Optics 046 · Interference, coherence, cavities, and metrology

LIGO Interferometer

A cutaway optical core links suspended test masses, partially transmitting arm mirrors and carrier recycling to complex storage response, a synthetic photodiode tone and an explicitly assumed differential-strain noise budget.

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 $\text{valid model regime}$

Validated scientific apparatus

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 arm changes cancel. Add Fabry–Pérot arms and sweep frequency: storage boosts phase but delays and attenuates fast motion. The recycling preset raises carrier power; its noise curves are assumptions, not LIGO data. Set the readout bias to zero to lose first-order DC detection. Drag to orbit; 3D motion is normalized and slowed, never an optical photon movie.

Ideal lossless resonant single-mode arms and linear readout with an effective phase bias. No signal recycling, feedback calibration, chirp, real detector spectrum, radiation-pressure quantum correlations or astrophysical inference. Arm lengths are compressed; mirror size and motion are schematic. Not a complete Advanced LIGO simulator.

CONTINUE EXPLORING

Understand, then measure

How does a tiny arm-length change become a signal at the output?

Physics tutorial

Audit the physics of LIGO Interferometer

BackgroundFor an overhead, optimally oriented plus-polarized wave in the long-wavelength limit, the differential change is $\Delta L=Lh$ . Each arm moves by half that amount in opposite directions. Here a common-motion mix is an injected disturbance, not another gravitational-wave polarization. The 3994.5 m arms and 1064 nm vacuum wavelength follow the 2015 Advanced LIGO instrument paper; this simplified optical core is not the full instrument.

Why it mattersHow does a power-recycled kilometer-scale interferometer turn strain into a dark-port signal?

Start with the essentials

Focus question: How does a power-recycled kilometer-scale interferometer turn strain into a dark-port signal?
One-sentence intuition: A perfectly dark intensity port is quadratic in a small phase. With a calibrated carrier bias, the first-order signal becomes $P_{\mathrm{AC}}\simeq\frac{P_{\mathrm{BS}}\sin\beta_0}{2}\,\delta\phi$ . Zero bias removes linear DC readout; no sensitivity value is then reported. Real LIGO produces its local oscillator by a small arm detuning and uses an output mode cleaner. This Lab treats that oscillator as an effective phase bias and does not solve the detuned cavities or output mode cleaner.

Core mathematical model

Define differential and common injection

\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 matched common motion. The input amplitude is a known synthetic tone. It is not reconstructed from noisy data, and common-mode amplitude is not labelled gravitational strain. Physical plots retain metre and watt units; 3D geometry uses fixed normalized motion and a four-second display cycle.

Storage is a coherent delayed sum

\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 lossless input mirror form a resonant single-ended arm. Each additional round trip contributes another delayed field weighted by r. The source convention is positive frequency with a complex response multiplying the tone. Lower transmission gives larger DC phase gain but a narrower band. The bare Michelson retains only the propagation delay. No optical loss, transverse modes or lock acquisition are solved.

Mismatch leaks common motion

\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}

Matched arms cancel common motion exactly in this ideal scalar model. Unequal input transmissions have unequal gain and lag, leaving a residual even when geometric differential motion is zero. The response plot and noise reference always use a pure differential calibration injection, while the photodiode trace uses the selected mix. The 10–1000 Hz band assumes wavelength much longer than the arms; the full directional gravitational-wave transfer is omitted.

Carrier recycling is 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}

The resonant, lossless recycling cavity reuses the bright-port carrier. Static reflected plus dark-port output power closes energy. Arm power uses the same beamsplitter power and resonant buildup. The small-signal calculation freezes carrier power at that DC equilibrium: it omits dynamic PRM coupling and bias-dependent amplitude feedback, including first-order terms. This is a block approximation, not the exact coupled optical transfer. Removing the mirror means 100 percent transmission. Ideal gain may exceed the published instrument gain. Signal recycling is absent and must not be confused with power recycling.

An assumed noise budget, not an observed 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}

Use an ideal unit-efficiency Poisson photodiode and one-sided amplitude spectral densities. The residual-motion and coating-shaped differential displacement ASDs are explicit teaching assumptions: 3e-20 metres per root hertz at 10 Hz with inverse-square rolloff, and 4e-20 metres per root hertz at 100 Hz with inverse-square-root rolloff. Divide them by arm length and add independent PSDs, not ASDs. No random noise trace, measured LIGO curve, radiation-pressure backaction, squeezing, detection threshold or event SNR is claimed.

Common difficulties

Dark does not mean linearly sensitive

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

Better mental modelAt exactly zero bias, first-order intensity modulation vanishes. A local oscillator enables signed linear power variation, but a real strain measurement also needs calibrated sensing, actuation and feedback. The 2016 calibration paper explicitly includes those functions and signal recycling. This Lab omits them. With 1.4 percent input transmission its arm-only half-power pole is about 42 Hz and full linewidth about 84 Hz, not the full Advanced LIGO sensing pole.

Run the experiment

01
Scene 1: Biased Michelson readout
Compare differential and common motion in the bare Michelson; set bias to zero and watch linear detection vanish.
What to observe: Partially transmitting input mirrors admit the beam; opaque end faces reflect it. The apparatus never puts an enlarged diagnostic plot in the optical path.
02
Scene 2: Arm storage and bandwidth
Add arm cavities. Sweep frequency and lower input transmission; compare gain and lag, then mismatch the two arms to leak common motion.
What to observe: Carrier power and cavity response are distinct. More storage increases low-frequency gain, not unlimited high-frequency sensitivity.
03
Scene 3: Power recycling and assumed noise
Enable recycling. Compare 3 percent with 100 percent transmission, raise laser power and vary the assumed disturbance multiplier.
What to observe: Noise spectra are one-sided ASDs; the clean waveform is a noiseless synthetic tone. Neither is an observation or a reconstructed gravitational-wave event.