By varying the geometrical filling factor from 8% to 12% and the hole radius from 3 to 5 µm, we investigate the interference pattern of a set of surface emitting, electrically pumped random lasers (RLs) at terahertz frequencies employing a surface pattern of random holes, studying the polarization of the emitted modes, the spectral emission, and the power extraction. By funneling the laser beams onto a wire grid polarizer, we demonstrate spectral reshaping of the RL emission and of the far-field profile, achieving highly collimated emission (5° × 3° angular divergence) and a spectral coverage of 340 GHz with up to 11 random lasing modes. The polarization analysis of the far-field and spectral characteristics of the laser offer an interesting tool to investigate the complex behavior of quantum cascade RLs, providing a route to engineer the emission of RLs in more detail.
INTRODUCTION
Photonic engineering of semiconductor heterostructures, combined with new resonator concepts, has recently enabled the performance of quantum cascade lasers (QCLs), operating at terahertz (THz) frequencies, to be designed with an incredible level of control, offering a flexible platform to tailor the emission spectrum, the output power, the beam profile, and its polarization simultaneously.1 The most widely exploited examples include third-order distributed feedback (DFB) lasers,2 bi-periodic DFB lasers,3 sinusoidal wire lasers,4 or graded photonic heterostructure lasers,5 all allowing one to generate a robust single-mode, linearly polarized beam, emitting in one direction, selected by design. Such a combination of specifications is ideal for a large variety of applications in astrophysics and remote sensing, since it combines the inherent low power consumption, small form factor, and continuous-wave (CW) operation, typical of the deeply sub-wavelength cavities of QCL wire lasers.6
Alternatively, one can conceive circular-grating DFB lasers,7 photonic crystal lasers,8–10 or quasi-crystal THz lasers;11–13 the latter, releasing the distinction between symmetric and anti-symmetric modes, can allow circumventing the typical photonic crystal issue of power cancellation in the far-field, achieving a remarkably high power extraction (240 mW), ≈720 mW/A slope efficiency,13 and good beam collimation (<10° divergence), mainly in a single-mode regime.11,12 In all these cases, the surface-emitting nature of the generated modes provides radial polarization, which is highly desirable when coupling light into a THz metal-wire waveguide,14 hollow waveguide,15 or the metallic tip of a near-field imaging system.16
Very recently, disordered random THz lasers have also been successfully developed, in either bi-dimensional17–20 or one-dimensional21 geometries. They strongly differ from conventional lasers, which traditionally comprise a gain medium enclosed in an optical cavity to produce feedback. Although random lasers (RLs) also require an active medium, this no longer needs to be embedded in a well-designed cavity, since lasing modes are confined by multiple scattering of light from randomly placed scatterers.22 In RLs, the emitted photons can be amplified and scattered many times in the gain medium, resulting in a rich interference scheme, with each optical mode having a different degree of localization. This complex interplay between the intrinsic disorder and the nonlinearity of a random active medium22,23 gives rise to many interesting physical phenomena, such as gain competition, nonlinear wave mixing of the optical modes,24 and Anderson localization.22 In addition, RLs also display a typically low spatial coherence,22 which, combined with the high temporal coherence, makes these lasers particularly suitable to applications in which the lack of speckle-induced artifacts is mandatory.25
THz QCL RLs have been demonstrated by employing different device geometries, exploiting air pillars17 [with filling fractions (FF) ranging from 8% to 34%], dielectric pillars18 (with FF ∼ 36.5%), or a combination of semiconductor and metal pillars19 (FF = 12%), all etched through the entire active region or through a disordered sequence of scatterers (with FF in the 5%–34% range) only implemented in the upper metal and highly doped semiconductor cladding,20 leaving the active region core unperturbed. Multimode emission, also in CW over an optical bandwidth of 450 GHz that perfectly matches the one of the core active region, has been demonstrated.20
As a common distinctive characteristic, all electrically pumped random QCLs developed so far display almost diffraction limited beams, indicating in-phase coherent emission from the laser surface, which reflects in Gaussian-like far-fields consisting of multiple spectral components and in an emission pattern that strictly depends on the implemented surface pattern of scatterers.16,20
Analyses of light polarization in semiconductor heterostructured lasers have been performed in the last years in resonators of different architectures.26–29
In this work, we investigate the polarization of the emitted beam, the interference pattern, the spectral emission, and the power extraction of RLs displaying very low FF (2%) and a different degree of short-range disorder. By shining the vertically emitted beam on a wire grid polarizer, we then spectrally reshape the random laser emission and the far-field profile achieving a richer sequence of random modes, with a highly collimated beam profile (5° × 3° angular divergence, in the best case), and a spectral coverage of 340 GHz with up to 11 random lasing modes.
To devise our random resonators, the QCL active medium is first sandwiched between two metallic cladding layers to create an Au–Au double-metal waveguide. Circular holes of radius r = 3 (sample A) and 5 µm (sample B) are then lithographically defined on the top metal layer and vertically etched through the metal and over the highly doped semiconductor cladding (see Methods). Consequently, THz photons undergo multiple elastic scattering and are confined and amplified inside the active material due to the high refractive index contrast between the gold-coated semiconductor and the air holes; furthermore, when the photon in-plane momentum is reduced to zero, light is extracted vertically through the holes and coupled into free space.
We fabricate two sets of resonators, employing a computer generated random arrangement of holes in a patterned surface of average side L, with an average inter-site distance a = L/, where N is the number of holes. The corresponding geometrical filling factor r/a varies from 8.5% (sample A) to 12% (sample B) as N varied from 90 (sample A) to 54 (sample B).
The hole pattern is generated through a Matlab code, positioning the holes in squares of side L = 325 µm, with a minimum edge-to-edge distance of 2 μm between each pair of holes in order to avoid overlap between the scatterers.
Figure 1(a) shows the schematic device layout. At small r/a, the propagation of the optical modes in the resonator is characterized by a quasi-ballistic regime,20 as an effect of the low hole density and the large average inter-site separation. Conversely, larger r/a provides a stronger light scattering mechanism, with a much richer interference, inducing resonating eigenmodes with a complex electric field distribution, as demonstrated in previous reports.20 The disordered photonic structure was then surrounded by an irregularly shaped absorbing chromium layer deposited on the mesa border featuring protrusions of typical size ≈25 μm [Fig. 1(a)]. The average width is therefore kept comparable with the wavelengths of the expected lasing modes in the semiconductor (≈35 μm for 3 THz radiation). Undesired electromagnetic modes, such as Fabry–Pérot (FP) or whispering-gallery modes, are then intentionally suppressed.
(a) 3D schematics of the random QCL sample. The random holes are marked in white. (b) Calculated autocorrelation functions for sample A (blue trace) and B (orange trace). The insets correspond to the random patterns of samples A and B. (c) Voltage–current density and light–current density characteristics of samples A and B measured by driving the device in pulsed mode with a pulse width of 1 µs (duty cycle 10%) at 15 K. The green dots mark the three operational regimes (505, 530, and 555 A cm−2 for laser A and 505, 530, 565 A cm−2 for laser B) selected to collect the emission spectra and the far field profiles. (d) and (e) Optical microscope images of samples A (d) and B (e).
(a) 3D schematics of the random QCL sample. The random holes are marked in white. (b) Calculated autocorrelation functions for sample A (blue trace) and B (orange trace). The insets correspond to the random patterns of samples A and B. (c) Voltage–current density and light–current density characteristics of samples A and B measured by driving the device in pulsed mode with a pulse width of 1 µs (duty cycle 10%) at 15 K. The green dots mark the three operational regimes (505, 530, and 555 A cm−2 for laser A and 505, 530, 565 A cm−2 for laser B) selected to collect the emission spectra and the far field profiles. (d) and (e) Optical microscope images of samples A (d) and B (e).
A preliminary analysis of the generated geometries is performed by calculating their spatial autocorrelation,20 whose sharper or smoother decay reflects the amount of short-range disorder of our structure.20 The spatial variation of the dielectric constant ε(r) can be characterized statistically by the correlator
where ⟨⟩ represents the ensemble average. When the random medium is isotropic, the width of K(r) is called the correlation radius Rc [Fig. 1(b)]. It reflects the length scale of spatial fluctuation of the dielectric constant. When Rc is smaller than the lasing wavelength in the material (∼35 μm for 3 THz radiation), light is scattered by short-range disorder. Comparing Rc with the average inter-site distance of the selected pattern, one can evaluate the amount of short-range disorder in the selected random structure. We have then calculated Rc20 and selected two random lasers having Rc = 0.14a (sample A) and Rc = 0.25a (sample B), i.e., a larger and a moderate short range disorder. Then, following the procedure described in Ref. 20, we have calculated the corresponding expected radiative out-coupling efficiency ηr that we assume is proportional to the ratio Q/Qvertical, where Q = (1/Qohmic + 1/Qvertical)−1 is the total quality factor; here, Qvertical accounts for the vertical radiative losses and Qohmic accounts for the in-plane losses. For the selected random patterns, the calculated average Q in the explored frequency range varies from Q = 80 in A to Q = 45 for laser B.
The computed photon loss rate due to surface emission of the main optically active modes,31 calculated as the time-averaged integrated power flow through the open air domains and normalizing it with respect to the resonator energy Eres is, γr ∼ 3.7 GHz, corresponding to a Qvertical ∼ 810 for laser A, and γr ∼ 3.4 GHz, corresponding to a Qvertical ∼ 910 for laser B. This results in predicted extraction efficiencies of ∼10% for laser A and ∼5% for laser B.
The voltage–current density (V–J) and the light–current density (P–J) characteristics of samples A and B [Fig. 1(c)] unveil threshold current densities Jth,A ∼ 440 A/cm2 and Jth,B ∼ 463 A/cm2, respectively, slightly larger than those retrieved in reference double-metal Fabry–Pérot cavities embedding the same active region (410 A/cm2)30 and in agreement with those extrapolated in random lasers having much larger r/a values (r/a = 21%–40%).20 The current density at the peak emission power is Jmax,A ∼ 555 A/cm2 and Jmax,B ∼ 565 A/cm2, meaning that the dynamic ranges Jmax/Jth remain almost comparable, varying from ∼1.26 in laser A to ∼1.21 in laser B. On the other hand, the peak optical power is significantly affected by the photonic pattern, varying from Ppeak = 14 mW in sample A to Ppeak = 5 mW in sample B, in agreement with the computed photon loss rates. Such behavior is in agreement with the reported decrease in the optical power with increasing filling factors and with decreasing r,20 as a consequence of the more efficient optical confinement achieved in resonators with the large metal-covered portion of the top contact (98% of the emitting surface [see Fig. 1(d)].
The evolution of the FTIR emission spectra [Figs. 2(a)–2(f)] and the far-field emission profiles [Figs. 2(g)–2(i) and 2(l)–2(n)] of laser A [Figs. 2(a)–2(c) and 2(g)–2(i)] and B [Figs. 2(d)–2(f) and 2(l)–2(n)], as a function of the driving current, collected under the same transport regimes [green dots in Fig. 1(c)], in pulsed mode (10% duty cycle, 1 µs pulse width) discloses visible differences between the two resonators.
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample A, measured in rapid scan mode using a deuterated triglycine sulfate (DTGS) pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 555 A cm−2 (c). (d)–(f) FTIR emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 565 A cm−2. (g)–(n) Far-field intensity patterns of sample A (g)–(i) and B (l)–(n), measured under the same experimental conditions of panels (a)–(f), while raster scanning a pyroelectric detector placed at a distance of ∼6 cm from the laser surface, in the plane orthogonal to the laser surface, and projecting the acquired 2D signal onto a spherical surface centered on the device.
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample A, measured in rapid scan mode using a deuterated triglycine sulfate (DTGS) pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 555 A cm−2 (c). (d)–(f) FTIR emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 565 A cm−2. (g)–(n) Far-field intensity patterns of sample A (g)–(i) and B (l)–(n), measured under the same experimental conditions of panels (a)–(f), while raster scanning a pyroelectric detector placed at a distance of ∼6 cm from the laser surface, in the plane orthogonal to the laser surface, and projecting the acquired 2D signal onto a spherical surface centered on the device.
The FTIR spectra of laser A [Figs. 2(a)–2(c)] show a rich multimode emission, with up to 12 optical modes covering the 2.91–3.25 THz range. While the spectral power is almost equally distributed among all modes at low bias, a progressive increase in the relative intensity of the dominant mode at ∼3.07 THz is visible while driving the QCL toward the peak optical power emission, where the dominant mode holds more than 80% of the total emitted power. Such power distribution is reflected in the corresponding far field profiles [Figs. 2(g)–2(i)]. At J = 505 A cm−2, the far-field map comprises two lobes covering a total angular size of ϕ ∼ 7° and θ ∼ 10°, encompassed by a lower intensity region with diffraction-ring-like intensity distribution related to the sub-diffraction limit size of the emitting scatterers. At J = 530 A cm−2, the two lobes diverge (ϕ ∼ 12° and θ ∼ 20°), as an effect of the increasing number of optically active modes with competing intensities; finally, at Jmax,A, the far-field reflects a highly collimated beam with a total divergence ϕ ∼ 10° and θ ∼ 12°, resulting either from the dominant mode centered at 3.07 THz that becomes 15 dB more intense that the others or from the interference of the 12 optically active modes.
The emission spectra of laser B [Figs. 2(d)–2(f)] comprise a lower number of emitted modes (5), with a dominant mode at 3.14 THz holding ∼90% of the total optical power. Different from sample A, the power distribution between the modes remains almost unperturbed when the driving current is increased. The bias dependent emission spectra show the characteristic RL multiplets20 that appear separated at low biases and more convoluted as the driving current is increased; in this latter case, three additional weak modes (showing a two-order of magnitude intensity decrease with respect to the dominant mode) at 3.200, 3.218, and 3.225 THz are activated at the onset of the negative differential resistance regime. The most remarkable differences between the two QCLs emerge from the comparison of the far-field patterns. In sample B, the intensity distribution is more irregular, evolving from a broad lobe distributed over a 20° × 20° angular region at J = 505 A cm−2 to a more irregular pattern, reflecting the incoherent superposition of the different optical modes in the far-field.
As a common characteristic, both lasers disclose a rich interference pattern in the far-field that is visibly related to the bias dependent spectral emission and discloses a specific polarization set by the geometrical configuration of the random scatterers. The holes can be indeed represented by randomly arranged and temporally coherent in-phase dipole emitters with a given polarization.
The far-field is the result of the incoherent superposition of the far-field intensities corresponding to different individual dipole orientations in a spatial dipole arrangement, which is the same for all orientations. A diffraction-limited beam is created if all dipole emitters are aligned in the same direction. A partial net polarization can lead to constructive interference for a fraction of the emission perpendicular to the sample surface, while in the totally unpolarized case, interference effects are expected to fully dominate the far-field patterns.
With the purpose to identify the correlation between the chosen random patterns and the polarization components of the surface emitting beams under the same transport regimes [green dots in Fig. 1(c)], we employ a wire-grid Mylar polarization filter positioned on a rotating stage. We first oriented the wire-grid polarizer at the angle (α in Fig. 1) that maximizes the transmitted signal at the peak emission and collect Ppol-peak. In laser A, where the random pattern is more homogeneously distributed, r/a is smaller, and r = 3 μm, such a direction is coincident with the in plane the x axis (α ∼ 0); on the contrary, in laser B (r = 5 μm, larger r/a), it falls at α ∼ 70°. We then turn the polarizer by 90°; as expected, the signal does not vanish, confirming that the random modes display different polarizations. The analysis of the ratio Ppol/P at α ∼ 0° (sample A) or α ∼ 70° (sample B), as a function of the current density [Figs. 3(a) and 3(b)] discloses a characteristic polarization behavior, which is distinctive of the specific random laser pattern.
(a) Ratio between the optical power of sample A measured with the polarizer at an orientation angle α = 0° (Ppol) and without (P) the polarizer, as a function of the current density. (b) Ratio between the optical power of sample B measured with the polarizer at an orientation angle α = 70° (Ppol) and without (P) the polarizer, as a function of the current density. (c) and (d) Angular polarization patterns measured in laser A at J = 505 (black dots), 530 (red dots), and 555 (blue dots) A cm−2. Angular polarization patterns measured in laser B at J = 505 (black dots), 530 (red dots), and 565 A cm−2 (blue dots).
(a) Ratio between the optical power of sample A measured with the polarizer at an orientation angle α = 0° (Ppol) and without (P) the polarizer, as a function of the current density. (b) Ratio between the optical power of sample B measured with the polarizer at an orientation angle α = 70° (Ppol) and without (P) the polarizer, as a function of the current density. (c) and (d) Angular polarization patterns measured in laser A at J = 505 (black dots), 530 (red dots), and 555 (blue dots) A cm−2. Angular polarization patterns measured in laser B at J = 505 (black dots), 530 (red dots), and 565 A cm−2 (blue dots).
In laser A [Fig. 3(a)], at J > 538 A cm−2, the power is distributed almost equally among the α = 0° (55%) and the α = 90° (45%) directions, while at lower current, it is larger at α = 90° (∼64%). Conversely, in laser B [Fig. 3(b)], the ratio Ppol/P is almost constant with the driving current with the linear component at α = 70° carrying 80% of the total emitted power.
The polarization-dependent intensity patterns, plotted as a function of α (Fig. 1), recorded in lasers A and B are shown in Figs. 3(c) and 3(d), respectively. In sample A, the polar intensity plots has an elliptical shape at Jmax that becomes more circular at lower currents; the main axis of the ellipse turns from α = 90° at J = 505 A cm−2 to α = 0° at Jmax, where the main mode is at 3.07 THz. On the other hand, the polar plots retrieved in sample B [Fig. 3(d)], strongly elongated in α = 70° direction, independent of the driving current, reflect a more pronounced net polarization in the surface emission patterns.
We then collect the emission spectra and the far-field intensity plots with the wire-grid polarizer oriented along the dominant polarization direction (α = 0° in sample A and α = 70° in sample B) and at 90° from that while varying the laser driving current.
The comparison between the spectra collected under those configurations in sample A [Figs. 4(a)–4(c), α = 0° and Figs. 4(d)–4(f), α = 90°] shows that while in both cases the rich multimodal emission is preserved, in the first case, the polarizer does not affect the intensity ratio between the emitting modes, with the most intense mode always at 3.07 THz, following the same bias dependent spectral evolution [Figs. 4(a)–4(c)], while in the second case (α = 90°), the spectral power is more equally distributed between all modes, independently from the applied bias and up to Jmax. These observation suggest that the mode at 3.07 THz is largely affected by the polarizer orientation, with the remaining modes becoming dominant while orienting the polarizer at α = 90° [Figs. 4(d)–4(f)].
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample A, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 555 A cm−2 (c) and with a wire grid polarizer oriented at α = 0°, according to the reference system on Fig. 1. (d)–(f) FTIR emission spectra of sample A, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (d), 530 (e), and 555 A cm−2 (f) and with a wire grid polarizer oriented at α = 90°, according to the reference system on Fig. 1. (g)–(i) Far-field intensity patterns of sample A, measured under the same experimental conditions of panels (a)–(c). (l)–(n) Far-field intensity patterns of sample A, measured under the same experimental conditions of panels (d)–(f). The pyroelectric detector is placed at a distance of ∼6 cm from the laser surface and raster scanned in the plane orthogonal to the laser surface. The acquired 2D signal is projected onto a spherical surface centered on the device.
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample A, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 555 A cm−2 (c) and with a wire grid polarizer oriented at α = 0°, according to the reference system on Fig. 1. (d)–(f) FTIR emission spectra of sample A, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (d), 530 (e), and 555 A cm−2 (f) and with a wire grid polarizer oriented at α = 90°, according to the reference system on Fig. 1. (g)–(i) Far-field intensity patterns of sample A, measured under the same experimental conditions of panels (a)–(c). (l)–(n) Far-field intensity patterns of sample A, measured under the same experimental conditions of panels (d)–(f). The pyroelectric detector is placed at a distance of ∼6 cm from the laser surface and raster scanned in the plane orthogonal to the laser surface. The acquired 2D signal is projected onto a spherical surface centered on the device.
The above effects are reflected in the corresponding far field profiles [Figs. 4(f)–4(m)]. At α ∼ 0°, the random laser shows a visibly collimated emission with a divergence varying from φ = 5°, θ = 3° at 505 A cm−2 [Fig. 4(f)] to φ, θ ≤ 10° at larger current densities [Figs. 4(g) and 4(h)]. In contrast, at α ∼ 90°, the low divergent nature of the emission profile is preserved (φ, θ ≤ 5°) only at 505 A cm−2 [Fig. 4(l)], while the dominant role of all modes, operated by the polarizer, is reflected in the distinctive spots retrieved in the far field profile [Figs. 4(m) and 4(n)].
Figure 5 shows the same experimental study performed on sample B. The bias dependent evolution of the FTIR emission spectra, acquired with the polarizer oriented at α = 70° [Figs. 5(a)–5(c)] and α = 160° [Figs. 5(d)–5(f)], reflects a dissimilar behavior with respect to sample A.
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 565 A cm−2 (c) and with a wire grid polarizer oriented at α = 70°, according to the reference system on Fig. 1. (d)–(f) FTIR emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (d), 530 (e), and 565 A cm−2 (f) and with a wire grid polarizer oriented at α = 160°, according to the reference system on Fig. 1. (g)–(i) Far-field intensity patterns of sample B, measured under the same experimental conditions of panels (a)–(c). (l)–(n) Far-field intensity patterns of sample B, measured under the same experimental conditions of panels (d)–(f). The pyroelectric detector is placed at a distance of ∼6 cm from the laser surface and raster scanned in the plane orthogonal to the laser surface. The acquired 2D signal is projected onto a spherical surface centered on the device.
(a)–(c) Fourier transform infrared (FTIR) emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (a), 530 (b), and 565 A cm−2 (c) and with a wire grid polarizer oriented at α = 70°, according to the reference system on Fig. 1. (d)–(f) FTIR emission spectra of sample B, measured in rapid scan mode using a DTGS pyroelectric detector while driving the laser in pulsed mode (10% duty cycle, 1 µs pulse width) at a heat-sink temperature of 15 K, at current densities of 505 (d), 530 (e), and 565 A cm−2 (f) and with a wire grid polarizer oriented at α = 160°, according to the reference system on Fig. 1. (g)–(i) Far-field intensity patterns of sample B, measured under the same experimental conditions of panels (a)–(c). (l)–(n) Far-field intensity patterns of sample B, measured under the same experimental conditions of panels (d)–(f). The pyroelectric detector is placed at a distance of ∼6 cm from the laser surface and raster scanned in the plane orthogonal to the laser surface. The acquired 2D signal is projected onto a spherical surface centered on the device.
The spectra do not show any visible polarization-induced spectral change, with the intensity ratio between the emitted optical modes mostly unperturbed by the polarizer orientation [Figs. 5(a)–5(f) and Figs. 2(d)–2(f)], with a 75% decrease in the emitted power [Fig. 3(b)] at α = 160°. Each peak appears as a convolution of an emission doublet, with two modes closed spaced in energy (<0.075 cm−1).
The comparison between the far-field intensity patterns without [Figs. 2(l)–2(n)] and with the polarizer [Figs. 5(g)–5(n)] reveals a visible reshaping of the intensity distribution pattern and a clear dependence from the polarized orientation angle. Below the negative differential resistance regime (J < 565 A cm−2), the comparison between the far-field profiles collected at α = 70° and α = 160° shows visibly complementary spots, which seems to be the effect of a dominant linear polarized emission of different optical mode. Interestingly, the highly divergent far-field profile [Figs. 2(l)–2(n)] is significantly reshaped in a ∼10° divergent emission profile [Fig. 5(h)] under a specific polarization (α = 70°). This suggest that each mode shows a net polarization along a specific orientation angle, meaning that the almost single lobe far-field profile results from the coherent superposition of the far-fields of different individual dipoles, associated with the imprinted random pattern.
It is worth mentioning that analogous experiments performed on a statistical set of four resonators having the same area and r/a of sample A and three resonators having the same area and r/a of sample B allow retrieval of similar results and the same conclusions. The measured device properties (frequencies and number of emitted optical modes, related intensity ratio, polarization, and far-field evolution) are therefore critically dependent on the specific scatterers’ size and distribution.
In summary, the study presented in the current work offers an interesting tool to investigate and “engineer” the complex behavior of QCL RLs. Through the polarization analysis of the far-field and spectral characteristics of the lasers, a deeper investigation on individual cavity modes can be drawn, providing a route to engineer the emission of RLs more in detail. Furthermore, the possibility to intensity and frequency modulate specific random modes via opto-mechanical coupling with an external mirror or grating mirror20 in continuous-wave31 opens up the way toward a new and versatile design principle for metrological-grade, broadband coherent light sources or low-spatial coherence sources for confocal microscopy applications32 and for near-field imaging applications requiring a fine control of the polarization of individual modes.
METHODS
Fabrication procedure
The GaAs/Al0.15Ga0.85As QCL heterostructure was grown by molecular beam epitaxy on an undoped GaAs substrate. The active region design is based on a three-quantum well resonant phonon scheme, and the layer sequence is 5.5/11.0/1.8/11.5/3.8/9.4/4.2/18.4 (in nm), where Al0.15Ga0.85As layers are shown in bold face, GaAs is in roman font, and the underlined number indicates a Si-doped layer with a density of 2 × 1016 cm−3. The active region growth is terminated by a 700 nm-thick highly doped (2 × 1018 cm−3) GaAs contact layer, with an Al0.5Ga0.5As etch-stop layer on the top. After growth, Au–Au thermo-compressive wafer bonding of the QCL wafer onto an n+-GaAs carrier wafer was performed. After removing the host GaAs substrate and the Al0.5Ga0.5As etch-stop layer by selective wet etching, the active region was coated with a top metal layer of Cr/Au (5/150 nm). By using a maskless laser writer for the optical lithography, the sample surface was patterned with air holes, whose center positions were placed with a uniform distribution in a square, which was defined using a MATLAB script. To reduce cavity losses, the 700-nm-thick n+ top contact layer was totally removed from below the etched holes by means of a reactive-ion etching (RIE) process. Strongly absorbing boundary conditions were then realized by adding an external 7-nm-thin Cr border imprinted on the active region using optical lithography, with irregular protrusions of average size ≈25 μm and an average width of ≈25 μm. Since the Cr border acts as a protective mask for the underlying active core during the RIE process (and will be partially removed by the process), the n+ top contact layer was not etched away in order to ensure the suppression of modes extending toward the edge of the devices. Finally, the mesa was etched down using a second RIE process, devised to optimize the vertical sidewalls of the border spikes and to avoid lateral current spreading. After processing, individual devices were indium soldered onto a copper bar and wire bonded regularly along the perimeter to ensure uniform current injection and minimize the perturbation of the far-field emission profile.
Optical characterization
For electrical and optical testing, the RLs were mounted on the cold finger of a liquid helium-flow cryostat to reach the desired operating heat-sink temperatures. The data shown in the manuscript have been corrected to take into account the 75%-absorption of the cryostat polyethylene window and the collection efficiency of the detector, retrieved by integrating the pyroelectric signal obtained by a far-field raster scan. Fourier transform infrared (FTIR) emission spectra, in rapid-scan mode with a frequency resolution of 0.075 cm−1, were acquired by focusing the laser radiation by means of an f/1 parabolic mirror through the FTIR interferometer (Bruker Vertex 80) onto an internal deuterated triglycine sulfate (DTGS) pyroelectric detector. The emitted power was measured by using a Thomas Keating absolute THz power-meter. The far-field emission patterns of the lasers were measured with a pyroelectric detector having a sensitive area of 7 mm2, which was scanned on a 2D square area at 6-cm-distance from the device. The retrieved x–y maps were then projected onto a spherical surface centered on the device, via geometrical transformation in ϑ, φ polar coordinates. The direction corresponding to φ = 0° and ϑ = 0° coincides with normal vector to the device surface (z axis). The piezoelectric driver used in the far field experiment was controlled with 20 V saw-tooth pulses with an average displacement of 0.18 μm/pulse at 15 K.
Simulations
A commercial finite element (FEM) solver was used to compute the eigenvalue solutions to Maxwell's equations for our system [illustrated schematically in Fig. 1(a)]. The top and bottom metal claddings are treated as perfect electrical conductors (PECs). Effective dielectric constants were calculated by solving the Helmholtz equation. The 10-μm-thick GaAs-based active medium is characterized by a bulk refractive index with n1 = 3.60, both under the metal and in the regions with holes, and the external mesa border is described with a complex refractive index n2 = 4.43 + i0.31, accounting for the thin absorbing chromium layer placed on the top of the QCL and providing smooth boundary conditions for the guided modes. Finally, the resonator is surrounded by air, which has an index n3 = 1, with scattering boundary conditions being imposed on all system boundaries to mimic free-space propagation. Since the FEM model includes losses due to both the vertical extraction of light and the lateral confinement, we can simulate the overall quality factors (Q) associated with the electromagnetic eigenmodes. The 3D simulation provides an estimate of the photon loss rate due to surface emission γr. This has been estimated by extracting from the 3D FEM simulation the time-averaged integrated power flow through the open air domains and normalized with respect to the resonator energy Eres,
where E and H represent the electric and magnetic fields, respectively, ϵ is the dielectric constant, and μ is the permeability. The corresponding quality factors have been derived from the relation Qvertical = ν/γr.
ACKNOWLEDGMENTS
This work was partly supported by the European Union ERC Consolidator Grant SPRINT (No. 681379).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.