1. Introduction

The Fresnel reflectivity of cleaved semiconductor waveguide facets, fabricated in the silicon-on-insulator or III-V materials systems, is typically on the order of 30%. Ability to modify this intrinsic value is essential for the functionality and performance of many planar waveguide devices. To achieve both antireflective (AR) and highly reflective (HR) waveguide facets, thin film coatings are traditionally used. Recently, we have demonstrated that an efficient AR effect can be achieved by patterning the facets with subwavelength gratings (SWGs) using standard nanofabrication techniques [1]. Indeed, SWGs have long been used to reduce the reflectivity of bulk optical surfaces, as an alternative to thin film coatings [2]. It has recently been demonstrated that surfaces, which are highly reflective over a surprisingly broad range of incident wavelengths, can also be created by SWGs [3]. Such patterned surfaces have been shown to be suitable as the top mirror of a vertical cavity surface emitting laser (VCSEL) [4]. Here we explore the possibility of applying the HR SWG concept to waveguide facets. As it was shown in [3], in order to obtain high reflectivity from a surface SWG, the grating needs to be separated from the high index substrate by a thin layer of low index material. In the VCSEL device this was achieved by fabricating a grating freely suspended above the substrate with an air gap of ~1 µm. An analogous structure for planar waveguide facets would consist of a row of vertical posts in front of a flat waveguide facet at a specific distance (equivalent to the air gap of the VCSEL structure). However, since in such a structure there is no vertical mode confinement in the air gap, out-of-plane radiative losses are a serious problem. We have found with finite difference time-domain (FDTD) simulations that for silicon-on-insulator (SOI) waveguides with typical dimensions of several micrometers or less, these radiative losses are prohibitive for practical devices.

In this study, we adapt to waveguide facets a different high reflectivity grating concept, which does not rely on an air gap, namely the so-called “giant reflectivity to order zero” (GIRO) gratings, which were first discussed and demonstrated by Goeman et al. [5]. These highly reflective gratings are not truly SWGs, since their period is large enough that the first order of diffraction is allowed inside the bulk silicon, but it can be effectively suppressed by the choice of the grating parameters. The reflective properties of such gratings on surfaces and waveguide facets are analyzed numerically using rigorous coupled wave analysis (RCWA) [6] and finite difference time-domain (FDTD) modeling. In particular, we investigate the effect of finite grating size on the reflectivity and present optical transmission measurements on SOI waveguides with patterned facets as an experimental demonstration of the HR facet grating effect.

2. RCWA and FDTD simulations

RCWA was used to calculate the reflectivity and diffraction efficiency of silicon surface gratings. These calculations were carried out using the freely distributed software package RODIS [7]. The modeled structure consists of a grating with rectangular teeth in a silicon surface, with a pitch Λ=0.7 µm, and a duty ratio of 54%, i.e. the width of one silicon tooth is 0.38 µm. These parameters are consistent with the design rules discussed in [8] and were optimized by numerical reflectivity calculations. For a plane wave with a free space wavelength of λ=1.55 µm, incident normal to the surface, such a grating is subwavelength on the air side, since Λ<λ_air, and hence, diffraction is suppressed. On the silicon side, however, the first order diffraction is allowed, since the wavelength of light in silicon λ_Si=λ_air/n_Si=0.45 µm<Λ, where n_Si=3.48 is the refractive index of bulk silicon.

The results of the RCWA calculations for the diffraction efficiency are shown in Fig. 1(a) and 1(b) as a function of the grating modulation depth (peak-to-peak). All results are for normally incident light polarized with the electric field parallel to the grating vector. For light incident from the silicon side (Fig. 1(a)), the allowed orders of diffraction are the 0th order transmission T₀ as well as the 0^th (specular) and ±1^st reflection orders R₀ and R_±1, respectively. The value of R₁ plotted in Fig. 1 is the sum of the diffraction efficiencies of the +1 and -1 orders, which are equal for normal incidence due to symmetry. A grating modulation depth equal to zero corresponds to a plane silicon-air interface with a Fresnel reflectivity of 31%. As the modulation depth is increased, the efficiencies T_0, R₀ and R₁ exhibit an interesting behavior. For a modulation depth of 470 nm, marked by an arrow in Fig. 1(a), the HR effect is observed: The specular reflection reaches a maximum of >99.99%, while the transmitted and diffracted orders are both efficiently suppressed. In Fig. 1(b), the analogous results for light normally incident on the grating from the air side are shown. In this case, the allowed orders of diffraction are the 0th reflection order and the 0th and ±1^st transmission orders. For the amplitude of 470 nm, where the reflection anomaly is observed in Fig. 1(a), approximately 98% of the incident light is diffracted into the ±1^st transmission orders. The remaining 2% of the light is almost completely reflected, with transmission in 0th order largely suppressed. The wavelength dependence of the reflection peak in Fig. 1(a) for a grating with a modulation depth of 470 nm is shown in Fig. 1(c). Reflectivities >95% and >99% are obtained in wavelength bands 1.3 µm<λ<1.7 µm and 1.45 µm<λ<1.6 µm, respectively.

 

Fig. 1. Diffraction efficiency of square silicon surface gratings with a period of 0.7 µm and a duty cycle of 54% for plane waves with a wavelength λ=1.55 µm at normal incidence according to RCWA as a function of grating modulation depth and wavelength. a) 0^th order transmission (T₀) and reflection (R₀) and 1^st order reflection (R₁) for waves incident from the silicon side and b) with light incident from air. c) 0^th order (specular) reflection for light incident from silicon as a function of wavelength for a modulation depth of 0.47 µm, corresponding to the reflection peak indicated by the arrow in a).

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To investigate the potential of the grating described above for use as an HR waveguide facet, 2D FDTD simulations for a silicon slab waveguide geometry were carried out. A simulation layout is shown in Fig. 2(a). A silicon slab waveguide of width 7 µm (blue, n_Si= 3.476) with SiO₂ cladding layers (green, n_SiO2=1.44) is terminated at the facet with a square grating. Grating parameters are identical to the ones used in the RCWA simulations with a modulation depth of 0.47 µm for maximum reflectivity. In Fig. 2(b) the simulation result for a waveguide mode incident on the facet from inside the waveguide is shown. A continuous wave (CW) waveguide mode with a free space wavelength of 1.55 µm and transverse magnetic (TM) polarization (electric field in the plane of the drawing) is launched at the excitation plane. The mesh size used for the simulation is 10 nm×10 nm. Figure 2(b) shows the field intensity map after 16,000 time steps of Δt=2.2×10^-17 s. It is observed that transmission through the patterned facet is efficiently suppressed. Between the excitation plane and the facet, the forward and backward propagating waveguide modes are superimposed, forming a standing wave pattern. To the left of the excitation plane the reflected mode propagates in the slab waveguide. A numerical estimate of the facet modal reflectivity is obtained by calculating the ratio of the intensities reflected from the facet and emitted by the source and multiplying this ratio by the overlap integral of the reflected intensity profile with the fundamental waveguide mode. The internal facet reflectivity obtained by this procedure is 96.4%, thus verifying the validity of the HR grating concept for waveguide facets. The situation is markedly different for external coupling to the waveguide, as shown in Fig. 2(c). In this case the FDTD source has a Gaussian intensity profile with a 1/e² full width of 10.4 µm (SMF-28 fiber mode) and is located at the excitation plane near the waveguide facet (Fig. 2(c)). The calculated field in the waveguide exhibits a strong lateral modulation with a period equal to one half of the grating period. As the light propagates in the waveguide, this modulation persists almost without perturbation. This lateral intensity modulation results from the coupling to the +1^st and -1^st diffraction orders, with the 0^th transmission order suppressed. The interference of the +1^st and -1^st diffraction orders, propagating at angles of approximately ±40° with respect to the waveguide axis, yields the observed intensity pattern. This effect, including zero order suppression, is commonly employed in phase masks used in the fabrication of fiber-Bragg gratings [9]. In the case of waveguide facets, obviously the coupling from the optical fiber to the fundamental waveguide mode is largely suppressed. Nevertheless, the FDTD calculations confirm that high internal facet reflectivity is achieved at the same time. The HR grating effect on waveguide facets has potentially interesting applications, e.g. for resonators.

 

Fig. 2. FDTD simulations of light propagation in waveguides terminated with high reflectivity grating facets. a) Simulation layout. b) Electric field map of the TM mode (electric field in the plane of the drawing) launched towards the right from the excitation plane and reflected by the facet. c) Simulation of fiber-to-waveguide coupling with a Gaussian beam launched towards the left from the excitation plane.

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By inspecting the phase front of the waves in the FDTD calculations some insight into the physical reason for the observed diffraction anomaly can be gained. In simple terms, the modulation depth of the grating is such that the phase difference for the light passing through the peaks and troughs is approximately π radians. Therefore the interference in the forward direction (0^th order diffraction) is destructive for light incident from either side of the grating. Depending on the duty cycle of the grating this can lead to the almost complete cancellation of the 0^th order transmitted intensity observed in Figs. 1(a) and 1(b). For light incident from the air side, this π-modulated phase front of the light wave just after passing through the grating region corresponds to a maximum in the ±1^st order diffraction efficiency. For light incident from the Si side, diffraction in transmission (in air) is suppressed due to the subwavelength nature of the grating. The reflected wave passes through the grating region for a second time (in the inverse direction). It then exits the grating with a phase shift of approximately 2π radians, i.e. with a flat phase front, which corresponds to a minimum in the diffraction efficiency in Si. These findings from the FDTD simulations are consistent with the general concept upon which the HR grating design rules are based [8].

3. Experimental results and analysis

Figure 3 shows the scanning electron microscope (SEM) image of a fabricated HR waveguide facet. Samples were fabricated on SOI wafers with a Si layer thickness of 1.5 µm and a buried oxide (BOX) layer thickness of 1 µm. The facets and ridge waveguide were defined in two separate patterning steps, each of which consisted of electron beam lithography and dry etching steps. The fabrication procedure is identical to one we have previously used to fabricate AR facets, as described in ref. [1]. The width of the ridge waveguide at the facet is 7 µm, as in the FDTD simulations. Away from the facet (not shown in Fig. 3), the waveguides are adiabatically tapered to a width of 1.5 µm. This ensures single-mode operation of the ridge waveguides at their nominal etch depth of 0.7 µm.

 

Fig. 3. Scanning electron micrograph of a SOI ridge waveguide facet patterned with an HR grating.

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Fig. 4. (a). Schematic top view of the waveguide test structure used in transmission measurements. (b). The optical model for a lossy asymmetric Fabry-Pérot cavity with an input mirror reflectivity of 31%, corresponding to the reflectivity of the flat input facet and a variable back mirror reflectivity. (c). The depth of the Fabry-Pérot fringes as a function of the back mirror reflectivity.

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The facet reflectivity can be obtained from measurements of the waveguide transmittance spectra, using the Fabry-Pérot method. Fiber-to-waveguide coupling through a facet with an HR grating is not possible according to the FDTD results discussed above. In fact, we have found that the transmittance of waveguides terminated with HR facets at both ends was too small to be measured experimentally (<-60 dB). Therefore, waveguides with a flat input facet and a patterned output facet were used for the experiments. A schematic of such a structure is shown in Fig. 4(a). The transmittance of our waveguides was modeled as an asymmetric Fabry-Pérot cavity with an input mirror reflectivity of 31% and an unknown output mirror reflectivity to be determined by the measurements. To account for waveguide propagation loss, a loss factor α is included in the Fabry-Pérot model, as indicated in Fig. 4(b). The loss factor is defined as the ratio of the electric field amplitude after and before a single trip through the waveguide. Thus, for zero propagation loss, α=1. The expected Fabry-Pérot fringe depth for α=0.85, the loss factor obtained from the measurements described below, is shown in Fig. 4(c). Peak-to-peak fringe amplitude varies from 0 for a perfect AR back facet to 7.5 dB for a HR back facet with 100% reflectivity.

 

Fig. 5. Left: Experimentally measured Fabry-Pérot fringes in the waveguide transmission spectra for three different back facets: a) Patterned AR with a nominal reflectivity of 3.6%, b) flat reference facet, and c) HR grating facet with a modulation depth of 0.35 µm and a duty ratio of 0.5. Right: Facet reflectivity as a function of grating modulation depth inferred from the Fabry-Pérot measurements compared to 3D FDTD simulations.

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Transmission measurements were carried out on waveguides terminated with HR gratings with varying grating parameters, AR SWG facets and flat facets, the latter with a nominal reflectivity of 31%. We used a lensed polarization maintaining optical fiber to couple light from a tunable laser with a wavelength near 1550 nm into the waveguides and an InGaAs photodetector to measure the output intensity. Transmission spectra for three waveguides are shown in Fig. 5 (left panel). The curves are offset for clarity. From the observed fringe depth of 4 dB in the spectrum of the reference waveguide with flat input and output facets, the intra-cavity loss factor was determined to be α=0.85, assuming flat facet reflectivity of 31%. For our 4.5 mm long waveguides, this corresponds to a propagation loss of 3.1 dB/cm. The loss is believed to be caused predominantly by sidewall scattering. The upper spectrum shown in Fig. 5 was obtained from an AR SWG back facet, as described in [1], with a nominal reflectivity of 3.6%. The observed reduction in fringe amplitude is in good agreement with the behavior expected from the Fabry-Pérot model shown in Fig. 4. The lower curve in Fig. 5 is the transmission spectrum of a waveguide with an HR back facet with a grating modulation depth of 0.35 µm and a duty ratio of 0.5. The measured fringe depth of 6.5 dB compared to 4 dB for the flat facet confirms the predicted HR effect of the grating. Assuming the same propagation loss as measured for the reference waveguides, according to the plot shown in Fig. 4, this fringe depth corresponds to a back facet reflectivity of 77%. The reflectivity of a series of waveguides with HR gratings of varying modulation depth was determined in the same fashion and the results are shown in Fig. 5 (right panel). The double peak curve obtained as a function of grating modulation depth is reminiscent of the behavior of the 0^th order reflection R₀ according to the RCWA calculations shown in Fig. 1(a). However, the peak reflectivity of 77% obtained experimentally is significantly lower than the values predicted by RCWA or 2D FDTD simulations discussed above. To interpret the experimental results, full 3D FDTD simulations were therefore carried out, using the nominal structures as in the experiments. The general procedure for these simulations is similar to the 2D simulations, but the vertical confinement of the mode inside the 1.5 µm thick Si layer is now accounted for. The mesh size used for the 3D simulations was 30 nm×30 nm×50 nm. The 3D FDTD simulation results for the facet reflectivity and the experimental results are overlaid in Fig. 5 and the agreement is excellent. The small deviations of experiment and FDTD are most likely due to differences between the nominal and actual grating parameters as well as corner rounding, as observed in the SEM picture in Fig. 3. For both experiment and simulations, the peak reflectivity >99% predicted by the RCWA calculations is reduced to below 80% in this 3D geometry.

4. Discussion

The discrepancy between the 2D and 3D FDTD simulations suggests that vertical mode confinement is the reason for the lower than expected experimental facet reflectivities. To verify this hypothesis, we carried out a series of 3D FDTD simulations with varying waveguide thickness. Since the 2D simulations correspond to infinite waveguide thickness, one would expect the 3D FDTD results to converge with the 2D results for increasing waveguide thickness. In the simulations we used waveguide thicknesses of 1.5, 2.5, 4, 6 and 10 µm. The corresponding calculated power reflectivities are R=91.2%, 96%, 97.5%, 97.6% and 98.6% and the mode overlap factors of the reflected power with the fundamental waveguide mode are γ=85.5%, 89%, 95.1%, 96.8% and 97.5%. Clearly, both the power reflectivity and the mode overlap increase montonically with the waveguide thickness, converging to the result of the 2D simulations. From these results, modal reflectivities of γR >90% are achieved for waveguide thicknesses ≥4 µm.

The effect of mode confinement on the reflectivity can be interpreted in terms of the dependence of the grating reflectivity on the angle of incidence. From Fourier analysis it is known that a mode propagating through a plane facet can be decomposed into a set of plane waves distributed over a finite range of incident angles. Let us consider the electric field of a slab waveguide mode, which is proportional to cos(k_yy)exp(iβz) inside the waveguide core, where y is the transverse coordinate and z is the light propagation direction. In a first order approximation, we can take the root mean square of the angular distribution of the incident light for a confined waveguide mode as $\tilde{\varphi }$=arctan(k_y/β). This mean angular width is plotted in the inset of Fig. 6(a) as a function of the waveguide core thickness for the fundamental mode of a SiO₂/Si/SiO₂ slab waveguide. As expected, the larger mode size in thicker waveguides corresponds to a smaller mean angle of incidence on the waveguide facet. Using this waveguide thickness to angle conversion, we plot the power reflectivities obtained from the 3D FDTD simulations as a function of mean angle of incidence in Fig. 6(a). This allows us to compare the FDTD results with RCWA calculations of a plane wave incident on a bulk surface grating as a function of the incident angle ϕ, as illustrated in Fig. 6(b). The RCWA calculations show that the reflectivity of surface gratings decreases rapidly for off-normal incidence, e.g., when ϕ is increased from 0° to 2°, the reflectivity drops from 99.99% to 98.5%. This angular dependence of the reflectivity is shown in Fig. 6(a) as a solid line. Considering the approximate nature of our argument, the agreement between RCWA and FDTD is rather good. Given the uncertainties inherent in FDTD simulations due to finite mesh size or other numerical errors, the combination of the RCWA and average angle of incidence calculations in Fig. 6(a) is an alternative and computationally less demanding way of estimating the expected reflectivity for thick waveguides. For example, to obtain a reflectivity >99.5%, an angular width of <1° is required, which corresponds to a waveguide thickness >12.5 µm. Interestingly, a comparable finite-size effect has been found for the case of HR SWGs with an air gap [10].

 

Fig. 6. (a). Grating specular reflectivity as a function of incident angle according to RCWA compared to 3D FDTD results for facet reflectivity as a function of waveguide thickness. Inset: Mean angle of incidence as a function of silicon slab waveguide thickness as discussed in the text. b) Grating illumination geometry for the RCWA calculations.

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HR diffraction gratings have been proposed as an alternative to multilayer distributed Bragg reflectors as the top mirror for long wavelength VCSELs [5], which usually require a reflectivity >99.5%. The obvious advantage of these gratings over the SWGs demonstrated in [4] is that no air gap is required to achieve the HR effect. Therefore, fabrication of the VCSEL top mirror can be greatly simplified. Our results indicate that, due to the mode size effect, the minimum lateral waveguide dimension for which such a high reflectivity can be achieved is approximately 12 µm. Typically, such large lateral sizes imply multimode operation for VCSELs; however, the mode size effect may be advantageously used to suppress the lasing of higher order modes, as grating reflectivity will be decreased for higher order modes with a larger mean angle of incidence.

4. Conclusion

We have demonstrated by RCWA calculations, FDTD simulations and experiments on SOI waveguides with patterned facets, that high specular reflectivity can be achieved with square gratings on waveguide facets. These gratings are subwavelength on the air side but allow the ±1^st diffraction orders inside the silicon. Plane wave reflectivity for light incident from inside the bulk silicon is >99.99% according to RCWA. For light incident from the air side, the grating acts as a zero-order suppressed phase mask, with most of the light diffracted into the ±1^st orders. Experimentally, we have demonstrated this HR effect on SOI waveguide facets. Modal reflectivities of up to 77% were achieved for 1.5 µm thick waveguides in good agreement with 3D FDTD simulations. The simulations predict reflectivities in excess of 90% for waveguide thicknesses larger than 4 µm. We have provided evidence that this mode size limited effect is related to the dependence of the reflectivity on the angle of incidence. These results have also interesting implications for the suggested suitability of these HR gratings as top reflectors for VCSELs.