Definition
Luminous flux (symbol:$\Phi_v$ ) is a photometric quantity that measures the light power perceived by the human eye. It represents the total luminous energy emitted by a light source per unit time, and directly reflects the overall brightness of the source.

Figure 1: Schematic illustration of luminous flux

(Image source: https://en.wikipedia.org/wiki/Radiant_flux#/media/File:Photometry_radiometry_units.svg)

Unit
The SI unit of luminous flux is the lumen (lm). One lumen is defined as the luminous flux of light produced by a light source that emits one candela of luminous intensity over a solid angle of one steradian.
$$1\ \text{lm} = 1\ \text{cd} \times 1\ \text{sr}$$

Calculation
Luminous flux is a photometric quantity derived from radiant flux ($lm$) by weighting the radiation according to the response of the CIE standard photometric observer. It can be calculated from the spectral radiant flux distribution using the following formula:

$$\Phi_{v}(\lambda)=K_{m}V(\lambda)\Phi_{e}(\lambda)$$ where $K_{m}$ is the maximum luminous efficacy, $\Phi_{v}(\lambda)$ is the spectral radiant flux,$V(\lambda)$ is the spectral luminous efficiency function, and $\lambda$ is the wavelength of light (the integral covers the visible spectrum from 380 nm to 780 nm, as the human eye is insensitive to radiation outside this range).

Measurement
According to GB/T 20178-2010 Methods for the Measurement of Luminous Flux, luminous flux can be measured by several established methods, with three core approaches outlined below: calculation from luminous intensity distribution, calculation from illuminance distribution, and measurement using an integrating sphere.

1.Calculation of Luminous Flux from Luminous Intensity Distribution
Luminous flux $\Phi_v$ can be derived from the spatial distribution of luminous intensity using the following formula:
$$ \Phi_v = \int_{(\Omega)} I_v \, d\Omega$$ where $\Omega = 4\pi\ \text{sr}$ is the total solid angle. The luminous intensity distribution is measured using a goniophotometer.

2.Calculation of Luminous Flux from Illuminance Distribution
By definition, given the illuminance distribution $E$ over a closed surface $A$ surrounding a light source, the luminous flux $\Phi_v$ can be obtained from the following formula:
$$\Phi_v = \int_{(A)} E \, dA$$ The illuminance distribution can be measured on a spherical surface centered on the light source using a goniophotometer. The light source does not need to be placed precisely at the center of the virtual sphere, but it is recommended to position it as close to the center as possible.

Due to mechanical constraints, the minimum distance from the sphere center to the photometer probe is determined by the maximum dimension of the test lamp. This distance can be less than the limit photometric distance, provided the photometer can correctly calculate the illuminance value (cosine response) based on orientation and other conditions.

3.Luminous Flux Measurement with an Integrating Sphere
The luminous flux of a light source can be obtained by comparative measurement against a standard lamp in an integrating sphere. During the test, the test source and the standard lamp are placed sequentially at the same position inside the sphere, and the indirect illuminance on the inner surface of the sphere is used as the metric for luminous flux.

The luminous flux of the test source is calculated as:
$$\Phi = \Phi_{\text{N}} \cdot \frac{Y}{Y_{\text{N}}} \cdot \frac{Y_{\text{HN}}}{Y_{\text{H}}}$$

where:

• $\Phi$ is the luminous flux of the test source,
• $\Phi_{\text{N}}$ is the known luminous flux of the standard lamp,
• $Y$ and $Y_{\text{N}}$ are the detector readings for the test source and standard lamp respectively,
• $Y_{\text{HN}}$ and $Y_{\text{H}}$ are the auxiliary lamp readings for calibration correction.

Figure 2: Layout for luminous flux measurement in an integrating sphere

where $L$ is the light source,$H$ is the auxiliary lamp with a baffle,$S$ is the baffle, $d$ is the sphere diameter, and $F$ is the photometer probe port.