Absorption spectroscopy and Stokes polarimetry in a $^{87}$Rb vapour in the Voigt geometry with a 1.5 T external magnetic field

We present a detailed spectroscopic investigation of a thermal $^{87}$Rb atomic vapour in a magnetic field of 1.5~T in the Voigt geometry. We fit experimental spectra for all Stokes parameters with our theoretical model \textit{ElecSus} and find very good quantitative agreement, with RMS errors of $\sim 1.5$\% in all cases. We extract the magnetic field strength and the angle between the polarisation of the light and the magnetic field from the atomic signal, and we measure the birefringence effects of the cell windows on the optical rotation signals. This allows us to carry out precise measurements at a high field strength and arbitrary geometries, allowing further development of possible areas of application for atomic magnetometers.


I. INTRODUCTION
Over the past decades atomic spectroscopy has cemented itself as one of the cornerstones of precision measurements. From helping define the standards for time-keeping [1][2][3] to measurement of electromagnetic fields [4][5][6][7][8][9], spectroscopic techniques have become key tools in high precision systems. In recent years, together with the push for new technologies in other areas, this has seen the field of atom-based sensors burgeon. Of particular interest are magnetic field sensors [10] which rely on the interaction of the nuclear and electronic spins with the external field and well-known spectroscopic signals. Applications of such atomic sensors span a variety of disciplines, ranging from explosives detection [11]; gyroscopes [12]; medical imaging of soft tissues [13][14][15][16]; microfluidics [17] and measurements on spin-active solidstate systems [18]. In order to improve and sustain the development of these technologies, atomic spectroscopy in the presence of external magnetic fields has become an area of wide interest [19][20][21][22][23]. This in turn has given rise to a variety of experiments and optical devices [24][25][26][27][28][29][30][31] that help demonstrate the depth of understanding of the physics involved in said interactions, furthering the reach and possible applications of this area. Nevertheless, this in-depth understanding of the interactions between atoms and an externally applied magnetic field has primarily occurred for fields up to ∼ 1 T.
The use of spectroscopic techniques in atomic systems has received less attention at the higher end (> 1 T) of the range of field strengths for different reasons. There is the matter of most methods used to obtain large fields being destructive in their nature, thus complicating the experimental reproducibility. In such experiments, the Zeeman splitting of energy levels in alkalimetal atoms is used to observe field strengths of the order of tens/hundreds of Tesla via spectroscopic techniques [32][33][34]. Non-destructive techniques for producing * francisco.s.ponciano-ojeda@durham.ac.uk these large fields also exist, and this has enabled work in large pulsed magnetic fields up to 58 T with similar alkaliatom systems [35,36]. Aside from the large Zeeman splitting produced at such large magnetic fields, there are other changes in the atoms that in turn allow additional effects to be observed. Of these changes the most relevant is the decoupling of the nuclear and electronic spins in the atom, with the external magnetic field now being a common axis for the precession of both. This is known as the hyperfine Paschen-Back regime [37][38][39][40][41][42][43] and, in contrast to the experiments at lower fields (< 10 mT) -where it is easily and directly measured [10] -the Larmor precession frequency is not of interest as it can be too high for conventional electronic systems. Rather, the characteristic absorption spectrum of the atomic system is a more straightforward way of obtaining information about the magnetic field.
Work has previously been done in atomic vapours of both low and high atomic densities [44][45][46], and other relevant magneto-optic effects have been the subject of extensive studies [47][48][49][50]. This atom-light interaction whilst in the presence of an external magnetic field has also been used in other systems such as [21,[51][52][53]. The majority of said work has been done in the Faraday configuration, where the k vector of the laser beam used is parallel to the direction of the external magnetic field B. This gives rise to characteristic absorption spectra that are highly symmetric [41,54] and reflect only the magnitude of the magnetic field [42]. However, there has been some work [55,56] that makes use of the less studied Voigt configuration where, rather than being parallel, the k vector of the light and the direction of the magnetic field B are perpendicular. The change in geometry results in a change in the atomic transitions permitted by the selection rules [57,58] and thus allows for information not only on the magnitude but also on the direction of the magnetic field to be experimentally observed.
Here we present detailed analysis of spectroscopic measurements of an atomic vapour at a magnetic field strength of 1.5 T in the Voigt geometry. These measurements include optical rotation signals, expressed in terms of the Stokes parameters. By fitting to our theoretical model ElecSus that describes the atomic susceptibility [59,60] we extract the magnetic field strength and compare it against expected values. We also show that we can extract, given a known input light polarisation, the relative orientation of the magnetic field. Finally, we demonstrate a method to measure the birefringence of the cell windows by looking at small changes in the expected optical rotation signals produced by the atomic medium in the presence a strong external magnetic field.
The rest of the manuscript is organised as follows. In section II we briefly summarise the theoretical model we have used in our analysis; section III discusses our experimental approach; in section IV we discuss the spectra of a vapour of 87 Rb in a large magnetic field and compare the experimental data to our model to extract a magnetic field strength; finally, in section V we use the optical rotation signals to extract information regarding the birefringence introduced by the cell windows.

II. THEORETICAL MODEL
The theoretical model we use for fitting our data, Elec-Sus, is described in detail in refs. [59,60]. Here we summarise only the main points relevant for this experiment. We begin by assuming an atom-light system operating in the weak-probe regime [61,62], although recent work [46] provides methods to generalise this into a strong-probe regime. The model used here is based on the complex electric susceptibility, χ(∆), of the atomic medium as a function of the optical frequency detuninḡ h∆ = (hω laser −hω 0 ) in the vicinity of the alkali D-lines, where ω laser is the angular frequency of the laser and ω 0 is the angular frequency of the atomic transition. Here we note that experimentally linear detuning, ∆/2π, is most commonly used. Using a matrix representation we construct the atomic states in the uncoupled m L , m S , m I basis, accounting for internal energy levels that come about as a result of the fine and hyperfine structure and include terms from the interaction with an external magnetic field via the Zeeman effect. From this matrix, the Hamiltonian of the atomic system, we proceed to calculate the transition energies and absolute line strengths. Finally, the Doppler effect is included by convolving a Voigt profile (convolution of the Lorentzian homogeneous linewidth and the Gaussian from the Maxwellian velocity distribution) with each of the atomic transitions as defined by their energy and strength. Thus, for a given global linear detuning, the total susceptibility is the sum over all transitions, where each transition has an effective detuning. This allows for the components of the susceptibility for the σ ± and π transitions to be calculated separately.
For a given total susceptibility the propagation of light through the atomic medium is done by solving the corresponding wave equation and finding the two propagation eigenmodes. Each eigenmode is associated with a Figure 1. General geometry for experiments involving an externally applied magnetic field and light; for the Voigt geometry where k ⊥ B, θB = π/2. The light is assumed to be linearly polarised with an electric field vector E that lies in the x − y plane, in this case taken to be vertically polarised along the x-axis. The external magnetic field makes an angle φB with respect to E, which determines the parallel and perpendicular components of E projected along B, which in turn are associated with the different refractive indices in the atomic medium and the π and σ ± transitions they drive. complex refractive index, which couples to the allowed atomic transitions in a distinct way. The exact coupling depends on the geometry of the system. In the experiment presented the system is set up in the Voigt geometry, in which the externally applied magnetic field vector B is perpendicular to the light wavevector k, as shown in figure 1. For simplicity we constrain the geometry by making the electric field vector of the light E to lie along the Cartesian x-axis, which leads to a laser beam that we assume as a linearly polarised plane-wave propagating along the z-axis and polarised in the x − y plane. In this manner, for the Voigt geometry we take the external magnetic field vector in the x − y plane, with an angle φ B defined as the direction of said vector relative to the x-axis ( E). This also determines the relative coupling of the light to the atomic electron transitions as a consequence of the projections of B onto E, which in turn change the angular-momentum algebra. For a projection of B parallel to the polarisation of the light (i.e. E) π transitions with ∆m J = 0 are driven, while for a projection of B perpendicular to the polarisation of the light the σ ± transitions, with ∆m J = ±1, are driven [63]. The relative phase between the two polarisation components is unimportant in the Voigt geometry -i.e. circularly polarised light of either handedness couples to the atoms in the same way as linearly polarised light with equal x and y components (φ = (2n − 1)π/4 with n an integer). Such considerations, including the oscillatory nature of light, mean that the direction of the applied field cannot be derived from a single measurement, meaning that equivalent solutions are found for changes in φ B of π radians.
As mentioned previously, each eigenmode propagates through the medium with its associated refractive index. In general these two refractive indices are different, and since they are complex, the medium is both dichroic and birefringent. To take these effects into account we calculate the electric field after propagating through the medium by transforming the input electric field into the eigenbasis coordinate system (in the Voigt geometry this is simply the Cartesian basis parallel and perpendicular to the external magnetic field vector), propagating each index n i for a distance L by multiplying by e inikL , and finally transforming the resulting field back to the lab coordinates which are most relevant. We then analyse the output via Stokes polarimetry [50,64], which provides a convenient set of parameters easily amenable to laboratory measurements that only require measuring the intensity in sets of orthogonal polarisation bases. For an incident beam of intensity I 0 the Stokes parameter S 0 , defined as represents the normalised total transmitted intensity, and is therefore independent of the measurement basis. The remaining Stokes parameters, give information on the optical rotation of light generated by the atomic medium by looking at the differences between the orthogonal polarisation components in the horizontal (I x , I y ), diagonal (I , I ) and circular bases (I RCP , I LCP ), respectively.

III. EXPERIMENTAL SETUP
The experimental setup used is shown in figure 2. The laser is a 780 nm distributed feedback (DFB) laser with a quoted linewidth of < 2 MHz. The frequency is tuned by changing the temperature of the laser chip, which allows a frequency mode-hop free tuning range over many hundreds of GHz. A typical scan of this size takes ∼ 2 seconds and is limited by the response time of the temperature control circuit of the laser. A fraction of the emitted light is sent to reference optics that allow us to calibrate the frequency of the laser scan; we use a Fabry-Pérot etalon to linearise the scan in parallel with a 75 mm natural abundance Rb reference cell which provides an absolute frequency reference, using the approach outlined in ref. [67] to obtain spectra as a function of linear detuning. The remainder of the light is then passed through several polarising elements that allow us to obtain a welldefined linear polarisation to use in our experiment cell. A circular polarisation can be obtained by replacing the half-wave retarder plate (λ/2) for a quarter-wave retarder Light from a distributed feedback (DFB) laser is split into two parts after passing through and optical isolator (OI). One part is sent to reference optics, composed of a Fabry-Pérot etalon made with two mirrors (M) and a 75 mm natural abundance Rb reference cell, and is used to calibrate the laser scan in zero magnetic field. The rest of the light is sent to the experiment cell via an acousto-optic modulator (AOM) that can be used to keep the power constant [65] and polarising beamsplitter (PBS) cubes and half-/quarterwave plates (λ/2 & λ/4, respectively) to ensure the input light is in the desired polarisation state. The experiment cell is a 1 mm 87 Rb isotopically enriched (99% 87 Rb) vapour cell, and is placed in a cylindrical magnet designed to give a primarily axial field of 1.62 T at its centre [66]. A polarisation-sensitive detection scheme is used after the experiment cell, consisting of a PBS, retarder wave plate and two balanced photodidodes (PD), that allow a voltage signal with information on the optical rotation and absorption of the medium to be measured. Plano-convex lenses (L) are used to resize the beam at different points in order to avoid significant clipping. plate (λ/4) before the light goes through the experiment cell.
The experiments are carried out in a cuboidal 1 mm long isotopically enriched (99% 87 Rb) vapour cell, which is heated to provide the required atomic density and hence optical depth for observing the desired phenomena. The cell and beam-steering optics (uncoated right-angle prisms) are mounted on a central bed of copper which also houses an internal heater, as seen in figure 3. During operation the temperature is set by applying a voltage to the heater to raise the temperature of the copper surrounding the cell. Temperature stability is passively maintained by fixing the voltage and allowing the cell mount to thermalise with the surroundings. The copper bed is also surrounded by a cylindrical PTFE shield that is only in weak thermal contact with it in order to ensure there are no sharp fluctuations in the temperature of the system. Due to the reduced footprint of the system imposed by the central bore size of the magnet it was not possible to include a suitable temperature sensor in this mount.
The external magnetic field is generated by a cylindrical permanent magnet, with a central bore of diameter Figure 3. Internal copper heater block used in the experiments in the Voigt geometry. Shown here is the main copper block, with an angled slot for the 1 mm cell to avoid backreflection from the cell windows and two spaces for 5 mm right-angle prisms. Light enters and exits the heater block through two optical access holes (seen in the upper-left) and passes through the cell perpendicular to its original direction of incidence. The larger hole (bottom-centre) houses a resistive heater element that serves to heat the entire block in order to raise the temperature of the vapour in the interior of the cell to the desired point. This block is covered by a copper lid and housed in a custom PTFE cylinder that preserves the optical access and cable feed-throughs necessary to carry out experiments when the whole assembly is located inside the permanent magnet.
22 mm along its axis, designed using the "magic sphere" configuration described in [66]. The maximum value of the field produced by the magnet is 1.62 T at its centre, with the field strength quickly falling radially outwards to the ends of the magnet. The cylindrical PTFE assembly housing the copper heater block with the experiment cell sits inside the magnet's bore. This particular design allows for a well-characterised field along the axis of the magnet, which in turn ensures field homogeneity across the length of the experiment cell, while maintaining the field on the outside of the magnet to below hundreds of mT. Further details on the construction and characterisation of the magnet can be found in ref. [66].
To avoid optical pumping, the optical power in the of the interrogating laser beam (i.e. the probe beam) is maintained below the saturation intensity, such that the atoms are in the weak-probe regime [62]; in practice this means around 1 µW of optical power with a beam waist (1/e 2 ) of approximately 0.7 mm. The effective spatial resolution of the field probe is set by the volume of atoms interrogated by the laser beam, which roughly comprises a cylinder of length 1 mm (i.e. the length of the experiment vapour cell) and radius 0.7 mm. It is worth nothing that the length of the cylindrical heater block assembly previously described (28 mm) is smaller than the length of the magnet (152 mm) and this generates difficulties in aligning the heater block axis relative to the laboratory frame of reference once inside the magnet. As such, the heater block is slightly rolled about the axis of the magnet (z) which in turn results in a relative orientation of the x, y axes of the atom and the laboratory x, y axes that can be described by an effective offset in the angle φ B .

IV. ANALYSIS OF ATOMIC ABSORPTION
SPECTRA AT HIGH FIELDS Figure 4 is created using our theoretical model as described in section II and combines the atomic energy level evolution as a function of magnetic field strength with the expected absorption signals at a 1.5 T external magnetic field. The top two panels show the calculated absorption for an isotopically enriched (99% purity 87 Rb) vapour cell at 100 • C, for π (upper panel) and σ ± (lower panel) transitions. Coloured vertical lines indicate the linear frequency at which the atomic resonance lines occur, while the colour of the line indicates the type of transition: π transitions are olive green, σ + transitions are blue and σ − transitions are purple. Below these panels, the energy diagram shows the states in the 5S 1/2 and 5P 3/2 manifolds at 1.5 T and the initial and final states involved in each individual transition. The state decompositions in the m J , m I basis are shown on the right side of the figure. The arrows, with colours again corresponding to the type of transition as previously mentioned, are semi-transparent to evidence the fact there are still some overlapping transitions due to a small remnant admixture in the state decomposition due to hyperfine interactions. This can also be seen in the respective detunings of the transitions on the top two panels, where π and σ ± have frequencies that coincide between them.
In an applied magnetic field of 1.5 T, the 5P 3/2 states strongly decouple into the m J , m I basis, leading to four groups of states organised by the m J = 3/2, 1/2, −1/2, −3/2 projection. Furthermore, at this magnetic field strength the ground state has a much more complete decoupling than that observed at lower field strengths [21,42,[51][52][53]56]. As such, we can treat the system as being completely in the Hyperfine Paschen-Back (HPB) regime. This means that the initial hyperfine ground states are now split into two distinct groups, corresponding to the projections m J = ±1/2, with the m I = 3/2, 1/2, −1/2, −3/2 states clearly defined despite the Doppler-broadening in the vapour. This gives us well-defined multiplets of four 'strong' transitions (|m J , m I → |m J , m I , with m J = m J (π), m J + 1 (σ + ), m J −1 (σ − )). It is worth adding that at this point there are still some 'weak' transitions present, which result from the ground states not being pure eigenstates in the m J , m I basis; in this case a small admixture of the opposite m J state (< 1%) remains in the state decomposition on the bottom right of fig. 4 (more details can be found in ref. [42]). Figures 5 and 6 show experimental data, averaged over five spectra, that have been fitted using ElecSus along with the residuals R, which have been multiplied by a factor of 100 for clarity. The RMS difference between theory and experiment for figure 5(6) is 1.2%, and, combined with the lack of any discernible structure in the residuals, indicates a very good fit [68]. The fit is carried out with three free parameters: φ B , the angle between the magnetic field and the direction of polarisation of the laser beam (see figure 1), taken to be linear along the x(y)-axis; B, the magnitude of the magnetic field the atoms are exposed to and T , the temperature of the atoms in the experiment cell. Other significant experimental parameters, such as those relating to the effects due to buffer gas in the cell, such as the amount of inhomogeneous broadening Γ Buff caused by collisions and a shift in the frequency of the transitions δ shift , are kept fixed. The values for these fixed parameters are obtained a priori by fitting other spectra similar to those averaged and shown in figures 5 and 6. Of the remaining parameters in the fit, the field angle θ B is fixed by the geometry of the experimental setup in the Voigt geometry (θ B = π/2). We attribute the significant buffer gas broadening in the spectra to He atoms trapped in the cell, after the cell was exposed to a He environment in previous experiments and note there is no significant shift in any of the resonance lines. Using the literature values of the broadening coefficient for He [69], we extract a pressure of ∼ 18 torr (∼ 24 mbar) for the amount of said buffer gas in our experiment cell. Note that while the time necessary to acquire a spectrum is on the order of a second, the time needed to analyse the data and generate a fit is on the order of minutes due to the complexity of the parameter space.
For the spectrum shown in figure 5 we obtain a value of φ B = (0.4491 ± 0.0007) radians ((25.74 ± 0.04) • ). Similarly, the fit shown in figure 6 we obtain a value of φ B = (2.0081 ± 0.0007) radians ((115.05 ± 0.04) • ). Both of these values differ from their corresponding expected values by ≈ 0.45 radians (∼ 25 • ), which we take as a systematic error due to the orientation of the cell heater block inside the bore of the cylindrical magnet used in the experiment. As a result, there is excitation of both π and σ ± transitions in both spectra shown due to the presence of parallel and perpendicular components of B along the direction of polarisation of the light. In this case, the difference in strength between the transitions is given as a simple factor of cos 2 (φ B ) for the parallel component and sin 2 (φ B ) for the perpendicular component. According to the difference in the expected value of φ B obtained from the fits, this results in an approximate 4 : 1 ratio in the strengths of the lines; this is clearly visible in both figure 5 and 6.
Similarly, for both spectra shown in figures 5 and 6 we obtain a value B = 1.52 T. The uncertainties in these values, 80 mT and 70 mT, respectively for the two spectra, can be mainly attributed to linearity of the laser scan in our experiment. The DFB laser used in our experiment allows for a large mode-hop-free scan (∼ 150 GHz) at the expense of a non-linearity that is introduced as the frequency is changed. This, together with other systematic errors in the scaling of the frequency axis, is the primary source of uncertainty in our measurements. In future work we plan to design the experiment so that this nonlinearity can be reduced in order to improve the precision in our measurement of the magnetic field strength.
Furthering the ideas and work presented in [56], we propose this as an atomic technique for measuring large magnetic fields and their relative orientation. With the system completely in the hyperfine Paschen-Back regime, Figure 6. Experimental data (blue circles; colour online) taken as a function of linear detuning for a vertically polarised input beam with the corresponding fit (purple line; coloured online) using our ElecSus model, with residuals shown (bottom panel). There is very good agreement between the data and fit (RMS error of 1.2%). For this spectrum the free parameters in the fit are: φB, the angle of the magnetic field with respect to the x-axis, B, the magnetic field strength and T , the temperature of the atoms. Average values of φB = (2.0082 ± 0.0007) rad, B = (1.52 ± 0.07) T and T = (110.23 ± 0.03)C are obtained from fitting five spectra. All other parameters for the system are fixed as follows: θB = π/2, Γ Buff = 350 MHz and δ shift = 50 MHz.
the Zeeman shift presented in all the resonance line positions allows for a better determination of the magnetic field strength. In addition to this the relative strength between sets of transitions, due to different coupling strengths, is better observed, which in turn allows a more precise determination of the relative direction between the external magnetic field and the direction of polarisation (i.e. the direction of the electric field vector) of the light. Thus, the present system and technique lead to a natural application of atomic-based spectroscopy in vector magnetometry.

V. SENSITIVITY OF OPTICAL ROTATION SIGNALS TO THE BIREFRINGENCE OF CELL WINDOWS
At the high magnetic field strength used for this work, optical rotation phenomena can provide additional information about the medium through which the light propagates [70]. In the case of an atomic medium measuring this optical rotation via the Stokes parameters, as mentioned in section II, proves to be of natural interest for understanding the interactions between the atoms and the external magnetic field. In this case, we have experimentally measured the dichroism and birefringence of the Figure 7. Experimental data (blue circles) taken as a function of linear detuning for a linearly polarised input beam with the corresponding fit (purple line) of the S1 parameter taking window birefringence into account, with residuals R shown. There is very good agreement between the data and fit (RMS error of ∼2%). In this case the fixed parameters are Γ Buff = 350 MHz, δ shift = 50 MHz and θB = π/2, while the fit allows T, B, φB to float; also included in the fit are the parameters to take into account the birefringence effects of the cell windows (θBR, φBR). A fit without the effect of the cell window birefringence (broken line) is included for comparison. atomic medium in the three bases corresponding to the S 1 , S 2 and S 3 parameters as shown in figures 7, 8 and 9, respectively.
In order to carry out these measurements a set of polarising optics (PBS+λ/2, λ/4) and two photodiodes was set up, as seen in figure 2, to measure the light transmitted through the experiment cell in terms of the linear detuning of the laser scan. This allows for two orthogonal polarisation components to be recorded simultaneously and then be processed into the corresponding Stokes parameter for the basis in question. We take the definitions of the Stokes parameters as used in references [59,60]. Figure 7 shows the S 1 parameter, taken as the difference between the orthogonal linear polarisations (i.e. horizontal and vertical) (eq. 2). Figure 8 shows the S 2 parameter, defined as the difference between the linear polarisations in a basis rotated by π/4, giving diagonal components in a Cartesian basis (eq. 3). Lastly, figure 9 shows the last Stokes parameter, S 3 , as the difference between orthogonal circular polarisations in the helicity basis, (i.e. left-hand and right-hand circular) (eq. 4). In these cases, given a well-defined input polarisation, the transmitted light gives information regarding the linear and circular birefringence of the atomic medium. Using our theoretical model, ElecSus, we proceed to fit the data to each of the three Stokes parameters mentioned above. We can see in figures 7, 8 and 9 that there is very good agreement between the data (solid purple Figure 8. Experimental data (blue circles) taken as a function of linear detuning for a linearly polarised input beam with the corresponding fit (purple line) of the S2 parameter taking window birefringence into account, with residuals R shown. There is very good agreement between the data and fit (RMS error of ∼3%). In this case the fixed parameters are Γ Buff = 350 MHz, δ shift = 50 MHz and θB = π/2, while the fit allows T, B, φB to float; also included in the fit are the parameters to take into account the birefringence effects of the cell windows (θBR, φBR). A fit without the effect of the cell window birefringence (broken line) is included for comparison. curve) and model [68]. Despite this, there are still slight discrepancies between the experimental data and the fit. We can try and remove some of these errors by taking into account the birefringence of the windows of our experiment cell. To do this, we include in our model the effects of two thin, birefringent windows interacting with the electric field of our laser beam twice: once before the light enters the atomic medium and once when the light has passed through the atomic medium and exits the cell. We carry out these calculations by using the Jones matrix formalism [60], so that in this case the transmitted electric field E out in our experiment can be written as where E in is the incident electric field, M θBR,φBR is the Jones matrix representing the birefringent window of the cell and J atoms is the Jones matrix representing the dichroic and birefringent atomic medium. The matrix M θBR,φBR has been included twice to account for the entry and exit windows of the experiment cell. This output electric field can be multiplied by the appropriate Jones' matrices to give the desired polarisation components to process into the form of the different Stokes' parameters.
From our fits we can see that the birefringence due to the cell windows is considerably small. Here we make an initial assumption that both of these windows are identical in their birefringent properties. Figures 7, 8 and 9 show there is very good agreement between the data (bro- Figure 9. Experimental data (blue circles) taken as a function of linear detuning for a linearly polarised input beam with the corresponding fit (purple line) of the S3 parameter taking window birefringence into account, with residuals R shown. There is very good agreement between the data and fit (RMS error of ∼2%). In this case the fixed parameters are Γ Buff = 350 MHz, δ shift = 50 MHz and θB = π/2, while the fit allows T, B, φB to float; also included in the fit are the parameters to take into account the birefringence effects of the cell windows (θBR, φBR). A fit without the effect of the cell window birefringence (broken line) is included for comparison. ken green curve) and our model [68]. The residuals R shown in these figures correspond to the results of the fit that includes the birefringence of both cell windows. We obtain average values of (0.96 ± 0.16) radians ((55 ± 9) • ) for the angle θ BR and of (0.06±0.03) radians ((3±2) • ) for the angle φ BR . It is worth noting that these values correspond to a fit of the effect both cell windows have on the electric field transmitted through the cell. Comparing the fits to the experimental data with and without the birefringence effects from the cell windows we can see that this effect is particularly evident in the S 3 parameter, as seen in figure 9. Due to the definition of said parameter (eq. 4) we can proceed to say that the cell windows have a predominantly circular birefringence. In particular, we exploit the sensitivity of the atomic system to optical rotation, in this case in the basis of orthogonal circular polarisation states, to obtain a signal that enhances these effects so that they are clearly visible. As such, this experimental system provides a tool to characterise these birefringence effects due to vapour cell windows in order to reduce systematic errors in future measurements.

VI. CONCLUSIONS
In conclusion, we have demonstrated a spectroscopybased technique to measure the absolute magnetic field strength and angle of polarisation using a thermal vapour of alkali-metal atoms in real time. Our results use an isotopically enriched sample of 87 Rb, but this technique is applicable to any alkali-metal atom. We have found very good agreement between our detailed spectroscopic data and our theoretical model of the transmission through the medium. We have used polarisation-sensitive detection in order to better constrain the polarisation angle measured, as well as measure the birefringence effects due to the vapour cell windows. Using this technique it is also possible to envisage a precise spectroscopy setup for atomic magnetometry in large (> 1 T) fields. Furthermore, the work here presented opens up new areas of research using atomic vapours, such as measurement of fundamental constants via precision thermometry using the ground-state populations.