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Abstract
An orbital angular momentum (OAM) dividable on-chip directional coupler design is proposed. To guide OAM modes of topological charge number l = ±1, a waveguide needs to support TE01 and TE10 modes with degeneracy. When a directional coupler is made with such an OAM mode waveguide, it is additionally required to equalize the horizontal-direction coupling strengths of those two OAM constitutive eigenmodes. Base on the coupled mode theory formulation, we have found that this requirement can hardly be satisfied and devised a modified cross-shaped waveguide structure to solve this problem. An example design of OAM mode directional coupler is demonstrated. The coupling length of the designed device is 670 µm, and our numerical simulation showed negligible degradation of OAM mode purity during the operation of complete optical power transfer between two waveguides. To the best of our knowledge, this is the first proposal of the on-chip OAM mode directional coupler. The proposed design approach can be applied to implement various devices for OAM mode-based photonic integrated circuits.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a new exploitable degree of freedom for quantum states of a photon, orbital angular momentum (OAM) attracts great attention in various fields in the present decade [1–5]. One of the unique characteristics of OAM is its helical phase wavefront expressed as exp(-ilϕ), where l is a topological charge number and ϕ is an azimuthal angle. The topological charge number l can take any integer, and OAM modes of different l are orthogonal to each other. For this distinctive property, OAM has great potential to increase a fiber-optic transmission capacity by introducing a new type of mode-division multiplexing.
To exploit this feature, especially in the fields of optical communications and quantum information, a lot of OAM based devices have been studied recently, including OAM-mode fibers [6,7], generators [8,9], detectors [10], (de)multiplexers [11–13], couplers [14,15], and manipulators [16]. In many applications requiring high scalability, such as quantum information and quantum computing [17–20], handling OAM in photonic integrated circuits (PIC) is essential. However, implementation of on-chip OAM devices compatible with typical semiconductor manufacturing processes are quite challenging since most of integrated optical waveguides do not have circular symmetry in their transverse geometries in contrast to optical fibers. Recently, an integrated waveguide structure to support high-order OAM modes up to l = 4 was proposed [21], but, still there are a lot more problems to be solved even in the integrated OAM waveguide itself, including guiding of OAM modes higher than l = 4 and more importantly, guiding multiple OAM modes simultaneously in the same waveguide. Although a number of OAM modes that can be handled in the on-chip devices is limited at this moment, OAM mode-based PICs may find useful applications in quantum information and computing if OAM of photon is adopted as a means to implement a qudit, extending the dimension of Hilbert-space [8].
In PICs, a directional coupler is one of the fundamental building blocks. In general, the directional coupler consists of two parallelized identical waveguides, and its design is rather simple for ordinary applications. However, for the OAM mode-based applications, the directional coupler design is not straightforward since the OAM mode is formed by combining two degenerate eigenmodes and their coupling behaviors should be equalized [14]. Although there were several reports about OAM mode directional couplers, all of those were based on fiber-optics, in which very weak coupling strength was adopted to avoid the imbalance between the coupling lengths of two constitutive modes of each OAM mode and thus, a large separation between two fibers results in a huge device dimension, i.e., of several tens of cm [14,15]. For a small footprint suitable for PIC applications, those previous fiber optics-based designs needs to be improved, especially a means to balance the coupling lengths of two constitutive modes of each OAM mode with an acceptably small waveguide separation should be devised [14].
In this paper, we propose an on-chip l = ±1 OAM mode directional coupler design with a specially designed waveguide structure of a cross-shaped cross-section, achieving a 670 µm coupling length at 1550 nm wavelength which is significantly shorter than the fiber-optics devices. This small footprint is attributed to the coupling behavior of the waveguide structure of a cross-shaped cross-section, which allows balancing the coupling lengths of two constitutive modes of each OAM mode with a small waveguide separation of ∼ 1.5 µm. The performance of the designed device was verified by a numerical simulation, showing acceptably small mode purity degradation. For an input OAM mode (l = 1), the topological charge number numerically calculated right after coupled into the device was 0.9286, which decreased to 0.913 at the end of the device. The mode purity degradation during the coupling into the device is attributed to the imperfect OAM mode purity of the cross-shaped waveguide, which is inevitable for waveguides of non-circular symmetric cross-section [21]. The slight additional mode purity degradation after passing through the device is due to the imperfect balancing between the coupling strengths of two constitutive eigenmodes. The mode purity degradation implies the modal property deformation from the perfect OAM mode. So, the degraded OAM mode can be purified when it is coupled to the OAM fibers which support near-perfect OAM mode (close to Laguerr-Gaussian mode). In this process, additional coupling loss occurs in return. All of calculation in this paper are performed at 1550 nm wavelength and used refractive index values are Si = 3.4 and SiO2 = 1.45 respectively.
2. Design principles of OAM mode waveguide and directional coupler
2.1 Rectangular waveguide based directional coupler
As shown in Fig. 1, a rectangular silicon waveguide can guide l = 1 OAM mode by exciting two degenerate LP-like eigenmodes (TE01 and TE10 modes) with relative π/2 phase difference [22,23]. The requirement in the design of such a waveguide is the degeneracy (the same effective indices) between TE01 and TE10 modes, which maintains the relative phase difference between those two eigenmodes as they propagate along the waveguide. This requirement can be achieved by a proper choice of the waveguide width and height. Figure 1(b) shows an example design of an OAMl=1 mode waveguide of W=0.72 µm and H=0.6 µm.
Fig. 1. (a) Rectangular waveguide cross-section; (b) Generation of OAMl=1 mode in rectangular waveguide.
When a directional coupler is made of two identical OAMl=1 mode waveguides as depicted in Fig. 2(a), the coupling behaviors of two eigenmodes may be different, resulting in OAM mode breakdown as optical power couples from one waveguide to the other. Figures 2(b)–2(d) show numerically calculated coupling coefficients of TE10 and TE01 modes as a function of separation distance for three different rectangular waveguides supporting OAMl=1 mode. In the directional couple, the optical power launched into one waveguide transfers to the other gradually as it propagates. Once the input optical power transfers completely to the other waveguide, then it transfers back to the input waveguide since the coupling between the waveguides occurs constantly. As a result, the optical power in one waveguide shows periodical change along the propagation direction. The half of the period is defined as a coupling length (Lπ). The coupling coefficient (κ) is related to the coupling length (Lπ) as κ = π/Lπ. Once two supermodes in the directional coupler of even and odd symmetry in the x direction are calculated, Lπ is given as Lπ = π/(kΔneff), where Δneff is an index difference between those two supermodes. In this work, all the mode calculation was conducted with the finite-difference method (FDM)-based commercial software (Lumerical mode solutions). So, the coupling coefficient is given as κ = kΔneff. As seen in Figs. 2(b)–2(d), the coupling coefficient of TE10 mode is always larger than that of TE01 mode for any separation distance, so that there is no chance to obtain the same coupling coefficients for TE01 and TE10 modes for the three considered waveguide structures. This implies that the directional coupling composed of those rectangular waveguides cannot work for OAMl=1 mode.
Fig. 2. Coupling coefficient of TE10 and TE01 modes (a) Cross-section of rectangular waveguide coupler; (b) In case of W=0.72 μm and H=0.6 μm; (c) In case of W=0.928 μm and H=0.8 μm; (d) In case of W=1.13 μm and H=1 μm.
According to the coupled mode theory (CMT) [24], the coupling coefficient κ of TE-like modes in x-axis aligned rectangular waveguides is approximately as:
2.2 Directional coupler based on waveguides of cross-shaped cross-section
In rectangular waveguides, the transverse wavenumber, kx and consequently, the coupling coefficient of each eigenmode sensitively varies with a waveguide width [25]. From this point of view, we can expect that a width variation along the vertical direction in a waveguide structure may have a chance to equalize the coupling coefficients of TE01 and TE10 modes. Of course, in such a waveguide structure, kx cannot be defined precisely and (1) is no longer valid. However, in our previous work, it was shown that a waveguide of a cross-shaped cross-section can support TE01- and TE10-like modes, forming OAMl=1 mode [21]. From the fact that the modal properties can be controlled with the cross-section shape change with keeping the mode profiles virtually the same, we may expect that equalizing the coupling coefficients of TE01 and TE10 modes can be achieved by the waveguide cross-section modification. In order to test this idea, we investigated the effect of the corner etching of the rectangular waveguide on the coupling coefficient. We started from one of the rectangular waveguide supporting OAMl=1 mode (in Fig. 2(d)) and numerically calculated κ as increasing the corner etching depth, t (Fig. 3(a)). Figure 3(b) shows the results, where one can see that TE10 and TE01 modes have the same coupling coefficient when t is around 490 µm. Since TE10 mode is more tightly confined in the x direction, the effect of the corner etching, which can be considered as reduction of an effective width, on κ is relatively small. On the contrary, the corner etching severely changes κ of TE01 mode.
Fig. 3. (a) Cross-section of cross shape waveguide coupler; (b) Coupling coefficient of TE10 and TE01 modes as a function of t.
Obviously, the introduction of the corner etching breaks down the degeneracy between TE10 and TE01 modes in the starting rectangular waveguide. So, in order to achieve κ equalization and degeneracy at the same time, H, W, and the dimensions of the corner etching in the cross-shaped waveguide need to be optimized simultaneously, which was conducted by using the particle swarm optimization (PSO) method [26] in this work. As shown shortly, the vertical and the horizontal dimensions of the corner etching were not kept the same in the waveguide optimization. The performance of the directional coupler composed of the optimized cross-shaped waveguide is presented in the next section.
3. Simulation results and analysis
The optimized cross-shaped waveguide structure and electric field (Ex) profiles of TE10 and TE01 modes are presented in Fig. 4. A fabrication scheme for this waveguide is described in [21]. The effective indices of TE10 and TE01 are 3.0903332 and 3.0903265, respectively. The effective index difference of those two modes of 6.7 × 10−6 is sufficiently small to maintain the relative phase difference as they propagate along ∼ cm- long waveguides, which is long enough for practical PIC applications. The topological charge number l of the designed waveguide was numerically calculated [27], resulting in l=0.9286, which is comparable to previous OAM waveguide designs [13,24].
Fig. 4. (a) Cross-section of the optimized cross shaped waveguide and corresponding mode profiles: (b) TE10 and (c) TE01.
Figure 5 shows the coupling behaviors of the designed cross-shaped waveguide as a function of D. When 1.45 µm ≤ D ≤ 1.5 µm, κTE10 / κTE01 is quite close to 1. Even for D > 1.5 µm, κTE10 / κTE01 increases very slowly, keeping it below 1.05 up to D = 1.8. This quite flat slope of κTE10 / κTE01 with respect to D is the evidence that the corner etching scheme is effective, that is, in the designed waveguide, the evanescent field profiles of TE10 and TE01 modes in the x direction are almost the same. Whereas, for D < 1.35 µm, as two waveguides get closer, κTE10 / κTE01 increases rapidly. This is due to non-negligible contribution of the field distribution in the core to the field overlap integral between two waveguides [24]. Furthermore, Fig. 5(b) shows that as D increases, the coupling length increases faster than eD. This feature is very useful because the coupling effect in the bending region at the input/output of the directional coupler becomes negligible, which can cause a serious problem in multimode waveguide based directional couplers [28].
Fig. 5. (a) Coupling coefficient ratio of κTE10 to κTE01 as a function of separation distance D; (b) Coupling length of TE10 and TE01 mode as a function of separation distance D.
The performance of the designed OAMl=1 mode directional coupler with D = 1.5 µm has been analyzed using the eigenmode expansion (EME) solver (Lumerical mode solutions). Figure 6(a) shows a schematic diagram of the device, and the calculated optical power coupling behavior is plotted as a function of the device length in Fig. 6(b), showing Lπ = 670 µm. The field distributions on x-z planes at y = 0 and 0.4 µm are shown in Figs. 6(c) and 6(d). One can see that the designed device functions as a typical directional coupler for the input of OAMl=1 mode.
Fig. 6. (a) Schematic diagram of cross-shaped waveguide based directional coupler, (b) Simulation results of output power at each directional coupler port with coupler length, field distributions on x-z plane at (c) y=0 μm, and (d) y=0.4 μm.
Finally, to confirm the conservation of OAM in both waveguides of the directional coupler during the optical power coupling operation, we examined topological charge number l at z = 335 µm, 407.5 µm and 670 µm, at which the ratio of optical powers in the input to the output waveguides are 1:1, 1:2, and 0:1(complete power transferred), respectively. The electric field (Ex) distributions and phase profiles at those locations are presented in Fig. 7. In Table 1, the normalized optical power and the topological charge number in each port at those locations are listed. One can see that the topological charge number degradation is negligible during the complete power transfer operation in the designed directional coupler, which verifies the operation of OAMl=1 mode directional coupler. The slight degradation of the topological charge number stems from the small effective index difference between TE01 and TE10 modes.
Fig. 7. Field (Ex, horizontal component) and phase distributions of the OAM mode in each coupler length case.
Table 1. Topological charge number of each port and normalized output power.
As seen in Fig. 5(b), the coupling length can be shorted by reducing the separation distance D. With D = 1.4 µm, the optimized waveguide design has a coupling length ∼ 200 µm, and the mode purity of 0.86 is achieved after the complete power transfer between the waveguides, which is still acceptable. It appears that there is a trade-off between the coupling length and the mode purity. For the smaller separation distance, the stronger coupling may bring about nonnegligible deformation of the constitutive eigenmode of each waveguide, resulting in the degradation of the mode purity.
4. Conclusions
In this work, based on the CMT formulation, we have found that an OAM mode directional coupler of practically fabricable dimensions can hardly be realized with a rectangular waveguide. From this finding, we have proposed on-chip OAM mode directional coupler design, in which a cross-shaped waveguide structure has been optimized to equalize the horizontal-direction coupling strengths of two constitutive eigenmodes of OAMl=1 mode. The proposed approach to control the horizontal-direction coupling strengths of the waveguide modes appeared to work effectively, so that two OAM constitutive eigenmodes optimized for a separation distance of 1.5 µm showed almost the same coupling behaviors over a remarkably wide separation distance range from 1.35 µm to 1.8 µm. The designed device has a coupling length of 670 µm, and shows negligible degradation of OAM mode purity during the complete optical power transfer operation. In that the directional coupler is one of key building blocks in PICs, we believe that our proposed design approach offers prospect of various OAM mode based on-chip device implementation, especially for gates of photon-based quantum information and computing.
However, the extension of the design approach to the higher order OAM modes needs further research, which might not be straightforward. To design a directional coupler for an OAM mode of a specific order up to l = 4, the cross-shaped waveguide, which has been shown to support the higher OAM mode by combining two constitutive eigenmodes [21], still can be adopted. For l larger than 4, however, the cross-section of the waveguide may be modified to support the higher-order OAM mode in the first place and equalize coupling strengths of its constitutive eigenmodes. Furthermore, if the directional coupler is to work for multiple OAM modes of different orders simultaneously, its design will be quite difficult. First of all, we need a waveguide to support the target OAM modes simultaneously. Although it has been shown that the cross-shaped waveguide can be designed to simultaneously support the OAM modes of l = 1, 2 [21], its extension to the other higher-order OAM modes is not clear. Secondly, the waveguide shape should have a number of design parameters enough to equalize the coupling strengths of all the constitutive eigenmodes corresponding to the target OAM modes, and the optimization of the waveguide shape to obtain the equalization will be quite difficult. Therefore, the extension of the proposed design approach to the higher-order OAM modes remains as a rich topic of further research at this moment.
Funding
Agency for Defense Development of Korea.
Acknowledgments
This work was supported by the research fund of Signal Intelligence Research Center, supervised by Defense Acquisition Program Administration and Agency for Defense Development of Korea.
Disclosures
The authors declare no conflicts of interest.
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