Small Radiometry FAQ

Let's define the radiometry and photometry units for the light source. Radiometry is the measurement of optical radiation, which is electromagnetic radiation within the frequency range between $3\times10^{11}$ and $3\times10^{16}$ Hz ($\lambda \in 0.01 .. 1000 \mu m$). Typical units encountered are Watt/$m^2$ and photons/s-steradian[1].

Photometry is the measurement of light, which is defined as electromagnetic radiation that is detectable by the human eye. It is restricted to the wavelength range from $\lambda \in 0.36 .. 0.83 \mu m$ [1]. Photometry is radiometry that is weighted by the spectral response of the eye.


Light sources
The light is generated by the source with known radiance and wavelength $\lambda$. Irradiance is known for the sensor in Watt/$m^2$ [2]. There are two types of light sources: lambertian and isotropic[3]. Both terms mean the same in all directions but they are not interchangeable.

Isotropic implies a spherical source that radiates the same in all directions, i.e., the intensity (W/sr) is the same in all directions. We often encounter the phrase ``isotropic point source'', however there can be no such thing because the energy density
would have to be infinite. But a small, uniform sphere comes very close. The best
example is a globular tungsten lamp with a milky white diffuse envelope. A distant star can be considered an isotropic point source.

Lambertian refers to a flat radiating surface. It can be an active surface or a passive, reflective surface. Here the intensity falls off as the cosine of the
observation angle with respect to the surface normal, that is Lambert's law[4]. The radiance
(W/$m^2$-sr) is independent of direction.



Radiometric units
Radiometric units can be divided into two conceptual areas: those having to do with
power or energy, and those that are geometric in nature. The first two are:

Energy is an SI derived unit, measured in joules (J). The recommended symbol for
energy is $Q$. An acceptable alternate is W[2].

Power (radiant flux) is another SI derived unit. It is the rate of flow
(derivative) of energy with respect to time, $dQ/dt$, and the unit is the watt (W). The
recommended symbol for power is $\Phi$. An acceptable alternate is P.

Now we incorporate power with the geometric quantities area and solid angle.

Irradiance (flux density) is another SI derived unit and is measured in W/$m^2$.
Irradiance is power per unit area incident from all directions in a hemisphere \textit{onto a
surface} that coincides with the base of that hemisphere. The symbol for irradiance is $E$ and the symbol for radiant exitance is $M$. Irradiance is the derivative of power
with respect to area, $d\Phi/dA$. The integral of irradiance or radiant exitance over
area is power.

Radiance is the last SI derived unit we need and is measured in W/$m^2$-sr. Radiance
is power per unit projected area per unit solid angle. The symbol is L. Radiance is
the derivative of power with respect to solid angle and projected area, $d\Phi/d\omega dA
cos(\theta)$, where $\theta$ is the angle between the surface normal and the specified direction. The integral of radiance over area and solid angle is power.

A great deal of confusion concerns the use and misuse of the term intensity. Some
use it for W/sr, some use it for W/$m^2$ and others use it for W/$m^2$-sr. It is quite
clearly defined in the SI system, in the definition of the base unit of luminous
intensity, the candela. For an extended discussion see Ref.[5].


Astronomical Magnitudes
In astronomy, absolute magnitude (also known as absolute visual magnitude when measured in the standard V photometric band) measures a celestial object's intrinsic brightness. To derive absolute magnitude from the observed apparent magnitude of a celestial object its value is corrected from distance to its observer. One can compute the absolute magnitude $M$, of an object given its apparent magnitude $m\,$ and luminosity distance $D_L\,$:

$M = m - 5 ((\log_{10}{D_L}) - 1)\, $

where $D_L\,$ is the star's luminosity distance in parsecs, wherein 1 parsec is approximately 3.2616 light-years. The dimmer an object appears, the higher its apparent magnitude.
The apparent magnitude in the band x can be defined as (noting that $\log_{\sqrt[5]{100}} F = \frac{\log_{10} F }{\log_{10} 100^{1/5}} = 2.5\log_{10} F$)


$m_{x}= -2.5 \log_{10} (F_x/F_x^0)\,$

where $F_x\,$ is the observed flux in the band $x$, and $F_x^0$ is a reference flux in the same band x, such as the Vega star's for example.


Conversion to photons
Photon quantities of light energy are also common. They are related to the radiometric quantities by the relationship $Q_p = hc/\lambda$ where $Q_p$ is the energy of a photon at wavelength $\lambda$, $h$ is Planck's constant and $c$ is the velocity of light. At a wavelength of $1 \mu m$, there are approximately $5\times10^{18}$ photons per second in a watt. Conversely, a single photon has an energy of $2\times10^{-19}$ joules (W s) at $\lambda = 1 \mu m$.

References

1

J.M. Palmer and L. Carroll.
Radiometry and photometry FAQ, 1999.

2

C. DeCusatis.
Handbook of applied photometry.
American Institute of Physics, 1997.

3

The basis of physical photometry.
Technical report, CIE Technical Report 18.2, 1983.

4

J.H. Lambert.
Photometria sive de mensura de gratibus luminis, colorum et umbrae.
Eberhard Klett, 2, 1760.

5

J.M. Palmer.
Getting intense on intensity.
Metrologia, 30:371, 1993.