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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 9 — Apr. 26, 2010
  • pp: 9341–9350
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Blue-shifted contra-directional coupling between a periodic and conventional dielectric waveguides

Linfang Shen, Xudong Chen, Xufeng Zhang, and Li Pan  »View Author Affiliations


Optics Express, Vol. 18, Issue 9, pp. 9341-9350 (2010)
http://dx.doi.org/10.1364/OE.18.009341


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Abstract

The interaction between a periodic and conventional dielectric waveguides is investigated theoretically for a two-dimensional model system. A modified coupled-mode theory is formulated for the considered system and found to agree well with rigorous numerical calculations. It is shown that in a certain wavelength range the contra-directional coupling between the two waveguides can be achieved with high efficiency. But the spectrum of the coupling efficiency is blue-shifted and thus the strongest coupling does not occur in the case when two individual waveguides have the same propagation constant. For such a contra-directional coupling system, the coupling efficiency grows with the coupling length and it tends to 100% (excluding insertion loss) when the coupling length is larger than a certain value, and the coupling window can be largely broaden by reducing the distance between the coupled waveguides.

© 2010 Optical Society of America

1. Introduction

Backward waves in left-handed materials (LHM) are of interest because they are the foundation for a variety of novel phenomena [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. 10, 509–514 (1968). [CrossRef]

]. Backward waves are such electromagnetic (EM) waves whose Poynting vector and wave vector are in opposite directions. In the waveguiding systems, the counterpart of backward waves is backward mode with antiparallel energy and phase flows, which can be supported by a waveguide containing LHM [2

2. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

, 3

3. B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003). [CrossRef]

]. It has been suggested that coupled LHM waveguide and conventional waveguide can realize contra-directional coupling [4–8

4. A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology , Washington, DC, Aug. 26-28, 2002, pp. 233–234. [CrossRef]

]. Such a contra-directional coupling is completely different from that achieved by grating-assisted couplers [9–12

9. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994). [CrossRef] [PubMed]

], because it has a broad bandwidth with high efficiency and is insensitive to coupling length. For the coupled LHM and conventional waveguides, the coupling efficiency can monotonously grow with coupling length and tends to 100% (exclude insertion loss) when the coupling length is larger than a certain value [8

8. W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008). [CrossRef]

]. This type of contra-directional coupling offers a new possibility in the design of optical components and circuits [13

13. W. Yan and L. F. Shen, “Open waveguide cavity using a negative index medium,” Opt. Lett. 33, 2806–2808 (2008). [CrossRef] [PubMed]

]. So far, however, the contra-directional coupling based on LHMs has been well demonstrated only in the microwave regime [5–7

5. R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004). [CrossRef]

] and its extension to the optical regime is severely restrained due to large dissipation and anisotropy in the LHMs [14

14. C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008). [CrossRef]

].

2. Modified coupled-mode theory

With the hope of eliciting essential physical properties, we choose to perform studies on two-dimensional (2-D) model systems, which are uniform in the y direction. Let us consider two dielectric guiding layers separated by a distance d in the x direction, as illustrated in the inset of Fig. 1(a). The upper layer is an array of rectangular dielectric columns of height a (a also represents the layer thickness), width b, and lattice constant p, while the lower one is a dielectric slab of thickness w. The columns with the relative permittivity εr 1 and slab with the relative permittivity εr 2 are surrounded by a third dielectric with the relative permittivity εr 3. As an illustrative example, we take the parameters of the waveguide system as follows: a = 0.26 μm, b = 0.28 μm, p = 0.34 μm, and w = 0.21 μm; εr 1 = 12.25 (Si), εr 2 = 6 (As2S3), and εr 3 = 2.1 (SiO2). Different values of d will be analyzed. In this waveguide system, waves travels along the z direction and they are assumed to be the E-polarization, i.e., the EM fields have the form of E = ŷ Ey and H = Hx + Hz. We investigate theoretically the contra-directional coupling between the two guiding layers, which correspond to a periodic dielectric waveguide (PDWG) and a conventional dielectric waveguide (CDWG), respectively. The dispersion relations for the individual PDWG (solid lines) and CDWG (dotted line) are shown in Fig. 1(a). The modes in the PDWG are solved by using Ho’s plane-wave expansion method (PWEM) [19

19. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

] and supercell technique [18

18. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995). [CrossRef]

]. In the numerical calculation, the period of the supercell in the x direction is chosen to be P = 6p+a. Ho’s PWEM has the property of fast convergence for the E-polarization [20

20. L. F. Shen and S. He, “Analysis for the convergence problem of the plane-wave expansion method for photonic crystals,” J. Opt. Soc. Am. A 19, 1021–1024 (2002). [CrossRef]

] and we employ 201×31 plane waves. To examine the accuracy of the obtained results, for the propagation constant β = 0.412(2π/p) we calculate the accurate values of the normalized frequencies of (three) modes in PDWG with a large plane wave number (Npw = 601×91) and then find the corresponding results shown in Fig. 1(a) to be accurate within 0.1%. For the PDWG, there exist four modes, and the second one with negative group velocity is of our interest, which is a backward mode with antiparallel energy and phase flows. The dispersion band of this backward mode intersects the dispersion curve for the CDWG at β = 0.412(2π/p), i.e., at a free-space wavelength of λ = 1.55 μm. In what follows, we restrict ourselves to the wavelength range of the second band for the PDWG.

Fig. 1. (a) Dispersion relations for the individual PDWG (solid lines) and CDWG (dotted d = 0.75p.

+Fc(x,zn+1)·ẑdx+Fc(x,zn)·ẑdx=iωε0znzn+1dz+dx(εsεm)E·Em*,
(1)

A1(n+1)A1(n)=iK12ei2Δβzn+1/2A2(n)+A2(n+1)2,
(2)
A2(n+1)A2(n)=iK21ei2Δβzn+1/2A1(n)+A1(n+1)2,
(3)

where A (n) 1 denotes A 1(zn), Δβ = (β 2-β 1)/2, and the coupling coefficients are given by

K12=ωε04N1(εr2εr3)xc2w/2xc2+w/2dxp/2p/2dz(eyuy*)ei2Δβz,
(4)
K21=ωε04N2(εr1εr3)xc1a/2xc1+a/2dxb/2b/2dz(uyey*)ei2Δβz,
(5)

As seen from Eq. (4) and Eq. (5), the integration in them is conducted with respect to both the transverse and longitudinal variables (x and z), therefore the coupling coefficients for the present CMT are conceptually different from those for the conventional CMT [21

21. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

], which corresponds to an integration only with respect to the transverse variable. The effective coupling often requires a phase matching between two coupled waveguides, i.e., β1 ≈ β2. In this situation, the quantities N 1 and N 2 have opposite signs. It is interesting if the coupling coefficients K 12 and K 21 then have opposite signs, as in the case of the coupled LHM waveguide and CDWG [8

8. W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008). [CrossRef]

]. To clarify this, the coupling coefficients for the case of d = 0.75p are plotted as a function of wavelength in Fig. 1(b). As expected, K 12 and K 21 have opposite signs and their magnitudes are almost equal in the neighborhood of λ = 1.55 μm, at which β 1 = β 2.

3. Supermodes in the entire structure

To validate the discrete coupled-mode equations, i.e., Eq. (2) and Eq. (3), we first use them to solve for the eigen modes of the entire structure (referred to as supermodes) and make a comparison with the accurate results obtained from the PWEM. Let A (n) 1 = Ā(n) 1 e iΔβzn and A (n) 2 = Ā(n) 2 e iΔβzn, then Eq. (2) and Eq. (3) are rewritten as

A̅1(n+1)ei2ΔβpA̅1(n)ei2Δβp=i2K12[A̅2(n)ei2Δβp+A̅2(n+1)ei2Δβp],
(6)
A̅1(n+1)ei2ΔβpA̅2(n)ei2Δβp=i2K21[A̅1(n)ei2Δβp+A̅1(n+1)ei2Δβp].
(7)

The coefficients in Eq. (6) and Eq. (7) are all constants independent on n. In terms of Ā1 and Ā2, the electric field in the entire structure is expressed as Ey(x, zn) = [Ā(n) 1 uy(x, zn)+Ā(n)2 ey(x)]e iβ̅zn, where β̅ = (β 1 +β 2)/2, thus for a supermode with propagation constant β we have Ā(n) m = Cm eiδβzn = Cmqn (m = 1, 2), where q = e βp and δ β = β - β¯. Substituting into Eq. (6) and Eq. (7) and eliminating the coefficients Cm, we obtain

q±=(1+K2)cos(Δβp)±4K2(1+K2)2sin2(Δβp)1K2,
(8)

where K=K12K21/2>0 . Evidently, there exist two supermodes in the entire structure, and β ±= β¯ +δ β ±= β¯ - i ln(q ±)/p.

Fig. 2. (a)-(c) Real part of the propagation constant of the supermodes. (d)-(f) Imaginary

From Eq. (8) we can easily analyze the guiding properties of the entire structure. In the case of our interest, where Δβ ≈ 0, Eq. (8) reduces to q ±= (1±K)/(1∓K), thus q + > 1 and q - = 1/q + < 1, indicating that δ β ± are equal and opposite imaginary numbers. Therefore, in this case two supermodes are a pair of evanescent modes that are decaying in the opposite directions [but with the same Re(β)]. Obviously, in a wavelength interval where ∣ sin(Δβ p)∣ < 2K/(1+K 2), the supermodes are always evanescent and the maximum of the decay rate ∣δ β∣ occurs at a wavelength for which Δβ = 0. In the case when ∣sin(Δβ p)∣ > 2K/(1+K 2), q ± become complex and we find ∣q ±∣ = 1, thus δ β ± are real numbers and correspondingly the supermodes are two propagating modes with different propagation constants. These guiding properties of the entire structure are well illustrated in Fig. 2, where solid and dotted lines respectively represent the real and imaginary parts of β calculated with Eq. (8).

To obtain the accurate results of the supermodes, we again adopt the PWEM and the supercell technique. But here, the PWEM is formulated as an eigenvalue problem in the form of ℋX = β X, where X is a vector composed of the discrete Fourier coefficients of both Ey and Hx. In the formulation, Maxwells equations are converted into algebraic equations in the discrete Fourier space, and the product of ε Ey is Fourier factorized by Laurents rule, so the formulated PWEM has a merit of fast convergence [22

22. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

]. With such a PWEM, we can even solve for evanescent supermodes with complex β. In the numerical calculation, the period of the supercell in the x direction is taken to be P = 6p+(a+ d + w) and we employ 251×31 plane waves. The obtained results are also plotted as (solid or open) circles in Fig. 2. To examine the convergence of the PWEM and the accuracy of the obtained results, we calculate the propagation constant (β) of the supermodes as a function of plane wave number (Npw) for the case of d = 0.75p and λ = 1.55 μm. The values of the real and imaginary parts of β converge quickly as Npw grows and they are almost constant when Npw is larger than 301×41, and the values at Npw =251×31 are found to be accurate within 1%. Evidently, the agreement of the results from the modified CMT and the PWEM are remarkable, especially in the case with d = p, for which the coupling between the two waveguides is weaker.

Fig. 3. Electric field amplitudes of the supermode with propagation constants β + [(a), (b)] and the backward mode of the individual PDWG (c) at λ = 1.5 μm. (a) corresponds to results from the modified CMT and (b) to those from the PWEM.

4. Coupling characteristics

In a general case with finite L, both supermodes are excited in the coupling region and the coupling process can be treated as the interference of two supermodes [12

12. S. Z. Zhang and T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 12, 2403–2413 (1996). [CrossRef]

, 21

21. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

]. This is mathematically reflected in Eq. (6) and Eq. (7), which have the general solution Ā(n) m = C m e β-zn +C + m e β+zn , where C m ± (m = 1, 2) are constants. From the boundary conditions of A (N) 1 = 0 and A (0) 2 = A 0, we find the coupled-mode solution for the coupled waveguide system

A1(n)=ττ+qNq+Nτ+q+NτqN[q(nN)q+(nN)]eiΔβznA0,
(9)
A2(n)=qNq+Nτ+q+NτqN[τ+q(nN)τq+(nN)]eiΔβznA0,
(10)

where τ ±= (i/2)K 12(e iΔβp/2 + q ± e iΔβp/2)/(q ± e iΔβp/2 - e iΔβp/2). Thus, the contra-directional coupling efficiency, which is defined as η = ∣N 1∣∣A (0) N 1||A (0) 1|2/[N 2|A (0) 2|2], is found to be

η=N1N2ττ+(q+NqN)τ+q+NτqN2.
(11)
Fig. 4. (a) Coupling efficiency versus coupling length for different wavelengths. The value of d is fixed at 0.75p. (b) Coupling efficiency versus coupling length for different d values. The wavelength is fixed at λ = 1.55 μm. Lines in (a) and (b) are calculated with Eq. (11) and circles in (b) are obtained from the rigorous numerical calculations.
Fig. 5. Simulated E field amplitude for the coupling between the PDWG and CDWG at different wavelengths. The initial power is injected into the left end of the CDWG.

Figure 4(a) shows the coupling efficiency as a function of L calculated with Eq. (11) for the case of d = 0.75p. When the supermodes are evanescent modes at λ = 1.55 μm, η grows with L and it tends to 100% when L ≥ 30 μm. When the supermodes are propagating modes at λ = 1.52 μm (or 1.58 μm), η varies periodically with L and its maximal value is much less than 100%. Figure 4(b) shows the dependence of the coupling efficiency on L for different d values, and the wavelength is fixed at λ = 1.55 μm. For smaller d, η grows quicker with L and it tends to 100% at a smaller L, e.g., in the case of d = 0.5p, η tends to 100% when L ≥ 20 μm, which is considerably smaller than that for the case of d = 0.75p. It is desired that the coupling efficiency can be accurately computed using the exact supermodes obtained from the PWEM. To do so, we spatially divide the fields of each supermode in such a way that the (absolute) values of the energy flows on both sides of the division interface reach their maximum. But this division method is feasible only for two evanescent supermodes as they almost have the same division interface. In this way, the coupling efficiency as a function of L for λ = 1.55 μm is calculated numerically and also plotted as circles in Fig. 4(b), and it agrees well with that obtained from Eq. (11) for each value of d. To demonstrate the coupling behaviors described above, we simulate a coupling system with d = 0.75p and L = 85p using the commercial finite element software COMSOL. In the simulation, the scattering boundary condition is used at the boundaries of the computation domain. We choose the left end of the CDWG as an input port and apply a source there by setting the amplitude of Ey to be unity. Note that the right end of the CDWG is tapered to avoid the end reflection. Figure 5 shows the amplitudes of the electric fields for the wavelengths λ =1.52, 1.55, and 1.58 μm. For the cases of λ =1.52 and 1.58 μm, a large fraction of power in the CDWG travels through the coupling region and finally outputs at the right end. In contrast, for λ = 1.55 μm, almost no power outputs at the right end of the CDWG. All these agree well with our above analysis.

The spectral information of the coupling efficiency is of particular interest, and this is displayed in Fig. 6, where solid lines correspond to the results obtained from Eq. (11) and dotted lines with circles to the accurate values from the numerical calculations. The coupling length is fixed at L = 44.2 μm and our numerical calculations show that at λ = 1.55 μm, the (accurate) coupling efficiency is η = 99% for d = p, η = 99.95% for d = 0.75p, and η = 100% for d = 0.5p. If we define a coupling window for which η ≥ 90%, then it has a width Δλ = 16.7 nm for d = p, Δλ = 26.4 nm for d = 0.75p, and Δλ = 38 nm for d = 0.5p. The coupling window is broadening when d is decreased. The center of the coupling window calculated from the modified CMT is always at λ = 1.55 μm and it is rather accurate for the case of d = p. But when d is reduced, the actual center is obviously blue-shifted and so is the coupling window, as seen in Fig. 6. The central wavelength of the window is actually λ c = 1.548 μm for d = 0.75p and λ c = 1.546 μm for d = 0.5p. Though the coupling window is blue-shifted, the wavelength of λ = 1.55 μm, at which β 1 = β 2, always lies within the window.

Fig. 6. Spectrum of coupling efficiency. Solid line corresponds to the results obtained from 11) and dotted line with circles to the accurate results from the numerical calculations. (a) d = p, (b) d = 0.75p, and (c) d = 0.5p.

5. Conclusion

A modified CMT has been formulated and with which the interaction between the PDWG and

Acknowledgements

This work was supported by the Ministry of Education (Singapore) under Grant No. R263000485112.

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. 10, 509–514 (1968). [CrossRef]

2.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003). [CrossRef]

3.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003). [CrossRef]

4.

A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology , Washington, DC, Aug. 26-28, 2002, pp. 233–234. [CrossRef]

5.

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004). [CrossRef]

6.

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004). [CrossRef]

7.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006). [CrossRef]

8.

W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008). [CrossRef]

9.

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994). [CrossRef] [PubMed]

10.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996). [CrossRef]

11.

S. S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add-drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22, 688–690 (1997). [CrossRef] [PubMed]

12.

S. Z. Zhang and T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 12, 2403–2413 (1996). [CrossRef]

13.

W. Yan and L. F. Shen, “Open waveguide cavity using a negative index medium,” Opt. Lett. 33, 2806–2808 (2008). [CrossRef] [PubMed]

14.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008). [CrossRef]

15.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

16.

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72, 165112 (2005). [CrossRef]

17.

W. Kuang, C. Kim, A. Stapleton, and J. D. O’Brien, “Grating-assisted coupling of optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1604–1606 (2002). [CrossRef]

18.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995). [CrossRef]

19.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

20.

L. F. Shen and S. He, “Analysis for the convergence problem of the plane-wave expansion method for photonic crystals,” J. Opt. Soc. Am. A 19, 1021–1024 (2002). [CrossRef]

21.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

22.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

OCIS Codes
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(130.2790) Integrated optics : Guided waves
(260.2030) Physical optics : Dispersion
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: January 4, 2010
Revised Manuscript: March 26, 2010
Manuscript Accepted: April 12, 2010
Published: April 20, 2010

Citation
Linfang Shen, Xudong Chen, Xufeng Zhang, and Li Pan, "Blue-shifted contra-directional coupling between a periodic and conventional dielectric waveguides," Opt. Express 18, 9341-9350 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9341


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ∑ and ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  2. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, "Guided modes in negative-refractive-index waveguides," Phys. Rev. E 67, 057602 (2003). [CrossRef]
  3. B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability," J. Appl. Phys. 93, 9386-9388 (2003). [CrossRef]
  4. A. Alu and N. Engheta, "Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability," in Proc. IEEE Nanotechnology,Washington, DC, Aug. 26-28, 2002, pp. 233-234. [CrossRef]
  5. R. Islam, F. Elek, and G. V. Eleftheriades, "Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow," Electron. Lett. 40, 315-317 (2004). [CrossRef]
  6. C. Caloz, A. Sanada, T. Itoh, "A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth," IEEE Trans. Microwave Theory Tech. 52, 980-992 (2004). [CrossRef]
  7. Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, "Backward coupling waveguide coupler using left-handed material," Appl. Phys. Lett. 88, 211903 (2006). [CrossRef]
  8. W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, "Interaction between negative and positive index medium waveguides," J. Lightwave Technol. 26, 3560-3566 (2008). [CrossRef]
  9. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, "Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber," Opt. Lett. 19, 180-182 (1994). [CrossRef] [PubMed]
  10. L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, "Novel add/drop filters for wavelength-Divisionmultiplexing optical fiber systems using a Bragg grating assisted mismatched coupler," IEEE Photon. Technol. Lett. 8, 1656-1658 (1996). [CrossRef]
  11. S. S. Orlov, A. Yariv, and S. V. Essen, "Coupled-mode analysis of fiber-optic add-drop filters for dense wavelength-division multiplexing," Opt. Lett. 22, 688-690 (1997). [CrossRef] [PubMed]
  12. S. Z. Zhang and T. Tamir, "Rigorous theory of grating-assisted couplers," J. Opt. Soc. Am. A 12, 2403-2413 (1996). [CrossRef]
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  14. C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, "The science of negative index materials," J. Phys.: Condens. Matter 20, 304217 (2008). [CrossRef]
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