## Guided modes in channel waveguides with a negative index of refraction

Optics Express, Vol. 11, Issue 20, pp. 2502-2510 (2003)

http://dx.doi.org/10.1364/OE.11.002502

Acrobat PDF (320 KB)

### Abstract

The guided modes of a negative refractive index channel waveguide have been numerically investigated. It has been found that the modes exhibit a number of unusual properties that differ considerably from those of a conventional waveguide. In particular, it has been shown that these waveguides can exhibit low or negative group velocity as well as extraordinarily large group velocity dispersion. Calculation of the Poynting vector reveals that it is possible to support a mode with a zero energy flux motivating a simple design for an optical trap.

© 2003 Optical Society of America

## 1. Introduction

*ε*and magnetic permeability

*µ*, were theoretically investigated by Veselago in 1968 [1] where he concluded that they would have dramatically different propagation characteristics. In particular, these NIMs can exhibit extraordinary properties such as negative refraction, antiparallel group and phase velocities (backwards waves) and negative energy fluxes (radiation tension). Despite the physical significance of his analysis, the results appeared to be of limited practical application due to the absence of naturally occurring NIMs. However, motivated by earlier investigations [2, 3], in 2000 the first NIM was demonstrated by Smith

*et al.*in a composite material consisting of periodic regions of negative

*ε*and negative

*µ*[4].

*ε*and

*µ*throughout, they have nonetheless been shown to exhibit similar anomalous light behavior to the composite negative

*ε*and

*µ*materials [8]. Recently a PC exhibiting negative refraction was observed experimentally [9] and despite the fact that this experiment was still conducted in the microwave regime, by using electrically poled crystals [10] more complex structures can be fabricated that should scale to the optical regime.

*ε*and

*µ*) core. Similar analysis has already been performed for a planar waveguide [11]. We show that, as in the planar case, the guided modes differ considerably from those in a conventional positive index waveguide. Typical features of these waveguides include the absence of the fundamental mode, possible double degeneracy of modes and backwards propagating waves with negative energy flux. By calculating the dispersion curves we find that the group velocity can become very low, or negative, whilst the group velocity dispersion can be many orders of magnitude larger than conventional materials. Furthermore, because the low group velocity should lead to reduced nonlinear thresholds, by combining the large dispersions with large nonlinearities it should also be possible to observe novel nonlinear phenomena in extremely short device lengths. We expect these waveguides to find wide applications in many optical technologies including optical data storage, quantum computing, dispersion management and optical soliton formation.

## 2. Guided mode solutions and their properties

*ω*is the angular frequency of the field,

*β*is the propagation constant and E(

*x,y*) and H(

*x,y*) are the spatially localized transverse mode profiles of the electric and magnetic fields, respectively.

*x*direction, i.e., the

*E*

_{x}and

*H*

_{y}as the principal field components. Here

*p*and

*q*are integers which correspond to the number of peaks of the optical power in the

*x*and

*y*directions, respectively. It follows from Maxwell’s equations that the wave equation in this representation is [13]:

*β*is then obtained from,

*k*becomes purely imaginary [11

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

*β*exceeds a critical value and have been likened to surface waves in metal films [14

14. G. I. Stegeman, J. J. Burke, and T. Tamir, “Surface-polaritonlike waves guided by thin, lossy metal films,” Opt. Lett. **8**, 383–385 (1983). [CrossRef] [PubMed]

*k*

_{x}

*L,γ*

_{x}

*L*) and (

*k*

_{y}

*L,γ*

_{y}

*L*) to include imaginary values of

*k*

_{x}and

*k*

_{y}by defining :

*κ*

_{x}=i

*k*

_{x}and

*κ*

_{y}=i

*k*

_{y}. Fig. 2 shows typical solutions for the

*x*and

*y*components of the field. The solid lines are the right-hand-sides of Eqs. (5) and (6) and the dashed lines are obtained from the right-hand-sides of Eqs. (7) and (8). Here the 3 dashed lines correspond to waveguides with the same ratio

*ε*

_{1}/

*ε*

_{2}, but of differing width

*L*. The points of intersection indicate the existence of guided modes. From these intersections we can construct six different solutions and to illustrate this examples of the mode profiles can be seen in Fig. 3. The possible

*α*) mode has an imaginary

*k*

_{x}but real

*k*

_{y}and the (b,

*β*) mode has both

*k*

_{x}and

*k*

_{y}real. The middle row shows the strongly (c,

*δ*) and weakly (d,

*δ*) localized

*η*) and weakly (d,

*η*) localized

*β*so that it decays exponentially as it propagates in

*z*. We note that a similar analysis for the

*y*direction) leads to

*H*

_{x}solutions with forms such as those shown in Fig. 2, but with the

*x*and

*y*components interchanged. In addition, as in Fig. 2 the solutions are still functions of

*L*, it is clear that we will find similar solutions for rectangular guides.

*ε*

_{2}

*µ*

_{2}>

*ε*

_{1}

*µ*

_{1}). This is in contrast to negative index slab waveguides [11] and is a consequence of the dispersion relations of the

*y*component of the field. In addition, as the dispersion relations do not allow for an imaginary

*k*

_{y}, only the

*x*component of the field can exhibit surface wave effects. Secondly, however, in accordance with the observations in negative index planar waveguides we again find that the conventional hierarchy of the fast modes disappears. In particular, (i) for

*p*=0 the right-hand-side of Eq. (5) is negative so that the fundamental mode,

*L*, solutions of

*γ*

_{x}

*L*associated with the first order

*ω*greater than a critical value; (iii) for certain modes, as the

*x*solutions in Fig. 2 decrease monotonically with

*k*

_{x}

*L*at different rates, there can be two solutions to Eq. (5) so that it is possible for two modes with the same number of nodes to coexist in a waveguide, as illustrated by solutions (c,

*δ*) and (d,

*δ*).

*ω*

_{p,i}are the electronic plasma frequencies,

*ω*

_{0,i}are the magnetic resonance frequencies and

*F*is a constant dependent on the material structure. Based on earlier analysis [4, 11], we choose

*ω*

_{p,2}/2

*π*=10GHz,

*ω*

_{0,2}/2

*π*=4GHz and

*F*=0.56. In this case, the region of simultaneously negative

*ε*

_{2}and

*µ*

_{2}ranges from 4GHz to 6GHz. The values of

*ω*

_{p,1}/2

*π*=2GHz and

*ω*

_{0,1}/2

*π*=1GHz were then chosen so that

*ε*

_{1}and

*µ*

_{1}are always positive and

*ε*

_{2}

*µ*

_{2}>

*ε*

_{1}

*µ*

_{1}in this range. The dispersion curves for the two

*L*=2cm. The solid line corresponds to the strongly localized mode (c,

*δ*) and the dashed line is the weakly localized mode (d,

*δ*). From this we see that as we increase the frequency the two solutions converge until they reach a cutoff frequency

*ν*

_{c}associated with their intersection.We also note that although the strongly localized mode exists for frequencies below those plotted here, the weakly localized mode does not as the right-hand-side of Eq. (5) crosses the x axis. Despite the similar appearance of the two modes in Fig. 3, clearly the opposite slopes of β indicate there will be a significant difference in the dispersion properties. Indeed, on calculating the group velocity,

*ν*

_{g}=1/

*β*

_{1}=d

*ω*/d

*β*, Fig. 4(b) shows that whilst for the weakly guided mode

*ν*

_{g}is positive, for the strongly guided mode

*v*

_{g}is negative, corresponding to backward wave propagation [15]. In both cases, however, as the frequency approaches

*ν*

_{c}

*, v*

_{g}approaches zero so that the mode propagation will be slowed considerably. Thus this waveguide offers an alternative to other methods of generating “slow” light (

*ν*

_{g}≪

*c*), “fast” light (

*ν*

_{g}is negative) and perhaps to even trap light. The possibility to slow or trap light has many potential applications such as optical data storage, optical memories and quantum computing. Furthermore, as the light-matter interaction is enhanced for low

*ν*

_{g}, slow light can be used to observe nonlinear processes such as harmonic generation and four-wave mixing in even weakly nonlinear materials [16

16. J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B **19**, 722–731 (2002). [CrossRef]

*β*

_{2}=d

^{2}

*β*/d

*ω*

^{2}. As seen in Fig. 4(c), as the frequency approaches

*ν*

_{c}(the region of low

*ν*

_{g}) the waveguide exhibits large anomalous (c,

*δ*) or normal (d,

*δ*) dispersion. This is typical behavior of the GVD at the band edges of PCs [17

17. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

^{2}km

^{-1}). This makes them idea for dispersion management and particularly for use in integrated circuits where short device lengths are favored. In addition, by exploiting the reduced nonlinear threshold/large GVD combination it should be possible to investigate nonlinear effects such as optical soliton formation so that negative index channel waveguides could offer a simple alternative to devices such as side-coupled integrated space of resonators (SCISSOR) [16

16. J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B **19**, 722–731 (2002). [CrossRef]

*z*component of the Poynting vector :

**S**=Re[

**E**×

**H***] [13]. Since for backwards waves the Poynting vector and the wave vector point in opposite directions, we expect that the energy flux of the modes will also have opposite signs [1]. The total power flux through the core and cladding regions of the waveguide are calculated as,

*P*=(

*P*

_{core}+

*P*

_{clad})/(|

*P*

_{core}|+|

*P*

_{clad}|) [11], Fig. 5 shows that total energy flows in a positive direction for the weakly localized mode and a negative direction for the strongly localized mode, in agreement with the signs of

*ν*

_{g}. We note that |

*P*| <1 and

*P*→1 as the mode becomes poorly confined and

*P*→-1 as the mode becomes tightly confined. The significant feature of this result is that as the solutions converge (at

*ν*

_{c}), the energy fluxes inside and outside the guide exactly cancel so that the total energy flux vanishes. Importantly, in their analysis for a negative index planar waveguide, Shadrivov

*et al.*showed that at

*P*=0 the energy flowed in a double-vortex structure so that most of the energy remained localized inside the wave packet [11]. Thus as the energy flux goes to zero the guided modes do not disintegrate and we can expect an analogous result for the modes in a channel waveguide.

*L*. Figure 6 shows (a) the propagation constant and (b) the normalized energy flux for a fixed frequency,

*ω*/2

*π*=5GHz, where again the solid line corresponds to the strongly localized mode (c,

*δ*) and the dashed line is the weakly localized mode (d,

*δ*). As expected, these have similar forms to the previous curves for varying frequency except this time we see that the two solutions converge as we decrease

*L*until they reach a cutoff length

*L*

_{c}. However, significantly these results suggest that we can slow the propagating mode simply by adiabatically decreasing the waveguide width. Furthermore, by decreasing the width to the critical length

*L*

_{c}it should be possible to stop the light completely. Thus we expect that a simple waveguide structure such as that shown in the inset of Fig. 6 should act as an optical trap where the frequency of light that can be trapped is determined by the range of the waveguide width.

## 3. Discussion and conclusions

*ε*and

*µ*) core, due to similarities in the geometries and the behavior of light in negative effective index PCs, we expect to obtain analogous results in an optical fiber which exhibits negative refraction. With the rapid developments in the fabrication of photonic crystal fibers (PCFs) [18

18. T. M. Monro and D. J. Richardson, “Holey optical fibres: Fundamental properties and device applications,” C. R. Physique **4**, 175–186 (2003). [CrossRef]

19. N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: Measurement and future opportunities,” Opt. Lett. **15**, 1395–1397 (1999). [CrossRef]

## References and links

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

2. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

4. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. |

5. | R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic left-handed metamaterial,” Appl. Phys. Lett. |

6. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

7. | M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B |

8. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B |

9. | E. Cubukcu, K. Aydin, E. Ozbay, S. Forteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: Negative refraction by photonic crystals,” Nature |

10. | M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First order quasi-phase-matched LiNbO |

11. | I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E |

12. | E. A. Marcatili, “Dieclectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. |

13. | K. Okamoto, “Fundamentals of optical waveguides,” Academic Press, 2000. |

14. | G. I. Stegeman, J. J. Burke, and T. Tamir, “Surface-polaritonlike waves guided by thin, lossy metal films,” Opt. Lett. |

15. | S. Ramo, J. R. Whinnery, and T. Van Duzer, “Fields and waves in communication electronics,” John Wiley & Sons, Inc., 1965. |

16. | J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B |

17. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

18. | T. M. Monro and D. J. Richardson, “Holey optical fibres: Fundamental properties and device applications,” C. R. Physique |

19. | N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: Measurement and future opportunities,” Opt. Lett. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(230.7380) Optical devices : Waveguides, channeled

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 8, 2003

Revised Manuscript: September 18, 2003

Published: October 6, 2003

**Citation**

A. Peacock and N. Broderick, "Guided modes in channel waveguides with a negative index of refraction," Opt. Express **11**, 2502-2510 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-20-2502

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### References

- V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Sov. Phys. Usp. 10, 509�??514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, �??Extremely low frequency plasmons in metallic mesostructures,�?? Phys. Rev. Lett. 76, 4773�??4776, (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075�??2084, (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, �??Composite Medium with Simultaneously Negative Permeability and Permittivity,�?? Phys. Rev. Lett. 84, 4184�??4187, (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489�??491, (2001). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77�??79, (2001). [CrossRef] [PubMed]
- 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 (2000). [CrossRef]
- C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative effective index,�?? Phys. Rev. B 65, 201104 (2002). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Forteinopoulou, and C. M. Soukoulis, �??Electromagnetic waves: Negative refraction by photonic crystals,�?? Nature 423, 604�??605 (2003). [CrossRef] [PubMed]
- M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, �??First order quasi-phase-matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,�?? Appl. Phys. Lett. 62, 435�??436 (1993). [CrossRef]
- I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, �??Guided modes in negative-refractive-index waveguides,�?? Phys. Rev. E 67, 057602 (2003). [CrossRef]
- E. A. Marcatili, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell Syst. Tech. J. 48, 2071�??2102 (1969).
- K. Okamoto, �??Fundamentals of optical waveguides,�?? Academic Press, 2000.
- G. I. Stegeman, J. J. Burke, and T. Tamir, �??Surface-polaritonlike waves guided by thin, lossy metal films,�?? Opt. Lett. 8, 383�??385 (1983). [CrossRef] [PubMed]
- S. Ramo, J. R. Whinnery, and T. Van Duzer, �??Fields and waves in communication electronics,�?? John Wiley & Sons, Inc., 1965.
- J. E. Heebner, R. W. Boyd, and Q-H. Park, �??SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,�?? J. Opt. Soc. Am. B 19, 722�??731 (2002). [CrossRef]
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, �??Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,�?? Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
- T. M. Monro and D. J. Richardson, �??Holey optical fibres: Fundamental properties and device applications,�?? C. R. Physique 4, 175�??186 (2003). [CrossRef]
- N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, �??Nonlinearity in holey optical fibers: Measurement and future opportunities,�?? Opt. Lett. 15, 1395�??1397 (1999). [CrossRef]

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