We describe fiber-based quadrature low-coherence interferometers that exploit the inherent phase shifts of 3 x 3 and higher-order fiber-optic couplers. We present a framework based on conservation of energy to account for the interferometric shifts in 3 x 3 interferometers, and we demonstrate that the resulting interferometers provide the entire complex interferometric signal instantaneously in homodyne and heterodyne systems. In heterodyne detection we demonstrate the capability for extraction of the magnitude and sign of Doppler shifts from the complex data. In homodyne detection we show the detection of subwavelength sample motion. N x N (N> 2) low-coherence interferometer topologies will be useful in Doppler optical coherence tomography (OCT), optical coherence microscopy, Fourier-domain OCT, optical frequency domain reflectometry, and phase-referenced interferometry

Choma, Michael A.

Izatt, Joseph A.

Yang, Changhuei

English

Caltech Authors - Main

2162 OPTICS LETTERS / Vol. 28, No. 22 / November 15, 2003Instantaneous quadrature low-coherence interferometry with
3 3 3 fiber-optic couplers
Michael A. Choma, Changhuei Yang, and Joseph A. Izatt
Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708Received May 30, 2003
We describe fiber-based quadrature low-coherence interferometers that exploit the inherent phase shifts of
3 3 3 and higher-order fiber-optic couplers. We present a framework based on conservation of energy to
account for the interferometric shifts in 3 3 3 interferometers, and we demonstrate that the resulting inter-
ferometers provide the entire complex interferometric signal instantaneously in homodyne and heterodyne
systems. In heterodyne detection we demonstrate the capability for extraction of the magnitude and sign
of Doppler shifts from the complex data. In homodyne detection we show the detection of subwavelength
sample motion. N 3 N (N . 2) low-coherence interferometer topologies will be useful in Doppler optical
coherence tomography (OCT), optical coherence microscopy, Fourier-domain OCT, optical frequency domain
ref lectometry, and phase-referenced interferometry. © 2003 Optical Society of America
OCIS codes: 110.4500, 120.3180.Interferometry with broadband light sources has
become a widely used technique for imaging of biologic
samples by use of time-domain optical coherence
tomography1 (OCT), optical coherence microscopy,2
Fourier-domain OCT,3 optical frequency domain ref lec-
tometry,4 color Doppler OCT,5 and phase-referenced
interferometry.6 In many of these applications, it is
useful or necessary to have access to the entire com-
plex interferometric signal to extract amplitude and
phase information encoding scatterer locations and
(or) motions. Unfortunately, the square-law detector
output obtained in conventional OCT systems yields
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 systems7).
Several methods are available that allow for instan-
taneous or sequential retrieval of both quadrature
components of the complex interferometric signal.
These include (1) polarization quadrature encoding,8
where orthogonal polarization states encode the real
and imaginary components; (2) phase stepping,3 where
the reference ref lector 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 detection5 and phase-locked
loops. Polarization quadrature encoding and phase
stepping can be generically called quadrature interfer-
ometry since the complex signal is optically generated.
As such, they are useful in both homodyne and het-
erodyne systems.
Each of these techniques suffers from shortcomings.
Polarization quadrature encoding is instantaneous, but
it requires a complicated setup, and it suffers from
polarization fading. Phase shifting requires a stable
and carefully calibrated reference-arm setup, is not in-
stantaneous, and is sensitive to interferometer drift
between phase-shifted acquisitions. Synchronous de-
tection is not instantaneous and depends on the pres-
ence of an electronic carrier frequency. Systems based
on synchronous detection are thus not useful in an
important class of homodyne systems such as en-face0146-9592/03/222162-03$15.00/0imaging schemes and those that take advantage of ar-
ray detection.
In this Letter we present a f iber-based quadrature
broadband interferometer that exploits the inherent
phase shifts of 3 3 3 fiber-optic couplers. We present
a simple argument to explain interferometric phase
shifts in 2 3 2 and 3 3 3 Michelson interferometers,
and we demonstrate that the outputs of a 3 3 3 inter-
ferometer provide instantaneous access to the entire
complex interferometric signal in homodyne and het-
erodyne systems.
Fused-fiber couplers rely on evanescent-wave cou-
pling to split an input electric field between output
fiber paths, according to coupled-mode theory.9,10
Here we describe a simpler formalism based on
conservation of energy that predicts phase shifts for
interferometers based on 2 3 2 and 3 3 3 couplers.
While higher-order couplers (e.g., 4 3 4) can be used
in a manner similar to that described below for 3 3 3s
to acquire the complex interferometric signal, the
phase shifts in these couplers can be explained only
by coupled-mode theory.
Consider the 2 3 2 and 3 3 3 Michelson interfer-
ometers illustrated in Fig. 1. The coefficient aab de-
scribes the power transfer from fiber a to f iber b. For
example, a 2 3 2 5050 coupler has a11  a12  1/2,
and a 33 3 333333 coupler has a11  a12  a13  1/3.
Ignoring the return loss from the source 5050 coupler,
the optical intensity incident on the nth detector that
is due to a single ref lection in the sample is
In  gI0a11a1n 1 a12a2n 1 2EDx a11a1na12a2n12
3 cos2kDx 1 fn . (1)
Here k is the optical wave number, Dx is the
path-length difference between ref lectors in the refer-
ence and sample arms, EDx is the interferometric
envelope (i.e., the magnitude of the complex signal),
and fn is the phase shift between the optically het-
erodyned fields when Dx  0. g ensures that the
total power incident on the reference and sample arms
is I0 and is 1 for a 2 3 2 and g  1a11 1 a12 for
a 3 3 3. Since the noninterferometric portions of© 2003 Optical Society of America
November 15, 2003 / Vol. 28, No. 22 / OPTICS LETTERS 2163Fig. 1. (A) Conventional differential low-coherence
Michelson interferometer based on 2 3 2 f iber couplers.
I0 is the total power incident on the reference and sample
arms. g  1. D, detector; F, fiber. (B) Michelson
interferometer based on a 3 3 3 f iber coupler.
g  1a11 1 a12. A circulator may be used in place
of the input couplers in both (A) and (B) for increased
eff iciency.7 (C) Graphic representation of Eq. (2) for the
coupling ratios and intrinsic phase shifts in a 3 3 3
interferometer.
In sum to I0, the interferometric portions must sum
to zero, independently of Dx. Assuming (1) perfect
reciprocity (i.e., aab  aba) and (2) lossless coupling
(
P
n aan  1) in the fiber coupler,
Re
2
4X
n
p
a1na2n exp j2kDxexp jfn
3
5  0 , (2a)
X
n
p
a1na2n cosfn
X
n
p
a1na2n sinfn 0 . (2b)
Because variations in kDx rotate the a1na2n12 3
exp jfn vector in the complex plane, Eq. (2a) is true if
and only if a1na2n12exp jfn sums to zero [Eq. (2b)].
Equations (2) demand that jf1 2 f2j  180± for 23 2
couplers regardless of the coupler splitting ratio, while
fm 2 fn is an explicit function of aab for 3 3 3 cou-
plers. Equation (2b) can be represented graphically
as the sum of three complex vectors with magnitude
a1na2n12 and phase fn [Fig. 1(C)]. Given a set of
measured values of aab, the interferometric phase
shifts between coupler arms fm 2 fn are uniquely
determined, since the three vectors correspond to the
sides of a triangle whose inner angles are related to
fm2fn. For example, if aab  1/3 for all a and b, then
the interferometric phase shift between any two out-
put ports of a 3 3 3 Michelson interferometer is 120±.
Because the phase shifts among In are not con-
strained to be 0± or 180± for interferometers based
on 3 3 3 couplers, the complex interferometric signal
can be obtained instantaneously from simultaneous
measurements of two or more detector outputs. It
should be noted that this can be accomplished withall higher-order N 3 N interferometers (N . 2) of
various topologies (e.g., Michelson, Mach–Zehnder).
A straightforward algorithm for reconstructing the
complex interferometric signal uses the signal from
any two photodetectors. Defining in as the inter-
ferometric portion of In, and assigning any in as
the real part of the complex signal (denoted iRe
below), we may then obtain the imaginary part,
iIm  2EDx a11a1na12a2n12sin2kDx 1 fn, by use
of the cosine sum rule as
iIm 
in cosfm 2 fn 2 bim
sinfm 2 fn
,
b 
µ
a11a1na12a2n
a11a1ma12a2m
∂12
. (3)
We constructed the setup illustrated in Fig. 1(B)
from a superluminescent diode source (OptoSpeed;
l0  1274 nm, Dl  28 nm) and an AC Photonics
3 3 3 truly fused-fiber coupler. To demonstrate
instantaneous quadrature signal acquisition in a
heterodyne experiment, we scanned the reference arm
at v  8.3 mms. The interferometric portions of all
three acquired photodetector outputs are illustrated
in Fig. 2. When experimentally measured values of
aab were used to solve for the theoretical interfero-
metric phase shifts, these theoretical shifts deviated
from the experimental shifts by #2.5%. We noted
that fm 2 fn drifted over the course of minutes to
hours, and we hypothesize that these drifts are due to
temperature sensitivity of the coupler splitting ratio.
If i1, i2, and i3 are parametrically plotted against
one another, a three-dimensional Lissajous curve is
formed [Fig. 3(A)]. Time is encoded by the color of the
curve: In this experiment the data were captured as
Dx increased with time, and thus as time progresses,
the curve spirals outward and transitions from green
to black to red [Fig. 3(D)]. The curve fills an ellipse
[Fig. 3(B)]. The real (i2) and imaginary parts [calcu-
lated from Eqs. (3)] of the interferogram are plotted
as a Lissajous curve in Fig. 3(C) and were used to cal-
culate the magnitude and phase as plotted in Fig. 4.
The time derivative of the phase yields the Doppler
shift fd  2vl generated by the scanning reference
mirror. The experimentally calculated Doppler shift
(Fig. 4) agrees well with the predicted value, based on
the calibrated scan velocity of the reference arm.
We also demonstrated 3 3 3 instantaneous quadra-
ture signal acquisition in a homodyne experiment by
positioning a nonscanning reference mirror (v  0) such
thatDx corresponded to the middle of an interferogram.
Fig. 2. Interferometric detector outputs of a 33 3 Michel-
son interferometer in a heterodyne experiment.
2164 OPTICS LETTERS / Vol. 28, No. 22 / November 15, 2003Fig. 3. (A) Three-dimensional Lissajous plots of the
interferometric signal from three photodetectors in a
3 3 3 Michelson interferometer. Time is color coded
(green–black–red). (B) Lissajous plot of i3 versus i2.
(C) Real and imaginary signals calculated from i2 and i3.
(D) i2 versus time plotted with color coding.
Fig. 4. Magnitude, phase, and Doppler shift of complex
heterodyne interferometric signal calculated from outputs
of the 3 3 3 interferometer. The Doppler shift is calcu-
lated from the numerical derivative of the phase and is in
good agreement with the predicted value of 13 kHz.
We then recorded the outputs i2 and i3 for 10 s and
used Eqs. (3) to reconstruct amplitude and phase. Be-
cause of interferometer arm-length drifts of the order
of l, the phase of the complex interferometric sig-
nal varied by 2p, whereas the magnitude remained
steady (mean 1.52, variance 1.96 3 1024; see Fig. 5).
This result demonstrates the potential utility of N 3
N interferometers in homodyne OCT approaches such
as optical coherence microscopy, Fourier-domain OCT,
and optical frequency-domain interferometry.
Although higher-order N 3 N interferometers are
less eff icient with source light than conventional 23 2
interferometers, they are more eff icient at collecting
light ref lected from the sample and may thus be advan-Fig. 5. Magnitude and phase of the complex interferomet-
ric signal in a homodyne experiment. The reference arm
was not scanned; small random motions of the reference
and sample mirrors led to variations in the interferomet-
ric phase. Application of the algorithm of Eqs. (3) allowed
for clean extraction of magnitude and phase from outputs
D2 and D3 of a 3 3 3 interferometer. The magnitude was
constant (mean 1.52, variance 1.963 1024), whereas small
mirror motions led to p in phase.
tageous in sample exposure-limited applications such
as retinal imaging. For example, the 3 3 3 interfer-
ometer in Fig. 1(b) delivers only 1/6 of the source light
to the sample but collects 1/3 1 1/3 1 1/3 1/2  83% of the
light returning from the sample. The same interfer-
ometer without the input 2 3 2 coupler is an attrac-
tive alternative that delivers 33% of the source light to
the sample, collects 66% of the ref lected light, and still
performs instantaneous quadrature detection with two
detector signals (D2 and D3) as inputs to Eq. (3).
In conclusion, N 3 N interferometers allow for in-
stantaneous optical extraction of magnitude and phase
information in a compact and simple design. Their
advantages are particularly compelling in Fourier-
domain OCT and optical frequency domain ref lectom-
etry, where they will allow for optical resolution of
complex-conjugate ambiguity without phase stepping.
This work was supported by National Institutes of
Health grant R24-EB000243. J. A. Izatt’s e-mail ad-
dress is jizatt@duke.edu.
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Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers

Puliaf ito,

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Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers

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