1. Introduction

Random lasers [1–4] are a class of lasers in which the laser light is trapped within a disordered optical medium through multiple scattering, rather than a traditional sharply-defined cavity. Originally predicted by Letokhov in 1967 [1], random lasers have been realized in many different settings, including colloids infiltrated with laser dye [5], semiconductor powders [6–8], and semiconductor slabs with etched holes acting as scattering centers [9]. Random lasers can be used to study a variety of interesting phenomena, such as light localization [10], and have promising technological applications as speckle-free (low spatial coherence) light sources for imaging [11, 12] and chemical sensing [13–15]. For applications, it is highly advantageous to focus on random lasers in the “slab” geometry, with the scattering taking place within a 2D plane, with vertical index confinement and either vertical or edge emission. This allows for electrical pumping, and easier integration with other photonics devices.

Cao and co-workers have recently performed detailed studies of the efficacy of light confinement (and hence laser performance) in disordered 2D media [16, 17]. They found that the modal Q factors can be dramatically affected by the presence of short-range order, which can be produced, for example, by the close-packing of air holes on a dielectric slab. Such structures, which possess short-range order but lack long-range order, are called “amorphous”. At high index contrasts, amorphous structures can exhibit dips in the photonic density of states (DOS) [18, 19], analogous to bandgaps in fully-ordered photonic crystals [20]. However, even for low index contrast and relatively weak short-range order, where the DOS has a barely-observable dip rather than a gap, the Q factors of the modes can still demonstrate pronounced peaks near the DOS dip [16, 17]. In order to achieve low-threshold, multi-modal operation for imaging and other applications, amorphous slab lasers should be designed so that the operating frequency range of the gain medium coincides with a DOS dip.

The studies of Cao et al. were focused on transverse electric (TE) modes [16, 17], because most lasing media, including semiconductor laser diodes, emit in the TE polarization. However, transverse magnetic (TM) emission dominates in some lasers, including strained quantum well lasers [21] and quantum cascade lasers (QCLs) [22]. Notably, QCLs are among the only high-power light sources available in the mid-infrared to terahertz frequency range [23]. Recently, the first random laser operating at the mid-infrared frequencies was demonstrated by Liang et al., using a QCL slab with etched air holes [13]. This device, however, exhibited rather low Q factors and did not reach the regime of multi-mode lasing.

In this paper, we present numerical studies of TM mode confinement in amorphous photonic structures. Previous studies of TE modes dealt with structures consisting of air holes in a dielectric medium [16], or a network of dielectric veins [17]. As we have mentioned, the random QCL realized by Liang et al. also adopted an air hole design [13], as did the subsequent numerical studies by Molardi et al. [24, 25]. However, air hole structures are unsuitable for hosting TM modes. In 2D photonic crystals, TM bandgaps are opened by having “isolated” high-dielectric regions [20]; likewise, in the disordered regime, previous researchers have found strong dips in the TM DOS in amorphous photonic structures consisting of unconnected dielectric rods [18, 19]. We verify this basic result below for an amorphous structure consisting of unconnected rods arranged in a disordered honeycomb-like lattice. Near the gap edge, the TM modes indeed experience substantial Q factor enhancements, of up to three orders of magnitude.

However, a design consisting of unconnected dielectric rods is problematic for realizing an electrically pumped random laser, such as a random QCL, since the pumping current needs to be passed through the dielectric gain medium. To resolve this difficulty, we turn to a “connected” structure consisting of dielectric rods joined by dielectric veins. This is quite similar to the structure studied by Steinhardt and co-workers [26–28]. Those authors, however, were primarily concerned with achieving deep and complete (TE and TM) bandgaps, rather than enhancing bulk mode Q-factors for lasing.

We demonstrate that connected amorphous structures can exhibit Q-factor enhancements induced by a DOS dip. For relatively high vein width, comparable to the rod radius, we find a weak dip in the DOS, but the TM Q-factors are enhanced by around two orders of magnitude. Notably, this is a much larger enhancement than what was previously reported for TE modes [16]. When the vein width is further increased, the structure becomes similar to a network of dielectric veins [17], and the TM Q-factor enhancement disappears. Our study provides a guide for the design of future TM-polarized electrically-pumped amorphous slab lasers, and particularly QCLs operating in the technologically important mid-infrared to terahertz regimes.

2. Design of the amorphous photonic structures

The process for generating the 2D amorphous structures is shown in Figs. 1(a)–1(d). A set of 1025 disks is jam-packed (into a large circular region) using a molecular dynamics simulation [16]. Using bidisperse disks with radius fraction 1.2, we can achieve packed configurations with filling fractions of ≈ 0.7. The disk centers, which form an amorphous lattice locally similar to a triangular lattice, are then connected by Delaunay triangulation, as shown in Fig. 1(b). The incenters of each triangle form an amorphous honeycomb-like lattice, and are used as the center-points of dielectric rods (disks) of radius r. The original disks that were used for packing are now discarded; the packing procedure ensures that the lattice points are well-separated and form an amorphous lattice. The result is shown in Fig. 1(c). Finally, we connect nearest-neighbor rods using veins of width w, as shown in Fig. 1(d).

 

Fig. 1 Generation of the amorphous photonic structure: (a) An ensemble of bidisperse disks (blue and black circles) is close-packed, so that the disk centers (red circles) form an amorphous triangle-like lattice. (b) Delaunay triangulation of the disk centers (green lines). (c) The triangle incenters (black circles form an amorphous honeycomb-like lattice of dielectric rods. (d) Neighboring rods are joined by dielectric veins to form a connected lattice. The packing is performed within a circular region, producing an approximately circular sample [see Fig. 2(a)]; these plots show a close-up view of the lattice. (e) Spatial correlation function C(Δr) for the lattice of unconnected rods in part (c), averaged over 100 disorder realizations. The vertical lines represent the values of the displacement Δr corresponding to the nearest-neighbor (n.n.) and next-nearest-neighbor (n.n.n.) positions in the ordered honeycomb lattice.

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In the following, we set the radius of the dielectric rods to r = 0.3655a, where a is the mean nearest-neighbor distance between rod centers. In the absence of the veins, the filling fraction of the dielectric regions is around 0.30. Fig. 1 (e) shows the spatial correlation function, C(Δr) ≡ 〈Θ(r + Δr)Θ(r)〉/〈Θ(r)〉² – 1, for the unconnected structure of Fig. 1(c). (For this calculation, we take Θ = 1 inside the rods, and Θ = 0 outside.) The correlation function exhibits clear peaks at Δr ≈ a and $\mathrm{\Delta }r\approx \sqrt{3}a$, corresponding to the nearest-neighbor and next-nearest neighbor spacings of the perfectly-ordered honeycomb lattice. This confirms that the structure contains substantial short-range order.

We now set the refractive index of the rods to n_rods = 3.35, consistent with QCL media [22, 23]. The surrounding medium will be refractive index 1 (air). If we target a mid-infrared free-space operating wavelength of 10 µm, then for DOS dips occurring at frequencies around 0.2c/a (see below), the rod radius will be r ≈ 0.75 µm and the mean lattice constant will be a ≈ 2.1 µm. For comparison, this choice of rod radius is approximately half the radius of the etched hole features in the QCL random laser reported in [13].

3. Simulation results

We first consider an amorphous lattice of unconnected rods (w = 0), as shown in Fig. 2(a). The 2D spatial Fourier transform of this structure (for a single realization) is shown in Fig. 2(b); it forms clear rings, consistent with the spatial correlation function plotted in Fig. 1(e).

 

Fig. 2 (a) Schematic of a disordered unconnected lattice, consisting of ≈ 1950 unconnected dielectric rods in a honeycomb-like pattern. The rod radius is r = 0.366a, where a is the mean distance between the centers of nearest-neighbor rods. (b) 2D Fourier transform of the lattice. (c) Transverse-Magnetic (TM) Density of States (DOS) of the disordered lattice, computed using 100 disorder realizations (dark blue); the TM DOS of a honeycomb photonic crystal with the same a is shown for comparison (gold). (d) Q-factors of the TM modes for a single disorder realization, computed using 2D finite-difference time-domain (FDTD) simulations. (e) Average Q-factors versus frequency for the TM modes, calculated from 100 disorder realizations and taking the mean Q-factor in each frequency bin.

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To determine the optical properties, we performed finite-difference time-domain (FDTD) simulations using the MEEP program [29]. Fig. 2(c) shows the TM DOS spectrum, averaged over 100 disorder realizations. This exhibits an extremely strong gap (3 orders of magnitude dip in the DOS) around f ≈ 0.25c/a; this corresponds almost perfectly with the photonic bandgap of a honeycomb lattice of dielectric rods having the same nearest-neighbor lattice constant a. This is also consistent with previous studies of such amorphous photonic structures [18, 19].

Figs. 2(d)–2(e) show the Q factors of the TM modes. These are again obtained from FDTD simulations, using the response to a broad-band TM source [29]. As expected, there are no modes present in the gap; the modes just below the gap, however, have extremely high Q factors which are enhanced by up to 3 orders of magnitude relative to the other modes.

We now consider connected structures, which are generated by joining nearest-neighbor rods with dielectric veins of width w. (The rods and veins have the same refractive index.) Relatively thick veins are desirable in order to reduce Ohmic heating from electrical pumping, and to maximize the area of gain material available. Fig. 3(a) shows a connected structure with vein width w = 0.9r. The dielectric filling fraction (i.e., the relative area covered by the dielectric region) can be estimated from the equivalent honeycomb lattice:

 

Fig. 3 (a) Schematic of a disordered connected lattice, consisting of dielectric rods in a honeycomb-like pattern. The rods have radius r = 0.366a, and are connected by dielectric veins of width w = 0.9r. (b) 2D Fourier transform of the lattice. (c) TM DOS for moderate vein width (w = 0.9r) and high vein width (w = 2r). (d)–(e) Q-factors of the TM modes. In (e), the results are obtained by taking the mean in each frequency band over 100 disorder realizations. The w = 0.9r case still exhibits a two order of magnitude enhancement in Q-factors at f ≈ 0.2c/a, near the DOS dip.

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For our chosen rod radius of r = 0.3655a, this filling fraction is ≈ 0.44. For the amorphous lattice, we numerically determined the filling fraction to be approximately 0.4220 ± 0.0003. The spatial Fourier transform, shown in Fig. 3(b), still exhibits a strong primary Bragg ring indicating short-range order. The TM DOS is shown in Fig. 3(c). The presence of the moderately wide veins has strongly suppressed the TM gap, compared to the unconnected case [Fig. 2(c)]; this is consistent with the principle that TM gaps require well-isolated high-dielectric regions. However, there remains a small dip at f ≈ 0.2c/a. Around frequency, the TM modes have strongly enhanced Q-factors, which are up to 2 orders of magnitude larger than the modes at other frequencies, as shown in Figs. 3(d)–3(e).

The presence of the DOS dip and the Q-factor enhancements do require the veins to be narrower than the rod diameter. In Figs. 3(c) and 3(e), we also show the case of w = 2r. In this case, the rods are not “visible”, and the lattice becomes a network of veins similar to that studied in [17] by Noh et al. Evidently, these TM modes all have similar Q-factors.

Fig. 4 shows the dependence of the Q-factor enhancement on the vein width w. Here, we calculate the mean Q-factors, averaging over 50 disorder realizations and sampling modes in the frequency range 0.198c/a < f < 0.226c/a (which is roughly the range in which the Q-factor enhancement occurs, as shown in Fig. 2(e) and Fig. 3(e)). As can be seen, these Q-factors decrease from ~ 10⁵ in the unconnected structure (w = 0), to the background level ~ 10³ when the vein width equals the rod diameter (w = 2r). The case shown in Fig. 3, w = 0.9r, coincides with the “crossover” region. (However, the crossover value of w/r also depends on our specific choice of r.)

 

Fig. 4 Mean Q-factor versus vein width w. The Q-factors are sampled using modes in the frequency range 0.198c/a < f < 0.226c/a, over 50 disorder realizations.

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It should be noted that changing the vein width w also changes the dielectric filling fraction, since the rod radius r is kept fixed. Specifically, increasing w from 0 to 0.9r increases the filling fraction from ≈ 0.3 to ≈ 0.42. However, this filling fraction increase does not significantly alter the frequency of the Q-factor peak, which shifts by only ≈ 3%. Similar behavior may be observed in the bandstructure of a ordered honeycomb lattice, in which increasing vein width barely shifts the frequency of the lower band edge.

4. Field distributions

Fig. 5 shows the field distributions in an unconnected and a connected structure, when excited by Gaussian sources near the frequency of the the high-Q modes. The excited modes are not extended over the entire structure, but are localized to regions that are multiple lattice spacings in size. In the unconnected structure, the modes are evidently strongly-concentrated in the dielectric rods. This behavior is favorable for lasing, for it increases the mode overlap with the gain medium (which occupies the dielectric regions). The connected structure exhibits similar behavior, though the concentration is less apparent.

 

Fig. 5 Field patterns for the out-of-plane electric field (E_z), in the unconnected and connected structures. The modes are excited by five Gaussian sources at positions [0, 0], [±14.6a, 0], and [0, ±14.6a]; all the sources have frequency center f₀ = 0.21c/a and frequency width Δf = 0.055c/a. The field snapshot is taken at time 1.54 × 10³/f₀ after the sources end.

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To quantify the mode concentration, we calculate the “concentration factor” $\mathcal{C}$, defined as

Here, $\epsilon (\overrightarrow{r})$ denotes the dielectric function at position $\overrightarrow{r}$, and ε_d = (3.35)² is the dielectric constant of the high-dielectric regions (rods and veins). The integral in the numerator is restricted only to the high-dielectric region, while the integral in the denominator is taken over all space. The limit where the modes are entirely concentrated in the high-dielectric regions corresponds to $\mathcal{C}\to 1$.

Table 1 shows the concentration factors calculated from FDTD simulations. The simulations are run until long after the exciting Gaussian sources are turned off, in order for low-Q modes to decay away. For each simulation, the concentration factors are computed using the mean of 18 successive “snapshots” (to average over the oscillation period), and the results are further averaged over 25 disorder realizations.

Table 1. Concentration factor $\mathcal{C}$, as defined in Eq. (2), for modes excited by Gaussian sources with different center frequencies: f₀ = 0.16c/a (low frequency, below the DOS dip), f₀ = 0.21c/a (mid frequency, near the DOS dip), and f₀ = 0.267c/a (high frequency, above the DOS dip). The sources are located at the same positions as in Fig. 5, and they all have frequency width Δf = 0.055c/a. Results are averaged over 25 disorder realizations.

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From these results, we see that for w = 0, modes are highly concentrated in the dielectric rods in the “mid-frequency” region, where the Q-factors are strongly enhanced. The concentration is less prominent at other frequencies. A similar trend is observed for w = 0.5r. However, for w = 0.9r, we find that C ≳ 0.9 in all three frequency regions. This is beneficial for lasing, since most of the electric field energy is concentrated in the dielectric region where the gain resides.

5. Discussion and Conclusion

In the paper, we have demonstrated that an amorphous lattice of dielectric rods connected by relatively wide dielectric veins can support high-Q TM modes. This is a promising design for electrically-pumped amorphous quantum cascade lasers (QCLs) operating in the infrared to terahertz frequency range. By contrast, the random QCL previously reported in [13], by Liang et al. consisted of a dielectric slab with random air holes; that class of lattice lacks any DOS gap or dip, and hence does not exhibit any short-range order induced enhancement in the Q-factors of the TM modes.

Because the connectivity of the structure is important for electrical pumping, it is noteworthy that an enhancement of two orders of magnitude in the TM mode Q factors can be obtained for relatively wide veins, with w = 0.9r and area filling fraction of ≈ 0.42. Whether this provides sufficient electrical contact for a practical QCL shall have to be determined empirically. If so, the relatively high concentration factors (with ~ 90% mode overlap with the high-dielectric region) implies that the resulting modes should be favorable for low-threshold lasing.

Our numerical study was based on 2D simulations, similar to previous numerical studies of disordered slab structures [16–19]. These simulations neglect vertical (out-of-plane) scattering and emission, which will in practice reduce the Q factors of the modes. However, these vertical scattering processes should be relatively independent of the DOS dip that arises from in-plane scattering. Hence, in the full 3D structures we should still expect a substantial Q-factor enhancement near the DOS dip, relative to the other modes.

Finally, we would like to comment on the relationship between our work and that of Steinhardt and co-workers [26–28], who have performed extensive studies of similar structures (rods connected by veins). Those authors were mainly concerned with optimizing the (TE and TM) bandgaps, and introduced veins in order to help open the TE gaps. For this purpose, relatively narrow veins of width w ≈ 0.15r were sufficient [26]. In our case, only the TM modes are of interest, and we are not necessarily interested in deep TM gaps. Hence, we assumed significantly wider veins w ≈ 0.9r, giving rise to a TM DOS dip rather than a gap, which turns out sufficient to produce large Q-factor enhancements.