## Characteristics of bound modes in coupled dielectric waveguides containing negative index media

Optics Express, Vol. 11, Issue 6, pp. 521-529 (2003)

http://dx.doi.org/10.1364/OE.11.000521

Acrobat PDF (236 KB)

### 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

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

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]

*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]

*T*junction NIM waveguide configuration, and a simple plane two-layered waveguide was shown [5] to have a slow-wave factor that tends to infinity at small frequencies.

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]

## 2. Method

_{0}, and µ

_{0}. The upper waveguide layer of width 2

*d*

_{1}has permittivity ∊

_{1}and permeability µ

_{1}, and the lower guide of width 2

*d*

_{2}has permittivity ∊

_{2}and permeability µ

_{2}. The two channels are separated by a central film that is 2

*d*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.

*k*

_{z}. To proceed, we assume a sinusoidal time dependence exp(-

*i*ω

*t*) 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*(

*k*

_{z}

*z*-ω

*t*)], where we have factored out the spatial variation of the field in the

*x*direction, and

*k*

_{z}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:

**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]

_{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]

*k*

_{z}, common to all the layers, serves as a connection to constrain the transverse components.

## 3. Results

### 3.1 NIM Waveguides

*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

*E*

_{z}and

*H*

_{y}at the interfaces, we arrive at the following eigenvalue equation or dispersion relation for the symmetric and antisymmetric modes:

*k*

_{z}normalized by a design frequency ω

_{0}/

*c*versus the dimensionless frequency ω/ω

_{0}. In the following, we take ω

_{e}=2.34ω

_{0}, and ω

_{m}=1.98ω

_{0}. All curves are bound by the light line,

*k*

_{z}dispersion pair which reside in the vicinity of the line

*d*) between the two waveguide cores has the trend of increasing the divide between the two modal bands with different symmetry. Although not visible in the plot, the zeroth order TM mode is cutoff in frequency, in contrast to the usual PIM case where there is no frequency cutoff. By way of comparison, we also show in Fig. 2, the dispersion curve for a single NIM waveguide. It is apparent that the symmetric and antisymmetric modes are centered about the single waveguide case. One of the most dramatic features of the dispersion relation is the bending back of the bands close to the light line. This is a behavior seen in general whenever a waveguide channel consisting of NIM is in proximity to a material with PIM characteristics. The inset of Fig. 2 shows a close up of the second modal branch. Some salient features to be noted are the change in slope of the dispersion curve, varying from positive to negative as one cycles through the points 1 – 3 respectively. We thus find some peculiarities in the dispersion diagram that clearly deviate from the well understood coupled waveguide system with conventional dielectric material. As will be discussed below, there is an abrupt reversal of energy flow in the

*z*-direction between the two different media, and this quite naturally is reflected in the allowed guided wave modes. In order to further investigate this anomalous behavior of the energy flow in the vicinity of a bend, it is necessary to calculate the Poynting vector, which requires the full electromagnetic fields.

*k*

_{z}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),

*L*≡2

*d*

_{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,

*S*

_{x}is imaginary, and thus does not contribute to the time-average flux of energy. The

*z*component however does contribute, and is given by,

*S*

_{z}depends upon the sign of the permittivity of the medium, and is discontinuous across the boundaries. On the other hand, the longitudinal wave vector

*k*

_{z}remains continuous across each layer, as required by translational invariance. To illustrate the spatial dependence of

*S*

_{z}, we show in Fig. 3, the time averaged Poynting vector throughout the waveguide system. Each of the three curves corresponds to an allowed

*k*

_{z}and ω pair identified by the three points within the inset of Fig. 2. The most visible behavior is the rapid decline in

*S*

_{z}in the regions characterized by material parameters ∊

_{0}and µ

_{0}, 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

*S*

_{z}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

*d*ω/

*dk*

_{z}changes sign, there must be points in the dispersion curve that result in

*S*

_{z}in the middle and outer layers exactly countering

*S*

_{z}in the two waveguide channels. This is shown by the intermediate red curve in Fig. 4. For this special point on the dispersion curve,

*d*ω/

*dk*

_{z}=0 (see point 2, inset of Fig. 2), and the net

*S*

_{z}is zero.

*P*

_{z}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,

*P*

_{z}can be simply expressed as

*P*

_{z}curves correspond to the lower branch in the dispersion diagram, below the point where

*d*ω/

*dk*

_{z}=0 (see Fig. 2). Likewise, the negative

*P*

_{z}curves correlate with those ω,

*k*

_{z}pairs above that point. Thus a clear relationship exists between the net power and the slope of the dispersion curves:

*P*

_{z}is always negative for values of

*k*

_{z}and ω that satisfy

*d*ω/

*dk*

_{z}<0, and is positive when

*d*ω/

*dk*

_{z}>0. For those values of

*k*

_{z}and ω where

*P*

_{z}=0, we again find consistency with the dispersion curves, where at those same points

*d*ω/

*dk*

_{z}=0. This result follows from the exact cancellation of negatively directed energy flow in the waveguides with the positive contributions from the outer PIM layers.

### 3.2 NIM/PIM Waveguides

_{1}=-∊

_{2}, and µ

_{1}=-µ

_{2}. After a lengthy calculation, the dispersion relation can be expressed as:

*k*

_{z}and ω for the present asymmetric waveguide configuration. The two differently colored curves shown correlate to the choice of sign in Eq. (9). Since there is no longer any symmetry in the

*x*direction, the overall behavior of the dispersion curves is more complex than the previous case, where for a given frequency range, each waveguide had identical negative permittivity and permeability values. The figure illustrates the unusual change in sign of

*d*ω/

*dk*

_{z}that occurs when traversing the two visible U-shaped branches. To proceed with the calculation of the Poynting vector, we again calculate the electromagnetic fields within the five different layers. We find,

## 4. Conclusion

*d*ω/

*dk*

_{z}. 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.

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and µ,” Sov. Phys. Usp. |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,Phys. Rev. Lett. |

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. |

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. |

7. | R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E |

**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

- V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of E and µ,�?? Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, �??Imaging the near field,�?? cond-mat/0207026 (unpublished).
- 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]
- I. S. Nefedov and S. A. Tretyakov, �??Waveguide containing a backward-wave slab,�?? condmat/0211185 (unpublished).
- 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]
- 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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