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Optics 048 · Interference, coherence, cavities, and metrology

Fabry–Pérot Passive Cavity

How can two highly reflective mirrors let all the light through, and what does loss do to the resonance? On resonance, the light leaking out after each round trip adds up in step on the far side and cancels the light reflected at the front. With two identical mirrors and no loss, nothing is reflected and everything gets through; no energy is created. Loss inside the cavity absorbs some light and shortens how long the light stays stored, which broadens the resonance.

Move one mirror of a 3D two-mirror cavity and sum its many coherent round trips. Photodiodes read the transmitted and reflected light; add distributed absorption, check the exact Airy linewidth and run 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}

Physics tutorial

Audit the physics of Fabry–Pérot Passive Cavity

BackgroundA partially transmitting mirror keeps sending new light into the cavity while light from earlier round trips comes back. The complex amplitudes add before they are squared, and 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}. Neither coating absorbs, but a separate loss spread through the cavity can absorb power. The model is a one-dimensional plane wave at normal incidence, not a stable resonator with a finite Gaussian mode.

Why it mattersHow can two highly reflective mirrors let all the light through, and what does loss do to the resonance?

Start with the essentials

Focus question
How can two highly reflective mirrors let all the light through, and what does loss do to the resonance?
One-sentence intuition
Full transmission at resonance does not contradict the mirror reflectivity, and it creates no energy. Keep track of T+R+A=1T+R+A=1. With loss, a matched input can cancel the reflection, yet part of the energy is absorbed instead of transmitted. The width of the Airy scan and the linewidth from energy decay are nearly equal only at high finesse.

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 takes a percentage, which is converted here to a fraction. The symbol a is the power that survives a single pass, so the amplitude factor for two passes contains a itself, not its square root. The integer mode is the one nearest the 632.8 nm design carrier. Changing the length also retunes the source to that mode plus the chosen normalized detuning, so this is not a length scan with a fixed laser. Light travels at c, and the attenuation is set independently of dispersion.

Returning fields add as amplitudes, not as 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-travelling power inside the cavity at the input face, divided by the incident power. It is not the total stored energy or a net output gain. The phasor panel draws the first 16 terms and, separately, the exact infinite sum, and it shows the field fraction left out. At high reflectivity, 16 returns fall far short. The phasor scale adapts automatically and does not show the absolute electric field.

Both detector ports and the absorbing medium

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}

The reflection comes from interference between light reflected straight off the input mirror and light leaking back out of the cavity. On resonance with identical lossless mirrors, the reflection vanishes and the transmission is one. The absorbing medium has a uniform effective power attenuation; neither coating absorbs. An ideal lossless three-port circulator sends the input into the cavity and sends the light returning along the axis to the reflection photodiode. It is not a reciprocal beam splitter or an opaque folding mirror.

When the exact Airy half-width exists

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 of the resonant peak, not half of the peak-to-valley difference. Below the feedback limit in the last line, the whole scan stays above half the peak, so no Airy FWHM or Airy finesse is reported. The spectrum adds extra samples at each narrow peak and has a separate zoom, so a uniform grid cannot skip a peak. The plot is an ideal frequency scan, not a measured laser spectrum.

Stored energy decays twice as fast as the field

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}

A separate switch-off experiment shows the exponential envelope of the stored energy, together with exact samples of what survives after whole numbers of round trips. The field amplitude decays half as fast. Fourier transforming that decaying field gives the Lorentzian mode width, which differs from the exact periodic Airy scan width at low finesse. Between round trips the curve is only an envelope; it is not a full Maxwell transient after an abrupt shutoff, nor a detector response. Moving the sampling-time slider does not switch 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 amplitude reflectivity of the input mirror must match the attenuated return amplitude. With loss, the matched input mirror is less reflective than the output mirror. The shortcut picks the nearest step of the input slider, so a general match can leave a small residual reflection; the 10 percent loss preset matches exactly. The model has no gain medium, population inversion, laser threshold or transverse-mode stability, and it does not reproduce the precision of a real instrument.

Common difficulties

More reflective does not always mean more transmitted

Typical misconceptionHigher reflectivity always strengthens transmission, and zero reflection proves the cavity is lossless.

Better mental modelAway from resonance, higher reflectivity reduces transmission. On resonance, only a matched lossless pair reaches full transmission. In the lossy critical-coupling preset the reflection still vanishes, but the absorption bar is not zero. Narrower resonances also store energy longer: you give up bandwidth to gain lifetime.

Run the experiment

  1. 01

    Scene 1: Lossless resonant transmission

    Start on resonance, then detune. Compare the light leaving along each physical route with the two detector fractions.

    What to observe: All the returns inside the cavity share one axis. The substrates of the partial mirrors are clear, and each detector body sits behind its receiving face.
  2. 02

    Scene 2: Narrow peaks and energy storage

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

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

    Scene 3: Lossy critical coupling

    Try lossy critical coupling, break the input match, then restore the nearest match. Set both reflectivities to their minimum and the Airy half-height width is no longer defined.

    What to observe: Play animates only the route markers. With reduced motion, the steady-state power, the spectrum and the separate ringdown data stay the same.

Related experiments and articles

Continue exploring

Build up, scan, switch off

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

  1. LabPassive Optical ResonatorFollow successive round trips, then switch off the input.
  2. You are hereFabry–Pérot Passive CavityCompare transmission, reflection, linewidth and ringdown.
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