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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23687–23694
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Analytical theory for the nonlinear optical response of a Kerr-type standing-wave cavity side-coupling to a MIM waveguide

Ye Liu, Fei Zhou, and Qinghe Mao  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23687-23694 (2013)
http://dx.doi.org/10.1364/OE.21.023687


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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 Q0 and Q1 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

2. Analytical theory

Fig. 1 Typical plasmonic structure with a Kerr nonlinear cavity side-coupling to a MIM waveguide.

If S+1 has a time dependence of ejωt, the steady-state transmission of the plasmonic structure can be expressed as
T=|S1S+1|2=(ωωc1)2+(1Q0)2(ωωc1)2+(1Q0+12Q1)2.
(2)
And, the total energy in the cavity |A|2 and the incident power |S+1|2 (or Pin) satisfies:
|A|2=12ωcQ1(ωωc1)2+(1Q0+12Q1)2|S+1|2.
(3)

From Eq. (3) it can be clearly seen that the energy in the cavity increases with the incident power, which results in the changes of nonlinear refractive index. Strictly, the resonant frequency ωc and the quality factors Q0 and Q1 may change with the nonlinear refractive index variation of the cavity. However, if only weak Kerr nonlinearity, i.e., the nonlinear refractive index Δn < 5%, is considered, the tiny changes of Q0 and Q1 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.

On the other hand, an arbitrary standing-wave cavity side-coupling to a MIM waveguide may be regarded as a Fabry-Perot cavity [24

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]

, 25

25. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., 1999).

], whose resonant wavelength can be described as 2(neffLeff + 2Lpen) = c(N = 1, 2, 3, ···), with Lpen, neff and Leff 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:
λc=λc0+2n2|Eb|2Leff=λc0+2n2Leff|A|2εS
(4)
where λc0 is the linear resonant wavelength, n2 is the Kerr nonlinear-index coefficient, and 〈|Eb|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.

Substituting Eq. (3) into Eq. (4), using λc = 2πc/ωc and |S+1|2 = Pin, we can get a cubic equation with λc after eliminating the term of |A|2:
λc3+Bλc2+(C1C2Pin)λc+D=0,
(5)
where B = −(2λ + λc0), C1 = λ2 [1 + (1/Q0 + 1/ (2Q1))2] + 2λλc0, C2 = n2Leffλ2/(2πcQ1εS), and D = −λc0λ2 [1 + (1/Q0 + 1/ (2Q1))2]. From Eq. (5), we can see that, for a given structure of the standing-wave cavity side-coupling to a MIM waveguide, λc exhibits strong nonlinear relationship with the wavelength λ and the power Pin of the incident light.

In order to further explore the nonlinear relationship of λc with λ and Pin, we may find the derivative of Pin with respect to λc from Eq. (5), and let dPin/c = 0, then we have
λc3+B2λc2D2=0.
(6)

Note that 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 λc1, λc2 and λc3 are respectively the three distinct real roots of Eq. (6), then, they must spontaneously satisfy: (1) λc1 + λc2 + λc3 = −B/2, which is constantly larger than zero, and (2) λc1λc2λc3 = D/2, which is always less than zero. This indicates that there must be two positive and a negative real numbers among λc1, λc2, and λc3 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 Pinλc curve, and thus, implies that there may exist some optical bistability phenomenon when λ is suitably chosen to ensure Δ < 0.

To determine the range of λ for the optical bistability, according to the critical condition Δ = 0, we have
x327ζ2+158x2+34x+18=0
(7)
where x = λ/λc0 and ζ = 1/Q0 + 1/(2Q1). We can then get the two positive real roots by solving Eq. (7)[26

26. R. S. Irving, Integers, Polynomials, and Rings (Springer, 2004).

]
x1,2=B3+e±i2π3q2+Δ3+ei2π3q2Δ3
where B′ = (27ζ2 + 15)/8, q′ = (27 – 459ζ2)/256, and Δ = −(27ζ/64)2. Since ζ is in the order of 10−2, the higher order terms of ζ may be ignorable. Then, we have
x1,2=1±3ζ
or
λλc0=±3λc0(1Q0+12Q1)=±3Δλ2
(8)
where Δλ = 2λc0 (1/Q0 + 1/(2Q1)) is the FWHM of the linear transmission spectrum, which can be easily obtained from Eq. (2).

Based on the discussion as the above, if the critical wavelength λM is defined as λM=λc0+3Δλ/2 (or λM=λc03Δλ/2) for the positive (or negative) nonlinearity, it can be found that, when λ > λM (or λ < λM), the discriminant of Eq. (6) is less than zero, and thus two different λc truly exist for a specific given Pin, leading to the occurrence of optical bistability phenomenon.

3. Results and discussions

Firstly, we consider a Kerr nonlinear cavity side-coupling to a MIM waveguide (see Fig. 1) with specifications of Q0 = 235, Q1 = 65, λc0 = 1556.9 nm, S = 3.6 × 104 nm2, Leff = 360 nm, and n2 = 1 × 10−8 cm2/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]

], respectively. These parameters correspond to a plasmonic structure of an Ag-SiO2-Ag waveguide with a side-coupling nonlinear polystyrene rectangular cavity shown in Fig. 1, whose structure parameters are wt = 50 nm, 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 λM = 1589.1 nm, the optical bistability phenomenon may occur. To test the validity of our prediction, the transmission as a function of Pin 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]

] for 1610 nm are also given in Fig. 2(b), which shows they are in good agreement with respect to each other, indicating the validation of our analytical model. Figure 2(a) clearly shows that the optical bistability with the hysteresis behavior occurs only when 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.

Fig. 2 (a) Transmissions as a function of incident power for different incident light wavelengths; (b) The comparisons of the bistable curves obtained by our analytical theory (solid line) and the FDTD numerical simulations (red circles).

The transmission variations shown in Fig. 2(a) may be explained with our analytical theory. According to Eq. (3), the energy coupled into the cavity is increased with the increasing of Pin. However, the increasing speed of the cavity energy with Pin depends on the incident light wavelength. When λ is less than λM, the difference between λ and λc0 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 λc. 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 λc respect with to λc0 has been made if the incident power is increased to 0.25 W from zero. This red-shift of λc can further strengthen the resonant coupling effect between the cavity and the waveguide, which in turn speeds up the red-shifted of λc 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 Pin and reaches to their minimum rapidly at which λc = λ. Such a property of the transmissions decreasing monotonously to their minimum by slightly changing Pin may be used for the design of all-optical switches, as mentioned before. If then Pin is further increased, λc 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 λc 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 λc, finally resulting that the transmissions increases slowly with the increase of Pin 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 λc.

Fig. 3 (a) Schematic diagram for the linear transmission properties when the incident wavelength is larger or less than λM respectively; (b) and (c) Transmissions (black line) and non-linear resonant wavelengths (red line) as a function of incident light power for the incident light wavelength of 1580 and 1610 nm, respectively. The linear resonant wavelength and the incident light wavelengths are displayed with blue dot lines in (b) and (c).

When λ is larger than λM, as shown in Fig. 3(a), λ is far away from the linear cavity resonance band near λc0 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 Pin from zero, and thus, the Kerr nonlinearity gives an ignorable red-shift for λc. For an example, when the incident wavelength is 1610 nm, as shown in Fig. 3(c), a 5 nm red-shift of λc cannot be obtained until the incident power is increased to 1.0 W. Such a red-shift of λc is not large enough to make λc being close to λ. Because of this reason, the transmissions for λ > λM keep staying in high levels until Pin 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 Pin is so-called the first turning point (FTP) power, where a sudden red-shift of λc 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]

]. Note that λc is larger than λ once the transmission has jumped down to the low branch, just as shown in Fig. 3(c) in which λc jumps from 1578 nm to 1620 nm at the FTP for λ = 1610 nm. After then, like the case of λ < λM, the negative feedback mechanism also helps λ stay in the cavity resonance band near λc, and the transmission increases slowly with the increase of Pin. However, if Pin is decreased from a high level, since λ is located in the cavity resonance band near λc in the beginning, the resonant energy in the cavity can keep in a high level with the negative feedback mechanism even if Pin has been decreased to a level less than the FTP. This negative feedback mechanism can be maintained until λc = λ, where the minimal transmission is reached. If Pin decreases furtherly, λc will continue to blue-shift and a positive feedback happen. When Pin decreases to a level called as the second turning point (STP), the positive feedback mechanism will result in a sudden blue-shift of λc [from 1606 nm to 1564 nm in Fig. 3(c)] with a tiny increase of Pin, which makes the transmission jumps up to the up-branch. After this jump, the transmission is increased slowly again with the decrease of Pin due to the tiny blue-shift of λc. Finally the optical bistability with the hysteresis behavior occurs for the case of λ > λM.

Obviously, the cavity energy storage capability, determined by Q0 and Q1, may affect the above hysteresis bistability behavior. Figures 4(a) and 4(b) show the critical wavelength λM as a function of Q0 and Q1, respectively. As seen, for a given Q1 of 65, λM decreases from 1605 nm to 1582 nm when Q0 increases from 100 to 400. When Q1 increases from 40 to 100, the same change for λM can be obtained for a given Q0 of 235. The decrease of λM is originated from the less energy attenuation in the cavity with larger Q0 or Q1. Figure 4(c) shows the transmission at 1610 nm as a function of the incident power for different Q0 when Q1 is fixed at 65. Here, we can see that, with the increase of Q0, 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 Q0 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 Q0, the FTP and the STP power may respectively be increased and decreased with the increase of Q1 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 Q1 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 Q0 and smaller Q1 are preferred. Considering that the width of the bistable region may be narrowed by decreasing Q1, an optimized way for reducing FTP power is to increase the value of Q0 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]

].

Fig. 4 The critical wavelength of optical bistability as a function of (a) Q0 and (b) Q1 for a fixed Q1 and Q0, respectively. Bistable properties at 1610 nm for (c) different Q0 and (d) different Q1 when Q1 and Q0 are fixed, respectively.

4. Conclusions

By combining the TCMT and the Kerr nonlinearity, an analytical theory for describing the dynamic response characteristics of an SPP waveguide-cavity structure formed by a nonlinear standing-wave cavity side-coupling to a MIM waveguide has been presented. The criterion that λ>λc0+3Δλ/2(n2>0) or λ<λc03Δλ/2(n2<0) given in the analytical theory can be used to predict whether the optical bistability with the hysteresis behavior takes place. The optical bistability evolutions and its dynamic physical mechanism can also be phenomenologically analyzed with the analytical theory. Moreover, by using our analytical theory, the influence of Q0 and Q1 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

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11104282, 11204317 and 61250017, and by the National Basic Research Program of China under Grant No. 2011CB934304.

References and links

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]

6.

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007). [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]

9.

Y. Hwang, J. Kim, and H. Y. Park, “Frequency selective metal-insulator-metal splitters for surface plasmons,” Opt. Comm. 284, 4778–4781 (2011). [CrossRef]

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, 103020 (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]

14.

N. Nozhat and N. Granpayeh, “Switching power reduction in the ultra-compact Kerr nonlinear plasmonic directional coupler,” Opt. Comm. 285, 1555–1559 (2012). [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]

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 19, 12885–12890 (2011). [CrossRef] [PubMed]

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 8, 1035–1041 (2013). [CrossRef]

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]

25.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., 1999).

26.

R. S. Irving, Integers, Polynomials, and Rings (Springer, 2004).

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]

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]

29.

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

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


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 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. B73, 035407 (2006). [CrossRef]
  4. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonic beyond the diffraction limit,” Nat. Photonics4, 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]
  6. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B75, 245405 (2007). [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. Express17, 7549–7555 (2009). [CrossRef]
  9. Y. Hwang, J. Kim, and H. Y. Park, “Frequency selective metal-insulator-metal splitters for surface plasmons,” Opt. Comm.284, 4778–4781 (2011). [CrossRef]
  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, 103020 (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. Express19, 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,” Plasmonics6, 753–759 (2011). [CrossRef]
  14. N. Nozhat and N. Granpayeh, “Switching power reduction in the ultra-compact Kerr nonlinear plasmonic directional coupler,” Opt. Comm.285, 1555–1559 (2012). [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. Express19, 2910–2915 (2011). [CrossRef] [PubMed]
  16. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics3, 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. B28, 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. Express19, 9594–9599 (2009). [CrossRef]
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