Appendix A: Spatial Fourier series of a periodic optical power function
In order to calculate an analytical expression for c
k
in Sect. 2.1, we proceed as follows. We rewrite (1) and (4) as
$$I(x)=\frac{hI_\text{max}}{1+h-\cos y}=\sum_{k=-\infty}^\infty \beta_k e^{jky}, $$
(47a)
where
Multiplying both sides of (47a) by 1+h-cosy, using the fact that cosy=(e
jy+e
-jy)/2, and separating terms corresponding to different harmonics, one finds
$$(1+h)\beta_k-\frac{1}{2}(\beta_{k-1}+\beta_{k+1})=\left \{ \begin{array}{l@{\quad}l}hI_\text{max}{:} & k=0, \\0{:} & k \neq0.\end{array}\right . $$
(48)
Note that β
k
=β
-k
, and β
k
are real because I(y) is a real even function. Assuming that β
k
can be represented as
$$\beta_k=I_\text{max}\chi\alpha^{|k|}, $$
(49)
where χ and α are real, and substituting (49) into (48) for positive values of k results in
$$\alpha^2-2(1+h)\alpha+1=0.$$
The solution which ensures series convergence by satisfying the condition 0<α<1 is
$$\alpha=1+h-\sqrt{(1+h)^2-1}. $$
(50a)
The value of χ can be found from (48) for the case in which k=0, giving
$$ \chi=\frac{h}{\sqrt{(1+h)^2-1}}.$$
(50b)
Finally, (47d) gives
$$c_k=I_\text{max}\chi\alpha^{|k|}e^{-j2\pi k\frac{x_0}{L}}. $$
(50c)
It is straightforward to show that if the finesse is bigger than unity, i.e., 
is of order of ten or higher, the truncation error in (6) is negligible if 
.
An example of several truncated Fourier series calculated using (50a)–(50c) for different values of \(k_{\text{max}}\) is shown in Fig. 12.
Appendix B: Equilibrium analysis of the equations of motion
In this section, we analyze the equilibrium position of the third-order autonomous nonlinear dynamical system defined by (7) and (9) where the external exciting force is zero (f
m
=0).
By defining new variables \(p=\dot{x}\) and ΔT=T-T
0, the equations of motion can be rewritten as
where parameters defined in Sect. 2.2 have been used.
The equilibrium position of the dynamical system (i.e., the fixed point) is readily obtained by setting the velocities (i.e., the left-hand side of (51a)–(51c)) to zero. This results in a transcendental function for the equilibrium displacement A
0s
,
$$\varOmega_s^2A_{0s}+\alpha_3A_{0s}^3-\nu I(A_{0s})-\theta \varDelta T_0=0, $$
(52a)
where the equilibrium temperature shift ΔT
0 is
$$\varDelta T_0=\frac{\eta}{\kappa}I(A_{0s}) , $$
(52b)
and the equilibrium mechanical resonance frequency is
$$\varOmega_s=\omega_0-\beta \varDelta T_0. $$
(52c)
In the limit of a very small equilibrium displacement A
0s
≈0 (i.e., the limit of very weak optomechanical forces), (52a)–(52c) converge to the similar equations (26a)–(26c) derived in Sect. 2.5.
In general, multiple solutions of (52a)–(52c) may coexist, corresponding to several stable and unstable fixed points under the same experimental conditions. However, in the case in which the thermal frequency shift, the radiation pressure and the thermal force are all considered small, the limiting case of (26a)–(26c) predicts a single stable fixed point with a small static displacement A
0s
≪Γ.
Stability of the equilibrium is obtained via a local perturbation of the system fixed point defined by (52a)–(52c), resulting in a linear variation
$$\left ( \begin{array}{c}\dot{x} \\\dot{p} \\\varDelta \dot{T}\end{array}\right ) = M \left ( \begin{array}{c}x-A_{0s} \\p \\\varDelta T-\varDelta T_0\end{array}\right ),$$
where M is the Jacobian matrix of the first derivatives of the system functions given by the right hand parts of (51a)–(51c). Thus, equilibrium stability can readily be obtained by evaluating the eigenvalues λ
1, λ
2, and λ
3 of M, which satisfy
$$\lambda^3+c_1\lambda^2+c_2\lambda+c_3=0, $$
where
and where a prime denotes differentiation with respect to the mechanical displacement x.
Asymptotic stability of the equilibrium (i.e., \(\operatorname {Re}\{\lambda_{i}\}<0\)) is defined by positive coefficients and a positive second Hurwitz determinant, namely, c
i
>0 and Δ
2=(c
1
c
2-c
3)>0. Loss of equilibrium stability is defined by a zero eigenvalue (c
3=0), or a Hopf bifurcation where the Jacobian matrix M has a pair of pure imaginary eigenvalues, i.e., λ
1,2=±iω
H
.
The zero eigenvalue condition c
3=0 can be rewritten in a differential form as
This equation can be readily understood as a condition of equality between the thermally dependent nonlinear elastic force (left-hand side terms) and the optomechanical forces (right-hand side terms). This condition describes a saddle-node bifurcation, which can be reached for the case of larger optomechanical coupling than considered in this work. Note that the validity of the assumptions made in Sect. 2.2, especially the linear temperature dependence of the mechanical frequency and the thermal force, has to be carefully assessed in this case.
The Hopf bifurcation, which implies that periodic limit cycle oscillations can occur near the bifurcation threshold [54], can readily be shown to correspond to a zero second Hurwitz determinant, i.e., c
1
c
2-c
3=0, with a positive Hopf frequency \(\omega_{H}=\sqrt{c_{2}}\). Using (53a)–(53c), we find the bifurcation threshold condition to be
If we assume the mechanical dissipation, the nonlinear effects and the optomechanical coupling to be weak, namely, we assume the thermal frequency shift, the static displacement, the nonlinear and dissipation terms, the radiation pressure and the thermal force to be small, and, therefore, neglect all the small terms of the second order and higher, then the right-hand side of (54) vanishes. In this limit, the Hopf bifurcation condition given in (54) coincides with the condition γ=0 discussed in Sect. 3.1 [see (32a) and (37)].
Under the same assumptions, the Hopf frequency becomes
$$\omega_H=\sqrt{c_2}\approx\omega_0-\varDelta \omega_s, $$
(55)
where Δω
s
is defined in (26a) and (32b). This result coincides with the limit cycle frequency expression given in (43) in the limit of vanishing limit cycle amplitude.
We note that the Hopf bifurcation can either be supercritical or subcritical, culminating with stable or unstable self-excited limit-cycle solutions which are discussed in Sect. 4.
Appendix C: Averaging of the equations of motion
Using (6) and (20), we write the optical power expression I as
$$I(x) \approx\sum_{k=-k_\text{max}}^{k_\text{max}}c_k e^{j2\pi\frac {k}{L}(A_0+A_1\cos\psi)}. $$
(56)
It is beneficial to use the Jacobi–Anger expansion
$$e^{jz\cos\xi}=J_0(z)+2\sum_{n=1}^{\infty}j^nJ_n(z)\cos n\xi,$$
(57)
where z and ξ are some real variables, and J
n
(z) is the Bessel function of n-th order. The optical power expression given in (56) can be rewritten as
$$I(x) \approx P_0+2\sum_{n=1}^\infty P_n\cos n\psi, $$
(58)
where P
n
are defined in (24).
Next, we proceed to write the integral in (13) explicitly. Slow envelope approximation implies that the amplitude A
1, and the phase \(\tilde{\phi}\) do not undergo significant changes at timescales comparable to \(\omega_{0}^{-1}\). It follows that A
1 and \(\tilde{\phi}\) can be regarded as constants at timescales of order \(\omega_{0}^{-1}\) and κ
-1, and terms involving K in (16) can be estimated using the approximate equality
$$\int_0^t f(\tau) g(\tau-t)\,d\tau\approx f(t)\int_0^t g(\tau-t) \,d\tau,$$
(59)
where g(τ-t) is either e
κ(τ-t), \(e^{(\kappa\pm j\omega _{0})(\tau-t)}\) or \(e^{(\kappa\pm j2\omega_{0})(\tau-t)}\), f(t) is a function of slow varying terms A
1 and \(\tilde{\phi}\), and all fast decaying terms in ∫g(τ-t) dτ should be neglected. The result is
In order to solve (16) under the conditions described above, we use the harmonic balance method followed by the Krylov–Bogoliubov averaging technique [56], and require
It follows that
$$\dot{A}_1\cos\psi-A_1\dot{\tilde{\phi}}\sin\psi=0. $$
(62)
Introducing (61a), (61b) into (16) results in
Collecting all nonharmonic terms in (63) gives the expression for A
0:
where Ω is defined in (23), and terms proportional to (βηK)2 have been neglected because the frequency correction due to heating is considered small, i.e., βηK(I)≪ω
0. The term Δω
0 can be identified as a small frequency correction due to the heating of the mirror averaged over one mechanical oscillation period.
Equation (64) can be further simplified by assuming the static displacement A
0 to be small and using the weak nonlinearity assumption, i.e., \(\alpha_{3}A_{0}^{2}\ll\omega_{0}^{2}\), giving rise to (22).
The remaining terms in (63) constitute the following relationship [see also (58) and (60)]:
$$\dot{A}_1\sin\psi+A_1\dot{\tilde{\phi}}\cos\psi=B, $$
(65)
where
Here, \(\operatorname{NST}\) denotes the non secular terms (i.e., higher harmonics).
Equations (62) and (65) can be rearranged as follows:
Averaging of (67a), (67b) over one period of ψ can be made under the assumption of slow varying envelope, namely,
Substituting (66) into (68a), (68b) yields the slow envelope evolution equations (25a), (25b).