1. INTRODUCTION

Whispering-gallery mode (WGM) microcavities, which confine light by total internal reflection with the advantages of high quality (Q) factors and small mode volumes, have attracted great attention in the research of fundamental physics and optoelectronics applications ^[1]. Circular microresonators in different geometries were successfully used in the demonstration of ultralow-threshold microlasers, but the rotational symmetry causes a significant difficulty to efficiently collect the emission light from the microlasers. Deformed microresonators such as spiral ^[2], limaçon ^[3], quadrupole ^[4], flattened quadrupole ^[5], and ellipse-shaped ^[6] have been proposed to break rotational symmetry for realizing directional emission. By directly connecting an output waveguide to the microresonators, directional emission could also be realized ^{[79" target="_self" style="display: inline;">–9]}. Recently, circular-side polygonal microresonators were also proposed and demonstrated for Q factor control and mode selection ^[10,11].

Besides the directional emission and Q factor control, dual-wavelength laser sources that could be used for microwave or terahertz generation have also attracted great attention. There are several ways of producing dual lasing sources, such as Fabry–Perot (FP) cavities integrated by Y junction ^[12], dual-section vertical-cavity surface-emitting lasers ^[13,14], dual-section distributed feedback (DFB) lasers ^[15], fiber lasers based on fiber Bragg gratings ^[16,17], and square microcavity lasers ^[10,18]. In this paper, we propose and numerically investigate a deformed optical variable curvature microresonator (VCM) with a mixture shape of microsquare and microdisk resonators for stable dual-mode lasing. This VCM can support two types of high Q factor modes, the four-bounce mode and the circular-like mode. By carefully modifying perimeter distribution and waveguide position, stable dual-mode lasing with directional emission can be achieved according to the numerical simulation results.

This paper is organized as follows. In Section 2, the mode characteristics of the VCM are simulated based on two-dimensional (2D) finite element method (FEM). A ray dynamics analysis is also performed for the high Q factor modes. In Section 3, dual-mode stability is analyzed for the VCM and circular and square microresonators theoretically. The modulation of the mode wavelengths and quality factors, based on graphic electrode and waveguide positions, are investigated numerically. Finally, the summary is given in Section 4.

2. MODE CHARACTERISTICS

The schematic diagram of a 2D VCM is given in Fig. 1. There are four zero-curvature points distributed in the midpoints of square edge, and the connecting arcs between these points have a linear variable curvature. The curvature of the arc in the first quadrant can be expressed as where ρ, r, and θ are the curvature, polar radius, and angle, respectively, and k is a parameter showing the changing speed of curvature along the boundary. The microresonator size is determined by the parameter d, which is half the distance between the two opposite zero-curvature points. The curvature in the other quadrants is a periodic function as ρ(θ−π/2)=ρ(θ).

Fig. 1. Schematic diagram of a 2D VCM. The curvature is linearly changed as the boundary starting from zero curvature points to the neighboring maximum-curvature points.

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The VCM in different sizes could be proved similar to each other with the product of maximum curvature ρmax and VCM size d equaling 19.82. The deformation degree ϵ is defined as ϵ=(rmax−rmin)/rmin, which is a fixed number of 0.065 for the VCM. With a continuous variation of the curvature along the boundary, the boundary is smooth and second-order differentiable, leading to the emergence of the circular-like mode despite large deformation compared with the circular resonator. The same symmetry properties between the VCM and microsquare resonators also herald the same periodic orbits of optical rays.

The mode characteristics of the VCM are simulated by 2D FEM (COMSOL Multiphysics 5.0). The refractive indices of the resonator and outside media are set to 3.2 and 1.54, respectively, for the InP-based VCM and surrounded bisbenzo cyclobutene. The microresonator size is 4 μm. A perfect matched layer absorbing boundary with a width of 0.5 μm is used to terminate the simulation area. The intensity spectrum for TE modes with a wavelength resolution of 0.08 nm is obtained and plotted in Fig. 2(a). The spectrum shows that there are three sets of longitudinal modes, ranging from 1.50 to 1.59 μm. The four kinds of high-Q modes marked by circles, squares, five-pointed stars, and prisms represent the circular-like mode, fundamental four-bounce mode, first-order four-bounce mode, and high-order hybrid mode, respectively, with the field distributions of |Hz| as presented in Figs. 2(b)–2(e). The circular-like modes have relatively higher Q factors up to 4.34×105 with the field distributing mostly along the boundary, similar to the WGMs in a circular microresonator. The reflection positions of the four-bounce modes concentrate on four zero-curvature points with reflection angles of around 45°. The photons of the four-bounce modes are more likely to reach leaky regions, leading to lower-mode Q factors in the order of 104. The longitudinal mode intervals around 1.55 μm are 29 and 33 nm for the circular-like modes and the fundamental four-bounce modes, respectively. The first-order four-bounce modes and high-order hybrid modes have mode Q factors in the order of 103. The details of mode wavelengths and Q factors are summarized in Table 1.

Fig. 2. (a) Mode intensity spectrum and mode field patterns |Hz| of the (b) circular-like mode, (c) fundamental four-bounce mode, (d) first-order four-bounce mode, and (e) high-order hybrid mode.

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The VCM could be regarded as a mixture of circular and square shapes. For the comparison between these three types of microresonators, we simulated the microresonator with gradually changed shapes from VCMs to microcircular and microsquare resonators separately. The middle states of the microresonator are defined as where α is the variation degree, varying from 0 to 1. rdef, rVCM, and rcircular(square) are the radii of deformed, VCM, and circular (square) resonators, respectively. The areas of the VCM, microcircular, and microsquare resonators remain unchanged.

As shown in Fig. 3(a), when the VCM is transformed into a microcircular resonator, the wavelength of the circular-like mode decreases from 1.5372 to 1.5275 μm, the Q factor increases from 1.68×105 to 6.73×1013, and the mode pattern gradually transforms to the fundamental WGM TE97,1 in the microcircular resonator with a radius of 4.15 μm. As a comparison, the wavelength of the fundamental four-bounce mode increases from 1.5428 to 1.5806 μm, the Q factor increases from 1.51×104 to 1.31×108, and the mode pattern gradually transforms to the second-order WGM TE74,3. The evolutions of mode wavelengths and Q factors are plotted in Fig. 3(b) for transformation from the VCM to microsquare resonator. The Q factor of the circular-like mode decreases significantly as the corners of the VCM become sharper. Finally, the circular-like mode vanishes when α exceeds 10%. As for the fundamental four-bounce mode, the wavelength decreases from 1.5428 to 1.4286 μm. The mode pattern gradually transforms to TE22,23, with a low Q factor of 980 in the square microresonator with a side length of 7.36 μm. The low Q factor is caused by strong scattering loss induced by the symmetric mode field distribution relative to two diagonal lines.

Fig. 3. Q factors and wavelengths for deformed resonator from VCM to (a) microcircular resonator and (b) microsquare resonator.

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The ray dynamics are also simulated for the VCM. Due to the boundary being defined by an equation with differential and integral terms, we use dense points to represent the VCM in the numerical simulation. χ is the incident ray angle, S is the distance from the leftmost point along the boundary for the clockwise direction, and Smax is the perimeter of the VCM. The critical condition of total inner reflection for the TE mode is given by sin χ>n2/n1, where n1 and n2 represent the refractive index of the dielectric resonator and the external region, respectively. We calculated 3000 rays with random initial conditions above the critical condition. The rays after entering the leakage region are neglected. From the billiard, the four narrow regular regions are surrounded by regular and quasi-regular regions as shown in Fig. 4(a). The regular and quasi-regular regions form four water droplet islands and one of them is zoomed in, as the right box shows. The droplets are surrounded by easy, leaky chaotic sea. The light ray trajectories with reflection times of 3000 and initial condition of (i) S=0 and χ=58.80° and (ii) S=0 and χ=44.55° are shown in the right (blue) and left (red) insets, respectively, of Fig. 4(a). Both of them form regular orbits in the phase space, corresponding to the fundamental and first-order four-bounce mode. In Fig. 4(b), we enlarge the top and the droplet centers of the SOS to show the details related to the circular-like modes and the fundamental four-bounce modes. The long-lived quasi-regular unstable points emerge on the top of the phase space and have a limited distribution at the zero-curvature points, corresponding to the rays nearly parallel to the boundary, while the droplet centers are formed by concentric circles corresponding to regular rays with four times of reflections at the zero-curvature points. The existence of rays corresponding to the circular-like and fundamental four-bounce modes is proven by the SOS providing an overview of the phase space.

Fig. 4. (a) Poincaré surface of sections (SOSs) of VCM; inset shows ray trajectories of two kinds of closed orbits with reflection times of 3000, which are marked by dots of the same color in the phase space. We zoom in on one of the droplets to show the details of the regular regions. (b) Top and droplet centers of the SOS; horizontal and vertical coordinate ranges are expressed on the left and bottom of each graph.

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The ray dynamics are limited due to the hypothesis of infinitely small wavelength. The mode properties of the circular-like mode and fundamental four-bounce mode are further investigated by calculating the Husimi projection onto the Poincaré surface of section (SOS) ^[19]. The circular-like mode and fundamental four-bounce mode projecting onto the SOS are simulated and plotted in Figs. 5(a) and 5(b), corresponding to the rightmost circular-like and four-bounce modes in Fig. 2(a). For circular-like mode, the Husimi function distributes continuously on the top of the phase space as WGM does. However, the difference between circular-like mode and WGM is that it has four highly distributed islands at χ=57.1° and S/Smax=1/8, 3/8, 5/8, and 7/8, implying relative higher reflection probabilities at the maximum curvature points with angles around 57.1°. As for the four-bounce mode, the Husimi distribution forms four islands at χ=45° and S/Smax=0, 1/4, 2/4, and 3/4, which is typically the same with the rays marked by red dots in Fig. 4(a). The Husimi distributions of both modes are above the total reflection angle of 28.81°, which guarantees the high-Q properties for both modes. Also, the combination of both modes fills the phase space above the critical angle and the overlap between them is very small, which implies small interactions between them at the boundary.

Fig. 5. Husimi projections of the (a) circular-like mode and (b) fundamental four-bounce mode. Insets show the field distributions |Hz| of both modes.

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3. DUAL-MODE MANUPLATION

If dual modes in a resonator have spatially separated fields, the mode competition would not be intense due to the consumed carriers of both modes being in different regions. Rate equation investigations on dual-mode lasing indicate that the dual-mode stationary solution is stable if only if the cross-saturation coefficient is smaller than the self-saturation coefficient ^[20]. The saturation coefficient comes from the quantum model of the gain medium, which is defined as ^[10,21] where m and k represent modes m and k, Fm(r) and Fk(r) are normalized field distributions of the respective modes m and k, c is a constant parameter related to the refractive index, oscillating frequencies, and intraband relaxing time of the dipole, and 1/(1+δm,k) comes from the number of combinations of field components. Generally, different transverse or longitudinal-order modes have different mode patterns leading to relatively small cross-saturation coefficients. Furthermore, a smaller cross-saturation coefficient is expected due to the space separation of the circular-like modes and the fundamental four-bounce modes.

In Table 2, the ratios of the cross-saturation coefficient to the self-saturation coefficient are calculated for the microcircular resonators, microsquare resonators, and VCMs, where r, a, and d represent the radii of the microcircular resonator, side lengths of the microsquare resonator, and sizes of the VCM. G00 represents the self-saturation coefficient of the fundamental modes, G10 represents the cross-saturation coefficient between the fundamental and first-order modes, Gcc represents the self-saturation coefficient of the circular-like modes, and Gsc represents the cross-saturation coefficient between the circular-like and fundamental four-bounce modes. The ratios of cross-saturation coefficient to self-saturation coefficient of the microcircular and microsquare resonators are calculated based on the fundamental and first-order modes, while the corresponding ratios of the VCMs are calculated utilizing the circular-like and fundamental four-bounce modes. The ratios are about 0.64, 1, and 0.5 for the microcircular resonators, microsquare resonators, and VCMs, respectively. For the VCM with d=5μm, the fundamental four-bounce mode vanishes due to the strong coupling with leaky modes (with close wavelength), leaving a blank in the tablet. The smaller ratios of the cross-saturation coefficient to self-saturation coefficient for VCMs indicate it is more suitable for stable dual-mode lasing than the microcircular and microsquare resonators.

As per the preceding discussion, the VCM maintains smooth boundary and the same symmetry as the square resonator, which is the key to supporting two types of high-Q modes. However, the modulation of the dual-mode wavelength difference for a VCM is sometimes urged for multiple applications. In Fig. 6(a), we calculated the mode wavelengths and Q factors for VCM with d=4.2μm and a refractive index step Δn. The refractive index in the light-colored region maintains a constant of 3.2 while the refractive index of the dark-colored ring region increases from 3.195 to 3.205. The wavelengths of the circular-like mode rise from 1552.1 to 1556.5 nm, and the Q factors increase from 5.04×105 to 1.23×106. The wavelengths of the fundamental four-bounce mode rise from 1553.0 to 1554.8 nm, and the Q factors decrease from 1.46×104 to 1.33×104. Wavelength difference between the circular-like mode and the fundamental four-bounce mode varies from −0.9 to 1.7 nm continuously, the corresponding beat frequency varies from 110 to −210 GHz as shown in Fig. 6(b), and the continuous tuning in the microwave range has been proven, where fc and fs are frequencies of the circular-like and fundamental four-bounce modes, respectively. The refractive index step can be realized experimentally by the carrier and temperature distributions modulated by graphic current injection window ^[18]. As the field distributions of the circular-like mode mainly distribute in the dark-colored ring region, the wavelength increases much faster than the fundamental four-bounce mode.

Fig. 6. (a) Wavelengths, Q factors, and (b) frequency differentials of circular-like and fundamental four-bounce modes versus the refractive index change of the dark-colored ring region.

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For the further increasing of the beat frequency, we calculated the mode frequencies for the circular-like modes and fundamental four-bounce modes in different VCM sizes. In Fig. 7, when the VCM size increases, the mode frequencies for the circular-like mode and fundamental four-bounce mode decrease continuously as the arrow shows. The beat frequency between both modes remains at about 0.32 THz. If we assume the gain region covers about 1550 nm in the spectrum, each of the pairs of the circular-like and fundamental four-bounce modes have the chance to pass through the gain region. The beat frequency could be 0.05, 0.41, 0.75, and 1.09 THz in VCM sizes from 4.2 to 4.5 μm, and the tuning ranges are around 300 GHz according to Fig. 6. The disturbance of two modes emitted in one cavity remain consistent to each other and the linewidth of the beat signal could be minimum, which is explained as the common-mode noise rejection effect ^[22]. Much effort has been put into dual-mode laser fabrication and terahertz generation, including a fiber-based system ^[23] and laser diode with frequency-selected external cavity ^[24]. Terahertz radiation generated by semiconductor lasers also has been widely reported in Fabry–Pérot ^[25], DFB ^[15,26], and distributed Bragg reflector lasers ^[27]. The deformed WGM laser we propose has the benefits of stable beat frequency, easy fabrication, and integration capability. The generation of terahertz radiation on an integrated dual-mode system is the subject of ongoing work.

Fig. 7. Frequencies of the circular-like and fundamental four-bounce modes versus VCM size.

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In order to realize directional emission and control the mode Q factors for dual-mode lasing, an output waveguide is introduced. The Q factors of the four-bounce and circular-like modes versus the waveguide position angle θ are presented in Fig. 8(a), where θ is defined as the angle between the waveguide midline and the connected line of the VCM center and zero-curvature point. The VCM size is 4 μm, the waveguide width is 1 μm, and the distance between the VCM center and end of the waveguide is 7 μm. The wavelengths of the circular-like and four-bounce modes are around 1.537 and 1.542 μm. The modes c1 and c2 represent the antisymmetric and symmetric circular-like modes at θ=0°, while modes s1 and s2 represent the antisymmetric and symmetric fundamental four-bounce modes at θ=45°. When θ increases from 0° to 45°, the Q factors of the symmetric and antisymmetric circular-like modes show a decreasing tendency with oscillation, while the four-bounce modes increase with a smaller oscillation.

Fig. 8. (a) Q factors of dual mode versus waveguide rotating angles and field distributions of |Hz| for the circular-like modes with rotating angles of (b) 0° and (c) 27°, and for the fundamental four-bounce modes with rotating angles of (d) 27° and (e) 45°.

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The circular-like mode distributes mostly along the boundary, and has four highly distributed islands at the maximum curvature point as shown in Fig. 5(a), so the scattering loss involved with the waveguide is obviously high at the same point (θ=45°). In contrast, the scattering loss is smallest at θ of 0°. For the fundamental four-bounce mode, the field distributes mostly at the zero-curvature point along the boundary as shown in Fig. 5(b), so the Q factor of the fundamental four-bounce mode is the lowest when θ is 0°. For a definite angle around 27°, the quality factors could be modulated close to each other for dual-mode lasing. The field distributions of |Hz| of the high-Q circular-like modes at θ=0° and 27° and the high-Q four-bounce modes at θ=27° and 45° are plotted in Figs. 8(b)–8(e). It is proved that the waveguide only involves slight deformation for both modes when the Q factors remain high.

4. CONCLUSIONS

In summary, we have proposed a VCM for realizing dual-mode manipulation in a small microcavity. By numerically calculating the resonant mode spectrum and field distributions, we have found that the microresonator could support high-Q WGMs of circular-like modes distributing along the boundary and fundamental four-bounce modes distributing in a regular orbit. The slightly varied cavities from the VCM to the microcircular and microsquare resonators imply that both modes can transform to the fundamental modes of circular and square resonators continuously. The ray dynamics show the existence of a regular orbit for the fundamental four-bounce modes and quasi-regular orbit for the circular-like modes. Furthermore, the Husimi projections of the circular-like mode and fundamental four-bounce mode indicate that the fundamental four-bounce mode has high probabilities in four islands while the circular-like mode is a mixture of regular orbit modes and WGMs. The dual-mode stabilities of the VCMs are proven to be superior to those of microcircular and microsquare resonators. For the modulations of the VCM, step refractive indices and directly connected waveguides are proved effective in changing the wavelength difference and balancing the Q factors for dual-mode lasing with controllable wavelength interval.