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

)cos( dtd

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