L05 · Gain and feedback

Passive Optical Resonator

Add the returning complex field to each injected field. Compare resonance, destructive interference and impedance matching; switch off the drive and measure the cavity memory.

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 $\text{—}$

Current frequency offset $\text{—}$

Last-step energy residual $\text{—}$

L05 · Experiment

How can two passive mirrors select a frequency 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 on resonance, change only the input phase, then turn off the drive. Use the phasor and energy ledger to explain each transient.

Single axial mode envelope sampled at the input mirror once per round trip. Lossless mirror scattering and symmetric distributed attenuation; reflectivities are power fractions. Powers in the last-step ledger share the same time sample. Frequency offset is relative to a reference longitudinal resonance at each selected length. The optical standing wave is schematic, not wavelength-resolved. No gain, transverse stability or frequency-dependent mirror dispersion.

Start with the essentials

Focus question: Does changing the global input phase change steady transmission or only its approach?
One-sentence intuition: Phase controls constructive addition. Loss controls how long the field persists. Reflection, transmission, internal loss and stored energy must close the same ledger.

Core mathematical model

The field recurrence

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}

The amplitude convention absorbs fixed mirror phases into the reference resonance. Distributed power survival per round trip is called S; input phase is a separate global drive phase.

Airy reference

\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}

Half-round-trip power survival is the square root of S. This is a steady monochromatic reference; a finite-speed scan may not reach it.

Frequency spacing and energy memory

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 exponential energy decay time sampled once per round trip. All selected lengths are vacuum optical lengths.

Exact linewidth and 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 reflectivity range ensures a half-maximum exists. On resonance, the matching condition cancels steady reflected power, even though internal loss can prevent unit transmission.

One packet energy ledger

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

Multiplication by the round-trip time turns this power ledger into the sampled packet-energy ledger. Stored-field power is sampled after the new input encounter; outgoing powers belong to the preceding circulating packet.

Common difficulties

Circulating power is stored energy flow

Typical misconceptionLarge circulating power means the cavity creates energy.

Better mental modelThe last-step ledger includes the change in stored field. Passive mirrors only redirect energy, and internal attenuation removes it.

Input phase and cavity phase differ

Typical misconceptionChanging global input phase changes the resonance frequency.

Better mental modelA constant global phase rotates the entire steady field. The relative round-trip phase determines resonance; an input-phase jump causes a transient.

Run the experiment

01
Build one encounter at a time
Select On resonance and step through several round trips. Observe the returned, injected and sum vectors, then advance 100 trips.
What to observe: Coherent additions build a circulating field; the prompt reflected field cancels the returning leakage in steady state.
02
Change phase without changing power
After buildup, change input phase by 180 degrees. Step and then advance several groups of 100 trips.
What to observe: The old stored field initially interferes with the changed drive. After settling, the transmitted power returns to its prior value.
03
Measure ring-down
Turn Drive on off after buildup. Compare powers separated by the same number of round trips.
What to observe: Stored power decreases by a constant survival factor per round trip; the phasor no longer receives an injected contribution.
04
Match or scan
Compare Impedance matched and Internal loss, then use Continuous scan.
What to observe: Matching suppresses reflected power on resonance. The dynamic scan trace has finite buildup time, while the Airy plot remains a labeled steady reference.