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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 12 — Jun. 7, 2010
  • pp: 12213–12225
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Negative index modes in surface plasmon waveguides: a study of the relations between lossless and lossy cases

Yuan Zhang, Xuejin Zhang, Ting Mei, and Michael Fiddy  »View Author Affiliations


Optics Express, Vol. 18, Issue 12, pp. 12213-12225 (2010)
http://dx.doi.org/10.1364/OE.18.012213


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Abstract

Surface plasmon modes in structures of metal-insulator-metal (MIM), insulator-insulator-metal (IIM) and insulator-metal-insulator (IMI) are studied theoretically for both lossless and lossy cases. Causality dictates which solutions of Maxwell’s equations we accept for these structures. We find that for both lossless and lossy cases, the negative index modes and positive index modes are independent and should be treated separately. For the lossless case, our results differ from some published papers. By studying in detail the lossy case, we demonstrate how the curves should look like.

© 2010 OSA

1. Introduction

Negative index materials (NIMs), which have the significant property of ‘negative refraction’, have attracted much attention, and promise to be very important for the technology of light control. So far, NIMs are all artificial structures not found in nature. The first concept of NIMs was proposed by Veselago [1

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

] and experimentally verified by Shelby et al at microwave frequencies by using periodic arrays of metal split ring resonators (SRRs) and dipoles [2

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

]. For the purpose of light control, these artificial structures have to be made smaller and smaller in scale in order to work at visible or infrared wavelengths [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]

8

8. M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009). [CrossRef] [PubMed]

], but it is increasingly difficult to make these elements smaller because of fabrication limitations. On the other hand, metal-dielectric (or semiconductor-dielectric) layered structures are also known to exhibit negative refraction for TM polarized waves [9

9. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]

11

11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

], because the permittivity of metal (and some kinds of semiconductor) layers is negative at visible and/or infrared frequencies. Such layered structures are useful, for example for super-resolution imaging systems [10

10. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]

12

12. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

] which may eliminate the diffraction limit associated with normal optical imaging instruments. In these layered structures, the surface wave at the interface of the metal and dielectric layers makes important contribution. More flexible in fabrication, are surface wave-based planar structures, such as some kind of planar waveguide containing metal layers which have attracted much interest [12

12. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

15

15. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]

]. Smolyaninov et al [12

12. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

] use a single interface of metal and dielectric to produce negative refraction and verified the theory of the hyperlens [11

11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

], while arguments exist such as Ref. 16

16. A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]

which indicates negative refraction cannot exist at single metal-dielectric interface. Later, Lezec et al experimentally showed negative refraction at visible frequencies [15

15. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]

] by using the surface mode of MIM (metal-insulator-metal) structures, but the phenomenon responsible was not very clear; Dionne et al solved the dispersion relations for such three layered structures to explain the phenomenon [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

]. As indicated in Ref. [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

], with specific configurations, three layer systems (MIM, IIM and IMI) can show a negative refraction property, although only the MIM structure supports a single propagating negative index mode. Taking the MIM structure for example, Shin et al explain in Ref. [13

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]

]. that negative refraction is caused by the negative slope (i.e. negative group velocity) of the second band of the dispersion curves (see Fig. 1 in Ref. [13

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]

].). In Refs. [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

]. and [18

18. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465. [CrossRef] [PubMed]

], the authors notice that when considering the loss in the metal layers, the dispersion curves should be obtained by ensuring that the imaginary parts of the wave vectors were positive; this explanation is reasonable, but there still results in some ambiguity and contradictions between the dispersion curves for the lossless and lossy cases in these papers.

2. Causality and equations

The structures of three-layer systems (MIM, IIM and IMI) are setup as shown in Fig. 1(a)
Fig. 1 (a) Schematic of an incident wave impinging on a three-layer structure from a normal medium. The direction of the incident wave is confined to the x-y plane. (b) and (c) are schematics where (b) shows the correct phase and energy velocities with negative refraction for normal and oblique incidence, while case (c) violates causality (see direction of energy flow).
with the thickness of the middle layer being d, and the substrate and cover are infinite. Since negative refraction is caused by surface plasmon polaritons, and the surface wave is always bounded to interfaces, we can treat these structures as plasmon waveguides. Given the symmetry of the structure, the field distribution is the same in the x and y directions, thus we can confine our discussion to the x-z plane, and consider surface modes of the TM polarization (i.e. only field components Ex, Ez and Hy are nonzero).

In many published papers, the dispersion relations for the three-layer lossless case are always shown like those black curves in Fig. 2
Fig. 2 Dispersion relations of lossless Drude model for three types of layered structures: (a) MIM, (b) IIM (here we use air as the cover layer), and (c) IMI. The black curves are typical of those published by others. The red curves are the branches consistent with causality, which we discuss in this paper.
(see for example those in Refs. [13

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]

,14

14. M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007). [CrossRef]

,17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

,18

18. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465. [CrossRef] [PubMed]

,20

20. H. Reather, Surface plasmon (Springer, Berlin, 1988), Chap.2.

]). We can see that these curves show a negative slope in some frequency ranges, which corresponds to a negative group velocity, indicating the phase and energy velocities are anti-parallel. When an incident plane wave passes through the interface between a normal medium and such a structure, this anti-parallel property corresponds to negative index and hence refraction [1

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

].

When negative refraction occurs, the causality condition must be satisfied [16

16. A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]

,17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

,19

19. I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007). [CrossRef]

]. When a plane wave is incident into a NIM from a normal medium, the energy flow should be towards the interface from the incident media and then away from the interface on the other side (the refractive side) and in this way, energy will not accumulate at the interface. On the other hand, if the wave vector is complex in the negative index region (caused by loss in the material), the imaginary part should point in the positive direction and lead to field decay as the wave propagates rather than enhancement which will violate causality. When negative refraction occurs and considering this anti-parallel property associated with negative index, we plot the possible directions of phase and energy flow in Fig. 1(b) and 1(c). Based on the principle of causality, we can readily see that the case in Fig. 1(c) is impossible. At the interface, there is an energy accumulation, and the conservation of transverse wave vector does not hold. For the case in Fig. 1(b), the directions of the phase and energy of the refractive wave are anti-parallel and causality is maintained; the energy flow goes into the interface and leaves from the other side, and the transverse wave vectors are conserved. However, Fig. 1(b) indicates that the phase directions will be negative which will lead to different dispersion curves (the red curves in Fig. 2) from those traditionally taken (see black curves with negative slope). After verifying causality, we study negative refraction in three-layer structures and find that the relations between the lossless and lossy cases correct the black curves in Fig. 2.

For the three types of three-layer system in Fig. 2, we suppose the wave propagation to be in the x direction and we just need to study the ω-kx dispersion curves, where ω is the angular frequency of the incident wave and kx is the x component of wave vector in the structure. We can find the dispersion relation for such structures from Eq. (1) [21

21. S. A. Maier, Plasmonics: fundamentals and applications (Springer, 2007), Chap. 2.

]:

e2k1d=k1/ε1+k2/ε2k1/ε1k2/ε2k1/ε1+k3/ε3k1/ε1k3/ε3
(1)

In the three distinct areas (numbered in Fig. 2), the wave vectors ki (i = 1, 2, 3) should fulfill:
ki2=kx2εik02
(2)
Where, k 0 is the wave vector in vacuum, and kx is the transverse wave vector which is conserved throughout the three distinct regions. εi is the permittivity in each area, and here we use εd = 4 (such as Si3N4) as the permittivity of the dielectric layers. For the metal, we first use the lossy Drude model to describe the permittivity, which takes the form:

εmetal=εωp2ω2+iωΓ
(3)

Taking ε = 1, ωp = 9eV, and Г = 0.2687 eV, we approximately represent the permittivity of silver). We note that the value of Г is a little larger than usual to make the dispersion curves clear but this does not affect the results. Taking a smaller value, the curves of the lossless and lossy cases are too close to each other to be able to distinguish them. The materials used here are non-magnetic, thus permeability μi = 1 (i = 1, 2, 3).

The time averaged Poynting vector ‹S› is also derived to study the energy transport in these layered structures. As we mainly study the bound surface modes, we just need to study the x component of ‹S› (represented by ‹Sx›), since ‹Sx› can be used to describe the propagation of energy flow in the structures. As in Eq. (4)Sx› is composed of three parts, the first one (‹S1x›) represents the energy flow in layer 1, and the last two (‹S2x› and ‹S3x›) are for the substrate and cover, respectively:
Sx=d/2+d/2S1xdz+d/2S2xdz++d/2+S3xdz
(4)
With the components:
d/2+d/2S1xdz=kx2ωε0ε1|eikxx|2d/2+d/2(|C|2|ek1z|2+|D|2|ek1z|2+CD*ek1z(ek1z)*+DC*ek1z(ek1z)*)dz
d/2S2xdz=   kx2ωε0ε2|B|2|eikxx|2d/2|ek2z|2dz
+d/2+S3xdz=kx2ωε0ε3|A|2|eikxx|2+a+|ek3z|2dz
Where the coefficients have the following form:

{A=(ek1a(k3ε1+ε3k1)(k3ε1ε3k1)ek1a)ek3aCB=(ek1a+(k2ε1+ε2k1)(k2ε1+ε2k1)ek1a)ek2aCD=(k3ε1+ε3k1)(k3ε1ε3k1)e2k1aC
(5)

We solve the nonlinear Eq. (1) using numerical methods to get the ω-kx curves. The mode refractive index is decided by:

n=kx/k0
(6)

Thus, negative (positive) kx indicates negative (positive) index in the structures. Generally, for a certain frequency, there is always an infinite number of solutions that solve Eq. (1) because of the periodic character of the formula and both the real and imaginary parts of kx can be positive or negative. Mathematically, these solutions all satisfy Eq. (1), thus we must choose the solutions which are physically correct. Ref. [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

]. uses the imaginary part of kx, where imag(kx) must always be positive, to decide which solution is reasonable. This is reliable when the metal is lossy and kx always has a non-zero imaginary part. In contrast, for the lossless case imag(kx) of the propagating mode is always zero of course, in which case this approach fails leading to a contradiction between Fig. 1 and Figs. 2 to 4 in Ref. [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

]. It is therefore necessary to study the implicit relationship between the lossless and lossy cases in order to establish how the negative index property originates. To do this we calculate the time averaged Poynting vector ‹Sx› to study the energy flow directions, which is more reliable and in compliance with causality. From this we can find out the relationship between lossless and lossy cases, and clearly see how negative refraction arises in these layered structures.

When we consider the loss in metal, the propagation constant kx is always complex. A large imaginary part of kx means heavy damping when propagating along x direction, thus we will mainly study those modes, the imaginary parts (imag(kx)) of which are small or comparable with the real part (real(kx)).

3. The MIM case

For the MIM structure, the middle layer should be thin enough to obtain the phenomenon of negative refraction [13

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]

]. Here we take d = 20nm, and ε1 = εmetal , ε2 = ε3 = 4.

Based on the discussion of symmetric and anti-symmetric modes, for both lossless and lossy cases, we know that the positive index property is dominant at lower frequencies, while the negative index property is dominant at higher frequencies, which is similar to that described in Ref. [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

].

There is an interesting area that should be further discussed for the anti-symmetric mode. In Fig. 4(a), although ω = ωsp asymptotically, as real(kx) increases, the black solid and dashed curves do not tend to it directly but first cross ω = ωsp and then tend to the asymptote. The ω-kx and ω-‹Sx› curves are shown in Fig. 5(a)
Fig. 5 Details of the dispersion curves of the lossless case shown in Fig. 4. Solid curves are for the propagating modes, and solutions which violate causality are shown as dashed curves. The dash-dot curves indicate non-propagating modes (the imaginary part shown in the same picture in blue). The curves are consistent in color.
and 5(b), respectively. Based on causality, the physical solutions are plotted in red solid (for negative index property) and light green solid (for positive index property) curves, and the dashed curves are pseudo or non-physical solutions (corresponding negative ‹Sx›) while the dash-dot curves are non-propagating modes like that in Fig. 4. This further indicates that negative index modes and positive modes should be treated separately, which also applies to the IIM and IMI cases.

4. The IIM and IMI cases

In this section we discuss the IIM and IMI cases. For the lossless case and when negative refraction occurs, the dispersion curves for IIM and IMI structures can always be plotted like those black curves in Fig. 2(b) and 2(c) (see also Fig. 1 in Ref [17

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

], Fig. 2.23 and 2.25 in Ref [20

20. H. Reather, Surface plasmon (Springer, Berlin, 1988), Chap.2.

].). These curves always appear as one continuous and smooth curve with parts having both positive and negative slopes, which are different from those for the MIM structure. Note that for the MIM structure, when the dielectric layer is thin enough, the positive and negative slope bands are always separated completely by the line ωsp=ωp/(1+εd) as in Fig. 2(a). For our study, we argue that these curves for lossless IIM and IMI cases are ambiguous and lead to self-contradictory interpretations between the lossless and lossy cases. In the following paragraph of this section, we show how they can be represented and indicate that as a result of our correction there is consistency between lossless and lossy cases.

For the IIM case, we use air as the cover medium (ε2 = 1) and ε1 = 4, ε3 = εmetal, and let d = 10nm since the negative index property will be more evident when d is thin. For the lossless case, the ω-kx curves are shown in Fig. 6(a)
Fig. 6 Dispersion relations of the IIM structure for both lossless and lossy cases. (a) and (b) show the real and imaginary parts of the wave vector, respectively; (c) shows the ω-Sx› curves. For clarity, we do not show the curves of the lossless case when kx is large. The curves are consistent in color.
and for clarity, we do not show the curves in same figure when kx is large, instead, we will show these details in Fig. 7
Fig. 7 Details of the dispersion curves for the lossless case shown in Fig. 6. Solid curves indicate the propagating modes, and solutions which violate causality are shown by dashed curves. The dash-dot curves indicate the non-propagating modes (the imaginary part shown in the same picture in blue). Based on causality, we plot positive ‹Sx› for the propagating modes in (b). The curves are consistent in color.
. The red (blue) solid curves illustrate negative (positive) propagating mode directions; the dashed curves are for pseudo-solutions. We can obtain two pairs of non-propagating solutions (the dash-dot curves) for higher and lower frequencies (see the insets for details), which are axially symmetric and have large imag(kx) and ‹Sx› = 0. Similar to the anti-symmetric mode case in the MIM structure, these non-propagating curves will be asymptotes for curves of the lossy case and the curves for different modes also lie on different sides of the dash-dot curves. Based on the value of imag(kx), we find that the IIM structure shows positive index properties at lower frequencies, both positive and negative index properties over the middle frequency range (as imag(kx) is smaller in this frequency range), while decaying at higher frequencies. For propagating modes, the ω-kx curves are no longer continuous, which means the negative index mode and the positive index mode are never coincident and therefore should be treated separately. For the lossy case, we find physical solutions based on causality and plot them in light green (negative index mode) and light blue (positive index mode). We can see that the curves for the lossy case have those of the lossless case as asymptotes. The orange curve is a high order solution of the lossy case to further prove that the dash-dot curves are asymptotes of the lossy case. At these higher frequencies, there is another high order non-propagating mode of the lossless case which will be asymptotic to the orange curve, but we do not plot them here as it is not pertinent to this paper.

For larger kx and the lossless case, details are shown in Fig. 7. The situation is similar to the anti-symmetric mode of the MIM structure: the curves also tend asymptotically to ω = ωsp, but first dip down, cross and then rise up to the asymptote. The right branches are consistent with causality in Fig. 7(b) and here we just plot the curves with ‹Sx›>0 and do not show the curves with negative value of ‹Sx› which correspond to the dashed curves and are axially symmetric with the solid curves in (b).

For the IMI case and because of the symmetry of the structure, there exist two modes, the symmetric and anti-symmetric mode (we still use Hy for definition as in MIM structure). The anti-symmetric mode shows no negative index property and imag(kx) is always large and so we just study the symmetric mode here. We set ε1 = εmetal, ε2 = ε3 = 4 and d = 10 nm. The ω-kx curves of the IMI system are very similar to those of IIM structure, and we just plot the physical curves in Fig. 8
Fig. 8 The physically meaningful dispersion curves of the IMI structure, all curves corresponding to causal solutions and the pseudo solutions are not shown. The details of large real(kx) are shown in the inset upper left in (a). All the curves are consistent in color.
in which the branches are picked based on consistency with causality and the case for large kx is shown in the upper left inset. Similarly, the negative and positive index modes should be treated separately for both lossless and lossy cases. For the lossless case, the red curve indicates the propagating negative index mode and the blue one is for positive; the dash-dot curves represent the non-propagating modes. For the lossy case, the light green curve represents the negative index mode and light blue represents positive and since the curves are very close to each other, the upper right inset shows more details. Considering the imaginary parts of kx for which a smaller value corresponds to a longer propagation length, we can see that the IMI structure shows negative and positive index properties simultaneously in certain frequency ranges.

5. The case of actual metal data

6. Conclusion

In this paper, we have studied the dispersion relations of MIM, IIM and IMI structures, and plot the ω-kx and ω-Sx› curves for both lossless and lossy cases. We initially used the lossy Drude model for the permittivity of metal layers, and then we used real data for silver for verification. The results are in agreement for both cases. Using the principle of causality and in particular that the energy flow is along the positive x direction, we were able to pick out the correct branches corresponding to solutions with physical meaning from the multiple mathematical solutions of Maxwell’s equations for these structures. We showed the relationship between the dispersion curves for the lossless and lossy cases, and explained how the curves vary as the loss increases. Based on this, for both the lossless and lossy cases, we see that the branches of negative and positive index modes are independent of each other, and should be treated separately. In particular for the lossless case, our results are different from previously published results. Real(kx) of the propagating modes, where negative refraction occurs, will be negative and for IIM and IMI structures, the curves will not be continuous anymore; this is illustrated by the red curves in Fig. 2 for clarity. This result explains previously published dispersion curves for propagating modes were confused with non-physical modes. To the best of our knowledge, there are no publications showing such physically reasonable and self-consistent results regarding the dispersion relations of MIM, IIM, and IMI structures for both the lossy and lossless cases when negative refraction occurs. The results presented here show evident differences from those previously published papers which are ambiguous and/or self-contradictory.

Acknowledgments

References and links

1.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

2.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

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.

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]

5.

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]

6.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]

7.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]

8.

M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009). [CrossRef] [PubMed]

9.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]

10.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]

11.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

12.

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

13.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]

14.

M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007). [CrossRef]

15.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]

16.

A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]

17.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

18.

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465. [CrossRef] [PubMed]

19.

I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007). [CrossRef]

20.

H. Reather, Surface plasmon (Springer, Berlin, 1988), Chap.2.

21.

S. A. Maier, Plasmonics: fundamentals and applications (Springer, 2007), Chap. 2.

22.

E. D. Palik, Handbook of Optical Constants in Solids (Boston, MA: Academic, 1991).

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials

ToC Category:
Optics at Surfaces

History
Original Manuscript: November 23, 2009
Revised Manuscript: March 18, 2010
Manuscript Accepted: March 18, 2010
Published: May 25, 2010

Citation
Yuan Zhang, Xuejin Zhang, Ting Mei, and Michael Fiddy, "Negative index modes in surface plasmon waveguides: a study of the relations between lossless and lossy cases," Opt. Express 18, 12213-12225 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-12-12213


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References

  1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
  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. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
  5. 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]
  6. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
  7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
  8. M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009). [CrossRef] [PubMed]
  9. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]
  10. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]
  11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]
  12. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]
  13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]
  14. M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007). [CrossRef]
  15. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]
  16. A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]
  17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001 . [CrossRef]
  18. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465 . [CrossRef] [PubMed]
  19. I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007). [CrossRef]
  20. H. Reather, Surface plasmon (Springer, Berlin, 1988), Chap.2.
  21. S. A. Maier, Plasmonics: fundamentals and applications (Springer, 2007), Chap. 2.
  22. E. D. Palik, Handbook of Optical Constants in Solids (Boston, MA: Academic, 1991).

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