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We observed resonance effects on the transmission of a pump beam in a chaotic microcavity in an optimal free-space optical-pumping configuration. The far-field pattern of cavity transmission was significantly modified when the pump laser was resonant with a scar mode. From the difference between the non-resonant and on-resonance transmission patterns, we obtained the efficiency of the pump coupling into the scar mode to be as high as 45%, which is consistent with the recent excitation spectroscopy results of Yang et al. [Phys. Rev. Lett. 104, 243601 (2010)].
© 2010 Optical Society of America
1. Introduction
Over the past years, dielectric microcavities have drawn much attention owing to their various practical applications based on high-Q factors and small sizes. In many applications optical pumping is employed for efficient field excitation in the microcavities [1–8]. In the case of circular and nearly spherical microcavities, evanescent wave coupling with a prism [9, 10] or a fiber-taper-based coupler [11] is often used for exciting whispering gallery modes (WGMs), even with a near-unity coupling efficiency, which is possible since WGMs escape the cavity through tunneling process. When the cavity is strongly deformed from rotational symmetry for directional emission, however, the evanescent wave coupling is no longer usable since the loss mechanism is changed from the tunneling to chaotic ray dynamics followed by refractive emission. Instead, efficient free-space coupling without such special couplers has been demonstrated based on chaotic ray dynamics under nonresonant condition [12,13] when a pump beam is injected into the cavity by refraction in a time reversed way with respect to its directional output.
In such optical pumping based on the chaotic ray transport, the transmission of a pump is of particular interest since it can provide information on the internal ray/wave processes. It was reported in Ref. [14] that the transmission spectrum of a pump beam refractively injected into a strong-deformed microcavity shows periodic modulations, which was then explained in terms of the interference of multiple beam components that lead to a refractive-coupled output. The modulated pump-transmission spectrum, however, has not been reproduced in other chaotic microcavities and thus its importance has not been fully appreciated.
In this paper, advancing one step further, we report observation of resonance peaks on the pump-transmission spectrum, accompanied by a periodic baseline modulation pattern, in the free-space resonant coupling of a pump in a strongly deformed chaotic microcavity. The far-field pattern of cavity transmission is significantly modified when the pump is resonant with a scar mode excited by refractive pump injection. From the difference between the transmission pattern in the nonresonant pumping and that in the resonant pumping cases, we could obtain the efficiency of pump coupling into the scar mode to be as high as 45%, which is consistent with the result of the recent excitation spectroscopy experiment of Ref. [15].
2. Theoretical consideration
Our cavity is a quadru-octupole-deformed microcavity whose shape is described in the polar coordinates as r(ϕ) ≃ a0(1 + η cos2ϕ +εη2 cos4ϕ), where a0 = (15 ± 0.1)μm, the mean radius of the cavity, and η =0.19±0.005, the deformation parameter with ε =0.42±0.05 [16]. The ray motion in this cavity is mostly chaotic due to the high deformation. As a result, via chaotic ray dynamics, specifically via the time-reversed turnstile process [13] to be discussed in Sec. 2.2, the cavity modes can be efficiently excited under nonresonant condition by a focused pump beam at a particular incident position with a particular incident angle as depicted with the orange-arrowed line in Fig. 1(a). The pump beam is then transmitted in the specific output angles labeled as f′ and d′ in Fig. 1(a) and also as shown in an intensity profile in red solid line in Fig. 1(c), obtained from ray simulations. Under this optimal pumping condition, we can examine the multiple-beam interference effect of Ref. [14] with the ray-tracing method introduced there by considering the so-called amplitude-weighted length distribution. The simulated interference pattern at angle f′, which turns out to be important in the resonant pumping case below, as a function of the pump wavelength is shown in Fig. 1(b), clearly showing equally-spaced modulations even without any mode resonances involved.
Fig. 1 (a) For optimal non-resonant pumping configuration (see the text for details), pump transmission occurs at two major angles denoted by f′ and d′ (red arrows). (b) For f′ angle, calculated transmission spectrum shows periodic modulations. (c) Calculated far-field emission patterns (blue dashed curve) of high-Q modes, when only the counterclockwise direction is considered, show four dominant peaks marked by d, f′, d′ and f, corresponding to four blue arrows in (a). The pump transmission of (a), on the other hand, has only two major peaks at d′ and f′ (red solid curve).
Our main question in this work is then what will happen to the pump transmission if the pump frequency is resonant to a high-Q cavity mode. This question is quite important since for further enhancement of the pumping efficiency by a resonant coupling we need to know the pump-coupling efficiency and for reliable determination of the coupling efficiency we then need to observe resonance effects on the pump transmission to begin with. Here the coupling efficiency is defined, as in the evanescent wave coupling case, as the fraction of the transmitted pump power of non-resonant pumping that is reduced in the case of the resonant coupling [17, 18].
2.1. How to determine pump coupling efficiency
The resonance effect on the pump transmission can be predicted by examining the far-field emission pattern of a cavity mode. It is because the transmission pattern of a pump beam with 100% coupling to a mode on resonance would be the same as the the far-field emission pattern of that mode. It is now well known that high-Q modes of a deformed microavity exhibit almost identical far-field emission patterns [19–21], in our case with four dominant peaks (d, f′, d′, f) shown as blue arrows in Fig. 1(a) and as a blue dashed curve in Fig. 1(c). Here, only the counterclockwise direction is considered. This universal output directionality of high-Q modes has been explained by wave manifestation of unstable manifolds of classical chaos [19, 22, 23]. Then by comparing the non-resonant pump transmission pattern [red solid curve in Fig. 1(c)] with the on-resonance (universal) far-field emission pattern (blue dashed curve), we can easily predict that the transmission intensity would be increased for angles d and f and decreased for angles d′ and f′ when the pump is resonant to a cavity mode. Since how much it is increased/decreased at those angles will depend on the actual coupling efficiency of the pump on resonance, we can determine the coupling efficiency in reverse from the observed pump transmission pattern on-resonance.
2.2. Ray simulation of emission patterns
Before going into experimental details, let us consider how the universal far-field pattern of high-Q modes and the non-resonant pump transmission pattern in Fig. 1(c) are calculated in ray simulations. Figure 2(a) shows the phase-space structure formed by the stable (in blue) and unstable (in red) manifolds of the unstable period-4 orbit depicted by fixed points Fn(n = 1, 2, 3, 4) when η = 0.19 [23] with its Poincaré surface of section(PSOS) shown in the background. The reason why the unstable period-4 orbit is important is that in our cavity it is the nearest unstable orbit to the line of critical angle (sinχc = 1/m = 0.73, with m the refractive index of the cavity medium), marked by a green line in Fig. 2, in the phase space and thus chaotic ray transport occurring around it is responsible for the output emission. For example, high-Q modes, mostly localized above the tangled structure of the stable and unstable manifolds, can escape the cavity via the ray transport process from one lobe to another [blue- and red-colored regions in Fig. 2(a)] in the sequence of a → b → c → d → e → f and also in its π-shifted sequence, a′ → b′ → c′ → .... Output emission then occurs through four lobes d, f′,d′, f, major portions of which are below the critical line. The blue curve in Fig. 1(c) is the resulting output angular distribution, obtained by ray simulation with initial ray bundles prepared in a region 0.85 < sinχ < 0.95, simulating adiabatic KAM curves associated with high-Q modes [24], well above the tangled structure in the phase space.
Fig. 2 (a) Phase space (Poincaré surface of section) of our quadru-octupole microcavity with η = 0.19 in Birkhoff’s coordinates (inset), in which a ray hit the cavity boundary at polar angle ϕ with an incident angle χ. The stable (blue) and unstable (red) manifolds of the unstable period-4 orbit with fixed points Fn(n = 1, 2, 3, 4) form a complex tangled structure. (b) A pump beam initially prepared in a red-colored region follows chaotic transport processes in the numbered and color-coded sequence, and then escapes the cavity at lobes d′ and f′.
The nonresonant pump transmission can also be obtained by ray simulation with initial rays prepared in the mirror-image region of lobe d′ with respect to ϕ = π below the critical line as shown in Fig. 2(b), corresponding to the optimal pumping condition explained above. The initial rays are then, by the so-called time-reversed turnstile transport [13], mapped to intermediate trajectories [color- and number-coded in Fig. 2(b)] in subsequent reflections off the cavity boundary and finally they escape the cavity through lobes d′ (the 9th) and f′ (the 11th in the sequence). Due to the asymmetric pump-beam injection, the intermediate trajectories of the pump overlap significantly only with lobes a′,b′,c′ ..., not with their π-shifted lobes a, b, c.... The resulting pump transmission pattern is shown as the red curve in Fig. 1(c), characterized by two main peaks d′ and f′.
3. Experiments and discussions
We have performed pump transmission measurements on the aforementioned two-dimensional quadru-octupole-deformed chaotic microcavity. The details of our microcavity apparatus are described in Ref. [25]. The cavity is made of a liquid jet of ethanol doped with Rhodamine-640 dye at a concentration of 0.1 mM/l for observing the cavity-modified fluorescence (CMF) spectrum to be compared with the pump-transmission spectrum for mode identification. The dye concentration was chosen rather high in order to make Qabs, the absorption Q of the medium, about 104, and thus only the resonance modes with radial mode order l = 4, 5 with Q(< Qabs) of 6 × 103 and 2 × 103, respectively, were dominantly observed in the CMF spectrum, shown in Fig. 3(a), obtained by flood pumping the microcavity with the 532-nm second-harmonic of a Nd:vanadate laser. These l = 4, 5 modes are scar modes [27], corresponding to unstable periodic orbits of period 6 and 5, respectively, in PSOS. High-Q modes (Q ≫ Qabs) with l = 1, 2, 3 are hardly seen in the CMF spectrum because of their low out-coupling efficiency [26].
Fig. 3 (a) Cavity-modified fluorescence spectrum. (b) Pump-transmission spectrum at θ = 45° (at d). Green curve is a fit while the red curve indicates the interference modulations in the background. (c) Spectrum after removing the background modulations (red curve) of (b). Blue curves are fits for resonance peaks while the orange curve is the sum of them. (d) and (e) are the counter parts for the pump-transmission spectrum at θ = 60° (at f′).
3.1. Pump transmission spectrum
We then measured the pump-transmission spectrum, the transmitted pump power as a function of the pump wavelength, with a linear charge-coupled device, for the two representative peaks at θ = 45° (d) and θ = 60° (f′) in Fig. 1. A frequency tunable dye laser with a linewidth of 0.025 nm near 590 nm was used as a pump laser, which was focused down to a beam waist of 0.8 μm on the cavity surface. The observed pump-transmission spectra are shown in Fig. 3(b) for θ = 45° and in Fig. 3(d) for θ = 60°, compared to the CMF spectrum in the same spectral range. We observe periodic modulations, the wave-interference modulation as explained above, in the background. These interferential modulations are fit by a red curve in Figs. 3(b) and 3(d). After subtracting this fit from the spectra, we obtain Fig. 3(c), clearly showing resonance peaks, and Fig. 3(e), exhibiting resonance valleys, aligned to the dashed lines corresponding to the wavelengths of l = 4, 5 modes. In other words, the transmitted power is increased on resonance at 45° while it is decreased at 60° on resonance with respect to the baseline modulation. These observations are in a good agreement with the prediction from the ray simulations discussed before, indicating that the pump laser is resonantly coupled to the cavity modes, modifying the pump transmission.
3.2. Output angular distribution and determination of pump coupling efficiency
We next measured the output angular distribution at a resonance wavelength of l = 5 mode, marked by a red arrow in Fig. 3(a). The black curve in Fig. 4(b) shows the resulting far-field pattern Pres of the pump transmission on resonance. The pump-coupling efficiency for l = 5 mode can then be obtained by comparing Pres with the non-resonant pump transmission pattern Pnon (zero coupling) along with the far-field emission pattern P5 of the l = 5 mode (100% coupling) in Fig. 4(a). Here Pnon was obtained by plotting the non-resonant baseline modulation (marked ‘×’ in Fig. 3) at the resonance wavelength of l = 5 mode as a function of the output angle θ whereas P5 was measured with the spectroscopic technique of Ref. [19]. Both Pnon and P5 have been scale to have the same integrated area in angle θ.
Fig. 4 (a) Non-resonant pump-transmission pattern Pnon (red curve) and the far-field emission pattern P5 of the l = 5 mode (blue curve). (b) On-resonant pump-transmission pattern Pres (black curve) observed at the resonance wavelength of l = 5 mode marked by a red arrow in Fig. 3(a) and its fit (red curve) given by 0.55Pnon + 0.45P5.
We then find that Pres is well fit by a superposition of 55% of Pnon and 45% of P5 with 15% of fitting error. In other words, the transmitted pump power is reduced by 45% from that of non-resonant pumping in the case of resonant pumping, and therefore, the coupling efficiency for the l = 5 mode is 0.45 ± 0.15. This coupling efficiency is consistent with the value obtained from the recent excitation spectroscopy experiment of Ref. [15] within the experimental error.
As a final remark, the coupling efficiency defined in this paper quantifies how much power is coupled to a resonance mode out of the power refractively put into the cavity. For practical applications, how much of the input power is coupled to a mode is important. When a microcavity is strongly deformed, a pump beam can be easily put into the cavity by refraction without requiring any resonance. This fraction is about 75%, with the rest 25% is reflected on the cavity surface in our experiment. In a device, the refracted portion can be made near unity by employing anti-reflection coating. Our experiment indicates that about 50% of the refracted portion is coupled to a high-Q mode when the pump frequency is resonant with the mode. Note that under non-resonant condition, the coupling efficiency is basically zero.
4. Conclusion
We have observed the resonance effect on the pump transmission that indicates the pump beam is resonantly coupled to scar modes in a strongly deformed chaotic microcavity with refractive pump injection. The observed resonant coupling efficiency is high as 0.45 and well consistent with the value obtained from independent measurement of excitation spectroscopy. Such free-space resonant coupling can be used for simple but efficient optical pumping of microcavity devices. The cavity transmission switching observed on resonance can also be applied to optoelectronic switching circuits.
Acknowledgments
This work was supported by WCU Grant ( R32-10045). SWK was supported by NRF Grant ( 2009-0087261). SYL was supported by NRF Grant ( 2010-0008669).
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