Two-photon transfer functions and a Fabry-Perot cavity

In this section we analyze a simple Fabry-Perot cavity where the ETM is a free mass of mass \(m\) and the ITM is fixed.

import numpy as np
import scipy.constants as scc
import finesse
import finesse.components as fc
import finesse.analysis.actions as fa
import finesse.detectors as fd
from finesse.plotting import bode

finesse.init_plotting()

L_arm = 4000
model = finesse.Model()
LASER = fc.Laser("Laser", P=5e3)
ETM = fc.Mirror("ETM", T=0, L=0, Rc=2245)
ITM = fc.Mirror("ITM", T=0.01, L=0, Rc=1940)
ITMAR = fc.Mirror("ITMAR", T=1, L=0, Rc=np.inf)

# Link and add all the components to the model
model.link(LASER, ITMAR.p2, ITMAR.p1, ITM.p2, ITM.p1, L_arm, ETM.p1)

cavARM = model.add(fc.Cavity("cavARM", model.ETM.p1))
ETM_sus = model.add(fc.FreeMass("ETM_sus", model.ETM.mech, mass=40))
model.add(fd.PowerDetector("Parm", model.ETM.p1.o))

<'Parm' @ 0x7e208eaf8e60 (PowerDetector)>

Since Finesse does not (yet) have real mirrors, you have to manually add an HR surface and a perfectly transmisive AR surface. Most realistic models do this. It is important in this case because the convention in Finesse is that transmission through each surface is imaginary \(\mathrm{i}t\). The convention for real thick mirrors is thus that transmission is real and negative \(-t\). Having the transmission be imaginary would lead to non-intuitive and non-standard two-photon transfer matrices.

Warning

The AR and HR surfaces should be appropriately linked. We are not moving or rotating the ITM or having radiation pressure act on it, so we can be lazy and not do it here, but to be safe the AR parameters phi, xbeta, and ybeta should be symbolically linked to the corresponding HR parameters and all of the mechanical degrees of freedom of the AR surface should be connected to the corresponding degrees of freedom of the HR surface. Thick components will eventually be added which will make this step unnecessary.

Todo

Add a section explaining how to do this being careful about the gains depending on whether the surfaces are pointing towards each other or not.

Calculation of the transfer functions

As described in Radiation Pressure at a Mirror, the light exerts a radiation pressure force on the free mirror. In the quadrature picture, this induces a coupling from the amplitude to the phase quadrature with strength

\[\mathcal{K} = \frac{4k(2R + L)\chi_0P}{c} \rightarrow - \frac{8kP}{m\omega^2c}, \]

where \(P\) is the power in the cavity, \(R\) and \(L\) are the reflectivity and loss of the mirror, respectively, \(\chi_0\) is the free susceptibility of the mirror, and \(k=2\pi/\lambda_0\) is the wavenumber of the laser. The mirror in this example is a free perfectly reflecting mirror with no loss.

If \(a_1\) and \(a_2\) are the quadratures of the fields incident on the ITM, \(b_1\) and \(b_2\) are the quadratures of the fields reflected from the ITM, \(x\) is the position of the ETM, and \(F\) is an external force exerted on the ETM, the transfer functions are

\[\begin{aligned} \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} &= \underbrace{\begin{bmatrix} r_a & 0 \\ -t_a^2 \mathcal{K} & r_a \end{bmatrix}}_{\texttt{FrequencyResponse3}} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} + \underbrace{2kt_a\sqrt{P} \begin{bmatrix} 0 \\ 1 \end{bmatrix}}_{\texttt{FrequencyResponse4}} x\\ x &= \underbrace{-\frac{4t_a\sqrt{P}}{m\omega^2 c} \begin{bmatrix} 1 & 0 \end{bmatrix}}_{\texttt{FrequencyResponse2}} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \underbrace{- \frac{1}{m\omega^2}}_{\texttt{FrequencyResponse}} F \end{aligned} \]

where \(r_a\) and \(t_a\) are the reflection from the arm and the transmission through the arm, respectively. When the cavity is strongly overcoupled, as it is here, and for frequencies far below the FSR

\[r_a = \frac{1 - \mathrm{i}\omega/\gamma}{1 + \mathrm{i}\omega/\gamma}, \qquad t_a = -\sqrt{\frac{2\mathcal{F}}{\pi}} \frac{1}{1 + \mathrm{i}\omega/\gamma} \]

where \(\gamma\) is the cavity pole and \(\mathcal{F}\) is the cavity finesse.

To compute these four transfer functions in Finesse we need all four of the FrequencyResponse actions. We will calculate the transfer functions for both a cavity on resonance and for a detuned cavity. We therefore define a function analysis that returns all necessary actions to simplify the calculation.

F_Hz = np.geomspace(1, 1e3, 200)
model.fsig.f = 1
fsig = model.fsig.f.ref

def analysis(key):
    """Actions needed for computing the two-photon transfer functions"""
    return [
        # for DC arm power
        fa.Noxaxis(name=f"DC_{key}"),
        # for field reflection off of and transmission through the cavity
        fa.FrequencyResponse3(
            F_Hz,
            [
                (ITMAR.p2.i, +fsig),
                (ITMAR.p2.i, -fsig),
                (ETM.p1.o, +fsig),
                (ETM.p1.o, -fsig),
            ],
            [
                (ITMAR.p2.o, +fsig),
                (ITMAR.p2.o, -fsig),
            ],
            name=f"fresp3_{key}",
        ),
        # for ETM motion to fields
        fa.FrequencyResponse4(
            F_Hz,
            [ETM.mech.z],
            [
                (ITMAR.p2.o, +fsig),
                (ITMAR.p2.o, -fsig),
            ],
            name=f"fresp4_{key}",
        ),
        # for mechanical modification
        fa.FrequencyResponse(
            F_Hz,
            [ETM.mech.z, ETM.mech.F_z],
            [ETM.mech.z],
            name=f"fresp_{key}",
        ),
        # for radiation pressure on mirror
        fa.FrequencyResponse2(
            F_Hz,
            [
                (ITMAR.p2.i, +fsig),
                (ITMAR.p2.i, -fsig),
            ],
            [ETM.mech.z],
            name=f"fresp2_{key}",
        ),
    ]

sol = model.run(
    fa.Series(
        fa.PrintModelAttr("ETM.phi"),
        fa.Change({ETM.phi: 0}),
        # transfer functions for the cavity on resonance
        *analysis("tuned"),
        # detune the cavity by 1 degree
        fa.Change({ETM.phi: -1}, relative=True),
        # transfer functions for the detuned cavity
        *analysis("detuned"),
        fa.PrintModelAttr("ETM.phi"),
    )
)

 ETM.phi=0.0
 ETM.phi=-1.0

Note that the state of the model has changed at the end of the calculation so that the cavity is now detuned. To prevent that, we could have made the detuned analysis temporary using either temporary(), Temporary, or TemporaryParameters, but did not in this case to simplify accessing the results.

The results of FrequencyResponse3 are a matrix of sideband transfer functions \(M_\mathrm{sb}\), those of FrequencyResponse4 a vector \(v_\mathrm{sb}\) of sideband transfer functions, those of FrequencyResponse2 an adjoint vector of sideband transfer functions, and that of FrequencyResponse a scalar not involving optical fields. (Actually two scalars in this case because we asked for the response of mirror motion to both mirror motion and an external force.) The conversions to two-photon transfer functions are

\[M_{2\mathrm{p}} = A M_\mathrm{sb} A^{-1}, \qquad v_{2\mathrm{p}} = A v_\mathrm{sb}, \qquad v_{2\mathrm{p}}^\dag = v_\mathrm{sb}^\dag A^{-1} \]

A2 = np.array([
    [1,  1],
    [-1j, 1j],
]) / np.sqrt(2)
A2i = np.array([
    [1, 1j],
    [1, -1j],
]) / np.sqrt(2)

def extract_2p_tfs(key):
    from types import SimpleNamespace
    tfs = dict(
        refl = A2 @ sol[f"fresp3_{key}"].out[..., :2, 0, 0] @ A2i,
        trans = A2 @ sol[f"fresp3_{key}"].out[..., 2:, 0, 0] @ A2i,
        motion = A2 @ sol[f"fresp4_{key}"].out[..., 0],
        mech = sol[f"fresp_{key}"].out,
        rp = sol[f"fresp2_{key}"].out[..., 0] @ A2i,
    )
    return SimpleNamespace(**tfs)

tuned = extract_2p_tfs("tuned")
detuned = extract_2p_tfs("detuned")

Cavity on resonance

First consider the behavior of a cavity on resonance described by the above equations. We did not need to add a Cavity to this model since we are not modeling HOMs and could easily calculate the cavity pole and finesse by hand, but added the Cavity anyway to illustrate how you can use it to quickly get these parameters (and others) from a more complicated model.

k = model.k0
M = ETM_sus.mass.value
Parm = sol["DC_tuned"]["Parm"]
Krp = -8 * k * Parm / (M * (2 * np.pi * F_Hz)**2 * scc.c)
Fp_Hz = cavARM.pole
Fa = cavARM.finesse
print(f"Arm power: {Parm * 1e-6:0.1f} MW")
print(f"Arm cavity pole: {Fp_Hz:0.1f} Hz")
print(f"Arm finesse: {Fa:0.0f}")
r_arm = (1 - 1j * F_Hz / Fp_Hz) / (1 + 1j * F_Hz / Fp_Hz)
t_arm = -np.sqrt(2 * Fa / np.pi) * 1 / (1 + 1j * F_Hz / Fp_Hz)

Arm power: 2.0 MW
Arm cavity pole: 30.0 Hz
Arm finesse: 625

First look at the reflection of fields from the cavity as calculated by FrequencyResponse3. The reflection of the phase to phase and amplitude to amplitude quadratures are the same. Amplitude fluctuations are converted to phase fluctuations through radiation pressure, but there is no conversion of phase fluctuations to amplitude fluctuations.

axs = bode(F_Hz, tuned.refl[..., 0, 0], db=False, label="simulation")
bode(F_Hz, r_arm, axs=axs, db=False, ls="--", label="theory")
axs[0].set_ylim(0.1, 10)
axs[0].set_ylabel("Magnitude [$\sqrt{\mathrm{W}}$ / $\sqrt{\mathrm{W}}$]")
axs[0].set_title("Cavity amplitude to amplitude reflection")

axs = bode(F_Hz, tuned.refl[..., 1, 0], db=False, label="simulation")
bode(F_Hz, -t_arm**2 * Krp, axs=axs, db=False, ls="--", label="theory")
axs[0].set_ylabel("Magnitude [$\sqrt{\mathrm{W}}$ / $\sqrt{\mathrm{W}}$]")
axs[0].set_title("Cavity amplitude to phase reflection")
print(
    "Cavity amplitude reflection equal to phase reflection?",
    np.allclose(tuned.refl[..., 1, 1], tuned.refl[..., 0, 0]),
)
print(
    "Cavity phase to amplitude reflection zero?",
    np.allclose(tuned.refl[..., 0, 1], 0),
)

Cavity amplitude reflection equal to phase reflection? True
Cavity phase to amplitude reflection zero? True

Next look at the response of the optical fields reflected from the cavity to mirror motion as calculated by FrequencyResponse4.

axs = bode(F_Hz, tuned.motion[..., 1, 0], db=False, label="simulation")
bode(F_Hz, -2 * k * t_arm * np.sqrt(Parm), axs=axs, db=False, ls="--", label="theory")
axs[0].set_ylabel("Magnitude [$\sqrt{\mathrm{W}}$ / m]")
axs[0].set_title("Mirror motion to reflected phase");

There is no modification to the mirror dynamics and so its susceptibility is still that of a free mass as shown by the results of FrequencyResponse:

print(
    "Unmodified?",
    np.allclose(tuned.mech[..., 0, 0], np.ones_like(tuned.mech[..., 0, 0])),
)
print(
    "Free mass?",
    np.allclose(tuned.mech[..., 0, 1], -1 / (M * (2 * np.pi * F_Hz)**2)),
)

Unmodified? True
Free mass? True

Finally, the radiation pressure from the amplitude quadrature of the incident field is calculated by FrequencyResponse4. The phase quadrature does not source any radiation pressure in this case.

axs = bode(F_Hz, tuned.rp[..., 0, 0], db=False, label="simulation")
bode(
    F_Hz, -4 * t_arm * np.sqrt(Parm) / (M * (2 * np.pi * F_Hz)**2 * scc.c),
    axs=axs, db=False, ls="--", label="theory",
)
axs[0].set_ylabel("Magnitude [m / $\sqrt{\mathrm{W}}$]")
axs[0].set_title("Amplitude to mirror motion")
print("Phase to mirror motion zero?", np.allclose(tuned.rp[..., 0, 1], 0))

Phase to mirror motion zero? True

Detuned cavity

Detuning the cavity creates an optical spring. Notably, the mechanics of the mirror are modified and the susceptibility is no longer that of a free mass

axs = bode(F_Hz, detuned.mech[..., 0, 0], db=False)
axs[0].set_title("Mechanical modification")
axs = bode(F_Hz, detuned.mech[..., 0, 1], db=False)
axs[0].set_ylabel("Magnitude [m / N]")
axs[0].set_title("Radiation pressured modified susceptibility");

Both quadratures now cause mirror motion

axs = bode(F_Hz, detuned.rp[..., 0, 0], db=False, label="amplitude")
bode(F_Hz, detuned.rp[..., 0, 1], axs=axs, db=False, label="phase")
axs[0].set_ylabel("Magnitude [m / $\sqrt{\mathrm{W}}$]")
axs[0].set_title("Field to mirror motion");

and radiation pressure causes both quadratures to mix on reflection of the cavity

axs = bode(
    F_Hz, detuned.refl[..., 1, 0], db=False, label="amplitude to phase",
)
bode(
    F_Hz, detuned.refl[..., 0, 1], axs=axs, db=False, ls="--",
    label="phase to amplitude",
)
axs[0].set_ylabel("Magnitude [$\sqrt{\mathrm{W}}$ / $\sqrt{\mathrm{W}}$]")
axs[0].set_title("Field reflection off of the cavity");