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.
Its strict definition is: under given photometric conditions, if visible light of two different wavelengths produces the same intensity of visual sensation, the ratio of the radiant flux $\Phi_{e,\lambda_m}$ at the peak sensitivity wavelength $\lambda_m$ to the radiant flux $\Phi_{e,\lambda}$ at the target wavelength $\lambda$ is $V(\lambda)$.

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

III. Luminous Efficiency Functions under Different Photometric Conditions
The spectral sensitivity of the human eye is affected by factors such as visual adaptation state, field of view, and the incident direction of the light source. The CIE has defined multiple sets of standard functions for different application scenarios. The core classifications are as follows:

Photopic Vision \(V(\lambda)\)

• Applicable Scenarios: Daytime or strongly illuminated environments with luminance $L > 5 cd·m⁻²$, such as offices, shopping malls, and sunny outdoors.
  
• Physiological Mechanism: Dominated by cone cells, containing three types of 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.
  

• Relevant Standards: ISO 23539:2005 (E) / CIE S 010
  
• On the visual axis:
  
  • $V(\lambda)$: CIE 1924 photopic spectral luminous efficiency (corresponding 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: The \(V(\lambda)\) CIE 1924 photopic spectral luminous efficiency function is used, with a peak at 555 nm (green light band);
  • Above $4^\circ$ viewing angle: \(V_{10}(\lambda)\) from CIE 1964 (CIE 165:2005), which is \(y_{10}(\lambda)\) in the \(10^\circ\) standard colorimetric observer, adapted for large field of view and off-axis visual tasks.
• Practical Applications: In LED lighting design, matching the 555 nm peak can improve the luminous efficacy of luminaires; automobile headlights use the photopic curve to optimize the white light spectrum and enhance road visibility.
  

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

• Applicable Scenarios: Extremely low luminance environments with $L < 0.005\ \mathrm{cd\cdot m^{-2}}$, such as unlit outdoors at night and darkrooms.
  
• Physiological Mechanism: Dominated by rod cells, containing only a single photosensitive substance (rhodopsin). They have extremely high light sensitivity but saturate easily in bright environments. They can only perceive light and dark, cannot distinguish colors, and are responsible for contour and motion perception under low light.
  

• Core Characteristics: $V'(\lambda)$ CIE 1951 scotopic spectral luminous efficiency function, with the peak shifted to 507 nm (blue-green light band).
  
• Relevant Standards: ISO 23539:2005 (E) / CIE S 010
  
• Practical Applications: Night vision devices and astronomical telescopes enhance night detection capabilities by matching the 507 nm blue-green light band; nighttime road signs use blue-green reflective materials to improve human eye recognition in low luminance environments.
  

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

• Applicable Scenarios: Transitional lighting environments with luminance $0.005\ \mathrm{cd\cdot m^{-2}} < L < 5\ \mathrm{cd\cdot m^{-2}}$, such as evening streets, underground garages, and cinemas.
  
• Physiological Mechanism: Co-dominated by rod and cone cells. Visual sensitivity dynamically adjusts with ambient luminance, and color vision gradually weakens as luminance decreases.
  
• Core Characteristics: The function is determined by the adaptation level coefficient m. The shape of the function changes dynamically with ambient luminance and is a weighted combination of the photopic and scotopic vision functions.
  
• Relevant Standards: CIE 191:2010
  
• Practical Applications: Urban nightscape lighting design needs to adjust the spectrum according to the mesopic curve to reduce energy consumption while ensuring visual comfort; smart streetlights dynamically adjust the spectrum by detecting ambient luminance in real time, adapting to the visual changes of the human eye from dusk to night.
  

Comparison Curves of Photopic/Scotopic Vision Functions
(Image source: https://commons.wikimedia.org/wiki/File:LuminosityCurve1.svg)
The figure above clearly shows the difference in spectral sensitivity 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, in nanometers (nm).

IV. Photometric Calculations Based on the Luminous Efficiency Function
The core 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 of the light source as perceived by the human eye.
- \(\varPhi_{\text{e},\lambda}(\lambda)\) is the spectral radiant flux (unit: W·nm⁻¹), describing the physical radiant power distribution of the light source at different wavelengths.
- \(K_{\text{m}}\) is the maximum luminous efficacy (unit: lm·W⁻¹), which is the maximum value of the luminous efficacy function, representing the conversion efficiency at the most sensitive wavelength.
- \(V(\lambda)\) is the photopic spectral luminous efficiency function. For scotopic or mesopic vision scenarios, it is replaced by \(V'(\lambda)\) or \(V_{\text{mes},m}(\lambda)\).

Taking photopic vision conditions as an example, the spectral luminous efficiency of the human eye for 490 nm monochromatic light is about 20% of that for 555 nm light. Therefore, when the radiant power of a 490 nm monochromatic light source is 5 times that of a 555 nm monochromatic light source, both will produce the same perception of brightness in the human eye.

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Luminous flux