---
title: "Integrating Sphere Radiometer Interactive Lab | Sandbox Physics"
description: "How can one detector compare total light output or a sample’s reflectance?"
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last_modified: "2026-11-03"
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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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## The sample also changes the sphere

**Background** A bright diffuse wall sends arriving light in many directions. Repeated visits let a small detector sample light that began elsewhere.

**Why it matters** A sample absorbs light on its first illumination and on later returns. Replacing a white reference can therefore change the whole sphere response.

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 misconception** More accumulated wall illumination means extra photons were produced.

**Better mental model** One 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 misconception** Light should escape through the visible front half of the drawing.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

A substitution ratio becomes a reflectance or relative flux only with a stated sphere response and matching calibration geometry.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a white sphere compares a lamp’s total output**](https://sandboxphysics.com/stories/instruments/integrating-sphere-radiometer/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)