## Analytical theory for the nonlinear optical response of a Kerr-type standing-wave cavity side-coupling to a MIM waveguide |

Optics Express, Vol. 21, Issue 20, pp. 23687-23694 (2013)

http://dx.doi.org/10.1364/OE.21.023687

Acrobat PDF (874 KB)

### Abstract

In this article, an analytical theory to describe the nonlinear dynamic response characteristics of a typical SPP waveguide-cavity structure formed by a Kerr-type standing-wave cavity side-coupling to a metal-insulator-metal (MIM) waveguide is proposed by combining the temporal coupled mode theory and the Kerr nonlinearity. With the analytical theory, the optical bistability with the hysteresis behavior is successfully predicted, and the optical bistability evolutions and its dynamic physical mechanism are also phenomenologically analyzed. Moreover, the influence of the quality factors *Q*_{0} and *Q*_{1} on the first-turnning point (FTP) power of optical bistability and the bistable region width, the approaches to decrease the FTP power and to broaden the bistable region are also discussed in detail with our analytical theory. This work can help us understand the physical mechanism of the nonlinear dynamical response at nanoscale, and may be useful to design nonlinear nanophotonic systems for applications in ultra-compact all-optical devices and storages.

© 2013 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

2. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**, 131–314 (2005). [CrossRef]

3. J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength scale localization,” Phys. Rev. B **73**, 035407 (2006). [CrossRef]

4. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonic beyond the diffraction limit,” Nat. Photonics **4**, 83–90 (2010). [CrossRef]

5. D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. **12**, 015002 (2010). [CrossRef]

7. G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. **101**, 013111 (2012). [CrossRef]

8. Q. Zhang, X. G. Huang, X. S. Lin, J. Tao, and X. P. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express **17**, 7549–7555 (2009). [CrossRef]

11. G. X. Wang, H. Lu, X. M. Liu, D. Mao, and L. N. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express **19**, 3513–3518 (2011). [CrossRef] [PubMed]

12. G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. **2**, 125–135 (2008). [CrossRef]

13. J. Tao, Q. J. Wang, and X. G. Huang, “All-optical plasmonic switches based on coupled nano-disk cavity structures containing nonlinear material,” Plasmonics **6**, 753–759 (2011). [CrossRef]

15. H. Lu, X. M. Liu, L. R. Wang, Y. K. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express **19**, 2910–2915 (2011). [CrossRef] [PubMed]

16. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics **3**, 55–58 (2009). [CrossRef]

17. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. **11**, 2933–2937 (2011). [CrossRef] [PubMed]

18. J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, H. Ming, and Q. W. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. **94**, 081117 (2009). [CrossRef]

19. A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B **28**, 2820–2826 (2011). [CrossRef]

20. X. S. Lin, J. H. Yan, Y. B. Zheng, L. J. Wu, and S. Lan, “Bistable switching in the lossy side-coupled plasmonic waveguide-cavity structrues,” Opt. Express **19**, 9594–9599 (2009). [CrossRef]

21. X. L. Wang, H. Q. Jiang, J. X. Chen, P. Wang, Y. H. Lu, and H. Ming, “Optical bistability effect in plasmonic racetrack resonator with high extinction ratio,” Opt. Express **19**, 19415–19421 (2011). [CrossRef] [PubMed]

## 2. Analytical theory

*S*

_{+1}has a time dependence of

*e*, the steady-state transmission of the plasmonic structure can be expressed as And, the total energy in the cavity |

^{jωt}*A*|

^{2}and the incident power |

*S*

_{+1}|

^{2}(or

*P*

_{in}) satisfies:

*ω*

_{c}and the quality factors

*Q*

_{0}and

*Q*

_{1}may change with the nonlinear refractive index variation of the cavity. However, if only weak Kerr nonlinearity, i.e., the nonlinear refractive index Δ

*n*< 5%, is considered, the tiny changes of

*Q*

_{0}and

*Q*

_{1}due to the Kerr nonlinearity can be ignored. Thus, in the TCMT treatment for the plasmonic structure shown in Fig. 1, only the variation of the resonant frequency caused by the Kerr nonlinearity may be taken into accounted.

24. L. Liu, X. Hao, Y. T. Ye, J. X. Liu, Z. L. Chen, Y. C. Song, Y. Luo, J. Zhang, and L. Tan, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm. **285**, 2558–2562 (2012). [CrossRef]

*n*

_{eff}

*L*

_{eff}+ 2

*L*

_{pen}) =

*Nλ*

_{c}(

*N*= 1, 2, 3, ···), with

*L*

_{pen},

*n*

_{eff}and

*L*

_{eff}respectively being the penetration depths at two ends, the effective refractive index and the effective length of the F-P cavity. For the weak Kerr nonlinearity cavity with only single resonant mode, the resonant wavelength of the cavity can be written as: where

*λ*

_{c0}is the linear resonant wavelength,

*n*

_{2}is the Kerr nonlinear-index coefficient, and 〈|

*E*|

_{b}^{2}〉 = |

*A*|

^{2}/ (

*εS*) is the average electric field intensity in the cavity with

*S*and

*ε*being the area and the permittivity of the cavity, respectively.

*λ*

_{c}= 2

*πc/ω*

_{c}and |

*S*

_{+1}|

^{2}=

*P*

_{in}, we can get a cubic equation with

*λ*

_{c}after eliminating the term of |

*A*|

^{2}: where

*B*= −(2

*λ*+

*λ*

_{c0}),

*C*

_{1}=

*λ*

^{2}[1 + (1/

*Q*

_{0}+ 1/ (2

*Q*

_{1}))

^{2}] + 2

*λλ*

_{c0},

*C*

_{2}=

*n*

_{2}

*L*

_{eff}

*λ*

^{2}/(2

*πcQ*

_{1}

*εS*), and

*D*= −

*λ*

_{c0}

*λ*

^{2}[1 + (1/

*Q*

_{0}+ 1/ (2

*Q*

_{1}))

^{2}]. From Eq. (5), we can see that, for a given structure of the standing-wave cavity side-coupling to a MIM waveguide,

*λ*

_{c}exhibits strong nonlinear relationship with the wavelength

*λ*and the power

*P*

_{in}of the incident light.

*λ*

_{c}with

*λ*and

*P*

_{in}, we may find the derivative of

*P*

_{in}with respect to

*λ*

_{c}from Eq. (5), and let

*dP*

_{in}/

*dλ*

_{c}= 0, then we have

*B*and

*D*are just functions of the incident light wavelength

*λ*for a given waveguide-cavity structure. In general, Eq. (6) may have three distinct real roots if its discriminant is less than zero, i.e., Δ < 0. If assuming

*λ*

_{c1},

*λ*

_{c2}and

*λ*

_{c3}are respectively the three distinct real roots of Eq. (6), then, they must spontaneously satisfy: (1)

*λ*

_{c1}+

*λ*

_{c2}+

*λ*

_{c3}= −

*B*/2, which is constantly larger than zero, and (2)

*λ*

_{c1}

*λ*

_{c2}

*λ*

_{c3}=

*D*/2, which is always less than zero. This indicates that there must be two positive and a negative real numbers among

*λ*

_{c1},

*λ*

_{c2}, and

*λ*

_{c3}when Δ < 0 for Eq. (6). In fact, the existence of the two positive real roots for Eq. (6) means that there are two knee points for the

*P*

_{in}∼

*λ*

_{c}curve, and thus, implies that there may exist some optical bistability phenomenon when

*λ*is suitably chosen to ensure Δ < 0.

*λ*for the optical bistability, according to the critical condition Δ = 0, we have where

*x*=

*λ/λ*

_{c0}and

*ζ*= 1/

*Q*

_{0}+ 1/(2

*Q*

_{1}). We can then get the two positive real roots by solving Eq. (7)[26] where

*B′*= (27

*ζ*

^{2}+ 15)/8,

*q′*= (27 – 459

*ζ*

^{2})/256, and Δ

*′*= −(27

*ζ*/64)

^{2}. Since

*ζ*is in the order of 10

^{−2}, the higher order terms of

*ζ*may be ignorable. Then, we have or where Δ

*λ*= 2

*λ*

_{c0}(1/

*Q*

_{0}+ 1/(2

*Q*

_{1})) is the FWHM of the linear transmission spectrum, which can be easily obtained from Eq. (2).

*λ*

_{M}is defined as

*λ*>

*λ*

_{M}(or

*λ*<

*λ*

_{M}), the discriminant of Eq. (6) is less than zero, and thus two different

*λ*

_{c}truly exist for a specific given

*P*

_{in}, leading to the occurrence of optical bistability phenomenon.

## 3. Results and discussions

*Q*

_{0}= 235,

*Q*

_{1}= 65,

*λ*

_{c0}= 1556.9 nm,

*S*= 3.6 × 10

^{4}nm

^{2},

*L*

_{eff}= 360 nm, and

*n*

_{2}= 1 × 10

^{−8}cm

^{2}/W [27

27. X. Y. Hu, P. Jiang, C. Y. Ding, H. Yang, and Q. H. Gong, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm. **285**, 2558–2562 (2012). [CrossRef]

_{2}-Ag waveguide with a side-coupling nonlinear polystyrene rectangular cavity shown in Fig. 1, whose structure parameters are

*w*= 50 nm,

_{t}*w*= 100 nm,

*L*= 360 nm, and

*g*= 15 nm. According to our theory described in the above, when the incident light wavelength is larger than

*λ*

_{M}= 1589.1 nm, the optical bistability phenomenon may occur. To test the validity of our prediction, the transmission as a function of

*P*

_{in}is calculated for different given

*λ*by using Eqs. (2) and (5), and the results are shown in Fig. 2(a). Note that the calculation results with our analytical model and the finite-difference time-domain (FDTD) technique [19

19. A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B **28**, 2820–2826 (2011). [CrossRef]

*λ*is larger than 1589.1 nm. Otherwise, no bistability phenomenon is exhibited. However, when the incident wavelength is smaller than 1589.1 nm, the transmission can be continuously adjusted in a large range by only slightly changing the incident power near the transmission dip, such a property may be used to design all-optical switches.

*P*

_{in}. However, the increasing speed of the cavity energy with

*P*

_{in}depends on the incident light wavelength. When

*λ*is less than

*λ*

_{M}, the difference between

*λ*and

*λ*

_{c0}is small, making

*λ*to be within the cavity resonance band in the beginning [see Fig. 3(a)]. Thus, even if the initial incident power is low, the proportion of the power coupled into the cavity is still high, which makes an obvious red-shift of

*λ*

_{c}. For instance, when the incident light wavelength is fixed at 1580 nm, as shown in Fig. 3(b), a red-shift about 5 nm for

*λ*

_{c}respect with to

*λ*

_{c0}has been made if the incident power is increased to 0.25 W from zero. This red-shift of

*λ*

_{c}can further strengthen the resonant coupling effect between the cavity and the waveguide, which in turn speeds up the red-shifted of

*λ*

_{c}to be close to

*λ*[see Fig. 3(b)], forming a positive feedback mechanism. This positive feedback mechanism makes the transmissions decreases monotonously with the increase of

*P*

_{in}and reaches to their minimum rapidly at which

*λ*

_{c}=

*λ*. Such a property of the transmissions decreasing monotonously to their minimum by slightly changing

*P*

_{in}may be used for the design of all-optical switches, as mentioned before. If then

*P*

_{in}is further increased,

*λ*

_{c}may be gradually larger than

*λ*because of the red-shift effect caused by the Kerr nonlinearity. This results in the weakened resonant coupling between the cavity and the waveguide which retards the red-shifted

*λ*

_{c}to be away from

*λ*, forming a negative feedback mechanism. This negative feedback mechanism can help

*λ*to stay in the cavity resonance band with the center wavelength of

*λ*

_{c}, finally resulting that the transmissions increases slowly with the increase of

*P*

_{in}from their minimal values. As the incident power decreases from a high level, a reverse process is exhibited, and no hysteresis behavior occurs due to the monotonous changes of

*λ*

_{c}.

*λ*is larger than

*λ*

_{M}, as shown in Fig. 3(a),

*λ*is far away from the linear cavity resonance band near

*λ*

_{c0}in the beginning, and the proportion of the power coupled into the cavity is very low. This causes that the energy in the cavity increases very slowly with the increase of

*P*

_{in}from zero, and thus, the Kerr nonlinearity gives an ignorable red-shift for

*λ*

_{c}. For an example, when the incident wavelength is 1610 nm, as shown in Fig. 3(c), a 5 nm red-shift of

*λ*

_{c}cannot be obtained until the incident power is increased to 1.0 W. Such a red-shift of

*λ*

_{c}is not large enough to make

*λ*

_{c}being close to

*λ*. Because of this reason, the transmissions for

*λ*>

*λ*

_{M}keep staying in high levels until

*P*

_{in}has been increased to high enough to trigger the positive feedback mechanism described in the above, which results that the transmissions jump down to the low branches shown in Fig. 2. This specific

*P*

_{in}is so-called the first turning point (FTP) power, where a sudden red-shift of

*λ*

_{c}takes place [28

28. Q. H. Mao and J. W. Y. Lit, “Optical bistability in an L-band dual-wavelength erbium-doped fiber laser with overlapping cavities,” IEEE Photon. Technol. Lett. **14**, 1252–1254 (2002). [CrossRef]

*λ*

_{c}is larger than

*λ*once the transmission has jumped down to the low branch, just as shown in Fig. 3(c) in which

*λ*

_{c}jumps from 1578 nm to 1620 nm at the FTP for

*λ*= 1610 nm. After then, like the case of

*λ*<

*λ*

_{M}, the negative feedback mechanism also helps

*λ*stay in the cavity resonance band near

*λ*

_{c}, and the transmission increases slowly with the increase of

*P*

_{in}. However, if

*P*

_{in}is decreased from a high level, since

*λ*is located in the cavity resonance band near

*λ*

_{c}in the beginning, the resonant energy in the cavity can keep in a high level with the negative feedback mechanism even if

*P*

_{in}has been decreased to a level less than the FTP. This negative feedback mechanism can be maintained until

*λ*

_{c}=

*λ*, where the minimal transmission is reached. If

*P*

_{in}decreases furtherly,

*λ*

_{c}will continue to blue-shift and a positive feedback happen. When

*P*

_{in}decreases to a level called as the second turning point (STP), the positive feedback mechanism will result in a sudden blue-shift of

*λ*

_{c}[from 1606 nm to 1564 nm in Fig. 3(c)] with a tiny increase of

*P*

_{in}, which makes the transmission jumps up to the up-branch. After this jump, the transmission is increased slowly again with the decrease of

*P*

_{in}due to the tiny blue-shift of

*λ*

_{c}. Finally the optical bistability with the hysteresis behavior occurs for the case of

*λ*>

*λ*

_{M}.

*Q*

_{0}and

*Q*

_{1}, may affect the above hysteresis bistability behavior. Figures 4(a) and 4(b) show the critical wavelength

*λ*

_{M}as a function of

*Q*

_{0}and

*Q*

_{1}, respectively. As seen, for a given

*Q*

_{1}of 65,

*λ*

_{M}decreases from 1605 nm to 1582 nm when

*Q*

_{0}increases from 100 to 400. When

*Q*

_{1}increases from 40 to 100, the same change for

*λ*

_{M}can be obtained for a given

*Q*

_{0}of 235. The decrease of

*λ*

_{M}is originated from the less energy attenuation in the cavity with larger

*Q*

_{0}or

*Q*

_{1}. Figure 4(c) shows the transmission at 1610 nm as a function of the incident power for different

*Q*

_{0}when

*Q*

_{1}is fixed at 65. Here, we can see that, with the increase of

*Q*

_{0}, i. e., the decrease of the intrinsic cavity loss, both the FTP and the STP powers are decreased, with STP being decreased more, making that the bistalbe regions become wider. When

*Q*

_{0}increases from 100 to 400, the FTP and the STP powers reduce from 3 W to 2 W, and from 2.78 W to 0.9 W, respectively. The corresponding bistalbe region is increased from 0.22 W to 1.1 W. However, for a given

*Q*

_{0}, the FTP and the STP power may respectively be increased and decreased with the increase of

*Q*

_{1}because the weakend coupling strength between the waveguide and the cavity causes the decreases of both the incoming and outcoming energies for the cavity, thus giving wider bistable regions. Figure 4(d) shows such situations. From this figure it can be seen that when

*Q*

_{1}increases from 40 to 100, the FTP power increases from 1.6 W to 2.6 W, and the STP power reduces from 1.55 W to 1.25 W, resulting that the bistable region is broadened from 0.05 W to 1.35 W. Therefore, in order to reduce the FTP power of optical bistability, larger

*Q*

_{0}and smaller

*Q*

_{1}are preferred. Considering that the width of the bistable region may be narrowed by decreasing

*Q*

_{1}, an optimized way for reducing FTP power is to increase the value of

*Q*

_{0}as much as possible, which may be achievable by using optical gain in the cavity to compensate its intrinsic loss [29

29. P. Berini and I. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nat. Photonics **6**, 16–23 (2012). [CrossRef]

## 4. Conclusions

*Q*

_{0}and

*Q*

_{1}on the FTP power and the bistable region width is clarified, and the approaches to decrease the FTP power and to broaden the bistable region are also obtained. Our results may be helpful to get deep insight into the fundamental physics of the nonlinear dynamic response in nanoscale plasmonic waveguide devices and may be beneficial in their designs and optimizations.

## Acknowledgments

## References and links

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

2. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

3. | J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength scale localization,” Phys. Rev. B |

4. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonic beyond the diffraction limit,” Nat. Photonics |

5. | D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. |

6. | T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B |

7. | G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. |

8. | Q. Zhang, X. G. Huang, X. S. Lin, J. Tao, and X. P. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express |

9. | Y. Hwang, J. Kim, and H. Y. Park, “Frequency selective metal-insulator-metal splitters for surface plasmons,” Opt. Comm. |

10. | A. Noual, A. Akjouj, Y. Pennec, J. N. Gillet, and B. D. Rouhani, “Modeling of two-dimensional nanoscale Y-bent plasmonic waveguides with cavities for demultiplexing of the telecommunication wavelengths,” New J. Phys. |

11. | G. X. Wang, H. Lu, X. M. Liu, D. Mao, and L. N. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express |

12. | G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. |

13. | J. Tao, Q. J. Wang, and X. G. Huang, “All-optical plasmonic switches based on coupled nano-disk cavity structures containing nonlinear material,” Plasmonics |

14. | N. Nozhat and N. Granpayeh, “Switching power reduction in the ultra-compact Kerr nonlinear plasmonic directional coupler,” Opt. Comm. |

15. | H. Lu, X. M. Liu, L. R. Wang, Y. K. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express |

16. | K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics |

17. | J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett. |

18. | J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, H. Ming, and Q. W. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett. |

19. | A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B |

20. | X. S. Lin, J. H. Yan, Y. B. Zheng, L. J. Wu, and S. Lan, “Bistable switching in the lossy side-coupled plasmonic waveguide-cavity structrues,” Opt. Express |

21. | X. L. Wang, H. Q. Jiang, J. X. Chen, P. Wang, Y. H. Lu, and H. Ming, “Optical bistability effect in plasmonic racetrack resonator with high extinction ratio,” Opt. Express |

22. | H. Lu, X. M. Liu, Y. K. Gong, D. Mao, and L. R. Wang, “Enhancement of transmission efficiency of nanoplasmonic wavelength demultiplexer based on channel drop filters and reflection nanocavities,” Opt. Express |

23. | Y. Liu, F. Zhou, B. Yao, J. Cao, and Q. H. Mao, “High-extinction-ratio and low-insertion-loss plasmonic filter with coherent coupled nano-cavity array in a MIM waveguide,” Plasmonics |

24. | L. Liu, X. Hao, Y. T. Ye, J. X. Liu, Z. L. Chen, Y. C. Song, Y. Luo, J. Zhang, and L. Tan, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm. |

25. | J. D. Jackson, |

26. | R. S. Irving, |

27. | X. Y. Hu, P. Jiang, C. Y. Ding, H. Yang, and Q. H. Gong, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm. |

28. | Q. H. Mao and J. W. Y. Lit, “Optical bistability in an L-band dual-wavelength erbium-doped fiber laser with overlapping cavities,” IEEE Photon. Technol. Lett. |

29. | P. Berini and I. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nat. Photonics |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(190.3270) Nonlinear optics : Kerr effect

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 2, 2013

Revised Manuscript: September 13, 2013

Manuscript Accepted: September 13, 2013

Published: September 27, 2013

**Citation**

Ye Liu, Fei Zhou, and Qinghe Mao, "Analytical theory for the nonlinear optical response of a Kerr-type standing-wave cavity side-coupling to a MIM waveguide," Opt. Express **21**, 23687-23694 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23687

Sort: Year | Journal | Reset

### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003). [CrossRef] [PubMed]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408, 131–314 (2005). [CrossRef]
- J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength scale localization,” Phys. Rev. B73, 035407 (2006). [CrossRef]
- D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonic beyond the diffraction limit,” Nat. Photonics4, 83–90 (2010). [CrossRef]
- D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt.12, 015002 (2010). [CrossRef]
- T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B75, 245405 (2007). [CrossRef]
- G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett.101, 013111 (2012). [CrossRef]
- Q. Zhang, X. G. Huang, X. S. Lin, J. Tao, and X. P. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express17, 7549–7555 (2009). [CrossRef]
- Y. Hwang, J. Kim, and H. Y. Park, “Frequency selective metal-insulator-metal splitters for surface plasmons,” Opt. Comm.284, 4778–4781 (2011). [CrossRef]
- A. Noual, A. Akjouj, Y. Pennec, J. N. Gillet, and B. D. Rouhani, “Modeling of two-dimensional nanoscale Y-bent plasmonic waveguides with cavities for demultiplexing of the telecommunication wavelengths,” New J. Phys.11, 103020 (2009). [CrossRef]
- G. X. Wang, H. Lu, X. M. Liu, D. Mao, and L. N. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express19, 3513–3518 (2011). [CrossRef] [PubMed]
- G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev.2, 125–135 (2008). [CrossRef]
- J. Tao, Q. J. Wang, and X. G. Huang, “All-optical plasmonic switches based on coupled nano-disk cavity structures containing nonlinear material,” Plasmonics6, 753–759 (2011). [CrossRef]
- N. Nozhat and N. Granpayeh, “Switching power reduction in the ultra-compact Kerr nonlinear plasmonic directional coupler,” Opt. Comm.285, 1555–1559 (2012). [CrossRef]
- H. Lu, X. M. Liu, L. R. Wang, Y. K. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express19, 2910–2915 (2011). [CrossRef] [PubMed]
- K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics3, 55–58 (2009). [CrossRef]
- J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly efficient all-optical control of surface-plasmon-polariton generation based on a compact asymmetric single slit,” Nano Lett.11, 2933–2937 (2011). [CrossRef] [PubMed]
- J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, H. Ming, and Q. W. Zhan, “Optical bistability enhanced by highly localized bulk plasmon polariton modes in subwavelength metal-nonlinear dielectric multilayer structure,” Appl. Phys. Lett.94, 081117 (2009). [CrossRef]
- A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B28, 2820–2826 (2011). [CrossRef]
- X. S. Lin, J. H. Yan, Y. B. Zheng, L. J. Wu, and S. Lan, “Bistable switching in the lossy side-coupled plasmonic waveguide-cavity structrues,” Opt. Express19, 9594–9599 (2009). [CrossRef]
- X. L. Wang, H. Q. Jiang, J. X. Chen, P. Wang, Y. H. Lu, and H. Ming, “Optical bistability effect in plasmonic racetrack resonator with high extinction ratio,” Opt. Express19, 19415–19421 (2011). [CrossRef] [PubMed]
- H. Lu, X. M. Liu, Y. K. Gong, D. Mao, and L. R. Wang, “Enhancement of transmission efficiency of nanoplasmonic wavelength demultiplexer based on channel drop filters and reflection nanocavities,” Opt. Express19, 12885–12890 (2011). [CrossRef] [PubMed]
- Y. Liu, F. Zhou, B. Yao, J. Cao, and Q. H. Mao, “High-extinction-ratio and low-insertion-loss plasmonic filter with coherent coupled nano-cavity array in a MIM waveguide,” Plasmonics8, 1035–1041 (2013). [CrossRef]
- L. Liu, X. Hao, Y. T. Ye, J. X. Liu, Z. L. Chen, Y. C. Song, Y. Luo, J. Zhang, and L. Tan, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm.285, 2558–2562 (2012). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., 1999).
- R. S. Irving, Integers, Polynomials, and Rings (Springer, 2004).
- X. Y. Hu, P. Jiang, C. Y. Ding, H. Yang, and Q. H. Gong, “Systematical research on the characteristics of a vertical coupled Fabry-Perot plasmonic filter,” Opt. Comm.285, 2558–2562 (2012). [CrossRef]
- Q. H. Mao and J. W. Y. Lit, “Optical bistability in an L-band dual-wavelength erbium-doped fiber laser with overlapping cavities,” IEEE Photon. Technol. Lett.14, 1252–1254 (2002). [CrossRef]
- P. Berini and I. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nat. Photonics6, 16–23 (2012). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.