Analysis of a photonic nanojet assuming a focused incident beam instead of a plane wave

2D photonic nanojet formed by a dielectric cylinder [1] and the sensitivity of its backscattered light to the presence of small particles was first reported in 2004 using FDTD analysis [1]. Using vector spherical harmonic function, 3D nanojets formed by various particle size and index were calculated as well as their sensitivity of backscattered light perturbed by nanoparticles [2]. Field distribution of nanoparticles on Si surface was studied [3]. These topics and potential applications of nanojets were reviewed in [4]. An application was the used of an array of micro-particles to create an array of blind holes in Si [5]. Another interesting application related to the sub-wavelength capability of nanojets formed by micro-particles is the super-resolution imaging capability of these micro-particles [6]. A host of other articles on the detail analysis and presentations of nanojets can be found in [7–12]. Photonic nanojets were indirectly observed from measurements of two-photon fluorescence enhancement of molecules within a solution with silica microspheres [13], and also in triangular photonic molecules [14]. It was directly measured by data deconvolution on 3D maps of light passing through the confocal pinhole of a confocal microscope system [15]. Sensitivity of backscattered light to perturbation from small nanoparticles in front of the nanojet has also been measured [16]. Nanojets were also formed by thin planar disks with diameter of 1–10 μm [17], and on chains of coupled spherical cavities [18].

The analysis of nanojets generally assumes that the incident field is a plane wave. In this work, we investigate the nanojet formed by a focused incident beam with a spot size smaller or comparable to the particle size. We find that the usage of the vector spherical harmonics expansion of the focused beam allows a rigorous calculation of the 3D nanojets for the case when the particle center is at or away from the focal point. Not surprisingly we find that the position of the nanojet beam waist with respect to the particle surface can be adjusted by moving the particle's position about the focal point with little change in the nanojet dimension. It is also found that the nanojet spot size is more symmetric with respect to the azimuthal angle than the case if the incident field is a plane wave. In reference [19], vector spherical harmonics were first used in the expansion of an incident convergent beam incident on a spherical microparticle the size of λ. However, the reported focal spot formed is inside the particle in order to mimic the focal spot formed by a hemispherical lens.

Figure 1 depicts the optical configuration and parameters used in this analysis. A collimated x-polarized Gaussian input beam with beam width W impinges on a microscope objective with a numerical aperture denoted by NA. The aperture at the back focal plane of the microscope objective is D. Any point in the image plane of the microscope objective is denoted in spherical coordinates as (r_p, θ_p, ϕ_p). The analysis is made for the particle at position A at the focal point and at position B away from the focal point as shown. For this analysis, the collimated beam diameter is fixed at W = 2 mm and D = 7.65 mm. NA = 0.4 or 0.8. The particle diameter is fixed at 2 μm, and its refractive index is taken as 1.47. The wavelength is 400 nm.

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According to Richards and Wolf [20], the electric field E_i(r_p, θ_p, ϕ_p) at any point in the image space if expressed in spherical coordinate is:

$$\begin{eqnarray*}&&{{E}_{i}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right)={{E}_{r}}\cdot \overset{\scriptscriptstyle\frown}{e}{}_{r}^{{}}\;_{{}}^{{}}+\;{{E}_{\theta }}\cdot {{\overset{\scriptscriptstyle\frown}{e}}_{\theta }}+{{E}_{\phi }}\cdot {{\overset{\scriptscriptstyle\frown}{e}}_{\phi }}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{E}_{r}} & = & -\pi {\rm cos} {{\phi }_{p}}\cdot \{\left[ {{I}_{0}}({{r}_{p}},{{\theta }_{p}})+{{I}_{2}}({{r}_{p}},{{\theta }_{p}}) \right]\cdot {\rm sin} {{\theta }_{p}} \\ {} & {} & -{\mkern 1mu} 2i{{I}_{1}}({{r}_{p}},{{\theta }_{p}}){\rm cos} {{\theta }_{p}}\} \\ {{E}_{\theta }} & = & -\pi {\rm cos} {{\phi }_{p}}\cdot \{\left[ {{I}_{0}}({{r}_{p}},{{\theta }_{p}})+{{I}_{2}}({{r}_{p}},{{\theta }_{p}}) \right]\cdot {\rm cos} {{\theta }_{p}} \\ {} & {} & +{\mkern 1mu} 2i{{I}_{1}}({{r}_{p}},{{\theta }_{p}}){\rm sin} {{\theta }_{p}}\} \\ {{E}_{\phi }} & = & \pi {\rm sin} {{\phi }_{p}}\cdot \left[ {{I}_{0}}({{r}_{p}},{{\theta }_{p}})-{{I}_{2}}({{r}_{p}},{{\theta }_{p}}) \right], \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where I₀, I₁ and I₂ are given by,

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{I}_{0}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \theta }{\rm sin} \theta (1+{\rm cos} \theta ) \\ {} & {} & \times {{J}_{0}}(k{}_{a}^{{}}r_{{}}^{{}}{}_{p}^{{}}s_{{}}^{{}}in\theta {\rm sin} {{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (ik{}_{a}^{{}}r_{p}^{{}}{\rm cos} \theta {\rm cos} {{\theta }_{p}})F(\theta ){\rm d}\theta \\ {{I}_{1}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \theta }{{{\rm sin} }^{2}}\theta \\ {} & {} & \times \;{{J}_{{\rm l}}}\;(k{}_{a}^{{}}r_{{}}^{{}}{}_{p}^{{}}s_{{}}^{{}}in\theta {\rm sin} {{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (ik{}_{a}^{{}}r_{p}^{{}}{\rm cos} \theta {\rm cos} {{\theta }_{p}})F(\theta ){\rm d}\theta \\ {{I}_{2}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \theta }{\rm sin} \theta (1-{\rm cos} \theta ) \\ {} & {} & \times \;{{J}_{2}}(k{}_{a}^{{}}r_{{}}^{{}}{}_{p}^{{}}s_{{}}^{{}}in\theta {\rm sin} {{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (ik{}_{a}^{{}}r_{p}^{{}}{\rm cos} \theta {\rm cos} {{\theta }_{p}})F(\theta ){\rm d}\theta \;. \\ \end{array}\end{eqnarray} \tag{ 2 }$$

J₀, J₁, J₂ are Bessel functions and k_a is the medium wavenumber. θ is integrated to α, where α = sin⁻¹(NA). F(θ) is the focusing function that we have added to the original formula to describe a collimated input Gaussian beam.

The focusing function F(θ) is given by [21],

$$\begin{eqnarray}&&F(\theta )={\rm exp} \left\{ -0.5\cdot {\rm ln} (2)\cdot {{(\frac{D}{W\cdot NA})}^{2}}\cdot {{{\rm sin} }^{2}}(\theta ) \right\}\;.\end{eqnarray} \tag{ 3 }$$

The expression for I₀, I₁ and I₂ given above is to be used when the focal spot coincide with the center of the particle. If the center of the particle is to be displaced from the center of the focal spot by z₀, then simple change r_pcos θ_p to r_pcos θ_p + z₀, with the origin at the particle's center.

It is noted that under paraxial conditions the focusing beam in the image space can be written as a series expansion with respect to the ratio of the beam waist versus diffraction length [22]. If only the first order term is retained then the focusing beam is Gaussian in shape. This Gaussian shape focusing beam at the particle position was used in reference [23] for calculating the nanojet formed by a dielectric particle. We use the Richard and Wolf formula [20] to describe the focusing beam. The input collimated beam before the microscope object is Gaussian, but the image field vector as describe by equations (1) and (2) in the focal region where the particle is located is not, in contrast with reference [23]. Furthermore, the Richard and Wolf formula is valid even for numerical aperture approaching one [24]. Using a microscope objective with NA = 0.85, the spatial profile of the focal spot predicted by the Richard and Wolf formula agreed very well with experimentally measured profile using a scanning NSOM probe [21].

The cos ϕ_p and/or sin ϕ_p dependence of E_r, E_θ, Eϕ given above and the cos (mϕ_p) or sin (mϕ_p) dependence of the vector spherical harmonics given by (A2) in the appendix indicates that the only surviving expansion coefficients are B_o1n and A_e1n with m = 1 only. This conclusion is discussed in the appendix. B_o1n and A_e1n are calculated using equation (A12) which shows that A_e1n = −iB_o1n. Therefore, the resultant incident field $E{{^{\prime} }_{i}}$ expanded in vector spherical harmonics can be written as:

$$\begin{eqnarray}&&E{{^{\prime} }_{i}}\left( {{r}_{p}},{{\theta }_{p}},\;{{\phi }_{p}} \right)=\sum \limits_{n=1}^{\infty } {{E}_{n}}\left( M_{o1n}^{(0)}-i\cdot N_{e1n}^{(0)} \right)\;.\end{eqnarray} \tag{ 4 }$$

The superscript (0) indicates that the spherical Bessel function of the first kind j_n is to be used. The coefficient E_n = B_o1n. Note that the form for the incident field $E{{^{\prime} }_{i}}$ is exactly like the plane wave case except that coefficients which need to be calculated from (A12) are different.

It is necessary to determine the minimum number of summation terms needed in equation (4) so that $E{{^{\prime} }_{i}}$ can faithfully recreates the expression for the exact incident field ${{E}_{i}}$ given by equation (1). For instance, the value of E_n is calculated using (A12) for a microscope objective with NA = 0.4 for medium focusing power, and 0.8 for strong focusing power. The collimated beam size is fixed at W = 2 mm, and objective's back focal plane aperture is fixed at D = 7.65 mm. Values of E_n that show the number of summation terms needed are given in table 1 in the appendix.

Figures 2(a') and (a) show the plot of ${{\left| E{{^{\prime} }_{i}}\left( {{r}_{p}},{{\theta }_{p}},\;{{\phi }_{p}} \right) \right|}^{2}}$and ${{\left| {{E}_{i}}\left( {{r}_{p}},{{\theta }_{p}},\;{{\phi }_{p}} \right) \right|}^{2}}$ versus r_p perpendicular to the optical axis $\left( {{\theta }_{p}}=\pi /2 \right)$ and along the optical axis $\left( {{\theta }_{p}}=0 \right)$ using the vector harmonics expansion of equation (4) and the Richard and Wolf formula of equation (1) respectively. The numerical aperture NA = 0.4. Figures 2(b') and (b) compare the same plot for NA = 0.8. If a sufficient number of summation terms is used, plots using equation (4) or equation (1) are indistinguishable as shown in figure 2, indicating that the vector harmonics expansion is a valid representation of the incident field.

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For the case of the particle displaced from the focal plane by 3.9 μm, calculated coefficients ${{E}_{n}}$ and the number of terms needed for both NA = 0.4 and 0.8 are also given in table 1 in the appendix. For NA = 0.4, figures 3(a') and (a) show the plot of ${{\left| E{{^{\prime} }_{i}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right) \right|}^{2}}$ and ${{\left| {{E}_{i}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right) \right|}^{2}}$ versus r_p for ${{\theta }_{p}}=\pi /2$ and ${{\theta }_{p}}=\pi$ using the vector harmonic expansion and the Richard and Wolf formula respectively. The coordinate origin at r_p = 0 is at the particle's center which is 3.9 μm away from the focal point. The FWHM of the beam spot size at the particle position is now 1.2 μm as indicated by the case corresponding to ${{\theta }_{p}}=\pi /2$ instead of 0.84 μm at the focal point. For the case corresponding to θ_p = π, the field intensity peaks at r_p = 3.9 μm which is the focal point. Figures 3(b') and (b) show the same comparison plot for NA = 0.8. The field intensity at the coordinate origin at r_p = 0, which is the position of the displaced particle center, is small because of the highly divergent beam for NA = 0.8. An expanded view of the field intensity for the case corresponding to ${{\theta }_{p}}=\pi /2$ shows the beam spot size at the particle position is 1.65 μm.

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With the validity of the expansion of the incident field in terms of the vector harmonics established, we proceed to write the expression for the scattered field E_a and the field inside the particle E_b, which are given in (A16). Coefficients α_an and β_an etc are calculated from equation (A19–22). These coefficients are obtained by applying the boundary conditions for the tangential components of the electric and magnetic fields at r = r_b, as given in (A15)

We present calculations of nanojets at a wavelength of 400 nm. The particle is assumed to be a fused silica particle with a diameter of 2 μm and a refractive index of 1.47 at 400 nm wavelength. Figures 4(a) and (a') show the nanojet formed when the focal point and the center of the particle are at the same point for the case corresponding to NA = 0.4. The width of the nanojet waist is a function of the azimuthal angle ϕ. Figure 4(a) is for the case $\phi =\pi /2$, which corresponds to the plane perpendicular to the plane formed by the incident electric field polarization and its propagation direction. The electric field intensity normalized to the focal intensity versus r_p is shown below for θ_p = 0 and for θ_p = 0.115 when the intensity drops by one-half at the nanojet waist. From θ_p = 0.115, the calculated FWHM of the waist is about 253 nm. Figure 4(a') is for the case ϕ = 0, which correspond to the plane formed by the incident field polarization and its propagation direction. The half intensity occurs at θ_p = 0.145 as shown, which indicates that the FWHM of the waist is 319 nm. Figures 4(b) and (b') show the nanojet formed when the center of the particle is 3.9 μm away from the focal point. Compared with figures 4(a) and (a'), it is observed that the nanojet moved away from the surface of the particle by 215 nm. For ϕ = π/2, the half intensity occurs at θ_p = 0.102 which indicates that the FWHM of the waist is about 245 nm. For ϕ = 0, the FWHM is 316 nm. Thus the width of the waist remains about the same for the same ϕ but the position of the waist is moved farther outside the particle. Note that the nanojet field intensity is slightly smaller for figures 4(b) and (b') than for figures 4(a) and (a') because when the particle is displaced from the focal point, the FWHM of the incident beam at the particle is 1.2 μm rather than 0.84 μm; i.e. the particle collects less light. The asymmetry of the nanojet waist with respect to ϕ should be alleviated for circular polarization because the x or the y incident field polarization component is largely maintained during propagation.

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Figures (c), (c'), (d), and (d') are cases when the NA = 0.8. For (c) and (c'), the particle is at the focal point. The nanojet waist is inside the particle. For (d) and (d'), the particle is 3.9 μm away from the focal point. The nanojet waist is now 335 nm outside the particle. For ϕ = π/2, the FWHM of the waist is 256 nm, and for ϕ = 0, it is 315 nm. Thus, the FWHM of the waist are roughly the same for NA = 0.4 and NA = 0.8.

In figures 4(e) and (e'), the result for the case of a plane wave incident field, as generally assumed, is also calculated and compared with the focused beam case. It is noted that for ϕ = π/2, the FWHM of the beam waist is as narrow as 150 nm and occurs at the surface of the particle. However, for ϕ = 0, the FWHM is 340 nm. Thus, the focused beam analysis produce a less asymmetric results compared with the plane wave analysis.

It is noted that nanojets intensity enhancement for focused beam shown in figures 4(a)–(d) are substantially smaller than for a plane wave beam shown in figure 4(e). This is reasonable because a nanoparticle collects much more photon for an incident focused beam than a plane wave, which theoretically is of infinite spatial extent. For instance, the intensity enhancement in figure 4(c) is only about one. If the enhancement were 100, the nanojet beam size would have to shrink to about 25 nm (factor of 10) since the definition of intensity is power/area.

We calculate nanojets formed by a focused incident beam instead of a plane wave. It is shown that the usage of vector spherical harmonics expansion of the focused beam allows a rigorous calculation of the 3D nanojets for two cases in which the particle's center is at or away from the focal point. We find that the position of the nanojet beam waist with respect to the particle surface can be positioned outside the particle by moving the particle's position away from the focal point with little change in the nanojet spot size. It is also found that for the case of a collimated linear polarized input beam focused by a microscope objective the nanojet spot size is more symmetric with respect to the azimuthal angle than the case when the incident field is a plane wave.

The incident field E_i is expanded in terms of the orthogonal vector spherical harmonics as follows,

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{E}_{i}} & =\sum \limits_{m=0}^{\infty } \sum \limits_{n=m}^{\infty } \left( {{B}_{emn}}\cdot {{M}_{emn}}+{{B}_{omn}}\cdot {{M}_{omn}} \right. \\ {} & \quad \left. +{{A}_{emn}}\cdot N{}_{emn}^{{}}+_{{}}^{{}}{{A}_{omn}}\cdot {{N}_{omn}} \right). \\ \end{array}\end{eqnarray} \tag{ A1 }$$

The full expression for the vector spherical harmonics M_emn, M_omn, A_emn, and A_omn as a function of spherical coordinates (r_p, θ_p, ϕ_p) are listed in [25] equations (4)–(20):

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{M}_{emn}} & = & -{\rm sin} m{{\phi }_{p}}\cdot \frac{mP_{n}^{m}}{{\rm sin} {{\theta }_{p}}}\cdot {{z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\theta }} \\ {} & {} & -{\rm cos} m{{\varphi }_{p}}\cdot \frac{{\rm d}P_{n}^{m}}{{\rm d}{{\theta }_{p}}}\cdot {{z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }} \\ {{M}_{omn}} & = & {\rm cos} m{{\phi }_{p}}\cdot \frac{mP_{n}^{m}}{{\rm sin} {{\theta }_{p}}}\cdot {{z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\theta }} \\ {} & {} & -{\rm sin} m{{\varphi }_{p}}\cdot \frac{{\rm d}P_{n}^{m}}{{\rm d}{{\theta }_{p}}}\cdot {{z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }} \\ {{N}_{emn}} & = & {\rm cos} m{{\phi }_{p}}n(n+1)\cdot P_{n}^{m}\frac{{{z}_{n}}(\rho )}{\rho }\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{r}} \\ {} & {} & +{\rm cos} m{{\phi }_{p}}\cdot \frac{{\rm d}P_{n}^{m}}{{\rm d}{{\theta }_{p}}}{{Z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\theta }} \\ {} & {} & -{\rm sin} m{{\phi }_{p}}\frac{mP_{n}^{m}}{{\rm sin} {{\theta }_{p}}}{{Z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }} \\ {{N}_{omn}} & = & {\rm sin} m{{\phi }_{p}}n(n+1)P_{n}^{m}\frac{{{z}_{n}}(\rho )}{\rho }\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{r}} \\ {} & {} & +{\rm sin} m{{\phi }_{p}}\frac{{\rm d}P_{n}^{m}}{{\rm d}{{\theta }_{p}}}{{Z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\theta }} \\ {} & {} & +{\rm cos} m{{\phi }_{p}}\frac{mP_{n}^{m}}{{\rm sin} {{\theta }_{p}}}{{Z}_{n}}(\rho )\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }} \\ {{Z}_{n}}(\rho ) & \equiv & \frac{1}{\rho }\cdot \frac{{\rm d}}{{\rm d}\rho }[\rho \cdot {{z}_{n}}(\rho )]. \\ \end{array}\end{eqnarray} \tag{ A2 }$$

P_n^m(θ_p) is the associate Legendre function The radial function z_n(ρ) is either the spherical Bessel function or the Hankel function ρ = kr_p, where k is the wavenumber. The orthogonal property between the vector spherical harmonics allows the coefficients such as, B_omn etc to be written as,

$$\begin{eqnarray}&&{{B}_{omn}}=\frac{\begin{array}{lllllllllllllll} \int \limits_{0}^{\infty } \int \limits_{0}^{2\pi } \int \limits_{0}^{\pi } {{E}_{i}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right)\cdot {{M}_{omn}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right) \\ \quad \cdot {\mkern 1mu} {\rm sin} \left( {{\theta }_{p}} \right)\cdot {\rm d}{{\theta }_{p}}\cdot {\rm d}{{\phi }_{p}}\cdot {\rm d}{{r}_{p}} \\ \end{array}}{\begin{array}{lllllllllllllll} \int \limits_{0}^{\infty } \int \limits_{0}^{2\pi } \int \limits_{0}^{\pi } \cdot {{\left| {{M}_{omn}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right) \right|}^{2}} \\ \quad \cdot {\mkern 1mu} {\rm sin} \left( {{\theta }_{p}} \right)\cdot {\rm d}{{\theta }_{p}}\cdot {\rm d}{{\phi }_{p}}\cdot {\rm d}{{r}_{p}} \\ \end{array}}\end{eqnarray} \tag{ A3 }$$

Contrast to reference [2, 19, 25], we provide an additional integration with respect to r_p. To describe the field E_i at any point (r_p, θ_p, ϕ_p) in the image space of a microscope objective, we use the formula given in reference [20] equations (2)–(32) with addition factors which are discussed below. If the input beam is polarized in the x-direction, then the focused incident field E_i on the particle in the image space of the objective are written in Cartesian coordinates [20] as:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{E}_{x}}=-\pi \left( {{I}_{0}}+{{I}_{2}}\cdot {\rm cos} 2{{\phi }_{p}} \right)\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{x}} \\ {{E}_{y}}=-\pi \cdot {{I}_{2}}\cdot {\rm sin} 2{{\phi }_{p}}\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{y}} \\ {{E}_{z}}=2i\pi \cdot {{I}_{1}}\cdot {\rm cos} {{\phi }_{p}}\cdot {{{\overset{\scriptscriptstyle\frown}{e}}}_{z}} \\ \end{array}\end{eqnarray} \tag{ A4 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{I}_{0}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \theta }\;{\rm sin} \;\theta (1+\;{\rm cos} \;\theta ) \\ {} & {} & \times {{J}_{0}}(k{}_{a}^{{}}r_{{}}^{{}}{}_{p}^{{}}{\rm s}_{{}}^{{}}{\rm in}{\mkern 1mu} \theta \;{\rm sin} \;{{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (ik{}_{a}^{{}}r_{p}^{{}}{\rm cos} \theta \;{\rm cos} \;{{\theta }_{p}})F(\theta ){\rm d}\theta \\ {{I}_{1}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \;\theta }{{{\rm sin} }^{2}}{\mkern 1mu} \theta \\ {} & {} & \times \;{{J}_{1}}(k{}_{a}^{{}}r_{{}}^{{}}{}_{p}^{{}}{\rm s}_{{}}^{{}}{\rm in}{\mkern 1mu} \theta \;{\rm sin} \;{{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (i{{k}_{a}}{{r}_{p}}\;{\rm cos} \;\theta \;{\rm cos} \;{{\theta }_{p}})F(\theta ){\rm d}\theta \\ {{I}_{2}}({{r}_{p}},{{\theta }_{p}}) & = & \int \limits_{0}^{\alpha } \sqrt{{\rm cos} \;\theta }\;{\rm sin} \;\theta (1-\;{\rm cos} \;\theta ) \\ {} & {} & \times {{J}_{2}}(k{}_{a}^{{}}r_{p}^{{}}\;{\rm sin} \;\theta \;{\rm sin} \;{{\theta }_{p}}) \\ {} & {} & \times {\rm exp} (ik{}_{a}^{{}}r_{p}^{{}}\;{\rm cos} \;\theta \;{\rm cos} \;{{\theta }_{p}})F(\theta ){\rm d}\theta . \\ \end{array}\end{eqnarray} \tag{ A5 }$$

Here, α, the limit of integration for θ is given by the numerical aperture NA of the microscope objective according to α = sin^-1(NA). The additional factor F(θ) describes the specified microscope objective focusing of an incoming collimated Gaussian beam. F(θ) is given by [21].

$$\begin{eqnarray}&&F\left( \theta \right)={\rm exp} \left\{ -0.5{\rm ln} \left( 2 \right){{\left( \frac{D}{W\;NA} \right)}^{2}}{{{\rm sin} }^{2}}\left( \theta \right) \right\},\end{eqnarray} \tag{ A6 }$$

where D is the diameter of the back focal aperture of the objective and W is the FWHM of the collimated incoming Gaussian beam (see figure 1). (A6) is not normalized with respect to the incident optical power, which is not an issue for calculations presented here.

Since the vector spherical harmonics are in spherical coordinates, it is necessary to transform E_i from Cartesian to spherical coordinates. Thus, unit vectors e_x, e_y and e_z are transformed into unit vectors e_r, e_θ, and e_ϕ in spherical coordinates according to familiar formulas, the results are:

$$\begin{eqnarray}&&{{E}_{i}}\left( {{r}_{p}},{{\theta }_{p}},{{\phi }_{p}} \right)={{E}_{r}}\cdot {{\overset{\scriptscriptstyle\frown}{e}}_{r}}+{{E}_{\theta }}\cdot {{\overset{\scriptscriptstyle\frown}{e}}_{\theta }}+{{E}_{\phi }}\cdot {{\overset{\scriptscriptstyle\frown}{e}}_{\phi }}\end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray*}&&\begin{array}{lllllllllllllll} {{E}_{r}}=-\pi {\rm cos} {{\phi }_{p}}\cdot \{[{{I}_{0}}({{r}_{p}},{{\theta }_{p}})+{{I}_{2}}({{r}_{p}},{{\theta }_{p}})]\cdot {\rm sin} {{\theta }_{p}}-2i{{I}_{1}}({{r}_{p}},{{\theta }_{p}}){\rm cos} {{\theta }_{p}}\} \\ {{E}_{\theta }}=-\pi {\rm cos} {{\phi }_{p}}\cdot \{[{{I}_{0}}({{r}_{p}},{{\theta }_{p}})+{{I}_{2}}({{r}_{p}},{{\theta }_{p}})]\cdot {\rm cos} {{\theta }_{p}}+2i{{I}_{1}}({{r}_{p}},{{\theta }_{p}}){\rm sin} {{\theta }_{p}}\} \\ {{E}_{\phi }}=\pi {\rm sin} {{\phi }_{p}}\cdot [{{I}_{0}}-{{I}_{2}}]\;. \\ \end{array}\end{eqnarray*}$$

With formulas given above and the numerical aperture of the microscope objective and the input beam parameters specified, it is shown that the focal spot spatial profile agrees very well with the measured profile using a NSOM tip in the collection mode of operation [21] If the focal plane is to be shifted away from the origin of the particle, then simply make the following change in the expression for I₀, I₁, and I₂:

$$\begin{eqnarray}&&{{r}_{p}}{\rm cos} {{\theta }_{p}}\to {{r}_{p}}{\rm cos} {{\theta }_{p}}+{{z}_{o}}\;.\end{eqnarray} \tag{ A8 }$$

If z₀ is positive, then the center of the particle at r_p = 0 is to the right of the focal plane, which means that the beam is expanding at the particle's coordinate. If z₀ is negative, the opposite is true.

Note that the ϕ_p dependence of all four vector harmonics in (A2) contains either cos(mϕ_p) or sin(mϕ_p) terms, whereas the ϕ_p dependence of E_r, E_θ, E_ϕ (A7) contains only cos ϕ_p or sin ϕ_p. Therefore, all expansion coefficients obtained from equation (A3) will be zero unless m = 1 since all coefficients are the product of vector harmonics and E_i. And ϕ_p is integrated from 0 to 2π. Moreover, with m = 1, coefficients B_e1n involving M_e1n and A_o1n involving N_o1n are zero because the integration over ϕ_p involving these terms are always the product of sin ϕ_p and cos ϕ_p which yields zero. Thus, the only non-zero coefficients in the expansion of E_i are B_o1n and A_e1n involving M_o1n and N_e1n respectively. We write $E{{^{\prime} }_{i}}$ as,

$$\begin{eqnarray}&&E_{i}^{\prime }=\sum \limits_{n=l}^{\infty } \left( {{B}_{oln}}M_{oln}^{\left( 0 \right)}+{{A}_{eln}}N_{eln}^{\left( 0 \right)} \right).\end{eqnarray} \tag{ A9 }$$

The superscripts (0) in the vector harmonics indicates that its radial components z_n(ρ) are given by the spherical Bessel functions of the first kind j_n(k_ar_p), where k_a is the medium wavenumber. Expressions for the spherical vector harmonics M_o1n and N_e1n for the case m = 1, after conversion from the associated Legendre function P¹_n to the Legendre function P_n, are given as,

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{M}_{o{\rm l}n}} & = & {\rm cos} {{\phi }_{p}}\frac{{\rm d}{{P}_{n}}(y)}{{\rm d}y}{{z}_{n}}({{\rho }_{p}}){{{\overset{\scriptscriptstyle\frown}{e}}}_{\theta }}-{\rm sin} {{\phi }_{p}} \\ {} & {} & \times \left[ y\frac{{\rm d}{{P}_{n}}}{{\rm d}y}-(1-{{y}^{2}})\frac{{{{\rm d}}^{2}}{{P}_{n}}}{{\rm d}{{y}^{2}}} \right]{{z}_{n}}({{\rho }_{p}}){{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }} \\ \end{array}\end{eqnarray} \tag{ A10 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{N}_{e1n}} & = & {\rm cos} {{\phi }_{p}}n\left( n+1 \right)\sqrt{1-{{y}^{2}}}\frac{{\rm d}{{P}_{n}}(y)}{{\rm d}y}\frac{{{z}_{n}}\left( {{\rho }_{p}} \right)}{{{\rho }_{p}}}{{{\overset{\scriptscriptstyle\frown}{e}}}_{r}} \\ {} & {} & +{\rm cos} {{\phi }_{p}}\left[ y\frac{{\rm d}{{P}_{n}}}{{\rm d}y}-\left( 1-{{y}^{2}} \right)\frac{{{{\rm d}}^{2}}{{P}_{n}}}{{\rm d}{{y}^{2}}} \right]\cdot \\ {} & {} & \times \left[ \left( n+1 \right)\frac{{{z}_{n}}\left( {{\rho }_{p}} \right)}{{{\rho }_{p}}}-{{z}_{n+1}}\left( {{\rho }_{p}} \right) \right] \\ {} & {} & \times e{{^{\prime} }_{\theta }}-{\rm sin} {{\phi }_{p}}\frac{{\rm d}P{}_{n}^{{}}(y)_{{}}^{{}}}{{\rm d}y} \\ {} & {} & \times \left. \left[ (n+1)\frac{{{z}_{n}}\left( {{\rho }_{p}} \right)}{{{\rho }_{p}}} \right.-{{z}_{n+1}}\left( {{\rho }_{p}} \right) \right]{{{\overset{\scriptscriptstyle\frown}{e}}}_{\phi }}. \\ \end{array}\end{eqnarray} \tag{ A11 }$$

Here, y = cos θ_p. According to (A2), (A3) and (A7), B_o1n and A_e1n can eventually be written as

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{B}_{o1n}}= \\ \frac{{{\pi }^{2}}\int \limits_{0}^{\infty } \int \limits_{0}^{\pi } \begin{array}{lllllllllllllll} \left\{ \begin{array}{lllllllllllllll} -2\;{\rm sin} \;{{\theta }_{p}}\frac{{\rm d}{{P}_{n}}}{{\rm d}y}\left[ {{I}_{2}}\;{\rm cos} \;{{\theta }_{p}}+i{{I}_{1}}\;{\rm sin} \;{{\theta }_{p}} \right] \\ \;-n\left( n+1 \right){{P}_{n}}\left( {\rm cos} \;{{\theta }_{p}} \right){\rm sin} \;{{\theta }_{p}}\left( {{I}_{0}}-{{I}_{2}} \right) \\ \end{array} \right\} \\ \;\;{\rm d}{{\theta }_{p}}{\rm d}{{\rho }_{p}} \\ \end{array}}{2\pi \frac{{{n}^{2}}{{\left( n+1 \right)}^{2}}}{2n+1}\cdot \int \limits_{0}^{\infty } j_{n}^{2}\left( {{\rho }_{p}} \right)\cdot {\rm d}{{\rho }_{p}}} \\ \end{array}\end{eqnarray} \tag{ A12 }$$

$$\begin{eqnarray*}&&\begin{array}{lllllllllllllll} {{A}_{e1n}}= \\ \frac{\int \limits_{0}^{\infty } \left\{ \begin{array}{lllllllllllllll} \frac{{{j}_{n}}\left( \rho \right)}{\rho }\cdot N{{1}_{n}}\left( \rho \right) \\ \;-\left[ \left( n+1 \right)\cdot \frac{{{j}_{n}}\left( \rho \right)}{\rho }-{{j}_{n+1}}(\rho ) \right]\cdot N{{2}_{n}}\left( \rho \right)\cdot {\rm d}\rho \\ \end{array} \right\}}{\begin{array}{lllllllllllllll} \pi {{\left[ n\left( n+1 \right) \right]}^{2}}\left( \frac{2}{2n+1} \right) \\ \left\{ \begin{array}{lllllllllllllll} \left( 2n+1 \right)\left( n+1 \right)\int \limits_{0}^{\infty } {{\left( \frac{{{j}_{n}}\left( \rho \right)}{\rho } \right)}^{2}} \\ {\rm d}\rho +\int \limits_{0}^{\infty } j_{n+1}^{2}\left( \rho \right){\rm d}\rho -2\left( n+1 \right) \\ \int \limits_{0}^{\infty } \frac{{{j}_{n}}(\rho )}{\rho }{{j}_{n+1}}\left( \rho \right){\rm d}\rho \\ \end{array} \right\} \\ \end{array}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} N{{1}_{n}}(\rho ) & = & {{\pi }^{2}}n(n+1)\cdot \int \limits_{0}^{\pi } \left\{ -{{{\rm sin} }^{3}}{\mkern 1mu} {{\theta }_{p}}\cdot \frac{{\rm d}{{P}_{n}}}{{\rm d}y}\cdot ({{I}_{0}}+{{I}_{2}}) \right. \\ {} & {} & \left. +2i\;{\rm sin} \;\theta {}_{p}^{{}}\cdot _{{}}^{{}}\;{\rm cos} \;{{\theta }_{p}}\cdot \frac{{\rm d}{{P}_{n}}}{{\rm d}y}\cdot {{I}_{1}} \right\}{\rm d}{{\theta }_{p}} \\ N2{}_{n}^{{}}(\rho )_{{}}^{{}} & = & {{\pi }^{2}}\int \limits_{0}^{\pi } \left\{ \left[ {\rm sin} \;{{\theta }_{p}}\;{\rm cos} \;{{\theta }_{p}}\frac{{\rm d}{{P}_{n}}}{{\rm d}y}-{{{\rm sin} }^{3}}{\mkern 1mu} {{\theta }_{p}}\frac{{{{\rm d}}^{2}}{{P}_{n}}}{{\rm d}{{y}^{2}}} \right] \right. \\ {} & {} & \times \left[ {\rm cos} \;{{\theta }_{p}}({{I}_{0}}+{{I}_{2}})+2i\;{\rm sin} \;{{\theta }_{p}}{{I}_{1}} \right] \\ {} & {} & \left. +\;{\rm sin} \;{{\theta }_{p}}\frac{{\rm d}{{P}_{n}}}{{\rm d}y}({{I}_{0}}-{{I}_{2}}) \right\}{\rm d}{{\theta }_{p}}, \\ \end{array}\end{eqnarray} \tag{ A13 }$$

where j_n(ρ) is the spherical Bessel funcion of order n, P_n(cos θ_p) is the Legendre function, y = cos θ_p, and I₀, I₁, and I₂ are focal functions given in equation (A5)–(A6) with k_ar_p define as ρ. Numerical calculations using the formula given above yield A_e1n = −iB_o1n like the plane wave case except that the value of B_o1n is different and changes with different degree of focusing. We define B_o1n as B_o1n ≡⃥ E_n, and rewrite $E{{^{\prime} }_{i}}$ as

$$\begin{eqnarray}&&E{{^{\prime} }_{i}}=\sum \limits_{n=1}^{\infty } {{E}_{n}}\left( M_{o1n}^{(0)}-i\cdot N_{e1n}^{(0)} \right)\;.\end{eqnarray} \tag{ A14 }$$

Denote the scattered electric and magnetic fields outside the particle as E_a and H_a. Inside the particle fields are denoted as E_b and H_b. When r_p is outside the particle, the total field consists of the incident field plus the scattered field. Tangential components of the electric and magnetic fields in the e_θ and e_ϕ direction must be equal at the particle-medium interface at r_p = r_b. This equality condition on E and H are expressed as:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\left( E_{i}^{\prime }+{{E}_{a}} \right)}_{\theta ,\phi }} & = & {{\left( {{E}_{b}} \right)}_{\theta ,\phi }} \\ {{\left( H_{i}^{\prime }+{{H}_{a}} \right)}_{\theta ,\phi }} & = & {{\left( {{H}_{b}} \right)}_{\theta ,\phi }}. \\ \end{array}\end{eqnarray} \tag{ A15 }$$

E_i consists of a linear combination of M_o1n and N_e1n, therefore based on (A14) E_a and E_b are also a linear combination of M_o1n and N_e1n. Thus, E_a and E_b are expressed as

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{E}_{a}}=\sum \limits_{n=1}^{\infty } {{E}_{n}}\cdot (i{{\alpha }_{an}}\cdot N_{e1n}^{(1)}-\beta {}_{an}^{{}}{\mkern 1mu} _{{}}^{{}}\cdot {\mkern 1mu} M_{o1n}^{(1)}) \\ {{E}_{b}}=\sum \limits_{n=1}^{\infty } {{E}_{n}}\cdot ({{\beta }_{bn}}\cdot M_{o1n}^{(0)}-i{{\alpha }_{bn}}\cdot N_{e1n}^{(0)}). \\ \end{array}\end{eqnarray} \tag{ A16 }$$

Since the scattered field resembles an outgoing wave, the appropriate general expression for z_n describing the scattered field is the spherical Hankel function of the first kind: z_n(ρ_p) = h_n(k_ar_p), with h_n(ρ_p) ≡⃥ j_n(ρ_p) + i y_n(ρ_p), where y_n(ρ_p) is the spherical Bessel function of the second kind. The superscript (1) indicates the usage of this spherical Hankel function of the first kind.

To describe the field inside the particle z_n(ρ_p) = j_n(k_br_p) since j_n does not diverge at the origin. Here, k_b is the wavenumber associated with the particle.

The magnetic field H_i, H_a, H_b can be obtained from

$$\begin{eqnarray}&&{{H}_{i,a,b}}=\frac{1}{i\omega \mu }\cdot \nabla \times {{E}_{i,a,b}}\;.\end{eqnarray} \tag{ A17 }$$

Using the relationship between the vector harmonics N and M as given by reference [25], H are written as

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} H{{^{\prime} }_{i}}=-\frac{{{k}_{a}}}{\omega \cdot \mu }\cdot \sum \limits_{n=1}^{\infty } {{E}_{n}}\cdot (M_{e1n}^{(0)}+i\cdot N_{o1n}^{(0)}) \\ {{H}_{a}}=-\frac{{{k}_{a}}}{\omega \cdot \mu }\cdot \sum \limits_{1}^{\infty } {{E}_{n}}\cdot (i\cdot {{\beta }_{an}}\cdot N_{o1n}^{(1)}+{{\alpha }_{an}}\cdot M_{e1n}^{(1)}) \\ {{H}_{b}}=-\frac{{{k}_{b}}}{\omega \cdot \mu }\cdot \sum \limits_{1}^{\infty } {{E}_{n}}\cdot (i\cdot {{\beta }_{bn}}\cdot N_{o1n}^{(0)}+{{\alpha }_{bn}}\cdot M_{e1n}^{(0)}) \\ \end{array}\end{eqnarray} \tag{ A18 }$$

with (A14) and (A16) and (A18) inserted into the boundary conditions, (A15) at r_p = r_b, one obtains four equations that solve for the four constants α_an, α_bn, β_an, β_bn. They are in determinate form,

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\alpha }_{an}}= \\ \quad \frac{\left[ \begin{array}{lllllllllllllll} \frac{{{k}_{a}}}{\mu }\cdot {{j}_{n}}({{k}_{a}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{j}_{n}}\left( {{k}_{a}}{{r}_{b}} \right)}{{{k}_{a}}{{r}_{b}}}-{{j}_{n+1}}\left( {{k}_{a}}{{r}_{b}} \right) \\ \frac{{{k}_{b}}}{\mu }\cdot {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ (n+1)\frac{{{j}_{n}}({{k}_{b}}{{r}_{b}})}{({{k}_{b}}{{r}_{b}})}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}}) \\ \end{array} \right]}{\left[ \begin{array}{lllllllllllllll} \frac{{{k}_{a}}}{\mu }\cdot {{h}_{n}}({{k}_{a}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{h}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{h}_{n+1}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{b}}}{\mu }\cdot {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{j}_{n}}({{k}_{b}}{{r}_{b}})}{{{k}_{b}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}}) \\ \end{array} \right]} \\ \end{array}\end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\alpha }_{bn}}= \\ \quad \frac{\left[ \begin{array}{lllllllllllllll} \frac{{{k}_{a}}}{\mu }\cdot {{h}_{n}}({{k}_{a}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{h}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{a}}}{\mu }\cdot {{j}_{n}}({{k}_{a}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{j}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{a}}{{r}_{b}}) \\ \end{array} \right]}{\left[ \begin{array}{lllllllllllllll} \frac{{{k}_{a}}}{\mu }\cdot {{h}_{n}}({{k}_{a}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{h}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{h}_{n+1}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{b}}}{\mu }\cdot {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ \left( n+1 \right)\frac{{{j}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{b}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}}) \\ \end{array} \right]} \\ \end{array}\end{eqnarray} \tag{ A20 }$$

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\beta }_{an}}= \\ \quad \frac{\left[ \begin{array}{lllllllllllllll} {{j}_{n}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{a}}}{\mu }[(n+1)\frac{{{j}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{a}}{{r}_{b}})] \\ {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ \frac{{{k}_{b}}}{\mu }[(n+1)\frac{{{j}_{n}}({{k}_{b}}{{r}_{b}})}{{{k}_{b}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}})] \\ \end{array} \right]}{\left[ \begin{array}{lllllllllllllll} h{}_{n}^{{}}({{k}_{a}}{{r}_{b}})_{{}}^{{}} \\ \frac{{{k}_{a}}}{\mu }\cdot [(n+1)\frac{h{}_{n}^{{}}({{k}_{a}}{{r}_{b}})_{{}}^{{}}}{{{k}_{a}}{{r}_{b}}}-{{h}_{n+1}}({{k}_{a}}{{r}_{b}})] \\ {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ \frac{{{k}_{b}}}{\mu }\cdot [(n+1)\frac{{{j}_{n}}({{k}_{b}}{{r}_{b}})}{{{k}_{b}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}})] \\ \end{array} \right]} \\ \end{array}\end{eqnarray} \tag{ A21 }$$

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\beta }_{bn}}= \\ \quad \frac{\left[ \begin{array}{lllllllllllllll} {{h}_{n}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{a}}}{\mu }[(n+1)\frac{{{h}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{h}_{n+1}}({{k}_{a}}{{r}_{b}})] \\ {{j}_{n}}({{k}_{a}}{{r}_{b}}) \\ \frac{{{k}_{a}}}{\mu }[(n+1)\frac{{{j}_{n}}({{k}_{a}}{{r}_{b}})}{{{k}_{a}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{a}}{{r}_{b}})] \\ \end{array} \right]}{\left[ \begin{array}{lllllllllllllll} h{}_{n}^{{}}({{k}_{a}}{{r}_{b}})_{{}}^{{}} \\ \frac{{{k}_{a}}}{\mu }\cdot [(n+1)\frac{h{}_{n}^{{}}({{k}_{a}}{{r}_{b}})_{{}}^{{}}}{{{k}_{a}}{{r}_{b}}}-{{h}_{n+1}}({{k}_{a}}{{r}_{b}})] \\ {{j}_{n}}({{k}_{b}}{{r}_{b}}) \\ \frac{{{k}_{b}}}{\mu }\cdot [(n+1)\frac{j{}_{n}^{{}}({{k}_{b}}{{r}_{b}})_{{}}^{{}}}{{{k}_{b}}{{r}_{b}}}-{{j}_{n+1}}({{k}_{b}}{{r}_{b}})] \\ \end{array} \right]} \\ \end{array}\end{eqnarray} \tag{ A22 }$$

with α_an, α_bn, β_an, β_bn calculated and E_n obtained from (A16) for the case of the focused beam, all electric and magnetic fields as a function of (r_p, θ_p, ϕ_p) are given.