I.Definition
Radiant flux ($\Phi_e$), also known as radiant power, is the time rate of change of radiant energy. It represents the total radiant energy emitted, transferred, or received per unit time. The quantity is denoted by $\Phi_e$ or $P$ and is defined as:
$$\Phi_\text{e} = \frac{\mathrm{d}Q_\text{e}}{\mathrm{d}t}$$

where:

• $Q_\text{e}$ is the radiant energy.
  
• $t$ is time.
  

SI Unit: Watt (W)

Example: The output power of a microwave oven corresponds to its radiant flux $\Phi_e$.

Image source: https://commons.wikimedia.org/wiki/File:TECO_YM2323CB_spec_tag_20151029.jpg

II.Measurement
The measurement of radiant flux ($\Phi_e$) is commonly performed in accordance with standards such as CIE 250:2022 and CIE 130:1998.

In practical measurements, two principal approaches are commonly employed: the integrating sphere method and the goniophotometric method.

1. Integrating Sphere Method
An integrating sphere is a hollow spherical cavity whose inner wall is coated with a highly reflective diffuse material. The light source and detector are typically mounted inside the sphere. After entering the sphere, radiation undergoes multiple diffuse reflections, producing an approximately uniform radiance field within the cavity. This enables integrated measurement of the total radiant energy emitted by the source. To reduce measurement errors caused by direct illumination of the detector, a baffle is usually installed inside the sphere.
(Image source: https://en.wikipedia.org/wiki/Integrating_sphere#/media/File:Luminance_Chamber.jpg)

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

• $S$ is the detector output signal;
• $K$ is the system calibration coefficient

When spectroradiometric measurement is employed, the total radiant flux is obtained by integrating the spectral radiant flux over the target wavelength range:
$$\Phi_e = \int_{\lambda_1}^{\lambda_2} \Phi_{e,\lambda}(\lambda) d\lambda$$

2. Goniophotometer Method
The goniophotometric method determines radiant flux by measuring the distribution of radiant intensity in different spatial directions and integrating it over the entire solid angle. During measurement, either the detector or the light source is rotated through specified angles to obtain the radiant intensity distribution $I(\theta,\varphi)$ in each direction.
For an isotropic ideal point source, the radiant flux is:
$$ \Phi = 4\pi I$$

For a practical light source with finite dimensions, the radiant flux is obtained by integrating over the full solid angle:
$$ \Phi = \int_{0}^{2\pi} \int_{0}^{\pi} I(\theta,\varphi) \sin\theta \, d\theta d\varphi$$

where:

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