## Compact slanted grating couplers

Optics Express, Vol. 12, Issue 15, pp. 3313-3326 (2004)

http://dx.doi.org/10.1364/OPEX.12.003313

Acrobat PDF (4107 KB)

### Abstract

We present a compact and efficient design for slanted grating couplers (SLGC’s) to vertically connect fibers and planar waveguides without intermediate optics. The proposed SLGC employs a strong index modulated slanted grating. With the help of a genetic algorithm-based rigorous design tool, a 20µm-long SLGC with 80.1% input coupling efficiency has been optimized. A rigorous mode analysis reveals that the phase-matching condition and Bragg condition are satisfied simultaneously with respect to the fundamental leaky mode supported by the optimized SLGC.

© 2004 Optical Society of America

## 1. Introduction

2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature **386** (6621), 143–149 (1997). [CrossRef]

4. C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. **17**, 1682 (1999). [CrossRef]

25. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. 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**, 949 (2002). [CrossRef]

_{x}single mode waveguide at a wavelength of 1550nm. The measured performance of a fabricated device was 19%. Their approach employed a symmetric, rectangular shaped grating for surface-normal operation. To break the grating symmetry and realize unidirectional coupling, an extra first-order grating reflector and a multilayer Bragg-mirror like bottom reflector were introduced, which complicates the device.

28. M. Li and S. J. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. **35**, 3101 (1996). [CrossRef]

31. A. V. Tishchenko, N. M. Lyndin, S. M. Loktev, V. A. Sychugov, and B. A. Usievich, “Unidirectional waveguide grating coupler by means of parallelogramic grooves,” SPIE **3099**, 269 (1997). [CrossRef]

33. J. Jiang, J. Cai, G. P. Nordin, and L. Li, “Parallel micro-genetic algorithm design of photonic crystal and waveguide structures,” Optics Letter. **28**, 2381 (2003). [CrossRef]

## 2. SLGC simulation and design method

34. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385 (1982). [CrossRef]

25. D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. 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**, 949 (2002). [CrossRef]

_{0}=1.55µm) is used as the ridge material for the grating. The high index contrast between the grating and waveguide materials together with a thick grating layer strongly disturbs the underlying waveguide mode in the coupling region. Because of this strong coupling mechanism, efficient coupling can occur with a short grating. The fiber we simulated is a single mode fiber with a core size of 8.3µm and core and cladding refractive indices of 1.470 and 1.4647, respectively. This fiber is simulated as a 2D slab waveguide. Berenger perfect matched layer (PML) boundary conditions [36

36. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185 (1994). [CrossRef]

*η*) back toward the input space, transmission (

_{R}*η*) into the lower cladding, and left and right coupling efficiencies (

_{T}*η*and

_{LCE}*η*) to the slab waveguide. These parameters are calculated from FDTD simulation results by:

_{RCE}*P*and

_{j}*P*are the total power detected on each monitor and

_{RCE}*P*is the power in the incident fiber mode. The

_{i}*MOI*term in Eq. (1) is the mode overlap integral between the actual FDTD field distribution detected on a monitor and the analytical mode profile of the waveguide at the specific location of the monitor, which indicates how much of the power detected on the monitor will actually be guided by the waveguide.

*η*of the SLGCs include grating period, Λ, in the x direction; the grating depth along the slanted direction, t; the fill factor (grating ridge width/period), f; the slant angle, θs; and the relative lateral position between the fiber and the slanted grating, d. To quickly and efficiently explore these parameters, we apply a parallel rigorous design tool recently developed by our group [33

_{RCE}33. J. Jiang, J. Cai, G. P. Nordin, and L. Li, “Parallel micro-genetic algorithm design of photonic crystal and waveguide structures,” Optics Letter. **28**, 2381 (2003). [CrossRef]

*c*is a positive scaling coefficient. The purpose of µGA optimization is to minimize

*f*and therefore maximize

*η*. Usually it takes approximately 100 µGA generations to converge to an optimal SLGC design.

_{RCE}## 3. Simulation results

### 3.1 Uniform SLGC

*η*,

_{RCE}*η*,

_{LCE}*η*and

_{R}*η*are 66.8%, 0.69%, 6.63%, and 18.48% respectively. We can see several salient features of SLGCs from this design. First, with a grating period of 1.0263µm, the slanted grating spans less than 20µm. It is well known that it is essentially impossible for traditional weak index-modulated grating couplers to achieve high coupling efficiency within such a short coupling length. Second, we notice that the slanted grating of SLGC greatly suppress the left coupled light. The left coupling efficiency (

_{T}*η*) is only 0.69%, which demonstrates the excellent unidirectional coupling capability of the SLGC. It is also interesting to note that the power coupled toward the right without considering the mode overlap integral (i.e., just the power ratio term in

_{LCE}*Eq. (1)*) is 74.2%. This means that 7.4% of the incident power (or 10% of the coupled power) radiates away from the waveguide along the propagation direction due to a mismatch between the monitored field and the waveguide mode. This loss originates from the mode mismatch between the coupled leaky mode and the output waveguide mode, and boundary scattering at the right edge of the slanted grating.

### 3.2 Non-uniform SLGC

32. R. Ulrich, “Optimum excitation of optical surface waves,” J. Opt. Soc. Am. **11**, 1467(1971). [CrossRef]

11. R. Ulrich, “Efficiency of optical-grating couplers,” J. Opt. Soc. Am. **63**, 419 (1973). [CrossRef]

*t*=1.6266µm, θ

_{s}=35.940 and

*d*=10.115µm. These values are very close to those of uniform SLGC except the lateral position d of the incident fiber. This SLGC has a right coupling efficiency (

*η*) of 80.1%, which is 13% greater than the uniform SLGC design. The improvement mainly comes from the decrease of the transmitted light (

_{RCE}*η*) to the substrate, which is 10.04%. Efficiency of

_{T}*η*and

_{R}*η*also decrease to 4.02% and 0.19% respectively.

_{LCE}## 4. Physical analysis and discussion

40. R. Petit, *Electromagnetic Theory of Gratings*, (Springer-Verlag, Berlin, 1980). [CrossRef]

*γ*=

_{m}*β*+

_{m}*iα*(

_{m}*m*is the mode index) of the leaky modes supported by the SLGC. The real part of the propagation constant,

*β*, is responsible for the phase-matching condition of SLGCs as will be discussed later and

_{m}*α*is the radiation factor of the leaky modes, which determines the coupling length of the grating coupler. We will skip the detail and tedious mathematical formulations of the RCWA procedure and refer the interested reader to the literature, such as Ref. [41

_{m}41. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. **70**, 804(1980). [CrossRef]

*β*are responsible for this coupling and will therefore be solved.

_{m}_{0}=1.55µm two lowest order leaky modes are found for this SLGC, a fundamental mode with

*γ*=

_{0}*β*+

_{0}*iα*=6.122037+i0.319629 and a higher order mode with

_{0}*γ*=

_{1}*β*+

_{1}*iα*=6.02677+i0.0464558. We can define the effective index of a leaky mode as:

_{1}*k*

_{0}is the wave propagation constant in vacuum. With this definition, the effective indices of the fundamental and higher order mode are 1.51025 and 1.48674. On the other hand, a simple mode analysis of the single mode 2D slab waveguide reveals that the effective index of its fundamental mode is 1.476.

*k*is the

_{ix}*x*component of the incident k vector, and Λ is the grating period in the x direction, and

*q*is the diffraction order of the slanted grating. For surface normal incidence,

*k*is zero. For the µGA optimized value Λ=1.02633µm, it is easy to see that the phase-matching condition of the SLGC can be satisfied with respect to the fundamental leaky mode when

_{ix}*q*=1. In other words, the +1 diffraction order of the grating provides the required phase matching between the incident fiber mode and the fundamental leaky mode of the SLGC. Note that this phase-matching mechanism is very different from that of conventional weak coupling grating couplers. In weak grating couplers, even though the +1 diffraction order is also responsible for the phase matching, the phase matching is always assumed to be realized for the unperturbed waveguide mode, and the period of the coupler can be analytically calculated. However, analytical determination of the grating period of our SLGC from the phase-matching condition is impossible because the fundamental leaky mode of SLGCs is a function of all of grating parameters, including the grating period itself. Consequently, realization of the phase-matching condition must be part of the SLGCs’ design process. In this sense, the design and optimization of SLGCs is far more complicated than weak grating couplers.

*k*

_{0}for convenience. As a zeroth order approximation, the slanted grating layer of the SLGC can be treated as a homogeneous layer with an average index defined as the volume average between the two materials forming the grating [28

28. M. Li and S. J. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. **35**, 3101 (1996). [CrossRef]

*K⃗*is the normally incident k-vector and the dotted slanted line refers to the orientation of the slanted grating ridges relative to the k

_{inc}_{y}axis, which is 34.98° in this case. The two dotted vertical lines L

_{1}and L

_{2}at

*k*=1.476 and 1.51025 correspond to the effective indices of the waveguide mode and the fundamental leaky mode, respectively. If we draw the grating vector

_{x}*K⃗*perpendicular to the orientation of the slanted ridges, the diffracted k-vector,

_{G}*k⃗*, which is equal to vector sum of (

_{final}*K⃗*+

_{inc}*K⃗*), terminates on L

_{G}_{2}because phase matching condition is satisfied. If Bragg condition is obeyed, the diffracted k-vector,

*k⃗*, will terminate on the circle. As seen in Fig. 9, this is the case. It is clear from this k-vector diagram that the Bragg diffraction condition is simultaneously satisfied for the +1 diffraction order of the slanted grating while phase matching between the incident fiber mode and the fundamental leaky mode of the SLGC is also satisfied. Bragg diffraction suppresses other diffraction orders and therefore enforces unidirectional coupling of the SLGC. For surface-normal operation, the occurrence of Bragg diffraction is the main advantage of an asymmetric parallelogramic grating shape compared to a symmetric grating. Note that finding a SLGC design that simultaneously satisfies phase matching and the Bragg condition for a grating that operates in the strong coupling regime strongly demonstrates the powerful optimization capability of our µGA-2D FDTD design tool.

_{final}*k⃗*in Fig. 9 is tilted at about 20° with respect to the

_{final}*k*axis. This implies that the phase front in the SLGC coupling region for the structure of Fig. 3 should also be tilted about 20° with respect to the x-axis. The tilted phase front can be clearly seen from the phase distribution of this SLGC calculated by 2D-FDTD, which is shown in Fig. 10. The tilted wavefront is quite flat when the light is coupled in by SLGC and is gradually rippled in the area around the right end interface of the slanted grating. The tilted wave front and the mismatch between the effective indices of the waveguide mode and the fundamental leaky mode of SLGC (the distance between L

_{x}_{1}and L

_{2}as shown in Fig. 9) will induce scattering loss at the boundary between the slanted grating region and the undisturbed waveguide region. In general this scattering loss is more severe than that of grating couplers that operate in the weak coupling regime where the mode mismatch is assumed to be negligible.

*γ*=

_{0}*β*+

_{0}*iα*=6.201752+0.3148340i (

_{0}*n*=1.52991) and a higher order mode with

_{eff}*γ*=

_{1}*β*+

_{1}*iα*=6.029924+0.040366i (

_{1}*n*=1.4875). By following a similar procedure, it is straightforward to show that both the phase-matching and Bragg conditions are satisfied for the fundamental leaky mode. The k-vector diagram and the phase distribution of the 2D FDTD simulation are shown in Figs. 11(a) and 11(b), respectively. Although these figures are very similar to those for the uniform SLGC case, we notice that the phase ripple near the right hand termination of the grating in Fig. 11(b) is much less than in Fig. 10, which means that the transition from the coupling grating to the output waveguide is smoother and hence the scattering loss is lessened.

_{eff}## 5. Conclusions

## Acknowledgments

## Reference and links

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: Putting a new twist on light,” Nature |

3. | Y. Hibino, “High contrast waveguide devices,” Conf. Opt. Fiber Commun. Tech. Dig. Ser.54, WB1/1 (2001). |

4. | C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. |

5. | B. Mersali, A. Ramdane, and A. Carenco, “Optical-mode transformer: A III–V circuit integration enabler,” IEEE J. Sel. Top. Quantum Electron. |

6. | I. Moerman, P. P Van Daele, and P.M. Demeester, “A review on fabrication technologies for the monolithic integration of tapers with III–V semiconductor devices,” IEEE J. Select. Topics Quantum Electron. |

7. | P.V. Studenkov, M.R. Gokhale, and S.R. Forrest, “Efficient coupling in integrated Twin-waveguide lasers using waveguide tapers,” IEEE Photon. Tech. Lett. |

8. | V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nano-taper for compact mode conversion,” Opt. Lett. |

9. | T. Tamir, |

10. | T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. |

11. | R. Ulrich, “Efficiency of optical-grating couplers,” J. Opt. Soc. Am. |

12. | R. Waldhäusl, B. Schnabel, P. Dannberg, E. Kley, A. Bräuer, and W. Karthe, “Efficient coupling into polymer waveguide by gratings,” Appl. Opt. |

13. | V. A. Sychugov, A. V. Tishchenko, B. A. Usievich, and O. Parriaux, “Optimization and control of grating coupling to or from a Silicon-based optical waveguide,” Opt. Eng. |

14. | J. C. Brazas and L. Li, “Analysis of input-grating coupler having finite lengths,” Appl. Opt. |

15. | R. W. Ziolkowski and T. Liang, “Design and characterization of a grating-assisted coupler enhanced by a photonic-band-gap structure for effective wavelength-division multiplexing,” Opt. Lett. |

16. | D. Pascal, R. Orobtchouk, A. Layadi, A. Koster, and S. Laval, “Optimized coupling of a gaussian beam into an optical waveguide with a grating coupler: comparison of experimental and theoretical Results,” Appl. Opt. |

17. | T. W. Ang, et al., “Effects of grating heights on highly efficient unibond SOI waveguide grating couplers,” IEEE Photon. Technol. Lett. |

18. | S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Volume grating preferential-order focusing waveguide coupler,” Opt. Lett. |

19. | S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, “Design of a high-efficiency volume grating coupler for line focusing,” Appl. Opt. |

20. | S.-D. Wu and E. N. Glytsis, “Volume holographic grating couplers: Rigorous analysis by use of the finite-difference frequency-domain method,” Appl. Opt. |

21. | J. K. Bulter, S. Nai-Hsiang, G. A. Evans, L. Pang, and P. Congdon, “Grating-assisted coupling of light between semiconductor glass wavwguides,” Journal of Lightwave Technology |

22. | 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. |

23. | G. Z. Masanovic, V. M. N. Passaro, and T. R. Graham, “Dual grating-assisted directional coupling between fibers and thin semiconductor waveguides,” IEEE Photonics Technology Letters |

24. | L. C. West, C. Roberts, J. Dunkel, G. Wojcik, and J. Mould Jr.,“Non uniform grating couplers for coupling of Gaussian beams to compact waveguides,” Integrated Photonics Research Technical Digest, Optical Society of America, 1994. |

25. | D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. 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. |

26. | D. M. Chambers, B. Wang, G. P. Nordin, and J. Jiang, “Stratified grating coupler for waveguide applications,” Integrated Photonics Research Topical Meeting in Washington, DC, USA on June 16–18, 98–100, 2003. |

27. | Personal communications with Dr. Mark Hoffbauer in Los Alamos National Laboratory about slant etches using atomic oxygen technique. |

28. | M. Li and S. J. Sheard, “Experimental study of waveguide grating couplers with parallelogramic tooth profiles,” Opt. Eng. |

29. | T. Liao, S. Sheard, M. Li, J. Zhuo, and P. Prewett, “High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles,” IEEE J. Lightwave Technology |

30. | M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. |

31. | A. V. Tishchenko, N. M. Lyndin, S. M. Loktev, V. A. Sychugov, and B. A. Usievich, “Unidirectional waveguide grating coupler by means of parallelogramic grooves,” SPIE |

32. | R. Ulrich, “Optimum excitation of optical surface waves,” J. Opt. Soc. Am. |

33. | J. Jiang, J. Cai, G. P. Nordin, and L. Li, “Parallel micro-genetic algorithm design of photonic crystal and waveguide structures,” Optics Letter. |

34. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. |

35. | A. Taflove, |

36. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

37. | K. Krishnakumar, “Micro-genetic algorithm for stationary and non-stationary function optimization,” SPIE |

38. | Z. Michalewicz, |

39. | D. E. Goldberg, |

40. | R. Petit, |

41. | K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. |

42. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1950) Diffraction and gratings : Diffraction gratings

(230.3120) Optical devices : Integrated optics devices

(230.3990) Optical devices : Micro-optical devices

(230.7370) Optical devices : Waveguides

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 7, 2004

Revised Manuscript: July 6, 2004

Published: July 26, 2004

**Citation**

Bin Wang, Jianhua Jiang, and Gregory Nordin, "Compact slanted grating couplers," Opt. Express **12**, 3313-3326 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3313

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: Molding the flow of light (Princeton University Press, Princeton, N.J., 1995).
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: Putting a new twist on light," Nature 386 (6621), 143-149 (1997). [CrossRef]
- Y. Hibino, �??High contrast waveguide devices,�?? Conf. Opt. Fiber Commun. Tech. Dig. Ser. 54, WB1/1 (2001).
- C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H.A.Haus, and J.D. Joannopoulos, �??High-density integrated optics,�?? J. Lightwave Technol. 17, 1682 (1999). [CrossRef]
- B. Mersali, A. Ramdane, and A. Carenco, �??Optical-mode transformer: A III-V circuit integration enabler,�?? IEEE J. Sel. Top. Quantum Electron. 3, 1321 (1997). [CrossRef]
- I. Moerman, P. P Van Daele, and P.M. Demeester, �??A review on fabrication technologies for the monolithic integration of tapers with III-V semiconductor devices,�?? IEEE J. Select. Topics Quantum Electron. 3, 1308 (1997). [CrossRef]
- P.V. Studenkov, M.R. Gokhale, and S.R. Forrest, �??Efficient coupling in integrated Twin-waveguide lasers using waveguide tapers,�?? IEEE Photon. Tech. Lett. 11, 1096 (1999). [CrossRef]
- V. R. Almeida, R. R. Panepucci, and M. Lipson, �??Nano-taper for compact mode conversion,�?? Opt. Lett. 28, 1302 (2003). [CrossRef] [PubMed]
- T. Tamir, Integrated Optics, (Springer Verlag, 1975).
- T.Tamir and S. T. Peng, �??Analysis and design of grating couplers,�?? Appl. Phys. 14, 235 (1977). [CrossRef]
- R. Ulrich, �??Efficiency of optical-grating couplers,�?? J. Opt. Soc. Am. 63, 419 (1973). [CrossRef]
- R. Waldhäusl, B. Schnabel, P. Dannberg, E. Kley, A. Bräuer, and W. Karthe, �??Efficient coupling into polymer waveguide by gratings,�?? Appl. Opt. 36, 9383 (1997). [CrossRef]
- V. A. Sychugov, A. V. Tishchenko, B. A. Usievich, and O. Parriaux, �??Optimization and control of grating coupling to or from a Silicon-based optical waveguide,�?? Opt. Eng. 35, 3092 (1996). [CrossRef]
- J. C. Brazas, and L. Li, �??Analysis of input-grating coupler having finite lengths,�?? Appl. Opt. 34, 3786 (1995). [CrossRef] [PubMed]
- R. W. Ziolkowski, and T. Liang, �??Design and characterization of a grating-assisted coupler enhanced by a photonic-band-gap structure for effective wavelength-division multiplexing,�?? Opt. Lett. 22, 1033 (1997). [CrossRef] [PubMed]
- D. Pascal, R. Orobtchouk, A. Layadi, A. Koster, and S. Laval, �??Optimized coupling of a gaussian beam into an optical waveguide with a grating coupler: comparison of experimental and theoretical Results,�?? Appl. Opt. 36, 2443 (1997). [CrossRef] [PubMed]
- T. W. Ang, et al., �??Effects of grating heights on highly efficient unibond SOI waveguide grating couplers,�?? IEEE Photon. Technol. Lett. 12, 59 (2000). [CrossRef]
- S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, �??Volume grating preferential-order focusing waveguide coupler,�?? Opt. Lett. 24, 1708 (1999). [CrossRef]
- S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, �??Design of a high-efficiency volume grating coupler for line focusing,�?? Appl. Opt. 37, 2278 (1998). [CrossRef]
- S.-D. Wu and E. N. Glytsis, "Volume holographic grating couplers: Rigorous analysis by use of the finite-difference frequency-domain method," Appl. Opt. 43, 1009 (2004). [CrossRef] [PubMed]
- J. K. Bulter, S. Nai-Hsiang, G. A. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor glass wavwguides," Journal of Lightwave Technology 16, 1038 (1998). [CrossRef]
- 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, 5773 (2000). [CrossRef]
- G. Z. Masanovic, V. M. N. Passaro, and T. R. Graham, "Dual grating-assisted directional coupling between fibers and thin semiconductor waveguides," IEEE Photonics Technology Letters 15, 1395 (2003). [CrossRef]
- L. C. West, C. Roberts, J. Dunkel, G. Wojcik, and J. Mould, Jr., �??Non uniform grating couplers for coupling of Gaussian beams to compact waveguides, �?? Integrated Photonics Research Technical Digest, Optical Society of America, 1994.
- D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. 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, 949 (2002). [CrossRef]
- D. M. Chambers, B. Wang, G. P. Nordin, and J. Jiang, �??Stratified grating coupler for waveguide applications,�?? Integrated Photonics Research Topical Meeting in Washington, DC, USA on June 16-18, 98-100, 2003.
- Personal communications with Dr. Mark Hoffbauer in Los Alamos National Laboratory about slant etches using atomic oxygen technique.
- M. Li, and S. J. Sheard, �??Experimental study of waveguide grating couplers with parallelogramic tooth profiles,�?? Opt. Eng. 35, 3101 (1996). [CrossRef]
- T. Liao, S. Sheard, M. Li, J. Zhuo, and P. Prewett, �??High-efficiency focusing waveguide grating coupler with parallelogramic groove profiles, �?? IEEE J. Lightwave Technology 15, 1142 (1997). [CrossRef]
- M. Li, S. J. Sheard, �??Waveguide couplers using parallelogramic-shaped blazed gratings,�?? Opt. Commun. 109, 239(1994). [CrossRef]
- A. V. Tishchenko, N. M. Lyndin, S. M. Loktev, V. A. Sychugov, and B. A. Usievich, �??Unidirectional waveguide grating coupler by means of parallelogramic grooves,�?? SPIE 3099, 269 (1997). [CrossRef]
- R.Ulrich, �??Optimum excitation of optical surface waves,�?? J. Opt. Soc. Am. 11, 1467(1971). [CrossRef]
- J. Jiang, J. Cai, G. P. Nordin, and L. Li, �??Parallel micro-genetic algorithm design of photonic crystal and waveguide structures,�?? Optics Letter. 28, 2381 (2003). [CrossRef]
- M. G. Moharam and T. K. Gaylord, �??Diffraction analysis of dielectric surface-relief gratings,�?? J. Opt. Soc. Am. 72, 1385 (1982). [CrossRef]
- A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Massachusetts, 1995).
- J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185 (1994). [CrossRef]
- K. Krishnakumar, �??Micro-genetic algorithm for stationary and non-stationary function optimization,�?? SPIE 1196, 289 (1989).
- Z. Michalewicz, Genetic Algorithm + Data Structures + Evolution Programs, (Springer-Verlag, Berlin, 1992).
- D. E. Goldberg, Genetic Algorithm in Search, Optimization, and Machine Learning, (Addison Wesley, Massachusetts, 1989).
- R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, Berlin, 1980). [CrossRef]
- K. C. Chang, V. Shah, and T. Tamir, �??Scattering and guiding of waves by dielectric gratings with arbitrary profiles,�?? J. Opt. Soc. Am. 70, 804(1980). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 90, (Cambridge University Press, 1996).

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