MICRODISK lasers supported by a pedestal to form strong

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 11, JUNE 1, Mode Coupling and Vertical Radiation Loss for Whispering-Gallery Modes in 3-D Microcavities Yong-Zhen Huang, Senior Member, IEEE, and Yue-De Yang Abstract The characteristics of whispering-gallery modes (WGMs) in 3-D cylindrical, square, and triangular microcavities with vertical optical confinement of semiconductors are numerically investigated by the finite-difference time-domain (FDTD) technique. For a microcylinder with a vertical refractive index 3.17/3.4/3.17 and a center layer thickness 0.2 m, -factors of transverse electric (TE) WGMs around wavelength 1550 nm are smaller than 10 3, as the radius 4 m and reach the orders of 10 4 and 10 6 as = 5 and 6 m, respectively. However, the Q-factor of transverse magnetic (TM) WGMs at wavelength m reaches as = 1 m. The mode coupling between the WGMs and vertical radiation modes in the cladding layer results in vertical radiation loss for the WGMs. In the microcylinder, the mode wavelength of TM WGM is larger than the cutoff wavelength of the vertical radiation mode with the same mode numbers, so TM WGMs cannot couple with the vertical radiation mode and have high -factor. In contrast, TE WGMs can couple with the corresponding vertical radiation mode in the 3-D microcylinder as 5 m. However, the mode wavelength of the TE WGM approaches (is larger than) the cutoff wavelength of the corresponding radiation modes at = 5 m(6 m), so TE WGMs have high Q-factors in such microcylinders too. The results show that a critical lateral size is required for obtaining high -factor TE WGMs in the 3-D microcylinder. For 3-D square and triangular microcavities, we also find that the Q-factor of TM WGM is larger than that of TE WGM. Index Terms Finite-difference time-domain technique (FDTD), microcavities, optical resonators, quality factor. I. INTRODUCTION MICRODISK lasers supported by a pedestal to form strong vertical optical confinement have attracted great attention since 1992 [1] [6]. However, the microdisk on a pedestal prevents the heat dissipation and the current injection efficiency. To improve heat design, an aluminium oxide-encased microdisk laser was fabricated with high vertical optical confinement provided by the refractive index difference between semiconductor and the aluminium oxide [4]. Microcavities with the vertical confinement of semiconductor materials will have better thermal conductivity and current injection efficiency than the microdisk on a pedestal, but with a low refractive index Manuscript received September 14, 2007; revised January 2, This work was supported by the National Nature Science Foundation of China under Grants and , and the Major State Basic Research Program under Grant 2006CB Y.-Z. Huang is with State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China, and also with College of Information Science and Engineering, Huaqiao University, Quanzhou , Fujian, China. Y.-D. Yang is with State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing , China. Digital Object Identifier /JLT contrast for the vertical optical confinement. High-efficiency microcavity lasers can be fabricated if whispering-gallery modes (WGMs) still have a high -factor in such microcavities. InAs/GaAs quantum-dot microcylinder lasers and InGaN multiple-quantum-well spiral-shaped micropillars were fabricated [7], [8] with vertical semiconductor confinements. The quantum-dot microcylinder laser with the radius of 5 m reached threshold; however, the quantum-well microcylinder lasers realized stimulated emission with the radius m [7]. However, the microdisk laser with submicrometer radius was realized at room temperature [2]. The results indicate that the optical loss, especially vertical radiation loss, plays an important role in the microcylinder lasers, in addition to the increase of the surface recombination with decreasing radius. Recently, 3-D finite-difference time-domain (FDTD) simulations indicated that the -factors of transverse magnetic (TM) WGMs are much larger than those of transverse electric (TE) WGMs in microcylinders with vertical optical confinement of semiconductors [9]. In the microcylinder, the WGMs are vertically confined by the refractive index difference between the center layer and the cladding layers. The WGMs will couple with the vertical radiation modes in the microrod of the cladding layer if they have nonzero overlap integral, as the WGM and the vertical radiation mode have the same angular and radial mode numbers. The mode wavelength of TM WGM is larger than the cutoff wavelength of the vertical radiation mode with the same mode numbers, so TM WGMs cannot couple with the corresponding radiation modes. The TM WGM at wavelength about 2.0 m can have a -factor as large as 10 in the semiconductor microcylinder with the radius of 1 m. We also found that the -factors of TM WGMs are much larger than those of TE WGMs in 3-D semiconductor square and triangle cavities by numerical simulation [10], [11], although the mode coupling is not strictly between the modes with the same mode numbers as in the microcylinder. In this paper, the mode characteristics are investigated for 3-D cylindrical, equilateral triangular and square microcavities with vertical optical confinement of semiconductors by 3-D FDTD simulation. In addition to high -factor TM WGMs, we find that TE WGMs can have high -factors in a microcylinder with vertical confinement of semiconductors as the microcylinder radius is larger than a critical value. In the microcylinder with a large radius, the mode wavelength of the TE WGM is larger than that of the vertical radiation mode with the same angular and radial mode numbers in the cladding layer. So the TE WGMs cannot couple with the corresponding vertical radiation mode too. This paper is organized as follows. In Section II, the intensity spectra with clear peaks of the WGM and the vertical radiation modes are compared for TE and TM WGMs in a microcylinder, and the mode Q-factors of WGMs versus the radius are investigated. In /$ IEEE

2 1412 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 11, JUNE 1, 2008 Fig. 1. Schematic of the cross section of a 3-D microcylnder. The refractive indices of center and cladding layers are n and n, and the thickness of the center layer is d. Section III, the numerical simulation results are presented for TE and TM WGMs in equilateral-triangular and square microcavites, and finally the conclusion is summarized in Section IV. II. CYLINDRICAL MICROCAVITIES A microcylinder with radius, center layer thickness, and the vertical refractive index distribution as shown in Fig. 1 are considered in this section. The refractive indices and are used in the numerical simulation with the refractive index difference easy to realize in GaAs and InP material systems. Although there are no real TE and TM modes in 3-D microcavities, we can assign confined modes as TE-like and TM-like WGMs based on their main polarizations in the -direction slab waveguide. In the microcylinder with symmetric vertical waveguide, TE-like and TM-like WGMs have symmetric electromagnetic field components and, respectively, relative to the middle plane at, where and are electric and magnetic field components in the 3-D cylinder coordinates. So we can simulate the TE-like and TM-like WGMs separately using exciting sources with different symmetries. Ignoring the vertical mode number, we mark the TE-like and TM-like WGMs in the microcylinder as TE and TM, where and are the angular and radial mode numbers, respectively. Applying the effective index approximation to the 3-D microcylinders, we can calculate the mode wavelengths from the complex roots of the following eigenvalue equation of the 2-D cylindrical microcavities [12]: where and are the Bessel and second-kind Hankel functions of order, is the wavenumber in a vacuum, is the effective index of the slab waveguide for the TE (TM) polarization, and equals to ( ) for the TE-like (TM-like) WGMs. The WGMs will couple with the vertical radiation modes and in the uniform microrod of the cladding layer. The cutoff wavelengths of and can be obtained as their vertical propagation constant, which are the same as the mode wavelength of the TE and TM WGMs obtained from (1) with the refractive (1) index. For a microcylinder with the radius of 1 m, we found that the mode wavelength of TM WGM is larger than the cutoff wavelengths of and, but the mode wavelength of TE WGM is smaller than the cutoff wavelength of as and [9]. The mode coupling will result in a large vertical radiation loss as the mode wavelength of the WGM is smaller than the cutoff wavelength of the corresponding mode with the same mode numbers. We simulate the mode characteristics for the microcylinder by 3-D FDTD technique [13]. Based on the circular symmetry, the 3-D problem of a microcylinder can be transformed into a 2-D one with the angular field dependence of [14], where is the angular mode number. The perfect matched layer (PML) absorbing boundary condition [13], [15] in circular cylindrical coordinates is used as the boundaries to terminate the FDTD computation window. The spatial steps and are set to be 10 and 20 nm, respectively, and the time step is chosen to satisfy the Courant condition. At the inner boundary, the condition is used for the and field components based on the asymptotic behavior of the Bessel function [14]. In the simulation, an exciting source with a cosine impulse modulated by a Gaussian function is added to electromagnetic fields at a point inside the microcylinder, where and are the times of the pulse center and the pulse half width, respectively, and is the center frequency of the pulse. Symmetric sources about the middle plane at on electric fields and and magnetic fields and are used to excite the TE-like and TM-like WGMs, respectively. The time variation of a selected field component at some points inside the microcylinder is recorded as a FDTD output, then the Padé approximation with Baker s algorithm [15] is used to transform the FDTD output from the time-domain to the frequency-domain, and finally the mode frequencies and -factors are calculated from the obtained intensity spectrum. Taking the impulse (2) at,, and THz, we first simulate the mode characteristics for the modes with mode index in the microcylinder with m, m,, and. The intensity spectra plotted in Fig. 2 are calculated from the last step FDTD output of step FDTD simulation by the Padé approximation. The -factors of TM and TE WGMs are and at the wavelengths of and m, respectively. The minor peaks at the wavelengths of and m can be attributed to the vertical radiation modes in the cladding layer with vertical propagation constant, which do not have energy transmission in the direction. The cutoff wavelengths of and obtained from (1) are and m, respectively, which agree very well with the aforementioned minor peak values. Next, we consider the microcylinders with m,,, and the radius and m. The wavelengths of TE-like and TM-like WGMs obtained by 3-D FDTD simulation and the cutoff wavelengths of the corresponding and modes are listed in Table I, where the mode index is chosen to keep the mode wavelength around (2)

3 HUANG AND YANG: MODE COUPLING AND VERTICAL RADIATION LOSS FOR WHISPERING-GALLERY MODES IN 3-D MICROCAVITIES 1413 TABLE I MODE WAVELENGTHS OF WGMs AND RADIATION MODES FOR MICROCYLINERS WITH VERTICAL REFRACTIVE INDICES 3.17/3.4/3.17 Fig. 2. Intensity spectra obtained by 3-D FDTD simulation and Padé approximation for a microcylinder with the radius of 1 m. The intensity spectra of TE and TM WGMs are obtained under exciting sources with different symmetry, respectively. Fig. 3. Intensity spectra for TE-like WGMs in the microcylinders with R = 3; 4; 5; and 6 m m. The cutoff wavelengths of and modes are obtained from (1) for TM and TE modes with the refractive index of 3.17, respectively. With the increase of the radius, we find that the wavelengths of the TE-like WGMs are larger than the cutoff wavelengths of the corresponding and modes at the radius between 5 and 6 m. So the TE-like WGMs can also have high -factor in the microcylinder as m. Because the factors of TM-like WGMs are very high and difficult to obtain from the intensity spectrum exactly, only the factors of the TE-like WGMs obtained from 3-D FDTD simulation are listed in Table I. In Fig. 3, the intensity spectrum of TE-like WGMs Fig. 4. Field distributions of E (z) at the field peak position in the r direction for TE-like WGMs in microcylinders with R =3; 4; 5; and 6 m. obtained by 3-D FDTD simulation and Padé approximation are plotted for the microcylinders with and m, respectively, with the corresponding angular mode index and. The intensity spectra are obtained from the last step FDTD output of , , , and steps FDTD simulations, respectively, with the impulse (2) at,, and THz. Using a long optical pulse with a narrow bandwidth to excite only one mode, we can obtain the mode field distribution by the FDTD simulation. Fig. 4 depicts the distributions of for TE-like WGMs at the field peak position in the direction, which is normalized to keep the peak value of unity at. Because the field distribution is symmetric about the plane, we only present the field distribution at. The whole field patterns of the electric field for TE-like WGMs in the microcylinders are shown in Fig. 5(a) and (b) at and m, respectively, where the intensity in the region of m is magnified ten times. The vertical field distributions are confined well at and m, so the modes have high Q-factors. However, the vertical field distribution oscillates in the cladding layer at and m, corresponding to a vertical radiation loss. The results indicate that a critical lateral size exists for obtaining high -factor TE WGMs, which agrees well with the experimental results [7]. III. EQUILATERAL-TRIANGULAR AND SQUARE MICROCAVITIES In fact, the aforementioned 3-D microcylinders are simulated by the 2-D FDTD technique. However, real 3-D FDTD simulation is required for 3-D equilateral-triangular and square microcavities. The schematic diagrams of equilateral-triangular

4 1414 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 11, JUNE 1, 2008 Fig. 7. Intensity spectra obtained by 3-D FDTD simulation and Padé approximation for the 3-D equilateral-triangular microcavity with a = 3 m under symmetric and anti-symmetric conditions relative to the x = 0plane for (a) TE-like and (b) TM-like modes. Fig. 5. Field patterns of the electric field E (r; z) for TE-like WGMs in the microcylinders at (a) R =3m and (b) R =6m. The intensity in the region of z>1:5 m is magnified ten times. Fig. 6. Schematic diagrams of equilateral triangular and square microcavities with the side length a. and square microcavities are shown in Fig. 6, where the vertical waveguide structures are similar to the microcylinders with,, and m, which only support the fundamental mode in the vertical direction. The spatial steps,, and are set to be 20 nm, and the time step is chosen to satisfy the Courant condition step FDTD simulation with the exciting source (2) is performed for a 3-D equilateral triangular and square microcavities with the side length of 3 and 2 m, respectively, and the intensity spectra are obtained from the FDTD output of the last steps. As in 2-D equilateral-triangular microcavities [17], [18], the confined TE and TM modes in 3-D equilateral-triangular microcavities are marked as TE and TM ignoring the vertical mode index, where and are the transverse and longitudinal mode numbers, respectively. The field distributions of TE (TM) modes can be expressed as symmetric and anti-symmetric magnetic (electric) field distributions relative to the plane. The symmetric and anti-symmetric modes marked by superscripts and are accidentally degenerate modes with different Q-factors as is a multiple of 3 [17]. The intensity spectra obtained by 3-D FDTD simulation and Padé approximation under different symmetric conditions relative to plane are shown in Fig. 7(a) and (b) for TE-like and TM-like modes, respectively. The first-order transverse modes are observed between two fundamental modes for the TM-like modes in Fig. 7(b). However, the peak of the first-order TE-like mode does not appear in Fig. 7(a) because the -factors of TE and TE are too small, which are less than 100 obtained by 2-D FDTD under the effective index approximation. In Table II, the mode -factors and wavelengths obtained by the 3-D FDTD simulation are compared with those of 2-D FDTD simulation under effective index approximation for the confined TE and TM modes. For the corresponding slab waveguide with the thickness of 0.2 m, the effective index of and obtained at the wavelength 1.55 m are used in the 2-D FDTD simulation for TE and TM modes, respectively. The -factors of 3-D FDTD simulation are smaller than those of the 2-D FDTD simulation, because the vertical radiation loss is not considered in the 2-D FDTD simulation. Assuming, where and are -factors obtained by the 3-D and 2-D FDTD simulations, we obtain and list determined by the vertical radiation loss in Table II too. of TM-like modes is much larger than that of TE-like modes, because the TM-like modes have smaller vertical loss

5 HUANG AND YANG: MODE COUPLING AND VERTICAL RADIATION LOSS FOR WHISPERING-GALLERY MODES IN 3-D MICROCAVITIES 1415 TABLE II MODE Q-FACTORS AND WAVELENGTHS FOR THE EQUILATERAL-TRIANGULAR MICROCAVITY WITH a = 3 m OBTAINED BY 3-D FDTD SIMULATION AND 2-D FDTD SIMULATION UNDER THE EFFECTIVE INDEX APPROXIMATION TABLE III MODE Q-FACTORS AND WAVELENGTHS FOR THE SQUARE MICROCAVITY WITH a = 2 m OBTAINED BY 3-D FDTD SIMULATION AND 2-D FDTD SIMULATION UNDER THE EFFECTIVE INDEX APPROXIMATION than TE-like modes in the equilateral triangular microcavity with the vertical optical confinement of semiconductors. Finally, we consider a 3-D square microcavity with the side length m. Ignoring the vertical mode index, we mark the TE-like and TM-like modes as TE and TM similar to 2-D square resonators [19], where the mode numbers and denote the number of wave nodes in the and directions, respectively. The high Q-factor modes in the microsquare are WG-like modes with - is a multiple of 2, which have antisymmetric magnetic (electric) field about the diagonal mirror plane of the square for TE-like (TM-like) WG-like modes. Mode Q-factors and wavelengths obtained by 3-D FDTD simulation and 2-D FDTD simulation under the effective index approximation are listed in Table III for WG-like modes in the square microcavity. 3-D FDTD simulation also gives smaller Q-factors than 2-D FDTD simulation. Similar to equilateral triangular microcavity, of TM-like modes is much larger than that of TE-like modes in the square microcavity. The results show that TM-like modes have less vertical radiation loss than TE-like modes in the square microcavity. In the microcylinder, the TM-like WGMs do not have vertical radiation loss, because the mode wavelength of TM-like WGM is always larger than the cutoff wavelength of the vertical radiation mode with the same angular mode numbers, and mode couplings are forbidden between the modes with different angular mode number. However, in the 3-D square microcavities, the WG-like mode can couple with the vertical radiation modes with the same symmetry [10], instead of the vertical radiation mode with the same mode index in the microcylinder. The break of the symmetries in the equilateral triangular and square leads to some changes in the mode characteristics [20]. The mode field distributions in equilateral triangular and square microcavities have many angular components, so the TM-like WGMs can couple with the vertical radiation modes of lower mode number but with the same angular components. However, through the simulation, we find that the -factors of TM-like modes still larger than those of TE-like modes. IV. CONCLUSION We investigate the mode characteristics for cylindrical, equilateral triangular and square microcavities by 3-D FDTD simulation, and verify that TM-like WG modes have much less vertical radiation loss and higher -factors than TE-like WG modes. Furthermore, we find that TE WG-like modes can have high Q-factors as the radius of the microcylinder is larger than a critical value in the 3-D microcylinder. An ultralow threshold microlaser can be fabricated with vertical semiconductor confinement based on the TM WG-like modes or TE WG-like modes in the cavities with a large lateral size. REFERENCES [1] S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Whispering-gallery mode microdisk lasers, Appl. Phys. Lett., vol. 60, pp , [2] A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, Room temperature operation of submicrometer radius disk laser, Electron. Lett., vol. 29, pp , [3] D. Y. Chu, M. K. Chin, W. G. Bi, H. Q. Hou, C. W. Tu, and S. T. Ho, Double-disk structure for output coupling in microdisk lasers, Appl. Phys. Lett., vol. 65, pp , [4] S. M. K. Thiyagarajan, D. A. Cohen, A. F. J. Levi, S. Ryu, R. Li, and P. D. Dapkus, Continuous room-temperature operation of microdisk laser diodes, Electron. Lett., vol. 35, pp , [5] M. Fujita, R. Ushigome, and T. Baba, Large spontaneous emission factor of 0.1 in a microdisk injection laser, IEEE Photon. Technol. Lett., vol. 13, no. 5, pp , May [6] K. Djordjev, S.- J. Choi, S.- J. Choi, and P. D. Dapkus, High-Q vertically coupled InP microdisk resonators, IEEE Photon. Technol. Lett., vol. 14, no. 3, pp , Mar [7] M. Arzberger, G. Böhm, M.-C. Amann, and G. Abstreiter, Continuous room-temperature operation of electrically pumped quantum dot microcylinder lasers, Appl. Phys. Lett., vol. 79, pp , [8] G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars, Appl. Phys. Lett., vol. 83, pp , 2003.

6 1416 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 11, JUNE 1, 2008 [9] Y. D. Yang, Y. Z. Huang, and Q. Chen, High-Q TM whisperinggallery modes in three-dimensional microcylinders, Phys. Rev. A, vol. 75, p , [10] Y. D. Yang, Y. Z. Huang, and Q. Chen, Comparison of Q-Factors between TE and TM modes in 3-D microsquares by FDTD simulation, IEEE Photon. Technol. Lett., vol. 19, no. 22, pp , Nov [11] Y. D. Yang and Y. Z. Huang, High Q-factor TM modes in threedimensional semiconductor microresonators, in Proc. CLEO/Pacific Rim, Seoul, Korea, 2007, ThP 072. [12] M. Hentschel and K. Richter, Quantum chaos in optical systems: The annular billiard, Phys. Rev. E, vol. 66, p , [13] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Boston, MA: Artech House, [14] B. J. Li and P. L. Liu, Numerical analysis of the whispering gallery modes by the finite-difference time-domain method, IEEE J. Quantum Electron., vol. 32, no. 9, pp , Sep [15] J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., vol. 114, pp , [16] W. H. Guo, W. J. Li, and Y. Z. Huang, Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation, IEEE Microw. Wireless Compon. Lett., vol. 11, no. 5, pp , May [17] Y. Z. Huang, Q. Chen, W. H. Guo, Q. Y. Lu, and L. J. Yu, Mode characteristics for equilateral triangle optical resonators, IEEE J. Sel. Quantum Electron., vol. 12, no. 1, pp , Jan./Feb [18] Q. Chen, Y. H. Hu, Y. Z. Huang, Y. Du, and Z. C. Fan, Equilateraltriangle-resonator injection lasers with directional emission, IEEE J. Quantum Electron., vol. 43, no. 6, pp , Jun [19] W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, Modes in square resonators, IEEE J. Quantum Electron., vol. 39, no. 12, pp , Dec [20] Y. D. Yang and Y. Z. Huang, Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators, Phys. Rev. A, vol. 76, p , Yong-Zhen Huang (M 95 SM 01) was born in Fujian Province, China, in He received the B.Sc., M.Sc., and Ph.D. degrees in physics from Peking University, Beijing, China, in 1983, 1986, and 1989, respectively. In 1989, he joined the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, where he worked on the tunneling time for quantum barriers, asymmetric Fabry Perot cavity light modulators, and VCSELs. In 1994, he was a Visiting Scholar at BT Laboratories, Ipswich, U.K., where he was involved in the fabrication of the 1550-nm InGaAsP VCSEL. Since 1997, he has been a Professor with the Institute of Semiconductors, Chinese Academy of Sciences, where he is also the Director of the Optoelectronic R&D Center. His current research interests include microcavity lasers, semiconductor optical amplifiers, and optical add-drop filters. Yue-De Yang was born in Hunan Province, China, in He received the B. Sc. degree in physics from Peking University, Beijing, China, in He is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, and studying the design and the fabrication of microcavity lasers and filters.