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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 6 — Mar. 24, 2003
  • pp: 521–529
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Characteristics of bound modes in coupled dielectric waveguides containing negative index media

Klaus Halterman, J. Merle Elson, and P. L. Overfelt  »View Author Affiliations


Optics Express, Vol. 11, Issue 6, pp. 521-529 (2003)
http://dx.doi.org/10.1364/OE.11.000521


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Abstract

We investigate the characteristics of guided wave modes in planar coupled waveguides. In particular, we calculate the dispersion relations for TM modes in which one or both of the guiding layers consists of negative index media (NIM)-where the permittivity and permeability are both negative.We find that the Poynting vector within the NIM waveguide axis can change sign and magnitude, a feature that is reflected in the dispersion curves.

© 2003 Optical Society of America

1. Introduction

With the continual refinement of laser arrays, and optical fibers, the role of coupled waveguide systems in integrated optics has magnified in the past few years. The fabrication of arrays which use parallel dielectrics as guiding structures has been demonstrated in millimeter-wave dielectric circuits. Furthermore, the coupling of waves in planar dielectric sheet waveguides provides a means of reflectionless signal transfer from one waveguide to another. For these systems, generally the constituent dielectric materials are presumed to have positive values of the electric permittivity, ∊ and magnetic permeability, µ. Recently, however there has been interest in researching materials which have the unusual property of both ∊ and µ being negative, a concept first proposed by Veselago [1

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

] long ago. These materials support electromagnetic wave propagation in which the phase velocity is antiparallel to the energy flow, or group velocity. Furthermore, if both ∊, and µ are negative in a medium, the refractive index has the extraordinary property of also being negative, therefore such media are given the name negative index media (NIM), compared with the more familiar positive index media (PIM)-where both µ and ∊ are positive. As discussed by Veselago [1

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

], a multitude of other anomalous characteristics of NIM follow from Maxwell’s equations, including reversal of the Doppler shift, counter-directed Cherenkov radiation cone, and the refocusing of EM waves from a point source. It was shown [2

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

] that a NIM slab can focus focus both near and far fields, attaining perfect resolution due to the amplification of the evanescent waves. Subsequently however, the original “perfect lens” model was modified [3

3. S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” cond-mat/0207026 (unpublished).

] to overcome limitations of imperfect materials by introducing a multilayer stack. While the study of various NIM structures has intensified during the past few years, the number of theoretical works involving waveguides with NIM components is less common. The S parameters and electromagnetic properties were investigated [4

4. C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys. 90, 5483 (2001). [CrossRef]

] numerically for a T junction NIM waveguide configuration, and a simple plane two-layered waveguide was shown [5

5. I. S. Nefedov and S. A. Tretyakov, “Waveguide containing a backward-wave slab,” condmat/ 0211185 unpublished).

] to have a slow-wave factor that tends to infinity at small frequencies.

Fig. 1. Coupled waveguide structure considered in this paper. The red and blue regions correspond to the waveguide channels with dispersive material parameters. The permittivity and permeability in the channels can take positive or negative values. The free space parameters take the values ∊00=1.

On the experimental side, the fabrication of devices constructed with NIM is less prolific, as there are no naturally occurring NIM in nature, and thus one must artificially engineer a composite system having the desired electromagnetic properties. One such medium was constructed [6

6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Masser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]

] in which the effective permeability and effective permittivity were shown to be simultaneously less than zero over a finite frequency band. These structures that simulated a NIM were comprised of a periodic array of unit cells, where each cell contained two split metallic rings located on a dielectric panel along with one copper wire. The overall medium is considered homogeneous since the elements of each cell varied over length scales much smaller than the wavelengths used in the experiments. The experimental situation involving these unconventional materials is thus in the early stages of development, and there exists a broad range of potential applications, including optical and microwave devices.

It is the purpose of this paper to investigate the electromagnetic properties of a coupled planar dielectric waveguide system. In order to study the interaction between NIM and PIM configurations, our method will allow for a wide range of material parameters. The paper is organized as follows. In Section 2, we introduce the geometry and method used to calculate the relevant quantities of interest. In Section 3, we calculate the dispersion relations, electromagnetic fields, and energy flow characteristics for differing waveguide structures. Finally in Section 4, we summarize our results.

2. Method

In this section we give a brief overview of the model used for our coupled waveguide system. The schematic of the planar dielectric waveguide is illustrated in Fig. 1. The structure consists of two waveguide channels embedded in a medium characterized by the free space parameters ∊0, and µ0. The upper waveguide layer of width 2d 1 has permittivity ∊1 and permeability µ1, and the lower guide of width 2d 2 has permittivity ∊2 and permeability µ2. The two channels are separated by a central film that is 2d in width and is divided in the plane at x=0. The z axis of the coordinate system coincides with the axis of the guide.

We are interested in guided wave modes, which are electromagnetic waves that propagate along the waveguide with a given phase and group velocity, intensity distribution, and polarization. These modes are relevant since the corresponding fields satisfy the wave equation throughout the structure, and the appropriate boundary conditions. Furthermore, each mode is characterized by its frequency ω and its wave vector kz . To proceed, we assume a sinusoidal time dependence exp(-iωt) for the fields. The dielectric media is linear and isotropic in the y and z directions. The translational invariance in the y and z direction allows, (for the case of TM modes) one to write the magnetic field H as, H=ŷ h(x) exp[i(kzzt)], where we have factored out the spatial variation of the field in the x direction, and kz is the z component of the wave vector. Employing Maxwell’s equations, one can immediately write down the reduced wave equation that must be solved in each of the layers:

[2x2kz2+μi(ω)i(ω)ω2c2]h(x)=0,i=0,1,2,
(1)

with an analogous expression for the E field. In what follows, we consider material parameters in the NIM regions to be dispersive, otherwise the energy density would be negative [1

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

]. We take the permittivity and permeability to have the form,

i(ω)=1ωe2ω2,μi(ω)=1ωm2ω2,
(2)

where ω e , and ω m are the effective electrical and magnetic plasma frequencies [2

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

], respectively. To simplify the model, we have neglected damping. Elementary solutions to Eq. (1) admit the following wave vectors,

ki=i(ω)μi(ω)ω2c2kz2i=0,1,2,
(3)

where the wave vectors are labeled according to the respective values of the permeability and permittivity in each region depicted in Fig. 1. It is clear that the longitudinal wave vector kz , common to all the layers, serves as a connection to constrain the transverse components.

3. Results

3.1 NIM Waveguides

We consider first the case where the permittivity and permeability are both simultaneously negative in each waveguide core, and each of the guiding layers has the same width d 1 = d 2 (see Fig. 1). To derive the TM modes, we proceed to seek elementary solutions of Eq. (1) that do not radiate away from the system. Upon invoking the continuity of Ez and Hy at the interfaces, we arrive at the following eigenvalue equation or dispersion relation for the symmetric and antisymmetric modes:

±e2k0dsin(2k1d1)sin(2k1d12ψ)=0,
(4)
Fig. 2. Dispersion curves for waveguide structure with material parameters given by Eqs. (2). The widths of the middle layer and the two waveguide channels are given by d=0.1λ0, and d 1=0.2λ0 respectively, where ω0=2πc0. The red curve corresponds to the symmetric modes, and the blue curve is for the asymmetric modes. The central green curves correspond to the dispersion relation for a single NIM waveguide of thickness 2d 1. The inset is a magnification of the same quantity localized around one of the bending features. The numbers label points of interest as discussed in the text.

where we have introduced a phase angle ψ,

ψarctan(k01k10).
(5)

Having obtained the set of kz and ω that are solutions to Eq. (4), the spatially varying electromagnetic fields can be straightforwardly calculated from Eq. (1). To demonstrate this, we write down the magnetic fields consistent with the dispersion relation in Eq. (4),

h(x)=cos(ψ)cos(k1d1ψ)ek0(xL),xL,
(6a)
=cos[k1(xL)+ψ]cos(k1d1ψ),dxL,
(6b)
=cos(2k1d1ψ)cos(k1d1ψ)cosh(k0d)cosh(k0x),xd,
(6c)
=cos[k1(x+L)ψ]cos(k1d1ψ),dxL,
(6d)
=cos(ψ)cos(k1d1ψ)ek0(x+L),xL,
(6e)

where L≡2d 1+d, and for brevity we show only the symmetric field, noting that the antisymmetric fields are calculated similarly. The magnetic field distribution in Eq. (6) reflects the fact that the fields are oscillatory within the two waveguide cores, and exponentially decaying in the outer regions. For a given set of material parameters, Eqs. (6) supplemented with Eq. (4) completely determine (apart from an unimportant overall constant amplitude factor) the electromagnetic properties of the structure. We are ultimately interested in the time-averaged power flow density however, as given by the real part of the Poynting vector: S=[c/8π](E×H*). Upon calculating the electric field distribution from Eq. (6) by taking the appropriate derivatives via Maxwell’s equations, we find that the transverse component, Sx is imaginary, and thus does not contribute to the time-average flux of energy. The z component however does contribute, and is given by,

Sz(x;kz,ω)=c28πkzωi(ω)[h(x)]2,i=0,1,2.
(7)

In this instance, the direction of Sz depends upon the sign of the permittivity of the medium, and is discontinuous across the boundaries. On the other hand, the longitudinal wave vector kz remains continuous across each layer, as required by translational invariance. To illustrate the spatial dependence of Sz , we show in Fig. 3, the time averaged Poynting vector throughout the waveguide system. Each of the three curves corresponds to an allowed kz and ω pair identified by the three points within the inset of Fig. 2. The most visible behavior is the rapid decline in Sz in the regions characterized by material parameters ∊0 and µ0, while the two NIM waveguide channels show relatively little variance. The blue curve reveals that more total energy is flowing in the central and two outer layers, while the parameters corresponding to the green curve shows that the net negative Sz in the waveguide regions is greater overall. This trend continues at other points, so that the sign of the slope of the dispersion relation correlates with that of the net energy flow. Since dω/dkz changes sign, there must be points in the dispersion curve that result in Sz in the middle and outer layers exactly countering Sz in the two waveguide channels. This is shown by the intermediate red curve in Fig. 4. For this special point on the dispersion curve, dω/dkz =0 (see point 2, inset of Fig. 2), and the net Sz is zero.

Fig. 3. The normalized z-component of the Poynting vector Sz as a function of dimensionless position. The continual decline in amplitude of Sz in the central film and the outer two layers from blue to green coincides (in order) with points 1–3 in the inset of Fig. 2 respectively. Meanwhile, the two waveguides have relatively little shift in amplitude. The vertical lines serve as guides to the eye.

The interesting properties of the net energy flow in our coupled waveguide system can be investigated further by calculating the time average of the power flow Pz across all the layers. In order to do this, we spatially integrate Eq. (7) over all of the spatial region encompassing the system. For the single layer case, Pz can be simply expressed as

Pz(kz,ω)dxSz(x;kz,ω)=c28πkzd1ω{cos2(k1d1)0k0d1+11[1+sin(2k1d1)2k1d1]}.
(8)

Fig. 4. The normalized z component of power, Pz , is plotted as a function of dimensionless frequency ω/ω0. The color labeling of the curves coincides with the dispersion curves displayed in the inset of Fig. 2. For comparison purposes, the green curve given by Eq. (8), is multiplied by a factor of 2.

3.2 NIM/PIM Waveguides

Next, we consider the case where the two waveguide channels possess opposite material characteristics, i.e., ∊1=-∊2, and µ1=-µ2. After a lengthy calculation, the dispersion relation can be expressed as:

1e4k0dsin(2k1d1)±sin(2ψ)=0.
(9)

h(x)=sin[2(k1d1+ψ)]sin(2k1d1)ek0(x2d),xL,
(10a)
=ek0(L+2d)sin(2k1d1)cos(ψ)sin[2(k1d1ψ)]cos[k1(xL)+ψ],dxL,
(10b)
=ek0Lsin(2ψ){ek0(x+d)sin[2(k1d1+ψ)]ek0(x+d)sin[2(k1d1)]},xd,
(10c)
=ek0Lcos(ψ)cos[k1(x+L)+ψ],dxL,
(10d)
=ek0x,xL.
(10e)

From the plot in Fig. 5, we select four points (kz , ω) on the dispersion curves intersected by the ω/ω0=1.24 line. Inserting these points into Eqs. (10) and then into (7) results in the normalized spatially dependent energy flow that is shown in Fig. 6. The upper panel in Fig. 6(a) depicts a relatively flat red line, which is actually three curves merged together, along with a dominant energy distribution (black curve) for the parameter values ω/ω0=1.24, and kz /(ω0/c)=2.22. Thus a feature arises in which three out of the four wave vector values have negligible contributions to Sz in this region. Proceeding to the lower panel (b), we exhibit Sz for the same parameter values as in (a), except we consider now the NIM region. In addition to the energy flow reversing direction, all four kz and ω pairs are now visible and have a relatively complicated asymmetry when compared to the case when the waveguide structure is physically symmetric, as in Fig. 3, where the guiding layers are both NIM media. It is clear therefore, that there can be cases where the energy flow is dominant in either the NIM or PIM slab, or is shared between both slabs.

Fig. 5. Dispersion relation for the coupled waveguide structure consisting of a NIM channel and a PIM channel. The widths of the middle layer and the two guide channels are given by d=0.1λ0, and d 1=0.2λ0 respectively. The red (blue) curves correspond to the plus (minus) sign in Eq. (9).

4. Conclusion

In this paper we have examined the geometrical and material dispersion properties of a coupled planar waveguide structure that contains negative index media. We calculated the TM guided wave modes and found a variety of unconventional results. For the case where both coupled waveguides have NIM characteristics and are embedded in PIM media, the dispersion relation revealed clear deviations from the well understood case involving positive material parameters. We found that due to the reversal of energy flow along the propagation direction, the dispersion curve reflected a striking behavior in curvature. This followed ultimately from the correlation between the net power in the system and the slope dω/dkz . Furthermore, decreasing the relative distance between each wave channel has the effect of further splitting the symmetric and antisymmetric dispersion curves. Finally, we presented results for a more complicated asymmetric waveguide structure comprised of both NIM and PIM guiding layers. In this case, the Poynting vector and dispersion characteristics exhibited nontrivial behavior. In particular, we found that there are cases when the energy flow reverses direction when going from the NIM to PIM layer, and that the relative distribution of power varied dramatically between the two, depending on the particular guided wave mode under consideration.

Fig. 6. The spatial dependence of the (normalized) Poynting vector Sz , in both waveguide cores, as a function of x0. The top panel (a) depicts Sz in the PIM guide, and the bottom panel (b) corresponds to the NIM guiding layer. Energy flow is always negative in the NIM waveguide, and positive in the PIM waveguide. For all four curves, ω/ω0=1.24 and kz /(ω0/c)=1.27 (blue), 1.59(green), 1.89 (red), and 2.22 (black). Upon comparing both panels, it is evident that for kz /(ω0/c)=2.22, the PIM waveguide layer bears most of the energy flux from the electromagnetic fields.

Acknowledgments

References and links

1.

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

2.

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

3.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” cond-mat/0207026 (unpublished).

4.

C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys. 90, 5483 (2001). [CrossRef]

5.

I. S. Nefedov and S. A. Tretyakov, “Waveguide containing a backward-wave slab,” condmat/ 0211185 unpublished).

6.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Masser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]

7.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.7390) Optical devices : Waveguides, planar
(310.2790) Thin films : Guided waves

ToC Category:
Research Papers

History
Original Manuscript: February 11, 2003
Revised Manuscript: March 11, 2003
Published: March 24, 2003

Citation
Klaus Halterman, J. Elson, and P. Overfelt, "Characteristics of bound modes in coupled dielectric waveguides containing negative index media," Opt. Express 11, 521-529 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-521


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References

  1. V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of E and µ,�?? Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
  2. J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
  3. S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, �??Imaging the near field,�?? cond-mat/0207026 (unpublished).
  4. C. Caloz, C.-C. Chang, and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483 (2001). [CrossRef]
  5. I. S. Nefedov and S. A. Tretyakov, �??Waveguide containing a backward-wave slab,�?? condmat/0211185 (unpublished).
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Masser, and S. Schultz, �??Composite Medium with Simultaneously Negative Permeability and Permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
  7. R. W. Ziolkowski, and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001). [CrossRef]

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