Calculating transfer functions of effective systems

Sometimes it is useful to simulate a simpler effective thermo-optomechanical system that emulates a more complex physical one, but that is not strictly physical itself. In such cases it may be difficult to simulate the DC fields needed to describe this effective system. In Finesse, the DC fields are known as the carrier_solution. Usually when calculating a FrequencyResponse, the DC fields are first computed and then used in the computation of the AC response. It is also possible to override this DC calculation by providing a carrier_solution where the DC fields have been manually set to their desired values. We illustrate this here with the effective three mirror coupled cavity, which describes some of the dynamics of modern and future gravitational wave detectors.

Warning

It is your responsibility to ensure that sensible DC fields are being set. Finesse will happily return garbage if you haven’t been careful when providing a carrier_solution to one of the FrequencyResponse actions.

You can manually provide a carrier_solution to any of the four FrequencyResponse actions, but we’ll demonstrate it here only with FrequencyResponse.

The DARM coupled cavity

The optical topology used by all current and future detectors is known as the dual-recycled Fabry-Perot Michelson (DRFPMI) and consists of five optical cavities made up of (at an absolute minimum) seven core optics. For many purposes the dynamics of the DRFPMI can be described by effective three mirror coupled cavities. The differential arm motion (DARM) is equivalent to a single effective arm cavity formed by a single ITM and ETM with the addition of a signal extraction mirror (SEM) to form the signal extraction cavity (SEC). Similarly, the common arm motion (CARM) is equivalent to a single arm cavity with the addition of a power recycling mirror to form the power recycling cavity (PRC). Here we build the DARM coupled cavity equivalent to the differential arm motion of LIGO.

import numpy as np
import finesse
import finesse.components as fc
import finesse.analysis.actions as fa
from finesse.utilities import set_DC_fields
from finesse.plotting import bode

finesse.init_plotting()

model = finesse.Model()
model.add(fc.Mirror("ETM", T=0, L=0, imaginary_transmission=False))
model.add(fc.Mirror("ITM", T=0.014, L=0, imaginary_transmission=False))
model.connect(model.ITM.p1, model.ETM.p1, 4e3)
model.add(fc.FreeMass("ITM_sus", model.ITM.mech, mass=40))
model.add(fc.FreeMass("ETM_sus", model.ETM.mech, mass=40))
model.add(fc.Mirror("SEM", T=0.325, L=0, imaginary_transmission=False))
model.connect(model.SEM.p1, model.ITM.p2, 55)
model.add(fc.BalancedHomodyneReadout("BHD", model.SEM.p2.o, phase=90))
model.modes("off")

By setting imaginary_transmission=False, we use the real transmission conventions for the mirrors so that the phases are intuitive. Thus, if we set the amplitude quadrature of the DC fields in the arm to be 0 deg, the BalancedHomodyneReadout phase is set to 90 deg in order to sense a phase signal excited by ETM motion.

Manually setting the DC fields

The system we want is one in which the arm cavity has 750 kW circulating in the fundamental mode and no power anywhere else. This almost means that we need the four nodes of the cavity (ITM incident and reflected and ETM incident and reflected) to have fields of power 750 kW. Since the mirrors are not perfectly reflecting, a field reflected from a mirror is reduced by the amplitude reflectivity. We also have to be careful about phases. So we’ll just extract the exact operators used to compute the real carrier_solution with an Operator action so that we don’t have to think about it.

Parm_W = 750e3
freq = 0  # carrier frequency, only 0 in this simple model
sol = model.run(fa.Series(
    fa.DCFields(name="DC"),
    fa.Operator(model.ETM.p1.i, model.ITM.p1.o, frequency=freq, name="op"),
))
DC = sol["DC"]
op = sol["op"]
DC.fields[:] = 0  # set the fields at every node to zero

op is an OperatorSolution which has all of the operators along the path from ETM.p1.i (the field incident on the front of the ETM) to ITM.p1.o (the field reflected from the front of the ITM) in its matrices attribute. You can see the entire path by

for idx, (fr, to) in enumerate(op.connections):
    print(f"operator {idx}: to {to} from {fr}")

operator 0: to ETM.p1.o from ETM.p1.i



operator 1: to ITM.p1.i from ETM.p1.o



operator 2: to ITM.p1.o from ITM.p1.i

These operators are especially simple in this example since we’re not simulating HOMs, but this method is very useful in more complicated situations. DC is now an empty DCFieldsSolution ready to be manually populated by the desired DC fields. We could do this by hand like this

f_idx = list(DC.fields).index(freq)  # frequency index of the carrier
# set the power in the fundamental to 750 kW
Efr_i = np.zeros(len(DC.homs), dtype=complex)
Efr_i[0] = np.sqrt(Parm_W)
Ec = DC["ETM.p1.i", f_idx] = Efr_i
Ec = DC["ETM.p1.o", f_idx] = op.matrices[0] @ Ec
Ec = DC["ITM.p1.i", f_idx] = op.matrices[1] @ Ec
Ec = DC["ITM.p1.o", f_idx] = op.matrices[2] @ Ec

But we can also use the finesse.utilities.frequency_response.set_DC_fields() helper function. This takes as arguments the initial field to be set at the first node of op, the frequency of the field to be set, the DCFieldsSolution to be populated, and the OperatorSolution containing the operators along the path.

DC[:] = 0
set_DC_fields(Efr_i, freq, DC, op)

You can use finesse.utilities.frequency_response.set_DC_fields() multiple times with different OperatorSolution to set the DC fields starting from multiple different nodes along different paths. You could also do more complicated things with this such as setting the power in the first eigenmode of a cavity with distortions and/or apertures by simply passing that eigenmode in instead of Efr_i.

Calculating the frequency response

We will compute the optomechanical plant for five cases to illustrate the dynamics of the DRFPMI using the simple DARM coupled cavity. Letting \(\phi_\mathrm{s}\) denote the extra one-way phase the carrier accumulates when propagating through the SEC, which is equivalent to SEM.phi, these cases are

  • No SEM present: this is just a single Fabry-Perot cavity which is equivalent to a gravitational wave detector without an SEC, like the initial gravitational wave detectors. If the SEM is misaligned it will not form a cavity and it is effectively absent (except it will still add loss if there was any). This is accomplished in finesse by setting SEM.misaligned=True.

  • Signal recycling (SR): when \(\phi_\mathrm{s}=0\), the bandwidth is narrowed and the DC gain is increased relative to the single FP cavity.

  • Resonant sideband extraction (RSE): when \(\phi_\mathrm{s} = \pi/2\), the bandwidth is broadened and the DC gain is reduced relative to the single FP cavity. This is how most gravitational wave detectors operate.

  • Optical spring: when \(\phi_\text{s}>\pi/2\), there is an anti-damped optical spring. This is how the low frequency Einstein telescope is planned to operate.

  • Optical anti-spring: when \(\phi_\text{s}<\pi/2\), there is a damped anti-spring.

Note that there is no standard convention and often \(\phi_\mathrm{s}\) is defined as SEM.phi - \pi/2 in the literature and other codes so that all of the above cases shift by \(\pi/2\).

We’ll compute the same transfer function for each of these cases and so define a simple function fresp which returns the same frequency response giving it a different name so that it can be retrieved easily from the solution. Then we’ll compute these five cases in a Series using Change between each to set the correct configuration. Since this will change the state of the model we do this in a temporary_parameters context manager so that the model reverts back to its initial state after the analysis.

F_Hz = np.geomspace(1, 10e3, 300)
fresp = lambda k: fa.FrequencyResponse(
    F_Hz, "ETM.mech.z", "BHD.DC", carrier_solution=DC, name=k,
)

with model.temporary_parameters():
    sol = model.run(fa.Series(
        fa.Change({"SEM.phi": 0}),
        fresp("SR"),
        fa.Change({"SEM.phi": 90}),
        fresp("RSE"),
        fa.Change({"SEM.phi": 98}),
        fresp("spring"),
        fa.Change({"SEM.phi": 82}),
        fresp("anti"),
        fa.Change({"SEM.misaligned": True}),
        fresp("arm"),
    ))

Note that detuning the arm is a situation where you would need to think carefully about manually setting the carrier_solution. There is now an extra phase introduced in the arm and so you would either need to retrieve the new operators with an Operator action (the most robust) or multiply all of the fields by the appropriate phase. There are no DC fields in the SEC so we did not need to worry about that in this case where we only detuned the SEC.

The effective coupled cavity transfer functions need to be divided by \(\sqrt{2}\) to account for the presence of the beamsplitter in the DRFPMI.

plt_fresp = lambda k: sol[k].out.squeeze() / np.sqrt(2)
lw = 2.5
axs = bode(F_Hz, plt_fresp("RSE"), db=False, label="RSE", lw=lw)
bode(F_Hz, plt_fresp("SR"), axs=axs, db=False, label="SR", lw=lw)
bode(
    F_Hz, plt_fresp("spring"), axs=axs, db=False, label="Optical spring", lw=lw,
)
bode(
    F_Hz, plt_fresp("anti"), axs=axs, db=False, label="Optical anti-spring",
    ls="--", lw=lw,
)
bode(F_Hz, plt_fresp("arm"), axs=axs, db=False, label="Fabry-Perot", lw=lw)
axs[0].set_ylabel("Magnitude [W/m]")

Text(0, 0.5, 'Magnitude [W/m]')
../../_images/effective_fresp_6_1.svg