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# Spectral Quantities

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

German: spektrale Größen

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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 Φe,ν is the radiant flux Φe per (infinitesimally small) unit frequency integral in fundamental units of W/Hz; similarly Φe,λ is the radiant flux per unit wavelength interval in units of W/m.

Some important examples of such quantities:

QuantitySymbolUnitsRemarks
spectral fluxΦe,ν
Φe,λ
W/Hz
W/nm
radiant flux per unit frequency or wavelength
spectral intensityIe,Ω,ν
Ie,Ω,λ
W sr−1 Hz−1
W sr−1 nm−1
radiant intensity per unit frequency or wavelength
Le,Ω,λ
W sr−1 m−2 Hz−1
W sr−1 m−2 nm−1
radiance per unit frequency or wavelength
Ee,λ
W m−2 Hz−1
W m−2 nm−1
irradiance per unit frequency or wavelength
Je,λ
W m−2 Hz−1
W m−2 nm−1
radiosity per unit frequency or wavelength
spectral exitanceMe,ν
Me,λ;
W m−2 Hz−1
W m−2 nm−1
radiant exitance per unit frequency or wavelength
spectral exposureHe,ν
He,λ
J m−2 Hz−1
J m−2 nm−1
radiant exposure per unit frequency or wavelength
spectral luminous fluxΦv,ν
Φv,λ
lm Hz−1
lm nm−1
luminous flux per unit frequency or wavelength
spectral luminous intensityIv,ν
Iv,λ
cd Hz−1
cd nm−1
luminous intensity per unit frequency or wavelength
spectral illuminanceEv,ν
Ev,λ
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 Φe,ν (a quantity referring to optical frequencies) over all wavelengths, simply using the argument ν = c / λ. 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 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. If you like this article, share it with your friends and colleagues, e.g. via social media: