threshold or even thresholdless semiconductor lasers [1]. For semiconductor microcavity lasers, the volume of the laser cavity should be decreased as much as ...
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Whispering-Gallery-Like Modes in Square Resonators Wei-Hua Guo, Yong-Zhen Huang, Senior Member, IEEE, Qiao-Yin Lu, and Li-Juan Yu
Abstract—The mode frequencies and field distributions of whispering-gallery (WG)-like modes of square resonators are obtained analytically, which agree very well with the numerical results calculated by the FDTD technique and Padé approximation method. In the analysis, a perfect electric wall for the transverse magnetic mode or perfect magnetic wall for the transverse electric mode is assumed at the diagonals of the square resonators, which not only provides the transverse mode confinement, but also requires the longitudinal mode number to be an even integer. The WG-like modes of square resonators are nondegenerate modes with high-quality factors, which make them suitable for fabricating single-mode low-threshold semiconductor microcavity lasers. Index Terms—Microcavities, semiconductor lasers, square resonators, whispering-gallery (WG) modes.
I. INTRODUCTION
S
EMICONDUCTOR microcavity lasers are of great interest recently because of their potential to realize ultra-low threshold or even thresholdless semiconductor lasers [1]. For semiconductor microcavity lasers, the volume of the laser cavity should be decreased as much as possible while the cavity quality factor should be still kept high. These conditions can be satisfied by using suitable cavity structures with high-index contrast or using defects in photonic crystals [2], [3]. Up to now various cavity structures have been proposed, of which the microdisk lasers based on whispering-gallery (WG) modes and are most successful. The threshold current as low as 40 the spontaneous emission factor as high as 0.1 were achieved [4], [5]. In order to realize directed laser output, deformed microdisk lasers were attempted [6], [7]. However, the WG modes of microdisks are degenerate modes. To realize single mode operation, etching gratings along the periphery of the microdisk was proposed to eliminate the degeneracy [8]. Polygon resonators are of great interest recently for their potential use in microlasers and add–drop filters [9]–[14]. Equilateral triangle resonators were analyzed and high-quality-factor modes were found in these resonators with incident angles larger than the critical angle onto the triangle side [9]. WG modes in hexagon resonators were analyzed by the wave-matching method [10] and the boundary element method [11]. Hexagonal microlasers based on organic dyes in nanoporous crystals were realized Manuscript received April 1, 2003; revised May 14, 2003. This work was supported by the National Nature Science Foundation of China under Grant 60225011 and by the Major State Basic Research Program under Grant G2000036606. The authors are with State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China. Digital Object Identifier 10.1109/JQE.2003.816090
[10]. Square resonators based on square-shaped silica fiber with the side length of were used as filters and high coupling efficiency from a prism to the WG-like modes of the square resonators was obtained [12]. The free spectral range (FSR) of the spectra radiated from the square-shaped fiber edge was in good agreement with that of the ray optics analysis [12], [13]. Conventionally, we consider the square resonator as a Fabry-Pérot (F-P) cavity, and do not think that it would have high quality factor modes as its length approaches micron-scale. However, numerical simulations show that high quality factor modes still exist if the electric field perpendicular to the two-dimensional (2-D) square resonator is null along the square diagonals [14]. In this paper, we investigate the mode characteristics for 2-D square resonators analytically and calculate the corresponding mode quality factors by the FDTD method [15] and the Padé approximation method [16]. The results show that some modes with high quality factors exist in the square resonators. Based on the mode distributions, we find that the high-quality-factor modes are nondegenerate modes of the square resonator, and the electric field of the transverse magnetic (TM) mode or magnetic field of the transverse electric (TE) mode perpendicular to the 2-D resonator is null along the square diagonals. Theoretical analysis indicates that the high-quality-factor modes are the WG-like modes of the square resonator, and analytical mode frequencies and distributions agree with the numerical results very well. A perfect electric wall for the TM mode or perfect magnetic wall for the TE mode is assumed to exist at the square diagonals, which provides not only the transverse mode confinement but also the longitudinal mode selection. The resulting WG-like modes of square resonators are nondegenerate high-quality-factor modes with the longitudinal mode spacing twice that determined by the cavity length. II. THEORETICAL ANALYSIS A 2-D square resonator with side length of as shown in and Fig. 1(a) is considered, with the refractive index of in the inner and outer of the resonator. The electric field of its TM mode and magnetic field of its TE mode, both defined in the -direction shown in Fig. 1(a), are assumed to be null along the square diagonals, which means that a perfect electric ) for the TM mode or a perfect magnetic wall wall ( ) for the TE mode is assumed to exist at the square ( diagonals. Considering the first quarter of the square resonator shown in Fig. 1(b), we assume one wave incident from OA and propagating in the -direction with the transverse distribution totally confined in the line OA. Omitting the time dependent
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permeability and permittivity, for the TM mode and for the TE mode. The magnitude of the longitudinal that can be regarded as a component is of the order of first-order small quantity relative to . In the following, we take the zeroth-order approximation to neglect the longitudinal component and consider the incident wave as a pure transverse electromagnetic (TEM) wave. The incident wave would be reflected by the square side AB and then propagated in the -direction to OB. The field and the transverse component of the field of the reflected wave are given by (6) (7) We have assumed implicitly that the reflection does not induce any loss and the total phase variation is . Based on the continuous condition of the field at the boundary AB, the field of the transmitted wave that is exponentially decayed in the -direction can be written as
(a)
(8) (b) Fig. 1. (a) Illustration of the square resonator. The operators of the point group as well as the coordinates are also indicated. (b) Illustration of the first quarter of the square resonator.
C
factor , the -direction field component of the incident wave can be written as
where the zeroth-order approximation is taken to neglect small . For satisfying the Helmholtz equation, the items containing decaying constant can be written as (9) field component in the -direction of the transmitted The wave is
(1)
(10)
where represents the electric field for the TM mode or magnetic field for the TE mode (called field in the following), is the longitudinal propagation constant, is the initial phase, and is the transverse distribution of the incident wave
and for the TM and TE modes, respectively. where From the boundary condition the tangential field at the AB boundary should also be continuous, which yields
(2)
(11) From (11), we can derive
, is the transverse mode number and where is the transverse propagation constant
(12) (3)
The propagation constants satisfy (4) is the wavenumber in vacuum. The magnetic where field of the TM mode or electric field of the TE mode (called field in the following) has a transverse component in the -direction and a longitudinal component in the -direction. The transverse component is (5) where for the TE mode,
for the TM mode and and are the vacuum
Considering the continual propagation of the wave inside the resonator, it will arrive at OA after four reflections and the accumulated phase shift should be a multiple of , i.e., (13) , is the longitudinal mode number. The where total field in the first quarter of the square resonator is: the sum of the incident and reflected wave inside the triangle OAB (in) and transmitted wave outside the triangle (indidicated by ). Hence, the field in this quarter can be written cated by as follows: (14)
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where the function has the value of 1 for points inside the indicated regime and 0 outside. The field of the whole resonator can be simply expressed in terms of the operators of the point that describes the symmetry of a square [17]. For group each group element , we define its corresponding operator as
TABLE I CHARACTER TABLE FOR THE A AND B IRREDUCIBLE REPRESENTATIONS OF THE POINT GROUP C
(15) where represents the 2 2 matrix corresponding to and is a scalar function. The field of the whole resonator can be given by (16) is the phase shift accumuwhere lated as the wave propagates from one quarter to the next quarter of the square resonator, the subscript “ ” denotes the field for the whole resonator instead of in the first quarter. Considering the wave propagation in the opposite direction, we can obtain the other distribution (17) (a)
The waves of (16) and (17) can form the following standing waves: (18) (19) where the upper indicators “ ” and “ ” mean even and odd modes, respectively. It can be proven that these two standing operator. In the waves have definite parities relative to the first quarter, we have
(b) Fig. 2. Intensity spectra for (a) TE modes and (b) TM modes in a 2- -side-length square resonator with the refractive index of 3.2. The spectra are calculated by the FDTD technique and Padé approximation method. The peaks are marked by (m; l) for indicating the transverse number m and longitudinal mode number l.
m
(20) Using the group operators, we can express the standing-wave distributions for the whole resonator as (21) (22)
As we know, the standing waves can stably exist only if the perfect electric or magnetic wall is on the wave nodes, which means that the square diagonals should be nodes of the standing-wave distributions. Considering the standing-wave distributions in (20)–(22), we find that the longitudinal mode and ( is an integer) number should have the form for the even and odd modes, i.e., cosine and sine functions in (20), respectively. For other , the standing waves cannot exist because the square diagonals are not located at the wave nodes. After simple manipulations of the mode distributions, irreducible reprewe find that the even modes form the and the odd modes form sentation of the point group representation [17], whose corresponding characters the are listed in Table I. Hence, for the WG-like modes in square resonators, the longitudinal mode number should be an even
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TABLE II MODE FREQUENCIES AND QUALITY FACTORS OF THE 2- -SIDE-LENGTH SQUARE RESONATOR CALCULATED ANALYTICALLY AND NUMERICALLY BY THE FDTD TECHNIQUE AND PADÉ APPROXIMATION METHOD
m
(a)
(b)
(c)
(d)
and (b) TE modes and the Fig. 3. Magnetic field for the (a) TE electric field for the (c) TM and (d) TM modes calculated by the FDTD technique in the square resonator with the side length of 2 m and refractive index of 3.2.
(a)
(b)
(c)
(d)
integer, which means that the longitudinal mode spacing is twice that determined by the cavity length. Furthermore, the corresponding modes form the one-dimensional irreducible . representations of the point group III. NUMERICAL SIMULATION Using the FDTD technique, we solve the Maxwell equations for a 2-D square resonator with the refractive index of 3.2 inside .A and 1.0 outside the resonator and the side length of 2 uniform mesh with cell size of 10 nm, a time step just equal to the Courant limit [15], and a 20-cell PML absorbing layer [18] are used in the simulation. First, we add pulses with wide spectra to points inside the square with low symmetries to excite all possible modes, and then we use the Padé approximation method [16] to calculate the intensity spectra, which are plotted in Fig. 2(a) and (b) for TE and TM modes, respectively. We find that there exist many low quality factor modes with the quality factors calculated to be around 200 and several high-qualityfactor modes whose mode frequencies and quality factors are listed in Table II. By combining (3), (4), (9), (12), and (13), we also calculate the frequencies of the WG-like modes of the square resonator analytically and compare them with the FDTD results. We find that the high-quality-factor modes correspond to the WG-like modes very well. The average difference between mode frequencies is about 0.9% for the fundamental transverse modes ( ) and 1.5% for the first-order transverse modes ) for the square resonator with side-length of 2 m. The ( detailed comparisons are presented in Table II.
and (b) Analytical results of the magnetic field for the (a) TE TE modes and the electric field for the (c) TM and (d) TM modes in the square resonator with the side length of 2 m and refractive index of 3.2. Fig. 4.
To calculate mode distributions by the finite difference time domain (FDTD) method, we set the central frequencies of the excitation pulses to the mode frequencies and compress the pulse spectrum width narrow enough to ensure that just one mode can be excited. In Fig. 3, we illustrate the magnetic field and modes in (a) and (b) and electric field for the and mode in (c) and (d). It is clearly for the shown that they have odd parities relative to the diagonal mirror planes. Furthermore, by manipulating the figures according to the group operators, we can prove that the (0, 14) mode form irreducible representation of and the (0, 16) mode the form the representation for both TE and TM modes. From (20)–(22), we can get the mode distributions analytically and illustrate the results in Fig. 4. It can be seen that they agree with the FDTD results very well, which proves that our model describes the WG-like modes of a square resonator well.
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IV. CONCLUSION A theoretical model that can give the spectral location and wave pattern of WG-like modes in square resonators is presented. This model is suitable for square resonators with side lengths approaching wavelength scale, which is not available for the ray optics analysis. The WG-like modes in square and irreducible representations resonators that form the of the point group are nondegenerate modes; furthermore, their mode spacing is twice that determined by the cavity length and their quality factors are much higher than other modes in square resonators. All of these characteristics make square resonators suitable for fabricating single-mode low-threshold semiconductor microcavity lasers. Considering that the flat square sidewall allows a longer interaction length, and therefore a wider gap distance for evanescent coupling between the cavity and the straight waveguide, square resonators are also suitable for fabricating add–drop filters [12]–[14].
[14] C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron., vol. 35, pp. 1322–1331, 1999. [15] A. Taflove, Computational Electrodynamics—The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1995. [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 Microwave Wireless Components Lett., vol. 11, pp. 223–225, 2001. [17] B. S. Wherrett, Group Theory for Atoms, Molecules and Solids. London, U.K.: Prentice-Hall, 1986. [18] J. P. Berenger, “A perfectly matched layer for the absorption of eletromagnetic waves,” J. Computational Phys., vol. 114, pp. 185–200, 1994.
Wei-Hua Guo was born in Hubei Province, China, in 1976. He received the B.Sc. degree in physics from Nanjing University, Nanjing, China, in 1998. He is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, studying the FDTD simulation and fabrication of photonic microcavities, phtonic crystals, and semiconductor optical amplifiers.
REFERENCES [1] T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Select. Topics Quantum Electron., vol. 3, pp. 808–830, 1997. [2] 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. 289–291, 1992. [3] O. Painter, J. Vuˇckovic´, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Amer. B, vol. 16, pp. 275–285, 1999. [4] M. Fujita, R. Ushigome, and T. Baba, “Continuous wave lasing in GaInAsP microdisk laser with threshold of 40 A,” Electron. Lett., vol. 27, pp. 790–791, 2000. , “Large spontaneous emission factor of 0.1 in a microdisk injection [5] laser,” IEEE Photon. Technol. Lett., vol. 13, pp. 403–405, 2001. [6] J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett., vol. 75, pp. 2678–2681, 1995. [7] C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science, vol. 280, pp. 1556–1564, 1998. [8] M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering galley mode microgear cavity,” IEEE J. Quantum Electron., vol. 37, pp. 1253–1258, 2001. [9] Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron., vol. 37, pp. 100–107, 2001. [10] I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, “Hexagonal microlasers based on organic dyes in nanoporous crystals,” Appl. Phys. B, vol. 70, pp. 335–343, 2000. [11] J. Wiersig, “Boundary element method for resonators in dielectric microcavities,” J. Opt. A: Pure Appl. Opt., vol. 5, pp. 53–60, 2003. [12] Y. L. Pan and R. K. Chang, “Highly efficient prism coupling to whispering gallery modes of a square cavity,” Appl. Phys. Lett., vol. 82, pp. 487–489, 2003. [13] A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett., vol. 26, pp. 632–634, 2001.
Yong-Zhen Huang (M’95–SM’01) was born in Fujian Province, China, in 1963. 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, China, where he worked on the tunneling time for quantum barriers, asymmetric Fabry–Pérot 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, and is the Vice Director of the Optoelectronic R& D Center. His current research interests involved microcavity lasers, semiconductor optical amplifiers, and photonic crystals.
Qiao-Yin Lu was born in Jiangsu Province, China, in 1974. She received the B.Sc. and M.Sc. degree in optics from Changchun Institute of Optics and Fine Mechanics, Changchun, China, in 1998 and 2001, respectively. She is currently working toward the Ph.D. degree at the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, studying the design and the fabrication of microcavity lasers.
Li-Juan Yu was born in Heilongjiang Province, China, in 1963. She received the B.S. and M.Sc. degrees in solid physics from Jilin University, Changchun, China, in 1986 and 1989, respectively, and the Ph.D. degree in microelectronics from Xi’an Jiaotong University, Xi’an, China, in 2000. She joined the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, in 2000, working on material growth by MOCVD and the fabrication process of semiconductor optical amplifier. She is currently an Associate Research Professor at the Institute of Semiconductors, Chinese Academy of Sciences, with research interests in the material growth and the process techniques for optoelectronic devices.
