Laser cavity and method of forming the same

Abstract

Various embodiments may relate to a laser cavity. The laser cavity may include a substrate, a contact layer in contact with the substrate, and an active layer in contact with the contact layer. The active layer may include a lattice including a plurality of hexagonal cells, and a plurality of daisy-like holes, each of the plurality of daisy-like holes including six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC). The laser cavity may additionally include a cladding layer on a portion of the active layer, a first electrode, and a second electrode such that the substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.

Description

LASER CAVITY AND METHOD OF FORMING THE SAMECROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority of Singapore application No. 10202401959T filed July 3, 2024, the contents of it being hereby incorporated by reference in its entirety for all purposes.TECHNICAL FIELD

[0002] Various embodiments of this disclosure may relate to a laser cavity. Various embodiments of this disclosure may relate to a method of forming a laser cavity.BACKGROUND

[0003] Low-threshold laser sources realized in ultra-compact cavities enabled by extremely tight mode confinement, exhibiting energy efficient and monolithic integrated characteristics, are highly desired in modern optoelectronic applications, such as on-chip communication, biological sensing and quantum information. Photonic bound states in the continuum (BIC) are peculiar nonradiative localized modes that theoretically have infinite lifetime within the radiation continuum, rendering them favorable candidates for pursuing single-mode, low- threshold and surface-emitting lasers. In large-scale photonic cavities, the symmetry-protected BIC mode resides at the center of the Brillouin zone (T-point), making it also function as a traditional band edge mode with a small group velocity that can simultaneously provide exceptional in-plane and out-of-plane field localization. However, in miniaturized devices, the continuous photonic band splits into a series of discrete modes due to momentum space quantization. The fundamental mode deviates from the T-point, resulting in the degeneration of its nonradiative and dispersionless feature. Consequently, both the group index and quality (Q) factor suffer significant reductions, accompanied by notably increased side-leakages and radiation losses. FIG. 1 shows (above) a comparison of angular frequency © / group index (in logscale) / Q factor (in logscale) between cavity modes in normal bound states in the continuum (BIC) devices with infinite- and finite-sizes; and (below) a schematic illustrating that substantial radiation losses, scattering losses, and side leakages exist in the cavity for a normal BIC mode in a finite-size device. The continuous band in an infinite-size cavity transforms intoa series of discrete modes (black dots) in a cavity with finite-size, with corresponding group indices and Q factors deviating from the values at T-point. Here the group index, n&= dv&, where vg= dto dk, can reflect the in-plane feedback and localization capabilities in a photonic crystal slab. A higher group index means a stronger in-plane feedback and localization.

[0004] A straightforward and commonly used approach to mitigate side leakages is to introduce photonic heterostructure confinement, such as the implementation of topological band-inversion or photonic bandgap boundaries. However, the symmetry mismatch induced mode confinement in these typical designs occurs only around the band edge. The Q factor still faces substantial compromise when a finite-size cavity is constructed with its mode deviating from the T-point. Furthermore, the imposition of external boundaries inevitably introduces mode scatterings and increases the overall cavity footprint.SUMMARY

[0005] Various embodiments may relate to a laser cavity. The laser cavity may include a substrate, a contact layer in contact with the substrate, and an active layer in contact with the contact layer. The active layer may include a lattice including a plurality of hexagonal cells, and a plurality of daisy-like holes, each of the plurality of daisy-like holes including six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC). The laser cavity may additionally include a cladding layer on a portion of the active layer, a first electrode, and a second electrode such that the substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.

[0006] Various embodiments may relate to a method of forming a laser cavity. The method may include forming a contact layer in contact with a substrate The method may also include forming an active layer in contact with the contact layer. The active layer may include a lattice including a plurality of hexagonal cells. The active layer may also include a plurality of daisylike holes, each of the plurality of daisy -like holes including six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC). The method may further include forming a cladding layer on a portion of the active layer. The method may also include forming a first electrode The method may additionally include forming a second electrode such thatthe substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.BRIEF DESCRIPTION OF THE DRAWINGS

[0007] In the drawings, like reference characters generally refer to the same parts throughout the different views. The drawings are not necessarily drawn to scale, emphasis instead generally being placed upon illustrating the principles of various embodiments. In the following description, various embodiments of the invention are described with reference to the following drawings.FIG. 1 shows (above) a comparison of angular frequency co / group index (in logscale) / Q factor (in logscale) between cavity modes in normal bound states in the continuum (BIC) devices with infinite- and finite-sizes; and (below) a schematic illustrating that substantial radiation losses, scattering losses, and side leakages exist in the cavity for a normal BIC mode in a finite-size device.FIG. 2 shows a general illustration of a laser cavity according to various embodiments.FIG. 3 shows a general illustration of a method of forming a laser cavity according to various embodiments.FIG. 4A shows (above) a three-dimensional plot of angular frequency co as a function of wave vector along x axis kxand wave vector along y axis kyillustrating a three-dimensional (3D) photonic band diagram of the hexagonal supercell with flat-band bound states in the continuum (BIC) device (Inset: a schematic of the polarization singularity of the flat-band with topological charge of q = +2 at the T point); and (below) a schematic illustrating that side leakages may be well suppressed enabled by the flat-band device according to various embodiments.FIG. 4B shows (above) a schematic illustration of the flat-band multi-bound states in the continuum (multi-BIC) device in momentum space, with the plus and minus signs represent topological charge of +1 and -1, respectively according to various embodiments; (below left) a schematic illustrating that radiation, scattering and side leakage losses are well suppressed when the multi-BIC is introduced to the flat-band cavity according to various embodiments; and (below right) a two dimensional quality factor (Q factor) and polarization map of super- BIC mode in momentum space around T point according to various embodiments.FIG. 4C shows a schematic of a laser cavity with a double-metal quantum cascade laser configuration according to various embodimentsFIG. 4D shows a table comparing the performance of the laser cavity according to various embodiments and reference cavity devices with similar device configuration and fabrication processes.FIG. 5A shows a conduction profile and electron wavefunctions of a single period of the active layer according to various embodiments with a bias voltage of 59 mV / module. The single period is indicated by the dashed box.FIG. 5B shows a plot of gain (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the calculated gain profile of the terahertz (THz) quantum cascade laser (QCL) according to various embodiments with a bias voltage of 59 mV / moduleFIG. 5C shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the emission spectra at different pump currents of a ridge laser fabricated on the same quantum cascade wafer according to various embodiments.FIG. 6 shows a schematic showing a method of forming a laser cavity according to various embodiments.FIG. 7 shows (a) electric field distribution of B, Ei, and E2 modes in a typical Cr, -symmetric lattice according to various embodiments; (b) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of ro for cavities with circular air hole; (c) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of ro for cavities with daisy-like airhole represented by r(c / >) = ro - rd cos(6<l>) according to various embodiments; and (d) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of ro for cavities with daisy-like airhole with 30° rotation represented by r(( / >) = ro + cos(6<j>). FIG. 8 shows (a) a plot of the real part of the B and E modes as a function of wave vector k illustrating the dispersion of the real part of the B and E modes according to various embodiments; and (b) a plot of the imaginary part of the B and E modes as a function of wave vector k illustrating the dispersion of the imaginary part of the B and E modes according to various embodiments.FIG. 9 shows a plot of quality (Q) factor as a function of wave vector k illustrating the calculated Q factors of B, Ei, and E2 band in a normal flat-band photonic crystal structurewithout the introduction of multiple bound states in continuum or multi -BIC (multi -BIC) according to various embodimentsFIG. 10A shows plots of angular frequency w as a function of wavevector k illustrating calculated band diagrams of the daisy-like air hole photonic crystal with “flat-band bound states in continuum (BIC)” (top) according to various embodiments, with “normal C -BIC” (achieved by rotating the air hole by 30°) (middle), and with “normal C4v-BIC” (achieved using a circular airhole) (bottom). Each BIC band in the diagram is denoted as B (B' and B"), Ei (E ) and E2 (E2').FIG. 10B shows (a) the unit cell and Brillouin zone of the flat-band bound states in continuum (BIC) structure according to various embodiments (top), and the full band structure of the flatband BIC structure according to various embodiments (bottom); (b) the unit cell and Brillouin zone of the “normal Cev-BIC” structure by rotating the daisy-like air hole in (a) with 30° (top), and the full band structure of the “normal Cev-BIC” structure (bottom); (c) the unit cell and Brillouin zone of the “normal Cw-BIC” structure (top), and the full band structure of the “normal C4v-BIC” structure (bottom), the inset showing the electric field EzofB" mode within a unit cell at the T point; (d) the electric field Ezof each mode in (a) with a unit cell at the T point according to various embodiments; and (e) the electric field Ezof each mode in (b) with a unit cell at the T pointFIG. 10C shows near-field profiles in finite-size cavities corresponding to the fundamental mode of Ei band (top), B' band (middle), and B" band (bottom) in FIG. 10A, respectively, according to various embodiments.FIG. 10D shows a plot of group index as a function of wavevector k illustrating group indices for different modes computed from the corresponding Bloch band structures according to various embodiments.FIG. 10E shows (a) a plot of quality factor (Q factor) as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profiles (1st order and 2nd order) of the B' band in a cavity with side length, / . of 5a (strong side leakage exist due to the weak in-plane confinement) according to various embodiments; and (b) a plot of quality factor (Q factor) as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profiles of the Ei' band in a cavity with side length, L of 5a (in-plane losses are significantly reduced but still inferior to Ei mode in such a small cavity because that the E band is not as “flat” as Ei band).FIG. 10F shows (a) a plot of quality Q factor as a function of wave vector k illustrating the Q factor of the B” band (top), and a plot of group index and a function of wave vector k comparing the group index between the designed flat-band BIC mode (Ei mode) according to various embodiments and the normal C4v-BIC mode (B" mode) (bottom); and (b) near-field profiles in finite-size cavities corresponding to the fundamental mode of B" band.FIG. 10G shows (a) a plot of effective index as a function of angular frequency <n illustrating the effective index of the two bands with linear dispersion (i.e., B and E2 bands, the effective indices approximate to zero around the Dirac point) in the flat-band bound states in the continuum (BIC) according to various embodiments; (b) the electric field distribution of the flat-band BIC mode (Ei mode) according to various embodiments; (c - d) the extracted magnitude (top) and electric field distribution (bottom) of the two bands with quadratic dispersion (B' and E2') in the normal Cev-BIC cavity (the magnitude displays a gaussian shape and the fields are tightly confined inside the cavity); and (e) electric field distributions of another degenerated mode of E2' (E2) in the normal Cev-BIC cavity.FIG. 10H shows (top) a near-field profile of the fundamental mode belonging to the B band with linear dispersion (side lengths of cavity, L = 13a) according to various embodiments; and (bottom) plot of intensity (in arbitrary units or a.u.) as a function of Y-axis (in micrometers or pm) illustrating the extracted electric field intensity along the dashed line within the cavity according to various embodiments.FIG. 101 shows (top) a near-field profile of the fundamental mode belonging to the E2 band with linear dispersion (side lengths of cavity, L = 1 a) according to various embodiments; and (bottom) plot of intensity (in arbitrary units or a.u.) as a function of X-axis (in micrometers or pm) illustrating the extracted electric field intensity along the dashed line within the cavity according to various embodiments.FIG. 10J shows (a) stimulated electric distribution of the B mode with air boundary condition according to various embodiments, and (b) a plot of intensity (in arbitrary units or a.u.) as a function of Y-axis (in micrometers or pm) illustrating the magnitude of the electric field along the dashed line shown in (a) according to various embodiments.FIG. 10K shows a comparison of the quality (Q) factors of B (B1), Ei, E2 (E?1) modes with distinct band dispersion features according to various embodiments.FIG. 10L shows (a) angular frequency w as a function of wave vector k illustrating band structure of the Dirac-like band after taking an |Aro| = 0.5 pm air hole radius perturbation intoconsideration according to various embodiments (A / v represents the radius deviation from the initial design (12 pm) during the fabrication process); (b) a plot of group index as a function of wave vector k illustrating the group index of each band as shown in (a) according to various embodiments; (c) electric distributions of each band as shown in (a) according to various embodiments, and (d) a comparison of the quality (Q) factors of each band in (a) according to various embodiments.FIG. 11A shows (a) a plot of lattice constant a (in micrometers or pm) as a function of radius ro (in micrometers or pm) illustrating the tuning of lattice parameter to form a tuning of the lattice parameters to ensure a Dirac-like band feature according to various embodiments; and (b) plots of quality (Q) factor as a function of wave vector k illustrating the calculated Q factors for the lattice with parameters of a = 28.8 pm, ro = 9.5 pm (left), a = 33 pm, ro = 12 pm (middle), and a = 37.2 pm, ro = 14.5 pm (right) (rd is set to 2 pm) according to various embodiments.FIG. 1 IB shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating a two-dimensional (2D) map of the Q factor and polarization distribution of the flat-band with super bound states in continuum (super-BIC) cavity states in momentum space according to various embodiments.FIG. 12A shows a plot of frequency (in terahertz or THz) as a function of wave vector k illustrating the calculated band diagram of a flat-band cavity with super bound states in continuum (super-BIC) (a = 33 pm, ro = 12 pm) and without super-BIC (a = 37.2 pm, ro = 14.5 pm) according to various embodiments, ra may be set to 2 pm for both cases.FIG. 12B shows a plot of group index as a function of wave vector k illustrating the calculated group index of the flat-band shown in FIG. 12A with super bound states in continuum (super- BIC) and without super-BIC according to various embodiments.FIG. 12C shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating two-dimensional (2D) mapping of the quality (Q) factor for the flat-band without super bound states in continuum (super-BIC), the map is darker at off-F point than the flatband with super-BIC states (shown in FIG. 11B) according to various embodiments, which means a significant lower quality (Q) factor in a finite-size cavity.FIG. 12D shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated Q factors and mode profile of the flat-band in a cavity without super bound states in continuum (super-BIC) according to various embodiments.FIG. 12E shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated Q factors and mode profde of the flat-band in a cavity with super bound states in continuum (super-BIC) according to various embodiments.FIG. 12F shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating polarization states of the flat-band mode in k space, from which a topological charge of q = +2 can be obtained according to various embodiments.FIG. 12G shows a plot of quality (Q) factor as a function of cavity area a2illustrating Q factor comparison between flat-band super bound states in continuum (super-BIC) and normal Cw BIC modes in cavities with different areas according to various embodiments.FIG. 13 A shows a plot of quality (Q) factor as a function of wavevector k illustrating the calculated Q factor of the flat-band cavities with and without super bound states in continuum (super-BIC) according to various embodiments.FIG. 13B shows (a) a schematic of an asymmetric hole acting as a fabrication imperfection (Ar, Ax, and Ay is set as 0.5 pm based on the actual fabrication accuracy) according to various embodiments; and (b) a plot of quality (Q) factor as a function of wave vector k illustrating simulated Q factors of the unit cell with disorder according to various embodiments.FIG. 13C shows (a) schematics of vertical sidewall (top) and undulating sidewall (bottom) profiles used in simulation according to various embodiments; (b) electric field distributions of the super bound states in continuum (super-BIC) modes with the vertical sidewall (top) and undulating sidewall (bottom) according to various embodiments; (c) beam profiles of the super bound states in continuum (super-BIC) modes with the vertical sidewall (top) and undulating sidewall (bottom) according to various embodiments; and (d) a comparison of the quality (Q) factor values for the flat-band mode with the two types of sidewalls according to various embodiments.FIG. 13D shows a plot of quality (Q) factor as a function of cavity area a2illustrating the calculated Q factors for the fundamental modes of the flat-band cavities with and without super bound states in continuum (super-BIC) mode, along the cavities with the “normal CA-B1C cavity” (B1mode), and the “normal C iv-BIC cavity” (B" mode) according to various embodiments.FIG. 13E compares the calculated quality (Q) factors of L = 5a devices with titanium (Ti) / gold (Au) (Drude Model) cladding according to various embodiments.FIG. 14A shows a scanning electron microscopy (SEM) image of a fabricated L = 5a laser cavity with daisy-like air holes according to various embodimentsFIG. 14B shows a magnified scanning electron microscopy (SEM) image of the fabricated laser cavity shown in FIG 14A. according to various embodimentsFIG. 14C shows a plot of voltage (in volts or V) / peak power (in milli-Watts or mW) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the fabricated laser cavities according to various embodiments.FIG. 14D shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current- voltage (L-I-V) curves of the fabricated laser cavity A = 5a according to various embodiments.FIG. 14E shows a plot of intensity (in arbitrary units or a.u.) as a function of current (in amperes or A) illustrating the light-current (L - I) curve in a L = 5a device according to various embodiments.FIG. 14F shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the laser cavity according to various embodiments at various pumping current densities.FIG. 14G shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the L = 5a laser cavity according to various embodiments at different frequencies and pumping current densities.FIG. 14H shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.27 kA / cm2of the L = 5a laser cavity with a measured side-mode suppression ratio (SMSR) ~20 dB according to various embodiments.FIG. 141 shows (a) a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode provide of a laser cavity with side length L = 9a according to various embodiments; and (b) a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode provide of a laser cavity with side length L = 17a according to various embodiments.FIG. 14J shows a scanning electron microscopy (SEM) image of a fabricated L = 9a laser cavity with daisy-like air holes according to various embodiments.FIG. 14K shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the L = 9a laser cavity according to various embodiments at different frequencies and pumping current densities.FIG. 14L shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.34 kA / cm2of the L = 9a laser cavity with a measured side-mode suppression ratio (SMSR) - 24 dB according to various embodiments.FIG. 14M shows a scanning electron microscopy (SEM) image of a fabricated L = 17a laser cavity with daisy-like air holes according to various embodiments.FIG. 14N shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the / . = 17a laser cavity according to various embodiments at different frequencies and pumping current densities.FIG. 140 shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.25 kA / cm2of the L = 17a laser cavity with a measured side-mode suppression ratio (SMSR) ~26 dB according to various embodiments.FIG. 15A shows a schematic of a setup for focusing the diverged beam of the quantum cascade lasers (QCLs) according to various embodiments.FIG. 15B shows a plot of power (in micro-Watts or pW) as a function of time (in seconds or s) illustrating characterization of a typical ridge laser.FIG. 15C shows a plot of power (in micro-Watts or pW) as a function of time (in seconds or s) illustrating characterization of a L = 9a laser cavity according to various embodiments.FIG. 15D shows a table comparing the radiative efficiency of the flat-band cavity design according to various embodiments with other photonic crystal-based cavities.FIG. 16A shows a simulated beam profile and far-field pattern of a laser beam of the designed super bound states in continuum (super-BIC) according to various embodiments.FIG. 16B shows (a) a schematic illustrating measuring far-field image of the as-prepared flatband laser without a focal lens according to various embodiments; (b) the measured far-field image generated by the setup in (a) according to various embodiments; and (c) a plot of intensity (in arbitrary units or a.u.) as a function of x position (in millimeters or mm) illustrating the intensity distribution of the far-field image in (b) according to various embodiments.FIG. 16C shows (a) a schematic illustrating measuring far-field image of the as-prepared flatband laser with a focal lens according to various embodiments; (b) the measured far-field image generated by the setup in (a) according to various embodiments; and (c) a plot of intensity (in arbitrary units or a.u.) as a function of x position (in millimeters or mm) illustrating the intensity distribution of the far-field image in (b) according to various embodiments.FIG. 17A shows a plot of voltage (in volts or V) / peak power (in milli-Watts or mW) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of a ridge laser.FIG. 17B shows a simulated Fabry-Perot (F-P) cavity mode cavity.FIG. 17C shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the ridge laser at various temperatures.FIG. 17D shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the super bound states in continuum (super-BIC) laser according to various embodiments at various temperatures.FIG. 18A shows a schematic of a photonic bandgap (shell)-confined bound states in continuum (BIC) mode (core) device.FIG. 18B shows a plot of frequency (in terahertz or THz) as a function of wave vector k illustrating the photonic band diagram of the CV -symmetry photonic crystals with a = 28 pm and rcore(r shell) = 10 pm (9 pm).FIG. 18C shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profile of the photonic bandgap-confined bound states in continuum (BIC) cavity.FIG. 18D shows a scanning electron microscopy (SEM) image of the fabricated core-shell hetero-structure with rCore (10 pm) > / 'shell (9 pm).FIG. 18E shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating lightcurrent-voltage (L-I-V) curves of the core-shell cavity with a lasing threshold of -0.28 kA / cm 2.FIG. 18F shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the core-shell device.FIG. 18G shows a scanning electron microscopy (SEM) image of the traditional “normal C4v- BIC” cavity without optimized core-shell structure, where rcore= rSheii (10 pm).FIG. 18H shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode profile of the traditional bound states in continuum (BIC) cavity.FIG. 181 shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the device based on traditional continuum (BIC) cavity.DESCRIPTION

[0008] The following detailed description refers to the accompanying drawings that show, by way of illustration, specific details and embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized and structural, logical, and electrical changes may be made without departing from the scope of the invention. The various embodiments are not necessarily mutually exclusive, as some embodiments can be combined with one or more other embodiments to form new embodiments.

[0009] Features that are described in the context of an embodiment may correspondingly be applicable to the same or similar features in the other embodiments. Features that are described in the context of an embodiment may correspondingly be applicable to the other embodiments, even if not explicitly described in these other embodiments. Furthermore, additions and / or combinations and / or alternatives as described for a feature in the context of an embodiment may correspondingly be applicable to the same or similar feature in the other embodiments.

[0010] In the context of various embodiments, the articles “a”, “an” and “the” as used with regard to a feature or element include a reference to one or more of the features or elements.

[0011] In the context of various embodiments, the term “about” or “approximately” as applied to a numeric value encompasses the exact value and a reasonable variance, e.g., within 10% of the specified value.

[0012] As used herein, the term “and / or” includes any and all combinations of one or more of the associated listed items.

[0013] By “comprising” it is meant including, but not limited to, whatever follows the word “comprising”. Thus, use of the term “comprising” indicates that the listed elements are required or mandatory, but that other elements are optional and may or may not be present.

[0014] By “consisting of’ it is meant including, and limited to, whatever follows the phrase “consisting of’. Thus, the phrase “consisting of’ indicates that the listed elements are required or mandatory, and that no other elements may be present.

[0015] Embodiments described in the context of one of the laser cavity are analogously valid for the other laser cavities. Similarly, embodiments described in the context of a method are analogously valid for a laser cavity, and vice versa.

[0016] FIG. 2 shows a general illustration of a laser cavity according to various embodiments. The laser cavity may include a substrate 202 and a contact layer 204 in contact with the substrate 202. The laser cavity may also include an active layer 206 (alternatively referred to as active region layer or active region) in contact with the contact layer 204. The active layer 206 may include a lattice including a plurality of hexagonal cells. The active layer 206 may further include a plurality of daisy-like holes, each of the plurality of daisy-like holes including six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC). The laser cavity may additionally include a cladding layer 208 on a portion of the active layer 206. The laser cavity may also include a first electrode (alternatively referred to as front electrode) 210. The laser cavity may further include a second electrode (alternatively referred to as back electrode) 212 such that the substrate 202, the contact layer 204, the cladding layer 208 and the active layer 206 are between the first electrode 210 and the second electrode 212.

[0017] In other words, various embodiments may relate to a laser cavity having an active layer 206 that is configured to achieve a flat-band bound states in continuum (BIC) during operation, i.e., when a potential difference is applied between the first electrode 210 and the second electrode 212. The active layer 206 may include atoms arranged in hexagonal cells. Daisy-like holes may be present in the active layer 206, with each hole in a respective hexagonal cell.

[0018] For avoidance of doubt, FIG. 2 is intended to illustrate some features of a laser cavity and are not intended to limit the dimensions (e.g., thicknesses, widths etc.), sizes, orientations etc. of the various features.

[0019] In various embodiments, the lattice (of the active layer 206) may include alternating layers of quantum barriers and quantum wells The alternating layers may be of different thicknesses.

[0020] The quantum barriers and quantum wells may be formed by any suitable materials. For instance, the quantum barriers may be formed by aluminum gallium arsenide (e.g., Alo.15Gao.85As), while the quantum wells may be formed by gallium arsenide (GaAs). In another example, the quantum barriers may be formed by indium aluminum arsenide (InAlAs), while the quantum wells may be formed by indium gallium arsenide (InGaAs). In yet another example, the quantum barriers may be formed of magnesium zinc oxide (MgxZm.x0, where 0 < x < 1), and the quantum wells may be formed of zinc oxide (ZnO).

[0021] In various embodiments, the plurality of daisy-like holes may extend through the active layer 206. The plurality of daisy-like holes may also extend through the first electrode (i.e., front electrode) 210. At least a portion of the plurality of daisy-like holes may also extend through the cladding layer 208.

[0022] Each of the plurality of daisy -like holes may include a circular hole and six air bulges extending from the circular hole, with a tip of each of the six air bulges coinciding with a respective comer of six comers of the hexagonal cell. Each daisy-like hole may be described by r(^) = ro -where refers to the angle and ranges from 0° to 360°, / 'u refers to the mean radius, while ra refers to the radius variations from the mean radius, the radius variations defining the troughs between neighboring air bulges, and apexes of the air bulges.

[0023] In various embodiments, the flat-band bound states in continuum (BIC) may be achieved by a flat Ei band intersecting a E2 band at an apex of the E2 band, and a B band at an apex of the B band. The Ei, E2 and B bands may refer to different photonic bands of a photonic crystal. All three bands may belong to different irreducible representations at the f-point and may be decoupled from free space The B and E2 bands, characterized by linear dispersion, may exhibit an exceedingly low group index (e.g., ~5.2) near the T point, which suggests weak in-plane feedback and limited localization effect, and may thus facilitate the single-mode lasing from the flat Ei mode, which has strong in-plane feedback. Also, substantial side leakages of the neighboring B and E2 modes may significantly enhance mode selectivity (notable Q contrast) between these two modes and the flat-band mode (Ei mode), facilitating the attainment of stable single-mode flat-band lasers.

[0024] In various embodiments, the laser cavity may achieve flat-band super bound states in continuum (super-BIC, alternatively referred to as multiple bound states in continuum or multi -BIC).

[0025] In various embodiments, flat-band super bound states in the continuum (super-BIC) may be achieved at predetermined lattice (e.g., lattice spacings or ad) and hole parameters (i.e., parameters defining the daisy-like hole, e.g., ro , ra).

[0026] In various embodiments, the laser cavity may achieve a quality factor (Q-factor) of a value above e.g., 1300, e.g., 1400, e.g. a value of 1440.

[0027] In various embodiments, the laser cavity may be a terahertz (THz) laser cavity.

[0028] FIG. 3 shows a general illustration of a method of forming a laser cavity according to various embodiments The method may include, in 302, forming a contact layer in contact with a substrate. The method may also include, in 304, forming an active layer in contact with the contact layer. The active layer may include a lattice including a plurality of hexagonal cells. The active layer may also include a plurality of daisy-like holes, each of the plurality of daisylike holes including six air bulges aligned substantially with comers of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC). The method may further include, in 306, forming a cladding layer on a portion of the active layer. The method may also include, in 08, forming a first electrode. The method may additionally include, in 310, forming a second electrode such that the substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.

[0029] For avoidance of doubt, FIG. 3 is intended to illustrate some steps according to various embodiments, and is not intended to limit the sequence of the various steps.

[0030] A continuous semiconductor layer (including alternating quantum barriers and quantum wells, e.g., aluminum gallium arsenide layers and gallium arsenide layers) may be formed on or over another substrate. The contact layer (e.g., gold or Au layer) may be formed by depositing one or more suitable metals on the substrate, depositing the one or more suitable metals on the continuous semiconductor layer on the other substrate, and bonding the substrate and the other substrate using the one or more suitable metals in a wafer bonding process (e.g., thermo-compression wafer bonding). The other substrate may be separated from the continuous semiconductor layer after the wafer bonding process.

[0031] The cladding layer may be formed by depositing a suitable cladding material on the continuous semiconductor layer, and removing a portion of the deposited cladding material using lithography and etching.

[0032] The first electrode may be formed by depositing a suitable photoresist material over the cladding layer and the continuous semiconductor layer, and patterning the suitable photoresist material to form a plurality of photoresist structures. The method may also include depositing one or more suitable materials over the photoresist structures, the cladding layer and the continuous semiconductor layer, and removing portions of the one or more suitable materials deposited on the photoresist structures in a lift off process. The one or more suitable materials may include metals such as titanium (Ti) and gold (Au) and dielectrics such as silicon oxide (SiCh) The method may further include etching remaining portions of the one or more suitable materials deposited on the cladding layer and the continuous semiconductor layer to form the first electrode. The dielectrics such as silicon oxide (SiCh) may be removed during etching.

[0033] The active layer may be formed by etching the continuous semiconductor layer using the remaining portions of the one or more suitable materials as a mask.

[0034] The substrate may be polished before the second electrode is formed in contact with the substrate using a back electrode deposition process.

[0035] In various embodiments, the flat-band bound states in continuum (BIC) may be achieved by a flat Ei band intersecting a E2 band at an apex of the E2 band, and a B band at an apex of the B band.

[0036] In various embodiments, the laser cavity may achieve flat-band super bound states in continuum (super-BIC) / multiple bound states in continuum (multi-BIC). Flat-band super bound states in continuum (super-BIC) / multiple bound states in continuum (multi-BIC) may be achieved at predetermined lattice and hole parameters

[0037] In various embodiments, the laser cavity may be a terahertz laser cavity.

[0038] Various embodiments may relate to a highly compact cavity design based on a flatband combined with multiple bound states above the light cone, denoted as “flat-band multi- BIC” or “flat-band super-BIC”.

[0039] FIG. 4 A shows (above) a three-dimensional plot of angular frequency co as a function of wave vector along x axis kxand wave vector along y axis kyillustrating a three-dimensional (3D) photonic band diagram of the hexagonal supercell with flat-band bound states in thecontinuum (BIC) device (Inset: a schematic of the polarization singularity of the flat-band with topological charge of q = +2 at the T point); and (below) a schematic illustrating that side leakages may be well suppressed enabled by the flat-band device according to various embodiments.

[0040] FIG. 4B shows (above) a schematic illustration of the flat-band multi-bound states in the continuum (multi-BIC) device in momentum space, with the plus and minus signs represent topological charge of +1 and -1, respectively according to various embodiments; (below left) a schematic illustrating that radiation, scattering and side leakage losses are well suppressed when the multi-BIC is introduced to the flat-band cavity according to various embodiments; and (below right) a two dimensional quality factor (Q factor) and polarization map of super-BIC mode in momentum space around T point according to various embodiments.

[0041] FIG. 4C shows a schematic of a laser cavity with a double-metal quantum cascade laser configuration according to various embodiments. The top and bottom gold (Au) layers may act as electrodes for current injection, as well as vertical confinement of the cavity mode.

[0042] FIG. 4D shows a table comparing the performance of the laser cavity according to various embodiments and reference cavity devices with similar device configuration and fabrication processes. FIG. 4D seeks to highlight the compactness and the high-Q of the flatband.

[0043] By localizing the photonic state in the in-plane (out-of-plane) directions through utilization of a flat-band (multi-BIC / super-BIC) feature, a high Q factor of -1440 (considered as high-Q in the platform of terahertz quantum cascade lasers) in an ultra-compact laser cavity (with a side length of L = 5a, see FIG. 4C) can be achieved.refers to the length of one side (i.e., of the 6 sides) of the hexagonal cell, while “a” refers to the lattice constant. This value may exceed normal BIC modes by approximately 20- to 40-fold and may also be significantly higher than other optimized cavity designs (see FIG. 4D). Experimentally, a high-performance terahertz laser exhibiting stable single-mode operation with a side mode suppression ratio (SMSR) of -20 dB and an ultra-low lasing threshold of -0.18 kA / cm2or -0.19 kA / cm2has been demonstrated. Notably, the lateral dimension of the cavity may just be ~3X, rendering it one of the most compact devices within the realm of photonic crystal -based lasers. Moreover, benefitting from the favorable mode selectivity associated with the Dirac-cone-like photonicband feature, the device may exhibit scalability without compromising its excellent singlemode performance.

[0044] Design of Flat-band Multi -BIC / Super-BIC Cavity

[0045] The laser cavity design may be constructed by etching periodic air holes through the active medium of a three-well resonant-phonon GaAs / Alo isGao ssAs designed terahertz QCL wafer to form a triangular photonic lattice structure. The air holes may feature a daisy-like shape, which may facilitate manipulation of the photonic band structure. The shape can be written as r(< / ))= 12 gm and rA= 2 gm. FIG. 5A shows a conduction profile and electron wavefunctions of a single period of the active layer according to various embodiments with a bias voltage of 59 mV / module. The single period is indicated by the dashed box. FIG. 5B shows a plot of gain (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the calculated gain profile of the terahertz (THz) quantum cascade laser (QCL) according to various embodiments with a bias voltage of 59 mV / module. FIG. 5C shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the emission spectra at different pump currents of a ridge laser fabricated on the same quantum cascade wafer according to various embodiments. The gain bandwidth is around 2.8-3.4 THz with gradually blue-shifted with increasing pumping, due to the Stark shift of the intersubband transition in the THz QCL medium. The overall thickness ofthe wafer is -18.3 gm. The active region consists of 131 repeat periods. Each period consists of a series of alternating barriers ( Alo.15Gao.s5As) and quantum wells (GaAs) with the thickness of 1 / 12.9 / 2 / 11,8 / 3 / 9,5 / 3 / 8.6 / 3 / 7.1 / 3 / 17 / 3 / 14.5 / 4 / 10.1 / 0.5 / 16.2 nm, where bold numbers refer to the Alo 15Gao 85As barriers. The central 17 nm GaAs well may be n-doped with silicon (Si) at 2.0 x 1016. The electron wavefunctions as well as the intersubband transition process in each period of the THz QCL in FIG. 5 A may be calculated by using the tight-binding model. The corresponding gain profile in each period may also be calculated with the same bias voltage of 59 mV.

[0046] FIG. 6 shows a schematic showing a method of forming a laser cavity according to various embodiments. The laser cavity may be a terahertz quantum cascade laser (THz QCL) cavity. The method may include, in step 1, depositing suitable metals (e.g., titanium (Ti) and gold (Au)) on the substrate 602 (e g., QCL wafer) to form metal layers 604a (e g., Ti (15 nm) / Au (700 nm) metal layers). The Ti layer may be formed on substrate, and the Au layer may be formed on the Ti layer. Suitable metals (e g., titanium (Ti) and gold (Au)) may also bedeposited on a continuous semiconductor layer 606a, which may in turn be on or over another substrate 614 (e.g., n+doped GaAs substrate), to form metal layers 604b (Ti (15 nm) / Au (700 nm) metal layers). The Ti layer may be formed on the continuous semiconductor layer 606a, and the Au layer may be formed on the Ti layer The suitable metals may be deposited over the substrate 602 and the other substrate 614 using an electro-beam evaporator. The method may also include, in step 2, bonding the substrate 602 and the other substrate 614 using the suitable metal in a wafer bonding process (e.g., Au / Au thermo-compression wafer bonding). In step 2, metal layers 604a, 604b may be bonded together to form contact layer 604. The gold layers of the metal layers 604a, 604b may be bonded together via thermos-compression wafer bonding. The method may further include, in step 3, separating the other substrate 614 (e.g., n+doped GaAs substrate) from the continuous semiconductor layer 606a. The other substrate 614 (e.g., n“ doped GaAs substrate), together with any etch stop layers (including e.g., AhoGasoAs) between the other substrate 614 and the continuous semiconductor layer 606a, may be separated or removed by a wet etching process, e.g., with citric acid / hydrogen peroxide H2O2 / deionized water H2O (100 g / 30 ml / 100 ml) and hydrofluoric acid (49%) solution, respectively (the transfer between different solutions may have to be fast to reduce the exposure time to air to achieve complete substrate removal for subsequent good ohmic contact with the top electrode). The substrate 602 (e.g., QCL wafer) may be thinned to be around ~50 pm using a polishing instrument (MULTIPREP™ POLISHING SYSTEM - 8", ALLIED, HIGH TECH PRODUCTS, iNC., USA).

[0047] The method may additionally include, in step 4, depositing a suitable cladding material (e.g., silicon oxide or SiO2) to form a continuous dielectric layer 608a (e.g., 200 nm SiO2 insulation layer) on or over the continuous semiconductor layer 606a. The method may also include, in step 5, defining a pump region using photolithography to avoid unexpected whisper-gallery modes caused by gain spreading outside the cavity. A suitable photoresist material, e.g., AZ5214, may be deposited on the continuous dielectric layer 608a, and patterned using photolithography to form photoresist structures 616. The method may further include, in step 6, removing a portion (i.e., at the cavity center) of the continuous dielectric layer 608a (i.e., the deposited cladding material) using dry etching (e.g., reactive-ion etching (RIE)), while a remaining portion (e.g., two periods the edge region) may be preserved to form a cladding layer 608.

[0048] The method may include, in step 7, defining the cavity via photolithography. A suitable photoresist material, e.g., AZ5214, may be deposited over the cladding layer 608 and the continuous semiconductor layer 606a. The suitable photoresist material may be patterned, using photolithography, to form a plurality of photoresist structures 618. The method may further include, in step 8, depositing suitable materials (e g., metals such as Ti and Au and dielectric material such as SiCh) over the photoresist structures 618, the cladding layer 608 and the continuous semiconductor layer 606a. Consequently, a continuous layer 620 may be formed including sub-layers of Ti (20 nm) / Au (350 nm) / Ti (20 nm) / SiO2 (710 nm), with the Ti (20 nm) sub-layer on the continuous semiconductor layer 606a, the Au (350 nm) sub-layer on the Ti (20 nm) sub-layer, another Ti (20 nm) sub-layer on the Au (350 nm) sub-layer, and the SiCh (710 nm) sub-layer on the other Ti (20 nm) sub-layer. Alternatively, in another example, the continuous layer 620 may include sub-layers of Ti (15 nm) / Au (550 nm) / Ti (15 nm) / SiO2 (850 nm). The method may also include, in step 9, removing portions of the suitable metal deposited (i.e., portions of the continuous layer 620) on the photoresist structures in a lift off process, with daisy-like holes defined by the remaining portions 610a of the suitable metal deposited (i.e., remaining portions 610a of the continuous layer 620). The method may further include, in step 10, etching the continuous semiconductor layer 606a using the remaining portions 610a as a mask (i.e., hard mask) to form the active layer 606 The etching on the continuous semiconductor layer 606a may be carried out via a reactive-ion dry etching (RIE) process (e.g., with a gas mixture of CI2 / BCI3 / CH4 = 5 / 100 / 20 standard cubic centimeters per minute). The cladding layer 608 between the continuous semiconductor layer 606a and the remaining portions of the suitable metal 610a may also be etched. The Ti (20 nm) / SiO2 (710 nm) sublayers (i.e., the other Ti (20 nm) sub-layer on the Au (350 nm) sub-layer, and the SiCh (710 nm) sub-layer on the other Ti (20 nm) sub-layer) may have ~ 300 nm residual after etching, and may be removed by the etching process to form a first electrode 610. The first electrode 610 may include the Ti (20 nm) / Au (350 nm) sublayers, and may be used for current injection. In the alternative example, the first electrode 610 may have a thickness of approximately 200 nm (15 nm Ti, 185 nm Au). The method may also include, in step 11, polishing the substrate 602. The substrate 602 may be thinned, e.g., to a thickness of -200 pm, to reduce the impedance for less heat generation. The method may also include, in step 12, forming the second electrode 612 in contact with the substrate 602 using a back electrode deposition process. The second electrode 612 may include Ti (20 nm) / Au (250 nm) layers. The as-prepared devices may thenbe cleaved, indium-soldered onto a copper pedestal, wire-bonded, and fixed onto a cold finger in a cryostat for measurement.

[0049] FIG. 7 shows (a) electric field distribution of B, Ei, and E2 modes in a typical Ce- symmetric lattice according to various embodiments; (b) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of ro for cavities with circular air hole; (c) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of ro for cavities with daisy -like airhole represented by r(cj>) = ro - rd cos(6(j>) according to various embodiments; and (d) a plot of angular frequency co as a function of mean radius ro (in micrometers or pm) illustrating the eigenfrequency of the three modes as a function of r for cavities with daisy-like airhole with 30° rotation represented by r(c / >) ro rd cos(6<j)). For the plots in FIG. 7, a is set to 33 pm and / is set to 2 pm.

[0050] The formation of the Dirac-like band feature requires the degeneracy of B mode, Ei mode, and E2 mode, which may be realized by manipulating the geometric parameters. The k- space engineering starts from a normal Cc,v structure with circular air hole. The electric field (Ez) of the three modes are shown in FIG. 7(a). As indicated in FIG. 7(b) for cavities with the conventional circular airholes, the eigenfrequency of each mode at the T point can be modulated by tuning the radius of air holes. When the modulation is not strong enough, the frequency of Ei and E2 modes are higher than B mode, and the degeneracy can only be achieved when the air hole is quite small (corresponding radius not shown in the figure), which may increase the fabrication difficulty in the deep etching process (~18.3 pm). The band aggregation may cause some mode competition issues.

[0051] Various embodiments may relate to a laser cavity with the daisy-like air hole (with the “air bulges” towards the corners) to help to manipulate the photonic band structure, as illustrated by FIG. 7(c). As the electric field of the B mode concentrates at the corner of a (V- symmetry lattice, the effective index, which can be described by ne^ — jje|ectrjcpermittivity of QCL, E representing theelectric field, and V representing volume. The B mode may be significantly reduced, resulting in increased eigenfrequency. Accordingly, the eigenfrequency of Ei and E2 mode would be reduced. Thus, the mode degeneracy can be achieved. As comparison as illustrated in FIG7(d), when the air holes are rotated by 30° to make the “air bulges” toward the sides, the situation would be opposite, and the degeneracy cannot be realized.

[0052] By forcing an accidental degeneracy between the isolated B mode and the doubly degenerated Ei and E2 modes, a distinct flat-band intersecting with two linear-dispersion bands (Dirac cone) at their apex can be achieved (as shown in FIG. 4A).

[0053] The slow light effect, arising from the dispersionless nature of the flat-band, may provide excellent in-plane field confinement, and significantly reduce side leakage losses, even as the mode shifts away from the T-point in finite-size cavities. All three selected bands belong to different irreducible representations at the T-point (B, Ei and E2) and are decoupled from free space, i.e., typical symmetry-protected BIC modes which may effectively mitigate radiation losses that may occur due to mode coupling when they come into proximity to each other. FIG. 8 shows (a) a plot of the real part of the B and E modes as a function of wave vector k illustrating the dispersion of the real part of the B and E modes according to various embodiments; and (b) a plot of the imaginary part of the B and E modes as a function of wave vector k illustrating the dispersion of the imaginary part of the B and E modes according to various embodiments. The brightness of the color indicates the dominance of the E or B mode in the current mode, e g., brighter colors representing a stronger presence of the E mode. The frequency detuning is e = 0.002 and the group velocity is (:> = 1 .

[0054] A brief discussion of the interaction between singlet B and doublet E modes near the T point is provided herein. Given that the two E modes are coupled with the B mode along the kxand kydirections, respectively, the interaction between these three modes into a mixing between one of the E modes and the B mode may be simplified as follows,where j is the imaginary number, (3 is the group velocity, e is the frequency detuning, AB^Eare the eigenvalues and eigenvectors of B and E modes at any given Bloch vector k. By taking the eigenvectors at T point, VB 0and VE 0, as the orthogonal bases, the eigenvectors of B and E modes at vector k can be expressed as,VB = +UFB ,0 + bVE 0VE— bVB 0+ al^ owhere coefficients a and b satisfy a2+ b2= 1 and can be obtained through orthogonality. FIG. 8 illustrates the dispersion of both the real and imaginary parts of the eigenvalues, with the color of the lines denoting the mixing of the modes. In FIG. 8(a), the changing colors on the two bands clearly indicate the mixing between E and B bands as the Bloch vector k drifts away from T point. In FIG. 8(b), the loss of B band is smaller than the E band, albeit they both increase as vector k drifts away.

[0055] FIG. 9 shows a plot of quality (Q) factor as a function of wave vector k illustrating the calculated Q factors of B, Ei, and E2 band in a normal flat-band photonic crystal structure without the introduction of multiple bound states in continuum or multi -BIC (multi -BIC) according to various embodiments. Here a = 37.2 pm, ro = 14.5 pm, and n = 2 pm.

[0056] The non-radiative modes B, Ei, E2 (with theoretically infinite Q factor) may be characterized by a polarization singularity (e.g., the flat-band) at the center of the Brillouin zone (i.e., at the T-point). By manipulating the lattice parameters, a series of accidental BICs may be engineering in the vicinity of the T-point, thereby leading to the BIC feature known as a “super-BIC” or “multi-BIC”. The “super-BIC” or “multi-BIC” with a broad high Q region may enable the cavity mode to stand robust against photonic finite-size effects, systematic disorders, and structural imperfection, significantly suppressing the radiation and scattering losses. The topological charge may be determined to be q = +2 by selecting a circle centered around the T point with |k| = 0.1 (27t / ^3 / a).

[0057] Various embodiments may relate to a laser structure including a substrate and an active layer (alternatively referred to as active region layer or active region) arranged on or over the substrate A plurality of airholes may be formed in the active layer. Each air hole may have a daisy-like shape. The daisy-like shaped air holes can provide the flat band. In various embodiments, the size (e.g. radius) of the air holes may be designed to provide super-BIC or multi-BIC.

[0058] Flat-band Enhanced In-plane Mode Confinement

[0059] FIG. 10A shows plots of angular frequency co as a function of wavevector k illustrating calculated band diagrams of the daisy-like air hole photonic crystal with “flat-band bound states in continuum (BIC)” (top) according to various embodiments, with “normal C<jv- BIC” (achieved by rotating the air hole by 30°) (middle), and with “normal Cn-BIC” (achieved using a circular airhole) (bottom). Each BIC band in the diagram is denoted as B (B' and B"),Ei (Ei') and E2 (E2'). Each BIC band in the diagram is denoted as B (B1and B"), Ei (Ef) and E2(E2').

[0060] The flat-band with neighboring Dirac-cone-like dispersion feature (FIG. 10A, top) may result from the accidental degeneracy occurring between the isolated B mode and the doubly degenerate Ei and E2 modes (FIG. 10A, middle). It may be worth noting that all these selected modes are non-radiative, owing to their inherent symmetry mismatch with plane waves, i.e., symmetry-protected BICs, to mitigate additional radiation losses that may occur due to mode coupling when they come into proximity to each other. The advantages of the flatband BIC mode (Ei mode in FIG. 10A, top) may be emphasized through a comparison to the conventional BIC modes, i.e., B' mode with Cev symmetry in the middle of FIG. 10A (alternatively referred to as the “normal C&V-BIC”), and B" mode with Chv symmetry in the bottom of FIG. 10A (alternatively referred to as the “normal Cw-BIC”). The full band structures and electric field distributions within a unit cell for these photonic lattices can be found in FIG. 10B. FIG. 10B shows (a) the unit cell and Brillouin zone of the flat -band bound states in continuum (BIC) structure according to various embodiments (top), and the full band structure of the flat-band BIC structure according to various embodiments (bottom); (b) the unit cell and Brillouin zone of the “normal Cev-BIC” structure by rotating the daisy-like air hole in (a) with 30° (top), and the full band structure of the “normal C v-BIC” structure (bottom); (c) the unit cell and Brillouin zone of the “normal C4v-BIC” structure (top), and the full band structure of the “normal Chv-BIC” structure (bottom), the inset showing the electric field of B" mode within a unit cell at the T point; (d) the electric field Ezof each mode in (a) with a unit cell at the T point according to various embodiments; and (e) the electric field Ezof each mode in (b) with a unit cell at the T point.

[0061] As illustrated, the Ei-irreducible photonic mode may display a remarkable flat-band feature throughout the entire region above the light cone, whereas the B' mode may feature a quadratic band dispersion. To investigate their disparities on mode localization within a miniaturized laser cavity, the fundamental modes of the bands with distinct dispersion behaviors may be calculated. FIG. 10C shows near-field profiles in finite-size cavities corresponding to the fundamental mode of Ei band (top), B' band (middle), and B" band (bottom) in FIG 10A, respectively, according to various embodiments The simulations (details to be illustrated below) are modeled strictly according to the actual device structure with finite size, incorporating a typical, classic unpumped absorption boundary (the outerunpattemed region), which also serves for wire bonding for pumping current injection. The electric field (|Ez|) associated with the Ei mode may be tightly confined to the center of the cavity, in stark contrast to the B' and B" modes, which may substantially leak out of the patterned region, forming a near-plane wavefront away from the interfaces. The effective inplane field confinement in the flat-band cavity may arise from the ultra-low group velocity with enhanced mode feedback.

[0062] All these three cases (i.e., Ei, B' and B") may demonstrate negligible mode extension on top of the photonic crystal slab attributed to the nonradiative nature of the BICs. The group indices associated with the bands featuring distinct dispersion are presented in FIG. 10D. FIG. 10D shows a plot of group index as a function of wavevector k illustrating group indices for different modes computed from the corresponding Bloch band structures according to various embodiments. The group index of the Ei band may exceed that of the B' and B" bands by approximately ten-fold as they deviate from the T point, which may give rise to an effective in-plane field confinement as demonstrated in FIG. 10C.

[0063] Further simulations focusing on bands with different dispersion behavior but within the same laser cavity (B1and El' mode) are detailed in FIG 10E. FIG. 10E shows (a) a plot of quality factor (Q factor) as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profiles (1 st order and 2nd order) of the B' band in a cavity with side length, L of 5a (strong side leakage exist due to the weak in-plane confinement) according to various embodiments; and (b) a plot of quality factor (Q factor) as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profiles of the Ef band in a cavity with side length, L of 5a (in-plane losses are significantly reduced but still inferior to Ei mode in such a small cavity because that the E band is not as “flat” as Ei band).

[0064] Further simulations based on a normal BIC in commonly used C4v-symmetric lattice structure as illustrated in FIG. 10B(c) are also performed. FIG. 10F shows (a) a plot of quality Q factor as a function of wave vector k illustrating the Q factor of the B” band (top), and a plot of group index and a function of wave vector k comparing the group index between the designed flat-band BIC mode (Ei mode) according to various embodiments and the normal C4v-BIC mode (B" mode) (bottom); and (b) near-field profiles in finite-size cavities corresponding to the fundamental mode of B" band The mode may present a non-localized feature (in-plane) with the electric field substantially leaking out of the cavity, because of its weak in-plane feedback.

[0065] The consistent results may collectively support the conclusion that a flat-band with minimal dispersion may indeed ensure effective in-plane confinement in miniaturized cavities.

[0066] The B and E2 band, characterized by linear dispersion, may be observed to exhibit an exceedingly low group index of ~5.2 near the T point, which suggests a weak in-plane feedback and limited localization effects. Direct examination of the effective index based on the band structure reveals a characteristic “zero-index” behavior. FIG. 10G shows (a) a plot of effective index as a function of angular frequency ro illustrating the effective index of the two bands with linear dispersion (i.e., B and E2 bands, the effective indices approximate to zero around the Dirac point) in the flat-band bound states in the continuum (BIC) according to various embodiments; (b) the electric field distribution of the flat-band BIC mode (Ei mode) according to various embodiments; (c - d) the extracted magnitude (top) and electric field distribution (bottom) of the two bands with quadratic dispersion (B' and E2') in the normal CTv- BIC cavity (the magnitude displays a gaussian shape and the fields are tightly confined inside the cavity); and (e) electric field distributions of another degenerated mode of E2' (E2) in the normal Cev-BIC cavity. Also, E2' mode with quadratic dispersion is tightly confined. While the E2 mode with linear dispersion shows a flat field distribution, lots of field components may be leaked out. The huge side leakage losses induced by the linear dispersion (large in-plane group velocity) may effectively increase the imaginary free spectral range (iFSR) between the flatband mode and other adjacent BIC modes, which may facilitate the realization of a single-mode operation laser.

[0067] Direct examination of the effective index based on the band structure reveals a characteristic “zero-index” behavior. These effective zero-index structures are renowned for their lossless propagating nature and the flat-envelope field distribution in previous research. To provide a more intuitive understanding of the electric field distribution resulting from this distinctive dispersion behavior, the fundamental modes within a larger cavity (L = 13a) may be examined.

[0068] FIG. 10H shows (top) a near-field profile of the fundamental mode belonging to the B band with linear dispersion (side lengths of cavity, L = 13a) according to various embodiments; and (bottom) plot of intensity (in arbitrary units or a.u.) as a function of Y-axis (in micrometers or pm) illustrating the extracted electric field intensity along the dashed line within the cavity according to various embodiments. FIG. 101 shows (top) a near-field profile of the fundamental mode belonging to the E2 band with linear dispersion (side lengths of cavity,L = 13a) according to various embodiments; and (bottom) plot of intensity (in arbitrary units or a.u.) as a function of X-axis (in micrometers or pm) illustrating the extracted electric field intensity along the dashed line within the cavity according to various embodiments. FIG. 10J shows (a) stimulated electric distribution of the B mode with air boundary condition according to various embodiments, and (b) a plot of intensity (in arbitrary units or a.u.) as a function of Y-axis (in micrometers or pm) illustrating the magnitude of the electric field along the dashed line shown in (a) according to various embodiments. The B mode may be well selected when the air boundary is introduced. The huge in-plane side leakage losses may still exist in the double-metal QCL case, where only traverse magnetic (TM) mode can be supported in the cavity. The uniform field profile across the entire cavity may display an unambiguous effective zero-index material feature, a typical characteristic of a Dirac-cone band structure. The results reveal that the electric field associated with the B modes may exhibit uniform distributions and extend beyond the cavity boundary (as depicted in FIG. 10H and FIG. 10J).

[0069] The substantial side leakage losses may result in a significantly lower Q factor of approximately ~39. FIG. 10K shows a comparison of the quality (Q) factors of B (B'), Ei, E2 (E2') modes with distinct band dispersion features according to various embodiments. A flatband BIC cavity with lattice parameters of a = 37.2 pm and ro = 14.5 pm is adopted here to exclude the influence of the “super-BIC” The unpattemed regions were set as lossy dielectrics with an imaginary part of 0.016, obtained by the realistic material absorption losses of = 20 cm-1for terahertz quantum cascade lasers. In contrast, the electric field distribution of the B' band, sharing the same Cevmode symmetry but featuring quadratic band dispersion, exhibits a typical Gaussian shape, with the mode being tightly confined within the cavity core (FIG. 10G(c)), leading to a relatively high Q factor of approximately -1100 (FIG. 10K). Comparisons of the mode distribution of the E2 and E2' band (FIG. 101 and FIG. 10G(d) and (e)) have also been conducted, and the results reveal a similar phenomenon. It is worth noting that the substantial side leakages of the neighboring B and E2 modes may significantly enhance the mode selectivity (notable Q contrast) between these two modes and the flat-band mode, facilitating the attainment of single-mode flat-band lasers, despite they are also nonradiative BIC modes.

[0070] The exceptional mode selectivity afforded by the Dirac-cone-like band design may exhibit notable robustness against the fabrication errors. FIG. 10L shows (a) angular frequency m as a function of wave vector k illustrating band structure of the Dirac-like band after takingan |Aro| = 0.5 pm air hole radius perturbation into consideration according to various embodiments (Arc represents the radius deviation from the initial design (12 pm) during the fabrication process); (b) a plot of group index as a function of wave vector k illustrating the group index of each band as shown in (a) according to various embodiments; (c) electric distributions of each band as shown in (a) according to various embodiments; and (d) a comparison of the quality (Q) factors of each band in (a) according to various embodiments. The results in FIG. 10L(b) and FIG. 10L(c) may be consistent with each other, i.e., a higher group index may induce a better in-plane mode confinement, and thus higher Q factor. In (a) of FIG. 10L, the photonic band structure may be calculated after taking in an air hole radius perturbation of |Aro| = 0.5 pm. The results show that despite the triple intercept point (at T = 0) being not a strictly stable point in the parameter space, the effect afforded by the Dirac-like design may possess acceptable robustness. The high group index of the Ei modes and low group index of the B and E2 modes are barely changed in FIG. 10L(b). It may also be noted that the destruction of the linear dispersion feature of B and E2 modes would cause a slight group index enhancement at the vicinity of the T = 0. While the enhancement only occurs near the T point, the flat-envelope mode profile with large side-leakages in a finite-size cavity may not be impacted, as shown in FIG. 10L(c). Thus, the excellent Q contrast between the flat-band mode (Ei) and the other two bands with near-linear dispersion (B and E2 modes) can be maintained, as shown in FIG. 10L(d).

[0071] Super- / Multi-BIC enabled Q factor enhancement

[0072] The significance of the super-BIC in the pursuit of ultra-compact lasers should be highlighted. Normally, two main issues introduce additional radiation losses in a finite-size cavity. First is the splitting of the modes into discrete points away from the T point due to band quantization, where the Q factor is dramatically decreased. Second, the BIC modes are mixed with the off-T low-Q modes because of the out-off plane scattering losses arising from fabrication imperfections and lattice disorders. Nevertheless, in the flat-band super-BIC cases, the construction of a series of accidental BICs around the T point may serve as an effective strategy to suppress the radiation losses induced by scattering and finite-size effects.

[0073] FIG. 11 A shows (a) a plot of lattice constant ci (in micrometers or pm) as a function of radius ro (in micrometers or pm) illustrating the tuning of lattice parameter to fonn a tuning of the lattice parameters to ensure a Dirac-like band feature according to various embodiments; and (b) plots of quality (Q) factor as a function of wave vector k illustrating the calculated Qfactors for the lattice with parameters of a = 28.8 pm, ro = 9.5 pm (left), a = 33 pm, ro = 12 pm (middle), and a = 37.2 pm, ro = 14.5 pm (right) (ra is set to 2 pm) according to various embodiments. FIG. HA(a) shows the super- or multi -BICs may be constructed by fine tuning the lattice constant a. The Q factors may be calculated in k space for the lattices with different parameters in FIG. HA(b). The results illustrate the emergence and disappearance of the accidental BICs at off-T point with the adjustment of lattice structure, accompanied with a significant Q factor enhancement in the situation with multi-BIC states (middle). Considering the high-Q factor and the gain range of the QCL wafer, the final parameter for achieving multi- BIC as shown in FIG. 11 A may be a = 33 pm, ro = 12 pm and ra = 2 pm. The values of a, ro and / or a for achieving multi-BIC may be obtained by sweeping through these parameters.

[0074] FIG. 1 IB shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating a two-dimensional (2D) map of the Q factor and polarization distribution of the flat-band with super bound states in continuum (super-BIC) cavity states in momentum space according to various embodiments. The color map represents the Q factor and the white dashed line represents the polarization vector. The topological charge was determined to be q = +2 by selecting a circle centered around the T point with |A| = 0.1 (2ji / 3 / a), as detailed below.

[0075] Q factor calculations are performed for flat-band cavities with and without super- BIC / multi-BIC feature. To control variables, both cavities share basically identical band features and group indices. FIG. 12A shows a plot of frequency (in terahertz or THz) as a function of wave vector k illustrating the calculated band diagram of a flat-band cavity with super bound states in continuum (super-BIC) (a = 33 pm, ro = 12 pm) and without super-BIC (a = 37.2 pm, ro = 14.5 pm) according to various embodiments, ra may be set to 2 pm for both cases. FIG. 12B shows a plot of group index as a function of wave vector k illustrating the calculated group index of the flat -band shown in FIG 12A with super bound states in continuum (super-BIC) and without super-BIC according to various embodiments. FIG. 12C shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating two-dimensional (2D) mapping of the quality (Q) factor for the flat-band without super bound states in continuum (super-BIC), the map is darker at off-T point than the flatband with super-BIC states (shown in FIG. 1 IB) according to various embodiments, which means a significant lower quality (Q) factor in a finite-size cavity. FIG. 12D shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated Qfactors and mode profile of the flat-band in a cavity without super bound states in continuum (super-BIC) according to various embodiments. FIG 12E shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated Q factors and mode profile of the flat-band in a cavity with super bound states in continuum (super-BIC) according to various embodiments. FIG. 12F shows a plot of wave vector along x axis kxas a function of wave vector along y axis kyillustrating polarization states of the flat-band mode in k space, from which a topological charge of q = +2 can be obtained according to various embodiments. FIG. 12G shows a plot of quality (Q) factor as a function of cavity area a2illustrating Q factor comparison between flat-band super bound states in continuum (super-BIC) and normal C4v- BIC modes in cavities with different areas according to various embodiments. The introduction of super-BIC may barely change the field profile because of the similar band dispersion behavior, but may significantly enhance the Q factor from -367 to -1440 in a small cavity with a side length of L = 5a.

[0076] The only distinction may lie in the intentional construction of unremarkable points featuring divergent Q factors at k = ±0.36 (2nN3 / a) accompanied by a larger polarization singularity core within the flat-band multi-BIC cavity

[0077] These modifications may establish a broad high-Q region in the Brillouin zone, endowing the cavity with greater tolerance to systematic disorders, structural imperfections, and photonic finite-size layouts, which may be highly desired in integrated photonics. FIG. 13A shows a plot of quality (Q) factor as a function of w avevector k illustrating the calculated Q factor of the flat-band cavities with and without super bound states in continuum (super- BIC) according to various embodiments. FIG. 13B shows (a) a schematic of an asymmetric hole acting as a fabrication imperfection (Ar, Ax, and Ay is set as 0.5 gm based on the actual fabrication accuracy) according to various embodiments, and (b) a plot of quality (Q) factor as a function of wave vector k illustrating simulated Q factors of the unit cell with disorder according to various embodiments. The Q factor of the flat-band mode with super-BIC states may be found to be robust across a range of k values.

[0078] The QCL wafer utilized may have a quite thick active layer, e.g., -18.3 gm. Thus, the deep etching process may be divided into several steps to facilitate checking of the etching progress, which may introduce undulating sidewall profile in the vertical dimension in some cases. To figure out the impact of the undulating sidewall on the super-band BIC mode, simulation within finite-size cavity may be conducted to figure out the impact of the undulatingsidewall on the optical properties in real devices. The envelope of the undulating sidewall was simulated by introducing a two-period sinusoidal function disturbance with amplitude of 0.4 pm. FIG. 13C shows (a) schematics of vertical sidewall (top) and undulating sidewall (bottom) profiles used in simulation according to various embodiments; (b) electric field distributions of the super bound states in continuum (super-BIC) modes with the vertical sidewall (top) and undulating sidewall (bottom) according to various embodiments; (c) beam profiles of the super bound states in continuum (super-BIC) modes with the vertical sidewall (top) and undulating sidewall (bottom) according to various embodiments; and (d) a comparison of the quality (Q) factor values for the flat-band mode with the two types of sidewalls according to various embodiments. The cavity is L = 5a. The electric field distribution (b), the beam shape (c), and the Q factors (d) as illustrated in FIG 13B may basically be the same

[0079] Thus, it may be concluded that the undulating sidewall profile may have no significant impact on the quality factor and optical properties of the super-band BIC mode, as the photonic lattice symmetry remains undamaged.

[0080] As a result, the Q factors of the flat-band super-BIC cavities may exhibit remarkable enhancement. Taking the cavity with a side length of L = 5a for instance, the Q factor can be as high as -1440. In contrast, the Q factors for the flat-band cavity without super-BIC, the “normal C&V-BIC cavity” (B' mode), and the “normal C4v-BIC cavity” (B" mode) may only reach values of -368, -60, -80, respectively, and may be submerged within bulk modes. FIG. 13D shows a plot of quality (Q) factor as a function of cavity area a2illustrating the calculated Q factors for the fundamental modes of the flat-band cavities with and without super bound states in continuum (super-BIC) mode, along the cavities with the “normal Cr, -BIC cavity” (B' mode), and the “normal C4V-BIC cavity” (B" mode) according to various embodiments. Here, the calculations are based on actual cavity structures with finite cavity sizes and double-metal device, incorporating a double-metal device configuration with consideration of the corresponding cavity losses. The solid lines are plotted as reference guides.

[0081] It should be noticed that the Q factor of flat-band cavity without super-BIC may still be approximately 5 or 10 times higher than the normal BIC mode (B1and B" mode), benefiting from the enhanced group index, even though they have similar Q factors in an infinite size cavity. The flat-band super-BIC design according to various embodiments may also outperform previous optimized cavity designs with similar device configuration (FIG. 4D).

[0082] FIG. 13E compares the calculated quality (Q) factors of L = 5a devices with titanium (Ti) / gold (Au) (Drude Model) cladding according to various embodiments. The Ti / Au layer may be modeled as lossy metal with refractive index 182.67 + 212. Hi. The thickness of the Ti / Au layer is -350 nm. The Q factor of the flat-band multi -BIC cavity is calculated to be around -170, which may still be significantly higher than the normal BIC cavity (-32) with the same cavity size.

[0083] Measurements of flat-band super- / multi-BIC lasers

[0084] Benefitting from the exceptional in-plane and out-of-plane field confinement offered by the flat-band multi-BIC mode as discussed, we successfully fabricated compact QCL devices with high performance. FIG. 14A shows a scanning electron microscopy (SEM) image of a fabricated / . = 5a laser cavity with daisy-like air holes according to various embodiments. FIG. 14B shows a magnified scanning electron microscopy (SEM) image of the fabricated laser cavity shown in FIG. 14A. according to various embodiments. As presented in FIGS. 14A - B, the lateral size of the L = 5a cavity is just 294 pm (-3 ), with the daisy -like air hole patterns imprinted onto the QCL slab. The surrounding absorption boundary (unpattemed) may experimentally be realized by inserting a thin SiO2 insulating layer between the metal and QCL medium layers (FIG. 6). The laser was characterized by using a customized setup (details below).

[0085] FIG. 14C shows a plot of voltage (in volts or V) / peak power (in milli-Watts or mW) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the fabricated laser cavities according to various embodiments. FIG. 14D shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the fabricated laser cavity L = 5a according to various embodiments. The light-current-voltage (L-I-V) curves (FIGS. 14C - D) demonstrate a distinct lasing threshold of approximately 0.18 or 0.19 kA / cm2, attributed to the enhanced Q factor that stems from the collective effect of the flat-band with high group-index and the super-BIC with a broad high-Q core.

[0086] The ultra-compact device footprint as well as the low lasing threshold may lead to a small pumping current, which is essential for pursuing low power consumption, low thermal generation THz QCLs. FIG. 14E shows a plot of intensity (in arbitrary units or a.u.) as a function of current (in amperes or A) illustrating the light-current (L - I) curve in a L = 5adevice according to various embodiments. Due to intrinsic current spreading in QCLs, the experimentally measured pumping current may be higher than anticipated. To further reduce power consumption, the device may be optimized by removing the highly doped layer in the unpattemed region to minimize current spreading, achieving a minimum current of —125 mA.

[0087] Moreover, distinct single-mode lasing emission can be identified across the entire dynamic range. FIG. 14F shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the laser cavity according to various embodiments at various pumping current densities. FIG. 14G shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the L = 5a laser cavity according to various embodiments at different frequencies and pumping current densities. The single mode performance is evaluated by calculating the side-mode suppression ratio (SMSR) of the device at a pumping current density of 0.27 kA / cm2, where the SMSR can reach up to ~20 dB. FIG. 14H shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.27 kA / cm2of the L = 5a laser cavity with a measured side-mode suppression ratio (SMSR) ~20 dB according to various embodiments. It is worth highlighting that this flat-band super-BIC QCL may represent the most compact single-mode photonic crystal laser. Additionally, the substantial group index contrast between the Ei band and the adjacent B (Ei) band may effectively increase the imaginary free spectral range (iFSR), thus facilitating single-mode operation in larger cavities.

[0088] Full-wave-simulations may be conducted to assess the mode distribution and Q factor for cavities with side lengths of L = 9a and L = 17a. FIG. 141 shows (a) a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode provide of a laser cavity with side length L = 9a according to various embodiments; and (b) a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode provide of a laser cavity with side length L = 17a according to various embodiments. The electric fields may be tightly confined within the cavities with Q factors of approximately l.Ox lO4and 5.7><104, respectively.

[0089] FIG. 14J shows a scanning electron microscopy (SEM) image of a fabricated L = 9a laser cavity with daisy -like air holes according to various embodiments. FIG. 14K shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the A = 9a lasercavity according to various embodiments at different frequencies and pumping current densities. FIG. 14L shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.34 kA / cm2of the L = 9a laser cavity with a measured side-mode suppression ratio (SMSR) ~24 dB according to various embodiments

[0090] FIG. 14M shows a scanning electron microscopy (SEM) image of a fabricated I. = 17a laser cavity with daisy -like air holes according to various embodiments. FIG. 14N shows a plot of current density (in kilo-Amperes per square centimeter or kA / cm2) as a function of frequency (in terahertz or THz) illustrating a two-dimensional (2D) spectra mapping of the L = 17a laser cavity according to various embodiments at different frequencies and pumping current densities. FIG. 140 shows a plot of signal (in decibels or dB) as a function of frequency (in terahertz or THz) illustrating the lasing spectrum at 0.25 kA / cm2of the A = 17a laser cavity with a measured side-mode suppression ratio (SMSR) ~26 dB according to various embodiments. Stable single-mode lasing emission can also be observed when continuously increasing the pumping current densities, with a measured SMSR of ~23 dB or ~24 dB and -25 or -26 dB for cavities with side lengths of L = 9a and L = Yla, respectively.

[0091] The far-field pattern of the designed flat-band multi-BIC laser was collected using a customized intensity scanner equipped with a THz Golay cell detector. FIG. 15A shows a schematic of a setup for focusing the diverged beam of the quantum cascade lasers (QCLs) according to various embodiments. The power of the QCLs were measured by a terahertz powermeter (Gentec-EO T-Rad with detector head of THZ9B-BL-DZ-D0). The THz detector has a collection aperture of 0.9 mm in diameter. To improve the collection efficiency, the laser emission was focused by two parabolic mirrors. The measurement was performed in a dry nitrogen purging circumstance. A correction factor of 2.1 was considered for this powermeter when calculating the peak power.

[0092] Due to the sensitivity limitations of the detector, the = 9a device with a peak power of 1.02 mW was characterized here. As a baseline, a typical ridge laser was first characterized. FIG. 15B shows a plot of power (in micro-Watts or pW) as a function of time (in seconds or s) illustrating characterization of a typical ridge laser. The ridge laser has a similar cavity area (0.14 mm2) as the L = 9a laser cavity (-0.11 mm2). The ridge length and width are 1000 pm and 140 pm, respectively. The measured average power is 1.87 pW, corresponding to a peak power of 0.78 mW.

[0093] FIG. 15C shows a plot of power (in micro-Watts or pW) as a function of time (in seconds or s) illustrating characterization of a L = 9a laser cavity according to various embodiments. The average power of the designed flat-band multi-BIC laser with similar effective pumping area (~0.l l mm2) was calculated to be 2.44 pW corresponding to a peak power of 1.02 mW. The measured flat-band multi-BIC laser may present a higher output power than the ridge laser because of its high collection efficiency brought by the good directivity.

[0094] The results show that the power of the designed laser, which is aimed at good mode confinement and a low current threshold, may indeed not be high. The primary limitation may lie in the quality of the QCL wafer used.

[0095] In addition, it may be noted that for higher output power application, the flat-band cavity without multi-BIC states, which still maintains a Q factor of -367 (compared to -60 for traditional band-edge modes with quadratic dispersion, which are unable to lase in such a small cavity), could be adopted. The flat-band feature only enhances the in-plane mode confinement without compromising the power out-coupling efficiency.

[0096] The radiative out-coupling efficiency of the flat-band cavities is calculated, which may be proportional to Qtotai / Qverticai, where Qtotai = (1 / Qverticai + l / Qin-piane)'1FIG. 15D shows a table comparing the radiative efficiency of the flat-band cavity design according to various embodiments with other photonic crystal -based cavities.

[0097] Qverticai can be derived from the equation:2irfQverticai—“Yrad where yrad represent the surface emission loss rate following the equation: H) - n dS+ pH ■ H) dV where Pswf represents the power dissipation due to surface emission, and can be estimated by extracting the time-averaged integrated power flow through an air domain upon the resonator, Etot is the total electromagnetic energy stored in the resonator, E and H represent the electric and magnetic fields respectively, e the permittivity and p the permeability of the material.

[0098] For the flat-band multi-BIC cavity (Z = 5a), the photon loss rate is calculated to be Yrad = 4.23 GHz, corresponding to a Qverticai - 4601. Considering the Qtotai - 1440 from the 3D simulation, an out-coupling efficiency of i]r= 31.3% can be obtained. As a comparison, theflat-band without multi-BIC design has a Qtotai ~ 367 and Qvenicai - 450, corresponding to a significantly higher out-coupling efficiency of / ,- = 81 6%.

[0099] Furthermore, when the absorption is considered, the overall Q factor would be Qtotai = ( 1 / Qvcrticai + 1 / Qin-pianc + 1 / Qabs)'1. We take the absorption loss ctabs = 20 cm'1(corresponding to a Q value of -117), the calculated radiative efficiency for the flat -band cavity with and without multi-BIC would be 2.3% and 20.2%, respectively.

[0100] Numerical calculations indicate that the cavity mode features a cylindrical vector beam profile with good directionality and a doughnut-shape far-field pattern. FIG. 16A shows a simulated beam profile and far-field pattern of a laser beam of the designed super bound states in continuum (super-BIC) according to various embodiments. Such a beam profile with a polarization winding number of |q| = 2 can be classified as a type of vector beam.

[0101] Experimentally, the far-field image was obtained by collecting the beam intensity with a THz Golay cell detector (TYDEX GC-1T). The Golay cell was mounted on a mechanical arm with a scan step of 1 mm. The detector was approximately 5 cm away from the QCL devices and has an aperture diameter of -5 mm. FIG. 16B shows (a) a schematic illustrating measuring far-field image of the as-prepared flat-band laser without a focal lens according to various embodiments; (b) the measured far-field image generated by the setup in (a) according to various embodiments; and (c) a plot of intensity (in arbitrary units or a.u.) as a function of x position (in millimeters or mm) illustrating the intensity distribution of the far-field image in (b) according to various embodiments. The measured far-field image shows a six-fold symmetric beam profile together with an intensity singularity at the center and a beam divergency approximately -15°, which matches well with the simulated far-field pattern. The asymmetric intensity distribution may come from the tilt of the test system.

[0102] A better result may be obtained with a focal lens. FIG. 16C shows (a) a schematic illustrating measuring far-field image of the as-prepared flat-band laser with a focal lens according to various embodiments; (b) the measured far-field image generated by the setup in (a) according to various embodiments; and (c) a plot of intensity (in arbitrary units or a.u.) as a function of x position (in millimeters or mm) illustrating the intensity distribution of the far- field image in (b) according to various embodiments. The outer pattern with circular shape may come from the scattering at the edge of the focal lens.

[0103] The experimentally obtained ncar-G, symmetric intensity profiles as shown in FIGS. 16B - C may be in good agreement with the simulation results, and the beam divergence may approximately be 15°.

[0104] FIG. 17A shows a plot of voltage (in volts or V) / peak power (in milli-Watts or mW) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-l-V) curves of a ridge laser. The double-metal ridge laser is measured to be 1000 pm in length, 140 pm in width and 18.3 pm in height. The L-I-V curves of the ridge laser demonstrates a lasing threshold current density of -233 A / cm2, which is -25% higher than that of the designed flat-band multi -BIC laser (-187 A / cm2). The maximum lasing temperature of designed flat-band multi -BIC laser (-89 K) is also higher than the double metal ridge laser (-70 K).

[0105] In the electrical pumping QCL system with inter-subband transitions, a certain part of voltage is required to establish correct band alignment. Thus, at the beginning, a portion of the injection current is actually due to leakage before the alignment of the structure is reached. After the correct band alignment and population inversion are established, the raised gain (g) will compensate the cavity losses, i.e., arand am. When the gain is equal to the total losses, g = cttot, the lasing action occurs. The current leakage and the absorption aniare two main intrinsic processes in QCLs. The flat-band multi-BIC design according to various embodiments may focus on the arpart, which may play an important role in a compact laser cavity based on traditional band edge or normal BIC mode (FIG. 4D). The radiative losses atin QCL can be written as,where n represents the refractive index, X is the working wavelength, and Q is the quality factor.

[0106] The calculated radiative loss of the flat-band multi-BIC mode (Q = 1440) is -1.6 cm'1, which may be much lower than the Fabry-Perot (F-P) waveguide mode of -7.7 cm'1and traditional band edge mode with normal CG-BIC state (-62.3 cm'1, corresponding to the Q factor of -37). FIG 17B shows a simulated Fabry-Perot (F-P) cavity mode cavity Benefiting from the excellent mode confinement in the flat-band multi-BIC cavity by suppressing the out- of-plane (radiative and scattering) losses and in-plane side leakages, various embodiments may achieve the reduced lasing threshold. FIG. 17C shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square31centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the ridge laser at various temperatures FIG. 17D shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating the light-current-voltage (L-I-V) curves of the super bound states in continuum (super-BIC) laser according to various embodiments at various temperatures.

[0107] Control devices based on an optimized photonic bandgap confined “normal C4v- BIC” design (B" mode) may be fabricated using the same QCL wafer. FIG. 18A shows a schematic of a photonic bandgap (shell)-confmed bound states in continuum (BIC) mode (core) device. FIG. 18B shows a plot of frequency (in terahertz or THz) as a function of wave vector k illustrating the photonic band diagram of the Qv-symmetry photonic crystals with a = 28 pm and / "core (r shell) = 10 pm (9 pm). FIG. 18C shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating calculated Q factors and mode profile of the photonic bandgap-confined bound states in continuum (BIC) cavity. FIG. 18D shows a scanning electron microscopy (SEM) image of the fabricated core-shell hetero-structure with rcore (10 pm) > / 'shell (9 pm). Only the core region (inside the white box) is selectively pumped. The overall cavity size is ~ 5 . FIG. 18E shows a plot of voltage (in volts or V) / intensity (in arbitrary units or a.u.) as a function of current density (in kilo-Amperes per square centimeter or kA / cm2) illustrating light-current-voltage (L-I-V) curves of the core-shell cavity with a lasing threshold of -0.28 kA / cm2. FIG. 18F shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the core-shell device. Single-mode lasing can be obtained because of the higher Q factor of the BIC mode than the surrounding bulk modes as illustrated in FIG. 18C. Notably, the observed Q factor is significantly lower (-265) than the flat-band multi-BIC mode, resulting in an enhanced lasing threshold of -0.28 kA / cm2, despite the overall side length being larger (-5X).

[0108] Moreover, the device based on the “normal C4v-BIC” design, without any optimization, can only operate with multi-modes. FIG. 18G shows a scanning electron microscopy (SEM) image of the traditional “normal C4V-B1C” cavity without optimized coreshell structure, where rcore= rsheii (10 pm). FIG. 18H shows a plot of quality (Q) factor as a function of frequency (in terahertz or THz) illustrating the calculated quality (Q) factors and mode profile of the traditional bound states in continuum (BIC) cavity. FIG. 181 shows a plot of intensity (in arbitrary units or a.u.) as a function of frequency (in terahertz or THz) illustrating the lasing spectra of the device based on traditional continuum (BIC) cavity. Multi-mode lasing can be observed because the BIC mode is submerged in the bulk modes as shown in FIG. 18H.

[0109] Methods

[0110] Numerical Calculation

[0111] A commercial finite element solver COMSOL Multiphysics software was used to perform the simulations in this work. The laser cavity includes a hexagonal lattice of daisy-like shaped holes drilled through the 18.3-pm-thick active medium of a THz QCL wafer. The three- dimensional band structures, eigenmodes, and Q factors of the photonic crystal structures were calculated by using the ‘eigenfrequency’ module in the frequency domain with periodic condition imposed in the x-y plane. The QCL active layer was modelled as a lossless dielectric with a refractive index of 3.6, the top and bottom boundaries of the dielectric medium were set as perfect electric conductor layers to extract the TM mode. For the full structure simulations with finite cavity size, scattering boundary conditions were used to make a boundary transparent and avoid back reflection for the outgoing waves. The surrounding unpumped region was realized by setting an imaginary refractive index of 0.016 in the simulations.

[0112] The topological charges (q) of the symmetry-protected BIC mode in FIG 11B is calculated to be +2. As determined by the winding number of the polarization vectors,where C is the closed path with a counterclockwise direction surrounding a BIC. (k) — arg [cx(k) + icy(k) ] represents the polarization angle at each k, and cxand cyare the coefficients of far-field radiation with electric fields along the x and y directions, respectively.

[0113] Device fabrication

[0114] The flat-band super-BIC lasers were designed on a broadband three-well resonantphonon GaAs / Alo.uGao.xsAs THz QCL wafer with a metal-semiconductor-metal device configuration The thickness of the QCL active region was measured to be around 18.3 pm The calculated gain bandwidth of the wafer is around 2.8-3.3 THz, with a refractive index of n = 3.6.

[0115] Device characterization

[0116] The fabricated devices were placed in a helium-gap-steam cryostat system with a temperature of ~8.6 K. For the spectra measurements, the lasers were pumped by an electrical pulse generator (repetition rate of 10 kHz and pulse width of 500 ns). The laser spectra werecollected by a Bruker Vertex 80 Fourier-transform infrared (FTIR) spectrometer with a roomtemperature deuterated-triglycine sulfate (DTGS) detector The spectra resolution is 0.08 cm’ i

[0117] Various embodiments may relate to a flat-band super-BIC cavity that can effectively suppress the side leakage and the radiation loss originating from the finite-size effects, as well as the out-of-plane scattering due to fabrication imperfections. The remarkable electric filed confinement may lead to a highly compact (~3 ) electrical pumping QCL device with an ultralow lasing threshold (~0. 18 kA / cm2or ~0. 19 kA / cm2) and excellent single-mode performance (~20 dB). Considering the laser cavity relies solely on dielectric constant modulations, various embodiments may readily be extended to other wavelength regimes, e.g., the near-infrared (IR) and the visible regions. Various embodiments may represent an effective approach in pursuing energy-efficient and monolithic integrated ultra-compact laser sources with low laser thresholds, which are highly desired for on-chip applications in advanced optoelectronics and integrated photonics. Various embodiments may not be limited to laser applications, but may also be extended to the realms such as sensing, modulation, and harmonic signal generation.

Claims

Claims1. A laser cavity comprising: a substrate; a contact layer in contact with the substrate; an active layer in contact with the contact layer, the active layer comprising: a lattice comprising a plurality of hexagonal cells; and a plurality of daisy-like holes, each of the plurality of daisy-like holes comprising six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC), a cladding layer on a portion of the active layer; a first electrode; and a second electrode such that the substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.

2. The laser cavity according to claim 1 , wherein the lattice comprises alternating layers of quantum barriers and quantum wells.

3. The laser cavity according to claim 2, wherein the quantum barriers are formed by aluminum gallium arsenide; and wherein the quantum wells are formed by gallium arsenide.

4. The laser cavity according to any one of claims 1 to 3, wherein the plurality of daisy-like holes extends through the active layer.

5. The laser cavity according to claim 4, wherein the plurality of daisy -like holes also extends through the first electrode.

6. The laser cavity according to any one of claims 1 to 5, wherein the flat-band bound states in continuum (BIC) are achieved by a flat Ei band intersecting a E2 band at an apex of the E2 band, and a B band at an apex of the B band.

7. The laser cavity according to any one of claims 1 to 6 wherein the laser cavity achieves flat-band super bound states in continuum (super-BIC).

8. The laser cavity according to claim 7, wherein flat-band superbound states in the continuum (super-BIC) are achieved at predetermined lattice and hole parameters.

9. The laser cavity according to claim 7 or claim 8, wherein the laser cavity achieves a quality factor (Q-factor) of 1440.

10. The laser cavity according to any one of claims 1 to 9, wherein the laser cavity is a terahertz laser cavity.

11. A method of forming a laser cavity, the method comprising: forming a contact layer in contact with a substrate; forming an active layer in contact with the contact layer, the active layer comprising: a lattice comprising a plurality of hexagonal cells; and a plurality of daisy-like holes, each of the plurality of daisy-like holes comprising six air bulges aligned substantially with corners of a respective hexagonal cell of the plurality of hexagonal cells such that the laser cavity achieves flat-band bound states in continuum (BIC); forming a cladding layer on a portion of the active layer; forming a first electrode; andforming a second electrode such that the substrate, the contact layer, the cladding layer and the active layer are between the first electrode and the second electrode.

12. The method according to claim 11, wherein a continuous semiconductor layer is formed on or over another substrate; wherein the contact layer is formed by: depositing one or more suitable metals on the substrate; depositing the one or more suitable metals on the continuous semiconductor layer on the other substrate; and bonding the substrate and the other substrate using the one or more suitable metals in a wafer bonding process; and wherein the other substrate is separated from the continuous semiconductor layer after the wafer bonding process.

13. The method according to claim 12, wherein the cladding layer is formed by: depositing a suitable cladding material on the continuous semiconductor layer; and removing a portion of the deposited cladding material using lithography and etching.

14. The method according to claim 13, wherein the first electrode is formed by: depositing a suitable photoresist material over the cladding layer and the continuous semiconductor layer; patterning the suitable photoresist material to form a plurality of photoresist structures; depositing one or more suitable materials over the photoresist structures, the cladding layer and the continuous semiconductor layer;removing portions of the one or more suitable materials deposited on the photoresist structures in a lift off process; and etching remaining portions of the one or more suitable materials deposited on the cladding layer and the continuous semiconductor layer to form the first electrode.

15. The method according to claim 14, wherein the active layer is formed by etching the continuous semiconductor layer using the remaining portions of the one or more suitable materials as a mask.

16. The method according to claim 15, wherein the substrate is polished before the second electrode is formed in contact with the substrate using a back electrode deposition process.

17. The method according to any one of claims 11 to 16, wherein the flat-band bound states in continuum (BIC) are achieved by a flat Ei band intersecting a E2 band at an apex of the E2 band, and a B band at an apex of the B band.

18. The method according to any one of claims 11 to 17, wherein the laser cavity achieves flat-band super bound states in continuum (super-BIC).

19. The method according to claim 18, wherein flat-band super bound states in the continuum (super-BIC) are achieved at predetermined lattice and hole parameters.

20. The method according to any one of claims 11 to 19, wherein the laser cavity is a terahertz laser cavity.

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