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Abstract
We propose an ultra-broadband, ultra-compact and a dynamically tunable $\mathrm{1\ \times 2}$ power splitter on a silicon on Insulator (SOI) platform with a 220 nm thick silicon light-guiding layer, using two multimode interference (MMI) couplers connected with graphene-based waveguides as the phase-tuning section through a Mach-Zehnder Interferometer (MZI) configuration. First, we theoretically present and demonstrate a novel design for the MMI couplers by combining the plane wave expansion method (PWEM) and the mode expansion conjecture concept. To verify the proposed theory, a center-fed $\mathrm{1\ \times 2}$ MMI coupler and a $\mathrm{2\ \times 2}$ MMI coupler, respectively, as the input and output sections of our proposed device, are designed and simulated. The simulation results achieved by Lumerical FDTD show good agreement with the design theory. Then, a highly tunable graphene-embedded silicon waveguide, for the highly efficient modulation of the effective mod index (EMI), is duly designed using Lumerical Mode Solutions. As the two MZI arms, a pair of the proposed waveguides is introduced into the middle of the cascaded MMI couplers. Accordingly, the integration properties of the analytically designed MMI couplers and the numerically designed waveguide is demonstrated through our proposed device for the aim of achieving any wanted power splitting ratio. To this end, we consider the case that the real part of the EMI of the waveguide in the lower MZI arm is modulated by varying the graphene Fermi level values, being the same for all the layers belonging to the same waveguide, while that of the upper arm is constant. The corresponding power splitting ratio can be dynamically tuned in the range of $\textrm{0}\textrm{.5:0}\textrm{.5 - 0}\textrm{.85:0}\textrm{.15}\textrm{.}$ All reported results assume TE polarization. The designed MZI-based splitter possesses a bandwidth of $\textrm{400 nm}$ over the wavelength range from $\textrm{1}{.35\, \mathrm{\mu}\mathrm{m}}$ to $\textrm{1}{.75\,\mathrm{\mu} \mathrm{m}}$ for various power splitting ratios, maintaining the averaged insertion loss and the averaged power imbalance, respectively, below as low as $\textrm{0}\textrm{.39\,dB}$ and $\textrm{0}\textrm{.48\,dB}\textrm{.}$ The overall footprint of the proposed device is also highly small, i.e., about $\mathrm{58\,\mathrm{\mu} m\ \times 3}\mathrm{.5\,\mathrm{\mu} m}\textrm{.}$
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical power splitter as one of the fundamental building blocks for splitting and combining optical signals in photonic integrated circuits (PICs), finds applications in realizing switches, modulators and multiplexers [1]. Although uniform power splitters have been extensively studied and developed, the exploration of an asymmetric power splitter with unequal splitting ratios is essential for many applications such as for optical power tap [2], power equalizers [3], ring lasers [4] and ladder-type optical filters [5]. There have been numerous demonstrations of asymmetric power splitters, including an asymmetrical Y branch [6], a T-junction structure [7], a photonic crystal power splitter [8], a circular-to-rectangular polymer-fiber converter [9], an asymmetrical MZI [10], an asymmetrical coupler with phase control sections [11] and the MMI based power splitters [12]. Among these approaches, MMI splitters are the most popular and have been widely investigated for the development of asymmetrical power splitters [13–16]. However, they suffer from large footprints and wavelength sensitivity. Besides, these passive approaches which are based on inducing a certain phase shift in the MMI section, for example with suitably designed tilts [17], suitably designed butterfly shapes [18], cuts [19], holograms [20] or by introducing a pair of unequal-widths waveguides as a phase shifter into the middle of cascaded MMI sections [14], are preferable when a given fixed splitting-ratio is desired. In other words, all these methods need to be restructured for achieving arbitrary splitting ratios. One way to obtain tunable and freely chosen power splitting ratio, without reconfiguration, is to exploit electro-optic [21] or thermo-optic [22] effect. However, the main drawback is here that the free-carrier absorption and temperature-effects can induce a performance degradation and an increased propagation loss [16]. Moreover, both effects have been investigated to induce a small perturbation of refractive index typically well below 0.01, resulting in devices with large footprint and increased energy consumption [23]. Consequently, broadband, compact and dynamically tunable schemes for power splitters with reasonable performance are still needed. Broad operating bandwidth and compactness conditions can be fulfilled by using inherent anisotropy of subwavelength grating (SWG) [24]. SWGs were used to engineer the dispersion properties of a conventional MMI by effectively flatten the wavelength dependence of the beat length to overcome the bandwidth limitation [24]. In addition, applied to MMIs, sub-wavelength index engineering can be utilized to design compact devices for integrated photonics [24]. More recently, a comparison between the bandwidth of different uniform power splitters at telecom wavelengths has been presented at Table 1 of [25] with the best performance (bandwidth of 325 nm and length of $\textrm{25}\mathrm{.4\mu m)}$ obtained by using intrinsic anisotropy of SWG MMI for TE polarization. However, this device is known to be capable of providing the fixed power splitting ratio of $\textrm{0}\textrm{.5:0}\textrm{.5}\textrm{.}$
Table 1. The effective indices of the TE-guided modes of our SWG MMI waveguide at ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.75}\,\mathrm{\mu m}$ as the width of the waveguide varies
Graphene, a single atom thick carbon sheet with atoms arranged in a hexagonal structure, has attracted growing attentions due to its exceptional optical and electrical attributes [26]. The permittivity of graphene can be actively controlled by dynamically tuning the Fermi energy $\textrm{(}{\textrm{E}_\textrm{f}})$ [27]. Moreover, due to its extremely high electron mobility [26,27], large electro-absorption [28,29], electro-refraction (ER) effects [30,31] and good compatibility with CMOS [32], graphene can be applied to realize broadband and compact different device functionalities including amplitude/phase modulators, polarizers and photodetectors SOI platform. However, to the best of our knowledge, there is no report on graphene-based dynamically tunable power splitter in silicon photonics. Graphene can provide a means for tuning the splitting ratio of power splitters at the cost of power consumption. The surface conductivity of graphene can be expressed by using Kubo formula as follows [33]:
Fig. 1. In-plane schematic diagram of our proposed dynamically tunable 1 × 2 power splitter based on two short SWG MMI sections interconnected by a pair of graphene-embedded silicon waveguides with their cross-section view shown in the middle.
2. Basic theories
The PWEM is illustrated in several papers [38,39]. The problem of expanding a field over the set of waveguide modes is also well known [40,41]. In the following subsections, we summarize both basic theories very briefly.
2.1 PWEM theory
Modal analysis is here done by solving source-free and non-permeable Maxwell’s curl equations:
In these equations, $\mathrm{\vec{\beta }}$ is called the Bloch wave vector. The integers $\textrm{p,}$ $\textrm{q}$ and $\textrm{r}$ are the indices of the plane waves in the expansion. ${\mathrm{\vec{S}}_{\textrm{pqr}}}$ and ${\mathrm{\vec{U}}_{\textrm{pqr}}}$ are the complex amplitudes of the electric and magnetic fields, respectively, for the $\textrm{pqr - th}$ plane wave. The reciprocal lattice vector (or grating vector) $\mathrm{\vec{K}(p,q,r)}$ is defined by:
2.2 Mode expansion conjecture concept
One of the fundamental concepts of guided-wave optics is that of the expansion of a given field distribution in terms of waveguide modes. It is well known that for the guide axis chosen to coincide with the z axis, the transverse components of an arbitrary input electromagnetic field $\textrm{E}_\textrm{x}^{\textrm{in}}\textrm{(x,y,z),}\,\,\textrm{E}_\textrm{y}^{\textrm{in}}\textrm{(x,y,z),}\,\,\textrm{H}_\textrm{x}^{\textrm{in}}\textrm{(x,y,z)}$ and $\textrm{H}_\textrm{y}^{\textrm{in}}\textrm{(x,y,z)}$ of frequency $\mathrm{\omega ,}$ can always be written as a linear superposition of the transverse components of the electromagnetic fields $\textrm{E}_\textrm{x}^\textrm{m}\textrm{(x,y),}\,\,\textrm{E}_\textrm{y}^\textrm{m}\textrm{(x,y),}\,\,\textrm{H}_\textrm{x}^\textrm{m}\textrm{(x,y)}$ and $\textrm{H}_\textrm{y}^\textrm{m}\textrm{(x,y)}$ of all the modes of a given MMI waveguide in every cross-sectional plane (z = constant) as follows [37,44]:
At a certain distance $\textrm{z = }{\textrm{z}_\textrm{0}}$, from the beginning of the MMI waveguide z = 0, a double image of the input field with the equal amplitudes 1/√2 (3 dB power divider) is created which may couple to well positioned output waveguides with minimal reflections. In this case, all light is coupled to the fundamental modes of the upper $\textrm{(}{\textrm{p}_\textrm{1}})$ and lower $\textrm{(}{\textrm{p}_2})$ output waveguides as $\mathrm{\varphi }_\textrm{0}^\textrm{U}\textrm{(x,y)}$ and $\mathrm{\varphi }_\textrm{0}^\textrm{L}\textrm{(x,y),}$ respectively and can be written as:
3. Description of TE-guided modes in SWG MMI waveguide
Using the idea of periodic repetition [45,46], the developed methods for analysis of periodic structures can be used for analysis of some kinds of non-periodic structures. In this section, we apply the PWEM for 3D modal analysis of a SWG-based multimode waveguide which is periodic only in one dimension and can be considered as arrays of rectangular silicon pillars with longitudinal period ${\mathrm{\Lambda }_\textrm{z}}\textrm{,}$ duty cycle D and width ${\textrm{W}_{\textrm{MMI}}}$ (see Fig. 2(a)). We use the coordinate frame in this figure, in which the z-direction coincides with the direction of propagation of the wave. The aim is here to determine propagation constants of the various propagating modes and their corresponding field distributions. For developing a 3D PWEM analysis of such open dielectric waveguide of finite width and height, a periodically repeated structure should be investigated. The periodic repetition will be applied in transverse directions such that the electromagnetic fields in the x- and y-directions completely vanish in one period. To this end, the periodicities should be set to be so large that each cell practically shows negligible effect on the adjacent cells. In other words, The transverse periods ${\mathrm{\Lambda }_\textrm{x}}$ and ${\mathrm{\Lambda }_y}$ for the repeated structure are properly chosen to guarantee that the fields become zero before the next period starts. As a result, the propagation characteristics of the repeated structure are equivalent to those of the original structure. Hereafter, the duty cycle D is set at 50% in order to maximize the minimum feature size of the device. Furthermore, ${\mathrm{\Lambda }_z}$ is chosen to be 180 nm, substantially smaller than the minimum wavelength to ensure that the structure operates in SWG regime with negligible Bragg reflections while facilitating the manufacturing process. As a design example, we examine the propagation characteristics of a SWG MMI waveguide of width ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m}$ at the central wavelength ${\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m}$ used in integrated optics. For this calculation, ${\mathrm{\Lambda }_\textrm{x}}\textrm{ = 3}\,{\textrm{W}_{\textrm{MMI}}}$ and ${\mathrm{\Lambda }_\textrm{y}}\textrm{ = 10}\,\textrm{H}$ have been set. The PWEM method focuses on a single unitcell consisting of a rectangular silicon pillar embedded in $\textrm{Si}{\textrm{O}_\textrm{2}}$ background and uses periodic boundary conditions. By assuming its periodicity in our expansions, we are essentially using periodic boundary conditions. Now, we begin to build a picture of our unitcell on a high-resolution grid in order to accurately compute its Fourier coefficients. By choosing a sufficient number of discrete data points, the mesh sizes in each coordinate directions are assumed to be $\textrm{dx = 13nm,}\,\,\textrm{dy = 7nm}$ and $\textrm{dz = 2nm}$ which are, respectively, ${\textrm{1} / {\textrm{100}}},$ ${\textrm{1} / {\textrm{200}}}$ and ${\textrm{1} / {\textrm{800}}}$ of the minimum wavelength of interest, i.e., $\mathrm{\lambda =\ 1}\mathrm{.35\mu m}$ and sufficiently low to resolve the local features of the unitcell with high accuracy. Figures 2(b) and (c) demonstrate two different views of the permittivity of the proposed unit cell via MATLAB on x-y and z-y planes. Once we have that really high resolution array defining the material property as a function of position, we then calculate Fourier coefficients through a fast Fourier transform (FFT) of the array. Using the concept of the complex Fourier series, the unit cell is decomposed into a set of planar gratings with infinite number according to Eq. (6). Each planar grating is described by a complex amplitude ${a_{pqr}}$ from the Fourier series and grating vector ${\mathrm{\vec{K}}_{\textrm{pqr}}}$ calculated analytically. We could reconstruct the original unitcell by adding all of the planar gratings whose their amplitudes ${a_{pqr}}$ are less than some thresholds. A threshold that works in many cases is one that is around 2% of the maximum ${a_{pqr}}$ [43] which results in a truncated number of planar gratings (as well plane waves) in the expansion of Eq. (6) (Eq. (4)) for x, y and z directions to be, respectively, $\textrm{P = 11,}\,\,\textrm{Q = 19}$ and $\textrm{R = 11}\textrm{.}$ When a higher number of terms are used in the expansion, better results are obtained but it is more computationally intensive. After truncation, we thus retain only a minimum number of planar gratings as seen in Figs. 3(a)-(c) for Fourier coefficients ${a_{pqr}}$ presented in three different views.
Fig. 2. (a) The 3D view of a SWG MMI waveguide composed of arrays of rectangular silicon pillars embedded in $\textrm{Si}{\textrm{O}_{\textrm{2}}}$ background. Relative permittivity of the materials throughout the unitcell of a SWG MMI waveguide with ${\mathrm{\Lambda }_{\textrm{z}}}\textrm{ = 180}\,\textrm{nm,}$ $\textrm{D = 50}\,{\%}$ and ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m}$ as a function of position is displayed on (b) x-y and (c) z-y planes, at the central wavelength of ${\mathrm{\lambda }_0} = 1\textrm{.55}\,\mathrm{\mu m}\textrm{.}$
Fig. 3. (a) – (c) Amplitude of the truncated Fourier coefficients through FFT of the proposed unitcell array of Figs. 2(b)-(c). (d) calculation and representation of the corresponding convolution matrix $[{{\mathrm{\varepsilon }_\textrm{r}}} ]\textrm{.}$
The remained Fourier coefficients are then used to populate the convolution matrix $[{{\mathrm{\varepsilon }_\textrm{r}}} ]$ shown in Fig. 3(d). The corresponding diagonal matrices of ${\mathbf{\tilde{G}}_\mathbf{x}}\mathbf{,}\,\,{\mathbf{\tilde{G}}_\mathbf{y}}$ and ${\mathbf{\tilde{G}}_\mathbf{z}}$ are also presented in Fig. 4. Finally, the coefficient matrix of Eq. (7) is constructed to define the eigenvalue problem which is solved numerically to calculate the eigenvalues and eigenvectors. Figure 5 shows the resulting dispersion curves (over the wavelength range of $\textrm{1}\mathrm{.35\mu m\ -\ 1}{.75\mathrm{\mu} \mathrm{m}}$) and the electric field profiles $\textrm{(at }{\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m)}$ of the five lowest order TE guided modes whose effective indices are greater than the refractive index of the surrounding medium, of the investigated SWG MMI waveguide.
Fig. 4. (a) $\mathbf{\tilde{G}}_{\mathbf{x}}\mathbf{,}$ (b) $\mathbf{\tilde{G}}_{\mathbf{y}}$ and (c) $\mathbf{\tilde{G}}_{\mathbf{z}}$ which are the x, y and z-components of the wave vector expansion, respectively, corresponding to proposed unitcell.
Fig. 5. (a) Wavelength dependence of the effective indices of the TE-guided modes and (b)-(f) their corresponding electric field distributions at ${\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m,}$ for our SWG MMI waveguide. The simulations are conducted via the PWEM method.
For MMI beam splitting applications, a sufficient width of the multi-mode waveguide is required in order to achieve a good image quality [44]. For example, For the SWG MMI splitters with the number of two output ports used in our proposed device, the multi-mode waveguide must support at least three excited guided modes for high confinement. Guided modes of SWG MMI waveguides of various widths at the end of the wavelength range of interest $\mathrm{\lambda =\ 1}{.75\mathrm{\mu} \mathrm{m}}$ are classified in Table. 1 using PWEM. In order for the contributing guided modes to self-imaging to be well-confined, the width of the multi-mode waveguide has to be ${\textrm{W}_{\textrm{MMI}}} \ge 3.5\,\mathrm{\mu m}$ as can be seen in this table. We set the width of the multi-mode waveguide to the minimum width ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m,\ }$ due to the compactness of the device, throughout this work.
4. Design and simulation of SWG MMI splitters
Inspired by combination of the PWEM theory and the mode expansion conjecture concept, we propose a novel design method of SWG MMI couplers which is explained and demonstrated by two design examples for a $\mathrm{1\ \times 2}$ SWG MMI coupler and a $\mathrm{2\ \times 2}$ SWG MMI coupler. Both couplers are then evaluated using full wave simulations.
4.1 $\mathrm{1\ \times 2}$SWG MMI coupler
In this subsection, we will introduce a new design method for a center-fed $\mathrm{1\ \times 2}$ SWG MMI coupler, as the input section of our proposed device, at ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m}$ and the numerical simulations achieved by Lumerical FDTD are here presented over the wavelength range of interest. Figure 6(a) shows schematically our broadband SWG MMI coupler which comprises two main sections: 1- a SWG multimode region with period ${\mathrm{\Lambda }_\textrm{z}}\textrm{ = 180}\,\textrm{nm,}$ duty cycle $\textrm{D = 50}\,{\%,}$ width ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m}$ and the unknown length ${\textrm{L}_\textrm{1}}$ which requires careful design to suitably image the light from the input port onto the output ports. 2- the input (output) adiabatic strip-to-SWG (SWG-to-strip) mode converter (s) with a linearly taper from ${\textrm{W}_{\textrm{in}}}\textrm{(}{\textrm{W}_\textrm{a}}\textrm{)}$ to ${\textrm{W}_\textrm{a}}\textrm{(}{\textrm{W}_{\textrm{in}}}\textrm{),}$ which its number of periods is ${\textrm{N}_\textrm{t}}\textrm{,}$ with the same period and duty cycle as the SWG MMI region. We set the width of interconnecting strip single mode waveguide ${\textrm{W}_{\textrm{in}}}\textrm{ = 500}\,\textrm{nm}\textrm{.}$ Moreover, We choose the width of (input and output) SWG access waveguides to be ${\textrm{W}_\textrm{a}}\textrm{ = 1}\mathrm{.6\mu m,}$ to control excitation of higher order modes of SWG MMI section while ensuring that most of the input optical power is coupled to the first few lower order modes with small modal phase errors to guarantee high quality imaging [47]. Utilizing this selection, the minimum separation between the two output access waveguides can be computed to be ${\textrm{W}_{\textrm{MMI}}}\textrm{ - 2}{\textrm{W}_\textrm{a}}\textrm{ = 0}\mathrm{.3\mu m}$ which is sufficiently physically decoupled for practical applications. The aim is here to design the required coupling length ${\textrm{L}_\textrm{1}}\textrm{,}$ for 3 dB power splitting. Simplifying the analysis, the effect of the tapered access waveguides is here ignored (as shown in Fig. 6(b)) and only the fundamental mode of the SWG access waveguides of width ${\textrm{W}_\textrm{a}}\textrm{ = 1}\mathrm{.6\mu m}$ is taken into account which has been solved by the technique described in this paper and the results of this 3D modal analysis are shown in Figs. 6(c) and (d). An incident wave in the input access waveguide propagating in positive z-direct, at the interface between the input access waveguide $\textrm{(width }{\textrm{W}_\textrm{a}}\textrm{ = 1}\mathrm{.6\mu m)}$ and the wide MMI waveguide (width ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m),}$ will be decomposed into the modal field distributions of MMI waveguide.
Fig. 6. Schematic for the top view of a $\mathrm{1\ \times 2}$ SWG MMI coupler (a) with tapers and (b) without tapers such that only the fundamental modes of the (input and output) access waveguides are taken into account as shown, and with sketches of modal field distributions of the guided modes $\textrm{m = 0,}\,\textrm{1,}\,\textrm{2}$ of the wide multimode waveguide. (c) and (d), respectively, the modal field profile (at ${\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m)}$ and the dispersion plot of the fundamental mode of the (input) access waveguide of width ${\textrm{W}_\textrm{a}}\textrm{ = 1}\mathrm{.6\mu m}$ performed by PWEM.
By substituting the matrix form of the modal field distributions of the wide MMI waveguide (shown in Figs. 5(b)-(f)) and that of the input access waveguide (shown in Fig. 6(c)) in Eq. (9), the modal coefficients are thus computed which their amplitudes are shown in Fig. 7(a) (5 guided modes are taken into account). Three observations are yielded from this figure: 1) only the even modes will be excited and then contribute to self-imaging due to the symmetry of excitation and the structure. 2) it is clear that the most of the input optical power is coupled to the first two even guided modes. 3) the summation over the power fractions coupled into guided modes becomes unit ($\sum\limits_{m = 0}^4 {|{a_m}{|^2}} = 1$) which indicates that the injected input power is fully coupled from the input access waveguide to the MMI waveguide. Once we have the complex modal coefficients ${a_m}(z)$ (from Fig. 7(a) in which the amplitudes are shown and the phases are not) and the propagation constants ${\mathrm{\beta }_\textrm{m}}$ (from Fig. 5(a)), we compute the field $\Omega (x,y,z)$ along the MMI waveguide using Eq. (10). This field, as can be seen in Eq. (11), may be written as a superposition of the fundamental modes of the upper and the lower output access waveguides $\mathrm{(\varphi }_\textrm{0}^\textrm{U}\mathrm{(x,y)\ and\ \varphi }_\textrm{0}^\textrm{L}\textrm{(x,y))}$ obtained by sufficiently circularly shift of the matrix form of the modal field distribution of the input access waveguide. $\mathrm{\varphi }_\textrm{0}^\textrm{U}\textrm{(x,y)}$ and $\mathrm{\varphi }_\textrm{0}^\textrm{L}\textrm{(x,y)}$ are shown in matrix form in Figs. 7(b) and (c). The complex amplitudes, ${\textrm{b}_{\textrm{1,2}}}$ and the insertion loss can be then determined by means of Eq. (12) and Eq. (13) respectively. Now, we use a numerical optimization algorithm which finds appropriate coupling length ${\textrm{L}_\textrm{1}}$ in order to minimize the cost function (defined as insertion loss) at design wavelength ${\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m}\textrm{.}$ To this end, we use the “fminsearch” function of MATLAB. “Fminsearch” uses the NelderMead simplex algorithm as described in [48]. By this algorithm, we reach an insertion loss of $\textrm{IL = 0}\textrm{.08}\,\textrm{dB}$ with the coupling length of ${\textrm{L}_\textrm{1}}\mathrm{\ =\ 5\mu m\ }$ so that the number of pitches for SWG multimode section is given by ${\textrm{N}_\textrm{1}}\textrm{ = }{{{\textrm{L}_\textrm{1}}} / {{\mathrm{\Lambda }_\textrm{z}}}}$ which is approximately ${\textrm{N}_1}\textrm{ = 27}$ pitches. As it can be seen from Fig. 7(d), combination of the PWEM and the mode expansion conjecture as our proposed design method predicts that at certain distance $\textrm{z = }{\textrm{L}_\textrm{1}}$ and wavelength ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m,}$ the input field of the MMI coupler will be imaged 2 times superimposed with each image shifted in the x-direction. To verify more exactly, we plot the insertion loss for the designed 3 dB SWG MMI coupler as a function of wavelength using full-wave simulation (FDTD Lumerical) and the proposed procedure in Fig. 8. As seen a good agreement is observed between the two results. This figure also confirms high-performance device operation over a 400 nm wavelength span at telecom wavelengths with insertion loss below as low as $\textrm{0}\textrm{.24}\,\textrm{dB}\textrm{.}$ It should be noted that for full-wave simulation, we found that tapers (mode converters between strip and SWG) with ${\textrm{N}_\textrm{t}}\textrm{ = 24}$ pitches exhibit negligible losses and obtained simulated insertion losses of less than $\textrm{0}\textrm{.02}\,\textrm{dB}$(by simulating a structure in which two tapers are connected back to back, not shown here). Furthermore, Fig. 9 shows the simulated light propagation (electric field component) at different wavelengths. These figures reveal that the proposed compact coupler provides uniform and wavelength independent output power throughout the wavelength range of interest. It can be also clearly seen that the positions of the two images are well-matched to the output ports at the end of the SWG multimode waveguide.
Fig. 7. (a) Modal amplitudes ${a_m}$ for $\textrm{m = 0,}\,\textrm{1,}\,\textrm{2,}\,\textrm{3,}\,\textrm{4}$ obtained by combination of the PWEM and the mode expansion conjecture. (b) and (c) Fundamental mode field profiles of the upper and the lower access waveguides, respectively, achieved by circularly shift of the matrix form of the modal field distribution of the input access waveguide of Fig. 6(c). (d) The total electric field $\Omega (x,y,z)$ is decomposed into two-fold images for particular $\mathrm{1\ \times 2}$ SWG MMI coupler length of ${\textrm{L}_\textrm{1}}\mathrm{\ =\ 5\mu m,}$ obtained by combination of the PWEM and the mode expansion conjecture, as well. The operation wavelength is also chosen to be ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m}\textrm{.}$
Fig. 8. Insertion loss spectrum for the designed $\mathrm{1\ \times 2}$ SWG MMI coupler whose parameters are ${\textrm{W}_\textrm{a}}\textrm{ = 1}\mathrm{.6\mu m,}$ ${\mathrm{\Lambda }_\textrm{z}}\textrm{ = 180}\,\textrm{nm,}\textrm{D = 50}\,{\%,}$ ${\textrm{W}_{\textrm{MMI}}}\textrm{ = 3}\textrm{.5}\,\mathrm{\mu m}$ and ${\textrm{L}_\textrm{1}}\mathrm{\ =\ 5\mu m}$ using the proposed design method and full-wave simulation (FDTD Lumerical).
Fig. 9. Light propagation (electric field component) in the designed coupler for wavelengths (a) $\mathrm{\lambda =\ 1}\mathrm{.35(\mu m),}$ (b) $\mathrm{\lambda =\ 1}\mathrm{.45(\mu m),}$ (c) $\mathrm{\lambda =\ 1}\mathrm{.55(\mu m)}$ and (d) $\mathrm{\lambda =\ 1}\mathrm{.75(\mu m)}\textrm{.}$
4.2 $\mathrm{2\ \times 2}$SWG MMI coupler
In this subsection, we consider the design and the simulation of a $\mathrm{2\ \times 2}$ SWG MMI coupler (shown in Fig. 10) in which the longitudinal period ${\mathrm{\Lambda }_\textrm{z}}\textrm{,}$ duty cycle $\textrm{D,}$ width ${\textrm{W}_{\textrm{MMI}}}$ and tapers remain unchanged. The number of pitches for the core of this coupler given by ${\textrm{N}_2}$ (and related to coupling length ${\textrm{L}_\textrm{2}}$ by ${\textrm{N}_\textrm{2}}\textrm{ = }{{{\textrm{L}_\textrm{2}}} / {{\mathrm{\Lambda }_\textrm{z}}}})$ is determined based on combination of the PWEM and the mode expansion conjecture concept like the detailed discussion presented about choosing proper value of the coupling length ${\textrm{L}_\textrm{1}}$ in the $\mathrm{1\ \times 2}$ SWG MMI coupler investigated in previous subsection. The difference is here that we assume the input fields (caused by incident waves) are inserted in the fundamental mode of both the upper (${\textrm{p}_\textrm{1}}$) and the lower (${\textrm{p}_4}$) input access waveguides with the phases of ${\mathrm{\varphi }_\textrm{1}}$ and ${\mathrm{\varphi }_4},$ respectively. For symmetric excitation (the phase difference of $\mathrm{\Delta \varphi }\,\textrm{ = }\,{\mathrm{\varphi }_\textrm{1}}\textrm{ - }{\mathrm{\varphi }_\textrm{4}}\,\textrm{ = 0)}$ and by iteratively minimizing the cost function (insertion loss) during optimization at ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m,}$ appropriate coupling length of ${\textrm{L}_2}\mathrm{\ =\ 9\mu m\ }$ will be found to split the power of the input access waveguides equally over the two output access waveguides with insertion loss of $\textrm{IL = 0}\textrm{.05dB}\textrm{.}$ Table. 2 summarizes the simulation results of the proposed $\mathrm{2\ \times 2}$ SWG MMI coupler carried out by Lumerical FDTD over the wavelength band of $\textrm{1}\mathrm{.35\mu m\ -\ 1}{.75\mathrm{\mu} \mathrm{m}}\textrm{.}$ It shows that the power splitting ratio can be controlled by varying the relative phase between the two simultaneously applied input fields. As it is apparent, this coupler enables low averaged insertion loss (< $\textrm{0}\textrm{.16dB)}$ and low averaged power imbalance (< $\textrm{0}\textrm{.26dB)}$ for different input phase shifts, making it suitable for our application across the wavelength band of interest.
Table 2. Simulation results of the proposed $\mathrm{2\ \times 2}$ SWG MMI coupler for different relative phases between two simultaneously applied input fields
Fig. 10. In-plane schematic of a $\mathrm{2\ \times 2}$ SWG MMI coupler of length ${\textrm{L}_\textrm{2}}$ and keeping fixed all other dimensions as the $\mathrm{1\ \times 2}$SWG MMI coupler in Fig. 6(a).
5. Design and simulation of the interconnecting graphene-embedded silicon waveguides
This section focuses on the optical properties of a $\textrm{500}\,\textrm{nm}$ wide silicon waveguide with graphene multilayer embedded (as shown in Fig. 1 as inset), in the telecom band. By embedding the graphene sheets into the silicon waveguide and flexibly tuning the Fermi level (${\textrm{E}_\textrm{f}})$ of each of which via applied bias voltage, the light-matter interaction is significantly enhanced, thereby the EMI of the fundamental propagating mode can be effectively tailored within this waveguide. The more layer number of graphene the larger variation of modal characteristics of the waveguide mode [49]. An eleven- layer graphene (all layers with the same ${\textrm{E}_\textrm{f}})$ is here considered to maximize the graphene-light interaction and symmetrically inserted into the waveguide, inducing twelve-layer silicon with equal thickness of $\textrm{15nm}\textrm{.}$ Although it may be a challenge to manufacture such waveguide, it can be accomplished by growing graphene layers in chemical vapor deposition (CVD) and transferring them mechanically onto the silicon waveguide [30,50]. In detail, it consists of eleven periods of alternating CVD graphene sheets and Si layers on a $\textrm{Si}{\textrm{O}_\textrm{2}}$ substrate. The CVD graphene is grown on copper foil (Graphenea Inc) and transferred to the substrate using the standard PMMA method [51]. The copper foil is etched using an ammonium persulfate solution. The Si dielectric layer is deposited by atomic layer deposition (ALD) using trimethylaluminium as the Al precursor and H2O as the oxygen precursor [51]. The number of cycles used in the ALD process is calibrated to grow 15 nm of Si on graphene, with the thickness characterized by an ellipsometer. The procedure is repeated to fabricate eleven periods of the graphene-Si unitcell. In general, the manufacturing of layered structures consisting of several graphene sheets separated by dielectric layers is technologically achievable by using, e.g., well-developed CVD, MBE methods and cleavage techniques [51,52]. A feasible scheme for tuning the conductivity of graphene, via electrical biasing, has been demonstrated by Chang et al. [51]., where the chemical potential of graphene was efficiently tuned by applying a bias voltage across each graphene layer. A similar scheme can be adopted, in our proposed graphene-embedded silicon waveguide, to electrically bias the graphene layer, sandwiched between two layers of Si. Now, we use the Lumerical Mode Solutions as a full-wave electromagnetic solver to numerically calculate and analyze the properties of the proposed graphene-embedded silicon waveguide. Each graphene layer is modeled by an ultrathin layer of thickness $\mathrm{\Delta =\ 1nm}$ whose relative permittivity is defined by Eq. (2). Figure 11(a) shows the propagation loss (unit: ${{\textrm{dB}} / {\mathrm{\mu m}}})$ of this waveguide versus the Fermi level of the graphene layers, ${\textrm{E}_\textrm{f}}\textrm{,}$ at three different wavelengths (the beginning of the band, the middle of the band and the end of the band). It clearly illustrates that the region of low propagation losses (lower than $\textrm{0}\textrm{.024}\,\mathrm{dB/\mu m)}$ is matched to the Fermi levels above ${\textrm{E}_\textrm{f}} = \,\textrm{0}\textrm{.6}\,\textrm{ev}$ for the entire wavelength band of interest. Furthermore, Figs. 11(b)-(d) depicts the dependence of the real part of the EMI (${\textrm{n}_{\textrm{eff}}}$) of the TE fundamental mode of our proposed waveguide on ${\textrm{E}_\textrm{f}}$ for different wavelengths. It is seen that when Fermi level changes from $\textrm{0}\textrm{.6}\,\textrm{ev}$ to $\textrm{1}\,\textrm{ev,}$ the variation of ${\textrm{n}_{\textrm{eff}}}$ can be as large as $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}}\textrm{ = 0}\textrm{.041,}\,\,\,\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}}\textrm{ = 0}\textrm{.049}$ and $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}}\textrm{ = 0}\textrm{.059}$ at the wavelengths of $\textrm{1}\textrm{.35}\,\mathrm{\mu m,\ 1}\textrm{.55}\,\mathrm{\mu m}\,$ and $\textrm{1}\mathrm{.75\mu m,}$ respectively. Note that $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}},$ in which traditional electro-optic materials can be induced, is only at the order of $\textrm{1}{\textrm{0}^{\textrm{ - 4}}}$ [30].
Fig. 11. (a) The propagation loss and (b)- (d) the real part of effective modal index variation under different graphene Fermi levels for the eleven-layer graphene embedded waveguide, at different wavelengths.
The impressive value of large $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}}$ is of great importance in ER-based devices such as our proposed MZI-based tunable power splitter (shown in the schematic picture of Fig. 1) which is demonstrated using the integration properties of the designed $\mathrm{1\ \times 2}$ SWG MMI coupler at the input and the designed $\mathrm{2\ \times 2}$ SWG MMI coupler at the output. The input field is equally split between the two phase modulators (arms of the MZI) which are a pair of the proposed graphene-embedded silicon waveguides of length ${\textrm{L}_\textrm{3}}.$ Different bias voltage signals (corresponding to different Fermi levels) can be applied on both arms to modify the phase difference between the optical signals propagating in the two arms. The optical signals are then fed into the output SWG MMI coupler. We first fix the Fermi level of the upper arm ${\textrm{E}_{\textrm{f1}}}\textrm{ = 0}\textrm{.6ev}$ then shift the other Fermi level ${\textrm{E}_{\textrm{f2}}}$ from $\textrm{0}\textrm{.6ev}$ to $\textrm{1ev}\textrm{.}$With ${\textrm{E}_{\textrm{f2}}}$ changes and setting the design wavelength to be ${\mathrm{\lambda }_0}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m,}$ the corresponding $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}}$ will be modified and a ${\textrm{n}_{\textrm{eff}}}$ variation of $\mathrm{\Delta }{\textrm{n}_{\textrm{eff}}} = 0.049$ which is a large quasi linear dynamic range will be achieved, thus the required arm length ${\textrm{L}_\textrm{3}}$ to reach the ${\mathrm{\pi } / \textrm{2}}$ phase shift is modified and calculated to be ${\textrm{L}_\textrm{3}}\textrm{ = 7}\textrm{.9}\,\mathrm{\mu m}$ through ${\mathrm{\pi } / \textrm{2}}\mathrm{\ =\ \Delta }{\textrm{n}_{\textrm{eff}}}\mathrm{\ \times }{\textrm{L}_\textrm{3}}.$ This is why the phase shift has been set to ${\mathrm{\pi } / \textrm{2}}$ in order to maximize the difference in relative powers coupled in upper (${\textrm{p}_\textrm{2}}$) and the lower (${\textrm{p}_\textrm{3}}$) output waveguides of the $\mathrm{2\ \times 2}$ SWG MMI coupler which are, respectively, $\textrm{0}\textrm{.85}$ and $\textrm{0}\textrm{.15}$ as illustrated in Table. 2.
6. Simulation results of the proposed splitter
In this section, we numerically investigate the propagation of the waves through our designed device. In the following as the Figs. 12–15 reveals, the ${\textrm{E}_{\textrm{f2}}}$ is tuned in the range of $\textrm{(0}\textrm{.6}\,\textrm{ev,}\,\textrm{1}\,\textrm{ev)}$ at several discrete points and accordingly the performance of the proposed MZI-based tunable power splitter with only $\textrm{7}\textrm{.9}\,\mathrm{\mu m\ -\ }$ long graphene is characterized by its transmittance, its insertion loss and its power imbalance versus wavelength. Figure 12 presents the results at ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.6ev}$ which is called balanced modulation state. Figure 12(a) shows that the resultant device in this situation provides uniform transmittance with equal wavelength averaged splitting ratio of $\textrm{0}\textrm{.4582}$ over the two output ports, due to the same applied Fermi levels on the two MZI arms. From Figs. 12(b)-(d), it can be clearly seen that the averaged insertion loss and the averaged power imbalance for the two output ports are as low as, respectively, $\textrm{0}\textrm{.38dB}$ and $\textrm{0}\textrm{.13dB}$ across the whole wavelength range of interest. Similarly, Figs. 13–15 corresponds to Fermi levels of ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.7ev,}\,\,{\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.83ev}$ and ${\textrm{E}_{\textrm{f2}}}\textrm{ = 1ev,}$ respectively, which are called unbalanced modulation states and result in achieving transmittances with unequal wavelength averaged splitting ratios of $0.66/0.34,$ $0.79/0.21,0.85/0.15$ and the averaged power imbalances of $\textrm{0}\textrm{.46dB/0}\textrm{.2dB,}$ $\textrm{0}\textrm{.47dB/0}\textrm{.25dB}$ and $\textrm{0}\textrm{.38dB/0}\textrm{.31dB}$(the numbers in the numerators the denominators correspond to ports ${\textrm{p}_\textrm{2}}$ and ${\textrm{p}_3}$ in order). With the above mentioned unbalanced modulation states, the wavelength averaged insertion losses of $\textrm{0}\textrm{.36dB,}\textrm{0}\textrm{.36dB}$ and $\textrm{0}\textrm{.35dB}$ are obtained, respectively. The remarkable features of our proposed device with further statistical details are summarized in Table. 3 where it is observed that both the averaged insertion loss and the averaged power imbalance of the designed device, for various modulation states, are respectively, kept below $\textrm{0}\textrm{.39}\,\textrm{dB}$ and $\textrm{0}\textrm{.48}\,\textrm{dB}$ over a 400 nm wavelength span at telecom wavelengths. Moreover, the Lumerical FDTD was used to investigate the power propagation along the different sections of the proposed device at ${\mathrm{\lambda }_\textrm{0}}\textrm{ = 1}\textrm{.55}\,\mathrm{\mu m}\textrm{.}$ In balanced modulation state, the device operates as a perfect 3 dB splitter, as shown in Fig. 16(a). By applying different Fermi levels on the two arms, the splitting ratio of the device can easily be altered as illustrates in Figs. 16(b)-(d). Additionally, Figs. 17(a) and (b) show the simulated transmission of the proposed 1 × 2 MZI-based power splitter at balanced modulation state as the wavelength deviate from the designed values (respectively, at $\mathrm{\lambda =\ 1}\mathrm{.1\mu m}$ and $\mathrm{\lambda =\ 2\mu m)}\textrm{.}$ However, from these figures, one sees that directly extending this device operation to the out of wavelength region of interest will lead to a significant deterioration in the propagation, due to the losses originating from scattering, diffraction and material absorption.
Table 3. Simulation results of the proposed dynamically tunable 1× 2 power splitter, with further statistical details for different Fermi levels applied on the lower arm over the wavelength band of $\textrm{1}\mathrm{.35\mu m\ -\ 1}{.75\mathrm{\mu} \mathrm{m}}\textrm{.}$
Fig. 12. (a) The transmittance, (b) the insertion loss, (c) and (d) the power imbalance of output ports of the designed 1 × 2 MZI-based power splitter as a function of the wavelength for balanced modulation state of ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.6ev}\textrm{.}$
Fig. 13. (a) The transmittance, (b) the insertion loss, (c) and (d) the power imbalance of output ports of the designed 1 × 2 MZI-based power splitter as a function of the wavelength for balanced modulation state of ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.7ev}\textrm{.}$
Fig. 14. (a) The transmittance, (b) the insertion loss, (c) and (d) the power imbalance of output ports of the designed 1 × 2 MZI-based power splitter as a function of the wavelength for balanced modulation state of ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.83ev}\textrm{.}$
Fig. 15. (a) The transmittance, (b) the insertion loss, (c) and (d) the power imbalance of output ports of the designed 1 × 2 MZI-based power splitter as a function of the wavelength for balanced modulation state of ${\textrm{E}_{\textrm{f2}}}\textrm{ = 1ev}\textrm{.}$
Fig. 16. Light propagation (electric field component) along the designed dynamically tunable power splitter at wavelength ${\mathrm{\lambda }_0}\textrm{ = 1}\mathrm{.55(\mu m),}$ for (a) ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.6ev,}$ (b) ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.7ev,}$ (c) ${\textrm{E}_{\textrm{f2}}}\textrm{ = 0}\textrm{.83ev}$ and (d) ${\textrm{E}_{\textrm{f2}}}\textrm{ = 1ev}\textrm{.}$
Fig. 17. Light propagation (electric field component) along the designed dynamically tunable power splitter at balanced modulation state as the wavelength deviate from the designed values, respectively, at (a) $\mathrm{\lambda =\ 1}\mathrm{.1\mu m}$ and (b) $\mathrm{\lambda =\ 2\mu m}\textrm{.}$
7. Conclusion
In summary, we propose a novel graphene-based dynamically tunable power splitter on SOI platform. First, a novel design method of SWG MMI couplers, based on combining PWEM and the mode expansion conjecture concept is proposed. Two design examples for a $\mathrm{1\ \times 2}$ SWG MMI coupler and a $\mathrm{2\ \times 2}$ SWG MMI coupler are exhibited to explain and demonstrate the method. Both couplers are then connected with a pair of straight silicon waveguides of equal length embedded with multilayer graphene, which is duly and numerically designed in such a way as to provide the required phase shift by dynamic tuning of the graphene layers’ Fermi level values. In this MZI-based structure as our proposed device, the Fermi level of one of the arms is fixed to be ${\textrm{E}_{\textrm{f1}}}\textrm{ = 0}\textrm{.6ev}$ while that of the other arm, ${\textrm{E}_{\textrm{f2}}},$ varies from $\textrm{0}\textrm{.6}\,\textrm{ev}$ to $\textrm{1}\,\textrm{ev}\textrm{.}$ Correspondingly, the calculated power splitter ratio varies in the range from $\textrm{0}\textrm{.5:0}\textrm{.5}$ to $\textrm{0}\textrm{.85:0}\textrm{.15}\textrm{.}$ The numerical results reveal that the averaged insertion loss and the averaged power imbalance of the designed device are, respectively, lower than $\textrm{0}\textrm{.39}\,\textrm{dB}$ and $\textrm{0}\textrm{.48}\,\textrm{dB}$ over the wavelength range from $\textrm{1350}\,\textrm{nm}$ to $\textrm{1750}\,\textrm{nm}$ which confirms high performance device operation. Additionally, the advantage of compact design footprint of $\textrm{58}\,\mathrm{\mu m}\,\,\mathrm{\ \times }\,\,\textrm{3}\textrm{.5}\,\mathrm{\mu m}$ as well as the CMOS compactable structure indicates its valuable capability to be integrated into photonic circuits. It should be also noted here that since more layers of graphene may add to difficulty of fabrication, we can reduce the layer number of graphene at the cost of longer MZI arm lengths while keeping the design performance remarkably high.
Disclosures
The authors declare no conflicts of interest.
Data availability
Simulation details and associated data are available free of charge from authors upon reasonable request.
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