Stimulated Brillouin Scattering (SBS) is a phenomenon in laser physics and nonlinear optics, which results from the coupling of coherent light and sound. This process has a rich history and has been extensively studied^[1-3]. Brillouin fiber laser (BFL) makes use of low noise and narrow bandwidth characteristic of Brillouin amplification, has been widely used in various applications, such as optical communication systems^[4-6], distributed/point fiber sensing^[7-8], and optical bistability^[9-11]. The single-longitudinal-mode (SLM) and narrow linewidths BFL has been achieved utilizing a variety of methods. For example, a BFL with a 20 Hz ultra-narrow linewidth based on a large mode-volume (2 meters) optical fiber resonator, has been demonstrated^[12]. Additionally, by confining a stabilized self-injected distributed feedback (DFB) laser diode into a high Q-factor 11-meter-long fiber ring cavity, the BFL's linewidth has been narrowed to 400 Hz^[13]. Injection laser locking and composite cavity techniques can also be used to reduce the SBS gain spectrum linewidth. However, the erbium-doped fiber amplifier's bandwidth imposes a restriction on BFL's tunability^[14]. For BFL with longer cavity^[15], SLM output is challenging due to mode competition and higher optical signal-to-noise ratio (OSNR) requires higher pump intensity. The use of compound cavity and ultra-narrow bandwidth filter undoubtedly increases the operation difficulty and implementation cost.

The concept of Parity-Time (PT) symmetry, derived from a quantum mechanical principle, has garnered substantial attention from researchers in the fields of optics and beyond^[16-18]. The ability to effectively select modes is a crucial attribute of PT symmetry, and has been applied to long cavity BFLs, yielding a SLM output. The use of PT symmetry in semiconductor or fiber lasers as well as optoelectronic oscillators (OEOs) has become more and more popular in recent years. To achieve OEOs under SLM status, researchers have employed various techniques, such as constituting a dual-polarization loop incorporating a LiNbO₃ phase modulator (PM) at 2 to 12 GHz tunable frequencies range^[19], utilizing a single Sagnac loop to construct two interconnected feedback loops based on spatial wavelength^[20], and constructing two same hybrid optoelectronic loops primarily composed of a common dual-parallel Mach-Zehnder modulator (DPMZM) and a photodetector, with a 60 dB side mode suppression ratio (SMSR)^[21]. Another frequently used method for mode selection in fiber lasers is PT symmetry. For instance, in Ref. [22], PT symmetry was leveraged to realize a fiber laser with 129 kHz 3-dB linewidth, using optical polarization diversity between two subspaces of a single spatial unit. In the design of PT symmetry lasers, fiber Bragg gratings (FBGs) are also broadly utilized. The development of a resonator with two opposing PT symmetry FBGs has enabled the fabrication of a SLM laser with a linewidth of 8.667 kHz^[23]. In order to accomplish PT symmetry in a fiber ring laser, a single Sagnac fiber loop was used, yielding a 41.9 dB SMSR and 390 Hz linewidth^[24]. However, it has been reported that BFLs based on PT symmetry have limited tunability and OSNR, with a mere 1 nm tunability range and 65 dB OSNR^[25-26]. Thus, it is imperative to further investigate various PT symmetric BFL structures and conduct comprehensive assessments of their performance in order to expand the adjustable range and OSNR.

In this paper, we present a widely-wavelength-tunable BFL with improved OSNR based on PT symmetric and saturable absorption (SA) effect by polarization-maintaining erbium-doped fiber (PM-EDF) Sagnac loop. Furthermore, we carry out experiment to verify its performance.

The experimental configuration for the proposed widely-wavelength-tunable BFL is depicted in Fig. 1. A tunable laser source (TLS: PPCL 550), characterized by a 10 kHz linewidth and a wavelength-tunable range of 1526 to 1565.41 nm, is employed as the pump for the BFL and is fed into the 10 km single-mode fiber (SMF) via a circulator (Cir1) and PC1. Upon surpassing the SBS threshold, the pump elicits the generation of Stokes light through SBS. The Stokes light then circulates clockwise in the ring resonator, which is comprising of the 90/10 OC1, Cir1, Cir2, 10 km SMF, and the PM-EDF Sagnac loop. Subsequently, the Stokes light is launched into the PM-EDF Sagnac loop through Cir2. The PM-EDF Sagnac loop comprises a 50/50 optical coupler (OC3), a PM-EDF that spans 50 cm in length, and two polarization controllers (PC2 and PC3). The inherent birefringence of the PM-EDF enables the formation of two mutually coupled loops, one with SBS gain and the other with loss of SA participation, arising from the splitting and clock-wise/counterclockwise motion of the incident light. By adjusting the PCs control the polarization state of the injected Stokes light, the SA participating SBS gain and the loss of the loops corresponding to the clockwise and counterclock-wise light can be precisely regulated. When the light waves go through the same PM-EDF Sagnac loop, the two loops lengths are nearly similar. The PT symmetry is broken when the SA participating SBS gain and loss are well matched and the gain exceeds the coupling coefficient, the output of a BFL is triggered in turn. The SA can enhance SBS gain contrast value, leading to a higher OSNR for the laser^[25-26]. The Stokes wave is then fed through Cir2 into the ring cavity and passes through many clockwise roundtrips. The Stokes wave is divided two beams by 90/10 OC1, 90% of the divided Stokes light circulates in the ring resonator, and 10% is used as the output laser for the BFL. The output of widely-wavelength-tunable BFL is split into two light by the 90/10 OC2. An optical spectrum analyzer (OSA) with 0.01 nm resolution is used to monitor the SLM laser in one beam, while the other beam is converted into an electrical signal by a photodetector (PD) and detected by an electrical spectrum analyzer (ESA) with a resolution of 1 Hz.

The SBS effect causes a frequency shift from the center frequency $ {f}_{c} $ to $ {f}_{c}-{f}_{B} $, which occurs within the 10-km SMF after the pump wave injected into the tunable laser source via the circulator. The magnitude of the Brillouin frequency shift $ {{f}}_{\mathrm{B}} $ is defined as $ (2{V}_{A}/c){v}_{p} $, where $ {V}_{A} $ represents the acoustic velocity in the medium, $ {c} $ denotes the velocity of light, and $ {{v}}_{{p}} $ represents the optical frequency of the pump beam. It is worth noting that when the wavelength is 1549.955 nm, $ {f}_{{\mathrm{B}}} $ is approximately 10.735 GHz. The Brillouin gain spectrum exhibits the linewidths of $ \Delta {f}_{{\mathrm{B}}}= $ 20 MHz.

Additionally, the free spectral range (FSR) for the 10 km SMF is also specified as well^[6],

$ \mathrm{F}\mathrm{S}\mathrm{R}=\frac{c}{n{L}_{m}}\quad, $ (1)

where $ L $ is the length of ring resonator. In Fig. 2(a) (color online), $ n $ equals to the effective index of the fiber, is 1.468, and $ L $ denoting the length of the ring cavity, the widely-wavelength-tunable BFL's FSR is determined to be 21 kHz.

The SA coefficient in the EDF may be written as^[27]：

$ {a}_{n}=\frac{{a}_{l}}{1+{I}_{n}/{I}_{{\mathrm{sat}}}}\quad,$ (2)

where represents the linear absorption coefficient of the medium, and denotes the saturation intensity. The inherent birefringence of PM-EDF is utilized to form two equivalent loops constituting the PT symmetry, with one loop exhibiting gain from SBS and the other displaying loss. The following is a representation of the coupling equations for the modes inside the cavity in the time domain^[28]:

$ \frac{{\mathrm{d}}}{{\mathrm{d}}t}\left[\begin{array}{c}{A}_{n}\\ {B}_{n}\end{array}\right]=\left[\begin{array}{cc}-i{\omega }_{n}+{g}_{B\_{A}_{n}}'& i{k}_{n}\\ i{k}_{n}& -i{\omega }_{n}+{g}_{B\_{B}_{n}}'\end{array}\right]\left[\begin{array}{c}{A}_{n}\\ {B}_{n}\end{array}\right]. $ (3)

The $ {n} $-th modes' amplitudes in the SA participating SBS gain and loss loops, expressed as $ {A}_{n} $ and $ {B}_{n} $, respectively. With the lacking PT symmetry, the longitudinal modes localized eigenfrequency are affected, designated as $ {\omega }_{n} $. The SA participating SBS gain and loss coefficients of the n-th mode, defined as $ {g}_{B\_{A}_{n}}'={g}_{B\_{A}_{n}}-{a}_{n} $ and $ {g}_{B\_{B}_{n}}'={g}_{B\_{B}_{n}}-{a}_{n} $. It's a result of the interaction between the SBS gain and the inherent loss of the laser cavity in our system. In the PT symmetric setup, for the $ n $-th mode, another key determinant is the coupling coefficient represented by $ {k}_{n} $ between the two coupled loops. The eigenfrequencies of the PT-symmetric system may be derived by solving the Eq. (4).

$ {\omega }_{n}^{\left(\mathrm{1,2}\right)}={\omega }_{n}+i\frac{{g}_{B\_{A}_{n}}'{+g}_{B\_{B}_{n}}'}{2}\mp \sqrt{{{k}}_{{n}}^{2}-{\left(\frac{{g}_{B\_{A}_{n}}'{-g}_{B\_{B}_{n}}'}{2}\right)}^{2}}. $ (4)

Assuming that the desired PT-symmetric condition can be met by adjusting the phase compensators, then a SLM output for the designated mode can be obtained, we have $ {g}_{B\_{A}_{n}}' $=$ {-g}_{B\_{B}_{n}}' $=$ {g}_{{B}_{n}}' $. Eq. (4) can be written as

$ {\omega }_{n}^{\left(\mathrm{1,2}\right)}={\omega }_{n}\mp \sqrt{{k}_{n}^{2}-{g}_{{B}_{n}}'}\quad. $ (5)

Under the conditions that $ {g}_{{B}_{n}}' < {k}_{n} $, the PT-symmetric system operates within an unbroken regime, resulting in the frequency division of a specific mode. Conversely, if $ {g}_{{B}_{n}}' > {k}_{n} $, the PT-symmetry is deemed broken and a conjugate pair of modes experiences a discernible deviation; one mode experiences SBS gain while the other encounters a loss, while the remaining modes remain unaffected, as demonstrated in Fig. 2(b).

Additionally, in a conventional BFL, the SLM operation can be achieved as long as the primary mode's SBS gain, specifically, the 0-th mode's $ {g}_{{B}_{0}} $, surpasses the oscillation threshold while the SBS gain of the other modes remains below the threshold. Under such circumstances, the maximal contrast of SBS gain is measured as follows:

$ {g}_{\max}'{=g}_{{B}_{0}}'-{g}_{{B}_{1}}' \quad.$ (6)

Under PT symmetry, the gain difference between $ {g}_{{B}_{0}}'={g}_{{B}_{0}}-{a}_{0} $ and $ {g}_{{B}_{1}}' $ can be calculated by Eq. (7). This results in a significantly greater gain contrast as compared to the regular BFL scenario, which enhances the stability of the SLM laser. The gain difference is calculated as follows:

$ {g}_{\max\_PT}'=\sqrt{{{g}_{{B}_{0}}'}^{2}-{{g}_{{B}_{1}}'}^{2}}\quad. $ (7)

The measure of the improvement in mode selection, known as the SBS gain enhancement or gain contrast ratio, is determined by comparing the gain difference between a conventional laser source and a laser source that possesses PT symmetry. This ratio is given by Eq. (8) and serves as a useful metric for assessing the PT-symmetric laser's performance in comparison to traditional laser.

$ {G}'=\frac{{g}_{{\mathrm{max}}\_{\mathrm{PT}}}'}{{g}_{{\mathrm{max}}}'}=\sqrt{\frac{{g}_{{B}_{0}}'/{g}_{{B}_{1}}'+1}{{g}_{{B}_{0}}'/{g}_{{B}_{1}}'-1}}\quad. $ (8)

Thus, with the utilization of the PT-symmetric configuration, the SLM operation can achieve a higher SBS gain contrast, resulting in a higher OSNR lasing.

The BFL is renowned for its exceptional narrow linewidth characteristic, which is able to transform a pump wave with broad linewidth into a Brillouin Stokes wave with a remarkably narrow linewidth. There exist two methods for determining the linewidth of BFL.

The linewidths of the BFL, $ \Delta {f}_{B} $, and the pump light, $ \Delta {v}_{p} $, have a proportional relationship that may be written as follows^[29]:

$ \Delta {f}_{B}=\frac{\Delta {v}_{p}}{{\left(1+{\text{π}} \Delta {v}_{B}/\left(-\dfrac{c\mathrm{ln}R}{nL}\right)\right)}^{2}}\quad. $ (9)

With the SBS gain spectrum intrinsic linewidth being $ \Delta {v}_{B}= $ 20 MHz, and the amplitude feedback coefficient of the widely-wavelength-tunable BFL being R = 0.5 × 0.9 = 0.45, Eq. (9) reveals that, given a pump light linewidth of $ \Delta {v}_{p}= $ 10 kHz, the linewidth of the BFL, $ \Delta {f}_{B} $ = 6.910 × 10⁻⁴ Hz.

Lasers with wide linewidths, such as semiconductor lasers, He-Ne lasers, fiber DFB lasers, and fiber DBR lasers, are characterized by their short cavity lengths. The Schawlow-Towns linewidth limit, which represents the maximum linewidth of the laser, is specified by^[30]:

$ \Delta {f}_{B}=\frac{2{\text{π}} {(\Delta {v}_{c})}^{2}h{f}_{c}}{{P}_{0}} \quad.$ (10)

The FWHM is represented by $ \Delta {v}_{c}=cL_{c}/2{\text{π}} L $, where L_c denotes the net loss of the cavity, $ {P}_{0} $ signifies the output intensity, and h symbolizes Planck’s constant. For a fiber ring laser operating at a wavelength of $ \lambda = $1549.996 nm with a 10 km long resonator, a net loss of 0.45, and an output intensity of 0.005 mW, the calculated linewidth limit is extremely small, at $ \Delta {f}_{B}=2.47\times {10}^{-15} $ Hz.

The narrow linewidth of BFL, resulting from its capability to convert wide linewidth pump light into Brillouin Stokes light, is a remarkable attribute. The linewidth of BFL can be calculated through two methods. The equation (9) provides the proportional correlation between the linewidths of the pump wave $ \Delta {{v}}_{{p}} $ and the BFL $ \Delta {{f}}_{{B}} $. Here, the calculated value of $ \Delta {f}_{B} $ is $ 6.910\times {10}^{-4} $ Hz, while the linewidth of the pump light is determined to be 10 kHz. Although, the theoretical BFL’s linewidth is immensely narrow, but the actual linewidth may be wider due to the inherent noise and instability of the setup.

The proposed BFL based on PT symmetric and SA effect was evaluated experimentally by utilizing the setup illustrated in Fig. 1. The pump wave at 1 549.996 nm emitted by the TLS, possessing a 10 kHz 3-dB linewidth in the experiment. Fig. 3(a) (color online) depicts the output optical spectra of the BFL and the TLS pump light, showcasing a remarkable OSNR of 77 dB. The output intensity of the BFL in relation to the pump intensity is illustrated in Fig. 3(b) (color online), where the pump intensity was measured at the TLS and the BFL output intensity was recorded at the 10% port of the OC1. The maximum 10 mW output intensity was achieved while preserving stable single-frequency operation.

The output wavelength of the BFL perfectly aligns with the Brillouin frequency shift of the pump laser, thereby allowing for the control of the lasing wavelength, resulting in a wide tunability range. The ability to adjust the output wavelength of the TLS within the range of 1526 to 1565.41 nm, enables the wavelength tuning of the BFL between 1526.088 to 1565.498 nm, as demonstrated in Fig. 4 (color online). The figure presents the stable tuning of the BFL's optical spectra within the frequency range of 1528−1564 nm, with a tuning step size of approximately 2 nm, all while preserving a high OSNR of 77 dB, which is increased by 12 dB with respect to the general PT symmetric BFL^[25-26]. This design not only has higher OSNR but also has a broader tunable range, as compared to existing BFL designs^[31]. Furthermore, this design is not limited by the bandwidth of the erbium-doped fiber amplifier (EDFA).

As depicted in Fig. 5 (color online), the spectrum of the frequency beat signal at the PD output when pump wavelength is set to 1550 nm is measured. The attainment of PT symmetry and SLM lasing are dependent upon the adjustment of SA participating SBS gain and loss to be matched, as evidenced by the generation of multiple frequency beat signals at 100 kHz to 4 MHz frequency range when PT symmetry is not achieved, as depicted in Fig. 5(a). Conversely, by precisely tuning the phase compensators PC1 and PC2 to achieve a well-matched SA participating SBS gain and absorption loss, exceeding the coupling coefficient and thus breaking PT symmetry, SLM BFL is achieved, as shown in Fig. 5(b). While, in Fig. 5(c) and (d), the zoomed-in view-ports of the unbroken and broken PT-symmetry at the frequency range of 100−300 kHz are presented, with a maximum SMSR of 16 dB corresponding to an FSR of 21 kHz, which, according to Eq. (1), corresponds to a cavity length of 10 km cavity length. These measurement results are consistent with the theoretical predictions. Similarly, the zoomed-in view-ports of the unbroken and broken PT-symmetry at 300−500 kHz frequency range are shown in Fig. 5(e). It can be seen that the maximum SMSR is 14 dB. Furthermore, Fig. 5(f) illustrates the maximum SMSR of 14 dB between unbroken and broken PT-symmetry at 500−700 kHz frequency range.

Fig. 6 shows the schematic diagram of the linewidth measurement setup to determine the linewidth of the BFL based on PT symmetric and SA effect. The output of the TLS is divided into two beams by a 50/50 OC1, which acting as the pump. In the upper branch, a second BFL is established, consisting of a Cir1, 10-km SMF and a 90/10 OC2. The pump output, which exceeds the SBS threshold, is injected into the 10 km SMF via Cir1 and generates Stokes light as a result of SBS. This Stokes light undergoes numerous CCW roundtrips, resulting in periodic resonances within the ring cavity, each of which possess a linewidth that is remarkably narrow. The laser output is then acquired via 10% of OC3 and combined with the output of the BFL based on PT symmetric and SA effect via a 1∶1 OC4. By detecting the beat frequency between the two BFLs via a PD, the laser linewidth can be measured with an ESA. The electrical spectrum at the PD output is depicted in Fig. 7. The theoretical linewidth of BFL, according to Eq. (8), is about 10⁻⁸ Hz, surpassing the ESA measurement accuracy with 1 Hz. The linewidth under the -20 dB intensity point was determined to be 2.81 kHz, equating to 140.5 Hz 3-dB linewidth.

In conclusion, we have proposed and validated experimentally a BFL based on PT symmetric and SA effect with the unique characteristics of widely tunable wavelengths and high OSNR, while still maintaining narrow linewidth SLM lasing. To attain PT symmetry, we utilized a PM-EDF Sagnac loop composed of two polarization controllers and a PM-EDF. After stimulating SBS gain in the 10 km SMF with the pump light, the Stokes wave was injected into the Sagnac loop. The inherent birefringence of the PM-EDF generated two interconnected feedback loops for polarized orthogonally wave, with one loop experiencing SBS gain and the other experiencing SBS loss. The gain and loss of SBS could be regulated precisely by varying polarization state of the injected Stokes wave. When the gain and loss of SBS involving SA were perfectly matched and the gain exceeded the coupling coefficient, PT symmetry was broken, resulting in a high coherent BFL. By adjusting center wavelength of the the pump light, the output wavelength of BFL could be tuned. It is worth mentioning that this design does not require an EDFA to amplify the pump intensity, thus, the tunability of this BFL is not limited by EDFA bandwidth, allowing for a wider tuning range of wavelengths. Furthermore, due to the advantage of SA in PM-EDF, there are a bigger gain contrast value and a higher OSNR. The wavelength of the BFL was set between 1526.088 to 1565.498 nm, and its OSNR is high, reaching 77 dB. Our experiments demonstrate 140.5 Hz linewidth of the BFL based on PT symmetric and SA effect, with a measured linewidth of 2.81 kHz at the -20 dB intensity point. With its advantages of widely tunable wavelengths and high OSNR, this design holds great potential for many application fields such as optical bistablity, ultrahigh-coherent optical communication and optical fiber sensing.