Transcript Stability and dynamics in Fabry-Perot cavities due to

Stability and Dynamics in Fabry-Perot
cavities due to combined photothermal and
radiation-pressure effects
Francesco Marino1, Maurizio De Rosa2, Francesco Marin3
1
2
3
Istituto Nazionale di Fisica Nucleare, Firenze
CNR-INOA, Sezione di Napoli
Dip. di Fisica, Università di Firenze and LENS
Radiation-Pressure Effects
Kerr Cavity: The optical path depends on the optical
intensity (via refractive index) that depends on the
optical path in the cavity
Radiation-Pressure-driven cavity : Radiation pressure
changes the cavity length, affecting the intracavity
field and hence the radiation pressure itself
d
    cos( )
dt
MultiStability
P. Meystre et al.
J. Opt. Soc. Am. B, 2, 1830, (1985)
The Photothermal Effect
The intracavity field changes the temperature of the mirrors via residual
optical absorption
Cavity length variation through thermal expansion (Photothermal Effect)
Experimental Characterization: P = Pm + P0sin( t)
Typical thermal
frequency
Frequency
Response of
length
variations
M. Cerdonio et al. Phys. Rev. D, 63, 082003, (2001)
M. De Rosa et al. Phys. Rev. Lett. 89, 237402, (2002)
Self-Oscillations: Physical Mechanism
Interplay between radiation pressure and photothermal effect
Optical injection on the long-wavelength side of the cavity resonance
1.
The radiation pressure tends to increase the cavity lenght respect to
the cold cavity value the intracavity optical power increases
2.
The increased intracavity power slowly varies the temperature of mirrors
 heating induces a decrease of the cavity length through
thermal expansion
P
P
l
l
Physical Model
We write the cavity lenght variations as
L(t) = Lrp(t) + Lth(t)
Radiation Pressure Effect
Limit of small displacements
Damped oscillator forced by the intracavity optical power
Photothermal Effect
Single-pole approximation
The temperature relaxes towards equilibrium at a rate  and Lth T
Intracavity optical power
Adiabatic approximation
The optical field instantaneously follows the cavity length
variations
Physical model
The stability domains and dynamics depends on the type of steady states bifurcations
Stationary solutions
Depending on the parameters the system can have either one or three fixed points
Bistability
Analyzing the cubic equation for  
On this curve two new fixed points (one stable and the other unstable)
are born in a saddle-node bifurcation
Changes in the control parameters can
produce abrupt jumps between the stable states
Resonance condition
The distance between the stable and the unstable
state defines the local stability domain
Stability domain
width
8/3 3
-8 / 3
3
Hopf Bifurcation
Single steady state solution: Stability Analysis
They admit nontrivial solutions K et for eigenvalues  given by
Hopf Bifurcation: The steady state solution loses stability in correspondence of a
critical value of 0 (other parameter are fixed) and a finite frequency limit cycle
starts to grow
Re  = 0 ; Im  = i 
Hopf Bifurcation Boundary
For sufficiently high power
the steady state solution loses
stability in correspondence of a
critical value of 0
Q = 1,  = 0.01, =4 Pin , =2.4 Pin
Further incresing of 0 leads to
the “inverse” bifurcation
Linear stability analysis is valid in the vicinity of the bifurcations
Far from the bifurcation ?
Relaxation oscillations
 small  separation of the system evolution in two time scales: O(1) and O()
We consider  = 0 and 1/Q » 
Evolution of  is slow
Fast Evolution
Fixed Points
()
By linearization we find that the stability
boundaries (F1,2) are given by C = -1
Relaxation oscillations
Slow Evolution
By means of the time-scale change  =  t and
putting  = 0
Defines the branches of slow motion
(Slow manifold)
Fixed point, p
If () > p  dt < 0
If () < p  dt > 0
G=0
At the critical points F1,2 the system
istantaneously jumps
Numerical Results
Q=1
Temporal evolution of the  variable and corresponding phase-portrait
as 0 is varied
Far from resonance:
Stationary behaviour
In correspondence of 0c:
Quasi-harmonic Hopf limit
cycle
Further change of 0:
Relaxation oscillations,
reverse sequence and a new
stable steady state is reached
Numerical Results
Q>1
Temporal evolution of the  variable (in the relaxation oscillations regime) and
corresponding phase-portrait as Q is incresed
Q= 5
Q=10
Q=20
Relaxation oscillations with damped oscillations when jumps between the stable
branches of the slow manifold occurs
Numerical Results
The competition between Hopf-frequency and damping frequency leads to a
period-doubling route to chaos
Between the self-oscillation regime
and the stable state  Chaotic spiking
Stability Domains: Oscillatory behaviour
Stability domain in presence of the Hopf bifurcation
Q»1
For high Q is critical
Stability domains for Q=1000,
10000,100000
Future Perspectives
•
Experiment on the interaction between radiation pressure and photothermal effect
•
The experimental data are affected by noise
 Theoretical study of noise effects in the detuning parameter
•
Time Delay Effects (the time taken for the field to adjust to its equilibrium value)
 Extend the model results to the case of gravitational wave detectors