October 29th, 2007

This report focuses on the experimental observation
of mechanical oscillations in optical microcavities, induced by
radiation pressure in ultra-high-Q optical microdisk cavities (we refer
to refs. [1-6]). The coupling of energy back and forth from the optical
to the mechanical domain gives rise to a so-called* parametric
oscillation* in which a steady build-up of mechanical vibration
occurs when an optical threshold energy has been surpassed.

Fig. 1: SEM micrographs of optical microdisk
oscillators (K. De Greve, 2007-05-29). |

Optical microdisk oscillators, as shown in Fig. 1, are essentially semiconductor/dielectric disks of sizes in the micrometer-regime. They have been extensively used over the last 20 years in the fields of micro/nanophotonics and quantum optics (see [8] for a thorough review of the field of optical oscillators). Beyond their remarkable optical properties, which will be briefly discussed in the next section, they are also interesting from a mechanical point of view, especially when actuated in their vibrational eigenmodes.

When a microdisk oscillator interacts
with a radially symmetric mechanical force F(t) (*cf.* Fig. 2a), it
will
expand resp.contract under the influence of that force and its
effective stiffness, K_{eff}.

In view of the radial symmetry, an effective 1D-model can be developed, as shown in Fig. 2b, which can also account for the dynamic properties of such a system under the influence of the time-dependent force F(t):

Fig. 2: Radially symmetric mechanical vibration of a
microdisk resonator (a) and its 1D equivalent model
(b). |

Fig. 3: Lorentzian lineshape, amplitude response of a driven,
damped harmonic oscillator. |

in which r(t) stands for the deviation form
equilibrium (the radial displacement), F_{eff}
(F/m_{eff}) for the effective mass-renormalized actuation force,
γ_{0} for a mass-renormalized friction constant, and
Ω =(K_{eff}/m_{eff})^{1/2}, the ratio of
the effective stiffness constant K_{eff} and the effective mass
m_{eff}) is the resonance frequency of the system.

When driven by an oscillatory excitation at frequency
ω_{1} (possibly detuned from the resonance frequency
Ω), the steady state response of the system, after an initial
transient, will be at the driving frequency ω_{1}:

which comes down to the well-known Lorentzian
response, consisting of both a phase and amplitude part, the amplitude
of which peaks at ω_{1} = Ω as shown in Fig. 3. Note
that we introduced in that figure the constant Q
(=Ω/γ_{0}), commonly denoted as *quality
factor*. From the Lorentzian lineshape, the full width at half
maximum Δω equals Ω/Q, defining an effective bandwidth
of the system.

The operation principle of optical microdisk
oscillators is described in Fig. 4. Through evanescent coupling,
light from a waveguide is coupled into the microdisk, in which
(lossless) total internal reflection confines the light within the
microdisk. For frequencies ω_{0} equalling c/nR, with R
the radius of the microdisk, the circulating light constructively
interferes, leading to an effective build-up of optical energy.

The energy build-up due to the incoupling of light
from the input waveguide is eventually balanced by the losses in the
system, described by an effective quality factor Q (note that as the
losses due to intrinsic absorption and input/output coupling can be
considered additive to first order, the effective quality factor Q
equals (Q_{int}^{-1} + Q_{in}^{-1} +
Q_{out}^{-1})^{-1}). Advances in micro- and
nanofabrication have led to Q factors on the order of 10^{8} to
be realized experimentally [8]. By equating the losses from a
circulating optical power P_{circ} to the (net) input power, an
effective increase of the circulating power P_{circ} over the
input power P_{in} of Q/2π (> 10^{7}) can be
realized.

When excited with incoming light at frequency
ω_{1}, the transmitted output power will show an inverse
Lorentzian lineshape defined by a linewidth Δω =
ω_{0}/Q, which for optical frequencies, and with Q-factors
in the 10^{7}-10^{8} range leads to linewidths in the
(tens of) MHz range.

From Einstein's theory of relativity, we know that a
photon with energy E will have a momentum p = E/c. When reflecting off a
surface, a net momentum change of p_{out} - p_{in} is
exerted on the photon. In view of Newton's second law, an equal and
opposite momentum change will be exerted on the wall. Without losses,
p_{out} = -p_{in}, and the net impulse on the wall will
be 2p_{in} = 2E_{in}/c.

A light field impinging on a wall can be regarded as
a stream of photons bouncing into and off that wall. When the incoming
optical power P_{in} contains photons of energy E, the number of
photons impinging per second equals P_{in}/E. As the net impulse
(per photon) exerted on the wall is equal to 2E/c, the total impulse
exerted on the wall per unit time will be 2nP_{in}/c (we account
for a reduced speed of light by an index of refraction n). By
definition, the net impulse per unit time equals the force exerted on
the wall (see Fig. 5 a)).

In optical microdisk oscillators, total internal reflection confines the light inside the oscillator, effectively bouncing the photons off the sidewall of the oscillator. Upon circulating halfway through the oscillator, there is a net force on the sidewall, equal to the integrated force f per unit length (in view of the radial symmetry, this per-unit-length force is uniform over the entire oscillator, see Fig. 5 b)). From this, the force per unit legth can be calculated. Integrating this force over the entire boundary, and renormalizing to the effective mass of the oscillator, the total force F equals [4]:

in which m_{eff} is the effective mass of the
microdisk, and P_{circ} the circulating optical power (amplified
by a factor of ~ Q/(2π) over the input power P_{in}). In view
of the Q/(2π) amplification, the circulating optical power can reach
values on the order of 100W for input powers in the mW regime [8], and
the total force exerted on the walls (this time non-mass-normalized)
10^{-5} N. With these force levels, mechanical effects caused by
the circulating optical power cannot be neglected, as will be shown
below.

An oscillating circulating optical field intensity will give rise to an oscillating radiation pressure on the sidewalls of the optical microdisks. When the oscillation frequency falls approximately within the mechanical resonance bandwidth, a significant mechanical vibration will be excited.

Conversely, a changing effective radius of the
optical microdisk will shift the resonance frequency according to the
formula ω_{0}(t)=c/n(R+r(t)), where r(t) denotes the
deviation from equilibrium of the microdisk radius.

A mechanical oscillation of the microdisk at
frequency Ω leads to a periodically varying input coupling of the
incoming light wave (note that the cavity field lineshape is Lorentzian,
and centered around ω_{0}; intuitively, one could say that
this Lorentzian lineshape will oscillate around ω_{0} at
frequency Ω) and hence a periodically varying circulating optical
power. Note the interdependence of these two phenomena: a mechanical
oscillation leads to an optical power oscillation, and vice versa.

Although the physical description is more
complicated than this intuitive reasoning, a fairly elucidating view on
this phenomenon involves the corpuscular theory of light. An input
photon of frequency ω_{1} is transformed to respectively a
red-shifted, Stokes-photon (ω_{1}-Ω) or a
blue-shifted anti-Stokes photon (ω_{1}+Ω) (see Fig.
6a. As long as the Stokes and anti-Stokes photons fall within the
optical bandwidth of the oscillator, their amplitudes will be
non-neglegible. The beating of the Stokes and anti-Stokes modes with the
fundamental leads then to an effective force at frequency Ω.

Blue-detuning of the input frequency
ω_{1} with regard to the (unmodulated) cavity frequency
ω_{0} (ω_{1} > ω_{0}), in
combination with the Lorentzian lineshape (we refer to Fig. 6b
guarantees net power flow ΔP from the optical domain to the
mechanical domain, which is subsequently absorbed by the effective
mechanical loss mechanisms. We note that, conversely, red detuning of
the input wave should then lead to net absorption of mechanical energy,
or *cooling*, as was indeed shown in [5].

The phenomenon of coupled mechano-optical
oscillations is a classic example of the well-known parametric oscillation process (see
e.g [9] for a thorough description of parametric processes in the
non-linear optical domain). In the textbook example hereof, a
sinusoidally (frequency ω_{1}) varying parameter of a
harmonic oscillator leads to a modulation of the oscillator response at
frequencies ω_{1}+ω_{2} resp.
ω_{1}-ω_{2,} as illustrated in Fig. 7. In
our system, the optical oscillation can be considered as the harmonic
oscillator, and its modulation due to the mechanical vibration as the
time-varying spring.

In view of the radial symmetry, an effective 1-D model can be invoked for the parametric oscillation, as developed in [7] and reproduced in Fig. 8. There, a Fabry-Perot oscillator is considered, where the recircurculation of the optical wave after bouncing back and forth from two mirrors (one semi-transparent: the input mirror) leads to the constructive interference and optical field build-up, similar to the case of optical microdisk cavities. One of the mirrors is attached to a spring, leading to the possibility of parametric oscillation due to mechanical vibration of one mirror. Apart from the type of optical resonator, this is exactly the mechanism of the parametric oscillation in the optical microdisk oscillators.

The slowly varying envelope approximation can be invoked, in which a slow variation of the field amplitudes is decoupled from the fast, oscillatory movement at the respective sinusoidal frequencies. This yields the following equations for the fields (4)-(7):

Here, a_{1}, a_{S}, a_{AS}
and a_{r} are the normalized time-dependent field amplitudes of
the input optical, Stokes/anti-Stokes and mechanical waves. Neglecting
the anti-Stokes component (justified in the case of blue-detuning,
where the anti-Stokes amplitude can be significantly smaller than the
Stokes amplitude, cf. Fig. 6, we obtain the following equations for the
Stokes and mechanical waves [7]:

where γ_{1} and
γ_{m} are the damping coefficients of the optical
(and Stokes) vibration and the optical field respectively, and R
represents the length of the Fabry-Perot resonator. Apart from
numerical factors, these same equations govern the behavior of the
mechanical and optical oscillations in optical microdisk
oscillators (cf. [2]).

Solutions of the coupled system of eqs. (8) and (9) can be found [7]:

leading to a growing amplitude (eventually counteracted
by non-linearities in the system, not-considered in this model) as of a
threshold circulating power P_{th}:

Fig. 9: a) Experimental setup, from [2]. b) Detected
output power, redrawn from [2]. c) Threshold behavior of the
Stokes mode, redrawn from [3]. |

The foregoing has been experimentally confirmed in
ultra-high Q (10^{6}-10^{7}) silica optical microdisk
oscillators (see refs. [1] to [6]). By melting the silica microdisks
under illumination with a high-power CO_{2 }laser, an
ultrasmooth surface can be created, significantly reducing the surface
scattering and associated losses [6].

The spectral analysis of the transmitted optical
power P_{out} was performed through electrical spectral analysis
of the output of a photodiode, in which the inherent quadratic
non-linearity leads to beat notes which are detected in an electrical
spectrum analyzer [3]. We refer to figs. 9 a) and 9 b). Note that,
besides the Stokes sideband, also higher harmonics were observed, due to
the non-linear transmission of the microdisk oscillator [3].

In Fig. 9c, the threshold behavior of the Stokes field amplitude and the mechanical oscillation can be observed.

In order to verify the dependence of the threshold power on the mechanical quality factor as described by equation (11), a setup was designed in which a microprobe was gradually pressed onto the microdisks [2]. By pressing the probe onto the microdisk, the damping of the mechanical vibration and hence the mechanical quality factor could be gradually changed, as shown in Fig. 10a.

A clear linear dependence could be observed of the threshold power on the inverse of the mechanical quality factor, effectively confirming the predicted result of eq. (11), and ruling out other possible parametric oscillation mechanisms besides mechanical vibrations [2].

The study of joint mechanical-optical oscillations, made possible through the creation of ultrahigh-Q optical microdisk oscillators, has confirmed the possibility of radiation pressure effects leading to parametric oscillations in optical microdisks.

Besides the results discussed in this report, other intriguing effects such as optical cooling of of microdisk oscillators have been reported (see [5]); on the technological side, it has been proposed to use this parametric effect in order to fabricate all-optical clocks (see [6] and references therein).

© 2007 Kristiaan De Greve. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] T. Carmon, H. Rokhsari *et al.*, Phys. Rev.
Lett. **94**, 223902 (2005).

[2] T.J. Kippenberg *et al.,* Phys. Rev. Lett.
**95, **033901 (2005).

[3] H. Rokhsari *et al.*, Opt. Express
**13**, 5293 (2005).

[4] H. Rokhsari, Appl. Phys. Lett. **89**, 261109
(2006).

[5] A. Schliesser *et al.*, Phys. Rev.
Lett. **97**, 243905 (2006).

[6] M. Hossein-Zadeh *et al.*, Phys. Rev. A
**74**, 023813 (2006).

[7] V.B. Braginsky *et al.*, Phys. Lett. A
**287**, 331 (2001).

[8] K. Vahala, Nature **424**, 839 (2003).

[9] R. Boyd, *Non-linear optics, 2nd ed.
*(Academic Press, 2003).