RELATIVE AND ABSOLUTE MEASUREMENTS OF
IRRADIANCE
TARAS PLAKHOTNIK
Irradiance I is the radiant power received by a surface area divided by the
area. SI units of this quantity are W · m−2 . Measurements of irradiance exploit
photodetectors. At each frequency of the electromagnetic radiation, the output
of a good photo detector U is proportional to the irradiance, that is U ∝ I but
unfortunately, sensitivity of every photodetector depends on the frequency of the
measured radiation. That is U = S(ν)σI, where σ is the photo sensitive area of the
photodetector and S(ν) is the frequency (wavelength) dependent sensitivity. Note
that if wavelengths λ is used instead of ν, then one should specify the media where
the wave travels because the wavelength depends on the speed of light which is
media dependent. If the light travels in air, the wavelength will be slightly shorter
then the wavelength of the same radiation in vacuum.
To use a photo detector for absolute measurements, one has to know functions
S(ν) or S(λ). This knowledge can be obtained in the process of calibration which
can be done by irradiating the photodetector with a light source for which the
spectral intensity at all relavent wavelengths is known. Note that this is different from irradiance. The spectral intensity L(ν, Ω) characterises the source and
tells how much energy is emitted in a unit time in a particular direction and per
unit range of its spectrum. The units of spectral intensity are W · sr−1 · Hz−1 or
W · sr−1 · m−1 . Such sources are quire expensive and are calibrated at a factory.
The manufacturer frequently specifies L(ν, Ω) only for a particular direction. But
the function L(ν, Ω) approximately (with a good accuracy for most practical applications but not for all) is a product of two functions LΩ (Ω)Lν (ν) and therefore
the spectral dependence and the directional dependence are separated.
When exposing the detector to light coming from the calibrated source, we
should consider the distance from the source to the detector. It is also possible
that some additional equipment is employed (e.g. a fibre may be used to collect
light and to direct it to the photodetector, etc). Such details are important to
know for calculations of the solid angle dΩ within which the radiation is collected
and directed onto the photo sensitive area of the photo detector. For example, for
a point-like source and a small-area collection optics, dΩ decreases proportionally
to R−2 , where R is the distance from the source to the point where we collect the
radiation. It is more complicated to extended sources. The solid angle in principle
can also depend on ν. The collecting optics can also cut out a spectral interval dν
(correspondingly dλ) out of the entire spectrum of the calibrated source and both
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TARAS PLAKHOTNIK
dν and dλ may depend on the the frequency (wavelength) but in some cases one
of them is constant over a relevant spectral range. For example, dλ is the same
across the entire working range of a diffraction grating spectrometer with a fixed
slit size.
Now we can write the following expression for the irradiance on the photo sensitive area of the photo detector as follows I(ν) = LΩ (Ω)Lν (ν)dΩdν or alternatively
I(λ) = LΩ (Ω)Lλ (λ)dΩdλ. The output of the photodetector is then
(1)
U = S(λ) × LΩ (Ω) × Lλ (λ) × σ × dΩ × dλ
and similar for the frequency. Some of the factors in the above expression are
difficult to measure and control accurately but if one needs only relative calibration
then the expression simplifies significantly
S(λ)
Lλ (λ)
U (λ)
=
×
U (λr )
S(λr ) Lλ (λr )
where it is assumed that dΩ, dλ and σ do not depend on the wavelength and λr is
a reference wavelength (arbitrary selected from the spectral range). Because the
ratio Lλ (λ)/Lλ (λr ) is easy to obtain using the data recorded at the factory for the
source while U and Ur are directly recorded by the photodetector, one can get the
ratio R = S(λr )/S(λ), which is called a relative sensitivity.
The recorded spectrum U (λ) should be multiplied by R(λ) to reproduce the
spectrum of a source correctly. This spectrum should be multiplied by a wavelength
independent scaling factor for conversion, for example, to the units of spectral flax
(W·nm−1 ) or units of spectral intensity. Absolute calibration at a single wavelength
is required for conversion from the relative units to absolute units.
(2)
(3)
Lλ (λ) = U (λ)R(λ)
Lλ (λr )
U (λr )