# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 4 Results and Discussions  
The response of the OPM to the magnetic-field modulation shown in Equation (18) is composed of the contributions from eight Lorentz peaks with different detuning \(\delta\omega_{\alpha\;\alpha-1}\). We firstly examine the influence of the modulation detuning on the response from a single Lorentz peak. To do so, we set the biasing magnetic field to be around 1.15 \(\upmu\)T, where the NLZ effect is negligible and the magnetic resonance can be viewed as a single Lorentz peak, and then measure the response of the OPM to magnetic-field modulations when the detuning \(\delta\omega=\gamma B_{0}-\omega_{\text{M}}\) is set to a different value. Figure 2 compares the experimental and simulated frequency responses of the Bell-Bloom cesium OPM at 1.15 \(\upmu\)T with different detunings. To get the simulated frequency responses, we scan the magnetic resonance signal and fit it with a single Lorentz peak for deriving its amplitude and linewidth (about 2 Hz). Subsequently, we use these fitting parameters to predict the OPM's response to magnetic-field modulation with the help of Equation (18). The figures in the left (right) column in Figure 2 indicate the amplitude (phase)-frequency responses. From the top to the bottom, the modulation detuning increases. All these curves are normalized by OPM's response to magnetic-field modulation at the low-frequency limit when the detuning \(\delta\omega\) equals 0. It is clear for a detuning frequency of 0, the frequency response is a Butterworth low-pass filter and the magnetometer has hence a larger response to the magnetic-field signals with lower frequencies. When the detuning \(\delta\omega\) increases, the overall response of the magnetometer decreases. But the range for which the magnetometer has the largest response changes towards higher frequencies. We also check our model when the NLZ effect is non-negligible in the Bell-Bloom OPM operated in the Earth's magnetic field. The magnetic resonances and frequency responses of the OPM with biasing field \(B_{0}\) of 50 \(\upmu\)T and 70 \(\upmu\)T are shown in Figure 3. We firstly fit the magnetic resonances with Equation (17) to get the central frequency, the amplitude, and the linewidth of each Lorentz curve. The fitted linewidth and amplitude of each Lorentz curve are listed in Table 1 (from lower central frequency ones to higher central frequency ones). The fitted linewidth with a biasing field of 70 \(\upmu\)T is larger than that with a biasing field of 50 \(\upmu\)T and both of them are larger than the 2 Hz linewidth with 1.15 \(\upmu\)T biasing field. The broadening of each Lorentz curve at a larger biasing field is mainly due to the increased magnetic gradient. The linewidth of a single Lorentz curve is about 3 times smaller than the experimental measured overall magnetic resonance linewidth for the corresponding biasing field, which is about 20 (40) Hz with a biasing field of 50 (70) \(\upmu\)T (\(\omega_{\text{NLZ}}\) at 50 and 70 \(\upmu\)T biasing field are about 6.7 and 13.1 Hz, respectively). This suggests that the broadening of the overall magnetic resonance is mainly due to the NLZ effect, rather than the magnetic field inhomogeneity. Finally, we use these fitting results to calculate the theoretical frequency responses according to Equation (20) which is then compared with the experimental results. During the experiment, the low-pass filter in the lock-in amplifier for 50 (70) \(\upmu\)T experiment is a second (third)-order Butterworth low-pass filter with a time constant of 1 ms, which is taken into consideration in the calculation of the simulated frequency responses. It is shown that the calculated results coincide with the experimental results for both the amplitude- and the phase-frequency responses. We also calculate the frequency responses according to the first-order Butterworth low-pass filter and present the results in Figure 3 (the yellow dashed lines), which have large deviations from the experimental results in the amplitude-frequency responses.