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Guided Plasmon Modes of Elliptical Cross Section Silver Nanoridges

Propagating two-dimensional plasmon modes guided by elliptical cross section silver nanoridges are investigated in this paper. Mode field profiles, dispersion curves, propagation distances, and figure-of-merits of the plasmon ridge modes are calculated for various elliptical cross sections. It is found that an elliptical cross section metal nanoridge, if designed properly, can support a well-confined plasmon mode with a longer propagation distance and a higher figure-of-merit than the flat-top nanoridge plasmon waveguide of the same width. The optimal ridge waveguide cross section is obtained when the elliptical ridge top has a semicircular cross section. When the curvature of the elliptical nanoridge is large, the mode approaches the tightly confined wedge plasmon mode.

Recently, experimental investigation concerning a gold nanoridge, fabricated with the focused ion beam milling technique [52], confirms that metal nanoridges can also support propagating surface plasmon modes. The 2D confined plasmon modes of flat-top silver nanoridges have been investigated recently [53].
This paper presents comprehensive numerical investigations of the plasmon modes supported by elliptical cross section silver nanoridges. We calculate the mode field distributions, mode indices, dispersion curves, propagation distances, mode sizes, and the figure-of-merits of these silver nanoridges with various surface curvatures as well as their dependence on the wavelength and the ridge width. The motivation for investigating such elliptical cross section silver nanoridge waveguides is to find an optimal geometry that provides longer propagation distance and higher figure-of-merit. In addition, as will be shown, elliptical cross section nanoridge waveguides provide more uniform distributions of the surface plasmon mode fields when compared to the flat-top and wedge plasmon waveguides. This feature is important in applications requiring an expanded interaction region such as in chemical and biological sensing.

WAVEGUIDES
The 3D view and cross section of such an elliptical silver nanoridge are shown in Fig. 1(a) and (b), respectively, where the nano-scale metal ridge is extended in the z-direction with its width (w) in the x-direction and the half ellipse height (h) in the y-direction. The plasmon wave propagates along the ridge top in the z-direction. We assume the height of the ridge is sufficiently large so that the substrate does not influence the ridge plasmon mode. Our previous work on the plasmon modes supported by the flat-top silver nanoridges [53] has found that a ridge width of 120 nm produces an optimal compromise between propagation distance and mode confinement. In this study, we thus fix the ridge width at 120 nm, for all elliptical cross section nanoridge waveguides investigated.
To describe the elliptical cross section, we use the curvature (κ), instead of the ellipse height (h). The curvature is the reciprocal of the radius of curvature. The radius of curvature (R) of the elliptical cross section ridge at the top of the ridge is R=w 2 /4h [54]. Thus, the curvature at the top of the ridge is κ=1/R=4h/w 2 . We consider the situation where the surrounding medium of the ridge waveguides is air (ε d = 1.0), and the ridges are made of silver with its electric permittivity ε m =-127. 5-5.3j at the telecommunication wavelength of 1.55 µm [55]. For the flat-top ridge with a width of 120 nm, the mode effective index and the attenuation coefficient are found to be n eff =1.018-0.00074j and 261.87 dB/cm, respectively. While this study investigates silver as the ridge material, the analysis can obviously be extended to other types of high electron density materials, such as heavily doped semiconductors [56][57][58] albeit at different wavelengths. Fig. 2 (a)-(c) are the mode profiles of the three electric field components (E x , E y , E z ) of the 120 nm wide elliptical silver nanoridge plasmon waveguide at the 1.55 µm wavelength. Fig. 2 (d)-(f) are the mode profiles of the three magnetic field components (H x , H y , H z ) of the elliptical nanoridge waveguide at the same wavelength. The major components of the electric field are the transverse components E x and E y , as expected. The longitudinal component of the electric field E z is seen to be roughly three orders of magnitude less than the transverse components. The major magnetic field components are also the transverse components H x and H y , while the longitudinal component of the magnetic field H z is also seen to be three orders of magnitude less than the transverse components. Based on these results, the elliptical cross section nanoridge plasmon modes can be considered as a quasi-transverse electromagnetic (TEM) mode. From Fig. 2, we can see that the E y mode profile and H x mode profile are symmetrical with respect to the center of the metal ridge (i.e. x=0 plane), while the E x and H y profiles are anti-symmetrical with respect to the x=0 plane as expected, for such a plasmon mode propagating along the top surface of an elliptical metal ridge.
We calculated the electric field intensity distributions of the fundamental plasmon propagating modes supported by elliptical nanoridges of different curvatures at the wavelength of 1.55 µm with width of the nanoridges set to 120 nm in all cases. Fig. 3(a) shows that the electric field intensity distribution of the flat-top nanoridge has two hot spots located at each corner of the ridge, which can be considered as a hybrid mode of two 90° wedge plasmon modes first described in Ref. [53]. Fig. 3(b-f) is the electric field intensity distribution of the elliptical nanoridge plasmon mode with the curvature of 15 µm -1 , 30 µm -1 , 70 µm -1 , 110 µm -1 , and 150 µm -1 , respectively. It can be seen that for the elliptical nanoridge with a curvature between 9 and 18 µm -1 , the mode spans the entire ridge. As the curvature increases, the mode progressively converges towards the tip of the ellipse with increasing mode confinement. As the curvature increases to above 18 µm -1 , the mode converges to only one hot spot at the tip, which can be considered as a small angle wedge mode. We also find that the mode energy is distributed over the entire ridge when the curvature is small. In this case, the elliptical nanoridge mode is seen to be insensitive to the change of curvature.
However, once the curvature becomes greater than 18 µm -1 , the plasmon mode is seen to be concentrated at the tip of the elliptical ridge, and is seen to rapidly intensify with the increasing curvature.
The mode dispersion curves for the elliptical silver nanoridges of different curvatures are calculated and shown in Fig. 4. The black solid line is the light line in air, and the black dashed line is the plasmon dispersion curve associated with the    1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59 silver-air flat interface. In this figure, for all the curvatures, as the frequency increases, the plasmon mode dispersion curves drift away from the light line in air, suggesting slower group velocity and tighter mode confinement. This effect is far more pronounced for ridges with larger curvature. This is in contrast to the effects of the curvature on the dispersion curve, which is more easily seen in the inset of Fig. 4 µm -1 and 150 µm -1 , respectively. It is also seen that the dispersion curves of ridges with curvatures between 9 and 18 µm -1 are relatively insensitive to curvature changes. While beyond a curvature of 18 µm -1 , the mode intensity is more concentrated to the tip of the elliptical ridge. The dispersion curve moves rapidly away from the light line when the curvature of the elliptical ridge continues to increase, and the ridge plasmon mode gradually approaches a wedge plasmon mode.
The real and imaginary parts of the elliptical nanoridge mode index versus the wavelength for several different curvatures (κ = 0 µm -1 , 15 µm -1 , 30 µm -1 , 90 µm -1 , and 150 µm -1 ) are plotted in Fig. 5, where the black dashed lines are those of the silver-air flat plasmon mode. As the wavelength increases, both the real and imaginary parts of the elliptical nanoridge mode index decrease, indicating a reduction in the mode confinement as well as the propagation loss. We can also see when the curvature increases from the zero, both the real and imaginary part of the elliptical nanoridge mode index first reduce, and further increase of the curvature increases those of the elliptical nanoridge mode index, which is consistent with the results shown in Figs. 3 and 4.
The real and imaginary parts of the mode index versus the curvature at 1.55 µm wavelength are shown in Fig. 6.
Interestingly, both the real and imaginary parts of the elliptical nanoridge mode index are first seen to decrease with an initial increase in the elliptical curvature, indicating a reduction in propagation attenuation and mode confinement, however, after reaching minimum values at 16.67 µm -1 , rapid increase in both real and imaginary parts of the mode index is observed, which corresponds to a quick rise in both attenuation and confinement. It is interesting to note that considering the 120 nm ridge width here, a semi-circle of 60 nm radius would have a curvature of 16.67 µm -1 . This suggests that a semicircular geometry is the optimal for reduced propagation loss and increased mode confinement.   In general, the propagation of surface plasmon modes can be characterized by a complex wave vector of the form β z =n eff k 0 =β -jα, along the z-direction, where β is the phase propagation constant of the mode, and α is the attenuation constant. The propagation distance is defined as the distance where the mode intensity attenuates to 1/e of its initial value, i.e. L p =1/(2α). The propagation distances are calculated for the elliptical nanoridge plasmon waveguide at different curvatures and different free space wavelengths. The results are shown in Fig. 7 (a) and (b). It can be seen that at any given wavelength as the curvature increases, the propagation distance first increases, and then reaches plateau for curvature between 9 and 18 µm -1 , after which it is seen to decrease rather rapidly. As the wavelength increases, the propagation distance also increases. As discussed in the Ref. [26], for the wedge plasmon mode, the propagation loss increases dramatically as the wedge angle decreases. When the curvature is between 9 and 18 µm -1 , it appears that the mode energy is distributed more evenly along the entire ridge, and the associated nanoridge plasmon mode can be considered as an infinite number of coupled large-angle wedge plasmon modes. Within this curvature range, these coupled wedge plasmon modes of the elliptical nanoridge have smaller loss, so the nanoridge plasmon mode has the longest propagation distance.
Outside the metal ridge in the surrounding dielectric medium, the transverse component of the complex wave vector is given by β⊥= γ -jδ, where γ and δ describe the field oscillation and decay in the transverse direction, respectively. It follows from Maxwell's equations that the complex propagation constants in the propagation direction and the transverse directions are related as: where k 0 is the mode propagation constant in the free space, and ε d is the dielectric constant of the surrounding dielectric. Solving (1), we can obtain γ and δ, which can be used to define the mode size as 1/2δ+1/2δ=1/δ. The mode size of the elliptical nanoridge waveguide versus the free space wavelength and the ellptical curvature is shown in Fig. 8(a), while Fig. 8(b) shows the mode size versus the free space wavelength at various elliptical curvatures. Here is seen that the mode size increases, as the wavelength increases. Once again, for the curvature smaller than 9 µm -1 , the mode size slightly increases, but when the curvature increases higher than 18 µm -1 , the mode size decreases rapidly.
While it is always desirable to obtain surface plasmon waveguides with tight confinement and low attenuation, the reality is that there is always a trade-off between the propagation attenuation and the mode confinement [12,59]. While tight mode confinement is the merit, attenuation is the cost. Figure-of-merits have been proposed to characterize this trade-off between attenuation and mode confinement [60,61]. Here, we define the figure-of-merit of the nanoridge plasmon waveguide as the ratio of the propagation distance over the mode size: α δ δ α   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  We calculated the figure-of-merit versus the wavelength and the curvature. The results are shown in Fig. 9(a). Fig. 9(b) shows the figure-of-merit versus the wavelength for several different curvatures. It can be seen from Fig. 9(a) and (b) that the figure-of-merit reaches a maximum at the 1.05 µm wavelength for all curvatures considered in this study. It also can be seen that the magnitude of the figure-of-merit peak is related to the curvature, with values between 9 and 18 µm -1 producing figure-of-merits that are larger than those obtained for even the zero curvature (the flat-top nanoridge).
We specifically calculated the propagation distance and the figure-of-merit for the 120 nm wide elliptical nanoridge waveguide versus the curvature at the wavelength of 1.55 µm.
The results are shown in Fig. 10. As the curvature increases, both the propagation distance and the figure-of-merit increase when the curvature is less than 9 µm -1 , and reach a plateau when the curvature is between 9 and 18 µm -1 , after which both are seen to decrease dramatically. It is concluded that for the 120 nm wide elliptical cross section silver nanoridge waveguides discussed here, the best performance can be obtained with a curvature between 9 and 18 µm -1 at all wavelengths, with the ultimate performance being obtained when the excitation wavelength is roughly 1.05 µm.    We also have calculated the propagation distance and the figure-of-merit for different elliptical height and nanoridge width at the 1.55 µm wavelength. The results are shown in Fig.   11. The horizontal axis is the ellipse height of the nanoridge from 0 to 1000 nm. The vertical axis is the width of the nanoridge from 50 to 500 nm. It can be seen that as the height of the elliptical ridge increases, the propagation distance and the figure-of-merit increase first, and then decrease. The white dashed lines in the Fig. 11 indicate the elliptical cross sections which become semi-circles. The propagation distance and figure-of-merit reach the maximal values along this line. This is because the plasmon mode energy is more uniformly distributed over the entire ridge, when the cross section of the nanoridge waveguide becomes a semi-circle.

III. CONCLUSIONS
We investigated the propagating plasmon modes supported by elliptical cross section silver nanoridges. We calculated the dispersion curves, propagation distances, mode sizes, and the figure-of-merits of these nanoridge plasmon modes. It is found that as the curvature of the elliptical nanoridge increases, both the propagation distance and the figure-of-merit first increase, reach a plateau, and then decrease dramatically. When the curvature falls between 9 and 18 µm -1 for 120 nm wide silver elliptical ridges, the ridge plasmon modes have the longest propagation distance and the highest figure-of-merit. The elliptical nanoridge waveguides with the curvature between 9 and 18 µm -1 have better performance in terms of the propagation distance and figure-of-merit than the flat-top nanoridge waveguides investigated previously [53]. It is found that the semicircular cross section nanoridge waveguides, which give more uniformly mode energy distribution over the entire ridge surfaces, represent an optimal trade-off between mode confinement and propagation distance. The elliptical cross section nanoridge waveguides investigated in this paper are very desirable for applications that require large surface interaction area, such as in chemical and biological sensing.