## Abstract

Numerical analysis predicts that continuous-wave Raman lasing is possible in silicon-on-insulator (SOI) waveguides, in spite of the detrimental presence of two-photon absorption and free-carrier absorption. We discuss in particular the dependence of the lasing characteristics of SOI Raman lasers on the effective lifetime of the free carriers generated by two-photon absorption. It is shown that the pump-power-dependent cavity losses lead to a rollover of the output-power characteristics at a certain pump-power level and that there exists an upper shutdown threshold at which the laser operation breaks down.

©2004 Optical Society of America

## 1. Introduction

Stimulated Raman scattering (SRS) in silicon-on-insulator (SOI) waveguides is an attractive way of amplifying optical signals in SOI planar lightwave circuits (PLCs), because it does not require the introduction of any dopants during fabrication of the SOI wafer and the amplified wavelength can be chosen almost freely by simply using pump light of an appropriate shifted wavelength. SRS on-off gains in SOI waveguides of up to 2.3dB have been observed in continuous-wave amplification experiments [1, 2, 3] and up to 12dB in pulsed-pumping configurations [4, 5, 6].

It has been recognized that the efficiency of the Raman-amplification process in an SOI waveguide can be degraded significantly by two effects [3–8]: First, two-photon absorption (TPA) in silicon noticeably depletes the pump wave at the power levels typically used (up to several Watts). Second, the accumulated free electrons and holes generated by TPA enhance the conductivity of the waveguide material, thus introducing additional optical losses through free-carrier absorption (FCA).

Recently, Raman lasing has been demonstrated in an SOI waveguide [9]. The authors used a pulsed pump laser with a pulse period much larger and a temporal pulse width much shorter than the lifetime of the carriers generated by TPA. Thus, free carriers can not accumulate significantly in the waveguide, and FCA is effectively suppressed.

In continuous-wave (CW) SOI Raman lasers, FCA can not be suppressed as easily, however, and the question arises whether CW-lasing operation is achievable at all. In this paper, we will show by numerical analysis that CW Raman lasing is possible in SOI waveguides in spite of the detrimental presence of TPA and FCA. An integrated-optical CW Raman laser could be an interesting option for a number of applications, such as a compact laser-wavelength converter, or for the on-chip conversion of the wavelength of one externally applied pump laser to almost arbitrary other wavelengths required for signal processing operations elsewhere on a PLC.

## 2. Modeling

#### 2.1. Mathematical model

Figure 1 shows the schematic of the SOI Raman laser setup we analyze in this paper. It consists of an SOI waveguide of length *L*, into the left-hand side of which pump-laser light at the wavelength *λ*_{p}
with power *P*_{0}
is coupled in with an efficiency of *T*_{p}
. Inside the waveguide, Raman scattering generates optical power at the Stokes wavelength *λ*_{s}
. We assume that the end surfaces of the chip have been coated such that the pump (“p”) and Stokes (“s”) wavelengths see reflectivities of *R*_{p,l}
, *R*_{p,r}
, *R*_{s,l}
and *R*_{s,r}
at the left-hand (“l”) and right-hand (“r”) sides of the chip, respectively.

In a straightforward extension of the models used previously to describe single-pass Raman amplification in SOI waveguides [2, 5, 7], we use a simple incoherent model for the SOI Raman laser, i. e., we describe the laser in terms of the longitudinally varying optical powers of forward-and backward-propagating waves and neglect any effects concerning phases. Thus, our model is similar to the standard model used to describe Raman fiber lasers [10], but will include additional terms describing the effects of TPA and FCA.

Given the assumptions above, we generalize the differential equations describing the longitudinal distributions of the pump and Stokes powers inside the SOI waveguide [5, 7] by taking into account pump depletion due to stimulated Raman scattering [10] and including counter-propagating waves, yielding the differential equations

for the forward- and backward-propagating pump waves ${P}_{p}^{+}$ and ${P}_{p}^{-}$ , respectively, and

for the Stokes waves ${P}_{s}^{\pm}$
. Here, *z* is the longitudinal coordinate, and *α* represents the linear waveguide losses (assumed equal for the pump and Stokes wavelengths). The effective Raman-gain constant *g*=*g*_{R}
/(1+Δ*ν*_{p}
/Δ*ν*_{R}
) is lower than the peak Raman-gain constant of silicon, *g*_{R}
, due to the convolution of the pump-beam spectrum of spectral width Δ*ν*_{p}
with the Raman-gain spectrum of spectral width Δ*ν*_{R}
. The modal effective area is defined as

where *I*(*x*,*y*) is the transverse intensity profile of the fundamental mode of the waveguide (assumed equal for the pump and Stokes wavelengths) and the integration is over the entire transversal plane. The constant *β* is the two-photon absorption coefficient of silicon, and *$\overline{\phi}$*=6.0×10^{-10} quantifies the efficiency of the FCA process [7, 11]. The effective charge-carrier density is

where we have used *ν*_{s}
≈*ν*_{p}
, *h* is Planck’s constant, and τ_{eff} is an effective charge-carrier lifetime. Typical values (measured and estimated) of τ_{eff} for SOI waveguides are in the range of 0.7…100ns [2–8]. Finally, the boundary conditions

model the reflection of the waves at the two end faces of the SOI waveguide (at *z*=0 and *z*=*L*) with the reflectivities *R*
_{{p,s},{l,r}} and the input coupling of the pump power *P*
_{0}.

#### 2.2. Verification of the model for the single-pass amplifier case

As continuous-wave Raman lasers have not yet been demonstrated experimentally, it is difficult to assess the validity of our complete model. However, our calculations show a very good agreement with the experimental results in the case of a recently demonstrated co-pumped CW amplifier in an SOI waveguide [3], which can be numerically treated simply by setting all reflectivities to zero and injecting a Stokes probe signal *P*
_{probe} together with the pump power *P*
_{0}, i. e., replacing the boundary conditions (5)–(6) with the initial values

The marks in Fig. 2 show the measured on-off gain for a Stokes signal as a function of the pump-laser power *P*
_{0} (data from Fig. 3 in Ref. [3]). The solid curve shows the result of our calculations. The simulation parameters are as given in Ref. [3]: launching efficiencies *T*_{p}
=*T*_{s}
=30%, *P*
_{probe}=2.6mW, *A*
_{eff}=1.57µm^{2}, *λ*_{s}
=1574.3nm, *λ *_{p}
=1455nm, *L*=4.8cm, *α*=0.22dB/cm, *β*=0.5cm/GW, *τ*=23ns. The only parameter that required fitting is the Raman-gain constant *g*, which was not given in Ref. [3]. The solid curve in Fig. 2 shows the numerically calculated gain for *g*=27cm/GW. This value is in accordance with the reported range for the peak Raman-gain constant in SOI structures of *g*_{R}
=20…76cm/GW [1, 2, 12].

## 3. Influence of laser parameters on the threshold and the input-output characteristics

#### 3.1. Simulation parameters

We initially concentrate on lasers with non-coated chip end-faces and assume that all reflectivities are due to the silicon/air interface, *R*_{p,l}
=*R*_{p,r}
=*R*_{s,l}
=*R*_{s,r}
=30%. Assuming ideal coupling of the pump power, *T*_{p}
=1-*R*_{p,l}
. For the value of the peak Raman-gain constant in SOI structures, values in the range of *g*_{R}
=20…76cm/GW have been reported [1, 2, 12]. We use a rather conservative value of *g*_{R}
=30cm/GW and a pump-laser linewidth of Δ*ν*_{p}
=50GHz [1]. Given the Raman-gain linewidth Δ*ν*_{R}
≈100GHz [1], we obtain an effective Raman-gain constant of *g*=20cm/GW. We assume a relatively large effective modal area of *A*
_{eff}=5µm^{2} [1]. The pump and Stokes wavelengths are *λ*_{p}
=1427nm and *λ*_{s}
=1542nm, respectively, separated by the silicon Stokes shift of 15.6THz.

#### 3.2. Numerical calculation of the threshold power

Around threshold, the Stokes powers are much smaller than the pump powers, and we can simplify Eqs. (1), (2) and (4) to

The laser is at threshold when the Stokes round-trip net gain equals the losses due to outcoupling at the left-hand and right-hand end faces with reflectivities *R*_{s,l}
and *R*_{s,r}
. From Eqs. (10) and (6) we thus obtain the threshold condition [10]

In order to find the threshold pump power *P*
_{th}, we numerically calculate the longitudinal pump-power distribution from Eqs. (9) and (5) for varying pump powers *P*
_{0} until ${P}_{p}^{+}$
(*z*) and ${P}_{p}^{-}$
(*z*) fulfill Eq. (12). The corresponding pump power *P*
_{0} is then the threshold pump power *P*
_{th}.

#### 3.3. Lasing threshold

We first look at the imaginary case of an SOI Raman laser in which both TPA and FCA are absent, i. e., we artifically set *β*=0 and τ_{eff}=0. The three solid curves in Fig. 3 show the threshold pump power of such a laser as a function of the chip length *L*, for three different loss coefficients *α*. The results suggest that it should be possible to pump an SOI Raman laser beyond threshold by using a pump laser with only a few Watts of output power, provided the effects of TPA and FCA are negligible.

Next we look at the influence that TPA has on the threshold pump power. The dashed curves in Fig. 3 show the threshold power as a function of the chip length *L*, when the TPA coefficient has the value *β*=0.7cm/GW as in Ref. [7], yet FCA is still assumed to be absent, i. e., all charge carriers are assumed to recombine instantaneously after generation and thus τ_{eff}=0. TPA evidently increases the required threshold pump powers, but only relatively weakly.

In contrast to the slight effect of TPA, the effect of FCA (τ_{eff}>0) can be much more dramatic, which is illustrated in Fig. 4. The linear waveguide losses are now fixed at *α*=1.0dB/cm. The dashed line again shows the threshold power as a function of the chip length for no TPA (*β*=0) for comparison, whereas the solid lines show the threshold powers in presence of TPA (*β*=0.7cm/GW) and several different charge-carrier lifetimes τ_{eff}. As expected, a larger τ_{eff} results in an increased threshold. Furthermore, there is a limited usable range of chip lengths outside of which the laser has no threshold at all. Outside this range, the waveguide will never start lasing, no matter how large the pump power is (e. g., for *L*=80mm and τ_{eff} larger than approximately 3.0ns, the device has no lasing threshold). For increasing τ_{eff}, the usable range becomes increasingly smaller, until at τ_{eff}≈3.2ns, it vanishes completely. In other words, there is a maximum effective carrier lifetime that can be tolerated for lasing. As the effective carrier lifetime τ_{eff} can vary relatively strongly from one waveguide to another waveguide even on the same wafer [5], it is important to consider the effect of a change in τ_{eff} and find a robust design which can tolerate such changes.

The origin of the limited usable chip-length range is the increase of the overall cavity losses with increasing pump power through the nonlinear absorption mechanisms TPA and FCA. If only linear losses were present, the overall cavity losses would remain constant with respect to the pump power, and for any given chip length *L* there would be a pump-power level above which the laser will start lasing [10].

#### 3.4. Input-output characteristics

Even for configurations inside the usable chip-length range, there is a continuing growth of the overall SOI cavity losses when increasing the pump-laser power beyond threshold. This can be seen in Fig. 5, where the input-output characteristics of several lasers with various effective carrier lifetimes τ_{eff} are plotted. These characteristics were calculated from the full model Eqs. (1)–(2), (5)–(6), and we defined the output power of the laser as *P*
_{out}=${P}_{s}^{+}$
(*L*)(1-*R*_{s,r}
). Directly above threshold, an increase of the pump power also increases the output power. However, there clearly exists a rollover point, i. e., a critical pump power beyond which a further increase of the pump power actually results in a decrease of the output power and, eventually, in a return to zero at the “shutdown threshold”. Thus, an increase of the pump power not only increases the Raman gain, but also increases the losses for both the pump and the Stokes waves through the nonlinear loss mechanisms TPA and FCA. This eventually leads to the breakdown of lasing operation at the shutdown threshold. Furthermore, Fig. 5 shows that for increasing τ_{eff}, the maximum conversion efficiency of the lasers dramatically decreases and the lasing and shutdown thresholds come closer to each other.

The shutdown-threshold power was in fact also obtained during the numerical threshold computations according to Sect. 3.2 - for every given chip length *L*, there is either no threshold at all or there are two threshold pump powers (assuming *β*,τ_{eff}>0). The solid curves in Fig. 4 show the lower of the two threshold powers (i. e., the lasing threshold), while the dotted curves show the upper threshold (i. e., the shutdown threshold). Lasing of the device can only take place between these two pump-power levels, with a maximum output power somewhere in between. The two solution branches merge at two limit points (the ends of the usable chip-length range), forming a closed egg-shaped curve, which narrows as τ_{eff} increases and eventually vanishes completely at the maximum tolerated effective carrier lifetime.

#### 3.5. Increasing the end-face reflectivities

As a last example we consider what happens when we apply coatings to the ends of the SOI chip in order to increase the reflectivities. Specifically, we chose left-hand and right-hand Stokes reflecitivities of 80% and left-hand and right-hand pump reflectivities of 0 and 100%, respectively, such that *T*_{p}
=100%. Figure 6 shows the calculated threshold power versus chip length and the input-output characteristics of the high-reflectivity SOI Raman laser for various effective carrier lifetimes. The thresholds are much lower than in the first laser (see Fig. 4). We attribute this to the increased Stokes reflectivities which result in lower cavity losses, and this together with the efficient pump-backreflection arrangement yields lower thresholds. Furthermore, the maximum tolerable carrier lifetime is now about twice as large as in Fig. 4, and the optimal chip length for minimum threshold power varies more strongly with τ_{eff}.

## 4. Conclusions

We have shown numerically that Raman lasing is possible in SOI waveguides with moderate pump-laser powers on the order of a fewWatts, in spite of the presence of the detrimental effects of two-photon absorption and free-carrier absorption. By using waveguides with smaller effective areas than the 5µm^{2} we have employed in our simulations and higher Stokes reflectivities, sub-Watt threshold pump powers should be possible.

## Acknowledgments

This work was funded by the Freie und Hansestadt Hamburg in the framework of the Verbund-projekt Hybride Mikrophotonik.

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