## Abstract

Needle beam is a guided beam with nanoscale beam size and significant power propagating in core area of a three-layer dielectric waveguide. Systematical numerical analyses of properties of the needle beam are presented. Properties of the fundamental mode of the needle beam, including field distribution, power distribution, and power concentration, are calculated for different waveguide parameters. It is shown that there is an optimum value of normalized frequency for maximum power concentration. Concentrated power is higher if the refractive index difference between the core and the middle layer is higher.

© 2011 OSA

## 1. Introduction

Evanescent waves are an interesting near-filed phenomenon and have many applications in macroscopic imaging, coupling, and resonant transmissions. There are many research activities to control and manipulate evanescent waves at optical interfaces and waveguides. The “needle beam” was proposed for concentrating power of evanescent waves on a nanometer scale in a three dielectric medium cylindrical waveguide [1,2]. This novel approach is different from other attempts to realize nanoscale optical beam propagations involving mainly plasmon resonance [3–8]. It has been a challenging task to obtain light propagation with nanoscale beam size and significant power. One approach was to use a tiny hole as an aperture to reduce the beam size. For light transmission through the tiny holes, the main problems are extremely low transmission and strong diffraction [9]. A number of methods have been proposed to overcome the diffraction limit of light transmitting through a tiny hole. Ideas of using plasmons formed at metal/dielectric interfaces have been widely exploited, including transmission through a single aperture in a metal film [3–7], the formation of a one-dimensional optical beam [10], and plasmon guiding through an array of closely spaced metal nanoparticles [8]. However, all these designs have propagation loss much larger than that in pure dielectric media. A different idea, without using metal, was proposed to guide and, at the same time, concentrate light in one-dimensional low-refractive index material [11,12]. Light was guided in a one-dimensional air slot between two high-refractive index rectangular regions. It utilizes the discontinuity of the normal component of the electric field of the quasi-TE mode at the interface with large difference in refractive index. Another realized approach was the subwavelength-diameter silica wires [13]. Although the propagation loss along the wires is not significant, there is a larger amount of power propagating outside the wire.

The needle beam has two interesting properties, i.e., high-power light concentration in a nanometer scale and guided beam propagation in cylindrical waveguide for practical applications. The results in [1,2] showed that there exists eigenmodes in the nanoscale central area in a three-layer cylindrical dielectric waveguide. Only TE_{01} and TM_{01} modes were considered in [1,2] as an example to illustrate the phenomenon of nanoscale light concentration. Systematic analyses of the light propagation in the three-medium waveguide are needed in order to fully understand the properties of the needle beam. In fact, to obtain the maximum concentrated power in a small area, it is necessary to consider the solution of all modes. This paper presents general solutions of all eigenmodes in a three-layer cylindrical dielectric waveguide and gives the corresponding light propagation characteristics, especially power concentrations in the core area, for nanoscale beam propagation.

## 2. Governing Equations and Mode Conditions

The structure and the refractive index profile of a three-layer cylindrical waveguide are shown in Fig. 1
. The refractive index of the middle medium is highest, that is, n_{2>}n_{1} and n_{2}>n_{3}.The relationship between n_{1} and n_{3} is not fixed. Since the waveguide is cylindrically symmetric, we use the cylindrical coordinate system in our analyses. The field components are E_{r}, E_{φ}, E_{z}, H_{r}, H_{φ}, H_{z}. Here, z is the propagation direction. It is well known that wave equations for the z components are [14]

^{2}is the Laplacian operator,

*k = 2π /*λ (λ is the wavelength). Solutions to Eq. (1) take the form

*l*= 0, 1, 2, 3, …,

*w*is the eigenmode frequency, and

*Ψ(r)*has different forms in each of three layers in the waveguide: In Eqs. (3) to (5), p

^{2}= β

^{2}-n

_{1}

^{2}k

_{0}

^{2}, h

^{2}= n

_{2}

^{2}k

_{0}

^{2}-β

^{2}, q

^{2}= β

^{2}-n

_{3}

^{2}k

_{0}

^{2},

*I*and

_{l}(x), J_{l}(x), Y_{l}(x),*K*are the Modified Bessel function of the first kind, Bessel function of the first kind, Bessel functions of the second kind, and Modified Bessel function of the second kind, respectively, of order

_{l}(x)*l*, and

*a*and

_{1}, a_{2}, b, c, d_{1},*d*are arbitrary constants.

_{2}In order to determine the constants *a _{1}, a_{2}, b, c, d_{1}, d_{2}*, we consider boundary conditions at r = 0 and r→∞. Fields have to be finite at

*r*= 0 and r→∞. Therefore,

*a*and

_{2}*d*must be zero. Thus, the fields in the core medium (Region I), the middle medium (Region II), and the outside medium (Region III) are described by I, J + Y, K, respectively, and we call the wave I-(J + Y)-K profile. Equations (3-5) can now be written as

_{1}Using the wave equations, we can obtain E_{r}, E_{φ}, H_{r}, and H_{φ}. In the core area (r < r_{1}), they are

The expressions for E_{r}, E_{φ}, H_{r}, and H_{φ} in the middle layer (r_{1}<r <r_{2}) are

In the outside layer (r>r_{2}), they are

In Eqs. (8) - (19), the prime symbol denotes the first derivative of the function, ε_{1} = n_{1}
^{2}, ε_{2} = n_{2}
^{2}, ε_{3} = n_{3}
^{2}, and _{µ} is the permeability of the medium. All the eigenmodes propagating in the three-medium waveguide must satisfy the boundary conditions that E_{z}, H_{z}, E_{φ}, and H_{φ} should be continuous at the I/II boundary (r = r_{1}) and II/III boundary (r = r_{2}). This leads to the corresponding eight equations. The eight equations can be expressed by the following matrix form

Here, M is an 8 × 8 matrix and its non-zero elements are:

In order to get non-trivial solution for coefficients A_{i}, B_{i}, C_{i}, D_{i} (i = 1, 2) in Eq. (20), the determinant of the above 8 × 8 matrix must vanish. We will use this requirement to obtain mode conditions. To solve the mode conditions, we use the parameters V, η, θ, defined as V^{2} = k_{0}
^{2} r_{2}
^{2} (n_{2}
^{2}-n_{3}
^{2}), η^{2} = (n_{2}
^{2}- n_{1}
^{2})/ (n_{2}
^{2}- n_{3}
^{2}) and θ = r_{1}/ r_{2} [1,2]. Here V is equivalent to the normalized frequency in conventional optical fiber [14]. For a fixed operation wavelength, V depends on waveguide parameters r_{2}, n_{2}, and n_{3}. θ is the ratio between r_{1} and r_{2}. If θ = 0, we have a conventional two-medium step-index optical fiber. Letting the determinant of M be zero under fixed values of V, η, and θ, we can calculate different hr_{2} values, which correspond to different eigenmodes. Then we can calculate other characteristic constants of these modes, such as cut off frequency and normalized propagation constant. We use Matlab software in our calculations. It should be noted that Eq. (20) can be simplified analytically first and then solved numerically to obtain the similar results of mode conditions [15]. It is well known that, in a cylindrical waveguide, eigenmodes in general have non-vanishing E_{r}, E_{φ}, E_{z}, H_{r}, H_{φ}, H_{z} components. The eigenvalues resulting from matrix equation (Eq. (20)) lead to the two classes of solutions corresponding to the conventionally designated EH or HE modes. When *l* = 0, HE and EH modes become TE and TM modes, respectively.

Figure 2
shows the relationship between normalized propagation constant β/*k*
_{0} and normalized frequency V with fixed waveguide parameters, θ = 0.729, n_{1} = 1.00, n_{2} = 3.48, and n_{3} = 1.48. With n_{1} = 1, we have a waveguide with a small hole in the core area. It appears from Fig. 2 that for V<2.5, only the fundamental HE_{11} mode can propagate, which is the single-mode waveguide condition. It should be noted that the cut off frequency of different mode depends on the ratio of r_{1} and r_{2}, as well as n_{1}, n_{2}, and n_{3}. This is different from two-medium step-index fiber, whose cut off frequency of different modes is decided by the core size and the refractive indices of the waveguide. In the following sections, we will use the HE_{11} mode as an example and analyze its fields and power characteristics.

## 3. Field and Power Characteristics of HE_{11} Mode

In the calculation of fields and power, we can normalize all other fields to one non-zero amplitude (e.g., A_{1} in HE_{11} mode). After obtained β/*k*
_{0} of an eigenmode and set input signal, we can calculate normalized amplitudes of all fields. Then the Poynting vector and power can be numerically obtained.

For HE_{11} mode, we set A_{1} = 1. Then we solve for the values of A_{2}, B_{1}, C_{1}, B_{2}, C_{2}, D_{1}, and D_{2} of different fields using Eq. (20). After the field functions (Eqs. (6) to (19)) are obtained, the power can be calculated. In the cylindrical coordinate system, the time-averaged Poynting vector along the waveguide is expressed by

The amount of power in core (P_{core}), middle layer (P_{mid}) and outside layer (P_{out}) are given by

We study the power characteristics in the core medium in two aspects, the power percentage and power distribution.

The magnitude of coefficients of the field components E_{r}, E_{φ}, E_{z}, H_{r}, H_{φ}, and H_{z} in the core medium are presented in Fig. 3a
to Fig. 3f with different n_{2} and other fixed waveguide parameters, θ = 0.729, n_{1} = 1, and n_{3} = 1.48. Again we consider a waveguide with a hole in the core area (n_{1} = 1). It should be noted that there is a π/2 phase difference between two sets of fields (E_{z}, E_{φ}, and H_{r}) and (H_{z}, E_{r}, and H_{φ}) because the amplitude coefficients of E_{z}, E_{φ}, and H_{r} are real numbers and those of H_{z}, E_{r}, and H_{φ} are imaginary numbers.

Figure 4
shows power distribution in the core medium (a nanoscale hole) with the same parameters as in Fig. 3. It can be seen that there is larger power in the core area when n_{2} is larger. Figure 5
shows different power percentage of HE_{11} mode with different V and fixed values of θ, n_{1}, n_{2} and n_{3}. Compared to TE_{01} mode in Refs. 1-2, the power percentage of HE_{11} mode has similar behavior when V changes. It also shows that there is an optimal V for maximum P_{1}. Here the optimal V is about 3.8, which corresponds to a maximum power percentage of 30% in core medium when θ = 0.729, n_{1} = 1, n_{2} = 3.48, and n_{3} = 1.48.

The power percentage in the core medium (nanoscale hole) varies with values of V, θ and η. Figure 6a
and Fig. 6b respectively show power percentage in the core medium varying with different values of V, θ and η. We can see from Fig. 6a that there is an optimal V value which results in maximum power percentage in the core medium. This interesting effect can be explained by considering the relation between V and r_{2}. With fixed θ value in Fig. 6a, a decrease in V corresponds to the decrease of r_{2}, and hence r_{1}. Decreasing in r_{1} causes the increase in power density, but decrease in total volume of core region. As a results, there is an optimal V for maximum power percentage in the core region, i.e., V (and, with fixed θ, hence r_{1}) cannot be too large or too small for maximum power concentration. The optimal V value becomes larger when n_{2} changes from 3.48 to 1.75.

Similarly, as we can see from Fig. 6b, there is an optimal θ value which can lead to maximum power percentage value in the core medium. Both Fig. 6a and Fig. 6b show that larger n_{2} leads to larger power percentage in the core medium.

Figure 7 shows the power distribution in all three media with fixed values of V and refractive indices when θ changes. It can be seen from Fig. 7a and Fig. 7e that the power density in the core medium nearly is double as that in the middle medium and the light is well confined in the core medium. From Figs. 7a to 7d, we can see that as θ becomes larger, the power density in core medium gradually approaches to that in the medium II and the difference in power density between the core medium and medium II is less significant. That is, the area of the large power intensity moves away from the center point of the circle. So the “needle beam” will disappear gradually when the θ value becomes larger. It should be noted that the power density in the outside medium is always small.

## 4. Discussions

We have presented typical values of fields and power of the eigenmodes in a three-medium purely dielectric coaxial cylindrical waveguide. From our numerical analyses, we can get the following general results.

Compare to two-medium step-index waveguide, mode conditions of our waveguide are determined not only by the refractive indices, but also by θ, the ratio of the core and the middle layer radii. It means that waveguide dispersion properties (Fig. 2) will change when either θ or refractive indices change, while in two-medium step index waveguide they are functions of refractive indices only.

Several parameters of the waveguide affect the power density and percentage of power distribution in three regions. For nanoscale power concentration, the most important parameter is the core size. For smaller core diameter, there is a higher power density in the core area, corresponding to a stronger “needle beam” effect. As shown in Fig. 7, the power density in the core medium can be nearly twice as much as that in the middle medium. However, it should be noted that the total amount of power concentration is not necessary higher for smaller core diameter. This is due to the fact that the core area gets smaller for smaller diameter.

Other waveguide parameters (V, θ, and refractive indices) have various degrees of effect on power percentage distributions in three media. Figure 6b shows that when V is fixed, P_{core} is larger for larger θ up to certain value. Note that θ is determined by the diameters of core and middle layer. Therefore, larger θ corresponds to relatively larger core size. If θ is too large, it is no longer a nanoscale effect.

When θ is fixed, there is a best V value for the maximum power in the core medium (Fig. 5 and Fig. 6a). For all the cases we have studied, it is evident that larger refractive index in the middle medium (n_{2}) leads to higher power percentage in the core medium. This can be explained based on the fact that the needle beam is an effect of evanescent wave at the boundary between core and middle layers [1,2].

It should be noted that core diameter r_{1}, n_{2,} V, and θ are all inter-related. Therefore it is very important to choose suitable parameters to confine more power in core medium for nanoscale beam propagation.

The above results are calculated for the fundamental mode HE_{11}. For high power concentration in a small area, we have also examined power distribution of other modes. Table 1
lists power percentage in core medium and middle medium for different eigenmodes of the waveguide with V = 20, θ = 0.729, n_{1} = 1, n_{2} = 3.48, and n_{3} = 1.48. From Table 1, we can see that power percentage of TE and HE modes in core medium are larger than that of TM and EH modes. This is a typical result of boundary conditions of dielectric waveguides.

## 5. Conclusions

We have systematically analyzed the properties of the purely dielectric three-medium cylindrical waveguide, and presented general equations, mode conditions, and light propagation characteristics of waveguide modes. The results of the numerical analysis show that it is feasible to realize nanoscale light transmission in a three-medium dielectric waveguide. It is found that power in core area is higher if the refractive index difference between the core and the middle layer is higher. There is optimum value of normalized frequency for maximum power in core medium. Power density in the core (nanoscale hole) area gets larger for smaller radius of the core area and there is an optimal value of core radius for maximum power concentration in the core region. Waveguide and mode parameters are all inter-related. Therefore, it is very important to choose suitable waveguide parameters in order to confine significant power in core medium.

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