PHYSICAL REVIEW A 80, 033806 共2009兲
Splitting of atom-cavity polariton peaks for three-level atoms in an optical cavity
Haibin Wu, J. Gea-Banacloche, and Min Xiao*
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
共Received 12 May 2009; published 3 September 2009兲
We explore the high-density, nonlinear regime of atom-cavity coupling for a gas of three-level atoms at high
temperature inside an optical cavity, interacting with the cavity field and a strong driving field. We find a region
of parameters where the two side peaks in the transmission spectrum 共which, in the linear regime, are associated with “bright polaritons”兲 split into two pairs of peaks. Experimental results are presented along with a
theoretical model that indicates that this phenomenon is unique to the three-level system in the large Dopplerbroadening regime.
DOI: 10.1103/PhysRevA.80.033806
PACS number共s兲: 42.50.Pq, 42.50.Nn, 32.80.Qk
I. INTRODUCTION
When an ensemble of two-level atoms interacts with a
cavity field, the cavity transmission spectrum can show a
double-peak structure when the coupling strength, measured
as 冑Ng, between the atoms and the cavity field is larger than
the atomic decay rate ␥ and cavity decay rate ␬ 关1兴 共Here, g
is the single atom-cavity coupling strength and N the number
of atoms inside the cavity mode volume.兲 This double-peak
structure has been called “vacuum Rabi splitting” or
“normal-mode splitting” in the literature. The frequency difference between the two peaks is 2冑Ng when both the atoms
and the cavity are on resonance with the laser field. In order
to overcome the Doppler effect, this normal-mode splitting
phenomenon has always been studied either in an atomic
beam 关2–4兴 or in a cold atomic cloud 关5兴. Recently, we have
shown that such a double-peak transmission spectrum can
even be observed in a Doppler-broadened atomic vapor, with
a relatively stronger intracavity field intensity, to saturate the
atomic absorption within the Doppler profile 关6兴.
When three-level atoms are placed inside the optical cavity 共i.e., an additional coupling laser beam is used to interact
with another transition, sharing a common level with the first
one兲, three-peak transmission spectra have been observed,
both in cold atoms 关7兴 and in a hot atomic cell 关8兴, with the
narrow central transmission peak being due to electromagnetically induced transparency 共EIT兲. The narrowness of this
central peak results from the use of a two-photon Dopplerfree configuration in the optical ring cavity 关9兴.
It was shown in 关8兴 that, at relatively low atomic densities
and moderate intracavity field intensities, the three-peak
transmission structure is stable, with the side-peaks moving
outwards as the temperature 共atomic density兲 or coupling
laser beam power increases. At a moderate intracavity field
intensity 共cavity input power of 0.8 mW, corresponding to a
Rabi frequency at the beam waist of 2␲ ⫻ 11 MHz, which is
about 0.65 times the cavity linewidth.兲, the atomic absorption
is saturated, but the dispersion is still in a “quasilinear” region, a phenomenon pointed out in 关8兴 and analyzed theoretically 共for a two-level system兲 in 关6兴.
In this work, we show that if the atomic cell temperature
is set higher 共above 73 ° C, corresponding to the atomic den-
*[email protected]
1050-2947/2009/80共3兲/033806共6兲
sity of 7 ⫻ 1011 cm3 or a single-pass absorption 共optical
depth兲 of 29.兲 with a higher intracavity field intensity 共cavity
input power larger than 1.5 mW兲, each normal-mode side
peak can split into two peaks as the temperature increases.
One of the split peaks moves outwards following the typical
normal-mode relation as given in Ref. 关8兴 with the temperature increase, and another one moves inwards at the same
time. The inward moving peaks 共one from each side兲 get
smaller as they move near the central peak, and eventually
disappear due to high absorption, so that the spectrum shows
the typical three-peak structure again. Since the absorption
and dispersion of the intracavity atomic medium depend sensitively on experimental parameters, such as temperature
共atomic density兲, intracavity intensity, and coupling beam intensity, the splitting of the normal-mode peaks only occurs in
certain parameter regions.
Hot atomic vapors are much easier to use and manipulate
than cold gases, without the need for a vacuum system, and
can easily reach high atomic densities. Macroscopic cavity
systems such as ours may be of interest for a number of
applications, such as atomic memories for quantum information processing. It is, therefore, important to achieve as full a
characterization, and understanding, of the properties of
these systems as possible, in all the different regimes. Moreover, as we shall show below, the fully nonlinear regime
investigated in this paper exhibits signs of optical bistability,
which is an interesting effect in itself and may also be associated with nonclassical properties of light, such as squeezing, that can have useful applications as well. In the following, we first describe our experimental observation of the
previously unobserved peak-splitting phenomenon, and then
present a simple qualitative theoretical explanation for it.
II. EXPERIMENTAL SETUP AND OBSERVATIONS
The experimental setup is shown in Fig. 1. Two commercial high-power CW diode lasers are used as the cavity input
共probe兲 and the coupling laser beams, with linewidths of
about 1 MHz. The input mirror M1 and output mirror M2 of
the ring cavity have 3% and 1.4% transmissivities, respectively, and M3 is a high reflector mounted on a PZT for
cavity frequency scanning and locking. The cavity length L
is about 37 cm. The rubidium vapor cell is l = 5 cm long with
Brewster windows, and is wrapped in ␮-metal sheets for
033806-1
©2009 The American Physical Society
PHYSICAL REVIEW A 80, 033806 共2009兲
APD
M2
P
C
|a>
Coupling
Beam
|c>
Coupling
Laser
Probe
Beam
|b>
0.4
M3
Probe
Laser
Rb cell
M1
Cavity transm
ission (arb. un
its
)
WU, GEA-BANACLOCHE, AND XIAO
PZT
PBS
0.0
FIG. 1. 共Color online兲 Experimental setup. PBS, polarizing cubic beam splitters; M1-M3, cavity mirrors; APD, avalanche photodiode detector; and PZT, piezoelectric transducer.
P (m
p
W)
2.0
-200
0
Prob
e det
uning
∆ (M
p
Hz)
200
2.2
FIG. 2. 共Color online兲 Cavity transmission spectra for different
cavity input power. The experimental parameters are ⌬c = 0, cavity
detuning ⌬␪ = 0, T = 73 ° C, and Pc = 7 mW.
powers, cavity transmission has the typical three-peak structure, but it shows a five-peak structure in a certain range of
cavity input powers, for a given atomic density 共temperature兲
and coupling laser power. The peak splittings are basically
symmetric on both sides of the transmission spectrum.
Next, the cavity input power is fixed at P P = 1.05 mW and
the coupling beam power at Pc = 7.4 mW. At T = 73 ° C, the
cavity transmission shows the typical three-peak structure.
As the cell temperature is increased from T = 73 ° C, the cavity transmission spectrum first shows a three-peak structure,
and then five peaks at T ⬇ 75.4 ° C, as shown in Fig. 4. Again
the inward peaks become smaller and eventually disappear as
they move closer to the middle due to larger absorption at
high atomic densities 共temperature兲. The outward peaks
200
Splitting frequency (MHz)
magnetic field shielding. The coupling beam is injected into
the cavity through the polarization beam splitter 共PBS兲 and
copropagates 共not circulating inside the cavity兲 through the
atomic vapor cell with the cavity field in order to eliminate
the first-order Doppler effect using the two-photon Dopplerfree configuration 关9兴. The beam waist radii are 100 and
600 ␮m for the probe and coupling beams, respectively. We
use a third diode laser to lock the cavity frequency 共not
shown in Fig. 1兲. The empty cavity finesse is about 100.
When the atomic cell is placed inside the optical cavity with
laser frequency tuned far away from the atomic resonances,
the cavity linewidth is increased to be about 17 MHz, corresponding to the cavity finesse of about 48. The Dopplerwidth of the atomic medium at the cell temperature of
67.7 ° C is about 535 MHz.
We first set the experimental conditions to be the same as
in 关8兴 in order to observe the three-peak transmission
spectrum with T = 67.7 ° C: the coupling beam power
Pc = 7 mW 共corresponding to a Rabi frequency of about
⍀ = 120 MHz兲, cavity input power P P ⬇ 0.8 mW 共at the
cavity input兲, and the coupling laser frequency detuning ⌬c
and cavity frequency detuning ⌬␪ 共from the atomic resonant
frequency兲 both equal to zero. When the temperature is increased to 73.5 ° C, the side peaks get much smaller due to
increased absorption. Larger cavity input power can compensate for and saturate such absorption to enhance the side
peaks. Figure 2 shows the transmission spectra for different
cavity input powers, for the given cell temperature. At the
large cavity input power 共first curve in Fig. 2 with
P P = 2.2 mW兲, the transmission spectrum is the typical threepeak structure, as in 关8兴. However, as the cavity input power
decreases, the side normal-mode peak on each side splits into
two, with one moving outward, similarly as the regular
normal-mode peak 关8兴, and another 共smaller one兲 moving
inward toward the center, as shown in the second and third
curves in the figure. This inward moving peak gets smaller as
it moves near the central peak where the absorption is large
共just outside the EIT window兲, and eventually disappears, as
shown in the last curve of Fig. 2.
The frequencies of the split peaks are plotted as a function
of the probe input power in Fig. 3, where the splitting frequency is measured from the frequency of the maximum
transmission of the side-peaks to the frequency of the central
peak. One can see that at both lower and higher cavity input
1.8
Pc=7.0 mW
T=73.5 0C
100
0
-100
-200
2
4
Probe power (mW)
FIG. 3. 共Color online兲 Splitting frequencies of the cavity transmission peaks as a function of the cavity input power. The experimental parameters are the same as in Fig. 2.
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dispersion and absorption(MHz)
Splitting frequency (MHz)
SPLITTING OF ATOM-CAVITY POLARITON PEAKS FOR…
Pc=7.4 mW
Pp=1.4 mW
0
-100
increasing coupling field strength
increasing (intracavity) probe strength
increasing atomic density
74
76
78
Temperature 0C
FIG. 4. 共Color online兲 Splitting frequencies of the cavity
transmission peaks as a function of the atomic cell temperature
共atomic density兲. The experimental parameters are: ⌬c = 0, ⌬␪ = 0,
Pc = 7.4 mW, and P P = 1.4 mW.
again follow the typical normal-mode relation as described
in 关8兴.
Finally, we keep the cell temperature and the cavity input
power fixed at T = 73 ° C and P P = 1.04 mW, respectively,
and change the coupling laser power. At Pc = 7 mW, which
is the value used in Figs. 3 and 4, only the typical three-peak
structure appears in the cavity transmission spectrum. When
the coupling beam power is increased from this point, only
the three-peak structure shows, following the normal-mode
relation as described in 关8兴. However, as Pc decreases to
around 5 mW, again a five-peak structure shows up in the
cavity transmission spectrum, as shown in Fig. 5. Such fivepeak spectra only exist for a small range of the coupling
beam powers. When the coupling beam power was set to
zero 共which is the two-level atom case兲 a five-peak cavity
Splitting frequency (MHz)
100
T=73.5 0C
Pp=1.05 mW
0
-100
FIG. 6. 共Color online兲 The medium dispersion 共␻0l / 2L兲␹⬘ 共solid
red line兲 and the negative of the probe detuning ⌬ 共dashed blue line兲
versus ⌬ for a driven, Doppler-broadened, three-level medium. The
dotted green line shows the effect of decreasing the atomic density.
The dash-dotted purple line shows the medium absorption
共␻0l / 2L兲␹⬙.
transmission spectrum could not be observed in the large
parameter 共for temperature and cavity input power兲 ranges
available under our current experimental conditions.
III. THEORETICAL DISCUSSION
Our observations can be qualitatively understood as follows. In the general, nonlinear regime, the intracavity intensity I is obtained as the solution of the self-consistency equation 共see 关3兴兲,
I=
TIin
1 + r2e−␻0l␹⬙/c − 2re−␻0l␹⬙/2c cos关共⌬ + 共␻0l/2L兲␹⬘兲L/c兴
共1兲
where T is the transmissivity of the input mirror M1; Iin is
the input intensity; r ⬍ 1 is a parameter describing the losses
due to reflection and absorption at the various optical elements; and ␹ = ␹⬘ + i␹⬙ is the medium’s complex susceptibility, which in general depends on I and ⌬, the probe detuning
from the cavity 共and the atomic兲 resonance.
We recall that, as we have recently pointed out 关6,8兴, for a
Doppler-broadened medium the absorption saturates much
more strongly than the dispersion as the intracavity field increases. Hence we can expect to gain some insight into the
peak-splitting phenomenon starting from a consideration of
the dispersive properties of the medium only, as given by ␹⬘,
the real part of the susceptibility.
Neglecting absorption in the medium altogether, Eq. 共1兲
becomes
I=
0
8
16
Coupling power (mW)
FIG. 5. 共Color online兲 Splitting frequencies of the cavity transmission peaks as a function of the coupling beam power. The experimental parameters are: ⌬c = 0, ⌬␪ = 0, T = 73.5 ° C, and
P P = 1.05 mW.
TIin
1 + r2 − 2r cos关共⌬ + 共␻0l/2L兲␹⬘兲L/c兴
共2兲
and one expects the transmission peaks to be given by the
maxima of cos关共⌬ + 共␻0l / 2L兲␹⬘兲L / c兴 in Eq. 共2兲, which is to
say 共since dispersion in our system is relatively weak, and
the only relevant maximum of the cosine function is the one
at zero兲, to correspond to values of ⌬ for which the difference between 共␻0l / 2L兲␹⬘ 共considered as a function of ⌬兲 and
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PHYSICAL REVIEW A 80, 033806 共2009兲
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2000
1500
1500
1000
x
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WU, GEA-BANACLOCHE, AND XIAO
x
500
0
50
1000
500
100
150
200
250
0
50
300
(MHz)
100
150
200
250
300
FIG. 7. 共Color online兲 Contour plots of the function f in Eq. 共3兲,
which correspond, except for a scaling, to the cavity transmission
curves predicted by the solutions of Eq. 共2兲 共for ease of visualization, the plot actually shows contours of log f兲. The horizontal
dashed line shows a typical cross-section for constant I; the crosses
mark minima of f as a function of ⌬ 共for constant I兲 which correspond to crossings such as those seen in Fig. 6 共see text for details兲.
The vertical dashed line shows how the transmission peak may
exhibit a sharp edge in the presence of bistability.
−⌬ is minimum. This would be unambiguous in the linear
regime, in which ␹⬘ does not depend on I, but, as we shall
show below, with some care this reasoning may still be used
to gain a qualitative understanding of the situation in the
nonlinear case as well.
Figure 6 shows, as a solid red line, the calculated dispersion function 共␻0l / 2L兲␹⬘ for a typical split-peak situation, in
which it crosses the dashed straight line −⌬ twice. 共For all
numerical plots in this section, we use a model described in
关8兴 that includes Doppler-broadening and treats both the coupling and probe field to all orders in field strength.兲 The plot
is for a fixed, constant I; increasing I generally causes 兩␹⬘兩 to
decrease, due to saturation of the medium; hence, eventually,
one would expect the crossings to disappear for sufficiently
large I, in qualitative agreement with the disappearance of
the peak splitting observed in Fig. 3.
To take into account the nonlinearity, note that Eq. 共2兲 can
be rewritten as
TIin = I共1 + r2 − 2r cos关共⌬ + 共␻0l/2L兲␹⬘兲L/c兴兲 ⬅ f共I,⌬兲
共3兲
where the last equality defines a function f of I and ⌬. Now,
the contour lines of f, if plotted with ⌬ on the horizontal
axis, are in fact, except for a scaling factor, the cavity transmission curves for different values of the input intensity Iin
共see Fig. 7兲. This is because every point on the contour corresponds to the value of the intracavity intensity I that satisfies Eq. 共2兲 for the given ⌬ and fixed Iin, and the cavity
transmission is obtained simply by multiplying I by the
FIG. 8. Contour plots of the function g in Eq. 共4兲, which correspond, except for a scaling, to the cavity transmission curves predicted by the solutions of the full Eq. 共1兲 共including the medium’s
absorption losses兲. As in Fig. 7, for ease of visualization, the plot
actually shows the contours of log g.
transmissivity of the output mirror, M2, and dividing by Iin,
which is constant across the contour.
As a function of ⌬, for constant I, the function f has
minima at the crossings identified in Fig. 6, that is, the points
where 共␻0l / 2L兲␹⬘ = −⌬, and if there are two crossings the
value of f is the same at both. Then, in a sufficiently small
neighborhood of each minimum, there must be two contour
lines 共i.e., four lines altogether兲 that belong to the same contour, and that means that the contour must exhibit two transmission peaks 共again, see Fig. 7兲. We conclude, then, that it
is possible to predict when a split peak will occur by studying the crossings in Fig. 6.
Note that the nonlinearity of the problem may result in
regions of dispersive bistability, such as those exhibited by
the transmission peak on the right-hand side of Fig. 7. The
vertical dashed line in the figure indicates the way the transmission might appear in an experiment, given the availability
of a stable lower transmission branch 共of course, the precise
point at which the jump from one branch to the other will
take place cannot really be predicted, given the unavoidable
noise in an experiment兲. Note that the sharp left edge of the
peak is, in fact, consistent with the experimental results seen
in Fig. 2.
It remains to be seen how the inclusion of medium absorption changes this picture. By analogy with the above, we
can define a function
g共I,⌬兲 = I共1 + r2e−␻0l␹⬙/c − 2re−␻0l␹⬙/2c
⫻cos兵关⌬ + 共␻0l/2L兲␹⬘兴L/c其兲,
共4兲
to go with Eq. 共1兲, and look at its contours. Figure 8 shows
this for the same parameters used for Fig. 7, namely,
⍀c = 100 MHz 共Rabi Frequency of the coupling beam兲 and
g冑N = 290 MHz 共recall g is the atom-cavity photon coupling
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SPLITTING OF ATOM-CAVITY POLARITON PEAKS FOR…
(a)
(b)
FIG. 9. 共Color online兲 The medium dispersion 共␻0l / 2L兲␹⬘ 共solid
red line兲 and the negative of the probe detuning ⌬ 共dashed blue line兲
versus ⌬ for 共a兲 a Doppler-broadened, two-level medium and 共b兲 a
driven, homogeneously-broadened three-level medium. The dotted
green line in 共a兲 shows the effect of decreasing the atomic density.
The dash-dotted purple line in 共b兲 shows the medium absorption
共␻0l / 2L兲␹⬙.
rate, and N the atomic density; see 关8兴 for further details of
our model兲. Clearly, the larger absorption at smaller ⌬ tends
to reduce 共or even completely eliminate兲 the small-⌬ transmission peak. Note that, for the case shown, the bistability of
the large- ⌬ peak persists.
With this background, we can use the simple picture provided by Fig. 6 to understand qualitatively the experimental
results, and in particular the effect of varying the experimental parameters. For instance, we can see, with reference to
that figure, that decreasing the atomic density naturally decreases ␹⬘, which eventually results in a dispersion curve
similar to the dotted green line, that no longer crosses the
blue −⌬ line; note that this still has a 共single兲 transmission
maximum for nonzero ⌬, corresponding to the point where
the green and blue lines are closest. Increasing the intracavity
field has an effect similar to decreasing N; i.e., it causes ␹⬘ to
decrease, due to saturation of the medium.
The dash-dotted purple line in Fig. 6 shows the medium
absorption rate, as given by 共␻0l / 2L兲␹⬙. It is clearly largest
in the neighborhood of the dressed-state resonance, ⌬ = ⍀ / 2
共the figure has been drawn for ⍀ = 90 MHz兲, where ␹⬘ = 0.
As indicated above, in connection with Fig. 8, this explains
why the transmission peak corresponding to the smaller
value of ⌬ is always smaller, and why eventually it disappears if either the density becomes too large 共Fig. 4兲 or the
intracavity field becomes too small 共Fig. 3兲: both of these
have the effect of increasing the overall absorption as well as
moving the small-⌬ peak closer to the high-absorption region. These observed features are related to the dramatic
modification of the linear 关9兴 and nonlinear 关10兴 dispersions
of the three-level atomic system, compared to the two-level
one, due to the induced atomic coherence by the coupling
laser beam in the EIT systems.
Figure 9共a兲 shows, for comparison, a similar plot for a
two-level medium 共no coupling field兲, such as the one considered in 关6兴 共it is, in fact, essentially the same as Fig. 2 of
关6兴 upside down兲. It shows that in this case it is only possible
to have one 共purely dispersive兲 side peak, since only one
crossing for ⌬ ⫽ 0 is possible, and if the curves do not cross
at all then the point where they are closest is at ⌬ = 0.
Figure 9共b兲 shows the situation for a three-level, homogeneously broadened medium 共e.g., a dense gas of cold atoms兲,
with a coupling field. The intracavity field has been assumed
to be, as in the other plots, about 20 times the saturation
intensity, which accounts for the large power-broadening of
the dressed-state resonance at ⍀ = ⌬ / 2. In this case, as in Fig.
6, there are two crossings, but one of them is always very
close to the high-absorption point ⌬ = ⍀ / 2 共again, the absorption is shown as the dash-dotted curve兲, which means
that this peak should be unobservable in practice. We conclude that the phenomenon of the splitting of the side
normal-mode peak is probably unique to the three-level,
Doppler-broadened system.
Finally, returning to Fig. 6, we may consider the effect of
changing the strength of the coupling field, ⍀. This mainly
shifts ␹⬘ to the right or left, by moving the dressed-state
resonance point ⌬ = ⍀ / 2. Clearly, for sufficiently large ⍀ this
will cause the crossings, and hence the splitting of the side
peak, to disappear, as we have observed in the experiment
共Fig. 5兲. On the other hand, for sufficiently small ⍀, the
small-⌬ split peak will be brought too close to the highabsorption region, and it will become unobservable in the
experiment. Hence, for this parameter also, as for the atomic
density and probe field, the split peak can only be observable
for a restricted range of values, as our experimental results
show.
We want to point out that such splitting of the cavity +
atoms normal modes only occurs for certain range of the
intracavity intensity. At either too low or too high cavity
input power, the splitting of the normal-mode peak would
not show up for any temperature or coupling laser strength.
This indicates that it is a genuine nonlinear effect, distinct
from the “quasilinear” regime explored in 关8兴. This is also
apparent from the shape of the split peaks in Fig. 2, which is
consistent with the underlying bistability revealed by the calculations in Figs. 7 and 8 共compare also with the similar
curves for the nonlinear two-level system in 关11兴兲. We have
not been able to observe hysteresis directly because of difficulties keeping the system stable, but it is well known that
nonclassical light effects, such as squeezing, can be observed
in systems exhibiting optical bistability 共see, e.g., 关12兴兲. It
would be interesting to explore whether this is the case also
in the present system.
IV. CONCLUSIONS
In summary, we have experimentally demonstrated the
splitting of the atom-cavity normal modes at certain values
of intracavity field intensity, coupling beam power, and
atomic density in a system of inhomogeneously-broadened
three-level atoms inside an optical ring cavity. Detailed parametric dependence of such split-peak structures has been ex-
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transverse patterns with high density atoms inside an optical
cavity.
We note that there has recently been a renewed interest in
the quantized-field version of these types of systems 共typically described by the Jaynes-Cummings model, or a suitable
extension兲 in the context of so-called “circuit QED.” In particular, a recent paper has reported a nonlinear splitting of the
“vacuum Rabi resonance” 关14兴 that exhibits a superficial resemblance to our results. We do not believe, however, that
the two results are directly related, in part because the splitting reported in 关14兴 was only visible via heterodyne detection and not by direct photon counting. Nonetheless, as
pointed out above, the search for possible quantum-field effects in our system remains an interesting possibility 共see
also 关15兴兲.
plored. The observed phenomena can be qualitatively explained by the interplay of the nonlinear dispersion and
absorption in the system, by considering the Doppler broadening and cavity field nonlinearity. Such normal-mode peak
splitting phenomenon seems to be a unique feature of the
three-level, inhomogeneously-broadened atomic system. It
would be interesting to explore the possibility of nonclassical
light effects in the neighborhood of this instability. We note
also that, because of the high density of the medium, the
average number of photons per atom is relatively low, so we
cannot in principle exclude the possibility of interesting
quantum-field effects underlying our observations 共see, e.g.,
关13兴兲. In the experiment, we sometimes observe higher-order
transverse modes, but at present we do not know whether
they are due to imperfect alignment of the cavity or to the
nonlinearity. We are currently working on a theoretical treatment taking into account the transverse profiles of the laser
beams that we hope may help us in answering these interesting questions on the formations of transverse instabilities and
Funding support from the National Science Foundation is
acknowledged.
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ACKNOWLEDGMENT
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