We propose a simple model to elucidate the dispersion behavior of spiraling modes on silver nanowire by finding correspondence parameters and building a simple equivalent relationship with the planar insulatormetalinsulator geometry. The characteristics approximated for the proposed structure are compared with the results from an exact solution obtained by solving Maxwell’s equation in cylindrical coordinates. The effective refractive index for our proposed equivalent model is in good agreement with that for the exact solution in the 4002000 nm wavelength range. In particular, when the radius of the silver nanowire is 100 nm, the calculated index shows typical improvements; the average percentage error for the real part of the effective refractive index is reduced to only 5% for the 0
^{th}
order mode (11.9% in previous results) and 1.5% for the 1
^{st}
order mode (24.8% in previous results) in the 400800 nm wavelength range. This equivalent model approach is expected to provide further insight into understanding the important behavior of nanowire waveguides.
I. INTRODUCTION
In realizing integrated nanophotonic circuits, a promising key is the use of surface plasmons, collective electron oscillations that allow the localization, confinement, and transmission of electromagnetic energy
[1
,
4]
. Various geometries for plasmonic waveguides and integrated components have been suggested; these include grooves and slots in metal films
[2

5]
, metal strips
[6

8]
, dielectric strips deposited on a metal film
[9]
, metal caps on a silicononinsulator rib waveguide
[10

12]
, and others
[13

15]
.
An alternative approach to achieve integrated nanophotonic circuits involves utilizing the nanowire (NW)based plasmonic waveguides, which are typical onedimensional waveguide structures with relatively low losses in visible and nearinfrared spectral ranges
[16

20]
. To understand the properties of NWbased waveguides, it is essential to investigate their modal dispersion behavior. Schmidt and Russell established a connection between the dispersion of propagating modes on NW geometry and the dispersion of those on planar metalinsulator geometry with a single interface
[17]
. For an NW radius less than 100 nm, however, there is wide discrepancy owing to interaction with the electromagnetic field.
In this study, we propose a simple model to elucidate the dispersion behavior of spiraling modes on silver nanowire by building a relationship with the planar insulatormetal insulator geometry. This approximated analysis is compared with a more exact solution obtained by solving Maxwell’s equation in cylindrical coordinates. Schmidt and Russel previously established a connection between the dispersion of propagating modes on an NW geometry and those on a planar metalinsulator geometry when the radius of the NW is much larger than the skin depth
[17]
. For a Ag NW with a radius of 100 nm, the average percentage error for the fundamental mode of the real component of the effective refractive index was reduced from 11.9% (previous result
[17]
) to 5% in the 400800 nm wavelength range. The average percentage error for the 1
^{st}
order mode was reduced from 24.8% (previous result
[17]
) to 1.5% in the 400800 nm wavelength range. This approach provides important insights into the behavior of NW waveguides. Further, it has much broader applicability than previous methods and it can be extended to provide insights into more complicated NWbased waveguides.
II. METHODS
An infinitely long and straight Ag NW (with radius
a
) was oriented along the
z
axis and surrounded by silica as illustrated in
Fig. 1(a)
. We found the modal dispersion relation of the Ag NW by building a relationship with the planar insulatormetalinsulator (IMI) geometry as shown in
Fig. 1(b)
. It is worth noting that there has been previous work to investigate the surface plasmon propagation and confinement in IMI geometry
[22]
. Solving Maxwell’s equations using the boundary conditions for the IMI geometry, we obtained the dispersion relation (details are given in the Appendix.),
(a) An infinitely long and straight Ag NW (radius a) is oriented along the zaxis and surrounded by silica. (b) The proposed equivalent model with planar insulatormetalinsulator (IMI) geometry.
where
,
k_{0}
=
ω/c
is the freespace wavevector,
^{ε}_{m}
,
^{ε}_{d}
are the relative permittivity in the Ag and SiO
_{2}
, respectively. The dielectric constant for Ag is taken from experimental data
[21]
, the realvalued dielectric constant for SiO
_{2}
is fixed to (1.47)
^{2}
. The effective thickness of the metal layer
^{D}ν
is expressed by
where
ν
is the mode order. The propagating mode in the Ag NW is influenced by the finite crosssectional area of NW, however, the propagating mode in the IMI geometry is infinitely extended in the xdirection. Therefore we needed to define the effective width along to the xaxis to build an equivalent relationship with Ag NW, and we assumed that the crosssectional area of the NW was equal to the effective metal area of the IMI geometry, described by
for the fundamental mode. Here,
^{W}_{eff}
is defined as a perimeter of an effective mode area for fundamental mode, that is to say
^{W}_{eff}
= 2
π
(
a
+
δ
), where
δ
represents the effective field radius and defined as the distance required for the electric field intensity to decay by a factor of 1/
e
along the radial direction of the Ag NW, as shown in the inset image of
Fig. 2
. We assumed that the propagating mode had its total wavevector determined by the IMI dispersion relation,
The effective field radius δ as a function of the radius and the wavelength. The black dots are obtained from an exact solution (Eq. (6)) and the black dashed lines are obtained from our analytical solution (Eq. (7)). The inset shows the radial dependence of the zcomponents of the electric field for ν =0 and a =100 nm at wavelengths of 400 nm and 800 nm.
where
β
is the component of the wavevector along the propagating axis of the Ag NW
[15
,
17]
. Combining Eqs. (1)(4), we obtain the equivalent dispersion relation for the Ag NW, described by
To obtain the effective field radius δ , we used a direct a direct solution of the dispersion relation of the Ag NW obtained by solving Maxwell’s equation in cylindrical coordinates, as follows
[23]
:
where
J_{ν}
is a Bessel function of the first kind,
is a derivative of
J_{ν}
,
K_{ ν}
is a modified Bessel function of the second kind,
is a derivative of
K_{ ν}
,
,
.
The propagation constant
β
was calculated by solving Eq. (6). Using these results, we obtained the radial dependence of the zcomponents of the electric field and the
δ
as shown in inset image of
Fig. 2
.
Figure 2
shows the effective field radius
δ
as a function of the radius a and the wavelength
λ
. The
δ
obtained from Eq. (6) is indicated by the black dots. The analytical expression indicated by the black dashed lines was obtained by a fitting of the
δ
obtained from Eq. (6) and is described by
where
A
=0.01349,
B
=33.55, and
τ_{a}
=611.6. Here,
f (a)
=
f_{∞}
(1
γ
_{1}
e
^{a/a01}

γ
_{2}
e
^{a/a02}
where
f_{∞}
=1.58587,
γ
_{1}
=0.4437,
a
_{01}
=1499.3,
γ
_{2}
=0.1271, and
a
_{02}
=66.2.
III. RESULTS AND DISCUSSION
The complex values of the effective refractive index
n_{e}
=
n_{eR}
+
in_{eI}
are yielded from Eq. (5) and (6).
Figure 3
shows the effective refractive index of the first three azimuthal mode orders as a function of the wavelength. The top panel shows the real and imaginary parts of the effective refractive index for a radius of 100 nm, and the bottom panel shows the same for a radius of 500 nm. For a radius of 100 nm, the real part of the effective refractive index for the exact solution was in good agreement with that for our approximate value in the 4001400 nm wavelength range. Small discrepancies were noticed between the exact and approximate solutions for the imaginary part of the effective refractive index (shown in the inset of the top panel of
Fig. 3
. For a radius of 500 nm, the real and imaginary parts of the effective refractive index for the exact solution were in good agreement with those for our approximate value in the 4001400 nm wavelength range.
The effective refractive index of the first three azimuthal mode orders as a function of the wavelength. The real and imaginary parts of the effective refractive index (top panel) for a radius of 100 nm and (bottom panel) for a radius of 500 nm. The grey solid line corresponds to the bulk material refractive index of SiO_{2}.
We now examine in detail the percentage error of the real part of the effective refractive index and the propagation length in the 400800 nm wavelength range as shown in Fig. 4. For the real part of the effective refractive index, the percentage error (
^{n}_{eR}
) is defined by 100×(
^{n}_{eR,exact}

^{n}_{eR, approx}
.  /
n_{eR,exact}
), and the approximated model proposed is lowvalue for the whole wavelength range compared with the previous planar metalinsulator geometry as shown in
Fig. 4(a)
[17]
. The average percentage error for the fundamental mode was reduced to 5% compared with the previous result of 11.9%, and that for the 1
^{st}
order mode was reduced to 1.5% compared with the previous result of 24.8%.
(a) The percentage error ( ^{n}_{eR} ) for the real part of the effective refractive index in the 400800 nm wavelength range. The dashed lines and the dotted lines correspond to the proposed IMI geometry and the previous metalinsulator geometry, respectively. Inset shows the effective refractive index as a function of the wavelength from the exact solution (solid lines), previous metalinsulator geometry (dotted lines), and proposed IMI geometry (dashed lines). (b) The percentage error (L_{P}) of the propagation length in the 400800 nm wavelength range. Inset shows the propagation length as a function of the wavelength from the exact solution (solid lines), previous metalinsulator geometry (dotted lines), and proposed IMI geometry (dashed lines).
Figure 4(b)
shows the percentage error (L
_{P}
) of the propagation length. The propagation length is given by
L_{P}
= 1 / (2
n_{el}k_{0}
) and the percentage error (L
_{P}
) is defined by 100×(
^{L}_{P,exact}

^{L}_{P, approx}
.  /
L_{P,exact}
). The average percentage error for the fundamental mode was reduced to 48% compared with the previous result of 121.5%, and that for the 1
^{st}
order mode was slightly increased 39.4% compared with the previous result of 36.2%.
IV. CONCLUSION
In summary, we proposed a simple model to elucidate the dispersion behavior of spiraling modes on Ag NW by building a relationship with the planar IMI geometry. The previous model with planar metalinsulator geometry has large discrepancies compared with the exact solution when the radius of NW is compatible with the skin depth, due to interactions with the electric field around the NW. Our equivalent model solved this problem by introducing plasmon modes that exist on thin metal films sandwiched between two regions of identical dielectrics, and the percentage error was remarkably reduced. This approach provides important insights into the behavior of NW waveguides. In addition, it has much broader applicability than the previous model and it can be extended to provide insights into more complicated NWbased waveguides.
Acknowledgements
This work was supported in part by Inha university and the MSIP (Ministry of Science, ICT & Future Planning), Korea, under the CITRC (Convergence Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA2013H0301131010).
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