I. Definition
The CIE luminous efficiency function $V(\lambda)$ is a standard response function established by the International Commission on Illumination (CIE) to describe the relative visual sensitivity of the standard observer to visible light of different wavelengths under specific photometric conditions.

Strict Definition：Under specific photometric conditions, if radiant flux $\Phi_{e,\lambda}$ at wavelength $\lambda$ and radiant flux $\Phi_{e,\lambda_m}$ at the peak sensitivity wavelength $\lambda_m$ produce an identical visual sensation of brightness, the ratio of these two fluxes, $\Phi_{e,\lambda_m}/\Phi_{e,\lambda}$, defines the spectral luminous efficiency $V(\lambda)$.

II. Unit
Spectral luminous efficiency is a dimensionless quantity with a unit of 1. It represents only the relative ratio of radiant flux and has no physical dimension.

III. Luminous Efficiency Functions
Human spectral sensitivity depends on visual adaptation, field of view (FOV), and the angle of incidence. The CIE defines several standard functions for different photometric conditions:

Photopic Vision \(V(\lambda)\)

• Operating Conditions:: $L > 5 cd·m⁻²$(e.g., daylight or bright artificial lighting).
  
• Physiological Mechanism: Mediated by cone cells (3 types, containing photopigments corresponding to red/green/blue perception). They have low light sensitivity and high noise in dark environments, functioning only in bright environments, and are responsible for color vision and detail resolution.
  

Figure 1: Schematic Diagram of Cone Cells
Source: https://en.wikipedia.org/wiki/Cone_cell#media/File:Cone_cell_eng.svg

• Relevant Standards: ISO 23539:2005 (E) / CIE S 010;
• On the visual axis:
  
  • $V(\lambda)$: CIE 1924 photopic spectral luminous efficiency (corresponds to the fovea of the retina);
  • $CIE 1988 (CIE 086-1990)$: Corrected the shortcomings of CIE 1924 at short wavelengths
• Off the visual axis:
  
  • Below $4^\circ$ viewing angle: \(V(\lambda)\) (CIE 1924, peak at 555 nm, green light band);
  • Above $4^\circ$ viewing angle: \(V_{10}(\lambda)\) (CIE 1964, CIE 165:2005), adapted for large field of view and off-axis visual tasks.
• Practical Applications: In LED lighting design, matching the 555 nm peak improves luminous efficacy of luminaires; automobile headlights use the photopic curve to optimize white light spectrum and enhance road visibility.
  

Scotopic Vision \(V'(\lambda)\)

• Applicable Scenarios: $L < 0.005\ \mathrm{cd\cdot m^{-2}}$(e.g., darkrooms, unlit night).
  
• Physiological Mechanism: Mediated by rod cells (rhodopsin). High sensitivity; achromatic perception; no color vision.
  

Figure 2:Schematic Diagram of Rod Cells
Source: https://en.wikipedia.org/wiki/Rod_cell#/media/File:Rod_Cell.svg

• Relevant Standards: ISO 23539:2005 (E) / CIE S 010
  
• Core Characteristics: The CIE 1951 scotopic spectral luminous efficiency function $V'(\lambda)$ peaks at 507 nm (blue-green spectrum).
  
• Applications: Night-vision and astronomical equipment (optimized for 507 nm); blue-green reflective materials in nighttime road signage to enhance low-light visibility.
  

Mesopic Vision \(V_{\text{mes},m}(\lambda)\)

• Operating Conditions:: $0.005\ \mathrm{cd\cdot m^{-2}} < L < 5\ \mathrm{cd\cdot m^{-2}}$(e.g., twilight, parking garages).
  
• Physiological Mechanism: Co-mediated by rods and cones. Sensitivity and color perception shift dynamically with luminance.
  
• Relevant Standards: CIE 191:2010
  
• Core Characteristics: A weighted combination of photopic and scotopic functions, defined by the adaptation coefficient $m$.
  
• Applications: Urban night lighting and smart streetlamps (balancing visibility and energy efficiency).
  

Comparison of Spectral Luminous Efficiency Functions
(Image source: https://commons.wikimedia.org/wiki/File:LuminosityCurve1.svg)
The figure clearly shows the spectral sensitivity differences between photopic and scotopic vision. Scotopic vision is more sensitive to blue - green light (507 nm), while photopic vision is most sensitive to green light (555 nm). The horizontal axis represents wavelength, with units of nanometers (nm).

IV. Photometric calculations based on the luminous efficiency function
The primary application of the luminous efficiency function is to convert radiometric quantities into photometric quantities. The general calculation model is:
$$\varPhi_{\text{v}} = K_{\text{m}} \int_{0}^{\infty} \varPhi_{\text{e},\lambda}(\lambda) \cdot V(\lambda) \, d\lambda$$
where:

• \(\varPhi_{\text{v}}\) is the luminous flux (unit: lm), representing the total light output perceived by the human visual system from the visible radiation emitted by a light source;
• \(\varPhi_{\text{e},\lambda}(\lambda)\) is the spectral radiant flux (unit: $\mathrm{W\cdot nm^{-1}}$), describing the radiant power distribution of the source per unit wavelength interval;
• $K_\mathrm{m}$ is the maximum luminous efficacy (unit: $\mathrm{lm\cdot W^{-1}}$), with a standard value of $683\ \mathrm{lm/W}$, corresponding to the upper limit of energy conversion efficiency at the peak wavelength of photopic vision ($555\ \mathrm{nm}$);
• $V(\lambda)$ is the spectral luminous efficiency function for photopic vision; for scotopic or mesopic conditions, it is replaced by $V'(\lambda)$ or $V_\mathrm{mes}(\lambda)$, respectively.

As an example, under photopic conditions, the spectral luminous efficiency of the human eye at $480\ \mathrm{nm}$ is approximately $20\%$ of that at $555\ \mathrm{nm}$. Therefore, a monochromatic source at $480\ \mathrm{nm}$ with five times the radiant power of a $555\ \mathrm{nm}$ source would yield identical luminous flux. Under identical viewing conditions (same emitting area, field of view and observation distance), the two sources produce the same perceived brightness.