IN24 · Let light visit the wall again

Integrating Sphere Radiometer

How can one detector compare total light output or a sample’s reflectance? Diffuse reflections spread light around a bright spherical enclosure, so a small detector can sample the light from many directions. Comparing a source or sample with a known reference gives a relative result. Port losses and absorption by the substituted object change the sphere response and must enter the correction.

Follow diffuse reflections, replace a known sample or source, and recover a relative quantity from retained detector counts.

Replace the reference

Inferred reflectance $0.554\pm0.0129$

Observed detector counts $3230$

Independent check error $3.47\,\%$

Mean detected reflections $14.6$

MeasurementUnknown reflectance

\rho_u

0.55

Wall reflectance

\rho_w

0.94

Open-port radius

r_e/\mathrm{mm}

12

Baffle radius

r_b/\mathrm{mm}

44

Exposure

t/\mathrm{s}

0.5

Drag the baffle edge. After changing the sphere geometry, measure fresh references.

Source, surfaces and transport assumptions

Sample radius

r_s/\mathrm{mm}

22

Baffle reflectance

\rho_b

0.85

Interior source position

x_s/\mathrm{mm}

-20

Source housing reflectance

\rho_{\mathrm{source}}

0.15

Source cone half-angle

\alpha/{}^\circ

35

Entrance beam tilt

\theta/{}^\circ

0

Acquisition seed

s

197

Interior source patternShot-count noise

Monochromatic Lambertian ray transport includes the flat sample, both baffle faces and edge, four port contents and finite source housing. The source mode uses a reflective entrance plug. The visual cutaway adds no loss port. Supports have zero modeled optical area and enlarged drawn thickness.

The reference and an independent check have exact chosen reflectances or photon fluxes. Relative flux requires the same source shape, position, pattern and spectrum. Model wall reflectances, geometry, monitor ratio and detector linearity are assumed exact.

The count model uses Poisson sampling below fifty mean counts and a rounded Gaussian approximation above it. The count bootstrap shares each dark realization across sample and reference. The separate transport-sampling estimate is conditional on this geometric model; standard uncertainty, drift, spectral mismatch and unknown surface scattering are omitted.

Model assumptions

A monochromatic 550-nanometre ray model includes Lambertian walls, a flat sample, finite two-sided baffle, open ports and an absorbing source housing. It uses 8192 seeded weighted paths, with explicit detector capture, absorption, escape and truncated remainder. Uniform-radiance sphere formulas are comparison limits; the actual baffle and port contents remain in ray transport. Reflectance inference corrects sample reillumination using the declared geometric model. Relative flux requires matching source shape, position and angular pattern. Known standards, reflectances, geometry and detector linearity are assumed exact; no absolute radiometric calibration or material certification is provided. Counts use low-mean Poisson/high-mean rounded-Gaussian noise. Count bootstrap and conditional transport sampling uncertainty omit BRDF mismatch, spectral effects, drift and standard uncertainty. Cutaway is visual only. Supports have ideal zero optical area and visibly enlarged thickness; ray markers have schematic speed and size.

A retained substitution measurement

Raw detector and entrance-monitor observations
Observation	Signal counts	Monitor counts	Exposure $t/\mathrm{s}$

References, model correction and export

Simple ratio estimate $0$

Count standard uncertainty $0$

Conditional transport ratio error $0$

Truncated transport remainder $0$

Counts are simulated detector observations. Transport balance and reflection-order plots are predictions of the stated ray model at the inferred quantity; they are not measured arrival-time distributions. Count uncertainty omits reference and model uncertainty.

Retained measurements
Record	Mode	Calibration	Supported

CSV and JSON retain sample, reference, independent check and shared dark counts; standard definitions; all port, source, sample and baffle configurations; units, versions and seeds. The transport response is separately labeled and reproducible from its geometry and seed. JSON also retains the separate ideal reference.

Start with the essentials

Focus question: How can one detector compare total light output or a sample’s reflectance?
One-sentence intuition: Compare retained detector observations with a known reference, including how the substituted object changes the circulating light.

Core mathematical model

The uniform-wall comparison limit

M=\frac{\rho}{1-\rho(1-f)}

$\rho$ is the diffuse wall reflectance and $f$ is black-port area fraction. This geometric-series result assumes a spherical wall, an initial wall reflection and no internal baffle. The actual instrument uses finite-surface ray transport instead.

Dark-corrected, monitor-normalized reflectance substitution

\frac{C_u/C_{m,u}}{C_r/C_{m,r}}=\frac{p_d(\rho_u)}{p_d(\rho_r)}

$C$ denotes dark-corrected count rate; $m$ denotes the entrance monitor. The detector-capture probability depends on the unknown sample reflectance because light can return to the sample. The geometric model supplies that nonlinear response; known reference reflectance fixes the comparison.

Common difficulties

A multiplier above one does not create power

Typical misconceptionMore accumulated wall illumination means extra photons were produced.

Better mental modelOne packet can visit the wall many times. Its final detector capture, absorption or escape occurs once, and their fractions plus the explicit truncated remainder sum to one.

An open cutaway is not an extra port

Typical misconceptionLight should escape through the visible front half of the drawing.

Better mental modelThe cutaway exposes the apparatus for viewing. Transport retains the complete spherical boundary; only the declared openings lose light.

Run the experiment

01
Replace a white reference
Change unknown reflectance and compare the corrected estimate with the simple count-ratio estimate in the data details.
What to observe: The sample changes both its first reflection and the later circulation. A dark sample lowers the sphere response.
02
Let light escape
Widen the open port, then measure fresh references.
What to observe: More packets terminate at an opening; detector counts and the average detected reflection order fall. Old calibration does not match the changed geometry.
03
Shield the detector
Drag the baffle edge or choose the unshielded preset.
What to observe: The finite plate blocks direct sample-to-detector paths and also absorbs some circulating light. Its two faces and rim enter the same transport.
04
Compare source output
Choose total-output comparison, change source flux and retain a measurement.
What to observe: Matched source shape, placement and angular pattern allow a detector-signal ratio to report flux relative to the known source. Returning light can be absorbed by the source housing.

Integrating Sphere Radiometer

A retained substitution measurement

The sample also changes the sphere

Core mathematical model

The uniform-wall comparison limit

Dark-corrected, monitor-normalized reflectance substitution

Common difficulties

A multiplier above one does not create power

An open cutaway is not an extra port

Run the experiment

Replace a white reference

Let light escape

Shield the detector

Compare source output

Related experiments and articles

How physics instruments measure