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Linewidth suppression mechanism of the self-injection locked single-frequency fiber laser (SFFL) is investigated theoretically and experimentally. An analytical model based on the semi-phenomenological approach is built up to characterize the optical feedback in SFFL. According to the theoretical prediction, the linewidth tends to be reduced with longer external cavity photon lifetime. Experimentally, a 200-Hz linewidth self-injection locked SFFL is achieved with 101 m long delay fiber, which agrees well with the theoretical simulation. The model provides a new perspective to understand the mechanism of linewidth reduction of self-injection locked SFFL.
© 2016 Optical Society of America
1. Introduction
Single-frequency fiber lasers (SFFLs) have attracted increasing interest due to their promising performances of compact all-fiber configuration, low noise and kHz-level linewidth, all of which make them versatile in the field of coherent communications, coherent beam combination, and high-precision sensing [1–5]. However, in some advanced fields, such as high-resolution laser spectroscopy [6], optical atomic clocks [7] and gravitational wave detection [8], where a narrower linewidth is of great concern, suppressing the linewidth of SFFL has become a matter of intense interest.
Up to now, numerous efforts have been devoted to suppressing the linewidth of SFFL [9–13]. Optical injection locking feedback is one of the well-developed linewidth reduction techniques. On the one hand, the intracavity loss of the injection locked SFFL is lower than those previously reported [12,13]. In such cases, intracavity optical devices including wave plate or filter are needed to achieve the linewidth suppression and thereby having a high intracavity loss. On the other hand, injection locking a SFFL with an external resonator, typically the whispering-gallery-mode resonator or high-finesse Fabry–Perot cavity, would significantly suppress the laser linewidth [14–17]. More recently, we have demonstrated a SFFL with a linewidth of 700 Hz by self-injection locking [18]. The reason for the linewidth reduction is generally ascribed to the high-quality (Q) factor of the external cavity. However, the linewidth would only reach MHz level even if the Q factor of the external cavity is as high as 108, which is much larger than the hundreds-Hz linewidth that was obtained by self-injection locking [18]. Therefore, there are still much works to be done to explore the linewidth suppression mechanism of self-injection locked SFFL. In fact, previous works have been done on the theoretical study of the self-injection locking of semiconductor lasers [19–21]. The evaluation of the optical feedback effect typically involves Langevin approach, with which an analytical expression for the linewidth of self-injection locked laser can be obtained [20–22]. Nevertheless, whether the available theory is capable of describing the self-injection locking process in fiber lasers is remained an open question.
In this paper, we derive an analytical model based on the semi-phenomenological approach to analyze the linewidth narrowing effect in a self-injection locked fiber laser. The obtained expression predicts that the linewidth tends to narrow with longer external cavity photon lifetime. The relation between the laser linewidth and the external cavity photon lifetime is investigated by lengthening the fiber in the external cavity. Ultimately, a 200-Hz linewidth SFFL is achieved by self-injection locking with 101 m long delay fiber, agreeing well with the semi-phenomenological approach. The extension of the photon lifetime is also verified by the low-frequency shifting of the relaxation oscillation frequency (ROF) and the suppression of the ROF peak, which is reduced by about 37 dB from −83 to −120 dB/Hz.
2. Principle
We start from the simplified two-level carrier density rate equations for SFFLs with the optical injection locking feedback, analogous with that in [20].
where n is the upper-level population, N is the number of the rare earth ion; P is the power of laser operation while Pp is the pump power (at the position of the output port); overlapping factors Γp, Γs are assumed to be 1; vp and vs represent the frequencies of the pump and signal, respectively; τ’ is the lifetime of the upper-level; A is the area of the gain fiber; σap, σa are absorption cross sections of the rare earth ion at the wavelengths of pump and laser output, σep, σe are emission cross sections of the rare earth ion at the wavelengths of pump and laser output; vg is the group velocity of the laser signal; α’ is the fiber attenuation coefficient; αp accounts for the gain (if positive) at the point of l = ls (the output boundary) and αp = (dP/dz)/P|ls when the laser power P is exponential along the fiber; Rs, accounting for the spontaneous emission, is negligible when evaluating the laser property (the detailed order of magnitudes are given in Table 1), but is crucial with respect to the calculation of phase noise. With regarding to the derivative dP/dt, two components dPosc/dt and dPinj/dt = 2κPfcos(ϕ0 + δϕ) are stemming from the main oscillator and self-injection contribute, where the prior is extracted from the regular model of the fiber laser while the latter can refer to the [21].
Table 1. Typical order of magnitude for the parameters used in the theory calculation.
Subsequently, we substitute the variable I = P/hvsAvg for the power P and Eq. (1) can be rewritten in another way; additionally, rate equation for phase shift ϕ of the cavity field E = I1/2e-iϕ is presented by virtue of a standard laser equation [19,21],
where exact expressions of C, S, G, γ and R are given in Table 1. α is the linewidth enhancement factor, which is associated with the refractive index change. w and w0 are the laser frequency in self-injection and free running operation, respectively. As to the term resulting from the external feedback, f = [I(t-τe)/I(t)]1/2 and δϕ = ϕ(t)-ϕ(t-τe) represent the effect of feedback coupling delayed by the external-cavity roundtrip time τe; for τe = 2ηle/c, η is the refractive index, le is the length of the self-injection locking optical path (delay fiber), c is the vacuum velocity of light. Via further introducing Langevin noise sources Fn(t), FI(t) and Fϕ(t) in Eq. (2), we derive the linewidth in terms of δf = <Δϕ2(t)>/2πt and refer the reader to [21] for detailed derivation. Typical orders of magnitudes are calculated based on the database of a heavily-doped gain fiber [23]. In this case, the approximations ΓIΓn/(GGnI), ΓI/G<10−2 used in [21] are also well satisfied in SFFLs and the self-injection locking effect on the linewidth δf is approximately given byδfo is the linewidth of the laser without the self-injection locking feedback, and frequency shift (w-w0) is revealed. In the absence of self-injection locking feedback, τe = 0, δf = δfo and w = w0. As validated in the following experiment, the linewidth of the self-injection locked fiber laser would be affected by the environmental noise [24]. To fit the exact narrowing course in fiber lasers, the coefficient κ characterizing the external feedback part develops to include a phenomenological parameter β; that is,where Rl is the facet power reflectivity, and Re is the external power reflection reflectivity. A notion of effective intra-cavity roundtrip time represented by β/τs is brought in, which influences the averaging process for the Strang splitting. Namely, from the physical point of view, the external noise is taken into account by assuming an effective cavity length. For β = 1, Eq. (3) reduces to its standard form in the case of noise free.In terms of Eq. (3), δf would be changed by the variation of τe, or le. The relation between δf and τe provides a new method to think of the linewidth suppression mechanism of the self-injection locked SFFL, which would be understood from the Schawlow-Townes theory [11, 25]: the linewidth of the SFFL is inversely proportional to the photon lifetime. Relevant works have validated that the linewidth would be reduced by longer photon lifetime [11,12]. With regard to the high Q factor self-injection locked laser, the Q factor can be expressed as: Q = 2πντ, τ is the photon lifetime, ν is the optical laser frequency. Longer photon lifetime would result in a higher Q factor. Therefore, the linewidth suppression mechanism of the self-injection locked fiber laser would be related to the photon lifetime. In what follows, we investigate the relation between the linewidth and the photon lifetime by using different length of the delay fiber. In addition, we validate the variation of the photon lifetime by the shift of the ROF and the change of the relative intensity noise (RIN).
3. Experimental setup and results
The experimental setup of the self-injection locking scheme of SFFL is shown in Fig. 1. The laser cavity is consisted by a piece of 1.3 cm-long highly Er3+/Yb3+-codoped phosphate fiber that acted as the gain fiber [26], a polarization-maintaining narrow-band fiber Bragg grating (NB-FBG), and a wide-band FBG (WB-FBG). The NB-FBG has a peak reflectivity of 60% and a 3 dB bandwidth of 0.08 nm; while that of the WB-FBG is > 99.95% with a 3 dB bandwidth of 0.3 nm. The spare end of the WB-FBG is cleaved to an angle facet to prevent the detrimental Fresnel reflection. The laser cavity is assembled into a copper tube and thermally stabilized through a thermoelectronic temperature controller (TEC) with a resolution of 0.05 °C to maintain a robust single-longitudinal-mode operation. The fiber laser is backward pumped by a 980 nm laser diode (LD) via a polarization-maintaining wavelength division multiplexer (PM-WDM). The output laser is then launched into the port 2 of apolarization-maintaining optical circulator (PM-OC) from the signal port of the PM-WDM. Then a 10/90 optical coupler is employed to split 10% of the laser power to the port 1 of the PM-OC fusion-splicing with different length of delay fiber to achieve the self-injection locking.
During the experiment, the LD is driven by a 200 mA current, and the laser output power is 12 mW at a free-running state. In addition, 10% of the laser is self-injected back into the laser cavity through the PM-OC and the output power from the 90% port of the coupler is 10 mW. As shown in Fig. 2, the single-longitudinal-mode operation is verified by the result of the scanning Fabry-Perot interferometer (FPI) and spectrum analyzer. From the laser signal of the FPI and frequency spectrum, no beating signal appears. It is concluded that the fiber laser operated stably in a single-longitudinal-mode status.
Fig. 2 Longitudinal mode characteristics of self-injected SFFL and spectrum analyzer. Inset: the result of the scanning Fabry-Perot interferometer.
3.1 Verification of the correction factor β
In order to investigate the influence of external noise on the SFFL, contrast experiments of the linewidth of the initial free-running laser and the self-injection locked laser in different conditions have been done respectively. The linewidth is measured with the loss-compensated recirculating self-heterodyne method, which involves a 48.8 km long fiber and a fiber coupled acoustic optical modulator with a frequency shift of 40 MHz.
The lineshape of the second-order heterodyne signals are demonstrated in Fig. 3, which show the linewidths of the initial free-running laser and the self-injection locked laser in different condition. As shown in Fig. 3(a), the linewidths of the initial free-running laser are almost the same (1.5 kHz) in both case, while that of the self-injection locked fiber laser (Fig. 3(b)) is extended from 900 Hz to 2 kHz with noise. The noise in the experiment is made by a loudspeaker, which is driven by a signal generator. It is concluded from the contrast experiments that the self-injection locked fiber laser would be affected by the external noise. Therefore, it is reasonable to add β in the theoretical model.
Fig. 3 Measured self-heterodyne spectra of the fiber laser in different conditions. (a) Initial free-running laser (b) Self-injection locked fiber laser.
3.2 Linewidth reduction by lengthening the delay fiber
As we mentioned before, δf would be changed by the variation of τe according to the theoretical analysis. Experimentally, we investigate the relation between the linewidth and τe by changing the length of the delay fiber le. Figure 4 shows the linewidth of the laser with 22, 36, 62, 101 m long delay fiber respectively, from which the linewidth reduction by lengthening the delay fiber is verified.
As shown in Table 2, the linewidth of the initial free-running laser is 1.5 kHz, while that of the self-injection locked laser is remarkably compressed by lengthening the delay fiber. The linewidth of the laser is about 200 Hz with 101 m long delay fiber. The linewidth suppression ratio in Table 2 is defined as the logarithmic ratio of δf to δfo: 10log(δf / δfo).
Table 2. Length of the delay fiber, corresponding linewidth and suppression ratio.
Ultimately, it can be revealed from Fig. 5(a) that the linewidth suppression ratio is in agreement with the theoretical simulation. Therefore, both the theoretical and experiment result validate that the laser linewidth compression can be achieved by extending the external cavity photon lifetime. The simulation is carried on by Eq. (3): firstly, the frequency shift (w-w0) is numerically solved through the nonlinear equation (the shift is presumably within a hundred MHz), resulting in the flipping nature illustrated in Fig. 5(a); then, the suppression ration of linewidth is calculated. The parameters used are: η = 1.5, β = 8.5 × 10−3, κ = 4.6 × 108 (τs = 0.36 ns, Rl = 60%, Re = 10%), w0 = 1.2161 × 103 THz, α = 3.5. Due to the resolution of our linewidth measurement system, the lineshape occurs coherence when longer optical fiber (>101 m) is used in the experiment. As a result, we can only know that the linewidth becomes narrower but can’t obtain the explicit value of the linewidth if longer optical fiber (>101 m) is exploited. In addition, when the external cavity mode separation gets close to the relaxation resonant frequency of the fiber laser, the laser is susceptible to mode hopping or multi-longitudinal-mode [27]. The relaxation resonant frequency of the fiber laser is about 700 kHz, corresponding to an external cavity length of 300 m. Hence, the external cavity length should be kept below this value to avoid mode hopping. In addition, in order to corroborate this standpoint, the beat frequency result of a single-frequency fiber laser and the self-injected fiber laser with an optical fiber of 465 m is given in Fig. 5(b). As a consequence, the multi-peak structure indicates that the self-injected fiber laser with a longer optical fiber is in multi-longitudinal-mode operation.
Fig. 5 (a) The linewidth suppression ratio of experiment data and the simulation. (b) Beat frequency result of a single-frequency fiber laser and the self-injected fiber laser with an optical fiber of 465 m.
3.3 Validation of the extension of the external photon lifetime
As demonstrated in [12,13], the extension of the photon lifetime would be verified by the low-shift ROF and the suppression of the RIN. The RIN of the fiber laser is measured by an InGaAs photoelectric detector with the bandwidth of 150 MHz and an electrical spectrum analyzer [28]. Figure 6 shows the RIN of the initial free-running laser and the self-injection locked laser. It can be revealed that the ROF gradually shifts to lower frequency by lengthening the delay fiber. It is concluded that the external photon lifetime, which is the time needed for the light field to travel back and forth the external cavity, is extended by the increase of the length of the delay fiber. In addition, a maximum noise suppression of more than 37 dB of the intensity noise peak around the ROF is achieved from −83 to −120 dB/Hz.
Fig. 6 The relative intensity noises spectrum of the self-injection locked fiber laser with different length of delay fiber.
4. Conclusion
In conclusion, we have derived a semi-phenomenological approach for the linewidth of self-injection locked fiber laser by means of the Langevin approach, which predicts that narrower linewidth would be obtained with longer external cavity photon lifetime. Experimentally, the relation between the linewidth and the external cavity photon lifetime is investigated with different length of fiber in the external cavity. A 200-Hz linewidth self-injection locked SFFL is achieved with 101 m long delay fiber, agreeing well with the theoretical predictions. Both the theoretical and experiment result suggest that the laser linewidth compression can be achieved by the extension of the external cavity photon lifetime, which provide a new perspective towards understanding the mechanism of linewidth reduction of self-injection locked SFFL.
Funding
China State 863 Hi-tech Program (2014AA041902); Natural National Science Foundation of China (NSFC) (61535014, 51132004, 51302086, 51322208, 11174085); Fundamental Research Funds for Central Universities (2015ZP013 and 2015ZM091); Guangdong Natural Science Foundation (S2011030001349, S20120011380, 2016A030310410); China National Funds for Distinguished Young Scientists (61325024); Science and Technology Project of Guangdong (2013B090500028, 2014B050505007); The cross and cooperative science and technology innovation team project of the CAS (2012-119), China.
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