1. Introduction

In the presence of a longitudinal magnetic field in a medium with non-zero vector polarizability, the linear polarization of light rotates proportional to the strength of the magnetic field, an effect known as the (linear) Faraday effect [1]. The magnetic field induces circular birefringence, which results in a phase shift between the circularly polarized components, leading to a linear polarization rotation. Atomic vapor has narrow resonant lines, compared to solids and liquids, which allows the optically polarized atom to retain the Larmor spin precession for a longer time and consequently cause sustained dispersion of light. Recent advances in paraffin coatings in vapor cells and using buffer gas have resulted in longer spin relaxation time, thus improving the sensitivity of a thermal atomic-based optical magnetometer [2–7]. Once laser-cooled, the alkali atoms essentially become free from Doppler and collisional broadening and thus offer great potential to sensitive magnetometry [8–11] as well as characterization and control of ultra-cold atomic systems [12,13].

In presence of longitudinal magnetic field, the Zeeman sublevels get shifted resulting in different resonance frequencies for the two circular polarization components. This circular birefringence results in polarization rotation of a linearly polarized probe. This effect is prominent in the range of magnetic field which causes the Zeeman shift of the order optical transition linewidth. This effect, for low probe power, can be described using linear susceptibility $\chi ^{(1)}$. At higher optical power, $\chi ^{(3)}$ effects can be observed. Even at magnetic fields one order of magnitude lower, magneto-optical rotations can be observed. Such effect has been described as a three stage process [14]. First, the optical field along with spontaneous emission redistributes the atomic population (pumping stage). The atomic polarization then evolves in presence of external magnetic field (evolution stage) followed by change in the transmission properties of the probe light (probing stage). This build-up of coherence between atomic Zeeman states aided by light results in the non-linear Faraday effect, which leads to narrow resonances, thus enabling magnetometry at very low fields [15]. The pump and the probe in the non-linear Faraday effect can either be the same optical beam or two separate beams. The non-linear Faraday effect has been demonstrated both in thermal atoms [16–22] as well as in ultra-cold atoms [23,24].

The polarization rotation observed in ultra-cold atom experiments involving the non-linear Faraday effect (such as [23]) uses detectors either in balanced or cross-polarizer configurations. In balanced detection, the incident light (say horizontally polarized), after interacting with the atoms, is projected onto orthogonal states that are equally overlapping with the incident polarization (say diagonal-antidiagonal basis or left-right circular basis). The two projections are detected, and the difference is amplified. In the cross-polarizer case, the projection is made with the state orthogonal to the incident polarization. Since only one port is used, unlike the balanced detection technique, we can only infer the magnitude of the polarization rotation and not its direction.

The use of detectors allows the study of temporal evolution as well as the observation of signals away from near-zero magnetic fields using modulated light [25,26]. The spatial resolution, however, is limited by the cold atom cloud or the beam size. Further, the detector integrates over the spatially non-uniform polarization rotation. The magneto-optically trapped atomic cloud shifts for cases involving a persistent transverse magnetic field. To ensure the interaction of the atoms with uniform light intensity, the beam waist needs to be much larger than the size of the atomic cloud. In such scenarios, this spatial averaging would lead to lower sensitivity. For high spatial resolution, one would need to use a camera, which would come at the cost of losing temporal resolution.

Imaging of the Faraday effect has been performed in configurations analogous to the balanced and cross-polarized methods with detectors replaced by the camera. The dual port polarization contrast imaging (DPPI) [27] uses the two output ports of a polarizing beam splitter (PBS) in a balanced configuration, which increases the signal and helps reject common mode fringe noise patterns. On the other hand, the dark field Faraday imaging (DFFI) [28] uses the cross polarizer method, which only allows the polarization component orthogonal to the incident beam up to the extinction ratio and imperfections to be imaged and thus offers higher contrast even if the signal is lower. In DFFI, the light incident on the camera is much lower compared to DPPI for the same incident light. For the linear Faraday effect, this is fine for both of the methods as the incident light can be attenuated or intensified to some extent without having much consequence on the atomic state evolution.

The single-beam non-linear Faraday effect, however, is very sensitive to incident power. So, the power of the beam can only be attenuated after the interaction of atoms. With DFFI, often, there would be too little light on the camera to obtain any Faraday rotation signal. Increasing exposure is not an option as it averages over temporal dynamics. Increasing the gain of the camera does not improve the contrast, as the background noise is amplified as well. With DPPI, there is too much light in the camera, and the sensor may saturate. This can be attenuated using neutral density filters, which, however, decreases the signal as well. Thus, either of the extreme projections, balanced or orthogonal, is not optimal in the context of both the signal and the contrast ratio for studying the non-linear Faraday effect.

In this work, we present a solution that is tunable from an effective DFFI configuration to an effective DPPI configuration, which can be adjusted depending on the power at which the non-linear Faraday effect needs to be studied. We retain the balanced detection feature of the DPPI and thus are able to eliminate the common mode spatial noise. At the same time, we can improve the contrast better than the equivalent DPPI method by optimizing settings to take maximum advantage of the dynamic range of the imaging device.

The next sections are organized as follows: In Sec. 2, we describe the experimental setup, followed by the description of the imaging system. Then, in Sec. 3, we present the method to infer the polarization rotation from the dual port images. We show the improvement in contrast and sensitivity over other imaging techniques. Next, we report the effect of transverse magnetic fields on the non-linear Faraday rotation signal.

2. Experimental methods

We study the non-linear faraday effect in a cold atomic system (see Supplement 1 for preparation and characterization of the cloud). We use three pairs of Helmholtz coils (X-Coil, Y-Coil, and Z-Coil) to control the external DC magnetic field, which is kept constant through a single run of the experiment, starting from loading of the atoms in the trap till the measurement of Faraday rotation. As a result, depending on the external magnetic field, the center of the atomic cloud shifts for different runs. To ensure that the atomic cloud interacts with a nearly uniform probe beam intensity even at different external magnetic fields, the probe beam size must be much larger than the atomic cloud and the shifts in the position of the trap. The probe beam is derived from the cooling laser (Cooler beam of ILS-780 laser from MuQuans company) and made resonant to $| {F = 2}\rangle \rightarrow | {F=3}\rangle$ ${}^{87}\text {Rb } D_2$ transition line using an acousto-optic modulator (AOMO-3080-122, Gooch $\&$ Housego) which is used in double pass configuration. To maintain the Gaussian beam profile, we pass the beam through a polarization maintaining single mode fiber(P3-780PM-FC-5, Thorlabs), and it is collimated using a fiber collimator (FC220APC-780, Thorlabs). The beam size is increased to $\approx 4 \ \text {mm}$ using the beam expander constructed using two lenses ($L_1$ (LA1131-B, Thorlabs) and $L_2$ (LA1708-B, Thorlabs)). The polarization of the beam is made nearly horizontal by passing the beam through the polarizing beam splitter ${PBS}_1$ (PBS122, Thorlabs). This is followed by a combination of half-wave plate ${HWP}_1$ (WPH10M-780, Thorlabs) and polarizer ${POL}$ (LPVISC100, Thorlabs) which allows us to control the polarization of the probe beam with an improved extinction ratio. The detuning and the duration of the probe beam are controlled using the same AOM, which is used to make the probe beam before coupling to the fiber.

We image the beam and the atomic cloud using two $2f-2f$ lens arrangements as shown in Fig. 1(b). Using lens $L_3$ (LA1708-B, Thorlabs), we image both the beam and the atoms onto the plane of the aperture. Since the size of the beam is much larger than the size of the MOT, which is needed in order to have uniform intensity distribution over the atomic cloud, we use the aperture $A$ to block the beam down to a size of $\approx 2 \ \text {mm}$ so that we can perform dual port imaging with the same camera $CCD$ (iXon Ultra 897, Andor). The lens $L_4$ (LA1708-B, Thorlabs) transfers the image on the plane of the aperture to the plane of the $CCD$ camera. The 50:50 (R: T) non-polarizing beam splitter $BS_1$ (BS005, Thorlabs) placed after $L_4$ splits the beam into two paths which are recombined at another 50:50 (R: T) non-polarizing beam splitter $BS_2$ (BS005, Thorlabs) forming a Mach-Zehnder interferometer configuration. The $BS_2$ is then laterally translated so that the two beams do not overlap and can be imaged on the $CCD$ without interference. Half wave plates $HWP_{+}$ and $HWP_{-}$ (WPH10-780, Thorlabs) are placed on each path respectively oriented at angle $+\theta _p$ and $-\theta _p$ from the horizontal. The polarizing beam splitter $PBS_2$ (PBS252, Thorlabs) selects the horizontal component of the beam rotated by the non-linear Faraday effect and the waveplates. The bandpass filter $BPF$ (FB780-10, Thorlabs) prevents stray light specifically during long exposure ($\approx 50-100 \ \text {ms}$) for fluorescence imaging. The angle $\theta _p$ (typically kept between $2- 12 \deg$) can be optimized depending on the power of the probe beam (varied between $100 \ \mu \text {W} - 1 \ \text {mW}$) at which we aim to study the non-linear faraday effect. The duration of the probe was kept typically between $100\ \mu s$ and $1 \ \text {ms}$. The repumper beam is always kept on throughtout the experiment.

 

Fig. 1. Experimental setup. (a) [Top] shows the preparation of the polarization state of the probe followed by the vacuum chamber, lasers, and coils for the laser cooling and magneto-optical trapping of the atoms. The three pairs of Helmholtz coils are always on to provide a controlled external magnetic field after canceling Earth’s magnetic field. (b) The lens and aperture arrangement for 2f-2f imaging of the atomic cloud onto the CCD. The components for dual port detection are placed between $L_4$ and $CCD$ (not shown in this diagram (b)). (c) The dual port detection scheme using the same camera. This particular arrangement causes a path difference between the two ports, which, in principle, can be corrected using parallel glass windows. Since the path difference is much less than the focal length, the size of the image does not change significantly.

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In some experiments like [23], the homogeneous magnetic field for which we aim to observe the Faraday effect was turned on, and the quadrupole magnetic field was turned off for a few milliseconds. However, usually, the unknown magnetic field that we aim to probe cannot be turned on and off at will. Therefore, we kept the homogenous magnetic field always on in this experiment. This resulted in shifting the position of trapped atoms depending on the transverse magnetic field. The large beam size and the controllable aperture $A$ ensured the imaging was possible for the range of transverse magnetic fields used. After loading the trap for about $10 \ s$, we obtain a fluorescence image, and then we turn off the quadrupole coil and wait for $20 \ \text {ms}$ for the eddy currents to damp when the sub-doppler cooling is going on. After switching off the cooling beam, we wait another $1 \ \text {ms}$, and then we expose the atoms to the beam for $1 \ \text {ms}$ and capture the image with atoms. After $50 \ \text {ms}$, we capture another image, of the probe without atom. The difference between these two images gives intensity change due to polarization rotation caused by the atoms. Given the detuning of about $8 \ \text {MHz}$ (red-detuned) and the probe power of $1 \ \text {mW}$, the fraction of light absorbed is negligible. Even if it were substantial, the differential image between the two ports would eliminate its effects on the polarization rotation. We take five such images for a given longitudinal magnetic field $B_z$ and a particular transverse magnetic field $B_x$ or $B_y$.

3. Results and discussion

Let $I_0^{\pm }$ be the intensity distribution in the port with the half-wave plate angle at $\pm \theta _p$ in the absence of the atoms. The intensity distribution when the atomic cloud is present is denoted by $I^{\pm }$ for the respective ports. With $\epsilon$ being the extinction ratio fraction [28], we can express the above quantities as follows:

Note that in the above expressions, both $\mathcal {I}$ - the incident light intensity and the Faraday rotation angle $\theta _{fr}$ are functions of transverse position and time. In the camera, we detect the $I^{\pm }$ and $I_0^{\pm }$ as a function of $x$ and $y$ while the time dependence of $\theta _{fr}$ is integrated over the exposure time. Thus, any inference of polarization rotation is an effective temporal average over the exposure time.

For an ideal polarizer, i.e., with $\epsilon = 0$, we can infer the effective faraday rotation angle from the change in intensity distribution can be obtained as $dI^{\pm } = I^{\pm } - I_0^{\pm }$.

The effective Faraday rotation $\theta _{fr}$ for a typical image with $\theta _p \approx 3.6 \deg$ is shown in Fig. 2 along with the images $I^{+}, I^{-}$ and $dI^{+}$ and $dI^{-}$.

 

Fig. 2. (a) The single raw image of the two ports. This image is taken in the presence of atoms. A similar image is captured without the atoms. (b) The difference between the image with atoms and without atoms. For a particular faraday rotation angle, the intensity after the polarization projection in the $PBS_2$ are different for the two different ports, because the angles of $HWP_{+}$ and $HWP_{-}$ are different i.e, $+\theta _p$ and $-\theta _p$ respectively ($\theta _p \approx 3.6 \deg$). This results in one port with negative change in intensity while the other port having positive change in intensity. (c) Spatial distribution of polarization rotation inferred from the intensity change. The region containing more atoms makes the light undergo more non-linear Faraday rotation.

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The image $dI^{\pm }$ helps eliminate noise common to $I^{\pm }$ and $I_0^{\pm }$, predominantly the noise in the beam shape. But, since these two images are taken at different times, this method often introduces fringe noise. Often, such noise are present in both the ports. Taking the average of the difference of $dI^{+}$ and $dI^{-}$ helps in eliminating spatio-temporal noise common to both ports. This quantity can be expressed in terms of $I^{+}$ and $I^{-}$ as follows:

Similarly, we can define $\Sigma I$, which is the average of the two images.

To use the camera’s full dynamic range, we should have $\Sigma I$ to be approximately half of the saturation intensity $I_s$. Here, it is useful to consider the dual port contrast ratio $\mathcal {C}_d$, which quantifies what fraction of the dynamic range is used to obtain the change in intensity due to the non-linear Faraday rotation.

In Fig. 3(a), we present the experimentally determined $\mathcal {C}_d$. The numerator $\Delta I$ is obtained as the amplitude of a 2D gaussian fit of the image in Fig. 2(c) expressed as intensity change i.e., without the conversion to Faraday rotation angle. The denominator is obtained as the weighted average over the two ports of probe intensity in presence of atoms. The weight function correspondonds to the 2D gaussian ditribution of atomic density. This quantity is plotted as a function of the angle $\theta _p$ by which the two wave plates are rotated (one by $+\theta _p$ and the other by $- \theta _p$). The angle $\theta _p$ is inferred using Malus’ law for transmission of the probe through $PBS_{2}$. The uncertainty in the angle is presented as the horizontal error bars in the Fig. 3(a). The vertical error bars denote the standard deviation over 20 images. The theory curve is computed using RHS of Eqn. (9) using the values of $\epsilon = 0.0054$ and $\theta _{fr}$ obtained from the experiment. The uncertainty band corresponds to maximum and minimum constrast obtained within the inter-quartile range of $\theta _{fr} = (0.211, 0.196) \deg$ and $\epsilon = (0.0065, 0.0036)$. The experimental data agrees with the theory within the uncertainty.

 

Fig. 3. Experimentally obtained dual port contrast ratio as a function of $\theta _p$ and non-linear Faraday rotation as a function of longitudinal magnetic field $B_z$. (a) The dual port contrast ratio as a function of the half wave plate angle $\theta _p$ is presented above. The theory is computed using RHS of Eqn. (9) with $\theta _{fr} = 0.205^{+0.006}_{-0.009} \deg$ and $\epsilon = 0.0054^{+0.0011}_{-0.0018}$ obtained from the experiment. The experimental values were computed using the definition $\Delta I / \Sigma I$. The green line is the inverse of the derivative of contrast with $\theta _{fr}$ computed by first taking the derivative of RHS of Eqn. (9) and then substituting the experimentally obtained values. (b) The peak value of the non-linear Faraday rotation, obtained from the 2D gaussian fit of Fig. 2(c), as a function of longitudinal magnetic field. We perform a spline fit to the curve and computed the derivative of the faraday rotation as a function of magnetic field. The slope at zero magnetic field is about $1.3 \deg / G$. The dynamic range over which the magnetic field can be computed uniquely from the Faraday rotation angle is the region between the two zero slopes which corresponds to $\approx 0.52 \ G$.

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The maximum contrast obtained is $0.048$ at $2.08 \deg$. The contrast with balanced detection technique would have been $0.007$ at $\theta _p=22.5 \deg$ as seen in the curve. Experimentally the contrast was too low for the Faraday rotation to be observed. Thus, using the tunable differential imaging technique, the contrast is improved by a factor of $7$ comapared to the balanced imaging. However, this improvement depends on $\epsilon$. Ideally, as $\epsilon \rightarrow 0$, the contrast of the cross-polarizer method is maximum. For finite $\epsilon$, the tunable differential imaging can give better dual port contrast.

The sensitivity to the magnetic field can be related to the dependence of the slope of the dual port contrast ratio as a function of the Faraday rotation (see Supplement 1 Sec 2 for a detailed discussion on the sensitivity).

The first term $\left (\frac {\partial \theta _{fr}}{\partial B_z} \right )^{-1}$ originates from the atom-light interaction and depends on the experimental parameters like atomic number density, detuning, etc. From Fig. 3(b), we obtain this number as $1/(1.3 \deg /G) \approx 4.4 \times 10^{-3} T/ \mathrm {rad}$. The second term $\frac {\delta (\Delta I)}{I_s}$ is the property of the imaging device. In the best possible case, i.e., if the shot to shot intensity fluctuation and we utilize the full dynamic range of the camera, this ratio is the read noise (about 100) divided by the maximum value the pixel can take ($2^{15}$). Thus, the ratio is about $3 \times 10^{-3}$.

The third term $\left (\frac {\partial \mathcal {C}_d}{\partial \theta _{fr}}\right )^{-1}$ depends on $\theta _p$ i.e. on the method of detection. In the Fig. 3(a), we can see the minima of this term occurs at $1.6 \deg$ with the value $0.074 \ \mathrm {rad}$. hence, using Eqn. (10), we obtain $\delta B$ for a single image as $4.4 \times 10^{-3} T/ \mathrm {rad} \times 3 \times 10^{-3} \times 0.074 \ \mathrm {rad} \approx 1 \ \mu T$. The image is taken with an effective exposure of $1 \ \mathrm {ms}$ and hence per 1 s, the random noise in the magnetic field can be decreased by $\sqrt {1000}$. Hence, we obtain the sensitivity of $\approx 33 \ nT/\sqrt {\mathrm {Hz}}$.

The improvement of sensitivity among various imaging methods depends solely on $\left (\frac {\partial \mathcal {C}_d}{\partial \theta _{fr}}\right )^{-1}$. As seen in Fig. 3(a), this quantity is $0.505$ with the balanced imaging technique where as we can tune it to $0.074$. Thus, the sensitivity improves by a factor of $\approx 7$. Hence, the sensitivity and contrast can be improved by an order of magnitude by using components with better extinction ratio. The spatial resolution for the experimental parameters in this experiment is limited by the pixel size of the camera which is $16 \ \mu m$.

4. Effect of transverse fields

In Fig. 3(b), we study the Faraday rotation as a function of longitudinal field by ensuring the best possible cancellation of any transverse fields. Practically, it may not be always possible to cancel unknown transverse fields which are persistent. To find the effect of the transverse magnetic field, say $B_x$ on the Faraday rotation signal, a simple theoretical model was proposed in [29] that relates the total polarization rotation $\Theta _{fr}(B_x)$ in presence of the transverse field to the function $\Theta ^z_{fr}$ which computes the Faraday rotation assuming the total magnitude of the field is along the longitudinal direction. It has been argued in [29] that these two quantities will be related by the factor $B_{x}/\sqrt {B_x^2 + B_z^2}$ as shown in the Eqn. (11) below.

We experimentally study the above relation by measuring Faraday rotation as a function of the transverse magnetic field $B_x$ (with $B_y \approx 0$) and $B_y$ (with $B_x \approx 0$) respectively (see Fig. 4(a) and (b)). We kept the longitudinal field $B_z$ constant at the point marked in Fig. 3(b) (indicated by olive coloured dashed vertical line). At higher transverse magnetic fields, the atomic cloud shifts, and the image is obscured by the aperture which limits the transverse magnetic field $B_x$ and $B_y$ in the given plot range. This is a limitation of the particular implementation as the aperture size can be increased in principle given a bigger imaging sensor or appropriate minification.

 

Fig. 4. (a) The Faraday rotation is a function of the applied transverse magnetic field along the $x$ direction. (b) The Faraday rotation is a function of the applied transverse magnetic field along $y$ direction. The uncertainty band is obtained as the area between the two $\Theta _{fr}$ (or two $\Phi _{fr}$) functions where the corresponding $\Theta _{fr}^z$ is obtained as a spline curve to the upper and lower 1$\sigma$ deviations, respectively. The data points are indicated as black dots and the mean value is indicated by the red dot.

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We empirically find that the data agrees better with the expression $\Phi _{fr}$ described in Eqn. (12) below. Using $\Theta _{fr}(B_x)$, we find that $\chi ^2(X^2 \approx 0.095) \approx 10^{-8}$ where as using $\Phi _{fr}(B_x)$, we obtain $\chi ^2(X^2 \approx 0.009) \approx 10^{-13}$ indicating a better goodness of fit [30].

In [29], it is argued that the contribution to the polarization rotation angle will be determined by the projection of the response function of the medium on a plane orthogonal to the propagation direction. In Eqn. (12), we have used the square of the projection instead. This can be motivated by the fact that since the amplitude of the response function will depend on the projection, the associated probability would depend on its square. However, a detailed theoretical model is needed to substantially confirm either model.

In summary, we study the effect of persistent magnetic fields on the non-linear Faraday rotation in a cold atom system. Since the position of the atomic cloud shifts in the presence of a persistent transverse field, we need a larger beam for the single beam non-linear Faraday effect to ensure that the atomic cloud ensures a uniform light field. However, this implies that, in general, the polarization rotation angle due to non-linear Faraday rotation is a function of both space and time. With a detector, we would average over the spatial part, while with a camera, we report the temporally averaged effective polarization rotation as a function of transverse position. Since the non-linear Faraday effect requires specific power, we find that either the cross-polarizer method gives a very low signal or the balanced detection method saturates the camera. We could use a gain or neutral density filter, but they do not improve the contrast ratio. Instead, we present a tunable dual port differential imaging method, in which, in one port, we project the polarization with the half-wave plate at angle $\theta _p$ and the other port with the wave plate at angle $-\theta _p$. Depending on the Faraday rotation $\theta _{fr}$, $\theta _p$ can be tuned to adjust the contrast ratio, sensitivity, and overall signal strength, thus presenting us with a tool to study non-linear Faraday imaging. We demonstrated the implementation of tunable differential imaging. Using the tunable differential imaging system, we can reach a sensitivity of $33 \ nT/\sqrt {Hz}$ along with the spatial resolution of $16 \ \mu m$. These numbers, however, depend on the atomic density and the noise figure of the imaging device and hence can be improved. We focus on improving contrast and sensitivity obtained due to the tunable differential imaging method over other imaging methods. In particular, we improve by a factor of 7 compared to balanced detection. This improvement enabled us to measure the non-linear Faraday effect using the camera. With the balanced imaging method, the camera would saturate. The use of ND filters decreased the overall signal.. Using the cross-polarizer method, the signal strength was too low. Hence, we could not obtain the non-linear Faraday rotation image using existing methods due to a lack of contrast or signal strength. The tunable differential imaging technique enabled us to study spatially resolved non-linear Faraday imaging by allowing us to choose appropriate contrast. We also demonstrated the method in the presence of a transverse magnetic field. Therefore, this technique could be helpful in the study of various non-linear magneto-optical effects in ultra-cold atomic systems. Further study on the effect of transverse field enabled by this technique would pave the way towards precision spatially resolved vector magnetometry.