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Abstract
To improve the transmission capacity of an optical system, different multiplexing schemes have been proposed, such as optical time division multiplexing (OTDM), wavelength division multiplexing (WDM), polarization division multiplexing (PolDM), spatial division multiplexing (SDM), etc. One kind of SDM technique to boost the capacity is through modifying the spatial phase structure of an optical beam, which is known as the orbital angular momentum (OAM) division multiplexing. Moreover, the OAM signal emitter can be produced by using mature and high-yield silicon photonic (SiPh) technology, without the need of using bulky optical components or expensive spatial light modulator (SLM). The SiPh-based micro-ring resonator is one of the promising OAM signal emitter candidates, since it is simple, compact and easy to fabricate. However, the device performance is highly subjected to the structural design, and the uniformity-induced phase distortion will significantly degrade the purities of OAM beams; hence, introducing severe OAM signal crosstalk during the OAM division multiplexing. In this work, a compact SiPh-based micro-ring resonator type OAM signal emitter with detailed design parameters is presented and the output signal uniformity issue is comprehensively investigated. Two kinds of the structural optimization are performed by adjusting the angular grating width as well as the grating height. The results indicate that a significant improvement in output OAM beam uniformity can be achieved, with the attenuation factor being improved over 88% at the price of acceptable 4 ∼ 5% coupling efficiency reduction. The variations of the transmission and the uniformity induced by the fabrication error are also analyzed.
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1. Introduction
In recent years, the demands for bandwidth are increasing rapidly due to the applications of Internet-of-Thing (IOT), on-line shopping and gaming, 4 K/8 K video streaming, B5G/6 G wireless communications, etc [1]. To support these huge increases in bandwidth demands, enhancing the transmission data rate and capacity are important. Increasing the transmission system bandwidths or using new optical transmission wavelengths are not an immediate process and could face many challenges. Hence, increasing the transmission system spectral efficiency (i.e. transmitting more data at a fixed bandwidth) is becoming more and more important [2,3]. To improve the transmission capacity of the optical networks, different multiplexing schemes have been proposed, such as wavelength division multiplexing (WDM) [4–9], optical time division multiplexing (OTDM) [10,11], polarization division multiplexing (PolDM) [12,13], spatial division multiplexing (SDM) [14–16], as well as advanced digital multiplexing schemes, e.g. orthogonal frequency division multiplexing (OFDM) [17–19], non-orthogonal multiple access (NOMA) [20,21], etc. Among these multiplexing techniques, SDM permits signals transmission in different parallel waveguides [22] or different orthogonal eigenmodes in a waveguide [23,24]. The latter is known as mode division multiplexing (MDM). In optical fiber communication, the SDM signal transmission in parallel waveguides or different orthogonal eigenmodes in a waveguide can be realized via multi-core fiber (MCF) or few-mode fiber (FMF) respectively. Recently, another SDM technique has been proposed to boost the capacity through modifying the spatial phase structure of an optical beam. This technique is known as the orbital angular momentum (OAM) division multiplexing [25]. The OAM beams can be described by the Laguerre–Gaussian (LG) modes, the solution of the paraxial wave equation in the cylindrical coordinates (ρ, θ, z). The LG modes can be derived in Eq. (1).
The OAM signal can be generated using bulky optical components [28] or expensive spatial light modulator (SLM) [29]. Recently, the silicon photonic (SiPh) technology has attracted many attentions. It allows the use of mature and high-yield complementary metal oxide semiconductor (CMOS) fabrication process to manufacture high performance photonic devices at low costs [30]. Besides, SiPh technology also allows the monolithic integration of different optical components within small footprints. Different SiPh-based OAM signal emitters have been proposed, such as micro-ring resonator [31–34], multiple arc waveguides and grating couplers [35,36], tunable-phase arrayed waveguides with circular grating [37], and 2D grating coupler ring structure [38]. Among these SiPh-based OAM emitting devices, the micro-ring resonator is simple, compact and easy to fabricate. There is no need of electrical tunable-phase waveguide to produce OAM signals.
The SiPh-based micro-ring resonator is one of the promising OAM signal emitter candidates; however, the device performance is highly subjected to structural design. The uniformity-induced phase distortion will significantly degrade the purities of OAM beams; hence, introducing severe OAM signal crosstalk during the OAM division multiplexing. Up to the authors’ knowledge, there is no comprehensive analysis of the uniformity at different structural design of OAM signal emitters in the micro-ring resonator in the literature. In this work, a compact SiPh-based micro-ring resonator type OAM signal emitter with detailed design parameters is presented and the output signal uniformity issue is comprehensively investigated. Two kinds of the structural optimization are performed by adjusting the angular grating width as well as the grating height. We first present the uniformity analysis by defining the OAM beam overlap ratio and the phase ambiguity at different topology charges. Then, we optimize the micro-ring resonator type OAM signal emitter with different angular grating widths and grating heights. The results indicate that a significant improvement in output OAM beam uniformity can be achieved, with the attenuation factor being improved over 88% at the price of acceptable 4 ∼ 5% coupling efficiency reduction. The variations of the transmission and the uniformity induced by the fabrication error are also analyzed.
2. Device architecture
Figure 1(a) shows the compact SiPh-based OAM signal emitter, similar to [31,34]. It includes an access waveguide, a micro-ring resonator and the angular grating structure inside the resonator. Figure 1(b) depicts the 3D view of the OAM signal emitter in silicon-on-insulator (SOI) platform. The top silicon layer is 220 nm thick and the buried oxide (BOX) layer and the top cladding layer have the thickness of 2 µm and 1 µm, respectively. The operation principle of the device follows typical resonator, in which only some specific wavelengths can enter into the ring and be excited via the grating. The spacing of these wavelengths, which is known as the free spectral range (FSR), is given by FSRλ = λ2 / neff L, where neff is the effective refractive index and L is the length of the resonator. The vortex topological charge l can be affiliated to these excited wavelengths, generating different OAM beams. The inner radius of the ring is set to be 10 µm and the width of the ring is 500 nm. The angular period (i.e. the period over the radius) is 2π / 106 and the grating is square shaped having width of 200 nm.
Fig. 1. Schematic diagram of the compact SiPh based OAM signal emitter from (a) the top view and (b) the 3D view.
Based on these design parameters, the effective refractive index calculated is 2.45 and the 106-order resonant wavelength is located around 1525 nm, representing the charge of $l = 0$. Moreover, in order to have high coupling efficiency, the asymmetric directional coupler (ADC) must be well-designed. Thus, the gap of the ADC and the width of the access waveguide are 150 nm and 437 nm respectively after the optimization. Figure 2 shows the optimized spectrum of the device, where blue line is the remained power in the access waveguide and the red line is the coupling efficiency measured at 1.5 µm far-field plane. The excited wavelengths of 1535.4 nm, 1544.4 nm, 1553.5 nm and 1563.6 nm correspond to the charges of l = -1, -2, -3 and -4, respectively. The coupling efficiencies for all excited wavelengths are larger than 23% except for l = 0, at which the second order Bragg diffraction degrades the performance. At the charge of l = 0 as shown in Fig. 2, we can observe that the transmission peak (i.e. red curve) shifts towards the l = 1. This asymmetric distribution behavior can be explained by the fact that the counter-clockwise mode (i.e. the mode coupled into the ring resonator) and the clockwise mode (i.e. the mode excited due to the second order Bragg diffraction) couple to each other and split into two different resonant modes appearing on both sides of the real resonant wavelength. As the counter-clockwise mode and the clockwise mode experience unequal quality factors, the extinctions and the linewidths of the resonant modes are different, resulting in asymmetric distribution of the spectrum. The coupling efficiency is related to the critical point condition, where it ensures high power transfer from the access waveguide to the ring resonator. The critical point condition is satisfied if the ratio of the power entering into the ring via the asymmetric directional coupler (ADC) equals to the square of the round-trip loss of the E-field induced by the ring resonator. However, the round-trip loss for each resonant wavelength is slightly different; hence, it is impossible to meet the critical point condition for every wavelengths. Besides, the intrinsic coupling efficiency of the grating may not be equal for these wavelengths. We have made our effort to reduce the difference of the coupling efficiencies for the excited wavelengths corresponding to the charges of l = -1 ∼ -4 during our optimization. As the result, the coupling efficiency of the charge of l = 1 is slightly higher than that of the charge of l = -1 ∼ -4.
Fig. 2. The transmission and coupling efficiency of the SiPh based OAM signal emitter at different wavelengths. Blue curve: remained power in the access waveguide. Red curve: coupling efficiency measured at 1.5 µm far-field plane.
The magnitudes of the E-fields with the charges of l = -1 ∼ -4 are plotted in Figs. 3(a)-(d). To clarify the phase distribution over these fields, digital signal processing (DSP), such as applying of the quarter-wave plates (QWPs) is necessary since the propagated mode in the SiPh device is TE mode; and thereby the excited field is azimuthally polarized. The application of the QWPs can be described mathematically in Eq. (2).
Fig. 3. The magnitudes of the E-fields at wavelengths of (a) 1535.4 nm (l = -1), (b) 1544.4 nm (l = -2), (c) 1553.5 nm (l = -3), (d) 1563.6 nm (l = -4). The corresponding phase profiles in two orthogonal polarizations (a1) l = 0, (a2) l = -2, (b1) l = -1, (b2) l = -3, (c1) l = -2, (c2) l = -4, (d1) l = -3, (d2) l = -5.
Therefore, the excited field with the charge of l can produce two orthogonal linearly polarized fields with the charges of l + 1 and l - 1. The phase profiles are shown in the right part of Figs. 3(a1)-(d2). For example, the E-field of wavelength 1535.4 nm (l = -1) as shown in Fig. 3(a) can be decoupled into phase distributions of l = 0 and l = -2 after the application of QWP as illustrated in Figs. 3(a1) and 3(a2) respectively. The others decoupled phase distribution patterns can also be obtained similarly as shown in Figs. 3(b1)-(d2) respectively.
3. Uniformity analysis
The propagation of OAM beams can be well-characterized with the help of the Hankel Transform (HT). The far-field of the ring shape of OAM beams follows the HT pair as shown in Eq. (3),
In order to study the non-uniformity induced intensity and phase distribution profiles, numerical analysis is performed at different charges l and different attenuation factors α, respectively. The definition of attenuation factor α will be described in next paragraph. Figure 4 illustrates the degradation of the intensity uniformity and phase distribution at different topological charges and different attenuation factors. The attenuation factor α is defined along the azimuthal direction and the Fourier Transform (FT) is performed to represent the far-field under propagation. It is observed that the higher the α-value, the more severe distortion over the phase distributions as well as the intensity distributions.
Fig. 4. The degradation of the intensity uniformity and phase distribution at different charges l and different attenuation factors α (the scale in millimeter).
In order to quantify these effects, we define the overlap ratio compared to the 0th-order beams and the phase ambiguity as shown in Eqs. (4) and (5) respectively,
Fig. 5. (a) The phase ambiguity and (b) the overlap ratio for different l = l1 + l2, where l1 is original excited charge and l2 is any additional charge.
Fig. 6. (a) The phase ambiguity and (b) the overlap ratio for different attenuation factors in the case of (l1, l2) labeled in the figure.
4. Optimization and improvement
4.1 Original structure
The above analysis reveals that the effect of the non-uniformity leads to significant degradation; hence, in this section, we will provide the possible solutions. The optimization can start by analyzing the E-fields in the Figs. 3. The distributions at the radius of 10.25 µm (i.e. measured between the inner radius and the outer radius of the micro-ring resonator waveguide) of 1.5 µm far-field plane are extracted and shown in the Figs. 7(a)-(d). The red curves fit the data in the interval of [0, π] and the exponents can stand for the attenuation factor α. Based on the fitting curve of exp(αθ), the α-values are calculated to be 0.2755, 0.2576, 0.2153 and 0.2330 for l = -1 ∼ -4, respectively, and these values illustrate that the OAM beams will have very high phase ambiguity and higher overlap ratio due to the non-uniformity according our analysis in last section (i.e. Section 3).
Fig. 7. The E-field distribution when the grating size of 200 nm and the fitting curve at the radius of 10.25 µm (a) 1535.4 nm (l = -1), (b) 1544.4 nm (l = -2), (c) 1553.5 nm (l = -3), (d) 1563.6 nm (l = -4).
4.2 Changing the size of angular grating
First, we optimize the uniformity by changing the size of the angular grating structure inside the micro-ring resonator. Figures 8(a) and 8(b) show the angular grating structures inside the micro-ring resonator at the original size and the reduced size of 150 nm, respectively. Due to the change of the effective refractive index and the round-trip loss, the critical point condition is re-adjusted and the highest coupling efficiency occurs when the width of the access waveguide is 428 nm. The optimized spectrum at 1.5 µm far-field plane is obtained in the Fig. 8(c). The peaks of the ring with the grating size of 150 nm are all blue-shift since the neff decreases and the FSRf (i.e. FSRf = c / neff L) increases. The excited wavelengths are now 1531.3 nm, 1540.4 nm, 1549.5 nm and 1558.6 nm, corresponding to the topological charge of l = -1 ∼ -4, respectively; and all of the coupling efficiencies are over 25% except for l = 0, as explained before. Figures 9(a)-(d) show the magnitudes of the E-fields with the charges of l = -1 ∼ -4. It can be clearly observed that the E-field uniformity has been improved compared with that shown in the Figs. 3(a)-(d). The charges for these excited wavelengths are also examined after the DSP and shown in Figs. 9(a1)-(d2). It can be also clearly observed that the charges of two orthogonally polarized beams for all excited wavelengths are well maintained.
Fig. 8. The angular grating size of (a) 200 nm × 200 nm (b) 150 nm × 150 nm and (c) the transmissions at different wavelengths of the modified OAM signal emitter at 1.5 µm far-field plane.
Fig. 9. The magnitudes of the E-fields when the angular grating is adjusted to 150 nm at wavelengths of (a) 1531.3 nm (l = -1), (b) 1540.4 nm (l = -2), (c) 1549.5 nm (l = -3), (d) 1558.6 nm (l = -4). The corresponding phase profiles in two orthogonal polarizations (a1) l = 0, (a2) l = -2, (b1) l = -1, (b2) l = -3, (c1) l = -2, (c2) l = -4, (d1) l = -3, (d2) l = -5.
Similarly, in order to quantify the uniformity, the E-fields with the charge of l = -1 ∼ -4 at the radius of 10.25 µm at 1.5 µm far-field plane are extracted and analyzed in the same manner as in Figs. 7(a)-(d); and the new results are shown in the Figs. 10(a)-(d). Based on the fitting curve of exp(αθ), the α-values are calculated to be 0.1010, 0.0924, 0.0779 and 0.0840 for l = -1 ∼ -4, respectively. It clearly indicates that the improvement for these charges is over 63% in the comparison of the original structure as illustrated in Figs. 7(a)-(d).
Fig. 10. The E-field distribution when the grating size of 150 nm and the fitting curve at the radius of 10.25 µm. (a) 1531.3 nm (l = -1), (b) 1540.4 nm (l = -2), (c) 1549.5 nm (l = -3), (d) 1558.6 nm (l = -4).
4.3 Changing the height of angular grating
In addition to adjusting the grating size, the etching depth provides another degree of freedom for the design of OAM signal emitter. The uniformity under different etched depths of the grating will be discussed in this subsection. Figures 11(a)-(c) show three types of grating height, including 220 nm, which is the typical value for the single mode operation in SiPh SOI devices, 150 nm (i.e. 70 nm-etched) and 60 nm (i.e. 160 nm-etched). The size of the grating is fixed at 200 nm × 200 nm for fair comparison. The widths of the access waveguide are adjusted to 431 nm and 419 nm for the grating height of 150 nm and 60 nm to meet the critical point condition. The transmissions against different wavelengths for these two types of the grating are shown in the Figs. 12(a) and (b), respectively, where the excited wavelengths equipped with the charges of l = -1 ∼ -4 are now 1532.3 nm, 1541.4 nm, 1550.5 nm and 1559.6 nm for the height of 150 nm; and 1526.5 nm, 1535.5 nm, 1544.5 nm and 1553.5 nm for the height of 60 nm, respectively. The coupling efficiencies for all excited wavelengths are over 21% for the former and over 19% for the latter except for l = 0.
Fig. 12. The transmission at different wavelengths of the modified OAM signal emitter at 1.5 um far-field plane with the grating heights of (a) 150 nm (b) 60 nm.
The magnitudes of the E-fields with the charges of l = -1 ∼ -4 for the height of 150 nm and 60 nm are shown in the Figs. 13(a)-(d) and Figs. 14(a)-(d), respectively. The results from both cases indicate more uniform E-field distributions than that of 220 nm. Furthermore, the phase profiles are shown in Figs. 13(a1)-(d2) and Figs. 14 (a1)-(d2), respectively. Similarly, the attenuation factors for the height of 150 nm and 60 nm are also well-analyzed and shown in the Figs. 15(a)-(d) and Figs. 16(a)-(d), respectively. Particularly, for l = -1 ∼ -4, the new α-values are 0.1471, 0.1404, 0.1152 and 0.1294 in the case of the grating height of 150 nm; and the new α-values are 0.0247, 0.0286, 0.0153 and 0.0243 in the case of the grating height of 60 nm. 44% and 88% improvements can be achieved when compared with the original one. This shows that the reduction of the grating height has great benefits in terms of the uniformity.
Fig. 13. The magnitudes of the E-fields when the grating height is 150 nm at wavelengths of (a) 1532.3 nm (l = -1), (b) 1541.4 nm (l = -2), (c) 1550.5 nm (l = -3), (d) 1559.6 nm (l = -4). The corresponding phase profiles in two orthogonal polarizations (a1) l = 0, (a2) l = -2, (b1) l = -1, (b2) l = -3, (c1) l = -2, (c2) l = -4, (d1) l = -3, (d2) l = -5.
Fig. 14. The magnitudes of the E-fields when the grating height is 60 nm at wavelengths of (a) 1526.5 nm (l = -1), (b) 1535.5 nm (l = -2), (c) 1544.5 nm (l = -3), (d) 1553.5 nm (l = -4). The corresponding phase profiles in two orthogonal polarizations (a1) l = 0, (a2) l = -2, (b1) l = -1, (b2) l = -3, (c1) l = -2, (c2) l = -4, (d1) l = -3, (d2) l = -5.
Fig. 15. The E-field distribution when the grating height of 150 nm and the fitting curve at the radius of 10.25 µm. (a) 1532.3 nm (l = -1), (b) 1541.4 nm (l = -2), (c) 1550.5 nm (l = -3), (d) 1559.6 nm (l = -4).
Fig. 16. The E-field distribution when the grating height of 60 nm and the fitting curve at the radius of 10.25 µm. (a) 1526.5 nm (l = -1), (b) 1535.5 nm (l = -2), (c) 1544.5 nm (l = -3), (d) 1553.5 nm (l = -4).
Table 1 summaries all of the results of all the structures mentioned in this work, indicating the excited wavelengths for each topological charge and the corresponding transmissions and attenuation factors. It can be clearly observed the attenuation factor can be significantly improved through the structural optimization of the SiPh micro-ring resonator OAM signal emitter. The attenuation factor can be promoted to almost zero and the OAM excitation turns into near perfectly uniform field, increasing the quality during the propagation, at the cost of only 4 ∼ 5% coupling efficiency reduction. It is worth to mention that the design parameters shown in Table 1 comply with the design rules provided by the foundry and could be used for fabrication.
Table 1. The summary of the resonant wavelengths, the transmissions and the attenuation factors for the topological charges from l = -1 to l = -4 using different types of the structure
Our proposed SiPh-based micro-ring type OAM signal emitter has the efficiency of less than 25%, which is similar to the experimental results using similar device structures demonstrated in [31] and [39]. Their OAM emitters have experimental coupling efficiencies of 3 - 13% and 3 - 6.5% respectively [31,39]. Moreover, in order to improve the coupling efficiency, one can introduce a back-side metal mirror, such as aluminum mirror below the SOI layer, and the coupling efficiency can be increased to about 37% [34].
In our SiPh-based micro-ring OAM emitter uniformity study, we do not take the tunability into the consideration. However, the tunability can be achieved with the help of thermal heater, which can change the effective refractive index of the micro-ring resonator. Therefore, the FSR can be changed accordingly. This gives the opportunity to change different resonant order for a specific wavelength, making the topological charge tunable. The experimental results of [40] show that the topological charges for a given wavelength can be tuned at least between l = -2 and l = 2 using similar OAM emitter with thermal tuning.
For the fixed grating size of 150 nm and the grating height of 220 nm, three kinds of the upper cladding thicknesses are simulated and compared. The thicknesses are 1 µm, 2 µm and 3.25 µm; and the corresponding measured planes are at 1.5 µm, 2.5 µm and 3.4 µm, respectively. Figure 17 shows the transmission spectra at different upper cladding thicknesses. The results indicate that there is almost no difference between these three kinds of the thicknesses.
Fig. 17. Transmission spectra at different upper cladding SiO2 thicknesses at grating size of 150 nm and the grating height of 220 nm
4.4 Consideration of fabrication error
We also analyze of the influence of the transmission performance by considering the fabrication error specified by the SiPh foundries. Figures 18(a)-(d) show the transmissions for the charges l = -1 ∼ -4 and the corresponding variations within the range of the grating height fabrication error specified by the foundries using our best optimized SiPh OAM emitter (i.e. the size of 200 nm and the height of 60 nm). The height of 60 nm is set to be the reference. It can be observed that the variations are relatively small and within the range of ∼3.1%, ∼4.0%, ∼4.4% and ∼5.8% for l = -1 ∼ -4, respectively.
Fig. 18. The transmission variation of the OAM signal emitter at 1.5 µm far-field plane with the grating heights of 60 nm and under different fabrication errors. (a) l = -1, (b) l = -2, (c) l = -3, (d) l = -4.
We also analyze of the signal uniformity by considering the fabrication error specified by the SiPh foundries. Figure 19 shows the attenuation factor α for the charges l = -1 ∼ -4 when the grating height fabrication relative error is from -5 to +10 nm using our best optimized SiPh OAM emitter (i.e. the size of 200 nm and the height of 60 nm). The height of 60 nm is set to be the reference. In the worst case, the α-values for l = -1 ∼ -4 increase to 0.0319, 0.0347, 0.0218 and 0.0312, respectively. It can be observed that, even in the worst case (i.e. large α-value), the improvement is still > 86% when compared with the non-optimized structure.
Fig. 19. The α-value against the fabrication error. The grating height on the x-axis is relative to 60 nm.
5. Conclusion
The SiPh-based micro-ring resonator is one of the promising OAM signal emitter candidates. In this work, a compact SiPh-based micro-ring resonator type OAM signal emitter with detailed design parameters was presented and the output signal uniformity issue was comprehensively investigated. It is shown that the uniformity-induced phase distortion will significantly degrade the purities of OAM beams; hence, introducing severe OAM signal crosstalk during the OAM division multiplexing. The numerical analysis of the uniformity was firstly presented through defining the OAM beam overlap ratio and phase ambiguity, illustrating how the non-uniformity is related to the degradation of the quality of the OAM beams at different topological charges. Specifically, the overlap ratio was 0.515 at | l | = 1 and 0.119 at | l | = 5 when the attenuation factor was 0.3. Then, the uniformity of the SiPh-based micro-ring type OAM signal emitter was analyzed, indicating the attenuation factors were 0.2153 ∼ 0.2755 in the original structure. Nevertheless, by adjusting the angular grating sizes and the heights, the uniformity was successfully optimized, with the attenuation factors being reduced to 0.0153 ∼ 0.0286 in the best case. It is demonstrated there was over 88% improvement by the structural optimization at the price of acceptable 4 ∼ 5% coupling efficiency reduction. The variations of the transmission and the uniformity induced by the fabrication error were also analyzed, illustrating the realization of the fabrication.
Funding
National Science and Technology Council, Taiwan (NSTC-109-2221-E-009-155-MY3, NSTC-110-2221-E-A49-057-MY3).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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