1. INTRODUCTION

A surface plasmon polariton (SPP) is a coupled state between electromagnetic (EM) field and electron plasma oscillations at the interface between a metal and a dielectric for which the EM field could be confined in subwavelength dimensions normal to the surface of the metal. Consequently, metallic cavities supporting SPP modes have been used to realize SPP lasers (also known as plasmonic lasers or spasers) with subwavelength dimensions [1–5]. The energy in a spaser can remain confined as coherent SPPs or it can be made to leak out from the spaser as radiation. In many targeted applications in integrated optics and nanophotonics, spasers are developed as nanoscale sources of coherent EM radiation and show interesting properties such as ultrafast dynamics for applications in high-speed optical communications. Parallel-plate metallic cavities supporting SPP modes are also utilized for terahertz quantum-cascade lasers (QCLs) [6] to achieve low-threshold and high-temperature performance [7] owing to the low loss of SPP modes at terahertz frequencies that are much smaller than the plasma frequency in metal. The most common type of plasmonic lasers with long-range SPPs, which include terahertz QCLs, utilize Fabry–Perot type cavities in which at least one dimension is longer than the wavelength inside the dielectric [8–11]. One of the most important challenges for such plasmonic lasers is their poor coupling to free-space radiation modes owing to the subwavelength mode confinement in the cavity, which leads to a small radiative efficiency and highly divergent radiation patterns. This problem is also severe for terahertz QCLs based on metallic cavities and leads to low-output power and undesirable omnidirectional radiation patterns from Fabry–Perot cavities [12,13].

A possible solution to achieve directionality of far-field emission from spasers is to utilize periodic structures with broad-area emission, which has been used for both short-wavelength spasers [14–19] as well as terahertz QCLs [20,21]. On chip phased-locked arrays [22,23] or metasurface reflectors composed of multiple cavities [24] have also been utilized for directional emission in terahertz QCLs. However, edge-emitting Fabry–Perot cavity structures with narrow cavity widths are more desirable, especially for electrically pumped spasers to achieve a small operating electrical power and better heat removal from the cavity (along the width of the cavity in the lateral dimension, through the substrate) for continuous wave (cw) operation. In this paper, we theoretically and experimentally demonstrate a new technique for implementing distributed feedback (DFB) in plasmonic lasers with Fabry–Perot cavities, which is termed an antenna-feedback scheme. This DFB scheme has no resemblance to the multitude of DFB methods that have been conventionally utilized for semiconductor lasers. The key concept is to couple the guided SPP mode in a spaser’s cavity to a single-sided SPP mode that can exist in its surrounding medium by periodic perturbation of the metallic cladding in the cavity. Such a coupling is possible by choosing a Bragg grating of appropriate periodicity in the metallic film. This leads to excitation of coherent single-sided SPPs on the metallic cladding of the spaser that couple to a narrow beam in the far field. The narrow-beam emission is due in part to the cavity acting like an end-fire phased-array antenna at microwave frequencies as well as due to the large spatial extent of a coherent single-sided SPP mode that is generated on the metal film as a result of the feedback scheme. Experimentally, the antenna-feedback method is implemented for terahertz QCLs for which the method is shown to be an improvement over the recently developed third-order DFB scheme for producing directional beams [25] since it does not require any specific design considerations for phase matching [26]. The emitted beam is more directional and the output power is also increased due to an increased radiative field by virtue of this specific scheme.

2. ANTENNA-FEEDBACK SCHEME FOR PLASMONIC LASERS

Single-mode operation in spasers with Fabry–Perot cavities could be implemented in a straightforward manner by periodically perturbing the metallic film that supports the resonant SPP modes. The schematic in Fig. 1(a) shows an example of a periodic grating in the top metal cladding for a parallel-plate metallic cavity that could be utilized to implement conventional $p$-th order DFB by choosing the appropriate periodicity. Since the SPP mode has a maximum amplitude at the interface of the metal and dielectric active medium, a periodic perturbation in the metal film could provide strong Bragg diffraction up to high orders for the counterpropagating SPP waves inside the active medium with incident and diffracted wave vectors $k_i$ and $k_d=-k_i$, respectively, such that

 

Fig. 1. Antenna-feedback concept for spasers. (a) The general principle of conventional DFB that could be implemented in a spaser by introducing periodicity in its metallic cladding. A parallel-plate metallic cavity is illustrated; however, the principle is equally applicable to spaser cavities with a single-metal cladding. (b) If the periodicity in (a) is implemented by making holes or slits in the metal cladding, the guided SPP wave diffracts out through the apertures and generates single-sided SPP waves on the cladding in the surrounding medium. The figure shows a phase mismatch between successive apertures for SPP waves on either side of the cladding. Coherent single-sided SPP waves in the surrounding medium cannot therefore be sustained owing to destructive interference with the guided SPP wave inside the cavity, as illustrated in (c). (d) The principle of an antenna-feedback grating. If the periodicity in the metal film allows the guided SPP mode to diffract outside the cavity, a grating period could be chosen that leads to the first-order Bragg diffraction in the opposite direction, but in the surrounding medium rather than inside the active medium itself. Similarly, the single-sided SPP mode in the surrounding medium undergoes first-order Bragg diffraction to couple with the guided SPP wave in the opposite direction inside the cavity. (e) The grating in (d) leads to a fixed phase condition at each aperture between counterpropagating SPP waves on the either side of metal cladding. First, this leads to a significant buildup of amplitude in the single-sided SPP wave in the surrounding medium, as illustrated in (f). Second, emission from each aperture adds constructively to couple to far-field radiation in the end-fire ($z$) direction. As argued in the text, both of these aspects lead to a narrow far-field emission profile in the $x–y$ plane.

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In contrast to conventional DFB methods in which periodic gratings couple forward and backward propagating waves inside the active medium itself, the antenna-feedback scheme couples a single-sided SPP wave that travels in the surrounding medium with the SPP wave traveling inside the active medium, as illustrated in Fig. 1(d). The SPP wave inside the active medium with incident wave vector $k_i\approx 2\pi n_{\mathrm{a}}/\lambda$ is diffracted in the opposite direction in the surrounding medium with wave vector $k_d\approx -2\pi n_{\mathrm{s}}/\lambda$. For the first-order diffraction grating ($p=1$), Eq. (1) results in

The antenna-feedback scheme leads to the excitation of a coherent single-sided SPP standing wave on the metallic cladding of the spaser, which is phase locked to the resonant-cavity SPP mode inside the active medium, as shown in Fig. 1(f). Both waves maintain the exact same phase relation at each aperture location, where they exchange EM energy with each other due to diffraction, as illustrated in Fig. 1(e). The SPP wave in the surrounding medium is excited due to scattering of the EM field at apertures that generate a combination of propagating quasi-cylindrical waves and SPPs [32,33] that propagate along the surface of the metal film. The scattered waves thus generated at each aperture superimpose constructively in only the end-fire ($z$) direction owing to the phase condition thus established at each aperture. For coupling to far-field radiation, the radiation is therefore analogous to that from an end-fire phased array antenna that produces a narrow beam in both the $x$ and $y$ directions.

A third-order DFB technique was recently shown to achieve emission in a narrow beam for terahertz QCLs with Fabry–Perot cavities [25]. It can achieve high directionality for the radiated beam in both directions perpendicular to propagation as long as the effective propagation index of the SPP wave inside the active medium could be made $\sim 3.0$ by complex deep dry etching in the slits [25] or lateral corrugated geometry [34]. The so-called phase-matching condition is possible for GaAs-based QCLs by cavity engineering [26] since the $n_{\mathrm{GaAs}}\sim 3.6$ is close to 3.0. The antenna-feedback technique in this work offers a similar outcome as a perfectly matched third-order DFB with improved directionality as well as a novel outcoupling mechanism of the radiated beam from terahertz QCLs. It is to be noted that the antenna-feedback scheme is automatically phase matched and hence it could be utilized for any type of spaser without any restrictions on the required index in the active medium.

3. SIMULATION RESULTS

Figure 2 shows a comparison of the eigenmode spectrum of a terahertz QCL cavity with a conventional DFB, taking a third-order DFB as an example versus antenna-feedback gratings computed using a finite-element solver [35]. The occurrence of bandgaps in the spectra is indicative of the DFB effect. In both cases, the lower-frequency band-edge mode is the lowest-loss mode by way of DFB action since the DFB modes result in a standing wave being established along the length of the cavity with an envelope shape that vanishes close to the longitudinal boundaries (end facets). For the antenna-feedback grating, a standing wave for the single-sided SPP wave is additionally established in air, as can be seen from the field plot of the band-edge mode in Fig. 2(b). In contrast, the third-order DFB leads to a negligible amplitude of the single-sided SPP wave in air, as mentioned previously and illustrated schematically in Fig. 1(c). For the dominant TM polarized (Ey) electric field of the antenna-feedback, the hybrid SPPs mode bound to the top metal layer consists of both quasi-cylindrical waves and SPPs, which are evanescent-fields with a free-space propagation constant. Particularly at long wavelengths, such as the mid infrared and THz regions, SPPs and quasi-cylindrical waves complexly mix with each other [32], contributing to the large spatial extent of the SPP mode in the surrounding medium, as shown in Fig. 2(b), which does not exist for any conventional DFB, such as first-order, second-order and third-order DFB. In addition, hybrid SPPs on top of metallic gratings and standing wave inside the laser cavity show clearly different periodicity, with the ratio of the free-space wavelength over the guided wavelength, which further confirms that the excitation of the coherent SPP wave on both sides of the top metallic surface contribute to the feedback and coupling mechanism with the antenna-feedback scheme. The absorbing boundaries at the longitudinal ends of the cavity [36] increase the relative loss of the modes that are further away from the band-edge mode, which helps in the mode discrimination and will lead to the excitation of the desired band-edge mode for the single-mode operation of the spaser. The active region and metal layers are modeled as lossless since the exact loss contribution due to each is not clear in literature for terahertz QCLs at cryogenic temperatures. If a lossy metal is used, the relative loss of various resonant modes for the DFB cavities are not impacted and neither are the mode shapes and the corresponding resonant frequencies. For the band-edge modes in Figs. 2(a) and 2(b), a loss of $\sim 5{\mathrm{cm}}^{-1}$ was estimated as a contribution from the absorbing boundaries. Consequently, the radiative (outcoupling) loss of the third-order DFB is $\sim 1.7{\mathrm{cm}}^{-1}$ as compared to $\sim 5.6{\mathrm{cm}}^{-1}$ for the antenna-feedback. The radiative loss of the third-order DFB is smaller since the band-edge mode has zeros of the radiative field ($E_z$) being located at each aperture since the grating period $\mathrm{\Lambda }$ is an integer multiple of half-wavelengths in the GaAs/AlGaAs active medium ($\mathrm{\Lambda }=3{\lambda }_{\mathrm{GaAs}}/2$, where ${\lambda }_{\mathrm{GaAs}}\equiv \lambda /n_{\mathrm{GaAs}}$). Such a low outcoupling efficiency is also existent in surface-emitting terahertz QCLs with a second-order DFB [29]. For the cavity with antenna-feedback, the radiative loss is higher because the grating period is not an integer multiple of half-wavelengths inside the active medium ($\mathrm{\Lambda }\sim 0.78{\lambda }_{\mathrm{GaAs}}$) that leads to large amplitudes of the radiative field ($E_z$) in alternating apertures, as shown in the figure. As a consequence, the output power from terahertz QCLs with antenna-feedback should be greater than that with conventional DFB gratings, which is an additional advantage of the antenna-feedback scheme for terahertz QCLs. This was also verified experimentally from the measured output power. The recently developed second-order DFB QCLs with graded periodicity [37] achieve high-power emission for the same reason, i.e., a nonzero radiative field under the metallic apertures.

 

Fig. 2. Comparison between conventional DFB (third-order DFB as an example) and antenna-feedback schemes for terahertz QCL cavities. The figure shows a SPP eigenmode spectrum and electric field for the eigenmode with lowest loss calculated by finite-element simulations of parallel-plate metallic cavities, as in Fig. 1, with GaAs as the dielectric ($n_{\mathrm{a}}=3.6$) and air as the surrounding medium ($n_{\mathrm{s}}=1$). Simulations are done in 2D (i.e., cavities of infinite width) for 10 μm thick and 1.4 mm long cavities, and metal and active layers are considered lossless. Lossy sections are implemented in the cavities at both longitudinal ends of the cavity as absorbing boundaries, which eliminates the reflection of guided SPP modes from the end facets. A periodic grating with apertures of (somewhat arbitrary) width $0.2\mathrm{\Lambda }$ in the top-metal cladding are implemented for DFB. $\mathrm{\Lambda }$ is chosen to excite the lowest-loss DFB mode at similar frequencies close to $\sim 3\mathrm{THz}$. The eigenmode spectrum shows frequencies and loss for the resonant-cavity modes, which reflects a combination of radiation loss and the loss at longitudinal absorbing regions. (a) The results from a third-order DFB grating with $\mathrm{\Lambda }=41.7\mathrm{\mu m}$ and (b) the results from antenna-feedback grating with $\mathrm{\Lambda }=21.7\mathrm{\mu m}$. Radiation loss occurs through diffraction from apertures, and the amplitude of the in-plane electric-field $E_z$ is indicative of the outcoupling efficiency. The major fraction of EM energy for the resonant modes exists in the TM polarized ($E_y$) electric field. A photonic bandgap in the eigenmode spectrum is indicative of the DFB effect due to the grating. The antenna-feedback grating excites a strong single-sided SPP standing wave on top of the metallic grating (in air), as also illustrated in Fig. 1(f). Also, the radiative loss for the third-order DFB grating is smaller since the lowest-loss eigenmode has zeros of $E_z$ under the apertures, which leads to a smaller net outcoupling of radiation. The loss is $\sim 6.7{\mathrm{cm}}^{-1}$ and $\sim 10.6{\mathrm{cm}}^{-1}$ for the lowest loss resonant cavity mode of the third-order DFB and antenna-feedback scheme, respectively.

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4. EXPERIMENTAL DEMONSTRATION OF ANTENNA-FEEDBACK FOR TERAHERTZ QCLs

Figure 3 shows the experimental results from terahertz QCLs implemented with antenna-feedback gratings. Details about the fabrication and measurement methods are presented in Supplement 1 (Section S1). Figure 3(b) shows representative $L$-$I$ curves versus the heat-sink temperature for a QCL with $\mathrm{\Lambda }=24\mathrm{\mu m}$. The QCL operated up to a temperature of 124 K. In comparison, multimode Fabry–Perot cavity QCLs on the same chip that did not include longitudinal or lateral absorbing boundaries operated up to $\sim 140\mathrm{K}$. Light-current characteristics and spectra at a different bias with the Fabry–Perot cavity are shown in the Supplement 1 (Section S3). The temperature degradation due to the absorbing boundaries is relatively small and similar to previous reports of DFB terahertz QCLs [29]. The inset shows the measured spectra at a different bias at 78 K. Most QCLs tested with different grating periods showed a robust single-mode operation except a close to peak bias when a second mode was excited for some devices at a shorter wavelength, which suggests that it is likely due to a higher-order lateral mode being excited due to spatial-hole burning in the cavity. A peak- power output of $\sim 1.5\mathrm{mW}$ was detected from the antenna-feedback QCL measured directly at the detector without using any collecting optics. For comparison, a terahertz QCL with a third-order DFB (without phase matching) and similar dimensions was also fabricated on the same chip, which operated up to a similar temperature of $\sim 124\mathrm{K}$ and emitted a peak-power output of $\sim 0.45\mathrm{mW}$ (see the Supplement 1, Section S3). The antenna-feedback gratings lead to a greater radiative outcoupling compared to conventional DFB schemes for terahertz QCLs, as discussed in the previous section.

 

Fig. 3. Lasing characteristics of terahertz QCLs with antenna-feedback. (a) The schematic on the left shows the QCL’s metallic cavity with an antenna-feedback grating implemented in the top metal cladding. The active medium is 10 μm thick and based on a $3\mathrm{THz}\mathrm{GaAs}/{\mathrm{Al}}_{0.10}{\mathrm{Ga}}_{0.90}\mathrm{As}$ QCL design (details in the Supplement 1, Section S1). A scanning electron microscope image of the fabricated QCLs is shown on the right. (b) Experimental light-current-voltage characteristics of a representative QCL with antenna-feedback of dimensions $1.4\mathrm{mm}\times 100\mathrm{\mu m}$ at different heat-sink temperatures. The QCL is biased with low duty cycle current pulses of 200 ns duration and 100 kHz repetition rate. Inset shows the lasing spectra for different biases where the spectral linewidth is limited by instrument’s resolution. The emitted optical power is measured without any cone collecting optics inside the cryostat. (c) Measured spectra for four different antenna-feedback QCLs with varying grating periods $\mathrm{\Lambda }$, but similar overall cavity dimensions. The QCLs are biased at a current density of $\sim 440\mathrm{A}/{\mathrm{cm}}^2$ at 78 K.

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Figure 3(c) shows the spectra measured from four different terahertz QCLs with antenna-feedback gratings of different grating periods $\mathrm{\Lambda }$. The single-mode spectra scales linearly with $\mathrm{\Lambda }$, which is the clearest proof that the feedback mechanism works as expected and the lower band-edge mode is selectively excited in each case. Using $\lambda =\mathrm{\Lambda }(n_{\mathrm{a}}+1)$ from Eq. (2), the effective propagation index of the SPP mode in the active medium $n_{\mathrm{a}}$ is calculated as 3.59, 3.53, 3.46, and 3.33 for QCLs with $\mathrm{\Lambda }$ of 21 μm, 22 μm, 23 μm, and 24 μm, respectively. The effective mode index $n_{\mathrm{a}}$ decreases because a larger $\mathrm{\Lambda }$ introduces larger sized apertures in the metal film since the grating duty cycle was kept the same for all devices. Consequently, a greater amount of field couples to the single-sided SPP mode in air for increasing $\mathrm{\Lambda }$, thereby reducing the modal confinement in the active medium that reduces the propagation index of the guided SPP mode further.

Experimental far-field beam patterns for antenna-feedback QCLs with varying designed parameters are shown in Fig. 4. Single-lobed beams in both lateral ($x$) and vertical ($y$) directions were measured for all QCLs. As shown in Fig. 4(b), the full-width half-maximum (FWHM) for the QCL with 70 μm width, $\mathrm{\Lambda }=21\mathrm{\mu m}$ is $\sim 4°\times 4°$, which is the narrowest reported beam profile from any terahertz QCL to date. In contrast, previous schemes for emission in a narrow beam have resulted in divergence angles of $6°\times 11°$ using very long ($>5\mathrm{mm}$) cavities and a phased-matched third-order DFB scheme [26] and $7°\times 10°$ using broad-area devices with 2D photonic crystals [31] for single-mode terahertz QCLs, and $4°\times 10°$ [38] and $12°\times 16°$ [39] for multimode QCLs using metamaterial collimators. Figures 4(c) and 4(d) show representative beam patterns from QCLs with wider cavities of 100 μm width for the smallest and largest $\mathrm{\Lambda }$ in the range of fabricated devices, respectively. The beam divergence is relatively independent of $\mathrm{\Lambda }$, as expected. More importantly, the measurements show that the wider cavities result in a slightly broader beam. Such a result is counter intuitive because typically a laser emits in a narrower beam as its cavity’s dimensions are increased due to an increase in the size of the emitting aperture. Such a behavior is unique for a spaser with antenna-feedback and is discussed along with the full-wave 3D FEM simulation of the beam pattern in the Supplement 1 (Section S2). It can be argued that the size of the beam could be further narrowed by utilizing narrower cavities for terahertz QCLs, which will be extremely beneficial to develop cw sources of narrow-beam coherent terahertz radiation.

 

Fig. 4. Far-field radiation patterns of terahertz QCLs with antenna-feedback. (a) Schematic showing the orientation of the QCLs and definition of angles. The QCLs were operated at 78 K in pulsed mode and biased at $\sim 440\mathrm{A}/{\mathrm{cm}}^2$ while lasing in single mode. The plots are for QCLs with $\sim 1.4\mathrm{mm}$ long cavities and (b) 70 μm width and $\mathrm{\Lambda }=21\mathrm{\mu m}$ grating emitting at $\sim 3.1\mathrm{THz}$, (c) 100 μm width and $\mathrm{\Lambda }=21\mathrm{\mu m}$ grating emitting at $\sim 3.1\mathrm{THz}$, and (d) 100 μm width and $\mathrm{\Lambda }=24\mathrm{\mu m}$ grating emitting at $\sim 2.9\mathrm{THz}$, respectively.

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5. CONCLUSION

In this paper, we have presented a novel antenna-feedback scheme to achieve single-mode operation and a highly directional far-field radiation pattern from plasmonic lasers with subwavelength apertures and Fabry–Perot type cavities. It is conceptually different from any other previously utilized DFB scheme for solid-state lasers, and is based on the phase locking of a single-sided SPP mode on (one of) the metal film(s) in the spaser’s cavity, with the guided SPP mode inside the spaser’s active medium. The phase locking is established due to strong Bragg diffraction of the SPP modes by periodically perforating the metal film in the form of a grating of holes or slits. The uniqueness of the method lies in the specific value of the grating’s period, which leads to the spaser’s cavity radiating like an end-fire phased-array antenna for the excited DFB mode. Additionally, coherent single-sided SPPs are also generated on the metal film that have a large spatial extent in the surrounding medium of the laser’s cavity, which could have important implications for applications in integrated plasmonics. Coherent SPPs with a large spatial extent could make it easy to couple SPP waves from the plasmonic lasers to other photonic components, and could also potentially be utilized for plasmonic sensing. Experimentally, the scheme is implemented in terahertz QCLs with subwavelength metallic cavities. A beam-divergence angle as small as $4°\times 4°$ is achieved for single-mode QCLs, which is narrower than that achieved with any other previously reported schemes for terahertz QCLs with periodic photonic structures. Compared with the third-order DFB method, the new antenna-feedback scheme is easier to implement for fabrication by standard lithography techniques without any other complex fabrication technique to precisely match a well-defined effective mode index, and achieves a superior radiative outcoupling owing to the fact that the grating period is not an integer multiple of half-wavelengths of the standing SPP wave inside the active medium. Terahertz QCLs with antenna-feedback could lead to the development of new modalities for terahertz spectroscopic sensing and wavelength tunability due to the access of a coherent terahertz SPP wave on top of the QCL’s cavity, possibilities of sensing and imaging at standoff distances of a few tens of meters, and the development of integrated terahertz laser arrays with a broad spectral coverage for applications in terahertz absorption spectroscopy.