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Sandbox Physics

Optics 017 · Ray worlds, boundaries, and natural optics

Illumination & Radiometry Studio

A radiometry studio measures point-source inverse-square falloff, receiver cosine response, and finite Lambertian area-source transfer on one physical panel. A live two-dimensional irradiance map and centerline detector trace expose near-field nonuniformity that a single center value would hide.

Interactive modelIllumination & Radiometry Studio
Center illuminance or irradiance E0E_00.500.50
Panel uniformity U0U_050%50\%
Source-model audit Rs\mathcal R_s0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

Audit the physics of Illumination & Radiometry Studio

BackgroundIlluminance from a far-field point source combines geometric spreading with receiver projection: E=Icos⁡θr2E=\frac{I\cos\theta}{r^2}. Across a finite panel both distance and incidence angle vary, so one center measurement cannot establish uniformity.

Why it mattersHow can source size, distance, and surface angle create uniform illumination?

Start with the essentials

Focus question
How can source size, distance, and surface angle create uniform illumination?
One-sentence intuition
A Lambertian area source must be integrated patch by patch using dE=Lcos⁡θscos⁡θrr2 dAs\mathrm dE=L\frac{\cos\theta_s\cos\theta_r}{r^2}\,\mathrm dA_s. Source radiance stays constant; apparent softening and uniformity come from the changing projected solid angle of all visible patches.

Core mathematical model

Point-source illuminance

E=Icos⁡θr2E=\frac{I\cos\theta}{r^2}

The inverse-square factor describes spherical spreading and the cosine describes projected receiver area.

Finite Lambertian transfer

E(xr)=∫AsLcos⁡θscos⁡θrr2 dAsE(\mathbf x_r)=\int_{A_s}L\frac{\cos\theta_s\cos\theta_r}{r^2}\,\mathrm dA_s

Every source patch has its own distance and pair of projection cosines relative to one receiver sample.

Minimum-to-mean uniformity

U0=Emin⁡⟨E⟩U_0=\frac{E_{\min}}{\langle E\rangle}

A spatial field is required: equal center values can still hide very different edge falloff and uniformity.

Common difficulties

Applying inverse square to every source at every distance

Typical misconceptionDoubling distance always quarters illumination, even when the emitter is a large nearby panel.

Better mental modelInverse square is the far-field point-source result. A nearby extended source must be integrated over its apparent solid angle and approaches point-source behavior only when its size is small compared with distance.

Run the experiment

  1. 01

    Scene 1: Inverse-square probe

    Move the point source through several distances and compare center illuminance with the full tiled receiver map.

    What to observe: The center follows inverse-square scaling while panel edges fall faster because their local paths are longer and more oblique.
  2. 02

    Scene 2: Lambertian cosine law

    Rotate the receiver while keeping source distance fixed and track both center cosine loss and edge asymmetry.

    What to observe: Tilting changes the physical normal and creates a directional gradient rather than a uniform opacity change.
  3. 03

    Scene 3: Soft-light design challenge

    Enter the area-source scene and compare its integrated heat map and uniformity with the point-source case at similar separation.

    What to observe: A larger nearby Lambertian source can improve uniformity because different patches fill different projected directions without increasing radiance.