# Multimode, Aperiodic Terahertz Surface-Emitting Laser Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design and Simulations

^{2}in order to allow a proper balance between the total dissipated electrical power and surface-related diffractive effects.

_{vertical}) and to validate the predictions of the 2D model.

_{1}= 3.6 and n

_{2}= 2.7 in the metallized and non-metallized regions, respectively. This approximation may seem rather crude for a patterned device, as the fields are no longer uniform in the vertical direction due to the presence of the apertures [21]. Indeed, it is known that it induces deviations in the calculation of the eigenfrequencies; nonetheless, it gives a good qualitative representation of the existing modes and their symmetries. The 2D simulations were performed after defining a surrounding domain with complex effective index n

_{3}= 3.6 + 0.36i, which was used to model the absorbing layer, providing smooth boundary conditions for the guided modes. Finally, an external domain having n

_{4}= 1.0 and scattering boundary conditions (SBC) was used to model the open boundaries outside the mesa. The three-dimensional (3D) model was conversely performed by considering the top and bottom metallization as perfect electric conductors (PECs). The etched GaAs/AlGaAs QCL heterostructure with the thin (7 nm) absorbing Cr border has been treated as a Cr-surrounded 10-µm-thick GaAs slab with a uniform refractive index of 4.43 + 0.31i. The thin chromium border allows suppressing any Fabry-Perot or whispering gallery mode. The SBC set over the air domain around the resonator mimic light out-coupling to the free space. A schematic diagram of the simulated device is shown in Figure 1.

_{2D}), which accounts only for the lateral mode confinement, without including the out-of-plane radiative losses, was initially simulated to provide indication of the effects of the filling factor on the scattered light intensity. Modes with higher Q factors can here arise only from a reduced overlap with the outer absorbing boundary, i.e., from a spatial distribution mainly localized in the device center; these modes are therefore the principal ones confined by the grating feedback.

## 3. Results and Discussion

_{2D}> 200 can be found (Figure 2a) over a scattered background of low-Q

_{2D}(mean value Q

_{2D}= 125 ± 34) optical modes. Conversely, the defect quasi-crystal type B geometry shows a less scattered Q

_{2D}frequency distribution, with maximum values in the 150–180 range (Figure 3a). The more regular pattern architecture in the designed type A quasi-crystals induces an enhanced confinement of a few optical modes in the QCL resonator cavity, which is nicely reflected in the form factor S(k) (Figure 2c), which shows a few sharp peaks.

_{2D}modes, lying in the spectral region of 2.9–3.4 THz.

_{2D}modes A,B,C and D,E,F, represented by the modulus of the electric field component in the vertical (z) direction, are reported in Figure 3b–d and Figure 4b–d, respectively, and provide an initial indication of the possible Bragg peaks responsible for the feedback [19]. Standing waves form in the crystal as a consequence of multiple diffractions on the main reciprocal lattice points, according to the relation Σ (

**k**−

**K**) = 0, where

_{j}**K**are reciprocal lattice Bragg points and

_{j}**k**is the optical mode wavevector.

_{2D}modes and achieve multicolor emissions, we selected as an active medium a hybrid bound-to-continuum QCL design combined with a single-quantum-well phonon extraction stage [22], characterized by a 700 GHz gain bandwidth, extending from 2.7 THz to 3.4 THz.

_{th}) strongly depends on the photonic-quasi-crystal characteristics providing an initial indication that different optical modes are active, depending on the applied boundary conditions. Specifically J

_{th}progressively increases at large FF and varies between 400 A/cm

^{2}and 450 A/cm

^{2}. Furthermore, by increasing a and keeping fixed FF, a further J

_{th}increase is induced, as shown from the comparison between panel 5a and 5b.

_{th}vs FF trend follows the same behavior observed in type A resonators, with slightly larger values in the range J

_{th}= 450–530 A/cm

^{2}. However, differently from the previous case, a further lattice constant a increase at fixed FF (Figure 5d) induces a J

_{th}reduction.

_{vert}of the implementing resonators, the wall-plug efficiency (η

_{WP}) is commonly a very practical parameter to take into account.

_{WP}~ 0.01%. Figure 5f shows the measured η

_{WP}in our set of quasi-crystal resonators. Both type A and type B architectures allow reaching η

_{WP}= 0.14%, which is comparable with standard double-metal Fabry-Perot resonators.

^{−1}. Figure 6a–d show the set of spectra collected in the type A and type B resonators, respectively, at current values corresponding to the peak optical power. Multimode emission with a maximum of 10 spectral lines spread over a 430 GHz bandwidth is achieved for type A quasi-crystals (Figure 6b). The mode number and related intensities are highly dependent from the different geometrical parameters, resulting in an uneven amplification of the allowed modes in the photonic structures. In the case of defect mode type B quasi-crystals, the highest number of amplified modes (six) is obtained for the lowest FF = 23.3% and FF = 25% and for the larger lattice quasi-periodicity (Figure 6d). In all cases, the frequency spacing between adjacent modes cannot be attributed to Fabry-Perot-like or whispering gallery modes, due to the absorbing boundary created by the outer chromium border.

_{3D}optical modes are present, which are indeed responsible for the emitted laser lines of the corresponding resonators (Figure 6b–d), within a <2% frequency discrepancy.

_{H}= 105 K, which corresponds to a lattice temperature of T

_{L}≈ 111 K [23].

_{H}= 110 K (i.e., T

_{L}≈ 116 K). In all cases, the threshold current density increases with the temperature following the phenomenological formula J

_{th}= J

_{0}·exp(T/T

_{0}), which allows extracting T

_{0}= (113 ± 9)K for a type A laser (Figure 8c) and T

_{0}= (114 ± 10)K for a type B device (Figure 8d), demonstrating that the imprinted photonic structures do not significantly modify the device thermal behavior, which indeed matches that of the reference edge emitting double-metal QCL [22].

## 4. Conclusions

_{WP}= 0.14% have been reached in both configurations, with an optical beam divergence <15°. The demonstrated architecture opens interesting perspectives for the realization of multi-frequency surface-emitting resonators with possible impacts on physical investigations on novel micro-cavity phenomena and related application perspectives on imaging, metrology and optical communications.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix: Fabrication Details

_{0.15}Ga

_{0.85}As heterostructure based on the design reported in Reference [22]. The layer sequence is 5.5/11.0/1.8/11.5/3.8/9.4/4.2/18.4 (in nm), where Al

_{0.15}Ga

_{0.85}As layers are in bold, GaAs in Roman, and the underlined number indicates a doped layer with Si with a density of 2 × 10

^{16}cm

^{−3}. After growth, the QCL wafer was thermo-compressively bonded with an Au-Au interface on an n

^{+}-GaAs carrier wafer. After selective removal of the host GaAs substrate by etching to and removing the Al

_{0.5}Ga

_{0.5}As etch-stop layer, the active region was coated with a top Cr/Au (5 nm/150 nm) metallization. Using optical lithographic techniques, holes were imprinted on this surface, reproducing the quasi-crystal patterns generated by a MATLAB script, implementing the generalized dual method. Two distributions of vertices were designed: one features perfect symmetry under 2π/seven-rotation (type A), the other one is characterized by some defect points increasing the degree of disorder of the system (named type B). In order to implement strong absorbing boundary conditions, the pattern was surrounded by a pre-defined thin Cr (7 nm) frame in the shape of a 14-sided polygon which was placed 30 µm around each photonic structure, partially overlapping with the Au border in order to implement strongly absorbing boundary conditions. This Cr border acted as a mask during the reactive ion etching (RIE) process, preventing the n

^{+}top contact layer from being etched away at the periphery of the Penrose pattern where the absorbing boundary is required. As a final processing step, decagonal mesa structures were etched down to the bottom metal using an H

_{2}SO

_{4}:H

_{2}O

_{2}:H

_{2}O (11:9:50) etching solution to avoid lateral current spreading. Individual devices were indium soldered onto a copper block and symmetrically wire-bonded around the tetradecagonal border in order to ensure uniform current injection through the mesa, while avoiding any perturbative effects in the far-field.

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**Figure 1.**Simulated tetradecagonal mesa geometry with the main dielectric area having refractive index n

_{1}= 3.6, filled by circular scatters having refractive index n

_{2}. A surrounding region with a 35 μm width and a complex effective index of n

_{3}= 3.6 + 0.36i was used to model the absorbing layer, defining smooth boundary conditions for the guided modes. An external region, having n

_{4}= 1.0 and being terminated by scattering boundary conditions, was used to model the open boundaries outside the mesa device.

**Figure 2.**(

**a**) Scanning electron microscope (SEM) image of one of the fabricated type A resonators, with a hole radius r = 8.0 µm, average intersite distance a = 29.9 µm and area 0.53 mm

^{2}; (

**b**) SEM image of one of the prototype type B resonators, with r = 7.5 µm, a = 30.0 µm and area 0.53 mm

^{2}; (

**c**,

**d**) Form factor S(k) of the type A (

**c**) and type B (

**d**) quasi-crystals. The red spots indicate the reciprocal vectors, with radii proportional to associated squared Fourier coefficients. The blue circles of radius k

_{p}show Bragg peaks of S(k) responsible for vertical extraction.

**Figure 3.**(

**a**) Quality factor Q of the computed optical modes as a function of the radiation frequency for a seven-fold (type A) resonator with r = 8.0 µm and a = 29.9 µm. Green vertical lines indicate the employed QCL gain bandwidth; (

**b**–

**d**) Computed 2D spatial profiles of the electric field modulus for the higher Q optical modes of Figure 2a: (

**b**) A; (

**c**) B; and (

**d**) C.

**Figure 4.**(

**a**) Quality factor Q of the computed optical modes as a function of the radiation frequency for a seven-fold (type B) quasi-crystal resonator with r = 7.6 µm, a = 30.0 µm. Green vertical lines indicate the QCL gain bandwidth; (

**b**–

**d**) Computed 2D spatial profiles of the electric field modulus for the higher Q optical modes of Figure 3a: (

**b**) A; (

**c**) B; and (

**d**) C.

**Figure 5.**(

**a**–

**d**) Power current density (LJ) and voltage current density (VJ) characteristics measured at 10 K, in pulsed mode with a 1% duty cycle, for the whole set of fabricated type A (

**a**,

**b**) and type B (

**c**,

**d**) resonators. Panel (a) refers to a type A resonator with a = 29.9 µm, panel (b) to a type A resonator with a = 32.5 µm, panel (c) to a type B resonator with a = 28.8 µm and panel (d) to a type B resonator with a = 30.5 µm. Optical power scales have been corrected to take into account the detector collection efficiency and the absorption of the cyclic olefin copolymer cryostat window; (

**e**,

**f**) Slope efficiency dP/dI (

**e**) and wall-plug efficiency η

_{WP}(f) plotted as a function of the filling factor. Red symbols refer to type B samples having a = 28.8 μm (•) and a = 30.0 μm (▪); blue symbols refer to type A samples with a = 28.8 μm (•) and a = 30.0 μm (▪).

**Figure 6.**Emission spectra of type A (

**a**,

**b**) and type B (

**c**,

**d**) quasi-crystal lasers collected via an Fourier transform infrared spectrometer in rapid scan acquisition mode at 10 K while driving the QCLs with a 1% duty cycle. Panel (a) refers to a type A resonator with a = 29.9 µm, panel (b) to a type A resonator with a = 32.5 µm, panel (c) to a type B resonator with a = 28.8 µm and panel (d) to a type B resonator with a = 30.0 µm; (

**e**) Q-factor of the computed optical modes as a function of the radiation frequency for a seven-fold (type A) resonator with r = 8.0 µm and a = 29.9 µm; (

**f**) Q-factor of the computed optical modes as a function of the radiation frequency for a seven-fold (type B) quasi-crystal resonator with r = 7.5 µm, a = 30.0 µm. Green vertical lines indicate the QCL gain bandwidth.

**Figure 7.**(

**a**,

**b**) Far-field emission patterns of the (

**a**) type A (FF = 26.8%, a = 32.5 μm) and (

**b**) type B (FF = 25%, a = 30 μm) obtained by scanning a pyroelectric detector at a distance of about 5 cm distance from the device surface.

**Figure 8.**(

**a**,

**b**) J-V and J-L characteristics of a type A (

**a**) (FF = 26.8%), and a type B (

**b**) (FF = 25.0%) quasi-crystal, driven at 0.3% duty cycle and different heat sink temperatures; (

**c**,

**d**) Threshold current density J

_{th}as a function of the heat sink temperature for the type A and type B resonators of panels (a) and (b). The blue line represents the fitting function J

_{th}= J

_{1}+ J

_{2}·exp(T/T

_{2}), while the dashed line is the linear fit to the data for T ≥ 80 K.

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**MDPI and ACS Style**

Biasco, S.; Li, L.; Linfield, E.H.; Davies, A.G.; Vitiello, M.S.
Multimode, Aperiodic Terahertz Surface-Emitting Laser Resonators. *Photonics* **2016**, *3*, 32.
https://doi.org/10.3390/photonics3020032

**AMA Style**

Biasco S, Li L, Linfield EH, Davies AG, Vitiello MS.
Multimode, Aperiodic Terahertz Surface-Emitting Laser Resonators. *Photonics*. 2016; 3(2):32.
https://doi.org/10.3390/photonics3020032

**Chicago/Turabian Style**

Biasco, Simone, Lianhe Li, Edmund H. Linfield, A. Giles Davies, and Miriam S. Vitiello.
2016. "Multimode, Aperiodic Terahertz Surface-Emitting Laser Resonators" *Photonics* 3, no. 2: 32.
https://doi.org/10.3390/photonics3020032