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We study the propagation of orbital angular momentum (OAM) modes in rectangular multimode waveguides. Due to the multimode interference effect, an OAM mode input forms self-images at certain propagation distances. As OAM modes can be decomposed as the superposition of a pair of quarter-wave phase-shifted even and odd modes, their symmetry properties lead to two different self-imaging categories – forming the OAM-maintaining and the field-splitting self-images. We analyze these phenomena using multimode interference theory, and establish the rules governing the OAM-maintaining self-imaging, which allows the multi-mode interference waveguides to be used as OAM mode splitters and couplers.
© 2015 Optical Society of America
1. Introduction
Lightwaves carrying angular orbital momentum (OAM) [1] can potentially provide an additional dimension for multiplexed communications [2, 3]. They attract interests from researchers seeking ways to break the spectral efficiency limit (Baud/Hz) for optical and RF communication technologies. Latest efforts in integrated OAM devices focus on emitters [4], multiplexers/demultiplexers [5–8], and receivers in surface-normal beam configurations. For transmission channels, ring core optical fibers supporting multiple orders of OAM modes have already been used in system-level transmission demonstrations [9]. However, to the best of our knowledge, there exists no reported study for OAM mode transmission in rectangular waveguides, which is the key element for planar integrated photonic circuits.
Here, we demonstrate that rectangular waveguides supporting multiple transverse modes and maintaining symmetry in two orthogonal transverse directions can be utilized for OAM mode transmission and coupling. The analysis is based on the well-known multimode interference (MMI) theory [10–16], which is developed to deal with the special case of OAM mode propagation. Such OAM-MMI waveguide enables devices to be designed as OAM power splitters and couplers. Potential applications include OAM transmission system, nano-imaging and optical manipulations.
The paper is organized as follows. Section 2 reviews the MMI theory. Section 3 develops the OAM self-image forming properties in the rectangular waveguide. Section 4 shows the simulation results of central excitation. Section 5 gives an example of general interference cases and illustrates the application of the OAM-MMI. Section 6 concludes the paper.
2. Review of MMI theory
In this section we briefly review the principles of the MMI waveguide, and summarize the one-dimensional self-imaging theory with emphasis on difference between symmetric and anti-symmetric inputs. Then we extend the theory into the two-dimensional case. We derive the imaging properties of OAM modes in rectangular MMI waveguides in Sec. 3.
Multimode interference theory in one-dimensional waveguides is well developed in previous literatures [10–12], which suggests self-images at certain lengths can be expressed as the superposition of replicated input fields with certain offsets and phase shifts. Mathematically, a physical input field f(x) with x ranged in (0, W) can be anti-symmetrically extended to f(x)−f(−x), and then extended 2W periodically over the entire x axis to form ψ(x, 0). The general field replication theory of one-dimensional MMI waveguide with width W is based on the following equation [10]:
where ψ(x, L) is the output field distribution at certain propagation length L, which can be written as a superposition of N copies of the input field ψ(x-xp,0)exp(jΦp), indexed by p with transverse offset xp, and relative phase Φp. The self-imaging process produces N number of replicated field and M number of times along the propagation direction only at propagation lengths of L = 3Lc(M/N), with no common divisors between integers M and N, where Lc is the coupling length between the two lowest order modes and can be approximately related to the waveguide material refractive index nf and working wavelength λ0 as Lc = 4nfW2/3λ0. The offset and the phase of the replicated field is given by For simplicity, in the following discussion, we choose M = 1 which is also a common practice for the shortest device length.For a central input (at W/2) field which is symmetric or anti-symmetric, the number of images at the length of L = 3Lc/N can be reduced if N = 2K, where K is an integer. The output field can be expressed as
It means that overlaps happen for the replicated field indexed by p and p + K at the position xp + K, and inside the physical waveguide, an upright f(x) will overlap with an inverted −f(−x). The phase term depends on the input field being symmetric or anti-symmetric. For symmetric input, the final field phase is the phase difference of the two overlapping replicated fields, whereas for the anti-symmetric input, it’s the phase sum of the two.For image length with odd integer K, overlapped fields interference leads to K self-images expressed as
in which image relative offset xp and image phase term φp is deduced from Eqs. (1)–(4). These K self-images are with identical image offset for both symmetric and antisymmetric inputs:where the square bracket pair means rounded to the nearest integer less or equal to the number inside.However, the image phase term φp is dependent on the symmetry of the input field. In the case of symmetric input field, the image phase term φSp (subscript S denotes symmetric input field) is given as
whereas in the case of anti-symmetric input field, φAp (subscript A denotes anti-symmetric input field) is instead given asIn both cases, the parameter a = 1 for even [p + (K-1)/2], and a = −1 for odd [p + (K-1)/2].If we indexed the physical images from left to right inside the waveguide domain (0, W) as s, its relation with p is given as
Since for 0≤p≤[K/2], the images appear in the right side of the effective area (W/2,W), and for [K/2] + 1≤p≤K−1, the images are out of (0,W) and need to be coordinate-translated back into the left side of the effective area (0, W/2).For image length with even integer K, the self-images overlap with a phase difference of 0 or π (shown as phase term in Eq. (4)). Thus every other image will vanish, reducing the total image number to K/2. If the input field is symmetric, exact self-images appear at the output offset by x’Sp, which is at the center of image cell. This property is widely exploited to design shorter power splitters [11, 12]. The anti-symmetric image offset x’Ap is basically a W/K shift relative to symmetric input case:
This W/K offset of x’Ap relative to x’Sp moves self-images centers to the edges of the image cells, thus these self-imaging fields split [11]. The application of such field-splitting imaging position for anti-symmetric input field is very limited as most of the input fields launched into the MMI device section are fundamental waveguide modes that are inherently symmetric. However, as we will discuss in Sec. 3, OAM modes always have the anti-symmetric field components, thus this phenomenon leads to field-splitting positions for OAM-MMI at even-K imaging lengths. Field-splitting is not suitable for OAM self-images, thus the following discussions focus on the odd-K imaging lengths.Extending the MMI theory into the two-dimensional (2D) case, where multiple modes are supported in both horizontal and vertical dimensions, one assumes separation of variables in input field function is approximately valid in highly confined 2D-MMI waveguides [13]. Thus the 2D self-imaging can be considered in the x and y transverse directions to be fX(x) and fY(y) respectively. Self-imaging in horizontal and vertical directions is determined by the propagation distance, which is fractions Nx and Ny to the 3Lcx and 3Lcy, and in turn proportional to the square of the waveguide width (Wx) and height (Wy) respectively. In order to have discrete image-forming positions for horizontal and vertical directions to overlap, one needs to design the aspect ratio between waveguide width and height as [13]
With proper waveguide aspect ratio and choosing odd-K and odd-J imaging length L, K × J output images for central input to the rectangular waveguide is given by the 2-D MMI self-imaging equation [13]:where the offset x, y and the associated phase φ and θ in the two transverse directions are labeled with subscript p and q respectively. All the offsets and phases terms can be readily calculated with the one dimensional Eqs. (6)-(9).3. OAM-MMI theory
General imaging properties of Gaussian input mode to the 2D-MMI waveguide have been experimentally demonstrated to support the theory [16]. Figure 1(a) schematically illustrates upon central Gaussian mode input, a single self-image forms at positions L = 3Lc/4 and L = 3Lc/2, which belong to K = J = 2 and K = J = 1 respectively. Since Gaussian mode is symmetric in both x and y directions, the even-K position L = 3Lc/4 does not exhibit field-splitting.
Fig. 1 Schematic of central inputs into a square cross-section MMI waveguide. (a) Gaussian mode input forming single image at propagation lengths of 3Lc/4 and 3Lc/2. (b) Decomposition of λ = 1 OAM mode into odd and even modes, each of them symmetric in one direction and anti-symmetric in the other. (c) Decomposition of λ = 2 OAM mode into odd and even modes, the odd mode is anti-symmetric in both directions and the even mode is symmetric in both directions. Both λ = 1 and λ = 2 central OAM mode inputs exhibit field-splitting at 3Lc/4 and OAM-maintaining image at 3Lc/2.
As shown in Figs. 1(b) and 1(c), an OAM mode can be represented as superposition of an odd mode and a quarter-wave shifted even mode. Decomposing any order of OAM mode ( ± λ) into odd- and even-mode field components, they can be generally expressed as [17]
where the ± sign is determined by the sign of the OAM order ( ± λ).The odd and even part of the OAM mode can be further represented into field functions in x and y directions. For odd-order OAM modes like λ = 1 case shown in Fig. 1(b), the odd-mode field fo(x,y) = fS(x)fA(y) is symmetric along the horizontal direction, and anti-symmetric along the vertical direction, whereas the even-mode field fe(x,y) = fA(x)fS(y) has these symmetry directions swapped. Thus the odd-order OAM mode input field is expressed as
In contrast, for even-order OAM modes like λ = 2 case shown in Fig. 1(c), along both horizontal and vertical directions, the odd-mode field fo(x,y) = fA(x)fA(y) is anti-symmetric whereas the even-mode field fe(x,y) = fS(x)fS(y) is symmetric. Thus the even-order OAM input field can be expressed asIn order to have self-images matched with input OAM given by Eq. (14) and (15), it is required that field-splitting should be avoided for all field components. However, even-K and even-J image lengths with anti-symmetric field-splitting violates such requirement. As shown in Fig. 1(b), at the field-splitting position L = 3Lc/4 where K = 2, the odd-mode field component of the λ = 1 OAM mode splits in the vertical direction and the even-mode field component splits in the horizontal direction. For λ = 2 OAM mode shown in Fig. 1(c), the odd-mode field component is anti-symmetric and hence split in both directions. The even-mode field component remains identical to the input, as it is symmetric in both directions like Gaussian mode. But the superposition of the odd- and even-mode field does not represent a self-image of the input OAM.
Odd-K and odd-J image lengths exhibit no field-splitting, with Eq. (12) and applying the linear superposition property, for odd-order OAM mode input, the output field is given as
where the offset x, y and the associated phase φ and θ in the two transverse directions are labeled with subscript S or A to represent symmetric or anti-symmetric input cases, and the subscript p and q are the replicated field index. All the offsets and phases terms can be readily calculated with the one dimensional Eqs. (6)-(9). Likewise, for even-order OAM mode input, the output field is then given asAs shown in Figs. 1(b) and 1(c), odd- and even-mode field component of the OAM each forms self-images at L = 3Lc/2 where K = 1 is an odd number.Comparing Eq. (16) with Eq. (14), and Eq. (17) with Eq. (15), reconstructing the OAM modes from the non-field-splitting self-images of odd- and even-mode field components requires that the odd- and even-mode fields maintain quarter-wave phase shifted ( ± ejπ/2) as that in the input OAM. This leads to the phase-matching requirement, which means that the phase term of odd- and even-mode fields need to have integer multiples of π differences, i.e. (φAp + θSq)−(φSp + θAq) = mπ or (φSp + θSq)−(φAp + θAq) = mπ for odd- and even-order OAM modes respectively, where m is an integer. Such phase-matching requirement is automatically fulfilled for odd-J and odd-K image lengths, as one can easily expand the phase term difference for odd-order OAM mode input with the help of Eq. (7) and (8) into:
whereas for input of even-order OAM mode, the phase term difference can be expanded asIn both cases, a and b are always ± 1 and follow the condition defined as the case in Eq. (8). The equation above reveals that for all p and q combinations, the phase term difference is always mπ. Thus the odd- and even-mode field components are always quarter-wave phase shifted with each other, as the case in the OAM input. However, m parity depends on p + q parity, or s + r parity if we use the physical image index, where r is the physical imaging index in y direction refer to Eq. (9). For even m, OAM self-image order sign is identical to the input OAM, and for odd m the sign of the OAM self-image reverses relative to the input.Therefore, it can be concluded that for a rectangular waveguide with properly designed width and height given by Eq. (11), upon central input one can form K × J OAM self-images at image positions L = 3Lcx/2K = 3Lcy/2J with odd K and J. Among these odd numbers of self-images, the central one has the OAM order identical to the input. Half of the remaining OAM self-images are identical to the input OAM mode while the other half have their signs reversed. Summing up the orders of all output OAM self-images, the total output OAM order remains unchanged as the input, the total OAM mode order remains unchanged compared to the input, although calculating the OAM per photon requires a more rigorous calculation.
4. Central input and OAM mode power splitter
To figuratively demonstrate the self-imaging properties of OAM modes, we use the beam propagation method (BPM) to numerically simulate OAM mode propagation inside a rectangular MMI waveguide, and analyze the results with the theory presented above. We use the commercially available simulation software package OptiBPM® [18]. All the simulations assume a vacuum wavelength of 1550 nm, and a silicon waveguide (nf = 3.45) surrounded by silica (nc = 1.45), which can be realized with the silicon on insulator (SOI) technology. The coordinate system is set such that the center of the waveguide is at the origin. Since most practical MMI devices use the MMI length of fraction of 3Lc, we define the propagation length L = Z(3Lc), and use Z as the fraction of the single general image length.
Figure 2 shows the simulation results of a rectangular silicon waveguide with dimension of Wx = 62.6 μm and Wy = 28.0 μm, and corresponding to K/J = 5/1 in Eq. (11). Figure 2(a) shows the amplitude and phase of the input when an λ = 1 OAM mode with waist radius of 2 μm is launched at the center of the waveguide. Figure 2(b) shows the output amplitude and phase at imaging length L = 3Lcx/10 = 3Lcy/2. It shows that 5 × 1 images are formed, all with donut-shaped amplitude distribution and azimuthal varying phase, suggesting all outputs are OAM self-images. The central and the edge images have their azimuthal phase change identical to the input (increasing clockwise and λ = 1), as they correspond to images with even p + q. The self-images between them have their azimuthally phase increasing counterclockwise, suggesting their OAM orders are opposite to the input (λ = −1) because they correspond to images with odd p + q.
Fig. 2 The amplitude and phase distributions with a central OAM input with λ = 1 in rectangular MMI waveguide. (a) z = 0, (b) z = 3Lcy/2 = 3Lcx/10.
Figure 3 shows the simulated field pattern at selected propagation lengths inside a MMI waveguide with Wx = 25 μm and Wy = 25 μm, excited by an OAM beam with λ = 1 and waist radius of 1.5 μm at the center of the MMI input facet (Z = 0) as shown in Fig. 3(a). At the propagation length of Z = 1/2, a single donut-shaped OAM replica is formed as shown in Fig. 3(h). From the phase distribution on the right, we observe that the self-image has the azimuthally varying phase from 0 to 2π (color code blue, green, yellow, red) in the identical clockwise direction as the input. Thus the output has exactly the same OAM order as the input. This indicates that square MMI waveguides is capable of self-imaging the OAM input field. However, the output OAM phase is rotated π/2 in counter-clockwise direction as compared to the input (see the position of the red/blue boundary).
Fig. 3 The amplitude and phase distributions at various z-position with a central OAM input with λ = 1 in square MMI waveguide. (a)Z = 0, (b)Z = 1/12, (c)Z = 1/8, (d)Z = 1/6, (e)Z = 1/4, (f)Z = 1/3, (g)Z = 3/8, (h)Z = 1/2.
Figure 3(e) shows that at Z = 1/4, the field exhibits field-splitting pattern where it is cleaved into four semi-circles, each aligns its straight edge to one of the waveguide side boundaries. In this case, N = 2K with K = 2, suggesting that the anti-symmetric field will form field-splitting patterns. This propagation length for Gaussian center input is the first single self-image position. However, here the separated semi-circular field pattern cannot be used for OAM power splitting. Similar field-splitting pattern also appears at the propagation lengths with integer fraction of 1/4, since those lengths also fulfill the N = 2K condition with even K. For example, Fig. 3(b) and 3(c) show that at Z = 1/12 and Z = 1/8 with K of 6 and 4 respectively, and thus 3 × 3 and 2 × 2 field-splitting patterns are formed. Figure 3(g) shows that when Z = 3/8, similar 2 × 2 field-splitting pattern is formed as Z = 1/8 propagation length, belonging to Z = M/2K with even K case (M = 3, and K = 4). Notice that, at these locations, Gaussian mode center input will result in K/2 × K/2 replicas.
Figure 3(d) shows the OAM-maintaining 3 × 3 self-images of the input field at Z = 1/6, which is suitable for power splitter applications. Employing the same method, we find N = 2K = 6 where K = 3. Because it belongs to the odd-K image-forming length, all the self-images are OAM-maintaining. Figure 3(f) shows another propagation length Z = 1/3 for generating 3 × 3 OAM outputs. Although the image field distribution is identical to that of Fig. 3(d), this case has a different image forming mechanism that belongs to the general interference case discussed in the next section, for in this case N = 3 and it cannot be divided into integer number by 2. Thus OAM input has 3 × 3 self-images, and they will not overlap with each other. Hence these images all replicate OAM input modes.
Notice that in both 3 × 3 OAM self-imaging cases, 5 images (at the center and the corners) has output λ identical to the input OAM, whereas the other 4 images have reversed OAM order of −λ. Additionally, the output images may possess various phase shifts relative to the input. These properties can be analyzed using Eq. (16).
Table 1 below summarizes the self-image analysis of OAM modes.
Table 1. self-image analysis of OAM modes
Although the field-splitting self-imaging is not suitable for power splitting purposes, it may serve the purpose of OAM order and odd/even mode discrimination. Figure 4 shows field pattern at Z = 1/4 for OAM modes with order λ ranges from 1 to 4. For odd azimuthal order OAM modes (λ = 1, 3, …), the fragments of the field-splitting distribution are on the four edges of the waveguides, each reproduces half of the even/odd modes. This is because the decomposed odd and even modes are symmetric along one direction and anti-symmetric along the other as in Fig. 1(b). Whereas for even-order OAM modes (λ = 2, 4, …), the fragments of the field-splitting are distributed at the center and corners of the waveguide, since the decomposed modes are all symmetric or all anti-symmetric in both directions as in Fig. 1(c). Thus if an image sensing device is capable of distinguishing edge and corner field, one can attach it to MMI output to identify the parity of the OAM orders. Such phenomenon can serve as a novel method to distinguish the orders. Compared to the interference methods which may involve dove prisms and beam splitters [18], square MMI waveguides can potentially be used for on-chip integration.
Fig. 4 The amplitude and phase distributions at 3Lc/4, with a central OAM input with (a) λ = 1, (b) λ = 2, (c) λ = 3, (d) λ = 4 in square MMI waveguide.
From above analysis, we can come to the conclusion that central input OAM mode power splitter only offers odd numbers of output. And similar to one-dimensional MMI devices, the central input in square waveguide only offers 1 × N2 power splitting, with no additional input port for coupling applications.
5. General interference couplers and OAM mode synthesis / analysis
General interference happens at positions L = 3Lc(M/N) and N images will be formed. Unlike the special central input case where only odd number of images can be obtained, we can get any number of images though general interference, as long as the offset values of the input field in both x and y direction is the same and no overlap occurs between the output images. Here an example of N = 2 with W/6 input offset is given. The OAM input mode self-image forming property at L = 3Lc/2 can be readily derived from 1D analysis Eqs. (1)-(3) of symmetric and anti-symmetric field pattern as follows, with the superscript 1/3 representing the input position in the waveguide
Thus, the phase difference between odd and even modes can be easily obtained and the spatial overlap and phase match requirements are well served in this case. Therefore, the OAM mode maintains.Figure 5(a) shows offsetting the λ = 1 OAM input launching position to sixth third of the waveguide center in both x and y direction, i.e. at position of (W/6, W/6). Figure 5(b) shows the output images at a MMI length of Z = 1/2. The output forms four self-images with the characteristic donut-shaped field distribution representing OAM-maintaining replicas at the locations of ( ± W/6, ± W/6). The phase plot clearly reveals that these self-images have the correct azimuthal phase variation. However, the azimuthal phase increasing direction is paired into diagonal groups, with anti-diagonal group identical to the input field with λ = 1, and the diagonal group reversing the input OAM order to λ = −1. The summation of OAM orders of each output self-images is zero, which is different from the case of central OAM input presented in Sec. 2.
Fig. 5 The amplitude and phase distributions for an OAM input with λ = 1 in square MMI waveguide. The input locates at position of (W/6, W/6). (a) z = 0, (b) z = 3Lc/2.
Such input configuration can be utilized to implement interference between a Gaussian mode and an OAM mode of unknown order, as Fig. 6 demonstrates. Figure 6(a) gives the input position of the OAM mode under analysis and the reference Gaussian mode. Figures 6(b)–6(c) are the output at z = 3Lc/2 for OAM modes λ = 3, 2, and 1 respectively. Similar to the interference analysis using bulk optics, we can analyze the OAM orders by counting the interference lobes in the output intensity pattern. Notice that, with the MMI device, to achieve proper interference relies on the length of the MMI waveguide only, which is controllable through the fabrication process, whereas careful calibration is required for the two interference arms and several beam splitters are involved in a bulk optics interferometry setup.
Fig. 6 Interferences between Gaussian mode and OAM mode with different orders in a square MMI waveguide. (a) the input pattern of Gaussian mode at the lower left and OAM mode at the upper right. The other three are interference intensity patterns at z = 3Lc/2 for (b) λ = 3, (c) λ = 2, and (d) λ = 1.
However, in 2D MMIs, OAM mode cannot maintain if there is launch offset in only one direction, as this situation violates the phase match requirement. Figure 7(a) shows offsetting the λ = 1 OAM input launching position in x direction by W/6, yet keeping it central in the y direction. Consequently, it has a general interference in x direction, and a central interference in y direction. Figure 7(b) shows the output at z = 3Lc/2. The general input interference in the x direction forms two images at ( ± W/6), and the central input interference in the y direction means that both symmetric and anti-symmetric modes form images at the center. The output field distribution reduces to two HE-mode-like field patterns, which is confirmed by the phase plots. This is due to the different phase modification for odd and even modes. The relative phase difference between even and odd modes can be calculated as follows using the same method as Eq. (18), which corresponds to their positions.
where φ and θ is the relative phase modification of images in x and y directions with subscript to S or A to represent symmetric or anti-symmetric input case. The superscript 1/3 or 1/2 represents the input position in the waveguide. The phase terms can be calculated with Eq. (20)-(21) and Eq. (7)-(8). It can be seen that there is a π/2 modified phase difference of even and odd modes at corresponding positions. With the addition of the original π/2, the phase difference between the even and odd modes become identical (right) or opposite (left). As a result, the image pattern turns into two diagonal even and odd modes as shown in Fig. 7(b).Fig. 7 The amplitude and phase distributions of OAM distortion and synthesis. (a) Input of OAM mode with l = 1 at (-W/6,0), (b) Output pattern at 3Lc/2, (c) Input of two diagonal odd and even modes with a π/2 phase shift at (-W/6,0) and (W/6,0), (d) Output pattern of an λ = −1 OAM mode at (-W/6,0).
Although such a configuration may not find application as OAM-maintaining couplers, it could be used for OAM synthesis. The reciprocal principle suggests that in any passive device, the input and output ports can be swapped. Figure 7(c) shows that we deliberately launch even and odd modes, both rotated 45° clockwise, into the two input ports of such a device, with π/2 phase difference between them. Such input configuration results in a synthetic OAM mode at the output as shown in Fig. 7(d). Both the field amplitude and phase plot confirms a successful generation of λ = −1 OAM mode at the port position (−W/6, 0). Although similar OAM synthesis from even and odd modes can be realized using bulk optics, a silicon device with several millimeters length can provide the same function, which is also potentially integration ready.
6. Conclusions
We have investigated OAM mode propagation inside a slab waveguide supporting multiple modes. By decomposing OAM modes into odd/even modes, and considering their symmetry properties while applying conventional MMI analysis, we have developed a theoretical framework that extensively describes the self-imaging properties of OAM modes in MMI waveguides. The general requirements for successful OAM-maintaining self-imaging have been revealed. The theoretical framework and the requirements can be used to guide the creation of various devices based on OAM-MMI.
Examples of such OAM-MMI device have been numerically demonstrated and fully explained by above theory. For central OAM mode input, odd numbers of output self-images can be created, useful for OAM mode power splitting. For general input OAM modes, an even number of OAM mode output can be realized. If the input is offset in one direction and central in another, the device functions as an OAM synthesizer. This work therefore paves the way towards potential planar integrated devices for OAM optics.
As we presented here the OAM input can form OAM-maintaining self-images or field-splitting images, future works could be to examine rigorously whether the whole output field possesses orbital angular momentum. This leads to the question of OAM conservation property in rectangular waveguides. Meanwhile, in order to make OAM-MMI practical to applications, the OAM purity of the self-images along with fabrication error tolerance requires further study. Another important direction is to study the vector field OAM propagation properties in rectangular MMI waveguides.
Acknowledgments
This work is supported by the National Basic Research Program of China (973 Program) (No. 2014CB340000), the Natural Science Foundations of China (Grant No. 61490715), and the Research Fund for the Doctoral Program of Higher Education of China (No. 2013300004115469). Zelin Ma and Kaiyi Wu would like to thank the support from the Undergraduate Research Program by the School of Physics and Engineering of Sun Yat-sen University.
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