1. Introduction

Electrically pumped sources based on intracavity nonlinear processes are attracting a large interest in the scientific community, given their compactness and convenience for applications. Such structures have been demonstrated e.g. for: 1) pulsed THz generation from a dual-wavelength mid-infrared Quantum Cascade Laser (QCL) [1, 2]; 2) Second Harmonic Generation (SHG) with a QCL pump around 6 µm [3]; and 3) parametric fluorescence around 2 µm in a quantum well laser [4].

Among the different materials employed in the fabrication of this type of sources, the semiconductor AlGaAs is particularly attractive due to its high non-resonant nonlinear coefficient (d₁₄ ≈100 pm/V in the near infrared), its good thermal and mechanical properties, and its mature processing techniques. However Phase Matching (PM), which is mandatory for having high conversion efficiencies, cannot be trivially obtained in AlGaAs due to its lack of birefringence.

In the last years, three techniques have provided robust answers to the problem of phase matching near-IR χ⁽²⁾ processes in AlGaAs waveguides: 1) form birefringence [5], which has been exploited for both Spontaneous Parametric Down Conversion (SPDC) around 2 µm [6] and SHG around 0.775 µm [7]; 2) Quasi Phase Matching, implemented e.g. in [8, 9]; and 3) modal PM [10, 11]. A variant of modal PM has been proposed in [12] and used in different contexts [4, 13]: in this case, the three-wave mixing involves a Bragg mode, confined by two distributed mirrors, and two Total Internal Reflection (TIR) modes. The same concept, but with one Bragg reflector, has been numerically demonstrated in [14] showing higher nonlinear efficiencies with respect to the double Bragg configuration.

Here we extend the approach of [14] and discuss an active nonlinear source based on SPDC, whose active medium is made up of InAs Quantum Dots (QDs) instead of Quantum Wells (QWs) in GaAs. The main advantage of QDs in this context is related to their ability to trap charge carriers and quench diffusion toward non-radiative recombination centers, as shown since the mid 90’s for QDs in dislocated material [15] or close to etched surfaces in photonic microstructures [16–18]. Concerning lasers, a strong reduction of the lateral diffusion length and the surface recombination velocity at etched sidewall surfaces has been experimentally observed for the charge carriers [19]. This unique property of QD active media allows the fabrication of 1) deeply etched narrow-stripe (2-3 µm) laser diodes [20–22] with threshold currents comparable to those of broad area devices [19]; and 2) whispering gallery mode QD microlasers [23].

In the present context of active nonlinear sources based on SPDC, QDs offer (unlike QWs) the possibility to exploit narrow deeply etched ridge waveguides. As shown later, the width of such waveguides provides a useful and efficient degree of freedom to ensure phase matching.

The rest of this article is organized as follows: in Section 2 we describe the structure, we model its electrical properties and discuss our design choices. In Section 3 we report the results of electromagnetic simulations, showing a promising nonlinear efficiency and wide tuning capabilities for the source. Finally, we conclude with a few perspectives.

2. Source Specifications

In Fig. 1 we report the details of the semiconductor stack composing the source. The structure adopts the geometry of [14], relying on a single-sided Bragg waveguide, where the pump mode is confined by total internal reflection from the upper cladding and by Bragg reflection from the lower periodic stack. At variance with [14] however, we consider an active device and exploit QDs as active medium.

 

Fig. 1 Sketch and detailed description of the device. For each layer we indicate the material, its thickness, and doping type and levels. Layers 10 and 11 are repeated five times.

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The first layer in Fig. 1 is a cap added to prevent the possible oxidation of the Al-rich cladding (layer 2). The bottommost layers (10 and 11) are repeated five times and provide, together with layers 8 and 9, the Bragg reflector for the pump field. The core (layers 3 to 7) includes a thin GaAs layer (5) hosting a single layer of InAs QDs emitting at around 0.95 µm. Please notice that layer 7 acts as matching layer [24], allowing to easily satisfy the quarter-wave condition.

The growth of such a QD layer by molecular beam epitaxy is nowadays well mastered. For typical values of QD areal density (2 10¹⁰ cm⁻²) and inhomogeneous broadening (50 meV), the available gain is around 25 cm⁻¹ (see [20] for a review). Laser diodes exploiting a single array of QDs exhibit a very low transparency current density (~10 A cm⁻²), and their threshold current density can be smaller than for QW lasers provided the cavity losses are kept at a sufficiently low value (few cm⁻¹). This can be achieved by either using long cavities (>1 mm) or implementing high reflectivity mirrors at the output facets.

In conventional separate-confinement heterostructure (SCH) laser diodes, the energy bandgap decreases from the cladding toward the active QW (or QD) layers, so as to favor the electron and hole transport. This is not the case for the structure sketched in Fig. 1, sincethe bandgap of the Al_0.3Ga_0.7As layers 4 and 6 is larger than the bandgap of the Al_0.1Ga_0.9As layers 3 and 7. As shown in the next section, this particular design ensures a preferential coupling of the emitter to the Bragg mode of the structure.

The design of the above device has been optimized via the simulation of its laser diode properties with the commercial software Nextnano³ [25]. In order to inject efficiently electrons and holes in the QDs despite the particular composition profile of the structure, we combine two main ideas. Firstly, we use a slow Al composition grading (over 200 nm) between layers 3 and 4 and between layers 7 and 8. Secondly, we dope slightly (n or p = 10¹⁷ cm⁻³) layers 3 and 7, as well as 4 and 6 except in the vicinity of the QD layer. Although necessary for efficient electron/hole transport, this doping level is kept as low as possible within the central part of the structure, where the interacting fields have their maximum, so as minimize Free-Carrier Absorption (FCA) losses. Finally, layer 1 is heavily n-doped to grant a good lateral transport and a homogeneous current injection through the ridge, and layer 9 is heavily p-doped to provide a good ohmic contact. Since we adopt intracavity contacts (see Fig. 1 and [26]), there is no need to dope the rest of the Bragg stack (layers 10-11 and below).

Figure 2 shows the calculated band profiles and free carrier densities in the active region when a 1.3 V forward bias is applied to the structure. For such a bias, the energy difference between quasi-Fermi levels is also very close to 1.3 eV, which corresponds to the transparency threshold for the QD active medium. (In our 1D simulation we have modeled the QD wetting layer by a thin QW, and we have replaced each QDs by a trap defect characterized by a 1 ns recombination time. In fact, a more accurate description of the QDs is not necessary here. Due to their low - between 2 and 4 10¹⁰ cm⁻² - areal density, the electric field and charge distribution in the pn junction are not modified when the QDs are taken into account). As shown in Section 3, this charge distribution in the active region corresponds to a low level of FCA losses.

 

Fig. 2 Profiles of the band structure (top panel) and electron/hole density (bottom panel) in the active region, at the transparency threshold. Z = 0 corresponds to the location of the QD layer and the layer numbering is the same as in Fig. 1. We plot the energies of the minima of the conduction band for the Γ (CB _Γ) and X (CB_X) points of the Brillouin zone, and of the maximum of the valence band. The quasi-Fermi levels for electrons and holes are marked by dashed lines.

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If we neglect the recombination on QD levels (no trap defects) the current flowing through the junction at the transparency threshold is around 20 A cm⁻². This current is mostly due to the electron-hole recombination in the wetting layer and in the GaAs surrounding layer (14 A cm⁻²), with a smaller contribution stemming from recombination in the low bandgap regions 3 and 7 (6 A cm⁻²). We therefore expect a small threshold-current penalty for our QD laser with respect to a standard QD laser design.

3. Electromagnetic simulations: Results and discussion

The modeling of the QD laser emission is strongly coupled to the full 2D electromagnetic study of the guided-wave linear and nonlinear optical aspects of the SPDC source.

Due to the symmetry properties of the AlGaAs χ⁽²⁾ tensor, in the following we will consider a SPDC process where a TE-polarized pump photon at wavelength λ_P generates two cross-polarized photons (signal and idler, s and i) at wavelengths λ_s = λ_i = 2λ_P. The PM condition for this process can be written as Δn ≡ n_TE(λ_P) – ½ [ n_TE(2λ_P) + n_TM(2λ_P) ] = 0, where n_TE and n_TM are the effective indices of the propagating modes. Also, we will assume a Bragg pump mode and two TIR fundamental modes for signal and idler.

In Fig. 3 we show a chart of Δn as a function of the pump wavelength λ_P and the ridge width w_R, calculated at a temperature T = 23 °C. The simulations were performed with a 2D Finite-Difference Frequency-Domain code developed following [27]. We see that by simply changing w_R from 2.5 to 5 µm, we are able to phase match degenerate SPDC processes with pump wavelengths ranging from about 940 nm to a little more than 960 nm. The ridge width therefore provides a somewhat coarse - but easily usable - degree of freedom that facilitates the fulfillment of the phase matching condition.

 

Fig. 3 Phase mismatch (Δn) chart as a function of pump wavelength (λ_P) and ridge width (w_R).

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For the following discussion, we will focus on the parameters represented by the black dot in Fig. 3, corresponding to λ_P = 950 nm and w_R = 3.25 µm. In Fig. 4 we report the vertical profiles of all the modes with wavelength λ_P sustained by such a structure. In the graph caption, we also provide their normalized intensity overlap with the GaAs layer containing the QDs (layer 5 in Fig. 1), showing that the Bragg mode has the highest overlap with the latter. In fact, we optimized the position of the QD layer to enforce this property, which facilitates the attainment of lasing action on the Bragg mode instead of on one of the two TIR modes.

 

Fig. 4 Vertical cuts of the modes sustained by the waveguide at the pump wavelength λ_P = 950 nm. Solid line: Bragg mode; dashed line: TE₀ mode; dash-dotted line: TE₁ mode. The thin solid line is the refractive index profile. The mode overlaps with the GaAs layer are calculated to be: 1.78% (Bragg mode), 1.10% (TE₀ mode), and 0.58% (TE₁ mode).

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We can readily estimate the maximum available modal gain for the Bragg mode from a comparison with state-of-the-art QD lasers emitting in the same wavelength range [20]. A modal gain around 25 cm⁻¹ has been obtained for a single layer of InGaAs QDs (areal density 2 10¹⁰ cm⁻², and 50 meV of inhomogeneous linewidth). The QD layer, located in the middle of a 10 nm thick GaAs layer, was inserted in an AlGaAs graded-index separate confinement heterostructure, providing an overlap of the guided mode with the GaAs layer on the order of 2.3%. Therefore, for our present design, we estimate that a single state-of-the-art layer of InGaAs QDs could provide a modal gain on the order of 18 cm⁻¹ (25 cm⁻¹ x 1.7/2.3).

Losses due to FCA were estimated by using the carrier densities computed at the transparency point of the laser. We obtain α = 0.5 cm⁻¹ for the pump Bragg mode and α = 0.15 cm⁻¹ for both TE and TM generated modes. These low values suggest that the doping levels used for the source do not have a dramatic detrimental effect on the nonlinear process.

In order to assess the performance of our device, we calculated the efficiency of the type II SHG where two cross-polarized pump photons at 2λ_P generate a second-harmonic TE photon at λ_P, finding η = 390% W⁻¹ cm⁻². This is only a factor 3 lower than the record experimental SHG efficiency reported for a passive device in [7] and almost a factor 2 higher than what we calculated for the structure discussed in [14]. However, we stress that both these references involve a SHG process at shorter wavelengths.

An example of the tightly confined 2D transverse distributions of the pump, signal and idler modes involved in the SPDC process is provided in Fig. 5.

 

Fig. 5 Intensity maps of the modes involved in the SPDC process. From left to right: pump Bragg mode, generated TE₀₀ mode, and generated TM₀₀ mode. In the charts, only half profile is shown due to the modes’ symmetry with respect to the ridge center.

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Finally, in Fig. 6 we show the tuning curve of our source with respect to temperature. We have calculated it by taking into account the temperature dependence of both the waveguide refractive indices and the emission wavelength of the QD laser. For the latter, we rely on experimental data displaying stable QD lasing on the ground state transition, with emission around 1 µm. We further assume that, at T = 23 °C, the laser emits at λ_L = 950 nm (i.e. in correspondence of the black dot in Fig. 2). This assumption is by no means restrictive since the lasing wavelength of the QDs can be controlled with growth parameters. From Fig. 5, we see that the SPDC operating temperature spans from T = 23 °C down to T = 3 °C, below which the Bragg mode becomes leaky. With a temperature variation of 20 °C we can therefore cover a wide range of generated wavelengths, from about 1.7 µm to 2.1 µm.

 

Fig. 6 Generated wavelengths versus temperature for the device sketched in Fig. 1 and emitting at λ_L = 950 nm for T = 23 ^o C.

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4. Conclusion

We have designed an integrated SPDC active source based on a modal phase matching with the laser medium consisting of QDs embedded in the waveguide core. Intracavity contacts and refractive index grading allow us to use low doping levels, which in turn result in low propagation losses and considerable conversion efficiencies. Such device exhibits an effective degree of freedom (the ridge width, instead of current injection) for the coarse minimization of phase mismatch. In addition, fine PM adjustment and wavelength tuning can be carried out by a slight temperature variation. This study represents a decisive step towards the goal of fabricating an integrated electrically pumped optical parametric oscillator.