Radiant flux is the rate of change of radiant energy with respect to time, representing the total radiant energy (power) emitted, transferred, or received per unit time. It is denoted by the symbol $\Phi$ or $P$, and its mathematical expression is:
$$\Phi_\text{e} = \frac{\mathrm{d}Q_\text{e}}{\mathrm{d}t}$$ where $Q_\text{e}$ is the radiant energy and $t$ is time.
Unit: Watt (W)
Application example: The output power of a microwave oven, which is the radiant flux $\Phi$.
(Image source: https://commons.wikimedia.org/wiki/File:TECO_YM2323CB_spec_tag_20151029.jpg)

Radiant Flux Emitted from a Point Source to a Disk
Analyzing the radiant flux emitted from a point source to a disk can be used to calculate the radiant flux received by an optical system or receiver at a certain distance from the point source. As shown in the figure below, a point source $O$ emits optical radiation, and at a distance $l_0$ from the point source, there is a disk of radius $R$ perpendicular to the direction of radiation. Since the disk has a finite size, the distances from the point source to various points on the disk are unequal, resulting in non-uniform irradiance on the disk.
The radiant flux received by a differential area element $\mathrm{d}A$ on the disk is:
$$\mathrm{d}P = E\mathrm{d}A = \frac{I\cos\alpha}{l^2}\mathrm{d}A \tag{1}$$ Since $\mathrm{d}A = \rho\mathrm{d}\theta\mathrm{d}\rho$ and $\cos\alpha = l_0/l = l_0/\sqrt{\rho^2 + l_0^2}$, substituting these into Eq. (1) and integrating over $\rho$ and $\theta$ yields the total radiant flux received by the disk of radius $R$:
$$P = \int\mathrm{d}P = Il_0\int_0^{2\pi}\mathrm{d}\theta\int_0^R\frac{\rho}{(\rho^2 + l_0^2)^{3/2}}\mathrm{d}\rho = 2\pi I\left\{1 - \left[1 + \left(\frac{R}{l_0}\right)^2\right]^{-1/2}\right\} \tag{2}$$ When the disk is sufficiently far from the point source, i.e., $l_0 \gg R$, $l \approx l_0$, and $\cos\alpha \approx 1$, the flux received by the disk is
$$ P = \frac{I}{l_0^2}\pi R^2 = \frac{I}{l_0^2}S \tag{3} $$ In this case, the disk can be considered a differential area element, and the irradiance is uniform across all points on the disk.

How to Measure Radiant Flux?
The measurement of radiant flux ($\Phi_e$) typically refers to standards such as CIE 250:2022 “Guide to Spectroradiometric Measurements” and CIE 130:1998 “Measurement of Radiant Flux of Light Sources”. In practical measurements, two main methods are used: the integrating sphere method and the goniophotometer method.

1. Integrating Sphere Method
An integrating sphere is a hollow sphere with its inner wall coated with a highly reflective, neutral diffuse reflection material. The light source and detector are usually mounted inside the sphere. Light entering the sphere undergoes multiple diffuse reflections, creating an approximately uniform radiation field inside, which enables the integral measurement of the radiant energy of the light source. To prevent errors caused by direct illumination of the detector by the light source, a baffle is often placed inside the sphere.
(Image source: https://en.wikipedia.org/wiki/Integrating_sphere#/media/File:Luminance_Chamber.jpg)

The output signal of the detector is proportional to the total radiant flux. After calibration with a standard light source, it can be expressed as: $$\Phi_e = K \cdot S$$ where:

• $S$: detector output signal;
• $K$: system calibration coefficient.

If spectral measurement is used, the total radiant flux is the integral of the spectral radiant flux over the target wavelength band:
$$\Phi_e = \int_{\lambda_1}^{\lambda_2} \Phi_{e,\lambda}(\lambda) d\lambda$$

2. Goniophotometer Method
A goniophotometer measures the radiant intensity distribution of a light source in different spatial directions and integrates it over the entire space to obtain the total radiant flux. During the measurement, the detector or the light source is rotated at set angles to acquire the radiant intensity $I(\theta,\varphi)$ in each direction.
For an isotropic ideal point light source, its radiant flux is:
$$ \Phi = 4\pi I$$

For a practical light source with finite dimensions, integration over the solid angle of the entire space is required:
$$ \Phi = \int_{0}^{2\pi} \int_{0}^{\pi} I(\theta,\varphi) \sin\theta \, d\theta d\varphi $$

where:

• $I(\theta,\varphi)$: radiant intensity in the direction $(\theta,\varphi)$
• $\theta$: zenith angle
• $\varphi$: azimuth angle