Open Access
Add to BibSonomyAbstract
We present numerical demonstrations that anti-phase reflection coatings (APRCs) on the core layers of grating couplers (GCs) return anti-phase field into the core layers and cancel the downward scattering from the gratings by destructive interference to improve the upward directionality of the GCs while the output power per unit length is reduced. Investigating simplified models of GC, we reveal the effect of APRCs semi-analytically and quantitatively. The APRCs can be combined with other enhancement measures, like deep gratings and backside mirrors, to tailor an appropriate output power per unit length while achieving high upward directionalities cooperatively.
© 2016 Optical Society of America
1. Introduction
Grating couplers (GCs) are suitable for coupling photonic chips to a number of optical fibers simultaneously, since their ability to vertically couple to the fibers allows dense arrangement of the GCs on the chips. They are often used in optical transceiver chips for multi-channel link systems [1], and an effort to achieve high coupling efficiencies of the GCs is crucial because the coupling efficiencies sometimes determine the link distance of the systems. In that effort, the mode field overlap between a GC and a fiber is an important figure of merit to increase, and, more fundamentally, the directionality of the GC must be sufficiently improved.
There are three ways to improve the upward directionality. One is by forming a deep grating in a thick core layer to emit most light upward as a result of internal interference in the core layer [2]. When it is made of silicon, the core layer is usually more than 300 nm thick, and the depth of the grating grooves exceed half of the core thickness. These structural requirements can conflict with those of other devices, including single-mode waveguides, when silicon overlay is added only on the GC [3]. The grating can be blazed for more directionality [4]. Another way to improve the upward directionality is by placing a backside mirror below the core layer of a GC to upward reflect the light that is diffracted downward by the grating. The mirror can be a metal [5], or can comprise multiple layers of dielectrics [6]. A GC in a silicon-on-insulator (SOI) wafer can be regarded as having the backside mirror of base silicon substrate when the thickness of the buried oxide (BOX) is optimized. This way is powerful, but wafers with sufficiently efficient mirrors are currently unavailable commercially [7]. The third way to improve the upward directionality is by applying an anti-reflection coating (ARC) on top of a GC [8]. The ARC can reduce the undesirable reflection at the top surface of the GC and thereby increase the amount of light emitted upward. If light is emitted into the air through an over cladding, optimizing the thickness of over cladding is part of ARC design [9]. Even with an ARC on top of the GC, however, the upward directionality may be insufficient because light is still emitted downward.
We demonstrated previously that the upward directionality of a GC was most improved by applying a structurally detuned “ARC” having an excessive number of high-index layers [10,11]. A paper by Narasimha and Yablonovitch suggested a similar coating but did not provide details [12]. In this article, we demonstrate that the detuned ARC is, actually, an anti-phase reflection coating (APRC) cancelling the light wave that would otherwise propagate downward from the grating. In the following section, we first numerically demonstrate the high directionality of GCs having APRCs and then investigate several GC models to clarify the effect of the APRCs semi-analytically and quantitatively.
2. Numerical demonstration of GC having APRC
We used the two-dimensional finite-difference time-domain (2D-FDTD) method to numerically simulate a GC consisting of a core layer, a high-index layer over it, and a buffer layer between them, all embedded between over and under claddings. The high-index layer in the GC plays the role of a reflector layer, and the reflector layer and the buffer layer composes an APRC. No base substrate like in an SOI wafer was placed beneath the under cladding. The refractive index of the core layer ncore was set to 3.505, representing Si. That of the high-index layer nhigh was set to 8.490. The refractive indices of the claddings and the buffer layer were both set to 1.447, representing SiO2. The depth d and pitch Λ of the grating grooves were respectively 50 and 505 nm, and the groove width w was 303 nm. The length of the grating region was 25 μm. The thickness of the core layer tcore was 200 nm, which was determined by the formula tcore = λ0/(2ncore cosϕSi) + ζ, where λ0 is a wavelength in a vacuum, ϕSi is the tilt angle of the light propagating inside the core layer measured from the thickness direction, and ζ is a correction term reflecting the effect of the grating depth [10]. The thickness of the high-index layer thigh was set to 39 nm, determined by thigh = λ0/(4nhigh cosϕSi), and that of the buffer layer tbuffer was 432 nm, determined by tbuffer = λ0/(2nSiO2 cosϕSiO2) – d/2. Figure 1(a) shows a calculated contour map of the electric field propagating around the GC when a transverse electric (TE) field of light at a wavelength of 1310 nm was input from the left and propagated to the right along the core layer. It can be seen in Fig. 1(a) that almost all of the electric field is emitted upward except at the starting position of the grating region. Around the starting position, clear scattering toward particularly the downward direction can be observed. It can also be seen that electric field between the core layer and the high-index layer is more intense than the field emitted upward.
Fig. 1 Contour maps of electric field calculated for GCs with APRCs including (a) single high-index layer and (b) Si/SiO2/Si layers. Field is normalized between –1 (blue) and 1 (red).
The GC structure shown in Fig. 1(a) is one of the simplest among those having APRCs, and is, thus, suitable to be observed for characteristic field caused by the APRC. Since one may mind the unrealistic refractive index of 8.490, we also show in Fig. 1(b) a contour map of electric field calculated for another GC in which the reflector of the single high index layer in the APRC in Fig. 1(a) has been replaced with multiple layers of existing dielectrics. Specifically, the layers and their thicknesses are Si at 93 nm, SiO2 at 114 nm, and Si at 93 nm from the bottom. Figures 1(a) and 1(b) were calculated at the same elapsed simulation times. It can be found that the field shown in Fig. 1(b) is identical to that in Fig. 1(a) except for inside the Si/SiO2/Si layers. As a matter of fact, we chose the refractive index of 8.490 to produce such the same distributions of field between Figs. 1(a) and 1(b). Further discussion on the APRC designs will be provided in Subsection 3.3.
We define two upward directionalities. One is given by ηup,tot = Pup,tot/(Pup,tot + Pdown,tot), where Pup,tot and Pdown,tot are respectively the total upward power and the total downward power monitored in the whole length of the GC [13]. The power monitor for Pup,tot was placed immediately on the high-index layer, and that for Pdown,tot was placed 1 μm below the core layer. The other upward directionality is given by ηup,intr = Pup,intr/(Pup,intr + Pdown,intr), where Pup,intr and Pdown,intr are the upward power and the downward power that are monitored only in a 5-μm length ranging from 15 μm inside the starting position of the grating to the right. This ηup,intr is a more intrinsic directionality than ηup,tot because it hardly includes the effect of scattering at the starting position of the grating. The values of ηup,tot and ηup,intr calculated for the GC having the APRC with the single-layer reflector are plotted against nhigh in Fig. 2. In the course of the calculation, the thickness thigh was adjusted to meet thigh = λ0/(4nhigh cosϕhigh) for each nhigh. As seen in Fig. 2, with increasing nhigh, ηup,tot first increases to a peak value of 0.845 (–0.731 dB) at an nhigh of 4.500 and then decreases. On the other hand, ηup,intr increases monotonically and seemingly becomes saturated to 0.950 (–0.225 dB) at an nhigh of 8.500. These results indicate that the total directionality ηup,tot is reduced from the intrinsic directionality ηup,intr by the effect of field scattering at the starting position of the grating.
Fig. 2 Upward directionalities calculated against refractive index of high-index layer (nhigh) from upward and downward powers monitored within length ranges including (ηup,tot) or excluding (ηup,intr) the vicinity of starting position of grating (see Fig. 1(a)).
3. Discussion
3.1 Modeling of GC having APRC
In this section, we try to identify the effect of APRCs as analytically and quantitatively as possible. Since light emission from GCs is a result of multiple interferences in the GCs, it is hard to discriminate the effect of APRCs from the effects of other structural components. Therefore the first thing to do is to build GC models appropriate for investigating the effect of APRCs. In an actual GC, grating grooves exist in a limited region of the core layer. Light is input to the grating region from an edge, and hence some scattering occurs at the input and, sometimes, output ends of the grating region. Furthermore, the GC emits light in a direction a little tilted from the normal to the surface. One would therefore expect the vertical interference near the input end of the grating to be imperfect, deteriorating the upward directionality of the GC. These two irregularities occurring around the grating ends prevent us from accurately evaluating the effect of APRCs that are applied. Figure 3 illustrates a cross-section of a GC having an APRC. We slice it at a position x by a length of ΔLx to use as a simplified model of GC, the region of which is surrounded by a dotted line in Fig. 3. Within this sliced GC, we can ignore scattering at the input and output ends of the GC when the sliced GC is considered to be far from the ends. The direction of light emitted from the sliced GC is assumed to be normal to the surface.
The power flow in the GC model shown in Fig. 3 is as follows. The power conservation law for the entire GC is expressed by Pin = Pref + Pthr + (Pup + Pdown), where Pin, Pref, and Pthr are the input power, the reflected power and the through power, respectively, and Pup and Pdown are the upward and downward emitted powers. As for the sliced region of the GC, the input power Px,in enters it, and the through power Px,thr passes through it. The powers Px,u and Px,d are emitted upward and downward, respectively, out of the region. It is satisfied then that Px,in = Px,thr + (Px,u + Px,d). The power sum, Px,u + Px,d, can vary depending on the vertical structure of the GC. If Px,u + Px,d is increased while Px,in is unchanged, Px,thr deceases, which immediately means decrease of input power into the neighboring sliced GC region at the position of x + ΔLx. Since our chief concern is the intrinsic directionality of the GC, we will mainly discuss Px,u and Px,d hereafter, and the other power parameters will hardly appear again. However, it should be noted that the energy conservation law for the sliced GC region as well as the entire GC is always satisfied as explained above.
We make five more assumptions in our simplified (sliced) GC model to increase simplicity of analysis. First, the length ΔLx is small enough for us to assume the period and the filling factor of the grating to be constant within the entire length. Second, the grating grooves are shallow enough to have a nearly zero thickness and be confined in the interface between the core layer and the over cladding. This assumption allows us to ignore vertical interference within a virtual layer containing the grating grooves. Third, only the –1st-order diffraction by the grating is considered for analysis, so that the grating grooves can be substituted simply with a field launcher. Fourth, we assume that for both sides (up and down) the launcher is an electric field source (not a current source used in [12]) that adds electric fields of equal amplitudes and phases to the existing fields on both up and down sides, i.e. |Ex,u| = |Ex,d|. No free current is considered. The magnetic field in a simulator is excited according to Maxwell’s equation rot H = ε (∂E/∂t), where E and H represent electric and magnetic fields, respectively, ε is the permittivity, and t is the elapsed simulation time. This field launcher is classified as a soft source [14], and causes no reflection or scattering. The fourth assumption can be a good approximation when the depth of grating grooves is close to zero. This is because the shapes of the grooves can be ignored and, thus, the grating grooves behave like dipole moments, which are excited by the input light and then excite electric field homogeneously around them. Last, it is assumed that the grating does not re-diffract the field that has been launched by the grating itself. We adopt all these assumptions for our GC model investigated, but they are merely for simplification of analysis and can be loosen when designing practical GCs.
Figure 4 depicts the layer structure of the simplified model of GC having an APRC. For later convenience, the grating in the interface between the core layer and the cladding is called grating plane. The APRC is inserted between the core layer and the over cladding. The APRC here consists of a single high-index layer (as a reflector layer) and a buffer layer. The thicknesses of the high-index layer and the buffer layer satisfies thigh = λ0/(4nhigh) and tbuffer = λ0/(2nbuffer), respectively, where nhigh and nbuffer are the refractive indices of the two layers. Note that angle-dependent terms like cosϕ are missing because ϕ = 0 in this model. Other notations in Fig. 4 are the thickness of the core layer tcore, and the refractive indices of the over cladding nover, the core layer ncore, and the under cladding nunder. As is often the case with actual GCs, it is assumed that ncore > nbuffer, and ncore > nunder. The symbols Ex,u and Ex,d represent the upward and downward electric fields added by the launcher. It is also assumed that |Ex,u| = |Ex,d|, following our earlier discussion. The symbols Px,u and Px,d denote the optical powers emitted upward and downward from the GC.
The function of the APRC is explained as follows [10]. The electro-magnetic field launched upward from the grating plane is partially reflected at the bottom surface of the high-index layer. The reflected field propagates downward, and some of it enters the core layer and cancels the field downward launched from the grating plane. Since the cancellation results from destructive interference, the thickness of the buffer layer for producing anti-phase reflection is crucial. If it is a partial cancellation, the remaining field will propagate to the bottom surface of the core layer, be partially reflected, and return into the buffer layer. A high directionality will be achieved when the interference between the returned field and the originally upward-launched field are constructive, which occurs when tcore = mcoreλ0/(2ncore), where mcore is an integer. We note, however, that the effect of the APRC is then enhanced by the returned field from the core bottom even if there is no additional backside mirror.
3.2 Characterization of APRC
The pure effect of the APRC can be estimated if the degree of directionality enhancement effect by the core bottom reflection is known. Therefore, in the first step of investigation, we started with a GC model in which the bottom surface of the core layer is removed, which is possible in our model because there is no need for light confinement in the core layer. Specifically, we assumed that ncore = nunder = 3.505, representing Si, in addition to setting nover = 1.447, representing SiO2. We also removed the APRC. In the resulting model shown in Fig. 5(a), the launcher exists between the SiO2 and Si regions and excites electric fields of equal amplitudes and phases into both regions. A 2D-FDTD simulation was performed at a wavelength λ0 = 1310 nm. The power sum Px,u + Px,d calculated for this investigation step was represented by P0, i.e., Px,u + Px,d = P0 to be later used as a reference for normalization. As listed beneath the illustration in Fig. 5(a), the results were that Px,u = P0 × 0.292, Px,d = P0 × 0.708, and ηx,u = Px,u/(Px,u + Px,d) = 0.292 (–5.346 dB). The upward directionality ηx,u can be calculated analytically as well; the power P of a plane wave in a material of a refractive index n is given by P = n (cε0A) E2, where c is the speed of light in a vacuum, ε0 is the permittivity of a vacuum, and A is the area of the power monitor. Since E was constant throughout the model and A was the same for up and down sides, ηx,u = nover/(nover + nunder).
Fig. 5 Illustrations of GC models simulated in the (a) first, (b) second and (c) third steps. Calculated results are shown below each model.
As shown in Fig. 5(b), in the second step we returned nunder to 1.447, representing SiO2, in order to estimate the effect of the bottom mirror of the core layer. The thickness of the core layer was determined to be 186.9 nm by the formula tcore = λ0/(2ncore). Performing a 2D-FDTD simulation, we found that Px,u + Px,d = P0 × 1.712, Px,u = Px,d = P0 × 0.856, and ηx,u = 0.500 (–3.010 dB). They all increased because of the bottom mirror, although the setting of the field launcher in the simulation was unchanged from that in the first step. This can happen because the launcher used is an electric field adder but is not a constant-power feeder. Consider a situation where a field launcher adds a new electric field Eadd in-phase to an existing field Eexi, causing constructive interference between them. Then the total electromagnetic power will be n (cε0A)(Eadd + Eexi)2, which is larger than the sum of individual powers n (cε0A)(Eadd2 + Eexi2) by 2n (cε0A) Eadd Eexi. Therefore it can be understood that the power added by the launcher increases with the amplitude of the existing field, while the total output power will decrease if destructive interference occurs instead. Getting back to our model, constructive interference occurred in both the core layer and the over cladding, which enhanced the output power of the launcher. This explanation does not violate the energy conservation law, as long as Px,in > Px,u + Px,d. The increase in Px,u + Px,d merely means the increase of output power per unit length of the grating but not the increase of Pup + Pdown of the whole GC. Besides, it turned out that, with no APRC, ηx,u is exactly 0.500 under the conditions that tcore = λ0/(2ncore) and |Ex,u| = |Ex,d|.
As shown in Fig. 5(c), in the third step we returned the APRC to the model, for which nhigh = 3.505 and nbuffer = 1.447. The thicknesses of the high-index layer and the buffer layer were determined by thigh = λ0/(4nhigh) and tbuffer = λ0/(2nbuffer), respectively. Specifically, thigh was 93.4 nm and tbuffer was 452.7 nm, since λ0 = 1310 nm. The setting of the field launcher was unchanged again. The simulation results were that Px,u + Px,d = P0 × 0.500, Px,u = P0 × 0.427, Px,d = P0 × 0.073, and ηx,u = 0.854 (–0.685 dB). Adding the APRC decreased the power sum Px,u + Px,d, Px,u, and Px,d. It increased only the upward directionality ηx,u. Particularly notable is the fact that Px,u decreased while ηx,u increased. This is a general characteristic of APRCs, and it clearly distinguishes APRCs from ARCs, which would increase Px,u by reducing surface reflection. We note that a long grating is needed to output most of the light power input to a GC when the grating shows a small output power per unit length [10,15]. The power sum Px,u + Px,d being exact half of P0 is not a general characteristic of APRCs but was due to the specific structural conditions of the model.
The upward directionality can be improved by increasing the refractive index of the high-index layer. We set nhigh at 8.490, as assumed for the GC model described in Section 2. Then thigh = 38.6 nm, according to thigh = λ0/(4nhigh), while other conditions were unchanged. As a result, ηx,u increased to 0.972 (–0.123 dB). The power sum Px,u + Px,d decreased to P0 × 0.097. Figures 6(a) and 6(b) show a 1D distribution and a 2D contour map of the electric field in this GC model. They were calculated at identical elapsed simulation times. Intense field in the buffer layer can be seen in both figures. Although the electric field inside the buffer layer became more intense at other elapsed times, these snapshots were chosen to clearly show the weak field outside the buffer layer after the field was normalized with the maximum.
Fig. 6 Electric field distribution in GC having anti-phase reflection coating: (a) 1D graph and (b) 2D contour map.
It should be noted that nhigh was much larger than ncore for the APRC in the last GC model. If it were an ARC instead of the APRC, nhigh would be smaller than nbuffer and be given by the formula nhigh = (ncore nover)½. As stated in the introduction, the APRC in our previous work was, at first, categorized as an ARC, although it was structurally detuned from the exact ARC to produce anti-phase reflection [10]. After our investigations described above, it is clear that APRCs are different from ARCs in terms of both function and structure.
3.3 Variations and limitations of APRC
An APRC uses part of the upward launched field to cancel the downward launched field, and the degree of the cancellation is determined by the electric field reflectivity at the bottom surface of the high-index layer. In this sense, the role of the high index-layer in the APRC is a reflector, while that of the buffer layer is a phase adjuster. The reflectivity of the high index-layer rh can be derived by the transfer matrix method [16] and is given by
Equation (1) indicates that |rh| increases with the refractive index of the high-index layer nhigh when nbuffer nover < nhigh2. Available refractive indices are restricted to those of existing materials. As demonstrated in Section 2, we can lift the restriction by using, alternatively, multiple high-index layers separated by intermediate low-index layers as the reflector. When the refractive index and thickness of the intermediate layers are denoted by nbetween and tbetween, respectively, tbetween = λ0/(4nbetween). Under the condition that nbuffer = nbetween = nover, for simplicity, we derived the equationwhere m is the number of the high-index layers. When nbuffer = nover = 1.447 and nhigh = 8.490 for Eq. (1), which was assumed in Section 2, rh = –0.944. This high reflectivity is also given by Eq. (2) by setting nover = 1.447, nhigh = 3.505, which are respectively the refractive indices of SiO2 and Si, and m = 1. Thus it is possible to make such an effective APRC by depositing SiO2 and Si. The reflectivity of an APRC can be made as close as possible to –1 by increasing m in Eq. (2). Then, the maximum amplitude of electric field that can enter the core layer after being reflected downward by the APRC is limited purely by the amplitude of the field originally launched upward.We classify the behaviors of ηx,u against nhigh by the ratio of |Ex,u| to |Ex,d|. Figure 7 shows three different behaviors of ηx,u versus nhigh calculated with the GC model having the single high-index layer under the conditions that |Ex,u| = |Ex,d|, |Ex,u| = |Ex,d| × 1.5 (|Ex,u| > |Ex,d|), and |Ex,u| × 1.5 = |Ex,d| (|Ex,u| < |Ex,d|). As seen in Fig. 7, if |Ex,u| = |Ex,d|, which we assumed in this section, ηx,u monotonically increases and becomes saturated toward 1 as nhigh is increased. If |Ex,u| > |Ex,d|, ηx,u can be made exactly 1 by choosing an appropriate value of nhigh, which is 4.0 when |Ex,u| = |Ex,d| × 1.5. Further increase in nhigh will lead to reduction of ηx,u. If |Ex,u| < |Ex,d|, ηx,u will increase to a certain value smaller than 1 and then decrease as nhigh is increased. Figure 7 also shows the intrinsic upward directionality ηup,intr versus nhigh calculated for the demonstrated GC in Section 2 (which is shown also in Fig. 2). It can readily be noticed from Fig. 7 that the behavior of ηup,intr against nhigh is very close to that of ηx,u calculated under the condition that |Ex,u| = |Ex,d|. We can, thus, reconfirm that the condition |Ex,u| = |Ex,d| was a good assumption to precisely describe scattering by the grating in the demonstrated GC.
Fig. 7 ηx,u versus nhigh calculated under the conditions that |Ex,u| = |Ex,d| (red), |Ex,u| > |Ex,d| (blue) and |Ex,u| < |Ex,d| (purple), and ηup,intr versus nhigh for demonstrated GC in Section 2 (black).
3.4 Influence of APRC on wavelength dependence
In Section 2, we pointed out that there is intense field particularly in the buffer layer of the demonstrated GC in Fig. 1(a). Seeing that field, one may recall a Fabry-Perot interferometer and worry about the wavelength bandwidth available for the GC. Figure 8 shows the wavelength dependence of ηup,intr calculated for the demonstrated GCs with the APRCs of the single-layer and multiple-layer reflectors, and that of the GC without any APRC. As seen in Fig. 8, ηup,intr of the GCs with either APRC are higher than that of the GC without any APRC in the entire range of the O-band, indicating the usefulness of the APRCs in practice.
Fig. 8 Wavelength dependence of upward directionalities (ηup,intr) calculated for GCs with APRCs including single-layer reflector (red) or multiple-layer reflector (blue), and for GC without any APRC (black).
As observed in Fig. 8, the wavelength dependence of ηup,intr of the GC with either APRC is not peaked, denying the possibility that the buffer layer works as a Fabry-Perot interferometer. The reason for that is understood as follows. The buffer layer of the APRC with the single-layer reflector is sandwiched between the high-index layer and the core layer. The high-index layer can work as a reflector since its thickness is designed to be a quarter of the wavelength in the material of the high-index layer. On the other hand, the reflectivity of the core layer is nearly zero because its thickness is designed to be half of the wavelength in the material of the core layer and that the materials on both sides (up and down) of the core layer are the same (SiO2). Thus, confinement of field like in a Fabry-Perot interferometer does not occur in the buffer layer. In Fig. 1(a), the field in the buffer layer looked more intense as compared to the emitted field, but, as a matter of fact, it is less intense than the field in the core layer, which might be a little hard to observe.
The same explanation can be applied to the wavelength dependence of ηup,intr of the GC with the APRC including the multiple-layer reflector shown in Fig. 1(b). One may point out that the bandwidth of ηup,intr in the case of multiple-layer reflector is narrower than that in the case of single-layer reflector. It just reflects that the wavelength dependence of the reflectivity of the multiple-layer reflector is narrower than that of the single-layer reflector.
The 1-dB bandwidth of ηup,intr of the GC having the APRC with the single layer reflector is 100 nm, and that of the GC having the APRC with the multiple-layer reflector is 77 nm, which is more important in practice. To be severer, the 0.5-dB bandwidths of ηup,intr are 66 nm and 52 nm, respectively. The bandwidths of coupling efficiencies of the GCs would be somewhat narrower than those as a result of change in the emission direction depending on the wavelength.
4. Summary
We discussed anti-phase reflection coatings (APRCs), new coverings improving the upward directionality of grating couplers (GCs). An APRC is different from an anti-reflection coating. It causes anti-phase reflection into the core layer and cancels the downward scattering from the grating. As a result, the upward directionality of the GC is improved and the output power per unit length is reduced, which we proved semi-analytically and quantitatively by investigating simplified GC models that we proposed. In conclusion, applying an APRC is a new way to improve the directionality of a GC, and the APRC can be combined with a deep grating and a backside mirror to tailor an appropriate output power per unit length while achieving a high directionality cooperatively.
Acknowledgments
This work was partly supported by the New Energy and Industrial Technology Development Organization (NEDO). The authors would like to thank all of the project members for their fruitful discussions.
References and links
1. K. Yashiki, Y. Suzuki, Y. Hagihara, M. Kurihara, M. Tokushima, J. Fujikata, A. Ukita, K. Takemura, T. Shimizu, D. Okamoto, J. Ushida, S. Takahashi, T. Uemura, M. Okano, J. Tsuchida, T. Nedachi, M. Fushimi, I. Ogura, J. Inasaka, and K. Kurata, “5 mW/Gbps hybrid-integrated Si-photonics-based optical I/O cores and their 25-Gbps/ch error-free operation with over 300-m MMF,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th1G.1. [CrossRef]
2. D. Vermeulen, S. Selvaraja, P. Verheyen, G. Lepage, W. Bogaerts, P. Absil, D. Van Thourhout, and G. Roelkens, “High-efficiency fiber-to-chip grating couplers realized using an advanced CMOS-compatible silicon-on-insulator platform,” Opt. Express 18(17), 18278–18283 (2010). [CrossRef] [PubMed]
3. G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14(24), 11622–11630 (2006). [CrossRef] [PubMed]
4. K. C. Chang and T. Tamir, “Simplified approach to surface-wave scattering by blazed dielectric gratings,” Appl. Opt. 19(2), 282–288 (1980). [CrossRef] [PubMed]
5. F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Lightwave Technol. 25(1), 151–156 (2007). [CrossRef]
6. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002). [CrossRef]
7. M. K. Emsley, O. Dosunmu, and M. Ünlü, “Silicon substrates with buried distributed Bragg reflectors for resonant cavity-enhanced optoelectronics,” IEEE Quantum Electron. 8(4), 948–955 (2002). [CrossRef]
8. R. Orobtchouk, A. Layadi, H. Gualous, D. Pascal, A. Koster, and S. Laval, “High-efficiency light coupling in a submicrometric silicon-on-insulator waveguide,” Appl. Opt. 39(31), 5773–5777 (2000). [CrossRef] [PubMed]
9. X. Xiao, B. Yang, T. Chu, J.-Z. Yu, Y.-D. Yu, and N. Anastasia, “Design and characterization of a top cladding for silicon-on-insulator grating coupler,” Chin. Phys. Lett. 29(11), 114213 (2012). [CrossRef]
10. M. Tokushima, J. Ushida, T. Uemura, and K. Kurata, “Shallow-grating coupler with optimized anti-reflection coating for high-efficiency optical output into multimode fiber,” Appl. Phys. Express 8(9), 092501 (2015). [CrossRef]
11. J. Ushida, M. Tokushima, T. Uemura, and K. Kurata, “Highly fabrication-tolerant shallow-grating coupler for robust coupling to multimode “optical pin”,” in Proceedings of The 2015 International Conference on Solid State Devices and Materials, Sapporo, Japan, 27–30 Sept. 2015, paper PS-7–14.
12. A. Narasimha and E. Yablonovitch, “Efficient optical coupling into single mode silicon-on-insulator thin films using a planar grating coupler embedded in a high index contrast dielectric stack,” in Conference on Lasers and Electro-Optics, 2003 OSA Technical Digest (Optical Society of America, 2003), CWA46.
13. T. W. Ang, G. T. Reed, A. Vonsovici, A. G. R. Evans, P. R. Routley, T. Blackburn, and M. R. Josey, “Grating couplers using silicon-on-insulator,” Proc. SPIE 3620, 79–86 (1999). [CrossRef]
14. A. P. Zhao, A. V. Raisanen, and S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microw. Guided Wave Lett. 5(10), 341–343 (1995). [CrossRef]
15. M. Tokushima, J. Ushida, T. Uemura, and K. Kurata, “High-efficiency folded shallow-grating coupler with minimal back reflection toward isolator-free optical integration,” inProceedings of European Conference on Optical Communication (IEEE, 2015), P.2.20. [CrossRef]
16. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, 1986), p. 62.
