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Abstract
The utilization of directional couplers (DCs) as power splitters and combiners is ubiquitous for the realization of integrated silicon photonic devices in the silicon-on-insulator (SOI) platform. Benefiting from the excellent modal modulation performance of graphene in silicon waveguides, we present its high flexibility to engineer and reduce the wavelength dependence of devices based on the DC. In this regard, a wavelength-insensitive 3-dB optical power coupler (3-dB OPC) is firstly proposed, in which various power splitting ratios are realized for the TE mode. Next, we design a broadband polarization beam splitter (PBS) consisting of a silicon waveguide (SW) and a graphene-silicon vertical slot waveguide (GSVSW). The simulation results show that the proposed PBS has high polarization extinction ratios (PERs) of 20.93 and 20.4 dB and low insertion losses (ILs) of 0.12 and 0.21 dB at 1550 nm for the thru and cross ports, respectively. In addition, the designed PBS demonstrates the ability to work over a wide range of wavelengths by changing the chemical potential of graphene.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent decades, exponential network-traffic growth has stimulated a growing interest in research aimed at the enhancement of transmission capacity and bandwidth of the optical communication systems [1,2]. In this way, the mode division multiplexing (MDM) technology emerges as a technique able to increase the capacity per fiber [3]. In order to activate an MDM system, one of the key challenges is the achievement of mode and polarization manipulation. To fulfill this aim, many devices including two indispensable functional elements called the polarization beam splitters (PBSs) and the optical power couplers (OPCs) have been proposed. The directional coupler (DC) as one of the most promising candidate constituents has widely exploited in these two elements of MDM systems. Generally, the DCs consist of two asymmetrical waveguides in a parallel configuration to achieve mode coupling from one waveguide to another [4].
So far, several different approaches have been reported to implement and break symmetry between DC waveguides in the PBSs and OPCs in order to manage the coupling ratios. For example, a few structures have used the hybrid-plasmonic waveguide (HPW) so as to break the symmetry of modes of DCs in ultra-compact dimensions, but the modes modified by the plasmonic effects suffer from inherent insertion losses [5–7]. The high polarization ratios in rather small dimensions were achieved by DCs based on grating-assisted structures but at the cost of a tight fabrication tolerance of grating duty cycles [8–10]. Exploiting partially etched taper waveguides to form asymmetric DC structures can compensate sensitivity to the wavelength fabrication variations; nevertheless, a major drawback of such structures is their long lengths [11–13]. Using slot waveguides to introduce asymmetric DCs provides approaches for building the PBSs and OPCs with great polarization extinction ratios, but the cost could be measured by the fabrication difficulties [14,15]. Thus, despite large efforts to overcome the trade-off between the criteria of DC-based devices, there still is the demand for the improvement of their performance.
Graphene, a single layer of carbon atoms arranged in a honeycomb lattice [16], has received tremendous attention in recent years owing to its excellent electrical and optical properties, such as the strong interaction with light, the tunable optical conductivity and the CMOS compatibility process [17]. In the graphene-embedded waveguides, the field component of TE and TM modes parallel to the graphene’s surface can effectively interact with graphene which leads to an increase in light absorption and mode loss, providing an opportunity for light modulation. Recently, a graphene-based PBS was demonstrated in which seven graphene layers were used to fulfill the demanding phase mismatching of TM mode between the waveguides [18]. But, in addition to the fact that graphene is anisotropic, it’s challenging to regrow crystalline silicon on the top of graphene stacks with available manufacturing technologies [19]. In fact, to design such embedded structures, it is required to use polysilicon deposited and then patterned, which makes the fabrication process complicated [20]. Moreover, the implementation of electrical contacts for more than two graphene layers is difficult. Here, using an asymmetric DC consisting of a single graphene layer interpolated in a SI waveguide, we employ the huge capability of graphene–its effectiveness in the manipulation of modal properties of silicon waveguides for the TE polarization states—to design a novel and tunable 3-dB OPC. It can be observed that the incorporation of graphene leads to an improvement on coupling length sensitivity and operation bandwidth of DCs. In the second section of paper, we continue to present a silicon-on-insulator (SOI)-based high-performance PBS by a similar strategy, which owns the superior tunable feature towards the other PBSs. The graphene sheet in these structures is vertically inserted in a silicon waveguide to manipulate the effective refractive index (ERI) of the TE mode. Since the TE polarization illustrates the higher fraction of its field parallel to graphene, resulting in a larger variation of the ERI than the TM mode.
2. Principles, simulation, and discussion
2.1. Permittivity of graphene
As we have discussed above, graphene should be treated as an anisotropic material which the in-plane permittivity (${\varepsilon _\parallel }$) can be dynamically tuned with altering its Fermi level (or chemical potential ${\varepsilon _c}$) whereas the out-of-plane permittivity (${\varepsilon _ \bot }$) remains constant at 2.5. Here, the graphene is modeled as a 3D anisotropic permittivity medium which its permittivity can be expressed as following tensor:
Fig. 1. (a) Calculated refractive index of graphene (real part, imaginary part and magnitude) versus chemical potential at $\lambda = 1550\; nm$. (b) Refractive index of graphene versus incident light wavelength under chemical potential of graphene ${\mu _c} = 0\; \textrm{eV}$
2.2. 3-dB optical power coupler (3-dB OPC)
3-dB OPCs are indispensable elements in the photonic systems for equally splitting two orthogonal polarization modes of a waveguide into two different waveguides. The simplest form for an OPC can be realized utilizing two parallel waveguides with a very narrow gap. The main drawback of such conventional OPCs is their extreme sensitive to operating wavelengths, so that most current efforts are concentrated on improving the bandwidth [13,14].
The three-dimensional (3D) schematics of our proposed OPC is demonstrated in Fig. 2(a), which would be fabricated in the practical processes [21,22] as follows. The fabrication starts from the silicon waveguide prepared from a SOI wafer using E-beam lithography (EBL) along with the inductively coupled plasma etching process. It is feasible to cover the sidewalls of the silicon waveguide with graphene in a practical fabrication process [21,22]. Toward this end, the graphene sheet can be prepared on Nickle or Copper foils by chemical vapor deposition (CVD), and then transferred on the fabricated Si waveguide through wet or dry transfer techniques. Subsequently, similar patterning and etching methods as the initial Si waveguide can be employed to establish the pattern of the final waveguides. Also, EBL and E-beam evaporation can be utilized to fabricate the metal electrodes.
Fig. 2. (a) 3D schematic configuration of proposed OPC. (b) Cross-sectional view of the asymmetric coupling region.
Therefore, the final configuration consists of two adjacent silicon waveguides with the cross-section of 350 × 350 nm2, while an asymmetric coupling section is created by incorporating a single-layer graphene film at the center of right waveguide. Figure 2(b) shows the cross-sectional view including the asymmetric coupling region. The graphene and metal backgate form a parallel plate capacitor separated by the Si and SiO2 dielectrics. The charges are induced on the graphene under bias voltage that can modify the chemical potential of graphene. The structure is designed to operate within the wavelength range of 1500 - 1600 nm for the TE mode, where the field profile for this mode is strongest in the middle of the waveguide. Hence, so as to realize the maximum variations on the ERI of the TE mode, the graphene layer must be vertically inserted in the center of the waveguide to enhance the coupling with light.
A finite-difference eigenmode method (FDEM, Lumerical Inc.) is employed to calculate the modal characteristics for all eigenmodes in the waveguides. The relative permittivities of Si and SiO2 are 11.937 and 2.088, respectively, and air is considered as the cover cladding with the refractive index of one. Figure 3(a) shows the dependence of real and imaginary part of the TE mode of the waveguide with graphene on wavelength. At µc = 0.53 eV, one can observe that the real part of the ERI of TE mode varies sharply around λ = 1615 nm, so that the variation lessens gradually for shorter wavelengths. As depicted in Fig. 3(b), such alterations produce a difference in the ERI between the waveguides in the asymmetric section, ranging from a maximum value of around 0.18 at λ = 1600 nm to a minimum value of around 0.04 at λ = 1500 nm.
Fig. 3. (a) The real and imaginary parts of the ERI of the TE mode for the graphene embedded slot waveguide at µc = 0.53 eV and (b) the ERI difference of the TE mode of two waveguides in the asymmetric region, versus the wavelength.
Inside a waveguide with fixed geometry, the modal confinements and/or evanescent field distributions are highly dependent on the wavelengths, such that the longer wavelength, the less confinement. In fact, the greater levels of evanescent tail of mode profile make it easier to couple a mode from a waveguide to other waveguides in the DC. Accordingly, it can be expected that the transmission in a conventional OPC escalates continuously when the wavelength increases.
Furthermore, the coupling strength relies on the mode index difference between two waveguides of the coupling section, it reaches a maximum value in difference of zero. Bearing this point in mind, it is now evident from Fig. 3(b) that the values of $\Delta n(\lambda ) = n_{eff}^S(\lambda ) - n_{eff}^{SG}(\lambda )$ can be altered in the presence of graphene, and is worth noting that Δn become larger as the wavelength increases. Consequently, unlike conventional couplers, such a form of alteration will give rise to the weaker coupling for higher wavelengths, which in turn can in part neutralize the linear relation between the power coupling and wavelength mentioned for conventional couplers. Based on this phenomenon, we allow for the addition of an asymmetric region consisting of graphene with a length of L1 before the symmetric region with a length of L2, which acts as an extra adjustable parameter to attain the desired transmission spectra. Figure 4(a) shows the calculated power splitting ratio of our 3-dB OPC against wavelength which is defined as $\eta = {P_A}/({{P_A} + {P_B}} )$, where η is the coupling ratio, PA and PB are the output powers at ports A and B, respectively. We also plot the coupling ratio of a conventional DC for the comparison.
Fig. 4. (a) The coupling power ratio as the function of wavelength for the present 3-dB OPC (red) at the chemical potential of 0.53 eV and the conventional power coupler (red). (b) The coupling ratios of our OPC at ${\mu _c} \approx 0.55\; eV$ and 0.9 eV for three wavelengths of 1500 (red), 1550 (blue), and 1600 nm (black). (c), (d) Field distribution of light along the present OPC for $\lambda = 1550\; nm$ at ${\mu _c} = 0.55\; eV$ and 0.9 eV, respectivily.
A 3D finite-difference time-domain (FDTD) method has been employed to conduct the simulation. The optimal lengths of coupler for a desirable broadband splitting ratio are L1 = 5 µm and L2 = 8.3 µm, as well as a separation distance of 200 nm is fixed between the two waveguides. As depicted in Fig. 4(a), with the choice of µc = 0.53 eV based on the optimized results of L1,2, the coupling ratio is close to 0.5, simply a deviation of ${\pm} 1.9\%$ for the wavelength range of 1535 to 1600 nm. In the asymmetric region, the coupled energy in longer wavelength region $(1570\; \textrm{nm} < \lambda \le 1600\; \textrm{nm})$ because of the rather large mode index difference is less than in shorter wavelength region, a way for compensation of the length-dependent which the modes with different wavelength exhibit. It should be considered that the propagation losses, as shown in Fig. 3(a), are large in the wavelengths which Δn is large and the low power is coupled, so that the losses decrease in the wavelengths with more coupling strength. Therefore, it can be expected that the propagation loss does not be a serious issue; however, it partially leads to a slight difference in the output power of ports.
In addition to an equal power splitting ratio, an unequal splitting ratio is also desired [23]. Figure 5(b) shows the coupling ratio for three wavelengths (1500, 1550, and 1600 nm) of the present OPC at two different µc. The results indicated with markers express the capability of working with mutable power coupling ratios, such that the chemical potential µc shifts from around 0.55 to 0.9 eV, the maximum coupling ratio changes by 20%, 35%, and 36% could be achieved for λ = 1500, 1550 and 1600 nm, respectively. For instance, the directed power distribution along the proposed structure at λ = 1550 nm and for µc = 0.55 and 0.9 eV are indicated in Figs. 4(c) & 4(d), respectively. It can be observed that the larger output power passes through port B when µc shifts from 0.55 to 0.9 eV. As a result, apart from the role of graphene in changing the modal characteristic of waveguide, controllable coupling is another privilege of this structure.
2.3. Polarization beam splitter (PBS)
The massive refractive index contrast between the Si core and SiO2 cladding in the SOI platforms introduces a large polarization sensitivity. PBSs, which are able to split or combine the TE and TM modes, have played an important role in solving this issue. DC-based PBSs are designed to completely launch only one polarization into an adjacent waveguide, and simultaneously, to maintain the other polarization in the input waveguide. Nevertheless, generally, in such configurations, the mode selected to launch into the new waveguide plays a limiting role for the achievement of a broadband operation window, while producing effective phase mismatching for the orthogonal polarization state is usually realized over a wide wavelength range.
Here, by utilizing of the electro-refraction effect induced in graphene, we present a DC-based tunable, high-performance PBS, which shows a broad operation bandwidth at both outputs. The 3D schematic configuration and its cross-sectional view of the proposed PBS are displayed in Figs. 5(a) &; 5(b), respectively. The asymmetric DC used for the PBS consists of a silicon waveguide (SW) on one side, and a graphene-silicon vertical slot waveguide (GSVSW) followed by another SW on the other side. The aim of introducing GSVSW with a length equal to Lg is to control optimally the phase matching condition for the TE mode in a way which it reduces the wavelength dependence of device. Indeed, similar to the previous section, the single-layer graphene vertically located at the center of the GSVSW provides an effective coupling of the TE mode between the waveguides over a wide spectral band by appropriately choosing the variation range of the length and µc of graphene. It should be mentioned that the TM mode propagation is unaffected by the GSVSW since the graphene layer has much higher effect on the TE mode than the TM mode. The waveguides have different thicknesses and widths.
Additionally, we fix the height of waveguide B at 180 nm to assure that the TM mode is cutoff, which can effectively minimize the performance degradation of the present PBS caused by the fabrication errors of the waveguide gap. Consequently, as it will be seen in later sections, this design allows us to attain an overly broadband bandwidth response in a compact configuration. The TE polarized light entered into the input port will be eventually and efficiently coupled to the adjacent waveguide by choosing the optimal length of the waveguides, while the launched TM mode keeps propagating along the input waveguide the SW with negligible influence from the coupling region. The relative permittivities of used materials are similar to the previous section, as well as the entire device is covered with a SiO2 cladding.
To choose optimal structural parameters and the suitable µc, a precise analysis of the modal properties of the present device is essential. The height of the waveguide A is chosen to be h1 = 430 nm. So as to select the appropriate widths for waveguides, achieving the TE matching and the TM mismatching in the coupling region, we calculate the ERI of TE and TM fundamental modes of the waveguides as a function of the width varied from 250 to 500 nm at 1550 nm, for h2 = 180 nm, close to the cutoff height for TM mode. From Figure (6(a)), one can observe that the phase matching condition for the TE mode is perfectly satisfied at the ${w_1} = 285$ and ${w_2} = 450\; nm$, while there is a substantial difference between the ERIs of TM. Even though the phase-matching condition for the TE mode has been satisfied at 1550 nm, the coupling strength at a certain coupler length is highly dependent upon wavelength so that a well-satisfied coupling for different wavelengths will occur in diverse lengths. Figure (6(b)) shows the dependence of coupling length, Lc, on wavelength. The coupling length is defined as ${L_c} = \pi /[{{k_0}({{n_{even}} - {n_{odd}}} )} ]$, where ${n_{even}}\; ({{n_{odd}}} )$ and ${k_0}$ are the effective mode index of even-like and odd-like supermodes for the TE mode and the wave number in air, respectively. The gap width in the coupling region is chosen 220 nm which can be reduced to zero due to the cutoff of the TM mode in waveguide B. One can see from the figure (6(b)) which the Lc becomes greater with the increase of wavelengths. Thus, the broadband performance of a traditional PBS is extremely restricted respect to the wavelength dependent.
Fig. 6. (a) The calculated effective indices of the fundamental modes for thin and thick silicon waveguides as a function of their width separately. Here, h1 = 430, h2 = 180, and λ = 1550 nm. The insets show waveguides A and B. (b) the required length for the maximum power transmission of TE mode in the coupling region with respect to wavelength.
To overcome this drawback, the GSVSW is introduced and located before the silicon waveguide B, aiming to compensate the wavelength-dependent transmission ratios. Figure (7(a)) shows the ERI of TE and TM fundamental modes with varying the wavelength for the GSVSW. For an optimal ${\mu _c} = 0.53\; eV$, as the wavelength changes, the ERI of TE mode for the fixed geometrical GSVSW undergoes considerable variations, even up to 0.2. In fact, with insetting vertically a graphene layer in the center of the SW, the electric field of TE mode can be squeezed more extreme into the graphene layer in accordance with Maxwell's boundary conditions, and as a result, the ERI of TE becomes more sensitive to wavelengths due to the enhanced graphene-light interaction. More importantly, the interesting feature of graphene is that its effect on the ERI of the TE mode is stronger than that of the TM mode, so that the effective refractive indices of the TM mode only undergoes a very slight variation (0.02) at the same chemical potential which the TE mode experiences intense alteration. The reality that the graphene layer influences unequally the modes of the SW can be easily perceived from the variation of plotted curves [ Figs. 7(c) and 7(d)]. After all, with the accurate adjustment of a chemical potential of graphene, it is found that the ERI of TE mode at some wavelengths can be considerably altered as compared to the SW. Such a property of graphene enables us to manipulate the coupling dependence of the TE mode on the wavelengths. As is depicted in Figure (7b), the presence of graphene increases the ERI difference of TE mode between two adjacent waveguides as the wavelength increases. It is noteworthy that such changes occur in a way that lead to weakening the coupling strength of the TE polarization within a wavelength range from 1550 nm to 1600 nm, while their effectiveness on the TE transmission in shorter wavelengths is insignificant. Therefore, the dependence of coupling strength on wavelength can be to a certain extent circumvented in the present PBS. In Figs. 7(c) and 7(d), plots are given of the electric field intensity distributions of the fundamental modes at the wavelength of 1600 nm for the GSVSW. These electrical field profiles emphasize the above discussed subjects. By contrast, while the electric field of TE mode experiences huge changes, the TM mode remains nearly unmodified after the introduction of the graphene layer.
Fig. 7. The modal characteristics for the GSVSW. (a) The real and imaginary parts of the effective refractive index of TE mode. (b) The difference between the effective refractive index of TE mode in GSVSW and the adjacent SW as a function of wavelength. (c), (d) the electric field profile of ${|{{E_y}} |^2}$ for the TM mode and ${|{{E_X}} |^2}$ for the TE mode. The simulation is conducted by the FDEM.
Based on the above analysis, we can expect that the phase-matching condition be simply satisfied for the TE mode in our proposed device for a relatively wide range of wavelength, and subsequently, for the TM mode, the power coupling can be highly prevented owing to a phase mismatching. To investigate the propagation characteristics of the designed PBS, simulations using the 3D FDTD method are conducted, while the representation of graphene sheet is performed by four mesh grids. Assuming a bias voltage is applied to the graphene layer and the bottom of device is grounded, µc, could be turned by this applied gate voltage, Vg, through $|{{\mu_c}} |= \hbar {v_F}\sqrt {\pi {a_0}|{{V_g} - {V_{Dirac}}} |} $, where VDirac is the voltage deviation brought by the natural doping, ${v_F} \approx {10^6}m/s$ is the Fermi velocity, ${a_0} = {\varepsilon _r}{\varepsilon _0}/de$ originates from a plain parallel plate capacitor model; ɛ0 and ɛr are the permittivity of air and the relative permittivity of capacitor dielectric, respectively, and de is the thickness of capacitor. At the output terminal of the waveguide A, a bending structure is exploited to create the two output waveguides separated in order to further enhance the device performance. A bending radius of 1.3 µm has been chosen to ensure the minimum interaction of the waveguides with each other.
Two of the most important transmission characteristics of the PBS are polarization extinction ratio (PER) and insertion loss (IL) [24]:
Figures 8(a) & 8(b) represent the PER as a function of the wavelength at the cross and thru ports for our PBS and a conventional PBS used for comparison. The calculated PERs at the wavelength of 1550 nm are 20.4 dB and 20.93 dB for the TE and TM modes, respectively. The gap separation, w0, is determined as 220 nm to fabricate a compact DC with satisfactory fabrication difficulty. Considering the fact that the broadband performance of the vast majority of PBSs is virtually the thru port limited, the presented PBS illustrates a remarkable improvement in the bandwidth of thru output, achieving a wide wavelength spectrum of 77 nm with $PER > 14\; dB$ at the optimized lengths of ${L_g} = 4\; \mu m$ and ${L_c} = 12.5\; \mu m$. As the comparison with a conventional PBS, the bandwidth is increased over two times. Also, the simulation results show that the PER at cross port is to be slightly flattened at higher wavelengths, as it can be observed from the Fig. 8(b). The calculated PER of cross port is larger than 18.8 dB over the wavelength range from 1500 to 1600 nm, showing a little dependence on the length of the coupling region owing to the cut off of the TM mode. Figure 8(c) indicates ILs as a function of the wavelength for the TM and TE modes, which are as low as 0.21 dB and 0.12 dB at $\lambda = 1550\; nm$, respectively. In addition, Fig. 8(d) demonstrates the transmission value of the TE mode at both the thru and cross ports for our PBS and a conventional PBS. From the figure, one can see that the transmission at both thru and cross ports is ameliorated by comparison.
Fig. 8. The dependence wavelength of the presented PBS compared with a standard conventional PBS. (a), (b) PER as function of λ for the thru and cross ports, respectively. (c), ILs versus λ for both the TE and TM mode. (d) the TE transmission as function of λ at both outputs. The chemical potential of graphene is selected 0.53 eV.
Figures 9(a) & 9(b) exhibit the mode propagation in the proposed PBS with $R = 1.3\; \mu m$ for TM and TE input modes, respectively. It can be noted that the TE mode entered into the input waveguide is predominantly coupled to the neighboring waveguide and then outputs from the cross port. On the other hand, the input TM mode propagates along the input waveguide virtually unaffected by the adjacent waveguide. At this point, the sensitivity of bending loss to the polarization of modes should be considered, such that the transmission of the TM fundamental mode undergoes higher bending loss than that of the TE fundamental mode.
The graphene tunable conductivity enables the proposed PBS to work in a wide range with high PER at the ports. Figure 10(a) presents that the variation of PER at both the cross and the true port versus µc at a constant wavelength of 1550 nm. The minimum PER for the cross ports occurs at around ${\mu _c} = 0.56\; eV$, whereas the PER value can be shifted with µc. Also, the PER changes at the thru port with the wavelengths for three different µc is depicted in Fig. 10(b). It is obvious that the value of thru port PER can be flexibly tuned by altering the µc of graphene.
Fig. 10. The tunable characteristics for the proposed PBS. (a) the dependence of PER to µc. (b) PER for the thru port versus wavelength at three different µc.
3. Conclusion
In summary, taking advantage of the high capability of graphene in modulating the modal characteristics of a DC, we demonstrate two novel broadband PBS and OPC devices based on DC. In order to effectively interact the graphene layer with the TE mode, the graphene layer is place in the center of the SW in the proposed configurations. Our 3-dB OPC has simple structure consisting of a symmetric section of the SW and an asymmetric section including graphene, with the ability of operation over broad wavelength range as large as 70 nm and the various power coupling ratios. Also, the proposed PBS utilizing an asymmetric DC with a GSVSW is exhibited a desirable PER of 20.4 and 20.93 dB and relatively low IL of 0.12 and 0.21 dB at 1550 nm for the cross and thru ports, respectively. In addition, under the chemical potential tuning of graphene, PBS is also able to operate with changeable splitting ratio power.
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 may be obtained from the authors upon reasonable request.
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