Principles of Spectroradiometry, Photometry and Colorimetry Calculation

1. Introduction

In the fields of optical measurement and illumination engineering, the quantitative description of "light" involves three interrelated yet independently meaningful physical quantity systems: Radiometry, Photometry and Colorimetry.

Radiometry takes pure physical energy as its yardstick, describing how electromagnetic radiation is transmitted, distributed and received in space; its quantities do not depend on any observer's perceptual characteristics. Photometry introduces the human-eye spectral luminous efficiency function $V(\lambda)$ on top of radiometry, weighting physical radiant energy into a measure of the "brightness" perceived by the human eye, thereby bridging the gap from objective energy to subjective perception. Colorimetry goes further: through the CIE standard colorimetric observer model, it maps the spectral power distribution into coordinates in a three-dimensional color space, allowing the subjective experience of "color" to be precisely quantified and communicated.

The core link among the three is the spectral power distribution $\Phi(\lambda)$ — the common starting point for converting radiometric quantities into photometric and colorimetric quantities. This article follows the logical path of "radiometric quantity → photometric quantity → colorimetric quantity", systematically explaining the definitions, calculation methods and internal relationships of each quantity, and illustrating their engineering application value in real spectral measurement scenarios.

2. Fundamentals of Radiometry

Radiometry covers the ultraviolet, visible and infrared bands. Its core quantities all use the subscript e (energetic) to distinguish them from the corresponding subscript v (visual) used in photometry.

2.1 Radiant Flux

Radiant flux $\Phi_e$ is defined as the radiant energy emitted, transmitted or received per unit time, with the unit watt (W). In spectral measurement, radiant flux is usually expressed as a function of wavelength — i.e., the spectral radiant flux $\Phi_e(\lambda)$, with the unit $\mathrm{W/nm}$ or $\mathrm{W/}\mu\mathrm{m}$.

The integrated radiant flux over a finite wavelength interval is:

$$ \Phi_e = \int_{\lambda_1}^{\lambda_2} \Phi_e(\lambda) \,\mathrm{d}\lambda $$

In practical spectral measurement, the integral degenerates into a discrete summation:

$$ \Phi_e = \sum \Phi_e(\lambda) \,\Delta\lambda $$

2.2 Radiant Intensity

Radiant intensity $I_e$ is defined as the radiant flux emitted by a point source per unit solid angle, with the unit $\mathrm{W/sr}$ (watt per steradian):

$$ I_e = \frac{\mathrm{d}\Phi_e}{\mathrm{d}\Omega} $$

Radiant intensity describes the degree of power concentration of a radiation source in a given direction in space; it is an important parameter characterizing the radiation characteristics of directional sources such as LEDs and lasers.

2.3 Radiance

Radiance $L_e$ is one of the most central quantities in radiometry. It is defined as the radiant flux emitted by a surface source per unit projected area per unit solid angle, with the unit $\mathrm{W/(sr \cdot m^2)}$:

$$ L_e = \frac{\mathrm{d}^2\Phi_e}{\mathrm{d}A \cdot \cos\theta \cdot \mathrm{d}\Omega} $$

where $\theta$ is the angle between the observation direction and the surface normal, and $\mathrm{d}A \cdot \cos\theta$ is the effective projected area.

For the surface sources commonly encountered in spectral measurement, radiance can be calculated from the spectral radiant flux:

$$ L_e = \frac{1}{\pi S} \sum \Phi_e(\lambda) \,\Delta\lambda $$

where $S$ is the emitting area, and the coefficient $1/\pi$ arises from the integration of a Lambertian radiator over the hemispherical space. An important property of radiance is that, for an ideal Lambertian radiator, $L_e$ is independent of the observation direction (i.e., the brightness is equal in all directions), which makes radiance the preferred parameter for characterizing the uniformity of displays and diffuse surface sources.

2.4 Irradiance

Irradiance $E_e$ is defined as the radiant flux incident on unit receiving area, with the unit $\mathrm{W/m^2}$:

$$ E_e = \frac{\mathrm{d}\Phi_e}{\mathrm{d}A} $$

Irradiance characterizes the density of radiant energy received by an "illuminated surface", complementing radiant intensity (which describes the "source") and radiance (which describes the "surface brightness of the source"). In scenarios such as solar radiation measurement and UV disinfection dose assessment, irradiance is the most directly relevant radiometric quantity.

2.5 Summary of Radiometric Quantities

Quantity Symbol Definition Unit
Radiant flux $\Phi_e$ Radiant energy per unit time $\mathrm{W}$
Radiant intensity $I_e$ Radiant flux per unit solid angle $\mathrm{W/sr}$
Radiance $L_e$ Radiant flux per unit projected area per unit solid angle $\mathrm{W/(sr \cdot m^2)}$
Irradiance $E_e$ Incident radiant flux per unit area $\mathrm{W/m^2}$

3. Fundamentals of Photometry

Photometry is the "human-eye-weighted" version of radiometry in the visible band (about 380-780 nm). It weights physical radiant energy according to the sensitivity of the human eye to different wavelengths and integrates, yielding quantities consistent with the subjective brightness perceived by the human eye. Photometric quantities use the subscript v (visual).

3.1 The Human-Eye Spectral Luminous Efficiency Function $V(\lambda)$

In 1924, the CIE (International Commission on Illumination) defined the photopic spectral luminous efficiency function $V(\lambda)$, which describes the relative sensitivity of the human eye to monochromatic light of different wavelengths under high-brightness (photopic, mainly cone cells working) conditions.

$V(\lambda)$ is a bell-shaped curve whose peak lies at 555 nm (yellow-green), with a peak value of 1. At wavelengths away from 555 nm, $V(\lambda) < 1$, meaning the eye perceives these wavelengths as relatively "dimmer". For example:

  • At 510 nm (cyan), $V(\lambda) \approx 0.503$, i.e., at the same radiant power, the subjective brightness of 510 nm light is about half that of 555 nm;
  • At 650 nm (red), $V(\lambda) \approx 0.107$, i.e., the radiant power needs to be nearly 10 times that at 555 nm to produce the same perceived brightness.

Under scotopic (low-brightness, rod cells working) conditions, the scotopic spectral luminous efficiency function $V'(\lambda)$ is used instead, with its peak shifted to 507 nm. The discussion below is based on photopic conditions.

In practical engineering calculations, $K(\lambda)$ (also written $K_m V(\lambda)$) is often used as the spectral luminous efficacy function, with its peak $K_m = 683 \,\mathrm{lm/W}$.

3.2 Luminous Flux

Luminous flux $\Phi_v$ is the result of weighting the radiant flux $\Phi_e(\lambda)$ by $V(\lambda)$ and integrating (multiplied by $K_m$ to convert to lumens), with the unit lumen ($\mathrm{lm}$):

$$ \Phi_v = K_m \int_{380}^{780} \Phi_e(\lambda) \, V(\lambda) \,\mathrm{d}\lambda $$

Its discrete form is:

$$ \Phi_v = 683 \times \sum \Phi_e(\lambda) \, V(\lambda) \,\Delta\lambda $$

Luminous flux is the most basic derived quantity in photometry; it describes the "total brightness sensation produced in the human eye" by a light source in the visible band.

3.3 The Origin of $K_m = 683 \,\mathrm{lm/W}$

$K_m = 683 \,\mathrm{lm/W}$ is the maximum spectral luminous efficacy under photopic conditions. Its definition can be traced back to the redefinition of the candela at the 16th CGPM (General Conference on Weights and Measures) in 1979:

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540 \times 10^{12}\,\mathrm{Hz}$ (corresponding to a wavelength of about 555 nm in vacuum) and that has a radiant intensity in that direction of $1/683 \,\mathrm{W/sr}$.

In other words, at the peak wavelength of $V(\lambda)$, 555 nm, a radiant flux of $1\,\mathrm{W}$ corresponds exactly to a luminous flux of $683\,\mathrm{lm}$. This value was confirmed at the 1979 CGPM and has been used ever since. It is the fixed conversion coefficient connecting the two quantity systems of radiometry (watt) and photometry (lumen).

3.4 Luminous Intensity

Luminous intensity $I_v$ is the luminous flux emitted by a point source per unit solid angle, with the unit candela ($\mathrm{cd}$), $\mathrm{1\,cd = 1\,lm/sr}$:

$$ I_v = \frac{\mathrm{d}\Phi_v}{\mathrm{d}\Omega} $$

The candela is one of the seven base units of the International System of Units (SI) and the cornerstone of the photometric system.

3.5 Luminance

Luminance $L_v$ is the luminous flux emitted by a surface source per unit projected area per unit solid angle, with the unit $\mathrm{cd/m^2}$ (sometimes called nit):

$$ L_v = \frac{\mathrm{d}^2\Phi_v}{\mathrm{d}A \cdot \cos\theta \cdot \mathrm{d}\Omega} $$

For a Lambertian radiator (single-sided emission), there is a concise conversion relationship between radiometric and photometric quantities:

$$ \Phi_v = 683 \times \Phi_e \times K \times 10^{-6} = \pi \times L_v \times S \times 10^{-6} $$

where $K$ is the luminous-efficiency-function weighted average coefficient of the source and $S$ is the emitting area. This formula has significant practical engineering value for evaluating the luminous flux of Lambertian surface sources such as LED displays and backlight modules.

3.6 Illuminance

Illuminance $E_v$ is the luminous flux incident on unit area, with the unit lux ($\mathrm{lx}$), $\mathrm{1\,lx = 1\,lm/m^2}$:

$$ E_v = \frac{\mathrm{d}\Phi_v}{\mathrm{d}A} $$

Illuminance is the most commonly used design index in illumination engineering; national standards specify clear illuminance requirements for different environments and task types.

3.7 Summary of Photometric Quantities

Quantity Symbol Definition Unit
Luminous flux $\Phi_v$ $V(\lambda)$-weighted integral of radiant flux $\mathrm{lm}$
Luminous intensity $I_v$ Luminous flux per unit solid angle $\mathrm{cd = lm/sr}$
Luminance $L_v$ Luminous flux per unit projected area per unit solid angle $\mathrm{cd/m^2}$
Illuminance $E_v$ Incident luminous flux per unit area $\mathrm{lx = lm/m^2}$

4. Conversion Between Radiometric and Photometric Quantities

The conversion between radiometric and photometric quantities is, in essence, weighting the physical energy spectrum $\Phi_e(\lambda)$ by the human-eye spectral luminous efficiency function $V(\lambda)$ and integrating, then multiplying by the maximum spectral luminous efficacy $K_m$. The general relationship is:

$$ \Phi_v(\lambda) = 683 \times V(\lambda) \times \Phi_e(\lambda) $$

Integrating over the entire visible band gives the cumulative luminous flux:

$$ \Phi_v = 683 \int_{380}^{780} \Phi_e(\lambda) \, V(\lambda) \,\mathrm{d}\lambda $$

Its discrete form is:

$$ \Phi_v = 683 \sum_{\lambda=380}^{780} \Phi_e(\lambda) \, V(\lambda) \,\Delta\lambda $$

4.1 Conversion Flow Diagram

Spectral radiant flux Φ_e(λ)  ──× V(λ)──▶  Weighted spectrum
                                     │
                                     ▼
                               × 683 lm/W
                                     │
                                     ▼
                               Spectral luminous flux Φ_v(λ)
                                     │
                               Σ (integral / summation)
                                     │
                                     ▼
                               Total luminous flux Φ_v (lm)

4.2 Spectral Luminous Efficacy $K(\lambda)$

The spectral luminous efficacy function $K(\lambda)$ defined by the CIE satisfies:

$$ K(\lambda) = 683 \times V(\lambda) \quad [\mathrm{lm/W}] $$

$K(\lambda)$ intuitively gives the conversion coefficient "how many lumens of luminous flux correspond to 1 W of radiant flux" at each wavelength. It reaches its maximum $683\,\mathrm{lm/W}$ at 555 nm and approaches zero at the edges of the visible band.

4.3 Conversion of Other Corresponding Radiometric-Photometric Quantities

There is a one-to-one correspondence between radiometric and photometric quantities, as shown in the table below:

Radiometric quantity (subscript e) Photometric quantity (subscript v) Conversion relationship
Radiant flux $\Phi_e$ Luminous flux $\Phi_v$ $\Phi_v = 683 \int \Phi_e(\lambda)V(\lambda)\mathrm{d}\lambda$
Radiant intensity $I_e$ Luminous intensity $I_v$ $I_v = 683 \int I_e(\lambda)V(\lambda)\mathrm{d}\lambda$
Radiance $L_e$ Luminance $L_v$ $L_v = 683 \int L_e(\lambda)V(\lambda)\mathrm{d}\lambda$
Irradiance $E_e$ Illuminance $E_v$ $E_v = 683 \int E_e(\lambda)V(\lambda)\mathrm{d}\lambda$

5. Fundamentals of Colorimetry

Colorimetry maps the spectral power distribution into coordinates in the space of human color perception. Its theoretical basis is the trichromatic theory (Young-Helmholtz theory) — the human retina contains three types of cone cells (L, M and S) with different sensitivities to different wavelengths, and any color perception can be described by a linear combination of the response intensities of the three.

5.1 CIE 1931 Standard Colorimetric Observer

In 1931, the CIE established the CIE 1931 Standard Colorimetric Observer, defining three color matching functions (CMFs): $\bar{x}(\lambda)$, $\bar{y}(\lambda)$ and $\bar{z}(\lambda)$.

These three functions were obtained by statistical averaging over color-matching experiments with a large number of observers. Notably, $\bar{y}(\lambda)$ was deliberately designed to be exactly identical to the photopic spectral luminous efficiency function $V(\lambda)$ — this ingenious design makes the $Y$ value in colorimetry (one of the tristimulus values) also carry the meaning of luminous flux in photometry, becoming the key node where radiometry, photometry and colorimetry meet.

The CIE 1931 Standard Colorimetric Observer is based on a 2° field of view (small field, corresponding to the cone-cell distribution in the foveal region of the retina). For applications requiring a 10° field of view (large field), the CIE 1964 Supplementary Standard Colorimetric Observer should be used; its color matching functions differ, giving higher weight to the short-wavelength region.

5.2 Tristimulus Values $X$, $Y$, $Z$

The tristimulus values are the results of integrating the spectral power distribution $\Phi(\lambda)$ with the three color matching functions respectively:

$$ X = k \int_{380}^{780} \Phi(\lambda) \, \bar{x}(\lambda) \,\mathrm{d}\lambda $$

$$ Y = k \int_{380}^{780} \Phi(\lambda) \, \bar{y}(\lambda) \,\mathrm{d}\lambda $$

$$ Z = k \int_{380}^{780} \Phi(\lambda) \, \bar{z}(\lambda) \,\mathrm{d}\lambda $$

where $k$ is a normalization coefficient. For source colors (self-luminous objects), $k = 683 \,\mathrm{lm/W}$, in which case $Y$ is the luminous flux; for object colors (reflecting or transmitting objects), $k$ is chosen so that a perfect diffuse reflector has $Y = 100$, in which case $\Phi(\lambda)$ is the relative spectral power distribution.

The discrete summation form is:

$$ X = k \sum \Phi(\lambda) \, \bar{x}(\lambda) \,\Delta\lambda $$

$$ Y = k \sum \Phi(\lambda) \, \bar{y}(\lambda) \,\Delta\lambda $$

$$ Z = k \sum \Phi(\lambda) \, \bar{z}(\lambda) \,\Delta\lambda $$

5.3 Chromaticity Coordinates $x$, $y$ and the CIE Chromaticity Diagram

The tristimulus values $X$, $Y$, $Z$ contain both the luminance and chromaticity information of a color. After separating the luminance component, one obtains the chromaticity coordinates that describe only the color quality (hue and saturation):

$$ x = \frac{X}{X + Y + Z} $$

$$ y = \frac{Y}{X + Y + Z} $$

$$ z = \frac{Z}{X + Y + Z} = 1 - x - y $$

By definition, $x$ and $y$ fully determine the chromaticity ($z$ is redundant information), and all physically realizable colors fall within a horseshoe-shaped region in the $(x, y)$ coordinate system — the famous CIE 1931 Chromaticity Diagram.

Several key features of the CIE chromaticity diagram:

  • Spectral locus (horseshoe boundary): the curve formed by connecting the chromaticity coordinates of monochromatic light (380-780 nm), containing the most saturated colors;
  • Purple line: the straight line connecting the short-wavelength end (380 nm) and long-wavelength end (780 nm) of the spectral locus; the colors on it are mixtures of the two extreme wavelengths, with no corresponding monochromatic light in nature;
  • Equal-energy white point (point E): $x = y = z = 1/3$, i.e., $x = 0.3333$, $y = 0.3333$;
  • Planckian locus: the curve formed by the chromaticity coordinates of a blackbody at different temperatures; the chromaticity of white light sources is usually described by the nearest blackbody color temperature (CCT, correlated color temperature).

5.4 Example of Chromaticity Coordinate Calculation

Given a set of simplified spectral data (sampled at equal intervals $\Delta\lambda = 10\,\mathrm{nm}$), the steps for calculating the tristimulus values are:

  1. Read the values of $\bar{x}(\lambda)$, $\bar{y}(\lambda)$, $\bar{z}(\lambda)$ at each wavelength (CIE standard data tables);
  2. Calculate the weighted products: $\Phi(\lambda) \cdot \bar{x}(\lambda)$, $\Phi(\lambda) \cdot \bar{y}(\lambda)$, $\Phi(\lambda) \cdot \bar{z}(\lambda)$;
  3. Sum and multiply by $k$ and $\Delta\lambda$ to obtain $X$, $Y$, $Z$;
  4. Calculate $x = X/(X+Y+Z)$, $y = Y/(X+Y+Z)$.

6. CIE 1976 $L^*a^*b^*$ Uniform Color Space

Although the CIE 1931 chromaticity diagram laid the foundation of colorimetry, its chromaticity coordinates $(x, y)$ are visually non-uniform: in different regions of the diagram, the same geometric distance does not correspond to the same perceived color difference. This causes serious inconvenience for color-difference calculation and chromaticity tolerance setting.

To solve this problem, in 1976 the CIE recommended the CIE 1976 $L^*a^*b^*$ color space (abbreviated CIELAB), which maps the tristimulus values $X$, $Y$, $Z$ into an approximately visually uniform three-dimensional space through nonlinear transformation.

6.1 Coordinate Definitions

The three coordinates of the CIELAB space are defined as:

Lightness:

$$ L^* = 116 \cdot f\left(\frac{Y}{Y_n}\right) - 16 $$

Red-green axis:

$$ a^* = 500 \left[ f\left(\frac{X}{X_n}\right) - f\left(\frac{Y}{Y_n}\right) \right] $$

Yellow-blue axis:

$$ b^* = 200 \left[ f\left(\frac{Y}{Y_n}\right) - f\left(\frac{Z}{Z_n}\right) \right] $$

where $X_n$, $Y_n$, $Z_n$ are the tristimulus values of the reference white point (such as the D65 standard illuminant), and the function $f(t)$ is a piecewise function:

$$ f(t) = \begin{cases} t^{1/3}, & t > 0.008856 \\ 7.787 \, t + \dfrac{16}{116}, & t \leq 0.008856 \end{cases} $$

6.2 Meaning of the Coordinates

  • $L^*$ (Lightness): range $0 \sim 100$, with $L^*=0$ being pure black and $L^*=100$ being the reference white;
  • $a^*$ (red-green axis): positive values tend toward red, negative values toward green;
  • $b^*$ (yellow-blue axis): positive values tend toward yellow, negative values toward blue.

6.3 Color-Difference Formula

The greatest practical value of the CIELAB space lies in color-difference calculation. The color difference $\Delta E^*_{ab}$ between two colors is defined as the Euclidean distance:

$$ \Delta E^*_{ab} = \sqrt{(L^*_1 - L^*_2)^2 + (a^*_1 - a^*_2)^2 + (b^*_1 - b^*_2)^2} $$

Generally, $\Delta E^*_{ab} < 1$ means the color difference is essentially imperceptible to the human eye; $\Delta E^*_{ab} \approx 2-3$ is a perceptible but small difference; $\Delta E^*_{ab} > 5$ is an obvious difference. This formula is widely used in industrial fields such as LED binning, display color-accuracy evaluation and printed-color-difference control.

7. Application Scenarios in Practical Spectral Measurement

7.1 Comprehensive Evaluation of LED Light Sources

The application of spectroradiometric, photometric and colorimetric calculation is most typical in the LED industry. By measuring the absolute spectral power distribution $\Phi_e(\lambda)$ of an LED with a spectroradiometer or an integrating-sphere-spectrometer system, all key parameters can be derived at once:

  • Radiometric quantities: radiant flux $\Phi_e$ ($\mathrm{W}$);
  • Photometric quantities: luminous flux $\Phi_v$ ($\mathrm{lm}$), luminous efficacy ($\mathrm{lm/W}$);
  • Colorimetric quantities: chromaticity coordinates $(x, y)$, correlated color temperature (CCT), color rendering index (CRI / $R_a$);
  • Color space coordinates: $L^*a^*b^*$ or $L^*u^*v^*$ coordinates, used for LED binning.

7.2 Measurement of Displays and Surface Sources

For surface sources such as OLED / LCD displays and backlight modules, radiance (luminance) measurement is the core requirement. The radiance $L_e$ (or luminance $L_v$) is measured with an imaging luminance meter or a spectroradiometer equipped with a telephoto lens, and the luminous flux is then calculated together with the emitting area.

Using the property of a Lambertian radiator ($\Phi_v = \pi \cdot L_v \cdot S$), the total luminous flux can be quickly estimated when the emitting area and luminance are known, which is of significant engineering convenience for the online inspection of large-area sources.

7.3 Color Quality Control and Color-Difference Evaluation

In industries such as textile dyeing, plastic coloring, automotive paint and printing/packaging, the CIELAB color difference $\Delta E^*_{ab}$ is the common language of color quality control. The spectral reflectance of a sample is measured with a spectrophotometer or spectroradiometer, $X$, $Y$, $Z$ and $L^*a^*b^*$ are calculated, and $\Delta E^*_{ab}$ is compared with the target color to determine whether the product color is within the allowable tolerance range.

7.4 Solar Radiation and UV Measurement

In fields such as photovoltaics, meteorology and photobiological safety, irradiance $E_e$ (especially irradiance in the UV band) is the core measurement parameter. The absolute spectral irradiance of solar or artificial sources is measured with a spectroradiometer equipped with a cosine corrector (diffuser), from which the integrated irradiance of the UV-A, UV-B and UV-C bands can be calculated for photobiological safety classification (IEC 62471) and photovoltaic cell efficiency evaluation.

7.5 Illumination Engineering Design and Evaluation

In illumination engineering, the measurement and calculation of illuminance $E_v$ and luminance $L_v$ are the basis of design. The spectral power distribution of a source is measured with a spectroradiometer to calculate luminous flux and luminous efficacy, and the spatial illuminance distribution is then evaluated in combination with the luminous intensity distribution curve. At the same time, the chromaticity coordinates and correlated color temperature (CCT) are used to evaluate the lighting quality (suitability of color temperature, color rendering), ensuring compliance with the lighting standards of different application scenarios (offices, shopping malls, roads, sports venues, etc.).

8. Summary

Spectroradiometric, photometric and colorimetric calculations together constitute the theoretical cornerstone of optical measurement and illumination engineering. The three take the spectral power distribution as their common input and output physical parameters of different dimensions along different paths:

  1. Radiometry provides an observer-independent, purely physical energy description and is the origin of all optical radiation measurement;
  2. Photometry converts the physical energy spectrum into a "brightness" measure perceived by the human eye through the spectral luminous efficiency function $V(\lambda)$ and the conversion coefficient $683\,\mathrm{lm/W}$;
  3. Colorimetry maps the spectrum into three-dimensional color space coordinates through the color matching functions of the CIE standard colorimetric observer, and realizes quantitative color-difference evaluation with the help of the CIELAB uniform color space.

Understanding the definitions of these three systems, the conversion logic between them and the engineering calculation methods of discrete integration is the prerequisite for the design of spectral measurement systems, data analysis and result interpretation. With the continuous progress of spectral measurement technology (such as the popularization of array spectrometers and the miniaturization of micro spectral sensors), integrated radiometric-photometric-colorimetric measurement has become the industry-standard configuration; a thorough grasp of the relevant principles will directly improve the rationality of measurement scheme design and the reliability of data interpretation.

Compiled and written by Pynect.