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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3818–3824
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Layered metal-dielectric waveguide: subwavelength guidance, leveraged modulation sensitivity in mode index, and reversed mode ordering

Min Yan, Lars Thylén, and Min Qiu  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3818-3824 (2011)
http://dx.doi.org/10.1364/OE.19.003818


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Abstract

We describe a layered metal-dielectric waveguide, whose fundamental mode has an effective index as high as 7.35 at 1.55μm, enabling subwavelength spatial confinement. The loss is found to be reasonable in relation to the confinement. The indefinite dielectric tensor of the stratified metamaterial core generally leads to multimode operation of the waveguide, exhibiting a “reversed” mode ordering contrary to conventional waveguides. The waveguide features a strong leveraging in modal index change subject to a change of index in the dielectric layers, opening the design possibilities of very compact active electro-optic devices.

© 2011 Optical Society of America

1. Introduction

Recent progress in integrated photonics primarily exploited the silicon-air material system, which has a relatively large refractive index contrast [1

1. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006). [CrossRef]

, 2

2. L. Thylén, S. He, L. Wosinski, and D. Dai, “The Moore’s law for photonic integrated circuits,” J. Zhejiang Univ. Sci. 7, 1961–1967 (2006). [CrossRef]

]. However, one can show that at a vacuum wavelength (λ0) of 1.55μm the minimum lateral field width for a planar silicon waveguide in air is about 300nm. Thus, any attempts to increase the nanophotonics integration should surpass this value; for this purpose, it is necessary to find a successor to the current silicon nanowire technology. Such successors seem to have to rely on materials with negative ɛ, notably metals [3

3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

]. Metals offer a possibility for increasing the integration density in photonic circuits in two ways: (1) As surface-plasmon polaritons (SPPs) are guided by metal-dielectric interfaces, subwavelength diffraction-free waveguides can be realized, exemplified the slot waveguide [4

4. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82, 1158–1160 (2003). [CrossRef]

,5

5. G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007). [CrossRef]

]; (2) One can artificially engineer metal-based metamaterials with very large effective refractive indices, which can then be deployed in waveguides to achieve tight mode confinement. In the literature, the second method above has been comparatively less explored. Although there have been studies on waveguides based on metamaterials [6

6. B. T. Schwartz and R. Piestun, “Waveguiding in air by total external reflection from ultralow index metamaterials,” Appl. Phys. Lett. 85, 1–3 (2004). [CrossRef]

8

8. M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17, 14851–14864 (2009). [CrossRef] [PubMed]

], they are mostly not directly intended for the photonic integration purpose.

2. Geometry and subwavelength confinement

The layered metal-dielectric medium is schematically shown in the inset of Fig. 1(a). It is characterized by the period a and the filling fraction of the metallic layer fm = dm/a, where dm is the thickness of the metal layer. A slab waveguide can be formed when one chooses N periods of the metamaterial along y (hence height of the core h = Na) and place it in a low-index cladding (air is always considered here). One can further truncate the slab to a finite width along x (core width, or w) to obtain a 3D stripe waveguide design. In all cases wave propagation is along z. Further, to make use of the plasmonic nature of the system, we consider wave propagation in the yz plane with its magnetic field perpendicular to the plane, i.e. with the so-called transverse-magnetic (TM) polarization [13

13. This picture is only absolutely valid when the waveguide under consideration is infinitely extending along x, or of slab geometry. When the waveguide width is finite, the guided mode is always hybrid with both TM and transverse-electric (TE) field components. A plasmonic mode usually has a larger portion of TM components and is therefore a TM-like mode.

]. Silver is used in this study, whose permittivity is described by the Drude model as ɛm=1ωp2/(ω2+iωγ), with the plasma frequency ωp = 1.38 × 1016rad/s and the collision frequency γ = 3.2258 × 1013rad/s [11

11. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. Our investigations do not rest on a specific dielectric material, but in general use a permittivity value of ɛd = 9. A practical medium with index around 3 can be silicon or gallium lanthanum sulphide (GLS) [12

12. Z. L. Sámson, S.-C. Yen, K. F. MacDonald, K. Knight, S. Li, D. W. Hewak, D.-P. Tsai, and N. I. Zheludev, “Chalcogenide glasses in active plasmonics,” Phys. Status Solidi (RRL) 4, 274–276 (2010). [CrossRef]

] etc.

Fig. 1 (a) Schematic diagram of the waveguide cross-section. The blue region (air) is the cladding. Inset shows the multilayer medium in bulk. (b) Indices ( and neff) versus wavelength, and (c) Losses versus wavelength for a layered medium in bulk when examined in the MGT limit, and for the fundamental modes guided repectively by a slab and a stripe waveguides based on the layered medium.

If the period of the layered medium is infinitesimal, one can use the Maxwell-Garnet theory (MGT) to treat the stack as an effective medium with its effective permittivity tensor components obtained as [9

9. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007). [CrossRef]

, 10

10. C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35, 610–612 (2010). [CrossRef] [PubMed]

]
ɛy=ɛmɛdfdɛm+fmɛd,
(1)
and
ɛx=ɛz=fmɛm+fdɛd,
(2)
with fd = 1 – fm. In such a homogenized effective medium a natural z-propagating mode solution is a y-polarized plane wave whose wave number is solely determined by its ɛy permittivity component. In other words, in this MGT limit we can, for the concerned polarization, simply obtain the medium’s effective refractive index after homogenization as n¯=ɛy. For a stack with fm = 0.8, we calculate its as shown by the black curve in Fig. 1(b). A feature significant to this study is that, beyond the resonant wavelength ∼ 0.83μm, the medium has a rather large , valued at 7.92 at λ0 = 1.55μm. Such a value far exceeds those in natural dielectric materials. The loss of the medium at λ0 = 1.55μm is at a reasonable level of 1.46dB/μm. Here we should point out that the portion of the black curve in Fig. 1(b) to the left of the resonant wavelength (shown as dotted line) is not meaningful for the current study since over there we have negative ɛy, implying evanescent field within the layered medium.

Motivated by the finding, in the following we calculate the real part of the effective mode index neff and the loss of a stripe waveguide with h = 5.5a (5 bilayers and an extra silver layer) with a = 20nm and w = 40nm. fm = 0.8 is used. The choice of the waveguide height does not affect the main conclusion of this study. Here we simply chose this value since the height is much smaller than that of a Si nanowire waveguide (∼450nm); at the same time the stack still shows properties agreeing with the MGT prediction. The width of the waveguide is kept small such that no high-order modes with a vertical (y-oriented) nodal line would appear. Nevertheless, such a waveguide is always multimode with x-oriented nodal lines present in mode fields, with the number of modes determined in general by the number of dielectric layers in the core stack [9

9. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007). [CrossRef]

]. The fundamental mode has the properties as shown by the solid red curves in Fig. 1(b) and 1(c). COMSOL Multiphysics is used for the simulation. The curves agree reasonably well with those based on the MGT predictions. The resonant wavelength is noticeably shifted to a shorter wavelength (slightly less than 0.5μm). We also present in Fig. 1(b) and 1(c) the curves (red-dashed) for the corresponding slab structure (infinitely extending along x), calculated with an analytical transfer-matrix method which takes full account of the nanostructured core. Very close agreement between the slab and stripe waveguide results is noticed. We point out that here we have used a = 20nm. When a smaller a and more periods are used, both the neff and the loss results for the slab and stripe waveguides asymptotically approach those of the MGT results (not shown).

For the stripe waveguide, most importantly, the high neff value of the mode is retained. Given neff = 7.35 at λ0 = 1.55μm, the transverse decay constant of the mode intensity in air is 1/(2k0neff21)=17nm. Therefore a rough estimation of the mode field size is about 0.01μm2, about 20 times smaller than that of a Si nanowire with a crosssectional size of 450×250nm2 placed in air. We plot in Fig. 2 distributions of three field components and the z-component of the Poynting vector Sz for the fundamental mode at λ0 = 1.55μm. The fields plotted are the non-vanishing components for the waveguide in its slab limit. The propagation loss at this wavelength is 3.3dB/μm. Even though the propagation distance of a signal along such a guide is limited to a few micrometers, many functional devices, such as splitters, ring resonators, modulators etc, can be realized owing to the small mode size.

Fig. 2 The field patterns for the fundamental mode at 1.55μm. Domain: 300 × 300nm2. Values are in SI units.

The intriguing characteristics of the higher-order modes merits a detailed, independent discussion, which will be presented in Section 4.

3. Leveraged modulation sensitivity in mode index

Due to the highly composite nature of the waveguide, its guidance properties are sensitive to a number of parameters. Among these parameters, the dielectric filling fraction fd and the index of the dielectric medium nd are of special interest. The effects of these two parameters can be easily revealed by calculating the homogenized index of the layered medium in the MGT limit. With default parameters of ɛd = 9 and fd = 0.2, we notice that when fd is increased from 0.2 to 0.25, a blueshift of the resonance [from the original 0.83μm as in Fig. 1(b)] is observed; the corresponding shift of the () curve leads to a decrease in (from 7.915 to 6.756 at 1.55μm). When nd decreases from 3 to 2.5, the value of the medium similarly decreases, from 7.915 to 6.233 at 1.55μm.

The second sensitivity study above shows that the metamaterial exhibits a change in index over three times as large as a corresponding change in the dielectric index. Such an unusually high sensitivity of the metamaterial’s effective index to nd can be exploited for enhanced light modulation. Based on the waveguide studied in Fig. 2, we numerically calculate the change in the fundamental mode index neffneff) when nd changes from 3 to 3.1. A 0.1 change in refractive index is possible in some reported materials, e.g. GLS [12

12. Z. L. Sámson, S.-C. Yen, K. F. MacDonald, K. Knight, S. Li, D. W. Hewak, D.-P. Tsai, and N. I. Zheludev, “Chalcogenide glasses in active plasmonics,” Phys. Status Solidi (RRL) 4, 274–276 (2010). [CrossRef]

] and some other chalcogenide glass [14

14. Y. Ikuma, Y. Shoji, M. Kuwahara, X. Wang, K. Kintaka, H. Kawashima, D. Tanaka, and H. Tsuda, “Small-sized optical gate switch using ge2sb2te5 phase-change material integrated with silicon waveguide,” Electron. Lett. 46, 368–369 (2010). [CrossRef]

]. In Fig. 3, we show Δneff versus wavelength. Δneff is in general over 3 times as large as the change in nd, affirming the MGT prediction. At λ0 = 1.55μm, we have Δneff = 0.36. The characteristic lengh L of a phase modulator is determined by the length over which a light signal experiences a π phase change compared to a reference, or L=λ2Δneff. Therefore, the length of a phase modulator based on the proposed waveguide can be as short as 2.2μm. Given such a length, the difference between losses in two branches of a Mach-Zehnder modulator is about 0.58dB, which can easily be corrected by controlling the power splitting in the modulator.

Fig. 3 Difference in the effective mode index when the dielectric index changes from 3 to 3.1.

4. Reversed mode ordering

The waveguide presented in Fig. 2 has a peculiar multimode guidance property, even with a cross-sectional dimension of 116 × 40nm2. Overall the waveguide guides five modes. The effective mode indices as well as their losses are shown in Figs. 4(a) and 4(b) respectively over the wavelength range of 0.5 ∼ 2μm. For all higher-order modes, x-oriented nodal lines appear, as shown by the Hx field plots in the insets of Fig. 4. A striking feature from Fig. 4(a) is that the higher-order modes have higher neff values than the fundamental one, as opposed to the situation in all-dielectric waveguides. In general, the loss of a higher order mode is higher.

Fig. 4 Dispersion (a) and loss (b) curves of 5 modes guided in the waveguide. The mode numbers, from the fundamental mode to the highest-order mode, are denoted as 1 to 5 correspondingly. Insets show the Hx field distributions for the high-order modes.

We explain the reversed mode ordering from the MGT limit by treating the waveguide as if its core material is homogeneous but with an effective permittivity tensor of the form diag(ɛx,ɛy,ɛz). To further simplify our analysis, we examine the problem in the slab waveguide limit, where the waveguide extends infinitely along x. According to MGT, at the wavelength range of 0.85∼2μm, the medium has a positive ɛy, while negative ɛx and ɛz. With such an indefinite permittivity tensor [15

15. A tensor is indefinite if its three principle tensor components are not of the same sign. A medium with such a permittivity and/or permeability tensor(s) is referred to as an indefinite medium [16].

,16

16. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]

], one can analytically obtain the guided TM modes (with Hx, Ey, Ez). The dispersion and loss curves (not shown) are found to be similar to those in the actual 3D waveguide case; in particular, the modes also have reversed ordering. To see the cause more clearly, one can, for the slab case, show that the wave equation in the core is
2Hxy2=ɛz(neff2ɛy1)k02Hx,
(3)
where k0 is the vacuum wavenumber. In a conventional isotropic waveguide with a positive permittivity, the expression in brackets is negative, giving a transversally oscillatory solution in the core. The higher the mode order, the lower the neff and the larger the transversal spatial frequency of the field. On the other hand, for a negative ɛz, as is the case here, the opposite is true: the requirement of an oscillatory solution in the core mandates a positive value of the expression in brackets; higher-order modes with higher transversal spatial frequencies correspond to higher neff values, just as calculated.

In practical applications, one can selectively couple light to the fundamental mode. Due to the fact that the neff values (correspondingly the propagation constants) of higher-order modes are separated from the fundamental one by a fairly large margin, the adverse coupling to the higher-order modes can be safely ignored as long as the fabricated waveguide does not show severe irregularities.

5. Comparison with other waveguides and discussion

We finally compare the proposed waveguide with the slot waveguide as well as the Si waveguide at λ0 = 1.55μm, all in their slab limits. According to Fig. 1(b) and 1(c), our slab waveguide has neff = 8.91 with a loss of 2.8dB/μm. Such a high neff value can only be achieved in a plasmonic slot waveguide if its dielectric gap (with index nd) is as small as 7nm with a loss of 3.2dB/μm in the case of nd = 3, or even smaller at 0.7nm with 3.6dB/μm in the case of an air gap. The requirement of an extremely narrow gap size will seemingly increase the mode mismatch issue with conventional waveguides. As for the sensitivity in mode index, our slab waveguide exhibits a Δneff = 0.40 subject to a Δnd = 0.1, whereas for the slot waveguide case (7nm gap filled with nd = 3) one has Δneff = 0.37. Though rather similar performances in the two cases, the waveguide presented here has a larger mode size which eases the coupling issue; at the same time, from the modulation perspective, it requires only the same voltage as the slot waveguide counterpart for obtaining the same electric field in the dielectric layers through operating the metal layers as capacitors in parallel. For a 250nm-thick Si slab in air background, its guided transverse-electric mode, apart from a small neff = 2.91, only has a Δneff = 0.10 when a 0.1 change in Si index occurs.

In Section 3, we have briefly mentioned the effect of the dielectric filling fraction fd. A value of fd = 0.2 is used consistently for the current investigation. An increase of fd will lead to a decreased effective index of the layered medium, which will in turn lead to a smaller neff of a waveguide based on the medium, or equivalently a less confined mode. One can on the other hand decrease fd for an even higher . However that will lead to a thinner dielectric layer (from the current 4nm), which would pose challenges to nanofabrication processes. To a certain degree, one can adopt a larger period for the layered medium to circumvent the problem. However care has to be taken to prevent the waveguide from getting bulky or being similar to weakly-coupled slot plasmonic waveguides.

6. Conclusion

In conclusion, we have proposed a type of layered metal-dielectric waveguide which exhibits an extraordinarily high effective mode index for achieving subwavelength light channeling. Waveguides based on such a metamaterial can potentially enable photonic integration with a footprint surpassing that based on notably the silicon photonics technology. One special property of such a waveguide is that the effective mode index of its guided mode shows a much greater variation (more than three times) than a variation in the refractive index in the dielectric layers. Such a leveraged mode index sensitivity can be potentially used for realizing, e.g. a phase modulator as short as 2.2μm at the telecommunication wavelength. The multimode operation and the reversed mode ordering found in such a waveguide are discussed and explained theoretically.

Acknowledgments

This work is supported by the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR).

References and links

1.

B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006). [CrossRef]

2.

L. Thylén, S. He, L. Wosinski, and D. Dai, “The Moore’s law for photonic integrated circuits,” J. Zhejiang Univ. Sci. 7, 1961–1967 (2006). [CrossRef]

3.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

4.

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82, 1158–1160 (2003). [CrossRef]

5.

G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. 25, 2511–2521 (2007). [CrossRef]

6.

B. T. Schwartz and R. Piestun, “Waveguiding in air by total external reflection from ultralow index metamaterials,” Appl. Phys. Lett. 85, 1–3 (2004). [CrossRef]

7.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005). [CrossRef] [PubMed]

8.

M. Yan and N. A. Mortensen, “Hollow-core infrared fiber incorporating metal-wire metamaterial,” Opt. Express 17, 14851–14864 (2009). [CrossRef] [PubMed]

9.

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomater. 2007, 79469 (2007). [CrossRef]

10.

C. H. Gan and P. Lalanne, “Well-confined surface plasmon polaritons for sensing applications in the near-infrared,” Opt. Lett. 35, 610–612 (2010). [CrossRef] [PubMed]

11.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

12.

Z. L. Sámson, S.-C. Yen, K. F. MacDonald, K. Knight, S. Li, D. W. Hewak, D.-P. Tsai, and N. I. Zheludev, “Chalcogenide glasses in active plasmonics,” Phys. Status Solidi (RRL) 4, 274–276 (2010). [CrossRef]

13.

This picture is only absolutely valid when the waveguide under consideration is infinitely extending along x, or of slab geometry. When the waveguide width is finite, the guided mode is always hybrid with both TM and transverse-electric (TE) field components. A plasmonic mode usually has a larger portion of TM components and is therefore a TM-like mode.

14.

Y. Ikuma, Y. Shoji, M. Kuwahara, X. Wang, K. Kintaka, H. Kawashima, D. Tanaka, and H. Tsuda, “Small-sized optical gate switch using ge2sb2te5 phase-change material integrated with silicon waveguide,” Electron. Lett. 46, 368–369 (2010). [CrossRef]

15.

A tensor is indefinite if its three principle tensor components are not of the same sign. A medium with such a permittivity and/or permeability tensor(s) is referred to as an indefinite medium [16].

16.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]

OCIS Codes
(230.7370) Optical devices : Waveguides
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Optics at Surfaces

History
Original Manuscript: January 13, 2011
Revised Manuscript: February 3, 2011
Manuscript Accepted: February 4, 2011
Published: February 11, 2011

Citation
Min Yan, Lars Thylén, and Min Qiu, "Layered metal-dielectric waveguide: subwavelength guidance, leveraged modulation sensitivity in mode index, and reversed mode ordering," Opt. Express 19, 3818-3824 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3818


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References

  1. B. Jalali, and S. Fathpour, "Silicon photonics," J. Lightwave Technol. 24, 4600-4615 (2006). [CrossRef]
  2. L. Thylén, S. He, L. Wosinski, and D. Dai, "The Moore’s law for photonic integrated circuits," J. Zhejiang Univ. Sci. 7, 1961-1967 (2006). [CrossRef]
  3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  4. K. Tanaka, and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003). [CrossRef]
  5. G. Veronis, and S. Fan, "Modes of subwavelength plasmonic slot waveguides," J. Lightwave Technol. 25, 2511-2521 (2007). [CrossRef]
  6. B. T. Schwartz, and R. Piestun, "Waveguiding in air by total external reflection from ultralow index metamaterials," Appl. Phys. Lett. 85, 1-3 (2004). [CrossRef]
  7. J. T. Shen, P. B. Catrysse, and S. Fan, "Mechanism for designing metallic metamaterials with a high index of refraction," Phys. Rev. Lett. 94, 197401 (2005). [CrossRef] [PubMed]
  8. M. Yan, and N. A. Mortensen, "Hollow-core infrared fiber incorporating metal-wire metamaterial," Opt. Express 17, 14851-14864 (2009). [CrossRef] [PubMed]
  9. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, "Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation," J. Nanomater. 2007, 79469 (2007). [CrossRef]
  10. C. H. Gan, and P. Lalanne, "Well-confined surface plasmon polaritons for sensing applications in the near infrared," Opt. Lett. 35, 610-612 (2010). [CrossRef] [PubMed]
  11. P. B. Johnson, and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
  12. Z. L. Sámson, S.-C. Yen, K. F. MacDonald, K. Knight, S. Li, D. W. Hewak, D.-P. Tsai, and N. I. Zheludev, "Chalcogenide glasses in active plasmonics," Phys. Status Solidi (RRL) 4, 274-276 (2010). [CrossRef]
  13. This picture is only absolutely valid when the waveguide under consideration is infinitely extending along x, or of slab geometry. When the waveguide width is finite, the guided mode is always hybrid with both TM and transverse-electric (TE) field components. A plasmonic mode usually has a larger portion of TM components and is therefore a TM-like mode.
  14. Y. Ikuma, Y. Shoji, M. Kuwahara, X. Wang, K. Kintaka, H. Kawashima, D. Tanaka, and H. Tsuda, "Small-sized optical gate switch using ge2sb2te5 phase-change material integrated with silicon waveguide," Electron. Lett. 46, 368-369 (2010). [CrossRef]
  15. A tensor is indefinite if its three principle tensor components are not of the same sign. A medium with such a permittivity and/or permeability tensor(s) is referred to as an indefinite medium [16].
  16. D. R. Smith, and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]

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