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This article was submitted to Nonlinear Optics, a section of the journal Frontiers in Photonics

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We investigate theoretically mid-infrared (MIR) generation via difference frequency generation in multimode AlGaAs-on insulator (AlGaAs-OI) waveguides. The large refractive index difference between the AlGaAs core and the silica cladding shrinks the modes size down to the sub-^{2} scale, and, together with AlGaAs strong second-order nonlinear polarization, empowers strong nonlinear effects. As a result, efficient MIR generation is obtained in few-cm long waveguides with sub-^{2} transverse section, where higher order modes are exploited to achieve the phase-matching condition. These observations suggest that multimode AlGaAs-OI waveguides could represent a novel promising platform for on-chip, compact MIR sources.

Mid-infrared (MIR) optical sources are key tools in a diverse range of fields spanning medicine, defence, and environmental gas sensing (

Highly nonlinear crystals bring a good compromise between wide MIR tunability and compactness. The light-matter interaction in these crystals leads to the conversion of an input pump frequency into a different frequency via second - or third - order nonlinear processes. Today, commercial lithium-niobate (LiNbO_{3}) few-cm long waveguides are widely employed for MIR generation through difference frequency generation (DFG) (

DFG is a second-order nonlinear process where a pump and a seed beam respectively at frequencies _{
p
} and _{
s
} interact to give rise to a new beam, the idler, at the frequency _{
i
} = _{
p
} − _{
s
}. In the DFG process, one pump photon is converted into a seed photon and an idler photon (

Sketch of the DFG process in a rectangular AlGaAs waveguide fully embedded in a SiO_{2} cladding. An intense pump (blue) and a weak seed (green) at the waveguide input interact inside the waveguide, leading to seed amplification and idler generation (red).

In order for the process to be efficient, it must be phase-matched. Several phase-matching schemes have been developed and have been incorporated into crystals and small devices alike; these range from the heating of the crystal (

Compared to LiNbO_{3}, Aluminum gallium arsenide exhibits some key features (_{14} ≈ 50 pm/V for AlGaAs at 1,550 nm (_{33} ≈ 25 pm/V for LiNBO_{3} at 1,550 nm (_{3}, being _{3} (^{2}. These features, along with a relatively low two-photon absorption (TPA) (

In this article, we explore this platform for efficient MIR generation through DFG theoretically and numerically. One of the major issues of the phase-matching condition is resolved by exploiting higher order guided modes in multimode waveguides. Indeed, while keeping the cross-section size smaller than 1 ^{2}, we can still find several guided modes thanks to the high-index contrast between the AlGaAs core and the SiO_{2} cladding. By playing on the Al_{
x
}Ga_{1−x
}As composition (percentage of Aluminum

The multimode approach has successfully been applied to second harmonic generation (SHG) in waveguides where typically nonrectangular geometries (e.g. M-shaped) assist the phase-matching (

AlGaAs has a zincblende structure. One of the most common cut directions is along the waveguide axis. In this case the second-order nonlinear tensor exhibits 3 main contributions _{14} = _{25} = _{36} = _{
eff
} (^{(2)} as a function of the electric field E:_{0} is the vacuum permittivity. An equation similar to _{
s
} and _{
i
}. It is evident that the nonlinear polarization is boosted by the presence of a large longitudinal (z) component of the modes in play, as outlined by _{
p
}/_{
core
} or smaller, _{
p
} being the pump wavelength and _{
core
} the AlGaAs refractive index. We simulate the propagation of light into the waveguide via two separate approaches, namely finite element method (FEM), and coupled amplitude equations (CAEs).

In the first case, we solve Maxwell’s equations using the finite-element-method solver of the commercial software COMSOL Multiphysics (_{
n
} = _{
p,s,i
} is one of the three frequencies into play and _{0} the free space wave number. The three equations for the pump, the seed, and the idler are coupled through the nonlinear polarization terms appearing on the right-hand side, defined in

In the case of CAEs, we assume the total electromagnetic field E(_{
n
}), H(_{
n
}) oscillating with frequency _{
n
} is coupled to one guided mode of the waveguide and is factorized as follows:

Here _{
n
}(_{
n
} are respectively the corresponding slowly varying amplitude and the propagation constant. The mode is normalized so that the integral of the Poynting vector along the propagation direction _{
n
}|^{2}(_{
n
}.

By following the steps outlined in

An equation similar to _{
p
} − _{
s
} − _{
i
} is the phase mismatch term, whereas _{
n
} (

Differently from _{
n
} = (_{2}(_{
n
})_{
n
})/(_{
eff
} (_{
n
})), where _{2} (_{
n
}) and _{
eff
} are respectively the nonlinear refractive index and the modal effective area at frequency _{
n
} (_{2} ∼ 3 × 10^{–17} W/m^{2} at 1,550 nm for AlGaAs (

Moreover, we aim to generate an idler in the MIR region between 3,000 nm and 4,000 nm. As previously mentioned, at least one waveguide dimension should be of the order of _{
p
}/_{
core
} or lower to boost the second-order nonlinear polarization. We then set the waveguide thickness in the range 240–300 nm. For such a small thickness the higher order modes at the targeted MIR wavelengths (3,000–4,000 nm) are weakly confined, leading to a low overlap coefficient

_{
p
} − _{
s
} − _{
i
} = 0 for a rectangular waveguide with a fixed thickness of _{
y
} = 265 nm, and width _{
x
} varying from 1,000 to 1,600 nm (colored lines). The propagation constants _{
p,s,i
} are calculated through a mode solver. It should be noted that the seed wavelength is fixed via the relation _{
s
} = _{
p
} − _{
i
}.

Phase-matching diagrams. The colored lines denote the pump and idler wavelengths where the phase-matching condition Δ_{
p
} − _{
s
} − _{
i
} =0 occurs. _{
y
} =265 nm kept fixed, _{
x
} varying from 1 to 1.6 _{
x
} = 1.2 _{
x
} =1.2 _{
y
} varying from 240 to 300 nm.

From

Summary of the waveguide dimensions and the pump, seed and idler wavelengths for the two selected cases.

Case 1 | Case 2 | |
---|---|---|

_{
x
} ( |
1.2 | 1.4 |

_{
y
} (nm) |
265 | 265 |

_{
p
} (nm) |
1,550 | 1,550 |

_{
s
} (nm) |
2,693 | 2,881 |

_{
i
} (nm) |
3,652 | 3,355 |

FEM simulations of _{
P
} = _{
S
} = _{
I
}). Note that prior to the numerical solution of

_{
p
} = |_{
p
} (0)|^{2} = 1 W, whereas _{
s
} = |_{
s
} (0)|^{2} = 1 mW for the seed. _{
p
} = _{
s
} = _{
i
} = 0). The two approaches exhibit very similar results, thus validating the CAEs model that is less computationally demanding and order of magnitudes faster than FEM simulations. In particular, only few seconds are required for solving the CAEs model, whereas several hours are requested to solve the FEM model. _{
opt
} as the propagation length for which the pump conversion (and, hence, the idler generation) is maximized (for the case displayed in _{
opt
} = 1.7 cm). The larger the pump power, the faster the conversion from the pump to the idler therefore reducing the optimal length. Increasing values of losses reduce the average pump power, which in turn increases the optimal length, as reported in

_{
x
} =1,200 nm, _{
y
} =265 nm, _{
p
} =1,550 nm, _{
s
} =2,693 nm, and _{
i
} =3,652 nm) in the absence of losses. The two methods provide very similar results. _{
opt
} =1.75 cm indicates the propagation length where maximum pump conversion is achieved.

_{
i
} = _{
p
} − _{
s
}.

_{
wvg
} = _{
opt
}. As expected, a peak of conversion is observed in correspondence of the phase-matching condition (seed wavelength ∼ 2,693 nm and ∼ 2,881 nm in cases 1 and 2, respectively). Moreover, we note that in both cases a loss coefficient of 5 dB/cm annihilates the DFG. For losses in the range 2–3 dB/cm (

_{
p
}/^{2} and 0.27 GW/cm^{2} for cases 1 and 2, respectively, where A is the cross-section surface of the waveguide. While a CW beam with such a power density is likely to damage the waveguide, pulses a few-hundred picoseconds wide are not (_{
wvg
}ΔIGV that may compromise the temporal overlap of the two pulses, and then the efficiency of the DFG, whenever ΔT is comparable with the pulse duration of the pump and the idler. We have however verified that in our simulations the ΔIGV is of the order of 4 × 10^{–9 }s/m, which correspond to a time delay of tens of picoseconds. This does not impair the DFG dynamics as long as the pump pulse width is hundreds of picoseconds or more.

It is worth noting that for such a pulse width, SPM has little to no effect. More precisely, SPM induces a nonlinear phase accumulation that can shift the phase-matching condition, and that occurs in both CW and pulsed regime. We have verified that this effect is negligible for pump powers up to 1W. In addition, SPM leads to spectral broadening and pulse reshaping in the pulsed regime, which can be simulated by adding dispersion terms (e.g. group velocity dispersion + third-order dispersion) to

In this article we have explored theoretically the possibility of developing tiny on-chip MIR optical sources based on DFG in AlGaAs-OI waveguides. By focusing on a basic rectangular geometry, phase-matching is not achievable when pump, seed, and idler are all coupled to the fundamental mode. However, it can be achieved when resorting to higher order modes. The large core-to-cladding refractive index difference along with the strong nonlinear polarization response leads to effective DFG in waveguides as short as a few cm and with sub-^{2} cross-section. For example, we have shown that in a 1.7 cm long waveguide with 265 × 1,200 nm^{2} cross-section pumped by a TM01-beam at wavelength 1,550 nm and seeded by a TE00-beam at wavelengths around 2,693 nm, it is possible to generate a MIR-idler around 3,652 nm with a pump-to-idler conversion efficiency up to 15

It is worth noting that our analysis is not exhaustive. The maximum output MIR (idler) power (

To conclude with, we believe the results discussed in this work shed a light on the bright potential of the AlGaAs-OI platform for the development of a new generation of extremely compact, on-chip MIR sources.

In this section we describe the FEM model developed with COMSOL Multiphysics. The waveguide was modeled with a block of width _{
x
}, thickness _{
y
}, and length _{
z
}—representing the core, surrounded by a much bigger block, of width _{
clad
}, thickness _{
clad
}, and length _{
z
}— mimicking the cladding.

The refractive index of Al_{0.25}Ga_{0.75}As (_{2} (

The solution of the model relies on two steps.

(i) We computed the waveguide modes at the pump, the seed, and idler wavelengths. This is done by solving a 2D problem on a cross-sectional plane of the waveguide. In this way, we evaluated the electric field profile of the modes TE00 (for the seed and the idler) and TM01 (for the pump) and their propagation constants _{
n
} (

(ii) We introduced three

The external polarizations, defined in ^{(2)} is negligible in the material composing the cladding (i.e. SiO_{2}).

The electric field of the computed modes is mainly confined along the waveguide. _{
clad
} was chosen large enough to have the external boundary truncating the cladding where the electric field is negligible. Hence, the external boundary of the computation domain does not alter the results. For the sake of solving the model, scattering boundary conditions were applied on the external faces of the cladding.

The waveguide modes were excited by means of ports, placed at the beginning of the waveguide. At the other side of the waveguide, a matched boundary condition was applied to truncate the domain without spurious reflections (i.e. this boundary condition allows the outflow of the light from the simulation domain as if the waveguide continued indefinitely after the end of the considered domain).

To perform the two aforementioned steps for the solution of the model, we introduced two studies in COMSOL Multiphysics. (i) To excite the desired modes for the pump, the seed, and the idler, we added a study with three boundary modes analysis steps (one for every wavelength that has to be considered). In each study step, the effective refractive index of the mode to be launched in the waveguide is provided. For this computation, the frequency imposed in the Electromagnetic Waves, Beam Envelopes nodes is dictated by the solver. By running this study, we compute the electric field profile (in the planes of the ports) of the desired modes and their propagation constants.

(ii) To compute the electric field in all the waveguide, another study is necessary. This study has to compute only the envelope function of the electric field, since the propagation constant of the modes, the electric field profiles, and phases are known from the previous study. The second study has a frequency domain study step. The frequencies to be provided in the Electromagnetic Waves, Beam Envelopes nodes have to be set as user defined: in each Electromagnetic Waves, Beam Envelope node the user has to write the correct frequency (of pump, seed, or idler).

We used this approach to study the waveguide of dimensions _{
x
} = 1,200 nm and _{
y
} = 265 nm (case 1 in _{
clad
} = 6_{
x
}. This is a good compromise to obtain the modes with the effective refractive index close to the target value and to limit the computational burden. Also the mesh size in the plane of the ports was optimized. In particular, the mesh size in the core domain was smaller than _{
p
}/20, whereas in the cladding the mesh was kept finer in the region close to the core, where the electric fields are not negligible. The mesh developed on the ports plane was swept along the waveguide length, setting 5 mesh nodes every 1 mm along the waveguide length. These settings for the mesh were necessary to achieve the model convergence and to obtain a reasonable result.

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

JH performed the theoretical and numerical analysis. MGa, YF, and MGu also contributed to numerical analysis throughout. All authors participated in analysis of the results and manuscript writing.

JH, YF, and MGu, acknowledge funding from the European Research Council under the H2020 Programme (ERC Starting Grant No. 802682, MODES project). The work of MGa and CDA is partially funded by the European Union Horizon 2020 Research and Innovation programme under Grant Agreement No. 899673; by the National Research Council Joint Laboratories program, Project No. SAC.AD002.026 (OMEN); and by the Italian Ministry of University and Research (MIUR) through the PRIN project NOMEN Grant No. 2017MP7F8F.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.