## Unidirectionally optical coupling from free space into silicon waveguide with wide flat-top angular efficiency |

Optics Express, Vol. 20, Issue 17, pp. 18545-18554 (2012)

http://dx.doi.org/10.1364/OE.20.018545

Acrobat PDF (1816 KB)

### Abstract

A grating coupling scheme from free-space light into silicon waveguide with a remarkable property of wide flat-top angular efficiency is proposed and theoretically investigated. The coupling structure is composed of two cascaded gratings with a proper distance between their peak angular efficiencies. A quantitative semi-analytical theory based on coupled-mode models is developed for performance prediction and validated with the fully vectorial aperiodic Fourier modal method (a-FMM). With the theory, wide flat-top angular response is achieved and the conditions are pointed out. Proof-of-principle demonstrations show that the −1 dB angular width, a figure of merit to evaluate the flat-top performance, is broadened to almost 3 to 4 times, and meanwhile the −3 dB angular width, i.e., angular-full-width-half-maximum (AFWHM), is widened to nearly more than twice, compared with the reference gratings composed of the same number of periodic defects. We believe this work will find applications in biological or chemical sensing and novel optical devices.

© 2012 OSA

## 1. Introduction

20. H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. **14**, 1522–1529 (2008). [CrossRef]

## 2. Quantitative semi-analytical theory

_{2}hard mask, the silicon overlay is locally defined by epitaxial silicon growth [2

2. G. Roelkens, D. Vermeulen, D. V. 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]

21. D. Taillaert, F. V. Laere, M. Ayre, W. Bogaerts, D. V. Thourhout, P. Bienstman, and R. Baets, “Grating couplers for coupling between optical fibers and nanophotonic waveguides,” Jpn. J. Appl. Phys. **45**, 6071–6077 (2006). [CrossRef]

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

8. F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. V. Thourhout, M. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Light-wave Technol. **25**, 151–156 (2007). [CrossRef]

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

24. S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett. **28**, 1150–1152 (2003). [CrossRef] [PubMed]

8. F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. V. Thourhout, M. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Light-wave Technol. **25**, 151–156 (2007). [CrossRef]

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

24. S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett. **28**, 1150–1152 (2003). [CrossRef] [PubMed]

*λ*= 1550 nm,

*t*= 240 nm,

_{s}*t*= 900 nm,

_{c}*n*= 3.518 (Si),

_{s}*n*= 1.46 (SiO

_{c}_{2}). In this case, the optical waveguide only support the fundamental TM mode. Note that it is clear that there are so many parameters that pure simulations using finite difference time domain (FDTD) method or finite element method (FEM) suffer from high numerical cost and aimless parameters scan. To circumvent these problems, here we develop a quantitative theory based on coupled-mode model to provide parameters optimization with a clear picture of physics. We emphasize that although TM polarized plane wave is used as the example throughout the paper, the concept and the theory also work for TE polarization.

*d*, as illustrated in Figs. 2(b)–2(d); and a nested model on the excitation, reflectance and transmittance coefficients of WMs by the ‘black boxes’, as shown in Figs. 2(e)–2(g). The nested theoretical models bridge the scattering coefficients of the finite-size protuberances to those of a single one, resulting in a great reduction of the computational cost.

*B*

_{1}(or

*B*

_{2}) and

*A*

_{1}(or

*A*

_{2}) are the respective complex amplitudes of magnetic field

*H*of the backward- and forward-going WMs away from the first (or the second) ‘black box’. The main elementary scattering processes involved in the proposed model are shown in Figs. 2(c) and 2(d):

_{y}*N*defects under the plane wave illumination, respectively;

_{m}*t*

_{Nm}is the transmittance coefficient. The coupled-mode equations lead to: where

*w*= exp[

*ik*

_{0}(

*p*

_{1}

*N*

_{1}−

*p*

_{1}+

*w*+

_{r}*d*)sin

*θ*] is the phase shift introduced by the incident plane wave. This is because the zero phase is assumed to be at the leftmost side of the first ‘black box’ for the calculation of

*x*=

*z*= 0); whereas it is at the leftmost side of the second ‘black box’ for the calculation of

*x*=

*p*

_{1}

*N*

_{1}−

*p*

_{1}+

*w*+

_{r}*d*,

*z*= 0).

*d*is the structural distance between the two ‘black boxes’.

*u*= exp(

*ik*

_{0}

*n*

_{eff}

*d*) with

*n*

_{eff}being the complex effective refractive index of the waveguide mode. Note that

*n*

_{eff}. To calculate the WMs excitation coefficients,

*A*

_{0}= 0; whereas to calculate the WMs reflectance and transmittance coefficients,

*r*

_{N1N2}=

*B*

_{1}/(

*uA*

_{0}) and

*t*

_{N1N2}=

*A*

_{2}/

*uA*

_{0}, one sets

*t*

_{N2}| is usually small when

*t*

_{N2}and

*u*play key roles of phase modulation of

*m*is an integer.

*N*aperiodic defects with arbitrary distances

*d*(

_{j}*j*= 1,...,

*N*− 1) as shown in Fig. 2(e). Following our pervious work [25

25. G. Li, F. Xiao, L. Cai, K. Alameh, and A. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects vis field decomposition,” New J. Phys. **13**, 073045 (2011). [CrossRef]

26. G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express **18**, 10487–10499 (2010). [CrossRef] [PubMed]

*N*defects are given by: Where

*u*

_{N}_{−1}= exp(

*ik*

_{0}

*n*

_{eff}

*d*

_{N}_{−1}),

*u*=

_{j}*u*,

*w*=

_{j}*w*, and

^{j}*j*= 1,··· ,

*N*− 1,

*t*

_{1}.

*d*

_{1}= ... =

*d*

_{N}_{−1}=

*p*−

*w*.

_{r}*r*and

_{N}*t*can be calculated recursively starting from

_{N}*t*

_{1}, thus the geometry optimization for

*N*periodic defects is reduced into that for a single one. The constructive interference condition of

*t*

_{1}) is quite important since it may be relatively large in this case.

## 3. Results and discussions

20. H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. **14**, 1522–1529 (2008). [CrossRef]

*D*being the grating cross section represents the ratio of the forward-going WMs power flux to the incident power flux that launches into the gratings. To evaluate the performance of wide flat-top angular coupling efficiency, we adopt −1 dB angular width Φ and −3 dB angular width Θ as the figures of merit (FoMs), which are defined as the angular range between two specified angle cut-off points that are −1 dB and −3 dB below the peak angular efficiency, respectively. Specifically, Φ is used to evaluate flat-top angular response while Θ is AFWHM.

*N*

_{1}+

*N*

_{2}defects of period

*p*

_{1}(blue-solid line) or

*p*

_{2}(black-solid line) in (b,e,h,k) with the model predictions (green-dashed line with circles) and the a-FMM data (red-solid line) on the above-mentioned

*t*

_{N2}.

*d*(a,b), comparable incoupling cross sections (c,d) and suitable angular interval (e,f). There will be no wide flat-top angular coupling efficiency if

*d*deviates from the model prediction given by Eq. (4) (a,b), peak values of incoupling cross sections are not comparable (c,d), or the angular interval between the peaks of

*t*

_{1}of a single defect instead of

*t*of

_{N}*N*defects as functions of

*w*and

_{r}*h*.

_{r}*t*for various grating numbers or periods are then obtained with the nested models at negligible computational cost. This cost reduction is especially remarkable when one needs to increase

_{N}*N*to improve the performance. More importantly, there are no restrictions on the defect’s geometry and refractive index profile in the theoretical model. Apart from the above mentioned triangular shaped defect, the proposed method is also versatile for a deep etched (Fig. 6) or fully etched (not shown) rectangular grating. As clearly seen from Figs. 6(a)–6(d), the scattering coefficients of each ‘black-box’ composed of periodic rectangular grooves is able to be predicted with a high accuracy, in both amplitudes and phases. As a result, a flat-top angular efficiency is also achieved using the theoretical model: Φ is broadened to 3–4 times and meanwhile Θ is widened to more than 3 times compared with the reference one (e) as depicted in Figs. 6(e)–6(f). We should note that as the theoretical model is developed for single-mode waveguide, it is only suitable for the case of single-mode operation.

## 4. Conclusions

## Acknowledgments

## References and links

1. | T. W. Ang, G. T. Reed, A. Vonsovici, A. G. R. Evans, P. R. Routley, and M. R. Josey, “Effects of grating heights on highly efficient unibond SOI waveguide grating couplers,” IEEE Photon. Technol. Lett. |

2. | G. Roelkens, D. Vermeulen, D. V. 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. |

3. | M. A. Basha, S. Chaudhuri, and S. Safavi-Naeini, “A study of coupling interactions in finite arbitrarily-shaped grooves in electromagnetic scattering problem,” Opt. Express |

4. | D. Vermeulen, S. Selvaraja, P. Verheyen, G. Lepage, W. Bogaerts, P. Absil, D. V. Thourhout, and G. Roelkens, “High-efficiency fiber-to-chip grating couplers realized using an advanced CMOS-compatible silicon-on-insulator platform,” Opt. Express |

5. | S. Scheerlinck, J. Schrauwen, F. V. Laere, D. Taillaert, D. V. Thourhout, and R. Baets, “Efficient, broadband and compact metal grating couplers for silicon-on-insulator waveguides,” Opt. Express |

6. | G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. |

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

8. | F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. V. Thourhout, M. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Light-wave Technol. |

9. | L. Vivien, D. Pascal, S. Lardenois, D. Marris-Morini, E. Cassan, F. Grillot, S. Laval, J. M. Fédéli, and L. E. Melhaoui, “Light injection in SOI microwaveguides using high-efficiency grating couplers,” J. Lightwave Technol. |

10. | G. Maire, L. Vivien, G. Sattler, A. Kazmierczak, B. Sanchez, K. B. Gylfason, A. Griol, D. Marris-Morini, E. Cassan, D. Giannone, H. Sohlström, and D. Hill, “High efficiency silicon nitride surface grating couplers,” Opt. Express |

11. | J. Wu, G. Zheng, Z. Li, and C. Yang, “Focal plane tuning in wide-field-of-view microscope with Talbot pattern illumination,” Opt. Lett. |

12. | J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol. |

13. | S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature |

14. | A. F. Coskun, I. Sencan, T. W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express |

15. | K. L. Lee, C. W. Lee, W. S. Wang, and P. K. Wei, “Sensitive biosensor array using surface plasmon resonance on metallic nanoslits,” J. Biomed. Opt. |

16. | E. D. Tommasi, L. D. Stefano, I. Rea, V. D. Sarno, L. Rotiroti, P. Arcari, A. Lamberti, C. Sanges, and I. Rendina, “Porous silicon based resonant mirrors for biochemical sensing,” Sensors |

17. | A. Sentenac and A. Fehrembach, “Angular tolerant resonant grating filters under oblique incidence,” J. Opt. Soc. Am. A |

18. | A. B. Greenwell, S. Boonruang, and M. G. Moharam, “Multiple wavelength resonant grating filters at oblique incidence with broad angular acceptance,” Opt. Express |

19. | M. Born and E. Wolf, |

20. | H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. |

21. | D. Taillaert, F. V. Laere, M. Ayre, W. Bogaerts, D. V. Thourhout, P. Bienstman, and R. Baets, “Grating couplers for coupling between optical fibers and nanophotonic waveguides,” Jpn. J. Appl. Phys. |

22. | L. Dong, S. Iyer, S. Popov, and A. Friberg, “3D fabrication of waveguide and grating coupler in SU-8 by optimized gray scale electron beam lithography,” in Proceedings of ACP (2010). |

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

24. | S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett. |

25. | G. Li, F. Xiao, L. Cai, K. Alameh, and A. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects vis field decomposition,” New J. Phys. |

26. | G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(130.3120) Integrated optics : Integrated optics devices

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: April 18, 2012

Revised Manuscript: July 20, 2012

Manuscript Accepted: July 22, 2012

Published: July 30, 2012

**Citation**

Kun Li, Guangyuan Li, Feng Xiao, Fan Lu, Zhonghua Wang, and Anshi Xu, "Unidirectionally optical coupling from free space into silicon waveguide with wide flat-top angular efficiency," Opt. Express **20**, 18545-18554 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-18545

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### References

- T. W. Ang, G. T. Reed, A. Vonsovici, A. G. R. Evans, P. R. Routley, and M. R. Josey, “Effects of grating heights on highly efficient unibond SOI waveguide grating couplers,” IEEE Photon. Technol. Lett.12, 59–61 (2000). [CrossRef]
- G. Roelkens, D. Vermeulen, D. V. 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]
- M. A. Basha, S. Chaudhuri, and S. Safavi-Naeini, “A study of coupling interactions in finite arbitrarily-shaped grooves in electromagnetic scattering problem,” Opt. Express18, 2743–2752 (2010). [CrossRef] [PubMed]
- D. Vermeulen, S. Selvaraja, P. Verheyen, G. Lepage, W. Bogaerts, P. Absil, D. V. Thourhout, and G. Roelkens, “High-efficiency fiber-to-chip grating couplers realized using an advanced CMOS-compatible silicon-on-insulator platform,” Opt. Express18, 18278–18283 (2010). [CrossRef] [PubMed]
- S. Scheerlinck, J. Schrauwen, F. V. Laere, D. Taillaert, D. V. Thourhout, and R. Baets, “Efficient, broadband and compact metal grating couplers for silicon-on-insulator waveguides,” Opt. Express15, 9625–9630 (2007). [CrossRef] [PubMed]
- G. Roelkens, D. V. 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]
- D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Verstuyft, K. D. 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, 949–955 (2002). [CrossRef]
- F. V. Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. V. Thourhout, M. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. Light-wave Technol.25, 151–156 (2007). [CrossRef]
- L. Vivien, D. Pascal, S. Lardenois, D. Marris-Morini, E. Cassan, F. Grillot, S. Laval, J. M. Fédéli, and L. E. Melhaoui, “Light injection in SOI microwaveguides using high-efficiency grating couplers,” J. Lightwave Technol.24, 3810–3815 (2006). [CrossRef]
- G. Maire, L. Vivien, G. Sattler, A. Kazmierczak, B. Sanchez, K. B. Gylfason, A. Griol, D. Marris-Morini, E. Cassan, D. Giannone, H. Sohlström, and D. Hill, “High efficiency silicon nitride surface grating couplers,” Opt. Express16, 328–333 (2008). [CrossRef] [PubMed]
- J. Wu, G. Zheng, Z. Li, and C. Yang, “Focal plane tuning in wide-field-of-view microscope with Talbot pattern illumination,” Opt. Lett.36, 2179–2181 (2011). [CrossRef] [PubMed]
- J. Ho, A. V. Parwani, D. M. Jukic, Y. Yagi, L. Anthony, and J. R. Gilbertson, “Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies,” Hum. Pathol.37, 322–331 (2006). [CrossRef] [PubMed]
- S. Nagrath, L. V. Sequist, S. Maheswaran, D. W. Bell, D. Irimia, L. Ulkus, M. R. Smith, E. L. Kwak, S. Digumarthy, A. Muzikansky, P. Ryan, U. J. Balis, R. G. Tompkins, D. A. Haber, and M. Toner, “Isolation of rare circulating tumour cells in cancer patients by microchip technology,” Nature450, 1235–1239 (2007). [CrossRef] [PubMed]
- A. F. Coskun, I. Sencan, T. W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express18, 10510–10523 (2010). [CrossRef] [PubMed]
- K. L. Lee, C. W. Lee, W. S. Wang, and P. K. Wei, “Sensitive biosensor array using surface plasmon resonance on metallic nanoslits,” J. Biomed. Opt.12, 044023 (2007). [CrossRef] [PubMed]
- E. D. Tommasi, L. D. Stefano, I. Rea, V. D. Sarno, L. Rotiroti, P. Arcari, A. Lamberti, C. Sanges, and I. Rendina, “Porous silicon based resonant mirrors for biochemical sensing,” Sensors8, 6549–6556 (2008). [CrossRef]
- A. Sentenac and A. Fehrembach, “Angular tolerant resonant grating filters under oblique incidence,” J. Opt. Soc. Am. A22, 475–480 (2005). [CrossRef]
- A. B. Greenwell, S. Boonruang, and M. G. Moharam, “Multiple wavelength resonant grating filters at oblique incidence with broad angular acceptance,” Opt. Express15, 8626–8638 (2007). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin, “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron.14, 1522–1529 (2008). [CrossRef]
- D. Taillaert, F. V. Laere, M. Ayre, W. Bogaerts, D. V. Thourhout, P. Bienstman, and R. Baets, “Grating couplers for coupling between optical fibers and nanophotonic waveguides,” Jpn. J. Appl. Phys.45, 6071–6077 (2006). [CrossRef]
- L. Dong, S. Iyer, S. Popov, and A. Friberg, “3D fabrication of waveguide and grating coupler in SU-8 by optimized gray scale electron beam lithography,” in Proceedings of ACP (2010).
- A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. W. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Opt. Express11, 3555–3561 (2003). [CrossRef] [PubMed]
- S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval, R. Orobtchouk, M. Heitzmann, N. Bouzaida, and L. Mollard, “Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors,” Opt. Lett.28, 1150–1152 (2003). [CrossRef] [PubMed]
- G. Li, F. Xiao, L. Cai, K. Alameh, and A. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects vis field decomposition,” New J. Phys.13, 073045 (2011). [CrossRef]
- G. Li, L. Cai, F. Xiao, Y. Pei, and A. Xu, “A quantitative theory and the generalized Bragg condition for surface plasmon Bragg reflectors,” Opt. Express18, 10487–10499 (2010). [CrossRef] [PubMed]

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