## Abstract

We present a propagation model for the dynamics of distributed feedback Brillouin lasers. The model is applied to the recently demonstrated DFB Brillouin laser based on a $\pi $-phase shifted grating in a highly nonlinear silica fiber. Steady state results agree with the experimental values for threshold and efficiency. We also simulate a DFB Brillouin laser in chalcogenide and find sub-milliwatt thresholds and the possibility of centimeter-long Brillouin-DFB’s.

©2013 Optical Society of America

## 1. Introduction

There is much recent interest in distributed feedback lasers that rely on the gain produced by stimulated scattering in optical fibers [1–7]. This is because of the possibility for true single frequency operation at wavelengths that are not constrained by a gain medium but depend only on the choice of pump wavelength. Low-threshold lasing by stimulated scattering has been observed using $\pi $-phase shifted gratings in silica fibers [4–7]. On the theoretical front, DFB lasers based on stimulated Raman scattering were first proposed and analyzed by Perlin and Winful in 2001 [1,2]. To date, however, no theoretical study of DFB Brillouin lasers has been presented. There are marked differences in the nature of the two gain processes that necessitate a separate treatment for DFB lasers based on Brillouin gain. While Raman gain exists for both forward and backward waves, significant Brillouin gain occurs only for backward waves in conventional optical fibers (see Section 6). Furthermore, while the Raman gain bandwidth of several terahertz is orders of magnitude greater than the typical grating bandwidth of a few gigahertz, the Brillouin gain bandwidth is only of order tens of megahertz. Finally, the acoustic phonon lifetime being a few nanoseconds requires that the dynamics of the acoustic wave be included in the theory of DFB Brillouin lasers.

In this paper we present a detailed theoretical study of the dynamics of DFB Brillouin lasers. The theory is applied both to the experimentally demonstrated DFB Brillouin laser in silica fiber and to a proposed on-chip realization in a chalcogenide waveguide. We show that DFB Brillouin lasing can be observed in a chalcogenide chip as short as 1 cm with threshold power less than 100 mW and that sub-milliwatt thresholds are attainable in longer gratings.

## 2. The model

The geometry of the DFB Brillouin laser is shown in Fig. 1. It consists of a uniform grating of length *L* containing a $\pi $phase shift located at a distance *L _{1}* from the input end. The grating is excited at

*z =*0 by a pump wave of power ${\left|{A}_{p}^{+}(0,t)\right|}^{2}$ watts. Inside the grating the slowlyvarying amplitudes of the pump, Stokes, and acoustic waves satisfy the following coupled-mode equations:

## 3. Results for pi-phase-shifted DFB Brillouin laser in silica fiber

We apply the model described above to the recently demonstrated DFB Brillouin laser which employed a 12.4-cm-long $\pi $- phase-shifted grating in a highly nonlinear silica fiber [7]. The grating coupling constant was estimated to be $\kappa =90$m^{−1}. Brillouin lasing could be obtained primarily in the forward or backward direction by displacing the $\pi $- phase shift from the exact center of the grating. Figure 2 shows the reflectivity of such a grating with the phase shift at the center (solid line) and offset by 8% from the center (dashed curve). The pump laser is detuned by the Brillouin shift of 9.44 GHz away from the resonance at the Bragg frequency. This corresponds to ${\delta}_{p}L=\mathrm{36.77.}$ From Fig. 2 it can be seen that the pump will suffer a reflectivity in excess of 20% and hence a backward propagating pump may need to be included in the analysis as done here.

The threshold pump power can be estimated by using a result of Kremp, et al [11] for a $\pi $- phase shifted grating with an offset of $l\equiv {L}_{1}-L/2.$ The threshold gain for a Raman DFB laser is given by

This result assumes that the signal experiences bidirectional gain. For a Brillouin laser with a unidirectional pump, only the backward-propagating signal experiences gain and hence the gain required to reach threshold should be doubled. For $\kappa L>>1,$ the threshold gain for a Brillouin DFB laser is thus given byTo calculate the threshold we use the parameters appropriate for the OFS high nonlinearity fiber:*L*= 12.4 cm, $\kappa =90{\text{m}}^{-1},$ ${g}_{B}=1.78\times {10}^{-11}$ m/W, offset

*l =*9.9 mm, $\alpha =0.0064{\text{m}}^{-1}$, and fiber effective area ${A}_{eff}=11\mu {\text{m}}^{2}$at a wavelength $\lambda =1.5\mu \text{m}\text{.}$ We find the threshold power to be ${P}_{th}={\alpha}_{th}^{B}{A}_{eff}/{g}_{B}=26.6\text{mW}$. This is not far from the measured threshold of 30 mW [7].

We first solve the full set of Eqs. (1) by integrating along characteristics in the manner described in [12]. In addition to the parameters listed above we take ${\tau}_{B}=3.4$ns, $\gamma =11.5\times {10}^{-3}{\text{(Wm)}}^{-1}$ and $n=1.5$. The boundary conditions are

Figure 3(a) shows the transmitted and reflected pump as well as the forward and backward Stokes power as a function of time for a step input pump. Initially the transmitted pump drops from the incident value of 100 mW to about 75 mW because of reflections. Then, after a Stokes build up time of order microseconds, the SBS threshold is reached and the transmitted pump is depleted. The output Stokes radiation is primarily in the forward direction because the $\pi $- phase shift is located outward from the center of the grating. The conversion efficiency from incident power to forward Stokes is about 27%, in agreement with the experiment. The full set of equations with pump reflections are rather computationally intensive, taking about two days’ computing time for an un-optimized MATLAB code running on a 2.6 GHz laptop. In many cases, especially when the grating is apodized, one may neglect pump reflections and reduce the computational load by at least 50% by setting $\left|{A}_{p}^{-}\right|$ and ${\tilde{Q}}_{-}$ equal to zero. Figure 3(b) shows the result obtained by using the reduced set of Eqs. (1) to compute the transmitted pump and generated Stokes power. Surprisingly the Stokes output is about the same as that obtained with the full set of equations. In the full set of equations, even though the transmitted pump is lower because of reflections, the reflected portion does make some contribution to the forward Stokes power. It is somewhat fortuitous that in this particular case both the reduced and full set of equations yield the same efficiency. This happy circumstance does not hold in general and one should check the strength of pump reflections (using the grating reflection spectrum) before employing the reduced set. For the rest of this paper we will use the reduced set.

Figure 4 shows the power distribution along the grating. The Stokes power density at the phase shift is more than three orders of magnitude higher than the input power. The enhanced field at the location of the phase shift can result in nonlinear refractive index changes due to heating and the Kerr effect.

In the results presented above, the Stokes output was at the Bragg frequency, characterized by a detuning ${\delta}_{s}=0$. In the experiment of Abedin, et al [7], lasing was observed over a 1 GHz range, accompanied by hysteresis. This was attributed to a shift of the DFB cavity resonance owing to the Kerr nonlinearity and to non-uniform heating. To examine the tuning range we kept the pump power at 100 mW, varied the detuning ${\delta}_{s}$, and recorded the forward Stokes power for each value of ${\delta}_{s}$. We found that lasing occurs only within a 100-MHz range of the Bragg resonance. Our simulations do take the Kerr nonlinearity into account and thus we believe that its contribution to the refractive index change is insufficient to account for the 1 GHz tuning range seen in the experiment. On the other hand a temperature-induced shift of the resonance by 0.1 pm/mW [13] is well within the expected value and would be sufficient to explain the gigahertz tuning range seen in the experiment. We have not modeled this effect since it is strongly dependent on the particular thermal environment of the grating.

## 4. DFB Brillouin lasing in chalcogenide glasses

The Brillouin gain coefficient in chalcogenide glasses is two orders of magnitude larger than that of silica [14,15]. This should enable the operation of centimeter-scale DFB Brillouin lasers with milliwatt and sub-milliwatt thresholds in these glasses. We first consider a 1-cm-long grating in a rib waveguide made of the chalcogenide glass As_{2}S_{3} [15]. A short (centimeter-scale) Brillouin laser requires a very high *Q* resonator in order to achieve threshold with sub-Watt pump power. As it turns out, strong photo-induced gratings with coupling constants larger than 10^{4} m^{−1} have already been demonstrated in chalcogenide waveguides [16,17]. Taking a grating with $\kappa =2\times {10}^{3}{\text{m}}^{-1}$ and length *L* = 1 cm yields a coupling strength $\kappa L=20$ and a bandwidth of about 78 GHz, assuming a refractive index *n* = 2.45. This large bandwidth means that if a $\pi $-phase shift creates a transmission resonance at the Bragg frequency, the pump laser frequency, which has to be detuned by the Stokes shift of 7.5 GHz, would lie in the stop band. The pump laser would thus be rejected and hence there would be no Brillouin lasing. Instead of a $\pi $-phase shifted grating we consider a uniform grating with the pump tuned by at least a Stokes shift above the high frequency band edge. In this case the lasing modes would fall at the edge of the stop band where the Stokes radiation is enhanced by the finite-length DFB resonances and slow-light effects. (An alternative approach is to use a phase shift that is a small fraction of $\pi $ in order to locate the transmission notch closer to the band edge. The *Q* of the resulting resonance is much reduced, however.)

In the following simulations we neglect pump reflections and explore the behavior of a Brillouin DFB laser using parameters appropriate for the As_{2}S_{3} rib waveguide of Ref. 15: ${g}_{B}=0.71\times {10}^{-9}$m/W, ${A}_{eff}=2.3\times {10}^{-12}{\text{m}}^{2}$, ${n}_{2}=2.5\times {10}^{-18}{\text{m}}^{2}/\text{W}$, $\alpha =0.2\text{dB/cm}$, and ${\tau}_{B}=10\text{ns}$. For a uniform grating the threshold gain is approximated by [1]

^{−1}and hence a threshold pump power ${\alpha}_{th}^{B}{A}_{eff}/{g}_{B}=36\text{mW}$. In the simulations we checked that the full and reduced sets of equations yield similar conversion efficiencies.

Figure 5(a) shows the backward Stokes, forward Stokes, and transmitted pump power versus time when a pump of 250 mW is tuned such that the laser output corresponds to the first transmission resonance (m = 1) at the band edge. The inset shows that the spatial distribution of power is consistent with the first order resonance, with a peak three times higher than the incident pump. The conversion efficiency from incident pump to the combined Stokes output is only about 10%. This rather low efficiency is a result of the Kerr nonlinearity which detunes the transmission resonance from the peak of the Brillouin gain and suppresses further build up of the Stokes intensity. To check this we set the nonlinearity to zero in Fig. 5(b). It is seen that the pump is now completely depleted, with a conversion efficiency to forward and backward Stokes of close to 100%.

To achieve sub-milliwatt thresholds it would be necessary to use longer gratings and lower loss waveguides. To that end we consider a 12.4-cm $\pi $- phase shifted grating with $\kappa =90{\text{m}}^{-1}$ in a single-mode As_{2}S_{3} fiber. The measured Brillouin gain coefficient is ${g}_{B}=3.9\times {10}^{-9}$m/W in a fiber of mode area ${A}_{eff}=13.9\times {10}^{-12}{\text{m}}^{2}$and linear loss 0.57 dB/m [14]. Using these values and ${n}_{2}=2.5\times {10}^{-18}{\text{m}}^{2}/\text{W}$ in Eq. (3) we obtain a threshold pump power of only 540$\mu \text{W}$. For input pump values close to the threshold, the dynamic equations take an exponentially long time to reach steady state. We therefore simulate a chalcogenide DFB Brillouin laser with a pump power of 900 $\mu \text{W}$to confirm the sub-milliwatt threshold behavior. Figure 6 shows the exponential growth of the forward and backward Stokes signals from noise, along with the depletion of the pump. Here too the Kerr nonlinearity limits the conversion efficiency to less than 10%. In future work we will explore the possibility of pre-chirping the grating in order to mitigate the effects of the nonlinear index. The sub-milliwatt thresholds seen here should enable direct pumping by semiconductor lasers.

## 6. Conclusion

In this paper we have presented a model for the dynamics of DFB lasers with Brillouin gain. The model has been applied to the recently demonstrated DFB Brillouin laser and the simulation results reproduce the experimental observations with regard to threshold pump power and conversion efficiency. The model has also been applied to a proposed centimeter-scale DFB laser in a chalcogenide chip. Sub-milliwatt thresholds should be achievable in 10-cm-long waveguides. While this work has focused on conventional waveguiding structures in which only backward SBS has significant gain, recent research on nanophotonic structures has yielded strong forward SBS in silicon nanoscale waveguides mediated by radiation pressure [18,19]. This raises the exciting prospect of nanophotonic DFB Brillouin lasers on silicon chips and their possible integration with CMOS signal processing technologies.

## Acknowledgments

This work was done while HGW was on sabbatical at CUDOS, School of Physics, University of Sydney. Funding from the Australian Research Council (ARC) through its Laureate Project FL120100029 is gratefully acknowledged. This research was also supported by the ARC Center of Excellence for Ultrahigh bandwidth Devices for Optical Systems (project number CE110001018).

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