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Nov 21, 2013 - Shumin Xiao,1 Zhiyuan Gu,2 Shuai Liu,2 and Qinghai Song2,3,*. 1Department of Material Science and Engineering, Shenzhen Graduate ...
PHYSICAL REVIEW A 88, 053833 (2013)

Direct modulation of microcavity emission via local perturbation Shumin Xiao,1 Zhiyuan Gu,2 Shuai Liu,2 and Qinghai Song2,3,* 1

Department of Material Science and Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China 2 Department of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China 3 National Key Laboratory of Tunable Laser Technology, Institute of Opto-Electronics, Harbin Institute of Technology, Harbin 150080, China (Received 24 May 2013; revised manuscript received 6 November 2013; published 21 November 2013) Next-generation integrated photonic circuits (PCs) require the combination of on-chip coherent light sources and modulators to increase the integration density and reduce the energy consumption. While significant advances have been accomplished in direct laser modulation and channeling the modulated emissions into waveguides, there is still an enormous challenge in distributing the signals to their designed ends. Here we show that the outputs from microcavities can be directly modulated by local perturbations. We demonstrate this concept in a waveguides connected quadruple cavity, where the generated signals can be optionally distributed to one or multiple designed waveguides by slightly modulating the refractive index on the order of 0.01. This concept should help the advances of high-density on-chip photonic circuits. DOI: 10.1103/PhysRevA.88.053833

PACS number(s): 42.55.Sa, 42.60.Da, 42.60.Fc, 42.65.Pc

I. INTRODUCTION

Due to their small footprint, low energy consumption, and cost-effective fabrication processes, microdisks have been widely accepted as prominent candidates for on-chip coherent light sources of integrated photonic circuits (PCs) [1–5]. Typical microdisk lasers generate continuous wave (cw) emissions, which are guided by optical waveguides and distributed to multiple ends through Y-branching splitters, followed by the translations from cw beams to pulsed signals via external modulators [6–8]. Recently, due to the great demand for low energy consumption and high integration density, combining the laser and modulator to form a directly modulated light source has become a rapidly developing research direction in next-generation on-chip PCs [1,2]. Several types of microlasers with modulation speed ∼10 GHz have been successfully demonstrated [2,4]. There is, however, an important limitation in distributing signals to multiple PCs. Using the conventional modulators or switches will either consume extra power consumption to fine tune the resonant wavelengthes [8] or occupy too much space [9], contradicting the goal of next-generation PCs. Directly modulated light sources can save energy and space [2,4,10]. However, their signals will be sent to all ends simultaneously, leading to crosstalk or leakage of information. Therefore, finding a new mechanism to dynamically control the outputs in different waveguides is highly desirable for developing next-generation on-chip PCs. Directional outputs have been widely studied for a long time [11,12]. Soon after the first demonstration of semiconductor microdisks [13], people realized its limitation in isotropic emission and made many attempts to improve the output directionality. Compared with the typical scheme using evanescent coupling that requires precise position control [14–16], tailoring the cavity boundary has been recognized as the most cost-effective way [11]. Many shapes such as quadruple [11], stadium [17], spiral [18], etc. have been

*

[email protected]

1050-2947/2013/88(5)/053833(6)

proposed and realized in past decades to get directional emission, unidirectional emission, or highly directional emission [19–21]. Eventually, the chaotic cavities show significant advantages [5,20,22–24]. By utilizing wave localization [24], unstable manifolds [22–24], and chaotic layers [5,11], chaotic microdisks have successfully generated high quality factors and unidirectional output simultaneously in free space [24] and have finally been realized in optical waveguides [5], avoiding the possible insertion loss and significantly advancing the applications in PCs. However, the emission mechanisms of chaotic microcavities are usually very robust with respect to the refractive index, cavity shape, etc., making the directional outputs hard to tune [5,22–24]. Up to now, the control of directional emission has only been realized in a few attempts, which usually operate far above the threshold and cost lots of energy [15,25,26]. In this article, we introduce a general and robust mechanism that makes it possible to dynamically control the far-field emission of microcavities. The key to our finding is the perturbation of quantum chaos, which is at the heart of fundamental problems of quantum mechanisms [27,28]. While quantum systems have been pointed out to be quite stable with respect to the perturbations to initial conditions [29], the local density of states can be significantly changed when a local perturbation is employed [27,28]. In optical microcavities, local perturbations introduce local “potentials” that will enhance or suppress the local field distributions [30–34]. Once the enhancement or suppression happens at some decay channels [5], the output properties of microcavities can be modified. II. RESULTS AND DISCUSSION

A microdisk is usually treated as a quasi-two-dimensional system by applying the effective refractive index n0 . We assume n0 = 3.3 for both transverse electric (TE) and transverse magnetic (TM) polarization. The selected cavity shape is the waveguide-connected quadruple cavity [see inset to Figs. 1(a) and 4(a)], which is defined as ρ(φ) = R(1 + ε cos φ) in polar coordinates [5]. The positions and widths of waveguides are defined as φc (i) and φ(i), respectively. We select the

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©2013 American Physical Society

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PHYSICAL REVIEW A 88, 053833 (2013)

FIG. 1. (Color online) (a) Relative intensity distributions (IB /IA ) in two channeling waveguides as a function n. The inset shows the schematic of a two-channel quadruple cavity with ε = 0.08, φ1 = φ2 = 0.1, and φc1 = φc2 − π = 0.633. The overlap between the green circle (radius r) and the cavity represents the perturbed area with n = n0 + n. Here r/R is 0.225. And no perturbation has been applied at port B. Panels (b)–(d) show the corresponding far-field patterns (FFPs) of modes marked i, ii, iii, respectively, in panel (a). Here the resonance is set at kR = 25.04.

refractive index around the joint position as the perturbations [see Fig. 1(a)]. Compared with the cavity shape that is fixed once the semiconductor microdisk has been fabricated, the refractive index can be tuned dynamically by carrier injection [35,36], the Kerr effect [37], the thermal effect [38], etc., making it possible to directly modulate the characteristics of passive microcavities. Meanwhile, because the perturbed area is independent of the stable islands, the possible absorption associated with the modulation of the refractive index will not significantly spoil the Q factor of long-lived resonances.

A. Control emissions of two-channel scheme

We first numerically study the influence of perturbations on the two-channel scheme in the inset to Fig. 1(a), which gives the simplest case of multiple directional outputs. All the results are calculated with commercial software (COMSOL MULTIPHYSICS 3.5a). Figure 1(a) shows the ratio of integrated intensity along port B over port A (IB /IA ) as a function of n. If there is no perturbation applied, the emissions along the two ports are almost identical [see Fig. 2(b) and its log-scale field pattern in the inset] and give IB /IA = 1.0006, which is marked as mode i in Fig. 1(a); the slight difference between the two ports is generated by the meshing problem in the numerical calculation. This is consistent with a previous report [5] and can easily be understood from the symmetry of two leaky ports formed by waveguides. When a positive perturbation (n > 0) is employed, the intensity along port

A (IA ) is increased and the FFP of the microcavity becomes asymmetric. With increasing n, the enhancement of IA will increase and finally become more than an order of magnitude larger than that in port B (IB ). The corresponding ratio IB /IA is thus decreased to around 0.04 at n = 0.05. On the contrary, if a negative perturbation is applied, IA will be suppressed. It will decrease more than an order of magnitude with large perturbation amplitude, e.g., giving IB /IA ∼ 20 at n = −0.05. Figures 1(c) and 1(d) show examples of FFPs with n = −0.05 and n = 0.05, respectively. The insets are their corresponding log-scale field distributions. We can see that the perturbation of the refractive index simply suppresses or enhances one decay channel and turns bidirectional outputs into unidirectional emission, despite two leaky ports being connected to the quadruple cavity.

FIG. 2. (Color online) Panels (a) and (b) show the clockwise Husimi maps of modes ii and iii from Fig. 1(a). It is clear that the distribution of the chaotic layer has been changed by the perturbation n at the joint position of port A.

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The directional emissions along two waveguides are generated by channeling the light in the chaotic layer into waveguides. The intensity along two ports are determined by the overlap between additional leaky channels and the chaotic layer [5,39]. Because the leaky channel relates the width and position of waveguide, it is usually fixed once the waveguide is fabricated. Therefore, the enhancement and suppression of IA formed by local perturbation can be considered as the attraction and repulsion of the distribution of the chaotic layer at the leaky channel. Such behavior can be qualitatively understood as follows. In a two-dimensional cavity system, by analog with the Helmholtz equation to the Sch¨odinger equation, the refractive index can directly relate to a potential [30–34]. A negative n will thus generate a repulsive “potential” to repel the distribution of the chaotic layer at port A. While such repulsion is quite small for a single pass, the accumulation of the intermediate propagation of the chaotic layer makes it dramatic and thus blocks the emission along port A. Similarly, a positive n should generate an attractive potential and enhance the emission along port A. To further understand the influences of perturbation, we projected the internal mode structure onto the Poincar´e surface of section (SOS) using the Husimi function [40,41]. The SOS displays phase-space trajectories as points in a twodimensional (2D) plot with coordinates φ and χ and gives

precise ray content of the resonance at the boundary in terms of a density of rays and their angle of incidence. Figures 2(a) and 2(b) show enhanced views of the Husimi maps of resonances ii and iii from Fig. 1(a). Whereas the main field distributions of two modes are still confined within stable islands [5], the other properties have been dramatically changed by the perturbations. Typically, the light that tunnels out of the islands propagates surrounding their original orbits in an intermediate time. Such motion has been described by a slow horizontal diffusion (in φ) that finally forms a continuous chaotic layer in phase space [5]. At n = −0.05, Fig. 2(a) shows that the continuous chaotic layer has been interrupted. The intensity at φ ∼ 0.633, where port A and the perturbation have been applied, is suppressed to be orders of magnitude lower than the other area. This is consistent with the blocking of light emission along port A in Fig. 1(c) and gives clearer evidence of the repulsive potential formed by perturbation. Similarly, a positive perturbation with n = 0.05 increases the intensity of the Husimi map at φ ∼ 0.633 by more than an order of magnitude and breaks the continuity of the chaotic layer at φ = π ± 0.633 and 2π − 0.633, clearly showing the attractive potential. Because the potentials are formed within the quantum regime, where the responses of the systems to perturbations are not as sensitive as their classical counterparts, we can expect

FIG. 3. (Color online) (a) The relative intensity ratio (IB /IA ) at different resonances. All the cavity parameters are the same as for Fig. 1(c). Except for the resonance coupling to nearby resonances, most of the resonances show similar FFPs. (b) Dependence of (IB /IA ) of mode with kR = 25.04 on size (blue squares) and shape (red circles, a and b are both axes of an ellipse in Cartesian coordinates) of perturbation area. Panels (c) and (d) show the resonant wavelength and intensity ratio along two ports as a function of shape-deformation parameter ε. The diamond mode maintain its far-field pattern (FFP) with IB  IA for a wide range of ε except that it couples to other resonances. The stars in panel (d) show the significant tolerance of the perturbed FFP on refractive index. The regions between the dashed lines in panels (a), (b), and (d) correspond to bidirectional emission. Except for the varied parameters, the others are the same as the parameters in Fig. 1. 053833-3

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FIG. 5. (Color online) Panels (a) and (b) show the FFPs of mode 3 and mode 7 at ε = 0.095 and ε = 0.07, respectively. IB  IA is recovered when the diamond modes are away from the crossing points.

FIG. 4. (Color online) Panels (a) and (c) show the mode patterns of triangle and rectangle resonances at ε = 0.07 and ε = 0.11 (from Fig. 3), respectively. The corresponding FFPs are shown in the bottom panels (b) and (d). For comparison, the FFPs of nearby diamond resonances are shown in the upper panels of (b) and (d). It is clear that the FFPs of high-Q modes are dominated by the low-Q modes via mode coupling. Here and below, the basic parameters of cavities are n0 = 3.3, n = −0.05, and ε = 0.08, except where specifically stated otherwise.

the universality of this mechanism to control the FFPs. We first study its dependence on the resonant wavelengths. Near the resonance at kR = 25.04, we have found a series of high-Q modes, which have similar diamond field patterns and can be excited in a lasing experiment. Figure 3(a) shows that most of them FFPs similar to those in Fig. 1(c) when the perturbation n = −0.05 is applied. Only a few resonances such as mode 3 and mode 7 show bidirectional emission as in Fig. 1(b) or reversed unidirectional emission as in Fig. 1(d) due to the coupling with nearby resonances. The control of directional output with local perturbation is also robust with respect to shape deformation. With the variation of cavity shape, the ratio IB /IA of mode 6 is almost constant from ε = 0.0775 to ε = 0.095 and reduces around ε = 0.07 and ε = 0.105 [see Fig. 3(d)]. Such reductions are also caused by mode coupling with low-Q resonances that are confined in the chaotic sea. The mode nearby mode 3 is a down-triangle resonance as shown in Fig. 4(a). Because the down-triangle mode has a bouncing point right at the position of port A, unidirectional emission can be expected at n = 0. When the perturbation is applied, the FFP does not show obvious change. It is understandable that the field distribution at port A is too strong to be perturbed by a slight change in refractive index. When mode 3 is close to the down triangle resonance, its far-field pattern is thus dominated by the down triangle mode via mode coupling [see Fig. 4(b)]. Similarly, mode 7 is close to a rectangle resonance as shown in Fig. 4(c). This resonance has two bouncing points on port A and port B simultaneously and is hard to be perturbed like the downtriangle mode. Thus the FFP of mode 7 is dominated by the rectangle resonance, too [see Fig. 4(d)]. Since the FFPs of mode 3 and mode 7 with n = −0.05 are mainly caused by the coupling with low-Q modes as in Figs. 3(c) and 3(d), we can expect that the unidirectional

emission with IB  IA will be recovered when the resonances are shifted way from the crossing points. This indeed happens and the results are shown in Fig. 5. When the shape deformation is ε = 0.095 instead of ε = 0.08, the emission of mode 3 along port A [Fig. 5(a)] has been significantly suppressed as in Fig. 1(c). Similar recovery has also been observed in Fig. 5(b) where the shape deformation of mode 7 is 0.07. Besides being insensitive to the wavelength and cavity shape, the new mechanism is also robust with respect to the refractive index. In the quadruple cavity with n = −0.05, we can see that the perturbed FFP with IB  IA can be obtained within a wide refractive index range 2 < n0 < 4, indicating its validity in most semiconductor materials ranging from GaN to InP. For further support of such robustness, we show some results (see Fig. 6) at n0 = 2.4, which is the typical refractive index of GaN or ZnO for UV lasers. Similar to the results in Fig. 3(a), unidirectional emission with IB  IA is obtained from 12 resonances in a wide range from kR ∼ 34 to kR > 40 (the modes in a larger kR region have not been calculated) if a perturbation n = −0.05 is applied on port A. There are three resonances that show bidirectional emissions that are also caused by the mode coupling like Fig. 4. Our results in Fig. 3(b) show that the FFP with IB  IA is also robust with respect to the size (squares) and shape (circles) of the perturbed area. It can be obtained by a circular shaped perturbation with a wide range of radii from r/R ∼ 0.17 to r/R ∼ 0.27 or a highly deformed elliptical

FIG. 6. (Color online) (a) Examples of FFP and corresponding field pattern off mode marked by arrow in panel (b). The perturbed area is the overlap between the quadruple cavity and the red circle. (b) Tolerance of new mechanism on the resonant wavelengths. As opposed to Fig. 3, here the refractive index n0 is 2.4 for GaN or ZnO and n = −0.05. It is clear that the FFPs with Ib  IA can be obtained in a wide kR range from 34 to 40. Several modes have bidirectional emissions between the dashed lines in panel (b) are formed by mode coupling as in Figs. 1 and 3.

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FIG. 7. (Color online) Control of FFPs of chaotic microcavities with small perturbation amplitude (nA = −0.005 and nB = 0.005). The suppression of IA happens at both (a) n0 = 3.3 and (b) n0 = 2.4. All the other parameters are the same as in Fig. 3.

perturbation area. Moreover, the amplitude of perturbation is mostly set at |n| = 0.05 to generate impressive enhancement or suppression of IA . In a real experiment, such modulation on n0 is of course possible [35,37] but not necessary. Typically, the typical intensity ratio around 2 is already good enough for switching the FFPs [26]. In the current structure, it can be easily obtained with n ∼ 0.01. If we apply two reversed perturbations on both channels, the amplitude can even be as low as  ∼ ±0.005, which can be easily obtained with the Kerr effect or carrier injection. The results are shown in Fig. 7(a), where IB /IA > 2 can be easily obtained with such small perturbations. Similar control in the far-field pattern has been obtained in microdisks with n0 = 2.4 as well. B. Control emissions of four-channel scheme

Finally, we demonstrate the control of FFPs in a fourwaveguide-connected microcavity. Different from the twochannel system, the four-channel scheme is more attractive because it can form an optical network. Without considering the perturbation, four waveguides form four leaky channels above critical lines, which intercept the chaotic layer and generate four-directional outputs. This scheme works well if four PCs that connected with the directly modulated light sources work simultaneously. For sending signals to arbitrary channels, the control of directional emissions along the waveguides must be considered. To distribute the signal to three channels, the simplest way is to apply a negative perturbation on the rest channel to block the emission along it. One example with nA = −0.05 and nB−D = 0 is shown in Fig. 8(f), where the emission along port A is significantly suppressed to be several times smaller than the others. The suppression at one of the other three channels can be expected from the symmetry of the structure. As what we have discussed above, such directional output can also be fulfilled by applying positive perturbations on three channels to enhance the emissions along them. Similarly, one can also get bidirectional emissions along two ports. Figures 8(c)–8(e) show three possible schemes of bidirectional emission by using n = −0.05 on two channels to block their emissions. The case of unidirectional emission is the reversed case of three-directional emission. One example is shown in Fig. 8(b), where the emission along one port is an order of magnitude larger than the others. Unlike the other figures, here the intensity difference between channels are more dramatic. This is because we have employed a group of perturbations with different amplitudes. This attempt gives more degrees

FIG. 8. (Color online) (a) Schematic of four-channel quadruple cavity. There are four perturbation areas with amplitude nA , nB , nC , nD . Here ε = 0.08, rA /R = rB /R = rC /R = rD /R = 0.225, φA = φB = φC = φD = 0.1, and φA = π − φB = φC − π = 2π − φD = 0.633. The selected resonance has kR = 45.835, which is around two times that of the modes in Figs. 1–3 to illustrate the robustness. (b) Possible unidirectional output along port B. Here nA = nB = −0.05, nC = −0.03, and nD = 0.02. Panels (c)–(e) show directional outputs in two ports. The perturbations with n = −0.05 are applied on the other two ports. (f) Directional outputs in three ports simultaneously except port A. Here nA = −0.05, and nB = nC = nD = 0. Due to the symmetries, the directional emission in one and three ports can be converted to all the other ports.

to tailor the FFPs in case nearly complete suppression of the emissions along undesired channels is required. III. SUMMARY

In summary, we have demonstrated a generic mechanism that can be used to control the FFPs of optical microcavities without operating at high energy. By applying local perturbations on the refractive index, the emission of long-lived resonances can be significantly modified and switched within different channels optionally. The demonstrated mechanism is a generic approach to control the FFPs of resonances as long as they are not coupled to other low-Q modes. It can be obtained in a wide range of cavity shapes, perturbation areas, and semiconductor materials, giving a large tolerance for real experiments. Moreover, because the whole process discussed here does not relate to the gain, the demonstrated mechanism can also be used as ultrafast optical switches in passive cavities. We believe that our finding should shed light on the development of the next generation of on-chip PCs for optical interconnects or computing.

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SHUMIN XIAO, ZHIYUAN GU, SHUAI LIU, AND QINGHAI SONG ACKNOWLEDGMENTS

This work was supported partly by the NSFC under the Grants No. 11204055 and No. 61222507 and by the Program

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