1. Introduction

As the most well-known magneto-optical effect, Faraday effect has been attracting a huge amount of research and development about its formation mechanism [1–7] and application [8–10]. It is a bridge that connects light polarization to the magnetic field and has extensive applications such as in the measurement of magnetic field [10–12] and noninvasive glucose concentration [13–18], as well as in the sensing of optoelectric current [19]. In particular, the Faraday modulator could manipulate the light polarization by modulating the magnitude of the magnetic field applied on the magneto-optical material, possessing great application prospects [20–24].

In the Faraday modulator, an important quantity is the magneto-optical rotation coefficient $V$, with a unit in rad (or degree)$\cdot$T$^{-1}\cdot$m$^{-1}$. The magneto-optical rotation coefficient represents the rotation capability of the magneto-optical media under a unit magnetic field and unit length. It is linked with the polarization rotation angle $(\theta )$ via the relation $\theta = BLV$, where $B$ is the magnetic field and $L$ is the length of the magneto-optical media.

Commercially available Faraday rotation media comprise various doped garnet single crystals, ceramic materials, and rare-earth-element-doped glass materials, such as Yttrium-iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG) [25,26], terbium gallium garnet (Tb$_3$Ga$_5$O$_{12}$, TGG) [25,27–30], terbium aluminum garnet (Tb$_3$Al$_5$O$_{12}$, TAG) [28,29], terbium scandium aluminum garnet (Tb$_{3-x}$Sc$_{x+y}$Al$_{5-y}$O$_{12}$, TSAG) [28,31] and diamagnetic ion glasses [32]. These materials exhibit magneto-optical rotation coefficients typically ranging from $10^1$ to $10^3\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ [10,25–33].

In addition to those materials, numerous new types of materials, including 1,3,5-tris[4-nitrophenylethynyl]-2,4,6-tris(n-decyloxy)-benzene (TTB) [33], Dy$^{3+}$-doped magnetite and cobalt ferrite nanoparticles [10], have been studied. The pursuit of achieving larger rotation angles, reducing the required magnetic field strength, or compacting the system [10,32] has made research on magneto-optical materials with high magneto-optical rotation coefficients a focal point of attention.

In the last century, Kastler had the idea of building up large differences of population between Zeeman sublevels or between hyperfine structure levels via exciting atoms with polarized light [34–36] and this method is widely known as optical pumping. Following this method and based on Dehmelt’s suggestion [37,38], Bell and Bloom successfully utilized optical pumping in detecting magnetic resonance in alkali metal vapor, constructing the prototype of the nonlinear magneto-optical rotation (NMOR) magnetometer [39,40]. In the past decades, NMOR magnetometers utilizing alkali metal atoms have undergone extensive study due to their highest sensitivity among all kinds of magnetometers and ease of implementation [11,41–47]. When resonant light interacts with alkali atoms in a magnetic field and induces $\Delta m=\pm 1$ transitions, the asymmetry in the excitation process between left- and right-circular components of the light occurs. This asymmetry leads to different dielectric constants experienced by the two components of light within the atomic media and thereby a modified polarization of the overall light field. The scale of the polarization rotation reflects the strength of the magnetic field. In the field of precision measurement of the magnetic field, researchers focus more on the sensitivity–which could achieve fT/$\sqrt {\text {Hz}}$ [48,49]–of the method, however, as discussed before, it is evident that if a material can exhibit a significant rotation angle under a weak magnetic field and short working lengths, it must possess a relatively large V. Fortunately, all these requirements can be achieved in NMOR process and it strongly suggests besides working as a detecting media, alkali metal atoms could also be compelling candidates for materials with high V values and be helpful in constructing Faraday modulator. On the other hand, traditional Faraday modulator works on a macroscale, but aiming for ultra-broad analog bandwidths, high speed, and low power consumption, photonic integrated circuits (PIC) designed on a microscale are the goal treated as a trend that holds great promise [50–53]. However, some facts like the existence of a magnetic field and lattice mismatching are obstructing the Faraday modulator from PIC. Utilizing Faraday rotation media requires working in a magnetic field which leads to the side effects on electronic circuits and unwanted material magnetization [54–56]. Additionally, traditional Faraday modulators employ various solid-state materials as magneto-optical rotation media. In contrast, PICs commonly use Si as the substrate. The lattice mismatch between those magneto-optical rotation materials and the Si substrate poses challenges to growing magneto-optical crystals on silicon-based materials, hindering the integration of magneto-optical materials in PIC devices [55,57–59]. Presently, on-chip optical polarization modulators induce changes in the refractive indices by altering the waveguide’s electrical characteristics, enabling the manipulation of polarization. As circuit dimensions shrink and electronic component density surges, fragility about electrical, magnetic, and thermal crosstalk arises. They usually originate from the presence of coupling capacitance, coupling inductance, as well as the thermal impact of electronic components [50,53,60–62], thereby impairing the operational efficacy of on-chip modulators. Consequently, all those existing problems in Faraday modulators and on-chip modulators indicate that the pursuit of all-optical polarization modulators that are magnetic-free, easy to produce, and without electric crosstalk becomes essential in the development of on-chip integrated polarization modulators.

It is noteworthy that alkali metal atoms can function as polarization rotation media not only in magnetic field environments but also as all-optical polarization modulators in the absence of a magnetic field. By exposing an ensemble of $^{87}$Rb atoms to a pump light field with specific polarization states, the spin distribution in the F=1 and F=2 ground states of the atoms can be modified, achieving spin polarization. Compared with the optical rotation process in NMOR, the pump light plays a role similar to that of the magnetic field, modulating atoms to exhibit differential absorption and refraction capabilities for the left- and right-circularly polarized components of the light, thereby resulting in polarization rotation. By adjusting the polarization state of the pump light, the polarization rotation angle of the light passing through the $^{87}$Rb atomic media can be modulated, achieving cross-polarization modulation of the two light fields.

In our experiments, we employ $^{87}$Rb atoms as optical rotation media and successfully realize polarization modulation in magnetic and magnetic-free environments. Within the magnetic field, a magneto-optical rotation coefficient of $1.74\times 10^{8}\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ is achieved through the utilization of the NMOR mechanism, which reveals the remarkable optical rotation capabilities of Rb atoms as magneto-optical media, providing fresh insights for achieving more compact and powerful magneto-optical modulators. In parallel, in magnetic-free conditions, we successfully achieve the cross-polarization modulation between the pump and probe light via Rb atoms and it unveils new opportunities for creating even more miniaturized and easier-to-product polarization modulators on PIC devices, along with novel techniques for all-optical polarization modulation.

2. Experimental setup

Our experimental setup is illustrated in Fig. 1. The cylindrical atomic cell with a diameter of 1 cm and a length of 1 cm is placed within a five-layer magnetic shield, containing isotopically enriched $^{87}$Rb atoms. A set of Helmholtz coils is positioned in proximity to the cell to generate a magnetic field along the z-direction which is chosen as the quantum axis in this research. The experiment employs three laser beams: pump, repump, and probe. The polarization of these lasers is adjusted along the y-axis after traversing a polarizer respectively. Then the linearly polarized pump and repump lasers are combined by a beam splitter (BS1 in Fig. 1(a)) and are subsequently transformed into left-circularly polarized using a quarter-wave plate. They are then combined with the probe laser by another beam splitter (BS2 in Fig. 1(a)) before entering the Rb cell. Ultimately, all three laser beams propagate along the z-direction, transmitting through the Rb cell.

 

Fig. 1. (a) Experimental Setup. POL: polarizer, BS: beam splitter, PBS: polarizer beam splitter, BPD: balanced photo-detector, $\lambda /2$: half-wave plate, $\lambda /4$: quarter-wave plate. (b) Detailed energy and population diagram of the system in the all-optical experiment. Atoms are polarized in $|5S_{1/2}, F=2, m_F=2\rangle$ state. Atoms can only interact with the $\sigma ^-$ component of the probe light because no state is coupled with $|5S_{1/2}, F=2,m_F=2\rangle$ state by the $\sigma ^+$ component.

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Concerning laser frequencies, the pump laser is resonant with the $^{87}$Rb D1 line $F=2$ to $F'=2$ transition, polarizing the atoms to the $|5S_{1/2}, F=2,m_F=2\rangle$ state. The repump laser is resonant with the $^{87}$Rb D1 line $F=1$ to $F'=2$ transition, exciting atoms from the $F=1$ ground state to enhance atoms’ spin polarization. The probe laser is 100 MHz blue detuned from the $^{87}$Rb D2 $F=2$ to $F'=2$ transition. A band-pass filter with a center wavelength of 780 nm is positioned behind the Rb cell. It is used to selectively filter out the (re)pump light, permitting only the probe light to reach the balanced photo-detector which measures the polarization rotation angle.

3. Theory model

To get the polarization rotation angle of the probe light, we elucidate it using the Maxwell equation:

Here, $k$ represents the wave vector of the light, $E$ signifies the electric field of the light, and $P$ denotes the atomic polarization induced by the electric field of the light. E is assumed to have the form of $E(z,t)=1/2\{A e^{i(zk-\omega t+\phi )} [(\text {cos}\alpha \cdot \text {cos}\epsilon -i \text {sin}\alpha \cdot \text {sin}\epsilon )\vec {e_x}+(\text {sin}\alpha \cdot \text {cos}\epsilon +i \text {cos}\alpha \cdot \text {sin}\epsilon ) \vec {e_y}]+c.c\}$ in which $A$ is the amplitude, $\alpha$ is the polarization angle with respect to the x axis, $\epsilon$ is the ellipticity. $P$ is assumed to follow the form of $P(z,t)=1/2\{ e^{i(zk-\omega t+\phi )} [(P_1-i P_2 )\vec {e_x}+(P_3-i P_4)\vec {e_y}]+c.c\}$ where the $P_i$ are the in-phase and quadrature components of the polarization. We applied the slow varying approximation to the equations, where second-order derivatives of parameters such as $d^2A/dz^2$, $d^2\phi /dz^2$, $d^2\alpha /dz^2$ and $d^2\epsilon /dz^2$ in the left side of the equation are neglected. Consequently, Eq. (1) simplifies into a set of first-order differential equations, with our focus being on the changes in polarization angle $d\alpha /dz=2\pi \omega /(Ac) P_4$[63]. The components of the polarization vector on the right side of the equation are determined by the atomic state through $P= n \textrm{Tr}(\hat {\rho }\hat {\text {d}})$ in which $n$ is the atom density, $\hat {d}$ is the dipole moment operator, and $\hat {\rho }$ is the density operator obtained by solving the atomic master equations

$\hat {H}$ is the Hamiltonian under rotating-wave approximation consisting of three components: $\hat {H}=\hat {H}_0+\hat {H}_E+\hat {H}_B$. $\hat {H}_0$ is the Hamiltonian of the atom, $\hat {H}_E$ is the Hamiltonian for the interaction between atoms and the optical field and $\hat {H}_B$ represents the interaction between atoms and the magnetic field. In particular, the block matrix form of the Hamiltonian can be expressed as

The subscript $g_1,g_2,e_1,e_2$ correspond to the states $|5S_{1/2}, F=1\rangle$, $|5S_{1/2}, F=2\rangle$, $|5P_{1/2}, F=2\rangle$, $|5P_{3/2}, F=2\rangle$ respectively. The diagonal terms are composed of ${H}_B$ and ${H}_0$: $H_{ii}=H_{B,i}+H_{0,i}$, here $i$ is $g_1,g_2,e_1,e_2$, $H_{0,g_{1}}=0$, $H_{0,g_{2}}=(-\Delta _1 + \Delta _2)\cdot \text {I}_5$, $H_{0,e_{1}}=-\Delta _1 \cdot \text {I}_5$, $H_{0,e_{2}}=(-\Delta _1 + \Delta _2 - \Delta_3)\cdot \text {I}_5$. $\Delta _1$, $\Delta _2$, $\Delta _3$ are detuning of repump, pump and probe light. $\text {I}_5$ is the 5-dimension identity matrix.The exact matrix form of $H_{B,i}$ are:

About the off-diagonal terms, they reflect the light-atom interactions $\hat {H}_E$. Considering that the pump beam couples the F=2 ground state with the $|5P_{1/2},F=2\rangle$ state, the repump beam couples the F=1 ground state with the $|5P_{1/2},F=2\rangle$ state, and the probe beam couples the F=2 ground state with the $|5P_{3/2},F=2\rangle$ state, only $H_{g_{1}e_{1}}$, $H_{g_{2}e_{1}}$ and $H_{g_{2}e_{2}}$ exist. In details, matrix form of these terms can be written as following:

In the atomic master Eq. (2), $\hat {\rho }$ is the density matrix. $\hat {\Gamma }$ and $\hat {\Lambda }$ are relaxation and repopulation term respectively. Atoms in excited states undergo spontaneous decay by rate $\Gamma$ and atoms in each state undergo decay by rate $\gamma$. The repopulation matrix ensures that the atomic density is normalized to unity. The master equations are a set of first-order differential equations with respect to time. In this theoretical modeling, 18 energy levels are included, so the master equations constitute a system of 324 coupled first-order differential equations. We performed their numerical solutions under steady-state conditions.

Solving the master equations provides us with the atomic state, which in turn allows us to determine the atom polarization ($P$). Subsequently, this enables the calculation of the polarization rotation angle $\alpha$ of the probe light. Besides, the Doppler effect can also have an impact on our final measurement results. All three involved lights propagate in the same direction, so we only need to consider the Doppler effects in one dimension. It causes atoms at different velocities to experience varying light frequencies, and what we detect is the weighted average of all atoms. The velocity distribution of atoms with mass $m$ under the temperature $T$ satisfies the Boltzmann Distribution $f(v)=\sqrt {v/2\pi mkT} exp(-mv^{2}/2kT)$, and the light frequency atoms experienced is $\omega (v)^{'}=\omega -k\cdot v$. Treating the numerical result $\alpha$ gotten from our model as the function of $\omega _{probe}^{'} (v),\omega _{pump}^{'} (v),\omega _{repump}^{'} (v)$, and using $f(v)$ as the weight function, we can get the final result $\alpha _{final}=\int f(v) \alpha [\omega _{probe}^{'} (v),\omega _{pump}^{'} (v),\omega _{repump}^{'} (v)] dv$.

This model can be used to describe both magnetic and all-optical models in our experiment, as long as reasonable values are assigned to $\Omega _R,\Omega _{R0},\Omega _{Rp},\Omega _L$ in specific simulation calculation.

4. Experimental results

Our experimental result consists of two parts. In the first part, we attain a substantial magneto-optical rotation coefficient for Rb atoms using the NMOR effect. This occurs in the presence of a magnetic field, with only a probe light utilized. In the second part, we accomplish all-optical polarization modulation in the absence of a magnetic field, incorporating both pump, repump, and probe light.

4.1 Magneto-optical rotation coefficient in Rb atoms

Figures 2(a) and (b) illustrate the polarization rotation angle obtained during the experiment, resulting from the NMOR mechanism. In Fig. 2(a), the prominent dispersion-like structure with a wide magnetic field distribution represents the rotation angle induced by the linear Faraday effect. Notably, there is a distinct and extremely sharp "peak" near the zero-field region, providing evidence of the presence of NMOR effects. To provide a more detailed depiction, an enlarged view of the yellow area in Fig. 2(a) is presented in Fig. 2(b). As shown in Fig. 2(b), under the NMOR effect, a sharp change in the rotation signal occurs near zero magnetic field. Lines of various colors represent results at temperatures ranging from 26$^{\circ }$C to 38$^{\circ }$C, with a fixed interval of 2$^{\circ }$C. As the temperature increases, the atom density within the cell rises and it enhances the atom-light interaction, resulting in a larger rotation angle. Within the range of approximately 4nT, the highest magneto-optical rotation coefficient was experimentally determined to be $1.74\times 10^8\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ under linear approximation at $38^{\circ }$C for a 1 cm-long Rb atomic cell. The magneto-optical rotation coefficient produced by this process represents the current state of the art, in comparison, the magneto-optical rotation coefficients of many known materials are typically within the range of $10^1\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ to $10^3\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ [10,25–33].

 

Fig. 2. (a) Polarization rotation signal obtained at $38^{\circ }$C. The wide-spread structure depicted in the blue background originates from the linear Faraday effect. The narrow and sharp structure depicted in the yellow background is what we utilize based on the NMOR effect. (b) Detailed structure within the yellow area in (a) but under various temperatures. (c) Illustration about linear Faraday effect. To have a simplifier expression, here we use a model in which the ground state is represented by $F$=1 and the excited state is $F$=0. Energy shift originating from Zeeman Effect causes different detuning with respect to $\sigma ^+$ and $\sigma ^{-}$ component of the probe light. (d) Illustration about NMOR effect. When the exciting rate caused by the light can not be neglected, the atoms’ state will be modified, causing coherence between the two ground states. The curve arrows in both (c) and (d) represent the decay process.

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Figure 2(c) elucidates the process behind the generation of the Faraday effect due to the interaction between light and atoms under conventional linear conditions.This process happens at the relatively large magnetic field where the probe light is far from resonant due to the manifest Zeeman splitting. In this scenario, probe light’s excitation effect on ground-state Rb atoms can be negligibly small. In the presence of an external magnetic field, the Zeeman effect causes shifts in energy levels, resulting in different detunings for the left-circularly and right-circularly components of the probe light. Consequently, the Rb atomic media exhibits different dielectric constants for these two components, leading to circular birefringence and polarization rotation of the light after exiting the Rb atomic cell. Different magnetic field strengths yield varying detunings for the left and right-circularly polarized components, resulting in the dispersion-like curve seen in the large structure in Fig. 2(a).

In contrast to the linear Faraday effect, the distinctive feature of the NMOR effect is that the population distribution of atoms in different energy levels is strongly influenced by the light. When the magnetic field is relatively weak, the optical pumping effect on the atomic system becomes more significant, resulting in the emergence of NMOR. Figure 2(d) illustrates the process of the NMOR effect.

4.2 All-optical polarization modulation

In the previously demonstrated process, the polarization modulation process is intricate and not intuitively straightforward due to the self-modification of the probe light. Our new experimental configuration introduces a circularly polarized pump light, which solely serves to adjust the atomic state. Simultaneously, the probe light is exclusively employed for detection. Remarkably, under such a configuration, the atomic system acquires the capability to exhibit optical rotation in an all-optical environment.

In an environment without an external magnetic field, although the five magnetic sublevels of $F=2$ are degenerate, we excited Rb atoms to the $|5P_{1/2}, F=2\rangle$ excited state using left-circularly polarized pump light. Under the influence of this light, the atoms undergo $m\rightarrow m'=m+1$ transitions. After that, atoms in the excited state undergo relaxation due to spin-exchange processes and optical power broadening, returning to the ground state. Because the $|5S_{1/2}, F=2, m_F=2\rangle$ state can only be the relaxation destination, Rb atoms are polarized in this state. What’s worth mentioning is, in the relaxation process, not only falling into the $|5S_{1/2}, F=2\rangle$ ground state, atoms may also enter the $|5S_{1/2}, F=1\rangle$ ground state. Atoms that enter the $F=1$ ground state are no longer driven by the pump light. Therefore, in addition to the pump light, we apply a repump light that connects the $F=$1 ground state to the $|5P_{1/2}, F=2\rangle$ excited state, to enhance the atoms’ spin polarization. Similar to the measurement of the NMOR effect, a linearly polarized light which couples the transition between the $|5S_{1/2}, F=2\rangle$ state and $|5P_{3/2}, F=2\rangle$ state works as the probe light. However, its power is much smaller than the power of the pump and repump light, and its influence on the atomic state could be neglected. Overall, among these laser beams, the polarization and power of the pump light primarily regulate the atomic state, with the repump light serving as an auxiliary component. By altering the polarization and power of the pump light, the polarization of the probe light is correspondingly changed, enabling the all-optical polarization modulation in a magnetic-free environment.

Figure 3 depicts the behavior of the detected signal from the balanced photo-detector as a function of the pump light power. The primary function of the pump light is to excite ground-state atoms. As the pump light power gradually increases, the pumping rate of atoms from the ground state to the excited state becomes less dominant. Therefore, in Fig. 3, it can be observed that the rotation signal rapidly increases at lower pump light power, and reaches saturation as the power is further raised. The red dots in Fig. 3 represent experimental data, while the blue curve corresponds to the result of theoretical simulations. The consistency in the trends between the two results confirms the accuracy of our model.

 

Fig. 3. Rotation signal changed with Rabi frequency of the pump light, which is experimentally related to the light’s power. The pumping rate increases along with the pump power at the beginning, causing the rotation signal to increase. After the pumping rate reaches saturation, the rotation signal will not change. In this case, the power of the repump and probe light are 3 mW and 50 $\mu$W, respectively.

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Figure 4 illustrates the behavior of the detected signal as a function of the repump light power. Similar to the dependence on pump light, as repump light power gradually increases, its effect on the polarization of the probe light also exhibits a saturation behavior. Unlike the behavior regarding the dependence on the pump light, even when the repump power approaches zero, the rotation is relatively more manifest. This is because the primary function of repump light is to prevent a significant accumulation of atoms in the $F=1$ ground state. Even without repump light, relying solely on pump light can still polarize the atomic system. This also implies that the power threshold required for repump light to reach saturation is much higher than that for pump light. In our experiment, it can be observed that the saturation thresholds for pump and repump light are $500\mu$W and 3mW, respectively.

 

Fig. 4. Rotation signal changed with repump light’s Rabi frequency. The repump light is used to enhance atoms’ spin polarization, so even when there’s no repump light, the rotation would not vanish. In this case, the power of the pump is 500$\mu$W and that of the probe light is 50$\mu$W.

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5. Conclusion

In conclusion, using $^{87}$Rb atoms as the rotating media, we achieve the state-of-the-art magneto-optical rotation coefficient at $1.74\times 10^8\ \text {rad}\cdot$T$^{-1}\cdot$m$^{-1}$ in a weak magnetic field via NMOR effect, and in a magnetic-free environment, the all-optical cross-polarization modulation via $^{87}$Rb atoms is also successfully realized. We notice that the bandwidth of our modulator is related to the time required for atoms to reach a steady state. Emerging from the atom-light interaction, these properties are better suited for utilization close to the resonance of alkali atoms, which enhances their applicability in atom-related applications, such as atomic magnetometers [64]. The outstanding magneto-optical rotation coefficient reveals the remarkable optical rotation capabilities of Rb atoms as a magneto-optical media, providing fresh insights for achieving more compact and powerful magneto-optical modulators. With respect to the all-optical configuration, it introduces inventive techniques free from any disruptions caused by electrical or magnetic interference, paving the way for exploring innovative possibilities in achieving even smaller and more condensed PIC devices.