## 1. Introduction

The tunability of currently available integrated sources is limited to a few tens of nanometers at most, via temperature or current control. While this is not a problem for most applications, certain fields like wavelength division multiplexing and spectroscopy are in demand of sources with broader tunability and choice of spectral range. Spectroscopy, especially, requires wide-band, continuously tunable sources with narrow emission lines. The 2–4 μm wavelength interval is of particular interest since it contains various peaks of atmospheric and hydrocarbon molecules, with important applications in environmental monitoring, security, and medicine [1–4]. This spectral region is at the frontier between the emission ranges of diodes and quantum cascade lasers (QCLs), and to date most existing sources around 3 μm, such as short-wavelength QCLs [5, 6] or GaSb diodes [7], are only available in laboratories. Interband cascade lasers (ICLs) are the only sources commercially available in this wavelength range, albeit at a high price [8–10]. Moreover, the tunability range of all these devices is limited to a few tens of nanometers. As a consequence, individual laser diodes are used for each spectroscopic line of absorption, which increases the price of a complete diagnosis based on several lines. In this context, nonlinear optics offers a solution for widely wavelength tunable sources, bulky tabletop optical parametric oscillators (OPOs) being commonly used to provide high-quality, tunable beams. Most miniaturized OPOs have been demonstrated in LiNbO_{3} [11, 12], but an OPO threshold has been achieved in a GaAs micrometric waveguide [13] with a potential span of 500 nm. Like GaAs, InP is an attractive material for its high χ^{(2)} and mature technology, especially for emission at 1.55 μm.

Here we report on the design of an InGaAsP/InP QD laser diode emitting at 1.55 μm, optimized for intracavity spontaneous parametric down conversion (SPDC) around 3.11 μm via modal phase matching. The use of QDs is justified by the choice of narrow, deeply etched structures insofar they have been shown to trap carriers and limit surface recombination [14], and narrow-ridge, low-threshold InAs/InP QD lasers have already been demonstrated [15]. In order to estimate the phase mismatch accurately, a precise knowledge of the refractive indices is critical at pump, signal, and idler wavelengths. While the index of InGaAsP lattice matched to InP is well known at 1.55 μm [16–20], to date only one publication deals with its measurement at longer wavelengths [21], and none exists at 3 μm. This makes it crucial to accurately characterize its refractive index up to 3.14 μm, outside of the scope covered by literature data.

## 2. Tunable source design

### 2.1. Laser diode design

We propose a 1.55 μm source optimized for SPDC around 3.11 μm. This design results from back-and-forth optimizations between optical and electrical simulations, to jointly facilitate electron-hole injection, increase the conversion efficiency, and reduce losses. The conduction band and composition profile of this structure are shown in **Figure 1**. To compensate for the material dispersion, the laser diode is conceived so as to favor lasing on the TE_{20} mode (the second order in the direction of growth). To achieve this, the refractive index kept small in the center of the waveguide. As a consequence, the TE_{20} mode confinement inside the active area is stronger than for the TE_{00} mode. **Figure 2** shows the modes TE_{00} and TE_{20} supported by the waveguide at a wavelength of 1.55 μm. In order to achieve an efficient electron injection despite the conduction band increase in the core center, we reduce the series resistance with two strategies. Firstly, we introduce compositional gradients at the interfaces. Secondly, the waveguide core is only lightly doped. **Figure 3** depicts the conduction band and doping profile of the structure. An electrical simulation of this device using the software Nextnano yields a transparency current of 26 A/cm^{2} at the transparency threshold.

### 2.2. Nonlinear properties

To achieve Type-II phase matching despite the error bars on the dispersion model and the fabrication tolerances, we use the ridge width as a crucial degree of freedom. **Figure 4** shows the phase mismatch at degeneracy vs. ridge width and pump wavelength, defined as

where the refractive indices are provided by our experimental data, presented in Part 2, and an interpolation of literature data [20]. By changing the ridge width from 3 to 7 μm, we are able to achieve phase matching for pump wavelengths of 1.50–1.60 μm. Furthermore, a variation in phase mismatch of ±0.02 can be compensated for by setting the correct ridge size. The ridge width thus acts as a gross parameter to meet the phase-matching condition, which can be set after wafers have been grown and characterized.

During operation, temperature provides a supplementary degree of freedom to tune the pump wavelength and reach a wide range of frequencies. **Figure 5** shows the wavelengths of signal and idler emitted beams, for a source of ridge width 3.3 μm, emitting at a pump wavelength of 1.55 μm at 20°C. The dependency of quantum dots wavelength emission with temperature was assumed to be 0.5 nm/K from [22].

**Figure 6** shows the profiles of the interacting modes, at a pump wavelength of 1.55 μm and signal and idler 3.11 μm. The expected conversion efficiency at a pump wavelength of 1.55 μm is 240% W^{−1} cm^{−2}. For an intracavity power of 100 mW, this corresponds to a parametric gain of 0.5 cm^{−1}. Since common InP QD lasers at 1.55 μm emit up to 20 mW outside the cavity without facet coatings [23], the above hypothesis on the intracavity power is very reasonable. The losses experimented by the mode at 3.11 μm are mainly expected to stem from free-carrier absorption in doped layers, which we estimate at about 0.34 cm^{−1} for the TE_{00} mode. Reaching the OPO threshold in this device is therefore a challenging but achievable task, if mirror losses are minimized as in [13]. **Figure 7** shows the intracavity pump power necessary to reach oscillation for different mirror reflectivities and cavity lengths, assuming a conversion efficiency of 240%W^{−1} cm^{−2} and propagation losses of 0.34 cm^{−1}.

## 3. Index measurement

### 3.1. Principle

The evaluation of refractive indices was performed through an m-lines setup. This measurement relies on the determination of the coupling angle inside a slab waveguide of the material of interest. Our samples consist of a layer of In_{1-x}Ga_{x}As_{y}P_{1-y} lattice matched to InP grown over InP. As a result of the index contrast, the quaternary alloy acts as a planar waveguide. A diffraction grating was deposited or etched on the surface. The samples are mounted vertically on a rotating mount, and a laser is shone horizontally on the grating (**Figure 8**). Whenever the angle of light diffracted into the surface layer matches the bounce angle of a guided mode, coupling occurs. The guided light is collected by a detector placed at the exit facet of the sample. Coupling angles are given by the conservation of momentum

where

This equation leads to

where N is the effective index of the mth propagating mode. This measurement therefore estimates precisely the effective indices of the guided modes, as long as the period, wavelength, and coupling angles are known.

### 3.2. Sample fabrication

The planar structures were grown by molecular beam epitaxy and characterized by X-ray diffraction and photoluminescence prior to processing. An electronic resist with average thickness 70 nm was deposited on the surface (MAN2401 spin coated at 6000 rpm), and diffraction gratings were written by electronic lithography. The samples were then either used as such or very shallowly etched in a chemical solution before removing the resist. Both methods lead to shallow gratings on the guiding layer surface.

### 3.3. Optical setup

Three laser beams are combined with a precision of 0.01°: (1) a visible one (λ = 543 nm) for alignment and period measurements, (2) a tunable fibered laser around 1.55 μm, and (3) a free-space laser diode emitting at either 3.14 or 3.17 μm. The sample is placed on the axis of a rotating stage driven by a motor of step 0.001° and is precisely aligned so that the diffraction grating lines lie in the vertical direction (**Figure 9**). The light output at its exit facet is collected with a detector placed after a slit (not shown here) and a focusing lens. All the detection setup is mounted on the stage, and it rotates rigidly with the sample. We used the same samples and gratings at 1.55 and 3 μm since the direction of the second-order diffracted beam at 1.55 μm roughly corresponds to the first order at 3.11 μm. To find the angle of normal incidence, measurements were done in a single long scan, with symmetric angles. Since all facets of a sample are cleaved and reflective, whenever θ_{1} is a coupling angle, the light couples as well in the other direction for θ_{2} = −θ_{1} and reflects on the back facet, reaching the detector (**Figure 10**). Thus the angle of normal incidence was taken to be (θ_{1} + θ_{2})/2. This reduces the error bar due to a possible misalignment of the beams and to the motor backlash.

In order to determine the grating period, the visible laser was shone on the sample. Under the Littrow condition, the diffracted beam exits through the entrance slit and is collected through a beam splitter on a photodiode (**Figure 11**). The grating period is retrieved through the equality

### 3.4. Results

A typical result of coupling measurement is shown in **Figure 12**. The position of the peaks is determined with a precision of 0.01°. Since the effective indices are a function of material index and thickness of the guiding layer, each measured value corresponds to a range of possible {material index, thickness} pairs. This is represented in **Figure 13**, where one line corresponds to the space of parameters that minimize the difference between measured and theoretical indexes. Since more than one effective index is measured, it is possible to determine the right pair {material index, thickness}, at the crossing point. **Figure 14** shows the average difference between measured and effective indices. Waveguides support three modes at 1.55 μm, two at 2.12 μm, and one at 3.14 μm. In order to derive the effective index at 3 μm, the guide thicknesses were estimated from the data at 1.55 μm (**Table 1**).

The lattice mismatch was measured by X-ray diffraction. Ga and As fraction are deduced through the model presented in [26].

**Figures 15**–**17** show the measured refractive indices as a function of As fraction, at wavelengths of 1.55, 2.12, and 3.14 μm. The data at 1.55 μm was compared to the model presented in [20]. At 2.12 and 3.14 μm, no model being available in the literature, we trace the data against a linear regression versus the molar fraction of As(y). The refractive index of InP from [24] was taken into account. **Figure 18** shows the refractive index versus wavelength, against a one-oscillator fit calculated from the Afromowitz model [25]. The fit parameters are presented in **Table 2**.

### 3.5. Discussion

The accuracy of this measurement is determined by several factors.

Values of the grating period were determined by repeated measurements with a precision of 0.1 nm. This leads to a 0.3 × 10

^{−3}error bar on the effective index.The uncertainty due to sample misalignment can be estimated at 0.5 × 10

^{−3}. This may be explained by local variations of the resist filling factor and depth.In order to estimate the impact of the thin layer of photoresist on the effective indices, we performed a set of measurements on a sample covered with a thin photoresist grating. Then we etched it shallowly, removed the resist, and took a new set of data. The estimated thickness diminishes by 11 nm. This is in agreement with a profilometry of the etched grating depth, yielding 15 nm. The estimated core index is raised by 0.7 × 10

^{−3}, a value lower than the possible variation of twice the experimental error. Thus we conclude that the resist has a negligible impact on the effective indices.While the laser beam alignment and position of the sample with respect to the rotating stage are adjusted in each measurement, one could point out that the axis of the rotating motor could be slightly misaligned with respect to the vertical axis and introduce a systematic error. A simple observation of the height of the beam reflection as the motor rotates indicates that the angle could be at most of 0.5 mrad. This leads, after a calculation, to an error on the effective index of 3 × 10

^{−5}.Finally, incertitude on the composition is the most important. It is determined by the photoluminescence and lattice mismatch of the samples, with a precision of ~1%, through the model described in [26]. This corresponds to an uncertainty on the effective index of 4 × 10

^{−3}. The deviation of our measurements with respect to literature and to a linear fit is in the range of 10^{−2}to 2 × 10^{−2}. This is in agreement with the observed variations of index due to the lattice mismatch observed in [16].