How to convert LED luminous flux φ and luminous intensity Iv to each other

The conversion between LED luminous flux (φ) and luminous intensity (Iv) is an important concept in LED optical design and application. To better understand the relationship between these two concepts, we need to first understand their respective definitions and their connections.
Luminous flux (φ)
Luminous flux is the energy transmitted by light radiation in a unit of time, usually measured in lumens (lm). It measures the total amount of light radiation, which is the total amount of light emitted by the light source. Luminous flux reflects the total amount of light emitted by the light source without considering the directionality and intensity distribution of the light.
Luminous intensity (Iv)
Luminous intensity (Iv), also known as luminous intensity density, refers to the luminous flux emitted per unit solid angle. Its unit is candela (cd). Luminous intensity describes the intensity of light emitted in a specific direction by the light source and is a directional quantity.
The relationship between luminous flux and luminous intensity
The relationship between luminous flux and luminous intensity can be described by the Stokes equation (Stokes’ Law), which states that for a uniformly emitting point light source, the luminous intensity within a unit solid angle is proportional to the luminous flux of the point light source and inversely proportional to the luminous area of the light source. The mathematical expression is:
[ I_v = \frac{\Phi}{4\pi A} ]
Where, (I_v) is the luminous intensity, (\Phi) is the luminous flux, (A) is the luminous area of the light source, and (4\pi) is the value of the unit solid angle (in spherical coordinate systems, the unit solid angle is 4π sr).
Interconversion between luminous flux and luminous intensity

Conversion from luminous flux to luminous intensity: According to the Stokes equation, if we know the luminous flux and the luminous area of the light source, we can directly calculate the luminous intensity of the light source in any direction using the above formula.

Conversion from luminous intensity to luminous flux: Similarly, if we know the luminous intensity of the light source and the luminous area of the light source in a specific direction, we can calculate the luminous flux of the light source in reverse by using the above formula.

Application Examples
Assuming there is an LED light source with a luminous flux of 1000 lumens and a luminous area of 0.01 square meters, we can calculate the luminous intensity in any direction (assuming the light source emits uniformly):
[ I_v = \frac{1000}{4\pi \times 0.01} = \frac{1000}{0.12566} \approx 7960 cd ]
This means that the luminous intensity of the LED light source is approximately 7960 cd in any direction.
Conclusion
Luminous flux and luminous intensity are important parameters describing the light radiation characteristics of LED light sources. The interconversion between them is crucial for the design, application, and performance evaluation of LEDs. By understanding these concepts and formulas, we can more accurately analyze and optimize the design of LED light sources, thereby improving their performance and efficiency.