Skip to main content
Sandbox Physics

Optics 048 · Interference, coherence, cavities, and metrology

Fabry–Pérot Passive Cavity

A movable two-mirror 3D cavity connects coherent multi-return sums, transmitted and reflected photodiodes, distributed absorption, exact Airy linewidth and a separate energy-ringdown experiment.

Interactive modelFabry–Pérot Passive Cavity
Transmitted / reflected / absorbed power PT,PR,PAP_T,P_R,P_A0.500.50
Mode spacing / exact Airy width / finesse FSR,ΔνA,FA\mathrm{FSR},\Delta\nu_A,\mathcal F_A50%50\%
Energy lifetime / Lorentzian width / matched input τE,ΔνL,R1,crit\tau_E,\Delta\nu_L,R_{1,\mathrm{crit}}0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

CONTINUE EXPLORING

Build up, scan, switch off

How are a narrow resonance and a long storage time connected?

Physics tutorial

Audit the physics of Fabry–Pérot Passive Cavity

BackgroundA partially transmitting mirror sends a new field into the cavity while earlier fields return. Their complex amplitudes add before squaring; the transmitted power fraction is T=(1−R1)(1−R2)a∣1−ρeiδ∣2T=\frac{(1-R_1)(1-R_2)a}{|1-\rho e^{i\delta}|^2}. Both coatings are lossless, but an independent distributed intracavity loss can absorb power. This is a one-dimensional normal-incidence plane-wave model, not a stable finite Gaussian-mode resonator.

Why it mattersWhy can two reflective mirrors transmit all light, and how do loss and lifetime change the resonance?

Start with the essentials

Focus question
Why can two reflective mirrors transmit all light, and how do loss and lifetime change the resonance?
One-sentence intuition
Perfect resonant transmission does not violate mirror reflectivity or create energy. Track T+R+A=1T+R+A=1. With loss, a matched input cancels reflection, but some energy is absorbed rather than transmitted. The Airy scan width and energy-decay linewidth become nearly equal only in the high-finesse limit.

Core mathematical model

Length, frequency and the round-trip factor

a=1−ℓ,ρ=R1R2 atRT=2L/c,FSR=c/(2L)νm=m FSR,δ=2πΔν/FSR\begin{aligned}a&=1-\ell,\quad \rho=\sqrt{R_1R_2}\,a\\t_{\mathrm{RT}}&=2L/c,\quad\mathrm{FSR}=c/(2L)\\\nu_m&=m\,\mathrm{FSR},\quad\delta=2\pi\Delta\nu/\mathrm{FSR}\end{aligned}

Reflectivities are power fractions. The loss control is a percentage, converted to a fraction here; a is single-pass power survival, so the two-pass amplitude multiplier contains a, not its square root. The selected integer mode is nearest the 632.8 nm design carrier. Changing length also retunes the source to that mode plus the selected normalized detuning; it is not a fixed-laser length scan. Propagation speed is c; attenuation is prescribed independently of dispersion.

Coherent returns, not independent ray powers

∑j=0∞(ρeiδ)j=11−ρeiδD=(1−ρ)2+4ρsin⁡2(δ/2)G+=(1−R1)/D\begin{aligned}\sum_{j=0}^{\infty}(\rho e^{i\delta})^j&=\frac1{1-\rho e^{i\delta}}\\D&=(1-\rho)^2+4\rho\sin^2(\delta/2)\\G_+&=(1-R_1)/D\end{aligned}

G is the forward intracavity travelling power at the input face divided by incident power, not the total stored energy or a net output gain. The phasor panel draws 16 terms and the exact infinite sum separately. Its omitted field fraction is explicitly displayed; high reflectivity makes 16 returns far from sufficient. Phasor scaling adapts and does not encode an absolute electric field.

Both detector ports and distributed absorption

T=(1−R1)(1−R2)a/DR=(R1+R2a2−2ρcos⁡δ)/DA=(1−R1)(1−a)(1+R2a)/DPT=PinT,T+R+A=1\begin{aligned}T&=(1-R_1)(1-R_2)a/D\\R&=(R_1+R_2a^2-2\rho\cos\delta)/D\\A&=(1-R_1)(1-a)(1+R_2a)/D\\P_T&=P_{\mathrm{in}}T,\quad T+R+A=1\end{aligned}

Reflection includes interference between the prompt reflection and cavity leakage. At resonance with identical lossless mirrors it vanishes and transmission is unity. The absorbing medium uses uniform effective power attenuation; neither coating absorbs. The ideal lossless three-port circulator routes input into the cavity and the coaxial return into the reflection photodiode. It is not a reciprocal beam splitter or an opaque folding mirror.

Exact Airy half-peak width has a boundary

ΔνAFSR=2πarcsin⁡1−ρ2ρFA=FSR/ΔνAρ≥3−22\begin{aligned}\frac{\Delta\nu_A}{\mathrm{FSR}}&=\frac2\pi\arcsin\frac{1-\rho}{2\sqrt\rho}\\\mathcal F_A&=\mathrm{FSR}/\Delta\nu_A\\\rho&\ge 3-2\sqrt2\end{aligned}

Half height means half the resonant peak, not half the difference between peak and valley. Below the stated feedback boundary the whole scan lies above half peak: no Airy FWHM or Airy finesse is reported. The displayed spectrum uses extra samples at each narrow peak and a separate zoom, avoiding uniform-grid aliasing. The plot is an ideal frequency scan, not an experimentally measured laser spectrum.

Energy lifetime is not amplitude lifetime

QRT=R1R2a2=ρ2τE=−tRT/ln⁡QRTU(t)/U0=e−t/τEΔνL=1/(2πτE)\begin{aligned}Q_{\mathrm{RT}}&=R_1R_2a^2=\rho^2\\\tau_E&=-t_{\mathrm{RT}}/\ln Q_{\mathrm{RT}}\\U(t)/U_0&=e^{-t/\tau_E}\\\Delta\nu_L&=1/(2\pi\tau_E)\end{aligned}

The independent switch-off experiment shows the exponential stored-energy envelope and exact integer-round-trip survival samples. Field amplitude decays twice as slowly. The Lorentzian modal width comes from Fourier transforming that decaying field; it is distinct from the exact periodic Airy scan width at low finesse. Between round trips this is an envelope, not a full abrupt-shutoff Maxwell transient or detector response. Moving the sampling-time slider does not turn off the driven 3D bench.

Critical coupling with an absorbing cavity

δ=0,R1,crit=R2a2R=0,T+A=1\begin{aligned}\delta&=0,\quad R_{1,\mathrm{crit}}=R_2a^2\\R&=0,\quad T+A=1\end{aligned}

The input amplitude reflectivity must match the attenuated return amplitude. Loss makes the matched input mirror less reflective than the output mirror. The shortcut picks the nearest input-slider step, so a general match can leave a small reflection residual; the 10 percent loss preset matches exactly. No gain medium, population inversion, laser threshold, transverse-mode stability or real-instrument precision is simulated.

Common difficulties

More reflective does not always mean more transmitted

Typical misconceptionHigh reflectivity always strengthens transmission, and zero reflection proves a lossless cavity.

Better mental modelAway from resonance, raising reflectivity suppresses transmission. On resonance only a matched lossless pair reaches unity. In the lossy critical-coupling preset reflection still vanishes, but the absorption bar is nonzero. Narrow resonances also store energy longer, trading bandwidth for lifetime.

Run the experiment

  1. 01

    Scene 1: Lossless resonant transmission

    Start on resonance, then detune; compare the physical output routes with the two detector fractions.

    What to observe: Cavity returns share one axis; partial-mirror substrates stay clear and detector bodies lie behind their receiving faces.
  2. 02

    Scene 2: Narrow peaks and energy storage

    Use the narrow-peak preset. Drag the gold output mirror: gap, FSR and lifetime change together while normalized detuning stays fixed.

    What to observe: All data panels sit outside the 3D apparatus. Beam brightness is input-normalized and logarithmic; input power changes absolute readouts, not normalized curves or beam brightness. Hardware sizes are schematic, not a calibrated photograph.
  3. 03

    Scene 3: Lossy critical coupling

    Try lossy critical coupling, mismatch the input, then restore the nearest match. Set both reflectivities to their minimum to lose the Airy half-height width.

    What to observe: Play animates route markers only. Reduced motion preserves the same steady-state power, spectrum and independent ringdown data.
FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy