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BACKGROUND OF THE INVENTION
The present invention relates to rotation sensors for use in, e.g.,
gyroscopes, and particularly to fiber optic rotation sensors.
Fiber optic rotation sensors typically comprise a loop of single-mode
optical fiber to which a pair of lightwaves are coupled for propagation in
opposite directions around a loop. If the loop is rotated, the
counter-propagating waves will undergo a phase shift, due to the
well-known Sagnac effect, yielding a phase difference between the waves
after traverse of the loop. By detecting this phase difference, a direct
indication of the rotation rate of the loop may be obtained.
If the optical path lengths about the loop for the counter-propagating
waves are equal when the loop is at rest, the interferometer is said to be
"reciprocal". In practice, however, fiber interferometer loops are
ordinarily not reciprocal, due to the fact that present, commercially
available optical fibers are not optically perfect, but are birefringent
(i.e., doubly refractive), resulting in two orthogonal polarization modes,
each of which propagates light at a different velocity. One of the
polarization modes, therefore, provides a "fast channel", while the other
provides a "slow channel." In addition, the fiber birefringence is
sensitive to environmental factors, such as temperature, pressure,
magnetic fields, etc., so that, at any given point along the fiber, the
birefringence can vary over time in an unpredictable manner. Birefringence
affects the counter-propagating waves in a complex way, however, the
effect may be viewed as causing a portion of the waves to be coupled from
one of the polarization modes to the other, i.e., from the "fast channel"
to the "slow channel or vice versa. The result of such coupling between
modes is that each of the counter-propagating waves may travel different
optical paths around the loop, and thus, require different time periods to
traverse the fiber loop, so that there is a phase difference between the
waves when the loop is at rest, thereby making the interferometer
non-reciprocal.
The foregoing may be more fully understood through a rather simplistic,
extreme example in which it is assumed that there is birefringence only at
one point in the fiber loop, and that this point is located near one end
of the loop. It is also assumed that such birefringence is sufficient to
cause light energy to be entirely coupled from one polarization mode to
the other, and that there is no coupling between modes anywhere else in
the fiber loop. If the counter-propagating waves are introduced into the
loop in the fast channel, one of the waves will immediately be coupled to
the slow channel while the other wave will traverse most of the loop
before being coupled to the slow channel. Thus, one of the waves will
traverse most of the loop in the fast channel, while the other will
traverse most of the loop in the slow channel, yielding a phase difference
between the waves when the loop is at rest. If this birefringence-induced
phase difference were constant, there would, of course, be no problem,
since the rotational induced Sagnac phase difference could be measured as
a deviation from this constant birefringence-induced phase difference.
Unfortunately, however, such birefringence-induced phase differences vary
with time, in an unpredictable manner, and thus, these
birefringence-induced phase differences are indistinguishable from
rotationally-induced, Sagnac phase differences. Thus, time varying changes
in birefringence are a major source of error in fiber optic rotation
sensors.
The prior art has addressed the problem of nonreciprocal,
birefringence-induced phase differences in a variety of ways. In one
approach, described by R. A. Bergh, H. C. Lefevre, and H. J. Shaw in
Optics Letters, Volume 6, No. 10 (October 1981), a fiber optical polarizer
is utilized to block light in one of the two orthogonal polarization modes
while passing light in the other. This insures that only a single optical
path is utilized, thereby providing reciprocity. This approach is also
described in International Patent Application No. PCT/US 82/00400
published Oct. 14, 1982, as Publication No. WO 82/03456, entitled "Fiber
Optic Rotation Sensor," and also in corresponding U.S. patent application
Ser. No. 307,095, filed Sept. 30, 1981, entitled "Fiber Optic Rotation
Sensor", which is a continuation-in-part of patent application Ser. No.
249,714, filed Mar. 31, 1981. Another approach involves utilizing
unpolarized light, which has been found to result in cancellation of
birefringence-induced phase differences upon combining the
counter-propagating waves after traverse of the loop. The degree of
cancellation is proportional to the degree to which the light waves are
unpolarized. This approach is described in detail in International Patent
Application No. PCT/US 82/00985, published Feb. 17, 1983 as Publication
No. 83/00552, and also in corresponding U.S. patent application, Ser. No.
288,212, filed Jul. 29, 1981, entitled "Fiber Optic Rotation Sensor
Utilizing Unpolarized Light".
It is also known in the art to utilize polarization-conserving fibers to
reduce coupling between the modes. Polarization-conserving fibers are
essentially high birefringence fibers, in which the fiber is mechanically
stressed during manufacture to increase the difference in the refractive
indicies of the two polarization modes. This reduces coupling between the
modes, since the high birefringence tends to preserve the polarization of
the light waves. In effect, changes in birefringence due to environmental
factors are overwhelmed by the stress-induced birefringence created during
manufacture of the fiber.
SUMMARY OF THE INVENTION
The present invention comprises a fiber optic Sagnac interferometer
employing high birefringence fiber, e.g., of the type described in
Electronics Letters, Volume 18, Number 24 (Nov. 25, 1982), pages 1306 to
1308. Such high birefringence fiber reduces the average optical power
transferred from one polarization mode to the other to about one percent
or less over 1 km of fiber. As an approximation, the maximum phase error
due to coupling between modes is equal to the fraction of power
transferred between the modes. Thus, for a 1-km fiber loop having a power
transfer rate of 1% per km, the maximum phase error would be 0.01 or
10.sup.-2 radians.
The present invention substantially reduces the maximum phase error by
utilizing a wide band, short coherence length laser source in combination
with the high birefringence fiber. The amount of reduction is dependent
upon the "fiber coherence length", which is a newly coined term that
should be distinguished from the coherence length of the source. As used
herein, the term "fiber coherence length" is defined as the length of
fiber required for the optical path length difference between the two
polarization modes to equal one coherence length of the light source. It
is approximately equal to the coherence length of the source divided by
the difference in refractive index between the polarization modes. In
general, the shorter the fiber coherence length, the greater the reduction
in phase error. More specifically, use of a short fiber coherence length
results in a phase error reduction which is proportional to 1/.sqroot.N,
where N is the loop length divided by the fiber coherence length.
The fiber loop may thus be considered as being divided into N segments,
each having a length of one fiber coherence length. Light coupled from one
polarization mode to another over one segment (fiber coherence length)
will add coherently over that segment but not thereafter. Further, after
the waves have traversed the fiber loop, and are recombined, the only
portions of the coupled light which will interfere with each other will be
those which were coupled at symmetric segments of the fiber loop.
Consequently, interference between lightwave components coupled between
polarization modes is reduced dramatically, thereby reducing the
birefringence-induced phase error. Through use of present, state of the
art components, such reduction in interference provides, e.g., an
additional factor of 100 improvement, so that the maximum phase error,
assuming a 1 km, high birefringence fiber having a power transfer rate of
1%, decreases from 10.sup.-2 radians to 10.sup.-4 radians.
Further improvement in phase error reduction may be obtained by launching
each of the orthogonal polarization modes with light that is uncorrelated
and of substantially equal intensity (i.e., unpolarized light). To the
extent that the intensities are equal and the phases are uncorrelated,
phase differences between interfering cross-coupled light wave components
will cancel, yielding a net non-rotationally-induced phase difference of
zero. Assuming that the intensities are equalized to within 1% of each
other, use of unpolarized light in combination with the high birefringence
fiber and short coherence length source provides a further improvement of
a factor of about 100 in the maximum phase error, reducing it to, e.g.,
10.sup.-6 radians.
Thus, the present invention substantially eliminates the effects of
birefringence-induced phase differences, permitting detection of the
rotationally induced Sagnac phase difference with a high degree of
accuracy.
In addition to reducing phase error, the present invention advantageously
improves the stability of the detected output signal. Those skilled in the
art will recognize that, even though an interferometer is perfectly
reciprocal and generates no phase errors, the output signal may
nevertheless vary in intensity. Such variations, in effect, change the
"scale factor" or "proportionality factor" between the detected intensity
and the rotation rate. In unpolarized light rotation sensors these
variations are caused, e.g., by inteference between lightwave components
which are coupled between polarization modes. Since the present invention
reduces interference between such coupled lightwave components, these
"scale factor" variations are reduced, thereby further improving
performance of the rotation sensor.
These and other advantages of the present invention are best understood
through reference to the drawings.
DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic drawing of the rotation sensor of the present
invention, showing a single, continuous strand of optical fiber, to which
light from a light source is coupled, and showing the multimode sensing
loop, formed from such single, continuous strand; in addition, FIG. 1
shows a detection system for detecting the phase difference between waves
counterpropagating through the fiber loop;
FIG. 2 is a schematic drawing illustrating a conceptual model of the fiber
loop, showing, for an exemplary pair of polarization modes, the electric
field components of the counterpropagating waves as they traverse the
fiber loop;
FIG. 3 is a schematic drawing of the conceptual model of FIG. 2, showing
the electric field components of the counterpropagating waves after they
have traversed the fiber loop;
FIG. 4 is a vector diagram of the optical output signal, showing a vector
directed along the real axis, which represents the vector sum of the "dc"
terms resulting from the electric field components shown in FIG. 3, and
another vector, rotating in the manner of a phasor, which represents the
vector sum of the interference terms resulting from the electric field
components shown in FIG. 3, and further illustrating the response of the
vector representing the interference terms to 1) the rotationally-induced
Sagnac phase difference, and 2) phase errors caused by non-rotationally
induced phase differences;
FIG. 5 is a graph, corresponding to the vector diagram of FIG. 4, of the
optical intensity, as measured by the detector, versus the Sagnac phase
difference, illustrating the effect of non-rotationally induced phase
errors;
FIG. 6 is a vector diagram of the interference terms resulting from Group
III electric field components;
FIG. 7 is a vector diagram showing a resultant vector which represents the
vector sum of the two vectors of FIG. 6, and illustrating the phase error
associated with such resultant vector sum;
FIG. 8 is a vector diagram showing the vectors of FIG. 6 equalized in
magnitude;
FIG. 9 is a vector diagram of a resultant vector, which represents the
vector sum of the vectors of FIG. 8, illustrating that phase errors may be
eliminated by equalizing the magnitudes of the vectors;
FIG. 10 is a graph of the optical intensity, as measured by the detector,
versus the Sagnac phase difference, illustrating the effect of changes in
the magnitude of the interference factor of FIG. 4, assuming a phase error
of zero;
FIG. 11 is a schematic drawing illustrating the fiber loop divided into two
segments, each having a length of one fiber coherent length;
FIGS. 12 and 13 are schematic drawings illustrating conceptual models of
the fiber loop, showing, for an exemplary pair of polarization modes, the
cross coupled electric field components of the counterpropagating waves as
they traverse the plural segment loop of FIG. 11;
FIG. 14 is a vector diagram of the interference term resulting from Group
III electric field components in the two segment fiber loop of FIGS. 11,
12, and 13, and illustrating that the vector in addition of such
components yield resultant vectors which are reduced in magnitude;
FIG. 15 is a vector diagram similar to that of FIG. 14, illustrating Group
III interference components for a 10 segment loop vectorially adding to
yield result invectors which are further reduced in magnitude;
FIG. 16 is a vector diagram of the optical output signal at the detector,
showing the interference vectors for Group I, Group II, and Group III
components vectorially adding to form the overall interference vector,
which represents the vector sum of all the interference terms, and further
illustrating the effect of the magnitude and phase of Group III
interference terms on the phase of the overall interference vector;
FIG. 17(a) and (b) are vector diagrams at times t.sub.1 and t.sub.2,
respectively, illustrating how variations in the phases of Group III
interference components for the two polarization modes can cause scale
factor problems through variations in Group III vector magnitude.
FIG. 18 is a sectional view of one embodiment of the fiber optical
directional coupler for use in the rotation sensor of FIG. 1; and
FIG. 19 is a sectional view of a fiber optic polarizer which may be
utilized in the rotation sensor of FIG. 1.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
In the preferred embodiment, shown in FIG. 1, the rotation sensor of the
present invention comprises a light source 10 for introducing a cw light
wave into a single, continuous length or strand of single mode optical
fiber 11. As used herein, "single mode fiber" means that the fiber
supports only one fundamental mode for the particular source light used,
as opposed to multimode fiber which supports more than one fundamental
mode. However, it will be recognized that a single mode fiber includes two
orthogonal polarization modes, each of which propogates light at a
different velocity.
The fiber 11 passes through ports, labeled A and C, of a first directional
coupler 12, and through ports, labeled A and C of a second directional
coupler 14. Thus, the fiber 11 extends from the light source 10 to port A
of the coupler 12 and extends from port C of the coupler 12 to port A of
the coupler 14 to form a line segment 15 of fiber between the source 10
and coupler 14. The portion of the fiber 11 extending from port C of the
coupler 14 is wound into a loop 16. By way of specific example, the loop
16 may comprise about 1400 turns, each bounding an area of about 150 sq.
cm for a total loop length of 600 meters. The end of the fiber 11, from
the loop 16, is passed through ports, labeled D and B, of the coupler 14,
with port D adjacent to the loop 16. A small portion 17 of the fiber 11
extends from port B of the coupler 14 and terminates nonreflectively,
without connection.
A second length of fiber 19 is passed through the ports labeled D and B of
the coupler 12. The portion of the fiber 19 projecting from port D
terminates nonreflectively, without connection. However, the portion of
the fiber 19 projecting from port B of the coupler 12 is optically coupled
to a photodetector 20, which produces an output signal proportional to the
intensity of the light impressed thereon.
The present invention also includes detection electronics 22, comprising a
lock-in amplifier 24, a signal generator 26, and a phase modulator 28. By
way of specific example, the phase modulator 28 may comprise a PZT
cylinder, having a diameter of e.g. about 1 to 2 inches, about which a
portion of the fiber loop 16 is wrapped, e.g., 4 to 10 times. The fiber is
bonded to the PZT cylinder 28 by a suitable adhesive, so that the fiber 11
will be stretched upon radial expansion of the cylinder 28. In this
regard, the phase modulator 28 is driven by an AC modulating signal,
having a frequency in the range of, e.g., 10-1000 kHz, which is provided
on a line 30 from the signal generator 26. For proper operation of the
detection electronics 22, it is important that the phase modulator 28 be
located on one side of the loop 16, e.g., adjacent to the port D of the
coupler 14, rather than at the center of the sensing loop 16.
The AC modulation signal from the generator 26 is also supplied on a line
32 to the lock-in amplifier 24. A line 34 connects the lock-in amplifier
24 to receive the detector 20 output signal. The amplifier utilizes the
modulation signal from the generator 26 as a reference for enabling the
amplifier 24 to synchronously detect the detector output signal at the
modulation frequency. Thus, the amplifier 24 effectively provides a band
pass filter at the fundamental frequency (i.e., the frequency of
modulation) of the phase modulator 28, blocking all other harmonics of
this frequency. It will be understood by those skilled in the art that the
magnitude of this harmonic component of the detector output signal is
proportional, through an operating range, to the rotation rate of the loop
16. The amplifier 24 outputs a signal which is proportional to this first
harmonic component, and thus, provides a direct indication of the rotation
rate.
Additional details of the detection electronics 22 are described in
international patent application No. PCT/U.S. 82/00400 published Oct. 14,
1982, as publication No. WO 82/03456, and entitled "Fiber Optic Rotation
Sensor", and in corresponding U.S. patent application Ser. No. 307,095,
filed Sep. 30, 1981, which is a continuation-in-part of U.S. patent
application Ser. No. 249,714, filed Mar. 31, 1981. These applications are
incorporated herein by reference. This detection system is also described
in Optics Letters, Vol. 6, No. 10, (October 1981) pp. 502-504.
In the embodiment shown, the fiber 11 comprises a highly birefringent
single mode fiber, e.g., of the type described in the article entitled
"Fabrication of Polarization Maintaining Fibres Using Gas-Phase Etching",
Electronics Letters, Vol. 18, No. 24, p. 1306 (Nov. 25, 1982).
The light source 10 should provide light which has a short coherence
length. A preferred light source for use as the source 10 is a
superradiance diode, e.g., of the type described in the article entitled
"High Power Low Divergence Superradiance Diode", Applied Physics Letters,
Vol. 41, No. 7 (Oct. 1, 1982).
The photodetector 20 is a standard pin or avalanchetype photodiode, which
has a sufficiently large surface area to intercept substantially all of
the light exiting the fiber 19, when positioned normal to the fiber axis.
The diameter of the photodetector 20 is typically in the range of about 1
millimeter, the exact size depending upon the diameter of the fiber 19,
the numerical aperture of the fiber 19 (which defines the divergence of
the light as it exits the fiber 19) and the distance between the end of
the fiber 19 and the photodetector 20.
In operation, a light wave W.sub.i is input from the light source 10 for
propagation through the fiber 11. As the wave W.sub.i passes through the
coupler 12, a portion of the light (e.g. 50 per cent) is lost through port
D. The remaining light propagates from port C of the coupler 12 to the
coupler 14, where the light is split evenly into two waves
W.sub.1,W.sub.2, which propagate in opposite directions about the loop 16.
After traverse of the loop 16, the waves W.sub.1,W.sub.2 are recombined by
the coupler 14 to form an optical output signal W.sub.0. A portion of the
recombined wave W.sub.0 may be lost through the port B of the coupler 14,
while the remaining portion travels from port A of the coupler 14 to port
C of the coupler 12, where it is again split, with a portion thereof
(e.g., 50%) transferred to the fiber 19. Upon exiting the end of the fiber
19, the wave W.sub.0 is impressed upon the photodetector 20, which outputs
an electrical signal that is proportional to the optical intensity of the
wave W.sub.0.
The intensity of this optical output signal will vary in proportion to the
type (i.e., constructive or destructive) and amount of interference
between the waves W.sub.1, W.sub.2, and thus, will be a function of the
phase difference between the waves W.sub.1,W.sub.2. Assuming, for the
moment, that the fiber 11 is "ideal" (i.e., that the fiber has no
birefringence, or that the birefringence does not change with time),
measurement of the optical output signal intensity will provide an
accurate indication of the rotationally induced Sagnac phase difference,
and thus, the rotation rate of the fiber loop 16.
As indicated above, present state-of-the-art, fibers are far from "ideal",
in that 1) they are birefringent, and 2) the birefringence is
environmentally sensitive and tends to vary, thus, yielding
nonrotationally induced phase differences (i.e., phase errors), which are
indistinguishable from the rotationally induced Sagnac phase difference.
The present invention utilizes three different techniques to reduce or
eliminate these phase errors, namely, 1) the use of a high birefringence
fiber to reduce coupling between the polarization modes; 2) the use of a
wideband, highly incoherent light source in combination with the high
birefringence fiber to reduce interference between lightwave components
which have been coupled between polarization modes; and 3) equalizing the
lightwave intensity in each of the two polarization modes to cause the
phase differences between interfering components of light which has been
coupled between polarization modes to cancel.
PHASE ERROR ANALYSIS
Such reduction or elimination of phase errors may be more fully understood
through reference to FIG. 2, which depicts a conceptual model of the two
orthogonal polarization modes of a single mode fiber. Each polarization
mode has a propogation velocity different from that of the other
polarization mode. Further, it is assumed that there is coupling of light
energy between modes, which may be caused e.g. by variations or
perturbations in the principal axes of birefringence of the fiber. Such
coupling of energy will be referred to herein as "cross coupling."
The conceptual fiber model of FIG. 2 will be utilized to represent the
sensing loop 16 (FIG. 1). The counterpropagating waves W.sub.1, W.sub.2,
are schematically represented as being coupled, by the coupler 14, to the
loop 16, by the dashed arrows. The two polarization modes of the single
mode optical fiber are schematically represented in FIG. 2 by a first
line, connecting a pair of terminals C' and D', and a second line,
parallel to the first line, connecting a second pair of terminals C" and
D". The terminals C' and C" on the left side of FIG. 2 correspond the port
C of the coupler 14, while the terminals D' and D" on the right side of
FIG. 2 correspond to the port D of the coupler 14. The above mentioned
first and second lines connecting the terminals will be used to represent
arbitrary modes i and j, respectively, of the fiber loop 16.
Cross coupling between the modes i and j is represented by a pair of lines,
labeled "Branch 1" and "Branch 2", respectively. Branch 1 represents cross
coupling between the terminials C" and D' while branch 2 represents cross
coupling between terminals C' and D". The intersection of branch 1 with
branch 2, designated by the referenced numeral 50, will be referred to as
the "coupling center". It will be understood that no coupling exist
between the two branches 1 and 2. The coupling center 50 is shown as being
offset from the center of the fiber loop 16 to illustrate that the
coupling between the polarization modes is not uniform along its length.
Therefore, cross coupled light will travel a longer path in one of the
modes than the other, yielding a nonrotationally induced phase difference
therebetween. Moreover, it will be understood that, in reality, the fiber
birefringence, being environmentally sensitive, varies with time, thus
making the optical paths travelled by the cross-coupled light also time
varying.
As shown in FIG. 2, the wave of W.sub.1 is coupled to the fiber loop 16 so
that the modes i and j are launched with electric field amplitudes
E.sub.i.sup.+ and E.sub.j.sup.+ respectively. Similarily, the wave W.sub.2
is coupled to launch each of the modes i and j with electric field
amplitudes E.sub.i.sup.- and E.sub.j.sup.-, respectively. The plus (+) and
minus (-) superscripts designate the direction of propegation, the
clockwise direction about the loop 16 being designated by the plus (+)
sign, and the counterclockwise direction around the loop 16 being
designated by the minus (-) sign.
As light in each of the modes i and j traverses the fiber loop 16, energy
is coupled between the modes, so that each electric field is divided into
two components, namely, a "straight through" component, designated by the
subscript "s", and a "cross coupled" component, designated by the
subscript "c". Thus, E.sub.i.sup.+ is divided into a straight through
component E.sub.is.sup.+ which remains in mode i during traverse of the
loop 16, and a cross coupled component E.sub.jc.sup.+, which is cross
coupled to mode j during traverse of the loop 16. Similarily,
E.sub.i.sup.- is divided into components E.sub.is.sup.- and E.sub.jc.sup.-
; E.sub.j.sup.+ is divided into components E.sub.ic.sup.+ and
E.sub.js.sup.+ ; and E.sub.j.sup.- is divided into components
E.sub.js.sup.- and E.sub.ic.sup.-.
After the light waves have traversed the fiber loop 16, the light at
terminal C' will comprise components E.sub.is.sup.- and E.sub.ic.sup.- ;
the light at terminal C" will comprise component E.sub.js.sup.- and
E.sub.jc.sup.- ; the light at terminal D' will comprise components
E.sub.is.sup.+ and E.sub.ic.sup.+ ; and the light at terminal D" will
comprise components E.sub.js.sup.+ and E.sub.jc.sup.+, as shown in FIG. 3.
These 8 electric field components are combined by the coupler 14 to form
the optical output signal W.sub.0. It will be recognized by those skilled
in the art that, in general, superposition of any two electric field
components, e.g., E.sub.is.sup.+ and E.sub.ic.sup.+ will yield a resultant
intensity (I), as measured by the detector 20, which may be defined as
follows:
I=.vertline.E.sub.is.sup.+ .vertline..sup.2 +.vertline.E.sub.ic.sup.+
.vertline..sup.2 +2.vertline.E.sub.is.sup.+
.vertline..vertline.E.sub.ic.sup.+ .vertline. cos .phi. (1)
where, in this particular example, .phi. is the phase difference between
field components E.sub.is.sup.+ and E.sub.ic.sup.+.
The first two terms of equation (1), namely .vertline.E.sub.is.sup.+
.vertline..sup.2 and .vertline.E.sub.ic.sup.+ .vertline..sup.2 are
steady-state or "d.c." terms, while the last term is an "interference"
term having a magnitude depending upon the phase difference .phi. between
the fields E.sub.is.sup.+ and E.sub.ic.sup.+.
In general, all 8 of the above fields
E.sub.is.sup.-, E.sub.ic.sup.-, E.sub.js.sup.-, E.sub.jc.sup.-,
E.sub.is.sup.+, E.sub.ic.sup.+, E.sub.js.sup.+ and E.sub.jc.sup.+,
will interfere with each other to provide an optical intensity at the
detector 20 (FIG. 1) comprised of 8 "dc" terms, which are not
phase-dependent, and 28 "interference" terms which are phase-dependant.
The number of combinations of phase-dependant terms is actually n(n-1) or
56 phase-dependent terms. However, one-half of these terms are simply the
re-ordered forms of the other half, yielding 28 non-redundant terms.
The 8 dc terms are shown in FIG. 4 as a single vector sum, labeled
I.sub.dc, while the 28 interference terms are shown in FIG. 4 as a single
vector, labeled I.sub.i. These vectors I.sub.dc and I.sub.i are plotted in
a complex plane. Upon rotation of the fiber loop 16 (FIG. 1) the
phase-dependent vector I.sub.i rotates, in the manner of a phasor, through
an angle equal to the rotationally reduced phase difference .phi..sub.s
due to the Sagnac effect. The projection of the interference vector
I.sub.i upon the real axis, when added to the vector I.sub.dc, yields the
total optical intensity I.sub.DET of the optical output signal W.sub.0, as
measured by the detector 20 (FIG. 1). In FIG. 5, this optical intensity
I.sub.DET is plotted as function of the Sagnac phase difference
.phi..sub.s, as illustrated by the curve 52.
As indicated above in reference to FIG. 2, cross coupling between the modes
i and j can cause the fiber loop 16 to be nonreciprocal, resulting in a
nonrotationally induced phase difference between the above described
electric field components, and yielding an accumulated phase error
.phi..sub.e, which is indistinguishable from the rotationally induced
Sagnac phase difference .phi..sub.s. The phase error .phi..sub.e causes
the phasor I.sub.i to be rotated, e.g., from the position shown in solid
lines to the position shown in dotted lines in FIG. 4. This results in the
curve 52 of FIG. 5 being translated by an amount .phi..sub.e e.g., from
the position shown in solid lines to the position shown in dotted lines in
FIG. 5.
Elimination or reduction of the accumulated phase error .phi..sub.e
requires an analysis of the 28 interference terms resulting from
superposition of the 8 electric field components discussed in reference to
FIG. 2. At the outset, it will be recognized that interference between
electric field components E.sub.is.sup.+ with E.sub.is.sup.-, and
E.sub.js.sup.+ with E.sub.js.sup.-, result in no phase error contribution,
since the light represented by these components is not cross coupled, and
traverses the loop in a single one of the modes. However, the remaining 26
interference terms can contribute to the accumulated phase error
.phi..sub.e. These 26 interference terms correspond to 26 pairs of
electric field components which may be classified into 3 groups, namely,
Group I, Group II, and Group III, as follows:
______________________________________
Group I Group II Group III
______________________________________
E.sub.is.sup.+ and E.sub.ic.sup.+
E.sub.is.sup.+ and E.sub.jc.sup.-
E.sub.ic.sup.+ and E.sub.ic.sup.-
E.sub.is.sup.+ and E.sub.ic.sup.-
E.sub.is.sup.+ and E.sub.js.sup.-
E.sub.jc.sup.+ and E.sub.jc.sup.-
E.sub.is.sup.- and E.sub.ic.sup.+
E.sub.is.sup.+ and E.sub.jc.sup.+
E.sub.is.sup.- and E.sub.ic.sup.-
E.sub.is.sup.+ and E.sub.js.sup.+
E.sub.js.sup.+ and E.sub.jc.sup.+
E.sub.ic.sup.+ and E.sub.js.sup.-
E.sub.js.sup.+ and E.sub.jc.sup.-
E.sub.ic.sup.+ and E.sub.js.sup.-
E.sub.js.sup.- and E.sub.jc.sup.+
E.sub.ic.sup.+ and E.sub.jc.sup.+
E.sub.js.sup.- and E.sub.jc.sup.-
E.sub.ic.sup.+ and E.sub.js.sup.+
E.sub.ic.sup.- and E.sub.jc.sup.-
E.sub.ic.sup.- and E.sub.js.sup.-
E.sub.ic.sup.- and E.sub.jc.sup.+
E.sub.ic.sup.- and E.sub.js.sup.+
E.sub.is.sup.- and E.sub.jc.sup.-
E.sub.is.sup.- and E.sub.js.sup.-
E.sub.is.sup.- and E.sub.jc.sup.+
E.sub.is.sup.- and E.sub.js.sup.+
______________________________________
Although only the interfering electric field components are listed above,
and not the interference terms themselves, it will be understood that the
interference term for each of the above listed pairs of components may be
readily calculated in accordance with the example provided in reference to
equation (1).
ELIMINATION OF GROUP I ERRORS
Group I includes those pairs of field components which originated in
different modes, but which are in the same mode upon reaching the coupler
14, after traversing the loop 16. For example, the first of Group I pair
of components comprises a straight-through component E.sub.is.sup.+, which
originated in mode i and remained in mode i during traverse of the loop
16, and a cross coupled component E.sub.ic.sup.+ which originated in mode
j but was cross coupled to mode i during traverse of the loop 16.
Ordinarily, these components would interfere with each other, as described
in reference to equation (1).
However, if the phase difference between these light wave components is
random, interference between the light wave components will be averaged to
zero in the detector 20. Accordingly, Group I interference terms can be
eliminated by insuring that, upon reaching the coupler 14, and thus the
loop 16, the light in each mode is incoherent, i.e., random in phase with
respect to the light in the other mode. Thus, for example, if the light in
mode i is incoherent with respect to light in mode j, the interference
between, e.g., the components E.sub.is.sup.+ and E.sub.ic.sup.+, will be
averaged to zero in the detector 20. Similarly, the interference between
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