1. Introduction

There has been significant progress in development of THz quantum cascade lasers (QCLs) [1, 2]. QCLs are the brightest available solid-state sources of coherent THz radiation that hold immense promise for applications in THz imaging, sensing, and molecular spectroscopy [3–5]. However, their use of sub-wavelength metallic cavities [6] results in poor radiation characteristics including low out-coupling efficiencies and divergent far-field beams that deviate significantly from the ideal fundamental Gaussian beam. There has been consistent prior effort toward development of distributed-feedback (DFB) and other out-coupling methods that include coupled-cavity techniques toward improving the spectral characteristics, beam quality, and output-power of THz QCLs [2].

The best radiative efficiencies and hence the highest output powers for single-mode THz QCLs have been reported using surface-emitting one-dimensional DFB schemes [7, 8]; however, such QCLs have divergent beams along the narrower lateral dimension of the cavities with the typical full-width at half-maximum (FWHM) divergence of $\gtrsim {20}^{\circ }$. There has been ongoing effort toward realizing THz QCLs with narrow beams including beam-shaping with metasurfaces [9, 10], and DFB structures such as those based on two-dimensional photonic-crystals [11], third-order DFB [12], antenna-feedback [13], and global mutual coupling of multiple-cavities [14]. However, the power output with all of these schemes has been lower compared to surface-emitting DFB QCLs. More recently, QCLs termed as VECSELs based on coupled-cavities and external feedback were demonstrated [15, 16] that achieve high power and low divergence simultaneously. However, DFB QCLs are arguably advantageous when robust single-mode operation is required at a specific desired frequency, or at a series of frequencies using arrays of lithographically designed QCLs on a single semiconductor chip.

Beam shaping of single-mode THz QCLs with optical components such as mirrors and lenses has not been explored in prior literature. For multi-mode ridge-cavity lasers, parabolic mirrors [17, 18] as well as high-density polyethylene (HDPE) [19] or polymethylpentene (TPX) [20, 21] lenses have been used for focusing. However, none of the reports mentioned focusing efficiency (fraction of incident power that could be focused at the output), which is likely to be much less than 50% based on the expected divergence of the original beam, the parameters of the components used in these reports, and the typically large ($\gtrsim 1$cm${ }^{-1}$) optical loss of common THz materials [22]. It may be noted that thick lenses made of high-resistivity Silicon have also been used for multi-mode THz QCLs with metallic cavities [23, 24]; however, the lens in that case is abutted to the facet of the laser requiring precise alignment and primarily functions as a beam condenser that improves the collection efficiency of the radiation from facets.

In this report, we show that single-mode THz QCLs with low-divergence and high-intensity beams with good beam-quality could be realized with precise choice and placement of a custom-made lens with a surface-emitting DFB QCL [8]. In contrast to commercial THz lenses that are expensive and have limited availability, plano-convex spherical lenses sawed from inexpensive industrial-grade HDPE balls are used to achieve some of the best reported characteristics for THz QCL beams. Comprehensive measurements are reported for focusing characteristics of a DFB THz QCL with various lenses. Through experiments and finite-element modeling (FEM), it is demonstrated that a thin lens of a diameter similar to that of the FWHM beam size of the source beam from the DFB QCL produces dual-lobed Airy beams at the output that move in a parabolic trajectory in opposite directions [25–27], and autofocus into an intense single-lobed beam with a large focusing efficiency and high-intensity contrast. While more complex techniques have been used previously to produce autofocusing Airy beams, this demonstration establishes an easy method to generate such beams with a simple lens that is especially convenient for long-wavelength lasers such as QCLs.

A $3.4\text{ THz}$ QCL is cooled to $62\text{ K}$ within the vacuum chamber of compact electrically operated Stirling cryocooler. A $f/1.2$ plano-convex HDPE lens when integrated with the QCL within the cryocooler realizes a single-lobed beam with the lowest divergence (FWHM ${1.4}^{\circ }\times {1.8}^{\circ }$) reported to-date for any THz QCL. In this case, a high focusing efficiency of 69% is realized with $118\text{ mW}$ peak optical power in the output beam. The same lens is also used to produce a beam with $113\text{ mW}$ peak power focused onto a FWHM spot size of $1.4\text{ mm}\times 0.9\text{ mm}$, which results in the highest demonstrated peak-intensity for THz QCLs. Beam propagation ratios (M${ }^2$ parameters) of $1.6$ and $2.5$ are experimentally estimated in the directions orthogonal to propagation, which leads to a brightness value of $5.4\times {10}^6$Wsr${ }^{-1}$ m${ }^{-2}$ for the QCL. This is only the second instance to our knowledge when brightness for THz QCLs has been reported, and it is approximately thrice as high as that reported for VECSELs in prior work [15].

 

Fig. 1 (a) Light-current (L-I), spectra (inset) and (b) Far-field radiation pattern characteristics from a 200 μm × 1.5 mm × 10 μm surface-emitting DFB QCL operating in pulsed mode (200 ns pulses repeated at 100 kHz) at a temperature of 62 K. (c) Cross-sectional plots of beam pattern in (b). The beam pattern was measured by scanning the pyroelectric detector in a plane orthogonal to the emission direction of the QCL at 45 mm distance as shown in the upper figure of (b).

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2. Surface-emitting DFB THz QCL

A surface-emitting DFB QCL with hybrid second- and fourth-order gratings as described in [8] is used as the THz radiation source. Such a DFB cavity leads to high output power in a single-lobed (yet divergent) far-field beam, and stable single-mode spectral output that is determined by the lithographically implemented period of the DFB gratings. Figure 1(a) shows the measured light-current (L-I) curve and emission spectra at different bias for the QCL used in this work in pulsed-mode of operation. This QCL is of dimensions $200\mu \mathrm{m}\times 1.5\text{ mm}\times 10\mu \mathrm{m}$ and is operated in a cryogen-free Stirling cooler [28] at a temperature of $62\text{ K}$. The QCL radiates in single-mode at $3.39\text{ THz}$ ($\lambda =88.4\mu \mathrm{m}$) across the entire bias range. Its peak optical power is measured with a power meter to be $170\pm 3\text{ mW}$. Figure 1(b) shows its far-field radiation pattern that was measured at a plane $45\text{ mm}$ away from the QCL in the surface-normal direction. The single-lobed beam has a FWHM divergence of $\sim 5^{\circ }\times {25}^{\circ }$, and the beam’s cross-section plots at that location are plotted in Fig. 1(c) that shows a much narrower and uniform radiation pattern in the longitudinal (x) direction. The reasons for spatial non-uniformity of radiation in the lateral (y) direction could not be ascertained, but are typical for such surface-emitting DFB QCLs [7] and could be due to one of multiple factors such as optical effects due to the TPX optical window of the cryocooler, spurious reflections from the metal casing surrounding the pyroelectric detector element used for beam measurements, thermal gradients in the cavity and/or non-uniform pumping in the active region.

 

Fig. 2 (a) Schematic view of a 3.39 THz QCL beam, modeled as a fundamental Gaussian beam, as it propagates through a thin plano-convex lens with a focal length of ∼22 mm. Calculated (b) focal point locations d₂ in logarithmic scale, (c) FWHM beam waists ω₂ at the focal point (z = d₂) and (d) FWHM divergence angles θ₂ after lens as a function of the distance from the laser to the lens d₁. Two dashed vertical lines indicate two specific cases where d₁ = 23 mm and d₁ = 30 mm that were used in experiments. An initial fundamental Gaussian beam (FWHM 5° × 25°) and an ideal thin lens are assumed in the calculation.

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3. Focusing with a thin plano-convex lens

The surface-emitting DFB QCL in Fig. 1 has a highly divergent and spatially asymmetric output beam. The primary motivation of this work was to determine if a lens could be used to produce a beam with a significantly lower divergence, and one that is much more spatially symmetric. It was also of interest to determine what the best focusing efficiency would be for such a lens, and to generally study the focusing characteristics of such a DFB THz QCL with a lens. Focusing of asymmetric THz laser beams with lenses has not been studied previously and neither is any published information available about typical focusing efficiencies of THz lenses for any type of THz QCL.

For a divergent beam, a fast lens that has a small f-number ($f/\#\equiv f/D$, where f is the focal length and D is the diameter of the lens) is required to collect and focus most of the incident radiation. For a given $f/\#$, the lens with a small focal length is typically thinner that reduces the propagation loss through lens material, which is typically large ($\gtrsim 1/$cm) for common THz optical materials. Commercial THz lenses are expensive, and thin lenses with small focal lengths are not readily available. As an alternative, industrial grade HDPE spheres were chosen to make plano-convex spherical lenses by manual sawing and polishing for this investigation. Such HDPE spheres are readily available with different radii to make lenses of different focal lengths and interface diameters. The data presented here is with HDPE lenses; however, TPX lenses are expected to provide even better focusing efficiency due to lower loss of TPX in the $3-4\text{ THz}$ spectral range (HDPE loss is $\sim 1.6$/cm and TPX loss is $\sim 1.1$/cm at $3.4\text{ THz}$) [22].

A plano-convex lens was carved out of a HDPE sphere of $0.5$-inch radius to yield a small focal length, $f=22.1\text{ mm}$, calculated by using the thin-lens equation $f=r/\left(n_{\text{HDPE}}-1\right)$ where r is the radius of the HDPE sphere and $n_{\text{HDPE}}\sim 1.57$ is the refractive-index of HDPE at $\sim 60\text{ K}$ [29]. The estimated fractional change in the linear dimension of the cold lens compared to room-temperature is negligible ($\sim 0.02$) based on the coefficient of linear expansion of HDPE. A f-number of $1.2$ is chosen for the lens, that makes the lens diameter slightly larger than the FWHM spot-size of the incident QCL beam in lateral (y) direction at the lens. The focusing characteristic of such a lens is described in further detail in section 6, which also describes generation of 1D autofocusing Airy beams in the lateral direction in which the beam overfills the lens.

For the aforementioned DFB QCL incident on a thin lens with large diameter, Fig. 2 shows various beam parameters after focusing calculated using ABCD matrices [30]. To simplify the calculation, the QCL is assumed to radiate in an ideal fundamental Gaussian beam at $3.4\text{ THz}$ with a FWHM divergence of $5^{\circ }\times {25}^{\circ }$ (with unity M² parameters in both x and y directions) and the plano-convex lens is modeled as an ideal thin-lens. Figures 2(b)-2(d) show the calculated focal-point location (d₂), FWHM beam waist (ω₂) at the focal point (z = d${ }_2$), and FWHM divergence angle (θ₂) after focusing respectively, as a function of the distance of the lens from the QCL (d₁).

Based on calculations shown in Fig. 2, two different focusing criteria are chosen to generate either a small spot-size or low-divergence beam for the focused beam, which is marked by the two different values of d₁ on the plots by dashed vertical lines. A value of large $d_1\sim 30\text{ mm}$ is chosen to achieve a small spot-size such that the beam gets focused in both x and y directions at approximately same location, while keeping d₁ small enough to limit the size of the incident beam at the location of the lens. To achieve a beam of low-divergence, a smaller d₁ is required such that the QCL is placed close to the focal point of the lens with $d_1\approx f$. A slightly larger value $d_1\sim 23\text{ mm}$ is however chosen, which allows the focused beam at the output to be more spatially symmetric (as seen from Fig. 2(c)) that also maintains the spatial symmetry as it propagates due to similar divergence in x and y dimensions (as seen from Fig. 2(d)).

A smaller f-number reduces the focusing efficiency due to thicker lens material, whereas a larger f-number makes the lens too small to capture the entire incident beam. A further reduction in f makes the lens too small based on considerations of mounting and securing the lens in the vacuum chamber in front of the QCL without interfering with incident beam. Table 1 shows the comparison of three lenses with thicknesses of $2.5$mm, $3.8$mm, and $5.7$mm that correspond to f-numbers of $1.46$, $1.22$, and $1.04$, and lens diameters of $15.1$mm, $18.1$mm, and $21.2$mm respectively when d${ }_1$ = 30 mm. From the result, we can see that the $2.5\text{ mm}$-thick lens has similar focusing efficiency to that of the $3.8\text{ mm}$-thick one, whereas the $5.7\text{ mm}$-thick lens has a smaller focusing efficiency despite a better collection efficiency due to a larger diameter. These measurements suggests that no further improvements are realized for the focusing efficiency once the lens diameter is made similar to the FWHM spot-size of the incident beam at the location of the lens, that still results in excellent focusing efficiency by the lens through the formation of autofocusing Airy beams at the output.

Table 1. HDPE Lenses of Different Thicknesses (d₁ = 30 mm).

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4. Experimental results

The active-medium of the THz-QCLs is based on a three-well resonant-phonon design with GaAs/${\text{Al}}_{0.15}{\text{Ga}}_{0.85}\text{As}$ superlattice (design RT3W221YR16A, wafer number VB0832, with a layer widths (starting from the injector barrier) of $57/18.5/31/9/28.5/16.5$ monolayers (ML, $1\text{ ML}=2.826$Å)). The superlattice was grown by molecular-beam epitaxy (MBE), with 221 cascaded periods, leading to an overall thickness of $10\mu \mathrm{m}$. Cu-Cu based metallic waveguides were fabricated using standard THz QCL fabrication techniques with thermocompression wafer bonding. Distributed-feedback (DFB) ridge cavities were processed by wet-etching, where the grating periodicity was defined lithographically. Details related to the MBE growth sequence and DFB QCL fabrication sequence that included implementation of lateral and longitudinal absorbing sections in the cavities to suppress undesired higher order cavity modes are described in [8].

The plano-convex HDPE lenses were sawed manually from 1-inch diameter commercial HDPE balls (from Precision Plastic Ball Co.). The cutting interface was polished to form a flat circular plane by a lapping machine with a silicon carbide paper (320 grit) followed by an alumina paper (360 grit). Upon visual inspection under the microscope at 20X magnification, the surface roughness was $\ll$ 10 um, which could be considered optically flat at the long wavelength of the QCL. The thickness of different lenses was varied by controlling the lapping time. Inside the Stirling cryocooler, the lens was mounted in front of QCLs by using home-made brass mounts.

A cleaved chip consisting of DFB QCLs was indium soldered on a copper block, the QCL to be tested was wire-bonded for electrical biasing, and the copper block was mounted on the cold-stage of the Stirling cryocooler. A $530\mu \mathrm{m}$ thick TPX window is used in the vacuum chamber to couple THz radiation outside of the cryocooler for measurements. The QCL is biased with $200$ ns wide rectangular current pulses repeated at $100$kHz at a heat-sink temperature of $62\text{ K}$. Far-field beam patterns were measured with a THz pyroelectric detector (Gentec THz 2I-BL-BNC, element size $2\text{ mm}\times 2\text{ mm}$) mounted on a 2D motorized scanning stage. When measuring narrow beam patterns focused by lenses, a pinhole with a diameter of $\sim 0.6\text{ mm}$ was mounted in front of the pyroelectric detector. This small pinhole was made on a copper plate using a milling machine. The absolute power was calibrated using a thermopile power meter (Scientech AC2500 with AC25H) and is reported without any corrections to the detected signal. This power was measured with QCL on a different copper mount such that the QCL was placed close to the TPX window of the dewar (within 10 mm distance) and the power meter was placed immediately adjacent to the TPX window outside the dewar, which should collect close to 100% of the radiated power except that lost due to transmission through the TPX window. Laser spectra were measured using a Fourier-transform infrared spectrometer (Bruker Vertex 70v).

 

Fig. 3 (a) Experimental setup of a HDPE lens mounted inside the cryocooler when d₁ = 30 mm. (b) Optical image of a f/1.2 plano-convex HDPE lens with an interface diameter of ∼ 18 mm. This lens is cut from a ∼1-inch diameter HDPE ball as shown in (c). (d) Experimentally measured FWHM beam sizes as a function of measurement distance z. Solid lines are the fitting curves corresponding to fundamental Gaussian-beam propagation after including the corresponding M² parameters. A multi-lobed region as marked exists prior to focus that is the region of propagation of autofocusing symmetric Airy beams. (e) Raster-scanned beams and their cross-sectional plots at z = 50 mm, 65 mm, 80 mm, 140 mm respectively.

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Figure 3(a) shows the experimental setup of the DFB QCL integrated with the home-made plano-convex HDPE lens within the cryocooler chamber. The entire assembly is mounted onto the copper plate of the cryocooler chamber, in which there is a $5.5$cm clearance from the top of the surface-emitting QCL to the lid of the cryocooler that also includes a TPX optical window to couple out the THz radiation (not shown in Fig. 3(a)). The plano-convex lens in Fig. 3(b) is sawed from an HDPE ball of 0.5-inch radius as shown in Fig. 3(c). The results in Fig. 3 are for $d_1=30\text{ mm}$ to realize a small-spot size after focusing. The FWHM spot size of the incident beam on the lens is expected to be $\sim 2.7\text{ mm}\times 13.5\text{ mm}$, and hence, a lens of slightly larger interface diameter of $\sim 18\text{ mm}$ is sawed that results in a $f/1.2$ lens. The lens is kept in place inside the vacuum chamber using a mounting plate and two screws that hold the lens at its edges. After the chamber is pumped down and cryocooler is cooled to $62\text{ K}$, beam patterns at different distance planes are measured that are shown in Fig. 3(e).

The FWHM beam sizes are estimated by using the method that is the equivalent of the commonly used knife-edge measurement technique [31, 32]. Section 5 provides additional details about this measurement. From the estimated beam sizes and curve-fitting of the Gaussian beam propagation data as plotted in Fig. 3(d), the minimum spot-size location for the focused beam occurs at $z\sim 80\text{ mm}$ with a FWHM spot-size of $1.4\text{ mm}\times 0.9\text{ mm}$. Beam propagation ratios $M_x^2$ and $M_y^2$ are calculated to be $1.6$ and $2.5$ respectively from the fitted relationship of beam-size as a function of z, as described in section 5. Therefore, we can estimate a brightness of $B_{\mathrm{r}}=5.4\times {10}^6$Wsr${ }^{-1}$m${ }^{-2}$ given by $B_{\mathrm{r}}=P/\left(M_x^2{M}_y^2{\lambda }^2\right)$, where P is the incident power and λ is the incident wavelength. The peak power collected after lens is $113\text{ mW}$ as measured by the power meter, which is measured right after the TPX window of the cryocooler. This power is entirely in the focused beam. The fraction of the beam that is not collected and focused by the lens is negligible ($<0.5\%$), as was verified through finite-element simulations. Hence, the focusing efficiency (ratio of absolute power after lens and the initial power of QCL) is estimated to be $\sim 67\%$ and the peak averaged intensity within the FWHM contour is $\sim 57.1\text{ mW}/{\text{mm}}^2$ at the plane of $z=80\text{ mm}$. The divergence angle of beam after lens is approximately ${2.5}^{\circ }\times {7.1}^{\circ }$. The divergence value in x direction agrees with the calculated result ($\sim 2^{\circ }$) as shown in Fig. 2(d); however, the FWHM divergence value in y direction is smaller than the calculated value ($\sim 9^{\circ }$) since the incident beam in y direction is non-uniform, which is unlike the ideal Gaussian beam assumed for calculation.

Figure 3(e) shows four separate beam patterns measured at locations $z=50\text{ mm}$, $65\text{ mm}$, $80\text{ mm}$, and $140\text{ mm}$ respectively. The beam patterns are dual-lobed along y direction (lateral to the cavity) prior to $z\sim 80\text{ mm}$ when the two lobes autofocus into a single-lobed beam. The two lobes move toward the optical axis with increasing propagation distance. The experimental setup does not allow beam measurements close to the lens prior to autofocus; however, FEM simulations as described in section 6 conclusively show that the dual-lobed beams are indeed autofocusing symmetric Airy beams for which the intensity in the main lobes increases by an order of magnitude as they follow a parabolic trajectory away from the lens to the point of collapse into a single-lobed beam at the focal plane. The beam becomes single-lobed close to and beyond the focal point for $z\gtrsim 80\text{ mm}$. The location of focus at $z\sim 80\text{ mm}$ agrees with the calculated value in Fig. 2(b), even though the calculations in Fig. 2(b) assume Gaussian beam propagation throughout using the ideal thin-lens formula. Several DFB QCLs with different grating periods were measured and all showed similar dual-lobed behaviors prior to focus, that collapse into single-lobed beam after focus.

 

Fig. 4 (a) FWHM beam sizes versus measurement distance z when the QCL is placed at an incident plane corresponding to d₁ = 23 mm. Solid lines are fitting curves to standard Gaussian beam propagation after including beam propagation ratios. A multi-lobed region exists prior to focus similar to that in Fig. 3. (b) Beam patterns at various locations along the optical axis that also experimentally show autofocusing Airy beams that turn into a predominantly single-lobed beam after focus.

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In the second case, a narrow divergence angle can be achieved by choosing $d_1\sim f=23\text{ mm}$. This case can be applied to beam combining and remote sensing [33, 34], in which it is desirable for the laser beam to propagate long distances with a small spot-size. Figures 4(a) and 4(b) show the beam sizes as a function of location z and several cross-sectional beam patterns measured after the autofocus point. Dual-lobed autofocusing beams are experimentally measured for this case as well prior to the location of the focal plane at $z\sim 150\text{ mm}$. By curve fitting similar to that for the case of $d_1\sim 30\text{ mm}$, a FWHM divergence of ${1.4}^{\circ }\times {1.8}^{\circ }$ is estimated. The power measured after lens is $118\text{ mW}$ leading to a focusing efficiency of 69%.

5. Beam quality factor (M² parameter)

The M² parameters of beam after lens in x and y directions are 1.6 and 2.5, which are obtained by using the method that is similar to the knife-edge measurement [31, 32]. Several instances of beam-patterns were measured along the optical axis, at increasing distances from $z=75\text{ mm}$ to $200\text{ mm}$ (as per the coordinate definitions in Figs. 1 and 2). Beam sizes at each z location were estimated by curve-fitting the integrated optical power P versus transverse dimension x (or y) using the standard equation

 

Fig. 5 Fitted power in x and y directions (as per coordinate definitions in Figs. 1 and 2) at (a) z = 75 mm, (b) z = 100 mm, (c) z = 140 mm, (d) z = 180 mm. The beam sizes ω_x and ω_y (radius where intensity reduces to 1/e² times its maximum value) deduced from fitting the curves to equation 1 are mentioned in each plot.

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6. Autofocusing symmetric Airy beams

Airy beams are optical waves that resist diffraction as they propagate, and they move freely by following a curved (parabolic) trajectory [25]. An exponentially truncated Airy beam with finite energy can be generated by imposing a cubic phase modulation on a Gaussian beam, and taking its Fourier transform (such as with a thin-lens). Such beams could be created in one-dimension (1D) or two-dimensions (2D) and the highest-intensity lobe could travel with same width for long distances, unlike that for a typical laser beam such as the Gaussian beam that expands as it travels. Conventional Airy beams are typically generated by complex optical devices such as spatial light modulators and phase masks, but have also been implemented with simpler optical elements such as cylindrical lenses using the inherent aberrations present in such elements [35].

 

Fig. 6 Two-dimensional finite-element simulation results of propagation of fundamental Gaussian beams that are incident on a f/1.2 HDPE plano-convex lens with f ∼ 22 mm, for a source distance of d₁ ∼ 30 mm from the lens. Intensity plots of beams with incident FWHM divergence of (a) 5° and (b) 25° are shown, where the beam propagation is in x direction and the electric-field is polarized in the y and z directions respectively corresponding to the DFB THz QCL used in experiments. Insets show zoomed in plots at locations close to focal points of the lens. The transmitted beam after the lens remains Gaussian in (a), whereas in (b), the resultant beam is a symmetric Airy beam that collapses beyond the focal plane into a beam with a single central lobe [27].

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Dual-lobed Airy beams can also exist, in which two non-diffracting lobes move in opposing directions and eventually autofocus into an intense single-lobed beam with a high-intensity contrast [26]. The first demonstration of such autofocusing phenomenon was done with radially symmetric Airy beams, however, more recently autofocusing Airy beams have also been realized under rectangular symmetry and were termed as symmetric Airy beams [27]. Previously, autofocusing Airy beams were realized by a complex technique where spatial light modulators are used for phase modulation. However, here generation of such beams is demonstrated using a much simpler method, where it is shown that a plano-convex lens when illuminated by a highly divergent Gaussian beam also produces autofocusing symmetric Airy beams.

Full-wave finite-element simulations were conducted in 2D [36] to simulate the propagation characteristics of the Gaussian beam after focusing with a plano-convex HDPE lens, and to demonstrate generation of the autofocusing symmetric Airy beams. Figures 6(a) and 6(b) show the electric-field intensity distribution of propagating Gaussian beams that model the terahertz beam incident from the $3.39\text{ THz}$ QCL in Fig. 1 in longitudinal (low divergence) and lateral (high divergence) dimensions of the QCL cavity respectively. A $f/1.2$ plano-convex HDPE lens with $f\sim 22\text{ mm}$ is implemented and the incident Gaussian beam source is placed with its minimum waist located at source distance $d_1=30\text{ mm}$ from the lens, propagating along the x direction. The source field is set to realize fundamental Gaussian beams with FWHM divergence angles of $\sim 5^{\circ }$ (Fig. 6(a)) and $\sim {25}^{\circ }$ (Fig. 6(b)) respectively. The polarization of electric-field is along y and z direction respectively corresponding to the actual polarization of the QCL beam. A $3.8\text{ mm}$-thick lens (carved out from a HDPE sphere of $0.5$in. radius) with a diameter of $\sim 18\text{ mm}$ is used with refractive index set to $1.57$ corresponding to that of HDPE. The lens is set to be lossless with air as its surrounding region. The inset plots show zoomed-in regions around focal points, that indicate that the focused beam remains single-lobed Gaussian for Fig. 6(a), whereas a two-lobed beam with a curved path is generated when the incident Gaussian beam is highly divergent as in Fig. 6(b). Similar results are obtained for both cases if an orthogonal polarization was used for the incident beam, which suggests that the focusing behavior is independent of the source polarization.

 

Fig. 7 (a) Two-dimensional finite-element simulation results for a fundamental Gaussian beam with FWHM divergence of 25° incident on f/1.2 HDPE plano-convex lens with f ∼ 22 mm and the source distance from lens d₁ = 30 mm, same as in Fig. 6(b). (b) Transverse intensity profile in the transmitted beam prior to reaching focal plane, that shows two main lobes separated by distance d along the transverse (y) dimension, centered symmetrically around the location of the optical axis. (c) Distance between the main lobes (that is twice the distance of a lobe from the optical axis in the transverse dimension) plotted as a function of propagation distance along the optical axis (x dimension). A fit of d as a parabolic function of x is also plotted (solid line). FWHM of (either of the) lobes as function of x is also plotted. (d), (e), and (f) show similar plots but for the case of d₁ = 23 mm.

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The full-size of the source beam (characterized by the beam diameter at which its intensity falls to $1/e^2$ of its peak value) is $\sim 4.5\text{ mm}\times 23\text{ mm}$ at the location of the lens on the source side for a source distance of $d_1=30\text{ mm}$ (the FWHM beam size is $2.7\text{ mm}\times 13.5\text{ mm}$). The beam size is greater than the size of the lens in the direction that is lateral to the QCL cavity as modeled in Fig. 6(b). The dual-lobed beam produced in this case, is a result of aberrations produced by the edges of the lens since significant beam intensity exists at the edges when it is highly divergent. Figure 7 analyzes the focused beam for two cases, source distances of $d_1=30\text{ mm}$ and $d_1=23\text{ mm}$ respectively to mimic the distances used in actual experiments as described in section 4. By analyzing the trajectory of the output beam that fits well to a parabolic curve as well as width of the main lobes in the beams that remain approximately constant (non-diffracting) as the beams propagate, it becomes clear that 1D symmetric autofocusing Airy beams are produced by the plano-convex lens in the dimension lateral to the QCL cavity, when the incident beam overfills the lens on the source side. The peak intensity in the main lobes of the dual Airy beams increases by a factor of $\sim 10$ from immediately outside the lens to the location of autofocus where the dual beams collapse into a single lobe, which demonstrates high intensity contrast as a function of position for the autofocusing beams similar to that reported in [27].

7. Conclusion

It was realized early in the development of THz QCLs that their poor radiation patterns posed a significant limitation to their usability due to the subwavelength metallic cavities. A large fraction of research on THz QCLs in the past decade has focused on improvement of their beam quality to primarily achieve low-divergence and single-lobed emission wherein [9–16] are only a small subset of variety of techniques that have been published in literature. Here we show that excellent beam characteristics for THz QCLs could instead be obtained by integrating an appropriate lens with single-lobed surface-emitting DFB THz QCLs [7, 8] within the vacuum housing of the cryocooler. Owing to the divergent incident beams and large optical loss of lens material, proximity of lens with the QCL and a precise choice of lens parameters are required to achieve large focusing efficiency. Such an investigation was made possible in this report due to an inexpensive method to make plano-convex lenses by sawing industrial-grade HDPE balls. Comprehensive measurements are reported for focusing characteristics of a DFB THz QCL with lenses of different parameters. With a $3.4\text{ THz}$

DFB QCL, record lowest beam divergence (${1.4}^{\circ }\times {1.8}^{\circ }$ with a focusing efficiency of 69%), as well as record highest brightness ($5.4\times {10}^6$Wsr${ }^{-1}$m${ }^{-2}$) and intensity ($57\text{ mW}/{\text{mm}}^2$ averaged in a spot-size of $1{\text{ mm}}^2$) are experimentally demonstrated for any THz QCL to-date. Finally, the combination of a highly divergent source laser beam when incident on a small diameter plano-convex lens is shown to produce dual-lobed symmetric autofocusing Airy beams [25, 27], in which the peak intensity in the lobes of beams increases by an order of magnitude as they autofocus into a single-lobed beam. Realization of QCL Airy beams is thus reported for the first time to the best of our knowledge, and a simple yet effective method to generate such autofocusing Airy beams is established compared to previous techniques [26, 27], which should work for lasers at any wavelength. These results hold important practical implications for targeting THz spectroscopy and sensing applications with QCLs.