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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 23711–23726
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Numerical investigation of mode characteristics of nanoscale surface plasmon-polaritons using a pseudospectral scheme

Chia-Chien Huang  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 23711-23726 (2010)
http://dx.doi.org/10.1364/OE.18.023711


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Abstract

This study uses a full vector pseudospectral scheme in the frequency domain to investigate the mode characteristics of surface plasmon-polariton (SPP) waveguides. The wave equations solved in this study are based on the transverse magnetic field components, and thus the spurious modes are removed due to the constraint of divergence-free magnetic vector. The waveguide dimension dependences on the mode confinement and propagation length of the dielectric-loaded surface plasmon-polariton waveguide (DLSPPW) are extensively studied and characterized. The numerical results of the DLSPPW show that the proposed scheme is highly efficient and yields accurate complex effective indices while requiring much less memory than the commonly used finite element method. This study also analyzes the propagation characteristics and figures of merit of an inverted metal slot waveguide (IMSW) in detail. The IMSW achieves a propagation loss an order of magnitude lower than nanoparticle chains with comparable degrees of lateral confinement.

© 2010 OSA

1. Introduction

The ability for fabricating the integrated circuits is now at dimensions below 100nm. Further scaling the electronic circuits encounters some limitations such as operating bandwidth, signal delay, and energy dissipation [1

1. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000). [CrossRef]

,2

2. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 (2007). [CrossRef] [PubMed]

]. For working with these problems, applying the photonic circuits offers an effective solution. However, using pure dielectrics to produce nanoscale photonic components is primarily restricted by the inherent diffraction limit. Therefore, it is difficult to make photonic integrated circuits fully compatible with the field of integrated electronic circuits. In recent years, exciting the surface plasmon polariton (SPP) is a promising way to squeeze the optical energy in nanometric cross section [3

3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

]. Theoretically, the SPP occurs at the interface between a dielectric and a metal due to the coupling of an electromagnetic wave and the free electrons of the metal [4

4. H. Raether, Surface plasmons on smooth and rough surfaces and on grating, Ch. 2, (Springer Verlag, New York. 1988).

]. Enlarging the SPP wavevector decreases the size of the resulting spatial confinement more than the conventional dielectric methods can. Many studies report various types of plasmonic structures to achieve the nanoguiding of light power [5

5. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef] [PubMed]

12

12. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]

]. The most essential characteristics of plasmonic waveguides include the dependences of mode confinement and propagation length on the geometric structure and the operating wavelength. Theoretically, the plasmonic waveguide structures need at least one interface composed of a dielectric and a metal to produce the SPP mode, and the SPP mode profile from the interface into the metal changes acutely (i.e., the penetration depth is only tens of nanometers). In numerical computations, an acute varying field is harder to be precisely captured than that in a pure, lossless dielectric waveguide.

2. Mathematical formulations

Assuming a monochromatic optical wave with a time dependence of exp(iωt) and propagating along the z direction in a medium with refractive index n(x,y,z), the vector wave equation derived from Maxwell’s equations based on the magnetic field vector H is
2H+k02n2H+n2n2×(×H)=0
(1)
where k 0 = 2π/λ 0 and λ 0 the wavelength in vacuum. For solving a mode problem, the waveguide structure in the longitudinal propagation direction is invariant. The refractive index profile only depends on the transverse directions, that is, n = n(x,y). Here, the wave with z dependence of the form exp(-ißz) is considered, where β = k 0 neff is the propagation constant and neff is the mode effective index. The general waveguide structures involve distinct materials with a high-contrast refractive index in the computational domain, especially for the plasmonic waveguides consisting of a dielectric medium and a metal medium with negative permittivity. This approach partitions the computational domain into a few subdomains with a uniform refractive index. The last term of Eq. (1) equals zero due to the homogeneous refractive index profile in each subdomain. This reduces the vector wave equation to two decoupled wave equations for the transverse magnetic field components Hx and Hy as follows:
2Hsx2+2Hsy2+k02(n2neff2)Hs=0,   (s=x,y).
(2)
However, the polarization dependence and the coupling effect can be recovered by incorporating the interfacial boundary conditions. These boundary conditions include the continuous normal and tangential components of magnetic fields at each intra-element boundary. Moreover, the continuities of the longitudinal components Hz and Ez provide the coupling effect of Hx and Hy through the relations ×H= jωε 0 n 2(x,y)E andH=0. For a horizontal interface, since the derivatives of Hx and Hy with respect to x on both sides of the interface are equal, the continuity of Ez yields
ny+2Hxy|yny2Hxy|y+=(ny+2ny2)Hyx
(3a)
and the continuity of Hz yields
Hyy|y+=Hyy|y
(3b)
where y + and y - refer to the locations infinitesimally above and below the horizontal interface, respectively. Likewise, for a vertical interface,
nx+2Hyx|xnx2Hyx|x+=(nx+2nx2)Hxy
(3c)
and
Hxx|x+=Hxx|x
(3d)
where x + and x - refer to the locations infinitesimally to the right and to the left of the vertical interface, respectively. Note that the spurious modes do not satisfy the divergence-free magnetic fields [40

40. B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984). [CrossRef]

] and then the spurious modes are completely eliminated in [41

41. A. J. Kobelansky and J. P. Webb, “Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields,” Electron. Lett. 22(11), 569–570 (1986). [CrossRef]

] by imposing the constraint Η=0. Importantly, the constraint Η=0 is explicitly imposed by the proposed scheme to patch the subdomains to effectively prohibit the spurious modes.

3. Numerical approaches

The pseudospectral approach first expands the dependent variables in each subdomain using a set of orthogonal basis functions [34

34. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, (Springer Verlag, New York. 1988).

,35

35. J. P. Boyd, Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, New York: (Springer, Verlag, 2nd Edition, 2000).

]. The product of basis functions and Fourier coefficients are then transferred to the individual Lagrange-type interpolation functions and the grid points of fields. This process transforms the unknown Fourier coefficients in the frequency domain to the field points in the physical domain to obtain the mode profiles directly. The pseudospectral scheme requires that the wave equations must be satisfied exactly at specific collocation points depending on the basis functions. The differential equations are then converted to system of linear equations. For the mode problems, the effective refractive index is found by solving a matrix eigenvalue equation. To preserve the exponential convergence behavior of the pseudospectral scheme, the computational window is partitioned into several subdomains with homogeneous refractive indices. Finally, the subdomains are assembled by patching the physical interfacial boundary conditions to the entire computational domain.

The other issue of the proposed scheme is to determine the basis functions to represent the mode fields. Distinct basis functions depending on the field behaviors are chosen for different subdomains in this work. Chebyshev polynomials are suitable to expand the optical fields in interior subdomains with finite boundaries because of their robustness to non-periodic structures. In contrast, Laguerre-Gaussian functions (LGFs) expand the exponential field profiles of guided modes in exterior subdomains with a semi-infinite boundary. For Chebyshev polynomials, the explicit form of the Lagrange-type interpolation function is as follows [35

35. J. P. Boyd, Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, New York: (Springer, Verlag, 2nd Edition, 2000).

],
θp(x)=(1)p+1(1x2)Tv'(x)cpn2(xxp),cp={2,ifp=0,  N1,if1pN-1
(9)
where Tv(x) is the general Chebyshev polynomial of order v and xp’s are the collocation points for Chebyshev polynomials. For LGFs, the explicit form is as follows [35

35. J. P. Boyd, Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, New York: (Springer, Verlag, 2nd Edition, 2000).

],
θp(αx)=xLv(αx)α(xxp)[xLv'(αx)]|x=xpeα(xxp)/2,
(10)
where Lv(αx) is the Laguerre polynomial of order v and xp’s are the collocation points for LGFs. Parameter α, called the scaling factor, and affects the accuracy for a given term of LGFs. The definite procedure to determine α has been derived in earlier research [37

37. C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005). [CrossRef]

].

4. Simulation results and discussion

This section investigates two plasmonic waveguides. First, the SPP modes in a DLSPPW structure are solved to demonstrate the computational merits of the proposed scheme compared with those of two reliable numerical schemes, the FEM [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

] and differential method (DM) [30

30. S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008). [CrossRef] [PubMed]

]. The mode confinement and figure of merit for DLSPPWs have also already been extensively studied in [42

42. J. Grandidier, S. Massenot, G. C. Francs, A. Bouhelier, J. C. Weeber, L. Markey, A. Dereux, J. Renger, M. U. González, and R. Quidant, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B 78(24), 245419 (2008). [CrossRef]

] by combining numerical simulations and leakage radiation microscopy. Moreover, the dependences of the mode confinement and propagation length by varying the geometric parameters of DLSPPW are extensively investigated. Secondly, this section studies the mode characteristics of a low-loss IMSW [43

43. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]

] applied to nanoscale circuits in detail, and discusses the figure of merit (FOM) [44

44. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

].

4.1 The dielectric-loaded surface plasmon-polariton waveguide (DLSPPW)

The mode characteristics of DLSPPW have been verified experimentally by near-field optical microscopy and leakage radiation microscopy, which are able to guide efficiently the SPP modes [45

45. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006). [CrossRef]

]. A variety of photonic components based on the DLSPPW structure involving ring resonators [46

46. T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009). [CrossRef] [PubMed]

], multimode interference splitter [47

47. G. H. Yuan, P. Wang, Y. H. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 17(15), 12594–12600 (2009). [CrossRef] [PubMed]

], active Bragg reflector [48

48. A. V. Krasavin and A. V. Zayats, “All-optical active components for dielectric-loaded plasmonic waveguides,” Opt. Commun. 283(8), 1581–1584 (2010). [CrossRef]

], and thermo-optics control Mach-Zehnder interferometers [49

49. J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010). [CrossRef] [PubMed]

] have been successively reported because the advantages including high localization, moderate propagation length (tens of micrometers), thermo-, electro-, all-optical functionalities, and easily integrated to optoelectronic devices. For the conventional integrated optics components; however, the efficiency of coupling has not been investigated yet. It would be useful to provide examples of more complex plasmonic circuits. The DLSPPW is constructed by depositing a dielectric polymer ridge with thickness t and width w on a gold film with thickness h covering a semi-infinite dielectric substrate. Figure 1(a)
Fig. 1 (a) The cross section of DLSPPW with refractive indices of core np, gold film ng, substrate nd, and air na.(b) the division of computational domain for the DLSPPW structure.
depicts the cross section of the DLSPPW. The refractive indices of materials are air na = 1, core np = 1.535, gold film ng = 0.55-11.5i (the form of complex refractive index depends on the plane-wave solution form chosen [50

50. E. D. Palik, Handbook of optical constant of solids II (Academic Press, New York. 1998).

], and this study uses the exp[i(ωt-βz)] solution), and substrate nd = 1.6 at an excitation wavelength of optical communication λ = 1.55μm. This study investigates the mode characteristics by varying the width and thickness of the polymer ridge at a fixed thickness of gold film h = 0.1μm, and then analyzes the wavelength dependence.

To compare the accuracy, Fig. 3(a) and (b)
Fig. 3 The relative errors of (a) the real parts of neff and (b) the propagation lengths Lc(μm) of the proposed scheme at t = 0.6μm versus different width t. The reference values are obtained by the FEM [17].
show the relative errors of the real parts of neff and L c's obtained by the proposed scheme using nx = 20 for various widths, respectively, relative to the reference values obtained by the FEM [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

].

Tables 3

Table 3. Comparisons of the real parts of effective index obtained by the Finite Element Method (FEM), the differential method (DM), and the proposed scheme (this work) for different widths w of ridge at t = 0.6μm.

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also lists the precise real parts of neff obtained by the proposed scheme using nx = 20 along with the FEM [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

] and DM [30

30. S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008). [CrossRef] [PubMed]

], and the L c's are shown in Table 4

Table 4. Comparisons of the propagation lengths Lc (μm) obtained by Finite Element Method (FEM), the differential method (DM), and the proposed scheme (this work) for different widths w of ridge at t = 0.6μm.

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.

Table 3 and 4 show good agreement of the mode characteristics between these approaches. The total number of unknowns in the proposed scheme with nx = 20 is in the order of 103. In contrast, the FEM [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

] requires much more unknowns with the order of 105. As a result, the proposed scheme shows higher computational efficiency (much less memory) than that of FEM. This study also computes the results of varying the thickness of the ridge at width w = 0.5μm. Figure 4(a) and (b)
Fig. 4 The relative errors of (a) the real parts of neff and (b) the propagation lengths Lc of the proposed scheme at w = 0.5μm versus different thicknesses t. The reference values are obtained by the FEM [17].
show the relative errors of the real parts of neff and L c's of the proposed scheme for various thicknesses, respectively, relative to the FEM [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

]. These calculated results by the proposed scheme and the FEM exhibit good agreement.

To study the mode characteristics of DLSPPW in detail, Fig. 5(a) and (b)
Fig. 5 (a) The real parts of effective index and (b) the propagation lengths of DLSPPW versus the ridge width w for several thicknesses t.
show the effective indices and the propagation lengths of the fundamental TM00 mode for various waveguide dimensions.

Figure 5(a) shows the real parts of effective index increase monotonously with the increase of the ridge width and thickness. A smaller thickness makes the mode characteristics approach the SPP mode at a metal-air interface. Considering the dependences of L c and the waveguide dimensions, L c increases monotonously as w decreases as shown in Fig. 5(b) because more power is confined in the ridge region. However, the dependence of L c and t is somewhat complicated. For the condition t<0.4µm, L c increases as t decreases for all w’s, but the reverse relations between L c and t are observed while t is greater than 0.4µm. In addition, the occurrence of cross points is resulted from the various relations between L c and the t. The result makes the optimum choice of the trade off between the confinement and propagation length. This study discusses two definitions of mode confinement. The first one is the lateral mode extent proposed in [17

17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

] (where the mode width is measured by the 1/e of peak value of power). The lateral mode extents are computed by the effective index method (EIM), which is a simple approximate approach but only accurate for the strongly guided mode. Here, the results calculated by the proposed scheme and those by the EIM (indicated as the dash lines) are shown in Fig. 6(a)
Fig. 6 (a) The lateral mode extents and (b) the confinement areas of DLSPPW versus the ridge width w for several thicknesses t.
.

The significant deviations of the width size for achieving the minimum lateral mode extents are found between the EIM the proposed scheme. For instance, the minimum width size obtained are w = 0.35μm by the proposed scheme but that are w = 0.5μm by the EIM for the conditions t = 0.4μm and t = 0.6μm. More deviations are found for t = 0.2μm and t = 0.3μm. These results show that an accurate numerical scheme is vital to design photonic devices. Different from the lateral mode extent, this study proposes a simple and intuitive definition to evaluate the confinement area of DLSPPW. The new definition computes the area of a rectangle consisted of the lateral and vertical mode extents. Figure 6(b) shows the calculated results. Except for t = 0.2μm, the dependence between confinement area and thickness in Fig. 6(b) show opposite tendency relative to those in Fig. 6(a). That is, increasing the ridge thickness should theoretically shrink the lateral mode extent, but instead stretches the vertical one. The degree of stretching in the vertical dimension is greater than the shrunken degree in the lateral dimension. Figure 7
Fig. 7 The mode intensity profiles |Hx|2 of DLSPPW at w = 0.8μm for (a) t = 0.6μm (b) t = 0.4μm (c) t = 0.3μm.
shows the mode intensity profiles |Hx|2 of the fundamental SPP modes at w = 0.8μm for different t's, and verifies the opposite tendency between the lateral mode extent and the confinement area. Note that the SPP mode contour plots shown in Fig. 7(a)-(c) indicate that the peak values in each plot are normalized to 1 and the outermost contour lines denote the 1/e of the peak value.

In particular, the confinement areas for the condition t = 0.2μm increases significantly while the width w was smaller than 0.4μm as shown in Fig. 6(b). The result can be observed in Fig. 8
Fig. 8 The mode intensity profiles |Hx|2 of DLSPPW at w = 0.2μm for (a) t = 0.6μm (b) t = 0.4μm (c) t = 0.3μm.
, which shows the mode intensity profiles at w = 0.2μm for different values of t. In Fig. 8(c), the largest mode spreading is observed at the condition t = 0.2μm.

From these results, some conclusions are given. In general, the smaller thickness and width allow the longer propagation length, and the confinement area decreases with the thickness decreases except the condition w<0.4μm for t = 0.2μm. Consequently, for obtaining the large values of FOM, the thin thickness and modest wide width around w = 0.4μm of the dielectric load are suitable for the optical communication wavelength λ = 1.55μm. In addition, the lateral mode extent as defined only provides the measurement of the bend angle with accepted bend losses. In contrast, the confinement area proposed in this study is to calculate the mode spreading along both the lateral and vertical extents. The confinement area can be used to evaluate the performances of the active plasmonics and to measure the cross-talk of powers while constructing the compact three-dimensional photonic components. This study also analyzes the dependence of the mode characteristics and excitation wavelength. The three wavelengths of λ = 1.55μm (ng = 0.55-11.5i), λ = 1.22μm (ng = 0.36-8.60i), and λ = 0.893μm (ng = 0.21-5.94i [50

50. E. D. Palik, Handbook of optical constant of solids II (Academic Press, New York. 1998).

]) at t = 0.3μm and w = 0.4μm are calculated, and Fig. 9
Fig. 9 The mode intensity profiles |Hx|2 of DLSPPW at w = 0.4μm and t = 0.3μm for (a) λ = 1.55μm (b) λ = 1.22μm (c) λ = 0.893μm.
shows the resulting mode profiles. The smaller wavelength λ = 0.893μm yields better confinement areas, and therefore results in shorter L c due to a longer penetration depth (with high material loss) of the field into the gold film.

4.2 The inverted metal slot waveguide

Recently, Chen et al. [43

43. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]

] reported experimentally that the IMSW can be efficiently coupled to a standard silicon dielectric waveguide with a high coupling efficiency of approximately 2.5dB per facet. In addition, compared with the degrees of mode confinement of nanoparticle chains, which achieve approximately λ/5 with losses of 30dB/μm (propagation length Lc = 0.15μm) [52

52. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

], the IMSW achieves a propagation loss one order of magnitude lower. However, the geometry and wavelength dependences of the propagation characteristics are not well defined or understood. Therefore, this study investigates in detail the dependences for this waveguide. Figure 10
Fig. 10 The cross section of the IMSW with refractive indices of core nSi, gold cladding nAu, substrate nSiO2, and silica cladding nSiO2.
depicts the cross section of this waveguide structure. To conveniently integrate with a silicon wire waveguide, this design used silicon (Si) as the core and gold (Au) as the cladding. To reduce the power loss produced by a metal, a thin silica (SiO2) layer was deposited on the top of silicon to isolate the optical field from the gold. SiO2 was also used as the substrate below the core region.

The refractive indices of Si and SiO2 are nSi = 3.48 and nSiO2 = 1.44, respectively, while that of gold is nAu = 0.55-11.5i under the excitation wavelength λ = 1.55μm. Figures 11(a)
Fig. 11 (a) The real parts of effective index and (b) the propagation lengths of the IMSW versus the core width w for several different core thicknesses t.
and 11(b) show the real parts of neff and propagation lengths Lc calculated by the proposed scheme as a function of core width w for several different thicknesses t for a SiO2 layer with a thickness of h = 80nm.

The mode behaviors of the IMSW are consistent with the physics of the coupled SPP mode at vertical metal-dielectric interfaces, indicating that the optical power has better confinement (larger real part of neff) and yields a shorter Lc (more power penetrates into metal leading to higher material loss) as the core width decreases. Figure 12
Fig. 12 The mode intensity profiles |Hy|2 of the IMSW at t = 250nm for (a) w = 150nm (b) w = 100nm (c) w = 50nm (d) w = 10nm.
illustrates the mode intensity profiles |Hy|2 at t = 250nm for several widths.

The mode sizes in both the lateral and vertical directions shrink as the width w decreases due to the stronger coupling of the two SPP modes at metal-dielectric interfaces. Accordingly, the spot size shrinks to several tens of nanometers while the cost is the decrease of Lc to several hundreds of nanometers. In particular, the Lc's for the case of t = 10nm increase markedly as the width increases. Figure 13
Fig. 13 The mode intensity profiles |Hy|2 of the IMSW at w = 150nm for (a) t = 230nm (b) t = 130nm (c) t = 50nm (d) t = 10nm.
illustrates the mode intensity profiles at w = 150nm for several different thicknesses. The effect of decreasing the thickness to the case of t = 10nm leads to a fairly strong spreading (less confinement) in the field from the core region to the silica regions, but the propagation length is much longer than those of the thicker core. This effect exists in pure dielectric waveguides as shown in Fig. 13(d) because the fundamental guided mode becomes a plane wave in the thin SiO2 layer.

Note that the coupled SPP mode of the two SPP modes determines the subwavelength confinement. Consequently, the mode characteristics are analogous while reducing the core width to w = 10nm with the strong coupling as Fig. 12(d). For further illustrating dependences between the mode characteristics and waveguide dimensions, Fig. 14(a)
Fig. 14 (a) The real parts of effective index and (b) the propagation lengths of the IMSW versus the core thickness t for three different core widths of w.
and 11(b) also show the real parts of neff and Lc’s versus the thickness t for three different widths. The Lc's are several tens of micrometers, while the thickness is smaller than 30nm for the case of w>100nm, as Fig. 14(b) shows. Beyond the width of w = 100nm, the differences of the mode characteristics are small.

In the IMSW, the confinement area is determined mainly by the beam height in vertical direction, and thus the degree of spreading produced by the index guiding in the y-direction is also analyzed. Figure 15(a)
Fig. 15 (a) Mode beam height in vertical direction versus the core thickness t for three different core width w and (b) the figure of merit (FOM) versus the core width w for several core thicknesses t of the IMSW.
shows the beam heights as a function of the thickness for several different widths. The minimum beam heights (about 60nm) for the cases w = 50nm and w = 100nm are both approximately at t = 40nm. The beam height is further reduced to 30nm while decreasing the width to w = 10nm. Compared to a nanoparticle plasmon waveguide with about Lc = 0.15μm [52

52. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

], the IMSW achieves several tens of micrometers Lc for a mode size of approximately λ/5 (here, ~310nm). In addition, this study adopts the definition of M 2 as a “benefit-to-cost” ratio of confinement and propagation length (M 2 = (real(neff)-n Si)/imag(neff)) as proposed by Berini [44

44. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

] to investigate the FOM of the IMSW. Figure 15(b) shows the resulting FOMs versus the w for several thicknesses t. The optimum values of FOMs are between 20 and 30, revealing a trade-off between the mode confinement and propagation distance.

Figure 16
Fig. 16 The mode intensity profiles |Hy|2 of the IMSW at the optimum FOMs of w = 30nm for (a) t = 250nm (b) t = 150nm (c) t = 50nm, and that at (d) w = 20nm and t = 10nm.
illustrates the mode intensity profiles |Hy|2 at four optimum FOMs. Figure 16(d) clearly shows a strong spreading of the field resulting from the diffraction limit, yielding a longer Lc.

Finally, this study analyzes dependence of the mode characteristics and excitation wavelength. Figure 17
Fig. 17 The mode intensity profiles |Hy|2 of the IMSW at t = 250nm and w = 150nm for (a) λ = 1.22μm (b) λ = 0.893μm.
shows the mode profiles for the two wavelengths, λ = 1.22μm and λ = 0.893μm, at t = 250nm and w = 150nm. The wavelength λ = 1.55μm has been shown in Fig. 12(a). The same conclusion as the DLSPPW is obtained, namely, a shorter wavelength yields shorter L c due to the longer penetration depth of the mode field into the gold cladding.

5. Conclusion

Acknowledgements

The author would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 98-2221-E-005-094.

References and links

1.

D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000). [CrossRef]

2.

J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 (2007). [CrossRef] [PubMed]

3.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

4.

H. Raether, Surface plasmons on smooth and rough surfaces and on grating, Ch. 2, (Springer Verlag, New York. 1988).

5.

J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef] [PubMed]

6.

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003). [CrossRef]

7.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

8.

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef] [PubMed]

9.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]

10.

G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30(24), 3359–3361 (2005). [CrossRef]

11.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104–1 (2006). [CrossRef]

12.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]

13.

S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron. 40(3), 325–329 (2004). [CrossRef]

14.

A. Hosseini, A. Nieuwoudt, and Y. Massoud, “Efficient simulation of subwavelength plasmonic waveguides using implicitly restarted Arnoldi,” Opt. Express 14(16), 7291–7298 (2006). [CrossRef] [PubMed]

15.

G. Veronis and S. Fan, “Modes of Subwavelength Plasmonic Slot Waveguides,” J. Lightwave Technol. 25(9), 2511–2521 (2007). [CrossRef]

16.

A. Hosseini, A. Nieuwoudt, and Y. Massoud, “On the design of dielectric strip plasmonic structures for subwavelength waveguiding applications,” IEEE Trans. NanoTechnol. 7(2), 189–196 (2008). [CrossRef]

17.

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]

18.

Y. Binfeng, H. Guohua, and C. Yiping, “Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides,” Opt. Express 17(5), 3610–3618 (2009). [CrossRef] [PubMed]

19.

D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009). [CrossRef] [PubMed]

20.

J. T. Kim, J. J. Ju, S. Park, M. S. Kim, S. K. Park, and S. Y. Shin, “Hybrid plasmonic waveguide for low-loss lightwave guiding,” Opt. Express 18(3), 2808–2813 (2010). [CrossRef] [PubMed]

21.

Y. Xiao-Li, “Light guiding in a slot waveguide that includes an additional confining core region,” Opt. Express 18(6), 6408–6416 (2010). [CrossRef] [PubMed]

22.

E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]

23.

S. H. Chang, T. C. Chiu, and C. Y. Tai, “Propagation characteristics of the supermode based on two coupled semi-infinite rib plasmonic waveguides,” Opt. Express 15(4), 1755–1761 (2007). [CrossRef] [PubMed]

24.

K. Y. Jung, F. L. Teixeira, and R. M. Reano, “Surface plasmon coplanar waveguides: mode characteristics and mode conversion losses,” IEEE Photon. Technol. Lett. 21(10), 630–632 (2009). [CrossRef]

25.

M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009). [CrossRef]

26.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

27.

P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express 7(10), 329–335 (2000). [CrossRef] [PubMed]

28.

T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008). [CrossRef] [PubMed]

29.

K. Tanaka, T. T. Minh, and M. Tanaka, “Analysis of propagation characteristics in the surface plasmon polariton gap waveguides by method of lines,” Opt. Express 17(2), 1078–1092 (2009). [CrossRef] [PubMed]

30.

S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008). [CrossRef] [PubMed]

31.

J. Grandidier, G. C. des Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009). [CrossRef] [PubMed]

32.

J. Shibayama, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam propagation method,” J. Lightwave Technol. 23(3), 1533–1539 (2005). [CrossRef]

33.

G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009). [CrossRef]

34.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, (Springer Verlag, New York. 1988).

35.

J. P. Boyd, Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, New York: (Springer, Verlag, 2nd Edition, 2000).

36.

C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21(10), 2284–2296 (2003). [CrossRef]

37.

C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005). [CrossRef]

38.

C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express 16(22), 17915–17934 (2008). [CrossRef] [PubMed]

39.

C. C. Huang, “Improved pseudospectral mode solver by prolate spheroidal wave functions for optical waveguides with step-index,” J. Lightwave Technol. 27(5), 597–605 (2009). [CrossRef]

40.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984). [CrossRef]

41.

A. J. Kobelansky and J. P. Webb, “Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields,” Electron. Lett. 22(11), 569–570 (1986). [CrossRef]

42.

J. Grandidier, S. Massenot, G. C. Francs, A. Bouhelier, J. C. Weeber, L. Markey, A. Dereux, J. Renger, M. U. González, and R. Quidant, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B 78(24), 245419 (2008). [CrossRef]

43.

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]

44.

P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]

45.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006). [CrossRef]

46.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009). [CrossRef] [PubMed]

47.

G. H. Yuan, P. Wang, Y. H. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 17(15), 12594–12600 (2009). [CrossRef] [PubMed]

48.

A. V. Krasavin and A. V. Zayats, “All-optical active components for dielectric-loaded plasmonic waveguides,” Opt. Commun. 283(8), 1581–1584 (2010). [CrossRef]

49.

J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010). [CrossRef] [PubMed]

50.

E. D. Palik, Handbook of optical constant of solids II (Academic Press, New York. 1998).

51.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996). [CrossRef]

52.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.7370) Optical devices : Waveguides
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: August 3, 2010
Revised Manuscript: October 8, 2010
Manuscript Accepted: October 19, 2010
Published: October 27, 2010

Citation
Chia-Chien Huang, "Numerical investigation of mode characteristics of nanoscale surface plasmon-polaritons using a pseudospectral scheme," Opt. Express 18, 23711-23726 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23711


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References

  1. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000). [CrossRef]
  2. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 (2007). [CrossRef] [PubMed]
  3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  4. H. Raether, Surface plasmons on smooth and rough surfaces and on grating, Ch. 2, (Springer Verlag, New York. 1988).
  5. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997). [CrossRef] [PubMed]
  6. T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003). [CrossRef]
  7. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]
  8. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef] [PubMed]
  9. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]
  10. G. Veronis and S. Fan, “Guided subwavelength plasmonic mode supported by a slot in a thin metal film,” Opt. Lett. 30(24), 3359–3361 (2005). [CrossRef]
  11. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104–1 (2006). [CrossRef]
  12. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]
  13. S. J. Al-Bader, “Optical transmission on metallic wires-fundamental modes,” IEEE J. Quantum Electron. 40(3), 325–329 (2004). [CrossRef]
  14. A. Hosseini, A. Nieuwoudt, and Y. Massoud, “Efficient simulation of subwavelength plasmonic waveguides using implicitly restarted Arnoldi,” Opt. Express 14(16), 7291–7298 (2006). [CrossRef] [PubMed]
  15. G. Veronis and S. Fan, “Modes of Subwavelength Plasmonic Slot Waveguides,” J. Lightwave Technol. 25(9), 2511–2521 (2007). [CrossRef]
  16. A. Hosseini, A. Nieuwoudt, and Y. Massoud, “On the design of dielectric strip plasmonic structures for subwavelength waveguiding applications,” IEEE Trans. NanoTechnol. 7(2), 189–196 (2008). [CrossRef]
  17. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]
  18. Y. Binfeng, H. Guohua, and C. Yiping, “Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides,” Opt. Express 17(5), 3610–3618 (2009). [CrossRef] [PubMed]
  19. D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009). [CrossRef] [PubMed]
  20. J. T. Kim, J. J. Ju, S. Park, M. S. Kim, S. K. Park, and S. Y. Shin, “Hybrid plasmonic waveguide for low-loss lightwave guiding,” Opt. Express 18(3), 2808–2813 (2010). [CrossRef] [PubMed]
  21. Y. Xiao-Li, “Light guiding in a slot waveguide that includes an additional confining core region,” Opt. Express 18(6), 6408–6416 (2010). [CrossRef] [PubMed]
  22. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]
  23. S. H. Chang, T. C. Chiu, and C. Y. Tai, “Propagation characteristics of the supermode based on two coupled semi-infinite rib plasmonic waveguides,” Opt. Express 15(4), 1755–1761 (2007). [CrossRef] [PubMed]
  24. K. Y. Jung, F. L. Teixeira, and R. M. Reano, “Surface plasmon coplanar waveguides: mode characteristics and mode conversion losses,” IEEE Photon. Technol. Lett. 21(10), 630–632 (2009). [CrossRef]
  25. M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009). [CrossRef]
  26. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
  27. P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express 7(10), 329–335 (2000). [CrossRef] [PubMed]
  28. T. T. Minh, K. Tanaka, and M. Tanaka, “Complex propagation constants of surface plasmon polariton rectangular waveguide by method of lines,” Opt. Express 16(13), 9378–9390 (2008). [CrossRef] [PubMed]
  29. K. Tanaka, T. T. Minh, and M. Tanaka, “Analysis of propagation characteristics in the surface plasmon polariton gap waveguides by method of lines,” Opt. Express 17(2), 1078–1092 (2009). [CrossRef] [PubMed]
  30. S. Massenot, J. C. Weeber, A. Bouhelier, G. Colas des Francs, J. Grandidier, L. Markey, and A. Dereux, “Differential method for modeling dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 16(22), 17599–17608 (2008). [CrossRef] [PubMed]
  31. J. Grandidier, G. C. des Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009). [CrossRef] [PubMed]
  32. J. Shibayama, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam propagation method,” J. Lightwave Technol. 23(3), 1533–1539 (2005). [CrossRef]
  33. G. Colas des Francs, J. Grandidier, S. Massenot, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Integrated plasmonic waveguides: a mode solver based on density of states formulation,” Phys. Rev. B 80(11), 115419 (2009). [CrossRef]
  34. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, (Springer Verlag, New York. 1988).
  35. J. P. Boyd, Chebyshev and Fourier spectral methods. Lecture Notes in Engineering, New York: (Springer, Verlag, 2nd Edition, 2000).
  36. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. 21(10), 2284–2296 (2003). [CrossRef]
  37. C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. 11(2), 457–465 (2005). [CrossRef]
  38. C. C. Huang, “Simulation of optical waveguides by novel full-vectorial pseudospectral-based imaginary-distance beam propagation method,” Opt. Express 16(22), 17915–17934 (2008). [CrossRef] [PubMed]
  39. C. C. Huang, “Improved pseudospectral mode solver by prolate spheroidal wave functions for optical waveguides with step-index,” J. Lightwave Technol. 27(5), 597–605 (2009). [CrossRef]
  40. B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microw. Theory Tech. 32(8), 922–928 (1984). [CrossRef]
  41. A. J. Kobelansky and J. P. Webb, “Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields,” Electron. Lett. 22(11), 569–570 (1986). [CrossRef]
  42. J. Grandidier, S. Massenot, G. C. Francs, A. Bouhelier, J. C. Weeber, L. Markey, A. Dereux, J. Renger, M. U. González, and R. Quidant, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B 78(24), 245419 (2008). [CrossRef]
  43. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]
  44. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14(26), 13030–13042 (2006). [CrossRef] [PubMed]
  45. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006). [CrossRef]
  46. T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009). [CrossRef] [PubMed]
  47. G. H. Yuan, P. Wang, Y. H. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon polariton waveguides,” Opt. Express 17(15), 12594–12600 (2009). [CrossRef] [PubMed]
  48. A. V. Krasavin and A. V. Zayats, “All-optical active components for dielectric-loaded plasmonic waveguides,” Opt. Commun. 283(8), 1581–1584 (2010). [CrossRef]
  49. J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, and A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010). [CrossRef] [PubMed]
  50. E. D. Palik, Handbook of optical constant of solids II (Academic Press, New York. 1998).
  51. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996). [CrossRef]
  52. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

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