OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 20 — Oct. 6, 2003
  • pp: 2502–2510
« Show journal navigation

Guided modes in channel waveguides with a negative index of refraction

A. C. Peacock and N. G. R. Broderick  »View Author Affiliations


Optics Express, Vol. 11, Issue 20, pp. 2502-2510 (2003)
http://dx.doi.org/10.1364/OE.11.002502


View Full Text Article

Acrobat PDF (320 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Negative index materials (NIM) offer a unique possibility to extend the experimental domain and investigate novel physical phenomena. Such materials, which possess simultaneously negative values of the dielectric permittivity 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].

Initial experimental investigations of negative refraction phenomena have been conducted in the microwave regime where the fabrication of such composite materials is possible [4, 5]. These experiments confirmed that the electromagnetic waves behave as predicted by the theory. It is, however, unlikely that these composite materials will scale to optical frequencies and instead photonic crystals (PC) have often been suggested as an alternative to extend the effects of negative refraction into the optical regime [6]. Indeed, in 2000 Notomi performed a detailed theoretical and numerical investigation of light propagation in a PC which showed that as the effective refractive index is determined by the photonic band structure it can in fact be less than unity or even negative [7]. Although these PCs may have positive 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.

In this paper, we numerically investigate the properties of the guided modes in a channel waveguide with a negative index (negative 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

E˜(x,y,z,t)=E(x,y)ei(ωtβz),
(1)
H˜(x,y,z,t)=H(x,y)ei(ωtβz),
(2)

where 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.

Fig. 1. Waveguide geometry and parameters.

To obtain a qualitative understanding of the these modes, our first analysis is based on Marcatili’s method for 2D optical waveguides [12

12. E. A. Marcatili, “Dieclectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

]. We choose the electric field to be polarized in the x direction, i.e., the Epqx mode, which has Ex and Hy 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

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

]:

2Hyx2+2Hyy2+(ω2c2εiμiβ)Hy=0,
(3)

Ex=ωμ0μiβHy+1ωε0εiβ2Hyx2.
(4)

We then assume that the electric and magnetic fields are confined to the core so that they decay exponentially in the cladding and are negligible in the shaded regions of Fig. 1. Solving for the field in each of the unshaded regions and applying the appropriate boundary conditions at the interfaces we obtain the following dispersion equations :

γxL=ε1ε2kxLtan(kxL(p1)π2),
(5)
γyL=kyLtan(kyL(q1)π2),
(6)

γx2=ω2c2(ε2μ2ε1μ1)kx2,
(7)
γy2=ω2c2(ε2μ2ε1μ1)ky2.
(8)

The propagation constant β is then obtained from,

β2=ω2c2ε2μ2(kx2+ky2).
(9)

Figures 2 and 3 illustrate some of the important properties of negative index channel waveguides. Firstly, the guided modes can only be supported in high-index waveguides (i.e., 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 ky , 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, H1,1y, does not exist; (ii) for a given width L, solutions of γxL associated with the first order H2,1y mode only exist for a particular range of ω greater than a critical value; (iii) for certain modes, as the x solutions in Fig. 2 decrease monotonically with kxL 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).

To investigate the frequency dispersion of the guided waves we need to consider the frequency dependence of both ε 2 and µ 2 in Eq. (9). Although the specific form of the refractive index of a photonic crystal depends of the band structure [7], here we will simply use the forms proposed by Veselago [1] as employed by other authors [4, 6, 11]:

Fig. 2. Typical solutions for the x (top) and y (bottom) components of the guided modes of a negative index channel waveguide. 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).
εi(ω)=1ωp,i2ω2,μi(ω)=1Fω2ω2ω0,i2.
(10)

Here ωp,i are the electronic plasma frequencies, ω0,i are the magnetic resonance frequencies and F is a constant dependent on the material structure. Based on earlier analysis [4, 11], we choose ωp,2 /2π=10GHz, ω 0,2/2π=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,1/2π=2GHz and ω 0,1/2π=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 H3,1y modes of Fig. 3, as found from Eq. (9), are plotted in Fig. 4(a) with 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 νc 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, νg =1/β 1=dω/dβ, Fig. 4(b) shows that whilst for the weakly guided mode νg is positive, for the strongly guided mode vg is negative, corresponding to backward wave propagation [15

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

]. In both cases, however, as the frequency approaches νc, vg approaches zero so that the mode propagation will be slowed considerably. Thus this waveguide offers an alternative to other methods of generating “slow” light (νgc), “fast” light (νg 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 νg , 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]

].

Fig. 3. Examples of the mode profiles for the solutions in Fig. 2. Top row : H2,1y solutions with L=0.1cm corresponding to intersections (a,α) and (b,β). Middle row : H3,1y solutions with L=1cm corresponding to intersections (c,δ) and (d,δ). Bottom row : H3,2y solutions with L=2cm corresponding to intersections (c,η) and (d,η).

The group velocity dispersion (GVD) of the guided modes is calculated via : β 2=d2 β/dω 2. As seen in Fig. 4(c), as the frequency approaches ν c (the region of low νg ) 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]

]. For these particular modes, the dispersion can be around 7 orders of magnitude larger than that of conventional materials such as silica fibers (20ps2km-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]

].

We have also calculated the energy flux of the guided modes in Fig. 4 which is characterized by the z component of the Poynting vector : S=Re[E×H*] [13

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

]. 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,

Pcore=coreSzdxdy,Pclad=cladSzdxdy.
(11)

For both modes we find that the power flux inside the core is opposite to that in the cladding (see middle row of Fig. 3). However, on calculating the normalized energy flux defined as P=(Pcore +Pclad )/(|Pcore |+|Pclad |) [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 νg . 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 νc ), 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.

Fig. 4. (a) Propagation constant, (b) group velocity, and (c) group velocity dispersion parameter of the H3,1y solutions from the middle row of Fig. 3. The solid and dashed lines correspond to the strongly (c,δ) and weakly (d,δ) localized modes, respectively.
Fig. 5. Normalized energy flux as calculated for the H3,1y solutions of Figs. 3 and 4.

Fig. 6. (a) Propagation constant and (b) normalized energy flux of the H3,1y solutions from the middle row of Fig. 3, as functions of the waveguide width L. The solid and dashed lines correspond to the strongly (c,δ) and weakly (d,δ) localized modes, respectively.

3. Discussion and conclusions

Although the analysis described here pertains specifically to channel waveguides with a negative index (negative 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]

], it should be possible to design fibers in the future that have photonic band structures such that the core has an effective index which is negative. This is of particular relevance to devices requiring large nonlinearities as these are readily obtained in small core PCFs [19

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. 10, 509–514 (1968). [CrossRef]

2.

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]

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. 47, 2075–2084, (1999). [CrossRef]

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. 84, 4184–4187, (2000). [CrossRef] [PubMed]

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. 78, 489–491, (2001). [CrossRef]

6.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79, (2001). [CrossRef] [PubMed]

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 62, 10696 (2000). [CrossRef]

8.

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]

9.

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]

10.

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]

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]

12.

E. A. Marcatili, “Dieclectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

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. 8, 383–385 (1983). [CrossRef] [PubMed]

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 19, 722–731 (2002). [CrossRef]

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]

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]

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


Sort:  Journal  |  Reset  

References

  1. V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Sov. Phys. Usp. 10, 509�??514 (1968). [CrossRef]
  2. 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]
  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. 47, 2075�??2084, (1999). [CrossRef]
  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. 84, 4184�??4187, (2000). [CrossRef] [PubMed]
  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. 78, 489�??491, (2001). [CrossRef]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77�??79, (2001). [CrossRef] [PubMed]
  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 62, 10696 (2000). [CrossRef]
  8. 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]
  9. 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]
  10. 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]
  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]
  12. E. A. Marcatili, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell Syst. Tech. J. 48, 2071�??2102 (1969).
  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. 8, 383�??385 (1983). [CrossRef] [PubMed]
  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 19, 722�??731 (2002). [CrossRef]
  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]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited