## Probing terahertz metamaterials with subwavelength optical fibers |

Optics Express, Vol. 21, Issue 14, pp. 17195-17211 (2013)

http://dx.doi.org/10.1364/OE.21.017195

Acrobat PDF (4966 KB)

### Abstract

Transmission through a subwavelength terahertz fiber, which is positioned in parallel to a frequency selective surface, is studied using several finite element tools. Both the band diagram technique and the port-based scattering matrix technique are used to explain the nature of various resonances in the fiber transmission spectrum. First, we observe that spectral positions of most of the transmission peaks in the port-based simulation can be related to the positions of Van Hove singularities in the band diagram of a corresponding infinite periodic system. Moreover, spectral shape of most of the features in the fiber transmission spectrum can be explained by superposition of several Fano-type resonances. We also show that center frequencies and bandwidths of these resonances and, as a consequence, spectral shape of the resulting transmission features can be tuned by varying the fiber-metamaterial separation.

© 2013 OSA

## 1. Introduction

1. Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B **84**(20), 205428 (2011). [CrossRef]

2. D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, “Two-dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antenn. Propag. **51**(10), 2713–2722 (2003). [CrossRef]

3. A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B **80**(24), 245115 (2009). [CrossRef]

4. I. A. I. Al-Naib, C. Jansen, N. Born, and M. Koch, “Polarization and angle independent terahertz metamaterials with high Q-factors,” Appl. Phys. Lett. **98**(9), 091107 (2011). [CrossRef]

5. C. Jansen, I. A. I. Al-Naib, N. Born, and M. Koch, “Terahertz metasurfaces with high Q-factors,” Appl. Phys. Lett. **98**(5), 051109 (2011). [CrossRef]

2. D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, “Two-dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antenn. Propag. **51**(10), 2713–2722 (2003). [CrossRef]

7. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. **56**(4), 554–557 (2009). [CrossRef]

8. K. Aydin, I. M. Pryce, and H. A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express **18**(13), 13407–13417 (2010). [CrossRef] [PubMed]

18. B. Ung, A. Mazhorova, A. Dupuis, M. Rozé, and M. Skorobogatiy, “Polymer microstructured optical fibers for terahertz wave guiding,” Opt. Express **19**(26), B848–B861 (2011). [CrossRef] [PubMed]

19. M. Consales, A. Ricciardi, A. Crescitelli, E. Esposito, A. Cutolo, and A. Cusano, “Lab-on-fiber technology: toward multifunctional optical nanoprobes,” ACS Nano **6**(4), 3163–3170 (2012). [CrossRef] [PubMed]

20. T. Srivastava, R. Das, and R. Jha, “Highly accurate and sensitive surface plasmon resonance sensor based on channel photonic crystal waveguides,” Sens. Actuators B Chem. **157**(1), 246–252 (2011). [CrossRef]

## 2. Geometry of a fiber-metamaterial coupler

*R = 200 µm*and it is made of a polymer with refractive index

*n = 1.55*. The fiber is suspended over metamaterial at a distance

*H*. The metamaterial is build on a

*700 µm*– thick fused silica substrate with refractive index

*n = 1.966*. The substrate is patterned with split ring resonators made of perfect electrical conductor with height

*h = 50 μm*. The period of metamaterial cell is

*Λ = 400 µm*. Finally, for each simulation presented in this work we specify the number of SRRs in the transverse direction

*N*, as well as the number of SRRs in the longitudinal direction

_{t}*N*.

_{l}**axis. Along this axis we either use the Floquet boundary condition for the band diagram calculation, or the port boundary condition for the transmission calculations. In the direction normal to the plane of FSS (**

*y***axis) a**

*z**6000 µm*air gap is added above the substrate as well as

*2000 µm*air gap is added below the substrate. In

**direction, the unit cell is terminated with the PML boundary conditions. The periodic boundary condition is applied in the transverse**

*z***direction, in order to simulate an infinitely wide metamaterial.**

*x**E*, see Fig. 1(d)). As a consequence, the minor

_{z}*E*component of the modal electric field is antisymmetric in the

_{x}**direction, which, in principal, can result in conflicting boundary conditions at the supercell boundary where we impose periodic boundary conditions. To prevent this from happening, the HE**

*x*_{11}modal fields (that constitute port boundary conditions) are somewhat modified to go to zero at the periodic boundaries.

*f>250 GHz*) when at least 3 SRRs are used in the transverse direction. At lower frequencies, transverse modal fields of the fundamental mode of a subwavelength fiber show very slow logarithmic decay [21] outside of the fiber core. Therefore, at such frequencies computational artifacts could be introduced due to coupling between the subwavelength fiber and its images.

*h*of the split ring resonators is much smaller than the one used in our simulations, and it is typically on the order of several microns. However, using such a small height is impractical even if finite element solver is used as such thin layers result in the intractable number of elements. For example, when performing transmission calculation using a unit cell consisting of only a single row of 3 SRRs, 50·10

^{3}elements are generated (6.4 GB) when

*h = 50 µm*, whereas 135·10

^{3}elements (17 GB) are generated when

*h = 5 µm*. A major problem arises when modeling longer systems. Thus, for

*h = 50 µm*and a 10 period-long supercell (total of 30 SRRs), 70 GB of memory and 40 minutes are required per single frequency calculation. Therefore, with 128 GB of memory and

*h = 50 µm*we are limited by 11 periods (33 SRRs).

## 3. Band diagram of the fiber-metamaterial coupler

*N*, and a single SRR in the longitudinal direction. Fiber-metamaterial separation is

_{t}= 3*h = 50 μm*. Floquet boundary conditions are used at the supercell boundaries terminating the fiber. The band diagram of an infinitely periodic fiber-metamaterial system is presented in Fig. 2(b). To show the effect of split ring resonators, we have repeated simulation by keeping exactly the same geometry, however, without the SRRs on the slab surface (see Fig. 2(a)). For the ease of comparison, resultant band diagram is presented within the same first Brillouin zone as in the case of a fiber-metamaterial system. Color code reflects the average value of the norm of a

**component of a Poynting vector taken over the volume of a fiber to the root mean square of the electric field in the whole computational cell. Thus, the blue color corresponds to low fraction of the modal fields in the fiber, while the red color corresponds to strong presence of the modal fields in the fiber.**

*y*_{11}mode is doubly degenerate. When bringing a planar silica substrate into the fiber vicinity, this degeneracy is lifted and the fiber modes can then be characterized as predominantly

**or**

*x***polarized depending on the leading transverse component of their electric fields. The perturbation seen by the subwavelength fiber is very strong due to a significant presence of its fields in the metamaterial substrate region. The fundamental HE**

*z*_{11}mode of an unperturbed fiber is, thus, absent from Fig. 2 and is replaced by the two slab-fiber supermodes. Moreover, fiber modes and slab modes can show strong hybridization in the vicinity of phase matching points, which results in standard avoiding crossing behavior of the modal dispersion relations (see a part of Fig. 2(a) within a large circle). For small fiber-slab separations, coupling between the modes of a fiber and a slab can be very strong, thus resulting in significant changes in the curvature of the dispersion relations of the hybrid modes in the vicinity of a phase matching point. This, in turn, leads to the high values of the group velocity dispersion of the hybrid modes. It is important to note from Fig. 2(a) that accidental crossing of the fiber and slab modes does not necessarily lead to avoiding crossing phenomenon or modal hybridization. In fact, when symmetries of the fiber and slab modes are not compatible this results in zero coupling strength, thus, no hybridization between the two modes occurs.

*k*. This means that in the vicinity of the edges of local bandgaps, the full dispersion relation of the allowed optical states is of either a parabolic or hyperbolic form

_{x}= 0## 4. Scattering theory

13. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**(3), 2257–2298 (2010). [CrossRef]

22. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. **80**(6), 908–910 (2002). [CrossRef]

_{1}) is broad (to represent a continuum of states), and the other one (characterized by γ

_{2}) is narrow (to represent a discrete state). We also assume that the narrow resonance is placed within the bandwidth of a broad resonance so that

*q*is the Fano asymmetry parameter, f

_{ϕ}is the Fano resonant frequency, and

## 5. Convergence of a supercell approximation

*N*

_{t}= [*3*

3. A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B **80**(24), 245115 (2009). [CrossRef]

*,*

*9*

9. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]

*,*

*15*

15. S. H. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**(23), 235112 (2002). [CrossRef]

*]*, while using only one SRR along the fiber length

*N*(see Fig. 3(a)). Transmission through various fiber-FSS supercells are shown in Fig. 3(b), where curves of different color correspond to different number of SRRs in the transverse direction. When comparing the transmission curves for supercells containing 3 and 9 SRRs we note that while many of the broader resonant features are present in both curves, their spectral positions are somewhat different. Moreover, the number of narrow peaks for a wider supercell is considerably larger than the number of peaks for a narrower supercell. When further increasing the width of a supercell to

_{l}= 1*N*(see Fig. 3(c)), one observes that position and shape of many broader peaks do not change, thus indicating convergence of these spectral features, while at the same time, many more narrow peaks appear in the spectrum.

_{t}= 15*N*. From this data it is clear that the number of resonant peaks is proportional to the supercell size. Based on these observations, one would wonder in what sense do these transmission results converge when increasing the width of a supercell, or how do these results relate to experimental measurements.

_{t}= 3,9,15*N*and

_{t}= 9*N*in the presence of losses. First, we note that in the absence of loss (Fig. 4(a)), while there is an overall correspondence between the positions and shapes of the wider resonances, there is clearly no such correspondence for narrow resonances. When introducing material losses for the substrate material into simulations (dashed curves) we observe that sharp peaks rapidly disappear when increasing material losses, while the positions and the widths of the broader peaks remain the same independently of the supercell size. This disappearance of the sharp features is consistent with our predictions from the classic scattering theory (see Eq. (9), and discussions of section 4). In our simulations in Fig. 4 we have used 0.02i (Fig. 4(b)) and 0.2i (Fig. 4(a)) for imaginary parts of the substrate permittivity, which correspond approximately to 1 cm

_{t}= 15^{−1}and 10 cm

^{−1}bulk material losses of the fused silica and other typical glasses.

## 6. Effect of the fiber-metamaterial separation on transmission spectrum

24. M. A. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express **14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

*H*.

25. H. Haus, W. P. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. **5**(1), 16–23 (1987). [CrossRef]

*H*. This means that the resonance position

*f*

_{0}the resonant frequency in the limit of zero coupling,

*~240 GHz*on the values of fiber-metamaterial separation

*H*. Position of the resonant frequency

*243 GHz*. As a consequence, there is significant interaction between the two resonant modes and, as a consequence, simple model (13) becomes inadequate in this regime.

_{z}| field distribution in the

**-**

*y***plane for a separation distance of**

*z**H = 50 µm*at the frequencies of minimal transmission (

*f = 238.4 GHz*) and maximal transmission (

*f = 242.9 GHz*). The imaging plain goes through the fiber center, therefore, in Figs. 6 (c) and 6(d) we see the fiber core (on the top) suspended over the metamaterial substrate (in the middle of the Fig.) with a single SRR positioned between the two. It is clear from Fig. 6(c) that in the case of zero transmission, the power lunched into the fiber through the input port (left side of the Fig.) is transferred completely into the metamaterial substrate by the time it arrives to the output port (right side of the Fig.). The field in the metamaterial substrate has a very small overlap with the mode of the output port, therefore, one detects minimum in the fiber transmission. On the other hand, in the case of a perfect transmission (see Fig. 6(d)), the power launched into the fiber at the input port is first transferred into the metamaterial substrate and then back into the fiber by the time it arrives to the output port, therefore, one detects maximum in the fiber transmission.

## 7. Effect of the fiber-material coupling length on transmission spectrum

*N*SRRs in the transverse direction, while in the longitudinal direction we vary the number of SRR periods from

_{t}= 3*N*to

_{l}= 1*N*. When increasing the number of SRR periods, the response of a finite-size metamaterial should eventually converge to that of an infinitely long periodic system. Therefore, we expect that in the limit or large

_{l}= 11*N*, there should be a direct correspondence between the band diagram structure (see Fig. 2) and the transmission spectra.

_{l}*N*periods. We note that the spectrum in Fig. 7 is calculated with resolution of

_{l}= 10*0.3 GHz*, which is not sufficient to resolve all the high-Q features in the fiber transmission spectrum. For example, in the spectral region around 249 GHz in Fig. 7 we observe presence of the transmission peak, however in order to see the detailed structure of this resonant lineshape one has to use resolution which is at least ten time higher (0.03 GHz) as presented in Fig. 8(b). In the same Fig. we mark (dotted lines) the spectral positions of various Van Hove singularities as found from the band diagram of the corresponding infinite system (see Fig. 2). To remind the reader, Van Hove singularities are found at frequencies at which dispersion relations of the optical bands show local maxima or minima, and as a consequence, optical density of states at such singularities have particularly high values. From Fig. 7 we note that a great majority of the resonant peaks in the transmission spectrum correspond to Van Hove singularities in the optical density of states. However, as seen from Fig. 7, not all Van Hove singularities found in Fig. 2 manifest themselves as transmission peaks. This is because optical modes at such singularities are either incompatible by symmetry with the fundamental fiber mode (port mode), or because field overlap between the fiber and metamaterial modes is too small. Additionally, resonances at frequencies of Van Hove singularities will not be seen in the fiber transmission spectrum if the value of the coupling strength γ of a corresponding resonant state of a metamaterial is either lower than the spectral resolution used in numerical simulations, or if the coupling strength is smaller than the losses of a resonant state (Γ parameter in Eq. (5)). Note that although there are no material losses in our calculations, however, resonances may still possess radiation losses that could, in turn, result in disappearance of the resonant peaks from the fiber transmission spectrum.

*~249 GHz*. In Fig. 8 we plot fiber transmission spectra for an increasing number of SSR periods. First, we note that the width of a resonance decreases rapidly when increasing the number of SRR periods from

*N*to

_{l}= 1*N*(see Fig. 8(a)). Further increase in the number of SRR periods does not lead to a significant change in the peak width as seen in Fig. 8(b) where we present transmission spectra for

_{l}= 7*N*. This behavior is expected as the peak width corresponds to the coupling strength between the fundamental fiber mode (port mode) and a particular metamaterial mode. This strength is, in turn, proportional to the overlap integral between the fields of the two modes. When increasing the number of SRR periods, the modal fields of a finite-size fiber-metamaterial coupler converge to those of an infinitely periodic system. Therefore one expects that the value of coupling strength, and, hence, the transmission peak bandwidth should also converge to a certain finite value.

_{l}= 8-11*N*. For this number of SRR periods, a second peak appears accidently in the vicinity of our main peak. In this case, the line shape (14) can no longer be used to perform the fit, therefore this point is omitted from Fig. 8. Unfortunately, calculations with the larger number of periods becomes impossible because of the virtual memory limitation (128G) of our machine.

_{l}= 10## 8. Resonance engineering using band diagram calculations

## 9. Conclusion

26. M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett. **35**(10), 1593–1595 (2010). [CrossRef] [PubMed]

**x**and

**z**polarized HE

_{11}fiber modes. In combination with actively tunable metamaterials, this can be used to realize THz beam polarization modulation. Addition of a second fiber underneath the metamaterial could produce a channel add-drop filter, provided that the metamaterial is carefully chosen. In general, tunable metamaterials (which have been proposed by many for THz wave modulation, see [7

7. J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. **56**(4), 554–557 (2009). [CrossRef]

27. H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics **3**(3), 148–151 (2009). [CrossRef]

## Acknowledgments

## References and links

1. | Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B |

2. | D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, “Two-dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antenn. Propag. |

3. | A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B |

4. | I. A. I. Al-Naib, C. Jansen, N. Born, and M. Koch, “Polarization and angle independent terahertz metamaterials with high Q-factors,” Appl. Phys. Lett. |

5. | C. Jansen, I. A. I. Al-Naib, N. Born, and M. Koch, “Terahertz metasurfaces with high Q-factors,” Appl. Phys. Lett. |

6. | M. Skorobogatiy and J. Yang, |

7. | J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt. |

8. | K. Aydin, I. M. Pryce, and H. A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express |

9. | Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

10. | D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A |

11. | Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. |

12. | S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A |

13. | A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. |

14. | L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. |

15. | S. H. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

16. | A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vucković, “Dipole induced transparency in waveguide coupled photonic crystal cavities,” Opt. Express |

17. | R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett. |

18. | B. Ung, A. Mazhorova, A. Dupuis, M. Rozé, and M. Skorobogatiy, “Polymer microstructured optical fibers for terahertz wave guiding,” Opt. Express |

19. | M. Consales, A. Ricciardi, A. Crescitelli, E. Esposito, A. Cutolo, and A. Cusano, “Lab-on-fiber technology: toward multifunctional optical nanoprobes,” ACS Nano |

20. | T. Srivastava, R. Das, and R. Jha, “Highly accurate and sensitive surface plasmon resonance sensor based on channel photonic crystal waveguides,” Sens. Actuators B Chem. |

21. | M. Skorobogatiy, |

22. | S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. |

23. | H. A. Haus, |

24. | M. A. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express |

25. | H. Haus, W. P. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. |

26. | M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett. |

27. | H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(160.3918) Materials : Metamaterials

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 22, 2013

Revised Manuscript: June 25, 2013

Manuscript Accepted: July 1, 2013

Published: July 11, 2013

**Citation**

Martin Girard and Maksim Skorobogatiy, "Probing terahertz metamaterials with subwavelength optical fibers," Opt. Express **21**, 17195-17211 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17195

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### References

- Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces,” Phys. Rev. B84(20), 205428 (2011). [CrossRef]
- D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tangonan, “Two-dimensional beam steering using an electrically tunable impedance surface,” IEEE Trans. Antenn. Propag.51(10), 2713–2722 (2003). [CrossRef]
- A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B80(24), 245115 (2009). [CrossRef]
- I. A. I. Al-Naib, C. Jansen, N. Born, and M. Koch, “Polarization and angle independent terahertz metamaterials with high Q-factors,” Appl. Phys. Lett.98(9), 091107 (2011). [CrossRef]
- C. Jansen, I. A. I. Al-Naib, N. Born, and M. Koch, “Terahertz metasurfaces with high Q-factors,” Appl. Phys. Lett.98(5), 051109 (2011). [CrossRef]
- M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2008).
- J. Han and A. Lakhtakia, “Semiconductor split-ring resonators for thermally tunable terahertz metamaterials,” J. Mod. Opt.56(4), 554–557 (2009). [CrossRef]
- K. Aydin, I. M. Pryce, and H. A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express18(13), 13407–13417 (2010). [CrossRef] [PubMed]
- Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]
- D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004). [CrossRef]
- Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett.96(12), 123901 (2006). [CrossRef] [PubMed]
- S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A20(3), 569–572 (2003). [CrossRef] [PubMed]
- A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys.82(3), 2257–2298 (2010). [CrossRef]
- L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: A coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett.108(8), 083902 (2012). [CrossRef] [PubMed]
- S. H. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65(23), 235112 (2002). [CrossRef]
- A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vucković, “Dipole induced transparency in waveguide coupled photonic crystal cavities,” Opt. Express16(16), 12154–12162 (2008). [CrossRef] [PubMed]
- R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett.104(24), 243902 (2010). [CrossRef] [PubMed]
- B. Ung, A. Mazhorova, A. Dupuis, M. Rozé, and M. Skorobogatiy, “Polymer microstructured optical fibers for terahertz wave guiding,” Opt. Express19(26), B848–B861 (2011). [CrossRef] [PubMed]
- M. Consales, A. Ricciardi, A. Crescitelli, E. Esposito, A. Cutolo, and A. Cusano, “Lab-on-fiber technology: toward multifunctional optical nanoprobes,” ACS Nano6(4), 3163–3170 (2012). [CrossRef] [PubMed]
- T. Srivastava, R. Das, and R. Jha, “Highly accurate and sensitive surface plasmon resonance sensor based on channel photonic crystal waveguides,” Sens. Actuators B Chem.157(1), 246–252 (2011). [CrossRef]
- M. Skorobogatiy, Nanostructured and Subwavelength Waveguides (Wiley, 2012).
- S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett.80(6), 908–910 (2002). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
- M. A. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express14(3), 1208–1222 (2006). [CrossRef] [PubMed]
- H. Haus, W. P. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol.5(1), 16–23 (1987). [CrossRef]
- M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ring-resonator photonic metamaterial with huge optical activity,” Opt. Lett.35(10), 1593–1595 (2010). [CrossRef] [PubMed]
- H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics3(3), 148–151 (2009). [CrossRef]

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