Lasing in ring resonators by stimulated Brillouin scattering in the presence of nonlinear loss
SAYYED REZA MIRNAZIRY,1,2,* CHRISTIAN WOLFF,1,2 M. J. STEEL,1,3 BLAIR MORRISON,1,4 BENJAMIN J. EGGLETON,1,4 AND CHRISTOPHER G. POULTON1,2
1Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), Australia
2School of Mathematical and Physical Sciences, University of Technology Sydney, NSW 2007, Australia 3MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University Sydney, NSW 2109, Australia
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4Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, NSW 2006, Australia
Abstract: We theoretically investigate lasing due to stimulated Brillouin scattering in integrated ring resonators. We give analytic expressions and numerical calculations for the lasing threshold for rings in the presence of for both linear and nonlinear loss. We demonstrate the operation of the ring in the different regimes of amplification and lasing, and show how these regimes depend on the coupling to the ring and on the nonlinear parameters. In the case of nonlinear losses, we find that there can exist an upper threshold to the lasing regime where the losses are dominated by free-carrier absorption. We also find that nonlinear losses can inhibit Brillouin lasing entirely for certain ranges of coupling parameters, and we show how the correct ranges of coupling parameters can be calculated and optimized for the design of integrated Brillouin lasers.
⃝c 2017 Optical Society of America
OCIS codes: (190.0190) Nonlinear optics; (190.2640) Stimulated scattering, modulation, etc.; (230.1040) Acousto-
optical devices.
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Journal © 2017 Received 7 Jul 2017; revised 23 Aug 2017; accepted 29 Aug 2017; published 18 Sep 2017
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1. Introduction
Stimulated Brillouin Scattering (SBS) is a strong nonlinear interaction between optical fields whereby energy is transferred between closely-spaced spectral lines by means of hypersonic waves [1]. SBS is important in several photonics applications, including the fast processing of radio-frequency signals [2], sensing [3], and the generation of ultra-narrow-linewidth sources [4, 5]. Of particular interest is the use of SBS in integrated platforms, which give significant advantages in terms of stability and device size [6]. Although the SBS gain can be significantly enhanced in nanowire waveguides [7], achieving useful levels of Stokes amplification still requires relatively high pump powers and waveguide lengths on the order of centimeters [8, 9]. One way of circumventing these limitations is to use high quality-factor integrated ring resonators, where the build-up of power in on-resonance pump and Stokes waves can dramatically improve input power requirements. Ring resonators are commonly employed to enhance nonlinear effects in a range of photonics applications [10–12], and recent experiments have demonstrated SBS in hybrid silicon-chalcogenide racetrack structures [13] and Whispering Gallery Mode (WGM) resonators [14]. The combination of a high-Q cavity and gain also opens up the possibility of SBS lasing, which is essential for many of the proposed SBS-based applications [15, 16]; SBS lasing has thus far been demonstrated in fibre ring structures [17, 18], and integrated ring resonators have recently been proposed both for SBS-based amplification [19] and for lasing [20, 21].
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Despite these recent studies, a quantitative picture of the physics of SBS lasing in integrated ring resonators, including such important effects as nonlinear losses, does not yet exist in the literature. Although several papers give expressions for the lasing threshold for the closely- related case of Raman scattering (see for example [22, 23]) and these expressions have been used (though without a formal derivation) in SBS lasing experiments [13], it is not known in which situations these formulas can be correctly applied, and a full derivation of the lasing threshold for SBS in integrated ring structures is currently lacking. As a result is sometimes not clear as to exactly when lasing occurs for SBS, as distinct from regimes where the Stokes signal is strongly amplified. This distinction is particularly problematic in semiconductor platforms, in which nonlinear losses, in particular Free-Carrier Absorption (FCA), can significantly affect the physics of the SBS interaction and will strongly affect the achievable SBS gain [24]. In the case of straight waveguides, nonlinear losses lead to the existence of an optimal waveguide length for SBS gain, as well as a maximum amplification of the Stokes signal; it is not immediately clear how these effects carry over to ring resonators, and how these losses affect the transition from SBS amplification to SBS lasing.
Here we theoretically and numerically investigate SBS lasing in ring resonators in the pres- ence of linear and nonlinear optical loss. This analysis provides a better understanding of the lasing mechanism in rings with materials such as silicon or germanium in which higher order optical losses are non-negligible. We adopt the formalism outlined in [19], in which techniques were given for the computation of SBS in the amplification regime. Building on that work, we here study the different regimes of operation of the ring while focussing on the transition between amplification and lasing, and compute the threshold powers for this transition in the presence of nonlinear losses. We derive analytic expressions for the lasing threshold and investigate the effect of nonlinear losses. This derivation follows that of [25], in which the threshold is derived without noise initiation of the Stokes. While the derivation of the lasing threshold follows from a small-signal approximation, we also provide and analyse a full model including the large signal terms, and compute the Stokes amplification and the Stokes output power for realistic ring resonator parameters. These results can therefore be used for resonator design and for comparison with experimental results. We also discuss the physics of SBS lasing in rings: in rings with nonlinear loss we show that there exists a finite power interval over which lasing occurs; we compute this interval and provide design parameters that can be used for SBS- based ring resonator structures. Our study is in particular useful for realizing SBS lasing in chip scale devices. It provides sufficient information about optimum cavity inputs/outputs as well as the required physical parameters, optical and acoustic properties in a ring configuration to be used in designing integrated racetrack resonators for SBS lasing.
2. Geometry and numerical computations
Following [19], we consider the geometry of a ring resonator sketched in Fig. 1. A ring of length
L is coupled to a single straight waveguide via a coupling region; for the sake of simplicity
we assume that the SBS gain occurs only in the ring section and not in the coupler. Here we
consider the backward SBS process, in which input pump Pin and contra-propagate; ps
the forward SBS process can be handled using the same formalism, appropriately modified to account for the different propagation direction. This modification however, does not lead to a change in the magnitude of resultant threshold powers or in output Stokes demonstrated in this paper, assuming identical gain, loss coefficients, coupling and physical properties (i.e. ring length) in the Forward SBS process to those of backward SBS.
Throughout this investigation we are assuming that the pump and the Stokes are aligned to two cavity modes, and that the free-spectral-range (FSR) of the ring is equal to an integer multiple of the Brillouin shift. Resonant coupling Brillouin lasers are in particular efficient — in terms of the required pump power to achieve lasing — for chip scale ring resonators in
Vol. 25, No. 20 | 2 Oct 2017 | OPTICS EXPRESS 23621
which pump and Stokes frequencies are designed to lie on close resonances of the cavity. In practice nonlinear dispersion (Kerr, free-carrier dispersion, and frequency pulling due the phase shift arising from the SBS process itself) as well as thermal effects can result in a change in the FSR of the cavity, however for typical power levels (as discussed in [26]) such changes are far smaller than the SBS linewidth and can be neglected. The absolute values of the cavity mode frequencies are however expected to shift markedly. We therefore assume that active stabilisation of the pump is used to track the frequency of the cavity mode; the shift of the Stokes frequency will then automatically continue to lie on-resonance. Note that this implicitly assumes that the variation in pump power is slow enough to accommodate thermal effects; throughout this paper we therefore operate in the quasi-CW regime, in which it is assumed that the pulse lengths are much longer than the phonon lifetime — in most platforms this is on the order of 10 ns. A study of full dynamic response, including the stability, of these devices we leave to future investigations.
The powers in the pump and Stokes within the ring are then governed by the equations [24]
where Pp and Ps are the circulating pump and Stokes powers, respectively; α, β and γ are the linear, TPA and FCA-induced loss coefficients respectively, and Γ is the SBS gain expressed in units of W−1m−1. We also note that pump and Stokes powers take positive values in our model, as opposed to the formalism where counter-propagating waves have negative powers [24]. We have assumed here also that both pump and Stokes are in the same optical mode, and so the nonlinear coefficients are identical in both equations; this is realistic given the close spectral spacing of pump and Stokes for the SBS interaction, but could be generalized at the expense of complicating the formulation. Furthermore, in Eqs. (1) and (2) we have not considered noise, as arising from spontaneous emission or from thermal phonons. As a result, this model does not allow noise-related predictions, such as the linewidth in the lasing regime. The focus of this work is to study the dependence of the lasing threshold on the various system parameters including coupling, nonlinear loss and pump power. A corresponding study of the noise properties, beginning from the coupled amplitude equations, is beyond the scope of this current work.
On resonance, the values of the pump power at the beginning (z = 0) and at the end (z = L) of the ring segment are related to the input pump power by [19]
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= −(α+βPp+γPp2)Pp−(2β+4γPp+γPs+Γ)PpPs, (1) dPs = (α + βPs + γPs2)Ps + (2β + 4γPs + γPp − Γ)PpPs, (2)
in 1 2 Pp = |κ|2 Pp(0) − |τ| Pp(L)
, (3) where κ and τ are the complex envelope coupling coefficients as depicted in Fig. 1, related by
|τ|2 + |κ|2 = 1 [25,27]. Similarly, the values of the Stokes are related via the coupling region by
in 1 2 Ps = |κ|2 Ps(L) − |τ| Ps(0)
In writing Eqs. (3) and (4) we have implicitly assumed the coupling coefficients κ and τ are the same for both pump and Stokes and do not change with frequency over very small (GHz) ranges studied here. The system of Eqs. (1)-(4) encapsulates the physics of the ring operation: the input pump is transferred to the ring at z = 0, experiences both linear and nonlinear loss as well as loss to the other mode, and then is partially transferred to the output. The Stokes on
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Fig. 1. Schematic of a ring resonator in vicinity of a straight coupler.
6 5 4 3 2 1 0
αL = 0.2 αL = 0.25 αL = 0.3
5 10 15 20 25 30 35 40 45
Pin [mW] p
R = 10−5 R = 10−7 R = 10−9 R = 10−11
P so u t [ m W ]
Psout [μW]
Fig. 2. Output Stokes power as a function of input pump power at the lasing region and resonant condition in the presence of (a) linear losses and (b) both linear and nonlinear losses.In(a)Γ= 500W−1m−1,R = 10−11 andκ = 0.31.In(b)αL = 0.2; γ = 1.8 × 105 W−2m−1, β = 10 W−1m−1, κ = 0.16 and Γ = 4000 W−1m−1. The length L = 10.879 mm corresponds a ring resonator with free spectral range equal to a Brillouin frequency shift of 10 GHz.
R = Psin in Pp
0 10 20 30 40 50 60
Pin [mW] (b)
the other hand is input at z = L and experiences gain with decreasing z, as well as linear and nonlinear loss.
Figure 2 shows the output Stokes power resulting from the numerical solution of Eq. (1)-(4),
for a ring resonator with (a) linear losses only, and (b) with both linear and nonlinear losses. To
solve these equations we apply the numerical approach presented in [19] in which pump and
Stokes are computed using an iterative shooting technique, by which the differential equations
are solved at each step of the iteration using a Runge-Kutta method, and the mismatch in the
boundary conditions becomes a measure of the closeness to the true solution. In Fig. 2(a) the
output Stokes is computed for a ring with Γ = 500 W−1m−1 keeping the coupling constant fixed
at κ = 0.31 and changing the values of αL in order to highlight the impact of the linear loss. A
clear threshold pump power can be seen, denoted by a sharp increase in Stokes power; the value
of this threshold increases with the linear loss. The effect of nonlinear losses can be seen in the
example shown in Fig. 2(b). Here we have assumed that αL = 0.2; γ = 1.8 × 105 W−2m−1,
L = 10.879 mm, Γ = 4000 W−1m−1, which are close to experimentally-realisable values for
a silicon nanophotonic waveguide, and we have neglected TPA (β = 0 W−1m−1), which has
a negligible direct effect on SBS in silicon. The Stokes power is depicted as a function of
input pump power for different values of the ratio between input pump and Stokes, denoted by
R = Pin/Pin. In order to have an idea for the order of magnitude of R in the lasing regime, we sp
can assume that only a single Stokes photon is initializing the lasing. Then Pin hf
where vg is the optical group velocity and h fs is the Stokes photon energy (h Plank’s constant and fs Stokes frequency). As in the linear case (Fig. 2(b)), the Stokes increases rapidly with the input pump beyond threshold. This trend however is reversed after the Stokes power rises to a maximum value; thereafter, the Stokes power decreases as nonlinear losses begin to dominate. For smaller power ratios R, the Stokes falls abruptly to negligible values once a second, higher threshold is crossed. For higher power ratios, we se
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