# Spectral Distribution of Radiometric Quantities

## Spectral Variables

### Frequency

Radiation is a composition of individual photons. Each of these photons is characterised entirely by the photon's frequency $\nu$. The frequency is an intrinsic property of the photon and does not change during the photon's lifetime.

### Other Spectral Variables

Besides that there are more equivalent spectral variables which are related to the frequency and more useful under specific circumstances. Here is an overview over some of them:^{[1]}

Name | Symbol | SI-units | Relation |
---|---|---|---|

frequency | $\nu$ | Hz | |

wavelength | $\lambda$ | m | $\lambda = \frac{c}{f}$ |

spectroscopic wavenumber | $\widetilde{\nu}$ | m^{-1} |
$\widetilde{\nu} = \frac{1}{\lambda}$ |

angular frequency | $\omega$ | rad s^{-1} |
$\omega = 2\pi \nu$ |

wavenumber | $k$ | rad m^{-1} |
$k = \frac{2\pi}{\lambda}$ |

As mentioned above, all these quantities are equivalent. However, in the following examples the wavelength $\lambda$ will be used as it is probably the most intuitive one. But all of the others work equally well!

## Spectral Distribution

Every real source of radiation has a continuous radiation spectrum, i.e. the emitted radiation is in some way distributed over all wavelengths $\lambda$. Indeed there might be special cases in which most of the emitted radiation is concentrated within a very narrow wavelength interval such that one may consider this radiation as *monochromatic* ("one color", only one wavelength). However, it is good to keep in mind that in reality there is no such thing as light of only one single wavelength. Likewise, every detector of radiation is sensitive to a finite wavelength *interval* rather than a single value only.

Thus, if one wants to specify the spectral distribution of a radiometric quantity (for instance the emitted flux $\Phi$), it is not meaningful
to search for a function that specifies how much flux $\Phi$ is emitted at a *single* specific wavelength $\lambda_\text{s}$.^{[2]}

### Definition: Spectral Density

Instead one should rather look at the flux $\Delta\Phi$ contributed by a *wavelength interval* $\Delta \lambda$ containing $\lambda_\text{s}$. When dividing $\Delta\Phi$ by the interval width $\Delta \lambda$ and in the limit $\lim_{\Delta \lambda \rightarrow 0}$ this becomes a physically meaningful quantity:
\begin{align}
\Phi_\lambda (\lambda_\text{s}) := \lim_{\Delta\lambda\rightarrow 0} \frac{\Delta \Phi (\lambda_\text{s})}{\Delta \lambda} = \frac{\partial \Phi (\lambda_\text{s})}{\partial \lambda}
\end{align}
The total flux emitted over any wavelength inverval $I=[\lambda_1,\lambda_2]$ can then be calculated by integration:
\begin{align}
\Phi(I) = \int_{\lambda_1}^{\lambda_2} \Phi_\lambda (\lambda) \D{\lambda}
\end{align}
In particular one could calculate the total emission by integrating over *all* wavelengths:
\begin{align}
\Phi_\text{tot} = \int_{0}^{\infty} \Phi_\lambda (\lambda) \D{\lambda}
\end{align}

^{[3]}

Any quantity defined in this manner is referred to as a spectral density. One can do the same for other quantities like the radiance or the energy density.^{[4]}

### Change of Spectral Variable

The same holds for the wavelength $\lambda$ which could be replaced by any spectral variable as for instance frequency $\nu$ or angular frequency $\omega$. In particular the change between two spectral variables affects the spectral density as follows: \begin{align} \Phi_\lambda &= \frac{\partial\Phi}{\partial\lambda} = \frac{\partial\Phi}{\partial\nu}\cdot\left|\frac{\partial\nu}{\partial\lambda}\right| \\ &= \Phi_\nu \cdot \left|\frac{\partial\nu}{\partial\lambda}\right| \end{align} Thus, it is not sufficient to substitute the variable, one also must take the differential nature into account.

Why is it necessary to take the absolute value of the derivative? When integrating any spectral density over some interval from the lower bound to the higher one, the result is supposed to be the same irrespective of the choice of the spectral variable. If the partial derivative however is negative, this means that during an integration over the same interval (in the new variable) the order of the bounds would actually be reversed and the sign of the result changes. Thus, the absolute value takes account of a consistent definition.

### Definition: Spectral Distribution

A function that relates the spectral density of a radiometric quantity to the corresponding spectral variable is called spectral distribution.

## References

[1] | Introduction to Radiometry SPIE Press 1998 (ch. 2.5) |

[2] | A First Course in Atmospheric Radiation Sundog Publishing 2006 (ch. 2.2.2) |

[3] | A First Course in Atmospheric Radiation Sundog Publishing 2006 (ch. 2.7.1) |

[4] | A First Course in Atmospheric Radiation Sundog Publishing 2006 (ch. 2.7.1) |