# Spectral Quantities

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: quantities in radiometry and photometry which describe the distribution e.g. of a radiant flux over different optical frequencies or wavelengths

In radiometry and photometry, some of the used quantities are *spectral quantities*, which generally depend on the optical frequency or wavelength. Some of them are simply frequency-dependent properties of materials or objects, such as a reflectance, transmittance or absorbance. Others describe the distribution e.g. of a radiant flux over different optical frequencies or wavelengths. Their symbols often contain “ν” (for the optical frequency) or “λ” (for the wavelength) in the subscript.

For example, the *spectral flux* <$\Phi_{\rm{e},\nu }$> is the radiant flux <$\Phi_\textrm{e}$> per (infinitesimally small) unit frequency interval in fundamental units of W/Hz; similarly <$\Phi_{\rm{e},\lambda }$> is the radiant flux per unit wavelength interval in units of W/m.

Some important examples of such quantities:

Quantity | Symbol | Units | Remarks |
---|---|---|---|

spectral flux | <$\Phi_{\rm{e},\nu}$> <$\Phi_{\rm{e},\lambda }$> | W/Hz W/nm | radiant flux per unit frequency or wavelength |

spectral intensity | <$I_{\rm{e},\Omega,\nu}$> <$I_{\rm{e},\Omega,\lambda}$> | W sr^{−1} Hz^{−1} W sr ^{−1} nm^{−1} | radiant intensity per unit frequency or wavelength |

spectral radiance | <$L_{\rm{e},\Omega,\nu}$> <$L_{\rm{e},\Omega,\lambda}$> | W sr^{−1} m^{−2} Hz^{−1} W sr ^{−1} m^{−2} nm^{−1} | radiance per unit frequency or wavelength |

spectral irradiance | <$E_{\rm{e},\nu}$> <$E_{\rm{e},\lambda }$> | W m^{−2} Hz^{−1} W m ^{−2} nm^{−1} | irradiance per unit frequency or wavelength |

spectral radiosity | <$J_{\rm{e},\nu}$> <$J_{\rm{e},\lambda }$> | W m^{−2} Hz^{−1} W m ^{−2} nm^{−1} | radiosity per unit frequency or wavelength |

spectral exitance | <$M_{\rm{e},\nu}$> <$M_{\rm{e},\lambda ;}$> | W m^{−2} Hz^{−1} W m ^{−2} nm^{−1} | radiant exitance per unit frequency or wavelength |

spectral exposure | <$H_{\rm{e},\nu}$> <$H_{\rm{e},\lambda }$> | J m^{−2} Hz^{−1} J m ^{−2} nm^{−1} | radiant exposure per unit frequency or wavelength |

spectral luminous flux | <$\Phi_{\rm{v},\nu }$> <$\Phi_{\rm{v},\lambda }$> | lm Hz^{−1} lm nm ^{−1} | luminous flux per unit frequency or wavelength |

spectral luminous intensity | <$I_{\rm{v},\nu }$> <$I_{\rm{v},\lambda }$> | cd Hz^{−1} cd nm ^{−1} | luminous intensity per unit frequency or wavelength |

spectral illuminance | <$E_{\rm{v},\nu }$> <$E_{\rm{v},\lambda }$> | lx Hz^{−1} lx nm ^{−1} | illuminance per unit frequency or wavelength |

(The subscript “e” stand for energy, indicating radiometric quantity, while “v” stands for “vision”, indicating photometric quantities.)

By integration of those quantities over all optical frequencies or wavelengths, respectively, one obtains the corresponding integral quantities. For example, the radiant flux equals the frequency- or wavelength-integrated spectral flux.

In most cases, the spectral distributions result from statistical processes. A notable exception is the generation or ultrashort pulses with mode-locked lasers, which is a highly deterministic process. Here, optical spectra maybe calculated from power spectral densities of field amplitudes.

## Conversion Between Frequency- and Wavelength-related Spectral Quantities

Note that it is *not* correct e.g. to integrate <$\Phi_{\rm{e},\nu}$> (a quantity referring to optical frequencies) over all *wavelengths*, simply using the argument <$\nu = c / \lambda$>. Even the resulting units would not be correct. One also needs to take into account the conversion from frequency to wavelength intervals. As we can calculate the integrated radiant flux as

we must conclude that

$${\Phi_\rm{e}} = \int {{\Phi_{{\rm{e,}}\nu }}(\nu )\;{\rm{d}}\nu } = \int {{\Phi_{{\rm{e,}}\lambda }}(\lambda )\;{\rm{d}}\lambda } $$where the conversion factor is wavelength-dependent.

A consequence of that is that the peak location of a spectral quantity in terms of optical frequency does *not* generally agree with <$c$> divided by the peak wavelength, calculated from the corresponding wavelength-related quantity. In cases with broad spectral distributions – for example, the spectrum of blackbody radiation –, that can make a substantial difference.

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