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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 20681–20689
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Guiding effects in waveguides with anti-symmetric refractive index layouts

Shlomo Ruschin  »View Author Affiliations


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

Negative refractive index materials (NIM) have being continuously promoting investigation over past years due to a host of remarkable effects, starting with the perfect imaging property pointed out by Pendry [‎1

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

]. Extensive experimental work is being carried at the present in order to artificially synthesize meta-materials with programmable optical parameters [2

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

], and advances in micro-fabrication are pushing the realization of these materials at shorter wavelengths down to the near infra-red and visible ranges [3

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

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]

]. The source of the outstanding optical effects reported in structures containing these novel materials relies on the drastic changes in the transmission and reflection properties taking place at the interfaces between negative and positive index materials, as compared with interfaces where the two materials have permeabilities of the same sign. Waveguiding structures partially possessing negative magnetic permeabilities or electric permittivities have attracted therefore much consideration [5

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

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]

]. Special attention is being invested nowadays on the applicability of waveguides containing negative index materials for slowing-down the speed of the propagating guiding light. Structures have been proposed to slow down modes down to zero group velocity and eventually immobilize or trap optical power by suitable tapering of the waveguide's thickness [9

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

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]

]. Configurations composed by pairs of planar waveguides with materials with refractive indices of different signs surrounded by perfectly conducting plates were studied in ‎ [12

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.

] and also allow controlling and slowing down the modal group velocity. In the present article alternative structures for controlling and stopping the speed of light are proposed.

In the following sections, waveguided modes are investigated in thin films located at the transition between a positive index material and a negative index material having the same index absolute value but opposite sign (see Fig. 1
Fig. 1 The basic layer disposition of material composition and thicknesses are shown for waveguide structures with opposite-signed refractive-index claddings and cores. Figure (a), (s = t) shows closed-loop ray trajectories, suggesting the possibility of localization of power. Figure (b) (s = t) shows the imaging property of this configuration. This scheme shows that rays emitted backwards also contribute to the image. In Figure (c) Quasi-circulating ray paths for core layers of different thicknesses are seen (s ≠ t). The net ray velocity can be controlled here by the parameter s-t.
). The thin films interfacing between the two materials have a raised absolute value of the refractive index providing support for propagating waveguided modes. Explicitly the electric and magnetic susceptibilities will be designated by [(μc, εc); (μg, εg); (−μg, −εg); (−μc, −εc)], and obey:

μc,μg,εc,εg>0εgμg>εcμc
(1)

The positive and negative interfacing films thicknesses are 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]

], the image here is expected to be inverted with respect to original one. The section following presents a modal analysis of the structure. The totally anti-symmetric case (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 > ko ng. 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 (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

Figure 1 shows ray traces for the totally antisymmetric arrangement (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 θ1 = −θ−1.

In Fig. 1(a) (t = s) rays are traced for the case of rays emerging at the x = 0 interface. The paths close back at the point of origin O, without power loss for any launching angle below the critical angle θcr = cos−1(|nc/ng|). The ray will continue to re-circulate on that closed loop creating a sort of ring-cavity. Moreover, it is straightforward to realize that the total phase accumulated in a round trip is zero, so that the cavity will resonate for any wavelength as long as the anti-symmetric scheme is preserved. Such a cavity (“white light cavity”) was proposed more than 20 years ago [ 13

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. 134(1-6), 431–439 (1997). [CrossRef]

] and is being object of considerably research nowadays [ 14

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. 99(13), 133601 (2007). [CrossRef] [PubMed]

]. The frequency regime obeying the required condition of oppositely-signed refractive indices is however always restricted by basic dispersion laws. The total zero-phase accumulation is due to the opposite sign of the refractive indices at the lower and upper portions of the path. Moreover, it is not disturbed by the Goos-Hänschen effect at the upper and lower interfaces which cancel mutually. The totally anti-symmetric sign-inverted waveguide of Fig. 1(a) presents an exceptional case where closed-ray loops can be achieved with one-dimensional (layered) structuring. Refs [ 9

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]

] and [11

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]

] also reported rays in closed loops in a tapered layout, but the condition there is based by the mutual cancellation of phases induced by propagation in the guiding layer and that induced by the Goos-Hänschen effect. On a field-type picture, one would talk here about 2D localization of light excitation in a linear 1D structure. The totally anti-symmetric scheme also displays an imaging property for off-axis objects as seen in Fig. 1(b). This property shares similarity to the well-known perfect-imaging 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]

], but distinctly to that case, the image here is inverted. Furthermore, in the present case also backwards-directed rays will contribute to the image, as seen by the dotted ray in Fig. 1(b). The imaging will however not be perfect here since power is lost due to finite transmission at angles beyond the critical value.

3. Electromagnetic modal analysis

3.1 Cores with different thicknesses (s ≠ t)

3.2 Totally anti-symmetric waveguide (t = s)

3.3 Interface-type modes

3.4 Power flow

The power flow in the configurations under study can also be calculated straightforwardly by integrating the Poynting vector across the different sections of the waveguide, explicitly for the m slab:
Pzm=12dmdm+1Ey(x)Hx(x)dx=β2μmωdmdm+1[Ey(x)]2dx
(14)
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 μm of the specific slab. It is obvious therefore that the totally antisymmetric structure will have zero total power flow in the 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:
Pztot=β4μgωA12(ts)
(15)
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 Pztot does not depend on the difference (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. Pztot is displayed in Fig. 5
Fig. 5 Normalized effective indices (dashed lines) and normalized power flow in the z-direction as a function of x = (t – s), for the first two modes in a waveguide structure with parameters ng = 2 and nc = 1.7. The total width, (t + s) is kept constant at a value of 2. The inset shows an enlargement of the threshold region of the TE1 mode to emphasize that the power flow is a continuous function of (t – s)
, 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.

4. Connection between the geometrical and modal pictures

5. Concluding remarks

References and links

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]

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 73(8), 085104 (2006). [CrossRef]

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. 281(4), 607–613 (2008). [CrossRef]

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 16(15), 11077–11082 (2008). [CrossRef] [PubMed]

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]

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]

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]

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. 134(1-6), 431–439 (1997). [CrossRef]

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. 99(13), 133601 (2007). [CrossRef] [PubMed]

15.

M. J. Adams, An Introduction to optical waveguides, (Wiley, 1981)

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

  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]
  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 73(8), 085104 (2006). [CrossRef]
  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. 281(4), 607–613 (2008). [CrossRef]
  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 16(15), 11077–11082 (2008). [CrossRef] [PubMed]
  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]
  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]
  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]
  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. 134(1-6), 431–439 (1997). [CrossRef]
  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. 99(13), 133601 (2007). [CrossRef] [PubMed]
  15. M. J. Adams, An Introduction to optical waveguides, (Wiley, 1981)

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