 2x2 fiber couplers in accordance with one embodiment of the
present invention; Fig. IB is a schematic diagram of a Michelson interferometer based on
a 3x3 fiber coupler in accordance with one embodiment of the present invention; and
Fig. 1C is a graphical representation of the intrinsic phase shifts in a 3x3 coupler-based
Michelson interferometer in accordance with one embodiment of the present invention;
Fig. 2 is a schematic diagram of a 3x3 coupler-based interferometer with all
potential detector locations populated in accordance with one embodiment of the present
invention;
Fig. 3 is a schematic diagram of a 3x3 coupler-based interferometer that provides
simultaneous detector outputs with 120 degree phase separation and a DC component
that includes source fluctuations in accordance with one embodiment of the present
invention;
Fig. 4 is a plot of the interferometric detector outputs of the 3x3 coupler-based
Michelson interferometer embodiment of Fig. IB during a heterodyne experiment, in
which amplitudes have been normalized, in accordance with one embodiment of the
present invention;
Fig. 5 A is a three-dimensional Lissajous plot of the interferometric signal from
three photodetectors in the 3x3 coupler-based Michelson interferometer of Fig. IB; Fig.
5B is a Lissajous plot of interferometric signal k vs. interferometric signal h for the 3x3
coupler-based Michelson interferometer of Fig. IB; Fig. 5C is a plot of real vs. imaginary-
signals calculated from interferometric signals h and h for the 3x3 coupler-based
Michelson interferometer of Fig. IB; and Fig. 5D is a plot of interferometric signal vs.
time for the 3x3 coupler-based Michelson interferometer of Fig. IB;
Fig. 6 is a plot of the magnitude, phase and Doppler shift of the complex
heterodyne interferometric signal calculated from outputs of the 3x3 coupled-based
Michelson interferometer of Fig. IB;
Figs. 7A-7D are graphs showing plots of the magnitudes and phases of the
complex interferometric signal for a homodyne experiment using the embodiment of Fig.
IB;
Fig. 8 is a diagram of an NxN coupler-based interferometer generalized to an
arbitrarily high order NxN coupler in accordance with one embodiment of the present
invention;
Fig. 9 is a diagram of an NxN coupler-based homodyne low coherence
interferometer in accordance with one embodiment of the present invention;
Fig. 10 is a schematic diagram of a Fourier-domain OCT system employing a 3x3
fiber coupler in accordance with one embodiment of the present invention;
Fig. 11 is a schematic diagram of an OCT system that employs the spectral-
domain technique of swept source OCT for depth ranging in accordance with one
embodiment of the present invention; and
Fig. 12 is a schematic diagram of a homodyne DLS system that employs a 3x3 (or
NxN, N>2) fiber coupler in accordance with one embodiment of the present invention.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Fused-fiber couplers rely on evanescent wave coupling to split an input electric
field between output fiber paths, according to coupled-mode theory. (It is to be
understood that an optical signal is one type of electric field that may be subject to the
present invention. In a broader sense, the embodiment of the present invention disclosed
herein may be applied to any type of electromagnetic radiation). A formalism based on
conservation of energy which predicts phase shifts for interferometers based on 2x2 and
3x3 couplers will now be described. While higher-order couplers (e.g. 4x4) can be used in
a manner similar to that described below for 3x3 couplers to acquire the complex
interferometric signal, the phase shifts in these couplers only can be explained by
coupled-mode theory.
Figs. IA and IB respectively show examples of 2x2 and 3x3 coupler-based
Michelson interferometers in accordance with the present invention. In these and
subsequent figures, coupling coefficients may be denoted by dab, which describes the
power transfer from one fiber or optical path α to another fiber or optical path b. For
example, the 2x2 coupler may have a 50/50 split ratio where cfti = an — Vz, and the 3x3
coupler may have a 33/33/33 ratio where an = an — an — 1/3. Alternative
embodiments of the invention may use couplers with disproportionate split ratios if
desired.
The Fig. IA interferometer includes a source 1, two fiber-optic couplers 2 and 3,
and reference and sample arms 4 and 5. The light source is preferably a broadband light
source (e.g., one having multiple wavelengths or modes), however a narrowband source
may be used if desired. The source is shown as 2γlo for reasons that will become more
apparent below. In this embodiment, the couplers are both 2x2 couplers. The first
coupler splits light from the source along two optical paths, one of which is input into
the second coupler. The second coupler splits this beam along fibers Fl and F2 which
respectively lead to the reference and sample arms. At least the reference arm terminates
with a reflector 6. The sample serves as the reflection source of the other arm.
Light beams reflected from the reference and sample arms are re-combined in
coupler 3. A portion of this re-combined light is then detected by detector D2, and
another portion of this light is input into coupler 2, where it is detected by detector Dl.
As will be described in greater detail below, the detector signals may be processed to
derive a complex interferometric signal in accordance with the present invention.
The Fig. IB interferometer includes a source 10, two fiber-optic couplers 12 and
13, and reference and sample arms 14 and 15. The light source is preferably a broadband
light source (e.g., one having multiple wavelengths or modes), however a narrowband
source may be used if desired. The source is shown as 2γlo for reasons that will become
more apparent below. In this embodiment, the first coupler is a 2x2 coupler and the
second coupler is an NxN coupler, where N ≥ 3. For illustrative purposes, N = 3 in the
figure.
The first coupler splits light from the source along two optical paths, one of which
is input into the second coupler. The second coupler splits this beam along fibers Fl and
F2 which respectively lead to the reference and sample arms. Light beams reflected from
these arms are re-combined in coupler 13. Portions of this re-combined light is then
detected by detectors D2 and D3, and another portion of the re-combined light is input
into coupler 12, where it is detected by detector Dl. As in the previous embodiment, the
reference arm may terminate with a reflector 16 and the sample may reflect the light in
the sample arm.
The detector signals may differ in respect to their phase, e.g., the phase of the
interference signal on detectors D2 and D3 may be different. For example, if the 3x3
coupler has a splitting ratio of 1/3:1/3:1/3, then the interference signals on D2 and D3
will be out of phase by 120 degrees. The amplitude of the interference signals may also be
different, if the splitting ratio is something other than 1/3:1/3:1/3. The amplitudes will
also be different if detectors D2 and D3 have different electronic gains.
In Figs. IA and IB, the first coupler may be used to allow for detection of that
portion of the interferometric signal which would otherwise return to the source. In
other embodiments of the invention, the first coupler may be replaced by an optical
circulator (if all returning components of the interferometric signal are required) or
eliminated entirely (if all but one of the returning components are sufficient for signal
processing). Ignoring the return loss from the source arm 50/50 coupler, the optical
intensity incident on the nΛ detector due to a single reflection in the sample is:
In = yI0(analn + ana2n +2E(Axyanalnaua2n cos(2kAx + φ,,)), (1)
where k is the optical wavenumber, Δx is the path length difference between reflectors in
the reference and sample arms, E(ΔΛΓ) is the interferometric envelope (i.e. the magnitude
of the complex signal), and φa is the phase shift between the detector photocurrents. This
phase shift is imparted by the evanescent wave coupling process. Also, γ ensures mat the
total power incident on the reference and sample arms is Io, e.g. γ is 1 for a 2x2 and
Since the non-interferometric portions of I
n sum to Io, the interferometric
portions must sum to zero, independent of Ax. Assuming perfect reciprocity (i.e.
<Xb=(%a) , losslessness (y i), and directionality (no internal coupling from input
---j ^ aann
back into input fibers) in die fiber coupler, this implies diat:
(2b)
Because variations in &dx rotate the
vector in the complex
plane (Eq. 2a), Eq. 2a is true if and only if
sum to zero (Eq. 2b). These
equations demand that |
for 2x2 couplers, regardless of the coupler splitting
ratio, while φm-φa is an explicit function of Oώ for 3x3 couplers.
Fig. 1C shows one way in which Eq. 2b may be graphically represented as a sum
of three complex vectors with magnitude (flfinCfen)1/2 and phase φa. Given a set of
measured values of Ωkb, the interferometric phase shifts between coupler arms φm-φn are
uniquely determined, since the three vectors correspond to the sides of a triangle whose
inner angles are related to φm-φa. For example, if
for all fibers a and b, then the
interferometric phase shift between any two output ports of a 3x3 Michelson
interferometer is 120°.
For 4x4 and higher order couplers, Eqs. (l)-(2b) are still valid, but are not
sufficient to uniquely predict the interferometric phase shifts between coupler arms φm-φn
as a function of the coupling ratios. However, 4x4 and higher order couplers may be
extremely useful for instantaneous quadrature interferometry, especially by specifically
designing them to give phase angles separated by 90°, in that they give all in-phase,
quadrature, and out-of-phase components (0o,90°,180o,270°) of the complex
interferometric signal without requiring any further signal processing.
Extracting Complex Interferometric Data from NxN Interferometers
Because the phase shifts among Jn in Eq. (1) are not constrained to be 0° or 180°
for interferometers based on 3x3 and higher-order couplers, the complex interferometric
signal can be obtained instantaneously from simultaneous measurements of any two or
more detector outputs. It should be appreciated that this can be accomplished with all
higher-order NxN interferometers (N>2) of various topologies (e.g. Michelson, Mach-
Zehnder, etc.). The detectors are preferably connected to a processor (not shown) which
performs computations for deriving the complex interferometric signal.
A method for extracting the real and imaginary components of the complex
interferometric signal from die interferometer detector outputs, corresponding to
detectors located on all NxN input fibers, and on any unused output detectors, will now
be described with reference to Fig. 2.
Fig. 2 shows a 3x3 interferometer in accordance wkh another embodiment of the
present invention. This interferometer may be constructed in a manner similar to Fig. IB,
except that a detector D4 is attached to optical fiber F3, an X-Y scanning mirror 30 is
used to scan and capture reflected light from a sample (which in this case is illustratively
shown as' insect), and a superluminescent diode (SLD) serves as the broadband light
source. An optional lens 31 may be included to focus the light into and out of the
scanning mirror. The Fig. 2 embodiment also includes a processor 35 which processes
signals from at least two detectors to compute a complex interferometric signal in
accordance with the present invention. (While only Fig. 2 shows this processor, it is
understood that other embodiments described herein also include a processor for
generating a complex interferometric signal from detector signals or pairs of detector
signals).
The first step in extraction of the complex interferometric data from NxN
interferometers is to separate out the interferometric components of at least two detector
signals from the non-interferometric or DC components. DC components, as used
herein, refer both to any non-interferometric light appearing on detector outputs due to
amplitude or polarization mismatch between sample and reference arms, as well as to any
autocorrelation terms resulting from multiple partial reflections occurring in either
sample or reference arms.
Removing DC components is critical to obtaining the complex interferometric
signal, and is also useful in optimizing signal-to-noise ratio in OCT systems, since these
DC components also contain light source fluctuations (excess noise) which are common
mode to light returning from the sample and reference arms. Cancellation of common-
mode source noise and removal of autocorrelation terms may be accomplished by some
of the techniques that will now be described.
In heterodyne systems, selection of interferometric components is done by AC
coupling or high-pass filtering all input fiber detector outputs (e.g., detectors Di through
D3 in Fig. 2) with a cutoff frequency below the heterodyne carrier frequency. An
alternatively approach would involve band-pass filtering around the heterodyne carrier
frequency. In homodyne systems, where AC coupling is not an option, AC coupling may
be simulated by separately measuring the DC level appearing on each input fiber detector
output (detectors Di through D3 in Fig. 2) due to the reference arm DC level and sample
arm DC level measured separately, then subsequently subtracting these DC levels from
each respective detector output during interferometric measurements. In the usual case in
OCT, where the quantity of light reflected from the sample is small compared to the
reference arm recoupling, this procedure may be approximated by subtracting only the
reference arm recoupled DC level from each of the detector outputs.
Simulating AC coupling by pre-measuring DC recoupling into each of the input
fiber detectors is sufficient for reconstruction of the complex interferometric signal, but
does not remove the common-mode source noise or autocorrelation terms. Two
techniques will now be described which accomplish all three tasks simultaneously.
If detector outputs are available from all input fibers of the NxN interferometer
(detectors Di through D3 in Fig. 2), then the DC level to be subtracted from each
detector, including source fluctuations and autocorrelation terms, may be obtained by
summing all detector outputs, scaled by any loss experienced on the path back to each
detector. For the embodiment of Fig. 2, the ideal total re-coupled power is given by
where the factor of 2 multiplying Di arises from the insertion loss of
the 2x2 coupler in the source arm. Given this measurement of DT, each detector output
may be effectively AC coupled by subtraction of the proper fraction of DT from that
output, i.e. Di
AC=Di-Dτ/6,
These equations should be corrected in view of die exact coupling ratios and any additional losses present
upstream of each detector in any system implementation.
A second technique for removal of DC components, including source fluctuations
(but not autocorrelation terms), from the detector outputs involves the use of any unused
output fiber of the NxN interferometer (e.g., detector D4 in Fig. 2). In the embodiment
of Fig. 2, the AC-coupled input fiber detector outputs are given approximately by
D
2-γ
2*D
2, and D
3 AC D
3-γ
3*D
3. Here, the quantities γ
n are
proportionality constants, which depend upon the specific recoupling ratios of light from
sample and reference arms back into the NxN coupler, as well as specific losses in each
detector arm, and would likely have to be separately characterized for each instrument.
Following the removal of DC components, source noise fluctuations, and
autocorrelation terms from each of the detector outputs in an NxN coupler-based
interferometer system, each detector output is preferably re-scaled to compensate for
specific losses in die optical path to each detector, as well as for variations in the gain
between different detectors, before proceeding to complex interferogram extraction. For
example, in the interferometer of Fig. 2, the output of detector Di must be multiplied by
a factor of 2 to compensate for the presence of the 2x2 coupler between detector Di and
die 3x3 coupler. The exact values for re-scaling of each detector output will depend on
the specific pattern and/or characteristics of the optical losses and electronic gains
associated with each detector's optical and electronic paths.
The complex interferometric signal may be exttacted from the AC-coupled and
re-scaled detector outputs. A technique for reconstructing the complex interferometric
signal uses the AC-coupled and re-scaled signal from any two detectors located on input
fibers of the NxN coupler (e.g. detectors D1-D3 in Fig. 2). Defining ia as the
interferometric portion of Jn, and assigning any in as the real part of the complex signal
(denoted zke below), then the imaginary part iιm =2Ε(Ax)(an(Xina\2(X2^)'ι^2sin(2kΔx+φa),
may be obtained by using the cosine sum rule, as:
The re-scaling of the m* detector output is expressed as an explicit function of
the coupling coefficients of the NxN coupler. The re-scaling parameter β may also be
obtained experimentally, as described above. Once zke and 2Ϊm are obtained according to
Eq. (3), then the magnitude and phase of the complex interferometric signal may be
obtained according to standard formulas from complex mathematics.
The technique summarized by Eq. (3) may be applied to any 2 detector outputs
(located on an input fiber to the NxN coupler) of an NxN interferometer. If multiple
pairs of outputs are available (e.g., detectors D1&D2, D2&D3, D1&D3 in Fig. 2), then an
improved estimate of the interferometric amplitude and phase may be applied by
averaging the amplitudes and phases obtained from the application of the technique of
Eq. (3) to each detector pak calculated separately.
Fig. 3 shows another particularly robust instantaneous quadrature interferometer
system in accordance with the present invention. This system takes advantage of the
technique described by Eq. (2) and includes a 3x3 coupler-based interferometer without a
2x2 coupler. In this embodiment, die broadband source is shown as λo, Δλ, where λo
corresponds to the source center wavelength and Δλ corresponds to the bandwidth. Also,
the optical signal reflected in the reference arm is generated by a 2-reference scan.
Through this arrangement, the quadrature output is simplified from Eq. (3) as l90°c(Ii20 -
Io cos(120°)), and A scans can then be processed based on this derived output and the
original 0 detector output.
In the case of a 4x4 coupler, quadrature and differential spectra are available
directly from the three unused input ports, akhough the overall power utilization
efficiency is slightiy less than in the 3x3 coupler case. In both 3x3 and 4x4 coupler cases,
all required signals are available from the unused input ports. Thus, a circulator or input
2x2 coupler is not required, as is included in the embodiments of Figs IA and IB. The
implementation of the system of Fig. 3 could be extremely robust, as no optics or
mechanics are required in the reference arm path at all. For example, the reference arm
path could simply be a length of fiber with a reflective end coating coiled in a box. Of
course, any mismatch in fiber lengths between the sample and reference arms will need to
be dispersion compensated using known techniques.
Instantaneous Quadrature Interferometry with NxN Interferometers
The embodiment of Fig IB has been tested using a superluminescent diode source
(OptoSpeed;
Δλ
:z:28nm) and an AC Photonics, Inc. 3x3 truly fused fiber
coupler. To demonstrate instantaneous quadrature signal acquisition in a heterodyne
experiment, the reference arm was scanned at ^=8.3 mm/sec. The interferometiϊc
portion of all three acquired photodetector outputs is illustrated in Fig 4. The phase shifts
between detector outputs were measured to be φι -
130.8° and
- φi — 255.4.
When experimentally measured values of c&b were used to solve for the theoretical
interferometric phase shifts (Eqs. 2a and 2b), these theoretical shifts deviated from the
experimental shifts by ≤ 2.5% (Fig 1C), φm-φn drifted over the course of minutes to
hours, and it is believed that these drifts are due to temperature sensitivity of the coupler
splitting ratio.
If Zi5 /2, and h are parametrically plotted against each other, a three-dimensional
Lissajous curve may be formed, as shown in the plot of Fig. 5A. Time is encoded by the
color of the curve: in this experiment the data was captured as Ax increased with time,
and the curve spirals outwards from the center and transitions from green to black to red.
(The green color corresponds to an inner portion including an inner circumferential
portion of the curve, the black color corresponds to an intermediate portion of the curve,
and the red color corresponds to an outer portion including an outer circumferential
portion of the curve). The curve fills an ellipse.
Fig. 5B is a Lissajous plot of interferometric signal h vs. interferometric signal Z2,
where the progression of color from green to black to red is similar to the pattern in Fig.
5A. Fig. 5D is a plot of interferometric signal vs. time. In this plot color progresses from
green to black to red with increasing time. The real (&) and imaginary parts (calculated
from Eq. 3) of the interferogram are plotted as a Lissajous curve in Fig. 5C with the same
color pattern as in Figs. 5A and B. The real and imaginary parts were used to calculate the
magnitude and phase as plotted in Fig 6. The time derivative of the phase yields the
Doppler shift fa. generated by the scanning reference mirror:
The experimentally-calculated Doppler shift (Fig 6) agrees well with the predicted
value of 13kHz, based on the calibrated scan velocity of the reference arm. This result
demonstrates the utility of NxN interferometers in color Doppler OCT.
A 3x3 instantaneous quadrature signal acquisition in a homodyne experiment has
also been demonstrated by positioning a non-scanning reference mirror (v=Q) such that
Δx corresponds to the middle of an interferogram. The outputs h and h (from detectors
D2 and D3) were then recorded for 10 seconds, and Eq. (3) was used to reconstruct
amplitude and phase. Due to interferometer drift on the order of ~λ, the phase of the
complex interferometric signal varied by ~2π, while the magnitude remained steady
(mean=1.52, variance=1.96xlθ"4. This is shown in Figs. 7A-7D, which are plots of the
magnitude and phase of the complex interferometric signal for the homodyne experiment
using the embodiment of Fig. IB. This result demonstrates the utility of NxN
interferometers in homodyne OCT setups such as optical coherence microscopy,
Fourier-domain OCT, and swept source OCT.
Although higher-order NxN (N>2) interferometers are less efficient with source
light than 2x2 interferometers, they are more efficient at collecting light reflected from
the sample, and may thus be advantageous in exposure-limited applications, such as
retinal imaging. For example, the 3x3 coupler-based interferometer of Fig. IB delivers
only 1/6 of the source light to the sample, but collects l/3+l/3+(l/3)(l/2)=83% of the
light returning from the sample. Replacing the first 2x2 coupler in Fig. IB with a
circulator would result in sample reflected light collection efficiency limited only by the
insertion loss of the circulator.
The simpler embodiment of Fig. 3 delivers 1/3 of the source light to the sample,
and collects 2/3 of the light returning from the sample. By modeling and analysis, it is
possible to optimize source utilization efficiency, sample return utilization efficiency, or
some combination by proper specification of NxN coupler splitting ratios.
As described above, NxN interferometers allow for instantaneous optical
extraction of magnitude and phase information simultaneously in a compact and simple
design. Their advantages are particularly compelling in FD-OCT and swept source OCT,
where they allow for optical resolution of the complex conjugate ambiguity without
phase stepping.
Higher Order Interferometric Topologies
The analysis described above, including the governing equations for NxN
interferometers (Eqs. 1-3) and the disclosed techniques for recovering the complex
interferometric signal from the raw detector outputs, are generalizable to NxN
interferometers based on arbitrarily high order NxN couplers.
Fig. 8 shows an exemplary embodiment of a generalized NxN interferometer
formed in accordance with the present invention. This interferometer includes a source
20 which is preferably a broadband source, an optical circulator 21, and an NxN coupler
22 having two outputs for inputting optical signals into reference and sample arms 23 and
24 along respective fibers Fl and F2. The sample arm terminates with a reflector 25.
Detectors Di - DN+I are connected to receive optical beams passing through the coupler.
In this embodiment, N ≥ 3.
While 3x3 coupler-based interferometers are attractive due to their simplicity and
efficiency with source light, and suitably designed 3x3 and 4x4 coupler-based
interferometers provide direct access to 90°-separated quadrature components of the
interferometric signal, higher-order NxN interferometers may be useful in situations
where more sophisticated phase extraction schemes than that described in Eq. (3) require
denser sampling of the complex plane, or where increased sample reflection utilization
efficiency is required in the absence of a circulator in the source arm.
Heterodyne Time-Domain Optical Coherence Tomography
An important potential application of NxN instantaneous quadrature
interferometry in heterodyne OCT systems is in Doppler OCT. In Doppler OCT, the
sign and magnitude of Doppler shifts arising from moving scatterers within a sample may
be recovered from time-frequency analysis of the interferometric signal. This analysis has
taken the form of windowed Fourier transform, wavelet transform, or Hubert transform
analysis of interferometric data, either digitized directly or electronically down-mixed
using synchronous detection. All of these approaches are non-instantaneous, and involve
considerable electronic and/or computational complexity.
Polarization encoding has been used to encode the real and imaginary parts of the
complex interferometric signal for Doppler processing. However, the polarization
encoding apparatus was complex and expensive and is subject to polarization fading. The
use of NxN interferometry for instantaneous recovery of the sign and magnitude of
Doppler shifts in Doppler OCT, as described above with reference to Fig. 6, greatly simplifies Doppler OCT.
Homodyne Time-Domain Optical Coherence Tomography
Use of NxN coupler-based low coherence interferometer systems of the present
invention allow for near instantaneous extraction of interference phase and amplitude.
One major advantage of such an approach is that it eliminates the need to generate a
heterodyne beat frequency through which the phase and amplitude information are
measured. The technical disadvantages of a heterodyne based system are varied, and
some major ones are now discussed.
The time window for extracting time varying amplitude and phase information is
limited by the period of the heterodyne beat. In other words, if the sample reflectance
where to change in phase shift or amplitude, the quickest the change can be resolved will
be in a time window that is at least larger than the heterodyne beat period.
However, NxN homodyne based systems can, in principle, render instantaneous
phase and amplitude measurements. In such a situation, the resolving time window and
the sensitivity of the measurements trade off with each other. An arbitrarily small time
window is achievable with a correspondingly arbitrarily low sensitivity. There is no longer
a hard limit on the resolving time window as with the heterodyne situation.
Heterodyne based phase detection is very prone to heterodyne beat frequency
fluctuations and finite data sampling rate. In an ideal world, where the heterodyne signal
can be sampled with infinite sampling frequency, any beat frequency fluctuations can be
detected and corrected for prior to amplitude and phase extraction. Realistically, this is
likely not possible and beat frequency fluctuations that are not corrected for will lead to
amplitude and phase errors.
Fig. 9 shows an NxN homodyne-based system in accordance with the present
invention which overcomes this problem in two ways. First, it does not require a moving
element, which is the most commonly used means for generating the heterodyne
frequency. As the non-uniformity of the velocity of the moving element is a major
contributor to the heterodyne beat frequency fluctuations, a homodyne-based system is
automatically free of this source of error. Second, even if a moving element is present in
the interferometer, such as for the purpose of making depth scanning, scan velocity
fluctuations can be detected and corrected for by setting the resolving time window at a
time scale that is shorter than the fluctuation time scale. (The Fig. 9 embodiment may
have a structure similar to the Fig. 8 embodiment except that the NxN coupler is
replaced by a 3x3 coupler 40, an X-Y scanning mirror 41 and lens 42 are included along
the sample signal path, a Z-scan is performed along the reference signal path, and four
detectors Di - D4 are included).
Heterodyne-based OCT systems generally acquire depth resolved scans (A-scans),
since the general incorporation of a moving element to induce the necessary heterodyne
beat generally generate a varying optical delay. This optical delay can be used to render
depth resolved scans. Heterodyne-based systems can disentangle depth scans from the
heterodyne beat generation process through the use of other optical elements, such as an
acousto-optic modulator or electro-optic modulator. Such distanglement allows for easier
en-face scan pattern (to create en-face images), by then scanning the focal point created
by the OCT horizontally within the sample.
However, such systems generally set a limit on the speed of en-face scanning.
Each pixel dwell time must be larger than the heterodyne frequency, which leads to slow
scan speed due to low heterodyne beat frequency, or high scan speed with a high
heterodyne beat frequency. The first situation is not ideal in terms of image acquisition
speed. The second requires fast data collection and processing and entails sophisticated
instrumentation .
However, an NxN homodyne-based system does not suffer from such limitations.
The point of interest can be as easily scanned in depth as it can be scanned laterally. A
shorter pixel acquisition time may be traded off by permitting lower sensitivity. The ease
of performing en-face scanning means that a homodyne-based en-face scanning optical
coherence microscope is possible that exhibits an image acquisition rate that is
unobtainable using a heterodyne-based system.
Fourier-Domain Optical Coherence Tomography With Array Detection
An alternate method of coherence gating that does not employ a scanning delay
line has been called various terms including spectral radar, Fourier domain OCT (FD-
OCT), and complex spectral OCT. In FD-OCT, the positions of reflectors in the sample
arm of a Michelson interferometer are resolved by acquiring the optical spectrum of
sample light interfered with light from a single, stationary (or phase modulated) reference
reflector. The power incident on the array of detectors is related to:
PD (* ) = (|E D (/c )|2 ^ S (k )R R + S (Jc )R s + 2 S (k y R R R S cos (2 /tΔx) (5)
where S(^) is the source spectral shape, ED(^) is the total field incident on the
spectrometer, Rs and RR are, respectively, the sample and reference reflectivities, and
PD(/&) has units of watts per wave number.
The double-pass position of the sample reflector is encoded in the frequency of
cosinusoidal variations of PD(^), while the sample reflectivity is encoded in the visibility
of these variations. The spectrally-indexed detector outputs, therefore, represent the real
part of the Fourier-transform of the A-scan. This is consistent with the assertion that
electronic photodetectors only give the real part of the interferometric signal.
Because Eq. (5) yields the real part of the Fourier transform, a reflector at +Ax
cannot be distinguished from a reflector at -Ax.
2 S(kyRRRs cos(2/cΔx) = 2 S(k)y]RRRs cos(- 2kΔx) . (6)
This ambiguity is called complex conjugate ambiguity, and can be resolved by additionally
collecting the imaginary part of the interferometric signal 2S(M)(RκRsy/2cos(2kAx),
noting that:
2 s(kyRRRs sin(2/cΔx) ≠ 2 S(k)jRRRs sin(- 2IcAx) (7)
Since the reference arm is not scanned in FD-OCT, the detector signals do not
contain an electronic carrier signal related to the phase of the interferometric signal. This
constitutes a homodyne detection system and, as such, electronic synchronous detection
methods cannot be used to extract the interferometric magnitude and phase.
Phase-shifting interferometry has been utilized to dither the phase of the reference
electric field, in order to collect real and imaginary parts on sequential scans. Phase
shifting, however, requires a stable and carefully calibrated reference arm step, is not
instantaneous, and is sensitive to interferometer drift between phase-shifted acquisitions.
Fig. 10 shows an FD-OCT system in accordance with the present invention that
uses the techniques described above to acquire a complex interferometric signal, and
which removes DC and autocorrelation terms for each spectrometer detector element.
This embodiment has a structure similar to Fig. 9, except that detectors Di through D4
are replaced by the spectrometer detector elements Speci through Spec4, respectively, and
the reference signal is not subject to a Z-scan. Also, in this embodiment, it must be
realized that since array detectors contain a finite number of elements that collect
photoelectrons in proportion to incident power and integration time, Eq. (5) must be
represented in a discrete form.
If an array contains M elements and interrogates an optical bandwidth of Ak, the
optical channel spacing is bk=Ak/M. The signal thus has values at M evenly spaced
wavenumbers K= {k\, ki,- . .,MM)- We can write Ne»[/έ»], the number of electrons collected
from the //* wavenumber on the «
ώ spectrometer as:
(8)
where A is an autocorrelation signal, S is source noise, At is the detector integration time,
η is the detector quantum efficiency, h is Planck's constant, and c is the vacuum speed of
light. It should be noted that since dispersive spectrometers operate in wavelength and
not wavenumber, Eq. (8) is experimentally obtained by numerically resampling the
spectrometer data.
As discussed above, the use of detectors (in this case, spectrometers) 1 through 3
allows for the removal of DC terms (i.e. those associated with RR. and Rs), autocorrelation
terms, and source noise, while the use of detectors 2-4 allows for the removal of DC
terms and source noise only. In either case, if ϋf>>R.R, autocorrelation terms are small
compared to the other terms in Eq. (8), so it is reasonable to assume that A~0 if
detectors 2-4 are used.
If is defined as the interferometric portion of Eq. (8), then any
can be defined as Ne
re[/£], the real part of the complex interferometric signal. Referring to
Eq. (3), and noting that n,me {1,2,3}, the imaginary part of the complex interferometric
signal is:
Λrim[/:]_ K:[k]cos(φm -<U-β NM \ auauana2n _ sinC^ -^J ' \ ccnaXmana2m '
The aim of FD-OCT is to obtain a depth-reflectivity profile of the sample arm.
This can be obtained with the discrete Fourier transform:
(10)
Optical Coherence Tomography With Optical Frequency Domain Reflectometry
(Swept-Source OCT) In optical frequency domain reflectometry, the A-scan Fourier transform is
acquired by sweeping a narrow-linewidth source over a broad spectral range. Fig. 11
shows an OCT embodiment that employs the frequency-domain technique of optical
frequency domain reflectometry for depth ranging, in accordance with one embodiment
of the present invention. Use of 3x3 (or more generally an NxN, N>2) coupler 70 allows
for acquisition of the entire complex inter ferometric signal. λ(t) is either a narrow-
linewidth source whose wavelength is tuned over time, or a broadband light source
filtered by a narrowband optical filter whose passband is tuned over time. Embodiments
which use only detectors D2 and D3, Di-D3, or D2-D4 are also possible, with reduced
immunity to source fluctuations or autocorrelation terms.
If source sweeping begins at t=0 and ends at At, then the detector output is
proportional to PD(^= Vt+kmhl) (Eq. 5). Here, kmin is the smallest wavenumber emitted by
the source, V has units of wavenumber per second, and V is such that
kndn— Ak. A-scans are obtained by Fourier transforming the time (and therefore
wavenumber) indexed detector output. Since the reference arm is not scanned in OFDR,
the detector signals do not contain an electronic carrier signal related to the phase of the
interferometric signal. This constitutes a homodyne detection system and, as such,
electronic synchronous detection methods cannot be used to extract the interferometric
magnitude and phase.
As with FD-OCT, a complex conjugate ambiguity is present if the entire complex
interferometric signal is not collected (see Eq. (6), k—Vt+km,!). The complex signal needs
to be extracted with an optical technique (e.g., polarization diversity, phase shifting). DC
and autocorrelation terms can be removed with the techniques described above. Unlike
FD-OCT, however, it also is possible to remove the DC components by AC coupling the
photodetectors. Defining i»(M) as the interferometric portion of L(M) (see Eq. (1),
k= Vt+kmn), and assigning any itt(β) as the real part of the complex signal (denoted i&dfi)
below), then the imaginary part iιm(^:=2&(J^x){an(XiQ(XnC(2^)'{/2sva(2kΔx+φ^)i may be
obtained by using the cosine sum rule, as (see Eq. (3) ):
• (j \ _ K {k)cos(φm -φn)-$im (k) I analnana ".2n
"2m
(11)
The complete complex interferometric signal is iκe+jhm, and the Fourier transform
of zke+jim yields the depth-resolved A-scan:
ASCAN(X) = l(iRe + jx ilm)exV(-jkx)dk
(12)
Homodyne Phase-Resolved Interferometry
A significant advantage of the NxN couplet-based interferometer systems of the
present invention are their ability to measure phase within an arbitrarily small time
window. However, phase measured through such means will simply be equal to the
optical path difference between the reference and sample interferometer arms. Any
vibrational jitters will still be encoded onto the phase as errors.
Fortunately, NxN coupler-based homodyne phase measurement can readily be
adapted for use with the numerous interferometer phase measurement techniques that
have been developed to circumvent the jitter induced error.
Phase noise considerations
Assuming negligible 1/f noise, one can arrive at an order of magnitude estimation
of the phase noise as obtained by the heterodyne approach. The shot noise of each
measurement is given approximately by sqrt(εloτ/e) (numerical scaling factor due to
coupling coefficients is presently ignored). The homodyne interference amplitude is equal
to 2sqrt(RIo 2)τ/e (numerical scaling factor due to coupling coefficients is presently
ignored). The maximum phase error that can be created is when the shot noise and
homodyne interference are orthogonal:
θerror = tan-i(sqϊt(εIoτ/e)/( 2sqϊt(R.Io2)τ/e)) » (l/2)sqrt(e/(εloτ)) ,
(13)
where, ε is the detector responsivity, I0 is the light power, τ is the acquisition time, e is
the electron charge, and R is the reflectivity of the sample.
When accounting for coupling coefficients, this error will be corrected slightly.
From the error analysis, we can see that higher intensity or longer acquisition time will
improve phase sensitivity.
Birefringence OCT
In birefringence OCT, two low coherence light components compose a pair of
orthogonal light polarization states. OCT signals are simultaneously acquired for both
polarization states and there relative phases can then be processed to determine the depth
resolved refractive index difference for the two polarization states within the target
sample. Common path interferometer jitters affects both phases equally and, thus, are
cancelled out during birefringence computations.
This OCT technique can be implemented in the homodyne mode by simply
inputting the two orthogonal polarization states into the OCT system and measuring the
homodyne signal at the two polarization states separately. One of the key advantages of
operating in homodyne mode is that it is very easy to acquire en-face images.
Wollaston prism based phase contrast OCT
In this form of phase contrast OCT, low coherence light of a pair of orthogonal
polarization states is physically displaced (but propagating in the same direction) prior to incidence on the sample. OCT signals are simultaneously acquired for both polarization
states and their relative phases can then be processed to determine the refractive index or
height variation between the two focal points.
This OCT technique can be implemented in the homodyne mode by simply
inputting the two orthogonal polarization states into the OCT system and measuring the
homodyne signal at the two polarization states separately. One of the key advantages of
operating in homodyne mode is that it will be very easy to acquire en-face images.
Phase dispersion OCT
For phase dispersion OCT, low coherence light of two different wavelengths is
used in the OCT system. OCT signals are simultaneously acquired for both wavelengths,
and their relative phases can then be processed to determine the wavelength dependent
refractive index through the depth of the target.
This OCT technique can be implemented in the NxN coupler-based homodyne
mode by simply inputting the two wavelengths into the OCT system and measuring the
homodyne phases for the two wavelengths.
One of the key advantages of operating in NxN coupler-based homodyne mode is
that it will be very easy to acquire en-face images. In addition, the 2 pi ambiguity problem
can be substantially mitigated by measuring phase variation with very high spatial
resolution. In such a situation, it is possible to depart from the condition that the two
wavelengths be harmonically matched, and yet still achieve superior 2 pi ambiguity free
measurements.
Phase referenced low coherence interferometry
For phase referenced low coherence interferometry, low coherence light of one
wavelength (or 1 polarization) and monochromatic light of a second wavelength (or the
orthogonal polarization) is used in the OCT system. An OCT signal and a
monochromatic interference signal are simultaneously acquired for both wavelengths,
and their relative phases can then be processed to determine optical path length changes
within the sample.
This OCT technique can be implemented in the NxN coupler-based homodyne
mode by simply inputting the two wavelengths into the OCT system and measuring the
homodyne phases for the two wavelengths.
One of the key advantages of operating in homodyne mode is that it will be very
easy to acquire en- face images. In addition, the 2 pi ambiguity problem can be
substantially mitigated by measuring phase variation with very high spatial resolution. In
such a situation, it is possible to depart from the condition that the two wavelengths be
harmonically matched, and yet still achieve superior 2 pi ambiguity free measurements.
This technique, which can be used to measure very small and slow motions in the
target (manifested as optical path length changes), has previously been limited in
sensitivity and time resolution by the first two problems described in the Homodyne Time-
Domain Optical Coherence Tomography section above. With the use of homodyne based
approaches, one no longer needs a heterodyne beat generator, and the monitoring of
optical path length changes can be made with an arbitrarily small time window.
Magnitude and Phase Resolved Dynamic Light Scattering
Dynamic light scattering (DLS) techniques characterize the time-variations in the
properties of light scattered from an illuminated sample. An important class of DLS
techniques involves the interferomettic mixing of the backscattered sample light with
unscattered sample light. The light source used in these techniques can be either CW laser
light or broadband light.
One advantage of using broadband light is that it allows for depth-localization
within an inhomogenous sample. In either case, the detector measures the real part of the
interferomettic signal. This is somewhat problematic because the magnitude and phase
variations in the interferomettic signal are due to different physical processes. For
example, the magnitude of the signal is influenced by the total number of particles within
the illuminated volume, while phase (speckle) variations are due to subwavelength
motions of particles within the illuminated volume. Unfortunately, it is impossible to tell
phase from magnitude variations from the real signal alone (see Fig. 7 and associated
discussion above).
Fig. 12 shows a homodyne DLS system that employs a 3x3 (or more generally an
NxN, N>2) fiber coupler 80 in accordance with one embodiment of the present
invention. This embodiment can be used with one or more sources 81 of any bandwidth
(e.g., CW where Δλ<<λ, broadband where Δλ~λ). In the case of a broadband light
source, the reference arm 82 is scanned to select a particular depth to interrogate in the
sample. If the sample arm light 83 is scanned, it is scanned in an en-face manner, as
described above. (Optional lenses 84 and 85 may be used to focus this light). This
embodiment is a homodyne detection system, and can extract the complex
interferometric signal with the techniques described above.
The foregoing embodiments and advantages are merely exemplary and are not to
be construed as limiting the present invention. The present teaching can be readily
applied to other types of apparatuses. The description of the present invention is
intended to be illustrative, and not to limit the scope of the claims.
For example, although fiber couplers are used as the NxN couplers in the
embodiments described above, it should be appreciated that bulk optical analog versions
of the NxN fiber couplers could be used while still falling within the scope of the present
invention. This would enable the extension of the techniques described above to two-
dimensional imaging geometries.
Many alternatives, modifications, and variations will be apparent to those skilled in
the art. In the claims, means-plus- function clauses are intended to cover the structures
described herein as performing the recited function and not only structural equivalents
but also equivalent structures.