Luminous efficiency function
The luminous efficiency function, also known as the luminosity function, quantifies the average relative sensitivity of the human visual system to electromagnetic radiation at different wavelengths, serving as the foundation for photometry by converting radiometric quantities (such as radiant flux) into photometric ones (such as luminous flux) that align with human perception of brightness.[1][2] It is defined across a spectral range typically from 360 nm to 830 nm, with the standard photopic function V(λ) peaking at 555 nm in the green-yellow spectrum, where the eye is most sensitive under well-lit conditions.[2][3] Established by the International Commission on Illumination (CIE) in 1924, the photopic V(λ) was derived from averaged experimental data using heterochromatic flicker photometry and other methods on observers with a 2° central visual field, providing a psychophysical analog to radiance for standard daylight-adapted vision.[1][3] This function, normalized to a maximum value of 1 at 555 nm, underpins the definition of the candela—the SI unit of luminous intensity—and enables the calculation of luminous efficacy, with a maximum spectral value of 683 lm/W at that wavelength.[2] Subsequent refinements, such as Judd's 1951 modifications for shorter wavelengths below 500 nm, improved accuracy without altering the core standard, which was reaffirmed by the CIE in 1983 and incorporated into international standards like those from the International Organization for Standardization (ISO).[1][2] For dim light conditions, the scotopic luminous efficiency function V'(λ)—adopted by the CIE in 1951—describes rod-dominated vision, peaking at 507 nm and extending sensitivity into the blue-green region, with values tabulated at 10 nm intervals from 380 nm to 780 nm.[1] This distinction reflects the eye's dual adaptation states: photopic for cone-mediated color vision in bright environments and scotopic for achromatic vision in low illumination.[2] In intermediate mesopic conditions, such as twilight, hybrid models interpolate between V(λ) and V'(λ) to account for mixed cone and rod contributions, though standardization remains complex due to variability in visual performance.[1] These functions are integral to fields like lighting design, display technology, and optical standards, ensuring measurements reflect human visual response rather than physical energy alone, and they continue to evolve with physiological research on cone fundamentals and spectral sensitivities.[1][3]Fundamentals
Definition and Purpose
The luminous efficiency function, also known as the luminosity function, is a standardized curve that describes the average human eye's relative sensitivity to light across different wavelengths under defined viewing conditions, such as photopic or scotopic vision.[4] It is inversely proportional to the ratio of the radiant energy required at a given wavelength to produce the same visual sensation as at the peak sensitivity wavelength (where V(λ) = 1), serving as the foundation for photometry by linking physical light measurements to human perception.[1] The primary purpose of the luminous efficiency function is to enable the conversion of radiometric quantities, like spectral radiant power, into photometric quantities, such as luminous flux, which quantify the perceived brightness of light sources.[4] By weighting a light source's spectral power distribution according to the eye's sensitivity, it allows for objective assessment of visual effectiveness in applications ranging from lighting design to display technology.[5] In photopic vision, corresponding to well-lit conditions, the function V(λ) reaches its peak sensitivity at 555 nm, where monochromatic light has the highest luminous efficacy.[4] The International Commission on Illumination (CIE) adopted the original V(λ) function in 1924 to standardize measurements of light for lighting and display standards, transitioning from subjective evaluations to a unified, objective system.[5] This conversion is mathematically expressed through the luminous flux equation: \Phi_v = 683 \int V(\lambda) P(\lambda) \, d\lambda where \Phi_v is the luminous flux in lumens (lm), V(\lambda) is the spectral luminous efficiency function (dimensionless, normalized to 1 at 555 nm), P(\lambda) is the spectral power distribution in watts per nanometer (W/nm), and 683 lm/W is the maximum luminous efficacy of monochromatic radiation at 555 nm under photopic conditions.[4]Photopic Luminosity Function
The photopic luminosity function, denoted as V(\lambda), represents the spectral luminous efficiency of the human visual system under photopic conditions, such as daylight or bright artificial lighting, as defined by the International Commission on Illumination (CIE) in its 1931 standard observer model. This function describes the relative sensitivity of the eye to monochromatic light at wavelength \lambda (in nanometers), based on averaged experimental data from observers viewing a 2° field. It is normalized such that V(555) = 1, reflecting maximum sensitivity at this green-yellow wavelength.[3][6] The curve of V(\lambda) is bell-shaped, spanning the visible spectrum from approximately 380 nm to 780 nm, with values approaching zero beyond these limits due to negligible cone photoreceptor response in the ultraviolet and near-infrared regions. Sensitivity is notably higher in the green-yellow band (around 500–600 nm), where the eye perceives light as brighter for equivalent radiant power compared to shorter blue or longer red wavelengths; for instance, at the half-maximum points, V(510) \approx 0.503 and V(610) \approx 0.503. This asymmetry and peaking reflect the combined response of long- (L-) and medium- (M-) wavelength cones dominating photopic vision.[7] Tabulated values of V(\lambda) are provided by the CIE at 1 nm intervals across the spectrum, as detailed in CIE Publication 018:2019 (Table 1), to facilitate precise interpolation for computations; coarser 5 nm or 10 nm tables are also available for general use. These tables ensure consistency in photometric standards, with the full dataset confirming the smooth, unimodal profile without secondary peaks. While Gaussian-like mathematical approximations, such as V(\lambda) \approx \exp\left( -\left( \frac{\lambda - 555}{\sigma} \right)^2 \right) for rough estimation (where \sigma approximates the curve's standard deviation around 50–60 nm), can model the overall shape, they introduce errors exceeding 5% outside the central range and are not suitable for accurate work—official tables or piecewise polynomial fits derived from them must be used instead.[8][9] In lighting design and photometry, V(\lambda) is essential for calculating perceived brightness, such as luminous flux (\Phi_v = 683 \int V(\lambda) P(\lambda) \, d\lambda lm, where P(\lambda) is spectral power and 683 lm/W is the maximum luminous efficacy at 555 nm), enabling optimization of sources for energy efficiency and visual comfort without overemphasizing invisible wavelengths.[9]Scotopic Luminosity Function
The scotopic luminosity function, denoted V'(λ), represents the average spectral sensitivity of the human visual system under low-light conditions dominated by rod photoreceptors. Adopted by the International Commission on Illumination (CIE) in 1951, it is based on empirical measurements from dark-adapted young observers and serves as the standard for calculating perceived brightness in scotopic vision, typically at luminances below 0.01 cd/m².[10][11] Unlike photopic vision, which relies on cone cells for color perception in brighter environments, scotopic vision is achromatic and more sensitive overall, with a maximum luminous efficacy of 1700 lm/W compared to 683 lm/W for photopic. The scotopic luminous flux Φ_v' is computed as \Phi_v' = K_m' \int_0^\infty V'(\lambda) \, P(\lambda) \, d\lambda, where P(λ) is the spectral power distribution of the incident radiation and K_m' = 1700 lm/W is the scotopic maximum luminous efficacy. This integral weights the radiant power by the eye's relative sensitivity, emphasizing wavelengths where rods are most responsive.[12][13] The V'(λ) curve peaks at 507 nm in the blue-green spectrum and is normalized to a maximum value of 1 at this wavelength, reflecting the absorption peak of rhodopsin, the rod photopigment. It is broader than the photopic counterpart and shifted toward shorter wavelengths, resulting in higher relative sensitivity in the blue region (around 400–500 nm) and lower sensitivity beyond 600 nm, where red light becomes nearly invisible. This configuration enhances detection of faint blue-green light sources at night while diminishing perception of warmer tones.[14][7] The CIE provides tabulated values of V'(λ) at 10 nm intervals from 380 nm to 780 nm, derived primarily from threshold measurements by Wald (1945) using small field sizes and brightness-matching experiments by Crawford (1949) with larger fields to minimize cone intrusion. Representative values illustrate the curve's shape and blue bias:| Wavelength (nm) | V'(λ) |
|---|---|
| 450 | 0.455 |
| 500 | 0.9817 |
| 507 | 1.0000 |
| 555 | 0.4048 |
| 600 | 0.0633 |