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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24813–24818
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Subwavelength electromagnetic switch: Bistable wave transmission of side-coupling nonlinear meta-atom

Yaqiong Ding, Chunhua Xue, Yong Sun, Haitao Jiang, Yunhui Li, Hongqiang Li, and Hong Chen  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24813-24818 (2012)
http://dx.doi.org/10.1364/OE.20.024813


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Abstract

We propose a scheme for subwavelength electromagnetic switch by employing nonlinear meta-atom. Bistable response is conceptually demonstrated on a microwave transmission line, which is side-coupled to a varactor-loaded split ring resonator acting as a nonlinear meta-atom. Calculations and experiments show that by applying conductive coupling instead of near-field interaction between the transmission line and the nonlinear meta-atom, switch performances are improved. The switch threshold of low to −5.8 dBm and the transmission contrast of up to 4.0 dB between the two bistable states were achieved. Subwavelength size of our switch should be useful for miniaturization of integrated optical nanocircuits.

© 2012 OSA

1. Introduction

In systems that display optical bistability, the output intensity is strongly depended on the input intensity, and might even display a hysteresis loop. As such, the property of optical bistability could be explored to develop a variety of nonlinear functional devices, such as all-optical switches, transistors, diodes, and logical gates. Bistable wave transmission, or optical switch effect, has been extensively investigated in nonlinear Fabry-Perot etalons [1

1. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36(19), 1135–1138 (1976). [CrossRef]

,2

2. T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry-Perot interferrometer,” Phys. Rev. A 19(3), 1169–1176 (1979). [CrossRef]

], and photonic crystals [3

3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35(5), 379–381 (1979). [CrossRef]

7

7. M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739 (2003). [CrossRef]

], for example, by utilizing the defect mode [4

4. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19(9), 2241–2249 (2002). [CrossRef]

], band edge [5

5. A. M. Yacomotti, F. Raineri, G. Vecchi, P. Monnier, R. Raj, A. Levenson, B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, L. Di Cioccio, and J. M. Fedeli, “All-optical bistable band-edge Bloch modes in a two-dimensional photonic crystal,” Appl. Phys. Lett. 88(23), 231107 (2006). [CrossRef]

], or resonant of optical microcavity [6

6. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601 (2002). [CrossRef] [PubMed]

,7

7. M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739 (2003). [CrossRef]

].

With the rapid development of fabrication technology, a new era of subwavelength nanophotonics is approaching. The realization of bistable transmission functionalities within a subwavelength volume is indispensable for all-optical components in integrated optical nanocircuits or metactronics [8

8. T. Ebbesen, C. Genet, and S. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). [CrossRef]

,9

9. N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]

]. Metamaterials refer to artificial materials with properties unattainable in nature [10

10. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

12

12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

]. These artificial materials usually consist of subwavelength-sized metallic resonant building blocks as meta-atoms [13

13. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

,14

14. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]

], which yield electric or/and magnetic responses. The local resonances of the meta-atoms can produce strong electromagnetic (EM) nonlinearity with assistance of strong localization and confinement of the EM field [15

15. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003). [CrossRef] [PubMed]

19

19. A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011). [CrossRef] [PubMed]

]. Interested Readers can also refer to a recent review article which presents deep analytical investigations of nonlinear meta-atoms [20

20. E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010). [CrossRef]

].

In this paper, we demonstrate a nonlinear meta-atom that can operate as a subwavelength EM switch. We employ a microwave transmission line side-coupled to a split ring resonator (SRR) and nonlinear medium inclusion to conceptually demonstrate the bistable response of nonlinear meta-atom for EM switch. Calculations show that by applying the conductive coupling instead of the near-field interaction between the transmission line and the meta-atom, the switch performances could be improved. Experiments show that the EM switch can achieve the threshold of low to −5.8 dBm and the transmission contrast of up to 4.0 dB between the two bistable states. The EM switch sample is only about one thirtieth of wavelength in the transmission line. The paper is organized as follows. In section 2 we present the theoretical model, optimization design, and experiment sample. Then in section 3 we experimentally investigate the bistable wave transmission of the suggested subwavelength EM switch. Finally, a brief summary is given in section 4.

2. Theoretical model, optimization design, and the sample

Figure 1(a)
Fig. 1 (a) Schematic of our all-optical subwavelength switch model. The nonlinear meta-atom, which has a resonance frequency of ωres, is side-coupled to a two-port waveguide. (b) Photograph of one sample. The varactor is mounted onto the slit of the SRR, and a slim metal strip is used to enhance the coupling between the SRR and microstrip, as shown in the inset.
schematically illustrates our theoretical model of all-optical subwavelength switch, which is a two-port network with a side-coupling nonlinear meta-atom. The meta-atom own a resonant (transition) frequency ωres, and the coupling strength between meta-atom and the waveguide is Ω. When the input power is very weak, nonlinearity doesn’t display. The linear transmission for this system could be written as T(ω)=(ωωres)2/[Ω2+(ωωres)2], which means a zero-transmission dip at ωres. In the meanwhile, the EM Field in the meta-atom is strongly enhanced at ωres. With the increase of input power, the system displays optical bistability. One can imagine that the resonant frequency of nonlinear meta-atom would change at first, and then bistable switching effect in transmission would be observed in the vicinity of the resonant frequency. One of the bistable states posses a near-zero transmission. Hence, the transmission contrast of the switch between OFF and ON states could be extremely large [12

12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

].

As a conceptual demonstration, we conduct a microwave experiment using one kind of magnetic meta-atoms, namely SRR [15

15. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003). [CrossRef] [PubMed]

19

19. A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011). [CrossRef] [PubMed]

]. Photograph of one sample is shown in Fig. 1(b). It is fabricated on the Rogers RT5880 (εr=2.2,tanδ=0.0009) substrate with the thickness of 0.787 mm. The split ring with a 0.5 mm slit etched at one side is made of 0.8 mm width copper strip, and its total dimension is 10 mm × 10 mm. A silicon hyperabrupt varactor (Infineon BBY52) acting as nonlinear medium is loaded in the slit of SRR. The nonlinear properties of varactor-loaded SRR in free space have been deeply investigated both in theory and experiment [18

18. B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express 16(20), 16058–16063 (2008). [CrossRef] [PubMed]

, 20

20. E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010). [CrossRef]

]. Here, SRR and the transmission line are conductively coupled through a slim copper strip (0.8 mm length, 0.2 mm width). Local details are shown in the inset in Fig. 1(b). We perform all numerical calculations with a finite-integration-technique based EM solver (CST Microwave Studio). In our experiments, transmission spectra are measured by an Agilent N5244A vector network analyzer (VNA), and microwave power is marked with an Agilent V3500a power meter.

We also calculation the surface current distributions to investigate the field enhancement of two samples with different coupling mechanism, as plotted in Figs. 2(b) and 2(c). It is found the amplitude of current in SRR with the slim metal strip is one order of magnitude stronger than that without it. It indicates that the conductive coupling not only helps to reduce the transmission at the resonance frequency, but also promote the localized EM field enhancement. Hence, conductive coupling must benefit the performance of the switch.

One may note there are two dips in transmission spectrum (0.77 GHz and 3.02 GHz) of the sample with the slim metal strip, corresponding to different resonant modes of SRR. The quality factors of two modes are distinct. To understand this difference, the surface current distribution of the second mode is also calculated, as shown in Fig. 2(d). Figure 2(b) depicts the strong currents oscillating in circular form along the ring, which implies that the first mode at 0.77 GHz is a magnetic resonance [21

21. P. Gay-Balmaz and O. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92(5), 2929–2936 (2002). [CrossRef]

,22

22. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93(10), 107402 (2004). [CrossRef] [PubMed]

]. The voltage across the slit is strongly enhanced, which contributes to reduce the switch energy. For the second mode at 3.02 GHz, the surface current oscillates in phase along the two sides of the SRR, which is known as an electrical resonance [21

21. P. Gay-Balmaz and O. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92(5), 2929–2936 (2002). [CrossRef]

,22

22. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93(10), 107402 (2004). [CrossRef] [PubMed]

]. Its amplitude is much smaller than that of the magnetic resonance, and the voltage across the slit is amplified not evidently. Therefore, the first mode is more suitable for exploiting the nonlinear property of the meta-atom.

3. Experimental investigation of the subwavelength EM switch

By loading the varactor acting as nonlinear medium in the sample, we experimentally investigated the bistable wave transmission and the switching effect in the following. Figure 3
Fig. 3 Measured transmission around the magnetic resonance with respect to the input power. The inset is the transmission around the electrical resonance.
presents the transmission spectra measured for different input power (from −25 dBm to 5 dBm). As expected, the transmission around the magnetic mode is strongly depended on the input intensity of power, while the transmission around the electrical mode is unchanged (see the inset in Fig. 3). The magnetic resonant frequency is shifted to red with increasing power. Quantitatively, the self-tuned frequency of magnetic resonance could change by 3% when the input power increases by 5 dB. In the meanwhile, the transmission dip becomes more and more asymmetric, which declare the nonlinear response in another aspect. Moreover, we find the transmission minimum gradually increases, and the resonant is broadened obviously. It is due to the increasing losses in the nonlinear meta-atom, when the input power increases.

In order to find the minimum bistable threshold, transmission spectra with respect to forward (black inverted open triangles) and backward (red open triangles) frequency sweep at different input level are plotted in Fig. 4
Fig. 4 Hysteresis effects in frequency sweeping at different input power levels. The lowest bistable threshold is only −5.8 dBm.
. Hysteresis loops are observed at the range around the resonance frequency, owing to the dynamic feedback in a nonlinear resonant system. The hysteresis loop disappears until the power value decrease to −5.8 dBm,which is much lower than the threshold value reported in Ref. 23

23. Y. Fan, Z. Wei, J. Han, X. Liu, and H. Li, “Nonlinear properties of meta-dimer comprised of coupled ring resonators,” J. Phys. D Appl. Phys. 44(42), 425303 (2011). [CrossRef]

(the threshold is 9 dBm for the structure comprising of the two coupled ring resonators loading the same varactor). It should be emphasized that −5.8 dBm can be achieved for normal microwave equipment without power amplifier, such as VNA and signal generator. Moreover, the transmission contrast between two bistable states exceeds 4 dB at the input power of −5.8 dBm.

To further demonstrate the switching response of the sample, we use a monochromatic input signal at f0 = 0.74 GHz with the power strength in the range of −6.0 dBm to −4.5 dBm. Figure 5
Fig. 5 Measured transmission spectra at 0.74 GHz as a function of input power from −6 dBm to −4.5 dBm. The black inverse-triangular and red triangular are respected to forward (increasing) and reverse (decreasing) power sweep, respectively.
shows the bistable hysteresis loop with respect to bidirectional sweep of input power. When the increasing (decreasing) input power meets its threshold, the transmission is abruptly down (up). Compared to conventional bistable switch in which the transmission jumps up with increasing power, this side-coupled system shows jump-down characteristic. This anti-form bistable switch may find application to avoid unexpected high power when Power Attenuator is not applicable. For the operating frequency of 0.74 GHz, the contrast in transmission between the two bistable states has a ratio of about 4 dB. This value is sufficient high to meet the general application.

5. Conclusion

In summary, a subwavelength EM switch based on nonlinear meta-atom is proposed and implemented on a microwave transmission line. Our experiments conceptually verify that the sample system can operate as a subwavelength EM switch. Varactor-loaded SRR is considered as the nonlinear meta-atom, and the size of it is about one thirtieth of wavelength in the transmission line, which means nonlinear light manipulation within a sub-wavelength scale. Strong localization and confinement of EM field in a subwavelength is original from the magnetic mode of SRR. In addition, the conductive coupling improves the switching performance. Ultra-low switching energy is realized (−5.8 dBm), and the transmission contrast exceeds 4 dB. Better performance can be expected if the losses in meta-atom could be reduced. It should be noted this model of subwavelength switch could be realized in optical nanocircuits by taking advantage of the resonances of quantum-dot materials [24

24. C. Arnold, V. Loo, A. Lemaître, I. Sagnes, O. Krebs, P. Voisin, P. Senellart, and L. Lanco, “Optical bistability in a quantum dots/micropillar device with a quality factor exceeding 200 000,” Appl. Phys. Lett. 100(11), 111111 (2012). [CrossRef]

], or localized surface plasmons [25

25. P. Chakraborty, “Metal nanoclusters in glasses as non-linear photonic materials,” J. Mater. Sci. 33(9), 2235–2249 (1998). [CrossRef]

], which is side-coupled to a subwavelength surface-plasmon waveguide [9

9. N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]

]. Our findings are beneficial for miniaturization of nonlinear optical components for metamaterial-inspired optical nanocircuits.

Acknowledgments

The authors thank Y. C. Fan and L. He for their help in experiments. The authors also acknowledge helpful assistance from Open Lab of Agilent Shanghai Center. This work was funded by National Basic Research Program of China (grant 2011CB922001), from the National Natural Science Foundation of China (Grants 51007064, 11074187, and 11204217), the Program of the Shanghai Science and Technology Committee (grant 11QA1406900) and the Shanghai Postdoctoral Scientific Program.

References and links

1.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. 36(19), 1135–1138 (1976). [CrossRef]

2.

T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry-Perot interferrometer,” Phys. Rev. A 19(3), 1169–1176 (1979). [CrossRef]

3.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35(5), 379–381 (1979). [CrossRef]

4.

S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19(9), 2241–2249 (2002). [CrossRef]

5.

A. M. Yacomotti, F. Raineri, G. Vecchi, P. Monnier, R. Raj, A. Levenson, B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, L. Di Cioccio, and J. M. Fedeli, “All-optical bistable band-edge Bloch modes in a two-dimensional photonic crystal,” Appl. Phys. Lett. 88(23), 231107 (2006). [CrossRef]

6.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601 (2002). [CrossRef] [PubMed]

7.

M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83(14), 2739 (2003). [CrossRef]

8.

T. Ebbesen, C. Genet, and S. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). [CrossRef]

9.

N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science 317(5845), 1698–1702 (2007). [CrossRef] [PubMed]

10.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

11.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

12.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

13.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

14.

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88(4), 041109 (2006). [CrossRef]

15.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003). [CrossRef] [PubMed]

16.

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express 14(20), 9344–9349 (2006). [CrossRef] [PubMed]

17.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007). [CrossRef] [PubMed]

18.

B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express 16(20), 16058–16063 (2008). [CrossRef] [PubMed]

19.

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011). [CrossRef] [PubMed]

20.

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys. 12(9), 093010 (2010). [CrossRef]

21.

P. Gay-Balmaz and O. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92(5), 2929–2936 (2002). [CrossRef]

22.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93(10), 107402 (2004). [CrossRef] [PubMed]

23.

Y. Fan, Z. Wei, J. Han, X. Liu, and H. Li, “Nonlinear properties of meta-dimer comprised of coupled ring resonators,” J. Phys. D Appl. Phys. 44(42), 425303 (2011). [CrossRef]

24.

C. Arnold, V. Loo, A. Lemaître, I. Sagnes, O. Krebs, P. Voisin, P. Senellart, and L. Lanco, “Optical bistability in a quantum dots/micropillar device with a quality factor exceeding 200 000,” Appl. Phys. Lett. 100(11), 111111 (2012). [CrossRef]

25.

P. Chakraborty, “Metal nanoclusters in glasses as non-linear photonic materials,” J. Mater. Sci. 33(9), 2235–2249 (1998). [CrossRef]

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(230.4320) Optical devices : Nonlinear optical devices
(160.3918) Materials : Metamaterials

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 30, 2012
Revised Manuscript: October 8, 2012
Manuscript Accepted: October 9, 2012
Published: October 15, 2012

Citation
Yaqiong Ding, Chunhua Xue, Yong Sun, Haitao Jiang, Yunhui Li, Hongqiang Li, and Hong Chen, "Subwavelength electromagnetic switch: Bistable wave transmission of side-coupling nonlinear meta-atom," Opt. Express 20, 24813-24818 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24813


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References

  1. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett.36(19), 1135–1138 (1976). [CrossRef]
  2. T. Bischofberger and Y. R. Shen, “Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry-Perot interferrometer,” Phys. Rev. A19(3), 1169–1176 (1979). [CrossRef]
  3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett.35(5), 379–381 (1979). [CrossRef]
  4. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B19(9), 2241–2249 (2002). [CrossRef]
  5. A. M. Yacomotti, F. Raineri, G. Vecchi, P. Monnier, R. Raj, A. Levenson, B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, L. Di Cioccio, and J. M. Fedeli, “All-optical bistable band-edge Bloch modes in a two-dimensional photonic crystal,” Appl. Phys. Lett.88(23), 231107 (2006). [CrossRef]
  6. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.66(5), 055601 (2002). [CrossRef] [PubMed]
  7. M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett.83(14), 2739 (2003). [CrossRef]
  8. T. Ebbesen, C. Genet, and S. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today61(5), 44–50 (2008). [CrossRef]
  9. N. Engheta, “Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials,” Science317(5845), 1698–1702 (2007). [CrossRef] [PubMed]
  10. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  11. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001). [CrossRef] [PubMed]
  12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  13. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999). [CrossRef]
  14. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett.88(4), 041109 (2006). [CrossRef]
  15. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett.91(3), 037401 (2003). [CrossRef] [PubMed]
  16. I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Express14(20), 9344–9349 (2006). [CrossRef] [PubMed]
  17. M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express15(8), 5238–5247 (2007). [CrossRef] [PubMed]
  18. B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Express16(20), 16058–16063 (2008). [CrossRef] [PubMed]
  19. A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett.107(6), 063902 (2011). [CrossRef] [PubMed]
  20. E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys.12(9), 093010 (2010). [CrossRef]
  21. P. Gay-Balmaz and O. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys.92(5), 2929–2936 (2002). [CrossRef]
  22. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett.93(10), 107402 (2004). [CrossRef] [PubMed]
  23. Y. Fan, Z. Wei, J. Han, X. Liu, and H. Li, “Nonlinear properties of meta-dimer comprised of coupled ring resonators,” J. Phys. D Appl. Phys.44(42), 425303 (2011). [CrossRef]
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