## Negative index Clarricoats-Waldron waveguides for terahertz and far infrared applications

Optics Express, Vol. 18, Issue 4, pp. 3626-3631 (2010)

http://dx.doi.org/10.1364/OE.18.003626

Acrobat PDF (465 KB)

### Abstract

We explore a class of dielectrically loaded metallic waveguides capable of supporting negative index modes in the far infrared and terahertz regime. Principles of operation, modal structure and appropriate coupling schemes are analytically and numerically investigated. The extreme simplicity of the proposed design, along with the non-conventional and counter intuitive electromagnetic properties of this family of waveguides, makes these structures excellent candidates for the practical realization of negative index far infrared and terahertz devices with new and interesting functionalities. Generalizations and extensions of the suggested design are also discussed.

© 2010 OSA

## 1. Introduction

2. 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**(18), 4184–4187 (2000). [CrossRef] [PubMed]

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

4. O. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antenn. Propag. **51**(10), 2619–2625 (2003). [CrossRef]

5. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics **1**(1), 41–48 (2007). [CrossRef]

6. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. **50**(12), 2702–2712 (2002). [CrossRef]

4. O. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antenn. Propag. **51**(10), 2619–2625 (2003). [CrossRef]

## 2. Field analysis of the Clarricoats-Waldron waveguide

_{r}= 30 and radius 24.9μm (grey cylinder), surrounded by another (empty in this case) dielectric region. Such geometry is advantageous since all the fields can be computed analytically, in very much the same way as done for ordinary optical fibers [14], with the sole difference that now the tangential electric fields have to vanish at the metallic boundary.

12. M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. **92**(6), 063903 (2004). [CrossRef] [PubMed]

_{0}/dβ) emerges for the fundamental mode (k

_{0}= ω/c). As a consequence the power flow and the phase velocity become antiparallel, in other words the mode exhibits negative effective index.

_{11}and TM

_{11}modes of a circular metallic waveguide [15

15. C. S. Lee, S. W. Lee, and S. L. Chuang, “Plot of Modal Field Distribution in Rectangular and Circular Waveguides,” IEEE Trans. Microw. Theory Tech. **33**(3), 271–274 (1985). [CrossRef]

_{11}mode, while the magnetic field retains the symmetry of the TM

_{11}mode.

17. S. Mokhov, R. El-Ganainy, and D. N. Christodoulides, “Power circulation via negative energy-flux wormholes in optical nanowaveguides,” Opt. Express **14**(8), 3255–3262 (2006). [CrossRef] [PubMed]

## 3. THz design of a Clarricoats-Waldron guide

18. P. H. Bolivar, M. Brucherseifer, J. Gómez Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the Dielectric Constant and Loss Tangent of High Dielectric-Constant materials at Terahertz Frequencies,” IEEE Trans. Microw. Theory Tech. **51**(4), 1062–1066 (2003). [CrossRef]

^{TM}.

20. S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express **17**(25), 22890–22897 (2009). [CrossRef]

^{−5}cm. The dimensions are chosen to be 80μm x 40μm for the metallic wall and 53μm x 26.5μm for the dielectric load, so as to achieve backward wave operation around 1THz. Based on these design parameters the field decay constant of the fundamental backward mode at 1THz is perturbatively estimated to be

## 4. Generalizations and extensions: Clarricoats-Waldron arrays

## 5. Conclusions

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509 (1968) (published in Russian in 1967). |

2. | 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. |

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

4. | O. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antenn. Propag. |

5. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

6. | G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. |

7. | P. J. B. Clarricoats and R. A. Waldron, “Non-periodic slow-wave and backward-wave structures,” Electron. Control |

8. | R. E. Collin, |

9. | P. J. B. Clarricoats, “Backward waves in waveguides containing dielectrics,” Proc. IEE |

10. | P. J. B. Clarricoats and A. B. Birtles, “Circular Waveguide Backward-wave Experiments,” J. Electron Contr. |

11. | R. A. Waldron, “Theory and potential applications of backward-waves in non-periodic inhomogeneous waveguides,” Proc. IEE |

12. | M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. |

13. | G. N. Tsandoulas, “Propagation in Dielectric-Lined Square Waveguides,” IEEE Trans. Microw. Theory Tech. |

14. | K. Okamoto, |

15. | C. S. Lee, S. W. Lee, and S. L. Chuang, “Plot of Modal Field Distribution in Rectangular and Circular Waveguides,” IEEE Trans. Microw. Theory Tech. |

16. | E. F. F. Gillespie, “Power flow and negative impedance in the dielectric rod waveguide,” Proc. Inst. Electr. Eng. |

17. | S. Mokhov, R. El-Ganainy, and D. N. Christodoulides, “Power circulation via negative energy-flux wormholes in optical nanowaveguides,” Opt. Express |

18. | P. H. Bolivar, M. Brucherseifer, J. Gómez Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the Dielectric Constant and Loss Tangent of High Dielectric-Constant materials at Terahertz Frequencies,” IEEE Trans. Microw. Theory Tech. |

19. | R. F. Potter, “Germanim (Ge),” in |

20. | S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(260.3090) Physical optics : Infrared, far

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 4, 2010

Revised Manuscript: February 1, 2010

Manuscript Accepted: February 1, 2010

Published: February 4, 2010

**Citation**

Alessandro Salandrino and Demetrios N. Christodoulides, "Negative index Clarricoats-Waldron waveguides for terahertz and far infrared applications," Opt. Express **18**, 3626-3631 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3626

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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 (1968) (published in Russian in 1967).
- 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(18), 4184–4187 (2000). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
- O. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antenn. Propag. 51(10), 2619–2625 (2003). [CrossRef]
- V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]
- G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50(12), 2702–2712 (2002). [CrossRef]
- P. J. B. Clarricoats and R. A. Waldron, “Non-periodic slow-wave and backward-wave structures,” Electron. Control 8, 455 (1960).
- R. E. Collin, Field Theory of Guided Waves, 2nd ed., (New York, IEEE Press, 1991) Chap. 9: periodic structures.
- P. J. B. Clarricoats, “Backward waves in waveguides containing dielectrics,” Proc. IEE 108, 496–501 (1961).
- P. J. B. Clarricoats and A. B. Birtles, “Circular Waveguide Backward-wave Experiments,” J. Electron Contr. 15, 325–330 (1963).
- R. A. Waldron, “Theory and potential applications of backward-waves in non-periodic inhomogeneous waveguides,” Proc. IEE 111, 1659–1667 (1964).
- M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004). [CrossRef] [PubMed]
- G. N. Tsandoulas, “Propagation in Dielectric-Lined Square Waveguides,” IEEE Trans. Microw. Theory Tech. 23(5), 406–410 (1975). [CrossRef]
- K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, New York, 2000), Chap. 3: optical fibers.
- C. S. Lee, S. W. Lee, and S. L. Chuang, “Plot of Modal Field Distribution in Rectangular and Circular Waveguides,” IEEE Trans. Microw. Theory Tech. 33(3), 271–274 (1985). [CrossRef]
- E. F. F. Gillespie, “Power flow and negative impedance in the dielectric rod waveguide,” Proc. Inst. Electr. Eng. 107c, 198–201 (1960).
- S. Mokhov, R. El-Ganainy, and D. N. Christodoulides, “Power circulation via negative energy-flux wormholes in optical nanowaveguides,” Opt. Express 14(8), 3255–3262 (2006). [CrossRef] [PubMed]
- P. H. Bolivar, M. Brucherseifer, J. Gómez Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the Dielectric Constant and Loss Tangent of High Dielectric-Constant materials at Terahertz Frequencies,” IEEE Trans. Microw. Theory Tech. 51(4), 1062–1066 (2003). [CrossRef]
- R. F. Potter, “Germanim (Ge),” in Handbook of Optical Constants of Solids, E.D. Palik, ed., (Academic, Orlando, Fla., 1985).
- S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express 17(25), 22890–22897 (2009). [CrossRef]

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