Optical properties of metal-multi-insulator-metal plasmonic waveguides

doi:10.1364/OE.20.012133

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Optical properties of metal-multi-insulator-metal plasmonic waveguides

Xiang-Tian Kong, Wei-Guo Yan, Zu-Bin Li, and Jian-Guo Tian »View Author Affiliations

Optics Express, Vol. 20, Issue 11, pp. 12133-12146 (2012)
http://dx.doi.org/10.1364/OE.20.012133

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Abstract

We theoretically study the plasmonic modes in metal-multi-insulator-metal (MMIM) waveguides. Two types of symmetric MMIM structures consisting of three insulators are investigated thoroughly. The effective refractive index, energy confinement, propagation length, and figure of merit are given in terms of geometric parameters. Due to the step index modulation, these properties of MMIM structures differ from the metal-insulator-metal (MIM) structure. Compared with the corresponding MIM structures, MMIM structures can possess either better energy confinement or larger propagation length, which depends on the geometric parameters and the index distribution. Propagation length of up to 10³ µm and a figure of merit of up to 10⁴ are observed for MMIM structure with core thickness of several hundred nanometers.

1. Introduction

Plasmonics, which is photonics based on surface plasmon polaritons (SPPs), has been investigated for over a decade as a promising candidate for merging photonics and electronics at nanoscale [1

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]

, 2

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010). [CrossRef] [PubMed]

]. SPPs, which are supported by metal surfaces, are electromagnetic waves coupled to collective oscillations of electron plasma in the metal. By coupling to SPPs, light at visible and near infrared frequencies can be routed and manipulated at nanoscale. A lot of plasmonic systems have been demonstrated and investigated to meet the requirement of both fast signal processing and miniaturization of devices. These include bends [3

J.-C. Weeber, M. U. Gonzalez, A.-L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” Appl. Phys. Lett. 87(22), 221101 (2005). [CrossRef]

–6

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

], splitters [4

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008). [CrossRef]

–6

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

], Bragg reflectors [7

J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express 16(1), 413–425 (2008). [CrossRef] [PubMed]

–9

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009). [CrossRef]

], nanocavities [10

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

–14

V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009). [CrossRef] [PubMed]

], interferometers [5

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

, 15

D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005). [CrossRef]

], ring resonators [5

, 9

], etc. In such systems, plasmonic waveguides play an essential role for guiding and confining SPPs [16

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]

Various types of metallic nanostructures have been proposed for guiding SPPs [16

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]

]. Generally, plasmonic waveguides that are compatible with planar technology and suitable for on-chip integration, can be modeled into three basic types: insulator-metal (IM) [4

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008). [CrossRef]

, 9

, 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. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B 83(16), 165412 (2011). [CrossRef]

], insulator-metal-insulator (IMI) [8

W. Mu, D. B. Buchholz, M. Sukharev, J. I. Jang, R. P. Chang, and J. B. Ketterson, “One-dimensional long-range plasmonic-photonic structures,” Opt. Lett. 35(4), 550–552 (2010). [CrossRef] [PubMed]

, 19

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

–22

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005). [CrossRef]

], and metal-insulator-metal (MIM) [6

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

, 7

J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express 16(1), 413–425 (2008). [CrossRef] [PubMed]

, 23

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]

–25

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95(1), 013504 (2009). [CrossRef]

]. In practice, both IMI and MIM waveguides are referred to symmetric heterostructures, where the SPPs that associated with the two metal surfaces interact with each other. Because of the geometric symmetry, the field profiles of the maintained modes resulting from the coupling effects are of either bilateral or central symmetry [26

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter 44(24), 13556–13572 (1991). [CrossRef] [PubMed]

, 27

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

There is a fundamental trade-off between mode confinement and propagation length. Typically, under visible or near infrared excitation, the SPPs supported by IM waveguides penetrate ~10 nm in the metals and ~10² nm in the dielectrics with propagation length on the order of 10 ~10² µm [17

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

]. IMI waveguides can support the long-ranging plasmonic modes, which penetrate several microns in the dielectrics with propagation length of 10² ~10⁴ µm [22

]. SPPs can be squeezed in a sub-100 nm insulator core of MIM waveguide, but the propagation length is limited to several microns [10

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

, 24

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [CrossRef] [PubMed]

]. As also theoretically discussed in [28

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef] [PubMed]

], only MIM waveguide can provide true sub-diffraction modal confinement. The high propagation losses suffered by MIM type waveguides can be either alleviated by carefully designing of the structures [29

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]

, 30

D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]

], or compensated by gain materials [31

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]

The performances of MIM waveguides can be enhanced by introducing step index modulation to the insulator cores [29

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]

, 30

D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]

, 32

Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010). [CrossRef] [PubMed]

–37

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]

], in which case the MIM structures change into metal-multi-insulator-metal (MMIM) structures. These structures bring more flexibility to the design of MIM type waveguides [29

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]

, 30

D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]

, 32

Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010). [CrossRef] [PubMed]

, 33

Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22(9), 634–636 (2010). [CrossRef]

], and have been proposed for efficient coupling between dielectric guides and MIM waveguides [25

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95(1), 013504 (2009). [CrossRef]

, 29

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]

], designing of CMOS-compatible plasmonic waveguides [30

D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]

, 34

M.-S. Kwon, “Metal-insulator-silicon-insulator-metal waveguides compatible with standard CMOS technology,” Opt. Express 19(9), 8379–8393 (2011). [CrossRef] [PubMed]

, 35

S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011). [CrossRef]

], nanocavities and nanolasing [36

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]

, 37

]. MMIM and MIM structures are similar in some features (for instance, both of them can squeeze SPPs in deep sub-wavelength insulator cores [10

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

, 35

]); however, due to the modulation effect of the step index insulator core of MMIM structure, they differ on the field profiles, energy distributions and dispersion properties.

Previous works have concentrated on the functionalities of various MMIM-based plasmonic devices, and also have provided the properties of the operating modes [29

N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]

, 30

D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]

, 32

Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010). [CrossRef] [PubMed]

–37

]. The characteristics of the operating modes in the devices can be well described by the properties of the corresponding plasmonic modes in planar MMIM waveguides. In these works, the discussions for planar MMIM structures are limited by the geometric parameter ranges, and not systematic or generic enough. Thus a thorough investigation is needed to provide a complete and generalized understanding of the plasmonic modes supported by MMIM structures.

In this paper, we first derive the characteristic equation of the guided modes in a general MMIM structure. Then we focus on the plasmonic modes in two types of symmetric MMIM structures. For simplicity, we only consider MMIM with three insulator layers, which are involved in most of the MMIM-based devices (e.g [32

Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010). [CrossRef] [PubMed]

–37

].). Certainly our study can be easily extended to MMIM structures with more dielectric layers. Depending on the geometry and index distribution, MMIM structures are found to exhibit either better energy confinement or larger propagation length than the MIM structures with the same core thickness.

2. Theory

Figure 1(a) illustrates an MMIM structure with three insulator slabs. The thicknesses of the three insulator slabs are denoted as d₁, d₂, and d₃, respectively. And the spatial distribution of dielectric constant is given by

ε (x) = {\begin{array}{l} ε_{0} = ε_{m_{1}}, & for x \leq a_{0} = 0, \\ ε_{p} = n_{p}^{2}, & for a_{p - 1} < x \leq a_{p}, p = 1, 2, 3, \\ ε_{4} = ε_{m_{2}}, & for x > a_{3} . \end{array}

(1)

The boundaries are given by x = 0, a₁, a₂, and a₃, respectively. All the materials are assumed to be nonmagnetic.

Fig. 1 Geometries and notations of a general MMIM structure with three insulators (a), and two symmetric candidates, MHLHM (b) and MLHLM (c). In (b) and (c), the letters H and L denote the high and low refractive index layers, respectively.

The guided modes are assumed to propagate along z direction. We focus on the guided plasmonic modes here. Since SPPs are intrinsically of transverse magnetic polarization, the electromagnetic fields are written as

\begin{array}{l} E = (E_{x}, 0, E_{z}) e^{i (β z - ω t)}, \\ H = \frac{1}{μ_{0} c} (0, H_{y}, 0) e^{i (β z - ω t)}, \end{array}

(2)

where β, ω, μ₀, and c are the propagation constant, the angular frequency, the permeability of free space and the speed of light in vacuum, respectively. The fields of

E

and

H

can be obtained by solving the vector wave equation under constraint of tangential field continuity. Uniqueness of the solutions is guaranteed by the Helmholtz theorem.

The transcendental characteristic equation is given by

\begin{array}{l} 0 = (N_{01} N_{23} + N_{12} N_{34}) T_{1} T_{2} T_{3} \\ + (N_{12} + N_{01} N_{24}) T_{1} T_{2} + (N_{13} + N_{01} N_{34}) T_{1} T_{3} + (N_{23} + N_{02} N_{34}) T_{2} T_{3} \\ + (N_{01} + N_{14}) T_{1} + (N_{02} + N_{24}) T_{2} + (N_{03} + N_{34}) T_{3} + (1 + N_{04}), \end{array}

(3)

with N_pq = ε_pU_q / ε_qU_p, T_p = tanh(d_pU_p), and U_p = k₀(n_eff² − ε_p)^1/2, where k₀ is the vacuum wavevector, n_eff = β / k₀ is the effective refractive index, and p, q = 0, 1, 2, 3, 4. The complex value of n_eff (hence β) can be obtained by solving Eq. (3). Once n_eff is known, it is straightforward to derive the field components of the corresponding mode. The field components and the calculating method are given in the appendix section.

Then we only discuss two types of symmetric MMIM structures, namely MHLHM [Fig. 1(b)] and MLHLM [Fig. 1(c)], where we denote the insulator slabs of high and low refractive indices by the letters H and L, respectively. In an MHLHM (or MLHLM), the first and the third insulator slabs share the same index that is greater (or less) than the index of the second insulator slab. And both MHLHM and MLHLM here are symmetric structures (d₁ = d₃). For symmetric structures, the eigenmodes are either (E_z-)anti-symmetric or (E_z-)symmetric. Following the terminologies used in [23

], we still call the plasmonic modes as bound modes.

We discuss the dependence of the modes' properties on the geometry, with fixing the operating wavelength and corresponding material parameters. We make the following assumptions for the physical parameters, unless otherwise specified: (1) the operating wavelength λ₀ is 1550 nm; (2) both the metals are silver with dielectric constant given by ε_Ag = −126 + 2.9i [38

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

]; (3) the indices of H and L slabs are given by n_H = 3.48 (silicon) and n_L = 1.44 (silica), respectively. Here we ignore the loss from the dielectrics, and only take the loss generated by the metal layers into account.

3. Cutoff properties and critical points of the modes

Both MHLHM and MLHLM waveguides can support anti-symmetric bound (a_b) modes and symmetric bound (s_b) modes as well as conventional modes. As in the MIM waveguide, the a_b modes exhibit no cutoff of the geometries, but the s_b modes and the conventional modes cut off for structures with thin insulator cores. The cutoff properties of MHLHM and MLHLM are shown in Figs. 2(a) and 2(d), respectively. In the regions 1, 2, 3, the number of supported modes is one, two, three, respectively. The s_b modes cut off at the blue lines and the conventional TM₁ modes cut off at the red lines. As the insulators become thicker, more conventional modes will emerge (not shown).

Fig. 2 Cutoff properties and Re(n_eff) of the eigenmodes in MHLHM [(a), (b), (c)] and MLHLM [(d), (e), (f)] in terms of the geometric parameters (d₁ and d₂). (a), (d) Number of solutions of the characteristic equation. The top and right axes are labeled by the optical thicknesses of the first and second insulator layers measured in λ₀, respectively. (b), (e) Re(n_eff) as a function of d₁ with d₂ fixed at 0 (black), 50 (blue), 100 (green), 200 (red) nm. (c), (f) Re(n_eff) as a function of d₂ with d₁ = 0 (black), 50 (blue), 100 (green), 200 (red) nm. In (b), (c), (e), (f), the curves for the a_b, s_b, and TM₁ modes are plotted in solid, dashed, and dotted lines, respectively. In (b), the gray lines started from the cutoff points of the s_b and TM₁ lines are extrapolated from the connected ones. The curves of the a_b modes in (c) and (e) intersect at points C (8.48, 3.66) and C₁ (4.77, 3.48), respectively, which correspond with the solid orange lines in (b) and (f), respectively. The curves of the s_b modes in (f) intersect at point C₂ (193.3, 1.452), which corresponds with the dashed orange line in (e).

Here we examine the real part of the effective refractive index of the bound modes in terms of the geometric parameters (d₁ and d₂). The bound modes are formed by the interaction of the individual SPPs associated with the two metal surfaces, whose effective refractive index is written as n_sp = [ε₁ε_Ag / (ε₁ + ε_Ag)]^1/2. For the M-H and M-L interfaces, the effective refractive indices of the individual SPPs are given by n_spH = 3.66 + 0.00452i and n_spL = 1.45 + 0.000282i, respectively. Figures 2(b) and 2(e) show Re(n_eff) as a function of d₁ with d₂ fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Figures 2(c) and 2(f) show Re(n_eff) as a function of d₂ with d₁ fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Note that when either d₁ or d₂ is equal to zero, the symmetric MMIM structures reduce to MIM structures.

For the a_b mode in both MHLHM and MLHLM, with fixing H layer thickness d_H, Re(n_eff) decreases monotonously with increasing L layer thickness d_L [solid lines in Figs. 2(c) and 2(e), respectively]. Whereas with fixing d_L, Re(n_eff) of the a_b mode can either decrease or increase with increasing d_H, and approaches a limit value when n_Hd_H >> λ₀, as shown in Figs. 2(b) and 2(f), respectively. The monotonicity depends on the value of L layer thickness. For MHLHM, when d_L is less (or greater) than a critical value d_C [shown as point C in Fig. 2(c)], Re(n_eff) decreases (or increases) with d_H; when d_L is equal to d_C, Re(n_eff) is independent on d_H, and is exactly equal to the limit value Re(n_spH) [the horizontal orange line in Fig. 2(b)]. For MLHLM, when d_L is less (or greater) than a critical value d_C₁ [shown as point C₁ in Fig. 2(e)], Re(n_eff) decreases (or increases) with d_H; when d_L is equal to d_C₁, Re(n_eff) is independent on d_H, and is equal to the limit value n_H [the horizontal orange line in Fig. 2(f)].

For the s_b mode in MHLHM, Re(n_eff) increases when either d_H or d_L increases [dashed lines in Figs. 2(b) and 2(c)]. For the s_b mode in MLHLM, with fixing d_L value, Re(n_eff) increases with increasing d_H [dashed lines in Fig. 2(f)]. Whereas with fixing d_H, Re(n_eff) can either increase or decrease with increasing d_L [dashed lines in Fig. 2(e)]. The monotonicity of Re(n_eff) as a function of d_L for the s_b mode in MLHLM depends on the H layer thickness d_H. When d_H is less (or greater) than a critical value d_C₂ [shown as point C₂ in Fig. 2(f)], Re(n_eff) increases (or decreases) with increasing d_L. When d_H is equal to d_C₂, Re(n_eff) remains a constant with d_L, and is equal to the limit value Re(n_spL) [the dashed orange line in Fig. 2(e)].

The existence of the critical points C, C₁, and C₂ can be explained by the monotonic and asymptotic properties of Re(n_eff). For the a_b mode in MHLHM (or MLHLM), since Re(n_eff) always approaches the limit value Re(n_spH) (or n_H) monotonously as d_H increases to infinity, Re(n_eff) remains unchanged if the condition Re(n_eff) = Re(n_spH) (or Re(n_eff) = n_H) is satisfied at d_H = 0 (M-L-M structure). For the s_b mode in MLHLM, since Re(n_eff) always approaches the limit value Re(n_spL) monotonously as d_L increases to infinity, Re(n_eff) remains a constant if the condition Re(n_eff) = Re(n_spL) is satisfied at d_L = 0 (M-H-M structure).

The critical values are determined by the material parameters, and therefore wavelength-dependent. For the MIM structures (here M-L-M and M-H-M), Re(n_eff) of the a_b (or s_b) modes decreases (or increases) with increasing insulator thicknesses and increases with increasing insulator indices. The effective refractive indices of the individual SPPs, n_spH and n_spL, also increase with increasing insulator indices, n_H and n_L. As a result, when we decrease n_H or increase n_L, all the critical values d_C, d_C₁, and d_C₂ increase, so that the conditions of the critical points can be still satisfied (Fig. 3 ).

Fig. 3 Dependence of the critical values d_C, d_C₁, and d_C₂ on (a) the L layer refractive index n_L with n_H = 3.48, and (b) the H layer refractive index n_H with n_L = 1.44. The gray dashed lines show the asymptotes for the curves of d_C and d_C₂; the red dashed line in (a) shows the asymptote for the curve of d_C₁.

As shown in Fig. 3, the critical values d_C, d_C₁, and d_C₂ increase sharply as n_L and n_H approach each other. The critical value d_C exists in all MHLHM structures, and d_C₂ exists in all MLHLM structures. When n_L and n_H are eventually equal, and the MMIM structures become MIM structures, the critical values d_C and d_C₂ go to infinity. We note that these observations are consistent with the properties of the modes in the MIM structures [the black lines in Figs. 2(b) and 2(e)]. For the bound modes in MIM, Re(n_eff) approaches Re(n_sp) with increasing insulator thickness. But Re(n_eff) is always greater (or less) than Re(n_sp) for the a_b (or s_b) mode. Thus for both the bound modes in MIM, the relation Re(n_eff) = Re(n_sp) can be satisfied only if the insulator becomes sufficiently thick, so that the two individual SPPs decouple. So, mathematically, the critical values d_C and d_C₂ for MIM structures are infinity.

The critical value d_C₁ only exists in MLHLM structures with n_L and n_H not very close to each other. We consider an MLHLM structure with d_H = 0 (M-L-M structure). For the a_b mode of the M-L-M structure, Re(n_eff) is always greater than Re(n_spL). So, when Re(n_spL) ≥ n_H, the relation Re(n_eff) = n_H is never satisfied. For the case shown in Fig. 3(a), as n_L approaches n_H, which is fixed at 3.48, Re(n_spL) increases. When n_L = n_Lc = 3.32 (the red dashed line), Re(n_spL) of the M-L-M structure is equal to n_H, and hence the critical point C₁ is no longer supported. Through similar analysis, for MLHLM with n_L fixed at 1.44 [Fig. 3(b)], when n_H ≤ Re(n_spL) = 1.452, the condition Re(n_eff) = n_H is never satisfied. d_C₁ exists only in the region n_H > Re(n_spL) = 1.452. Note that the asymptote n_H = 1.452 is very close to the line n_H = n_L = 1.44, so we do not plot it in Fig. 3(b). When n_L = n_Lc = 3.32 with n_H fixed at 3.48 [Fig. 3(a)] or n_H = 1.452 with n_L fixed at 1.44 [Fig. 3(b)], the condition Re(n_eff) = n_H can be satisfied only if the two individual SPPs decouple, so that Re(n_eff) = Re(n_spL). So, the d_C₁ value is mathematically infinity at the asymptotes.

We can gain an insight of the critical points from the properties of MIM type waveguide. For the a_b modes supported in the MMIM structures, as the H layer thickness increases, the mean refractive index of the insulator core increases, which leads to increasing of Re(n_eff), while the total thickness of the insulator core increases as well, which brings decreasing of Re(n_eff). And the two reverse mechanisms can balance at a particular L layer thickness (d_C for MHLHM or d_C₁ for MLHLM). For the s_b modes in MLHLM, when the L layer thickness increases, the mean refractive index of insulator core decreases, which causes decreasing of Re(n_eff), while the total thickness of the insulators increases, which causes increasing of Re(n_eff). The two reverse mechanisms can balance at a particular H layer thickness, d_C₂. However, the same balance cannot be found by the s_b modes of MHLHM. Because the energy of the bound modes in MHLHM are always bound at the metal surfaces, and affected much less by the middle insulator thickness, d_L, than the H layer thickness d_H.

We note that the limit values of the bound modes in MHLHM and MLHLM are attributed to the corresponding energy confinement behaviors, which are discussed in the next section.

4. Field profiles and energy confinement behaviors

The electric field component E_z is directly relevant to drive the plasma oscillations. Typical E_z profiles of the bound modes are shown in Figs. 4(a) -4(c) for the MHLHM and MLHLM structures. The field profiles of an MIM structure are also plotted in Fig. 4(a) (black lines) for comparison.

Fig. 4 Typical field profiles of Re(E_z) for the a_b and s_b modes in the (a) MHLHM and (b), (c) MLHLM structures. Solid lines: a_b modes; dashed lines: s_b modes. In (a), d₁ = d₂ = d₃ = 120 nm. The black lines show Re(E_z) for the MIM structure with core refractive index equal to n_H and core thickness equal to 3d₁. In (b), d₁ = d₂ = d₃ = 150 nm. In (c), d₁ = d₂ = d₃ = 1 (blue), 4.77 (red), 20 (green) nm. The scale of x axis varies for different curves.

Unlike the MIM structure, as a result of the step refractive index modulation, Re(E_z) of the bound modes in MMIM have inflection points at the interfaces of H and L layers. For the a_b modes in both MHLHM and MLHLM, the slope of Re(E_z) can be either positive or negative within the H layers. Especially for MLHLM, the slope is determined by the L layer thickness. As shown in Fig. 4(c), when d_L is less (or greater) than d_C₁ (4.77 nm), the slope in the H layer is negative (or positive). When d_L = d_C₁, Re(E_z) is zero in the H layer, and moreover the profiles of E_x and H_y are flat in the H layer (not shown). Namely the a_b mode exhibits transverse electromagnetic behavior in the H layer. Meanwhile, the E_z field in the metal and L layers coincides exactly with that of the M-L-M structure with core thickness equal to 2d_C₁. Consequently, when d_L = d_C₁, all fractions of the field of the a_b mode experience the same effective index n_H. Hence, Re(n_eff) of the a_b mode is equal to n_H at the critical point C₁.

Then we examine the energy confinement behaviors of the bound modes. The energy density is defined by [39

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

]

W = \frac{1}{2} {Re [\frac{d (ω ε_{0} ε)}{d ω}] E \cdot E^{*} + μ_{0} H \cdot H^{*}} .

(4)

Generally, W is discontinuous at the boundaries as a result of the discontinuity of the normal electric field. For both the a_b and s_b modes in MHLHM, and the a_b mode in MLHLM, we find that the energies in each layers of MMIM vary significantly with the H layer thickness, d_H, however slowly with the L layer thickness, d_L. In contrast, for the s_b mode in MLHLM, the energies in each layers vary significantly with d_L, but slowly with d_H. The energy proportions of each layers of MHLHM and MLHLM with respect to the sensitive geometric parameters are shown in Figs. 5(a) -5(c).

Fig. 5 Energy proportions in the metal, H, and L layers of (a) the a_b and s_b modes in MHLHM with respect to d_H, (b) the a_b mode in MLHLM with respect to d_H, and (c) the s_b mode in MLHLM with respect to d_L. Solid lines: a_b mode; dashed lines: s_b mode. In (a), (b), d_L is fixed at 50 (blue), 100 (green), and 200 (red) nm. In (c), d_H is fixed at 50 (blue), 100 (green), and 200 (red) nm.

For the bound modes in MHLHM [Fig. 5(a)], the energy is dominant in the L layer when d_H is small. As d_H increases, the coupling strength of the two individual SPPs decreases and more energy is confined to the metal surfaces, so the energy proportion decreases in the middle insulator (L), while increases in the H layers. When d_H is sufficiently large, the two individual SPPs tend to decouple, and the energy proportion in the L layer approaches zero. Hence, when d_H is sufficiently large, the limit value of Re(n_eff) as a function of d_L is equal to the effective refractive index of the individual SPP that associated with the M-H interface, Re(n_spH) [Fig. 2(b)].

The energy confinement behaviors differ greatly between the a_b and s_b modes in MLHLM. For the a_b mode in MLHLM [Fig. 5(b)], when d_H is small, the energy is dominant in the L layers. As d_H increases, the energy proportion decreases in the L layers, while increases in the H layer. When d_H is sufficiently large, the energy proportion in the L and metal layers approaches zero. Thus the energy is dominant in the middle layer and the a_b mode is less affected by the metal claddings. Hence, when d_H is sufficiently large, the limit value of Re(n_eff) of the a_b mode as a function of d_L is equal to the refractive index of the H layer, n_H [Fig. 2(f)].

For the s_b mode in MLHLM [Fig. 5(c)], when d_L is small, the energy is dominant in the H layer. As d_L increases, more energy is confined to the metal surfaces. The energy proportion decreases in the H layer, and eventually approaches zero. Hence, when d_L is sufficiently large, the limit value of Re(n_eff) as a function of d_L is equal to the effective refractive index of the individual SPP that associated with the M-L interface, Re(n_spL) [Fig. 2(e)].

We define an effective mode width to measure the energy confinement. The effective mode width is given by [39

]

A_{eff} = \frac{1}{W_{\max}} \int_{- \infty}^{\infty} W (x) d x,

(5)

where W_max is the maximum energy density in the core (0 < x < a₃). Figures 6(a) -6(d) show A_eff of the a_b and s_b modes in MHLHM and MLHLM.

Fig. 6 (a)-(d) Effective mode width (A_eff) of the a_b [(a), (c)] and s_b [(b), (d)] modes in MHLHM [(a), (b)] and MLHLM [(c), (d)]. The maximum energy density (W_max) can appear at different positions within the core. The regions 1, 2, …, 5 are divided according to the positions of W_max, by the gray lines. (e)-(i) Typical energy density profiles of the supported modes in MHLHM [(e), (f)] and MLHLM [(g)-(h)] corresponding with the regions 1, 2, …, 5 in (a)-(d). Solid lines: a_b modes; dashed lines: s_b modes. In (e)-(i), d₁ = d₂ = 50, 120, 20, 100, 180 nm, respectively. A_eff of both MHLHM and MLHLM is less than that of M-L-M structure with the same core thickness, except for the bottom right region bounded by the white lines in (c).

The maximum energy density in the core can appear at different positions. The regions 1, 2, …, 5 bounded by the gray lines in Figs. 6(a)-6(c) are divided according to the positions of W_max of the a_b and s_b modes, and correspond to the typical energy density profiles shown in Figs. 6(e)-6(i), respectively.

For MIM structure, the energy confinement is improved when the core refractive index increases. However, for MHLHM, the bound modes can present better energy confinement than the M-H-M structure with the same core thickness (2d₁ + d₂), albeit the mean refractive index of the core of MHLHM is less than n_H. For example, for the a_b mode of MHLHM with d₁ = d₂ = 120 nm, A_eff is equal to 171 nm, which is ~0.1λ₀ and less than 80% of A_eff of the a_b mode in the M-H-M structure with core thickness equal to 360 nm. The s_b mode presents slightly better confinement than the a_b mode in the same MHLHM structure. For the s_b mode of MHLHM with d₁ = d₂ = 120 nm, A_eff is equal to 169 nm. In these cases, the bound modes are confined to the metal surfaces (x = 0⁺ and a₃⁻).

Light can also be enhanced and confined in the middle insulator (L) of MHLHM by utilizing the a_b mode [Fig. 6(e), and region 1 in Fig. 6(a)]. Similar phenomenon has been found in a type of slot waveguide [40

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

], which contains a low-index slot embedded between two high-index slabs. The mode in the slot waveguide is formed by the interaction between the fundamental eigenmodes of the individual slabs and the mechanism is based on total internal reflection [40

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

]. The MHLHM structure can be considered as the slot waveguide sandwiched between the two metal claddings. However the field confinement in the L layer of MHLHM is due to the interaction between the two individual SPPs. Moreover, for practical applications, as a result of the screening of the metal layers, the integration density of MHLHM-based devices can be significantly improved compared with the slot waveguide.

Compared with MHLHM, the bound modes in MLHLM present relatively low energy confinement, which is due to the smaller mean refractive index of core as well as the energy distribution. The effective mode width A_eff of the a_b mode of MLHLM can even be greater than that of the M-L-M structure with the same core thickness [bottom right region bounded by the white lines in Fig. 6(c)], albeit the mean refractive index of MLHLM is greater than n_L. For the a_b mode in MLHLM, high energy confinement is only found at small d_L, in which case the energy is highly confined in the low-index layers between metal and H layers. For the s_b mode in MLHLM, high energy confinement is observed at small d_L as well as the region near the cutoff line [Fig. 6(d)].

We would like to compare the MLHLM structure with a type of hybrid plasmonic waveguide [39

, 41

Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 18(12), 13173–13179 (2010). [CrossRef] [PubMed]

] consisting of low-index material sandwiched between metal and high-index material. The hybrid plasmonic waveguide can be considered as a half of the MLHLM structure and can support the so called hybrid mode [39

]. Like the a_b mode, the hybrid mode can exhibit strong confinement of energy within the low index region. Meanwhile, the energy distribution of the hybrid mode in the low index region is analogous to the a_b mode in MLHLM. We can consider that the a_b mode in MLHLM is formed by the interaction between the two hybrid modes associated with the two halves of the MLHLM structure.

5. Propagation length and figure of merit

The propagation length L_p is defined as the 1 / e decay length of the energy, and is given by L_p = 1 / 2Im(β). L_p of the a_b and s_b modes supported by the MHLHM and MLHLM structures is shown in Fig. 7 , in terms of d₁ and d₂. For the bound modes of MIM structure, L_p increases with increasing insulator thickness and decreasing insulator index. For MMIM, L_p is additionally affected by distribution of core index. In order to make clear the effect of the contrasted core index, we compare L_p of MHLHM and MLHLM with that of the M-H-M and M-L-M structures, which is calculated from Eq. (3) and not given here for simplicity. In this section, the insulator thicknesses of both M-H-M and M-L-M are assumed to be 2d₁ + d₂.

Fig. 7 Propagation length (L_p) of the a_b [(a), (c)] and s_b [(b), (d)] modes of MHLHM [(a), (b)] and MLHLM [(c), (d)] in terms of d₁ and d₂. In (a), at the left (or right) side of the white line, L_p of the a_b mode is greater (or less) than that of the corresponding M-H-M structure. In (c), at the left (or right) side of the white line, L_p of the a_b mode is less (or greater) than that of the corresponding M-L-M structure.

L_p of the a_b mode of MHLHM is shown in Fig. 7(a). At the left side of the white line, L_p of the a_b mode of MHLHM exceeds that of the M-H-M structure. In contrast, at the right side, L_p of the a_b mode of MHLHM is less than that of the M-H-M structure, albeit the mean insulator index of MHLHM is less than the M-H-M structure.

For the a_b mode of MLHLM, as shown in Fig. 7(c), at the left side of the white line, L_p of the a_b mode of MLHLM is comparable with but less than that of the M-L-M structure. When d₁ is large enough, the a_b mode is less affected by the metal claddings. At the right side, L_p of the a_b mode of MLHLM is greater than that of the M-L-M structure and increases exponentially with increasing d₁ or d₂. As an example, when (d₁, d₂) = (300, 250) nm, L_p is as high as 4719 µm, which is over 20 times of L_p of the a_b mode of the M-L-M structure.

High losses are observed for the s_b modes. For both MHLHM and MLHLM, the propagation length of the s_b modes increases with the insulator thickness. Within our parameter range, the maxima of L_p of the s_b mode of MHLHM and MLHLM are 17 µm and 182 µm, respectively, and they are found at the upper right corners of Figs. 7(b) and 7(d), respectively. L_p of the s_b mode never exceeds that of the a_b mode that supported by the same structure. As d₁ or d₂ increase, L_p of the a_b and s_b modes approach each other for the MHLHM structure, however keep apart from each other for the MLHLM structure.

The real part of the effective index of the a_b mode is greater than that of the s_b mode (Fig. 2), so the phase velocity of the a_b mode is less than that of the s_b mode. Whereas high loss of the s_b mode compared with the a_b mode indicates that the group velocity of the a_b mode is greater than that of the s_b mode.

As shown in Figs. 6 and 7, there is a trade-off between mode confinement and propagation length. We define a figure of merit (FOM) to give a comprehensive evaluation of the MMIM structures. The FOM is defined as the ratio of propagation length to effective mode width (FOM = L_p / A_eff). Figures 8(a) -8(d) show FOM of the a_b and s_b modes in the MHLHM and MLHLM structures in terms of d₁ and d₂.

Fig. 8 Figure of merit (FOM) of the a_b [(a), (c)] and s_b [(b), (d)] modes of MHLHM [(a), (b)] and MLHLM [(c), (d)] in terms of d₁ and d₂.

For the a_b mode in MHLHM [Fig. 8(a)], largely, FOM decreases as the H layer thickness increases. For the a_b mode in MLHLM [Fig. 8(c)], FOM increases exponentially as core thickness increases. Extremely high FOM (~10⁴) can be achieved by the a_b mode of MLHLM with core thickness of several hundred nanometers. For example, when (d₁, d₂) = (300, 250) nm, FOM is as high as 2.16 × 10⁴, which is 66 times of FOM of the a_b mode in the M-L-M structure.

For the s_b mode in MHLHM [Fig. 8(b)], FOM increases with increasing core thickness. For the s_b mode in MLHLM [Fig. 8(d)], when the geometric parameters are far away from the cutoff line [region 5 in Fig. 6(d)], FOM increases with increasing core thickness. However, when the geometric parameters are near the cutoff line [region 4 in Fig. 6(d)], generally, with fixing d₁ value, FOM increases and reaches a maximum then decreases as d₂ increases. The maximum also increases as d₁ increases. And when d₁ is large enough (greater than ~300 nm), the maximum appears at d₂ = 0, hence the s_b mode of the M-L-M structure with core thickness equal to 2d₁ works better than that of MLHLM with geometric parameters near the cutoff line. It is noted that FOM of the s_b modes in both MHLHM and MLHLM is less than that of the a_b modes supported by the same structures.

6. Conclusions

In conclusion, the optical properties of the guided modes in MMIM waveguide have been discussed thoroughly. The characteristic equation of the MMIM structure has been derived. With fixing the operating wavelength and the corresponding material parameters, the cutoff property, effective refractive index, energy confinement behavior, propagation length, and figure of merit have been presented with respect to the geometric parameters for two symmetric MMIM structures, MHLHM and MLHLM.

The plasmonic modes supported by MMIM exhibit some unique properties that could not be observed in the MIM structures. In particular, for the a_b mode in MHLHM (or MLHLM), Re(n_eff) can be independent on the H layer thickness at a critical L layer thickness d_C (or d_C₁). And for the s_b mode in MLHLM, Re(n_eff) can be independent on the L layer thickness at a critical H layer thickness d_C₂. The existence conditions of the critical values are given according to the monotonic and asymptotic properties of Re(n_eff). The critical values d_C and d_C₂ exist for all the material parameters n_L and n_H. In contrast, the critical value d_C₁ exists only when n_H is greater than the effective index of the individual SPPs associated with the M-L interface. These phenomena are explained by the properties of the plasmonic modes in MIM waveguide.

The MHLHM structure shows better energy confinement than the MLHLM structure, yet suffers relatively high loss compared with the MLHLM structure. For the a_b mode of the MLHLM structure, since the field is less affected by the metal claddings when the d_L is large, long propagation length of up to millimeters can be observed for core thickness of several hundred nanometers. Overall, the figure of merit of the a_b modes is higher than that of the s_b modes supported by the same structures; and the figure of merit of the MLHLM structure is higher than that of the MHLHM structure. The a_b mode in the MLHLM structure is also superior to that in the MIM structure in terms of propagation length and the figure of merit. Therefore, the a_b mode in the MLHLM structure is of the most applicability.

These results provide physical insights into the guided plasmonic modes in the MMIM structures, and are useful for flexible designing of MIM type waveguides and MIM-based devices.

Appendices

Appendix: field components and calculating method

The nonzero field components are written as follows:

E_{x} = {\begin{array}{l} - i A \frac{n_{e f f} k_{0}}{U_{0}} e^{U_{0} x}, & x \leq 0 \\ - i \frac{n_{e f f} k_{0}}{U_{p}} [A_{p 1} \cosh (U_{p} x) + A_{p 2} \sinh (U_{p} x)], & a_{p - 1} \leq x \leq a_{p}, (p = 1, 2, 3) \\ i A_{4} \frac{n_{e f f} k_{0}}{U_{4}} e^{- U_{4} (x - a_{3})}, & x \geq a_{3} \end{array};

(6)

E_{z} = {\begin{array}{l} A e^{U_{0} x}, & x \leq 0 \\ A_{p 1} \sinh (U_{p} x) + A_{p 2} \cosh (U_{p} x), & a_{p - 1} \leq x \leq a_{p}, (p = 1, 2, 3) \\ A_{4} e^{- U_{4} (x - a_{3})}, & x \geq a_{3} \end{array};

(7)

H_{y} = {\begin{array}{l} - i A \frac{ε_{0} k_{0}}{U_{0}} e^{U_{0} x}, & x \leq 0 \\ - i \frac{ε_{p} k_{0}}{U_{p}} [A_{p 1} \cosh (U_{p} x) + A_{p 2} \sin h (U_{p} x)], & a_{p - 1} \leq x \leq a_{p}, (p = 1, 2, 3) . \\ i A_{4} \frac{ε_{4} k_{0}}{U_{4}} e^{- U_{4} (x - a_{3})}, & x \geq a_{3} \end{array}

(8)

The amplitude coefficients are given by

[\begin{matrix} A_{11} \\ A_{12} \end{matrix}] = [\begin{matrix} N_{01} \\ 1 \end{matrix}] A,

(9)

[\begin{matrix} A_{21} \\ A_{22} \end{matrix}] = [\begin{matrix} - S_{12} & C_{12} \\ C_{12} & - S_{12} \end{matrix}] [\begin{matrix} S_{11} & C_{11} \\ N_{12} C_{11} & N_{12} S_{11} \end{matrix}] [\begin{matrix} A_{11} \\ A_{12} \end{matrix}],

(10)

[\begin{matrix} A_{31} \\ A_{32} \end{matrix}] = [\begin{matrix} - S_{23} & C_{23} \\ C_{23} & - S_{23} \end{matrix}] [\begin{matrix} S_{22} & C_{22} \\ N_{23} C_{22} & N_{23} S_{22} \end{matrix}] [\begin{matrix} A_{21} \\ A_{22} \end{matrix}],

(11)

[\begin{matrix} A_{4} \\ A_{4} \end{matrix}] = [\begin{matrix} S_{33} & C_{33} \\ - N_{34} C_{33} & - N_{34} S_{33} \end{matrix}] [\begin{matrix} A_{31} \\ A_{32} \end{matrix}] .

(12)

Here S_pq = sinh(a_pU_q), C_pq = cosh(a_pU_q), with p, q = 1, 2, 3.

In this work, the complex transcendental characteristic equation is solved by an iterative algorithm. Taking the right side of Eq. (3) equal to f, the iteration formula is written as

\begin{array}{l} f_{r}^{(p)} {n_{r}^{(p)}; n_{i}^{(p - 1)}} & = & 0, \\ f_{i}^{(p)} {n_{r}^{(p)}; n_{i}^{(p)}} & = & 0, \end{array}

(13)

where f_r = Re{f}, f_i = Im{f}, n_r = Re(n_eff), n_i = Im(n_eff), and p = 1, 2, ... is the iteration number. The initial value n_i⁽⁰⁾ is set to zero in the iterative procedure. In our work, the residues of Re{f} and Im{f} are controlled to be less than 10⁻¹⁴ and 10⁻¹⁶, respectively, except for the cases that the modes nearly cut off (when a mode cuts off, Im(n_eff) approaches infinity, so strictly no solution exists at a cutoff point).

Acknowledgments

This research is supported by the Chinese National Key Basic Research Special Fund (2011CB922003), the National Natural Science Foundation of China (11074132), the Fundamental Research Funds for the Central Universities, Tianjin Natural Science Foundation (09JCYBJC01600), and the 111 Project (B07013).

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13.	Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). [CrossRef]
14.	V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett. 9(10), 3489–3493 (2009). [CrossRef] [PubMed]
15.	D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett. 86(16), 161101 (2005). [CrossRef]
16.	D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [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. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B 83(16), 165412 (2011). [CrossRef]
19.	D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]
20.	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]
21.	A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35(13), 2118–2120 (2010). [CrossRef] [PubMed]
22.	J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72(7), 075405 (2005). [CrossRef]
23.	J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]
24.	J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [CrossRef] [PubMed]
25.	J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95(1), 013504 (2009). [CrossRef]
26.	B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter 44(24), 13556–13572 (1991). [CrossRef] [PubMed]
27.	E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
28.	R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef] [PubMed]
29.	N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett. 32(21), 3086–3088 (2007). [CrossRef] [PubMed]
30.	D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express 18(17), 17958–17966 (2010). [CrossRef] [PubMed]
31.	S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006). [CrossRef]
32.	Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express 18(12), 13258–13270 (2010). [CrossRef] [PubMed]
33.	Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett. 22(9), 634–636 (2010). [CrossRef]
34.	M.-S. Kwon, “Metal-insulator-silicon-insulator-metal waveguides compatible with standard CMOS technology,” Opt. Express 19(9), 8379–8393 (2011). [CrossRef] [PubMed]
35.	S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett. 98(2), 021107 (2011). [CrossRef]
36.	M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007). [CrossRef]
37.	M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]
38.	E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
39.	R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]
40.	V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]
41.	Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 18(12), 13173–13179 (2010). [CrossRef] [PubMed]

OCIS Codes
(230.4170) Optical devices : Multilayers
(230.7390) Optical devices : Waveguides, planar
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 21, 2012
Revised Manuscript: April 15, 2012
Manuscript Accepted: May 7, 2012
Published: May 14, 2012

Citation
Xiang-Tian Kong, Wei-Guo Yan, Zu-Bin Li, and Jian-Guo Tian, "Optical properties of metal-multi-insulator-metal plasmonic waveguides," Opt. Express 20, 12133-12146 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12133

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References

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M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science328(5977), 440–441 (2010). [CrossRef] [PubMed]
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A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B78(4), 045425 (2008). [CrossRef]
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T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett.94(5), 051111 (2009). [CrossRef]
H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett.96(9), 097401 (2006). [CrossRef] [PubMed]
X. Zhu, J. Zhang, J. Xu, and D. Yu, “Vertical plasmonic resonant nanocavities,” Nano Lett.11(3), 1117–1121 (2011). [CrossRef] [PubMed]
J.-C. Weeber, A. Bouhelier, G. Colas des Francs, S. Massenot, J. Grandidier, L. Markey, and A. Dereux, “Surface-plasmon hopping along coupled coplanar cavities,” Phys. Rev. B76(11), 113405 (2007). [CrossRef]
Y. Gong and J. Vuckovic, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett.90(3), 033113 (2007). [CrossRef]
V. J. Sorger, R. F. Oulton, J. Yao, G. Bartal, and X. Zhang, “Plasmonic fabry-pérot nanocavity,” Nano Lett.9(10), 3489–3493 (2009). [CrossRef] [PubMed]
D. F. P. Pile and D. K. Gramotnev, “Nanoscale fabry-prot interferometer using channel plasmon-polaritons in triangular metallic grooves,” Appl. Phys. Lett.86(16), 161101 (2005). [CrossRef]
D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010). [CrossRef]
T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B75(24), 245405 (2007). [CrossRef]
Y. A. Akimov and H. S. Chu, “Plasmon coupling effect on propagation of surface plasmon polaritons at a continuous metal/dielectric interface,” Phys. Rev. B83(16), 165412 (2011). [CrossRef]
D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett.47(26), 1927–1930 (1981). [CrossRef]
P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B61(15), 10484–10503 (2000). [CrossRef]
A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett.35(13), 2118–2120 (2010). [CrossRef] [PubMed]
J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B72(7), 075405 (2005). [CrossRef]
J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B73(3), 035407 (2006). [CrossRef]
J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett.6(9), 1928–1932 (2006). [CrossRef] [PubMed]
J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett.95(1), 013504 (2009). [CrossRef]
B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B Condens. Matter44(24), 13556–13572 (1991). [CrossRef] [PubMed]
E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182(2), 539–554 (1969). [CrossRef]
R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A21(12), 2442–2446 (2004). [CrossRef] [PubMed]
N.-N. Feng and L. Dal Negro, “Plasmon mode transformation in modulated-index metal-dielectric slot waveguides,” Opt. Lett.32(21), 3086–3088 (2007). [CrossRef] [PubMed]
D. Dai and S. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express18(17), 17958–17966 (2010). [CrossRef] [PubMed]
S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun.258(2), 295–299 (2006). [CrossRef]
Y.-J. Chang, “Design and analysis of metal/multi-insulator/metal waveguide plasmonic Bragg grating,” Opt. Express18(12), 13258–13270 (2010). [CrossRef] [PubMed]
Y.-J. Chang and G.-Y. Lo, “A narrow band metal-multi-insulator-metal waveguide plasmonic Bragg grating,” IEEE Photon. Technol. Lett.22(9), 634–636 (2010). [CrossRef]
M.-S. Kwon, “Metal-insulator-silicon-insulator-metal waveguides compatible with standard CMOS technology,” Opt. Express19(9), 8379–8393 (2011). [CrossRef] [PubMed]
S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Fully complementary metal-oxide-semiconductor compatible nanoplasmonic slot waveguides for silicon electronic photonic integrated circuits,” Appl. Phys. Lett.98(2), 021107 (2011). [CrossRef]
M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics1(10), 589–594 (2007). [CrossRef]
M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express17(13), 11107–11112 (2009). [CrossRef] [PubMed]
E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008). [CrossRef]
V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett.29(11), 1209–1211 (2004). [CrossRef] [PubMed]
Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express18(12), 13173–13179 (2010). [CrossRef] [PubMed]

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