1. Introduction

Semiconductor microtubes fabricated using strain induced self-rolling mechanism were first introduced in 2000 [1]. These microtubes can be patterned and processed in 2-dimension (2D) by standard lithography and epitaxial growth techniques, after strain release and self-rolling the final devices function as 3-dimension (3D) structures [2, 3]. This novel structure has attracted much attention in both photonics and electronics. By carefully engineering the strain in the deposited layers and the geometry of the epitaxially grown patterns, researchers can now fabricate microtubes in a well-controlled manner and with great accuracy in terms of revolution numbers (winding numbers), tube diameters and tube orientation (of the rolling axis) on a planar substrate [4]. Moreover, these microtubes can be easily scaled up into large array with excellent uniformity among them, which makes them the natural building block for integrated devices and systems. On the other hand, the wall of the microtube is formed by epitaxial plane. It can be atomically smooth, which overcomes the side-wall roughness issue associated with the planar waveguides fabricated by lithographic technology. This is a very much desirable property for photonic applications. Researchers have demonstrated that the microtube supports resonant modes similar to the ones found in the planar ring resonator [5–7]. In addition, there are a wide range of materials that can be chosen to form the microtubes [8], which provides great flexibility in selecting material properties according to the specific application requirements. It is possible to embed advanced structures such as quantum wells (QWs), quantum wires and quantum dots (QDs) in the self-rolled-up microtubes. One interesting feature found in these microtube based active devices is that the band gap of these confined structures may shift as a function of the tube diameter, which opens a new dimension for energy band engineering [9]. Microtube based light emitting diode (LED) and lasers have been successfully demonstrated [10, 11].

Intuitively, the microtube can be thought of as a planar ring resonator rotated 90° about one of the in-plane axis of the Cartesian coordinate system. However, there are some obvious differences between these two. The microtube is not enclosed by concentric cylindrical surfaces but a spiral surface instead. This breaking of symmetry introduces additional loss to the electromagnetic wave propagating inside and preferential angular emission of the resonant mode at the inner edge of the microtube [12]. The length of the microtube is typically much longer than its diameter, while it is the opposite for typical planar ring resonator. It has been demonstrated that structural variation along the tube rotational axis may introduce axial light confinement and resonant mode splitting of the microtube [13]. When coupling to a planar waveguide perpendicular to the rotational axis, the coupling is typically more efficient in TM polarization for ring resonator, while it is more efficient in TE polarization for the microtube. This can be inferred from the relative geometry of the waveguide and the ring resonator and microtube.

The microtube can be monolithically fabricated on top of a planar photonic integrated circuit (PIC) and server as a vertical resonator cavity. If two planar waveguides placed very close to the microtube from both the top and the bottom, the microtube may act as a vertical coupler between these two waveguides. This 3D integration has been demonstrated experimentally in free space using two tapered fibers as the waveguide by Schmidt and et al. [14] and on silicon wafer by mechanically transferring the microtube to the top of a silicon waveguide using a tapered optical fiber inserted into the hollow core of the microtube [15]. In order to achieve truly monolithically fabricated 3D photonic integration using microtube as the vertical coupler between two adjacent photonic device planes, it is vital to have in-depth physical understanding of the resonant modes supported by the microtube and how these modes coupled to the planar waveguide above and beneath it. There are a few existing theoretical models to describe the resonant modes of the microtube. One model uses wave equation in cylindrical coordinate system and solves the Eigen equations determined by the boundary conditions at the tube surfaces [13]. This is an accurate model. But the drawback is that the boundaries of the microtube is very complicated, so only numerical solutions can be obtained. It is difficult to derive physical intuition out of this method. Another simpler model treats the microtube as planar waveguide and applies periodic boundary condition to it [13]. This model is straightforward and some important physics can be understood through the analytical solution. But this model suffers from too much simplification, some subtle but critical features were ignored such as the bending loss and the loss coming from breaking the rotational symmetry. In this paper, I propose an alternative model to find the resonant modes of the microtube. I start from the wave equation in cylindrical coordinate system. Then I apply conformal transformation to convert the structure into an equivalent planar parallel-piped waveguide, which preserve all the physical properties of the original structure. The propagation modes of the equivalent planar waveguide is solved with some simple approximations and periodic boundary conditions at each end to give analytical solutions. These solution can be easily transformed back to give the resonant modes of the microtubes. Due to the space limitation, this paper will focus on obtaining the analytical solution for the resonant modes of the microtube waveguide and how the resonant frequency shift as a function of change in some structural parameters. Detailed analysis of the coupling between the microtube and one or two planar waveguides will be provided in a separate paper.

2. Device structure and theoretical formulation

The structure of an exemplary device under investigation is illustrated in Fig. 1. Figure 1(a) is a 3 dimensional cartoon of the self-rolled-up microtube. It is typically fabricated using a U-shape mesa. After removing the sacrificial layer, the 2D membrane will self-assemble into a tubular structure driven by the mismatched strained within the epitaxial layers. Because of the U-shaped mesa, the two arms of the mesa will consists more turns than the center part. The final device have two pedals on both end to support and suspend the central tube section above the substrate. The length of the central tube section, L, is typically much larger than its diameter, d. The analysis of this paper is focused on the optical properties observed in the center tube section. Figure 1(b) is a cross-sectional view of the central tube section. The black lines defines the external boundaries of the rolled-up tube. The blue lines show the boundary of each individual layer of the original 2D membrane. In the ideal case, there’s no gap between adjacent layers of the rolled-up tube and the cross-section of the microtube can be mathematically described by a spiral in the polar coordinate system. The external boundaries of the rolled-up tube can be defined as:

 

Fig. 1 (a) The 3D structure of a typical rolled-up microtube. (b) The cross-sectional view of the rolled-up tube perpendicular to the rolling axis.

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In this study, I focus on the optical properties of the rolled-up tube as a vertical waveguide. I assume that there is no free charge inside the rolled-up tube. Therefore, I can use the wave equation without source to describe the behavior of light propagation in the rolled-up tube. Due to the rotational nature of this structure, cylindrical coordinate system is used with the z-axis coincident with the rotation axis of the tube. The wave equation in cylindrical coordinate system can be written as:

The axial modes of the rolled-up tube had been studied by Strelow and et al. [13], which will not be covered in this manuscript. As mentioned previously, the length of the tube is much longer than the diameter of it, $L\gg d$. To simplify the theoretical analysis, let’s further assume that the tube structure is invariant along the z-axis and the incident light is linearly polarized along z-direction such that optical field will not propagate along the rotational axis. As a result, the light propagation inside the rolled-up tube is reduced to a 2-dimensional (2D) problem in a polar coordinate system. In this study, I will focus on the azimuthal modes circling around the microtube cavity. Let us define $E(r,\phi ,z)=F(r,\phi ){\widehat{e}}_z$, $F(r,\phi )$ is the cross sectional field function, which satisfies the following 2D wave equation:

3. Conformal transformation and equivalent waveguide structure

Equation (3) can be solved in polar coordinate system and the field distribution $F(r,\phi )$ can be expressed as a superposition of a series of Bessel functions. However, because the boundaries of the structure shown in Fig. 1(b) deviate from concentric circles, an analytical solution of $F(r,\phi )$cannot be obtained using this method. Almost all previous studies of the resonant modes of rolled-up microtubes make the approximation to simply the structure into a perfect ring. In this study, I take a different route. First, I use conformal transformation to obtain an equivalent parallel piped waveguide structure. Then I try to obtain an approximate analytical solution of the field distribution in the equivalent waveguide. This solution can then be transformed back into the cylindrical coordinate system to represent the field inside the rolled-up tube. With such an analytical solution at hand, one may take further steps to study the coupling of the rolled-up tube with other integrated photonic elements, such as a straight waveguide placed very close to and beneath the rolled-up tube. This is would be first step leads to use the rolled-up tube as a 3D photonic coupler in an integrated photonic system.

Conformal transformation had been applied to analyze the optical field distribution in curved waveguide structures, such as waveguide bend [16] and ring resonator [17]. We use the following conformal mapping scheme similar to the ones used in [16]:

Applying the conformal transformation introduced in Eq. (4) to the boundaries of the rolled-up tube as defined in Eq. (1). One may find the boundaries of the converted waveguide in u-v plane are:

It is instructive to visualize the converted structure after the conformal transformation in the u-v plane as shown in Fig. 2. Because of the periodicity of rotation, $\phi \in [0,2\pi )$ is sufficient to cover the whole structure. As a result, $v$ has a primary domain of $[0,2\pi R_0)$. In Fig. 2(a) and 2(b), I plotted two transformed tubes in the primary domain, the first one has an integer winding number of 5, and the second one has a non-integer winding number of 4.75. Any value of v outside the primary domain can be translated back into this domain by subtract $2t\pi R_0$ from it, where t is some integer that satisfies $(v-2t\pi R_0)\in \left[0,2\pi R_0\right)$. When solving the field distribution inside the transformed waveguides, we may either enforce the periodic boundary condition for v, or unfold the structure within the primary domain repeatedly to cover the entire v-axis.

 

Fig. 2 Equivalent waveguide structures of the microtubes with integer (a) and fractional (b) winding numbers; N_t = 5 for (a) and N_t = 4.75 for (b); and the refractive index profiles of the corresponding waveguide (c) and (d), respectively.

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In this study, I assume a step-index profile of the original structure:

4. Analytical solution of the resonant modes of the microtube

In order to obtain an approximate analytical solution of the wave equation after the conformal transformation, I will have to make a few more assumptions. First, I will linearize the boundaries of the transformed waveguide. When $s<<R_0$, one may take the Tylor expansion of the log term on the right-hand-side of Eq. (6) and only keep the linear term. The result is:

 

Fig. 3 Equivalent waveguide structure for microtube with N_t = 5. The three different left boundaries shown in different colors are the exact (red), the linearized (black) and the approximated (green). The inset is a close-up view of these boundaries, which are very close to each other even in this view.

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The waveguide in the u-v plane is tilted and it makes an angle of $\theta ={\tan }^{-1}\left(s/2\pi R_0\right)$ with the vertical axis v. To further simplify the theoretical discussion below, one may rotate the $u-v$coordinate system counterclockwise by $\theta$ to make the boundaries of the transformed waveguide parallel to the axes of the rotated coordinate system $u\text{'}-v\text{'}$. The relationship between $u-v$ and $u\text{'}-v\text{'}$is:

To simplify the discussion, we will only focus on the microtube with winding number being integers. (For fractional winding numbers, one may divide the structure into two sections with different thickness, apply the same method to both sections and combine them together using the boundary conditions at the interface of these two sections along the v’-axis.) In the $u\text{'}-v\text{'}$coordinate system, one may apply the method of separation of variables and consider the periodicity of structure on $v\text{'}$ to assume

Insert Eq. (12) into Eq. (11) and reorganize the result, one may have

In region I of Fig. 3, as $u\text{'}$ goes to $-\infty$, $Z$goes to $+\infty$and $Bi(Z)$becomes unbounded. So the physically meaningful solution in this region must have $b=0$. On the other hand, for any propagation mode (or pseudo-mode) supported by the rolled-up tube, the optical field in region III ($u\text{'}>0$) should decay from the boundary. In the vicinity of the boundary, $Bi(Z)$ decays while $Ai(Z)$ increases. Therefore, similar to [17], we only keep $Bi(Z)$ and set $a=0$. The solution of all three regions can be summarized as:

In order to find a set of nontrivial coefficients, we may set the determinant of the matrix on the right hand side of Eq. (17) to zero, which leads to the following Eigen equation:

5. Results and discussion

The analytical solution given in Eq. (12) describe the resonant mode of the microtube waveguide. It is very useful to visualize the radial field distribution of the resonant modes. Because the determinant of the coefficient matrix in Eq. (17) is zero, these four equations are degenerate. In order to find the field distribution, one may fix any one of the variables and find the relative values of the remaining three with respect to the fixed one. In principle, the value of the fixe variable can be determined by the intensity of the incident optical field. Figure 4(a)-4(c) show the radial field distribution of the 40th mode of the first order (m = 1, q = 40) for tubes with the same out radius, R₀ = 2µm, but increasing single layer thickness, s, from 50 nm to 100 nm. The refractive index of the waveguide is chosen to be n₂ = 1.98, the refractive indices of the surrounding media are n₁ = n₃ = 1.0. The intensity of the field is normalized so that the maximum intensity is always 1. The two short vertical bar illustrate the radius of the inner and out boundaries of each tube. One can see that as the tube wall thickness increases, the field confinement improves, i.e., more electromagnetic energy is confined inside the tube. Comparing Fig. 4(a) and 4(b), one may notice some small shift of the peak position to the left (away from the outer rim of the tube). But the position and profile of the radial field don’t change much between Fig. 4(b) and 4(c). This illustrates that for a given tube material and radius, after the thickness of the tube wall reaches certain value, the mode becomes a whisper gallery mode. For the case shown in Fig. 4, once single layer thickness passes 80nm, the field barely touches the inner rim of the tube. This finding validates the previous claim that the exact position and shape of the inner boundary is less important for thicker tube.

 

Fig. 4 Radial field distribution of the 40th mode of the first order for three different tubes with the same outer radius, R₀ = 2µm, but increasing single layer thickness. The refractive index of the tube is chosen to be 1.98, the outside media is chosen to be air so that n₁ = n₃ = 1.0. The single layer thickness of the tubes are (a) 50 nm, (b) 80 nm, and (c) 100 nm.

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Now let’s take a look at the resonant modes of different azimuthal orders. Figure 5 shows the 40th resonant mode of the first 3 orders. The radius of the tube is R₀ = 2µm, the single layer thickness is s = 50 nm. The refractive indices are the same as used in Fig. 4, and the field intensity is normalized in the same way. We can see that as the azimuthal mode order, m, increases, one may find increasing number of local maximum/minimum points along the radial direction. This is similar to the modes found in a planar rectangular waveguide. For higher order mode, the peak closer to the inner rim normally has higher intensity because the effective index is lower at positions closer to the inner boundary.

 

Fig. 5 Radial field distribution of the 40th mode of the first 3 orders for the same microtube with R₀ = 2µm, single layer thickness s = 50 nm. The refractive index of the tube is chosen to be 1.98, the outside media is chosen to be air so that n₁ = n₃ = 1.0.

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With this tool at hand, we can set off to study the dependence of the resonant frequency as a function of some important design parameters. At first, let’s investigate the resonant frequency shift when the microtube radius, $R_0$, varies. Figure 6(a) shows the calculated resonant frequencies of the lowest 40 modes of the first azimuthal order for a silicon nitride microtubes (n₂ = 1.98) with radius varying from 2$\mu m$ to 3$\mu m$. The winding number is 5 and the single layer thickness is 50 nm for all these microtubes. We can see that as the tube radius increases, the resonant frequency decreases for a given mode number. This is clearly shown in Fig. 6(b), which takes the 15th mode as an example. As the microtube radius $R_0$increases, $k_{v\text{'}}\propto 1/R_0$ decreases. Because$k_{v\text{'}}=2n_{eff}\pi /\lambda =n_{eff}2\pi \nu /c$, the resonant frequency$\nu$decreases accordingly. This can easily be understood from simple geometric consideration. For a given resonant mode, the optical path should be integer number of the wavelength. As the radius increases, it takes longer wavelength to satisfy this condition for the same integer number, which corresponds to a lower resonant frequency. In Fig. 6, I investigate tubes with radius up to 3μm. If the tube radius continue to increases, s/R₀ will decrease accordingly and the assumptions I made in the previous section to use Taylor expansion to linearize the boundary and the refractive index become even more accurate. Similar behavior will be observed as shown in Fig. 6(a) for larger tube but with smaller slope.

 

Fig. 6 (a) The resonant frequencies up to the 40th azimuthal mode of silicon nitride microtubes with different radius from 2µm to 3µm. (b) The resonant frequencies of the 15th azimuthal mode for silicon nitride microtubes with radius varying from 2µm to 3µm.

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To validity of this model, I use it to calculated the resonant modes for a microtube with radius $R_0=2.625\mu m$, wall thickness $N_{t}s=200nm$, and refractive index$n_2=3.46$, which are the parameters used in [19]. As pointed out in [19], the free spectral range (FSR) is more important than the exact location of resonant modes. The calculated FSR is 0.022eV for the resonant modes where the photon energy falls into between 1.14eV and 1.26eV, which matches very well with the experimental results (~0.02eV) shown in [19].

As a comparison, I plotted the resonant frequencies calculated using COMSOL Multiphysics software package for the waveguide with outer radius R₀ = 2µm in Fig. 7. All the other parameters are the same as described in Fig. 6. The incident light is linearly polarized along the rolling axis. One may find that there’s no resonant mode for frequencies lower than 6 × 10¹³ Hz. This is in good agreement with the results obtained from this model as illustrated by the blue curve in Fig. 6(a). As the light frequency increases, higher order mode will be supported. Therefore, it would be easier to find FSR in the region between 2.0 × 10¹⁴ Hz and 4.0 × 10¹⁴ Hz where single resonant peak can be clearly identified. The FSR in this region is found to be 1.29 × 10¹³ Hz from COMSOL simulation, where the FSR obtained using the model developed in this manuscript is 1.31 × 10¹³ Hz. These results also show very good agreement.

 

Fig. 7 COMSOL Multiphysics simulation result of the resonant frequencies for the microtube with radius R₀ = 2µm, single layer thickness s = 50 nm. The refractive index of the tube is chosen to be 1.98, the outside media is chosen to be air so that n₁ = n₃ = 1.0.

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An interest application of the microtube is the optofluidic sensing [20]. When the refractive index of the surrounding media changes, the resonant frequencies of the microtube will shift accordingly. Figure 8 shows the relationship between the resonant frequency and the refractive indices outside the tube. The microtube we investigated in this study has the radius $R_0=2\mu m,$ single layer thickness s = 50nm and the winding number N_t = 5. The refractive index of the tube is chosen to be 1.98. The mode shown here is the 10th mode of the first azimuthal order (m = 1, q = 10) of the microtube. There are three cases shown in Fig. 8. The blue curve illustrates the frequency shift as the outer refractive index, n₃, varies while the refractive index of the tube and that of the hollow core remain the same. In this case, we see the resonant frequency increase monotonically with n₃. The red curve illustrates the frequency shift as the refractive index of the hollow core, n₁, varies. One may see that the resonant frequency initially decreases as n₁ increases. It reaches a minimum around n₁ = 1.55 and reverse the trend to increase steadily. The green curve shows the case where the refractive indices of both the outer media and the hollow core change simultaneously. It follows a similar pattern much like the second case described above. Before the refractive index of the outside media reaches 1.45, the resonant frequency decreases. This trend is in agreement with the experimental finding of resonant mode redshift as the outside refractive index increases from 1.0 to 1.3 in [20]. However, this model predicts that this resonant frequency redshift will reverse when the outside refractive indices increase further beyond 1.45. The resonant frequency will eventually blue-shift after the refractive indices reaches 1.6 and this trend accelerates when the outside refractive indices continue to approach that of the microtube waveguide.

 

Fig. 8 The resonant frequency of the microtube varies when the refractive indices of the media outside the microtube changes. The blue line illustrate the case that the refractive index of the inner core is kept fixed at 1.0 while the refractive index of the out media varies. The red line shows the opposite case. The green line shows the case that both of these refractive indices vary simultaneously.

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6. Conclusion

In this paper, I introduced an analytical model based on conformal mapping to describe the resonant modes of the self-rolled-up microtube waveguide. The equivalent parallel-piped waveguide can be used to better understand the physics of the wave propagation inside the curved microtube which lacks the rotational symmetry. With a few simple and straightforward approximations, an analytical solution of the wave equation is obtained, which can be used in further analysis of the coupling between the microtube waveguide with other devices. The results calculated from this model matched very well with published experimental data and with simulation result obtained using COMSOL Multiphysics software package.

The self-rolled-up microtube can be served as a vertical coupler for 3D photonic integration between 2 adjacent photonic device planes. Compared to conventional directional coupler based 3D integration scheme, the microtube based 3D integration has the advantages of smaller footprint of the coupler, seamless fabrication compatibility with existing 2D planar processing technology [21], flexibility in the choice of physical properties for the coupler and readily incorporated active and electronic tuning capability. One may envision of using this vertical couplers as a building block to naturally connect a III-V based active optoelectronics device plane with a passive silicon photonic device plane. This may open up tremendous opportunities in photonic integrated circuit (PIC) design and applications.