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Taper-microsphere coupling with numerical calculation of coupled-mode theory Chang-Ling Zou, Yong Yang, Chun-Hua Dong, Yun-Feng Xiao, Xiao-Wei Wu, Zheng-Fu Han,* and Guang-Can Guo Key Laboratory of Quantum Information, University of Science & Technology of China, Hefei 230026, Anhui, China *Corresponding author:
[email protected] Received July 14, 2008; revised August 30, 2008; accepted September 9, 2008; posted September 10, 2008 (Doc. ID 98738); published October 27, 2008 We have numerically studied a taper-microsphere coupling system with coupled-mode theory by introducing a mode-coupling mechanism. Transmission and reflection coefficients have been obtained directly through numerical calculation for a spherical silica microcavity. In the presence of the modal coupling mechanism, the optimal condition for transferring the light energy into the microsphere is no longer the critical coupling point. Instead, it should be in the under-coupling regime and can be reached by optimizing the size of the fiber taper and the taper-microsphere gap. © 2008 Optical Society of America OCIS codes: 140.3948, 230.4555.
1. INTRODUCTION High-quality-factor whispering gallery modes (WGMs) in microspheres [1] have exhibited more and more applications in the past few years, ranging from low-threshold lasers [2–5], polarization transmission [6], coupledresonator-induced transparency [7,8], and nonlinear optics [9,10] to cavity quantum electrodynamics (QED) [11–13]. To effectively excite WGMs, near-field couplers are usually required, such as a high-index prism [14], angle-polished fiber [15], and tapered fiber [16]. Among them, tapered fiber has been widely used in WGM experiments because of easy operation and the highest coupling efficiency [17]. Coupling efficiency is a crucial physical parameter in coupling systems and plays an important role in both theory and in experiments. In experiments, coupling efficiency can be obtained by measuring transmission or reflection spectra [16–19]. According to the coupling strength and the resulting transmission rate, the taper-microsphere system can be divided into three kinds of coupling regimes: under coupling, critical coupling, and over coupling. Critical coupling is considered as the optimal condition since the input light energy is fully coupled into the microsphere cavity. This is valid only when one traveling WGM is excited. In a realistic microcavity, backscattering is always present because it is induced by surface roughness and internal inhomogeneities of the microcavity, and it forms the well-known modal coupling [18–20] between a pair of WGMs (with opposite propagating directions). In other words, both clockwise (CW) and counterclockwise (CCW) WGMs can be excited even though the CW WGM is first coupled by a fiber taper, which produces both transmission and reflection spectra of the taper. In previous work [16], critical coupling is assumed to be obtained in the presence of phase matching. Recently, some numerical results based on the finite element method [21] 0740-3224/08/111895-4/$15.00
and the coupled-mode theory [22] show that critical coupling can be obtained for certain fiber taper radius or taper-microsphere gaps without phase matching. In this paper, a numerical simulation has been developed to study taper-microsphere coupling based on the coupled-mode theory. We obtain the transmission and reflection rates of the fiber taper, which are easy to measure directly in an experiment.
2. MODEL To study a taper-microsphere coupling system shown in Fig. 1, the coupled-mode theory [23] has been widely employed since it requires relatively lower computational resource consumption and possesses clear physics, where only the field distributions of the waveguide and microsphere are required. The coupling coefficient is achievable in an integral formula equivalent to that presented in [22,24–27]: * = sf = fs
⌬⑀ 4
冕冕冕
ជ ·E ជ * exp共i⌬z兲dv, E f s
共1兲
Vs
where sf共fs兲 is the coupling coefficient of a fiber taper (microsphere) coupled to a microsphere (fiber taper). ⌬ denotes the permittivity difference between the microsphere (the same material as the taper) and the air, and ⌬ = f − s defines the difference in propagation constants ជ and E ជ are the norof the taper and the microsphere. E f s malized fields of the fiber and the microsphere, which are ជ 兩2dxdy = 1, and V is volume of defined by 1 / 2兰兰冑 / 0兩E f共s兲 s the microsphere. The modes in a microsphere can be analytically solved in Maxwell equations due to the integrable spherical boundary and have been investigated extensively [25]. Without losing generality, here we consider a fundamen© 2008 Optical Society of America
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[18–20]. Thus, the coupled-mode equations for CW and CCW modes can be expressed as dEccw dt dEcw dt
= − 共0 + 1 + i⌬兲Eccw + igEcw + i
sf
Ein ,
= − 共0 + 1 + i⌬兲Ecw + igEccw ,
Er = ifsEcw , Et = 冑1 − 兩sf兩2Ein + ifsEccw ,
Fig. 1. (Color online) Schematic of a microsphere coupled to a fiber taper. At the coupling region, an adiabatically stretched fiber taper is simplified as a cylinder. Ei is the input electrical field into the coupling region, Ecw, and Eccw are the clockwise and counterclockwise WGM fields, which are distributed near the surface of the microsphere and Er, and Et are the reflected and transmitted fields after coupling. is the coupling strength between the cavity and fiber taper modes.
tal TE mode (mode numbers: q = 1, m = l = 192, in a microsphere with diameter D = 35 m), which couples to the HE11 mode of the fiber taper. The modes in the microsphere have been investigated extensively [25], since Maxwell equations can be analytical solved. Here we only are concerned with the fundamental TE modes (192, l, 192) with the smallest volume, which couple to the HE11 modes of the fiber taper. Due to the symmetry, coupling coefficients of CCW WGMs are the same as with the CW WGMs. In a silica microsphere, the intrinsic quality factor Q0 is mainly affected by the radiative loss, material absorption, and surface scattering loss. In the following calculation, the radiative loss can be ignored for a microsphere with the diameter of 35 m [12], so that Q0 is dominated by surface scattering and material absorption. In this bulk cavity without gain medium, the field 共E兲 is decayed with time exponentially Es ⬀ e−共0+1兲t. In a circulation time for the light traveling inside the microsphere, the energy attenuation can be approximately expressed as intrinsic cavity (material) decay and coupling loss: ⌬Es Es
= 冑1 − 兩fs兩2e/2Q0 ,
共2兲
where = 2nsrs / c is the circulation time for the light traveling inside the microsphere, ns is refractive index, rs is the radius of microsphere, and c is the light speed in vacuum. So, the intrinsic cavity decay rate 0 can be written as 0 = / 2Q0, while the coupling strength 1 between the fiber taper and the microsphere can be calculated from 1 = −ln共冑1 − 兩fs兩2兲 / . Here = 0 + 1 denotes the total cavity dissipation rate, and the loaded quality factor is defined as QL = / 2. In an actual system, backscattering will induce the modal coupling between the CW and CCW WGMs
共3兲
where ⌬ = − c is the carrier frequency detuning from the degenerated mode frequency and g describes the modal coupling strength between the CW and CCW modes. E with different subscripts is the electric field corresponding to signs in Fig. 1. In the steady case, the normalized transmission and reflection rates from Eq. (3) can be calculated as T=
R=
冏 冏 冏 冏 冏 冏 Et
2
=
Ein Er
Ein
冑1 − 兩sf兩2 −
共0 + 1 + i⌬兲2 + g2
2 1g
2
=
21共0 + 1 + i⌬兲
g2 + 共0 + 1 + i⌬兲2
冏
冏
2
,
2
.
共4兲
In a certain system in practice, 0 and g are usually fixed and T and R are tunable by changing the coupling strength 1, which can be realized by adjusting the radius of the fiber taper and the taper-microsphere gap.
3. SIMULATION After describing the theoretical analysis, now we will discuss some numerical results. First, in the absence of the backscattering, i.e., g = 0, it is obvious that no light will be reflected 共R = 0兲. The transmission T with different taper radius and taper-microsphere gap, which can be directly obtained from Eq. (4), is presented in Fig. 2(a). The whitest area 共T ⬍ 0.05兲 describes the taper-microsphere coupling approaches and even reaches the critical coupling point. Below this critical coupling area, the tapermicrosphere system enters the over-coupling regime, while above the critical-coupling area, the system works in the under-coupling regime. Remarkably, for the fiber taper radius 0.25 m ⬍ rf ⬍ 1.8 m, critical coupling always exists for a certain air gap. In other words, if the radius of the fiber taper is small enough, the system is initially in the over-coupling regime when the tapermicrosphere is in contact. Then critical coupling can certainly be achieved by just increasing the gap, which is consistent with the experiments. Taking backscattering into the simulation, the normalized energy transmission rate 共T兲, reflection rate 共R兲, and the energy transferring into the sphere 共L = 1-T-R兲 are obtained. The transmissions are shown in Figs. 2(a), 2(c), 2(e), and 2(g) with g = 0, 0.10, 0, and 100, respectively. The reflection is shown in Fig. 2(b) with g = 100, while the energy transferrences are shown in Figs. 2(d), 2(f), and 2(h) with g = 0.10, 0, and 100, respectively. From
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Fig. 3. (Color online) Normalized energy transmission (solid curve), reflection (dashed curve), and transferring into microsphere (dotted curve) versus the fiber taper radius when microspheres are in contact with taper with modal coupling strength  = 100. The inflection is due to the mode splitting disappearing. Other parameters are the same as in Fig. 2.
Fig. 2. Energy transport properties of the fiber taper. The brightness represents transmission rate; (a), (c), (e), and (g) are transmission with different fiber tapers, radius, and gaps in microsphere; (b) is reflection; (d), (f), and (h) are ratio of energy coupled into the microsphere. Modal coupling coefficients are: (a), g = 0; (c) and (d), g = 0.10; (e) and (f), g = 0; (c), (g), and (h), g = 100. Other parameters are: microsphere diameter D = 35 m, working wavelength = 780 nm, Q0 = 107.
transmission figures, it is found that when the modal coupling strength g increases, the critical coupling point shifts and the over-coupling regime shrinks obviously. Comparing the reflection [Fig. 2(b)] and transmission [Fig. 2(g)] with  = 100, the new critical-coupling regime actually corresponds to that of the input light is almost reflected due to the backscattering. Comparing Figs. 2(d), 2(f), and 2(h), the coupling position for the maximal fiberto-microsphere energy transferrence does not change, even in different presences of the backscattering, but the intensity is dramatically decreased. In Eq. (1), it indicates that the coupling coefficients 共sf , fs兲 are varied with the overlap and phase mismatching of the mode fields of taper and microsphere. However, since the overlap of the field decreases by increasing the gap, critical coupling can always be reached by increasing the gap in the over-coupling regime. The propagation constant difference is dominantly affected by the fiber taper radius. For a fixed taper-microsphere gap, 1 changes with the taper radius and exists as a maximum. Backscattering induces extra attenuation to the coupling system, which means stronger coupling strength is required for critical coupling; therefore, the over-coupling regime shrinks with the increasing g.
Due to backscattering, part of the incident energy will be reflected back, and the energy transferred into the microsphere cannot be estimated only by the transmission rate. Here, Fig. 3 is plotted to further explain the coupling when modal coupling exists, g = 100, and the fiber taper is in contact with the microsphere. The transmission, reflection, and the energy transferrance are plotted as the functions of radius with the solid, dashed, and dotted curves, respectively. Critical coupling is achieved when the taper radius reduces to about 1 m. At the same time R has a maximum value of 82%, while the energy transferred into microsphere is only 18%. In order to obtain the highest field intensity inside the microsphere, the system has to be under-coupled. From Fig. 3, the maximum energy stored can reach 50% when the taper radius is about 2 m. For applications such as low threshold lasers and nonlinear optics, high intracavity energy density is expected. The highest storage energy rate is always achieved when QL = Q0 / 2, not necessarily the critical coupling 共T = 0兲. Figure 3 shows that the optimal energy transfer can be achieved when the system is in the under-coupled regime. In experiments, such optimal coupling can be realized by controlling the coupling conditions, i.e., the gap between the taper and the microsphere.
4. CONCLUSION In summary, we numerically investigated a tapermicrosphere coupling system in the presence of backscattering-induced modal coupling. With numerical simulation, the transmission and reflection were analyzed with various taper radii and air gaps. Due to the backscattering, critical coupling is found not to be an optimal coupling condition for obtaining the highest energy storage in a microsphere. That is of importance in practice if we want to obtain high coupling efficiency to a microsphere for applications involving microlaser and nonlinear optical interactions.
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ACKNOWLEDGMENTS This work was supported by the National Fundamental Research Program of China under grants 2006CB921900; the National Science Foundation of China under grant 60537020 and 60621064; and The Knowledge Innovation Project of the Chinese Academy of Sciences and International Partnership Project. Y.-F Xiao was also funded by the China Postdoctoral Science Foundation. Chang-Ling Zou and Yong Yang contributed equally.
13. 14.
15.
16.
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