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Abstract
The focusing apodized subwavelength grating coupler (F-ASGC) has advantages of high coupling efficiency, small footprint and simple fabrication process, which make it a popular component for chip-scale coupling and testing of integrated optical circuit. However, the design of F-ASGC based on effective medium theory lacks accuracy, causing the drawbacks of peak wavelength deviation and performance degradation. In this work, we propose a deterministic design method of F-ASGC. Our grating coupler is formed by assembling various subwavelength grating units according to their complex effective indexes. The complex effective indexes of these grating units are accurately obtained by the weak form calculation. Then combining with transformation optics, we strictly analyze the F-ASGC for the first time. The simulation results show that the deterministically designed F-ASGC has high coupling efficiency of -2.51 dB, 3 dB bandwidth of 51 nm, and accurate central wavelength of 1553.1 nm. And we also fabricated it on the commercial SOI wafer. The measured maximum efficiency is -3.10 dB, the 3 dB bandwidth is 55 nm, and the central wavelength is 1551.5 nm.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Grating coupler utilizes the diffraction of grating to couple the light from the on-chip waveguide to the optical fiber [1–4]. The fabrication and packaging process of grating coupler is relatively easier than edge couplers, not requiring polishing of facets. And the grating coupler can be placed at any position on the chip without the need of waveguide extending to the edge of the chip, which makes the wafer-scale testing possible. However, the efficiency of a grating coupler is commonly lower than an edge coupler. In theory, this problem is mainly caused by the poor directionality (only part of the light diffracted upward to the fiber) and mode mismatch (the field profiles of diffracted light deviating from the fiber mode). Furthermore, shrinking the size of the grating coupler is also important for high-density integration. Researchers have proposed various design methods of grating coupler to improve the above three aspects, which will be commented in the following paragraphs.
The grating coupler usually diffracts light toward upper and lower cladding at the same time due to the grating equation satisfied in both of the claddings. Enhancing the upward diffraction can be realized by multi-layer grating [5–7], introducing metal [8,9] or Bragg mirrors [10], optimizing the diffraction angle [11] and so on. The multi-layer grating can coherently enhance the upward diffraction, but their fabrications require multi-step lithography and etching depth control. The metal or Bragg mirror on the substrate can reflect almost all the downward light back to the upper cladding, which improves the coupling efficiency dramatically. But the fabrication of metal or Bragg mirror is complicated. Through optimizing the diffraction angle, the downward light which will be partially reflected back by the substrate interface is utilized to coherently enhance the upward diffraction. This simple method can improve the coupling efficiency to a certain extent.
The fundamental mode of the optical fiber is approximately Gaussian. To improve the coupling efficiency of the grating coupler, the diffraction field should also be adjusted to the Gaussian profile [12]. Theoretically, it can be deduced that the grating coupler needs to be designed in an apodized form. Among varies designs [9,11,13–20], subwavelength grating coupler can fulfill both the apodized form and grating equation simply by tuning the size of subwavelength structure. It can be fabricated in one lithography step and has low back reflection. However, the subwavelength structure design based on the effective medium theory (EMT) lacks accuracy, resulting in the peak wavelength shift and coupling efficiency reduction.
Since the mode field diameter of the single-mode optical fiber is more than ten microns, the width of the grating coupler is also large. To connect to the on-chip narrow single-mode waveguide with low loss, a taper with a length of 250 µm is required. For shrinking the length of grating coupler, some compact taper structures were proposed [21–23]. A fabrication-easier solution is to bend the gratings to a cluster of elliptic curves [24–32]. Then the light can be high-efficiently focused on the single-mode waveguide in a remarkable short length. However, it is difficult to directly design the grating couplers combining focusing grating with apodized subwavelength grating, which usually relies on time-consuming numerical optimization.
In this paper, we propose a deterministic design method of focusing apodized subwavelength grating coupler (F-ASGC). Our grating coupler is composed of various subwavelength grating units (SGUs) according to their complex effective indexes. These indexes can be obtained from their complex bands calculated by weak form. Comparing the simulation results, we prove that our weak form method is more accurate than the traditional EMT method. Through combining the weak form calculation with transformation optics, we realize a rigorous analysis of the focusing subwavelength grating unit (FSGU) for the first time. From the vector grating equation, we proposed an algorithm to generate the layout of F-ASGC. The simulation results show that its maximum efficiency is 56.1% (-2.51 dB), the 3dB bandwidth is 51 nm, and the central wavelength is 1553.1 nm. When the minimum feature size of the grating coupler is restricted to 100 nm to meet the requirements of the CMOS foundry, the restricted F-ASGC has maximum efficiency of 50.7% (-2.95 dB), 3 dB bandwidth of 58 nm, and center wavelength of 1555.1 nm. We fabricated this F-ASGC on the commercial SOI wafer and tested its performance. The measured maximum efficiency is -3.10 dB, the 3 dB bandwidth is 55 nm, and the center wavelength is 1551.5 nm.
2. Subwavelength grating unit (SGU)
To explain our design method more clearly, we begin with the designing of a slab apodized subwavelength grating coupler (S-ASGC) working at the C band (1530 nm ∼ 1565 nm) for TE polarization, which can be fabricated on SOI wafer with 220 nm thick top silicon layer and air upper cladding. The layout of S-ASGC is shown in Fig. 1(a), which can be treated as the composition of various SGUs. Figure 1(b) shows the structure of the SGU. Since the size of SGU changes gradually along z direction, we assume that each SGU can be analyzed and calculated separately, like the unit cell of 2D photonic crystal. The SGU follows the Bloch period boundary condition in the y and z direction and the open boundary in the x direction.
Fig. 1. Design the slab apodized subwavelength grating coupler (S-ASGC) through assembling various subwavelength grating units (SGUs) according to their complex effective indexes. The real part and the imaginary part of the complex effective index are ${n_r}$ and ${n_i}$. (a) The layout of S-ASGC. (b) The structure of SGU. ${n_r}$ of every SGU must obey the grating equation Eq. (1) for the same diffraction angle $\theta $ consistently. The ${n_c}$ is the refractive index of upper cladding, p is the period length of the SGU, and $2\pi /p$ is the value of grating vector. And ${n_i}$ of every SGU at different position must follow the apodized form ${n_i}(z)$ as Eq. (2) to realize a Gaussian profile of the diffracted light.
To design the S-ASGC, the complex band of the SGU should be calculated [7]. The real part of the band presents the phase accumulation in one SGU, while the imaginary part presents the leakage characteristics of the light. Then we can obtain the complex effective index of SGU from the complex band by the formula ${n_{eff}} = k/{k_0}$, where ${k_0}$ is the wave vector in the vaccum and k is the eigen wave vector in the SGU. The real part of ${n_{eff}}$ is the effective refractive index ${n_r}$, and the imaginary part is the diffraction coefficient ${n_i}$.
For the grating coupler with diffraction order of $m = 1$, ${n_r}$ of every SGU should obey the grating equation for the same diffraction angle $\theta $ consistently, which is:
where ${n_c}$ is the refractive index of upper cladding, p is the period length of the SGU, and $2\pi /p$ is the value of grating vector. On the other hand, ${n_i}$ of every SGU at different position must follow the apodized form ${n_i}(z)$ to realize a Gaussian profile of the diffracted light. ${n_i}(z)$ is given by the following equation:2.1. Weak form of SGU
In this section, we will briefly introduce a weak form method for solving complex bands. The weak form is commonly utilized in the finite element simulation [33–35], which can solve the differential equation numerically with high accuracy. The commercial finite element software COMSOL (https://www.comsol.com) provides the weak form module to solve the differential equations by their corresponding weak forms.
For the SGU, the open boundary in the x direction can be realized by the perfectly matching layer (PML) [36], which is set at the edge of the simulation area. The wave vector of the light in PML is added with an imaginary part during numerical calculation to realize the attenuation. Hence the PML can absorb the light transmitted to the edge of the simulation area, effectively preventing the light from being reflected by the boundary and forming a fake resonance. Since the weak form module in COMSOL does not have PML, we need to find a way to realize PML which is also compatible with the weak form module. Based on the complex coordinate transformation method [35,37], PML can be achieved by changing the permittivity $\varepsilon $ and permeability $\mu $ to $\varepsilon ^{\prime} = \varepsilon {\mathbf \Lambda }$ and $\mu ^{\prime} = \mu {\mathbf \Lambda }$. And the matrix ${\mathbf \Lambda }$ is:
Hence, the vector wave equation of magnetic field including PML in frequency domain is:
2.2. Complex band calculation of the SGU
Figure 2 shows the simulation structure of the SGU in COMSOL. The upper cladding is air with thickness of ${h_c} = \textrm{3 }\mathrm{\mu }\textrm{m}$. The refractive index and thickness of the silicon core layer are ${n_{core}} = 3.476$ and ${h_s} = \textrm{220 nm}$. The buried oxide (BOX) lower cladding has refractive index of ${n_{box}} = 1.444$ and height of ${h_b} = \textrm{3 }\mathrm{\mu }\textrm{m}$. We set the length of the rectangular hole to be half of the grating period, that is, ${w_z} = {p_z}/2$. And we set the period in y direction of SGU as a constant ${p_y} = \textrm{450 nm}$. Therefore, the effective index ${n_r}$ and diffraction coefficient ${n_i}$ of the SGU are controlled by the width of rectangular hole ${w_y}$ and the grating period ${p_z}$. Then the structure of SGU is modeled in COMSOL as Fig. 2(a). We set the boundaries in the y and z direction as periodic boundaries. The PMLs are shown as the green areas in Fig. 2(a) with thickness ${h_p} = \textrm{0}\textrm{.5 }\mathrm{\mu }\textrm{m}$.
Fig. 2. Calculating the complex band of the SGU by the weak form in COMSOL. (a) The structure of SGU. The upper cladding layer is air. The refractive indexes of the buried oxide lower cladding and top silicon layer are ${n_{box}} = 1.444$ and ${n_{core}} = 3.476$. And the thicknesses of the buried oxide lower cladding, top silicon layer and upper cladding are ${h_b} = \textrm{3 }\mathrm{\mu }\textrm{m}$, ${h_s} = \textrm{220 nm}$ and ${h_c} = \textrm{3 }\mathrm{\mu }\textrm{m}$ respectively. We set ${w_z} = {p_z}/2$and ${p_y} = \textrm{450 nm}$. The green area is the PML with thickness ${h_p} = \textrm{0}\textrm{.5 }\mathrm{\mu }\textrm{m}$. Therefore, the effective index ${n_r}$ and diffraction coefficient ${n_i}$ of the SGU are controlled by the ${w_y}$ and ${p_z}$. (b) and (c) are the ${H_x}$ distribution in the xy plane and xz plane of the SGU with ${w_y} = \textrm{300 nm}$ and ${p_z} = \textrm{1050 nm}$.
The weak form Eq. (10) is calculated with the weak form module of COMSOL. The Bloch wave vectors are ${k_x} = 0$, ${k_y} = 0$, and ${k_z} = k$, where k is the eigenvalue. We set the permittivity and permeability of medium as $\varepsilon {\mathbf \Lambda }$ and $\mu {\mathbf \Lambda }$. The maximum element size is 100 nm, and the resolution of narrow regions is set to 5. To guarantee the surface integral in Eq. (1) 0 vanished in numerical calculation, the periodic boundaries need to be set as the same meshes by “copy face”. Type the integral in Eq. (10) into the weak form module, and then run the program to calculate the complex band of the SGU.
The complex effective index is calculated by ${n_{eff}} = k/{k_0}$. And the magnetic field ${{\mathbf H}_x}$ distribution of the SGU can be plotted in COMSOL. Figures 2(b) and 2(c) are the ${{\mathbf H}_x}$ distribution in the x-y plane and x-z plane of the SGU with ${w_y} = \textrm{300 nm}$ and ${p_z} = \textrm{1050 nm}$. It can be found that the SGU diffracting light both upward and downward. According to ${n_r}$ of the SGU, its diffraction angle $\theta $ can be calculated to be $\theta = 28^\circ $ from the grating equation Eq. (1), which is consistent with the direction angle of the ${{\mathbf H}_x}$ wavefront in Fig. 2(c).
To find the proper design parameters, we scan ${w_y}$ and ${p_z}$ of the SGU to get the corresponding ${n_r}$ and ${n_i}$. The scanning range of ${w_y}$ is from 10 nm to 300 nm, and the range of ${p_z}$ is from 550 nm to 1200 nm. According to Eq. (1), ${p_z}$ is varied with ${w_y}$ and diffraction angle $\theta $, as shown in Fig. 3(a). Then we can further get ${n_i}$ varying with $\theta $ and ${w_y}$, as shown in Fig. 3(b).
Fig. 3. (a) The grating period ${p_z}$ and (b) the diffraction coefficient ${n_i}$ varying with the diffraction angle θ and the width of the rectangular hole ${w_y}$. The black dotted lines are the contours of ${p_z}$ and ${n_i}$ for clarity.
2.3. Comparison between the weak form method and EMT method for S-ASGC design
According to Ref. [11], we set $\theta = 34^\circ $ as the optimized diffraction angle. And we interpolate the data of Figs. 3(a) and 3(b) to obtain the relationship curve of ${n_r}$ and ${n_i}$ varying with ${w_y}$, which are shown as the blue curves in Figs. 4(a) and 4(b). For comparison, we also calculated the relationship curve of ${n_r}$ and ${n_i}$ with ${w_y}$ using the EMT method. The idea of EMT method is to approximate the subwavelength structures in the lateral direction (y direction) as a uniform medium with the effective index ${n_g}$ [18,19,32]. To calculate ${n_r}$ of the SGU, we use the eigen frequency calculation in optics module of COMSOL with various ${n_g}$ and ${p_z}$ [14]. To obtain ${n_i}$ of the SGU, we first need to simulate the uniform grating coupler composed of the same SGU with various ${n_g}$ and ${p_z}$ [14,20]. Through detecting the attenuation coefficient $\alpha $ of the light field in the uniform grating coupler, we can obtain ${n_i}$ by ${n_i} = \alpha /{k_0}$. Similarly, through interpolation, the relationship curve of ${n_r}$ and ${n_i}$ with ${w_y}$ for $\theta = 34^\circ $ by the EMT method can be obtained, shown as the red curves in Figs. 4(a) and 4(b). The results of these two methods are significantly different. In this work, 2D FDTD is used to simulate the uniform grating coupler for calculating the ${n_i}$ of the SGU by the EMT method. While 3D FDTD is used to compare the performances of the grating couplers designed by EMT method and our weak form method. The simulation results will distinguish which method is more accurate. All the FDTD simulations were executed by the Lumerical FDTD software (http://www.lumerical.com).
Fig. 4. Comparison of ${n_r} - {w_y}$ curves in subplot (a) and ${n_i} - {w_y}$curves in subplot (b) for $\theta = 34^\circ $ calculated by the weak form method and the EMT method.
Before simulation, we need to generate the layout of the S-ASGC. We set ${z_1} = \textrm{0 }\mathrm{\mu }\textrm{m}$ as the position of the first SGU. Then we can obtain the required ${w_{y,1}}$ and ${p_{z,1}}$ of the first SGU according to ${n_i}({z_1})$, the grating equation Eq. (1) and the relationship curves in Fig. 4. The position of the next SGU is the position of the previous SGU plus its grating period, that is, ${z_{i + 1}} = {z_i} + {p_{z,i}}$. By repeating these iterations, ${z_i}$, ${w_{y,i}}$ and ${p_{z,i}}$ of each SGU can be obtained. Obviously, these two methods will lead to two different sets of ${z_i}$, ${w_{y,i}}$ and ${p_{z,i}}$.
In the 3D FDTD simulation, we set the light to be inputted from the optical fiber to the S-ASGC. A Gaussian light source tilting at $\theta = 34^\circ $ with beam radius ${W_g} = \textrm{7}\textrm{.5 }\mathrm{\mu }\textrm{m}$ is used to represent the light emitted from the optical fiber. And we detect the light coupled into the waveguide to obtain the coupling efficiency. To guarantee adequate simulation accuracy, we set the FDTD mesh sizes as dx=25 nm, dy=25 nm and dz=22 nm. The PML is used as boundary condition to absorb the light propagated to the edge of the FDTD simulation area. Figure 5 shows the coupling spectra of the S-ASGCs designed by the weak form method and EMT method, which are labeled as blue and red lines. Obviously, the spectrum of the S-ASGC by the EMT method has a significant deviation from the C band. Its center wavelength has deviated to 1512.0 nm. And its maximum coupling efficiency is only 44.1% (-3.55 dB). For the S-ASGC by the weak form method, the maximum coupling efficiency is 60.0% (-2.22 dB). And its center wavelength is 1553.1 nm, which accurately locates at the C band. Hence for the design of S-ASGC, our proposed weak form method is more accurate than the traditional EMT method.
Fig. 5. The coupling spectrums of the S-ASGCs designed by the weak form and EMT method, which are labeled as blue and red lines.
3. Focusing subwavelength grating unit (FSGU)
Figure 6(a) shows the layout of the F-ASGC working at C band for TE polarization. Similarly, it can be regarded as the composition of individual FSGUs. Figure 6(b) presents the structure of a FSGU. Different from the SGU, FSGU resembles the shape of sector ring rather than rectangle in yz plane. The FSGU also has the open boundary in x direction. But we cannot directly apply the Bloch periodic boundary conditions on the FSGU in the radial and angular direction. We need to find a way to approximately utilize the Bloch theorem, which is also compatible with our weak form method. Furthermore, ${n_r}$ of every FSGU follows the vector grating equation for the same diffraction angle $\theta $ consistently. The vector grating equation [38] is
Fig. 6. Design the focusing apodized subwavelength grating coupler (F-ASGC) through assembling various focusing subwavelength grating units (FSGUs). (a) The layout of F-ASGC. (b) The structure of FSGU. ${n_r}$ of every FSGU follows the vector grating equation Eq. (11) for the same diffraction angle $\theta $ consistently. The deviation angle between the wave direction of the focusing light ${\hat{{\mathbf n}}_{\mathbf r}}$ and the normal vector of the focusing grating curve ${\hat{{\mathbf n}}_{\mathbf g}}$ is $\xi $.
3.1. Analysis of the FSGU by weak form with transformation optics
Transformation optics utilizes the spatial coordinate transformation to flexibly change the shape of optical device, which is proposed by Pendry et al. [39] and Leonhardt [40]. Based on the form invariance of the Maxwell equations, the transformed device can inherit the same optical performance from the original device after adjusting the original permittivity $\varepsilon $ and permeability $\mu $ to $\varepsilon ^{\prime}$ and $\mu ^{\prime}$ as the equations below [41]:
Using transformation optics, the FSGU can be changed back to a cube, so that we can apply Bloch's theorem on it. And it only needs replacing $\varepsilon $ and $\mu $ with $\varepsilon ^{\prime}$ and $\mu ^{\prime}$, without adjusting the weak form. We set the curvature radius at the center of FSGU as ${R_c}$, so the inner radius is ${R_c} - {p_z}/2$ and the outer radius is ${R_c} + {p_z}/2$. Fortunately, an analytical conformal transformation shown in Fig. 7 can realize the transformation from the sector ring in $({z,y} )$ coordinates to the rectangle in $({u,v} )$ coordinates [42]. This conformal mapping f is ${z_c} = A\exp (mw)$, where ${z_c} = z + iy$ and $w = u + iv$. Through setting two unchanged points $({{R_c} - {p_z}/2,0} )$ and $({{R_c} + {p_z}/2,0} )$, we can determine the values of A and m. Based on the Eqs. (12) and (13), $\varepsilon ^{\prime}$ and $\mu ^{\prime}$ are:
Fig. 7. The transformation from the sector ring in $({z,y} )$ coordinates shown in subplot (a) to the rectangle in $({u,v} )$ coordinates shown in subplot (b), realized by an analytical conformal mapping f. This conformal mapping f is ${z_c} = A\exp ({mw} )$, where ${z_c} = z + iy$ and $w = u + iv$. Through setting two unchanged points $({{R_c} - {p_z}/2,0} )$ and $({{R_c} + {p_z}/2,0} )$, the values of A and m can be determined. The color distribution in subplot (b) presents the profile of the ratio ${\varepsilon ^{\prime}_z}/{\varepsilon _z}$. All the grids in subplots (a) and (b) represent coordinates $({u,v} )$.
Hence, we can calculate the complex band of the transformed FSGU with $\varepsilon ^{\prime}$ and $\mu ^{\prime}$ in $({u,v} )$ coordinates. And the boundaries in u and v direction can be set as the Bloch periodic boundary conditions, avoiding the therotical difficulty mentioned above in $({z,y} )$ coordinates. But the complex effective index ${n_{eff}}^\prime $ in $({u,v} )$ coordinates need to be transformed back to ${n_{eff}}$ in $({z,y} )$ coordinates. The Bloch eigenmode for TE polarization is:
At the center $({{z_0},{y_0}} )$ of the transformed FSGU, we apply the Taylor expansion on the phase term and only leave the terms below the first order derivative. Thus, we have:3.2. Calculation of the FSGU
We take the structure of SGU in Fig. 2(a) as an example, which is ${w_y} = \textrm{0}\textrm{.3 }\mathrm{\mu }\textrm{m}$ and ${p_z} = \textrm{1050 nm}$. Assuming that this SGU is transformed from some FSGUs, we will evaluate the influence of various curvature radius ${R_c}$ and deviation angle $\xi $. Different from before, we set the medium permittivity and permeability as $\varepsilon ^{\prime}$ and $\mu ^{\prime}$ according to Eq. (14) and Eq. (15) to include the effect caused by the curvature radius ${R_c}$. The Bloch vector needs to be set as ${k_x} = 0$, ${k_y} = k\sin \xi $ and ${k_z} = k\cos \xi $ to include the effect caused by $\xi $, where k is set as the eigenvalue. After the weak form calculation, we will have ${n_{eff}}^\prime = k/{k_0}$. Then ${n_{eff}}$ will be obtained based on Eq. (19).
Figure 8 presents the parameter scanning results of ${R_c}$ and $\xi $ respectively. The range of ${R_c}$ is from 5 µm to 30 µm, and the range of $\xi $ is from 0° to 16°. The orange line is ${n_r}$ of the original SGU, which is ${n_r} = 1.9462$. It can be found that ${n_r}$ is almost unaffected if ${R_c} > \textrm{15 }\mathrm{\mu }\textrm{m}$. And the change of ${n_r}$ is less than 0.01 if $\xi < 9^\circ $. Therefore, we can approximately treat the FSGU as the SGU in theory only if these two conditions are fulfilled.
Fig. 8. The effective index ${n_r}$ varied with (a) the curvature radius ${R_c}$ and (b) deviation angle $\xi $ respectively. The orange line is ${n_r}$ of the original SGU, which is ${n_r} = 1.9462$. It can be found that ${n_r}$ is almost unaffected if ${R_c} > \textrm{15 }\mathrm{\mu }\textrm{m}$. And the change of ${n_r}$ will be within 0.01 if $\xi < 9^\circ $.
4. Focusing apodized subwavelength grating coupler
4.1. Focusing grating curves generating
Based on the analysis above, if we fulfill ${R_c} > \textrm{15 }\mathrm{\mu }\textrm{m}$ and $\xi < 9^\circ $, the F-ASGC can be assembled by the SGU directly. We utilize the previous data presented in Fig. 3 without additional weak form calculation. But the layout of the F-ASGC is far more complicated than the S-ASGC’s because of the light focusing function. As shown in Fig. 9(a), the light emitted from the focal point O is coupled to the fiber through the SGUs. Each SGU is arranged along a focusing grating curve. Direct usage of elliptic curve for the F-ASGC is not rigorous in theory. We have proved in the Appendix that the elliptic curve can be used only if ${n_r}$ distribution of the focusing grating coupler is constant.
Fig. 9. (a) The light emitted from the focal point O is coupled to fiber through the SGU. Each SGU of the F-ASGC is arranged along a focusing grating curve. (b) The F-ASGC should follow the apodized form ${n_i}(z)$ at the z axis, which is shown as the orange curve. Through the relationship between ${n_i}$ and ${n_r}$ of the SGU at $\theta = 34^\circ $, we can obtain an equivalent apodized form ${n_r}(z)$, which is shown as the blue curve.
Notice that on the z axis ($\xi = 0^\circ $), the vector grating equation Eq. (11) for the F-ASGC is equivalent to the scalar grating equation Eq. (1) for the S-ASGC. On the other hand, the F-ASGC also need to follow the apodized form ${n_i}(z)$ on the z axis, which is shown as the orange curve in Fig. 9(b). Hence, the F-ASGC and S-ASGC should share the same layout on the z axis. Through the relationship between ${n_i}$ and ${n_r}$ of the SGU at $\theta = 34^\circ $, we can obtain an equivalent apodized form ${n_r}(z)$, which is shown as the blue curve in Fig. 9(b). The value of ${n_r}(z)$ is obviously not a constant, which means the elliptic curve cannot be used. Since the vector grating equation Eq. (11) is the basis of the derivation in the Appendix, we propose to directly generate the focusing grating curves numerically.
To ensure the light wavefront in the F-ASGC is a cylindrical wave, the ${n_r}$ distribution should be varied only along the vertical direction of the wavefront, that is, the radial direction. Therefore, the ${n_r}$ distribution of the entire F-ASGC can be immediately obtained from ${n_r}(z)$ on the z axis. We set the distance from the first grating curve to the focal point O as ${R_0} = \textrm{20 }\mathrm{\mu }\textrm{m}$ and the fan angle of the F-ASGC as $20^\circ $, which can include the mode spot of the entire Gaussian light source and fulfill the conditions of ${R_c} > \textrm{15 }\mathrm{\mu }\textrm{m}$ and $\xi < 9^\circ $. For vector grating equation Eq. (11), we set ${\hat{{\mathbf n}}_{\mathbf g}} = {n_{gz}}\hat{{\mathbf z}} + {n_{gy}}\hat{{\mathbf y}}$ as the normal unit vector of the grating curves. The tangent vector of grating curves is ${\hat{{\mathbf \tau }}_{\mathbf g}} = {\tau _{gz}}\hat{{\mathbf z}} + {\tau _{gy}}\hat{{\mathbf y}}$, which has ${\tau _{gz}} ={-} {n_{gy}}$ and ${\tau _{gy}} = {n_{gz}}$. Note that the length of tangent vector is unit, we can obtain that:
4.2. Simulation and experiment of focusing apodized subwavelength grating coupler
After the layout generation completed, we import the layout of F-ASGC into the 3D FDTD simulation, which is connected to a single-mode waveguide. We set the central position of the Gaussian light source as ${z_{gf}} = {z_g} + {R_0} = \textrm{29 }\mathrm{\mu }\textrm{m}$. The rest of the simulation settings are the same as the S-ASGC’s. Figure 11 presents the electric field ${E_y}$ distribution of the F-ASGC, wherein Fig. 11(a) is the top view and Fig. 11(b) is the side view. The ${E_y}$ profile in the top view demonstrates a good focusing effect, which proves the correctness of our layout generation algorithm. In the side view, the green line shows the position of the Gaussian light source. Most of the light enters the single-mode waveguide on the left, while part of the light continues to propagate downward and forms a reflection on the substrate interface. The reflected light can coherently enhance the light coupled into the waveguide due to the optimized diffraction angle $\theta = 34^\circ $.
Fig. 11. The electric field ${E_y}$ distribution of the F-ASGC. (a) the top view and (b) the side view. The ${E_y}$ in the top view shows a good focusing effect, which proves our layout generation algorithm is correct. In the side view, the green line represents the Gaussian light source. Most of the light enters the single-mode waveguide on the left, while part of the light continues to propagate downward and forms a reflection on the substrate interface. The reflected light can coherently enhance the light coupled into the waveguide due to the optimized diffraction angle $\theta = 34^\circ $.
The simulated spectrum of the F-ASGC is shown as the blue curve in Fig. 12, with maximum coupling efficiency of 56.1% (-2.51 dB), 3dB bandwidth of 51 nm and center wavelength of 1553.1 nm. The performance of the F-ASGC is slightly worse than the S-ASGC. The light cannot be completely converged to the focal point due to the diffraction effect. Thus, there is a small transition loss between the F-ASGC and the single-mode waveguide. Meanwhile, since this design method takes priority to ensure that the wavefront of the focusing cylindrical wave is well preserved, the distribution of ${n_r}$ is a radial gradient. Hence the distribution of ${n_i}$ also changes to a radial gradient. In the region away from the z axis, the profile of the diffracted light will deviate from the Gaussian distribution, which also produces an additional drop in coupling efficiency.
Fig. 12. The simulated and measured spectrums of the F-ASGC. The blue curve is the simulated spectrum of the F-ASGC, while the orange curve is the simulated spectrum of the feature-size restricted F-ASGC. The measured spectrum of this restricted F-ASGC is shown as the yellow curve, which coincides well with the orange curve.
Considering that the feature size of some subwavelength structures in our F-ASGC are below 100 nm, we need to replace them with other SGUs above the 100 nm. Different from the SGU we designed before, we set the ${w_z}$ and ${p_z}$ as the variables while fixing ${w_y} = \textrm{100 nm}$. Through some additional weak form calculations, a F-ASGC with feature size above 100 nm can be formed. The orange curve in Fig. 12 is the simulated spectrum of this restricted F-ASGC with maximum efficiency of 50.7% (-2.95 dB), 3dB bandwidth of 58 nm (1 dB bandwidth of 34 nm) and center wavelength of 1555.1 nm.
The electron-beam lithography and inductively coupled plasma (ICP) dry etching were applied to fabricate this restricted F-ASGC on a commercial SOI wafer with 220 nm thick top silicon layer and 3 µm thick buried oxide (BOX) layer. Figures 13(a) and 13(b) show the overall and magnified SEM photographs of this restricted F-ASGC respectively. The measured spectrum is shown as the yellow curve in the Fig. 12 with maximum efficiency of -3.10 dB, 3dB bandwidth of 55 nm (1 dB bandwidth of 28 nm) and center wavelength of 1551.5 nm. The measured spectrum coincides well with the simulated spectrum.
Fig. 13. SEM photographs of the restricted F-ASGC (a) the overall view; and (b) the magnified view of the subwavelength grating.
Table 1 compares the performances of the reported TE polarization focusing grating couplers on silicon platform. The designs of previous focusing grating couplers rely on time-consuming optimization process, while our method is deterministic. The advantages of our design method lie in three aspects. First, the weak form method is more accurate than the traditional EMT method. We can directly obtain accurate SGU parameters, without the need of numerical optimization processes for searching and calibration. Second, we achieve a rigorous analysis of FSGU by associating weak forms with transformation optics. According to the analysis, we can choose the appropriate F-ASGC size parameters to avoid performance degradation. Third, we propose an algorithm to generate F-ASGC layout from vector grating equation.
Table 1. Comparison of several reported TE polarization focusing grating couplers on silicon platform.a
5. Conclusion
In this paper, we propose a deterministic design method of F-ASGC. The grating coupler is composed of various SGUs which can be accurately designed by weak form calculation. We proved that our proposed weak form method is much more accurate than the traditional EMT method for grating coupler design. This advantage is very important for removing the abundant numerical optimizing process to correct the deviation of peak wavelength and degradation of coupling performance. Through associating the weak form with transformation optics, we realize the rigorous analysis of the FSGU for the first time. From the vector grating equation, we proposed an algorithm to generate the layout of F-ASGC. The simulation results show that the maximum coupling efficiency is 56.1% (-2.51 dB) with 3dB bandwidth of 51 nm. After restricting the feature size of the subwavelength structures to more than 100 nm, we fabricated this grating coupler on the commercial SOI wafer. The measured maximum coupling efficiency is -3.10 dB with 3dB bandwidth of 55 nm. The measured spectrum is in good agreement with the simulated spectrum. Our weak form method offers a useful tool for designing focusing grating couplers rapidly and accurately. And this design method can be extended to other kinds of material platform, such as silicon nitride, lithium niobate, indium phosphate and so on. Furthermore, the combination between weak form and transformation optics will inspire more designs and applications for the photonic crystals in the future.
Appendix
Elliptic curves for uniform focusing grating
Assuming that the function of the m-th grating curve is $({{z_m}(\varphi ),{y_m}(\varphi )} )$, each SGU on the grating curve satisfies the vector grating equation:
We express this curve in a polar coordinate form, there is ${z_m}(\varphi ) = {r_m}(\varphi )\cos \varphi $ and ${y_m}(\varphi ) = {r_m}(\varphi )\sin \varphi $. Hence, we can obtain:
Funding
Wuhan National Laboratory for Optoelectronics (Innovation Fund); National Natural Science Foundation of China (11504435, 61975062); National Key Research and Development Program of China (2019YFB2205202).
Acknowledgments
We thank Dr. Cheng Zeng and the engineer Mr. Pan Li in the Center of Micro-Fabrication and Characterization (CMFC) of WNLO for the support in electron-beam lithography and ICP dry etching.
Disclosures
The authors declare no conflicts of interest.
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