We theoretically present in-plane multiple guided emissions of metallically confined microdisk lasers which can be applied to drive multiple elements in compact photonic integration at the same time. Two to four-port microdisks with transverse magnetic and electric polarizations are investigated based on finite difference time domain simulation and padé approximation. Modes filtering of coupling ports are verified by the calculated mode quality factors (Q) which are decided by the matching of coupling ports with the energy density distribution of corresponding modes. Single mode lasing operation of semiconductor microdisk with guided emissions is possibly realized by selectively pumping.
©2011 Optical Society of America
Since the first microdisk laser was fabricated nearly two decades ago , the studies of microlasers have achieved great successes and the performances of microlasers are essentially stepped forward. On the hand of temperature, the microlaser operated from first liquid nitrogen environment to the followed room temperature , and the dimensions of dielectric disk subsequently decreased to submicro scale . From the perspectives of practical applications of microlasers as light sources, directional or guided emissions are indispensable for building transmitting channel with neighboring opto-electric elements by air or on-chip dielectric waveguides . In order to obtain the directional emission, asymmetric microcavities, such as spiral and limçon shaped, were proposed and cylinder microlasers demonstrated promising performances [5–7]. But from the view of the integration of optical elements, guided emission may be much more significant while it can be directly applied to the add port of filter or modulator etc. Through vertically and evanescently coupling with a bus waveguide, microdisk lasers with guided emissions were realized, and then InP-based microdisk lasers vertically integrated with silicon waveguide on silica were also fabricated, both schemes kept in-plane geometry of cavity undeformed [8,9]. In-plane couplings for guided emission were proposed too, circular microlasers with a waveguide in radial direction were reported and relatively low threshold currents were obtained [10–12]. However, even though the microlasers’ operation benefited from the large contrast of refractive index of semiconductor gain medium and surrounding air, the roughness of etched reflecting wall, the spaces extended by evanescent wave and the interferences from neighboring environment would weaken the performances of microlasers on power consumption, on-chip space and the stability of photonic circuits. Motivated by enhancing the modes confinement and insulating from other optical elements, metallo-dielectric confining structure was introduced and the dimensions of semiconductor lasers stepped into nano scale [13–15]. By exploiting strong light confinement of metallo-dielectric structure, we investigate metallically confined microdisk with in-plane multiple guided emissions, focusing on their modes characteristics while multiple ports are introduced for driving several optoelectronic elements .
Electrically pumped microdisk lasers, fabricated from a laser wafer (AlGaInAs-InP for instance ) by standard photolithography and inductively-coupled-plasma etching techniques, with the active region consisting of multi quantum wells and sustained by cladding layers and substrate, are potential light souses of optoelectronic circuits. It is generally expected that introducing defects at the edge of microdisk would collapse the mode confinement while the intense energy density locates at the fringe of cavity . The influences of vertical cladding layers have been studied . In this express, the two-dimensional (2-D) problems in transverse plane based on the effective indices are considered. We present metallically confined microdisks with two to four guided emissions, which are escaped in radial directions with included angles of 2π/2, 2π/3 and 2π/4. The modes characteristics are investigated by finite difference time domain (FDTD) method and padé approximation [19,20]. Firstly, Transverse magnetic modes are considered, the intensity spectra are calculated for the opened or defected microdisks. The so called high-order WG modes and coupled modes survive in resonator with multiple guided emissions of wide ports. In addition, the influences of the jointed ports on the modes Q-factors are studied and modes filtering of ports are verified. And then transverse electric modes are investigated too.
2. Transverse magnetic (TM) modes investigations
We firstly consider TM polarization of metallically confined microdisks with fixed radius R = 3 μm. The schematic diagram of a two-port microdisk is shown in Fig. 1(a) , where the two waveguides, described by dashed lines, are extended in two opposite radial directions, the width of ports is defined as w and the nonzero electric and magnetic fields are E z, H y and H x. Based on the model displayed above, the FDTD simulations are carried out with uniform mesh of dx = dy = 0.02 μm. The effective refractive indices of semiconductor medium, insulating dielectric and p-electrode metal are set as 3.2, 1.45 and 0.18 + 10.2i respectively, corresponding to InGaAsP/InP, silica and gold . The Drude dispersion model is used to model the performance of the electromagnetic filed in the metal layer [19,22]. To excite various resonant modes, Gaussian-modulated cosine impulse p(x0, y0, t) = exp[-(t-t0)2 /tw 2] cos(2πft) is added to Ez component at a point (x0, y0) of low symmetry inside the microdisk, where t0 and tw are the times of center and half-width of the pulse and f is the center frequency of the pulse. The time-domain outputs are recorded at arbitrarily selected monitor point (x 1, y 1) inside the disk as a FDTD output. At last, the Padé approximation with Baker's algorithm  is used to transform the FDTD output from the time-domain to the frequency-domain, then the mode intensity spectra are calculated, and finally the mode frequency and Q-factor are obtained from the peak frequency and the ratio of the peak to the corresponding 3-dB bandwidth by fitting the peak of the intensity spectra with a Lorentzian function.
As depicted in , the dominate modes of metallo-dielectric confined microdisk with wide waveguide are coupled modes and high-order modes, but since more waveguides are added, the modes are selectively filtered. The spectra of two, three and four-port microdisks are plotted in Fig. 1(b) by solid, dashed and dotted lines with uniform w of 0.6 μm, both the thicknesses of silica and gold are 0.2 μm (the same below). The modes have been denoted by the azimuthal and radial number (m, l). In two-port microdisk, the TM(2, 12), and the coupled mode of TM(18,5) and TM(28,2) have high Q factors of 1.6 × 104 and 3.4 × 104, the attenuating factor α is 2π/(1.6 × 104 × T) and 2π/(3.4 × 104 × T) if the power stored in cavity varies with e - αt, T and t are the resonant cycle and time . But for three-port microdisk, high-order WG modes are filtered and the coupled modes of TM(18,5) and TM(22,4), TM(19,5) and TM(21,4) survive, where the difference of azimuthal number is the times of 3. The four-port microdisk presents the high capacity of mode filtering, only TM(2, 12) survives. It means that, as the waveguides introduced, much more power losses out of the ports in a resonant cycle for some modes while the other modes are nearly not affected. In Fig. 2 , we display the energy density patterns of TM(2,12) in (a) two- and (b) four-port microdisks, and the coupled modes of TM(18,5) and TM(28,2), TM(18,5) and TM(21,4) in (c) two and (d) three-port microdisks with nearly the same resonant wavelength of 1.607 μm. The intense energy density of the high-order WG modes focuses at the center region of microdisks. On the contrary, that of coupled modes has the zero value near the center. The energy density of coupled mode of TM(18,5) and TM(21,4) demonstrates equilateral triangle shape due to wave interferences. Through selectively pumping, single mode lasing with guided emissions is possibly realized for electrically driving microlasers where the current injecting area of the p-electrode can be easily defined to the needed pattern by standard optical lithography and the major fraction of the high energy density of the resonant mode overlaps with the current injecting area, which is energy efficient for mode lasing and other modes could be suppressed.
The influences of the jointed ports on the modes Q-factors are considered. The Q-factors of TM(2,12) in two to four-port microdisks versus w is shown in the Fig. 3(a) . Once w increases beyond 0.2 μm, the Q-factor of TM(2,12) in three-port microdisk rapidly deteriorated while those of two- and four-port still keep on the amplitude of 104 until 0.7 μm. The introduced coupling ports have selective influences on the resonant modes and modes filtering are verified. The Q-factors of coupled modes versus w are shown in Fig. 3(b), modes coupling dose not happen while w is less than 0.2 μm, however, once w increases beyond 0.25 μm, the coupled modes of TM(18,5) and TM(28,2), TM(18,5) and TM(22,4) survive in two- and three-port microdisk respectively and the Q-factors are not deteriorated until w increases to near 0.7 μm. Wide w not only provides channels of lasing emissions, but also plays a role of modes filtering and makes for single mode lasing.
3. Transverse magnetic (TE) modes investigations and discussions
In this section, we studied TE modes with the same dimension, the nonzero electric and magnetic fields are E y, E x, and H z. Based on the investigation of TM modes, here, we armed at the influences of the introduced ports on the high Q TE modes. Figure 4 shows the intensity spectra of TE modes confined in two-, three- and four-port microdisk, where w is 0.6 μm. Obviously, only TE modes with small azimuthal number m are observed while those low-order WG modes have been collapsed by the introduced output ports. For the two-port microdisks, the high-order WG modes play much more important role on the mode lasing than those of low-order WG modes, even though they are predominant in un-defected dielectric microdisk. The Q-factors of TE(0,14), TE(1,13), TE(2,13), TE(3,12) and TE(5,11) are 4.2 × 103, 7.3 × 103, 6.4 × 103, 3 × 103 and 2.6 × 103. Except for that of TE(0,14) which have no zero nodes around the edge and are easily effected by the jointed ports, the Q-factors adversely decrease as azimuthal number m increase. Figures 5(a) and 5(b) display the energy density of highest order modes TE(0,14) and TE(1,13), the bow-tie shape of high energy density are obviously observed near the center region of microdisk. The emissions in waveguides are amplified by a factor of 50. The TE(0,14), TE(3,12) and TE(5,11) are effectively filtered by the added two ports for four-port microdisk while their energy density patterns contradict with the symmetry of defected cavity. The TE(1,13) has two degenerate states, their energy densities are shown in Figs. 5(c) and 5(d). Just like the mirrors of fabry-pérot resonant cavity, one pair of curve insulator-metal bi-layer mirrors can also form standing-wave like modes, the difference is that the energy densities display the characteristics of higher-in-center, smaller-in-both sides. Since the resonant wavelengths of WG modes are decided by the dimensions of microdisk, and the coupled mode comes from the two WG modes which have nearly the same resonant wavelength. For the TE modes, no WG modes with nearly the same wavelength are observed in the range of 1.50 μm to 1.60 μm. The spectrum of three-port microdisk shows only TE(3,12) while that of coupled modes are not structured, the Q-factor of 1.2 × 103 is one-order less than that of TM coupled modes.
At last, the influences of w on high-Q modes are considered. The Q-factors of TE(2,13) in two-, TE(3,12) in three-, TE(1,13) in four-port microdisk versus w are calculated and the results are shown in Fig. 5(e). The Q-factors of TE(2,13) in two-port cavities decrease gradually from 1.4 × 104 to 4 × 103 while that of TE(3,12) and TE(1,13) in three- and four-port cavities decrease rapidly to the order of 1 × 103.
Actually, the energy leaking from the ports is mainly decided by the distributions of energy density of the corresponding modes comprised of the electric field density εE 2/2 and magnetic field density μH 2/2, ε and μ are the permittivity and permeability of corresponding mediums. For modes with the same m and l, the fractions of the electric field density and magnetic field density occupying the total energy density are different, the polarization dependent influences of the ports on Q-factors may be explained. The relation of the fraction of power losses in a resonant cycle from the ports accounting for the total power stored in the cavity and mode Q-factors can be expressed as, where the and are the mode Q-factor of the cavity without and with ports, and is dissipation losses fraction of the metal near the ports jointed region and is a small quantity. Take the TE(2,13) in two-, TE(3,12) in three- and TE(1,13) in four-port microdisk for example, the fraction β are about 1.1 × 10−3, 5.7 × 10−3 and 6.2 × 10−3 as w increase to 1 μm. If is comparable with, the emission losses is small and the ports play the role of collecting scattering light from the cavity.
In conclusion, we have demonstrated metallically confined microdisks with multiple guided emissions. Mode characteristics of transverse magnetic and electric polarizations are studied by FDTD simulations and padé approximation respectively. Modes filtering are verified by the added waveguides which are used for extracting lasing emissions. The modes, such as high-order WG modes and coupled modes survive in metallically confined microdisks with wide output ports. By selectively pumping, single mode operating microlasers with multiple guided emissions are expected, and two to four optoelectronic elements may be driven by only one light source.
This work is supported by the Fundamental Research Funds for the Central Universities (FRFCU) under Grant No. 2011121048 and the National Natural Science Foundation of China (No. 61107045 and No. 50802080).
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