Atomic optical stimulated amplifier with optical filtering of ultra-narrow bandwidth

Taking advantages of ultra-narrow bandwidth and high noise rejection performance of the Faraday anomalous dispersion optical filter (FADOF), simultaneously with the coherent amplification of atomic stimulated emission, we propose a stimulated amplified Faraday anomalous dispersion optical filter (SAFADOF) at cesium 1470 nm. The SAFADOF is able to significantly amplify very weak laser signals and reject noise in order to obtain clean signals in strong background. We show that for a weak signal of 50 pW, the gain factor can be larger than 25000 (44 dB) within a bandwidth as narrow as 13 MHz. Having the ability to amplify weak signals with low background contribution, the SAFADOF finds outstanding potential applications in weak signal detections.

A 459 nm laser stabilized to the Cs 6S 1/2 (F = 4) − 7P 1/2 (F = 3) transition by the saturated absorption spectroscopy (SAS) pumps the Cs atoms inside a 10 cm-long quartz cell. The pumping laser corresponds to a weak transition whereafter the spontaneous decay of the excited state will occur via multiple intermediate states, and the analogous energy structure has been studied in various systems [34][35][36][37] . After pumping, the Cs atoms are population inverted between 7S 1/2 (F = 4) and 6P 3/2 (F = 5) states 38 . Hence with the function of the 1470 nm probing laser (coincide with the pumping laser), stimulated emission between the two states is generated, and thus the probing laser is significantly amplified. The Cs cell is placed between a pair of orthogonal Glan-Taylor prisms G1 and G2, of which the extinction ratio is 1 × 10 5 and drops to 6 × 10 4 with the vapor cell and the dichroic mirror between them. This also determines the out-of-band noise rejection ratio of the SAFADOF. The ring magnets outside the cell produce an axial magnetic field of about 8 Gauss, where we experimentally get the largest gain. An optical chopper together with a lock-in amplifier are used to eliminate the influences of the fluorescence generated by static superradiance 39,40 . Theory calculation of the gain factor. Considering SA operation, the stimulated emission can enhance the input signal with a factor of G SA . Combined with the function of two crossed Glan-Taylor prisms, the SAFADOF gain is given by: where the rotation angle is given by Here the relaxation rate Γ = 55 MHz, considering the natural broadening as well as the Doppler broadening caused by the saturated pumping 38 . The calculation method is described in refs 38,41 , and the detailed meaning of the parameters in Eq. (2) is given in ref. 42 . For ϕ ≤ π/2, sin 2 ϕ has the maximum value when g F μ B B/ħ = Γ/2, thus we have B ≈ 7.8 G. Experimentally we get the maximum gain at B ≈ 8 G, and the transmitted ratio is closed to 100% at 135 °C, meaning that we get almost the same gain factor with or without the two crossed Glan-Taylor prims. For simplicity, we keep the magnetic field to be optimal and assume: when calculating the gain factor at 135 °C.
For the interaction of a two-level atomic system with a radiation field, the transition probability is given by For the atoms with average lifetime τ, the distribution function of their interaction time with the radiation field is represented in the form again Γ is the relaxation rate considering the spontaneous emission and the Doppler broadening 38 . To match our experimental conditions, considering the length of the Cs cell and the probing laser with waist w 0 , the variation of signal power dP during a length of dL is given by where η = 1/τ cyc represents the pumping rate, with τ cyc being the atomic cycling time. Analogous to ref. 44  with Γ 23 , Γ 24 , Γ 35 , Γ 36 , Γ 51 corresponding to the relaxation rates of transitions 7P 1/2 − 7S 1/2 , 7P 1/2 − 5D 3/2 , 7S 1/2 − 6P 3/2 , 7S 1/2 − 6P 1/2 , and 6P 1/2 − 6S 1/2 , respectively. Some of the energy levels are not displayed in Fig. 1(b), referring to ref. 38 . In our experiment η is calculated to be 3.  , is the saturation intensity, and P 0 is the input probing light power. Considering G = P/P 0 , we have By Eqs (6) and (7) we obtain the theoretical transmitted power as well as the gain factor depending on the probe power at 135 °C as depicted in Fig. 3(d), which will be analyzed in the following.

Results
Superradiance background. Due to the collective behavior of static superradiance 39,40 , population reversed atom ensemble will radiate spontaneously from the 7S 1/2 (F = 4) state to the 6P 3/2 (F = 5) state, which is much faster and stronger than that of individual atoms, and exhibit well defined direction. In our system the 1470 nm static superradiance has been observed experimentally 39 .
The static superradiance light, of which the amplitude varies with the pumping power and temperature, cannot be optically filtered and will contribute to the background noise, as shown in Fig. 2(a). Such influence is eliminated by a synchronous modulation method, where the probing light is pre-modulated by an optical chopper, with a modulation frequency of 1.5 kHz. Then the detected transmission light is demodulated by a lock-in amplifier synchronized to the chopper. So that the transmission signal derived from the probing laser is well separated from the static superradiance and independently detected. Figure 2(b) illustrates the gain spectrum before and after modulation, as well as the demodulated signal in which the background is effectively suppressed. This method is proposed to improve the SNR of the SAFADOF, and is also applicative in other systems such as lamp-based atomic filters 24,47 , where the fluorescence has non-negligible influence.
Gain factor. In the context of weak optical communication, we are interested in obtaining long communication distance and high accuracy, which requires a high transmittance of the filter to reduce the loss, or possibly, a high gain factor. Compared to the above-mentioned Raman amplified atomic filter, where the Raman gain is transformed from the coupling laser without population inversion, the SAFADOF provides much more effective amplification. Figure 3(a) displays the calculated transmitted power at resonance (purple, dashed) and gain factor (red, solid) as a function of the probing power at 135 °C with 3.5 mW 459 nm pumping power. We see that the transmitted power quickly tends to a saturation value due to the limited output capability of the atoms, thereafter the gain factor decreases in an approximate inverse proportional relationship to the probing power. Experimentally the measured transmitted power (green, dashed) and gain factor (blue, solid) are also depicted. For probing powers relatively large, the measured results agree well with the calculation, while for ultra weak probing powers the measured gain factor undergoes a sharp decline. It may because that in this case the superradiance effect plays a larger role and the lock-in technique is not sufficient to separate the two effects. The largest gain factor of more than 25000 (44 dB) is obtained at 50 pW. For various probing powers and temperatures, the gain spectrums are density plotted in Fig. 3(b,c) respectively. While the gain factor decreases for lower temperature due to the reduction of the Cs atomic density in the cell, for higher temperature the increased collisions between atoms decrease the coherence time of the 7S 1/2 state, thus decreasing the gain factor. Such characteristics have also been reported in hydrogen maser 48 , Rb and Cs atomic systems 38,39,49 . Gain bandwidth. The gain bandwidth is of interest in particular regarding suppression requirement of the background noise. In the SAFADOF, the gain bandwidth is approximate to the natural linewidth of the atomic transition, for the zero-velocity selection of the atoms by Doppler-free stabilized pumping laser. However, as the power of the pumping laser increases, the saturation effect results in a velocity distribution of Cs atoms pumped to the 7P 1/2 state. These atoms decays to the 7S 1/2 state and participate in the stimulation emission, and finally broadens the gain bandwidth of the SAFADOF through the Doppler effect. Figure 3(d) shows the dependency on pumping power of the gain factor and the gain bandwidth. We see that with the pumping power increasing, a larger gain factor is obtained. Meanwhile the gain bandwidth is broadened, which indicates that there is some optimal pumping power depending on how large the gain factor is required. The preferred pumping power will depend on the particular application.

Conclusion
In summary, we have experimentally investigated a SAFADOF at 1470 nm based on population inversion. The SAFADOF provides a gain factor larger than 25000 (44 dB) and an ultra-narrow bandwidth of 13 MHz, and it opens the possibility of applications in weak optical communication.
To eliminate the fluorescence background caused by superradiance of the Cs atoms, we propose a synchronous modulation method, which experimentally suppressed the background, and the method can be further expanded to other lamp-based atomic filters 24,47 . We also studied the gain factor and gain bandwidth characteristics of the SAFADOF under different probing laser powers, pumping laser powers, and temperatures. The gain factor has an approximate inverse proportional relationship with the probing power, and increases with the pumping power, while the gain bandwidth mainly increases with the pumping power. Hence a trade-off between large gain factor and narrow bandwidth must be made when determining the pumping power in practice.