1. Introduction

Dispersion is one of the most important properties of optical waveguides. Controlling light propagation behavior by tailoring waveguide dispersion is widely used in fields such as optical communication and nonlinear optics [1–5]. Recent research shows that free-standing optical nanowires fabricated by either physical drawing or chemical reaction are promising for low-loss optical wave guiding [6–10]. When being used as subwavelength-diameter waveguides, these wires exhibit great potential for a variety of applications ranging from optical communications, optical sensing, nonlinear optics to atom trapping [11–22]. Previous work has shown that [23], guiding light with air-clad subwavelength-diameter nanowires provides the opportunity for obtaining waveguide dispersion one or more magnitudes larger than those of conventional weakly guided waveguides, making these wire waveguides potentially useful in applications such as nonlinear effects in fibrous materials. Generally, dispersion of a free-standing nanowire waveguide depends on a few parameters including diameter and refractive index of the nanowire and the wavelength of the guided light. Therefore, the dispersion of an as-fabricated nanowire is almost determined, though a certain dispersion or dispersion shift is desired in some applications. Based on numerical simulations of a 3-layer-structured cylindrical waveguide, here we show that coating a nanowire with a thin non-dissipative dielectric material may lead to considerable dispersion shift, and is therefore promsing for shifting dispersion of an optical nanowire with minimum change in its geometric dimension.

2. Mathematic model for dielectric coated nanowires

The mathematic model in our simulation is schematically illustrated in Fig. 1. A long straight nanowire with coat and air-clad is a cylindrical structure of translation symmetry involving three regions in the cross section (Fig. 1(a)): a circular dielectric core (e.g. silica or silicon nanowire) with radius ρ and a cylinder dielectric coat with thickness d_c , is embedded in the infinite air cladding. Refractive indices of the core, coat and air is assumed to be n_s , n_c and n_a , respectively (Fig. 1(b)). Solving Maxwell’s equations in cylindrical coordinates (r, θ, z) leads to the following expressions for the components of the electromagnetic field for the mth mode [24, 25]:

where the index j=s denotes the components inside the dielectric core ( r < ρ ), j=c, the cylinder coat (ρ < r < ρ + d_c ), and j=a, the infinite air ( r > ρ+d_c ), such that $Z_m^s$ (x) ≡ J_m (x), the Bessel function of order m, $Z_m^a$ (x) ≡ $H_m^{\left(1\right)}$(x), the Hankel function of the first kind of order m, $Z_m^c$ (x) ≡ c ₁ j_m (x) + c ₂ $H_m^{\left(1\right)}$(x), the linear combination of the Bessel function and the Hankel function, and the prime denotes differentiation with respect to the argument x ≡ k_j∙r. β is the propagation constant and k ₀ is the free space wave number: k ₀=ω/c. $a_m^j$ and $b_m^j$ are complex coefficients determined from the boundary conditions. F_m is the exponential factor,

and k_j is the transverse wave number in the respective medium,

By applying the boundary conditions, that the tangential components of the electromagnetic field E⃗ and H⃗ must be continuous at the inner and outer cylinder surfaces (r=ρ and r=ρ+d_c ), a system of eight linear homogeneous equations is obtained that is satisfied by the eight coefficients. The system admits a nontrivial solution only in case its determinant is zero. The propagation constant β is determined by the condition that the determinant of the system of linear equations shall vanish:

where M is the resulting matrix of the system of equations.

 

Fig. 1. Mathematic model of an optical nanowire with a cylinder coat. (a) Cross-section view and (b) refractive index profile of the nanowire.

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Usually, subwavelength-diameter optical nanowires are designed and desired for working as single-mode waveguides [23], therefore, here we consider the fundamental modes and thus set m=1 in Eqs.(1) and (2).

With propagation constants (β) obtained by numerically solving Eq.(4), the group velocities (v_g ) and waveguide dispersions (D_w ) can be obtained as [26]:

where λ is the wavelength and c the light speed in vacuum.

For numerical simulations, we choose silica and silicon as typical moderate- and high-index materials under the following considerations: (1) silica and silicon are among the most important photonic materials within the visible and near-infrared ranges; (2) both silica and silicon nanowires have been successfully fabricated [6–8, 27–28]; (3) they have typical values of moderate and high refractive indices (about 1.45 for silica and 3.4 for silicon). In addition, considering that material dispersion is usually orders of magnitude lower than waveguide dispersion in air-clad nanowire waveguides at their transparent (low-loss) spectral ranges [23], we ignore the material dispersion of the core and coat for simplicity.

3. Dispersion shifts in silica nanowires with high-index coat

For silica nanowires with moderate refractive index around 1.45, the single-mode cut-off diameter is about 400 nm at the wavelength of 550 nm and larger thereafter [23], therefore, we use a 400-nm-diameter silica nanowire (always single-mode when the wavelength exceeds 550 nm) as a typical situation for calculation.

Assuming the thickness (d_c ) of the coat is 5 nm, calculated dispersion of a dielectric-coated 400-nm-diameter silica nanowire are shown in Fig. 2, in which 5-nm-thickness coats with indices (n_c ) of 2.20 (e.g. of PbCl₂) and 2.70 (e.g. of TiO₂) are used. It shows that, starting from the short wavelength side, the waveguide dispersion goes through a maximum around 500-nm wavelength and a minimum around 900-nm wavelength, and then approaches zero at the IR edge. The dispersion increases with the increasing of the coat’s index before the minimum point (around 900 nm), and decreases with the increasing of the coat’s index afterward. Within the broad spectral range (e.g. from 500-nm to 1500-nm wavelength), considerable dispersion shift (compared to that of the bare nanowire) is produced by adding a thin high-index coat, and the shift increases with the index of the coating. For example, at 500-nm wavelength, dispersion of the fundamental modes of a bare 400-nm-diameter silica nanowire is about -480 ps∙nm^-1∙km^-1, well below the zero-dispersion level. When a 5-nm-thickness coating with index of 2.7 is added, a positive 485 ps∙nm^-1∙km^-1 shift is generated, which shifts the dispersion of the nanowires beyond the zero-dispersion point. Considerable dispersion shift of about 700 ps∙nm^-1∙km^-1 is observed within 560-800 wavelength range with a minimum (zero shift) at about 920-nm wavelength. For comparison, dispersion of a 410-nm-diameter silica nanowire (that is, a 5-nm-thickness silica coat with index of 1.45) is also provided.

 

Fig. 2. Modified dispersion of 400-nm-diameter silica wire with different coatings (d_c =5nm). Dashed line: dispersion of 400-nm-diameter silica wire without coat.

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We have also studied the dependence of dispersion shift with respect to the coat thickness. Calculated dispersions of a 400-nm-diameter silica nanowires is shown in Fig. 3(a), in which the wire is assumed to be coated with a high-index (n_c =2.7) film with thickness of 2, 5, and 10 nm, respectively. It shows that, although the changes in wire diameters (due to the thin coat) are very small, the shifts in dispersion are considerably large. For example, a 2-nm-thickness coat on a 400-nm-diameter wire (that is, 1% increase in wire diameter) leads to a 415 ps∙nm^-1∙km^-1 shift in dispersion at 633-nm wavelength. Figure 3(b) shows the coat-thickness-dependant dispersion of a 400-nm-diameter wire at the wavelength of 633 nm. Refractive index of the coat is assumed to be 2.7. The dispersion increases continuously and smoothly with the increasing thickness of the coat, indicating the possibility for fine modification of the dispersion of a nanowire waveguide by the thickness of the coat. For comparison, dispersion shift caused by solely increasing the diameter of the same wire (that is, coating the 400-nm-diameter silica wire using silica layers with an index of 1.45) is also provided. It shows that, for dispersion modification, coating a high-index layer is much more efficient than increasing the wire diameter. For example, adding a 5-nm-thickness high-index coat leads to a dispersion shift of about 950 ps∙nm^-1∙km^-1, whereas increasing the diameter to the same thickness brings a dispersion shift of only 250 ps∙nm^-1∙km^-1.

 

Fig. 3. (a) Modified dispersion of a 400-nm-diameter silica wire with different coating thickness. (b) Coat-thickness-dependent dispersion of a 400-nm-diameter silica wire at the wavelength of 633 nm. Refractive indices of the coats are assumed to be 2.7 and 1.45, respectively.

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4. Dispersion shifts in silicon nanowires with low-index silica coat

For silicon nanowires with short-wavelength absorption edge at 1200 nm and a notable high index of about 3.4, the single-mode cut-off diameter is about 350 nm at 1550-nm wavelength [23]. Therefore, here we use a 300-nm-diameter silicon nanowire for the simulation.

Generally, silicon nanowires are likely to be oxidized in the synthesis or exposed in air afterwards [27, 28], resulting in silicon oxides shell with thickness up to tens of nanometers. For simplicity, here we use SiO₂ (e.g. silica ), the dioxide (also the stable oxide) of silicon as a low-index (1.45 v.s. 3.4) coat.

When the oxidized coat is thin (compares with the thickness of the silicon core), the thickness of the SiO₂ coat (d_c ) can be obtained as

where d_Si is the thickness reduction of the silicon core, ρ_Si =2.35g/cm³ and ρ_SiO2 = 2.2g/cm³ are densities of Si and SiO₂ [27], M_Si =28.1 and M_SiO2 =60.1 are molecular mass of Si and SiO₂, respectively.

Calculated dispersions of SiO₂-coated silicon nanowires are shown in Fig. 4. The nanowires, with different-thickness SiO₂ layers, are assumed to be oxidized from the same pure silicon nanowire with a diameter of 300 nm. The simulation covers the overlapping area of transparent spectral ranges of silicon and silica (Fig. 4(a)). The thickness reduction of the silicon core is assumed to be 1, 2, 5 and 10 nm, corresponding to 2.3, 4.6, 11.4 and 22.8 nm thickness of SiO₂ coat, respectively. As shown in Fig. 4(a), large dispersion shift is generated with a very thin layer of oxidation and the dispersion shift increases with the thickness of the coat (silica layer). For example, at 1450-nm wavelength, a 1% decrease in the diameter of silicon nanowire (by oxidization of silicon into SiO₂ shell) lead to a 6849 ps∙nm^-1∙km^-1 dispersion shift (that is 34% decrease). Fig. 4(b) gives the oxidization-induced dispersion shift with respect to the reduction in silicon core at 1450-nm wavelength. The dispersion decreases smoothly and almost linearly with the thickness reduction of the silicon core. Practically, oxidization of silicon nanowires can be well controlled by environmental conditions such as temperature and reaction time [28], therefore, shifting waveguide dispersion of a silicon nanowire by oxidization may be a feasible and efficient way.

 

Fig. 4. (a) Modified dispersion of 300-nm-diameter silicon wire with thickness reduction in silicon core by oxidization. (b) Thickness-reduction-dependent dispersion of a 300-nm-diameter silicon nanowire at the wavelength of 1450 nm.

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5. Conclusion

In conclusion, using a 3-layer-structured cylindrical waveguide model, we’ve numerically investigated dispersion shifts in dielectric-coated silica and silicon nanowires. It shows that, with a thin dielectric coat, considerable dispersion shift of the guided light can be obtained. Since the thickness of the coat is much smaller than the diameter of the nanowires, the coat does not obviously change the geometric dimension and single-mode condition of the nanowire waveguide. In addition, the dielectric coat used in this work is non-dissipative and commonly available, making it simple for both theoretical modeling and experimental implementation.