Skip to main content
Sandbox Physics

Optics 090 · Waveguides, structured light, and modern optics

Gaussian Beam & Gouy Phase

An independently initialized three-dimensional apparatus connects Free-space waist, Lens focus transform, Mode mismatch and Gouy phase. Two dimensional physical controls, direct probe dragging, a detector trace, and three quantitative checks are recalculated from the stated equation.

Interactive modelGaussian Beam & Gouy Phase
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 Gaussian Beam & Gouy Phase

BackgroundGaussian Beam & Gouy Phase is one independently initialized apparatus with three linked investigations: Free-space waist, Lens focus transform, Mode mismatch and Gouy phase. Its two controls—Waist radius and Axial position—feed the governing relation w(z)=w01+(zzR)2w(z)=w_0\sqrt{1+\left(\frac{z}{z_{\mathrm R}}\right)^2}. The page uses the stated modal, coupled-mode, effective-medium, or envelope approximation and marks its breakdown instead of presenting it as a full-wave result.

Why it mattersHow do waist, divergence, lens transformation, and Gouy phase describe one focused beam?

Start with the essentials

Focus question
How do waist, divergence, lens transformation, and Gouy phase describe one focused beam?
One-sentence intuition
The detector curve and all three numerical readouts are recomputed from w(z)=w01+(zzR)2w(z)=w_0\sqrt{1+\left(\frac{z}{z_{\mathrm R}}\right)^2}. 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

w(z)=w01+(zzR)2w(z)=w_0\sqrt{1+\left(\frac{z}{z_{\mathrm R}}\right)^2}

The implementation evaluates this relation with dimensional inputs and an executable analytic or numerical benchmark. The page uses the stated modal, coupled-mode, effective-medium, or envelope approximation and marks its breakdown instead of presenting it as a full-wave result.

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: Free-space waist

    Select Free-space waist. Sweep Waist radius, hold Axial position 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: Lens focus transform

    Select Lens focus transform. Sweep Waist radius, hold Axial position 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: Mode mismatch and Gouy phase

    Select Mode mismatch and Gouy phase. Sweep Waist radius, hold Axial position 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.