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Mode frequency shifts and Q-factor Changes in 2D microflower cavity and its deformed cavity
Optics Communications 277 (2007) 406–410 www.elsevier.com/locate/optcom
Mode frequency shifts and Q-factor Changes in 2D microflower cavity and its deformed cavity Shanliang Qiu a b
a,*
, Jiaxing Cai a, Yongping Li
a,b,*
, Zhengfu Han
b
Department of Physics, University of Science and Technology of China, Hefei 230026, China Key Laboratory of Quantum Information, Chinese Academy of Science, Hefei 230026, China Received 4 February 2007; received in revised form 13 May 2007; accepted 14 May 2007
Abstract Mode frequency shifts and Q-factor changes in 2D microflower cavity and its deformed cavity are analyzed. The effective mode-splitting of double-degenerate WG modes is obtained and the Q-factor changes of matched and mismatched modes are discussed for the microflower cavity. The Q-factor stability of the splitted WGH(8,1) modes due to two types of local deformations is studied, showing that the local deformations can badly spoil the mode Q-factor if the deformations are not controlled properly. The output directionality of the splitted WGH(8,1) modes due to the local deformations also is presented, and a basically unidirectional light output of OO mode under local deformation DA (deformation happens at one ‘‘valley’’ of the microflower cavity) is obtained. 2007 Elsevier B.V. All rights reserved. Keywords: Microflower cavity; Deformed cavity; Q-factor stability; Directional output; Microcavity; Whispering-gallery modes
1. Introduction Dielectric microcavities have long become essential components for a variety of applications, such as low-threshold semiconductor lasers [1–3], narrow-linewidth wavelengthselective filters [4], sensitive optical sensors [4–6], cavity quantum electro-dynamics (QED) [7] and so on. A very large range of microcavities with different geometric shapes have been studied theoretically and experimentally in resent years [1–4,8–29]. Microdisk and microcylinder cavity, because of their simple geometries, have been studied widely, and are well known for their whispering-gallery modes (WGMs) with high Q-factor. The WGMs propagate along the circumference of the cavity denoted by WG(m,n) (characterized by two indices, one is radial mode number n, the other azimuthal mode number m). Due to rotational symmetry of the cavity, the WGMs are double-degenerate *
Corresponding authors. E-mail addresses: [email protected] (S. Qiu), [email protected] (Y. Li). 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.037
(except for m = 0). The double-degeneracy and the multi high Q-factor WGMs in microdisk cavity cause high lasing threshold and low lasing efficiency. So Fujita and Baba proposed a new structure which was added a Bragg grating with an azimuthal order twice the order m of the WGM [28], this structure is shown in Fig. 1a, and they termed it microgear cavity. Mode-selecting power and mode-splitting of the double-degenerate WGM for microgear cavity have been demonstrated by the finite-different time-domain method (FDTD) [28] and a frequency-domain method [29]. But mode Q-factor stability and output directionality of the cavity modes have not been demonstrated, which is important in view of the use of microgear cavity as microlasers. Usually, the scattering loss due to surface roughness (including sidewall roughness) introduced by fabrication implementation not only is a dominant loss mechanism in semiconductor microcavities [3] but also determinates the output direction in the rotationally symmetric cavities. The sidewall roughness can be modeled as small shape deformation of the cavity boundary. Mode Q-factor stability and output directionality of cavity modes under small
S. Qiu et al. / Optics Communications 277 (2007) 406–410
407
shapes that can be deduced analytically, all other cavities need numerical analysis. In our microflower cavity case, the analytical analysis is impossible, so we use a high efficient and accurate numerical method developed by author Boriskina et al. [29]. The microflower cavity has a boundary described by: qðuÞ ¼ að1 þ d cosðmuÞÞ
Fig. 1. (a) Shape of microgear cavity. (b) Shape of microflower cavity. (c) The resonant frequency shifts and the Q-factor variations of WGMs vs. normalized modulation depth. The curves marked by black points are for the splitted WGH(8,1) modes (where H stands for a transverse magnetic field, so the field is TM polarized), the curves marked by circles is for the WGH(9,1) mode; the curves marked by black squares is for the WGH(7,1) mode; the Q-factors of the three WGMs at zero modulation are Q = 9.176 · 104, Q = 4.319 · 105, Q = 1.988 · 104, respectively.
cavity shape deformation should be studied before cavities are used. In this Letter, we study the microflower cavity in detail, which has been used as a model for microgear cavity [4,29] and demonstrated that it can split the double-degenerate WGMs effectively. We investigate the mode-splitting of the matched WGM and Q-factors’ spoilage of the mismatched WGMs (or mode-selecting power) for the microflower cavity. Also, we study the mode Q-factor stability and the output directionality for the microflower cavity with two types of small local boundary deformations. 2. Mode-splitting and mode-selecting in microflower cavity Cavity mode problem is a three-dimensional (3D) electromagnetic field problem, but with the help of effectiveindex approximation [2,29], the initial 3D problem of thin microcavity can be simplified to a 2D problem. A 2D electromagnetic field problem has two polarization types. One has its electric field in the 2D plane denoted by TE (transverse electric field) polarization and the other its magnetic field in the 2D plane denoted by TM polarization (transverse magnetic field). Except some cavities with simple
ð1Þ
where q, u are shown in Fig. 1b, a the characteristic size of cavity, d the normalized corrugation depth or modulation depth (as d = 0, the cavity is an ideal diskcavity or circular cavity), and m is the corrugation frequency or modulation frequency and it should satisfy: m = 2m (m is the azimuthal mode number of WGMs) [28]. In our analysis, we focus on TM polarization, the analysis for TE polarization is similar, and we adopt a typical effective refractive index for TM polarization of thin microcavities used in Ref. [29]: nc = 3.2, and choose the characteristic size of cavity a = 0.85 lm so that the wavelength of WG(m = 8, n = 1)mode (k = 1.5539 lm) closes to communication wavelength 1.55 lm as the cavity has no modulation. Fig. 1c, and d gives the Q-factor variations (changes) and resonant wavelength shifts of the first radial WG modes with the normalized modulation depth, which have wavelength in the range of 1.3–1.8 lm at zero modulation. A notable mode-splitting between the double-degenerate WGH(8,1) mode can be found from Fig. 1c as the modulation depth is large enough. The spiltted WGMs have been classified into two types according to the symmetry of field distribution with respect to x axis and y axis. One has an even–even symmetry and denoted by EE mode, the other an odd–odd symmetry and denoted by OO mode, all of these can be easily found from the field intensity distribution patterns of the two splitted WGMs shown in Fig. 2. When the modulation depth reaches 0.1, about 157 nm wavelength-splitting can be achieved. The splitting always increases with the modulation depth, this is different from the TE polarization given in Refs. [4,28,29], so Q-factor variations exhibit different features obviously, there is no improvement of Q-factor for EE mode. But here, the improvement of Q-factor of EE mode is not important, the spoilage of Q-factors is not serious and Q-factor of the double-degenerate WGH(8,1) mode is high enough for applications, the more important thing is to get an effective splitting of the double-degenerate mode. Which one of the two splitted modes is more suitable for using as the working mode in practice, the fabrication accuracy and requirement to desired Q-factor should be considered together, the one has a better Q-factor stability to surface roughness may easily realizes high Q-factor resonant mode practically. The mismatched modes WGH(7,1) and WGH(9,1) are also considered in Fig. 1c. Surprisingly the two WGMs remain double degenerate within allowed numerical error. The Q-factor spoilage of WGH(9,1)mode can be easily seen from the figure. The spoilage to WGH(7,1) mode is not obvious, but it is not important to practical applications
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Fig. 2. Intensity patterns for z component of electric filed of splitted WGH(8,1) modes, the former is for EE mode, the latter is for OO mode, and the figure is given in d = 0.2.
because the Q-factor of this mode is smaller about 80% than WGH(8,1) mode’s at zero modulation. Obviously, the mode-selecting power is strong in microflower cavity as long as the modulation depth is large enough. 3. Q-factor stability and output directionality in deformed microflower cavity Due to the importance of Q-factor stability of cavity modes, we introduce some local deformations on the boundary of microflower cavity to simulate sidewall surface roughness of cavity boundary. We also hope it can lead to directional output of cavity modes. Here, two kinds of local deformations are studied, one has a deformation at the protuberant ‘‘petal’’ of the microflower cavity, it can cause the ‘‘petal’’ become longer or shorter, as shown in Fig. 3a; the other a deformation at the ‘‘valley’’ between two adjacent ‘‘petals’’, it can cause the ‘‘valley’’ to become deeper or shallower, as shown in Fig. 3b. For convenience, we denote the two deformations DA and DB, respectively. The two deformations can be described by a set of unified equations: qðuÞ ¼ a½1 þ dðuÞ cosðmuÞ n u 2N o 2pu 2N db ðuÞ ¼ d0 þ d1 eð lÞ þ eð l Þ
ð2Þ
u 2 ½0; 2pÞ dðuÞ ¼ db ðu aÞ where q(u), u, and m are similar to those in Eq. (1); d0 are the modulation depth, and it causes a diskcavity to be a microflower cavity, here d0 = 0.1; d1 is the local deformation parameter that introduces a additional local deformation on the boundary of microflower cavity as we have discussed above; N the order of Super-Gauss function in the expression of db(u), we have made N = 1 in our computation so that a smooth enough boundary can be generated, and it is necessary in using the numerical method
Fig. 3. (a) DA deformation (deformation happen at the protuberant ‘‘petal’’ of microflower cavity). (b) DB deformation (deformation happen at the ‘‘valley’’ between two adjacent ‘‘petals’’ of the microflower cavity). (c) The Q-factor variations and resonant wavelength shifts of the splitted WGH(8,1) modes with the local deformation parameter, the curves marked by black points are for DA deformation, the curves marked by black squares are for DB deformation, the upper is for EE mode, the lower is for OO mode.
presented in Ref. [29]; a the translation parameter, when a = 0, deformation DA is obtained, when a = p/m, deformation DB is given; and l is the angular width for Super-Gauss function, here it has been chosen at l = p/2m
S. Qiu et al. / Optics Communications 277 (2007) 406–410
so that deformations only happen at ‘‘teeth’’ of the microgear cavity. The Q-factor variations and resonant wavelength shifts of the splitted WGH(8,1) modes with the local deformation parameter are shown in Fig. 3c. The Q-factor stability for the two splitted WGH(8,1) modes can be easily seen. When deformation DA happens, the EE mode has bad Q-factor stability, and about 5% (2%) deformation of this type can lead to 95% (75%) Q-factor reduction, but the OO mode has good Q-factor stability; on the contrary, when deformation DB happens, the EE mode has good Q-factor stability, even some improvement can be achieved, but the OO mode has bad Q-factor stability; about 5% (2%) deformation of this type can lead to 82% (47%) reduction. A little steadier of Q-factor of OO mode than EE mode’s under the two kinds of deformations can be concluded from above analysis and Fig. 3c. Although surface roughness introduced by fabrication implementation is very complex (surface roughness may happen in every ‘‘petal’’ and every ‘‘valley’’), some halfquantitative conclusions can be obtained based on above analysis. In order to keep the Q-factor spoilage less than a magnitude, at least fabrication error should be controlled within several percent (5%). The resonant frequency shifts for the two splitted WGH(8,1) modes are not obvious under the local deformations as shown in Fig. 3c, and this
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agrees the expectation. Besides, we also find the doubledegenerate mismatched modes WGH(7,1) and WGH(9,1) happen small splitting under the two local deformations, but this is not presented in our Fig. 3. As we have mentioned above, a small surface roughness or local boundary deformation can lead to directional light output of the cavity modes, this can be demonstrated again in our work. Fig. 4 gives the farfield output directionality of the two splitted WGH(8,1) modes among the above two types of deformations. We have deliberately given the figures in different deformation parameters so that the Q-factors have no severe spoilage. From the results, we find when the local deformations have weak influence on cavity modes, output directionality is bad (Fig. 4b and c), and when deformations have strong influence on cavity modes, a better output directionality can be obtained (Fig. 4a and d). Especially, Fig. 4d shows that under deformation DB, OO mode can obtain a basically unidirectional output (about 30 divergence at the principal output peak), and this could be applied to the microlasers design. It is very hard to get a desired output directionality though the deformation principle is easy to understand. Limit to paper space, further research on output directionality of deformed microgear cavity will not be discussed here.
Fig. 4. Farfield output directionality of the splitted WGH(8,1) modes under the two local deformations. (a) Output directionality of EE mode under deformation DA at d1 = 0.03. (b) Output directionality of OO mode under deformation DA at d1 = 0.1. (c) Output directionality of EE mode under deformation DB at d1 = 0.1. (d) Output directionality of OO mode under deformation DB at d1 = 0.06.
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4. Summary In conclusion, we analyzed the resonant frequency shifts and the Q-factor changes for microflower cavity and its deformed cavity. A notable wavelength-splitting has been found in TM polarization for microflower cavity, and the Q-factor improvement for EE mode does not happen in this cavity in TM polarization. The result of Q-factor stability of splitted WGH(8,1) modes under two types of local deformations indicates that local deformation can lead to a severe spoilage of Q-factor if the local deformation has an obvious influence on the mode, about several percent deformation (5%) can lead to a magnitude reduction of Q-factors. Some directional outputs of cavity modes also are obtained under the local deformations, especially in the deformation at the ‘‘valley’’ (deformation DA), OO mode obtains about 30 unidirectional light emission at d1 = 0.06. Acknowledgements The authors thank Prof. Yongping Li and Prof. Zhengfu Han for helpful discussions. References [1] S.L. McCall, A.F.J. Levi, R.E. Slusher, S.J. Pearton, R.A. Logan, Appl. Phys. Lett. 60 (1992) 289. [2] M.K. Chin, D.Y. Chu, S.T. Ho, J. Appl. Phys. 75 (1994) 3302. [3] R.E. Slusher, A.F.J. Levi, S.L. McCall, S.J. Pearton, R.A. Logan, Appl. Phys. Lett. 63 (1993) 1310. [4] S.V. Boriskina, T.M. Benson, P. Sewell, A.I. Nosich, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 1175. [5] J. Yang, L.J. Guo, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 143.
[6] R.W. Boyd, J.E. Heebner, Appl. Opt. 40 (2001) 572. [7] K.J. Vahala, Nature 424 (2003) 839. [8] Y.Z. Huang, W.H. Guo, Q.M. Wang, IEEE Quantum Electron. 37 (2001) 100. [9] Q.Y. Lu, X.H. Chen, W.H. Guo, L.J. Yu, Y.Z. Huang, J. Wang, Y. Luo, IEEE Photon. Technol. Lett. 16 (2004) 359. [10] M.S. Kurdoglyan, S.Y. Lee, S. Rim, C.M. Kim, Opt. Lett. 29 (2004) 2758. [11] A.W. Poon, F. Courvoisier, R.K. Chang, Opt. Lett. 26 (2001) 632. [12] W.H. Guo, Y.Z. Huang, Q.Y. Li, L.J. Yu, IEEE J. Quantum Electron. 39 (2003) 1563. [13] T. Ling, L.Y. Liu, Q.H. Song, L. Xu, W.C. Wang, Opt. Lett. 28 (2003) 1784. [14] J.U. No¨ckel, A.D. Stone, Nature 385 (1997) 45. [15] H.G.L. Schwefel, N.B. Rex, H.E. Tureci, R.K. Chang, A.D. Stone, et al., J. Opt. Soc. Am. B 21 (2004) 923. [16] V. A. Podolskiy, E. E. Narimanov 30 (2005) 474. [17] J.U. No¨ckel, A.D. Stone, G. Chen, H.L. Grossman, R.K. Chang, Opt. Lett. 21 (1996) 1609. [18] S.Y. Lee, S. Rim, J.W. Ryu, T.Y. Kwon, M. Choi, C.M. Kim, Phys. Rev. Lett. 93 (2004) 164102. [19] G.D. Chern, H.E. Tureci, A.D. Stone, R.K. Chang, Appl. Phys. Lett. 83 (2003) 1710. [20] W. Fang, A. Yamilov, H. Cao, Phys. Rev. A 72 (2005) 023815. [21] J. Wiersig, M. Hentschel, Phys. Rev. A 73 (2006) 031802. [22] M.L. Gorodetsky, A.A. Savchenkov, V.S. Ilchenko, Opt. Lett. 21 (1996) 453. [23] Y. Han, L. Me´e`s, G. Gouesbet, Z. Wu, G. Gre´han, J. Opt. Soc. Am. B 23 (2006) 1390. [24] T. Nobis, M. Grundmann, Phys. Rev. A 72 (2005) 063806. [25] I. Braun, G. Ihlein, F. Laeri, J.U. No¨ckel, G. Schulz-Ekloff, F. ¨ . Weiß, D. Wo¨hrle, Appl. Phys. B 70 (2000) Schu¨th, U. Vietze, O 335. [26] G. Gouesbet, S. Meunier-Guttin-Cluzel, G. Gre´han, Opt. Commun. 201 (2002) 223. [27] G. Gouesbet, S. Meunier-Guttin-Cluzel, G. Grehan, Phys. Rev. E 65 (2001) 016212. [28] M. Fujita, T. Baba, IEEE J. Quantum Electron. 37 (2001) 1253. [29] S.V. Boriskina, P. Sewell, T.M. Benson, A.I. Nosich, J. Opt. Soc. Am. A 21 (2004) 393.
Mode frequency shifts and Q-factor Changes in 2D microflower cavity and its deformed cavity Shanliang Qiu a b
a,*
, Jiaxing Cai a, Yongping Li
a,b,*
, Zhengfu Han
b
Department of Physics, University of Science and Technology of China, Hefei 230026, China Key Laboratory of Quantum Information, Chinese Academy of Science, Hefei 230026, China Received 4 February 2007; received in revised form 13 May 2007; accepted 14 May 2007
Abstract Mode frequency shifts and Q-factor changes in 2D microflower cavity and its deformed cavity are analyzed. The effective mode-splitting of double-degenerate WG modes is obtained and the Q-factor changes of matched and mismatched modes are discussed for the microflower cavity. The Q-factor stability of the splitted WGH(8,1) modes due to two types of local deformations is studied, showing that the local deformations can badly spoil the mode Q-factor if the deformations are not controlled properly. The output directionality of the splitted WGH(8,1) modes due to the local deformations also is presented, and a basically unidirectional light output of OO mode under local deformation DA (deformation happens at one ‘‘valley’’ of the microflower cavity) is obtained. 2007 Elsevier B.V. All rights reserved. Keywords: Microflower cavity; Deformed cavity; Q-factor stability; Directional output; Microcavity; Whispering-gallery modes
1. Introduction Dielectric microcavities have long become essential components for a variety of applications, such as low-threshold semiconductor lasers [1–3], narrow-linewidth wavelengthselective filters [4], sensitive optical sensors [4–6], cavity quantum electro-dynamics (QED) [7] and so on. A very large range of microcavities with different geometric shapes have been studied theoretically and experimentally in resent years [1–4,8–29]. Microdisk and microcylinder cavity, because of their simple geometries, have been studied widely, and are well known for their whispering-gallery modes (WGMs) with high Q-factor. The WGMs propagate along the circumference of the cavity denoted by WG(m,n) (characterized by two indices, one is radial mode number n, the other azimuthal mode number m). Due to rotational symmetry of the cavity, the WGMs are double-degenerate *
Corresponding authors. E-mail addresses: [email protected] (S. Qiu), [email protected] (Y. Li). 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.037
(except for m = 0). The double-degeneracy and the multi high Q-factor WGMs in microdisk cavity cause high lasing threshold and low lasing efficiency. So Fujita and Baba proposed a new structure which was added a Bragg grating with an azimuthal order twice the order m of the WGM [28], this structure is shown in Fig. 1a, and they termed it microgear cavity. Mode-selecting power and mode-splitting of the double-degenerate WGM for microgear cavity have been demonstrated by the finite-different time-domain method (FDTD) [28] and a frequency-domain method [29]. But mode Q-factor stability and output directionality of the cavity modes have not been demonstrated, which is important in view of the use of microgear cavity as microlasers. Usually, the scattering loss due to surface roughness (including sidewall roughness) introduced by fabrication implementation not only is a dominant loss mechanism in semiconductor microcavities [3] but also determinates the output direction in the rotationally symmetric cavities. The sidewall roughness can be modeled as small shape deformation of the cavity boundary. Mode Q-factor stability and output directionality of cavity modes under small
S. Qiu et al. / Optics Communications 277 (2007) 406–410
407
shapes that can be deduced analytically, all other cavities need numerical analysis. In our microflower cavity case, the analytical analysis is impossible, so we use a high efficient and accurate numerical method developed by author Boriskina et al. [29]. The microflower cavity has a boundary described by: qðuÞ ¼ að1 þ d cosðmuÞÞ
Fig. 1. (a) Shape of microgear cavity. (b) Shape of microflower cavity. (c) The resonant frequency shifts and the Q-factor variations of WGMs vs. normalized modulation depth. The curves marked by black points are for the splitted WGH(8,1) modes (where H stands for a transverse magnetic field, so the field is TM polarized), the curves marked by circles is for the WGH(9,1) mode; the curves marked by black squares is for the WGH(7,1) mode; the Q-factors of the three WGMs at zero modulation are Q = 9.176 · 104, Q = 4.319 · 105, Q = 1.988 · 104, respectively.
cavity shape deformation should be studied before cavities are used. In this Letter, we study the microflower cavity in detail, which has been used as a model for microgear cavity [4,29] and demonstrated that it can split the double-degenerate WGMs effectively. We investigate the mode-splitting of the matched WGM and Q-factors’ spoilage of the mismatched WGMs (or mode-selecting power) for the microflower cavity. Also, we study the mode Q-factor stability and the output directionality for the microflower cavity with two types of small local boundary deformations. 2. Mode-splitting and mode-selecting in microflower cavity Cavity mode problem is a three-dimensional (3D) electromagnetic field problem, but with the help of effectiveindex approximation [2,29], the initial 3D problem of thin microcavity can be simplified to a 2D problem. A 2D electromagnetic field problem has two polarization types. One has its electric field in the 2D plane denoted by TE (transverse electric field) polarization and the other its magnetic field in the 2D plane denoted by TM polarization (transverse magnetic field). Except some cavities with simple
ð1Þ
where q, u are shown in Fig. 1b, a the characteristic size of cavity, d the normalized corrugation depth or modulation depth (as d = 0, the cavity is an ideal diskcavity or circular cavity), and m is the corrugation frequency or modulation frequency and it should satisfy: m = 2m (m is the azimuthal mode number of WGMs) [28]. In our analysis, we focus on TM polarization, the analysis for TE polarization is similar, and we adopt a typical effective refractive index for TM polarization of thin microcavities used in Ref. [29]: nc = 3.2, and choose the characteristic size of cavity a = 0.85 lm so that the wavelength of WG(m = 8, n = 1)mode (k = 1.5539 lm) closes to communication wavelength 1.55 lm as the cavity has no modulation. Fig. 1c, and d gives the Q-factor variations (changes) and resonant wavelength shifts of the first radial WG modes with the normalized modulation depth, which have wavelength in the range of 1.3–1.8 lm at zero modulation. A notable mode-splitting between the double-degenerate WGH(8,1) mode can be found from Fig. 1c as the modulation depth is large enough. The spiltted WGMs have been classified into two types according to the symmetry of field distribution with respect to x axis and y axis. One has an even–even symmetry and denoted by EE mode, the other an odd–odd symmetry and denoted by OO mode, all of these can be easily found from the field intensity distribution patterns of the two splitted WGMs shown in Fig. 2. When the modulation depth reaches 0.1, about 157 nm wavelength-splitting can be achieved. The splitting always increases with the modulation depth, this is different from the TE polarization given in Refs. [4,28,29], so Q-factor variations exhibit different features obviously, there is no improvement of Q-factor for EE mode. But here, the improvement of Q-factor of EE mode is not important, the spoilage of Q-factors is not serious and Q-factor of the double-degenerate WGH(8,1) mode is high enough for applications, the more important thing is to get an effective splitting of the double-degenerate mode. Which one of the two splitted modes is more suitable for using as the working mode in practice, the fabrication accuracy and requirement to desired Q-factor should be considered together, the one has a better Q-factor stability to surface roughness may easily realizes high Q-factor resonant mode practically. The mismatched modes WGH(7,1) and WGH(9,1) are also considered in Fig. 1c. Surprisingly the two WGMs remain double degenerate within allowed numerical error. The Q-factor spoilage of WGH(9,1)mode can be easily seen from the figure. The spoilage to WGH(7,1) mode is not obvious, but it is not important to practical applications
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Fig. 2. Intensity patterns for z component of electric filed of splitted WGH(8,1) modes, the former is for EE mode, the latter is for OO mode, and the figure is given in d = 0.2.
because the Q-factor of this mode is smaller about 80% than WGH(8,1) mode’s at zero modulation. Obviously, the mode-selecting power is strong in microflower cavity as long as the modulation depth is large enough. 3. Q-factor stability and output directionality in deformed microflower cavity Due to the importance of Q-factor stability of cavity modes, we introduce some local deformations on the boundary of microflower cavity to simulate sidewall surface roughness of cavity boundary. We also hope it can lead to directional output of cavity modes. Here, two kinds of local deformations are studied, one has a deformation at the protuberant ‘‘petal’’ of the microflower cavity, it can cause the ‘‘petal’’ become longer or shorter, as shown in Fig. 3a; the other a deformation at the ‘‘valley’’ between two adjacent ‘‘petals’’, it can cause the ‘‘valley’’ to become deeper or shallower, as shown in Fig. 3b. For convenience, we denote the two deformations DA and DB, respectively. The two deformations can be described by a set of unified equations: qðuÞ ¼ a½1 þ dðuÞ cosðmuÞ n u 2N o 2pu 2N db ðuÞ ¼ d0 þ d1 eð lÞ þ eð l Þ
ð2Þ
u 2 ½0; 2pÞ dðuÞ ¼ db ðu aÞ where q(u), u, and m are similar to those in Eq. (1); d0 are the modulation depth, and it causes a diskcavity to be a microflower cavity, here d0 = 0.1; d1 is the local deformation parameter that introduces a additional local deformation on the boundary of microflower cavity as we have discussed above; N the order of Super-Gauss function in the expression of db(u), we have made N = 1 in our computation so that a smooth enough boundary can be generated, and it is necessary in using the numerical method
Fig. 3. (a) DA deformation (deformation happen at the protuberant ‘‘petal’’ of microflower cavity). (b) DB deformation (deformation happen at the ‘‘valley’’ between two adjacent ‘‘petals’’ of the microflower cavity). (c) The Q-factor variations and resonant wavelength shifts of the splitted WGH(8,1) modes with the local deformation parameter, the curves marked by black points are for DA deformation, the curves marked by black squares are for DB deformation, the upper is for EE mode, the lower is for OO mode.
presented in Ref. [29]; a the translation parameter, when a = 0, deformation DA is obtained, when a = p/m, deformation DB is given; and l is the angular width for Super-Gauss function, here it has been chosen at l = p/2m
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so that deformations only happen at ‘‘teeth’’ of the microgear cavity. The Q-factor variations and resonant wavelength shifts of the splitted WGH(8,1) modes with the local deformation parameter are shown in Fig. 3c. The Q-factor stability for the two splitted WGH(8,1) modes can be easily seen. When deformation DA happens, the EE mode has bad Q-factor stability, and about 5% (2%) deformation of this type can lead to 95% (75%) Q-factor reduction, but the OO mode has good Q-factor stability; on the contrary, when deformation DB happens, the EE mode has good Q-factor stability, even some improvement can be achieved, but the OO mode has bad Q-factor stability; about 5% (2%) deformation of this type can lead to 82% (47%) reduction. A little steadier of Q-factor of OO mode than EE mode’s under the two kinds of deformations can be concluded from above analysis and Fig. 3c. Although surface roughness introduced by fabrication implementation is very complex (surface roughness may happen in every ‘‘petal’’ and every ‘‘valley’’), some halfquantitative conclusions can be obtained based on above analysis. In order to keep the Q-factor spoilage less than a magnitude, at least fabrication error should be controlled within several percent (5%). The resonant frequency shifts for the two splitted WGH(8,1) modes are not obvious under the local deformations as shown in Fig. 3c, and this
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agrees the expectation. Besides, we also find the doubledegenerate mismatched modes WGH(7,1) and WGH(9,1) happen small splitting under the two local deformations, but this is not presented in our Fig. 3. As we have mentioned above, a small surface roughness or local boundary deformation can lead to directional light output of the cavity modes, this can be demonstrated again in our work. Fig. 4 gives the farfield output directionality of the two splitted WGH(8,1) modes among the above two types of deformations. We have deliberately given the figures in different deformation parameters so that the Q-factors have no severe spoilage. From the results, we find when the local deformations have weak influence on cavity modes, output directionality is bad (Fig. 4b and c), and when deformations have strong influence on cavity modes, a better output directionality can be obtained (Fig. 4a and d). Especially, Fig. 4d shows that under deformation DB, OO mode can obtain a basically unidirectional output (about 30 divergence at the principal output peak), and this could be applied to the microlasers design. It is very hard to get a desired output directionality though the deformation principle is easy to understand. Limit to paper space, further research on output directionality of deformed microgear cavity will not be discussed here.
Fig. 4. Farfield output directionality of the splitted WGH(8,1) modes under the two local deformations. (a) Output directionality of EE mode under deformation DA at d1 = 0.03. (b) Output directionality of OO mode under deformation DA at d1 = 0.1. (c) Output directionality of EE mode under deformation DB at d1 = 0.1. (d) Output directionality of OO mode under deformation DB at d1 = 0.06.
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4. Summary In conclusion, we analyzed the resonant frequency shifts and the Q-factor changes for microflower cavity and its deformed cavity. A notable wavelength-splitting has been found in TM polarization for microflower cavity, and the Q-factor improvement for EE mode does not happen in this cavity in TM polarization. The result of Q-factor stability of splitted WGH(8,1) modes under two types of local deformations indicates that local deformation can lead to a severe spoilage of Q-factor if the local deformation has an obvious influence on the mode, about several percent deformation (5%) can lead to a magnitude reduction of Q-factors. Some directional outputs of cavity modes also are obtained under the local deformations, especially in the deformation at the ‘‘valley’’ (deformation DA), OO mode obtains about 30 unidirectional light emission at d1 = 0.06. Acknowledgements The authors thank Prof. Yongping Li and Prof. Zhengfu Han for helpful discussions. References [1] S.L. McCall, A.F.J. Levi, R.E. Slusher, S.J. Pearton, R.A. Logan, Appl. Phys. Lett. 60 (1992) 289. [2] M.K. Chin, D.Y. Chu, S.T. Ho, J. Appl. Phys. 75 (1994) 3302. [3] R.E. Slusher, A.F.J. Levi, S.L. McCall, S.J. Pearton, R.A. Logan, Appl. Phys. Lett. 63 (1993) 1310. [4] S.V. Boriskina, T.M. Benson, P. Sewell, A.I. Nosich, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 1175. [5] J. Yang, L.J. Guo, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 143.
[6] R.W. Boyd, J.E. Heebner, Appl. Opt. 40 (2001) 572. [7] K.J. Vahala, Nature 424 (2003) 839. [8] Y.Z. Huang, W.H. Guo, Q.M. Wang, IEEE Quantum Electron. 37 (2001) 100. [9] Q.Y. Lu, X.H. Chen, W.H. Guo, L.J. Yu, Y.Z. Huang, J. Wang, Y. Luo, IEEE Photon. Technol. Lett. 16 (2004) 359. [10] M.S. Kurdoglyan, S.Y. Lee, S. Rim, C.M. Kim, Opt. Lett. 29 (2004) 2758. [11] A.W. Poon, F. Courvoisier, R.K. Chang, Opt. Lett. 26 (2001) 632. [12] W.H. Guo, Y.Z. Huang, Q.Y. Li, L.J. Yu, IEEE J. Quantum Electron. 39 (2003) 1563. [13] T. Ling, L.Y. Liu, Q.H. Song, L. Xu, W.C. Wang, Opt. Lett. 28 (2003) 1784. [14] J.U. No¨ckel, A.D. Stone, Nature 385 (1997) 45. [15] H.G.L. Schwefel, N.B. Rex, H.E. Tureci, R.K. Chang, A.D. Stone, et al., J. Opt. Soc. Am. B 21 (2004) 923. [16] V. A. Podolskiy, E. E. Narimanov 30 (2005) 474. [17] J.U. No¨ckel, A.D. Stone, G. Chen, H.L. Grossman, R.K. Chang, Opt. Lett. 21 (1996) 1609. [18] S.Y. Lee, S. Rim, J.W. Ryu, T.Y. Kwon, M. Choi, C.M. Kim, Phys. Rev. Lett. 93 (2004) 164102. [19] G.D. Chern, H.E. Tureci, A.D. Stone, R.K. Chang, Appl. Phys. Lett. 83 (2003) 1710. [20] W. Fang, A. Yamilov, H. Cao, Phys. Rev. A 72 (2005) 023815. [21] J. Wiersig, M. Hentschel, Phys. Rev. A 73 (2006) 031802. [22] M.L. Gorodetsky, A.A. Savchenkov, V.S. Ilchenko, Opt. Lett. 21 (1996) 453. [23] Y. Han, L. Me´e`s, G. Gouesbet, Z. Wu, G. Gre´han, J. Opt. Soc. Am. B 23 (2006) 1390. [24] T. Nobis, M. Grundmann, Phys. Rev. A 72 (2005) 063806. [25] I. Braun, G. Ihlein, F. Laeri, J.U. No¨ckel, G. Schulz-Ekloff, F. ¨ . Weiß, D. Wo¨hrle, Appl. Phys. B 70 (2000) Schu¨th, U. Vietze, O 335. [26] G. Gouesbet, S. Meunier-Guttin-Cluzel, G. Gre´han, Opt. Commun. 201 (2002) 223. [27] G. Gouesbet, S. Meunier-Guttin-Cluzel, G. Grehan, Phys. Rev. E 65 (2001) 016212. [28] M. Fujita, T. Baba, IEEE J. Quantum Electron. 37 (2001) 1253. [29] S.V. Boriskina, P. Sewell, T.M. Benson, A.I. Nosich, J. Opt. Soc. Am. A 21 (2004) 393.