## Thin-film stack based integrated GRIN coupler with aberration-free focusing and super-high NA for efficient fiber-to-nanophotonic-chip coupling

Optics Express, Vol. 18, Issue 5, pp. 4574-4589 (2010)

http://dx.doi.org/10.1364/OE.18.004574

Acrobat PDF (582 KB)

### Abstract

Nanophotonic chip coupling using an optical thin-film stack forming a micro graded-refractive-index (GRIN) lens with a super-high numerical aperture (NA) that is highly compact (tens of micron long) and can be directly integrated is presented. We explore the lens’ integration on the surface of Silicon-On-Insulator (SOI) platform with an asymmetric GRIN profile. We show that to achieve high efficiency for optical coupling between an optical fiber and a nanophotonic waveguide with a sub-wavelength (λ/n) beam size, conventional asymmetric parabolic GRIN profile is no longer adequate due to the super-high NA needed (>3.1), which results in severe spatial beam aberration at the focal plane. We present an efficient algorithm to computationally generate the ideal GRIN profile that is completely aberration free even at super-high NA, which improves the coupling efficiency from ~66% (parabolic case) to ~95%. A design example involving an optical thin-film stack using an improved dual-material approach is given. The performance of the thin-film stack is analyzed. This thin-film stack based GRIN lens is shown to be high in coupling efficiency, wavelength insensitive and compatible with standard thin-film process.

© 2010 OSA

## 1. Introduction

7. D. Dai, S. He, and H. Tsang, “Bilevel mode converter between a silicon nanowire waveguide and a larger waveguide,” J. Lightwave Technol. **24**(6), 2428–2433 (2006). [CrossRef]

8. Y. Huang and S. T. Ho, “Superhigh numerical aperture (NA>1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. **30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

8. Y. Huang and S. T. Ho, “Superhigh numerical aperture (NA>1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. **30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

_{H}and n

_{L}) can be used to realize a GRIN profile with an effective arbitrary refractive index variation from n

_{0}to n

_{R}, which is referred to as the dual-material approach [see Fig. 1(b)]. With a proper design, the super-GRIN lens realized with the dual material approach can have a high numerical aperture [8

8. Y. Huang and S. T. Ho, “Superhigh numerical aperture (NA>1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. **30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

10. A. Delâge, S. Janz, B. Lamontagne, A. Bogdanov, D. Dalacu, D.-X. Xu, and K. P. Yap, “Monolithically integrated asymmetric graded and step-index couplers for microphotonic waveguides,” Opt. Express **14**(1), 148–161 (2006). [CrossRef] [PubMed]

**30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

**30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

## 2. Numerical aperture requirement, aberration issue and coupling between nanophotonic waveguide and optical fiber

^{2}of the beam intensity) at the wavelength of 1550 nm under different thickness of the nanophotonic waveguide (from 800 nm to 300 nm with a decrement of 100 nm). From Table 1, we see that when the thickness of the waveguide is 300nm (it is a singlemode waveguide used in nanophotonic integration with a waveguide width ~450 nm), the mode-field diameter is 380 nm and the divergence angle is as large as 64° (to encompass the 95% energy point).

_{R}at the top of the GRIN lens shall be given by

*D*of the GRIN lens provided that the refractive index contrast between n

_{0}and n

_{R}is given). The numerical aperture of a GRIN lens is given by

_{R}and the corresponding numerical aperture of the GRIN lens for different mode-field diameter of the waveguide mode is presented in Table 1, from which it can be seen that when the MFD is decreased to 380 nm, which corresponds to the sub-wavelength case (

_{R}is 1.53 and the corresponding NA of the GRIN lens is ~3.15.

_{0}=3.5, n

_{R}=1.5 (NA=3.16) and thickness D=10 μm. Light propagation in the parabolic asymmetric GRIN lens are investigated using ray tracing as well as full electromagnetic simulation using two-dimensional finite-difference-time-domain (FDTD) method respectively [11]. Figure 3(a) shows the ray-tracing in the asymmetric GRIN lens (from −60° to 60°, with a 5° step), from which we can see the rays are bent in the GRIN lens due to the super-high NA but there is substantial spatial aberration of the focused spot. The large-angle rays have shorter focusing length and they do not focus at the same point as the small-angle rays. In particular, the different focusing lengths make the paths of different rays to cross with each other after they are bent down from the peak. In the wave picture for an ideal GRIN lens, the beam shall form a “beam waist” with plane wavefront at both the nano-waveguide and the lens facet. However, the full electromagnetic simulation using FDTD method in the GRIN lens [see Fig. 3(b)] shows the severe interference while light beam propagating in the GRIN lens due to the different rays crossing with each other in the ray picture. In particular, the wavefront when the beam expended to near its maximum size never becomes a perfect vertical plane wavefront. The aberration in the wavefront results in poor coupling of the beam energy into the optical fiber.

*D*and length

*L*of the GRIN lens and also the fiber position (center of the fiber core)

*H*from the bottom of the GRIN lens. The location of the beam’s maximum size is approximately where the lens facet shall be or where one shall couple into an optical fiber. Using the coupling between a standard singlemode fiber (SMF28) and nanophotonic waveguide with a thickness of 300 nm as an example. The parameters for SMF28 fiber used are n

_{co}=1.4554, n

_{cl}=1.4447 and R=4.15 μm and with the parameter-space scanning approach, it is found that the maximal coupling efficiency can be achieved when D=15 μm, H=5.3 μm, and L=24.18 μm. With the optimized parameters, the corresponding normalized transverse light intensity profiles at the GRIN lens facet and the eigenmode of the optical fiber are shown in Fig. 4 . The severe ripples of the intensity profile at the facet of the GRIN lens are caused by the interference of light while propagating, as discussed above, due to the non-ideal parabolic graded refractive index profile. The maximal coupling efficiency from the nanophotonic waveguide to the optical fiber using this parabolic GRIN lens is estimated to be only 66%.

## 3. Aberration-free asymmetric super-GRIN lens: design algorithm and fiber to nanophotonic waveguide coupling

*P*. As shown in Fig. 5 , the design is to generate an ideal graded refractive index profile so that each ray emitting from the point source will be bent to be parallel exactly at the lens facet. This will occur at the half distance of the so-called “beat length” for a GRIN lens. This half-distance of the beat length will be referred to as the focusing length L

_{f}below.

*N*layers with a constant thickness of

*h*(

*h=D/N*) for each layer [see Fig. 5(a)]. Each layer is sufficiently thin so that the refractive index for each layer is regarded as a constant and denoted by n

_{i}(i=1,2,3…) for the i-th layer. The entire GRIN lens is designed if the refractive index of each layer n

_{i}is designed and then we let the layer thickness

*h*to become arbitrarily small to result in the design of a continuous index profile for the GRIN lens. For the asymmetric GRIN lens, the GRIN profile shall have gradually decreasing refractive index from the nano-waveguide to the lens’ top surface so that n

_{i}>n

_{i + 1}.

*n*of the first layer is first chosen to be equal to the refractive index of the nano-waveguide core, which is the refractive index of the material medium on the left of the GRIN lens containing the light source

_{1}*P*. This is done so that the optical reflection between the GRIN lens and nano-waveguide medium on its left is minimized when light propagates from the left side into the GRIN lens. After the first layer’s

*n*is chosen, the refractive index

_{1}*n*for the second layer can be determined using the condition that the ray shall become horizontal at distance

_{2}*L*from the point source [see Fig. 5(b)], thus its incident angle from layer

_{f}*n*to layer

_{1}*n*shall be at the critical angle

_{2}_{1}for the ray in layer

*n*shall be given by

_{1}*n*and layer n

_{1}_{2}(Note that this is equivalent to

*n*, which is obtained from Snell’s law at the critical angle). From Fig. 5(b), we can also see that

_{1}cos (θ_{1})=n_{2}cos(0=n_{2}*n*can be directly calculated by using this equation to solve for

_{2}*n*for the

_{i}*i-th*layer is presented in Fig. 5(c) and the incidence angle from the layer

*n*to layer

_{i-1}*n*is at the critical angle. We also have the relationship

_{i}*n*for the

_{i}*i*-th layer can be calculated with the equation below:

*D*,

*D*giving the GRIN lens’ height to be 10 μm and the desired focal length of the GRIN lens to be

*L*=10 μm. The layer numbers of the GRIN lens is chosen to be N=1000 (which is verified to be sufficiently large to converge to a continuous curve). After solving Eq. (1) for each layer, the refractive index profile obtained for the asymmetric GRIN lens is presented in Fig. 6 , for which the refractive index variation is from 3.5 to 1.458. As a comparison, the refractive index distribution based on a parabolic profile with the same refractive index variation range is also plotted in Fig. 6, from which it can be seen that the refractive index profile generated differs distinctly from the conventional parabolic profile.

_{f}*D*, length L and fiber position

*H*). In the design, the coupling efficiency to the SMF28 optical fiber is estimated using the overlap integral between the eigenmode of SMF28 and the expanded field through a 2D wide-angle beam propagation method (WA-BPM) [12

12. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. **17**(24), 1743–1745 (1992). [CrossRef] [PubMed]

_{f}until the computed coupling efficiency is the highest. The WA-BPM method, while good for use in parameter space optimization, however, cannot account for reflection losses, which can be accounted for by subsequently re-calculated with the FDTD method. For this numerical example, it is found that the maximal coupling efficiency is achieved when using the refractive index profile generated by D=15.6 μm and L

_{f}=16 μm. It has a non-parabolic refractive index profile ranging from 3.5 to 1.49 with a corresponding numerical aperture of NA=3.167. The length of GRIN lens is L=16.85 μm, and the vertical position of the center of the fiber core is 4.6 μm from the bottom of the GRIN lens. With this optimized parameters the corresponding maximal coupling efficiency is 96.4% according to the WA-BPM simulation.

## 4. Practical implementation of the super-GRIN lens using a thin-film stack, effect of layer thickness, and wavelength sensitivity

**30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

*n*such as silicon and the other material has a low refractive index

_{H}*n*such as silicon dioxide. The thickness of each layer is ranged from a few nanometers to tens of nanometers depending on the desired equivalent refractive index we want to achieve.

_{L}*f*can be different for different discretized layer

_{j}*j*and

*f*is chosen so that

_{Tj}*(f*, where

_{j}/f_{Tj})=C_{j}*C*is an integer describing the number of dual material pair within the j

_{j}^{th}discretized layer. There are thus many ways to implement the dual-material design for a given GRIN profile

*n(y).*To further simplify the design algorithm, in an algorithm we adopt, we choose

*C*=1 so each discretized layer is approximated only by one dual material pair and we further choose

_{j}*f*to be the same value for each of the discretized layer. Thus if we have

_{j}=f*M*discretized layer then

*f=D/M*. The thin-film stack designed has a total layer number of

*N*as each discretized layer is replaced by two thin-film layers.

_{f}=2M**30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

*f*(different thin-film layer number N

_{f}). Light propagation within the thin-film stack based GRIN lens is simulated using 2D FDTD. Corresponding light propagation patterns are shown in Fig. 10 from (a) to (d) When

*f=300 nm*(

*N*),

_{f}=104*f=250 nm (N*). These numerical results indicate that the two-material thin-film stack act as a GRIN lens and expands the light beam with a sub-wavelength spotsize from nanophotonic waveguide as expected.

_{f}=124), 200 nm (N_{f}=156), 150 nm (N_{f}=208**30**(11), 1291–1293 (2005). [CrossRef] [PubMed]

*f*is around 150 nm, the coupling efficiency reaches the maximal value of ~92%. When increasing the

*f*value to be over 250 nm, i.e., using a thin-film stack with a thicker layer but fewer layer number N

_{f}, the coupling efficiency decreases because of the scattering loss. According to these simulation results, the thin-film stack with a coupling efficiency >90% requires 100~200 layer, which is compatible with current optical thin-film (such as DWDM device) coating technology. To further increase the layer number of the thin-film stack will result in less scattering loss, but will increase the fabrication challenge. Hence, there is an optimal point due to practical consideration. Based on this layout design and performance evaluation, we can choose the optimal layout for this example i.e., f=150 nm and the corresponding binary refractive index profile is presented in Fig. 12 , with which the coupling between the nanophotonic waveguide and optical fiber is ~92%.

## 5 Conclusion

## Reference and links

1. | V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Optic. Lett. Vol. 28, pp. 1302–1304, 2003. |

2. | L. Vivien, X. L. Roux, S. Laval, E. Cassan, and D. Marris-Morini, “Design, Realization, and characterization of 3-D Taper for Fiber/Micro- Waveguide Coupling,” IEEE J. Sel. Top. Quantum Electron. |

3. | 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 coupling for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. |

4. | D. Tailaert, W. Bogaerts, and R. Baets, “Efficient coupling between submicron SOI-waveguides and single-mode fibers”, Proc. Symposium IEEE/LEOS, 2003. |

5. | B. Wang, J. Jiang, and G. P. Nordin, “Embedded slanted grating for vertical coupling between fibers and silicon-insulator planar waveguides,” IEEE Photon. Technol. Lett. |

6. | N. Izhaky, M. T. Morse, S. Koehl, O. Cohen, D. Rubin, A. Barkai, G. Sarid, R. Cohen, and M. J. Paniccia, “Development of CMOS-Compatible integrated silicon photonics devices,” IEEE Sel. Top. Quantum Electron. |

7. | D. Dai, S. He, and H. Tsang, “Bilevel mode converter between a silicon nanowire waveguide and a larger waveguide,” J. Lightwave Technol. |

8. | Y. Huang and S. T. Ho, “Superhigh numerical aperture (NA>1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. |

9. | R. Sun, V. Nguyen, A. Agarwal, C. Hong, J. Yasaitis, and L. Kimerling. AndJ. Michel, “High performance asymmetric graded index coupler with integrated lens for high index waveguides,” Appl. Phys. Lett. |

10. | A. Delâge, S. Janz, B. Lamontagne, A. Bogdanov, D. Dalacu, D.-X. Xu, and K. P. Yap, “Monolithically integrated asymmetric graded and step-index couplers for microphotonic waveguides,” Opt. Express |

11. | A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House Inc, 2000). |

12. | G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 4, 2009

Revised Manuscript: November 12, 2009

Manuscript Accepted: November 17, 2009

Published: February 22, 2010

**Citation**

Qian Wang, Yingyan Huang, Ter-Hoe Loh, Doris Keh Ting Ng, and Seng-Tiong Ho, "Thin-film stack based integrated GRIN coupler with aberration-free focusing and super-high NA for efficient fiber-to-nanophotonic-chip coupling," Opt. Express **18**, 4574-4589 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4574

Sort: Year | Journal | Reset

### References

- V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Optic. Lett. Vol. 28, pp. 1302–1304, 2003.
- L. Vivien, X. L. Roux, S. Laval, E. Cassan, and D. Marris-Morini, “Design, Realization, and characterization of 3-D Taper for Fiber/Micro- Waveguide Coupling,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1354–1358 (2006). [CrossRef]
- 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 coupling for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38(7), 949–955 (2002). [CrossRef]
- D. Tailaert, W. Bogaerts, and R. Baets, “Efficient coupling between submicron SOI-waveguides and single-mode fibers”, Proc. Symposium IEEE/LEOS, 2003.
- B. Wang, J. Jiang, and G. P. Nordin, “Embedded slanted grating for vertical coupling between fibers and silicon-insulator planar waveguides,” IEEE Photon. Technol. Lett. 17(9), 1884–1886 (2005). [CrossRef]
- N. Izhaky, M. T. Morse, S. Koehl, O. Cohen, D. Rubin, A. Barkai, G. Sarid, R. Cohen, and M. J. Paniccia, “Development of CMOS-Compatible integrated silicon photonics devices,” IEEE Sel. Top. Quantum Electron. 12(6), 1688–1698 (2006). [CrossRef]
- D. Dai, S. He, and H. Tsang, “Bilevel mode converter between a silicon nanowire waveguide and a larger waveguide,” J. Lightwave Technol. 24(6), 2428–2433 (2006). [CrossRef]
- Y. Huang and S. T. Ho, “Superhigh numerical aperture (NA>1.5) micro gradient-index lens based on a dual-material approach,” Opt. Lett. 30(11), 1291–1293 (2005). [CrossRef] [PubMed]
- R. Sun, V. Nguyen, A. Agarwal, C. Hong, J. Yasaitis, and L. Kimerling. AndJ. Michel, “High performance asymmetric graded index coupler with integrated lens for high index waveguides,” Appl. Phys. Lett. 90, 1–3 (2007).
- A. Delâge, S. Janz, B. Lamontagne, A. Bogdanov, D. Dalacu, D.-X. Xu, and K. P. Yap, “Monolithically integrated asymmetric graded and step-index couplers for microphotonic waveguides,” Opt. Express 14(1), 148–161 (2006). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech House Inc, 2000).
- G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17(24), 1743–1745 (1992). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1 |
Fig. 2 |
Fig. 3 |

Fig. 4 |
Fig. 5 |
Fig. 6 |

Fig. 7 |
Fig. 8 |
Fig. 9 |

Fig. 10 |
Fig. 11 |
Fig. 12 |

Fig. 13 |
||

« Previous Article | Next Article »

OSA is a member of CrossRef.