Quantum mode-locked Faraday laser

Over the past few decades, mode-locked lasers have been extensively utilized in frequency comb generation,¹ optical communication,² precision metrology,³ and material processing,⁴ to name just a few. Previous research on mode-locked lasers mainly focuses on generating ultra-short pulses with certain broad optical spectra and many comb lines. Following the recent development of saturated absorption spectroscopy (SAS)⁵ and velocity-grating atomic spectroscopy,^6,7 applying dual-frequency or multi-frequency lasers to interact with atoms has proved to greatly enhance the amplitude and signal-to-noise ratio (SNR) of the spectral signal. In both applications, if conventional broad-spectrum mode-locked lasers are used, the laser power of comb lines that fall outside the atomic resonance bandwidth is not only redundant but can introduce frequency shifts of atomic transitions, ultimately reducing the SNR of the spectral signal. In addition, the frequency spacing between laser fields also has a significant impact on the amplitude of the spectral signal.^5,7 Consequently, there is a pressing need for a narrow-spectrum mode-locked laser with a tunable repetition rate and a central wavelength corresponding to the quantum transition line.

The saturable absorber (SA) is crucial for achieving mode-locking. SAs are categorized as physical [e.g., the semiconductor saturable absorption mirror (SESAM),⁸ carbon-nanotube (CNT),⁹ graphene,¹⁰ and black phosphorus (BP)¹¹] and artificial [e.g., the nonlinear-optical loop mirror (NOLM),¹² the nonlinear amplifying loop mirror (NALM),¹³ the nonlinear polarization rotation (NPR),¹⁴ and the single mode-graded index multimode-single mode fiber (SMF-GIMF-SMF) device].¹⁵ The spectral widths of mode-locked pulses, generated by the aforementioned SAs, are consistently beyond the nanometer scale. To achieve pulses with a spectral width at the gigahertz order or less, scholars proposed the method of inserting a narrowband filter into a mode-locked laser resonator, for example, ultra-narrow band fiber Bragg grating (FBG),^16,17 the commercial tunable bandpass filter (TBF),¹⁸ and saturable dynamic induced grating (SDIG).¹⁹ However, the output pulse wavelengths of these devices are around the 1550 nm band, which do not correspond directly to atomic quantum transition lines. More importantly, these narrow-bandwidth mode-locking setups require the joint utilization of a discrete SA and narrowband filter, increasing the complexity of the laser system.

The Faraday laser is an external cavity semiconductor laser that incorporates a Faraday atomic filter within its cavity.^20–23 The Faraday atomic filter comprises two orthogonal polarized devices and a vapor cell subjected to a longitudinal magnetic field.^24,25 It employs circular birefringence to induce the Faraday rotation of light at different wavelengths, therefore forming a narrowband filter suitable for laser frequency selection. In addition, due to the significant Kerr nonlinear refractive index (⁠ $n2$⁠) of alkali metal vapor under intense light incidence,^26,27 the transmittance of atomic filters are intensity dependent,^28,29 similar to SAs. This SA, unlike those previously reported, exhibits nonlinear transmission behavior solely toward light in proximity to the atomic quantum transition line, thereby offering the dual functionalities of saturable absorption and narrowband filtering. However, previous reports on Faraday lasers primarily concentrate on single-^20,21 or dual-frequency^22,23 operations utilizing the wavelength-dependent loss of atomic filters, while studies on the mode-locking operation are lacking. In addition, the significant dispersion present in off-resonant atomic systems can introduce slow light effects during the propagation of optical pulses, resulting in pulse delays^30,31 and distortions,³² which affect the repetition rate and pulse profile of the mode-locking operation. Hence, it is imperative to investigate the repetition-rate characteristics of a mode-locked Faraday laser that incorporates a slow-light medium within its cavity.

In this Letter, we present a narrow-bandwidth quantum mode-locked Faraday laser using a 5-Torr-Ar-mixed ⁸⁵Rb atomic filter as the SA. This unique SA is characterized by its transmission center wavelength corresponding to the ⁸⁷Rb $(F=2)$ component of the $D2$ quantum transition line, a transmission bandwidth of gigahertz order, and a modulation depth of approximately 14  $%$⁠, which cause the mode-locking operation to occur solely near the atomic quantum transition frequency. The influence of the Faraday slow-light effect in rubidium-dispersive medium on the mode-locking repetition rate is investigated experimentally. By adjusting the temperature of the atomic vapor cell from 97 to 101 °C, the repetition rate of the mode-locked pulses decreases from 261 to 228 MHz, which is significantly lower than the fundamental frequency (470 MHz) of the physical cavity length. In both scenarios, the full width at half maximum (FWHM) of the pulse spectrum is approximately 1.3 and 1.85 GHz, respectively. Furthermore, pulse delay tests show that the fractional delays³⁰ (the ratio of the delay to the pulse duration) of both 261 and 228-MHz pulses exceed 3, but the profile distortion of the 261-MHz pulse is smaller than that of the 228-MHz pulse. This study provides valuable insights into the dynamic behavior of quantum mode-locked Faraday lasers.

The schematic of the experimental setup is shown in Fig. 1, which includes two components, namely, a quantum mode-locked Faraday laser and a pulse delay test system. The quantum mode-locked Faraday laser contains an anti-reflection coated LD (ARLD), a collimating lens (CL), an atomic filter-based SA, and a high reflective mirror (HRM). The atomic filter-based SA is formed by a 29-mm-long ⁸⁵Rb cell mixed with 5 Torr buffer gas Ar (cell_1) and two Yttrium vanadate (YVO₄) beam displacers^33,34 with optical axes orthogonal to each other and principal sections parallel to each other. Cell_1 is surrounded by eight permanent magnets (not shown) to create a 530-G axial magnetic field and is temperature stabilized to achieve the desired atomic density. The light emitted by the ARLD is collimated by a CL with a focal length f of 3.1 mm and a numerical aperture (NA) of 0.55. The collimated light is subsequently incident on YVO₄1 and only one beam of light (⁠ $I0$⁠) emanates from the crystal. After cell_1, the polarization direction rotates, as depicted in the inset of Fig. 1, resulting in two beams with different polarization (⁠ $I⊥$ and $I∥$⁠) exiting YVO₄2. The $I∥$ light is reflected by HRM with a reflectivity of 99  $%$ and dimensions of 8 × 8 × 1 mm³, while the $I⊥$ light serves as the output. The mode-locked pulsed light, after a 30-dB isolator, is split into two components by combining a half-wave plate (HWP) and a polarization beam splitter (PBS). One component is used for the measurements of the mode-locking characteristics and the other for pulse delay tests. The pulses propagate in an atomic cell (cell_2), configured identically to cell_1, and the transmitted pulses are detected using a fast photodiode and recorded on an oscilloscope.

Figure 2(a) shows the filter transmission (⁠ $I∥/I0$⁠) as a function of frequency detuning at the cell_1 temperature of 97 °C (blue curve) and 101 °C (pink red curve) for an incident power of 1 mW. A saturated absorption spectroscopy (SAS) of the natural Rb cell (orange curve) is also plotted for reference in Fig. 2(a). The transmission spectra at 97 and 101 °C both exhibit a double-peak structure, with the highest transmission peak corresponding to the ⁸⁷Rb (F = 2) component of the $D2$ quantum transition line. As the temperature increases, the left (P_L) and right (P_R) peaks shift outwardly, accompanied by an increase in transmission and bandwidth. At 97 °C, the transmission of P_L and P_R is 46  $%$ and 42  $%$⁠, respectively. When the P_L transmission decreases to 6  $%$⁠, the corresponding transmitted bandwidth of P_L is about 3.2 GHz. At 101 °C, the P_L and P_R transmission are 50  $%$ and 47  $%$⁠, respectively, and the corresponding transmitted bandwidth of P_L is about 4 GHz. We further investigate the nonlinear transmission of the filter for the cell_1 temperature of 97 °C, as plotted in Fig. 2(b). When the intensity exceeds 7 mW/mm², the P_L transmission increases with increasing light intensity, with a modulated depth of approximately 14  $%$⁠, while P_R is slightly elevated and significantly lower than P_R.

The stable mode-locking operation is observed within the LD current range of 118–123 mA at the cell_l temperatures of 97 and 101 °C (Fig. 4). At these two temperatures, the output pulse intervals are approximately 3.8 and 4.38 ns, as illustrated in Figs. 4(a) and 4(d), corresponding to optical cavity lengths of 577 and 658 mm, respectively. This difference in pulse periods is attributed to the increased vapor temperature, which enhances the group refractive index.³¹ The insets in Figs. 4(a) and 4(d) show pulse trains exhibiting stable intensity within a 10  $μ$s time span. Figures 4(b) and 4(e) present the output longitudinal-mode spectra measured by a Fabry–Pérot interferometer (FPI) with a free spectral range (FSR) of 10 GHz at the cell_l temperatures of 97 and 101 °C, respectively. The results clearly show that there are approximately 8 and 12 longitudinal modes in each pulse spectrum at these two temperatures, respectively. Through nonlinear fitting, the FWHM values of the mode-locked pulse spectrum in these two scenarios are approximately 1.3 and 1.85 GHz, respectively. Notably, the broadening of the mode-locked pulse spectrum aligns with the transmission characteristics of the filter, as shown in Fig. 2(a). In addition, Figs. 4(c) and 4(f) depict the corresponding radio frequency (RF) spectra within a 6 GHz frequency span with a 51-kHz resolution bandwidth (RBW) for these two conditions. In both RF spectra, the frequencies of the first comb lines are 261 and 228 MHz, respectively, accompanied by SNRs of 43 and 50 dBm, respectively. Further tests were conducted on the first comb lines in these two scenarios. At the RBW of 300 Hz, the FWHM values of the first comb lines are approximately 560 and 490 Hz by Lorentz fitting on the experimental data, as shown in the insets of Figs. 4(c) and 4(f), respectively.

To further demonstrate the effect of the slow light effect in the atomic vapor on the mode-locked repetition rate, we perform pulse delay tests. The 260-MHz pulse, generated at a cell_1 temperature of 97 °C experiences optical delays of 0.68 and 1 ns when propagated through cell_2 at temperatures of 96 and 98 °C, respectively, as shown in Fig. 5(a). Both delay values are located with either sides of the calculated intracavity delay of 0.857 ns at 97 °C in Fig. 3. In Fig. 5(a), the pulse shape essentially remains unchanged, yet due to relative group refractive index dispersion, the pulse broadens. Similarly, the 228-MHz pulse, generated at a cell_1 temperature of 101 °C, experiences delays of 0.59 and 1.1 ns when propagating through cell_2 at 98 and 101 °C, respectively, as shown in Fig. 5(b). The measured value of 1.1 ns at 98 °C is essentially equal to the calculated value of 1.11 ns in Fig. 3. The durations of the reference pulses in Figs. 5(a) and 5(b), measured by an oscilloscope, are approximately 250 and 240 ps, respectively. This indicates that all the fractional delays (ratio of the delay to the pulse duration) of the pulses in the resonator exceed 3. However, a significant distortion is observed in the 228-MHz pulse at the cell_2 temperature of 101 °C, with the main pulse commencing to split. This distortion is attributed to the increasing difference in the relative group refractive index between left- and right-handed rotations as the temperature rises, leading to the spatial separation of pulses.³² This phenomenon also accounts for the presence of relatively strong sub-pulses in the 228-MHz pulses emitted by the Faraday laser. Simultaneously, Figs. 5(a) and 5(b) reveal that when the cell_2 temperature is set at 98 °C, the delay of the 261-MHz pulse slightly exceeds that of the 228-MHz pulse. This discrepancy arises because the output center wavelength (780.2468 nm) of the 261-MHz pulse is closer to the resonance absorption peak of ⁸⁵Rb $52$ $S1/2F$ = 3  $→52$P_3/2 (780.244 nm) than that (780.2477 nm) of the 228-MHz pulse. Given that the temperature of the slow light medium remains constant, a smaller frequency detuning corresponds to a higher group refractive index.

In conclusion, we present a quantum mode-locked Faraday laser based on the saturable absorption and narrowband filtering of the 5-Torr-Ar-mixed ⁸⁵Rb atomic filter, with its output spectrum solely covering the ⁸⁷Rb (F = 2) component of the $D2$ quantum transition line. We experimentally investigate the impact of the slow light effect on the repetition rate of the mode-locking operation. The results reveal that the repetition rate of the mode-locked Faraday laser is strongly influenced by the atomic vapor temperature, and the spectral width of the output pulses is constrained by the transmission spectrum of the atomic filter. Compared with other narrowband mode-locking techniques using a discrete SA and narrowband filter, the Faraday-atomic-filter-based mode-locking technique not only reduces the number of intracavity elements but also outputs narrow spectral band pulses with a central wavelength corresponding to the quantum transition.

In the future, we will attach a piezoelectric transducer (PZT) to the cavity mirror to adjust the cavity length and tune the output wavelength. When the multi-frequency coherent laser emitted by the Faraday laser interacts with rubidium atoms,^5–7 the presence of many longitudinal modes (⁠ $≥$8) near the ⁸⁷Rb (F = 2) component of the $D2$ quantum transition line together with the velocity-selective resonance effect results in an increase in spectral signal amplitude. Compared with the previously reported multi-frequency lasers obtained using many electro-optic modulators (EOMs)^6,7 or an acousto-optic modulator (AOM)⁵ modulation, the scheme of this laser is simpler and more cost-effective. In addition, we plan to integrate all optical components into a sealed metal box to further improve the stability of the locking mode and to increase transportability. The mode-locked laser presented here has potential applications in the fields of atomic multi-frequency saturated-absorption spectroscopy, velocity-grating atomic spectroscopy, power amplification, and quantum precision metrology.

The work was supported by the China Postdoctoral Science Foundation (No. BX2021020), the Wenzhou Major Science and Technology Innovation Key Project (No. ZG2020046), and the Innovation Program for Quantum Science and Technology (No. 2021ZD0303200).