Enhancement of Doppler spectroscopy to transverse direction by using optical vortex

Tunable diode laser absorption spectroscopy (TDLAS) is a valuable method for measuring particle flow velocities in plasma. However, conventional TDLAS using a plane-wave beam is sensitive only to the laser propagation direction. This limitation is particularly unfavorable for the observation of the particle transportation perpendicularly incident on the material in the plasma–material interaction. In this paper, we show for the first time that flow measurements perpendicular to the beam direction are possible by replacing the probe beam with an optical vortex beam. Because an optical vortex has a helical wavefront, particles moving in its field experience an azimuthal Doppler shift in addition to the translational Doppler shift. Assuming a uniform gas flow across the optical vortex, the azimuthal Doppler shift of the absorption spectrum observed in the beam cross-section varies sinusoidally in the azimuthal direction. The transverse flow velocity is derived from the amplitude of this sinusoidal variation. At transverse velocities above 70 m/s, the measurement errors are found to be less than 15%, with a mean absolute percentage error of less than 8%.

beam, and the resonance absorption frequency was obtained from the Lamb dip in the saturated absorption spectrum [S1].Figures S2(a) and 2(b) show the saturated absorption spectrum and the absorption spectrum observed with a 300-ccm gas flow, respectively.The absorption spectrum is shifted by 155 MHz from the resonance frequency, and the gas flow velocity is determined to be 117 m/s considering that the incidence angle of the probe beam to the gas flow was 0.39 rad.The detuning frequency for the absorption spectrum from the resonance frequency was calibrated by FPI as described in Methods.This experiment was conducted under a constant pumping velocity, which means that in addition to changes in the flow velocity, the pressure varies with gas flow rate.
Furthermore, the product of pressure and flow velocity is proportional to the gas flow rate.The results of gas velocity measurements at various gas flow rates, ranging from 50 to 500 ccm, are presented in Fig. S3.The relationship between the gas flow rate and the flow velocity is nonlinear because the gas pressure changes simultaneously.

Translational Doppler shift component in OVLAS
In our experimental system, the OVLAS probe beam is injected almost perpendicular to the gas flow; however, this alignment is not perfect.Therefore, the observed Doppler shift also includes a small translational Doppler shift component, which is excluded in Fig. 3 to show the azimuthal Doppler shift distribution.The translational Doppler shift is obtained from the offset of the -directional variation of the Doppler shift distribution.Figure S4 shows the dependence of the translational Doppler shift on the gas flow velocity.The translational Doppler shift varied linearly with gas flow velocity, as expected, and from this relation, the deviation of the probe beam from the vertical direction was estimated to be 0.027 radians (1.55°).While evaluation of the translational Doppler shift depends on the calibrated laser wavelength, measurement of the transverse Doppler shift does not require such a calibration.Thus, in the OVLAS experiment, the laser wavelength was calibrated by saturation absorption spectroscopy only as a supplementary measure, resulting in a relatively large MAPE of approximately 20% for the translational Doppler shift.

OVLAS measurement errors
In this study, we evaluated the OVLAS measurement errors in relation to the calibrated gas flow velocity using the measurement system shown in Fig. S1; the results are given in Table 1.The transverse flow velocities, denoted by  + and  − for optical vortex beams with ℓ = +10 and ℓ = −10, respectively, were measured for gas velocities (  ) ranging from 48 to 147 m/s, respectively.
Here, Δ represents the difference between  + and  − from   .Excluding the case for   = 48 m/s, the value of Δ was distributed below 15%.The MAPE for  + and  − was 5% and 8%, respectively.While the evaluation of the translational Doppler shift in Fig. S4 depends on the calibrated laser wavelength, the transverse flow velocity is determined only by the relative change in the Doppler shift in the azimuthal direction, leading to reduced errors.

Figure S1 .
Figure S1.Gas flow velocity measurement using TDLAS and SAS.

Figure S3 .
Figure S3.Dependence of gas flow velocity on gas flow rate.

Figure S4 .
Figure S4.Dependence of translational Doppler shift on gas flow velocity.