1. Introduction

Single-frequency fiber lasers (SFFL) have been widely concerned and studied in coherent optical communication, coherent beam combining, and Doppler lidar, benefitting from the advantages of all-fiber compact structure, good beam quality, kHz-level linewidth, and convenient thermal management [1–3]. Among multiple laser parameters, the intensity noise of SFFL is a key metric for frontier researches on precise measurements, such as high-precision spectroscopy, precision time-frequency transfer, optical pump magnetometer, and single atom trapping [4–7]. To this end, it is of great importance to explore efficient noise suppression techniques to fulfill ultra-low-noise single-frequency laser emission.

In general, intensity noise presented in SFFL includes three primary parts: technical noise, relaxation oscillation noise, and quantum noise. Among them, relaxation oscillation noise constitutes the most prominent source, which derives from the gain dynamics and radiation in the fiber cavity. The arisen relaxation oscillation peak seriously deteriorates the relative intensity noise (RIN) level by over 60 dB [8,9]. Regarding intensity noise suppression, several techniques have been developed, such as photoelectric feedback, injection locking, mode cleaning, and gain saturation [7–14]. Among them, the gain saturation method based on semiconductor optical amplifier (SOA), as an effective and all-optical scheme for noise reduction, has attracted increasing research interest [15–17]. Especially in recent years, the intensity noise suppression technology based on SOA has been widely used in high-speed optical communication, interferometric fiber optic gyroscopes and other fields, which has effectively improved the signal-to-noise ratio (SNR) in communication system and gyroscope detection accuracy [18–21]. Benefiting from a sub-ns carrier lifetime, SOA can respond to high-frequency intensity fluctuations, implying a RIN reduction bandwidth of GHz scale [22]. However, the RIN suppression can hardly exceed 20 dB, which is limited by the small-signal gain of SOA [15,23]. Significantly, we have demonstrated a maximum noise suppression of over 50 dB around the relaxation oscillation frequencies and an effective bandwidth of up to 50 MHz [24]. This phenomenon is obviously beyond the previous prediction [25,26], which requires answering from a theoretical viewpoint. In the meantime, the explicit RIN-reduction bandwidth of the SOA-based method is hitherto unexplored.

In this article, we intend to shed light on the aforesaid questions by theoretically and experimentally studying SOA-based noise suppression technique. First, a vectorial model is exploited to characterize the orthogonally-polarized components propagating in the SOA, and reveals an induced relative phase shift between the intensity oscillations of the two polarization states. It predicts a noise-suppression mechanism underpinned by out-of-phase polarization mixing (OPM), then establishes a synthetic optical field with stable amplitude and dynamic polarization, wherein the fluctuations cancel out by summing up the orthogonally-polarized intensity oscillations with a relative phase shift of π. Then, we evidence the OPM phenomenon in SOA-based noise-reduction experiment by fine adjusting the input power and polarization state of SFFL. Such a mechanism can fully remove the relaxation oscillation peak and implement a maximum noise suppression of over 75 dB by breaking the intrinsic gain limitation of SOA. Complementary to the OPM, the gain saturation effect (GSE) simultaneously enables a wideband reduction of excess intensity noise. The joint contribution of these two mechanisms results in ultra-low noise property of SFFL, i.e., RIN of -160 dB/Hz in the wide frequency ranging from 0.5 MHz to 10 GHz — only 1.9 dB above the shot-noise-limit level.

2. Vector dynamics of semiconductor optical amplifier

Basically, the laser signal launched into the SOA can be decomposed into a transverse electric field (TE) component distributed parallel to the active layer and a transverse magnetic field (TM) component distributed perpendicular to the active layer. Both TE and TM components propagate along the laser propagation direction (LPD), as shown in Fig. 1. From these perspectives, it is important to investigate the vector dynamics in SOA and unravel the mechanism of RIN reduction existed by extending scalar framework [27] to a vectorial one.

 

Fig. 1. Schematic diagram showing active layer of SOA and definition of coordinate axes. The x, y axes designate the orientations of transverse electric (TE) and transverse magnetic (TM) modes, respectively; and z axis represents the laser propagation direction (LPD).

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We start from the propagation equations, of which the vectorial electric fields (A^S, A^F) polarized at slow and fast axes are assumed to align to TE and TM modes of the SOA, i.e., (A^E, A^M). According to the model in Ref. [28], we have

Theoretically, the two polarization components propagate independently, except for the indirect interaction through the carrier dynamics in the active layer that is interpreted as [28]:

To depict the time-varying numbers of holes n_x and n_y, we utilize rate equations as [28],

To investigate RIN, we account for the perturbed photon number S^E/M(z,t) by adding time-varying perturbations, i.e.,

Here, we adopt oscillatory perturbations in the subsequent analysis, i.e., cosine-type S^E/M(z, t). By means of the linear stability analysis, we characterize the fluctuations of the photon numbers with respect to TE and TM modes after the propagation in SOA via

Detailed derivation of Eq. (7) and approximations used are provided in Appendix 1. The ratio $\Re$, being complex, can allow a relative phase shift between the fluctuated photon numbers of TE and TM modes. Notably, there is a particular case wherein the ratio $\Re$ satisfies the formula:

Under this circumstance, the out-of-phase fluctuations with equal fluctuating amplitude can cancel out each other, and the amplitude of the synthetic electric field remains stable, only affect the polarization direction of the output laser. It should be noted that the intensity noise of output light is discussed only for the synthetic vector fields, so that the stable synthetic vector electric field amplitude means the extremely low intensity noise. This mechanism enables RIN reduction complementary to the GSE-based scheme. In the subsequent context, we term this approach as out-of-phase polarization mixing (OPM).

To demonstrate, we add a cosinoidal perturbation to the CW component at the input of SOA [ Fig. 2(a)], and preset the net gain $\overline {{G^E}} $ as well as the ratio of photon number $\overline {{S^M}} /\overline {{S^E}} $ to associate with pump current of SOA and incident polarization state in a practical situation, i.e.,

Hence, we can calculate the oscillatory perturbation after amplification of SOA in terms of

By summing up the resultant oscillatory photo numbers $\widetilde {\delta {S^E}}(L )$ and $\widetilde {\delta {S^M}}(L )$ which become out-of-phase [Fig. 2(b)], the cancellation of the fluctuations is thus obtained [Fig. 2(c)]. At this condition, this dynamic cancellation process is the essence of OPM implementation intensity noise. Also, the OPM-mediated noise suppression is visualized in the frequency domain, and the result given in Fig. 2(d) verifies a full elimination of the oscillation peak at 1 MHz and showcases the potential ability of the OPM mechanism in suppressing the relaxation oscillation peak of SFFL. More details about parameter setting and numerical computation are given in Appendix 2.

 

Fig. 2. Schematic diagram of OPM. (a) Superposition of photon numbers of TE and TM modes with the presence of cosinoidal perturbation at the input of SOA. (b) Out-of-phase oscillations of photon numbers with respect to TE and TM modes. (c) Superposition of photon numbers at the output of SOA. (d) Fourier transforms of t-varying waveforms in (a) and (c).

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3. Experimental validation of the theoretical model

The experimental setup to validate the above theoretical model is illustrated in Fig. 3. The SFFL cavity is constructed by connecting a segment of high-gain phosphate glass fiber [29,30] and a pair of 1550 nm fiber Bragg gratings (FBGs), and the structure of which is similar with the one used in our previous work [9]. Specifically, one of the FBG pair is a polarization-maintaining (PM) narrow-band FBG with a peak reflectivity of 65%, and the other is a wide-band FBG that has a high reflectivity of 99.95%. The 3-dB bandwidths of the PM narrow-band FBG and the wide-band FBG are 0.06 and 0.4 nm, respectively. The spare end of the wide-band FBG is cleaved to an 8° angle facet to prevent the detrimental Fresnel reflection. The laser cavity is temperature-controlled with a precision of 0.05 ℃, such that a steady single-longitudinal-mode operation is guaranteed. It is counter-pumped by a 976 nm laser diode (LD) through a PM wavelength division multiplexer (PM-WDM). A PM isolator (PM-ISO) is employed to prevent back reflection, and a subsequent PM variable optical attenuator (PM-VOA) is utilized to adjust the signal power injected into the SOA. Before launching the signal into the SOA, two options (orange dashed box), i.e., polarization controller (PC) and polarization beam splitter (PBS 1), are available to respectively obtain the input with arbitrary polarization state and linear polarization. The SOA used is a commercial type with a small signal gain of 20 dB (@ -20 dBm input power) and a working wavelength range of 1528-1562 nm, and its PM fiber pigtail (i.e., PM1550) largely prevents possible polarization evolution irrelevant with the vector dynamics of SOA. At the output, another PBS (PBS 2, purple dashed box) is exploited for polarization resolvable measurement. Alternatively, PBS 2 can be removed when the OPM, i.e., intensity superposition between the orthogonally-polarized components, is examined.

 

Fig. 3. Experiment setup to verify the theoretically-predicted vector dynamics of SOA.

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3.1 Identification of the GSE in SOA-based noise suppression

By fixing the driving current of SOA at 500 mA and adjusting PC, the maximum and minimum output powers varying against different input powers are recorded to identify the GSE in SOA. As displayed in Fig. 4(a), an obvious gain saturation phenomenon arises when the incident power exceeds 0 dBm. It is more straightforward to see in Fig. 4(b) that the relevant gain reduces from the small-signal gain (i.e., ∼19.6 dB for -20 dBm input power) to a saturated level which shows polarization dependence to some extent. The gain of SOA is found to be more susceptible to the polarization state of the incident signal in saturation regime, that is, a maximum 1.27 dB gain variation for 5.93 dBm input power in comparison with that of 0.28 dB for -20 dBm incident power. This 1.27 dB gain variation accords with the polarization dependent gain of the SOA given in the datasheet (Typical 1 dB, Max 2.5 dB). This polarization dependency is due to several factors including the waveguide structure, the polarization dependent nature of anti-reflection coatings and the gain material [31]. Hence, it is more preferred to scale the input power to enter the saturation regime for harnessing the polarization-dependent gain.

 

Fig. 4. Identification of GSE in SOA. (a) Output power and (b) gain with varying input power.

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To analyze the GSE on intensity noise suppression, we substitute PBS 1 for PC because the linearly-polarized input can well circumvent the influence of OPM. To vary the extent of GSE, we tune the input power via adjusting the VOA. Then, GSE-based intensity noise suppression is identified, and corresponding results are exhibited in Fig. 5, by connecting the input port of SOA with the slow-axis and fast-axis port of PBS 1, separately. The resolution bandwidth, scan time, and number of averages of the electrical spectrum analyzer are 10 kHz, 380 ms, and 10 respectively. In spite of the polarization state of the input signal, the tendency of noise reduction for increasing signal power is similar. Typically, we evaluate the ability of GSE by referring to the noise suppression at the relaxation oscillation frequencies. For 10 dBm input power along slow-axis orientation, the intensity noise level is decreased from -27.2 dBm to -45.7 dBm, yielding a reduction amplitude of 18.5 dB. While for fast-axis input with identical incident power, the reduction amplitude is 19.8 dB. To refine the contrast, the statistics SOA average RIN reduction for different input powers is shown in Table 1. The input light with different polarization directions has induced different amplitude of RIN reduction. Such discrepancy may derive from the polarization-dependent gain level in the saturation regime. Nonetheless, both are below the small-signal gain of SOA (i.e., 20 dB), suggesting the performance of the GSE-based method is intrinsically limited by the gain ability of SOA. In this context, further exploration of SOA noise reduction capabilities is particularly important.

 

Fig. 5. Demonstration of the GSE-based intensity noise suppression. Measured results of intensity noise with different incident powers for (a) slow-axis input and (b) fast-axis input.

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Table 1. SOA average RIN reduction for different input powers

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3.2 Identification of the OPM in SOA-based noise suppression

In theoretical analysis, the OPM requires a joint control of parameters $\overline {{G^E}} $, ${S^E}(0 )$, and $\overline {{S^M}} /\overline {{S^E}} $ which is closely associated with the driving current of SOA, incident power, and polarization state, respectively. Thus, we use the PC instead of PBS 1 for polarization manipulation; meanwhile, we remove PBS 2 to check the occurrence of the OPM. For parameter adjustment in experiment, we fine-tune the power and polarization state of the SFFL by simultaneously monitoring the radio-frequency (RF) spectrum of the output. In this process, the evidence of OPM can be found at the incident power of 7 dBm, and related properties in time and RF domain are presented in Fig. 6. For comparison, the oscilloscope waveform before noise reduction is provided in Fig. 6(a). As seen, the presence of relaxation oscillation peak imprints a visible fluctuation on the CW component. The period of 0.79 μs corresponds to the relaxation oscillation frequency of ∼1.27 MHz. Remarkably, by passing through the SOA the oscillatory structure of the oscilloscopic trace is fully eliminated, as described in Fig. 6(b). To further confirm its relation with OPM, PBS 2 is adopted at the output for polarization resolvable measurement. As for the orthogonal polarization states, measured temporal waveforms shown in Fig. 6(c) identify the out-of-phase oscillations, which are in good agreement with the temporal features of OPM predicted in theory [see Figs. 2(a)-2(c)]. The performances of intensity noise corresponding to the cases depicted in Figs. 6(a)-6(c) are plotted in Fig. 6(d).

 

Fig. 6. Experimental identification of the OPM. (a) Oscilloscopic trace of the laser signal before noise reduction. (b) Oscilloscope trace directly measured at the output of SOA. (c) Oscilloscope traces measured at slow-axis and fast-axis ports of PBS 2. (d) Performances of intensity noise for the cases in (a)-(c). Noise background of photodetector (PD) is also given for comparison.

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In comparison with the data registered by the oscilloscope, measurement in the RF domain has a much higher dynamic range, and thereby facilitates us to dissect the mechanism of SOA-based noise suppression. For the intensity noises measured with the involvement of PBS 2 [magenta and green curves in Fig. 6(d)], the OPM is prohibited, indicating that the noise suppression amplitude of ∼16 dB is attributed to the GSE of SOA. This phenomenon indicates that once the polarization decomposition is complete, the process of canceling out the dynamic noise of the light field in the two polarization directions is broken, and the intensity noise of the laser light field will be revealed again. Furthermore, the introduction of OPM can completely eliminate the whole relaxation oscillation peak (red curve), resulting in an intensity noise level close to the photodetector (PD) noise (black curve). The achieved maximum suppression amplitude of ∼68 dB, akin to the result reported in [24], dramatically exceeds the small-signal gain of ∼20 dB. This result presents the action of OPM enables effective suppression of the intensity jitter in two beams in different polarization directions.

In summary, the use of SOA provides not only GSE but also complementary OPM. The mechanism of GSE offers a relatively steady intensity noise suppression within a broad response bandwidth (higher than a ten of MHz), and its typical suppression amplitude is constrained by the gain limitation of SOA; while, the OPM particularly contributes to the suppression of relaxation oscillation noise with a suppression amplitude much larger than the intrinsic gain of SOA.

4. Realization of wideband ultra-low intensity noise suppression

In this section, we showcase the advantages of this SOA-based, all-optical scheme for noise reduction by demonstrating a wideband ultra-low-intensity-noise SFFL at 1550 nm. The experimental setup is shown in Fig. 7, which slightly differs from the configuration in Fig. 3 by adding an optical circulator (CIR) and a wideband high-reflective fiber Bragg grating (WB-HR-FBG) at the output. The combination of CIR with WB-HR-FBG is employed to filter out the amplified spontaneous emission (ASE) introduced by the SOA. The central wavelength of WB-HR-FBG is consistent with that of the SFFL, and its 3 dB reflection bandwidth and peak reflectivity are 1 nm and 99.9%, respectively.

 

Fig. 7. Schematic diagram of the wideband ultra-low-intensity-noise SFFL at 1550 nm.

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To fully leverage two mechanisms of the SOA-based intensity noise suppression method, the driving current of SOA is set to 500 mA, and the power launched into the SOA is adjusted to 7 dBm. Accordingly, the state of the PC is tuned to realize the OPM. For a consistent RIN measurement, the laser power injected into the PD is controlled to 6 dBm. The measured results are illustrated in Fig. 8, wherein the RINs before and after noise reduction are simultaneously plotted for comparison. The mutual effect of GSE and OPM well suppresses the relaxation oscillation noise at around 1.27 MHz and leads to a RIN level that approaches the shot noise limit (i.e., -161.9 dB/Hz) in the range of 0.5 MHz∼10 GHz. Specifically, the RIN within this frequency band is -160 dB/Hz which is only 1.9 dB above the shot noise limit, and the RIN at frequency of > 30 kHz is also below -150 dB/Hz. Remarkably, the maximum suppression amplitude reaches 75 dB. In the low-frequency range, the RIN gradually raises due to the presence of electronic noise of the SOA driver. From this perspective, we anticipate further noise suppression via introducing an optoelectronic feedback loop [24].

 

Fig. 8. RIN results of 1550 nm SFFL before and after SOA, along with the shot noise limit.

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The optical spectra of this SFFL are tested at a resolution of 0.02 nm, as exhibited in Fig. 9(a). Relatively severe ASE is induced after the SOA. Fortunately, the vast majority of ASE components is well filtered by applying the CIR and WB-HR-FBG, and the resultant laser has presented a favorable optical spectrum. In addition, the frequency noise is measured based on the phase generation carrier technology by combining an unbalanced fiber Michelson interferometer [32]. As shown in Fig. 9(b), the frequency noise before and after the SOA well overlap with each other, indicating that there is almost no degradation of frequency noise during the intensity noise suppression. The frequency noise about 5 kHz of the laser shows a deeper slop, which is closely related to the changing trend of the RIN [33]. It should be noted that SOA as an extra-cavity noise reduction technology does not change the intra-cavity state, so the frequency noise of the laser after SOA is basically unchanged for the frequency below 10 MHz [16]. This is further confirmed by the linewidth characterization, the self-heterodyne spectra, measured by a 48.8 km fiber delayed Mach-Zehnder interferometer and a 40 MHz fiber coupled acoustic optical modulator, are given in the inset of Fig. 9(b). To evaluate the laser linewidth, the second-order beat signal is used; the 20-dB bandwidth of 50.9 kHz manifests the linewidth of 2.55 kHz for the near-shot-noise-limited SFFL.

 

Fig. 9. (a) Optical spectra measured at different positions of the 1550 nm near-shot-noise-limited SFFL. (b) Measured results of laser frequency noise before and after the SOA. The inset shows the corresponding self-heterodyne spectra for laser linewidth characterization.

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5. Conclusions

In conclusion, the mechanism of SOA-based intensity noise suppression is investigated from both theoretical and experimental perspectives. By leveraging the vectorial model of SOA, a process that cancels out the fluctuation via the intensity superposition of out-of-phase orthogonally-polarized components is theoretically predicted. It has established a synthetic optical field with stable amplitude and dynamic polarization. This mechanism for noise reduction is termed as OPM and works cooperatively with the GSE to implement the SOA-based intensity noise suppression. Experimental verification is performed, which reveals that the action of OPM is able to completely remove the relaxation oscillation peak of the SFFL with suppression amplitude exceeding the small-signal gain of SOA. To showcase the advantage of this all-optical method, we demonstrate an ultra-low-noise SFFL whose RIN reaches -160 dB/Hz in the range of 0.5 MHz∼10 GHz. This RIN level is only 1.9 dB above the shot noise limit, the corresponding maximum suppression amplitude is over 75 dB. The present technique fully exploits the dynamics of SOA and thus serves as a general, promising solution of intensity noise suppression for single-frequency lasers.

Appendix 1 – Derivation of the vector model of SOA

Considering the perturbations of photon numbers and holes number in SOA, we insert Eq. (5) into Eq. (2), yielding

(11a)$$\left( {\frac{\partial }{{\partial t}} + v_g^E\frac{\partial }{{\partial z}}} \right)\delta {S^E}({z,t} )= {\Gamma ^E}[{{g^E}\delta {S^E}({z,t} )+ {S^E}(z )\delta {g^E}} ]- {a_l}\delta {S^E}({z,t} ),$$

(11b)$$\left( {\frac{\partial }{{\partial t}} + v_g^M\frac{\partial }{{\partial z}}} \right)\delta {S^M}({z,t} )= {\Gamma ^M}[{{g^M}\delta {S^M}({z,t} )+ {S^M}(z )\delta {g^M}} ]- {a_l}\delta {S^M}({z,t} ).$$

To tackle with the PDFs above, we present Eq. (11) in Fourier domain, i.e.,

(12a)$$v_g^E\frac{\partial }{{\partial z}}\widetilde {\delta {S^E}} = ({i\omega - {a_l}} )\widetilde {\delta {S^E}} + {\Gamma ^E}[{{g^E}\widetilde {\delta {S^E}} + {S^E}(z )\widetilde {\delta {g^E}}} ],$$

(12b)$$v_g^M\frac{\partial }{{\partial z}}\widetilde {\delta {S^M}} = ({i\omega - {a_l}} )\widetilde {\delta {S^M}} + {\Gamma ^M}[{{g^M}\widetilde {\delta {S^M}} + {S^M}(z )\widetilde {\delta {g^M}}} ],$$

where $\widetilde {\delta S}(\omega )$ is Fourier transform of δS(t). So ordinary differential equation (ODE) of δn is

(13a)$$\frac{{\partial \delta {n_x}(t )}}{{\partial t}} ={-} \frac{{\delta {n_x}(t )}}{{{T_1}}} - \frac{{\delta {n_x}(t )- \delta {n_y}(t )}}{{{T_2}}} - \delta {g^E}{S^E}(z )- {g^E}\delta {S^E}({z,t} ),$$

(13b)$$\frac{{\partial \delta {n_y}(t )}}{{\partial t}} ={-} \frac{{\delta {n_y}(t )}}{{{T_1}}} - \frac{{\delta {n_y}(t )- \delta {n_x}(t )}}{{{T_2}}} - \delta {g^M}{S^M}(z )- {g^M}\delta {S^M}({z,t} ).$$

By substituting the hole’s population (δn_x and δn_y) for the variations of gain variations (δg^E and δg^M), Eq. (13) is rewritten as

(14a)$$\left[ { - i\omega + \frac{1}{{{T_1}}} + \frac{1}{{{T_2}}} + 2{S^E}(z ){\xi^E}} \right]\widetilde {\delta {n_x}} + \left[ {{S^E}(z ){\xi^E} - \frac{1}{{{T_2}}}} \right]\widetilde {\delta {n_y}} ={-} {g^E}\widetilde {\delta {S^E}},$$

(14b)$$\left[ {{S^M}(z ){\xi^M} - \frac{1}{{{T_2}}}} \right]{\widetilde {\delta n}_x} + \left[ { - i\mathrm{\omega } + \frac{1}{{{T_1}}} + \frac{1}{{{T_2}}} + 2{S^M}(z ){\xi^M}} \right]{\widetilde {\delta n}_y} ={-} {g^M}{\widetilde {\delta S}^M},$$

where $\widetilde {\delta n}(\omega )$ is the Fourier transform of $\delta n(t)$. Equation (12) allows us to write the analytical forms of $\widetilde {\delta {n_x}}$ and $\widetilde {\delta {n_y}}$, i.e.,

(15a)$$\widetilde {\delta {n_x}} = {{\left[ {{g^M}\widetilde {\delta {S^M}}\left( {{S^E}{\xi^E} - \frac{1}{{{T_2}}}} \right) - {g^E}\widetilde {\delta {S^E}}\left( { - i\omega + \frac{1}{{{T_1}}} + \frac{1}{{{T_2}}} + 2{S^M}{\xi^M}} \right)} \right]} {\bigg /} K},$$

(15b)$$\widetilde {\delta {n_y}} = {{\left[ {{g^E}\widetilde {\delta {S^E}}\left( {{S^M}{\xi^M} - \frac{1}{{{T_2}}}} \right) - {g^M}\widetilde {\delta {S^M}}\left( { - i\omega + \frac{1}{{{T_1}}} + \frac{1}{{{T_2}}} + 2{S^E}{\xi^E}} \right)} \right]} {\bigg /} K},$$

$$\textrm{with, }K = \left( { - i\omega\,{+}\,\frac{1}{{{T_1}}}\,{+}\,\frac{1}{{{T_2}}}\,{+}\,2{S^E}{\xi^E}} \right) \times \left( { - i\omega\,{+}\,\frac{1}{{{T_1}}} + \frac{1}{{{T_2}}}\,{+}\,2{S^M}{\xi^M}} \right){-}\left( {\frac{1}{{{T_2}}}\,{-}\,{S^E}{\xi^E}} \right)\!\left( {\frac{1}{{{T_2}}}\,{-}\,{S^M}{\xi^M}} \right)\!,$$

as well as the gain variations,

(16a)$$\widetilde {\delta {g^E}} = {\xi ^E}{{\left[ {{g^M}\widetilde {\delta {S^M}}\left( {i\omega - \frac{1}{{{T_1}}} - \frac{3}{{{T_2}}}} \right) + {g^E}\widetilde {\delta {S^E}}\left( {2i\omega - \frac{2}{{{T_1}}} - \frac{3}{{{T_2}}} - 3{S^M}{\xi^M}} \right)} \right]} {\bigg /} K},$$

(16b)$$\widetilde {\delta {g^M}} = {\xi ^M}{{\left[ {{g^E}\widetilde {\delta {S^E}}\left( {i\omega - \frac{1}{{{T_1}}} - \frac{3}{{{T_2}}}} \right) + {g^M}\widetilde {\delta {S^M}}\left( {2i\omega - \frac{2}{{{T_1}}} - \frac{3}{{{T_2}}} - 3{S^E}{\xi^E}} \right)} \right]} {\bigg /} K}.$$

To avoid misunderstanding, we address that in Eq. (16) S^E/M is $z$-dependent, i.e., S^E/M(z) while gain function g^E/^M is calculated via the hole’s population in equilibrium, i.e., ${g^{E/M}}({\bar{n}_x},{\bar{n}_y})$.

Before solving Eq. (11), it is useful to first analyze the so-called gain-phase dynamics, that is the phase imparted by dynamical gain in terms of Kramers-Kronig relation, i.e.,

(17)$$\begin{aligned} &{\phi _E} \sim {a_E}[{{g^E} + \delta {g^E}(t )} ],\\ &{\phi _M} \sim {a_M}[{{g^M} + \delta {g^M}(t )} ], \end{aligned}$$

where a_E/M ∼1/2α^E⁄MΓ^E⁄M. Based on the experiment results (see in the next section), we have found a posteriori that relative phase between the two polarization states can keep relatively unchanged (time invariant), otherwise there should exist harmonics of 1 MHz with respect to the oscilloscopic trace in Fig. 6. It thus imposes constraint of the parameters, i.e.,

(18)$$\begin{array}{c} \Delta \phi (t )= {\phi _E} - {\phi _M} = {a_E}{g^E} - {a_M}{g^M} + {a_E}\delta {g^E}(t )- {a_M}\delta {g^M}(t ),\\ \Rightarrow \widetilde {\Delta \phi }(\omega )= \underbrace{{({{a_E}{g^E} - {a_M}{g^M}} )\delta (\omega )}}_{{DC - like}} + \underbrace{{({{a_E}\widetilde {\delta {g^E}} - {a_M}\widetilde {\delta {g^M}}} )\delta ({\omega - {\omega_R}} )}}_{{AC - like}},\\ \Rightarrow ({{a_E}\widetilde {\delta {g^E}} - {a_M}\widetilde {\delta {g^M}}} )= 0, \end{array}$$

where δ(ω) is the Dirac delta function. It corresponds to a case close to the practical situation in which the relaxation oscillations (ROs) primarily contribute to the RIN. To eliminate the AC-like term, we obtain the following relations:

(19)$$2({{a_E}{\xi^E}{g^E}\widetilde {\delta {S^E}} - {a_M}{\xi^M}{g^M}\widetilde {\delta {S^M}}} )= {a_M}{\xi ^M}{g^E}\widetilde {\delta {S^E}} - {a_E}{\xi ^E}{g^M}\widetilde {\delta {S^M}}.$$

(20)$$\begin{aligned} &\left( {\frac{1}{{{T_1}}} + \frac{3}{{{T_2}}}} \right)({{a_M}{\xi^M}{g^E}\widetilde {\delta {S^E}} - {a_E}{\xi^E}{g^M}\widetilde {\delta {S^M}}} )\\ &\quad + \left( {\frac{2}{{{T_1}}} + \frac{3}{{{T_2}}} + 3{S^E}{\xi^E}} \right)({{a_M}{\xi^M}{g^M}\widetilde {\delta {S^M}} - {a_E}{\xi^E}{g^E}\widetilde {\delta {S^E}}} )= 0. \end{aligned}$$

Equations (16) and (18) can be subsequently summarized as

(21a)$${a_E}{\xi ^E} = {a_M}{\xi ^M},$$

(21b)$${g^E}\widetilde {\delta {S^E}} = {g^M}\widetilde {\delta {S^M}}.$$

Equation (21) serves as a prerequisite for the subsequent derivation. Now, we proceed with solving $\delta {S^{E/M}}({z,t} )$. By inserting Eqs. (16) and (21) into Eq. (11), ODEs of the $\widetilde {\delta S}$ are expressed as,

(22a)$$v_{^g}^E\frac{\partial }{{\partial z}}\widetilde {\delta {S^E}} = ({i\omega - {a_l}} )\widetilde {\delta {S^E}} + {\Gamma ^E}{g^E}\widetilde {\delta {S^E}} + 3{\Gamma ^E}{g^E}{S^E}{\xi ^E}\widetilde {\delta {S^E}}\frac{{i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^M}{\xi ^M}}}{K},$$

(22b)$$v_{^g}^M\frac{\partial }{{\partial z}}\widetilde {\delta {S^M}} = ({i\omega - {a_l}} )\widetilde {\delta {S^M}} + {\Gamma ^M}{g^M}\widetilde {\delta {S^M}} + 3{\Gamma ^M}{g^M}{S^M}{\xi ^M}\widetilde {\delta {S^M}}\frac{{i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^E}{\xi ^E}}}{K}.$$

The Jacobian matrix of Eq. (22) is diagonal, so analytical solutions of $\widetilde {\delta {S^E}}(L )$, $\widetilde {\delta {S^M}}(L )$, i.e.,

(23a)$$\frac{{\widetilde {\delta {S^E}}(L )}}{{\widetilde {\delta {S^E}}(0 )}} = exp\left[ {\frac{L}{{v_g^E}}({i\omega - {a_l}} )} \right] \times exp\left[ {\int_0^L {\frac{{dz}}{{v_g^E}}{\Gamma ^E}{g^E}\left( {1 + 3{S^E}{\xi^E}\frac{{i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^M}{\xi^M}}}{K}} \right)} } \right],$$

(23b)$$\frac{{\widetilde {\delta {S^M}}(L )}}{{\widetilde {\delta {S^M}}(0 )}} = exp\left[ {\frac{L}{{v_g^M}}({i\omega - {a_l}} )} \right] \times exp\left[ {\int_0^L {\frac{{dz}}{{v_g^M}}{\Gamma ^M}{g^M}\left( {1 + 3{S^M}{\xi^M}\frac{{i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^E}{\xi^E}}}{K}} \right)} } \right].$$

To evaluate the RIN, unperturbed photon number S^E(L) at the output of the SOA is calculated by solving Eq. (2), namely,

(24a)$$\frac{{{S^E}(L )}}{{{S^E}(0 )}} = exp\left( {\frac{{ - {a_l}L}}{{v_g^E}}} \right)exp\left( {\int_0^L {\frac{{{\Gamma ^E}{g^E}dz}}{{v_g^E}}} } \right),$$

(24b)$$\frac{{{S^M}(L )}}{{{S^M}(0 )}} = exp\left( {\frac{{ - {a_l}L}}{{v_g^M}}} \right)exp\left( {\int_0^L {\frac{{{\Gamma ^M}{g^M}dz}}{{v_g^M}}} } \right).$$

By combining Eqs. (23) and (24), we present the RIN variation after passing through SOA as

(25)$$\begin{aligned} \frac{{\widetilde {\delta {S^E}}(L )/{S^E}(L )}}{{rin(0 )}} &= exp\left[ {\frac{{i\omega L}}{{v_g^E}} + \int_0^L {\frac{{{\Gamma ^E}{g^E}dz}}{{v_g^E}}\frac{{3{S^E}{\xi^E}\left( {i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^M}{\xi^M}} \right)}}{K}} } \right],\\ \frac{{\widetilde {\delta {S^M}}(L )/{S^M}(L )}}{{rin(0 )}} &= exp\left[ {\frac{{i\omega L}}{{v_g^M}} + \int_0^L {\frac{{{\Gamma ^M}{g^M}dz}}{{v_g^M}}\frac{{3{S^M}{\xi^M}\left( {i\omega - \frac{1}{{{T_1}}} - \frac{2}{{{T_2}}} - {S^E}{\xi^E}} \right)}}{K}} } \right], \end{aligned}$$

where $rin(0 )= \widetilde {\delta {S^E}}(0 )/{S^E}(0 )= \widetilde {\delta {S^M}}(0 )/{S^M}(0 ).$

It is noteworthy that rin, defined as power (proportional to the photon number) fluctuation divided by averaged power, differs from the usual RIN power spectral density (PSD). In the case of a uniform group velocity v_g (by omitting the walk-off effect caused by the group velocity difference), Eq. (23) gives rise to the ratio between output RINs of TE and TM modes, i.e.,

(26)$$\begin{aligned} &\quad\quad\quad\quad\quad\quad \frac{{\widetilde {\delta {S^E}}(L )/{S^E}(L )}}{{\widetilde {\delta {S^M}}(L )/{S^M}(L )}} = exp(\Re ),\\ &\Re = \int_0^L {\frac{{dz}}{K}} \left[ {\frac{{3{\Gamma ^M}{g^M}{S^M}{\xi^M}}}{{{v_g}}}\left( { - i\omega + \frac{1}{{{T_1}}} + \frac{2}{{{T_2}}} + {S^E}{\xi^E}} \right)} \right.\left. { - \frac{{3{\Gamma ^E}{g^E}{S^E}{\xi^E}}}{{{v_g}}}\left( { - i\omega + \frac{1}{{{T_1}}} + \frac{2}{{{T_2}}} + {S^M}{\xi^M}} \right)} \right]. \end{aligned}$$

Appendix 2 – Paradigm of out-of-phase polarization mixing in SOA

As a preliminary, we simplify the expression of ratio $\Re$ as

(27)$$\Re = \int_0^L {\frac{{dz}}{{K^{\prime}}}\left[ {3{G^M}{S^M}{\xi^M}\left( { - i\omega + \frac{2}{{{T_2}}} + {S^E}{\xi^E}} \right) - 3{G^E}{S^E}{\xi^E}\left( { - i\omega + \frac{2}{{{T_2}}} + {S^M}{\xi^M}} \right)} \right]} ,$$

with ${G^{E/M}} = {\Gamma ^{E/M}}{g^{E/M}}/{v_g},$

(28)$$K^{\prime} = \left( { - i\omega + \frac{1}{{{T_2}}} + 2{S^E}{\xi^E}} \right) \times \left( { - i\omega + \frac{1}{{{T_2}}} + 2{S^M}{\xi^M}} \right) - \left( {\frac{1}{{{T_2}}} - {S^E}{\xi^E}} \right)\left( {\frac{1}{{{T_2}}} - {S^M}{\xi^M}} \right),$$

due to the limit 1/T₂ >> 1/T₁. To further facilitate a quantitative evaluation, we exploit an averaging scheme based on twofold points:

① Gain parameter G^E/M is approximately invariant during amplification;

② Averaged parameter is estimated via the trapezoidal method in which the spatial step is the length of the SOA.

In this way, Eq. (27) is rewritten as,

(29)$$\Re = \frac{{\left[ {3\overline {{G^M}{S^M}} {\xi^M}\left( { - i\omega + \frac{2}{{{T_2}}} + \overline {{S^E}} {\xi^E}} \right) - 3\overline {{G^E}{S^E}} {\xi^E}\left( { - i\omega + \frac{2}{{{T_2}}} + \overline {{S^M}} {\xi^M}} \right)} \right]}}{{K^{\prime}}},$$

$$\textrm{with }\overline {{G^{E/M}}} = \int_0^L {dz{\Gamma ^{E/M}}{g^{E/M}}} /{v_g} \sim {\Gamma ^{E/M}}{g^{E/M}}L/{v_g},$$

$$\overline {{S^{E/M}}} = \frac{{{S^{E/M}}(0 )+ {S^{E/M}}(L )}}{2} \sim {S^{E/M}}(0 )\left( {1 + \frac{{\overline {{G^{E/M}}} }}{2}} \right).$$

Likewise, the averaged fluctuation of the photon number is given by

(30)$$\overline {\delta {S^{E/M}}} = \frac{{\delta {S^{E/M}}(0 )+ \delta {S^{E/M}}(L )}}{2}.$$

For convenience, parameters adopted in the RIN calculation are listed in Table 2.

Table 2. Parameters used in the RIN calculation

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Then, we search for the parameter region that meets the request of OPM, i.e., Eq. (9) by scanning the photon number S^E(0) ranging from 10² to 10⁶. Meanwhile, we fix the net gain about the TE mode as well as ratio between photon numbers of two polarization components:

(31a)$$\overline {{G^E}} = 1.5,$$

(31b)$$\overline {{S^M}} /\overline {{S^E}} = 6.$$

From a practical viewpoint, a fixed gain can correspond to a certain driving current of SOA. In this regard, we have to keep in mind that net gain, in spite of the polarization, should maintain below the small-signal gain (e.g., 20 dB). Meanwhile, fixing ratio $\overline {{S^M}} /\overline {{S^E}}$ associates with specific power ratio between TE and TM modes at the input of SOA, which determines the polarization state of the incident light (linear polarization is assumed here). Hence, what we do here is to find appropriate incident powers (equivalent to the initial photon numbers S(0)) for a preset polarization state and driving current of SOA. Since $\overline {{S^E}}$, $\overline {{S^M}}$, and $\overline {{G^E}}$ are given for

(32)$$\overline {{S^E}} = {S^E}(0 )({1 + \overline {{G^E}} /2} ),$$

$\overline {{G^M}}$ can be computed via Eq. (9a) to result in

(33)$$\overline {{G^M}} = \frac{{\overline {{G^E}} {S^E}{\xi ^E}[{3\omega Re({K^{\prime}} )+ 6Im({K^{\prime}} )/{T_2} + 3Im({K^{\prime}} ){S^M}{\xi^M}} ]- \pi {{|{K^{\prime}} |}^2}}}{{{S^M}{\xi ^M}[{3\omega Re({K^{\prime}} )+ 6Im({K^{\prime}} )/{T_2} + 3Im({K^{\prime}} ){S^E}{\xi^E}} ]}}.$$

Subsequently, Eq. (9b) yields the function in the form of

(34)$$exp[{Re(\Re )} ]- \frac{{\overline {{S^M}} ({1 + \overline {{G^M}} } )}}{{{S^E}(0 )({1 + \overline {{G^E}} } )({1 + \overline {{G^M}} /2} )}} = 0,$$

and consequently determines the value(s) of S^E(0) suitable for OPM, as illustrated in Figs. 10(a) and 10(b). In the case of OPM, the uniform fluctuating amplitude of the photon number at the output of SOA can be calculated from Eq. (21b):

$${\textrm{g}^E}\widetilde {\overline {\delta {S^E}} } = {\textrm{g}^M}\widetilde {\overline {\delta {S^M}} } \Rightarrow \widetilde {\delta {S^E}}(0 )+ \widetilde {\delta S}(L )= \frac{{\overline {{G^M}} }}{{\overline {{G^E}} }}[{\widetilde {\delta {S^M}}(0 )- \widetilde {\delta S}(L )} ]$$

(35)$$\Rightarrow \widetilde {\delta S}(L )= \frac{{\overline {{G^M}} \widetilde {\delta {S^M}}(0 )/\overline {{G^E}} - \widetilde {\delta {S^E}}(0 )}}{{1 + \overline {{G^M}} /\overline {{G^E}} }}.$$

 

Fig. 10. (a),(b) Averaged gain $\overline {{G^M}} $ and ${S^M}(L )/{S^E}(L )$ with varying photon number ${S^E}(0 )$. (c) Spectral response of the SOA-based intensity noise suppression. (d) Dynamics schematic diagram of OPM effect in SOA (amplitude not to scale).

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With the determined S^E(0) [designated by red dot in Fig. 10(b)], we can calculate the ω-dependent perturbation after the amplification of SOA in terms of

(36a)$${{\widetilde {\delta {S^E}}(L,\omega )} / {{S^E}}}(L) = ({{{\widetilde {\delta S}(0)} / {S(0)}}} )exp\left[ {\frac{{3{\xi^E}\overline {{G^E}{S^E}} }}{{K^{\prime}}}\left( {i\omega - \frac{2}{{{T_2}}} - {\xi^M}\overline {{S^M}} } \right)} \right],$$

(36b)$${{\widetilde {\delta {S^M}}(L,\omega )} / {{S^M}}}(L) = ({{{\widetilde {\delta S}(0)} / {S(0)}}} )exp\left[ {\frac{{3{\xi^M}\overline {{G^M}{S^M}} }}{{K^{\prime}}}\left( {i\omega - \frac{2}{{{T_2}}} - {\xi^E}\overline {{S^E}} } \right)} \right].$$

Equation (36) can access the spectral response of the SOA-based intensity noise suppression. As displayed in Fig. 10(c), the sole GSE offers a steady suppression amplitude over a broad band, while the OPM is relatively narrowband in comparison with the working bandwidth of GSE mechanism and is particularly response for the suppression of relaxation oscillation noise. It is worth noting that the maximum noise suppression frequency in Fig. 10(c) is deliberately regulated at the relaxation oscillation frequency of the fiber laser to highlight the noise suppression effect of this technique. Figure 10(d) has shown the dynamics schematic diagram of OPM effect in SOA (amplitude not to scale), which is a special condition result only used to visualize the OPM state.

Funding

National Natural Science Foundation of China (U22A6003, 12204180, 62035015, 62275082); National Key Research and Development Program of China (2022YFB3606400); Key-Area Research and Development Program of Guangdong Province (2020B090922006); Major Program of the National Natural Science Foundation of China (61790582); Fundamental Research Funds for the Central Universities (D6223090); Leading talents of science and technology innovation of Guangdong Special Support Plan (2019TX05Z344); China Postdoctoral Science Foundation (2021M701256); Basic and Applied Basic Research Foundation of Guangdong Province (2022A1515012594); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X137); Guangzhou Basic and Applied Basic Research Foundation (202201010003); Open Project Program of Shanxi Key Laboratory of Advanced Semiconductor Optoelectronic Devices and Integrated Systems (2022SZKF02).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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