1. Introduction

2. Materials and Methods

2.3. Light Coupler Design

Size mismatch between the optical fiber (i. e. SMF-28) and photonic waveguide is significant, Figure 3. In our specific setup, the former have a core diameter of 8.2 µm and the later a core height (thickness) of 0.22 µm. The optical fiber mode field diameter is approximately 10.4 µm.

Several coupling schemes can be employed to overcome this issue, we highlight [33]:

• Adiabatic couplers [33,35];
• Grating couplers [42,43,44,45];
• Lensed fibers [46];
• Spot size converters [33,47,48,49,50].

The 1st and 3rd methods, Figure 4(a) and (c), require tailor-made optical fiber terminations which demand fiber manipulation techniques, needing expensive equipment that are not commonly found in most optical laboratory setups. The 1st, 4th and 5th schemes, Figure 4(a), (d) and (e) typically require a very good matching between the optical fiber termination and the waveguide’s coupling region(s). These designs require very high precision lithographic equipment (e. g. based on Electron Beam), which is very expensive and not compatible with CMOS technology. Some spot size converters also rely on vertical structures which can be very complex and expensive to manufacture, like for example 3D tapers [48,49,50], Figure 4(d) and vertical couplers [46]. Unlike its crystalline counterparts, amorphous silicon cannot be subjected to anisotropic wet etching [36], making the development of 3D structures more complex. Some grating couplers can have minimum feature sizes on the orders of dozens of nanometers or even a few hundred nanometers, making these devices feasible by photolithographic processes, such as deep UV (DUV) lithography [42,51], employing excimer lasers.

There are several variations of grating couplers [52,53], specifically designed to improve coupling efficiency [42,43,54,55,56] and bandwidth [56,57,58], such as:

• Dual-level grating couplers [43];
• Gratings featuring stacked layers [53,59];
• Grating couplers with distributed Bragg reflectors (DBR) [57,58,60];
• Grating couplers based on metamaterials [44];
• Gratings with bottom metal reflector [44,61,62,63];
• Grating couplers with bottom grating reflector [56].

Designs presented in Figure 5(a), (b), (c), (e) and (f) can achieve remarkable performance, however, some of these (b), (c), (e) and (f) require multiple deposition steps, increasing fabrication complexity and cost. Gratings based on metamaterials, Figure 5(d) typically require a specific set of equipment that is not found in most typical semiconductor laboratory setups.

To simplify the etching process, we designed a fully etched grating coupler (i.e., the silicon layer is etched to the substrate/under cladding) in alternative to the more common partially etched designs, which require two lithographic processes/masks [42,58,65] or even more [43,51,56]. With the increased number of lithographic masks, misalignment impacts must be studied, since these can have an impact on the grating coupler’s performance [43].

The main design parameters of a grating coupler are: period ($\mathsf{\Lambda }$), coupling angle ($\theta$) and fill-factor ($F$). A grating coupler can have a constant effective refractive index (fill-factor and period) throughout its length, in a non-apodized design, or have a variation of the effective refractive index (fill-factor, period or both) over its span in an apodized variant.

Figure 6. Simplified representation of an optical fiber to silicon waveguide light coupling scheme, showing some of the grating coupler design parameters, coupling angle (θ), grating period ($\mathsf{\Lambda }$), etched length ($L_E$) and unetched length ($L_U$). This figure does not represent a functional coupling scheme and is not up to scale (e. g. the size mismatch between the optical fiber and the grating coupler is significantly greater).

Figure 6. Simplified representation of an optical fiber to silicon waveguide light coupling scheme, showing some of the grating coupler design parameters, coupling angle (θ), grating period ($\mathsf{\Lambda }$), etched length ($L_E$) and unetched length ($L_U$). This figure does not represent a functional coupling scheme and is not up to scale (e. g. the size mismatch between the optical fiber and the grating coupler is significantly greater).

The fill factor corresponds to the ratio between the unetched length ($L_U$) and the grating period ($\mathsf{\Lambda }$), which can be expressed as a percentage (1):

$$F=\frac{L_U}{\mathsf{\Lambda }}.$$

(1)

The effective refractive index of a diffraction grating coupler ($n_{eff}$) can be obtained from its fill-factor ($F$), the refractive index of the etched ($n_E$) and unetched segments ($n_U$) of the grating, as described in expression (2) [54]:

$$n_{eff}=F\times n_U+\left(1-F\right)\times n_E,$$

(2)

in this case, $n_E$ and $n_U$, correspond to the refractive indexes of air and of the waveguide’s fundamental mode of propagation, the transverse electric (TE) mode. Having the effective refractive index, $n_{eff}$, it is possible to calculate the required grating coupler period ($\mathsf{\Lambda }$), for a given coupling angle (θ), typically represented in degrees and free space wavelength (${\lambda }_0$) as described in equation (3) [66]:

$$\mathsf{\Lambda }=\frac{{\lambda }_0}{n_{eff}-\sin \theta },$$

(3)

the resulting grating period has the same units as the free space wavelength, typically both are represented in units of nanometer (nm).

Having the main design expressions, it is possible to begin the design of a grating coupler in its simplest form. Defining the fill-factor ($F$) has 0.5 (50%), while trying to keep the value of the denominator of expression (3) as low as possible, to achieve a higher period and consequently a large feature size. These measures can significantly decrease production costs if successfully implemented. The grating’s refractive index, $n_{eff}$, cannot be varied without changing the materials composing the grating coupler, waveguide’s height or width, fill-factor or etch depth, so this value is regarded as a constant. The coupling angle, $\theta$, was set to a very high value of 22.5°. Operating wavelength is 1550 nm, a typical value used in optical fiber communications, situated close to the middle of the optical C-band. Since the grating is fully etched, without upper cladding and simulated in an environment filled with air, $n_E$ can be assumed as 1, $n_U$ was obtained using the Finite Element Method (FEM), having a value of approximately 2.8372. The effective refractive index of the grating coupler can now be obtained from (2), having a value of about 1.9186. In the presence of all variables it is now possible to calculate the grating’s period ($\mathsf{\Lambda }$), equation (3), which is approximately 1009.2 nm (1.0092 µm), resulting in a feature size of about 505 nm, a value close to the required width of a silicon single-mode strip waveguide operating at 1550 nm [30,31]. A grating coupler with these characteristics has a relatively narrow diffracted field. The grating coupler was simulated using the 2D-FDTD method and the coupling efficiency (SMF-28 optical fiber) was average, measuring under -9 dB. There was some discrepancy in the value of the coupling angle which was found (in simulation) to be about 16.5° for optimum coupling.

2.5. Linear Refractive Index Variation

A linear variation of the grating’s refractive index can be achieved by varying both the period and fill-factor over the grating coupler’s length, Figure 8, while maintaining the coupling angle constant.

The expression for the linear variation of the effective refractive index over a grating’s length can be written as (5):

$$n_{eff}\left(z\right)=n_{eff}\left(0\right)+m\times z,$$

(5)

where $n_{eff}\left(0\right)$ is the effective refractive index at the start of the grating coupler (the effective refractive index of the first grating period), $m$ corresponds to the slope of the refractive index variation, in µm^-1 and $z$ is the position measured from the start of the grating.

Following optimization in 2D-FDTD software, expression (6), was obtained, for a 220 nm thick fully etched grating coupler of hydrogenated amorphous silicon (a-Si:H), operating at a wavelength of 1550 nm:

$$n_{eff}\left(z\right)=2.65348-0.0288z.$$

(6)

The value of $n_{eff}\left(0\right)$, 2.65348, was obtained considering a maximum fill-factor of 0.9 (90%), an etched section refractive index, $n_E$, of 1 and an unetched section effective refractive index, $n_U$, of 2.8372, refer to expression (2). The slope was adjusted considering a correlation which was observed between the length of the diffracted field distribution (gaussian like function) and the total variation of the refractive index over the grating coupler’s length. It was found on 2D-FDTD simulation, that for maximum coupling efficiency, the variation of the refractive index, $∆n_{eff}$, over a length of one optical fiber MFD (10.4 µm), $∆z$, had to be of about -0.3, resulting in a slope, $m$, of approximately -0.0288 µm^-1. Refer to expression (7) and plot of Figure 9.

$$m=\frac{∆n_{eff}}{∆\mathrm{z}}\left[{\mu m}^{-1}\right].$$

(7)

The plot of function (6) is represented in Figure 9, for a 20 µm grating coupler.

Figure 9. Proposed linear variation of the refractive index over the grating coupler’s length (represented in red).

Figure 9. Proposed linear variation of the refractive index over the grating coupler’s length (represented in red).

Accounting for the variation of the refractive index over the length of the grating coupler it is possible to rearrange (2), to obtain the fill-factor for each grating position (8):

$$F\left(z\right)=\frac{n_{eff}\left(z\right)-n_E}{n_U-n_E}.$$

(8)

The expression which gives the grating period (3) can also be rewritten to account for the variation of the effective refractive index over the grating’s length (9):

$$\Lambda \left(z\right)=\frac{{\lambda }_0\left[\mu m\right]}{n_{eff}\left(z\right)-\sin \theta }[\mu m].$$

(9)

A computational algorithm, implemented in MATLAB (The MathWorks, Inc.), obtains the effective refractive index values at the begin of each grating period, using expression (6) and calculates the fill factor, F from function (8), and period, Λ, using expression (9) at each iteration, until the end of the grating coupler. The diffraction angle was defined as 15°. The pseudo-code algorithm is presented in Appendix A. The resulting grating coupler is presented in Figure 10. The length can overshoot the predefined value, in this case, for example, it is approximately 20.6 µm, instead of 20 µm, it was decided that it is preferable to have a slightly longer grating coupler than the opposite.

The fill-factor (F) and period (Λ) for each grating segment, numbered from the beginning to the end of the grating coupler (or from left to right in Figure 10), considering a total of 28 segments. are presented in Table 2.

3. Results

3.7. Overlapped Micrometric Design

The coupling efficiency results for the overlapped micrometric period gratings (1500 nm) are presented herein. The two superimposed gratings with an offset of -400 nm form a grating coupler with a period of 1500 nm and a fill-factor of 0.7(6) (76.(6) %). In Figure 29, the contour map of the diffracted electric field is presented.

Two monitors are included, one centered at position (x, z) = (0, 5), measuring the power near the output of the waveguide and another centered at position (x, z) = (6, 11.9), which measures the overlap between the diffracted field and the fundamental mode of the optical fiber. This second monitor is placed at an angle of 22° (optimized coupling angle), its center is at a distance of 6 µm from the grating center axis (due to the higher coupling angle) and at a distance of 5.9 µm measured from the grating coupler’s starting point. The access waveguide is 6 µm long.

Figure 29. Representation on the XZ plane of the diffracted electric field, the green (transparent filled) horizontal bar is the waveguide’s monitor and the slanted green (transparent filled) slanted bar is the diffracted field monitor. Image obtained from 2D-FDTD.

Figure 29. Representation on the XZ plane of the diffracted electric field, the green (transparent filled) horizontal bar is the waveguide’s monitor and the slanted green (transparent filled) slanted bar is the diffracted field monitor. Image obtained from 2D-FDTD.

In Figure 29, it is possible to notice that the diffracted field intensity (exponentially) decays along the coupler’s length, a characteristic of non-apodized gratting couplers. By optimizing the fill-factor, monitor angle and position, it was possible to maximize the coupling efficiency.

In Figure 30, coupling efficiency results are shown, it is possible to attest that the waveguide monitor overlap (with the launch field, or fundamental quasi-TE₀₀ mode), remains nearly constant throughout the integration time. Coupling efficiency achieved at the time of convergence is slightly over -7.5 dB (~ 20.0%). The overlap with the diffracted field (i.e., the percentage of power of the diffracted field passing through the monitor which is used to excite the SMF-28 fundamental mode) is approximately 80.8%. The ratio between the diffracted power reaching the monitor and waveguide’ power is slightly under -6.5 dB (~ 22.3%).

Coupling efficiency was measured for wavelengths in vacuum between 1530 nm and 1565 nm, Figure 31. The -1 dB bandwidth is approximately 22 nm. Peak efficiency (center wavelength) occurs at 1549 nm. The optical C-band (1530-1565 nm) is covered with a maximum 6.3 dB penalty.

4. Discussion

5. Conclusions

6. Future Work and Improvement

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