Angular radiation pressure

Here we will create a model of a Fabry-Perot cavity with its two mirrors suspended from a simple pendulum. We will then drive some angular motions and see how the dynamics of the mirror are altered due to radiation pressure effects. This is commonly referred to in the gravitational wave community as the HARD and SOFT rotational modes. This refers to the fact that the optical torques modify the two suspension modes making one mode harder (higher in resonant frequency) and the other softer (lower in resonant frequency).

More can be read up on regarding angular instabilities shown here in [19], [20], and more on Advanced LIGO angular control design in [21].

First let us import the various features we will need. For this example we will use the Python API rather than KatScript as a demonstration.

import cmath
import numpy as np
import finesse
import finesse.components as fc
import finesse.detectors as det
from finesse.analysis.actions import FrequencyResponse


Next we build out Fabry-Perot model and attach some simple Pendulum dynamics. The pendulum has a resonance at 0.6 Hz and a moment of inertia of 0.757 kg.m^2.

L = 3994.5 # [m]
I = 0.757  # [kg.m^2]
c = 299792458 # [m.s^-1]
f_sus = 0.6 # [Hz]

model = finesse.Model()
model.fsig.f = 1 # set some initial signal frequency
model.modes(maxtem=1) # first order modes for modelling alignment signals

LASER = model.add(fc.Laser("LASER", P=1000))
# Add two mirrors for the cavity and attach a pendulum mechanics
M1 = model.add(fc.Mirror("M1", R=0.986, T=0.014, Rc=1934))
        "M1_sus", model.M1.mech, mass=np.inf, I_yaw=np.inf, I_pitch=I, fpitch=f_sus
M2 = model.add(fc.Mirror("M2", R=1, T=0, Rc=2245))
        "M2_sus", model.M2.mech, mass=np.inf, I_yaw=np.inf, I_pitch=I, fpitch=f_sus
model.connect(M1.p1, M2.p1, L=L)
model.connect(LASER.p1, M1.p2)
model.add(fc.Cavity('cavARM', M2.p1.o))
model.add(det.PowerDetector('P', M1.p1.o)); # cavity power

Next we need analyse the geometry of our cavity and determine how much of each mirror motion makes up HARD and SOFT, this requires computing the $r$ factor [21]

# Now we compute the decomposition of HARD and SOFT modes into motions of M1 and M2
g_1 = 1 - float(L / np.abs(M1.Rcx.value)) # input mirror g-factor
g_2 = 1 - float(L / np.abs(M2.Rcx.value)) # end mirror g-factor

r = model.add_parameter(
    2 / ((g_1 - g_2) + np.sqrt((g_2 - g_1) ** 2 + 4)),
    description='r term in eq.3.5 in T0900511',

Above we have added a new parameter to the model using model.add_parameter so that later we can change its value and any other element that uses model.r will automatically update itself. This also means we can vary r with an action such as Xaxis to sweep through different values easily.

Now we can define a new global degree of freedom in the model that combines the local motions of each cavity mirror. Here we use r.ref to make a reference to the new parameter we added.

HARD = model.add(fc.DegreeOfFreedom("HARD", M1.dofs.F_pitch, -1, M2.dofs.F_pitch, +r.ref))
SOFT = model.add(fc.DegreeOfFreedom("SOFT", M1.dofs.F_pitch, +r.ref, M2.dofs.F_pitch, +1))

[-1 ❮Symbolic='+r' @ 0x7f37f3d08680❯]
[❮Symbolic='+r' @ 0x7f37f24d8ec0❯ 1]

We can see our amplitudes now depend on a symbolic reference to r. You can see what the current numerical value is by simply casting this array to a float type which forces an evaluation of the

print("model.r = ", model.r.eval())
print("Current HARD amplutudes:", HARD.amplitudes.astype(float))
model.r =  1.1532419233817137
Current HARD amplutudes: [-1.          1.15324192]

It is worth stopping here to note that we are defining the HARD and SOFT mode to drive a pitch torque by using M1.dofs.F_pitch, a finesse.components.general.LocalDegreeOfFreedom. A local degree of freedom collects together the static (DC) and small signal (AC) states of an element. In this case it’s a degree of freedom that drives a pitch force (or torque, all forces/torque nodes are prepended with F_).

We can see what this local degree of freedom is by outputing its AC_IN, AC_OUT, DC attributes.

❮SignalNode M1.mech.F_pitch @ 0x7f37f3ce7590❯
❮SignalNode M1.mech.pitch @ 0x7f37f3ce74d0❯
❮M1.ybeta=((-HARD.DC)+(SOFT.DC*r)) @ 0x7f37f3cd1180❯

The AC_IN defines which signal node will be excited by this DOF, for example HARD.AC.i. The AC_OUT defines which nodes is used to generate the output of the DOF, HARD.AC.o. The DC attribute defines sets the static state of the, in this case it is M1.ybeta, the static pitch misalignment of the the mirror M1. By setting the HARD.DC value you can set what static DOF value is, relative to any DC parameter there is on the component itself.

Using the Python API it is possible to define your own finesse.components.general.LocalDegreeOfFreedom and mix and match any DC and AC inputs and outputs as you need. Commonly used ones are typically defined in an elements .dofs attribute.

Now to continue with the original example, we will set the power to get around 600kW in the cavity and perform a multiple-input-multiple-output (MIMO) frequency response analysis exciting the HARD and SOFT mode inputs to their outputs.

LASER.P = 1410 * 3/2 # get to roughly 600kW
freq_resp = FrequencyResponse(
    np.geomspace(0.1, 10, 2000),
    [HARD.AC.i, SOFT.AC.i],
    [HARD.AC.o, SOFT.AC.o],
sol =
# Alsp compute a single data point of the detectors to compute the power
# circulating in the cavity
out =
(2000, 2, 2)

The output of the frequqency response analysis is a (2000, 2, 2) array, as we have 2000 frequencies and a 2x2 MIMO matrix at each.

We can now plot the results and compare this against equation 2 in [20] which states how much the HARD and SOFT modes are altered by radiation pressure.

omega_0 = 2 * np.pi * f_sus
P = out['P']
# Eq 2 from
omega_plus = np.sqrt(
    omega_0**2 + P * L / (I * c) * (-(g_1 + g_2) + cmath.sqrt(4+(g_1 - g_2)**2))/ (1 - g_1*g_2)
omega_minus = np.sqrt(
    omega_0**2 + P * L / (I * c) * (-(g_1 + g_2) - cmath.sqrt(4+(g_1 - g_2)**2))/ (1 - g_1*g_2)

axs = finesse.plotting.bode(sol.f, sol['HARD.AC.i', 'HARD.AC.o'], label='HARD')
axs = finesse.plotting.bode(sol.f, sol['SOFT.AC.i', 'SOFT.AC.o'], label='SOFT', axs=axs)
axs[0].vlines(omega_0/2/np.pi, -70, 60, ls='--', label=f'$\\omega_{{0}} = {omega_0.real/2/np.pi:0.2f}$Hz', color='k', zorder=-100)
axs[0].vlines(omega_plus.real/2/np.pi, -70, 60, ls='--', label=f'$\\omega_{{+}} = {omega_plus.real/2/np.pi:0.2f}$Hz', zorder=-100)
axs[0].vlines(omega_minus.real/2/np.pi, -70, 60, color='red', ls='--', label=f'$\\omega_{{-}} = {omega_minus.real/2/np.pi:0.2f}$Hz', zorder=-100)
axs[0].set_title(f"E.Hirose, Appl. Opt. 49, 3474-3484 (2010)\nEq.2 vs FINESSE3, P={P/1e3:0.0f}kW")
axs[0].set_ylabel("Magnitude [dB - rad/Nm]")
Text(0, 0.5, 'Magnitude [dB - rad/Nm]')

If we reduce the power we can see the opto-mechanical effects are reduced and the HARD and SOFT resonances beging to return to the nominal suspension resonance frequency

LASER.P = 100
sol =
axs = finesse.plotting.bode(sol.f, sol['HARD.AC.i', 'HARD.AC.o'], label='HARD')
finesse.plotting.bode(sol.f, sol['SOFT.AC.i', 'SOFT.AC.o'], label='SOFT', axs=axs)
array([<Axes: ylabel='Magnitude [dB]'>,
       <Axes: xlabel='Frequency [Hz]', ylabel='Phase [Deg]'>],

As a last point to make, due to the asymmetric cavity mirror curvatures the actuation must be properly diagonalised. Here we can see if we simply just set the M1 and M2 actuation amplitude to a unit value we now see that when we drive HARD or SOFT we excite the other mode as well.

LASER.P = 1410 * 3/2 # get to roughly 600kW
model.r = 1
# Or alternatively you can replace the amplitudes directly with something like
# HARD.amplitudes[:] = [-1, +1]
# SOFT.amplitudes[:] = [+1, +1]
sol =
axs = finesse.plotting.bode(sol.f, sol['HARD.AC.i', 'HARD.AC.o'], label='HARD')
finesse.plotting.bode(sol.f, sol['SOFT.AC.i', 'SOFT.AC.o'], label='SOFT', axs=axs)
array([<Axes: ylabel='Magnitude [dB]'>,
       <Axes: xlabel='Frequency [Hz]', ylabel='Phase [Deg]'>],

Click to download example as python script

Click to download example as Jupyter notebook