## Dispersion Engineering with Leaky-Mode Resonant Photonic Lattices

Optics Express, Vol. 18, Issue 1, pp. 108-116 (2010)

http://dx.doi.org/10.1364/OE.18.000108

Acrobat PDF (634 KB)

### Abstract

We investigate the dispersion properties of leaky-mode resonance elements with emphasis on slow-light applications. Using particle swarm optimization, we design three exemplary bandpass leaky-mode devices. A single-layer silicon-on-insulator leaky-mode element shows a time-delay peak of ~8 ps at the resonance wavelength. A double membrane element exhibits an average delay of ~6 ps over ~0.75 nm spectral bandwidth with a relatively flat dispersion response. By cascading five double-membrane elements, we achieve an accumulative delay of ~30 ps with a very flat dispersion response over ~0.5 nm bandwidth. Thus, we show that delay elements based on leaky-mode resonance, by proper design, exhibit large amount of delay yet very flat dispersion over appreciable spectral bandwidths, making them potential candidates for optical buffers, delay lines, and switches.

© 2010 OSA

## 1. Introduction and background

14. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. **37**(4), 525–532 (2001). [CrossRef]

14. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. **37**(4), 525–532 (2001). [CrossRef]

15. C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. **10**(7), 994–996 (1998). [CrossRef]

18. M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. Le Grange, and S. S. Patel, “Integrated resonance-enhanced variable optical delay lines,” IEEE Photon. Technol. Lett. **17**(4), 834–836 (2005). [CrossRef]

20. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**(12), 741–747 (2008). [CrossRef]

21. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics **1**(1), 65–71 (2007). [CrossRef]

20. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**(12), 741–747 (2008). [CrossRef]

22. F. Schreier, M. Schmitz, and O. Bryngdahl, “Pulse delay at diffractive structures under resonance conditions,” Opt. Lett. **23**(17), 1337–1339 (1998). [CrossRef] [PubMed]

22. F. Schreier, M. Schmitz, and O. Bryngdahl, “Pulse delay at diffractive structures under resonance conditions,” Opt. Lett. **23**(17), 1337–1339 (1998). [CrossRef] [PubMed]

23. M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Shi, and X. Gao, “Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. **39**(17), 2871–2879 (2000). [CrossRef]

24. S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A **14**(7), 1617–1626 (1997). [CrossRef]

23. M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Shi, and X. Gao, “Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. **39**(17), 2871–2879 (2000). [CrossRef]

25. W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. **84**(24), 4905–4907 (2004). [CrossRef]

26. W. Nakagawa, R. Tyan, P. Sun, F. Xu, and Y. Fainman, “Ultrashort pulse propagation in near-field periodic diffractive structures by use of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A **18**(5), 1072–1081 (2001). [CrossRef]

27. T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express **10**(16), 840–843 (2002). [PubMed]

## 2. Computational basics

28. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE **73**(5), 894–937 (1985). [CrossRef]

29. 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**(5), 1077–1086 (1995). [CrossRef]

_{0}is the amplitude of the pulse; T is the temporal pulse width (T =Δτ(2ln2)

^{-1/}

*; Δτ is the full width at half maximum (FWHM) of |Е*

^{2}_{y}(t)|

^{2}); t

_{0}is the pulse-peak offset; ω

_{0}= 2πc/λ

_{0}is the central angular frequency and c and λ

_{0}are the speed of light and the wavelength in vacuum, respectively. To use RCWA for analysis, the incident Gaussian pulse is decomposed into its monochromatic Fourier components (plane waves), which is performed by the Fourier transformation and discretization. These discrete monochromatic components are then treated independently by our established RCWA analysis technique, which at a given incident angle provides the complex reflection coefficients R(ω

_{n}) (or R(λ

_{n})) and complex transmission coefficients T(ω

_{n}) (or T(λ

_{n})) of each diffraction order. In addition, the independent analysis of each monochromatic component can facilitate the inclusion of material dispersion effects. The reflected pulse Е

_{R}(ω

_{n}) and transmitted pulse Е

_{T}(ω

_{n}) in the frequency domain for a specific diffraction order are thus given by

_{R}(t; ω

_{n}) and Е

_{T}(t; ω

_{n}).

14. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. **37**(4), 525–532 (2001). [CrossRef]

## 3. Leaky-mode resonance dispersive device examples

31. M. Shokooh-Saremi and R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. **32**(8), 894–896 (2007). [CrossRef] [PubMed]

_{1}, F

_{2}, F

_{3}, F

_{4}] = [0.071, 0.275, 0.399, 0.255]. Also, n

_{H}= 3.48, n

_{S}= 1.48, and n

_{L}= n

_{inc}= 1.0 (air). Figure 3(a) shows the transmittance, phase response, delay, and dispersion of this filter under normal incidence with TE polarization. This filter provides delays as high as ~10 ps at the transmission resonance; however, the dispersion width is narrow and zero dispersion is obtainable only near 1524.5 nm. Figures 3(b) and (c) display the response of this filter to excitation with a pulse in the spectral (wavelength) and time domains, respectively. The input pulse has a width of 30 ps (FWHM) in time. The output pulse experiences a delay of ~8.25 ps with respect to the input pulse. It has reduced amplitude on account of the incomplete transmission and limited passband noted in Fig. 3(b).

_{1}, F

_{2}, F

_{3}, F

_{4}] = [0.0626, 0.3013, 0.4576, 0.1785], and d

_{Cavity}= 2000 nm. Figure 4(b) illustrates the transmittance, phase, delay, and dispersion of this device. This element shows a flat-top transmission bandwidth, which actually is a result of merging two adjacent narrow transmission resonances. In addition, the delay response exhibits an average of ~7 ps in the transmission band. In comparison to the previous case, the dispersion is flatter. Figures 5(a) and (b) show the pulse response of this filter in wavelength and time domains, respectively. The input pulse has a full-width half-maximum (FWHM) of 20 ps in time and spectrally fits well inside the transmission bandwidth of the filter. The input pulse is delayed by ~6.1 ps by being transmitted through this filter in good agreement with the delay in Fig. 4(b).

20. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**(12), 741–747 (2008). [CrossRef]

_{Cavity}= 5) with spacing d

_{B}= 5.0 μm. Figure 6(a) shows the computed results. Although the high-transmission bandwidth is smaller than it is for the single-cavity structure, cascading the cells results in a flat delay response of ~30 ps over a ~0.5 nm wavelength band. Moreover, the flat low-dispersion response illustrates that such structures are promising for imposing constant (and almost dispersion-free) delays on optical pulses. Theoretically, this ~30 ps group delay for the ~34 μm long structure designed here corresponds to a group velocity of ~0.0038c. Figures 6(b) and (c) display the response of this filter to pulse excitation. The input pulse has a FWHM of 30 ps, and the output pulse preserves its shape with a delay of ~30 ps with respect to the input pulse. For comparison, Notomi et al. reported 75 ps delay with 60 cavities each being 2100 nm in diameter; the total structure length was 175 μm [20

**2**(12), 741–747 (2008). [CrossRef]

_{D}is large, the device functions as a bulk element. Each sub-ridge layer in Fig. 7 is a GMR element like in Fig. 2. If we design the sublayer to operate as a bandpass filter, the input light will resonate transversely and be reradiated forward to the next GMR layer. This idea can also be implemented with a series of GMR filters on numerous substrates cascaded as a stack. The concept in Fig. 7 is convenient in that a large number of cascaded resonant units can be fabricated in a few steps by e-beam lithography (EBL) and deep reactive ion etching (DRIE), resulting in a compact system of resonant delay units. Certainly, the dimensions of the device and the input beam size should be specified with practical limitations in mind. On the other hand, if thickness d

_{D}is small, such as on the order of 100-300 nm, this can be a waveguide device. In that case, the functionality of the device employs waveguiding is a dual sense. First, there is the waveguide that guides light from one resonant layer to the next. For that to work, the structure requires a higher average refractive index than that of the surrounding media as usual. A membrane in air will satisfy this requirement with additional considerations if the device sits on a substrate. Second, each GMR cell forms a resonant waveguide, again similar to the one shown in Fig. 2. In principle, we can cascade a large number of these GMR cells to achieve a specified delay. Indeed, multiple-cell cascading is the basis for the new coupled-resonator optical waveguide (CROW) technology being developed [20

**2**(12), 741–747 (2008). [CrossRef]

## 4. Conclusions

## Acknowledgements

## References and links

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

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

3. | K. Sakoda, |

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

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

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

7. | 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. |

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

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

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

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

12. | Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express |

13. | M. Shokooh-Saremi and R. Magnusson, “Wideband leaky-mode resonance reflectors: influence of grating profile and sublayers,” Opt. Express |

14. | G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. |

15. | C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. |

16. | G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control,” J. Lightwave Technol. |

17. | C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. |

18. | M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. Le Grange, and S. S. Patel, “Integrated resonance-enhanced variable optical delay lines,” IEEE Photon. Technol. Lett. |

19. | E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photon. News, 40–45 (Nov. 2007). |

20. | M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics |

21. | F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics |

22. | F. Schreier, M. Schmitz, and O. Bryngdahl, “Pulse delay at diffractive structures under resonance conditions,” Opt. Lett. |

23. | M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Shi, and X. Gao, “Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. |

24. | S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A |

25. | W. Suh and S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. |

26. | W. Nakagawa, R. Tyan, P. Sun, F. Xu, and Y. Fainman, “Ultrashort pulse propagation in near-field periodic diffractive structures by use of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A |

27. | T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express |

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

29. | 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 |

30. | R. Eberhart, and J. Kennedy, “Particle swarm optimization,” in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995) 1942–1948. |

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

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(130.2790) Integrated optics : Guided waves

(260.2030) Physical optics : Dispersion

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 17, 2009

Revised Manuscript: October 20, 2009

Manuscript Accepted: November 5, 2009

Published: December 22, 2009

**Citation**

Robert Magnusson, Mehrdad Shokooh-Saremi, and Xin Wang, "Dispersion Engineering with Leaky-Mode Resonant Photonic Lattices," Opt. Express **18**, 108-116 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-1-108

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton, 1995.
- A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th edition, Oxford University Press, New York, 2007.
- K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Berlin, 2001.
- P. Vincent, M. Neviere, “Corrugated dielectric waveguides: A numerical study of the second-order stop bands,” Appl. Phys. (Berl.) 20(4), 345–351 (1979). [CrossRef]
- L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55(6), 377–380 (1985). [CrossRef]
- E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta (Lond.) 33, 607–619 (1986).
- G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15(7), 886–887 (1985). [CrossRef]
- I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36(11), 1527–1539 (1989). [CrossRef]
- R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992). [CrossRef]
- S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993). [CrossRef] [PubMed]
- Y. Ding, R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12(23), 5661–5674 (2004). [CrossRef] [PubMed]
- Y. Ding, R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12(9), 1885–1891 (2004). [CrossRef] [PubMed]
- M. Shokooh-Saremi, R. Magnusson, “Wideband leaky-mode resonance reflectors: influence of grating profile and sublayers,” Opt. Express 16(22), 18249–18263 (2008). [CrossRef] [PubMed]
- G. Lenz, B. J. Eggleton, C. K. Madsen, R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001). [CrossRef]
- C. K. Madsen, G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10(7), 994–996 (1998). [CrossRef]
- G. Lenz, C. K. Madsen, “General optical all-pass filter structures for dispersion control,” J. Lightwave Technol. 17(7), 1248–1254 (1999). [CrossRef]
- C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11(12), 1623–1625 (1999). [CrossRef]
- M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. Le Grange, S. S. Patel, “Integrated resonance-enhanced variable optical delay lines,” IEEE Photon. Technol. Lett. 17(4), 834–836 (2005). [CrossRef]
- E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photon. News, 40–45 (Nov. 2007).
- M. Notomi, E. Kuramochi, T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]
- F. Xia, L. Sekaric, Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]
- F. Schreier, M. Schmitz, O. Bryngdahl, “Pulse delay at diffractive structures under resonance conditions,” Opt. Lett. 23(17), 1337–1339 (1998). [CrossRef] [PubMed]
- M. S. Mirotznik, D. W. Prather, J. N. Mait, W. A. Beck, S. Shi, X. Gao, “Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method,” Appl. Opt. 39(17), 2871–2879 (2000). [CrossRef]
- S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14(7), 1617–1626 (1997). [CrossRef]
- W. Suh, S. Fan, “All-pass transmission or flattop reflection filters using a single photonic crystal slab,” Appl. Phys. Lett. 84(24), 4905–4907 (2004). [CrossRef]
- W. Nakagawa, R. Tyan, P. Sun, F. Xu, Y. Fainman, “Ultrashort pulse propagation in near-field periodic diffractive structures by use of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 18(5), 1072–1081 (2001). [CrossRef]
- T. Vallius, P. Vahimaa, J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express 10(16), 840–843 (2002). [PubMed]
- T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, 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(5), 1077–1086 (1995). [CrossRef]
- R. Eberhart, and J. Kennedy, “Particle swarm optimization,” in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995) 1942–1948.
- M. Shokooh-Saremi, R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. 32(8), 894–896 (2007). [CrossRef] [PubMed]

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