Fabry–Pérot Arms and Optical Cavities

A Fabry–Pérot arm cavity makes the light sample a differential arm displacement many times, increasing the low-frequency optical response without providing unlimited bandwidth. The same storage that builds signal also averages a rapidly varying displacement. A simple resonant cavity therefore behaves as a gain followed by a pole; a complete interferometer adds recycling cavities, readout quadrature, losses, and feedback. This page derives the cavity scale and keeps those layers distinct.

Required background. The Wave Equation and Characteristics supplies phase accumulation, interference, and resonant boundary conditions.

Helpful background. Gravitational Radiation and Waveforms fixes the strain convention. Probability and Inference for Gravity is useful when optical parameters are treated as uncertain rather than exact.

Resonance in one arm

Take a lossless two-mirror cavity of length $L$. Let $r=r_1r_2$ be the product of the mirrors’ real amplitude reflectivities for one round trip, with $0<r<1$, and let $\Phi=2\omega_0L/c$ be the round-trip carrier phase. Summing the returning fields gives

$$E_{\mathrm{cav}} \propto 1+r e^{i\Phi}+r^2e^{2i\Phi}+\cdots =\frac{1}{1-r e^{i\Phi}}.$$

The positive propagation phase follows from the time dependence $e^{-i\omega_0t}$: a returning field has sampled the input one round trip earlier. Resonance occurs at $\Phi=2\pi m$. Neighboring resonances are separated by the free spectral range

$$f_{\mathrm{FSR}}=\frac{c}{2L}.$$

For high reflectivity, the finesse $\mathcal F$ is the free spectral range divided by the full width at half maximum of the power resonance. In the present idealization,

$$\mathcal F\simeq\frac{\pi\sqrt r}{1-r}, \qquad f_p\simeq\frac{f_{\mathrm{FSR}}}{2\mathcal F} =\frac{c}{4L\mathcal F},$$

where $f_p$ is the half-width, or simple-cavity pole. Stating “linewidth” without saying full width or half width creates a factor-of-two ambiguity.

The energy stored in the cavity decays after the input is removed. After $k$ round trips the field amplitude is multiplied by $r^k$. Its exponential envelope therefore has decay time $\tau_a=-(2L/c)/\ln r$, while the stored energy decays with time constant $\tau_a/2$. In the high-finesse limit, $\tau_a\simeq(2\pi f_p)^{-1}$. Literature definitions of “storage time” can refer to either amplitude or energy, so the defining exponential must accompany the symbol.

Signal buildup and bandwidth

Let a small differential arm displacement $x(t)$ drive a resonant cavity, and let $y(t)$ denote its optical signal. At sideband frequencies small compared with $f_{\mathrm{FSR}}$, the high-finesse cavity is approximated by

$$\tau_a\frac{dy}{dt}+y=G_0x, \qquad y(t)=\frac{G_0}{\tau_a}\int_0^\infty e^{-s/\tau_a}x(t-s)\,ds.$$

The second form displays the physics: the cavity averages earlier displacements with an exponentially decaying weight. With $\widetilde x(f)=\int x(t)e^{2\pi ift}\,dt$, differentiation gives $-2\pi if\widetilde x$. Hence the response is

$$H_{\mathrm{arm}}(f) =\frac{G_0}{1-i f/f_p},$$

where $f_p=(2\pi\tau_a)^{-1}$ within this one-pole approximation and $G_0>0$ fixes the readout sign. Additional propagation or electronics delays must be included separately. Thus

$$\lvert H_{\mathrm{arm}}(f)\rvert =\frac{G_0}{\sqrt{1+(f/f_p)^2}}, \qquad \arg H_{\mathrm{arm}}=+\arctan(f/f_p).$$

Below the pole, successive samples of the displacement add almost coherently. Above it, but still well below the free spectral range, the response falls and lags. A positive Fourier phase denotes a physical lag in this convention: $\operatorname{Re}[e^{i\varphi}e^{-2\pi ift}]=\cos(2\pi ft-\varphi)$. The pole lies at $f=-if_p$, as required for a decaying causal response. The cavity has mapped strain to a frequency-dependent optical signal.

The response is not set by finesse alone. A larger finesse increases the number of coherent samples for a given arm length, but narrows the simple arm bandwidth. Input power, optical loss, mirror thermal noise, angular stability, radiation-pressure effects, and the recycling cavities enter the instrument-level optimization. The Advanced LIGO design makes these tradeoffs explicit in LIGO Scientific Collaboration 2015, §§2.1–2.4 and Table 1, PDF.

From arm cavity to differential readout

Two arm cavities are held near resonance and embedded in a Michelson. A differential gravitational-wave displacement produces oppositely signed arm phase shifts at the antisymmetric port. Several optical structures then act on that sideband:

Structure	Principal role	Why it cannot be replaced by a single gain
Arm cavities	Build differential phase over repeated traversals	Their storage produces a pole and frequency-dependent phase
Power recycling	Returns carrier light toward the interferometer	It changes circulating carrier power, not just signal storage
Signal recycling	Shapes the signal-sideband response	Its tuning and transmission alter the coupled bandwidth
Output mode cleaner and homodyne readout	Select the detected spatial mode and quadrature	The measured quadrature determines which optical field combination reaches the photodiode
Length and alignment controls	Hold the optical operating point	Feedback suppresses residual motion and introduces sensing/actuation transfer functions

The documented Advanced LIGO configuration uses Fabry–Pérot arms, power and signal recycling, and DC readout with an output mode cleaner LIGO Scientific Collaboration 2015, §§2.1–2.3, PDF. That paper describes a design and commissioning architecture, not a timeless observing state. The applicable state and response product must be identified separately on the configuration page.

What the effective pole does not mean

It is often useful to summarize a calibrated detector response by an effective pole or rational transfer function. That summary is conditional on an operating point, control model, sensing function, actuation model, and frequency band. Signal recycling can move the coupled optical response away from the bare arm-cavity value. Losses and detuning can introduce additional scales; calibration filters can compensate a known response only within their validated band.

The distinction is operational. If $d_{\mathrm{err}}(f)$ is a digital error signal, then the conversion to strain has the schematic form

$$\widehat h(f)=R(f)d_{\mathrm{err}}(f),$$

where $R(f)$ reconstructs strain from the coupled sensing and feedback model. The forward optical response $H_{\mathrm{arm}}$ maps displacement to optical signal; the calibration response $R$ maps a recorded digital signal back to strain. They have different roles and units, so substituting one for the other is not a calibration model. The reconstruction is developed on Calibration Response and Uncertainty.

Checks and limits of the results

Three checks discipline the cavity model:

1. $f_{\mathrm{FSR}}$ and $f_p$ must have units of frequency and scale as $L^{-1}$.
2. $H_{\mathrm{arm}}\to G_0$ as $f/f_p\to0$, while its magnitude falls as $f^{-1}$ far above the pole.
3. The impulse response $G_0(2\pi f_p)e^{-2\pi f_p t}\Theta(t)$ transforms to $G_0/(1-if/f_p)$ with the declared Fourier pair.

The stable equations use an ideal, resonant, lossless cavity. Any empirical statement must additionally bind the detector, observing interval, optical state, response version, calibration uncertainty, selections, and frequency band. The public observational interface for this volume is frozen to the GWOSC O4 technical documentation available on 2026-08-18. No current sensitivity is inferred from the design finesse or from the simple pole.

Common pitfalls

Equating optical gain with sensitivity. Sensitivity is set by signal response relative to a frequency-dependent noise budget. Increasing circulating power or finesse changes both signal and several noises.

Using the bare cavity pole as the detector response. Recycling, readout, feedback, and calibration shape the coupled response. The bare pole is a controlled teaching model and a useful limit, not a response file.

Quoting a storage time without its convention. Field amplitude and stored energy decay with different exponents. Define which quantity decays before comparing values.

Exercises

1. A simple 4 km arm. For $L=3994.5\,\mathrm{m}$ and $\mathcal F=450$, compute $f_{\mathrm{FSR}}$ and the ideal simple-cavity pole $f_p$. Why should the latter not be quoted as the coupled observing-state bandwidth?

Solution

Using $c=299\,792\,458\,\mathrm{m\,s^{-1}}$,

$$f_{\mathrm{FSR}}=\frac{c}{2L}\simeq 3.75\times10^4\,\mathrm{Hz},$$

and

$$f_p=\frac{f_{\mathrm{FSR}}}{2\mathcal F} \simeq 41.7\,\mathrm{Hz}.$$

This is the pole of the stated ideal arm model. A dual-recycled, controlled interferometer has a coupled optical response whose effective bandwidth depends on the signal-recycling state, readout, losses, and calibration model.

2. Gain at the pole. Evaluate the magnitude and phase of $H_{\mathrm{arm}}$ at $f=f_p$. What time-domain feature produces the phase lag?

Solution

At the pole,

$$H_{\mathrm{arm}}(f_p)=\frac{G_0}{1-i} =\frac{G_0}{\sqrt2}e^{i\pi/4}.$$

The amplitude is lower than its zero-frequency value by $1/\sqrt2$ and the physical sinusoid lags by $45^\circ$, represented here by positive Fourier phase. The lag is the signature of causal storage: the output at time $t$ contains exponentially weighted information from earlier arm displacements.

References

• Gravitational Wave Open Science Center. “O4 Technical Details.” Version available 2026-08-18. Official documentation.
• LIGO Scientific Collaboration. “Advanced LIGO.” Classical and Quantum Gravity 32 (2015): 074001. DOI. Open PDF.