L05 · Gain and feedback

Passive Optical Resonator

How can two mirrors with no gain between them pick out frequencies and store light? Returning light interferes with the new drive. Frequencies that return in step build the strongest stored field; detuning weakens the buildup. When the drive stops, the stored energy loses a fixed fraction each round trip. Smaller total losses give a longer memory, without any gain.

Sweep through resonance and watch recorded light build up between partially transmitting mirrors. Switch off the drive to compare the decay.

Round trips $\text{—}$

Elapsed model time $\text{—}$

Circulating power after input encounter $\text{—}$

Last-step transmitted power $\text{—}$

Last-step reflected power $\text{—}$

Free spectral range $\text{—}$

Exact Airy linewidth $\text{—}$

Finesse $\text{—}$

Energy ring-down lifetime $\text{—}$

Next-encounter frequency offset $\text{—}$

Last-step energy residual $\text{—}$

L05 · Experiment

How can two mirrors with no gain between them pick out frequencies and store light?

Drive onTurn off after buildup to observe ring-down.Sweep frequencyAdvance by 0.003 free spectral ranges per round trip.Vacuum cavity length

L

50\,\mathrm{cm}

Input power reflectivity

R_1

95\,\mathrm{\%}

Output power reflectivity

R_2

95\,\mathrm{\%}

Internal round-trip power loss

\ell

0\,\mathrm{\%}

Frequency offset

\Delta\nu

0\,\mathrm{MHz}

Input phase

\varphi_{\mathrm{in}}

0\,\mathrm{^\circ}

Drive power

P_{\mathrm{in}}

10\,\mathrm{mW}

Build up the field on resonance, change only the input phase, then turn off the drive. Use the phasor diagram and the energy balance to explain each transient.

Entry and changes acquire 300 actual round trips. Changing phase or turning the drive off continues the stored field; other physical settings start a fresh calculation. A fresh drive-off case first charges for 300 trips at the selected fixed frequency. The last acquisition replays over eight display seconds, with endpoint readouts frozen. Counterpropagating channel markers do not resolve a standing-wave wavelength. Brightness uses fixed logarithmic references, 100 mW externally and 2000 mW inside; it is not a calibrated image. Source and return share a prescribed boundary port; no optical circulator or feedback into the source is simulated. The separate Airy curve is a steady reference, and the phase/energy diagram describes the last encounter. Powers from the old circulating packet and the new stored packet remain distinguished. Markers use port powers; attenuation along the path is not spatially resolved, and markers are not individual photons.

The model follows the envelope of a single axial mode, sampled at the input mirror once per round trip. The mirrors scatter without absorbing, and the distributed attenuation is symmetric; reflectivities are power fractions. All powers in the last-step energy balance come from the same time sample. The frequency offset is measured from a reference longitudinal resonance at each chosen length. Counterpropagating channel markers are schematic and do not resolve the optical wavelength. Gain, transverse stability and frequency-dependent mirror dispersion are not included.

Start with the essentials

Focus question: If you shift the overall phase of the input, does the steady transmission change, or only the way it gets there?
One-sentence intuition: Phase decides whether each return adds constructively, and loss decides how long the field lasts. Reflected, transmitted and lost power, plus the change in stored energy, always add up to the input.

Core mathematical model

The field after each round trip

a_{n+1}=\sqrt{R_1R_2S}\,e^{i\phi_n}a_n+\sqrt{1-R_1}\,d_n,\qquad\phi_n=2\pi\Delta\nu_n/\mathrm{FSR}

Fixed mirror phases are folded into the reference resonance. S is the fraction of power that survives the distributed loss in one round trip. The input phase is a separate, overall phase of the drive.

Steady Airy transmission

\frac{P_t}{P_{\mathrm{in}}}=\frac{(1-R_1)(1-R_2)\sqrt S}{1+r^2-2r\cos\phi},\qquad r=\sqrt{R_1R_2S}

Over half a round trip, the surviving power fraction is the square root of S. The curve is the steady response at a single frequency; a scan at finite speed may not reach it.

Mode spacing and photon lifetime

T_R=\frac{2L}{c},\quad\mathrm{FSR}=T_R^{-1},\quad\tau_p=-\frac{T_R}{\ln(R_1R_2S)}

The photon lifetime is the exact time constant of the exponential energy decay, sampled once per round trip. All cavity lengths are vacuum optical lengths.

Exact linewidth and impedance matching

\delta\nu=\frac{2\,\mathrm{FSR}}{\pi}\arcsin\frac{1-r}{2\sqrt r},\quad\mathcal F=\frac{\mathrm{FSR}}{\delta\nu},\quad R_1=R_2S

The allowed reflectivity range guarantees that each resonance has a half-maximum point. On resonance, the matching condition cancels the steady reflected power, although internal loss can still keep transmission below one.

Energy balance for one round trip

P_{\mathrm{in},n}=P_{r,n}+P_{t,n}+P_{\mathrm{loss},n}+\left(|a_{n+1}|^2-|a_n|^2\right)

Multiplying by the round-trip time turns this power balance into an energy balance for one sampled packet. The stored power is sampled just after the new input arrives; the outgoing powers come from the packet that circulated just before.

Common difficulties

Circulating power is stored energy moving around

Typical misconceptionIf the circulating power is large, the cavity must be creating energy.

Better mental modelThe energy balance for each step includes the change in stored energy. Passive mirrors only redirect energy, and internal attenuation removes it.

Input phase is not the cavity phase

Typical misconceptionChanging the overall input phase moves the resonance frequency.

Better mental modelA constant overall phase just rotates the whole steady field. Resonance depends on the phase picked up in one round trip; a sudden jump in input phase causes a transient.

Run the experiment

01
Build the field one round trip at a time
Reset to On resonance and replay the recorded buildup. Expand Last encounter to inspect the returned, injected and summed vectors; Advanced controls also offers Record one actual round trip.
What to observe: Each in-phase addition makes the circulating field larger. In steady state, the light reflected straight off the input mirror cancels the light leaking back out of the cavity.
02
Change phase without changing power
In Advanced controls, shift Input phase by 180 degrees. The new 300-trip acquisition continues the stored field; compare its transient with the endpoint readouts.
What to observe: At first the old stored field interferes with the shifted drive. Once it settles, the transmitted power is back at its earlier value.
03
Measure ring-down
After buildup, uncheck Drive on. Compare powers taken the same number of round trips apart.
What to observe: The stored power drops by the same survival factor every round trip, because nothing new is injected into the phasor.
04
Impedance matching and frequency scans
Compare the Impedance matched and Internal loss presets, then try Continuous scan.
What to observe: Matching suppresses the reflected power on resonance. The scan trace includes the finite time the field needs to build up; the Airy plot stays as a labeled steady reference.

Passive Optical Resonator

Add fields before measuring power

Core mathematical model

The field after each round trip

Steady Airy transmission

Mode spacing and photon lifetime

Exact linewidth and impedance matching

Energy balance for one round trip

Common difficulties

Circulating power is stored energy moving around

Input phase is not the cavity phase

Run the experiment

Build the field one round trip at a time

Change phase without changing power

Measure ring-down

Impedance matching and frequency scans

Related experiments and articles

Build up, scan, switch off

Controlled light and precision measurement