1. Introduction

Surface plasmon-polaritons (SPPs) have attracted a lot of interest and been topics of numerous studies due to their sub-diffraction-limit confinement of light which opens huge opportunity to subwavelength optics [1]. The strong light confinement of the SPP results in field enhancement and hence, enhanced light-matter interaction such as optical nonlinearity. Energy-efficient nonlinear-optic device operations can also be potential applications of the SPP.

To implement compact planar routing means in integrated optical circuits, various SPP-based waveguide structures have been proposed [2–15]. Among them the channel plasmon polaritons(CPPs) or the wedge plasmon polaritons (WPPs) in V-shaped grooves or wedges seems promising in terms of compact confinement, that is, small modal area [8–15]. In general, however, the SPP-based waveguides show a trade-off between the field confinement and the propagation length: a waveguide of the smaller modal area have the shorter propagation length. The modal area and the propagation length can be tailored by coupling SPP modes in multiple metal-dielectric interface structures such as metal-dielectric-metal (MDM) or dielectric-metal-dielectric (DMD) structures [2,16–18]. Particularly, in the DMD structures, two SPP supermodes of even and odd symmetries are formed, which are dubbed long range(LR)- and short range(SR)-SPP modes, respectively. The LRSPP shows a larger mode size and a several orders of magnitude longer propagation length than the SPP at the single interface between a dielectric medium and a metal. On the other hand, the mode size of the SRSPP is smaller and the propagation length is much shorter than the single interface SPP. The mode sizes and the propagation lengths of the LR-SPP and the SR-SPP depend on the thickness of the metal film and their opposite tendencies intensify as the metal film thickness decreases. If the mode size can be compromised, very useful functionality can be obtained using the LR-SPP-based waveguides. The waveguides based on the LR-SPP show another useful feature that the boundness of the modes is very sensitive to the index difference between the surrounding dielectric media in the DMD structures and the bound modes turn into radiative modes if the index difference is large enough [18]. This feature has been used for implementation of tunable attenuators in conjunction with thermo-optic or electro-optic materials [19,20]. In these devices, interferometric structures are not needed since the radiative nature of the modes in the asymmetric dielectric media directly provides intensity modulation of an input signal controlled by the index asymmetry. In contrast to the LR-SPP, the SR-SPP does not show cutoff with an index asymmetry in the low frequency limit [18].

Recently, the effect of a finite metal film thickness on the propagation characteristics of the CPP and the WPP modes in a V-groove of a metal film has been numerically studied, and it has been shown that mode sizes and the propagation length decrease for both types of modes as the metal thickness decreases [21]. However, their work on the CPP mode was focused only on the CPP mode originated from the SR-SPP, which is dubbed SR-CPP in this paper, in the interest of compact light confinement, and the effect of the finite film thickness is not fully explored. In this work, we numerically investigated the characteristics of another type of the CPP mode originated from the LR-SPP in the thin metal film V-groove structure, which is dubbed LR-CPP in this paper. As expected from the properties of the LR-SPP, the LR-CPP mode shows a larger mode size and an extended propagation length compared to the SR-CPP mode. Moreover, when asymmetry of indices in the surrounding dielectric media is introduced, the LR-CPP mode shows rather different mode cutoff mechanisms and thus, more sensitive radiative behavior variation compared to the SR-CPP mode. This is a quite useful feature enabling all-optical intensity modulation without interferometric structures in conjunction with nonlinear-optic medium and the unique geometry of the thin film V-groove. The modal analysis has been carried out by using a commercial software based on the finite element method (FEM), COMSOL. In this work, our investigation is more focused on the modal characteristics of the LR-CPP at optical communication wavelengths, and the characteristics of the SR-CPP and the WPP modes are analyzed for the purpose of comparison to the LR-CPP.

2. Guiding principle of thin metal film V-grooves: two types of channel plasmon polariton modes

The thin metal(gold) film V-groove waveguide considered in this work is depicted in Fig. 1(a) which is similar to the structure considered in ref [21]. All the corners are rounded to avoid field singularities in the same way as in ref [21]: the radius of curvature of the inner corner is 10 nm and that of the outer corner is (t + 10) nm. For the surrounding medium, chalcogenide glass (As₂S₃, n = 2.38) is selected. Asymmetric surrounding media cases (n₁ ≠ n₂) as well as symmetric surrounding medium cases (n₁ = n₂) are considered. θ and h are the angle and the depth of the groove, respectively. The wave propagates along the z-direction and complex permittivity ε_Au of gold is extracted directly from Palik’s experimental data in wavelength range of 0.7~2.0 μm [22], and for the wavelength longer 2.0 μm, a modified Drude-Lorentz model is used since experimental data are not available [23].

 

Fig. 1 (a) Geometry of thin metal film V-groove waveguide. Electric field profiles, E_x for (b) Symmetrically-coupled LR-SPPs, (c) Symmetrically-coupled SR-SPPs, (d) Antisymmetrically-coupled LR-SPPs, and (e) Antisymmetrically-coupled SR-SPPs in one-dimensional dielectric/metal/dielectric/metal/dielectric (D/M/D/M/D) structure. Also, (f) and (g) shows the complex effective index as functions of gap distance for (b) and (c) cases, respectively: t = 10nm, n₁ = n₂ = 2.38, and λ = 1.55μm.

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The guiding principle of the thin metal film V-groove can be explained in terms of the SPP mode coupling between two metal films and its dependence on the gap distance. When two metal films are closely placed in a uniform dielectric medium, there are four possible SPP supermodes as illustrated in Fig. 1(b)-(e) [16,24]. Each metal film has two modes of even and odd symmetries, that is, LR-SPP and SR-SPP modes. Coupling of each type of mode between two films forms two supermodes of even and odd symmetries. The modes in Fig. 1(b) and (c) correspond to the symmetrically-coupled LR-SPP and SR-SPP modes. The anti-symmetrically-coupled LR-SPP and SR-SPP are depicted in Fig. 1(d) and (e). These four types of modes can be calculated by semi-analytically solving one-dimensional Maxwell’s equations [16] and the calculated complex effective indices (N_eff) of the symmetrically-coupled LR-SPP and SR-SPP modes for different gap distances (d) are plotted in Fig. 1(f) and (g). The complex effective index (N _eff) is defined as ${\beta }_z\lambda /2\pi$, where ${\beta }_z$ is the modal wave vector. Gold films (ε_Au = −131.95 + 12.65j) of t = 10 nm in a uniform medium of n = 2.38 is assumed and the calculation is carried out for the operating wavelength of λ = 1.55 μm. The real part of the complex effective indices of both symmetrically-coupled modes decrease as d increases, which is consistent with the previous research [24], whereas the anti-symmetrically-coupled modes show increasing effective indices as d increases, which are not plotted here. Therefore, in the thin metal film V-groove, both symmetrically-coupled modes can be confined inside the groove and form CPP modes, but the anti-symmetrically-coupled modes cannot. The guided mode in the thin metal film V-groove originated from the symmetrically-coupled LR-SPP is expected to have a longer propagation length than the one originated from the SR-SPP. From now on, the former mode is dubbed LR-CPP and the latter, SR-CPP. In ref [21], only the SR-CPP was investigated. This guiding principle is valid regardless of an operating wavelength, so that an infinite thin metal film V-groove is expected to show no cutoff for the LR-CPP and the SR-CPP in a long wavelength limit as the infinite bulk V-groove [13].

As reported in ref [21], there is another type of mode in the thin metal V-groove originated from WPP, whose guiding principle is quite different from those of the CPP modes. In this work, however, the WPP mode is not investigated in depth since our focus is the radiative behavior of the LR-CPP at the optical communication wavelengths.

3. Characteristics of long-range channel plasmon polaritons

Two types of guided modes in the V-grooves of various metal film thicknesses (t) were calculated numerically by using the FEM. In the FEM, Maxwell’s equations are solved in frequency domain, and discretization of equations with a triangular mesh results in an algebraic eigenvalue problem. The eigenvalues are related to wave vectors of modes, ${\beta }_z$. The complex effective index (N _eff) and the propagation length are defined as ${\beta }_z\lambda /2\pi$ and $1/2\mathrm{Im}({\beta }_z)$, respectively. The depth and the angle of the groove are h = 4 μm and θ = 30°, respectively, a symmetric dielectric medium (n₁ = n₂ = 2.38) is assumed, and the operating wavelength is λ = 1.55 μm. In the calculation using the FEM, the mesh was set densely near the metal film, so that the maximum mesh size was less than 5 nm. Far away from the metal film, rather coarse mesh was used to save memories, so that the maximum mesh size was less than 200 nm. The calculation domain was set large enough to be 30 μm x 30 μm. The calculated effective indices and the propagation lengths are plotted in Fig. 2(a) and (b) , respectively. The propagation length of the LR-CPP mode is about three orders of magnitude larger than that of the SR-CPP mode for t = 10 nm. As seen in Fig. 2(c) and (d), the mode size of the LR-CPP mode is much larger than that of the SR-CPP mode. A modal area can be defined as the ratios of the total energy and the peak energy density as in ref [7]:

 

Fig. 2 (a) Effective index and (b) propagation length with metal thickness for LR-CPP and SR-CPP modes for various metal thickness. Dominant electric field profiles, E_x for (c) LR-CPP with t = 10nm, (d) SR-CPP with t = 10nm, (e) LR-CPP with t = 100nm, and (f) SR-CPP with t = 100nm. (h = 4μm, θ = 30°, λ = 1.55μm, n₁ = n₂ = 2.38in all calculations.)

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The calculated modal areas of the LR-CPP and the SR-CPP for t = 10 nm at λ = 1.55 μm shown in Fig. 2(c) and (d) are approximately 8.02 x 10⁻⁶ mm² and 4.27 x 10⁻⁹ mm², respectively. For these modes, the figure-of-merits (FOMs), which are defined as FOM = (propagation length)/(modal area)^1/2 as in ref [7], are calculated to be approximately 207.4 and 11.1, respectively. It appears that for very thin metal V-grooves, the LR-CPP shows better performance at optical wavelengths in terms of the FOM. As the metal thickness increases, the propagation lengths of the LR-CPP and the SR-CPP modes change in opposite ways and asymptotically tend to the propagation lengths of the single interface SPP and the CPP in the bulk groove, respectively. For the metal film thickness larger than the skin depth, the coupling between the SPP modes on two surfaces in the metal film disappears. As a result, the LR-CPP tends to the SPP bound at the outer surfaces of the groove as seen in the mode profile for t = 100 nm (Fig. 2(e)), and the SR-CPP tends to the CPP bound inside the groove as seen in Fig. 2(f). The effective indices of the LR-CPP and SR-CPP also tend to those of the single interface SPP and the CPP in bulk groove.

The dependences of the propagation characteristics of LR-CPP on groove angle, groove depth, and wavelength are investigated. In all calculation, the metal thickness of t = 10 nm is assumed. In Fig. 3(a) , the effective index and the propagation length of the LR-CPP in a thin film V-groove of h = 5 μm are plotted as functions of θ at λ = 1.55 μm. As the groove angle increases, the effective index monotonously decreases, whereas the propagation length shows a maximum value for the angle of about 30°. The field distributions of the LR-CPP modes in grooves of θ = 10°, 30°, and 60° at λ = 1.55 μm are illustrated in Fig. 3(d), (e), and (f). As the angle increases, the coupling between LR-SPP modes of two sides of the groove gradually becomes weaker and the propagation length increases up to a certain angle as a result of the increased mode size. For very large angles, eventually the LR-CPP mode is decoupled into LR-SPP modes bound to the individual sides of the groove. Since the decoupled LR-SPP mode is weakly confined in lateral direction, the field experiences more metal and consequently, the propagation length is reduced. Figure 3(b) shows the plots of the effective index and the propagation length as functions of h for the groove of θ = 30° at λ = 1.55 μm, and both of them increase as h increases. The increase of the effective index stems from the increase of the amount of the metal with h. The field distribution of the LR-CPP mode in the groove of h = 8 μm and θ = 30° is shown in Fig. 3(g). By comparing Fig. 3(e) and (g), one can see that as h increases, the mode size gets larger and the portion of the field confined in the metal gets relatively smaller. This explains the increase of the propagation length with h. Figure 3(c) shows the plots of the effective index and the propagation length as functions of λ for the groove of h = 5 μm and θ = 30° and Fig. 3(h) shows the field distribution of the LR-CPP mode in the same groove at λ = 0.729 μm. Wavelength dependence of the LR-CPPs in the thin metal film V-groove is the same as general metal-base waveguides. The shorter operating wavelength causes the larger effective index, the smaller modal area, and the shorter propagation length.

 

Fig. 3 Effective index and propagation length for LR-CPP as functions of (a) groove angle (h = 5μm, λ = 1.55μm), (b) groove depth (θ = 30°, λ = 1.55μm), and (c) wavelength (h = 5μm, θ = 30°). Dominant electric field profiles, E_x for LR-CPP with different compositions: (d) θ = 10°, h = 5μm, λ = 1.55μm, (e) θ = 30°, h = 5μm, λ = 1.55μm, (f) θ = 60°, h = 5μm, λ = 1.55μm, (g) θ = 30°, h = 8μm, λ = 1.55μm, (h) θ = 30°, h = 5μm, λ = 0.729μm. (t = 10nm, n₁ = n₂ = 2.38 in all calculations.)

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4. Radiative behavior of long-range channel plasmon polaritons in asymmetric environment

In this section, the mode cutoff behavior of the LR-CPP is investigated in comparison to that of the SR-CPP. In order to investigate the cutoff behaviors, dispersion curves are calculated for the thin metal (gold) V-groove of h = 5 μm, θ = 30°, and n ₁ = n ₂ = 2.38. Figure 4(a) and 4(b) show the calculated dispersion curves for t = 10 nm and t = 20 nm, respectively. For reference, the dispersion curve of the WPP is also included. As seen in Fig. 4(a) and 4(b), all the modes in V-grooves with uniform dielectric medium show cutoff due to the finite groove size. In our calculation, the mode cutoff is defined as an occasion that the real part of N _eff becomes equal to the highest value of the surrounding dielectric media. For the wavelengths longer than the cutoff wavelengths, mode becomes radiative since the effective index of the bound mode cannot be smaller than the index of the surrounding dielectric medium. For t = 10 nm, cutoff energies (wavelengths) for the LR-CPP, the SR-CPP, and the WPP are approximately 0.667 eV (1.86 μm), 0.105 eV (11.84 μm), and 0.077 eV (16.06 μm), respectively. For t = 20 nm, cutoff energies (wavelengths) for the LR-CPP, the SR-CPP, and WPP are approximately 0.506 eV (2.45 μm), 0.137 eV (9.05 μm), and 0.097 eV (12.75 μm), respectively. The LR-CPPs show much shorter cutoff wavelengths (higher cutoff energies) than the SR-CPP and the WPP and most sensitive change to the film thickness change because of the weakest mode confinement. As the metal film gets thinner, the dispersion curve of the LR-CPP becomes closer to the light line and its cutoff energy increases since the confinement of the LR-CPP gets weaker. In contrast, the SR-CPP is more strongly confined for the thinner metal film, and thus, its dispersion curve and cutoff energy show opposite tendencies. These opposite tendencies of the LR-CPP and the SR-CPP can be understood from the same behaviors of the LR-SPP and the SR-SPP in the flat metal film for the film thickness change. Somehow, the WPP shows similar tendency to the SR-CPP. Figure 4(c) shows the calculated propagation lengths for t = 10 nm. The propagation lengths monotonically increase as photon energy gets closer to the cutoff energy.

 

Fig. 4 Dispersion curves for LR-CPP, SR-CPP, and WPP modes with (a) t = 10nm and (b) t = 20nm in a uniform medium of n₁ = n₂ = 2.38. The dashed line represents the light line of n = 2.38. (c) The propagation length of the modes for t = 10nm. (θ = 30°, h = 5μm in all calculations.)

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In order to investigate the effect of index asymmetry, the dispersion curves for the CPP modes are calculated for the V-groove of h = 5 μm, θ = 30°, and t = 10 nm with various index asymmetries. Figure 5(a) and 5(b) show the calculated dispersion curves near the cutoff energy for n₁ = 2.384 and 2.39 with n₂ = 2.38. The cutoff energy (wavelength) of the LR-CPP varies to 0.827 eV (1.5 μm) and 0.984 eV (1.26 μm) for n₁ = 2.384 and 2.39, respectively. For the SR-CPP shown in Fig. 5(b), the cutoff energy (wavelength) varies to 0.1055 eV (11.75 μm) and 0.106 eV (11.7 μm) for n₁ = 2.384 and 2.39, respectively. Another type of index asymmetry with a higher-index lower clad (n₂ > n₁ = 2.38) is also considered, and the calculated dispersion curves are plotted in Fig. 5(c) and 5(d). The cutoff energy (wavelength) of the LR-CPP varies to 0.713 eV (1.74 μm) and 0.821 eV (1.51 μm) for n₂ = 2.384 and 2.39, respectively. For the SR-CPP shown in Fig. 5(d), the cutoff energy (wavelength) varies to 0.107 eV (11.6 μm) and 0.112 eV (11.1 μm) for n₂ = 2.384 and 2.39, respectively.

 

Fig. 5 Dispersion curves for two CPP modes with various index asymmetries; (a) LR-CPP and (b) SR-CPP with n₁>n₂ = 2.38, (c) LR-CPP and (d) SR-CPP with n₂>n₁ = 2.38. The dashed line represents the light line of n = 2.38. The number in each legend denotes the index in the higher-index region. Dispersion curves for symmetric environment are also plotted for reference. (t = 10nm, θ = 30°, h = 5μm in all calculations.)

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Overall, the LR-CPP shows more sensitive cutoff energy changes for the index asymmetries than the SR-CPP. For the SR-CPP, cutoff occurs due to the finite size of the groove, and the index asymmetry does not considerably affect dispersion characteristic and mainly changes the cutoff conditions. (Note that the mode cutoff is defined as an occasion that the real part of N _eff becomes equal to the highest value of the surrounding dielectric media.) Consequently, the index asymmetry causes the cutoff energy change. In contrast to the SR-CPP, the cutoff behavior of the LR-CPP is greatly affected by the index asymmetry via two mechanisms. For the case of n₁ > n₂, the cutoff mechanism induced by the index asymmetry in the LR-CPP can be explained by the one-dimensional model used for the guiding principle of the CPP modes in section 2. By using the semi-analytic method [16], the guided modes are calculated for two coupled flat metal (gold) films of t = 10 nm where the index of the core (n₁) is different from that of the clad (n₂ = 2.38). For comparison, both symmetrically-coupled LR-SPP and SR-SPP are considered. Figure 6 shows the calculated complex effective indices as functions of the gap distance. The operating wavelengths close to the cutoff wavelengths of the LR-CPP and the SR-CPP are chosen, which are 1.55 μm and 11.8 μm, respectively. One can see that for the symmetrically-coupled LR-SPP mode, the trend of the effective index changes completely with a small index difference between the core and the clad. The effective index is almost a constant in the case n₁ = 2.39 and it increases as d increases in the case of n₁ = 2.40. This implies that if the index of the core is larger than 2.39, the symmetrically-coupled LR-SPP mode cannot be confined inside the groove so that the LR-CPP mode becomes radiative even in the infinite V-groove. On the other hand, for the symmetrically-coupled SR-SPP mode, the effective index shows the same decreasing trend even for n₁ = 3.38 with increasing d. Therefore, even with introduction of a large index asymmetry, radiative behavior is not expected for the SR-CPP mode in the infinite groove. Although this one-dimensional model directly applies only to an infinite V-groove, it reveals that besides the finite size effect there is another cutoff mechanism for the LR-CPP induced by the index asymmetry of the higher-index core (n₁ > n₂). It is surmised that this is why the LR-CPP shows much larger cutoff wavelength change for such index asymmetry.

 

Fig. 6 Dependence of complex effective indices on gap distance in two coupled metal films with different core indices, n₁: (a) real and (b) imaginary part of effective index for symmetrically-coupled LR-SPP, (c) real (d) imaginary part of effective index for symmetrically-coupled SR-SPP. The insets show corresponding mode shapes. The operating wavelengths are chosen close to the cutoff wavelengths of the LR-CPP and the SR-CPP: λ = 1.55μm in (a) and (b), and λ = 11.8 μm in (c) and (d). (t = 10nm, and n₂ = 2.38 in all calculations.)

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For the large enough index asymmetry of the higher-index clad (n₁ < n₂), the cutoff of the LR-CPP is caused by the cutoff of the LR-SPP in each side of the V-groove. (Note that in the flat metal film, the index asymmetry induces only the cutoff of the LR-SPP [18].) This also makes the cutoff characteristic of the LR-CPP more sensitive to the index asymmetry (n₁ < n₂) than the SR-CPP.

Those two cutoff mechanisms for the LR-CPP in a finite V-groove are discussed in detail later in this section.

For t = 20 nm case, which is not show here, the cutoff energy of the LR-CPP is less sensitive to the index asymmetry than t = 10 nm case, but still much more sensitive than the SR-CPP.

Figure 7(a) and 7(b) shows the calculated dispersion curves of the WPP with index asymmetries. The WPP appears to show considerable cutoff energy changes (from 0.0772 eV to 0.089 eV for n ₁ = 2.39) with the index asymmetry of the higher-index core, while very small cutoff energy changes (from 0.0772 eV to 0.0775 eV for n ₂ = 2.39) are observed with the index asymmetry of the higher-index clad. Physical origin of this phenomenon is not clear at this moment. As seen in Fig. 7(c), fields are confined completely near the outer surface of the groove and there is almost no field in the core in the WPP mode. Therefore, it is expected that the core index change would not affect the property of the WPP. However, this is quite opposite to our observation. This counter-intuitive behavior of the WPP remains as a topic of future research at this moment.

 

Fig. 7 Dispersion curves for the WPP modes with index asymmetries of (a) n₁>n₂ = 2.38 and (b) n₂>n₁ = 2.38. (t = 10nm, θ = 30°, h = 5μm) The dashed line represents the light line of n = 2.38. The number in each legend denotes the index in the higher-index region. Dispersion curves for symmetric environment are also plotted for reference. (c) Dominant electric field profiles, E_x for the WPP of n₁ = 2.39, n₂ = 2.38, t = 10nm θ = 30°, h = 5μm, λ = 14μm (0.089 eV).

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Two types of the index asymmetry induced cutoff mechanisms for the LR-CPP discussed above are investigated in detail. Since the cutoff energy of the LR-CPP mode in thin metal (gold) film V-grooves of h = 5 μm, θ = 30°, and t = 10 nm changes from 0.667 eV (1.86 μm) to 0.827 eV (1.5 μm) for index asymmetry of 4 x 10⁻³ as shown in Fig. 5(a), the modal characteristic change from bound to radiative modes is expected for the LR-CPP at λ = 1.55 μm. The radiative characteristics of the LR-CPP are investigated in detail with various index asymmetries at λ = 1.55 μm, which are plotted in Fig. 8 . As seen in Fig. 8(a), as n₁ increases from 2.38 with n₂ fixed to 2.38 (n₁ > n₂ = 2.38), the effective index increases slowly and reaches the value of n₁ when n₁ ≈2.383. If the core index is increased further (n₁ > 2.383), mode cutoff occurs and the LR-CPP mode becomes radiative as expected. Unfortunately, by using numerical mode solving method including the FEM, we can hardly find the exact solution for the radiative modes since the numerical methods are supposed to find only bound modes accurately. Although the accuracy of the calculated eigenvalues, which are related to the complex effective index, is rather doubtful, the radiative nature of the mode can be observed from the calculated field distribution in the cutoff regime with adopting the perfectly matched layer (PML) absorbing boundary condition [25]. The field distributions of the LR-CPP calculated in the cutoff regime (n₁ = 2.384 and 2.385, n₂ = 2.38) are shown in Fig. 8(d) and 8(e). For clear observation of the radiative nature, three-dimensional plots are used, and for reference, Fig. 8(c) shows the field distribution of the LR-CPP in the groove with the symmetric medium (n₁ = n₂ = 2.38). In Fig. 8(d) and 8(e), we can clearly see that the fields radiate into upside region of the groove and the radiation intensifies for the larger index asymmetry. One can see that the index asymmetry of 4 x 10⁻³ is enough to make the LR-CPP mode radiative at λ = 1.55 μm as expected. The calculated propagation length is plotted in Fig. 8(b). The propagation length decreases as the core index increases before the mode cutoff. This is because the mode experiences more metal as the mode center shifts upward. Once the mode turns radiative, propagation length increase is expected and this is conformed by the calculation in the cutoff regime (red dots in Fig. 8(b)).

 

Fig. 8 Propagation characteristics of LR-CPP in asymmetric environment. (a) Effective index and (b) propagation length for LR-CPP as functions of index asymmetry. Dominant electric field profiles, E_x for LR-CPP with different index compositions: (c) n₁ = 2.38, n₂ = 2.38, (d) n₁ = 2.384, n₂ = 2.38, (e) n₁ = 2.385, n₂ = 2.38, (f) n₁ = 2.38, n₂ = 2.392. Because of even symmetry, in x direction, only the half of the mode profile is plotted. (t = 10nm, h = 5μm, θ = 30°, λ = 1.55μm and calculation domain is 15μm x 60μm in all calculations.)

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The radiative behavior of the LR-CPP is also observed in the index asymmetry of higher index clad case (n₂ > n₁ = 2.38). As seen in Fig. 8(a), for n₂ > n₁ = 2.38, the effective index appears to reach n₂ for n₂ = 2.389 and the mode cutoff occurs. As seen in the field distribution calculated in the cutoff regime (n₁ = 2.38, n₂ = 2.392) illustrated in Fig. 8(f), the field radiates in a downward direction of the groove. This can be explained from the cutoff behavior of the LR-SPP mode in a flat metal film that fields radiate into the higher index dielectric medium if an index asymmetry is large enough [18]. Therefore, it can be said that in the higher index clad case, the mode cutoff of the LR-CPP stems from the cutoff of the LR-SPP bound in both sides of the thin metal film groove. As seen in Fig. 8(b), the propagation length increases as the clad index increases since the mode gets to experience less metal as the asymmetry increases.

5. Discussion and conclusion

The LR-CPP mode in the thin metal film V-groove show quite different characteristics from the SR-CPP mode as discussed in the previous sections. This difference stems from the fact that the LR-CPP is originated from the LR-SPP while the SR-CPP, from the SR-SPP. The rather large mode size of the LR-CPP may be disadvantageous in terms of integration density, but the large propagation length and more sensitive cutoff variation to an asymmetric environment may find many applications. Especially, the radiative characteristic of the LR-CPP under the index asymmetry with the higher core index appears promising for the applications in conjunction with electro-optic or thermo-optic media since it is more sensitive to the index asymmetry. Moreover, the vertically asymmetric geometry of the thin metal film V-groove enables all-optical intensity modulation without introduction of interferometric structures if a proper optical nonlinear medium of a positive Kerr coefficient is selected as a surrounding medium. Since the LR-CPP mode confines most of light inside the groove as seen in Fig. 2(c), increasing optical intensity launched into the groove will cause the increase of the core index and result in transmission modulation through radiation due to the index asymmetry. We believe this is a quite unique and useful functionality of the LR-CPP mode in the thin metal film V-groove.

Similar control of mode boundness can also be obtained for the SR-CPP and the WPP at the operating wavelength near the cutoff wavelengths. To observe the cutoff behavior of the SR-CPP and the WPP at optical communication wavelengths, the dimension of the V-groove should be much smaller than the case of the LR-CPP, since the cutoff wavelengths of the SR-CPP and the WPP are much longer than that of the LR-CPP. This small waveguide dimension may be advantageous in terms of integration density, but disadvantageous in terms of short propagation lengths and large insertion loss for optical fiber coupling.

In conclusion, we have numerically investigated the propagation characteristics of the LR-CPP mode in the thin metal film V-groove and compared to the previously studied SR-CPP mode. In contrast to the SR-CPP mode, the LR-CPP mode is originated from the LR-SPP so that it shows much longer propagation length and its boundness can be more sensitively controlled by the index asymmetry. This LR-SPP-like feature in conjunction with the unique geometric feature will find useful applications.