9.8.1 Retroreflection Quality
The fundamentals of measuring the geometric retroreflection quality are discussed pre-viously (Section 7.3), as well as in the experimental setup (Section 9.6). The following discussion aims to put a number on the frequency uncertainties originating from imper-fect retroreflection. The retroreflection quality is dependent on the angle between the two counterpropagating beams, the amplitude of each beam and the transverse mode structure.
The angle between the counterpropagating beams depends on the precision of the pointing alignment through the 0.8µm aperture (Figure 9.13). The alignment is carried out based on the light intensity passing through the aperture. The transverse position of the aperture is optimized for the forward propagating beam. Afterwards, the beam passes a distance over 2 m to reach the vacuum chamber and is reflected off the surface of multiple mirrors before propagating back through the same path and the same aperture. The combined sensitivity of the aperture and mirror alignment makes it possible to estimate that the displacement at the aperture between the two beams is within 0.1 mm. The 0.1 mm offset corresponds to an angle ofα = 40µrad between the two beams, taking the
9.8. Systematic Effects lens (F = 2 m) along the propagation path into account. Thus, the resulting maximum frequency shift of the measured transition is around 23 kHz, assuming the offset is along the propagation direction of the molecular beam. Since the laser system on a floating optics table can shift slightly over the course of a few hours, frequent optimization of the pointing is required. Realigning the laser between frequency scans randomizes the sign and the magnitude of the frequency shift. Additionally, the pointing error is not necessarily along the axis of the molecular beam, which represents the worst case. No frequency shift occurs if the pointing error is perpendicular to the propagation direction of the molecular beam. Considering these arguments, the estimated frequency uncertainty of an averaged set of measurements is less than 10 kHz.
The amplitude difference between the counterpropagating wave and the initial wave at the interaction point with the molecules also causes a frequency shift. After the in-teraction with the OH, the forwards propagating beam passes a fused silica window, is reflected and passes the same window a second time. The measured transmission of the Brewster windows is 98.5 % and the reflectivity of the UV-enhanced aluminum coated mirror is 93 % at 308 nm. Thus, the intensity of the reflected beam is approximately 90 % of the intensity of its forward propagating counterpart. The frequency shift caused by the different intensities is reduced by aligning both beams perpendicular to the molecular beam. This alignment of the retroreflection mirror is carried out with an iterative pro-cedure. First, the laser frequency is scanned through a strong absorption line of the OH with and without retroreflection. Afterwards, all mirrors are optimized, so that the center position of the transition line without retroreflection is the same as the center position with retroreflection. When both measurements show no difference in the line shape, the molecular velocity distribution along the laser propagation direction is interpreted as being maximally symmetric. Therefore, our estimate of the additional error by the amplitude mismatch is less then 5 kHz, so we assign an overall uncorrelated error of 10 kHz for each transition.
The laser beam profile also causes a frequency shifts if it deviates from the Gaussian TEM00 mode. The retroreflection quality depends on the intensity measurements behind the aperture. Higher Hermite-Gaussian modes TEM0n shift the center of maximum in-tensity, corrupting the alignment and decreasing the retroreflection quality. A knife edge measurement predicts around 80 % of the beam intensity inside the TEM0n modes of our UV spectroscopy laser. The remaining 20 % intensity is located in the TEM1n modes.
Additionally, around 50 % of the counterpropagating beam passes the 0.8 mm aperture a second time. Combining both observations leads to a potential offset of 0.27 mm between the center of the maximum and the true beam center at the position of the aperture. The 0.27 mm offset corresponds to a 60 kHz frequency shift of the measured transitions. This
Chapter 9. Experiment
frequency offset is the same for all measurements, with the assumption of a constant mode structure and the same alignment procedure of the optical beam path.
9.8.2 Zeeman Shift
The splitting of each hyperfine state inside a magnetic field has been discussed previously (Section 2.2.3.1). All measurements of electronic transitions in OH or OD are carried out in the ambient magnetic field of the laboratory. Thus, estimating the field strength is essential in predicting the corresponding frequency shift. For this purpose, the vacuum chamber was vented, and two Hall-effect probes (Lake Shore Cryotronics HMMT-6J04-VR and HMNA-1904-VR) were placed through an open CF40 flange into the region where the spectroscopy laser and molecules normally interact. To justify this measurement technique, we measured the magnetic field with the turbomolecular pumps on and off.
Since we measured no change of the magnetic field strength, we conclude a negligible contribution from the pumps and the associated electronics. The strongest magnetic field strength in the spectroscopy region is along the vertical axis with 75µT. A much weaker field is along the propagation axis of the spectroscopy laser of 14µT. A negligible field of 2µT is along the third axis, perpendicular to the previous two axes. Thus the magnetic field is, in good approximation, vertically oriented. Considering the horizontal polarization of the spectroscopy laser, the transitions expected are those with ∆MF =±1. For weak magnetic shifts the ∆MF =−1 and the ∆MF = +1 transitions are equally shifted, but with opposite signs, resulting in zero net shift. However, with an increasing magnetic field, the corresponding states mix with other hyperfine components. The resulting transitions are shifted by a different amount and have different transition strengths. This effect is rather subtle and is not immediately visible in the blended lines. The previous calculation of a transition cluster in OD highlights the splitting of the hyperfine lines inside an ambient magnetic field of 75µT (Figure 2.6a-2.6b). Thus, assigning an uncertainty caused solely by the magnetic field has proven to be complicated. Nonetheless, a later theoretical analysis provides an estimate of the contribution of the Zeeman effect on the uncertainty of individual line positions (Section 10.2).
9.8.3 AC Stark Shift
The time-varying electric field of the spectroscopy laser causes an additional shift since transitions between multiple hyperfine components can interact with a single laser fre-quency. The individual lines experience a laser power dependent shift, called the alternating current (AC) Stark shift (Section 8.2). The individual transitions also experience satura-tion with increasing laser power. Both effects depend on laser power and are difficult to analyze separately. As with the Zeeman effect, a later theoretical analysis estimate the contribution of these effects to the uncertainty of each line position (Section 10.2).
9.8. Systematic Effects
Chapter 10
Analysis
A full quantum mechanical (QM) fit is used to determine the zero-field line positions for measured electronic transitions in the hydroxyl radical (OH) and the deuterated hydroxyl radical (OD) (Section 9). The analysis is based on an effective Hamiltonian model, computed with the program PGOPHER[56]. Statistical uncertainties are assigned to each transition based on the uncertainties determined in the individual fits and the spread of fitted transition frequencies. Since the uncertainty is at the order of magnitude of the Zeeman effect and the alternating current (AC) Stark shift, it is important to account for these effects when determining the overall line position. To quantify the contributions of these shifts to the extracted line positions a simplified second model was used to determine the line positions ignoring Zeeman and/or AC Stark shifts (Section 10.2).