Analytical expression for the nth Fourier coefficient of a modulated Lorentzian dispersion lineshape function

Abstract

Modulated spectroscopic detection techniques that rely on dispersion, e.g.Faraday modulation/rotation spectroscopy and wavelength-modulated noiseimmune cavity-enhanced optical heterodyne molecular spectroscopy, are often described in terms of Fourier coefficients of the modulated dispersion lineshape function. This work derives a non-complex analytical expression for the n th Fourier coefficient of a modulated Lorentzian dispersion lineshape function, which is easier to implement, and orders of magnitude faster to execute, than previous approaches involving numerical calculations of integrals.The first six Fourier coefficients are illustrated for their optimum modulation amplitudes. An analytical expression for the first Fourier coefficient is explicitly given.

Laser-Induced Dispersion With Stimulated Raman Scattering in Gas-Filled Optical Fiber

“…3(a) shows the measured 2f signal for pure hydrogen with the gas pressure of 3.5 bar inside the hollow-core with a pump power of ~100 mW delivered to the HC-PCF, corresponding to a peak pump intensity of 0.314 MW/cm 2 at the center of the hollow-core. For comparison, the 2f signal is numerically calculated based on the formulations in [26] and shown in Fig. 3 (b), agreeing well with the measured result.…”

“…The second harmonic (2f) of the MZI output, which is linearly proportional to the 2f component of the probe light phase modulation, is lock-in detected and used as the system output. The depth of pump wavelength modulation is set to 2.84 to maximize the 2f output signal [26] (see appendix.). Balanced detection using the two outputs from the MZI ensures that the small induced Raman gain would not affect the dispersion measurement [27].…”

Atmospheric Measurements by Ultra-Light SpEctrometer (AMULSE) Dedicated to Vertical Profile in Situ Measurements of Carbon Dioxide (CO2) Under Weather Balloons: Instrumental Development and Field Application

“…At the absorption line center, the WMS-1 f term is dominated by the laser intensity contribution [16]. Generally, the second harmonic (2 f ) is used because it is strongly dependent on spectral parameters and gas properties and can therefore be compared with spectral simulations to infer gas properties.…”

“…Assuming linear intensity modulation with a phase shift of π, the 1 f -normalized WMS-2 f signal simplifies to the following equation [11,12]: $\frac{2f}{1f}=\frac{S(T)\times P\times x_i\times L}{i_0\times \pi }{\displaystyle \underset{-\pi }{\overset{+\pi }{\int }}\mathrm{sans-serif\Phi }\left({\nu }_{peak}+a\cos \left(\theta \right)\right)}\times \cos 2\theta d\theta$ where S ( T ) is the line strength at temperature T of the absorption transition, P is the partial pressure of the target species (CO 2 in this case), x i is the target species concentration (to be fitted), L is the path length of laser beam travel through the uniform absorbing medium, i 0 is the amplitude of the linear term of the laser intensity modulation (to be determined beforehand experimentally in the laboratory) [16], ν peak is the peak central frequency position, a is the amplitude of the frequency (wavelength) modulation, θ corresponds to ωt or 2π ft and Φ is the line shape function at the absorbing frequency (in our case we used a Lorentzian profile since we are working at pressures greater than 200 mbar).…”

High-precision and high-accuracy rovibrational spectroscopy of molecular ions

“…We fit the data to both the third Fourier coefficient for a Lorentzian dispersion lineshape 27 and a simple second derivative of a Lorentzian. While neither fit perfectly reproduces the observed lineshape, the derived linecenters compare very favorably with the previous work of Takahata et al 24 The linecenter as determined by the second derivative fit is 5 ± 22 kHz less than Takahata's, and that from the third Fourier coefficient fit was 10 ± 30 kHz greater than Takahata's.…”