## Guiding effects in waveguides with anti-symmetric refractive index layouts |

Optics Express, Vol. 18, Issue 20, pp. 20681-20689 (2010)

http://dx.doi.org/10.1364/OE.18.020681

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### Abstract

This paper presents and analyzes configurations of waveguides composed by both, regular dielectrics and negative-index materials disposed in an anti-symmetric way with respect to the optical axis. In its basic form, the configuration includes two guiding layers and two cladding media, where both the cladding and the guiding pairs have opposite-signed refractive index. A geometrical analysis shows that paths are closed, hinting the possibility of localized modes or light trapping with zero ray velocity. The ray model shows also a quasi-perfect imaging effect for off-axis objects. A modal approach shows that trapping is broadband and the propagation constant spectrum is continuous. When cores are allowed different widths, control of the group velocity is possible. More general anti-symmetrically mirrored layouts are also addressed.

© 2010 OSA

## 1. Introduction

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

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

3. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. **95**(13), 137404 (2005). [CrossRef] [PubMed]

4. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. **32**(1), 53–55 (2007). [CrossRef]

5. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(5), 057602 (2003). [CrossRef] [PubMed]

10. Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. **245**(1-6), 125–135 (2005). [CrossRef]

9. J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. **16**(2), 96–98 (2006). [CrossRef]

11. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature **450**(7168), 397–401 (2007). [CrossRef] [PubMed]

*[(μ*, and obey:

_{c}, ε_{c}); (μ_{g}, ε_{g}); (−μ_{g}, −ε_{g}); (−μ_{c}, −ε_{c})]*t*and

*s*respectively. In case these two thicknesses are equal (

*s = t*), the structure is totally anti-symmetric, or

*inverse-signed mirrored*with respect to the plane

*x = 0*. In the more general situation, I also analyze the case

*s ≠ t*, which also displays special outcomes. The analysis starts with a simple ray-optics approach to the waveguiding within the interfacing films. Ray trajectories will show the possibility of controlling the ray velocity by the asymmetry parameters, while the totally asymmetric case features zero average ray velocity and closed-loop trajectories. This structure also displays a self-imaging property, but distinctly to the thru-propagating case in NIM slabs [1

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

*s = t*) is shown to possess the exceptional property of a non-discrete (continuous) spectrum of values for the propagation constants. This configuration also supports “surface” or “SPP-type” modes, fulfilling

*β > k*. The fact that this structure supports a continuum of modes without an upper limit for the propagation constant reinforces the property of quasi-perfect imaging deduced form the ray model. For the more general condition (

_{o}n_{g}*s ≠ t*), the eigenvalue equation for the propagation constant is also exceptionally simple and solvable. The propagation constants are now discrete and formally similar to those of a standard planar waveguide surrounded by perfect conductors, but distinctly to that case, the number of bounded modes here is reduced and depends on the refractive index of both the waveguides and claddings. The article continues with power considerations and finishes with a discussion on the connection between the modal and geometrical pictures, and an extension to more general inverse-signed mirrored configurations

## 2. Ray approach

*(t = s)*and for the more general case

*(s ≠ t).*Ray tracing is particularly simple for the given choice of refractive indices, since besides the total internal reflection taking place at the interfaces between media of the same index sign, the refraction at the

*z = 0*plane simply obeys the law

*θ*

_{1}= −θ_{−1}.*(s ≠ t)*is also of interest, and ray traces are shown in Fig. 1(c) The paths here do not close any more, but due to the quasi circulating motion, the net ray velocity component in the z direction will be reduced by the back-and-forth motion as compared to that of a normal positive-index waveguide. By simple path calculation, this velocity is explicitly given by:where

*θ*is the launching angle. Also by simple geometry on realizes that the average power propagation direction is determined by the thicker among the two waveguide cores. One can conclude that the velocity of light and its direction can be controlled by determining the structural parameters

*s*and

*t*of the composed waveguide. This conclusion is also supported by the electromagnetic modal analysis of the following section.

## 3. Electromagnetic modal analysis

### 3.1 Cores with different thicknesses (s ≠ t)

*β*is in the range:

*k*, Eq. (8) is now identical to the dispersion equation of an equivalent simple single slab waveguide of refractive index

_{0}^{2}ε_{c}μ_{c}< β^{2}< k_{0}^{2}ε_{g}μ_{g}*n*and thickness

_{g}*| t - s |*surrounded by two perfectly conducting walls. The mode number of the equivalent slab will be

*N*= |

*n*+

_{1}*n*, and should obey

_{-1}|*N ≠ 0*for all

*s ≠ t.*The explicit solution for the propagation constant of mode Eq. (12) is given by:The number of guided modes

*M*is however smaller than that of a metallic-wall equivalent guide and is given by:In the last expression the refractive index of the cladding media does enter explicitly. To be consistent with conventional waveguide notation, we symbolize the corresponding modes as TE

_{N-1}. Examples of the electrical field in the mode TE

_{0}, and TE

_{1}are depicted in Fig. 2 . The similarity between the modal dispersion in the structure discussed here, and that of a single slab media surrounded by perfect mirrors suggests the possibility of occurrence of imaging and power splitting phenomena by means the Talbot effect.

### 3.2 Totally anti-symmetric waveguide (t = s)

*(t = s)*shown in Fig. 1(a) also needs special consideration in the electromagnetic treatment. In that case, both the clad and guiding section are antisymmetrically mirrored with respect to the plane

*x = 0*. Now Eq. (9) is not valid anymore, but Eq. (8) still holds if

*n*

_{1}

*+ n-*

_{1}

*= N = 0*, explicitly:For

*t = s*this last condition holds for

*any value*of

*β*for which

*p*The conclusion is that the mode propagation constants

_{g}(β) ≠ 0.*β*are not a discrete set and their spectrum and is

*continuous*within the entire confinement range:

*k*. The structure disclosed here seems to be the only reported case where a waveguide of finite width can support an infinite number of guided modes with a continuous spectrum. To illustrate that behavior, several modes are plotted in Fig. 3 , where deliberately arbitrary values of

_{0}^{2}n_{c}^{2}> β^{2}> k_{0}^{2}n_{g}^{2}*β*were chosen within the confinement range. It is apparent that the modes obey rigorously the wave equation and boundary values. This means that any radiation, once coupled into the waveguide within its numerical aperture will be guided without power loss to radiation modes. Furthermore, within the continuous spectrum of the propagation constant

*β*, values are found for which the boundary phases at the interface

*z = 0 (φ*) are integral multiples of

_{1}= -φ_{-1}*2π.*In that case there is no discontinuity at

*z = 0*, and the mode is identical to that of a corresponding symmetric waveguide of positive indices and thickness

*2t*. This would provide a perfect matching situation between a conventional waveguide mode and a mode of an anti-symmetric layout. The lack of guided mode quantization in the totally anti-symmetric situation is also consistent with the quasi-perfect imaging effects of the ray picture discussed in the previous section. It is actually a property of all anti-symmetrically mirrored waveguide profiles. This generalization is further commented in the concluding section

## 3.3 Interface-type modes

*x = 0*, I examine the possibility of their existence. Their functional character is exponential (or hyperbolic) everywhere and they evidently exist at the limiting case

*s = t = 0,*where they reduce to standard SPP waves. We start examining therefore the case

*s*≠

*t*≠

*0*in the range:The calculation procedure for this case will follow that of Eqs. (5)-(7) above, rendering instead of (8), the following modal equation for the propagation constant

*β*:This last equation cannot be fulfilled in the case:

*s ≠ t*by any real value of

*β*. Explicitly, the conclusion is that the waveguides with anti-symmetric disposition of refractive indices do not support interface modes for cores of unequal widths (

^{2}> k_{0}^{2}n_{g}^{2}*s ≠ t*). In totally antisymmetric case of:

*s = t*, a continuum of surface modes is supported again, this time without an upper limit in the value of the propagation constant

*β*. The structure of such a mode is visualized in Fig. 4 .

### 3.4 Power flow

*m*slab:From observing this last expression it is apparent that the sign of power flow depends on both the propagation constant

*β*of the mode and the permeability constant

*μ*of the specific slab. It is obvious therefore that the totally antisymmetric structure will have zero total power flow in the

_{m}*z*direction regardless of the width of the slab and the frequency

*ω*of the propagating light. For the more general case where

*s ≠ t*, the net propagating power reduces simply to:In that case, the total normalized net power is a function of the difference (

*t- s)*which determines also the direction of the power flow. The power component

*P*does not depend on the difference (

_{z}^{tot}*t- s)*in such a simple way as suggested by Eq. (15) since the width difference (

*t – s)*also enters implicitly in the value of

*β*for different modes and the normalization constant

*A*

_{1}.

*P*is displayed in Fig. 5 , together with the dispersion curves of the modes. The normalization is done with respect of the sum of the absolute powers flowing in all the four media.

_{z}^{tot}## 4. Connection between the geometrical and modal pictures

*x = t, -s.*This means that the phase jumps at reflection adds to the phase accumulated by the rays propagating at the upper and lower part of the waveguides which are mutually opposite in sign, and the conclusion is that they mutually cancel within a single loop. The clad parameters on the other side do affect the net power flow via the normalization constants. For the totally antisymmetric structures the sum of the accumulated phase is identically zero for all ray traces, and the transversal resonance condition is automatically fulfilled. This situation will hold also for general antisymmetrically mirrored multilayered structures and even for the cases where the index is graded within each of the half-planes delimited by x = 0.

## 5. Concluding remarks

*s ≠ t*), the structure allows the control of the phase and group velocities of guided radiation by changing the structural parameters. Modes within this configuration have dispersion relations which are equivalent to modes in a dielectric slab surrounded by perfect conductors. Unique is the case of totally antisymmetric structures: Here the geometrical ray paths are closed, hinting the possibility of localized modes in perfectly linear media. The ray model also shows a quasi-perfect imaging effect for off-axis objects in that case. Following an electro-magnetic approach, the modes of the totally anti-symmetric case seem exceptional in the sense that their spectrum is continuous. This property can also be translated to the frequency domain: A specific anti-symmetric structure will support localized modes for a wider frequency range as long as the chromatic dispersion doesn’t destroy the asymmetry of the refractive indices. In a pictorial description one may describe this effect as “white light localization”. Structures containing both positive and negative index analyzed in the previous literature showed already the possibility of slowing-down the group velocity of light attain eventually conditions of reducing this velocity down to zero, but this was shown to happen only for a definite frequency. Slow light has been advocated as a means of storing optical power for optical processing and buffering data. In the scheme described here, the storing would be possible for multi-mode and eventually wide-band radiation. With respect to polarization, only TE modes were described, and from duality considerations, the properties of the TM counterparts can be deduced. In a case where half of the space contains double negative permeability and permittivity materials, TE and TM modes will be degenerated in their propagation constants but have distinct modal distributions. Regarding the physical realization of these effects, actual devices trying to approach the ideal conditions required here will have necessarily deviations in the asymmetry property requirements, both electrical and geometrical. Dispersion and finite losses in the materials will also disturb the strict conditions of opposite sign in the permeability constants. The propagation behavior in these non-ideal cases is worth investigating. The mathematical properties of the set of eigenmodes of the antisymmetric structure are also unusual and deserve further investigation: they possess a continuous spectrum and are evidently a non-orthogonal, yet normalizable set. The structures studied here are actually special examples of more a class of general

*anti-symmetrically mirrored waveguided structures*, which may be composed of an arbitrarily large number of guiding layers and even be of graded-index nature.

## References and links

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

2. | V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

3. | S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. |

4. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. |

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

6. | K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B |

7. | Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. |

8. | J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express |

9. | J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. |

10. | Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. |

11. | K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature |

12. | A. Alù and N.Engheta, “Guided Modes in a Waveguide Filled With a Pair of Single-Negative (SNG), Double-Negative (DNG), and/or Double-Positive (DPS) Layers,” IEEE Trans. MTT, V.52, No. 1, 2004. |

13. | A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. |

14. | G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. |

15. | M. J. Adams, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 6, 2010

Revised Manuscript: August 12, 2010

Manuscript Accepted: August 13, 2010

Published: September 15, 2010

**Citation**

Shlomo Ruschin, "Guiding effects in waveguides with anti-symmetric refractive index layouts," Opt. Express **18**, 20681-20689 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20681

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### References

- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- V. R. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]
- S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007). [CrossRef]
- I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 057602 (2003). [CrossRef] [PubMed]
- K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006). [CrossRef]
- Z. H. Wang, Z. Y. Xiao, and S. P. Li, “Guided modes in slab waveguides with a left-handed material cover or substrate,” Opt. Commun. 281(4), 607–613 (2008). [CrossRef]
- J. He, Y. Jin, Z. Hong, and S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008). [CrossRef] [PubMed]
- J. He and S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006). [CrossRef]
- Y. He, Z. Cao, and Q. Shen, “Guided optical modes in asymmetric left-handed waveguides,” Opt. Commun. 245(1-6), 125–135 (2005). [CrossRef]
- K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow' storage of light in metamaterials,” Nature 450(7168), 397–401 (2007). [CrossRef] [PubMed]
- A. Alù and N.Engheta, “Guided Modes in a Waveguide Filled With a Pair of Single-Negative (SNG), Double-Negative (DNG), and/or Double-Positive (DPS) Layers,” IEEE Trans. MTT, V.52, No. 1, 2004.
- A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Müllera, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997). [CrossRef]
- G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007). [CrossRef] [PubMed]
- M. J. Adams, An Introduction to optical waveguides, (Wiley, 1981)

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