Open Access
Add to BibSonomy
Abstract
Quantum-cascade (QC) vertical-cavity surface-emitting lasers (VCSELs) could combine the single longitudinal mode operation, low threshold currents, circular output beam, and on-wafer testing associated with VCSEL configuration and the unprecedented flexibility of QCs in terms of wavelength emission tuning in the infrared spectral range. The key component of QC VCSEL is the monolithic high-contrast grating (MHCG) inducing light polarization, which is required for stimulated emission in unipolar quantum wells. In this paper, we demonstrate a numerical model of the threshold operation of a QC VCSEL under the pulse regime. We discuss the physical phenomena that determine the architecture of QC VCSELs. We also explore mechanisms that influence QC VCSEL operation, with particular emphasis on voltage-driven gain cumulation as the primary mechanism limiting QC VCSEL efficiency. By numerical simulations, we perform a thorough analysis of the threshold operation of QC VCSELs. We consider the influence of optical and electrical aperture dimensions and reveal the range of aperture values that enable single transversal mode operation as well as low threshold currents.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vertical-cavity surface-emitting diode lasers (VCSELs) [1] are attracting great interest, due to their properties such as low-threshold and single-fundamental-mode operation, small volumes, and low-divergent, symmetrical, and astigmatism-free output beam, which make them suitable for a wide range of applications in telecommunications and autonomous systems. Conventional VCSELs are composed of a single wavelength or few wavelengths long cavity and two distributed Bragg reflectors (DBRs) composed of a large number of quarter-wavelength layer pairs exhibiting noticeable contrast between their refractive indices. The power reflectance of one of the DBRs should be close to $100\%$. The power reflectance of the other DBR, which is responsible for light emission, can vary from $95\%$ to $99\%$. Such DBRs are usually several times thicker than the cavity in which the active region is embedded. In the case of possible VCSELs emitting mid and far infra-red radiation, the required thickness of the DBRs can be tens or even hundreds of micrometers, making their epitaxial growth highly challenging and efficient heatsinking problematic.
Subwavelength grating mirrors offer an alternative to DBRs. They are composed of equally spaced parallel stripes of a high-refractive-index material implemented on a cladding made of a low-refractive-index material. This configuration is known as a high refractive-index contrast grating (HCG) [2]. When the cladding is made of the same material as the stripes, this configuration is known as a monolithic HCG (MHCG). Broadband optical reflection of up to $100\%$ can be achieved from both an HCG [3] and an MHCG [4]. This is attributable to destructive interference of the grating modes [5,6], which may also be interpreted as Fano resonance, involving the discrete grating modes and the continuum of free-space incident waves [7].
In conventional VCSELs, stimulated emission results from band-to-band electron transitions. This does not impose specific requirements on the light polarization of stimulated emission [8]. However, the process of electron-hole recombination imposes a long wavelength limit, due to the semiconductor bandgap. In QCs, the emitted radiation is a result of electron transitions within the energy band between the successive energy levels of coupled quantum wells (QWs). Adequate energy separation between energy levels can be achieved by tuning the geometrical parameters of QWs, enabling a broad range of possible emission from mid-infrared to the terahertz region [9]. Radiative transitions in QCs are governed by quantum selection rules that enable the emission of photons with polarization normal to the plane of the epitaxial layers. The absence of this polarity in VCSEL optical modes precludes stimulated emission from QCs embedded in the conventional VCSEL configuration. In [10], we proposed a VCSEL configuration operating in the mid-infrared regime, in which one of the DBRs is replaced with a highly reflecting MHCG integrated with the QCs. Transverse magnetic (TM) polarization of the mode, when the magnetic component of the electromagnetic field is parallel to the stripes of the MHCG, induces an electric field of electromagnetic wave component normal to the plane of the QC layers embedded in MHCG, facilitating stimulated emission. Optical resonance is governed by the Fabry-Perot cavity, which is created by the parallel MHCG and DBR. The same mechanism of vertical optical resonance and stimulated emission in a subwavelength grating was proposed and experimentally demonstrated in [11] for a QC vertical-external-cavity surface-emitting laser (VECSEL) operating in the terahertz spectral range. The difference from our fully semiconductor configuration concerns the grating that is integrated with the metal. Exploitation of such grating in the midinfrared range introduces optical absorption that prevents laser emission.
In [10], we introduced the general principles for designing a QC VCSEL, related to the constraints imposed by optical phenomena. We determined the levels of the possible threshold currents and emitted power, assuming a simplified homogeneous gain distribution in the active region. Here, by taking into account electrical phenomena and a gain model, we design a scheme for electrical injection. We evaluate the influence of a nonuniform voltage distribution in QCs, which strongly affects the efficiency of electron tunneling and leads to voltage-driven gain cumulation (VDGC), affecting mode selection. Although this phenomenon seems analogous to the current crowding effect in broad area VCSELs [12], the origin and influence of VDGC on the modes in a QC VCSEL is nontrivial. We also determine the spatial dimensions of the MHCG and electrical aperture that enable single mode operation, and we evaluate the corresponding threshold current levels.
In Section 2, we describe numerical models of electrical, gain, and optical phenomena. Optical simulations requiring a vectorial three-dimensional model and gain simulations based on first principles are very time consuming, and their direct implementation in a self-consistent threshold current model would be ineffective. Therefore, in Section 3 we describe a reduced self-consistent approach that combines a reduced electrical model, interpolated results of gain model, and a spatially separated optical model. In Section 4, we describe the design of the QC VCSEL and provide a detailed physical interpretation of the elements that constitute the configuration of the laser. Finally, in Section 5 we perform threshold current analysis concerning the influence of the optical and electrical apertures on the level of threshold current and modal properties. We also evaluate the influence of VDGC on QC VCSEL operation.
2. Numerical model of QC VCSEL
In what follows, by ’electric field’ we refer to the electric field $F$ induced by the potential difference applied to the laser electrodes. Whereas the vector of the electric field of an electromagnetic wave will be denoted by $\mathbf {E}$ and by $E_x$, $E_y$, $E_z$ we indicate spatial components of the $\mathbf {E}$ vector.
Let us begin by considering the QC VCSEL configuration depicted in Fig. 1. This configuration will be discussed in more detail in Section 4. For now, we shall assume that the QCs are embedded within an MHCG that is implemented above the DBR. In Fig. 1, the QC VCSEL configuration is displayed with a cartesian coordinate system in which the $z$-axis is perpendicular to the plane of the epitaxial layers ($x$-$y$) and the $x$-axis is along the MHCG stripes. We consider only TM polarization of the electromagnetic modes of the QC VCSEL, in which the electric field contains $E_y$ and $E_z$ components. We consider the pulse operation of the laser, assuming short current pulses and sufficient time between the pulses to justify neglecting thermal processes. Therefore, QC VCSEL operation is assumed to be governed by interrelated optical, gain, and electrical phenomena, as described in greater detail in [13,14].
Fig. 1. Schematic image of the QC VCSEL, showing the main elements of the configuration and the coordinate system.
2.1 Electrical model
Electrical processes inside the bulk layers of QC VCSELs follow from drift of electron current within the structure. In particular, three-dimensional (3D) current density profiles $j(x,y,z)$ can be calculated from the differential Ohm’s law [14]
where $V(x,y,z)$ and $\sigma (x,y,z)$ stand for the 3D profiles of potential and electrical conductivity, respectively. Potential distribution is in turn governed by the Laplace equation: which is solved using the finite-element method. Determined distribution of $V$ enables straightforward determination of electric field $F$ distribution in the volume of the laser. Determination of the electrical conductivity in QC is described in section 2.3.2.2 Optical model
Our optical numerical model uses a fully vectorial plane wave admittance method (PWAM) [15]. The electromagnetic fields, as well as the magnetic and electrical permittivities, are decomposed on the orthonormal and complete basis of exponential functions in the $x-y$ plane. The basis functions are defined as the product of two spatially independent functions, which satisfy the orthonormality and completeness of the basis:
Using these assumptions, the set of Maxwell’s equations for each epitaxial layer is derived to the form in which fields and permittivities are replaced with coefficients of exponential expansions providing the relations of the field transformation between and within the layers of the structure. The relations are employed to determine the characteristic equation by forcing boundary conditions — i.e., by a zeroing of the fields on the upper-most and lower-most layer boundaries within computational window. This provides the complex value of resonant wavelength $\lambda$ and the corresponding distribution of the components of the electromagnetic wave. From the complex value of $\lambda$ the quality ($Q$) factor of the mode in the cavity can be determined according to the formula [5]:
where $\lambda _\mathrm {re}$ and $\lambda _\mathrm {im}$ are the real and imaginary parts of $\lambda$, respectively.2.3 Model of electron transport through QC and optical gain
The nonequilibrium Green’s function (NEGF) method [17] is used to describe electron transport and emission processes governed by coherent interactions such as quantum tunneling, electron correlation, and confinement, as well as incoherent interactions represented by scattering. There are no phenomenological assumptions in the model.
A quantum cascade (QC) is a unipolar structure that maintains translational invariance in the plane of the layers. It can be reasonably described as a one-band one dimensional Hamiltonian in the $z$ direction [18]:
The Hamiltonian is parametrized for the in-plane momentum modulus, $k=k_{xy}$. The $V(z)$ potential includes both the profile of the conduction-band edge $\mathcal {E}(z)$ and the Hartree potential, which can be calculated by solving the Poisson equation self-consistently with the NEGF equations. Interaction with valence and remote bands is accounted for energy dependent effective mass, $m(\mathcal {E},z) = m^*(z)\left ( 1 + \frac {\mathcal {E}-\mathcal {E}_c(z)}{\mathcal {E}_g(z)} \right )$, where $\mathcal {E}_g$ is the effective bandgap energy [19]. It has been shown that such a Hamiltonian gives results that match very well with the results obtained using an eight-band $k\cdot p$ model [20].
The equations of the NEGF formalism applied to the Hamiltonian in (5) are solved numerically for the steady state in the real space (position basis). The scattering self-energies in the formulations suitable for the above Hamiltonian are given e.g. in [18]. They include scattering by optical and acoustic phonons, ionized impurities, alloy disorder, and interface roughness. Due to periodicity of the QC structure, the range of analysis is limited to slightly more than one QC module (period) and the remaining part of the cascade is mimicked by suitable contact self-energies [18]. The gain/absorption is calculated in the Coulomb gauge based on the full theory outlined in [21], adapted for the case of energy-dependent effective mass [22]. As stressed in [23], the use of a full theory which accounts for the fluctuations of the self-energies caused by the optical field is crucial for the correct treatment of nondiagonal scattering, as for example in the case of longitudinal optical (LO) phonons.
3. Reduced model of the QC VCSEL
Calculations of the whole three-dimensional QC VCSEL are of high complexity and are very demanding in terms of time and computer memory, since threshold calculations of laser operation require self-consistent convergence of the sub-models. In brief, based on a given bias applied to the structure, the distribution of the current entering the active region and the distribution of the electric field in the active region are calculated. This enables determination of the electron transport and gain calculations in the gain model. The imaginary refractive indices are calculated based on the distribution of the gain in the active region, and then used in the optical model. The optical model determines complex resonant wavelengths. The imaginary part indicates whether the active region gain is sufficient to balance all optical losses and reach the lasing threshold ($\lambda _\mathrm {im} = 0$). If not, to achieve the lasing threshold a new voltage value is applied, based on a convergent approach. This convergence procedure requires several tens of steps. Each step is expected to last more than a day. Therefore, we reduce the complexity of the electrical model, reduce the dimensionality of the optical model, and interpolate the results of the gain model.
To reduce the complexity of the electrical model, we assume an equal difference in voltage between the metal fingers of the top and at the bottom electrodes for all MHCG stripes. Then we calculate the distribution of the voltage and the current for a single MHCG stripe and assume the same voltage and current distribution in all the MHCG stripes that is a consequence of the structure geometry (see Fig. 1).
To reduce the dimensionality of the optical model and provide the high efficiency required for selfconsistent calculations, we calculate the imaginary resonant wavelength ($\lambda _\mathrm {im2D}$) without gain in the active region for a two-dimensional configuration in the $y$-$z$ plane with absorbing boundary conditions. Then we calculate the resonant wavelength of two-dimensional structure in the $y$-$z$ plane with a single MHCG stripe and a periodic boundary condition ($\lambda _\mathrm {im1D}$). We introduce artificial absorption to the structure to achieve the same value of $\lambda _\mathrm {im1D}$ as $\lambda _\mathrm {im2D}$. This accounts for lateral optical losses in $y$ direction in the model with absorbing boundary conditions. In both cases gain is zero in QCs. Finally, we calculate the imaginary resonant wavelength $\lambda _\mathrm {im}$ in structure with a single MHCG stripe in three dimensions. In this case periodic boundary conditions are implemented in the $y$ direction. Absorbing boundary conditions are implemented in all other directions. Artificial losses are given from the previous step, and the gain distribution is as determined by the gain model. This approach introduces inaccuracy related to spatial separation of the Maxwell equation solution. This method is limited by small aperture sizes comparable with resonant wavelength. However, such an approach is typically used in effective optical methods in the case of lasers with optical apertures significantly larger than resonant wavelength, which reveal high accuracy compared to methods without the separation of spatial variables [24,25]. Exemplary results obtained with the model are presented in the Supplement 1 S2–S4.
We next interpolate the results of the electron transport and gain model. Using the results of NEGF (see Supplement 1 S4), we calculate the conductivity of the QC in the $z$ direction and gain spectra at different representative values of electric fields applied to the QC. We tune the electric field in the range from the level corresponding to $0$ gain to the value associated with maximal emission efficiency and slightly above. However, in the selfconsistent model of threshold operation, conductivity and gain are required as continuous functions of the electric field ($F$) in the QC. Conductivity and gain are related to tuning of the energy states in QC. Therefore both processes can be expressed by Gaussian functions of electric field. Figure 2 presents a fitted electrical conductivity ($\sigma$) and material gain ($g$) with points calculated by NEGF, showing very good agreement. The fitting function of electrical conductivity is given by formula:
where $F$ is electric field and values of the coefficients $a$, $b$, $c$, $d$ are collected in Table 1. The relation presented in the reduced model is also in good agreement with experimental results for conductivity as a function of the applied electrical field [26,27] (see Supplement 1 S2).Fig. 2. Electrical conductivity (blue color and right vertical axis) and material gain (red color and left vertical axis) as a function of the electric field in the QC active region. Crosses and dots are calculated with NEGF, solid lines are calculated based on formulas (6) and (7).
Gain ($g$) as a function of the electric field is approximated by formula:
and corresponding parameters are presented in Table 1.Exemplary calculations for a benchmark structure and the parameters are presented in the Supplement 1 S1.
4. QC VCSEL structure
The model described above was employed to design a QC VCSEL structure with a resonant wavelength of $9.5 \,\mathrm{\mu}\mathrm{m}$. The QC VCSEL differs from a bipolar MHCG VCSEL [28] that its active region is not embedded in the optical cavity between two mirrors. Nonetheless, the thickness of the layer positioned between the DBR and MHCG must still enable a standing wave [29], expressed by round-trip phase shift of light in the cavity that is an integer multiple of $2\pi$. In QC VCSELs, we call this layer a phase-tunning layer. In our configuration, we propose an MHCG and a phase-tunning layer composed of InP lattice-matched materials that in particular enable efficient infrared light generation from the QCs [30]. This part of the structure is conductive and both metal electrodes are implemented within it. The wafer-bounded DBR is composed of 16 pairs of GaAs and $\mathrm {Al_{0.95}Ga_{0.05}As}$ quarter-wave layers, which enable high refractive index contrast and relatively high thermal conductivity. The DBR is integrated with a golden layer at the bottom. The golden layer enhances the optical reflectivity of the DBR, enabling nearly total optical reflection of $99.9\%$. The total thickness of the DBR is $24.6\,\mathrm{\mu}\mathrm{m}$.
Nearly total power reflectance can be realized by many designs of MHCG with different periods, duty cycles and heights of the MHCG stripes. From among the multitude of MHCG parameters, the set with the lowest height of MHCG stripes enables the broadest reflection stopband [4,31] and the least processing effort. In this MHCG configuration, the light distribution in the stripe is composed of a single antinode and no nodes in the $z$-direction. The active region embedded in the MHCG stripes must be supplied with current, which is injected from the electrodes positioned in the top part of the stripes (see Figs. 1, 3). The electrodes are positioned at the extreme ends of the MHCG stripes, on a high doping contact layer that is selectively etched on the surface of the stripe (Fig. 3). Another highly doped layer is positioned between the top electrodes and the active region to uniformize current distribution in the active layer. Its higher doping and increased thickness improve the homogeneity of current supplying the active region but also enhance light absorption. Therefore, the highly doped layer should be positioned in the node of the standing wave (see Fig. 4(c)) that is absent in the configuration with the smallest height of MHCG stripe. In this configuration, the active region may be placed less than $1\,\mathrm{\mu}\mathrm{m}$ from the top surface of the MHCG stripe which causes nonuniform current injection into the active region.
Fig. 3. (a) Image of the QC VCSEL showing a cross-section in $x-z$ plane with graphical definitions of the electrical and optical apertures. (b) Image of three-dimensional cut of the QC VCSEL and description of the composing elements.
Fig. 4. Light intensity ($\mathbf {EE}^*$) distributions in the $y-z$ plane of the QC VCSEL (a) across the whole structure composed of 40 stripes, (b) for single stripe in $y$-$z$ plane with periodic boundary conditions and (c) calculated in $x$-$z$ plane by three dimensional model for single stripe with periodic boundary conditions imposed on $y$ axis and absorbing boundary conditions implemented on $x$ axis. $E_zE_z^*$ distribution in the $y-z$ plane of the QC VCSEL (d) across the whole structure composed of 40 stripes and (e) across the four central MHCG stripes.
The smallest height of MHCG stripe that enables a single node of the standing wave within the stripe is $8.6\,\mathrm{\mu}\mathrm{m}$. For such a structure, the light intensity distribution $\mathbf {EE}^*$ and the intensity of the $z$ component of the electric field $E_zE_z^*$ in the $y-z$ plane is shown in Fig. 4. (The light intensity and the phase of the propagating light out of the structure is presented in Fig. S7 of Supplement 1). As can be seen, light intensity has a single node in the grating layer in which the highly doped layer is implemented. The thickness of $100\,\mathrm {nm}$ of this layer is an optimal compromise between high conductivity and low light absorption. The $E_z$ component distribution required for stimulated emission is present in the grating area and demonstrates two distinct maxima, in which two active regions are embedded. Each active region is composed of $\mathrm {Al_{0.48}In_{0.52}As}$/$\mathrm {In_{0.53}Ga_{0.47}As}$ QC segments that are repeated 18 times. Positioning the active regions and highly doped layers requires careful design, as their refractive indices differ from the refractive index of InP. Reflection and light distribution in the configuration require optimization with respect to maximal power reflectance and maximal light interaction with the active region. The power reflectance of the infinite MHCG is $99.4\%$ without gain in QCs. In the case of finite MHCG incorporated in QC VCSEL the effective reflectivity additionally slightly decreases enabling top emission as seen in Figs. 4(a–c). The dimensions of the optimized MHCG stripes and reflectivity spectra are detailed in the Supplement 1 S3. The electrodes are made of gold, which has a real dielectric constant with a large negative imaginary part value [32] (Table S2 in Supplement 1). This introduces a waveguiding effect in the $x$ direction that reduces the interaction of light with the electrodes, as well as possible absorption and scattering.
Bottom electrodes are implemented on the sides of the MHCG (see Figs. 1, 3(a)). They are etched in the lower doped InP region into the highly doped contact InP layer of $100\,\mathrm {nm}$ thickness (see Fig. 3). The contact layer is positioned in the phase-matching layer in the node of the optical mode. In the configuration the electrical and optical apertures are separate. The current funneling is realized by proton implantation (Fig. 3(a)) which reduces electrical conductivity below the top contacts and enables current flow through central part of the active region without introducing additional absorption loses [33–35].
The optical aperture is determined by the distance between the top electrodes in the $x$-direction and by the size of the MHCG in the $y$-direction (Fig. 3(a)). Both distances are equal. The three last outer stripes have a modified duty cycle to prevent lateral light leakage in the $y$-direction. The top view of the MHCG is square-shaped to enable a circular-shaped emitted beam in the far field.
5. Results
5.1 Electrical phenomena
In edge-emitting QC lasers, the electrical contact implemented uniformly above the active region induces a nearly uniform electric field along the laser cavity, which is followed by nearly uniform optical gain throughout the active region. In QC VCSELs, the contacts are situated on each end of the MHCG stripes, which results in substantial electric field ($F$) inhomogeneity alongside the active region, reaching several tens of $\mathrm {kV/cm}$, as illustrated in Fig. 5(a). The full width at half maximum of the gain function versus electric field ranges from nearly $40\,\mathrm {kV/cm}$ to slightly above $60\,\mathrm {kV/cm}$ (see Fig. 2). This range of electric fields can be in the portion of the active region. Increasing the applied voltage modifies the profile of the electric field within the active region (Fig. 5(a)), which contributes to move the gain maximum from the $x$-border of the active region to the center, as depicted in Fig. 5(b). Such gain modification affects the modal structure of the emitted light. The gain maximum, positioned closer to the center of the active region promotes fundamental mode operation while the outer position leads to the excitation of higher-order modes.
Fig. 5. Electric field ($F$) (a) and gain (b) distributions in the active region for various voltages applied to the QC-VCSEL structure. 0 at horizontal axis corresponds to the center of the MHCG stripe.
5.2 Optical phenomena
The optical properties of the QC VCSEL cavity can be characterized by the quality factor ($Q$-factor), which is determined for each mode in the structure without gain in the active region and with all possible internal absorptions. Figure 6 illustrates the $Q$-factor as a function of the lateral size of the square-like MHCG footprint. The $Q$-factor increases with increases in the size of the MHCG, nearly linearly in the range from 30 to 50 stripes in the case of the fundamental mode. It approaches $10^4$ for an MHCG composed of 80 or more stripes.
Fig. 6. Quality factor of successive lateral modes in QC VCSEL without gain as a function of stripes number and corresponding optical apertures with depicted near-field distributions of the modes in $x$-$y$ plane.
Higher order lateral modes, with distributions that differ in the $x$ direction and fundamental-like distribution in the $y$ direction, reveal lower $Q$-factors. However, they display a similar tendency to the fundamental mode with increasing numbers of stripes. This shows that the optical structure favors the fundamental mode in a broad range of aperture sizes. Such behavior is expected in the case of MHCG-based cavities that are discriminative with respect to higher order modes, revealing the significant contribution of transversal components of wavevectors. Discontinuities in the function of the $Q$-factor are related to the emergence of low quality factor resonances that arise due to reflections between gold electrodes.
5.3 Exemplary results of self-consistent model
Figures 7(a), 7(b) demonstrate distributions of $E_zE_z^*$ in the $x$-$z$ plane within the central stripe, together with the electric field, current density, and gain profiles within the top and bottom active regions. Two examples of QC VCSEL configurations with relatively small ($60\,\mathrm{\mu}\mathrm{m}$, Fig. 7(a)) and large ($300\,\mathrm{\mu}\mathrm{m}$, Fig. 7(b)) electrical apertures are shown. Both examples represent the threshold of QC VCSEL operation. Despite the strongly anisotropic conductivity of the QCs promoting lateral current flow, the size of the QC active region is such that the current crowding effect is present in both configurations. The current nonuniformity is due to the lateral scheme of the current injection, which is funneled to the active region by selective proton implantation. The profile of the electric field within the active region reveals a nonuniform distribution with the minimal value in the center of the active region (Figs. 7(c), 7(d)). The range of the electric field ($F$) that enables high gain should overlap with regions of high current density. The result of overlapping of current density and electric field ($F$) is the gain distribution depicted for both configurations. In the example of a small aperture QC VCSEL, the gain distribution is relatively uniform and the size of the active region is small enough to enable positive gain in the whole aperture (Fig. 7(g)). This configuration enables fundamental mode emission in the threshold. In the case of a large aperture QC VCSEL, the nonuniformity of the electric field ($F$) and current density (Figs. 7(d), 7(f)) precludes high gain in the whole active region (Fig. 7(h)). The gain distribution promotes higher order lateral mode operation in the $x$ direction, while the fundamental mode distribution is preserved in the $y$ direction, since the distribution of the current in each stripe is the same. When applied bias increases, the position of the maximal gain shifts towards the central part of the active region as ilustrated in Fig. 5(b). In the case of extremely broad apertures, as in the example presented in Fig. 7(b), the maximum gain profile will not shift to the center of the active region, due to too low carrier density. Voltage-driven gain cumulation (VDGC) is the effect of maximum gain traveling in the active region driven by applied voltage.
Fig. 7. (a), (b) $E_zE_z^*$ distribution in the MHCG stripe in the $x-z$ plane. Distributions of electric field (c), (d), current density (e), (f) and material gain (g), (h) in the upper (dashed line) and lower (solid line) active regions. (a), (c), (e), (g) correspond to an optical aperture of $60\,\mathrm{\mu}\mathrm{m}$ and (b), (d), (f), (h) to an optical aperture of $300\,\mathrm{\mu}\mathrm{m}$.
5.4 Threshold operation of the QC VCSEL
Figure 8 summarizes the influence of the size of the electrical and optical apertures on the threshold current and the threshold current density. The operational region is limited by the line representing equality of electrical and optical apertures indicated by black dashed line in Figs. 8(a) and 8(b). When the electrical aperture is larger than the optical aperture, the electromagnetic field is scattered and absorbed by the gold electrode. This induces very high optical losses and makes stimulated emission impossible. Below the line, the gold contacts induce a waveguiding mechanism that confines the light between the electrodes in the $x$ direction. An optical aperture smaller than $50\,\mathrm{\mu}\mathrm{m}$ induces wavevectors with a significant lateral component. MHCG reflectivity is sensitive to the presence of lateral components in the wavevector, which lead to a significant increase in the threshold current density. The simulated emission from the smallest electrical aperture of around $20\,\mathrm{\mu}\mathrm{m}$ is prevented by the small overlap between the optical field with the gain region. Reaching the threshold in the case of the smallest optical and electrical apertures requires increasing the applied bias, which leads to values for the electrical field in the QCs exceeding the limit range in which high optical gain is possible. This effect is visible in the abrupt increase of the current density, although it is almost insignificant in the threshold current map for devices with small electrical aperture. Therefore, the QC VCSELs that operate at the lowest threshold currents are those with electrical apertures of $30\,\mathrm{\mu}\mathrm{m}$.
Fig. 8. Threshold currents (a) and threshold current densities (b) determined for QC VCSELs with various optical and electrical apertures. Circles represent fundamental modes, squares higher-order modes at the threshold. Black dashed line represents equality of optical and electrical apertures.
For a given optical aperture, the minimum threshold current density requires electrical apertures that are roughly half the size. This ratio is the result of the tradeoff between having an electrical aperture that is too small for the high gain that is required to balance all the optical losses of the mode, and having an electrical aperture that is too large with respect to the optical aperture, which contributes to strong light scattering and absorption by the metal electrode. QC VCSELs operating at the lowest threshold current density are an attractive option for possible continuous wave operation, due to the minimized density of heat sources.
The electrical aperture size is also responsible for fundamental mode operation, as discussed previously in section 5.3. An electrical aperture size of $180\,\mathrm{\mu}\mathrm{m}$ is the upper limit for fundamental mode operation at the threshold. Single fundamental mode operation enables low divergent emission and a single spot in the far field.
6. Conclusions
In this work, we have demonstrated by numerical simulations the physical phenomena that determine the architecture of quantum-cascade vertical-cavity surface-emitting lasers (QC VCSELs). We also explored mechanisms that influence QC VCSEL operation and determined the map of the threshold current and the current density. In particular, we discussed the key components that affect the operation of QC VCSELs: the position of the active region, which is split into two regions overlapping maxima of the $E_z$ component of the electromagnetic mode; the implementation of metal electrodes, and the position of contact layers to enable the lowest optical absorption by minimizing their overlap with the optical mode. We indicated spatial shifting of the gain mechanism which induced by nonuniform distribution of the electrical field across the active region. To ensure efficient emission, it is necessary to design a QC VCSEL with a high density of current that overlap with the required electrical field range. This tunning of carrier concentration and electrical field overlap is required exclusively in QC VCSELs and influences their efficiency significantly.
We also performed threshold analysis of QC VCSELs with a broad range of optical and electrical apertures. Threshold current is strongly related to the size of the electrical aperture, which should be smaller than optical aperture (determined by the distance between top gold electrodes). The threshold current dramatically increases when the size of the electrical aperture approaches the size of the optical aperture. Minimal averaged current density is achieved for an electrical aperture $30\%$ smaller than the optical aperture. This configuration minimizes the scattering losses at the metal contacts and enables efficient lateral overlapping of the mode with the gain region. Such configurations are expected to generate relatively small amounts of heat and are therefore promising for use as devices working under the continuous wave regime.
When comparing the minimum threshold current densities of $2.8\,\mathrm {kA/cm^2}$ and $4\,\mathrm {kA/cm^2}$ for the QC VCSEL and QCL (on the basis of which the active area was calibrated [26]), respectively, we can conclude that the cavity losses and light-electron interaction in QC have improved in QC VCSEL. However, this result is still far from the QCLs with the lowest current densities of $1.5\,\mathrm {kA/cm^2}$ for QCL emitting at the wavelength of $4.6\,\mathrm{\mu}\mathrm{m}$ [30] mainly (besides the wavelength) due to internal losses and light leakage in lateral directions in QC VCSEL. The minimal value of the threshold current of the QC VCSEL was $200\,\mathrm {mA}$, which is 10 times smaller than the threshold current of benchmark QCL which is a result of 40 times smaller electrical aperture in QC VCSEL with respect to benchmark QCL.
The proposed design suffers from the inherent difficulties of efficient heat dissipation. It is known that the efficiency of conventional edge emitting QCLs is severely limited by large heat generation in the active region. In QC-VCSEL active regions are embedded in narrow stripes of the grating, with thick enough layer above the active regions responsible for heat spreading and thermal flow is additionally deteriorated by periodic structure of the DBR. Therefore only operation under pulse regime is possible in the proposed configuration. Another issue is a complex fabrication of the grating with metal contacts and wafer bounding of the DBR. Nevertheless, possibility of threshold current reduction, together with the possibility of single longitudinal and lateral mode emission with low beam divergence and the possibility of two-dimensional array fabrication with additional possible feature of resonant wavelength tuning by manipulation of MHCG lateral parameters [29], makes QC VCSELs an attractive alternative to conventional QCLs in spectroscopic and free-space communications applications.
Funding
Narodowe Centrum Nauki (2017/25/B/ST7/02380, 2020/37/B/ST7/01830).
Acknowledgments
This work is supported by the National Science Center, Poland within the projects OPUS No 2017/25/B/ST7/02380. The gain calculations were supported by the National Science Center, Poland, under the Grant OPUS-19 No 2020/37/B/ST7/01830.
This work has been completed while the first author was the Doctoral Candidate in the Interdisciplinary Doctoral School at the Lodz University of Technology, Poland.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. A. Wong, “VCSELs and Their Use in Emerging Applications,” in SPIE Photonics West Industry Events, vol. 11768 (SPIE, 2021), vol. 11768, p. 117680P.
2. C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4(3), 379–440 (2012). [CrossRef]
3. M. Amann, “Tuning triumph,” Nat. Photonics 2(3), 134–135 (2008). [CrossRef]
4. M. Marciniak, T.-S. Chang, T.-C. Lu, F. Hjort, Å. Haglund, Ł. Marona, M. Gramala, P. Modrzyński, R. Kudrawiec, K. Sawicki, R. Bożek, W. Pacuski, J. Suffczyński, M. Geębski, A. Broda, J. Muszalski, J. A. Lott, and T. Czyszanowski, “Impact of Stripe Shape on the Reflectivity of Monolithic High Contrast Gratings,” ACS Photonics 8(11), 3173–3184 (2021). [CrossRef]
5. V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18(16), 16973–16988 (2010). [CrossRef]
6. V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36(9), 1704–1706 (2011). [CrossRef]
7. U. Fano, “The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves),” J. Opt. Soc. Am. 31(3), 213–222 (1941). [CrossRef]
8. W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer, 1994).
9. M. S. Vitiello, G. Scalari, B. Williams, and P. De Natale, “Quantum cascade lasers: 20 years of challenges,” Opt. Express 23(4), 5167–5182 (2015). [CrossRef]
10. T. Czyszanowski, “Quantum-Cascade Vertical-Cavity Surface-Emitting Laser,” IEEE Photonics Technol. Lett. 30(4), 351–354 (2018). [CrossRef]
11. L. Xu, C. A. Curwen, J. L. Reno, and B. S. Williams, “High performance terahertz metasurface quantum-cascade VECSEL with an intra-cryostat cavity,” Appl. Phys. Lett. 111(10), 101101 (2017). [CrossRef]
12. A. Krishnamoorthy, L. Chirovsky, W. Hobson, J. Lopata, J. Shah, R. Rozier, J. Cunningham, and L. d’Asaro, “Small-signal characteristics of bottom-emitting intracavity contacted VCSEL’s,” IEEE Photonics Technol. Lett. 12(6), 609–611 (2000). [CrossRef]
13. R. P. Sarzała. and W. Nakwaski, “Optimization of 1.3 $\mathrm{\mu}$m GaAs-based oxide-confined (GaIn)(NAs) vertical-cavity surface-emitting lasers for low-threshold room-temperature operation,” J. Phys.: Condens. Matter 16(31), S3121–S3140 (2004). [CrossRef]
14. R. Sarzała, T. Czyszanowski, M. Wasiak, M. Dems, Ł. Piskorski, W. Nakwaski, and K. Panajotov, “Numerical Self-Consistent Analysis of VCSELs,” Adv. Opt. Technol. 2012, 1–17 (2012). [CrossRef]
15. M. Dems, R. Kotynski, and K. Panajotov, “PlaneWave Admittance Method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13(9), 3196–3207 (2005). [CrossRef]
16. R. Mittra and U. Pekel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microw. Guid. Wave Lett. 5(3), 84–86 (1995). [CrossRef]
17. C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1(1), 011307 (2014). [CrossRef]
18. T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B 79(19), 195323 (2009). [CrossRef]
19. C. Sirtori, F. Capasso, J. Faist, and S. Scandolo, “Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells,” Phys. Rev. B 50(12), 8663–8674 (1994). [CrossRef]
20. J. Faist, Quantum Cascade Lasers (Oxford University, 2013).
21. A. Wacker, “Gain in quantum cascade lasers and superlattices: A quantum transport theory,” Phys. Rev. B 66(8), 085326 (2002). [CrossRef]
22. A. Kolek, “Nonequilibrium Green’s function formulation of intersubband absorption for nonparabolic single-band effective mass Hamiltonian,” Appl. Phys. Lett. 106(18), 181102 (2015). [CrossRef]
23. F. Banit, S.-C. Lee, A. Knorr, and A. Wacker, “Self-consistent theory of the gain linewidth for quantum-cascade lasers,” Appl. Phys. Lett. 86(4), 041108 (2005). [CrossRef]
24. T. Czyszanowski and W. Nakwaski, “Comparison of Exactness of Scalar and Vectorial Optical Methods Used to Model a VCSEL Operation,” IEEE J. Quantum Electron. 43(5), 399–406 (2006). [CrossRef]
25. T. Czyszanowski and W. Nakwaski, “Usability limits of the scalar effective frequency method used to determine modes distributions in oxide-confined vertical-cavity surface-emitting diode lasers,” J. Phys. D: Appl. Phys. 39(1), 30–35 (2005). [CrossRef]
26. M. Bugajski, P. Gutowski, P. Karbownik, A. Kolek, G. Hałdaś, K. Pierściński, D. Pierścińska, J. Kubacka-Traczyk, I. Sankowska, A. Trajnerowicz, K. Kosiel, A. Szerling, J. Grzonka, K. Kurzydłowski, T. Slight, and W. Meredith, “Mid-IR quantum cascade lasers: Device technology and non-equilibrium Green’s function modeling of electro-optical characteristics,” Phys. Status Solidi B 251(6), 1144–1157 (2014). [CrossRef]
27. O. Hassan, R. Faria, F. Hayee, S. Sohel, A. Ahmed, and M. Talukder, “Bias dependence of gain spectrum and output emission characteristics of two phonon resonance design quantum cascade lasers,” in 2012 7th International Conference on Electrical and Computer Engineering, ICECE 2012, (2012).
28. M. Gębski, J. A. Lott, and T. Czyszanowski, “Electrically injected VCSEL with a composite DBR and MHCG reflector,” Opt. Express 27(5), 7139–7146 (2019). [CrossRef]
29. E. Haglund, J. S. Gustavsson, J. Bengtsson, Å. Haglund, A. Larsson, D. Fattal, W. Sorin, and M. Tan, “Demonstration of post-growth wavelength setting of VCSELs using high-contrast gratings,” Opt. Express 24(3), 1999–2005 (2016). [CrossRef]
30. F. Cheng, F. Cheng, F. Cheng, J. Zhang, J. Zhang, Y. Guan, Y. Guan, P. Yang, P. Yang, N. Zhuo, S. Zhai, J. Liu, J. Liu, L. Wang, L. Wang, S. Liu, S. Liu, F. Liu, F. Liu, F. Liu, F. Liu, Z. Wang, and Z. Wang, “Ultralow power consumption of a quantum cascade laser operating in continuous-wave mode at room temperature,” Opt. Express 28(24), 36497–36504 (2020). [CrossRef]
31. M. Marciniak, M. Gębski, M. Dems, E. Haglund, A. Larsson, M. Riaziat, J. A. Lott, and T. Czyszanowski, “Optimal parameters of monolithic high-contrast grating mirrors,” Opt. Lett. 41(15), 3495–3498 (2016). [CrossRef]
32. R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of gold,” Phys. Rev. B 86(23), 235147 (2012). [CrossRef]
33. A. Szerling, K. Kosiel, M. Kozubal, M. Myśliwiec, R. Jakieła, M. Kuc, T. Czyszanowski, R. Kruszka, K. Pągowska, P. Karbownik, A. Barcz, E. Kamińska, and A. Piotrowska, “Proton implantation for the isolation of AlGaAs/GaAs quantum cascade lasers,” Semicond. Sci. Technol. 31(7), 075010 (2016). [CrossRef]
34. T. Miyamoto and H. Sakamoto, “Proton implantation in just above active region for improvement of power conversion efficiency of VCSELs,” Electron. Lett. 57(15), 587–588 (2021). [CrossRef]
35. P. Ressel, H. Strusny, S. Gramlich, U. Zeimer, J. Sebastian, and K. Vogel, “Optimised proton implantation step for vertical-cavity surface-emitting lasers,” Electron. Lett. 29(10), 918–919 (1993). [CrossRef]
