## Optical bistability in subwavelength metallic grating coated by nonlinear material

Optics Express, Vol. 15, Issue 19, pp. 12368-12373 (2007)

http://dx.doi.org/10.1364/OE.15.012368

Acrobat PDF (189 KB)

### Abstract

A developed two-dimensional Finite Difference Time Domain (FDTD) method has been performed to investigate the optical bistability in a subwavelength metallic grating coated by nonlinear material. Different bistability loops have been shown to depend on parameters of the structure. The influences of two key parameters, thickness of nonlinear material and slit width of metallic grating, have been studied in detail. The effect of optical bistability in the structure is explained by Surface Plasmons (SPs) mode and resonant waveguide theory.

© 2007 Optical Society of America

## 1. Introduction

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) **391**, 667–669 (1998) [CrossRef]

2. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,“Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. **86**, 1114–1117 (2001). [CrossRef] [PubMed]

6. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature **440**, 508–511 (2006). [CrossRef] [PubMed]

7. E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science **311**, 189–193(2006). [CrossRef] [PubMed]

8. I. I. Smolyaninov, “Quantum Fluctuations of the Refractive Index near the Interface between a metal and a Nonlinear Dielectric,” Phys. Rev. Lett. **94**, 057403 (2005). [CrossRef] [PubMed]

11. C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, “Beam manipulating by metallic nano-optic lens containing nonlinear media,” Opt. Express **15**, 9541–9546 (2007) [CrossRef] [PubMed]

## 2. Simulation method and model

*p*=1.44,

*w*=0.3 and

*h*=0.1 µm. A TM-polarized plane wave vertically illuminates the structure from the top with the wavelength at

*λ*=1.55µm, which is the typical wavelength used in telecommunications. The metallic grating is covered by a Kerr nonlinear material layer (thickness

*d*), whose dielectric constant

*ε*depends on the intensity of electric field |

_{d}*E*|

^{2}:

*ε*is the linear dielectric constant and

_{l}*χ*

^{(3)}is the third-order nonlinear susceptibility. In what follows, the linear dielectric constant is chosen to be

*ε*=2.25; the third-order nonlinear susceptibility is chosen as a typical value of nonlinear optical materials, such as InGaAsP, that is

_{l}*χ*

^{(3)}=1×10

^{-18}m

^{2}/V

^{2}(≈1.4×10

^{-10}esu) [12

12. M. Fujii, C. Koos, C. Poulton, I. Sakagami, J. Leuthold, and W. Freude, “A simple and rigorous verification technique for nonlinear FDTD algorithms by optical parametric four-wave mixing,” Microwave Opt. Technol. Lett. **48**, 88–91(2005). [CrossRef]

13. J. B. Jubkins and R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. **A. 12**, 1974–1983 (1995). [CrossRef]

*P*=

_{nl}*ε*

_{0}

*χ*

^{(3)}

*E*

^{3}[12

12. M. Fujii, C. Koos, C. Poulton, I. Sakagami, J. Leuthold, and W. Freude, “A simple and rigorous verification technique for nonlinear FDTD algorithms by optical parametric four-wave mixing,” Microwave Opt. Technol. Lett. **48**, 88–91(2005). [CrossRef]

*ε*

_{∞}is the relative dielectric constant at infinite frequency in the Lorentz model,

*P*is the linear polarization vector generated by the Lorentz model. Calculating with increasing or decreasing intensity of incident light, the nonlinear FDTD program results in two different transmission spectra, which compose a whole bistability loop. The periodic boundary conditions (PBC) [14

_{l}14. P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propagat. **42**, 1317–1324 (1994). [CrossRef]

*ε*=-86.64+i8.74 at the wavelength of 1.55µm, other dielectric constants are taken from values in [16]

_{m}## 3. Simulation results and discussion

*d*=0.88µm. Figure 2(a) shows the far-field transmission spectra corresponding to SPs excitation at different intensities of incident light. The intensities are chosen as

*I*=|

*E*|

^{2}=1×10

^{14}, 2.25×10

^{16}and 4×10

^{16}V

^{2}/m

^{2}respectively. In Fig. 2(a), it is obvious that the transmission peak red-shifts as the intensity increases, which can be explained by the SPs theory. The wavelength of transmission peak enhanced by SPs in metallic grating is approximately obtained from [18]:

*ε*is the dielectric constant of metal,

_{m}*ε*is the dielectric constant of nonlinear material on metal surface,

_{d}*θ*is the incident angle of the light hitting the metal surface. Although the light is incident vertically in our structure,

*θ*is not ignorable for scattering light on metal surface. According to Eq. (1) and Eq. (3), when the incident intensity grows,

*ε*is augmented by nonlinear response and then makes the wavelength λ

_{d}*larger, hence the transmission peak red-shifts, as shown in Fig. 2(a).*

_{sp}*λ*=1.55µm. T1 and T2 in Fig. 2(b) denote the far-field transmissions obtained from the increasing and decreasing intensity of incident light, respectively. When the intensity increases, the transmission T1 jumps to a higher value in a discontinuous manner around

*I*=1.7×10

^{16}V

^{2}/m

^{2}. By contrast, when the incident intensity decreases, the transmission T2 keeps a high transmittance situation and follows the upper branch in Fig. 2(b).

*w*, which can be controlled to enhance the bistability effect. The influence of thickness

*d*is introduced firstly. Figure 3(a) shows the far-field transmission spectra versus thickness

*d*at different intensities of incident light. The incident wavelength is

*λ*=1.55µm with intensities

*I*=1×10

^{14}, 2.25×10

^{16}and 4×10

^{16}V

^{2}/m

^{2}. In Fig. 3(a), three sharp peaks can be observed at all intensities, which presents the periodic situation of far-field transmission. The value of transmission period is about Δ

*d*≈0.74µm at low intensity (

*I*=1×10

^{14}V

^{2}/m

^{2}), and it decreases when the intensity grows to high values.

*k*

_{0}=2

*π*/

*λ*is the wave vector of incident light,

*k*=2

*π*/

*p*is the wave vector of the metallic grating. Φ

_{12}and Φ

_{23}are the accompanied phase changes respective at the air/dielectric and dielectric/metal interfaces, both independent to the thickness

*d*of nonlinear layer. When Eq. (4) is satisfied by periodic values of thickness d, the resonant waveguide mode is formed in the nonlinear layer. Hence more incident energy is restricted in the nonlinear layer and coupled with SPs, which strongly enhance the far-field transmission, as shown in Fig. 3(a). The transmission period Δ

*d*can be concluded from Eq. (4) as follow:

*ε*≈

_{d}*ε*=2.25, then we can obtain the value of period Δ

_{l}*d*≈0.742µm, coincident with the result Δ

*d*≈0.74µm in Fig. 3(a) at low intensity (

*I*=1×10

^{14}V

^{2}/m

^{2}). When the intensity grows to high values,

*ε*

_{d}increases and the transmission period Δ

*d*decreases according to Eq. (5), also shown in Fig. 3(a).

*d*=0.88µm, which is close to the middle transmission peak in Fig. 3(a). Next, we consider the bistability effect associated with other two peaks in Fig. 3(a), shown in Figs. 3(b) and 3(c). The thicknesses of nonlinear layer are respectively chosen as

*d*=0.16 and 1.62 µm, which result in different bistability effects. In Fig. 3(b), the bistability loop is hard to be observed, for the upper and lower branches almost overlap together. By contrast, the bistability loop at

*d*=1.62µm is rather obvious, which shows larger difference between upper and lower branches.

*d*=1.62µm. The physical origin of the bistability and the influence of nonlinear layer thickness can be understood taking into account the optical resonant waveguide mode. It is well known that optical bistability often appears at some optical resonant structure containing nonlinear material, for example, the Fabry-Pérot resonant cavity. In our structure, the bistability originates from the resonant waveguide mode in the nonlinear layer. The waveguide mode can gather the EM-field energy of incident light to enhance nonlinear effect in the nonlinear layer and excite SPs on metal surface. The role of SPs is to enhance nonlinear effect around metal surface and transmit energy to slits. If the thickness d becomes larger, more energy can be gathered in nonlinear layer and enhance the total nonlinear effect, hence the nonlinear dielectric constant and transmission both keep at a high situation with decreasing intensity, resulting in a more significant bistability loop.

*d*=0.88µm and wavelength

*λ*=1.55µm. The bistability loops at

*w*=0.1, 0.2 and 0.4 µm are shown in Figs. 4(a), 4(b) and 4(c) respectively. The bistability loop at

*w*=0.2µm is most significant, whose bistable region is wider than

*w*=0.1 and 0.4 µm. The relationship between slit width and bistability loop is further studied in Fig. 4(d), which shows the transmission ratio (T2/T1) at different slit widths, from

*w*=0.1 to 0.5 µm. It is easy to find that the best result appears at

*w*=0.16µm with a peak more than 20, and nearly no peak can be observed at the slit width more than

*w*=0.4µm. Moreover, the position of peak shifts to high intensity when the slit width increases in Fig. 4(d), which means optical bistability can be carried out at a low intensity with suitable slit width.

*w*=0.4µm, the resonant waveguide mode is destroyed, and more energy transits the slits directly instead of transmitted by SPs, hence the bistability loop is hard to be observed, as shown in Fig. 4(c). However, if the slit width is too small (e.g., w=0.1µm), the energy of incident light is hard to transit the slits, which also results in a weaker bistability effect, as shown in Fig. 4(a). So there should be an optimal slit width for optical bistability in the structure, which can supply both high transmission and enough waveguide region.

^{-6}esu and show ultrafast time response, which can be used in the structure for a high-sensitive ultrafast all-optical switching.

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) |

2. | L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,“Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. |

3. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

4. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, F. Martin-Moreno, L. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

5. | X. Jiao, P. Wang, L. Tang, Y. Lu, Q. Li, D. Zhang, P. Yao, H. Ming, and J. Xie, “Fabry-Pérot-like phenomenon in the surface plasmons resonant transmission of metallic gratings with very narrow slits,” Appl. Phys. |

6. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature |

7. | E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science |

8. | I. I. Smolyaninov, “Quantum Fluctuations of the Refractive Index near the Interface between a metal and a Nonlinear Dielectric,” Phys. Rev. Lett. |

9. | J. A. Porto, L. Martin-Moreno, and F. J. Garcia-Vidal, “Optical bistability in subwavelength slit apertures containing nonlinear media,” Phys. Rev. |

10. | G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals,” Phys. Rev. Lett. |

11. | C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, “Beam manipulating by metallic nano-optic lens containing nonlinear media,” Opt. Express |

12. | M. Fujii, C. Koos, C. Poulton, I. Sakagami, J. Leuthold, and W. Freude, “A simple and rigorous verification technique for nonlinear FDTD algorithms by optical parametric four-wave mixing,” Microwave Opt. Technol. Lett. |

13. | J. B. Jubkins and R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. |

14. | P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propagat. |

15. | A. Taflove and S. Hagness, |

16. | E. D. Palik, |

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

18. | H. Raether, |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 23, 2007

Revised Manuscript: September 10, 2007

Manuscript Accepted: September 11, 2007

Published: September 13, 2007

**Citation**

Changjun Min, Pei Wang, Xiaojin Jiao, Yan Deng, and Hai Ming, "Optical bistability in subwavelength metallic grating coated by nonlinear material," Opt. Express **15**, 12368-12373 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12368

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
- L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen,"Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001). [CrossRef] [PubMed]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, F. Martin-Moreno, L. J. Garcia-Vidal, and T. W. Ebbesen,"Beaming light from a subwavelength aperture," Science 297, 220-222 (2002). [CrossRef]
- X. Jiao, P. Wang, L. Tang, Y. Lu, Q. Li, D. Zhang, P. Yao, H. Ming, and J. Xie, "Fabry-Pérot-like phenomenon in the surface plasmons resonant transmission of metallic gratings with very narrow slits," Appl. Phys. B 80, 301-305 (2005).
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006). [CrossRef] [PubMed]
- E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006). [CrossRef] [PubMed]
- I. I. Smolyaninov, "Quantum Fluctuations of the Refractive Index near the Interface between a metal and a Nonlinear Dielectric," Phys. Rev. Lett. 94, 057403 (2005). [CrossRef] [PubMed]
- J. A. Porto, L. Martin-Moreno, and F. J. Garcia-Vidal, "Optical bistability in subwavelength slit apertures containing nonlinear media," Phys. Rev. B 70, 081402 (2004).
- G. A. Wurtz, R. Pollard, and A. V. Zayats, "Optical Bistability in Nonlinear Surface-Plasmon Polaritonic Crystals," Phys. Rev. Lett. 97, 057402 (2006). [CrossRef] [PubMed]
- C. Min, P. Wang, X. Jiao, Y. Deng, and H. Ming, "Beam manipulating by metallic nano-optic lens containing nonlinear media, " Opt. Express 15, 9541-9546 (2007) [CrossRef] [PubMed]
- M. Fujii, C. Koos, C. Poulton, I. Sakagami, J. Leuthold and W. Freude, "A simple and rigorous verification technique for nonlinear FDTD algorithms by optical parametric four-wave mixing," Microwave Opt. Technol. Lett. 48, 88-91 (2005). [CrossRef]
- J. B. Jubkins and R. W. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A. 12, 1974-1983 (1995). [CrossRef]
- P. Harms, R. Mittra, and W. Ko, "Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures," IEEE Trans. Antennas Propagat. 42, 1317-1324 (1994). [CrossRef]
- A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., (Artech House, Boston, MA 2000).
- E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, London 1985).
- M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1975).
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, Berlin 1988).

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