We calculate the oscillation conditions and the eigen-frequencies for phase-conjugating-resonator-normal-resonator coupled optical systems. With an eye toward applications to interferometry, we choose specific examples for which it is shown that the conditions for oscillation and the eigenfrequencies depend on the normal-resonator path length. The examples include both linear displacement and rotation sensing (Sagnac) resonant interferometers. Our results suggest that if the distortion-correcting and self-aligning properties of the phase-conjugating resonator are retained in the more complicated system, then these hybrid resonators may offer some advantages over their conventional counterparts

Anderson, Dana Z.

English

Caltech Authors - Main

September 1984 / Vol. 9, No. 9 / OPTICS LETTERS 417
Coupled resonators employing phase-conjugating and ordinary
mirrors
Dana Z. Anderson
Division of Physics, Mathematics, and Astronomy, California Institute of Technology 130-33, Pasadena, California 91125
Received February 9, 1984; accepted May 30, 1984
We calculate the oscillation conditions and the eigenfrequencies for phase-conjugating-resonator-normal-resona-
tor coupled optical systems. With an eye toward applications to interferometry, we choose specific examples for
which it is shown that the conditions for oscillation and the eigenfrequencies depend on the normal-resonator path
length. The examples include both linear displacement and rotation sensing (Sagnac) resonant interferometers.
Our results suggest that if the distortion-correcting and self-aligning properties of the phase-conjugating resonator
are retained in the more complicated system, then these hybrid resonators may offer some advantages over their
conventional counterparts.
Optical resonators with phase-conjugating mirrors
(PCM's) utilizing four-wave mixing have demonstrated
some fascinating properties, such as the correction of
intracavity phase aberrationsl-3 and self-alignment.2' 4
However, the fundamental resonance frequency of the
phase-conjugating resonators (PCR's) demonstrated
so far are insensitive to resonator length.lA7 This
feature does not make them a good choice for dis-
placement sensors. The primary motivation for this
research stems from a desire to incorporate the useful
characteristics of the PCR into the realm of conven-
tional resonant interferometry.
To this end we study the consequences of coupling
phase-conjugating and normal resonators. Specifically,
we calculate the eigenfrequencies and the oscillation
conditions for three such hybrid structures. These
structures are shown in Fig. 1. The first (Fig. la) is the
simplest configuration (in terms of the number of mir-
rors). Here, the PCR formed by the PCM and M1 is
coupled through the transmission of M1 to the normal
resonator formed by M1 and M2. This structure reveals
an unusual mode-frequency spectrum. In Fig. lb the
PCR formed by the PCM and M2 is coupled to the
folded Fabry-Perot resonator formed by M1, M2, and
M3. In Fig. Ic the PCR formed by PCMA and PCMB
is coupled to the Sagnac (or ring) resonator formed by
M1 , M2, M3 , and M4 . In each case we derive the
eigenfrequencies and the oscillation conditions for the
structure by replacing all the normal mirrors by a single
complex mirror whose electric-field amplitude reflection
and transmission coefficients are strongly dependent
on the incident-wave frequency (Fig. id). Thus the
problem is reduced to that of a single PCR having one
complex mirror and one or two PCM's.
In the following discussion it is assumed that all the
normal mirrors are lossless. The PCM's are pumped
by fields having frequency Po, and it is presumed that
they provide gain, i.e., they conjugate and amplify an
incident wave.5' 7 We shall neglect the transverse-mode
distribution, that is, we shall suppose that the modes are
plane waves so that the electric field between the mir-
rors in Fig. Id satisfies the one-dimensional free-space
wave equation:
02E(z, t) 1 82E(z, t) =
0z 2 C2 0t2 (1)
subject to the appropriate boundary conditions at the
PCM and at the complex mirror. In Eq. (1), z is a
coordinate along the optical path and c is the speed of
light. The nature of phase conjugation by means of
four-wave mixing is such that an incident wave having
frequency vo + 3v is reflected with frequency vo - 3v.7
With this in mind one can guess that an eigenmode may
consist of fields having two frequencies that are sym-
metrically located about the pump frequency. A gen-
eral eigenmode satisfying Eq. (1) would then have the
form
E(z, t) = [Au exp(ikuz) + Bu exp(-ikuz)]
X exp[-i(vo + ev)t]
+ [AL exp(ikLZ) + BL exp(-ikLZ)I
X exp[-i(vo- -P)t], (2)
with kuL = (vo ± 3v)/c. Associated the PCM is a gain
g(Qv) that may be dependent on the incident-wave
frequency but that, we will suppose, is symmetric about
the pump frequency so thatg(bv) = g(-bv). Then the
PCM located at z = 0 imposes the following boundary
conditions:
AL = g(bv)B u,
Au = g(bw)BL.
(3a)
(3b)
Here, we have considered the PCM to be ideal and not
to induce any additional phase shifts of its own.8'9 The
generic complex mirror has electric-field amplitude
reflection and transmission coefficients
r(v) = p(v) exp[ir(v) ,
t(v) = r(v) exp[ipt(v)],
(4a)
(4b)
0146-9592/84/090417-03$2.00/0 © 1984, Optical Society of America
418 OPTICS LETTERS / Vol. 9, No. 9 / September 1984
,0
C
( VO
PC MA
IMI I
L2 ,;K LI 
L - E
b 1/0 d v/,
POM M, M 2 PC
e-L 2 ~4 M3
subject to the energy-conserving requirement that p 2
+ 'r2 = 1. Thus the mirror at z = L imposes the
boundary conditions
BU = p(v0 + bz)Au expli[2huL + Yor(vo + tiv)]},
BL = p(O - 6P)AL expli[2kLL + 'pr(Vo - bp)]I.
(5a)
(5b)
In order for the field amplitudes to meet the boundary
conditions [Eqs. (3) and (5)], two conditions must be
satisfied:
(I) g2 (bV)p(v0 + bv)p(Vo - by) = 1
and
(II) 49(po + by) - y(vo - by) + 4bkL = 2nr
(n = 0, ±1, 2, ...
where 6k = by/c. Condition (I) is a familiar require-
ment for sustained oscillation; the saturated gain sup-
plied by the gain medium must equal the resonator
losses. Condition (II) says that the round-trip accu-
mulated phase must be an integer multiple of 2wr. This
condition determines the eigenfrequencies of the reso-
nator. Condition (II) is always satisfied for by = 0. As
an example, consider a simple mirror, that is, one with
p and 'Pr independent of frequency. A mode always
occurs at the pump frequencyvo. In addition, according
to condition (II), there will be side modes flanking this
fundamental mode with a frequency spacing of ni7rc/2L.
This is in agreement with previous findings.3 -6
Now consider the coupled resonator of Fig. la. Let
the intensity reflection coefficient of M2 be P22 = 1.
One can find the electric-field reflection coefficient for
the complex mirror formed by the pair by assuming an
incident field on M1 and summing contributions from
successive round trips within the resonator8' 9:
rao') 1W - piei Pi, (6)
where
E = (v - ao)(2L1/c) (7)
and a,0 is a resonance frequency of the two-mirrored
cavity. The magnitude is Irol = po = 1, and the phase
fCOMPLEX
A MIRROR
I
Z = L
Fig. 1. Coupled resonator configura-
tions. a, PCR coupled to a Fabry-Perot
resonator; b, PCR coupled to a folded
Fabry-Perot resonator; c, PCR coupled
to a ring resonator; d, PCR coupled to a
normal resonator represented as a
complex mirror.
'pr (z) varies rapidly near resonance. Now suppose that
the resonator length L1 is tuned to the pump frequency
(or vice versa), that is, vo = a0 . Condition (II) will be
satisfied not only for by = 0 but also for a value near 0.
The case in which L1 = L2 E L is identical with that
treated by Spencer and Lamb for a normal coupled
resonator.10 In this case, condition (II) is satisfied by
a frequency
bv - rc/2L.
Unlike the normal coupled resonator, however, the
frequency spectrum of this hybrid resonator is charac-
terized by a frequency triplet in the region of the pump
frequency and by doublets spaced nwc/2L from the
pump frequency. (The spectrum is more complicated
if L1 FD L2.) Notice that the splitting frequency TCI2L
is dependent on both L2 and L1. Thus a change in the
PCM distance from M1 will cause a change in the
splitting frequency.
The folded normal resonator of Fig. lb can also be
replaced by a single complex mirror. We set the
reflectivities of M2 and M3 to unity. The reflectivity(back toward the PCM) and transmission (away from
the PCM and up in the figure) of the complex mirror
are
(8a)r( - T12
t(v) = P1 1 Tl2 ei -E 1)1 pi2e ie
where e is now (v - wo)(2L 1 /c) and L1 is the unfolded
cavity length. Notice that the reflectivity magnitude
reaches one only at resonance. Therefore the PCM can
sustain oscillations only when the pump frequency is at
or near the cavity resonance frequency. In this sense,
the present configuration behaves like a normal reso-
nator with an internal atomic gain medium in which the
cavity length must be tuned to the atomic resonance in
order for lasing action to occur.
As a final example we treat the case of two PCM's
coupled to a ring interferometer in order to reveal the
rotation-sensing properties of the interferometer.'1 Ml
and M2 are considered identical, and M3 and M4 have
(8b)
1 I
September 1984 / Vol. 9, No. 9 / OPTICS LETTERS 419
unit reflection coefficients. Again the normal mirrors
forming the ring are represented as a single complex
mirror located at z = 0 flanked by PCM's at z = +L/2.
The left-hand side of the complex mirror represents
mirror M1 and the right-hand side replaces M2. By
anticipating the properties of the ring resonator, we
allow for different transmission coefficients for waves
traveling to the right (s°+, toward PCMA) and to the left
(so_, toward PCMB). Provided that there is no back-
scattering from the normal mirrors of the ring, the
complex mirror has zero reflectivity (that is, no light is
reflected back toward the PCM's). The assumed field
is divided into the two regions on either side of the
complex mirror:
E(z, t) = [A, exp(ikvz) + B, exp(-ikpz)]
X exp[-i(vo + bv)t] + [AL exp(ikLZ) + BL
X exp(-ikLz)] exp[-i(vo - ev)t],
z > O, (9)
with a similar expression for z < 0. The field is subject
to the appropriate boundary conditions at the four
mirror surfaces. In order to find the eigenfrequencies
for this arrangement of mirrors, the boundary require-
ments are combined, and they provide two separate
conditions:
26kL + sp+(vo + P) - p_(o- 5v) = 2nir, (1Oa)
23kL + sp-(vo + v) - p+(vo- 5v) = 2n7r. (lOb)
In order for field amplitudes Au and BL to nonzero, Eq.
(1Oa) must be satisfied. In order for amplitudes AL and
BU to be nonzero, Eq. (lOb) must hold. (There are
constraints on the gains as well, but we are interested
here in the eigenfrequencies.) In general, the mode-
frequency spectrum will be rather complicated, as it is
for the hybrid resonator of Fig. la. We focus on the
rotation sensitivity of this device by supposing that the
pump frequency of the PCM's is tuned to a stationary
ring-cavity resonance and center our attention around
frequencies near this resonance (n = 0). If the ring is
stationary, then both Eqs. (lOa) and (lOb) are satisfied
for bv = 0. When the ring is rotating, the resonance
frequencies for oppositely directed traveling waves
shift:
W+ = COO T as, (11)
where As is given by the Sagnac formula":
4xrA (12)
where A is the area enclosed by the ring, P is the pe-
rimeter of the ring, X is the wavelength, and Q is the
rotation rate in radians per second. Thus the trans-
mission phases become
so+(v)= sno(p - 6O), (13a)
so_{v) = YPo(" + act), (13b)
where
po0 (v) = arg{t) = arg f1 - pi2ei
and E is given by Eq. (7) with 2L1 replaced by P, the
perimeter of the ring. loo is a sharply varying function
of frequency near resonance for a ring cavity of high
finesse. In this case, we can drop the first term ap-
pearing on the right-hand sides of Eqs. (1Oa) and (lob).
If Q is defined to be positive for counterclockwise
rotation of the ring in Fig. 1c, then for Q > 0 only con-
dition (lOb) can be satisfied, and for Q < 0 only condi-
tion (1Oa) can be satisfied. In both cases, the required
conditions hold for 6v = 5co. Thus the eigenmodes of
this coupled system are oppositely directed traveling
waves having a difference frequency equal to that for
a normal ring resonator.
We have discussed three hybrid coupled-resonator
structures. The mode spectra of the coupled systems
were shown to depend on the normal resonator optical
length. Although the first of the three systems con-
sidered has some interesting aspects, the other two
should probably be considered more practical for ap-
plications in resonant interferometry. In each of the
hybrid configurations discussed, we specialized to the
case in which the pump frequency was tuned to normal
cavity resonance or vice versa. To do this in an active
way so that the pump and the resonator remain tuned
together for an indefinite length of time requires some
additional optics (and electronics), which were omitted
from both the figures and the discussion.12 In practical
applications, then, these hybrid configurations used as
interferometers become essentially equivalent to their
normal counterparts wherein the PCM and its pump are
replaced by a source laser. The distortion correcting,
aligning, and self-oscillation properties of the PCM have
yet to be demonstrated experimentally. Therefore
whether the hybrid interferometer has any real ad-
vantages to offer remains to be seen.
The author thanks R. W. P. Drever for his enthusi-
astic support. This research was supported by National
Science Foundation grant NSF PHY8204056.
References
1. J. F. Lam and W. P. Brown, Opt. Lett. 5, 61 (1980).
2. J. Feinberg and R. W. Hellwarth, Opt. Lett. 5,519 (1980);
erratum, Opt. Lett. 6, 257 (1981).
3. R. C. Lind and D. G. Steel, Opt. Lett. 6, 554 (1981).
4. J. 0. White and A. Yariv, Opt. Eng. 21, 224 (1982).
5. J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE
J. Quantum Electron. QE-15, 1180 (1979).
6. P. A. Belanger, Opt. Eng. 21, 266 (1982).
7. D. M. Pepper and A. Yariv, in Optical Phase Conjuga-
tion, R. A. Fisher, ed. (Academic, New York, 1982), pp.
42-48.
8. M. Born and E. Wolf, Principles of Optics (Pergamon,
New York, 1970), pp. 323-341.
9. In Eqs. (6), (7), and (13) we have omitted, for clarity, some
inconsequential constant phases.
10. M. B. Spencer and W. E. Lamb, Jr., Phys. Rev. A 5,893
(1972).
11. E. J. Post, Rev. Mod. Phys. 39, 475 (1967).
12. For optical frequency-locking techniques, see, for exam-
ple, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough,
G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31,
97 (1983).


Coupled resonators employing phase-conjugating and ordinary mirrors

September 1984 / Vol. 9, No. 9 / OPTICS LETTERS 417
Coupled resonators employing phase-conjugating and ordinary
mirrors
Dana Z. Anderson
Division of Physics, Mathematics, and Astronomy, California Institute of Technology 130-33, Pasadena, California 91125
Received February 9, 1984; accepted May 30, 1984
We calculate the oscillation conditions and the eigenfrequencies for phase-conjugating-resonator-normal-resona-
tor coupled optical systems. With an eye toward applications to interferometry, we choose specific examples for
which it is shown that the conditions for oscillation and the eigenfrequencies depend on the normal-resonator path
length. The examples include both linear displacement and rotation sensing (Sagnac) resonant interferometers.
Our results suggest that if the distortion-correcting and self-aligning properties of the phase-conjugating resonator
are retained in the more complicated system, then these hybrid resonators may offer some advantages over their
conventional counterparts.
Optical resonators with phase-conjugating mirrors
(PCM's) utilizing four-wave mixing have demonstrated
some fascinating properties, such as the correction of
intracavity phase aberrationsl-3 and self-alignment.2' 4
However, the fundamental resonance frequency of the
phase-conjugating resonators (PCR's) demonstrated
so far are insensitive to resonator length.lA7 This
feature does not make them a good choice for dis-
placement sensors. The primary motivation for this
research stems from a desire to incorporate the useful
characteristics of the PCR into the realm of conven-
tional resonant interferometry.
To this end we study the consequences of coupling
phase-conjugating and normal resonators. Specifically,
we calculate the eigenfrequencies and the oscillation
conditions for three such hybrid structures. These
structures are shown in Fig. 1. The first (Fig. la) is the
simplest configuration (in terms of the number of mir-
rors). Here, the PCR formed by the PCM and M1 is
coupled through the transmission of M1 to the normal
resonator formed by M1 and M2. This structure reveals
an unusual mode-frequency spectrum. In Fig. lb the
PCR formed by the PCM and M2 is coupled to the
folded Fabry-Perot resonator formed by M1, M2, and
M3. In Fig. Ic the PCR formed by PCMA and PCMB
is coupled to the Sagnac (or ring) resonator formed by
M1 , M2, M3 , and M4 . In each case we derive the
eigenfrequencies and the oscillation conditions for the
structure by replacing all the normal mirrors by a single
complex mirror whose electric-field amplitude reflection
and transmission coefficients are strongly dependent
on the incident-wave frequency (Fig. id). Thus the
problem is reduced to that of a single PCR having one
complex mirror and one or two PCM's.
In the following discussion it is assumed that all the
normal mirrors are lossless. The PCM's are pumped
by fields having frequency Po, and it is presumed that
they provide gain, i.e., they conjugate and amplify an
incident wave.5' 7 We shall neglect the transverse-mode
distribution, that is, we shall suppose that the modes are
plane waves so that the electric field between the mir-
rors in Fig. Id satisfies the one-dimensional free-space
wave equation:
02E(z, t) 1 82E(z, t) =
0z 2 C2 0t2 (1)
subject to the appropriate boundary conditions at the
PCM and at the complex mirror. In Eq. (1), z is a
coordinate along the optical path and c is the speed of
light. The nature of phase conjugation by means of
four-wave mixing is such that an incident wave having
frequency vo + 3v is reflected with frequency vo - 3v.7
With this in mind one can guess that an eigenmode may
consist of fields having two frequencies that are sym-
metrically located about the pump frequency. A gen-
eral eigenmode satisfying Eq. (1) would then have the
form
E(z, t) = [Au exp(ikuz) + Bu exp(-ikuz)]
X exp[-i(vo + ev)t]
+ [AL exp(ikLZ) + BL exp(-ikLZ)I
X exp[-i(vo- -P)t], (2)
with kuL = (vo ± 3v)/c. Associated the PCM is a gain
g(Qv) that may be dependent on the incident-wave
frequency but that, we will suppose, is symmetric about
the pump frequency so thatg(bv) = g(-bv). Then the
PCM located at z = 0 imposes the following boundary
conditions:
AL = g(bv)B u,
Au = g(bw)BL.
(3a)
(3b)
Here, we have considered the PCM to be ideal and not
to induce any additional phase shifts of its own.8'9 The
generic complex mirror has electric-field amplitude
reflection and transmission coefficients
r(v) = p(v) exp[ir(v) ,
t(v) = r(v) exp[ipt(v)],
(4a)
(4b)
0146-9592/84/090417-03$2.00/0 © 1984, Optical Society of America
418 OPTICS LETTERS / Vol. 9, No. 9 / September 1984
,0
C
( VO
PC MA
IMI I
L2 ,;K LI 
L - E
b 1/0 d v/,
POM M, M 2 PC
e-L 2 ~4 M3
subject to the energy-conserving requirement that p 2
+ 'r2 = 1. Thus the mirror at z = L imposes the
boundary conditions
BU = p(v0 + bz)Au expli[2huL + Yor(vo + tiv)]},
BL = p(O - 6P)AL expli[2kLL + 'pr(Vo - bp)]I.
(5a)
(5b)
In order for the field amplitudes to meet the boundary
conditions [Eqs. (3) and (5)], two conditions must be
satisfied:
(I) g2 (bV)p(v0 + bv)p(Vo - by) = 1
and
(II) 49(po + by) - y(vo - by) + 4bkL = 2nr
(n = 0, ±1, 2, ...
where 6k = by/c. Condition (I) is a familiar require-
ment for sustained oscillation; the saturated gain sup-
plied by the gain medium must equal the resonator
losses. Condition (II) says that the round-trip accu-
mulated phase must be an integer multiple of 2wr. This
condition determines the eigenfrequencies of the reso-
nator. Condition (II) is always satisfied for by = 0. As
an example, consider a simple mirror, that is, one with
p and 'Pr independent of frequency. A mode always
occurs at the pump frequencyvo. In addition, according
to condition (II), there will be side modes flanking this
fundamental mode with a frequency spacing of ni7rc/2L.
This is in agreement with previous findings.3 -6
Now consider the coupled resonator of Fig. la. Let
the intensity reflection coefficient of M2 be P22 = 1.
One can find the electric-field reflection coefficient for
the complex mirror formed by the pair by assuming an
incident field on M1 and summing contributions from
successive round trips within the resonator8' 9:
rao') 1W - piei Pi, (6)
where
E = (v - ao)(2L1/c) (7)
and a,0 is a resonance frequency of the two-mirrored
cavity. The magnitude is Irol = po = 1, and the phase
fCOMPLEX
A MIRROR
I
Z = L
Fig. 1. Coupled resonator configura-
tions. a, PCR coupled to a Fabry-Perot
resonator; b, PCR coupled to a folded
Fabry-Perot resonator; c, PCR coupled
to a ring resonator; d, PCR coupled to a
normal resonator represented as a
complex mirror.
'pr (z) varies rapidly near resonance. Now suppose that
the resonator length L1 is tuned to the pump frequency
(or vice versa), that is, vo = a0 . Condition (II) will be
satisfied not only for by = 0 but also for a value near 0.
The case in which L1 = L2 E L is identical with that
treated by Spencer and Lamb for a normal coupled
resonator.10 In this case, condition (II) is satisfied by
a frequency
bv - rc/2L.
Unlike the normal coupled resonator, however, the
frequency spectrum of this hybrid resonator is charac-
terized by a frequency triplet in the region of the pump
frequency and by doublets spaced nwc/2L from the
pump frequency. (The spectrum is more complicated
if L1 FD L2.) Notice that the splitting frequency TCI2L
is dependent on both L2 and L1. Thus a change in the
PCM distance from M1 will cause a change in the
splitting frequency.
The folded normal resonator of Fig. lb can also be
replaced by a single complex mirror. We set the
reflectivities of M2 and M3 to unity. The reflectivity
(back toward the PCM) and transmission (away from
the PCM and up in the figure) of the complex mirror
are
(8a)r( - T12
t(v) = P1 1 Tl2 ei -E 1)1 pi2e ie
where e is now (v - wo)(2L 1 /c) and L1 is the unfolded
cavity length. Notice that the reflectivity magnitude
reaches one only at resonance. Therefore the PCM can
sustain oscillations only when the pump frequency is at
or near the cavity resonance frequency. In this sense,
the present configuration behaves like a normal reso-
nator with an internal atomic gain medium in which the
cavity length must be tuned to the atomic resonance in
order for lasing action to occur.
As a final example we treat the case of two PCM's
coupled to a ring interferometer in order to reveal the
rotation-sensing properties of the interferometer.'1 Ml
and M2 are considered identical, and M3 and M4 have
(8b)
1 I
September 1984 / Vol. 9, No. 9 / OPTICS LETTERS 419
unit reflection coefficients. Again the normal mirrors
forming the ring are represented as a single complex
mirror located at z = 0 flanked by PCM's at z = +L/2.
The left-hand side of the complex mirror represents
mirror M1 and the right-hand side replaces M2. By
anticipating the properties of the ring resonator, we
allow for different transmission coefficients for waves
traveling to the right (s°+, toward PCMA) and to the left
(so_, toward PCMB). Provided that there is no back-
scattering from the normal mirrors of the ring, the
complex mirror has zero reflectivity (that is, no light is
reflected back toward the PCM's). The assumed field
is divided into the two regions on either side of the
complex mirror:
E(z, t) = [A, exp(ikvz) + B, exp(-ikpz)]
X exp[-i(vo + bv)t] + [AL exp(ikLZ) + BL
X exp(-ikLz)] exp[-i(vo - ev)t],
z > O, (9)
with a similar expression for z < 0. The field is subject
to the appropriate boundary conditions at the four
mirror surfaces. In order to find the eigenfrequencies
for this arrangement of mirrors, the boundary require-
ments are combined, and they provide two separate
conditions:
26kL + sp+(vo + P) - p_(o- 5v) = 2nir, (1Oa)
23kL + sp-(vo + v) - p+(vo- 5v) = 2n7r. (lOb)
In order for field amplitudes Au and BL to nonzero, Eq.
(1Oa) must be satisfied. In order for amplitudes AL and
BU to be nonzero, Eq. (lOb) must hold. (There are
constraints on the gains as well, but we are interested
here in the eigenfrequencies.) In general, the mode-
frequency spectrum will be rather complicated, as it is
for the hybrid resonator of Fig. la. We focus on the
rotation sensitivity of this device by supposing that the
pump frequency of the PCM's is tuned to a stationary
ring-cavity resonance and center our attention around
frequencies near this resonance (n = 0). If the ring is
stationary, then both Eqs. (lOa) and (lOb) are satisfied
for bv = 0. When the ring is rotating, the resonance
frequencies for oppositely directed traveling waves
shift:
W+ = COO T as, (11)
where As is given by the Sagnac formula":
4xrA (12)
where A is the area enclosed by the ring, P is the pe-
rimeter of the ring, X is the wavelength, and Q is the
rotation rate in radians per second. Thus the trans-
mission phases become
so+(v)= sno(p - 6O), (13a)
so_{v) = YPo(" + act), (13b)
where
po0 (v) = arg{t) = arg f1 - pi2ei
and E is given by Eq. (7) with 2L1 replaced by P, the
perimeter of the ring. loo is a sharply varying function
of frequency near resonance for a ring cavity of high
finesse. In this case, we can drop the first term ap-
pearing on the right-hand sides of Eqs. (1Oa) and (lob).
If Q is defined to be positive for counterclockwise
rotation of the ring in Fig. 1c, then for Q > 0 only con-
dition (lOb) can be satisfied, and for Q < 0 only condi-
tion (1Oa) can be satisfied. In both cases, the required
conditions hold for 6v = 5co. Thus the eigenmodes of
this coupled system are oppositely directed traveling
waves having a difference frequency equal to that for
a normal ring resonator.
We have discussed three hybrid coupled-resonator
structures. The mode spectra of the coupled systems
were shown to depend on the normal resonator optical
length. Although the first of the three systems con-
sidered has some interesting aspects, the other two
should probably be considered more practical for ap-
plications in resonant interferometry. In each of the
hybrid configurations discussed, we specialized to the
case in which the pump frequency was tuned to normal
cavity resonance or vice versa. To do this in an active
way so that the pump and the resonator remain tuned
together for an indefinite length of time requires some
additional optics (and electronics), which were omitted
from both the figures and the discussion.12 In practical
applications, then, these hybrid configurations used as
interferometers become essentially equivalent to their
normal counterparts wherein the PCM and its pump are
replaced by a source laser. The distortion correcting,
aligning, and self-oscillation properties of the PCM have
yet to be demonstrated experimentally. Therefore
whether the hybrid interferometer has any real ad-
vantages to offer remains to be seen.
The author thanks R. W. P. Drever for his enthusi-
astic support. This research was supported by National
Science Foundation grant NSF PHY8204056.
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Coupled resonators employing phase-conjugating and ordinary mirrors

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