Luminous Efficiency Functions¶
The luminous efficiency function is the ratio of radiant flux weighted according to $V(\lambda)$ to the corresponding radiant flux. [1] It characterizes the average spectral sensitivity of human visual perception of brightness. Brightness is defined as the attribute of a visual perception according to which an area appears to emit, or reflect, more or less light. [2]
Photometry is the measurement of quantities referring to radiation as evaluated according to a given spectral luminous efficiency function, e.g. $V(\lambda)$ or $V^\prime(\lambda)$. [3]
Photometry & Radiometry¶
Radiometry is defined as the measurement of the quantities associated with optical radiation. [4]
Photometric quantities are weighted accordingly to human visual system spectral sensitivity within the wavelength range 360-780 nanometres ($nm$) while radiometric quantities represent unweighted absolute power within the wavelength range 0.01-1000 micrometres ($\mu m$).
Given the following terms and units table: [5]
| Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Frequency | $v$ | $s^{-1}$ | Hertz | ||
| Wavelength | $\lambda$ | $\lambda=\cfrac{c}{v}$ | $m$ | Metre | $c$: Velocity of radiant energy in vacuum |
| Wavenumber | $m$ | $m=\cfrac{1}{\lambda}$ | $m^{-1}$ | Metre | |
| Solid angle | $\omega$ | $\omega=\cfrac{S}{r^2}$ | $sr$ | Steradian | $S$: Portion of sphere surface $r$: Radius of sphere, also distance between (1) of $dA_1$ and (2) of $dA_2$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ }$ |
Radiometry uses the following quantities, terms and units: [5]
| Quantity / Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Radiant energy | $Q_e$ | $J$ | Joule | ||
| Radiant flux (or power) | $P_e$ | $W$ | Watt ($J\cdot sr^{-1}$) |
$d^2P_e=L_e\cfrac{dA_1\cos\varepsilon_1dA_2\cos\varepsilon_2}{r^2}$ | |
| Radiant exitance | $M_e$ | $M_e=\cfrac{dP_e}{dA_1}$ | $W\cdot m^{-2}$ | Watt per square metre | $dA_1$: Surface element of source |
| Irradiance | $E_e$ | $E_e=\cfrac{dP_e}{dA_2}$ | $W\cdot m^{-2}$ | Watt per square metre | $dA_1$: Surface element of receiver |
| Radiant intensity | $I_e$ | $I_e=\cfrac{dP_e}{d\omega_1}$ | $W\cdot sr^{-1}$ | Watt per steradian | $d\omega_1$: Element of solid angle with apex (2) at surface of source |
| Radiance | $L_e$ | $L_e=\cfrac{d^2P_e}{dA_1\cos\varepsilon_1d\omega_1}$ $L_e=\cfrac{d^2E_e}{dA_2\cos\varepsilon_2d\omega_2}$ $L_e=\cfrac{d(E_e)_n}{d\omega_2}$ |
$W\cdot m^{-2} \cdot sr^{-1}$ | Watt per square metre per steradian | $\varepsilon_1$: Angle between direction (1)-(2) and normal $n_1$ of $dA_1$ $\varepsilon_2$: Angle between direction (1)-(2) and normal $n_2$ of $dA_2$ $d\omega_2$: Element of solid angle with apex (2) at surface of receiver $dA_1\cos\varepsilon_1$: $dA_1$ orthogonally projected on plane perpendicular to direction (1)-(2) $dA_2\cos\varepsilon_2$: $dA_2$ orthogonally projected on plane perpendicular to direction (1)-(2) $d(E_e)_n=\cfrac{dE_e}{dA_2\cos\varepsilon_2}$ $d\omega_1=\cfrac{dA_1\cos\varepsilon_1}{r^2}$ $d\omega_2=\cfrac{dA_2\cos\varepsilon_2}{r^2}$ |
Photometry uses the following quantities, terms and units: [5]
| Quantity / Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Luminous energy | $Q_v$ | $lm\cdot s$ | Lumen second | ||
| Luminous flux (or power) | $F_v$ (or $P_v$) | $F_v=KP_e$ $F_v=K_m\int_\lambda P_{e,\lambda}V(\lambda)d\lambda$ |
$lm$ | Lumen ($cd\cdot sr$) |
$P_e$: Radiant flux ($W$) $K$: Luminous efficacy ($lm\cdot W^{-1}$) |
| Luminous exitance | $Mv$ | $M=\cfrac{dF_v}{dA_1}$ | $lx$ | Lumen per square metre (or lux) ($lm\cdot m^{-2}$) |
|
| Illuminance | $E_v$ | $E_v=\cfrac{dF_v}{dA_2}$ | $lx$ | Lumen per square metre (or lux) ($lm\cdot m^{-2}$) |
$dA_2$: Surface element of receiver |
| Luminous intensity | $I_v$ | $I_v=\cfrac{dF_v}{d\omega_I}$ | $cd$ | Candela ($lm\cdot sr^{-1}$) |
Luminous flux per unit solid angle $d\omega_I$: Element of solid angle with apex (1) at surface of source |
| Luminance | $L_v$ | $L_v=\cfrac{d^2F_v}{dA_1\cos\varepsilon_1d\omega_1}$ $L_v=\cfrac{dI_v}{dA_1\cos\varepsilon_1}$ $L_v=\cfrac{d^2E_v}{dA_2\cos\varepsilon_2d\omega_2}$ $L_v=\cfrac{dE_{v,n}}{d\omega_2}$ |
$cd\cdot m^{-2}$ | Candela per square metre (or nits) ($lm\cdot m^{-2}\cdot sr^{-1}$) |
$dA_1$: Surface element of source $\varepsilon_1$ Angle between direction (1)-(2) and normal $n_1$ of $dA_1$ $dA_1\cos\varepsilon_1$: $dA_1$ orthogonally projected on plane perpendicular to direction (1)-(2) $dE_{v, n}$: Illuminance on $dA_2$ normal to the direction (1)-(2) $d\omega_2$: Element of solid angle with apex (2) at surface of receiver |
| Luminous efficacy function | $K(\lambda)$ | $K(\lambda)=K_mV(\lambda)$ with $K_m$=683 |
$V(\lambda)$: Relative photopic luminous efficiency function |
The following figures illustrate the notes from the above tables: [5]
from IPython.core.display import Image
Image(filename='resources/images/Photometric_Quantities_001.png')
from IPython.core.display import Image
Image(filename='resources/images/Solid_Angle_001.png')