Optical resonators pumped by a monochromatic light source can be calibrated straightforward by analyzing the cavity ring down decay of a light pulse. The decay time is a direct measure of the resonator quality and the corresponding mirror reflectivity
R. This procedure is commonly used in cavity ring-down spectroscopy (CRDS) with monochromatic light sources, and usually as a calibration method for cavity enhanced absorption spectroscopy (CEAS) as well.
For an incoherent broadband light source (e.g., LED, Xe-Arc lamp), the properties of the resonator become wavelength dependent. Since the mirror reflectivity varies with wavelength, the wavelength integrated decay measured at the exit mirror of the cavity is the summation of decays at all wavelengths emitted by the light source [
]. Since the effect varies continuously with wavelength, this sum can be written as integral over the wavelength:
) is the wavelength dependent mirror reflectivity,
the transmitted intensity at
) is the normalized emission spectrum of the light source,
) is the extinction coefficient, and
is a baseline. Generally,
) exhibits a complex multiexponential decay and its deviation from an single exponential decay increases with cavity finesse. Usually, the bandwidth of dielectric mirrors decreases with increasing reflectivity. This effect is demonstrated in Fig.
. The resonator of the TRAPS apparatus was equipped with two different pairs of mirrors with peak reflectivity at 405 nm (
=0.99980) and 415 nm (
=0.99995). These mirrors allow for effective path lengths of 4050 m and 16200 m at maximum reflectivity, respectively. From Fig.
, it can be seen that the cavity with high finesse exhibits a strong multi-exponential decay, whereas the decay of the cavity with low finesse can be modeled reasonably by a single exponential function with
=4.4 μs. For an empty resonator or a resonator filled with a species that exhibits an extinction that varies slowly with wavelength, it was shown previously that Eq. (
) can be modeled by a double exponential decay with minor overall error [
]. However, in practice, this approach is cumbersome and adds substantial uncertainty to the method.
Decay of two cavities of different finesse pumped by the same incoherent broadband light source.
Green: R=0.99980 mirror. Blue: R=0.99995 mirror
Here, an alternative approach for the calibration of a CEAS resonator pumped by an incoherent broadband light source is presented. As in monochromatic CEAS, a well-defined concentration of reference scatterers is used. The well-known Rayleigh extinction of N
is suitable in this case since it is of the right order of magnitude for CEAS (molecular extinction cross section
]. Moreover, the extinction can be conveniently tuned by varying the N
partial pressure inside the cavity over a pressure range of five orders of magnitude. The calibration is based on the CEAS-equation [
represents the extinction coefficient of the gas inside the resonator,
is the transmitted intensity for the evacuated resonator (
is the transmitted intensity at a given partial pressure of nitrogen
. All parameters have to be interpreted as being integrated over the wavelength region covered by the light source. The extinction coefficient
can be rewritten using ideal gas law
and thus the intensity relation
The mirror reflectivity
is usually strongly dependent on wavelength, peaking at
is the nominal mirror reflectivity usually given by the manufacturer. The shape of the reflectivity function can be approximated by a parabolic curve, its bandwidth is commonly defined as the region where 1−
is within a factor of two of the center wavelength peak reflectivity. If (1−
) is the wavelength at peak reflectivity
is half of the bandwidth, the reflectivity curve can be written as
The wavelength dependent Rayleigh extinction
) of N
is taken from the parameterization of Sneep and Ubachs [
]. In the wavelength region relevant for this manuscript (400 nm–430 nm), it can be approximated using the coefficients
Substituting Eq. (
) and Eq. (
) into Eq. (
) leads to the full wavelength dependent description of the CEAS transmitted light intensity
) must be integrated over the emission spectrum of the light source (
). For the mirror/light-source combination used herein, the bandwidth of the mirror coating can be used as integration interval since no relevant amount of light is emitted by the LED outside this interval (see Fig.
), and thus (
). The normalization has to satisfy the constraint
=1. Thus, the measurement is fully described by
) was used to describe the measured CEAS intensities obtained during a slow pressure reduction experiment of the TRAPS apparatus. A typical dataset is shown in Fig.
. The only free parameter in this fit was the peak reflectivity of the mirrors which could therefore be determined with high precision to be
Typical dataset of a mirror calibration experiment. Nitrogen pressure is varied inside the CEAS cavity and the transmitted intensity is measured. The data is fitted by Eq. (
). The fit results in 7 R 0=0.999842. The nominal peak reflectivity given by the manufacturer is R 0=0.99995
Using the method described above,
R 0 of a set of mirrors can be determined. With known R 0 the extinction coefficient α of the particles cloud inside the cavity can be calculated using the same assumptions as made for the derivation of Eq. ( ). Thus, this calibration method strongly depends on the reliability of the mirror parameters 7 λ 0 and Δ provided by the manufacturer. Since these parameters are subject to change upon mirror degradation, a direct calibration method avoiding a fit to a mirror-reflectivity model is preferable, if the value of R 0 has not to be known for other purposes. Using the above mentioned relation α= σ
p ( k
b T) −1, the pressure in Fig. can be expressed as an extinction coefficient 7 α( S)= α( I 0/ I).