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Sandbox Physics

Optics 104 · How lasers are generated and controlled

Stable Resonator & Transverse Modes

An independently initialized three-dimensional apparatus connects Stability-map design, Hermite–Gaussian modes, Laguerre–Gaussian modes. Two dimensional physical controls, direct probe dragging, a detector trace, and three quantitative checks are recalculated from the stated equation.

Interactive modelStable Resonator & Transverse Modes
Primary prediction P1\mathcal P_10.500.50
Physical scale P2\mathcal P_250%50\%
Limit check V\mathcal V0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

How to investigate Stable Resonator & Transverse Modes

BackgroundStable Resonator & Transverse Modes is one independently initialized apparatus with three linked investigations: Stability-map design, Hermite–Gaussian modes, Laguerre–Gaussian modes. Its two controls—Mirror radius and Mode order—feed the governing relation 0<g1g2<1,gi=1−LRi0<g_1g_2<1,\qquad g_i=1-\frac{L}{R_i}. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Why it mattersWhich mirror geometry produces a stable low-loss cavity dominated by the fundamental transverse mode?

Start with the essentials

Focus question
Which mirror geometry produces a stable low-loss cavity dominated by the fundamental transverse mode?
One-sentence intuition
The detector curve and all three numerical readouts are recomputed from 0<g1g2<1,gi=1−LRi0<g_1g_2<1,\qquad g_i=1-\frac{L}{R_i}. Geometry and glow are presentation encodings; the equation, units, conservation or limit check, and validity indicator are the quantitative evidence.

Core mathematical model

Governing relation

0<g1g2<1,gi=1−LRi0<g_1g_2<1,\qquad g_i=1-\frac{L}{R_i}

The implementation evaluates this relation with dimensional inputs and an executable analytic or numerical benchmark. The validity indicator marks the paraxial, lossless, weak-field, or steady-state assumption used by this apparatus.

Common difficulties

Mistaking glow for measured power

Typical misconceptionA brighter cinematic trail must represent proportionally more optical power.

Better mental modelUse the detector and normalized readouts for comparison. Glow is deliberately nonlinear so weak structure stays visible.

Run the experiment

  1. 01

    Scene 1: Stability-map design

    Select Stability-map design. Sweep Mirror radius, hold Mode order fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  2. 02

    Scene 2: Hermite–Gaussian modes

    Select Hermite–Gaussian modes. Sweep Mirror radius, hold Mode order fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.
  3. 03

    Scene 3: Laguerre–Gaussian modes

    Select Laguerre–Gaussian modes. Sweep Mirror radius, hold Mode order fixed, and then reverse the roles. Drag the stage probe to repeat the first sweep directly.

    What to observe: Read the primary prediction, physical scale, limit check, and validity indicator together. Record where the approximation boundary changes.