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We propose and numerically verify a scheme for compact optical modulation which can enable complex directional switching of signals in integrated micro-optical circuits within hundreds of femtoseconds. The scheme is based on a trimer comprised of two identical silica whispering gallery mode (WGM) microresonators spaced by a central non-linear WGM resonator. The non-linear resonator is in the form of a silica cylinder with a thin coating of an ultrafast Kerr nonlinear material (a J-aggregate of cyanine dye). Using a two-dimensional finite-difference time-domain method and realistic material and structural parameters, we investigated the near-field coupling from a waveguide to the trimer and the subsequent switching process. In our scheme the sandwiched central control resonator has a resonant frequency that is mismatched to that of the input and output resonators. Therefore the optical energy is coupled from the waveguide into only the primary resonator in linear operation. However, for control light intensities of more than ~10−2 W/μm the effective index and hence eigenfrequency of the central resonator can be shifted to match that of its neighbors and hence the optical energy can be redirected.
©2009 Optical Society of America
1. Introduction
Optical micro resonators in the form of a micro disk [1,2], microsphere [3,4] or micro cylinder [5] are receiving much interest because such structures can have a high quality (Q) factor due to the whispering gallery mode (WGM). As a result, WGMs have seen exciting applications in micro-lasers [5,6], nanoparticle and molecular sensors [7–9], and so forth. The combination of micro resonators with nonlinear optical material is expected to provide efficient and compact optical switching and optical bistable devices [10–14]. Previously, we have fabricated a micrometer-sized nonlinear optical resonator constituted of a dielectric microsphere coated with nonlinear optical material [15] and succeeded in the observation of a nonlinear optical response [16]. Therefore, we suggested a new optical device using the WGM as shown in Fig. 1 . The conceptual scheme from Fig. 1 is combined linear and nonlinear sphere. In here, input optical signal from input port propagate through the spheres as a WGM to output port. When the input optical signal reaches the nonlinear sphere without a control light, the WGM is not excited because the refractive index and the diameter of the sphere are not satisfied with the condition of the existence of WGM. Namely, no light appears at the output port. Then the control light incident from control port, the refractive index of nonlinear sphere will be changed. Consequently, the WGM resonance puts off into on, and input optical signal propagates to output port.
The optical device using a WGM has many advantages i.e. it can be constructed freely with various spheres and totally device size is tens of microns. Furthermore, this micro photonic device is expected high-density integrated optical circuits. However, for practical devices we require larger ON/OFF ratios with even smaller input powers. Recently researchers have highlighted by numerical simulation and experiments the potential and enhanced characteristics of coupled WGM systems [3,4,17–19].
In this paper, we numerically investigate the nonlinear optical characteristics of a trimer WGM switch in which one nonlinear cylindrical resonator is sandwiched by two linear cylinder resonators. By modulating the eigenfrequency of the central mediating control cylinder, power sensitive switching of the signal is shown below.
2. Numerical methods
We evaluated the wavelength dependence of electric field intensity and scattered light intensity on each a single cylinder individually and also the combined cylinders by numerical simulation using the two dimension finite-difference time-domain (2D-FDTD).
Figures 2(a) and 2(b) show (respectively) the calculation geometry for the single silica cylinder (Cylinder 1) and the single silica cylinder coated with the nonlinear optical material (Cylinder 2). We evaluated the characteristics of the WGM excitation with TE polarized light incident upon Cylinders 1 and 2 beside a waveguide with the angle of incidence of 55° and beam radius of 2 μm. Cylinder 1 had a diameter of 4.5 μm, while Cylinder 2 had a diameter of 3.0 μm plus and additional thickness, owing to the nonlinear optical material coating, of 100 nm (i.e. the total diameter is 3.2 μm). The refractive index of the waveguide was assumed to be 1.59 (e.g. SU-8 photoresist) and that of the cylinders to be 1.39 (e.g. silica). The nonlinear optical material considered here is silica doped J-aggregates for which the refractive index at around 800 nm and the third-order nonlinear susceptibility χ(3) are 2.39 and 1.15 × 10−2 μm2/V2, respectively [16]. We assumed that the absorption loss of the material was expressed by the imaginary part of Lorentz model [20,21]. A WGM of the cylinders was excited by the evanescent field of incident light in the waveguide penetrating to outside and we then calculated the wavelength dependence of electric field intensity at the observation point [as marked in Fig. 2 (a) and (b)].
Fig. 2 Geometry for the numerical simulations. (a) Cylinder 1: Single silica cylinder, (b) Cylinder 2: Single silica cylinder coated with the nonlinear optical material. In the 2D-FDTD calculations a unit cell size of 25 nm was used throughout the paper.
Figure 3 shows the numerical geometry for the trimer structure of interest here: a single silica cylinder (Cylinder 2) coated with the nonlinear optical material is sandwiched by two silica cylinders (Cylinder 1 and 3). Cylinders 3 has the same parameters as Cylinder 1 (4.5 μm in diameter with a refractive index of 1.39) and Cylinder 2 has the same properties as mentioned above. The waveguide is structured so as to touch Cylinders 1 and 2 in order to couple the signal and control light into the cylinders efficiently using the evanescent coupling (although using two separate waveguides is also feasible). The structure was designed with two main goals in mind: (1) the ability to efficiently couple to both Cylinders 1 (signal) and 2 (control) to allow power efficient deep modulation; (2) the reduction of scattered light which would otherwise interfere with the output signal and degrade the signal-to-noise ratio.
The geometry of the trimer structure used for the calculations below is shown in Fig. 3. The TE-polarized signal and control beams (both with 2 μm radius) irradiate at a 55° in angle of incidence with respect to the normal to the waveguide interface (θi, θc) upon Cylinders 1 and 2 which is approximately the optimum coupling angle for the considered structure. We calculated the wavelength dependence of electric field intensity which is present for the observation points (OP) on each cylinder, and the output light spectra on the observation lines (Fig. 3). The output intensity and its spectrum were evaluated using the sum of pointing vector components. Below we also calculate the dependence of the output light intensity on the control light intensity. In all cases the nonlinear response of the material was assumed to be instantaneous.
3. Results and discussions
If the resonant wavelengths of all three WGM resonators are matched the trimer switch is effectively in the “on” case. I.e., the signal energy that entered into Cylinder 1 can couple to Cylinders 2 and 3 and output can be observed. However, in our design, Cylinder 2 deliberately has a different resonant frequency to the other cylinders (which is implemented by different diameter and the presence of the non-linear coating material) and hence the signal is not efficiently coupled to the output (Cylinder 3) (in the absence of the non-linear effect). Fortunately, the resonant frequency of Cylinder 2 can be manipulated by the intensity of the control beam which affects the non-linear layer’s refractive index and the effective index of the WGM mode. This is the effect we exploit to turn the switch on and off and the numerical analysis follows.
At first we investigate the wavelength dependence of electric field intensity of the chosen resonators (with slightly different resonant frequencies) in the absence of the control beam – see Fig. 4 for the response of Cylinder 1 (black line) and Cylinder 2 (red line). The WGM resonances are indicated by the peak wavelengths in Fig. 4.
Fig. 4 The wavelength dependency of electric field intensity for the single cylinders excited by the waveguide at the input cylinder (black line) and the control cylinder (red line) in absence of the control beam. The intensities are taken at the observation points noted in Fig. 2.
After determining the structural parameters to give resonators that deliberately have slightly different resonant wavelengths at around 800 nm we then can investigate the entire trimer structure. Figure 5 shows the wavelength dependence of electric field intensity for the timer structure (at the observation points noted in Fig. 3) in the absence of the control field and as can be seen by comparing Fig. 5 OP 1 (black line) and OP 2 (red line) the excitation wavelength has changed slightly due to the combination of the structures.
Fig. 5 The wavelength dependency of electric field intensity for the trimer structure at the input Cylinder 1 (black line), control Cylinder 2 (red line) and output Cylinder 3 (blue line) in the absence of the control beam. The intensities are taken at the observation points noted in Fig. 3.
We have to indicate how to decide the optimum angle of incident light. We observed electric field intensity depends on the incident angle from 45° to 60° at the observation point using the geometry for the numerical simulations with cylinder (E) and without cylinder (Ei) of Fig. 2. We determined the optimum angle as the angle of incidence to obtain the maximum intensity at the observation point. Figure 6(a) and 6(b) shows the wavelength dependency of electric field intensity of difference in the incident angle for Cylinder 1 and 2 (as shown in Fig. 2), respectively. In Fig. 6 the black, the red, the blue and the green line show the electric field intensity of different incident angle of 45°, 50°, 55° and 60°, respectively. The WGM excitation occurs at the wavelength corresponding to the peaks of electric filed intensity. It is clear that the incident angle of 55° in Cylinder 1 is the largest electric field intensity at 756.7 nm as shown in Fig. 6(a). For this reason, the wavelength of input signal use 756.7 nm for the characteristics of optical switching as shown below. In the same way, the incident angle of 60° in Cylinder 2 is the largest electric field intensity at 783.7 nm as shown in Fig. 6(b). However, we used the incident angle of 55° in this research because it is difficult to get a large incident angle more than 55° in the experiment.
Fig. 6 Variation of the electric field intensity spectra depending on the incident angle for a single cylinder at observation points in Fig. 2. Figures 6(a) and 6(b) are for Cylinder 1 and Cylinder 2, respectively. The black, the red, the blue and the green line show incident angle of 45°, 50°, 55° and 60°, respectively.
Now we concern ourselves with the output signal and switching. The output light spectra from Cylinder 3 in the linear response are shown in Fig. 7 (In the inset, enlarged to show clearly the wavelength range from 735 to 765 nm). The arrows indicate the wavelength of the signal (756.7 nm) used to excite a WGM and that of the control light (783.7 nm) used in the non-linear calculations below (i.e. Fig. 7 is with no control beam).
Fig. 7 Calculated output light intensity spectra on the observation lines without the control beam. In the inset, enlarged from 735 to 765 nm.
Here we introduce the control beam which is used to match the resonant wavelength of the control cylinder to that of the input and output cylinders. In Fig. 8 , the dependence of output light coupling efficiency on control light intensity is shown. The normalized output light efficiencies η were expressed by
where P is output light intensity without control light, P1 is output light intensity when the control light is arbitrarily-fixed. When the control light intensity was increased from 10−3 W/μm to 10−2 W/μm, η decreased to about 99% on observation line. For the case of stronger control light intensities, η increased rapidly and achieved the maximum value of 138% at 5.4 × 10−2 W/μm on observation line. For control intensity over 10−1 W/μm, η rapidly decreased. This is derived from the fact that the refractive index of the nonlinear optical material on Cylinder 2 is changed along the path A → B → C → D as a result of the varying control light intensity. In other words, the refractive index increases due to the control beam’s intensity in Cylinder 2 which induces the shift of the WGM resonance wavelength towards shorter wavelengths. In here, an amount of permittivity change Δε were expressed bywhere, χ(3) is the third-order nonlinear susceptibility, E is the peak value of the electric field intensity within nonlinear optical material. We can estimate a permittivity variation Δε from temporal averaged electric filed intensity in Fig. 8, taking into account the value ot χ(3), i.e., 1.15 × 10−2 μm2/V2. For example, Δε at C (E = 4.476 V/μm) and D (E = 14.24 V/μm) in Fig. 8 were estimated to be 0.067 and 0.214, respectively. Additionally, these permittivity variations happened locally on the cylindrical surface. Therefore the dielectric permittivity of the non-linear coating can change by more than 0.5, when the control light intensity increases more than 5 × 10−1 W/μm. As a result, the effect cannot only shift onto resonance, but also off resonance, as shown in Fig. 7. The points A’ → B’ → C’ → D’ in the inset of Fig. 7 suggest the corresponding resonance shift induced by the changing effective index of Cylinder 2 with varying control beam intensity (e.g. corresponding to A → B → C → D in Fig. 8).Subsequently, we discussed how the electric field distribution of output signal depends on the control beam intensity. The input beam intensity was fixed to 9.68 × 10−4 W/μm and the control beam intensity was considered for three different values: 5.42 × 10−3 W/μm, 5.42 × 10−2 W/μm and 5.42 × 10−1 W/μm. When the control light intensity is 5.42 × 10−3 W/μm [Fig. 9(a) ] light coupling to Cylinder 3 is weak, i.e., an off resonance state is achieved as far as output light is concerned. When the control light intensity is increased to 5.42 × 10−2 W/μm [Fig. 9 (b)], coupling to Cylinder 3 is seen to be strong, i.e., an on resonance state is achieved. Moreover, when the control beam intensity is as strong as 5.42 × 10−1 W/μm [Fig. 9 (c)], field distribution changed significant due to too large a change of the effective index of Cylinder 2 (by the same process that results in the change to D, D’ in Figs. 7 and 8).
Fig. 9 Snap shots of Electric field distribution of the trimer in a cross section at signal light intensity of 9.68 × 10−4 W/μm and control light intensity of (a) 5.42 × 10−3 W/μm, (b) 5.42 × 10−2 W/μm and (c) 5.42 × 10−1 W/μm.
Finally, we have also estimated the on-off response time, which is defined here as a required time for the output intensity to stabilize after the control beam intensity is changed. Here, we have estimated the on-off response time when the control beam intensity changed from (i) to (ii) or from (ii) to (i) in Fig. 8. Figure 10(a) and 10(b) shows the temporal response of the output electric field intensity at the observation line noted in Fig. 3 when the control light intensities are increased and decreased, respectively. Moreover, the control light intensity was changed after we confirmed that the control light was stabled in transit time. The wavelength and intensity of the input light were 756.7 nm and 5.42 × 10−4 W/μm, respectively. For Fig. 10(a) and 10(b), the control light intensity, set to be 2.71 × 10−2 W/μm and 5.42 × 10−2 W/μm initially, were increased to 5.42 × 10−2 W/μm at 1.96 ps and decreased to 2.71 × 10−2 W/μm at 1.96 ps, respectively. From Fig. 10(a), the on response time was evaluated to be about 1 pico-second. On the other hand, the off response time was evaluated to be 800 femto-seconds from Fig. 10(b).
Fig. 10 The on-off response time of the output electric field intensity. (a) from off to on state at 1.96 ps with control beam intensity 5.42 × 10−2, (b) from on to off state at 1.96 ps with control beam intensity 2.71 × 10−2. The intensities are taken at the observation line noted in Fig. 3.
4. Conclusions
Nonlinear optical responses were predicted in a trimer of WGMs with an extremely high-speed response of 1 pico-secand (on response time) and 800 femto-seconds (off response time), respectively. The on-off response time of single sphere coated with nonlinear material (i.e. 600 fs) is slower than the trimer structure [16]. The main reason for this delay is the large-scale structure analysis. The fast switching is enabled by both the response of the J-aggregates and the efficiency and sensitivity of the trimer structure. In addition to the predicted practical power efficiency the trimer system is much more compact than modulation based on non-linear waveguide modulation schemes (with non-linearity in one of the waveguides or mediated by a single WGM resonator). Such a trimer is a high-speed and compact optical switching element that can provide a building block for compact optical logic circuits.
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