## Physical basis for wideband resonant reflectors

Optics Express, Vol. 16, Issue 5, pp. 3456-3462 (2008)

http://dx.doi.org/10.1364/OE.16.003456

Acrobat PDF (263 KB)

### Abstract

In this paper, we address resonant leaky-mode reflectors made with a periodic silicon layer on an insulating substrate. Our objective is to explain the physical basis for their operation and to quantify the bandwidth provided by a single resonant layer by illustrative examples for both TE and TM polarized incident light. We find that the number of participating leaky modes and their excitation conditions affect the bandwidth. We show that recently reported experimental [

© 2008 Optical Society of America

## 1. Introduction

3. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. **20**, 345–351 (1979). [CrossRef]

10. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express **12**, 5661–5674 (2004). [CrossRef] [PubMed]

12. A. E. Willner, “All mirrors are not created equal,” Nature Photonics **1**, 87–88 (2007). [CrossRef]

_{H}=n

_{Si}=3.48) on a SiO

_{2}substrate (n

_{S}=n

_{SiO2}=1.48). It is necessary that the structure form a waveguide grating such that the periodic layer (Si) possesses higher refractive index than the adjacent regions (air, silica). The reflector works under a guided-mode resonance (GMR), which arises when the incident wave couples to a leaky waveguide mode by phase matching with the second-order grating [14

14. A. Hardy, D. F. Welch, and W. Streifer, “Analysis of second-order gratings,” IEEE J. Quantum Electron. **25**, 2096–2105 (1989). [CrossRef]

15. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express **15**, 680–694 (2007). [CrossRef] [PubMed]

16. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**, 2038–2059 (1997). [CrossRef]

15. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express **15**, 680–694 (2007). [CrossRef] [PubMed]

17. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. **21**, 144–150 (1985). [CrossRef]

3. P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. **20**, 345–351 (1979). [CrossRef]

_{R}+jβ

_{I}is the complex propagation constant of the leaky mode, β

_{I}=0 at one edge, implying that no leakage is possible at that edge. In this paper, for clarity, we treat resonance elements with two-part periods which can only have symmetric profiles.

## 2. Numerical methods and assumptions

18. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE **73**, 894–937 (1985). [CrossRef]

19. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach,” J. Opt. Soc. Am. A **12**, 1077–1086 (1995). [CrossRef]

20. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. **23**, 123–133 (1975). [CrossRef]

9. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. **32**, 2606–2613 (1993). [CrossRef] [PubMed]

22. M. Shokooh-Saremi and R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. **32**, 894–896 (2007). [CrossRef] [PubMed]

## 3. Results

_{0}=1.0 over the 1.45–2.0 µm wavelength band in TM polarization. The parameters that are optimized to achieve the target reflectance are period, thickness, and fill factor. Figure 2(a) shows the reflectance and transmittance of the designed element in which Λ=0.766 µm, d=0.490 µm, F=0.7264, n

_{H}=3.48, n

_{L}=1.0, and n

_{S}=1.48. The spectrum for which R

_{0}>0.99 exhibits a bandwidth of ~520 nm. There are three transmittance dips inside the high-reflectance band, each of which corresponding to a guided-mode resonance at which the transmittance approaches zero. Thus, this high-reflection band is supported by a blend of three leaky modes. Figure 2(b) illustrates the amplitudes of the magnetic modal fields inside the grating structure and in the surrounding media at the center resonance which arises at 1.620 µm. The amplitude of the incident zero-order wave is denoted by S

_{0}whereas the first evanescent wave has amplitude S

_{1}. This low-Q resonance raises the internal field strength of the first-order leaky mode by ~×3 relative to the excitation field. The S

_{1}field shows a predominant TM

_{0}shape that nevertheless also exhibits TM

_{2}features with two nulls in the low-amplitude region near the input edge. There is an appreciable second-order field (S

_{2}) in the structure as seen in the figure. It is important to note that the zero-order field fits well into the layer which is ~one wavelength in thickness at resonance (compare d=0.490 µm and λ/N=1.620/3.0=0.540 µm using the second-order effective-medium-theory refractive index of the grating layer N=3.0). This sets up an efficient excitation of the leaky modes; i.e. the input wave “pumps” the structure well, a key condition in achieving broad reflectance spectra.

_{0}>0.99. Figure 4(b) displays the electric field distribution pattern at the resonance wavelength. A dominant TE

_{0}leaky mode is generated by the first evanescent diffraction order. Moreover, by dividing the period into four parts ({Si, air, Si, air} with corresponding fill factors {F

_{1}, F

_{2}, F

_{3}, F

_{4}; F

_{1}+F

_{2}+F

_{3}+F

_{4}=1.0}), very broad reflection bandwidths with R

_{0}>0.99 are realizable; these may exceed ~600 nm in both TE and TM polarization for a single Si layer without substrate enhancement [10

10. Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express **12**, 5661–5674 (2004). [CrossRef] [PubMed]

1. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. **16**, 1676–1678 (2004). [CrossRef]

23. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladding subwavelength grating,” IEEE Photon. Technol. Lett. **16**, 518–520 (2004). [CrossRef]

_{L}=0.83 µm, and F=0.75. We use this set in Fig. 5(a) which shows the spectra of this reflector including a sharp leaky-mode transmission minimum consistent with the results presented above. The computed reflection bandwidth is ~467 nm for R

_{0}>0.99 with TM polarization. Figure 5(b) shows the effect of the sublayer/substrate combination on the reflectance spectrum. Without the sublayer, the bandwidth for this design is ~385 nm. Therefore, in this case, the sublayer extends the flat band by ~80 nm or ~20%.

## 4. Conclusions

1. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. **16**, 1676–1678 (2004). [CrossRef]

## Acknowledgements

1. | C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. |

2. | M. C. Y Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-indexcontrast subwavelength grating,” Nature Photonics |

3. | P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. |

4. | L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Comm. |

5. | E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta |

6. | G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. |

7. | I. A. Avrutsky and V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. |

8. | R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. |

9. | S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. |

10. | Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express |

11. | H. A. Macleod, |

12. | A. E. Willner, “All mirrors are not created equal,” Nature Photonics |

13. | A. Yariv and P. Yeh, |

14. | A. Hardy, D. F. Welch, and W. Streifer, “Analysis of second-order gratings,” IEEE J. Quantum Electron. |

15. | Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express |

16. | D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. |

17. | R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation loss,” IEEE J. Quantum Electron. |

18. | T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE |

19. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

20. | S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. |

21. | R. Eberhart and J. Kennedy, “Particle swarm optimization,” in |

22. | M. Shokooh-Saremi and R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. |

23. | C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladding subwavelength grating,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(130.2790) Integrated optics : Guided waves

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 10, 2007

Revised Manuscript: February 23, 2008

Manuscript Accepted: February 24, 2008

Published: February 29, 2008

**Citation**

Robert Magnusson and Mehrdad Shokooh-Saremi, "Physical basis for wideband resonant reflectors," Opt. Express **16**, 3456-3462 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3456

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### References

- C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 ?m) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004). [CrossRef]
- M. C. Y Huang, Y. Zhou, and C. J. Chang-Hasnain, "A surface-emitting laser incorporating a high-index- contrast subwavelength grating," Nature Photonics 1, 119-122 (2007). [CrossRef]
- P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979). [CrossRef]
- L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Comm. 55, 377-380 (1985). [CrossRef]
- E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986). [CrossRef]
- G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985). [CrossRef]
- I. A. Avrutsky and V. A. Sychugov, "Reflection of a beam of finite size from a corrugated waveguide," J. Mod. Opt. 36, 1527-1539 (1989). [CrossRef]
- R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61,1022-1024 (1992). [CrossRef]
- S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32,2606-2613 (1993). [CrossRef] [PubMed]
- Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004). [CrossRef] [PubMed]
- H. A. Macleod, Thin-Film Optical Filters, (McGraw-Hill, New York, 1989).
- Q4. A. E. Willner, "All mirrors are not created equal," Nature Photonics 1, 87-88 (2007). [CrossRef]
- A. Yariv and P. Yeh., Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, New York, 2007).
- A. Hardy, D. F. Welch, and W. Streifer, "Analysis of second-order gratings," IEEE J. Quantum Electron. 25, 2096-2105 (1989). [CrossRef]
- Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express 15, 680-694 (2007). [CrossRef] [PubMed]
- D. Rosenblatt, A. Sharon, and A. A. Friesem, "Resonant grating waveguide structures," IEEE J. Quantum Electron. 33, 2038-2059 (1997). [CrossRef]
- R. F. Kazarinov and C. H. Henry, "Second-order distributed feedback lasers with mode selection provided by first-order radiation loss," IEEE J. Quantum Electron. 21, 144-150 (1985). [CrossRef]
- T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995). [CrossRef]
- S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975). [CrossRef]
- R. Eberhart and J. Kennedy, "Particle swarm optimization," in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995) 1942-1948.
- M. Shokooh-Saremi and R. Magnusson, "Particle swarm optimization and its application to the design of diffraction grating filters," Opt. Lett. 32, 894-896 (2007). [CrossRef] [PubMed]
- C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004). [CrossRef]

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