## Monolithically integrated asymmetric graded and step-index couplers for microphotonic waveguides

Optics Express, Vol. 14, Issue 1, pp. 148-161 (2006)

http://dx.doi.org/10.1364/OPEX.14.000148

Acrobat PDF (767 KB)

### Abstract

A monolithically integrated asymmetric graded index (GRIN) or step-index (GRIN) mode converters for microphotonic waveguides are proposed and described. The design parameters and tolerances are calculated for amorphous silicon (a-Si) couplers integrated with silicon-on-insulator waveguides. The GRIN and step-index couplers operate over a wide wavelength range with low polarization dependence, and the lithographic resolution needed is only ±1 μm. Finally, experimental results are presented for a single layer 3 μm thick step-index a-Si coupler integrated on a 0.8 μm thick SOI waveguide. The measured variation of coupling efficiency with coupler length is in agreement with theory, with an optimal coupling length of 15 μm for this device.

© 2006 Optical Society of America

## 1. Introduction

_{x}N

_{y}waveguides, the two commonly used platforms for compact high index contrast waveguide circuits and photonic crystal devices, the waveguide mode diameter is often of the order of one micron or less. In the absence of an effective mode converter, the mode mismatch losses between the waveguide and typical fiber mode or free space beam are high. Many coupler designs have been proposed and demonstrated, based for example on two dimensional structures such as inverse tapers [1–3

1. V.R. Almeida, R.R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**, 1302–1304 (2003). [CrossRef] [PubMed]

4. G.Z. Masanovic, V.M.N. Passaro, and G.T. Reed, “Dual grating-assisted directional coupling between fibers and thin semiconductor waveguides,” IEEE Photonics Technol. Lett. **15**, 1395–1397 (2003). [CrossRef]

5. A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Optics Express **11**, 3555–3561 (2003). [CrossRef] [PubMed]

2. T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. **38**, 1669–1670 (2002). [CrossRef]

3. K.K. Lee, D.R. Lim, D. Pan, C. Hoepfner, W.-Y. Oh, K. Wada, L.C Kimmerling, K.P. Yap, and M.T. Doan, “Mode transformer for miniaturized optical circuits,” Opt. Lett. **30**, 498–500 (2005). [CrossRef] [PubMed]

6. C. Manolatou and H.A. Haus, *Passive components for Dense Optical Integration* (Kluwer Academic Publishers, Boston, 2002), Chapter 6. [CrossRef]

_{x}O

_{y}structure [7]. Such GRIN structures are the planar analogue of the cylindrical GRIN lens used in fiber optic and optoelectronics packaging. Conventional and planar GRIN lenses ideally will have a quadratic index profile peaked at the center on the lens [8]. Coupling is achieved by aligning the center plane of the symmetric GRIN structure with the waveguide core of the device, either in a hybrid integration scheme [7], or by the overgrowth of the lens structure in a previously formed trench with the correct depth to achieve the required alignment [6

6. C. Manolatou and H.A. Haus, *Passive components for Dense Optical Integration* (Kluwer Academic Publishers, Boston, 2002), Chapter 6. [CrossRef]

10. A. Delâge, S. Janz, D.-X. Xu, D. Dalacu, B. Lamontagne, and A. Bogdanov, in *Photonics North 2004: Optical Components and Devices*,
J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. “Graded-index coupler for microphotonic waveguides,” Proc. SPIE **Vol. 5577**, 204–212 (2004). [CrossRef]

*n*< 3.47 at λ = 1550 nm) or for Si

_{3}N

_{4}waveguides (1.5 <

*n*< 2.0 at λ = 1550 nm) are easily obtained using well-calibrated PECVD deposition of amorphous silicon (a-Si) or SiO

_{x}N

_{y}. Finally we present an experimental demonstration of a single layer amorphous silicon (a-Si) step-index waveguide coupler monolithically integrated with an SOI waveguide.

## 2. Theory

_{0}and x

_{0}determine the magnitude and spatial variation of the refractive index. A waveguide with this index profile will support orthogonal waveguide modes with field profiles given by Hermite-Gaussian functions

_{v}(x) is the Hermite polynomial of order

*v*and

*k*

_{0}is the free space wavevector [8]. The corresponding mode wavevectors are given by

*c*are the expansion coefficients determined by initial conditions (i.e. the input field profile launched into the structure), while the term exp(-

_{v}*iβ*) gives the phase evolution of the field along the z-axis or propagation direction. The focusing properties of a quadratic index waveguide or GRIN lens result from the evolution of the relative phase of these modes as they propagate.

_{v}z*c*= 0 for odd values of

_{v}*v*. If the combined expansion coefficients and phase term,

*c*exp(-

_{v}*iβ*), in Eq. (4) are all real and positive, then the mode superposition generates the tightly focused field profile shown in Fig. 2(a). Note that in this pedagogic example the expansion coefficients were arbitrarily chosen to satisfy

_{v}z*c*= (1/

_{v}*v*!). On the other hand when the phase of

*c*(-

_{v}exp*iβ*) for each successive symmetric mode differs by π (i.e. successive terms alternate in sign) the broad field distribution in Fig. 2(a) is generated for the same

_{v}z*c*values. For the lower order modes where

_{v}*v*<<

*k*, the wavevectors of successive even modes obtained from Eq. (3) are spaced by approximately equal increments of

_{0}n_{0}x_{0}*Δβ*~ 2/

*x*. Hence, if either one of the two field distributions shown in Fig. 2(a) are launched into the GRIN waveguide, as the light propagates the total field profile will periodically oscillate between the these two distributions with a period

_{0}*ΔL*= π

*x*, giving an effective lens focal length of

_{0}*f*= π

*x*/2. This is the same focal length obtained by the usual ray tracing calculation for a GRIN lens [10

_{0}10. A. Delâge, S. Janz, D.-X. Xu, D. Dalacu, B. Lamontagne, and A. Bogdanov, in *Photonics North 2004: Optical Components and Devices*,
J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. “Graded-index coupler for microphotonic waveguides,” Proc. SPIE **Vol. 5577**, 204–212 (2004). [CrossRef]

*k*~ 100, and the assumption of equally spaced modes is valid for at least the first five modes.

_{0}n_{0}x_{0}*x*= 0 as in Fig. 2(b), only the odd modes in Eq. (4) are allowed waveguide modes. When the weighting coefficients of these odd modes are all real and in phase, the narrow field distribution near the bottom edge of the waveguide in Fig. 2(b) is produced. When the successive modes are π out of phase, the broad distribution is generated. As in the case for the full GRIN structure, as the modes propagate in the asymmetric-GRIN structure the field distribution will oscillate between the focused and wide distribution shown in Fig. 2(b) with the same focal length

*f*= π

*x*/2 as the symmetric GRIN lens, and the starting field profile is reproduced at the self-imaging length 2π

_{0}*x*.

_{0}*M*are now the normalized orthogonal modes of the combined coupler-waveguide structure. The amplitude of each excited mode

_{v}(x)*c*is determined from a field overlap integral of the orthogonal waveguide modes with the source field

_{v}*S(x)*that is launched into the waveguide.

*c*are all real provided the launched field profile

_{v}*S(x)*is real at

*z*= 0, the input coupling efficiency

*η*is just the sum of the light power coupled to each waveguide mode, and can be expressed as a sum of the squared mode expansion coefficients:

_{in}*η*into the output waveguide is given by the overlap integral of the field

_{c}*E(x,L)*with the final output waveguide mode profile

*G(x)*at the end of the coupler.

*g*are the expansion coefficients of the output mode profile G(x) in terms of the GRIN waveguide modes M

_{v}_{v}(x). As discussed above, in an ideal quadratic index waveguide the mode coefficients

*c*required to match the wide input field profile

_{v}*S(x)*will tend to have alternating signs with incrementing mode number

*v*, while the coefficients

*g*needed to match the narrow output waveguide mode

_{v}*G(x)*will have the same sign. As a result the product

*c*in Eq. (7) will also alternate in sign as the mode index

_{v}g_{v}*v*increments. Hence from Eq. (7) the maximum coupling efficiency is obtained if the wave vectors

*β*are equally spaced by

_{v}*Δβ*=

*π/L*. Here the coupler length

*L*=

*π/Δβ*is just the effective focal length

*f*of the GRIN lens. On the other hand, in a realizable waveguide the lower waveguide boundary is not a perfect reflector, and the electric field will have an evanescent tail that extends into the lower cladding. Finally, the target waveguide under the GRIN layers will usually have a uniform refractive index which does not match the lower edge of the GRIN lens structure.

*β*plotted in Fig. 4. Examination of the mode shapes clearly shows that a linear in-phase superposition of modes must result in a tightly focussed field distribution, while an anti-phase superposition will yield a broad profile. The similarity in the modes in Fig. 3(a), (b) and (c), and those for an ideal quadratic index lens occurs because the quadratic index gradient confines the lower order modes to a region well away from the upper cladding interface, while the small evanescent tail at the bottom Si – SiO

_{v}_{2}interface represents a negligible perturbation to the mode shape and wave vector. The wave vector differences between successive modes

*β*in Fig. 4 are nearly constant for the first several modes because

_{v}*n*~ 100, with obvious deviations from linearity only occurring for

_{0}x_{0}k_{0}*v*> 6. This satisfies the criteria for focusing and imaging. This comparison demonstrates that the deviations from a quadratic index structure necessary to create a monolithic asymmetric-GRIN waveguide coupler of Fig. 3(c) have a negligible effect on the mode profiles and effective indices of the lowest order modes in a GRIN structure. In summary, when the lowest order modes dominate the launched field profile, the asymmetric-GRIN coupler waveguides in Fig. 3 retain the useful imaging and focusing properties of the ideal symmetric and asymmetric quadratic index structures.

*S(x)*having a Gaussian profile with a FWHM of 3.5 μm, centered on the coupler/waveguide structure. Only the power in the fundamental mode of the 0.5 μm thick output waveguide is considered. A coupling efficiency of 19% is obtained by direct coupling of this input mode to the 0.5 μm waveguide. Losses due to reflection at the waveguide facet are not included in these calculations.

*c*and

_{v}*g*for the first 10 modes in the truncated quadratic index structure and the GRIN waveguide coupler in Figs. 3((b) and (c) respectively. Also given are the calculated (using Eq. (7)) coupling efficiencies

_{v}*η*from the input mode to the coupler structure,

_{in}*η*from the coupler structure to the 0.5 μm silicon output waveguide (

_{out}*η*=∑

_{out}*g*

_{v}^{2}), and the overall coupling

*η*from input mode to the output waveguide. Both structures described in Table 1 behave much the same way as the idealized quadratic index lenses discussed previously. Since the signs of successive coefficients

_{c}*c*alternate, while the coefficients

_{v}*g*all have the same sign, Eq. (7) requires that successive values of the phase

_{v}*β*must increment by π to achieve efficient coupling. This condition is approximately satisfied for only the first six modes at the optimum coupler lengths of L = 11 μm for the truncated quadratic index structure and L = 12 μm for the GRIN coupler, but the power in the higher modes is small and contributes a negligible amount to the overall coupling efficiency. A coupling efficiency of

_{v}L*η*= 88% is calculated for the ideal truncated quadratic index structure of Fig. 3(b), while for the optimized asymmetric-GRIN coupler of Fig. 3(c), a coupling efficiency into the Si waveguide of

_{c}*η*= 78% is predicted.

_{c}10. A. Delâge, S. Janz, D.-X. Xu, D. Dalacu, B. Lamontagne, and A. Bogdanov, in *Photonics North 2004: Optical Components and Devices*,
J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. “Graded-index coupler for microphotonic waveguides,” Proc. SPIE **Vol. 5577**, 204–212 (2004). [CrossRef]

*c*and

_{v}*g*for the 5 first modes for the same input mode

_{v}*S(x)*used in the preceding section, as the index step Δn between the coupler layer and the silicon waveguide is varied. From the table, two observations are evident: the incident field couples primarily with the two first modes shown in Fig. 3(d), while higher modes (

*v*≥2) contribute little to the power coupled to the output waveguide. Equation (7) simplifies to

*η*~ |

_{c}*c*| + |

_{0}g_{0}*c*| and the coupling length L depends on the modes spacing

_{1}g_{1}*L*=

*π/k*where

_{0}Δβ*Δβ*can here be taken as the mode spacing between the first two modes. Despite the simplicity of this structure, a coupling efficiency of η

_{c}= 45% is achieved at

*Δn*= 0.10. As the number of layers forming the GRIN region is increased to more fully approximate a true quadratic index profile, the coupling efficiency will improve. Our previous simulations [10

*Photonics North 2004: Optical Components and Devices*,
J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. “Graded-index coupler for microphotonic waveguides,” Proc. SPIE **Vol. 5577**, 204–212 (2004). [CrossRef]

13. R.U. Ahmad, F. Pizzuto, G.S. Camarda, R.L. Espinola, H. Rao, and R.M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. **14**, 65–67 (2002). [CrossRef]

*Photonics North 2004: Optical Components and Devices*,
J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. “Graded-index coupler for microphotonic waveguides,” Proc. SPIE **Vol. 5577**, 204–212 (2004). [CrossRef]

## 3. Alignment and fabrication tolerances

## 4. Experiment

_{2}layer. Ridges were etched using reactive ion etching, to an etch depth of approximately 0.5 μm. The ridge waveguide width was 2.0 μm, but 400 μm long adiabatic tapers expanded to a 10 μm wide waveguide at the input facets shown in Fig. 7(b). Three-dimensional BPM simulations indicate that the coupling of such adiabatically tapered GRIN lenses is almost identical to that predicted by calculations for the case of a slab waveguide with an overlying GRIN layer.

_{2}was first deposited on top of the SOI waveguide layer by PECVD. The waveguide coupler sections were initially defined by etching a window in this SiO

_{2}layer over the waveguide adjacent to the eventual input facet position. Finally a 3 μm layer of a-Si was deposited by PECVD over the entire wafer, and subsequently removed everywhere except over the coupler sections. The remaining a-Si regions slightly overlap the boundaries of the oxide window so that the exact coupler length is defined by the oxide window dimensions, as indicated in Fig. 7(a). Since the dimensional uncertainty in patterning windows in the thin SiO

_{2}layer is much less than in the removal of the 3 μm thick a-Si layer, it is much easier to fabricate couplers of a specified length using this window masking process. Couplers of lengths ranging from 5 μm to 200 μm were fabricated, so that the dependence of coupling efficiency on coupler length could be measured. Although the monolithic GRIN coupling scheme does not require specialized lithography, it is clear from Fig. 6 that the final coupler length should be within a few microns of the optimum value. To achieve this objective, the vertical input facets were fabricated using a lithographically defined two step vertical etch process. A 10 μm deep inductively coupled plasma (ICP) etch was used to create the vertical waveguide facets, and a second deep etch was used to facilitate dicing. The details of facet fabrication will be described elsewhere. Once the facets were formed, the Si wafers were diced into 4 mm wide bars to expose the etched facet and allow coupling of light into the waveguide from an optical fiber.

*n*= 3.36 to 3.40. Using an assumed a-Si refractive index of n = 3.365, the calculated variation of coupling efficiencies with coupler length shown in Fig. 8 is in qualitative agreement with experiment with respect to both the coupling periodicity and variation of coupling magnitude. Future work will be directed at optimizing the a-Si deposition conditions and facet fabrication process in order to reduce the number of the defects, improved calibration of the a-Si refractive index of a-Si deposition, and finally the fabrication of multilayer GRIN lens for optimized fiber to waveguide coupling.

## 5. Summary

_{x}N

_{y}waveguides, and does not depend on high-resolution lithography or 3-dimensional fabrication techniques. The index range required to fabricate the asymmetric GRIN and step-index couplers is available using PECVD deposition of a-Si and a-Si

_{x}O

_{y}[7], as well as SiO

_{x}N

_{y}for which a similar step-index structure has recently been reported [14

14. V. Nguyen, T. Montalbo, C. Manolatou, A. Agarwal, Yasaitis, L.C. Kimmerling, and J. Michel, “Compact 3dB single mode fibre-to-waveguide coupler,” in *Proceedings of the 2nd International Conference on Group IV Photonics*, 195–197 (Institute of Electrical and Electronics Engineers, Piscataway, NJ, 2005).

## References and links

1. | V.R. Almeida, R.R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. |

2. | T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. |

3. | K.K. Lee, D.R. Lim, D. Pan, C. Hoepfner, W.-Y. Oh, K. Wada, L.C Kimmerling, K.P. Yap, and M.T. Doan, “Mode transformer for miniaturized optical circuits,” Opt. Lett. |

4. | G.Z. Masanovic, V.M.N. Passaro, and G.T. Reed, “Dual grating-assisted directional coupling between fibers and thin semiconductor waveguides,” IEEE Photonics Technol. Lett. |

5. | A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Optics Express |

6. | C. Manolatou and H.A. Haus, |

7. | K. Shiraishi, C.S. Tsai, H. Yoda, and K. Minagawa, “A micro-GRIN slab tip for integrating coupling between superfine-core waveguides and single mode fibers,” in |

8. | K. Iizuka, |

9. | H. Kogelnik, “Theory of Optical Waveguides,” in |

10. | A. Delâge, S. Janz, D.-X. Xu, D. Dalacu, B. Lamontagne, and A. Bogdanov, in |

11. | L.B. Soldano and E.C.M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. |

12. | R. Gordon, “Harmonic oscillation in a spatially finite array waveguide,” Opt. Lett. |

13. | R.U. Ahmad, F. Pizzuto, G.S. Camarda, R.L. Espinola, H. Rao, and R.M. Osgood, “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,” IEEE Photon. Technol. Lett. |

14. | V. Nguyen, T. Montalbo, C. Manolatou, A. Agarwal, Yasaitis, L.C. Kimmerling, and J. Michel, “Compact 3dB single mode fibre-to-waveguide coupler,” in |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(130.5990) Integrated optics : Semiconductors

(230.7370) Optical devices : Waveguides

**ToC Category:**

Integrated Optics

**Citation**

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**, 148-161 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-148

Sort: Journal | Reset

### References

- V.R. Almeida, R.R. Panepucci, and M. Lipson, "Nanotaper for compact mode conversion," Opt. Lett. 28, 1302-1304 (2003). [CrossRef] [PubMed]
- T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, "Low loss mode size converter from 0.3 µm square Si wire waveguides to singlemode fibres," Electron. Lett. 38, 1669-1670 (2002). [CrossRef]
- K.K. Lee, D.R. Lim, D. Pan, C. Hoepfner, W.-Y. Oh, K. Wada, L.C Kimmerling, K.P. Yap and M.T. Doan, "Mode transformer for miniaturized optical circuits," Opt. Lett. 30, 498-500 (2005). [CrossRef] [PubMed]
- G.Z. Masanovic, V.M.N. Passaro, and G.T. Reed, "Dual grating-assisted directional coupling between fibers and thin semiconductor waveguides," IEEE Photonics Technol. Lett. 15, 1395-1397 (2003). [CrossRef]
- A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. Prather, "Fabrication and characterization of three-dimensional silicon tapers," Optics Express 11, 3555-3561 (2003). [CrossRef] [PubMed]
- C. Manolatou and H.A. Haus, Passive components for Dense Optical Integration (Kluwer Academic Publishers, Boston, 2002), Chapter 6. [CrossRef]
- K. Shiraishi, C.S. Tsai, H. Yoda, and K. Minagawa, "A micro-GRIN slab tip for integrating coupling between superfine-core waveguides and single mode fibers," in Proceedings of CLEO/Pacific RIM 2003, CD-ROM (Institute of Electrical and Electronics Engineers, Piscataway, NJ, 2003).
- K. Iizuka, Engineering Optics, 2nd Edition (Springer-Verlag, Berlin 1987), Chapter 5.
- H. Kogelnik, "Theory of Optical Waveguides," in Guided-wave Optoelectronics, T. Tamir, ed., 7-87 (Springer Verlag, Berlin 1990)
- A. Delâge, S. Janz, D.-X. Xu, D. Dalacu, B. Lamontagne, and A. Bogdanov, in Photonics North 2004: Optical Components and Devices, J.C. Armitage, S. Fafard, R.A. Lessard, and G.A Lamprpoulos, eds. "Graded-index coupler for microphotonic waveguides," Proc. SPIE Vol. 5577, 204-212 (2004). [CrossRef]
- L.B. Soldano and E.C.M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]
- R. Gordon, "Harmonic oscillation in a spatially finite array waveguide," Opt. Lett. 29, 2752-2755 (2004). [CrossRef] [PubMed]
- R.U. Ahmad, F. Pizzuto, G.S. Camarda, R.L. Espinola, H. Rao, and R.M. Osgood, "Ultracompact corner-mirrors and T-branches in silicon-on-insulator," IEEE Photon. Technol. Lett. 14, 65-67 (2002). [CrossRef]
- V. Nguyen, T. Montalbo, C. Manolatou, A. Agarwal, Yasaitis, L.C. Kimmerling, and J. Michel, "Compact 3dB single mode fibre-to-waveguide coupler," in Proceedings of the 2nd International Conference on Group IV Photonics, 195-197 (Institute of Electrical and Electronics Engineers, Piscataway, NJ, 2005).

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.