1 INTRODUCTION

Birefringence and dichroism are fundamental optical phenomena in anisotropic media and are widely applied in medicine, biology, nanotechnologies, and other fields of science and technology. These phenomena ensure the operation of numerous optical devices such as polarizers, phase plates, and optical isolators, where anisotropy can be intrinsic, as in crystals, or induced by an external action. Birefringence and dichroism are associated with the anisotropy of the refractive index and the absorption coefficient, respectively. These phenomena underlie some precise measurement methods such as ellipsometry [1] and magnetometry [2, 3].

The polarimetric technique for the detection of resonances associated with magnetic birefringence or dichroism in alkali metal vapor is widely used in atomic magnetometry [4, 5]. In this case, a change in the polarization of light after its interaction with a medium is detected depending on the magnetic field. In the general case, both birefringence and dichroism affect the polarization. However, only one of two phenomena decisively affects magneto-optical resonances at the appropriate choice of the frequency of optical radiation.

In particular, the authors of [6] demonstrated an atomic magnetometer with a record sensitivity of 0.16 fT Hz–1/2 based on circular birefringence and the related magneto-optical rotation (MOR) of the linear polarization of light. In this case, it is convenient to represent the linearly polarized wave at the input of the cell with atoms as the superposition of two waves with opposite circular polarizations \({{\sigma }^{ + }}\) and \({{\sigma }^{ - }}\). The refractive indices \({{n}_{ + }}\) and \({{n}_{ - }}\) for these waves are different because of circular birefringence, which results in the phase shift between these field components at the output of the cell and in the rotation of the linear polarization of the total field.

In [7], a linearly polarized light beam resonant with the optical transition in the 85Rb D2 line led to the alignment of atomic spins, which is equivalent to the appearance of linear dichroism in the medium. In this case, the axis of alignment (dichroism) precessed at the Larmor frequency in the external magnetic field. These oscillations were detected through MORs at the output of the cell and could be used to measure the magnetic field. Circular dichroism induced in Cs vapor by an elliptically polarized wave resonant with the D1 line was used in [8]. Dichroism in this scheme changes the ellipticity of the polarization, which also carries information on the magnetic field.

Two geometries of the laser field are mainly used in MOR magnetometry. In the first geometry, two spatially separated beams (pump and probe) intersect each other in a gas cell at a right angle (see, e.g., [6, 9]). The second geometry, which is more appropriate for compact sensors, involves a single beam, which can also be represented as the sum of the pump and probe beams separated in polarization [10] or in time [11]. Other geometries of the field, e.g., with counterpropagating laser beams, are also possible [12, 13].

At the same time, to improve the sensitivity of measurements in any geometry of the field, it is also necessary to develop methods of detection of MORs such that the rotation angle of the linear polarization is maximal. A high sensitivity of the rotation angle to the variation of the magnetic field, i.e., a small width of the resonance, should be ensured. The achievement of a noticeable rotation angle in a short length of the medium is also important for many applications. In particular, this requirement is of fundamental importance for the fabrication of compact magnetic sensors, particularly needed for medical diagnostics [14, 15]. The quality parameter of MORs for such sensors can be taken in the form

$$Q = \frac{{{{\varphi }_{{\max }}}}}{{\Delta \times {{L}_{{{\text{cell}}}}}}},$$
(1)

where φmax is the maximum rotation angle detected in the experiment, Δ is the FWHM of the resonance, and Lcell is the length of the cell.

Rotation angles in MOR magnetometers usually range from fractions to several tens of milliradians. For  example, a spherical cell 30 mm in diameter with  the antirelaxation coating of walls filled 87Rb vapor was used in [16] to detect an MOR. The amplitude and width of the MOR were about 0.4 mrad and 7 nT, respectively. Despite the relatively narrow resonance, the rotation angle was also small, so that Q ≈ 2 mrad μT–1 mm–1. The cell 100 mm in diameter with coating filled with 85Rb vapor was also used in [7], where the resonance had a width of about 0.4 nT and an amplitude of about 1.8 mrad; correspondingly, Q ≈ 45 mrad μT–1 mm–1. The authors of [13] recently detected a giant MOR in 87Rb vapor in a 60-mm-long cell with a buffer gas. In spite of a large rotation angle of 120 mrad, the resonance had a relatively large width of ≈1 μT; correspondingly, Q ≈ 2 mrad μT–1 mm–1.

Multipass schemes, where the probe beam passes through the cell several times, which is equivalent to an increase in the length of the medium, are used in some works to significantly increase the rotation angle. In particular, the probe pulse in [17] was reflected more than 100 times from mirrors placed inside a 23-mm-long rubidium cell heated to a temperature of 133°C. Magneto-optical resonances were observed in the form of oscillations after the action of a π/2 RF pulse (60 kHz). Although extremely large angles (100 rad) were detected in [17], this scheme of observation of MORs, as well as the signal processing algorithm, seems complicated for the implementation in a compact magnetic sensor. Meanwhile, the rotation angle per pass was about 1 rad, which is also large.

In this work, a quite simple configuration of the light field, which does not require any RF field and consists of counterpropagating light waves with polarizations making an angle of 45° with each other, is used to observe MORs in 87Rb vapor. One of the waves has a higher intensity and is the pump wave, whereas the second, weaker, wave is the probe wave. Our geometry scheme is similar to that proposed in [13]. However, a fundamental difference is that both waves in our case are in resonance with the Fg = 2 → Fe = 1 transition in the D1 line, whereas the pump and probe waves in [13] were in resonance with the D2 and D1 lines, respectively. In addition, the waves used in [13] were redshifted from the centers of absorption lines (from ≈150 to 5000 МHz). Thus, our scheme for the observation of MORs is much simpler. Furthermore, experiments show that a much larger rotation angle of about 390 mrad (22°) at a width of the resonance of about 300 nT is observed in our scheme at approximately the same temperature of vapor (Tcell ≈ 80°C), but at the length of the cell shorter by a factor of 3. The  maximum parameter Q in our experiments is ≈90 mrad μT–1 mm–1. These results demonstrate that the proposed scheme for the detection of MORs is promising for the fabrication of compact atomic magnetometers to measure ultraweak magnetic fields.

2 EXPERIMENT

Figure 1 presents the scheme of the experimental setup for the detection of MORs with an external-cavity diode laser (ECDL) with a wavelength of λ ≈ 795 nm (87Rb D1 line) and a linewidth of less than 1 МHz in the Littrow geometry [18]. The wavelength of radiation was smoothly varied with a piezoceramic plate equipped with a diffraction grating. The optical frequency was controlled by means of a WS7 (Angstrem) wavelength meter with a resolution of 500 kHz. This control was sufficient because the laser frequency during the experiment did not significantly drift and was in resonance with the Fg = 2 → Fe = 1 transition (see Fig. 2), for which the spontaneous relaxation rate of the excited state is γ ≈ 5.6 МHz.

Fig. 1.
figure 1

(Color online) Layout of the experimental setup described in the main text.

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Fig. 2.
figure 2

Transmission of a light wave with an intensity of I ≈ 180 mW/cm2 through the cell with 87Rb vapor at 80°C. Digits are the angular momenta of energy levels in the D1 line, which are indicated in the inset, where the arrow marks the optical transition used in this work.

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Laser radiation passed through a Faraday optical isolator (OI) to eliminate the effect of parasitic back reflections. The phase half-wave plate (λ/2) matched the linear polarization of radiation with the fast axis of the polarization-maintaining optical fiber (PM fiber). The beam at the output of the fiber was extended to a 1/e2 diameter of 5 mm by a telescope formed by the output collimator of the fiber and an additional lens. A polarizing beam splitter (PBS) splits the laser beam into the pump (Ec) and probe (Ep) beam. The λ/2 plate in front of the splitter allowed the redistribution of the optical power between beams so that Ic ≫ Ip, where Ic and Ip are the intensities of the pump and probe laser pulses, respectively. The optical power of the pump beam was additionally controlled by a set of neutral density filters (NDF). The λ/2 plate placed in the channel of the pump wave allowed the rotation of the linear polarization of the field so that the polarizations of two counterpropagating waves in the cell made an angle of ≈45°. A beam splitter (BS) guided the pump beam to the cell without significantly affecting the polarization of the probe wave, which also passed through this splitter. The probe wave after passing though the beam splitter reached a polarimeter consisting of a λ/2 plate, a Wollaston prism (WP), and a balanced photodetector (BPD). The λ/2 plate allowed us to ensure zero signal from the differential port of the balanced photodetector far from the nonlinear resonance. Thus, a signal at the output of the differential port of the balanced photodetector indicated the rotation of the linear polarization of the probe wave.

The 20-mm-long cylindrical borosilicate cell 25 mm in diameter was filled with isotopically enriched 87Rb vapor. The ends of the cell are inclined at a small angle to reduce the effect of reflections. The cell was placed in a nonmagnetic thermostat (Thermochamber). The heating of the thermostat was switched off during the measurements to remove the effect of the parasitic magnetic field. The longitudinal magnetic field (Bz) was generated by a solenoid. The cell, thermostat, and solenoid were surrounded by a three-layer magnetic shield, which suppressed the laboratory magnetic field in the cell in the center of the shield to ≤20 nT.

The cell was filled with argon buffer gas at a pressure of about 12 Torr, which increased the lifetime of the polarization of the atom in the ground state and reduced the width of a magneto-optical resonance. Optical effects selective in velocities of atoms (e.g., saturated absorption resonance) are not observed at this pressure because the collisional broadening of the D1 absorption line is ≈230 МHz [19], which is close to the Doppler HWHM (≈225 МHz at T = 75°C). Using data and formulas from [20–22] and a value of na ≈ 1012 cm–3 for the density of rubidium atoms at T = 75°C, we obtained an estimate of Γ ≈ 450 Hz for the relaxation rate of the ground state to the isotropic distribution over magnetic sublevels. In the weak optical field limit, this Γ value determines the minimum width ≈120 nT of the MOR.

Figure 3 shows signals from the channels of the balanced photodetector and the polarization rotation resonance signal from the differential port of the balanced photodetector, which were obtained under the slow variation of the longitudinal magnetic field around zero (fscan = 10 Hz). The polarization rotation angle was calculated by the formula

$$\varphi = \frac{1}{2}\left| {\arcsin \frac{{{{P}_{1}} - {{P}_{2}}}}{{{{P}_{1}} + {{P}_{2}}}}} \right|,$$
(2)

where \({{P}_{1}}\) and \({{P}_{2}}\) are the signals from channels 1 and 2 of the balanced photodetector, respectively, and \({{P}_{1}} - {{P}_{2}}\) is the signal from the differential port of the photodetector. As seen in Fig. 3, electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) resonances are observed in channels 1 and 2, respectively. The mechanism of formation of these resonances is discussed in the next section. The comparison of the resonances in Figs. 3a–3b shows that noise in the differential port is much weaker than that in each of the channels obviously because noise common for both channels vanishes at the differential port. In contrast to noise, the desired signal (nonlinear resonance) increases in the difference of the signals \({{P}_{1}}\) and \({{P}_{2}}\) because resonances in different channels have opposite signs. These features ensure the advantage of the polarimetric technique for the detection of resonances compared to the detection of the total intensity of light transmitted through the cell.

Fig. 3.
figure 3

Magneto-optical resonances detected in the following channels of the balanced photodetector: (a) channel 1, (b) channel 2, and (c) the differential port, including the rotation angle of the polarization of the probe wave calculated by Eq. (2) at Ip ≈ 0.2 mW/cm2, Ic ≈ 5.5 mW/cm2, and T ≈ 82°C.

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Magneto-optical resonances that are associated with birefringence and are observed far from the optical resonance with the medium, where dichroism is hardly manifested, have a dispersion shape [5]. In our case, dichroism prevails over birefringence, which results in the Lorentzian shape of the resonance. Meanwhile, the resonance in Fig. 3 is asymmetric, which can be attributed to the residual effect of birefringence, in particular, to nonlinear Faraday rotation [4, 5].

Figure 4 presents the parameters of the MORs measured at different temperatures of the rubidium cell. As seen in Fig. 4a, a maximum rotation angle of  ≈390 mrad is observed at T ≈ 82°C and Ic ≈ 5.5 mW/cm2. The width of the resonances, which is shown in Fig. 4b, is a linear function of the intensity of the pump wave, which is typical of EIA and EIT resonances in the presence of the buffer gas when the collision broadening of the spectral line is comparable to or exceeds the Doppler broadening. The minimum measured width is about 120 nT, which coincides with the above estimate.

Fig. 4.
figure 4

(Color online) (a) Maximum rotation angle, (b) FWHM (see Fig. 3c), and (c) quality parameter of magneto-optical resonances versus the intensity of the pump wave at Ip ≈ 0.2 mW/cm2 and at four temperatures of rubidium vapor. Solid lines are guides for eye.

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The maximum quality parameter of ≈90 mrad μT‒1 mm–1 (Fig. 4c) occurs at quite low intensities of the pump wave, which is due to the fast saturation of the dependence φmax(Ic). It is also noteworthy that Q increases with the temperature of vapor. At the same time, the probe wave at the intensity of the pump wave below 3 mW/cm2 is almost completely absorbed in the cell at T ≈ 82°C and above, which did not allow the measurement of the parameters of the resonances (see lines with triangles in Fig. 4). However, the increase in the intensity of the pump wave leads to the increase in transparency of the medium, which results in the appearance of the signals on the photodetector.

3 QUALITATIVE THEORY

In this work, we only qualitatively explain the observed effect. The probe wave (Ep) in the considered scheme is assumed to be rather weak to neglect nonlinear effects associated with this wave. At the same time, the polarization rotation effect observed in the experiments can also be attributed to the nonlinear MOR, but nonlinearity in this case is due to the pump wave (Ec).

It is convenient to represent the linear polarization of the probe wave in the form of the sum of two orthogonal linear components (Fig. 5) one of which is parallel to the polarization of the pump wave (E||) and the other is perpendicular to it (\({{E}_{ \bot }}\)). Since the polarimeter is adjusted so that the waves with the polarizations E|| and E⊥ enter channels 1 and 2 of the balanced photodetector, it is sufficient to consider the absorption of the E|| and E⊥ components separately to explain the plots in Fig. 3.

Fig. 5.
figure 5

(Color online) Configuration of the electromagnetic field for the detection of magneto-optical resonances. The blue arrows indicate the wave vectors of counterpropagating waves having the same magnitude. The green arrow marks the polarization of the pump wave Ec. The pink arrows indicate the polarizations of the probe wave Ep and its two components E|| and E⊥. The angle between the vectors Ec and Ep is 45°. The orange left–right arrow marks the scanning direction of the magnetic field.

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Let the quantization z axis be directed along the polarization of the pump wave (Fig. 5). Figure 6a demonstrates the equilibrium distribution of populations over magnetic sublevels of the ground state. This state is sometimes called isotropic. Next, the optical pump in the absence of the longitudinal magnetic field Bx leads to the appearance of the second-rank polarization moment of atoms in the ground state, which is called alignment, because of selection rules, as shown in Fig. 6b [5]. In other words, a series of stimulated and spontaneous optical transitions involving the Fg = 2 level result in the drift of a significant part of populations of magnetic sublevels toward extreme sublevels with the magnetic quantum numbers \(m = \pm 2\). Some fraction of populations of sublevels is also transferred to the lower level of the ground state with the total angular moment Fg = 1. If the pressure of the buffer gas is low, so that the hyperfine components of the excited state are spectrally resolved, as in Fig. 2, the pump wave does not interact with the \(m = \pm 2\) sublevels and the medium becomes almost transparent to this wave with the conservation of a significant fraction of atoms in these sublevels.

Fig. 6.
figure 6

(Color online) Scheme of levels in the 87Rb D1 line. The hyperfine levels in the 2S1/2 ground state with angular momenta of 1 and 2 are indicated by digits 1 and 2, respectively, whereas similar energy levels in the 2P1/2 excited state are marked as 1' and 2'. Green circles schematically represent the distribution of populations over magnetic sublevels (a) in the absence of the electromagnetic field, (b) in the presence of only the pump wave Ec, and (c, e) in the presence of the pump wave Ec and the (c) E|| and or (e) \({{{\mathbf{E}}}_{ \bot }}\) component of the probe field. Panels (d) and (f) are the same as panels (c) and (e), respectively, but with the inclusion of the magnetic field, which leads to the mixing of magnetic sublevels (orange left–right arrows), which is shown only on the Fg = 2 level for simplicity. Spontaneous transitions are not shown, except for panel (b), where wavy arrows mark two spontaneous transitions, and optical transitions associated with the field E⊥ are shown in panels (e) and (f) only partially.

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According to Fig. 6c, the E|| component of the probe field is only a small addition to the pump field. Thus, the wave E|| is also weakly absorbed in the medium, as is the wave Ec. The longitudinal magnetic field Bx || k does not linearly shift magnetic sublevels but mixes them, which prevents optical pumping the \(m = \pm 2\) sublevels. This process can be described in terms of polarization moments, which means in this case that the axis of alignment of atoms precesses in the presence of the magnetic field, and the joint action of the magnetic and pump fields destroys alignment. For this reason, the absorption of the wave E|| increases in the presence of the magnetic field. However, this increase is limited because some atoms are transferred to the Fg = 1 level, where they do not interact with the laser field and, thereby, are not involved in the resonant interaction with the field E||. Thus, the EIT resonance is observed under the variation of the field Bx in channel 1 of the balanced photodetector (Fig. 3a).

It is seen in Fig. 6e that the wave \({{{\mathbf{E}}}_{ \bot }}\) is strongly absorbed in the medium in the absence of the magnetic field because it interacts with the densely populated \(m = \pm 2\) sublevels. In other words, atoms are aligned by the pump wave in such a way that the probe wave strongly interacts with them. In the presence of the magnetic field (see Fig. 6f), a significant number of atoms are transferred by the field Ec to the Fg = 1 level, and the medium becomes transparent to the wave \({{{\mathbf{E}}}_{ \bot }}\). Thus, scanning the magnetic field around zero is accompanied by the formation of the EIA resonance in channel 2 of the balanced photodetector (Fig. 3b). Such a change in the sign of magneto-optical resonances (EIT ↔ EIA) depending on the dire-ction of the linear polarization of the probe wave (\({{{\mathbf{E}}}_{ \bot }}\) or E||) was studied in [23, 24].

The above qualitative analysis shows that the E⊥ component is absorbed much more strongly than the E|| component in the absence of the magnetic field when the pump wave transfers the medium to the linear dichroism state. As a result, the linear polarization of the total probe field Ep at the output of the cell is rotated toward the linear polarization of the pump wave. In a fairly strong magnetic field, i.e., in the wings of magneto-optical EIT and EIA resonances (Figs. 3a and 3b), the E|| and E⊥ components of the probe wave are absorbed in the medium identically. This means that the polarization of the field Ep at the output of the cell is not rotated, as seen in Fig. 3c.

The role of the buffer gas and the open system of levels in the considered method for the detection of MORs is noteworthy. The buffer gas not only affects the width of resonances but also increases the efficiency of optical pumping the \(m = \pm 2\) sublevels, thus increasing the amplitude of observed EIA resonances. For this reason, the pressure of the buffer gas in the case of 87Rb should be no more than 15–20 Torr (depending on the composition of the buffer gas) for the pump field to insignificantly excite the Fe = 2 level. In other words, it is important in our case that hyperfine components of the excited level be spectrally resolved (see Fig. 2). The effect of neighboring excited levels, such as Fe = 2 in our case, on EIA and EIT resonances was studied in more detail, e.g., in [25, 26]. The possibility of the transfer of atoms to the nonresonant Fg = 1 level in the open system of levels shown in Fig. 6 also positively affects the amplitude of EIA resonances, as mentioned in [8, 27–29].

4 CONCLUSIONS

To summarize, a scheme has been proposed and experimentally verified to detect magneto-optical resonances of polarization rotation of a probe wave in the presence of a pump wave. A quite large rotation angle of ≈390 mrad in a fairly short length of a medium (20 mm) in a cell filled with 87Rb vapor and a buffer gas has been detected. The full width of the corresponding resonance is ≈300 nT. According to the qualitative theoretical analysis, the observed large polarization rotation angle can be explained by linear dichroism induced in the medium by the pump wave.

The parameters of the observed resonances demonstrate that the proposed scheme is promising for applications in quantum magnetometry. To determine the sensitivity of measurements, it is necessary to examine noise in signals in our experiment, which is beyond the scope of this work. At the same time, the maximum achievable variation sensitivity for the case of photon shot noise can be estimated by the simple formula \(\delta B \approx \Delta {\text{/}}SNR\). Here, SNR is the signal-to-noise ratio in a band of 1 Hz, which is \(\sqrt N \) in the considered limit, where \(N\) is the number of photons in the probe wave per 1 s. Using the experimental values \({{\Delta }_{{\min }}} \approx 120{\kern 1pt} \) nT and \({{P}_{{\text{p}}}} = \pi {{d}^{2}}{{I}_{{\text{p}}}}{\text{/}}4 \approx 40\) μW, we obtain \(\delta B \approx 10{\kern 1pt} \) fT Hz–1/2.

Some features of the proposed scheme for the detection MORs distinguish it from a number of other known schemes. In particular, either a quite long medium (≈50–100 mm), as, e.g., in [7, 13], or an increased temperature of vapor (≈150–200°C) for the achievement of a high density of working atoms is usually used to observe large rotation angles [10, 17]. In our scheme, large rotation angles at a width of the resonance of about 100 nT are observed in a 20-mm-long cell at a temperature of rubidium vapor of ≤80°C. Therefore, the proposed scheme can be used to fabricate high-sensitivity compact sensors of the magnetic field with a low heat release. Furthermore, our scheme does not involve the microwave or RF field, as in some other types of magnetometers (see, e.g., review [30]). This facilitates the removal of the cross-talk effect of several nearby sensors. Finally, since the suppressed spin-exchange relaxation regime [9], which limits the measurable magnetic field (≤50 nT), is not used in our experiments, the dynamic range of a sensor, which is determined in our case by the width of the magneto-optical resonance, can be much higher (≈0.5–1 μT).