Inferring actual M-values during polarisation build-up in presence of the pump laser

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Longitudinal probe absorption measurements

In order to illustrate the conditions for the use of a longitudinal probe laser at fixed frequency in the present work, computed absorption spectra of 3He for different light polarisations in B = 0 − 30 mT are presented in figure 4.2.
Energies and intensities of the optical transitions in 3He at B = 0 are represented as well as computed absorption spectra at B = 30 mT for σ−, π and σ+ light polarisations. The shifts of the σ− and σ+ spectra at B = 30 mT compared to the transition frequencies at B = 0 are clearly visible. The Zeeman shifts up to 30 mT remain below Doppler width at room temperature, so that the low field approach mentioned in the preliminary remarks of this chapter, namely to fix the probe laser frequency to the transition frequency in B = 0 in order to monitor both light polarisations simultaneously is appropriate. Furthermore, the computed spectra show that only the two resolved lines C8 and C9 are appropriate to be used as probe transitions.
Figure 4.3 shows experimental absorption spectra for σ+ and σ− light polarisations at B = 30 mT for M = 0 and M = 0.5 (the corresponding computed spectrum at M = 0 is represented in figure 2.4). The difference between the σ+ and σ− peaks due to Zeeman splitting of energy levels amounts to approximately 1.05 GHz or 1.45 GHz at B = 30 mT in the C8 or C9 resonances respectively which is inferior to the Doppler width of 3He gas at room temperature (cf. caption of figure 4.2). Hence, in most of the cases throughout this work (low and intermediate field 0-30 mT), the probe laser frequency has been locked to the transition frequency at B = 0 so that the evolution of σ+ and σ− peak intensities can be monitored as function of time which is equivalent to polarisation during build-up. Independently of the magnetic field, it is also possible just as well to sweep the probe laser frequency as demonstrated in figure 4.3 (more experimental examples of probe laser frequency sweeps, see section 6.2).
The probe beam components (with circular polarisations σ+and σ−) used in our experiments are obtained from a low power monochromatic laser beam that is substantially expanded and subsequently diaphragmed (see section 3.2). Therefore we  egitimately consider here a probe beam with uniform light intensity I that is weak enough for the absorbed intensity to be proportional to the local intensity at any point along the beam path (linear regime). The variation of the probe beam intensity at this point can thus be written as: dI dl = −ka(l) I.

Polarisation measurements in the absence of OP light (during polarisation decay)

In section 2.8.2, it is discussed in detail, that during polarisation decay in absence of OP, the departure of the population distribution in 23S from a ST distribution induced by 23S relaxation is very small. Hence the populations a no OP i can be safely replaced by the aST i ones during polarisation decay.
The measured absorption signals, Aσ+ and Aσ− in the longitudinal probe scheme, are the amplitudes of the modulation depth (i.e., the ratios of ac amplitudes to dc components) of the corresponding probe powers exiting the cell [Tal11]. Each ratio is robust against fluctuations of the incident power I0 and directly proportional to the absorbance −ln Ts, with a coefficient that only depends on the plasma response to the rf excitation [Cou02, Cou01]. Neglecting all constant prefactors, A is given by: A(M) ∝ nS m(M) X i,j ~ωij Tij(B) e−(δij L /D)2 aST i (M).

Influence of residual π-light on longitudinal probe absorption measurements

The considerations in this subsection are generally relevant for probe absorption signals, independently of the chosen transition C8 or C9: The influence of a small fraction of π-polarisation within the probe light, originating on the one hand from the nonparallel probe beam incidence compared to the magnetic holding field axis (inclined probe configuration, see chapter 3.2), and emerging on the other hand from field components of the terrestrial magnetic field non-parallel to the holding field. This influence of the earth field results in a slight change of the overall magnetic field axis compared to the pump and probe laser beams. Here, only the impact on the determination of M by the probe beam is presented.
A ’contamination’ of the measured σ+- and σ−-probe absorption signals by a small fraction of π-light signifies that not only one 23S sublevel (or one set of sublevels for probe C9) is monitored, but also a second one (or a second, different set of sublevels for probe C9).
In order to quantify the error arising from this effect, computed absorption signals  for the three light polarisations σ+, σ− and π by the model for MEOP-kinetics at given probe frequency and magnetic field as function of polarisation are used to create ’contaminated’ σ+- and σ−-probe absorption rates by different fractions of π- light, keeping the overall beam intensity constant. Using these ’contaminated’ probe absorption rates to infer M8 and M9 in the habitual way (see sections 4.2.1 and 4.2.2) yields the following relative errors in the determined nuclear polarisation presented in figure 4.17 for probe C8 and C9 in resonance (B = 0) as function of M, at B = 1 mT and 30 mT.

Polarisation measurements with OP light (during polarisation build-up)

As discussed in section 2.8.3, the distribution of metastable atoms between 23S sublevels is deeply modified when a pump laser drives 23S-23P transitions. Moreover, a significant fraction of He atoms are promoted to the 23P level. These two effects simultaneously occur and both contribute to change the probe light absorption signals:
1/ The populations ai are no longer distributed according to the spin temperature distribution associated to the current nuclear polarisation M.
2/ The absorption rate for the Ai → Bj component varies, since it is proportional to (ai − bj) and bj 6= 0. We introduce the apparent polarisations Ma 8 and Ma 9 as defined by the values inferred from the reduced ratios R8 and R9 using the spin-temperature formulas (equations (4.13) for probe C8 and (4.15) for probe C9). Due to the two above-mentioned effects, these apparent polarisations can significantly differ from the actual nuclear polarisation M of the ground state. They depend on various experimental conditions (pump laser tuning, polarisation, power, transverse profile; probe line; gas pressure).

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Table of contents :

List of Figures
List of Tables
List of Symbols, Constants, He OP lines, Abbreviations
1 Introduction 
2 MEOP basics and OP model 
2.1 Short introduction to MEOP
2.2 He level structure at low magnetic field
2.3 Optical transition rates
2.3.1 Monochromatic excitation
2.3.2 Broadband excitation
2.4 Rate equations for OP and ME in pure 3He gas
2.4.1 Generic OP rate equations
2.4.2 Generic ME rate equations
2.5 The improved 2-class OP model
2.5.1 Relaxation processes
2.5.2 Velocity-dependent light excitation
2.5.3 Inhomogeneous light excitation and atomic response: 1-D model
2.6 Comparison of all rates relevant for MEOP in 3He gas
2.7 Numerical computation of MEOP dynamics
2.7.1 Equation solving strategy
2.7.2 Full rate equations
2.7.3 Numerical implementation
2.7.4 Input data and input parameters
2.8 MEOP dynamics
2.8.1 ME-driven spin temperature distribution
2.8.2 Dynamics of polarisation decay (no OP)
2.8.3 OP-driven MEOP dynamics
2.8.4 Angular momentum budget
2.9 Photon efficiencies for low field MEOP
2.10 Final discussions
2.10.1 Comparison with previous OP models
2.10.2 Robustness of our computed OP results
2.10.3 Conclusion
3 Experimental setup 
3.1 Magnetic field, cells and rf discharge
3.2 Optical setup
3.3 Measurement and acquisition system
4 Optical measurement of nuclear polarisation 
4.1 Longitudinal probe absorption measurements
4.2 Polarisation measurements in the absence of OP light (during polarisation decay)
4.2.1 Determination of M by C8 probe
4.2.2 Determination of M by C9 probe
4.2.3 Influence of residual π-light on longitudinal probe absorption measurements
4.3 Polarisation measurements with OP light (during polarisation build-up)
4.4 Measurements of metastable density nS m
4.4.1 Determination of nS m for single-component and multi-component transitions: Examples of C8 and C9 probe
4.4.2 Influence of collisional broadening and lifetime
5 Methods of data processing and data reduction 
5.1 Introduction to data reduction
5.2 Polarisation build-up and decay
5.2.1 Transmitted probe signals
5.2.2 Demodulated probe signals
5.3 Dedicated experiments to account for perturbations of 23S- and 23Ppopulations in presence of the pump laser
5.4 Inferring actual M-values during polarisation build-up in presence of the pump laser
5.5 Analysis of polarisation build-up kinetics
5.6 Pump output signals
5.6.1 Transmitted pump signals
5.6.2 Demodulated pump signals
5.7 Laser-enhanced relaxation
5.7.1 Deriving polarisation loss rates ΓR using the MEOP model
5.7.2 Deriving polarisation loss rates ΓR from a detailed balance of angular momentum
6 Results 
6.1 Characterisation of the plasma without OP light
6.1.1 Key plasma parameters for MEOP: nm(M = 0) and ΓD
6.1.2 Transverse distribution of 23S atoms
6.1.3 Variation of metastable density with nuclear polarisation
6.2 Results of dedicated experiments to account for perturbations of 23Sand 23P-populations
6.2.1 Influence of probe detuning on apparent polarisation at M = 0 . 197
6.2.2 Computed Ma at M = 0 with probe laser in resonance as function of pump laser power
6.2.3 Example of apparent polarisation as function of actual polarisation
6.2.4 Reproducibility of apparent polarisation in fixed OP conditions 208
6.2.5 Examples of Ma measured by probe on the C9-transition
6.2.6 Perturbations of 23S and 23P populations in higher magnetic field: B = 30 mT versus 1 mT
6.2.7 Scaling of Ma with incident pump laser power and pressure
6.2.8 Effect of pump beam diameter and probe parameter xs on apparent polarisation
6.2.9 Conclusion
6.3 Optical Pumping results at 1 mT
6.3.1 Results at B = 1 mT and M = 0: Relaxation-free data to test and validate the model for MEOP kinetics
6.3.2 Empirical determination of the intrinsic relaxation rate in the 23P state
6.3.3 Results at Meq (B = 1 mT) and evidence of laser-enhanced relaxation
6.3.4 Results as function of M (B = 1 mT) and further characterisation of laser-induced relaxation
6.4 Effects of magnetic field on OP performances
6.5 Discussion of laser-enhanced relaxation effects
6.5.1 Comparison of ΓL rates of different works
6.5.2 Radiation trapping
6.5.3 OP-induced plasma ’poisoning’ (e.g., by metastable He molecules)292
7 Conclusion and Outlook 
A Tables and matrices for low magnetic field (B < 0.162 T): Transition frequencies and intensities for 3He and 4He, Zeeman shifts, hyperfine mixing parameters, vector and matrix operators 
B MEOP rate equations and angular momentum budget in the improved OP model 
B.1 The improved OP model
B.2 Two-class partition and description of local OP rates
B.3 Local rate equations for 23S and 23P populations
B.4 Rate equation for M, MEOP dynamics, and global angular momentum budget
C Computation of the average pump light intensity inside the cell for the improved OP model 
D Computation of photon efficiencies of C8 and C9 lines at null polarisation and in zero magnetic field, for the limits of no and full collisional mixing in the 23P state (Kastler and Dehmelt OP regimes) 
E Numerical demodulation of signals 
E.1 Requirements and characteristic functionalities
E.2 Rician noise in the context of lock-in detection
F Validation of methodological approach in analysis of polarisation build-up kinetics using synthetic data 
G Influence of stimulated emission on scaling of Ma with incident pump laser power and pressure 
Bibliography 

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