Post on 18-Feb-2016
description
Stability and Dynamics in Fabry-Perot cavities due to combined photothermal and
radiation-pressure effects
Francesco Marino1, Maurizio De Rosa2, Francesco Marin3
1 Istituto Nazionale di Fisica Nucleare, Firenze 2 CNR-INOA, Sezione di Napoli
3 Dip. di Fisica, Università di Firenze and LENS
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Radiation-Pressure EffectsRadiation-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
MultiStability
P. Meystre et al.J. Opt. Soc. Am. B, 2, 1830, (1985)
The Photothermal EffectThe 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)
M. Cerdonio et al. Phys. Rev. D, 63, 082003, (2001)M. De Rosa et al. Phys. Rev. Lett. 89, 237402, (2002)
Typical thermalfrequency
Frequency Response of length variations
Self-Oscillations: Physical MechanismSelf-Oscillations: Physical Mechanism
P
P
1. The radiation pressure tends to increase the cavity lenght respect to the cold cavity value the intracavity optical power increases
Optical injection on the long-wavelength side of the cavity resonance
2. The increased intracavity power slowly varies the temperature of mirrors heating induces a decrease of the cavity length
through thermal expansion
Interplay between radiation pressure and photothermal effect
Physical ModelPhysical Model
Limit of small displacements Damped oscillator forced by the intracavity optical power
Radiation Pressure Effect
Photothermal Effect
Intracavity optical power
Single-pole approximationThe temperature relaxes towards equilibrium at a rate and Lth T
Adiabatic approximationThe optical field instantaneously follows the cavity length variations
L(t) = Lrp(t) + Lth(t)We write the cavity lenght variations as
Physical modelPhysical model
Stationary solutions
The stability domains and dynamics depends on the type of steady states bifurcations
Depending on the parameters the system can have either one or three fixed points
BistabilityBistability
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
8 / 3 3
Changes in the control parameters canproduce abrupt jumps between the stable states
The distance between the stable and the unstablestate defines the local stability domain
Resonance condition
-8 / 3 3
Stability domainwidth
Hopf BifurcationHopf Bifurcation
They admit nontrivial solutions K et for eigenvalues given by
Single steady state solution: Stability Analysis
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 BoundaryHopf Bifurcation Boundary
Linear stability analysis is valid in the vicinity of the bifurcationsFar from the bifurcation ?
Q = 1, = 0.01, =4 Pin , =2.4 Pin For sufficiently high powerthe steady state solution losesstability in correspondence of a critical value of 0
Further incresing of 0 leads to the “inverse” bifurcation
Relaxation oscillationsRelaxation oscillations
We consider = 0 and 1/Q »
By linearization we find that the stability boundaries (F1,2) are given by C = -1
()
Evolution of is slow
Fast Evolution
small separation of the system evolution in two time scales: O(1) and O()
Fixed Points
Relaxation oscillationsRelaxation oscillations
By means of the time-scale change = t andputting = 0
G=0
Fixed point, p
If () > p dt < 0 If () < p dt > 0
Slow Evolution
Defines the branches of slow motion(Slow manifold)
At the critical points F1,2 the systemistantaneously jumps
Numerical ResultsNumerical Results
Q = 1Temporal evolution of the variable and corresponding phase-portraitas 0 is varied
Far from resonance:Stationary behaviour
In correspondence of 0c:
Quasi-harmonic Hopf limitcycle
Further change of 0:Relaxation oscillations,reverse sequence and a newstable steady state is reached
Numerical ResultsNumerical Results
Q > 1Temporal 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 ResultsNumerical Results
Between the self-oscillation regime and the stable state Chaotic spiking
The competition between Hopf-frequency and damping frequency leads to aperiod-doubling route to chaos
Stability Domains: Oscillatory behaviourStability Domains: Oscillatory behaviour
Q » 1
Stability domains for Q=1000,10000,100000
Stability domain in presence of the Hopf bifurcation
For high Q is critical
Future PerspectivesFuture 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