Optics 046 · Interference, coherence, cavities, and metrology

LIGO Interferometer

How does a kilometer-long interferometer turn a passing gravitational wave into a measurable signal? A gravitational wave stretches one arm and squeezes the other, so the two returning beams shift in phase. Arm cavities bounce the light many times to build up that shift, and power recycling sends light heading back toward the laser in again, raising the stored power. The output sits slightly off perfect darkness, so the leaking light changes in step with the phase.

A cutaway of the optical core shows suspended test masses, partially transmitting arm mirrors and carrier recycling. Follow the complex response of the stored light, a synthetic photodiode tone and explicitly assumed noise curves for differential strain.

Injected differential motion / phase amplitude

A_{x-y},A_\phi

$0.50$

Photodiode carrier / first-order signal

P_{\mathrm{DC}},A_{\mathrm{AC}}

$50\%$

One-arm carrier power / assumed differential sensitivity

P_x,\sqrt{S_h}(f)

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

LIGO Interferometer

Reference injection amplitude exponent

e_h

-21

Signal frequency

\nu

120\,\mathrm{Hz}

Common-mode mix (zero differential, one common)

q

0

Arm input-mirror power transmission

T_{\mathrm{ITM}}

1.40\%

Relative arm transmission mismatch

\epsilon_T

0\%

Recycling-mirror transmission (100 percent removes it)

T_{\mathrm{PRM}}

3\%

Calibrated effective dark-port phase bias

\beta_0

30\,\mathrm{mrad}

Input laser power

P_{\mathrm{in}}

25\,\mathrm W

Assumed displacement ASD multiplier

a_n

1

Start with the bare Michelson. Mix toward common motion: equal changes in both arms cancel. Add Fabry–Pérot arm cavities and sweep the frequency: the stored light boosts the phase but delays fast motion and weakens its signal. The recycling preset raises the carrier power; its noise curves are assumptions, not LIGO data. Set the readout bias to zero and first-order DC detection is lost. Drag to orbit. The 3D motion is normalized and slowed; it does not show photons.

The arms are ideal, lossless, resonant and single-mode, and the readout is linear with an effective phase bias. Signal recycling, feedback calibration, chirps, a real detector spectrum, radiation-pressure quantum correlations and astrophysical inference are not included. Arm lengths are compressed, and mirror size and motion are schematic. This is not a complete Advanced LIGO simulator.

Physics tutorial

Audit the physics of LIGO Interferometer

BackgroundTake a plus-polarized gravitational wave arriving from directly overhead, oriented for the strongest response, with a wavelength much longer than the arms. It changes the difference in arm length by $\Delta L=Lh$ . One arm stretches by half that amount and the other shrinks by the same half. The common motion you can mix in is an injected disturbance, not a second gravitational-wave polarization. The 3994.5 m arm length and 1064 nm vacuum wavelength come from the 2015 Advanced LIGO instrument paper, but this simplified optical core is not the full instrument.

Why it mattersHow does a kilometer-long interferometer turn a passing gravitational wave into a measurable signal?

Start with the essentials

Focus question: How does a kilometer-long interferometer turn a passing gravitational wave into a measurable signal?
One-sentence intuition: At a perfectly dark output port, the intensity responds only to the square of a small phase change. A calibrated carrier bias gives a signal that is linear in the phase: $P_{\mathrm{AC}}\simeq\frac{P_{\mathrm{BS}}\sin\beta_0}{2}\,\delta\phi$ . At zero bias the linear DC readout is lost, and no sensitivity value is reported. Real LIGO makes its local oscillator by slightly detuning the arms and also uses an output mode cleaner. Here the local oscillator is an effective phase bias, and neither the detuned cavities nor the output mode cleaner is solved.

Core mathematical model

Injected differential and common arm motion

\begin{aligned}\delta L_x(t)&=\tfrac12 Lh_0\sin(2\pi ft)\\\delta L_y(t)&=(2q-1)\delta L_x(t)\\\Delta L(t)&=(1-q)Lh_0\sin(2\pi ft)\end{aligned}

The mix q runs from pure differential motion to equal motion of both arms in the same direction. The input is a synthetic tone with known amplitude, not something recovered from noisy data. Common-mode amplitude is not called gravitational strain. The physical plots keep units of metres and watts; the 3D drawing uses a fixed, normalized motion that repeats every four seconds.

Arm storage adds delayed fields coherently

\begin{aligned}r&=\sqrt{1-T},\quad\tau=2L/c\\H(f)&=\frac{(1+r)e^{-i\pi f\tau}}{1-re^{-i2\pi f\tau}}\\H(0)&=\frac{1+r}{1-r}\end{aligned}

An ideal, perfectly reflecting end mirror and a lossless input mirror form a resonant arm that light enters and leaves only through the input mirror. Each extra round trip adds another delayed copy of the field, weighted by a further factor of r. The convention uses positive frequency, with the complex response multiplying the tone. Lower input transmission gives more phase gain at zero frequency but a narrower band. The bare Michelson keeps only the travel delay. Optical loss, transverse modes and lock acquisition are not solved.

Unequal arms let common motion leak through

\begin{aligned}T_x&=T(1+\epsilon_T),\ T_y=T(1-\epsilon_T)\\H_D&=\tfrac12[H_x-(2q-1)H_y]\\\delta\phi(f)&=\frac{4\pi L}{\lambda_0}H_D(f)h(f)\end{aligned}

In this ideal scalar model, matched arms cancel common motion exactly. If the two input mirrors transmit differently, the arms have different gain and lag, so a residual signal remains even when the geometric differential motion is zero. The response plot and the noise reference always use a pure differential calibration signal; the photodiode trace uses the mix you choose. The 10–1000 Hz band assumes the gravitational wavelength is much longer than the arms, and the full direction-dependent response to a gravitational wave is left out.

Power recycling raises carrier power, not phase gain

\begin{aligned}B_P&=\frac{T_P}{[1-\sqrt{1-T_P}\cos(\beta_0/2)]^2}\\P_{\mathrm{BS}}&=B_PP_{\mathrm{in}}\\P_{\mathrm{DC}}&=P_{\mathrm{BS}}\sin^2(\beta_0/2)\end{aligned}

A resonant, lossless recycling cavity sends the carrier from the bright port back in. In steady state, the reflected power and the dark-port power add up to the input power. The arm power uses the same beamsplitter power and resonant buildup. The small-signal calculation holds the carrier power at this DC equilibrium. It leaves out dynamic coupling to the recycling mirror and amplitude feedback that depends on bias, including first-order terms. This is a block-by-block approximation, not the exact coupled optical response. Setting the transmission to 100 percent removes the mirror. The ideal gain can exceed the gain published for the real instrument. Signal recycling is not included, and it is a different technique from power recycling.

Assumed noise sources, not a measured spectrum

\begin{aligned}\sqrt{S_{P,\mathrm{shot}}}&=\sqrt{2(h_Pc/\lambda_0)P_{\mathrm{DC}}}\\\sqrt{S_{h,\mathrm{shot}}}&=\frac{\sqrt{S_{P,\mathrm{shot}}}}{|\partial P_{\mathrm{AC}}/\partial h|}\\S_{h,\mathrm{total}}&=S_{h,\mathrm{shot}}+S_{h,\mathrm{res}}+S_{h,\mathrm{coat}}\end{aligned}

The photodiode is ideal, with unit efficiency and Poisson statistics, and all spectra are one-sided amplitude spectral densities (ASDs). The residual-motion and coating-shaped differential displacement ASDs are stated teaching assumptions: 3e-20 metres per root hertz at 10 Hz, falling as the inverse square of frequency, and 4e-20 metres per root hertz at 100 Hz, falling as the inverse square root of frequency. Each is divided by the arm length, and independent noises add as power spectral densities, not as ASDs. The Lab shows no random noise trace, measured LIGO curve, radiation-pressure backaction, squeezing, detection threshold or event SNR.

Common difficulties

A dark port is not a linear detector

Typical misconceptionMake the output perfectly black and any tiny strain becomes directly visible.

Better mental modelAt exactly zero bias, the intensity has no first-order response. A local oscillator gives a linear power change whose sign follows the strain, but a real strain measurement also needs calibrated sensing, actuation and feedback. The 2016 calibration paper includes those functions and signal recycling; this Lab leaves them out. With 1.4 percent input transmission, the arm alone has a half-power pole near 42 Hz and a full linewidth near 84 Hz. That is not the sensing pole of the full Advanced LIGO detector.

Run the experiment

01
Scene 1: Biased Michelson readout
In the bare Michelson, compare differential and common motion. Then set the bias to zero and watch the linear signal disappear.
What to observe: The partly transmitting input mirrors let the beam in, and the opaque end mirrors reflect it back. No enlarged diagnostic plot is drawn inside the optical path.
02
Scene 2: Arm storage and bandwidth
Add the arm cavities. Sweep the frequency, lower the input transmission and compare the gain and lag. Then make the two arms unequal and watch common motion leak into the signal.
What to observe: Carrier power and cavity response are different quantities. Storing light longer raises the gain at low frequency; it does not give unlimited sensitivity at high frequency.
03
Scene 3: Power recycling and assumed noise
Turn on power recycling. Compare 3 percent with 100 percent transmission, raise the laser power and change the assumed disturbance multiplier.
What to observe: The noise spectra are one-sided ASDs, and the clean waveform is a synthetic tone without noise. Neither one is an observed or reconstructed gravitational-wave event.

LIGO Interferometer

Audit the physics of LIGO Interferometer

Core mathematical model

Injected differential and common arm motion

Arm storage adds delayed fields coherently

Unequal arms let common motion leak through

Power recycling raises carrier power, not phase gain

Assumed noise sources, not a measured spectrum

Common difficulties

A dark port is not a linear detector

Run the experiment

Scene 1: Biased Michelson readout

Scene 2: Arm storage and bandwidth

Scene 3: Power recycling and assumed noise

Related experiments and articles

Understand, then measure

Gravity leaves a trace