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• Abstract

SECTION 1

## INTRODUCTION

Mid-infrared (mid-IR) silicon photonics, especially in the range of 2–6 $\mu\hbox{m}$, has potential applications in chemical sensing, spectroscopy, the study of kinetic processes in drug delivery, and defense applications [1], [2]. Since two-photon absorption (TPA) does not occur in the mid-IR wavelength band [3], [4], [5], [6], it may be possible to exploit nonlinear optical effects for device applications without any limitations from absorption of TPA-generated free carriers. However, unlike the situation in the telecommunication band, the strong absorption of silicon dioxide above 2.6 $\mu\hbox{m}$ limits the performance of conventional silicon-on-insulator (SOI) wafers in the mid-IR spectral range [7], [8]. Silicon-on-sapphire (SOS) was therefore proposed as an alternative platform for integrated silicon photonics for the mid-IR wavelength [9], [10], [11], [12]. Integration of waveguides and ring resonators has been demonstrated on SOS wafers [10], [11]. Optical loss of about 1.9 dB/cm was recently reported for SOS waveguides at 5.08 $\mu\hbox{m}$ wavelength [12].

Butt-coupled technique has been employed for fiber-chip coupling for mid-IR studies [10], [11], [12]. Despite the relaxed alignment tolerance made possible by the longer wavelengths, the coupling efficiency, however, is still much lower than the results achieved in telecommunication band. Mechanical polishing of SOS waveguide facets is also typically needed for end-fire coupling.

Waveguide grating couplers are an alternative technique for fiber-chip coupling and have already been well developed for applications in the telecommunication band. They offer the advantages of simpler back-end processing, potentially high coupling efficiency and convenience for coupling light in/out at any location on the chip [13], [14], [15], [16]. However, there has to date been no detailed theoretical study on the coupling efficiency that may be achieved at mid-IR wavelengths, where the longer wavelengths will introduce the additional challenge of having even fewer periods of waveguide grating for the same fiber core diameter of 9 $\mu\hbox{m}$.

In this paper, we present the theoretical analysis of the shallow-etched uniform grating, full-etched subwavelength grating, and apodized grating for coupling to the transverse-electric (TE) mode and transverse-magnetic (TM) mode SOS waveguides at 2.75 $\mu\hbox{m}$ wavelength in detail.

For TE mode gratings, theoretical simulation shows that the directionality (defined as the percentage of up-coupled light toward fiber over the total diffracted light power) of shallow-etched grating is much larger than that of full-etched subwavelength grating. After optimization, the finite-difference time-domain (FDTD) simulations predict a coupling efficiency of 80.6% from a shallow-etched apodized grating. Despite the longer wavelengths and fewer periods of grating for fiber-waveguide coupling, the theoretical coupling efficiency is only marginally lower than the 84% predicted for apodized near-infrared grating couplers [16].

For TM mode waveguides, one problem is the large lateral leakage of light power into the slab modes in the rib waveguide [17] can introduce wavelength-dependent loss. Hence, strip waveguide is a better choice for TM mode applications. In order to fabricate the grating couplers with the same full-etched process as the waveguide devices, we use subwavelength structure to engineer the coupling strength while reducing the back reflection [18], [19], [20]. The simulation results show that the coupling efficiency of full-etched subwavelength grating is comparable with shallow-etched ones. With the apodized subwavelength structure, the maximum TM mode coupling efficiency of 39.6% is predicted theoretically.

In experiments, we used an $\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped mid-IR fiber laser for characterization of SOS devices. The TE mode shallow-etched uniform grating and TM mode full-etched subwavelength grating are fabricated using standard semiconductor processing techniques. Dependence of coupling efficiency on etch depth, grating period, fill factor (defined as the ratio between etched length and grating period), and incident angle are also investigated both experimentally and theoretically.

SECTION 2

## DESIGN AND FABRICATION

### 2.1. Simulation Results of Shallow-Etched 1-D Grating

The SOS wafers we used consist of 600 nm thick silicon layer on top [10], [11]. Shallow-etched uniform 1-D grating is etched on the 10-$\mu\hbox{m}$-wide silicon waveguide, as shown in Fig. 1(a). The angle between fiber axis and the surface normal is denoted as $\theta$ ($\theta = 10^{\circ}$ for the following simulations). The grating parameters can be determined by the phase matching condition [18] and the coupling efficiency can be calculated by evaluating the overlap integral [14]. Two-dimensional FDTD simulations are carried out with the perfect matched layer absorbing boundary condition using Rsoft (www.Rsoft.com).

Fig. 1. (a) Schematic illustration of grating for coupling between photonics wire waveguide and fiber. (b) Top view of subwavelength grating couplers. Square-shape subwavelength structure is replaced by nanoholes with the same area. (c) Scanning electron microscope (SEM) image of shallow-etched uniform grating on the 10 $\mu\hbox{m}$ wide waveguide. (d) SEM image of shallow-etched uniform grating with 405 nm etch depth, 0.4 fill factor, and 1120-nm period. (e) SEM image of full-etched nanoholes subwavelength grating on the 10-$\mu\hbox{m}$-wide waveguide. (f) SEM image of full-etched nanoholes subwavelength grating with 600-nm etch depth, 253-nm nanoholes radius, and 1250-nm period.

As illustrated in Fig. 2(a), the maximum directionality of TE mode shallow-etched uniform grating couplers is obtained with an etch depth of 380 nm. The maximum achievable coupling efficiency is 46.8% with $\sim$140 nm 1 dB bandwidth, using uniform grating when the fill factor $f_{y} = 0.6$, as plotted in Fig. 2(c). To further improve the coupling efficiency, apodized grating can be used to form a Gaussian-shaped profile of the diffracted light through engineering the coupling strength $\alpha (y)$, which is given by the following equation [14], [16]: TeX Source $$\alpha (y) = {0. 5G^{2} (y) \over 1 - \int \limits_{0}^{y} G^{2} (t) dt}\eqno{\hbox{(1)}}$$ where $G(y)$ is a normalized Gaussian field profile. The diameter of mid-IR fiber core is 9 $\mu\hbox{m}$ (provided by IRphotonics) with a mode field diameter of 10.6 $\mu\hbox{m}$ at wavelength of 2.75 $\mu\hbox{m}$ (simulated by Rsoft). According to (1), the coupling strength required along the $y$-axis of the grating coupler is calculated and plotted in Fig. 3(a). The coupling strength can be tuned by changing the fill factor ${f}_{y}$. Thus, by fixing the etch depth at 380 nm, we obtain the coupling strength $\alpha (z)$ for different fill factors, as presented in Fig. 2(b). The coupling strength of mid-IR SOS gratings can be tuned from 0.011 to 0.16 by changing the fill factor from 0.05 to 0.42 at 2.75 $\mu\hbox{m}$ wavelength. If we further increase fill factor, the coupling strength begins to decrease slowly.

Fig. 2. (a) Dependence of directionality on etch depth and fill factor. (b) The dependence of coupling strength on fill factor. (c) TE mode coupling efficiency and back reflection as a function of wavelength for various designs by simulations. (d) TM mode coupling efficiency and back reflection as a function of wavelength.
Fig. 3. (a) Calculation to achieve Gaussian-shape output profile with engineered coupling strength. (b) The mode profile from ZBLAN fiber at 2.75 $\mu\hbox{m}$ and output diffraction beam profiles from grating. (c) Coupling efficiency of the TE mode apodized grating as a function of incident angle.

We designed the grating structure according to the trend of theoretical results. However, the maximum coupling strength achievable at 380 nm etch depth is lower than the theoretical requirement. In order to diffract as much power out as possible from the grating, we use apodized structure in the front section of the grating, and uniform gratings with maximum coupling strength for the rear section. The grating period (nm) and fill factor (in the bracket) we used are 970 (0.05), 977 (0.06), 1001 (0.12), 1017 (0.16), 1052 (0.24), 1073 (0.29), 1101 (0.36), and 18 periods of uniform grating with 1124 nm period and 0.42 fill factor, as presented in Fig. 3(a).

After optimization $(\theta = 12. 3^{\circ})$, the back reflection and transmission are suppressed to 0.54% and 0.41%, respectively. The coupling efficiency can be improved to 80.6% between SOS chip and fiber as present in Fig. 2(c), which is corresponding to an overlap of 95.4% between the diffracted light and Gaussian-shape profile, as shown in Fig. 3(b). This result is comparable to the result obtained in the near-infrared range [16]. The optical field by FDTD simulation is presented in Fig. 4(b).

Fig. 4. Simulated output field of (a) a uniform grating coupler and (b) an apodized grating coupler.

The performances of TM mode uniform shallow-etched grating couplers are also simulated as shown in Fig. 2(d). The maximum directionality can be obtained with an etch depth of 300 nm. After optimization, the maximum coupling efficiency of TM mode shallow-etched uniform grating only reaches to 31.7% with a 300-nm etch depth and 0.4 fill factor, when the diffraction angle is set to be 10 °.

### 2.2. Simulation Results of Full-Etched Subwavelength Gratings

At 2.75 $\mu\hbox{m}$ wavelength, the effective refractive indexes (RIs) of fundamental TE and TM polarization mode for a 600-nm-thick silicon waveguide layer are calculated as $n_{{TE}0{\rm eff}} = 3.07$ and $n_{{\rm TM}0{\rm eff}} = 2.74$ using effective index method (EIM), and the corresponding lateral period $(\Lambda_{X})$ of subwavelength structure should, therefore, be less than 890 nm for the TE mode and less than 1000 nm for the TM mode, respectively. Three different lateral periods are selected, and the RIs of subwavelength grating structure with different fill factors in $x$-direction $(f_{x})$ are calculated with (2) and (3), which are the zeroth- and second-order approximation of the effective medium theory (EMT) [21] TeX Source \eqalignno{n_{\rm TM}^{(2)} = &\,n_{\rm TM}^{(0)}\left[1 + {\pi^{2} \over 3}\left({\Lambda_{x} \over \lambda} \right)^{2}f_{x}^{2}(1 - f_{x})^{2}\left(n_{\rm silicon}^{2} - n_{\rm hole(air)}^{2}\right)^{2}{1 \over \left(n_{\rm TM}^{(0)}\right)^{2}}\right]^{{1 \over 2}}\cr\noalign{\vskip4pt} n_{\rm TE}^{(2)} = &\,n_{\rm TE}^{(0)}\left[1 + {\pi^{2} \over 3}\left({\Lambda_{x} \over \lambda}\right)^{2}f_{x}^{2}(1 - f_{x})^{2}\left(n_{\rm silicon}^{2} - n_{\rm hole (air)}^{2}\right)^{2}\left(n_{\rm TM}^{(0)}\right)^{2}\left({\left(n_{\rm TE}^{(0)}\right)^{2} \over n_{\rm silicon}^{2}n_{\rm hole (air)}^{2}}\right)^{2}\right]^{{1 \over 2}}&\hbox{(2)}} where TeX Source $$n_{\rm TM}^{(0)} = \left[f_{x}n_{\rm hole (air)}^{2} + (1 - f_{x})n_{\rm silicon}^{2}\right]^{{1 \over 2}},\quad {1 \over n_{\rm TE}^{(0)}} = &\,\left[{f_{x} \over n_{\rm hole (air)}^{2}} + {(1 - f_{x}) \over n_{\rm silicon}^{2}}\right]^{-{1 \over 2}}\eqno{\hbox{(3)}}$$ where $n_{\rm TM}^{(0)}$ and $n_{\rm TE}^{(0)}$ are TM mode and TE mode RIs derived by the zeroth-order approximation, $n_{\rm TM}^{(2)}$ and $n_{\rm TE}^{(2)}$ are TM mode and TE mode RIs derived by the second-order approximation, $\lambda$ is the center wavelength of grating coupler. $n_{\rm silicon} = 3. 43$ and $n_{\rm hole (air)} = 1$ are calculated by the Sellmeier equation.

The calculation results of zeroth- and second-order approximation of EMT are plotted in Fig. 5(a). For the zeroth-order EMT, the RI is independent with $\Lambda_{X}$, while the RI of second-order EMT depends on both $\Lambda_{X}$ and ${f}_{x}$, which is more accurate than the zeroth-order approximation. The calculation shows that for a given subwavelength period, the RI of subwavelength grating structure can be tuned from 1 to 3.43 by changing the lateral fill factor ${f}_{x}$. In the following simulation, we will fix $\Lambda_{X} = 700 \ \hbox{nm}$ and engineer the RI by ${f}_{x}$. The 2-D FDTD simulation results are shown in Fig. 5.

Fig. 5. (a) RIs of subwavelength structure with different lateral fill factors. (b) The dependence of directionality and coupling strength on fill factors for TM mode gratings. (c) The dependence of coupling efficiency and back reflection of TM mode uniform subwavelength gratings on longitudinal fill factors. (d) Simulation results of TM mode apodized subwavelength gratings and square-shape subwavelength gratings (W is the side length).

After optimization, the maximum coupling efficiency of TE mode uniform subwavelength grating coupler can only reach 33.0%, which is lower than the shallow-etched uniform grating coupler, mainly because of the low directionality (less than 50%). However, the coupling efficiency of TM mode uniform subwavelength grating coupler is 32.7% with a 1 dB bandwidth of $\sim$130 nm, which is comparable with the uniform grating with additional shallow-etched step, as presented in Fig. 5(b).

In order to further improve the coupling efficiency of TM mode subwavelength grating, apodized structure is applied. The lateral period and fill factor are fixed as 700 nm and 0.7; the coupling strength can be engineered by tuning the longitudinal fill factor $({f}_{y})$, as presented in Fig. 5(b). According to the (3), we choose the periods (nm) and fill factors (in the bracket) as 1108 (0.05), 1126 (0.09), 1202 (0.22), 1244 (0.29), 1285 (0.35), 1322 (0.42), 1362 (0.47), and 1415 (0.55), followed by the uniform grating 1450 (0.6). With an incident angle of 12.2°, the maximum coupling efficiency can be improved to 39.6% corresponding to 94.3% power overlap and 0.33% back reflection. The optical field is plotted in Fig. 6(b).

Fig. 6. Simulated output field of (a) the uniform subwavelength grating and (b) the apodized subwavelength grating.

The maximum coupling efficiency is 10% lower than that obtained in the near-infrared spectral range [18]. This is because the directionality of the subwavelength structure on SOS is lower than that on SOI. One reason for the lower directionality is the RI of sapphire is $\sim$1.72 at 2.75 $\mu \hbox{m}$, which is larger than the buried oxide ($\sim$1.44) at 1.55 $\mu\hbox{m}$. Moreover, there is power reflection on the surface between silicon substrate and buried oxide for SOI.

For the ease of fabrication, we use the 2-D array of circular nanoholes instead of square-shape structures to reduce the fabrication error. This will introduce insignificant difference according to our pervious results [17]. The approximation is illustrated in Fig. 1(b). In Fig. 5(d), the coupling efficiency and back reflection of subwavelength grating with square-shape holes are presented. We also fix $\Lambda_{X} = 700 \ \hbox{nm}$ and choose the square with side length of 450 nm and 550 nm (denoted as $W$), respectively. The IR of lateral subwavelength structure is calculated by (2) and (3) first. Then, the corresponding ${f}_{y}$ and period $(\Lambda_{y})$ are calculated by phase matching condition, and the performance of subwavelength grating with the square-shape structure is simulated by 2-D FDTD. As shown in Fig. 5(d), the maximum coupling efficiency can be achieved is 32.5% with 3.4% back reflection into waveguide, when the diffraction angle is set to be 10°.

### 2.3. SOS Waveguides Design and Fabrication

Fig. 1(a) shows the device structure and experiment setup. The grating couplers are connected to a 0.1-mm-long single mode waveguide through 1-mm-long tapers. The waveguide modes are calculated using the finite element method (FEM) mode solver. The waveguide with cross-sectional dimensions of 850 nm × 600 nm can only support the fundamental TE and TM mode. The higher order modes are cutoff at wavelength of 2.75 $\mu\hbox{m}$, as shown in Fig. 7(a).

Fig. 7. (a) Effective refractive index if SOS waveguide with different waveguide widths, solved by FEM solver. (b) The single mode waveguide used to connect two grating couplers. (c) The cross-sectional SEM of an end-cleaved SOS waveguide.

The SOS waveguides and grating couplers are fabricated using electron-beam lithography followed by dry etching using C4F8 and SF6 gas in an inductively coupled plasma chamber (Oxford Instruments). Top view SEM images of grating couplers and the cross-sectional SEM image of SOS waveguide are shown in Figs. 1(c)(f) and 7(c), respectively.

SECTION 3

## EXPERIMENTAL RESULTS

$\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped zirconium, barium, lanthanum, aluminum, and sodium fluoride (ZBLAN) fiber laser has been demonstrated to be a competitive light source in the mid-IR range of 2.7–2.9 $\mu\hbox{m}$, because of its high output power, stability and widely tunable range [22], [23], [24]. In our experiment, commercially available single mode optical fibers made from fluorozirconate glass, which is formed by a mixture of ZBLAN were used to create and transport the mid-IR light. The experimental setup is shown in Fig. 8. An $\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped ZBLAN fiber (IRphotonics, Inc.) was end-pumped by a 976 nm multimode laser diode as the light source. The concentration of $\hbox{Er}^{3+}$ and $\hbox{Pr}^{3+}$ ion in the core of the $\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped ZBLAN double-cladding fiber are 30 000 ppm and 5000 ppm, respectively. The ZBLAN fiber has 300 $\mu\hbox{m}$ outer cladding diameter, 140 $\mu\hbox{m}$ inner cladding diameter, and 9 $\mu\hbox{m}$ core diameter. Sapphire lenses were used to focus light into and out from the ZBLAN fiber. Single-end backward pumping structure with Fresnel reflection ($\sim$4%) against one fiber end was adopted in the experiment and fiber length in the cavity is $\sim$1.5 m. A dichroic mirror, which has a reflectivity of 0.1% at 2.75 $\mu \hbox{m}$ and a reflectivity of 90.6% at 976 nm, was set with 10° angle for free space coupling of the mid-IR light into an undoped single mode ZBLAN fibers. The mid-IR laser was measured by a thermoelectrically cooled HgCdZnTe detector (PVI-2TE-4 Vigo System) with a bandwidth of 50 MHz as a reference signal after filtering by a monochromator (WDG30, Beijing Optical Instrument Factory). Then, the light was coupled into/out from the SOS waveguides through grating couplers to determine the coupling efficiency. The loss of waveguide was neglected, because the waveguide length between two gratings is quite short (0.1 mm). To enhance the signal to noise ratio, the mid-IR signal was modulated to 1480 Hz by a mechanical chopper and detected by a lock-in amplifier. A computer with a LabView (National Instruments) script was connected to the lock-in amplifier and used to collect data.

Fig. 8. Experimental setup. 1 Laser Diode. 2 Fiber holder. 3 Collimated lens. 4 Chopper. 5 Dichroic mirror. 6 Sapphire lens. 7 $\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped ZBLAN fiber. 8 Un-doped ZBLAN fiber. 9 SOS Sample. 10 Monochromator. 11 Mid-infrared detector. 12 Oscilloscope. 13 Lock-in amplifier. 14 Computer.

In the following experiment, the slits of monochromator were fixed as 40 $\mu\hbox{m}$. After filtering by monochromator, the mid-IR laser power of $\sim\!\! 2.4 \ \mu\hbox{W}$ was measured with noise level of $\sim$0.01 nW under the lock-in amplifier integration time of 100 ms. Fifty-four TE mode shallow-etched uniform gratings and twenty-four TM mode full-etched subwavelength gratings with different periods and fill factors were tested experimentally.

Fig. 9 shows experimentally measured coupling efficiency of TE mode shallow-etched uniform grating couplers with different etch depths, fill factors, and periods. It can be seen that we can achieve higher coupling efficiency by increasing etch depth and fill factor. Under the 10 ° incident angle, the maximum coupling efficiency we measured is 32.6% with 405 nm etch depth, 0.6 fill factor, and 1190 nm period. Moreover, the measured curve is flatter with 0.6 fill factor, which means that the optical bandwidth with fill factor at 0.6 is larger than that when $f_{y} = 0.4 \ \hbox{and} \ 0.5$. It agrees well with our simulation results.

Fig. 9. Experimentally measured coupling efficiency for TE mode shallow-etched uniform gratings with different etch depths, fill factors, and periods. (a) $\hbox{Etch depth} = 300 \ \hbox{nm}$. (b) $\hbox{Etch depth} = 355 \ \hbox{nm}$. (c) $\hbox{Etch depth} = 405 \ \hbox{nm}$.

The preliminary experimental results of nanoholes subwavelength grating couplers with four different radii are presented in Fig. 10(a). With the incident angle equals to 10 °, the coupling efficiency is 9.0% with 253 nm radius and 1270 nm period. The performance of grating couplers is also measured under different incident angles, as shown in Fig. 10(b). The maximum coupling efficiency is demonstrated to be 11.6% with the incident angle equals to 12.5°. The experimental measurement for TM mode gratings is $\sim$20% lower than the simulation results. We believe the difference mainly comes from the additional loss induced by the TM mode grating taper, because the mode conversion from the fundamental TM-mode to high-order TE-mode may exist in the asymmetrical waveguide taper [22].

Fig. 10. (a) Experimental measured coupling efficiency of TM mode subwavelength nanohole gratings with different radii and periods. (b) Experimental measured and simulated coupling efficiency with various fiber incident angle $\theta$, when ${R} = 253 \ \hbox{nm}$ and $\Lambda_{y} = 1270 \ \hbox{nm}$. (d) The spectra of light directly from the ZBLAN fiber laser and the light coupled out from TM mode grating couplers: ${R} = 253 \ \hbox{nm}$ and $\Lambda_{y} = 1270 \ \hbox{nm}$.

Because we are so far unable to control the polarization of the ZBLAN fiber laser to optimize the coupling efficiency, the measured results in this paper may be underestimated. Further studies will be carried on.

SECTION 4

## CONCLUSION

In this paper, shallow-etched uniform grating couplers, full-etched subwavelength grating couplers and apodized grating couplers are studied theoretically on SOS waveguides at mid-IR spectral range for the first time. 80.6% efficiency for TE mode shallow-etched apodized gratings and 39.6% efficiency for TM mode apodized subwavelength gratings are predicted by FDTD simulations. A coupling efficiency of 32.6% for TE mode shallow-etched uniform grating coupler and 11.6% coupling efficiency for TM mode subwavelength grating coupler are experimentally demonstrated at the wavelength of 2.75 $\mu\hbox{m}$, with an $\hbox{Er}^{3+} - \hbox{Pr}^{3+}$ co-doped ZBLAN fiber laser.

## Footnotes

This work was supported by Innovation and Technology Fund Grant ITF433/09. X. Chen was supported by the Royal Society through the Newton International Fellowship. Corresponding author: Z. Cheng (e-mail: zzcheng@ee.cuhk.edu.hk).

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