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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, the investigation and detailed modeling of a cascaded Raman laser, operating in the midwave infrared region, is described. The device is based on silicon-on-insulator optical waveguides and a coupled resonant microcavity. Theoretical results are compared with recent experiments, demonstrating a very good agreement. Design criteria are derived for cascaded Raman lasers working as continuous wave light sources to simultaneously sense two types of gases, namely C_{2}H_{6} and CO_{2}, at a moderate power level of 130 mW.

For over two decades compact, broadly tunable, energy efficient midwave infrared (MWIR) and longwave infrared (LWIR) sources and devices have been the topic of active research [

Laser-based gas sensing is attractive because it can provide a way to achieve highly sensitive, real-time, _{4}, C_{2}H_{6}, CO_{2}, NH_{3}, N_{2}O, SO_{2}, H_{2}O, can be detected using lasers operating in the mid-IR wavelength range 2–10 μm. Thus, the use of mid-IR lasers is expected to greatly increase the sensitivity of gas sensing and reduce the optical path length and system sizes.

Recently, mid-IR laser sources based on difference frequency generation in quasi-phase-matched (QPM) LiNbO_{3} have been studied as promising candidates for these applications because they can provide continuous wave (CW) mid-IR light in the wavelength range 2–5 μm at room temperature. However, in spite of their excellent sensitivity, the low conversion efficiency of the conventional QPM-LiNbO_{3} devices has limited their use because large, expansive, high power lasers must be also used to achieve a reasonable amount of mid-IR optical output [

Another approach recently proposed to realise mid-IR light sources is based on Stimulated Raman Scattering (SRS) effect. In fact, one of the major advantages of Raman lasers is their ability to generate coherent light in wavelength regions that are not easily accessible with other types of lasers [

Therefore, in this work we theoretically analyze the possibility to realise a Raman cascaded laser for simultaneously sensing two different gases having their absorption peaks in the wavelength range 3–5 μm. Our choice is motivated by two concurrent aspects. First, the cascade Raman emission can result attractive in the range 3–5 μm, since two photon absorption (TPA) and free carrier absorption (FCA) effects are completely eliminated. Second, the cascade Raman laser could represent an efficient answer to the drawbacks of interband diode lasers and quantum cascade lasers. In fact, the ability to induce the cascaded lasing by means of SRS effect can potentially produce several wavelengths in the range 2–5 μm, at the same time too long for interband diode lasers to be reached due to Auger recombination, and usually too short for quantum cascaded lasers owing to the finite-conduction-band offset at room temperature [

This paper is organized as follows. In Section 2 we derive the mathematical model to study the nonlinear effects in a resonant microcavity coupled to an external waveguide, following a different approach from literature [

In this section, a very accurate physical model proposed in our previous works [

Modeling of cascaded Raman laser in mid-IR region has been already proposed in literature [

In our analysis we assume the architecture as sketched in _{p}_{coup}

The scheme of

Hereinafter, we assume a SOI waveguide as sketched in _{s}

In a single-mode SOI waveguide two propagating modes are typically confined, one quasi-TE (dominant x-component of electric field) and one quasi-TM (dominant y-component). Our coupled-mode approach describes the power transfer among pump wave (_{1}) and higher-order Stokes waves. Two higher-order Stokes waves (_{2},_{3}) and both polarizations are considered.

Under assumption of translational invariance along the propagation direction (z), due to large radius ^{jk̄z}where ^{(}^{TE}^{)}, 2 = ^{(}^{TM}^{)}, 3 = s_{1}^{(}^{TE}^{)}, 4 = _{1}^{(}^{TM}^{)} 5 = _{2} ^{(}^{TE}^{)} 6 = _{2}^{(}^{TM}^{)} 7 = _{3}^{(}^{TE}^{)} 8 = s_{3}^{(}^{TM}^{)} In this equation, summations in _{p}_{,}_{m}_{s1}_{,}_{n}, ω_{s2}_{,}_{r}, ω_{s3}_{,}_{q}_{i,m}

As detailed in [_{p}_{p}_{s}= ω_{p}_{R}_{R}_{p}_{s1}_{p}n_{eff},_{p}L_{Cavity}/_{s}_{1} _{eff},_{s}_{1} _{Cavity}/_{s}_{2} _{eff},_{s}_{2} _{Cavity}/_{s}_{3} _{eff},_{s}_{3} _{Cavity}/_{eff}_{,}_{p}_{eff}_{,}_{s}_{1}, _{eff}_{,}_{s}_{2}, _{eff}_{,}_{s}_{3}) the effective index of pump (Stokes) wave inside the racetrack resonator, c the light velocity in vacuum, and _{p}, ω_{s1}, ω_{s2}, ω_{s3}

Moreover, the previous consideration holds if the cavity free spectral range (FSR) is larger than the input pulse bandwidth, δ_{pulse}_{FWHM}_{pluse}_{cavity}_{FWHM}_{NL}

Then, we have for quasi-TE and quasi-TM first (

In _{k}=ω_{s}_{1}, _{k}_{ωs1,n}μ_{k}_{s2}, ω̄_{k}_{s2r̄} μ_{p,m}_{p}_{s}_{1,}_{n̄} _{,}_{s}_{1}), (_{s}_{2,}_{r̄,} − _{s}_{2}), and (_{s}_{3,}_{q̄}_{,}_{s}_{3}) designate the mismatch from the resonance condition of input pump and Stokes waves originated inside the resonator, respectively. The term _{ζ}_{κ}, τ_{ρ}_{l}_{,}_{ζ}_{c}_{,}_{ζ}_{g,ζ}_{κ}_{ρ}

In _{R}_{s}_{1}_{0}^{R}_{R}_{eff,p}n_{eff,s}_{1}) [_{0} the magnetic susceptibility, _{eff,p}_{eff,s1}^{R}_{R}_{R}_{R}

Moreover, the coefficients _{i,i}_{2}_{i}f_{i,i}_{i,j}_{2}_{i}f_{i,j}_{2} the nonlinear refractive index [_{i,j,k,l}_{2}_{i}f_{i,j,k,l}/c, i,j_{i,j}_{i,j,k,l}

In particular,
_{i,i,j,j}_{i,i,j,j}_{i,j,k,l}

However, since these terms can be considered as negligible they are not included in

For completeness, group velocity dispersion (GVD) and third-order dispersion (TOD) effects are also included in the model, as indicated by the terms proportional to _{2}_{,i}_{3}_{,i}_{FWHM}_{c,ζ}_{FWHM}_{c,ζ}

Finally, it is worth to note that TPA and FCA effects are not included in our equations due to excellent transmission of silicon in mid IR because of the absence of TPA for wavelengths longer than 2.2 μm [

To test the analytical formulas and physical assumptions for Raman effect into optical microcavities under CW operation, we have compared our numerical results with some experiments proposed in literature. A very interesting set of comparisons with experimental results involves the CW cascaded Raman laser based on SOI resonator. The architecture used in the experimental setup [_{loss}_{eff}_{R}^{TPA}^{TPA}

It is very important to consider also TPA and FCA effects in _{c}_{e}_{h}_{0} = 1.45 × 10^{−17} cm^{−2} [_{i}_{eff}^{TPA}_{g,ζ}β^{TPA}f_{ζ,ζ}_{ζ}^{2}_{ζ}_{g,κ}β^{TPA}f_{κ,κ}_{ζ}^{2}_{κ}_{g,ρ}β^{TPA}f_{ρ,ρ}_{ζ}^{2}_{ρ}

The solid lines designate our numerical results as evaluated by solving the coupled equations proposed in the model. Our numerical calculations demonstrate the same group velocity inside the resonator for pump and Stokes waves, and thus the partial differential _{g,p}_{g,s}_{1} ≅ ν_{g, s}_{2} = ν_{g}^{7} m/s. The plots show a very good agreement with experimental data in terms of threshold values, output powers above threshold, external efficiencies and output saturation. In particular, the good agreement with first Stokes output saturation confirms how the Raman cascaded lasing is well described by the mathematical model proposed in this paper.

To the best of our knowledge, systematic investigations have been not yet presented in literature for optical properties of SOI waveguides in mid-IR region, although this is a topic of increasing interest [

As it is well known, a single-mode waveguide can in general support two modes polarized in orthogonal directions. Under ideal conditions, a mode excited with its dominant polarization, i.e., in x direction (quasi-TE mode), would not coupled to the mode with orthogonal dominant y-polarization state (quasi-TM). However, in real waveguides random variations of cross section shape and stress-induced anisotropy result in a mixing of two polarization states. Thus, the two modes exchange their powers in a periodic fashion as they propagate inside the waveguides with period
_{i,j,k,l}_{2} and to the products between four electric field components. Some of these terms are responsible for SPM and XPM effects, while the remaining terms in _{i,j,k,l}

_{s}

The plots show that for each value of the rib width, the modal birefringence changes sign with increasing _{eff}_{B}_{cavity}_{B}_{i,j,k,l}_{cavity}_{B}_{i,i,j,j}_{B}_{i,j,k,l}_{B}_{cavity}_{B}

Another important optical characterization of SOI waveguides in mid-IR region can be made in terms of group velocity mismatch between quasi-TE and quasi-TM modes.

As a general trend, the spectral curves of

Recently, efficient Raman lasing in silicon has been demonstrated in NIR region [

Without any loss of generality, hereinafter we focus our attention on the design of a Raman laser source for ethane gas sensing. In _{s}_{1} = 3.3485 μm for ethane gas detection. This implies a laser pump tuned at _{p}_{s}_{2} = 4.0544 μm and _{s}_{3} = 5.1376 μm, respectively.

In the following analysis, we guess that the pump wave is aligned with quasi-TE polarized mode (as usually occurs in most experimental set-ups). Under this assumption, _{2} is zero and then all terms in _{i,j,k,l}_{2} have to be set to zero in

For the following discussion, we assume that the first-order Stokes wave grows-up being mainly aligned as a quasi-TM mode. This assumption will be verified in the next sub-section, due to smaller optical area and thus larger Raman modal gain of quasi-TM over quasi-TE mode. Under this assumption, we can evaluate as the walk-off parameter between quasi-TM first-order Stokes wave and quasi-TE pump is influenced by the waveguide sizes. Then,

As a general trend,

As demonstrated in [_{w}_{0}/‖_{TM,TE}‖ could be shorter than the cavity length (_{0} = _{FWHM}_{w}_{cavity}_{w}

_{B}_{B}_{B}^{st}

Thus, an optical resonator with _{cavity}

Further, we show in

Now, both optical waveguides and minimum cavity length are designed and estimated. Another very important aspect is to find the design guidelines for the directional coupling. In fact, as shown by a formula derived in [_{ζ}_{,max}/_{ζ}_{ζ}_{,max} the pump amplitude maximum inside the cavity and _{ζ}

Due to the absence of detrimental effects induced by FCA and TPA contributions in mid-IR, it is possible to estimate _{l}_{,}_{ζ}_{c}_{,}_{ζ}

The previous guideline is rigorous if we are mainly focusing on first-order Stokes wave emission. In the case of cascaded Raman laser [

A number of 3D simulations based on BPM show a good trade-off between previous requirements when we consider a gap _{coup}

The plot also shows a quasi-TE pump wave coupling factor of 5% at _{coup}_{cavity}

Using the waveguide sizes and coupling factors, we can progress to the cascaded Raman laser simulation in the range 2.85 ÷ 4.0544 μm. According with some experimental works (i.e., [_{R}_{2} = 5 × 10 ^{−5} cm^{2}/GW, ^{TPA}

The pump is assumed aligned as a quasi-TE mode with _{eff}_{,1} = 3.2911, _{eff}_{,1} = 2.4585 μm^{2} and ν_{g}_{,1} = 8.3424 × 10^{7} m/s. The optical waveguide parameters, related to Stokes waves and obtained by means of full-vectorial FEM method [

To complete the cascaded Raman laser design, some comments about the absorption losses in the mid-IR region are needed. The total propagation loss inside the cavity

Thus, according with results proposed in [_{Si}_{SiO2}_{s}_{p}_{s}_{2} =4.0544 μm, where silicon oxide presents large absorption values

_{s}_{Bus}

In the plot we can observe the output power for first and second-order Stokes waves considering both polarizations. Several comments are worthy to give. First, the quasi-TE second-order Stokes wave presents a large value of lasing threshold, thus it cannot be lasing with an input power less than 150 mW. On the contrary, the quasi-TM second-order Stokes presents similar lasing threshold and external output efficiency, independently from the polarization state of exciting first order Stokes. It seems very interesting the first-order Stokes wave lasing. In fact, in this case the coupling factors for quasi-TE and quasi-TM polarizations do not represent a discriminating factor for the threshold level, since it has the same order of magnitude. In any case, above threshold (equal for both polarizations) the quasi-TM mode presents an output power level and an external efficiency larger than that for quasi-TE mode, as induced by smaller core area and larger coupling factor. Thus, the quasi-TM polarization is well suitable for the simultaneous lasing of the first two Stokes waves, allowing detection of two gases.

Moreover,

In particular, the Raman laser is designed to detect two different types of gases, i.e., C_{2}H_{6} and CO_{2} [_{s}_{3} = 5.1376 μm could be also obtained, giving for example the possibility to detect a third gas, such as NO [

The generalized model presented in this paper allows to accurately predict the time dynamics of pump and Stokes waves in cascaded Raman lasers based on a SOI microcavity resonator and operating in mid-IR region. In addition, the theoretical study has allowed to demonstrate high performance of cascaded Raman laser in terms of reduced power consumption, stability, wavelength tunability and polarizations selectivity, offering unique advantages as mid-IR optical sources and thus competing with complex and bulk solid-state laser systems. Finally, a detailed design of a cascaded Raman laser for gas sensing of C_{2}H_{6} and CO_{2} has been developed under CW operation. This approach could be also used for gas sensing at longer mid-IR wavelengths with improved sensitivity.

Schematic architecture of a cascaded racetrack-resonator SOI Raman laser.

Cross-section of SOI waveguide.

Comparison between experimental data in literature and modeling of this paper in terms of CW cascaded Raman emission versus input pump power (_{p}_{s}_{1} = 1.686 μm, _{s}_{2} = 1.848 μm).

Modal birefringence spectra for different values of

Walk-off parameter spectra for different values of

Ethane spectrum around 3.35 μm.

Walk-off parameter between first-order Stokes and pump waves for different values of _{p}_{s1} = 3.3485 μm).

Modal birefringence for pump and first-order Stokes waves versus waveguide width for different values of _{p}_{s}_{1} = 3.3485 μm).

GVD and TOD coefficients versus waveguide width for pump and Stokes waves (_{p}_{s}_{1} = 3.3485 μm).

Power exchange in the directional coupler for pump wave (_{p}

Laser output versus pump power for different combinations of polarization states.

Time dynamics for both pump and Stokes waves (_{p}_{s}_{1} = 3.3485 μm, _{s}_{2} = 4.0544 μm).

Coupling factors for Stokes waves.

^{2} (quasi-TE) (%) |
^{2} (quasi-TM) (%) | |
---|---|---|

1^{st} Stokes wave (_{s}_{1} = 3.3485 μm) |
1.81 | 3.13 |

2^{nd} Stokes wave (_{s}_{2} =4.0544 μm) |
65.14 | 2.47 |

Optical waveguide parameters for Raman laser simulations.

| ||||||
---|---|---|---|---|---|---|

_{eff} |
_{eff}^{2}) |
_{g} |
_{eff} |
_{eff}^{2}) |
_{g} | |

1^{st} Stokes wave (λ_{s}_{1} = 3.3485 μm) |
3.2363 | 2.6289 | 8.2892×10^{7} |
3.2555 | 2.3418 | 8.3424×10^{7} |

2^{nd} Stokes wave (λ_{s}_{2} = 4.0544 μm) |
3.1518 | 2.9302 | 8.1974×10^{7} |
3.1802 | 2.4776 | 8.2537×10^{7} |

3^{rd} Stokes wave (λ_{s}_{3} = 5.1376 μm) |
3.0088 | 3.5665 | 8.0725×10^{7} |
3.0497 | 2.7383 | 8.0812×10^{7} |

Total absorption coefficients.

| ||||||
---|---|---|---|---|---|---|

Γ_{Si} |
Γ_{SiO2} |
Γ_{Si} |
Γ_{SiO2} |
|||

Pump wave (_{p} |
0.9917 | 0.0041 | 0.042 | 0.9910 | 4.95×10^{-4} |
0.0059 |

2^{nd} Stokes wave (_{s2} |
0.9731 | 0.0137 | 0.096 | 0.9776 | 0.0015 | 0.011 |