Definition
Luminous intensity (symbol:$I_v$ ) describes the intensity of a light source in a given direction, defined as the luminous flux emitted per unit solid angle. It is an inherent property of the light source, independent of the source area and observation distance.

Figure 1: Schematic illustration of luminous intensity

Source:https://en.wikipedia.org/wiki/Radiant_flux#/media/File:Photometry_radiometry_units.svg

Its mathematical expression is: $$ I_v = \frac{\mathrm{d}\Phi_v}{\mathrm{d}\Omega} $$ where $I_v$ is the luminous intensity in the specified direction, $\Phi_v$ is the luminous flux emitted in the specified direction (unit: lm, lumen), and $\Omega$ is the solid angle containing the direction (unit: sr, steradian).

Unit
The candela (\(\mathrm{cd}\)) is the SI unit of luminous intensity. It is defined by fixing the numerical value of the luminous efficacy of monochromatic radiation of frequency \(540\times 10^{12}\ \mathrm{Hz}\), denoted \(K_{\mathrm{cd}}\), to be \[ K_{\mathrm{cd}} = 683\ \mathrm{lm\,W^{-1}}. \]

From the relationship between luminous flux \(\Phi_v\) and luminous intensity \(I_v\),the unit equivalence follows: \[ 1\ \mathrm{cd} = 1\ \mathrm{lm\,sr^{-1}}. \]

Thus, a light source emitting a luminous flux of \(1\ \mathrm{lm}\) into a solid angle of \(1\ \mathrm{sr}\) has a luminous intensity of \(1\ \mathrm{cd}\) in that direction.

Relationship to Radiometry
The luminous intensity $I_v$ is a photometric quantity derived from the radiometric quantity spectral radiant intensity $I_{e,\lambda}(\lambda)$. It is calculated by weighting the spectral distribution of the radiant power with the human eye's spectral sensitivity, as defined by the CIE standard observer. The relationship is given by the following integral: \[I_v = K_m \int_{380}^{780} I_{e,\lambda}(\lambda) V(\lambda) d\lambda\] where:

• $I_v$ is the luminous intensity in candelas (cd);
• $K_m$ is the maximum spectral luminous efficacy, which is exactly 683 lm/W for photopic vision at a frequency of $540 \times 10^{12}$ Hz;
• $I_{e,\lambda}(\lambda)$ is the spectral radiant intensity in watts per steradian per nanometer (W·sr\(^{-1}\)·nm\(^{-1}\));
• $V(\lambda)$ is the photopic luminous efficiency function, a dimensionless function that describes the average spectral sensitivity of human visual perception.

Relation to Luminous Flux
The luminous flux $\Phi_v$ emitted by a light source within a specific solid angle $\Omega$ can be determined from the distribution of its luminous intensity $I_v$ as a function of the emission direction $(\theta, \varphi)$. The relationship is defined by the following surface integral: \[\Phi_v = \iint_{\Omega} I_v(\theta, \varphi) \sin\theta d\varphi d\theta\] where:

• $\Phi_v$ is the luminous flux in lumens (lm);
• $I_v(\theta, \varphi)$ is the luminous intensity in candelas (cd) in the direction specified by the polar angle $\theta$ and the azimuthal angle $\varphi$;
• $\Omega$ represents the solid angle in steradians (sr) over which the flux is integrated;
• $\sin\theta d\varphi d\theta$ is the differential element of the solid angle $d\Omega$ in spherical coordinates.