We report on theoretical and experimental investigation of azimuthal and longitudinal modes in rolled-up microtubes at telecom wavelengths. These microtubes are fabricated by selectively releasing a coherently strained InGaAs/GaAs bilayer. We apply planar waveguide method and a quasi-potential model to analyze the azimuthal and longitudinal modes in the microtubes near 1550 nm. Then we demonstrate these modes in transmission spectrum by evanescent light coupling. The experimental observations agree well with the calculated results. Surface-scattering-induced mode splitting is also observed in both transmission and reflection spectra at ~1600 nm. The mode splitting is in essence the non-degeneracy of clockwise and counter-clockwise whispering-gallery modes of the microtubes. This study is significant for understanding the physics of modes in microtubes and other microcavities with three-dimensional optical confinement, as well as for potential applications such as microtube-based photonic integrated devices and sensing purposes.
©2013 Optical Society of America
The rapid development of chip-level information transmission using photonic integrated circuits (PICs) calls for novel components to meet the requirements of size, cost, performance, and ease of operation. Rolled-up microtubes, formed when two strained nano-layers are selectively released from the host substrate [1–5], have recently emerged as a promising candidate for both active and passive applications in PICs. By embedding active media (e.g., quantum dots and quantum wells) into one of the strained layers, microtubes can function as on-chip coherent light sources [6–12]. Other novel microtube-based PIC components, such as filters, modulators, directional couplers, and phototransceivers have also been realized [13–15]. The applications of microtubes are not limited to photonic integrated devices. With a naturally built-in inner hollow region as a microfluidic channel, microtubes can be used for sensing and optofluidic detection [16–19]. Catalytic microtubular engines for applications in chemistry and biology have also been reported .
Similar to conventional circle-shaped optical cavities (e.g., rings, disks, toroids, and spheres), light in microtubes is confined by total internal reflection and resonates as whispering-gallery modes (WGMs) . These modes are confined in three dimensions (radial, azimuthal and longitudinal directions) and behave differently from modes in two-dimensional (2-D) cavities (e.g., microrings). Most previous studies have characterized the three-dimensional (3-D) modes in InGaAs/GaAs microtubes by photoluminescence (PL) measurements in the energy range of ~0.95-1.40 eV (wavelength range of ~0.9-1.3 μm) [8,9,22–24], which is the typical emission range of embedded quantum dots (QDs). The low absorption coefficient of GaAs near the 1.55 μm telecom wavelength window is crucial to applications of microtubes for chip-level information transmission on PICs. However, PL characterization of 3-D modes near 1.55 μm is limited due to the weak emission of QDs and the large optical leakage from microtubes to their GaAs substrate. Recently, we proposed an approach to precisely transfer microtubes from their host substrate to any other platforms . This approach makes it feasible to characterize microtubes by evanescent light coupling at telecom wavelengths. Up to now, we have measured a Q-factor of ~1.5 × 105 (the highest reported to date on microtubes) by integration with a silicon-on-insulator (SOI) waveguide . We have selectively excited both the TE and the TM polarized modes in microtubes with an adiabatically tapered fiber . We have also demonstrated both the clockwise and counter-clockwise WGMs in microtubes  and realized an external cavity fiber laser utilizing these modes . Very recently, lateral probing of the longitudinal modes in SiOx-based microtubes has been investigated by PL measurements at wavelengths much shorter than 1.55 μm . A transmission add-drop filter configuration using the azimuthal modes in SiOx-based microtubes has been demonstrated near 1.55 μm . However, a thorough study of the azimuthal and longitudinal modes in III-V semiconductor microtubes at telecom wavelengths has not been reported.
In this paper, we present theoretical and experimental characterization of the azimuthal and longitudinal modes in rolled-up InGaAs/GaAs microtubes at telecom wavelengths. First, planar waveguide method and a quasi-potential model are utilized to investigate the azimuthal and longitudinal modes. Next, we demonstrate these modes in the microtube transmission spectrum by evanescent light coupling using an adiabatic fiber taper. Experimental observations match well the calculated results. An interesting phenomenon of mode splitting is also observed at ~1600 nm. The splitting is in essence the non-degeneracy of clockwise and counter-clockwise WGMs induced by localized scattering centers on the microtube surfaces. These results are useful for explaining the mode behaviors of microtube and other 3-D microcavities, as well as for potential applications such as microtube-based photonic integrated devices and sensing purposes [31,32].
2. Microtube fabrication
The InGaAs/GaAs microtube fabrication process is illustrated in Fig. 1. As shown in Fig. 1(a), a 50-nm AlAs sacrificial layer was first deposited on a GaAs substrate. Then an InGaAs/GaAs heterostructure bilayer was grown on the sacrificial layer. The bottom of the bilayer is 20-nm thick In0.18Ga0.82As, coherently strained and capped with a 30-nm GaAs layer on the top. Two layers of self-organized InAs QDs were incorporated in the top GaAs layer for active applications [8,9,23]. In order to achieve a free-standing cavity, a U-shaped mesa [8,9,23] was defined by lithographic process, as shown in Fig. 2(b). With selective etching of the AlAs sacrificial layer, the InGaAs/GaAs bilayer can roll upon itself to release strain, and finally forms a tube structure. Figure 1(c) is an optical microscopy image showing a fully rolled-up microtube. Due to the defined U-shaped mesa, the tube has two thick ends attached to the substrate and a thin free-standing region isolated from the substrate, ensuring ideal optical properties of the circular cavity in the free-standing region. By defining corrugations along the inner edge of the U-shaped mesa shown in Fig. 1(b), the outer surface geometry of the tube free-standing region can be precisely controlled. Figure 1(d) is a scanning electron microscopy (SEM) image showing the engineered outer surface geometry of a rolled-up microtube. We designed parabolic lobe-like patterns, which can provide effective optical confinement along the tube’s longitudinal direction, leading to 3-D modes in the microtube cavity.
The geometry of microtubes can be controlled in the fabrication process. The tube diameter is largely determined by the composition and thickness of the InGaAs/GaAs bilayer. The tube wall thickness is related to the bilayer thickness and the number of rolled-up layers (winding number), which can be controlled by the etching time. The tube length and surface-lobe shape are defined by the lithographic mask of the U-shaped mesa. Geometric parameters of the microtubes can be observed in microscopy images. The tube in Fig. 1(d) has a diameter d of ~6 μm and a surface-lobe width w of ~8 µm. The tube in Fig. 1(c) has a length z of ~140 μm. The tube wall thickness can be estimated as follows. For a diameter value of d ~6 μm, the circumference is c = πd ~19 μm. The rolling path lengths of the free-standing region and the thick ends in Fig. 1(c) are estimated to be l1 ~40 μm and l2 ~180 μm, leading to winding numbers of n1 = l1/c ~2 and n2 = l2/c ~ 9, respectively. One single bilayer thickness is t ~50 nm (30 nm GaAs and 20 nm InGaAs), hence the wall thicknesses of the free-standing region and the thick ends are tf = n1t ~100 nm and te = n2t ~450 nm, respectively. More observations show that the fabricated microtubes have a typical diameter of 5-7 μm, a surface-lobe width of 5-10 µm, a length of 50-150 μm, and a free-standing region wall thickness of 50-300 nm.
3. Modeling of the azimuthal and longitudinal modes in microtubes
In order to theoretically study the azimuthal and longitudinal modes of microtubes, we define cylindrical coordinates (r, φ, z) in a simplified microtube model, shown in Fig. 2(a). The three dimensions of mode confinement are in radial (r), azimuthal (φ) and longitudinal (z) directions. The tube free-standing region wall thickness (~100 nm) is much smaller than the confined light wavelength (~1550 nm), thus only the fundamental mode is guided along the radial (r) direction. In the ring-like azimuthal (φ) direction, mode can be supported when the round-trip optical path is a multiple number of the light wavelength. Periodic modes will appear in a series of azimuthal orders, with resonant wavelengths separated by free-spectral range (FSR). Along the longitudinal (z) direction, optical confinement is provided by the microtube surface-corrugations. Several orders of longitudinal modes can exist, depending largely on the geometry of the engineered surface-corrugations.
First, we adopt an approximate planar waveguide model to study the mode polarization states of the microtubes. The transverse-electric (TE) and transverse-magnetic (TM) polarized modes are defined with electric and magnetic field vectors polarized along the tube axis (z), respectively. As discussed above, the fundamental mode dominates in the radial (r) direction, thus we focus on the fundamental polarized modes TE1 and TM1. By using a telecom wavelength of λ = 1550 nm and a constant material refractive index of n = 3.5, the relation between mode effective refractive indices and waveguide thickness can be calculated , as shown in Fig. 3. For microtubes used in this work (wall thickness ~150 nm), the effective refractive index of TM1 mode is less than 1.1, while the value for TE1 mode is above 2.5. Thus, TM1 mode is a leaky mode but TE1 mode can be well guided in the microtubes. To date, the excitation of TM1 mode has only been reported in  and , both using thick-wall microtubes.
When a microtube is formed by the strain-induced self-rolling, the inner and outer edges are usually not overlapped, as depicted in Fig. 2(b). Hence the tube winding number is non-integer and the wall thickness is not uniform along the azimuthal (φ) direction. To study 2-D modes in the (r, φ) plane of the microtubes, we apply a modified planar waveguide model to take the wall thickness difference into consideration, as shown in Fig. 2(c). In this model, the waveguide consists of two segments with different thicknesses (t1, t2), lengths (c1, c2), and effective refractive indices (neff1, neff2). Modes are guided if the round-trip optical path can be divided by the propagating light wavelength:Fig. 3. Therefore, λmr,φ can be calculated for a given tube diameter d and wall thickness (t1 or t2) with a variable (c1 or c2).
Along the longitudinal (z) direction, the tube winding number and the wall thickness are also not uniform due to the existence of designed surface-corrugations. It has been shown that the mode energy is linearly dependant on the winding number . Thus the surface-corrugations can form quasi-potentials for modes along the longitudinal (z) direction, and the potential shape follows the shape of corrugations. In this study, the surface-corrugations are shaped as parabolic lobes (see Fig. 1(d)). Therefore, if we define z = 0 at a lobe center, the quasi-potential can be expressed asEq. (2) in the wave equation along the longitudinal (z) direction and solving the eigenfunction problem, the mode fields along the longitudinal (z) direction can be expressed by a series of quasi-Hermite-Gaussian functions :Eq. (2), and v is defined as the longitudinal mode order.
By using the typical tube geometric parameters discussed in Section 2, we can calculate the microtube mode resonant wavelengths in the telecom wavelength range, as shown in Fig. 4(a). It can be observed that the resonant wavelength separation between adjacent modes is nearly equal: ~5 nm for modes with the same azimuthal order but different longitudinal orders, and ~40 nm for modes with the same longitudinal order but different azimuthal orders. By determining the parameters in Eq. (3), the first four orders of axial modes with the same azimuthal order m of 32 are simulated and shown in Fig. 4(b). It can be observed that the fundamental longitudinal mode is located in the center of the lobe, and higher order modes have fields away from the center.
4. Experimental characterization of the azimuthal and longitudinal modes in microtubes
In order to characterize the azimuthal and longitudinal modes in microtubes at telecom wavelengths, we utilize an optical fiber adiabatic taper to evanescently couple light into the microtubes to measure the transmission spectra. The optical fiber adiabatic taper was fabricated by a butane flame to achieve a waist diameter of ~1 µm (less than the light wavelength ~1.55 µm). After fabrication, the adiabatic taper was fixed on a thin glass slide with wax on non-tapered regions at both ends. To transfer the microtube, an optical fiber was abruptly tapered down by a splicer machine to a tip diameter of ~2 µm (less than the tube diameter) with a tapered length of ~300 µm (larger than the tube length). The abrupt taper then functions as a probe to extract the microtube from its substrate and transfer it close to the adiabatic taper on the glass slide. The detailed transfer method is described in .
The experimental setup for characterization of the azimuthal and longitudinal modes in the microtube transmission spectrum is illustrated in Fig. 5. A tunable laser with wavelength range of 1490-1650 nm and an optical power meter are used as source and detector, respectively. For automation measurement purpose, the source and detector are connected to a computer through general purpose interface bus (GPIB) ports. A polarization controller (PC) is placed after the source and in front of an adiabatic taper to adjust input light polarization. The microtube is transferred and held by an abrupt taper, which is mounted on a computer-controlled micro-positioning stage (MPS) with a step of 0.1 µm. The coupling between the microtube and adiabatic taper can be optimized by moving the abrupt taper along three linear directions (x, y, z axes) controlled by the MPS.
Figure 6 shows the measured transmission spectra displaying the azimuthal and longitudinal modes of a microtube. In the wavelength range of 1490-1650 nm, there are four major mode groups, corresponding to the 2-D modes in the (r, φ) plane. The four mode groups appear periodically and are separated nearly equally by the FSR discussed in Section 3. It is observed that the FSR is slightly larger for longer resonant wavelength (FSR1 ~37 nm, FSR2 ~39 nm, FSR3 ~43 nm). When compared with Fig. 4(a), the resonant wavelengths of the four major mode groups are found to be in accordance with the azimuthal order number m of 31-34, and the FSRs are close to the calculated value (~40 nm). This agreement between experimental observation and calculated results proves the WGM-like behavior of 2-D modes in the (r, φ) plane of the microtubes, and we refer to them as 2-D azimuthal WGM modes in the following discussion.
By moving the microtube along its longitudinal (z) direction, the transmission spectrum changes and different mode profiles are observed at different positions. Figures 6 (a) and (b) show the spectra at two positions where three and four axial modes associated with the 2-D azimuthal WGM modes are observed, respectively. It can be observed that the resonant wavelengths of longitudinal modes with the same azimuthal order are nearly equally separated, with a separation value of ~6 nm. This number is close to the calculated value (~5 nm) in Fig. 4(a), validating the longitudinal quasi-potential model. It should be noted that the numbers (0, 1, 2) and (0, 1, 2, 3) labeled in Fig. 6 are not necessarily the longitudinal mode orders. The reason is that Fig. 4(b) shows longitudinal modes for an ideal parabolic lobe, while the lobe shape of a real microtube is non-ideal (see Fig. 1(d)). Therefore, the positions and orders of longitudinal modes are complex and may not correspond exactly to what Fig. 4(b) predicts. The unique 3-D mode profiles in Fig. 6 cannot be observed in 2-D microcavities like microrings. It is the surface-lobes along the longitudinal (z) direction that provide an additional dimension of optical confinement, making the modes of microtubes behave differently from modes in 2-D microcavities.
The transmission spectra of different microtubes have been measured to observe various 3-D mode profiles. When a microtube different from the one measured in Fig. 6 is positioned at a certain place along its longitudinal (z) direction, each 2-D azimuthal WGM mode has only one component, as shown in Fig. 7(a). It is noted that this microtube shows five azimuthal modes in the wavelength range of 1490-1650 nm, and the FSR is smaller compared to Fig. 6, due to a larger tube diameter. Also, a very interesting phenomenon of mode splitting is observed at the mode near 1600 nm. To verify this phenomenon, we changed the setup in Fig. 5 by adding a circulator between the PC and adiabatic taper to measure the transmission and reflection spectra at the same time [27,28]. Figure 7(b) shows the simultaneously measured transmission and reflection spectra of this microtube at the same coupling position. The clockwise (CW) and counter-clockwise (CCW) WGMs exist in the transmission and reflection spectra, respectively. The pair of CW and CCW WGMs near 1600 nm split into doublets correspondingly, as clearly shown in the magnified inset on the right of Fig. 7(b).
The phenomenon of mode splitting has been widely investigated in other microcavities such as microspheres , microtoroids [32,37], and microdisks . In these studies, mode splitting is induced by external perturbations (e.g, nano-probes/particles) placed close to the cavity surface. The splitting of modes can be controlled and tuned for applications in sensing, filtering, opto-mechanics, and tunable dual wavelength lasing [31,32,37]. Mode splitting in microtubes has first been predicted in , and observed in PL measurements in the wavelength range of ~0.9-1.3 μm [8,24]. However, this wavelength range is shorter than the 1.55 μm telecom wavelength window, where microtubes have important applications for on-chip information transmission. To our knowledge, Fig. 7(b) gives the first experimental demonstration of mode splitting in microtube transmission and reflection spectra at telecom wavelengths. The splitting of modes in microtubes can be explained by the existence of localized scattering centers on microtube surfaces (e.g., surface defects, inner/outer edges shown in Fig. 2(b)), which break the circular symmetry of light propagation in the (r, φ) plane, causing coupling and non-degeneracy of the pair of CW and CCW WGMs. When the scattering rate exceeds the cavity decay rate, the single resonance splits into a doublet. The splitting frequency is related to the scattering rate by 24] and , where the finite-difference time-domain (FDTD) method is applied to model the mode splitting in microtubes.
In Fig. 7, the mode splitting is only observed at ~1600 nm. Similarly, in PL measurement , mode splitting has been observed only at a certain photon energy point. These results indicate that the surface-scattering in microtubes is wavelength sensitive, which agrees with Eq. (4). Another interesting observation from Fig. 7 is that compared to the non-split modes, each of the doublets has a smaller linewidth, and thus a higher Q-factor. As labeled in the right inset of Fig. 7(b), the left lobe has a 3-dB linewidth of ~0.8 nm, leading to a Q-factor of ~2 × 103. The Q-factors of modes in Figs. 6 and 7 are low, compared to a value of ~1.5 × 105 for the same material of microtubes when the light coupling medium is a waveguide . The reason is that nearly critical coupling can be achieved with a waveguide, while the microtube is over-coupled with an adiabatic tapered fiber in this work. If the mode splitting is measured using a waveguide, a Q-factor higher than 1.5 × 105 can be obtained. The mode-splitting-induced higher Q-factor suggests possible applications of microtubes such as wavelength selecting for narrow linewidth lasers and lasing stabilization techniques .
In summary, we have presented the theoretical and experimental investigation of azimuthal and longitudinal modes in rolled-up InGaAs/GaAs microtubes at telecom wavelengths. We have applied planar waveguide method and a quasi-potential model to study these modes. Mode resonant wavelengths near telecom wavelengths have been calculated and quasi-Hermite-Gaussian longitudinal mode profiles have also been simulated. These modes have been demonstrated in the microtube transmission spectrum by using an adiabatic fiber taper for evanescent light coupling. The experimental observations are in excellent agreement with calculation and simulation results. We have also observed mode splitting in both transmission and reflection spectra at ~1600nm. The mode splitting is induced by localized scattering centers on the microtube surfaces, bringing the non-degeneracy of CW and CCW WGMs. This investigation of azimuthal and longitudinal modes at telecom wavelengths is important to microtube-based photonic integrated devices for chip-level information transmission as well as sensing purposes.
The authors thank Dr. Songrui Zhao at McGill University for help in taking the SEM images of the microtube samples. This work is being supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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