Abstract

A new generation of Silicon-on-Insulator fiber-to-chip grating couplers which use a silicon overlay to enhance the directionality and thereby the coupling efficiency is presented. Devices are realized on a 200mm wafer in a CMOS pilot line. The fabricated fiber couplers show a coupling efficiency of −1.6dB and a 3dB bandwidth of 80nm.

© 2010 OSA

1. Introduction

In the last decade much research has focused on silicon photonic integrated circuits, due to the scaling in size that can be accomplished because of the high refractive index contrast available on the SOI material platform. However, the scaling of the device cross-section complicates the interfacing with a single mode optical fiber. Several schemes have been proposed to tackle this problem. We focused particularly on fiber-to-chip coupling by means of subwavelength structured surfaces such as line gratings or photonic crystals. With these grating couplers one can couple out-of-plane and thus test photonic circuits on a wafer-scale without the need of any post-processing or cleaving. Furthermore because of a relaxed 1dB alignment sensitivity of 2µm, it is even possible to align several optical fibers at the same time by using fiber arrays for wafer-scale testing or packaging. Despite the fact that these grating couplers are a very good candidate for fiber interfacing for the low cost and high volume oriented silicon photonics framework, they suffer from relatively high coupling loss, making them only useful for research purposes or specific applications with a relaxed power budget.

In 2002 we reported an SOI grating coupler with a fiber coupling efficiency of −7dB [1]. Since then, a lot of attempts were made to improve the coupling efficiency significantly. In general there are three factors that contribute to the poor coupling efficiency of standard grating couplers. Most of the light 35%−45% is lost because of diffraction towards the substrate and around 20% is lost because of poor mode-matching to the Gaussian shaped mode of the fiber. For perfectly vertical coupling another 30% is reflected back into the waveguide due to second order Bragg diffraction. This last issue can be easily solved by coupling under a small angle which breaks the symmetry and eliminates the reflection [2]. An alternative method is to etch a deep slit in front of the grating which acts together with the grating as a Fabry-Perot cavity [3]. By chirping the grating, back reflections [4] and losses due to mode-mismatch [2] can be minimized. Most attempts to increase the directionality of the grating all involve substrate engineering [5]. Other approaches are either based on amorphous silicon, which limits the thermal budget [6], or require a lot of post-processing [7]. Alternatively, one can use exotic gratings with for example slanted facets, in analogy to a blazed grating, to enhance the coupling to the first diffraction order in the superstrate [8]. In 2006 however, we proposed a rather simple and elegant way to adapt the grating and increase the coupling efficiency drastically, i.e. by defining a silicon overlay prior to etching the grating [9]. Early attempts which involved epitaxial silicon growth were promising and showed efficiencies up to −2.6dB [10]. In this paper we demonstrate high efficiency grating couplers, fabricated in a CMOS pilot line, with −1.6dB coupling efficiency, approaching the coupling efficiencies required in practical applications.

2. Fiber Coupler Grating Design

For designing a fiber-to-chip grating coupler, which is inherently highly efficient, several design issues have to be taken into account. These are summarized in Fig. 1. The first order diffraction of a grating is described by the Bragg condition βK = kz where the propagation constant of the guided mode β=2πλneff , K=2πΛ and the projected wave vector of the incident mode kz=2πλsinθ with λ the wavelength of the light, Λ the period of the grating and θ the angle of the fiber with respect to the surface normal of the grating. In order to avoid a high second order reflection, it is convenient to couple in under a positive fiber angle as shown in Fig. 1(a). In general the effective refractive index neff of the grating is a function of the grating parameters shown in Fig. 2(a). However, since for a given grating period and given fiber angle it is almost always possible to find a wavelength at which the Bragg condition is fulfilled, this dependency will have a minor influence on the maximal coupling efficiency of the grating coupler but rather on the central wavelength of the coupling spectrum. In the rest of the analysis we will assume that the Bragg condition is satisfied and that the fiber coupling angle θ is close to zero.

 

Fig. 1. Schematic illustrations of fiber-to-chip high index contrast grating coupler optimization factors: the Bragg condition (a), upwards directionality (b), 2D fiber mode-matching in z- and x-direction (c,d) and mode-matching between the waveguide and Bloch mode of the grating (e).

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The physical principle behind a highly directional grating as shown in Fig. 1(b) can be understood as follows. Considering the diffracted field pattern as the superposition of the fields emitted by an array of scattering centers (which have a π phase shift with respect to each other for a perfect vertically coupling grating), constructive interference towards the superstrate (and hence the optical fiber) can be achieved by realizing an additional π phase shift during the propagation towards the superstrate, since light is propagating either in air (in the etched slit) or in silicon (in the grating tooth). This directionality is thus a function of the waveguide thickness h, the etch depth e in the waveguide and the silicon overlay thickness o (see Fig. 2). Furthermore, in order to reach high fiber coupling efficiency, a highly directional grating is required while at the same time the grating strength needs to be optimized for maximal overlap with the Gaussian fiber mode, shown in Fig. 1(c). This mode-matching requirement is a function of the same three parameters h, e and o as the high directionality condition. For high index contrast gratings, the strength of the alternating scattering centers is in general not equal, resulting in an imbalanced interference to the superstrate. The duty cycle of the grating can be used as a tuning parameter to compensate and maximize the grating directionality. Three different classes of grating cases shown in Fig. 2 can be considered which can be favorable, depending on the considered photonics platform, robustness and cost considerations.

The first case Fig. 2(b) is a grating without silicon overlay where high directionality and an optimal grating strength are only achieved for a certain waveguide thickness h and etch depth e. In [11] this grating type is optimized and a coupling efficiency of −1.2dB is achieved for a waveguide thickness h of 340nm and a deep waveguide etch depth e of 200nm. Important drawbacks are the fixed waveguide thickness which is multimode and the deep waveguide etch depth which introduces a large mode mismatch between the waveguide mode and the Bloch mode of the grating resulting in high back reflections [see Fig. 1(e)]. The last issue is improved by apodizing the grating, leading to reduced reflections and a better mode matching with the Gaussian fiber mode. However, an apodized grating requires feature sizes which are beyond the limit of current 193nm DUV lithography.

 

Fig. 2. Parameterized highly directional grating couplers: the general case (a), without silicon overlay (b), with silicon overlay but no waveguide etch (c) and both silicon overlay and waveguide etch (d).

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For case Fig. 2(c), no waveguide etch e is required. This kind of grating coupler has been theoretically reported in [12], where a transferred silicon nanomenbrane is proposed in order to fabricate the grating. For a waveguide thickness h of 220nm and a silicon overlay thickness o of 240nm the simulated efficiency was 64%. We optimized this grating type and simulated for a uniform grating with h = 150nm and o = 220nm around 80% coupling efficiency, which is the theoretical upper limit because of the mode mismatch between the exponential shaped electrical field profile of the uniform grating and the Gaussian shaped mode of the optical fiber. Unfortunately, a waveguide thickness of h = 150nm leads to a lower confinement of the TE optical mode, and does not support any TM polarized waveguide modes. We also like to point out that the coupling strength of a grating with no waveguide etch is solely dependent on the silicon overlay and thus on the evanescent field of the waveguide mode. Because this evanescent field is different for orthogonal polarized waveguide modes, it is unlikely that there exists a grating which is highly efficient for both polarizations. Despite these drawbacks, this grating coupler type could be an ideal candidate for using in a single-polarization low-cost photonics platform.

The most general fiber coupler grating using a silicon overlay is case Fig. 2(d) where both a silicon overlay o and waveguide etch e is used to define the grating. This gives us the freedom to choose an appropriate waveguide thickness h of for example 220nm. This high efficiency grating coupler type is theoretically described in [9] and shows around 80% of coupling efficiency according to simulations. Although it requires more process steps than the previous cases, it is compatible with any high index contrast photonic platform. Moreover, the introduced additional thickness levels could be used to optimize other integrated photonic devices.

An optimization issue that is unrelated to the previous discussion is the mode-matching between the Gaussian fiber mode and the waveguide mode in the x-direction as is shown in Fig. 1(d). For a 220nm thick waveguide, the optimal waveguide width w is 15 µm. Since the taper length scales quadratic with the waveguide width, a long taper section (≈ 500 µm) would be needed to adiabatically transform the broad waveguide of the grating into a single-mode optical waveguide. This can be easily solved by curving the grating as is demonstrated in [13], without decreasing the grating coupling efficiency.

3. Fabrication

We developed an advanced Silicon-on-Insulator process flow which is capable of realizing the most general case in Fig. 2(d) of the high efficiency grating couplers. The complete fabrication process is done in the 200mm imec CMOS pilot line with 193nm DUV lithography. We start from a 200mm SOI wafer with a buried oxide layer thickness of 2µm and a crystalline silicon layer thickness of 220nm. Simulations show that the thickness of the buried oxide layer is of minimal importance for the coupling efficiency, since the grating intrinsically is very directional such that only a small fraction of the light is reflected at the interface of the buried oxide and silicon substrate [9]. In a second step an amorphous silicon layer of 160nm is deposited over the whole wafer and a deep etch is performed through the amorphous silicon reaching 70nm into the crystalline silicon layer. This is used to define the slits of the high efficiency grating couplers. Next, the amorphous silicon is removed where necessary by dry etching using a combination of a chlorine/fluorine-based and bromine-based chemistry. The width of the first tooth will not affect the grating coupling efficiency if it is narrower than the subsequent teeth [9]. By contrast, larger first tooth widths introduce a rapid decrease of the coupling efficiency. The consequences of possible mask misalignment between the silicon overlay and the grating coupler are anticipated by designing the first tooth of the grating narrower. This can be clearly seen in Fig. 3 where a bird’s eye view image of the fabricated silicon overlay grating is shown. In the final step the strip waveguides are defined in the crystalline silicon layer. The measured propagation loss for a straight single-mode strip waveguide with a height of 220nm and a width of 450nm was 1.8dB/cm, demonstrating the high process quality of this advanced SOI fabrication process.

 

Fig. 3. Bird’s eye view of a silicon overlay grating fiber coupler.

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4. Experimental Results

The measurement results are shown in Fig. 4. All measurements were performed with a single-mode fiber tilted under an angle of 13 degrees relative to the surface normal of the grating. Index matching fluid was applied between the grating and fiber facet to reduce Fresnel reflections. TE polarized light is used in all experiments, because the gratings are polarization dependent. The coupling efficiency was determined by measuring the fiber-to-fiber insertion loss of two grating couplers connected with a 300µm long and 15µm broad waveguide. Figure 4(a) shows the central wavelength of the fiber coupling spectrum as a function of the grating period and this for several duty cycles. The central wavelength scales almost linearly with the grating period because the effective refractive index of the grating is only weakly dependent upon the wavelength. In Fig. 4(b), the coupling efficiency for different grating periods is plotted versus the duty cycle. It seems that a duty cycle of 35% (refers to the unetched silicon part) is optimal and we measured a maximum efficiency of −1.6dB with a 1dB and 3dB bandwidth of respectively 44nm and 80nm for a grating period of 690nm and central wavelength of 1530nm. This is the highest fiber coupling efficiency reported to date for CMOS-compatible fiber-to-chip grating couplers.

 

Fig. 4. Measurement results of a sweep of 1D fiber couplers (a-Si overlay) for different duty cycles (a) and different grating periods (b).

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5. Conclusion

Different types of high efficiency grating couplers which are inherently highly directional are discussed. The most general case of a grating coupler with silicon overlay and etch into the optical waveguide is fabricated using 193nm DUV lithography. We have shown experimentally a coupling efficiency of −1.6dB which is the highest fiber coupling efficiency for grating couplers fabricated in a CMOS pilot line. The optimal duty cycle of the grating is found to be 35%. The amorphous silicon is stable up to a temperature of 500°C. As an alternative a polysilicon overlay could be used which is stable up to a temperature of 1000°C. A silicon overlay could also be beneficial for more advanced grating couplers which incorporate polarization diversity and wavelength demultiplexing [14]. Besides the grating couplers, the silicon overlay can be used in other components on the SOI chip as an extra degree of design freedom.

Acknowledgements

Diedrik Vermeulen thanks the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT) for a grant. W. Bogaerts and G. Roelkens acknowledge the Fund for Scientific Research Vlaanderen (FWO) for a postdoctoral grant. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no 224312 HELIOS and from Ghent University through the IOF STARCHIP grant.

References and links

1. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002). [CrossRef]  

2. D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29, 2749–2751 (2004). [CrossRef]   [PubMed]  

3. G. Roelkens, D. Van Thourhout, and R. Baets, “High Efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32, 1495–1497 (2007). [CrossRef]   [PubMed]  

4. X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008). [CrossRef]  

5. L. Vivien, D. Pascal, S. Lardenois, D. Marris-Morini, E. Cassan, F. Grillot, S. Laval, J.M. Fédéli, and L. El Melhaoui, “Light injection in SOI microwaveguides using high-efficiency grating couplers,” J. Lightwave Technol. 24, 3810–3815 (2006). [CrossRef]  

6. S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

7. F. Van 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, 151–156 (2007). [CrossRef]  

8. J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007). [CrossRef]  

9. G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14, 11622–11630 (2006). [CrossRef]   [PubMed]  

10. G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008). [CrossRef]  

11. X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

12. T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009). [CrossRef]  

13. F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

14. C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009). [CrossRef]  

References

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  1. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
    [Crossref]
  2. D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29, 2749–2751 (2004).
    [Crossref] [PubMed]
  3. G. Roelkens, D. Van Thourhout, and R. Baets, “High Efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32, 1495–1497 (2007).
    [Crossref] [PubMed]
  4. X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
    [Crossref]
  5. L. Vivien, D. Pascal, S. Lardenois, D. Marris-Morini, E. Cassan, F. Grillot, S. Laval, J.M. Fédéli, and L. El Melhaoui, “Light injection in SOI microwaveguides using high-efficiency grating couplers,” J. Lightwave Technol. 24, 3810–3815 (2006).
    [Crossref]
  6. S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).
  7. F. Van 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, 151–156 (2007).
    [Crossref]
  8. J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
    [Crossref]
  9. G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14, 11622–11630 (2006).
    [Crossref] [PubMed]
  10. G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
    [Crossref]
  11. X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).
  12. T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009).
    [Crossref]
  13. F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).
  14. C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
    [Crossref]

2009 (4)

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009).
[Crossref]

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

2008 (2)

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
[Crossref]

2007 (3)

2006 (2)

2004 (1)

2002 (1)

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Ayre, M.

Baets, R.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
[Crossref]

F. Van 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, 151–156 (2007).
[Crossref]

G. Roelkens, D. Van Thourhout, and R. Baets, “High Efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32, 1495–1497 (2007).
[Crossref] [PubMed]

G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29, 2749–2751 (2004).
[Crossref] [PubMed]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Bienstman, P.

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29, 2749–2751 (2004).
[Crossref] [PubMed]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Bogaerts, W.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Brision, S.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

Cassan, E.

Chen, L.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

Chen, X.

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
[Crossref]

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

Chen, Y. K.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

Claes, T.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

De Mesel, K.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Doerr, C. R.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

Dumon, P.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

Earnshaw, M.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

El Melhaoui, L.

Fédéli, J. M.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

Fédéli, J.M.

Fung, K.

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

Gautier, P.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

Grillot, F.

Krauss, T. F.

F. Van 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, 151–156 (2007).
[Crossref]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Lardenois, S.

Laval, S.

Li, C.

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
[Crossref]

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

Lo, S. M. G.

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

Lyan, P.

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

Marris-Morini, D.

Moerman, I.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

O’Faolain, L.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

Pascal, D.

Rasras, M.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

Roelkens, G.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

F. Van 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, 151–156 (2007).
[Crossref]

G. Roelkens, D. Van Thourhout, and R. Baets, “High Efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32, 1495–1497 (2007).
[Crossref] [PubMed]

G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

Saha, T. K.

T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009).
[Crossref]

Schaekers, M.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

Scheerlinck, S.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

Schrauwen, J.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
[Crossref]

F. Van 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, 151–156 (2007).
[Crossref]

Selvaraja, S. K.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

Sleeckx, E.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

Taillaert, D.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

F. Van 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, 151–156 (2007).
[Crossref]

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Lett. 29, 2749–2751 (2004).
[Crossref] [PubMed]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Tsang, H. K.

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
[Crossref]

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

Van Dale, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Van Laere, F.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
[Crossref]

F. Van 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, 151–156 (2007).
[Crossref]

Van Thourhout, D.

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
[Crossref]

F. Van 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, 151–156 (2007).
[Crossref]

G. Roelkens, D. Van Thourhout, and R. Baets, “High Efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32, 1495–1497 (2007).
[Crossref] [PubMed]

G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

Vermeulen, D.

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

Verstuyft, S.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

Vivien, L.

Weiner, J.

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

Zhou, W.

T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009).
[Crossref]

Appl. Phys. Lett. (1)

G. Roelkens, D. Vermeulen, D. Van Thourhout, R. Baets, S. Brision, P. Lyan, P. Gautier, and J. M. Fédéli, “High efficiency diffractive grating couplers for interfacing a single mode optical fiber with a nanophotonic silicon-on-insulator waveguide circuit,” Appl. Phys. Lett. 92, 131101 (2008).
[Crossref]

IEEE J. Quantum Electron. (1)

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Dale, 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. 14, 949–955 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (4)

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20, 1914–1916 (2008).
[Crossref]

J. Schrauwen, F. Van Laere, D. Van Thourhout, and R. Baets, “Focused-ion-beam fabrication of slanted grating couplers in silicon-on-insulator waveguides,” IEEE Photon. Technol. Lett. 19, 816–818 (2007).
[Crossref]

F. Van Laere, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, D. Taillaert, L. O’Faolain, D. Van Thourhout, and R. Baets, “Compact focusing grating couplers for silicon-on-insulator integrated circuits,” IEEE Photon. Technol. Lett. 27, 612–618 (2009).

C. R. Doerr, L. Chen, M. Rasras, Y. K. Chen, J. Weiner, and M. Earnshaw, “Diplexer With Integrated Filters and Photodetector in Ge/Si Using Γ − X and Γ − M Directions in a Grating Coupler,” IEEE Photon. Technol. Lett. 21, 1698–1700 (2009).
[Crossref]

J. Lightwave Technol. (2)

J. Phys. D: Appl. Phys. (1)

T. K. Saha and W. Zhou, “High efficiency diffractive grating coupler based on transferred silicon nanomembrane overlay on photonic waveguide,” J. Phys. D: Appl. Phys. 42, 085115 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Other (2)

S. K. Selvaraja, D. Vermeulen, M. Schaekers, E. Sleeckx, W. Bogaerts, G. Roelkens, P. Dumon, D. Van Thourhout, and R. Baets, “Highly efficient grating coupler between optical fiber and silicon photonic circuit,” in Lasers and Electro-Optics/International Quantum Electronics Conference, (2009).

X. Chen, C. Li, S. M. G. Lo, K. Fung, and H. K. Tsang, “Silicon Waveguide Grating Couplers with Engineered Coupling Strength for Optimized Coupling,” in European Conference on Integrated Optics, (2010).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic illustrations of fiber-to-chip high index contrast grating coupler optimization factors: the Bragg condition (a), upwards directionality (b), 2D fiber mode-matching in z- and x-direction (c,d) and mode-matching between the waveguide and Bloch mode of the grating (e).

Fig. 2.
Fig. 2.

Parameterized highly directional grating couplers: the general case (a), without silicon overlay (b), with silicon overlay but no waveguide etch (c) and both silicon overlay and waveguide etch (d).

Fig. 3.
Fig. 3.

Bird’s eye view of a silicon overlay grating fiber coupler.

Fig. 4.
Fig. 4.

Measurement results of a sweep of 1D fiber couplers (a-Si overlay) for different duty cycles (a) and different grating periods (b).

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