Full-Wave Analytical Solution of Second-Harmonic Generation in Metal Nanospheres

Abstract

We present a full-wave analytical solution for the problem of second-harmonic generation from spherical nanoparticles. The sources of the second-harmonic radiation are represented through an effective nonlinear polarization. The solution is derived in the framework of the Mie theory by expanding the pump field, the nonlinear sources, and the second-harmonic fields in series of spherical vector wave functions. We use the proposed solution for studying the second-harmonic radiation generated from gold nanospheres as a function of the pump wavelength and the particle size, in the framework of the Rudnick–Stern model. We demonstrate the importance of high-order multipolar contributions to the second-harmonic radiated power. Moreover, we investigate the dependence of the p- and s-components of the second-harmonic radiation on the Rudnick–Stern parameters. This approach provides a rigorous methodology to understand second-order optical processes in nanostructured metals and to design novel nanoplasmonic devices in the nonlinear regime.

Appendices

A Vector Spherical Harmonics

The vector spherical harmonics X _mn are [32, 33] as follows:

$$\begin{array}{@{}rcl@{}} {\mathbf{X}_{mn}}( {\theta ,\phi } ) &=& i\frac{1}{{\sqrt {n( {n + 1} )} }}\sqrt {\frac{{2n + 1}}{{4\pi }}\frac{{( {n - m} )!}}{{( {n + m} )!}}} \\ &&\cdot \left[ {i{\pi_{mn}}( {\cos \theta } )\hat \theta - {\tau_{mn}}( {\cos \theta } )\hat \phi } \right]{e^{im\phi }}\;, \end{array} $$

where

$$\begin{array}{lll} {\pi_{mn}}( {\cos \theta } ) &=& \frac{m}{{\sin \theta }}P_{n}^{m}( {\cos \theta } )\\ {\tau_{mn}}( {\cos \theta } ) &=& \frac{d}{{d\theta }}P_{n}^{m}( {\cos \theta } ), \end{array} $$

and $P_{n}^{m} = P_{n}^{m}( u )$ is the associated Legendre function of the first kind and of degree n and m.

B Calculation of $\left \{{p_{mn}^{( \omega )},q_{mn}^{( \omega )}}\right \}$

The expansion coefficients in Eq. 10a, for a linearly polarized plane wave propagating along the z − axis with the electric field parallel to the x − axis (Fig. 1a), are

$$\begin{array}{@{}rcl@{}} p_{mn}^{( \omega )} &=& q_{mn}^{( \omega )} = 0 \mathrm{,\; for}\, | m | \ne 1\\ p_{1n}^{( \omega )} &=& q_{1n}^{( \omega )} = -p_{-1n}^{( \omega )} = q_{-1n}^{( \omega )} = \frac{1}{2}{{(-i)}^{n}}\sqrt{{4\pi(2n + 1)}}. \end{array} $$

C Calculation of $\left \{ {a_{mn}^{( \omega )},b_{mn}^{( \omega )}},{c_{mn}^{( \omega )},d_{mn}^{( \omega )}} \right \}$

The coefficients $\left \{ {a_{mn}^{( \omega )},b_{mn}^{( \omega )}},{c_{mn}^{( \omega )},d_{mn}^{( \omega )}} \right \}$ are expressed as

$$\begin{array}{@{}rcl@{}} {{a_{mn}^{(\omega )}}} &=& {{a_{n}^{(\omega )}}} {{p_{mn}^{(\omega )}}},\quad {{b_{mn}^{(\omega )}}} = {{b_{n}^{(\omega )}}} {{q_{mn}^{(\omega )}}},\\ {{c_{mn}^{(\omega )}}} &=& {{c_{n}^{(\omega )}}} {{q_{mn}^{(\omega )}}},\quad {{d_{mn}^{(\omega )}}} = {{d_{n}^{(\omega )}}} {{p_{mn}^{(\omega )}}}, \end{array} $$

where

$$\begin{array}{@{}rcl@{}} {{a_{n}^{(\omega )}}} {}&=&{} \frac {{\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{e}^{(\omega )}\right) {}-{} \;{\psi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right)}} {{\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \xi }_{n}}\left(x_{e}^{(\omega )}\right) {}-{} \;{\xi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right)}},\\ {{b_{n}^{(\omega )}}} {}&=&{} \frac {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\psi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right) {}-{} {\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{e}^{(\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\xi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right) - {\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \xi }_{n}}\left(x_{e}^{(\omega )}\right)},\\ {{c_{n}^{(\omega )}}} &=& \frac {-i\frac{{{k_{i}}(\omega )}}{{{k_{e}}(\omega )}}} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\xi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right) {}-{} \;{\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \xi }_{n}}\left(x_{e}^{(\omega )}\right)},\\ {{d_{n}^{(\omega )}}} &=& \frac {{i\frac{{{k_{i}}(\omega )}}{{{k_{e}}(\omega )}}}} {{\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(\omega )}}\;{\psi_{n}}\left(x_{i}^{(\omega )}\right)\;{{\dot \xi }_{n}}\left(x_{e}^{(\omega )}\right) {}-{} \;{\xi_{n}}\left(x_{e}^{(\omega )}\right)\;{{\dot \psi }_{n}}\left(x_{i}^{(\omega )}\right)}}, \end{array} $$

$x_{e}^{(\omega )} = k_{e}^{(\omega )}R,\;\;x_{i}^{(\omega )} = k_{i}^{(\omega )}$ and ${\psi _{n}} = {\psi _{n}}( \rho )\;$, ${\xi _{n}} = {\xi _{n}}( \rho )$ are the Riccati–Bessel functions defined as ${\psi _{n}}(\rho ) = \rho \;{j_{n}}(\rho ),\;\;\;{\xi _{n}} = \rho \;h_{n}^{(1)}(\rho )$. $\dot \zeta$ denotes the first derivative of ζ = ζ(ρ) with respect to ρ.

D Selvedge Region

The selvedge region (Fig. 1b) is a layer of infinitesimal depth $\delta$ at the interface metal–vacuum. In this region, there is a volumetric current density $\mathbf {J}_{\perp }^{( {2\omega } )} = i2\omega \left ( {\mathbf {P}_{s}^{\left ( {2\omega } \right )} \cdot \hat {\mathbf {n}}/\delta } \right )\hat {\mathbf {n}}$, which is exactly compensated by the normal component of the displacement current density, $\mathbf {J}_{\perp }^{( {2\omega } )} + i2\omega \mathbf {D}_ \perp ^{( {2\omega } )} = {\bf {0}}$; otherwise, there would be an unbounded magnetic field. Therefore, in the selvedge region, $\mathbf {D}_ \perp ^{( {2\omega } )} = - \left ( {\mathbf {P}_{s}^{\left ( {2\omega } \right )} \cdot \hat {\mathbf {n}}/\delta } \right )\hat {\mathbf {n}}$. From the Faraday–Neumann law, applied to the elementary curve $\Delta l$ shown in Fig. 1b, we have $\left ({\mathbf {E}_{i}^{\left ({2\omega } \right )} - \mathbf {E}_{e}^{({2\omega })}}\right )\Big |_{\Sigma } \cdot \Delta {l_{\parallel }} = {u_{2}} - {u_{1}}$, where ${u_{(\alpha )}} = \int _{\Delta l_{\perp }^{(\alpha )}}{\mathbf {E}^{( {2\omega } )}}\cdot {d{ l}} = \left . ({\mathbf {P}_{s}^{( {2\omega } )} \cdot \hat {\mathbf {n}}})/\varepsilon '\right |_{Q^{(\alpha )}}$ and α = 1, 2. By combining these relations, we obtain the equations $\hat {\mathbf {n}} \times \left . \left ( {\mathbf {E}_{i}^{\left ( {2\omega } \right )} - \mathbf {E}_{e}^{( {2\omega } )}} \right )\right | {_{\Sigma }} = \hat {\mathbf {n}} \times {\nabla _{s}}\left ( {\mathbf {P}_{s}^{\left ( {2\omega } \right )} \cdot \hat {\mathbf {n}}} \right )/\varepsilon '$.

E Calculation of$\left \{ {a_{mn}^{( {2\omega } )},b_{mn}^{( {2\omega } )}},{c_{mn}^{( {2\omega } )},d_{mn}^{( {2\omega } )}} \right \}$

The coefficients $\left \{{a_{mn}^{( {2\omega } )},b_{mn}^{( {2\omega } )}},c_{mn}^{( {2\omega } )},d_{mn}^{( {2\omega })} \right \}$ are given by

$$\begin{array}{@{}rcl@{}} a_{mn}^{( {2\omega } )} = {a' }_{n}^{( {2\omega } )} {u' }_{mn}^{( {2\omega } )} + {a^{\prime\prime}}_{n}^{( {2\omega } )} {u^{\prime\prime}}_{mn}^{( {2\omega } )}, \\ b_{mn}^{( {2\omega } )} = {b' }_{n}^{({2\omega } )} {v' }_{mn}^{( {2\omega } )} + {b^{\prime\prime}}_{n}^{( {2\omega } )} {v^{\prime\prime}}_{mn}^{( {2\omega } )}, \\ c_{mn}^{( {2\omega } )} = {c' }_{n}^{({2\omega } )} {v' }_{mn}^{( {2\omega } )} + {c^{\prime\prime}}_{n}^{( {2\omega } )} {v^{\prime\prime}}_{mn}^{( {2\omega } )}, \\ d_{mn}^{( {2\omega } )} = {d' }_{n}^{({2\omega } )} {u' }_{mn}^{( {2\omega } )} + {d^{\prime\prime}}_{n}^{( {2\omega } )} {u^{\prime\prime}}_{mn}^{( {2\omega } )}, \end{array} $$

where the coefficients $\left \{ {{u'}_{mn}^{(2\omega )},{v'}_{mn}^{(2\omega )}} \right \}$ and $\left \{ {{u^{\prime \prime }}_{mn}^{(2\omega )},{v^{\prime \prime }}_{mn}^{(2\omega )}} \right \}$ are given in Appendix F.

We denote with one apex the contribution due to the tangential surface SH sources and with two apices the contributions of both the normal surface SH sources and the γ bulk SH sources.

For the tangential surface SH sources, we have

$$\begin{array}{@{}rcl@{}} {{{a'}_{n}^{(2\omega )}}} = \frac {-x_{e}^{(2\omega )} {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right) - {\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right)},\\ {{{b'}_{n}^{(2\omega )}}} = \frac {{-x_{e}^{(2\omega )} {\psi_{n}}\left(x_{i}^{(2\omega )}\right)}} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right) - {\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right)},\\ {{{c'}_{n}^{(2\omega )}}} = \frac {x_{i}^{(2\omega )} {\xi_{n}}\left(x_{e}^{(2\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\xi_{n}}(x_{e}^{(2\omega )}) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right) - {\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right)},\\ {{{d'}_{n}^{(2\omega )}}} = \frac {{x_{i}^{(2\omega )} {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right)}} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right) - {\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right)}, \end{array} $$

and for the contribution of both the normal surface SH sources and the γ bulk SH sources, we have

$$\begin{array}{@{}rcl@{}} {{{a^{\prime\prime}}_{n}^{(2\omega )}}} = \frac {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}x_{e}^{(2\omega )} {\psi_{n}}\left(x_{i}^{(2\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right) - {\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right)},\\ {{{b^{\prime\prime}}_{n}^{(2\omega )}}} = \frac {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}x_{e}^{(2\omega )} {\dot \psi_{n}}\left(x_{i}^{(2\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right) - {\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right)},\\ {{{c^{\prime\prime}}_{n}^{(2\omega )}}} = \frac {-x_{i}^{(2\omega )} {\dot \xi_{n}}(x_{e}^{(2\omega )})} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right) - {\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right)},\\ {{{d^{\prime\prime}}_{n}^{(2\omega )}}} = \frac {-x_{i}^{(2\omega )} {\xi_{n}}\left(x_{e}^{(2\omega )}\right)} {\frac{{{\zeta_{e}}}}{{{\zeta_{i}}(2\omega )}}{\psi_{n}}\left(x_{i}^{(2\omega )}\right) {{\dot \xi }_{n}}\left(x_{e}^{(2\omega )}\right) - {\xi_{n}}\left(x_{e}^{(2\omega )}\right) {{\dot \psi }_{n}}\left(x_{i}^{(2\omega )}\right)}, \end{array} $$

where $x_{e}^{(2\omega )} = k_{e}^{(2\omega )}R,\;\;x_{i}^{(2\omega )} = k_{i}^{(2\omega )}R,$ ${\psi _{n}} = {\psi _{n}}( \rho ),$ and ξ _n = ξ _n (ρ) are the Riccati–Bessel functions.

F Calculation of $\left \{ {{u'}_{mn}^{(2\omega )},{v'}_{mn}^{(2\omega )}} \right \},\left \{ {{u^{\prime \prime }}_{mn}^{(2\omega )},{v^{\prime \prime }}_{mn}^{(2\omega )}} \right \}$

The coefficients $\left \{ {{u'}_{mn}^{(2\omega )},{v'}_{mn}^{(2\omega )}} \right \}$ for the surface tangential source can be expressed as

$$\begin{array}{@{}rcl@{}} {u'}_{mn}^{(2\omega )} &=& i 2 \left( \frac{\chi_{\parallel\perp\parallel}}{\chi^{(2)}_{0}} \right) \frac{\zeta_{e}}{\zeta_{0}} \sum\limits_{{n_{1}}}^{\infty} \sum\limits_{{m_{1}} = - {n_{1}}}^{{n_{1}}} \sum\limits_{{n_{2}}}^{\infty} \sum\limits_{{m_{2}} = - {n_{2}}}^{n_{2}}\\ &&\times \left[A_{{m_{1}}{n_{1}}}^{(1)}A_{{m_{2}}{n_{2}}}^{( - 1)}C_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(1,0, - 1)}\right.\\ &&\left.\quad\; + A_{{m_{1}}{n_{1}}}^{(0)}A_{{m_{2}}{n_{2}}}^{( - 1)}C_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(1,1, - 1)}\right] \\[3.5pt] {v'}_{mn}^{(2\omega )} &=& 2 \left( \frac{\chi_{\parallel \perp \parallel}}{\chi_{0}^{(2)}} \right) \frac{{{\zeta_{e}}}}{\zeta_{0}}\sum\limits_{{n_{1}}}^{\infty} \sum\limits_{{m_{1}} = - {n_{1}}}^{n_{1}} \sum\limits_{{n_{2}}}^{\infty} \sum\limits_{{m_{2}} = - {n_{2}}}^{{n_{2}}}\\ &&\times \left[A_{{m_{1}}{n_{1}}}^{(1)}A_{{m_{2}}{n_{2}}}^{( - 1)}C_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(0,0, - 1)}\right.\\ && \left.\quad\;+ A_{{m_{1}}{n_{1}}}^{(0)}A_{{m_{2}}{n_{2}}}^{( - 1)}C_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(0,1, - 1)}\right] \end{array} $$

and the coefficients $\left \{ {{u^{\prime \prime }}_{mn}^{(2\omega )},{v^{\prime \prime }}_{mn}^{(2\omega )}} \right \}$ for both the γ bulk and the surface normal polarization source can be expressed as

$$\begin{array}{@{}rcl@{}} {u^{\prime\prime}}_{mn}^{(2\omega )} &=&\left( \frac{\chi_{\perp \perp \perp}}{\chi^{(2)}_{0}} \right) \frac{i \sqrt {n(n + 1)}}{{k_{0}}(\omega )R} \sum\limits_{{n_{1}}}^{\infty} \sum\limits_{{m_{1}} = - {n_{1}}}^{{n_{1}}} \sum\limits_{{n_{2}}}^{\infty} \sum\limits_{{m_{2}} = - {n_{2}}}^{{n_{2}}} A_{{m_{1}}{n_{1}}}^{(-1)}A_{{m_{2}}{n_{2}}}^{(-1)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(-1,-1)}\\ &&+\left( \frac{\gamma}{\chi^{(2)}_{0}} \frac{{{{\varepsilon}_{0}}}}{{{{\varepsilon}_{i}}(2\omega )}} \right) \frac{i\sqrt {n(n + 1)}}{{{k_{0}}(\omega )}R} \sum\limits_{{n_{1}}}^{\infty} \sum\limits_{{m_{1}} = - {n_{1}}}^{{n_{1}}} \sum\limits_{{n_{2}}}^{\infty} \sum\limits_{{m_{2}} = - {n_{2}}}^{{n_{2}}}{} \left[ A_{{m_{1}}{n_{1}}}^{(1)} A_{{m_{2}}{n_{2}}}^{(1)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(1,1)} + A_{{m_{1}}{n_{1}}}^{(0)} A_{{m_{2}}{n_{2}}}^{(0)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(0,0)}\right.\\ &&{\kern18.5pc} + A_{{m_{1}}{n_{1}}}^{(1)} A_{{m_{2}}{n_{2}}}^{(0)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(1,0)} + A_{{m_{1}}{n_{1}}}^{(0)} A_{{m_{2}}{n_{2}}}^{(1)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(0,1)}\\ &&{\kern18.5pc}\left.+ A_{{m_{1}}{n_{1}}}^{(-1)} A_{{m_{2}}{n_{2}}}^{(-1)} W_{{n_{1}}{m_{1}}{n_{2}}{m_{2}}nm}^{(-1,-1)} \right]\\ {v^{\prime\prime}}_{mn}^{(2\omega )} &=& 0\\ \end{array} $$

where

$$\begin{array}{@{}rcl@{}} A_{mn}^{( 0)}{} ={} c_{mn}^{(\omega )}{\left. {{j_{n}}({k_{i}}(\omega )r)} \right|_{r = R}}; A_{mn}^{( 1)} {}={} d_{mn}^{(\omega )}{\left. {i\frac{1}{{{k_{i}}(\omega )}}\left( {\frac{\partial }{{\partial r}} {}+{} \frac{1}{r}} \right){j_{n}}( {{k_{i}}(\omega )r} )} \right|_{r = R}}; A_{mn}^{(-1)} {}={} d_{mn}^{(\omega )}{\left. {i\sqrt {n(n + 1)} \frac{1}{{{k_{i}}(\omega )r}}{j_{n}}( {{k_{i}}(\omega )r} )} \right|_{r = R}} \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{} C_{J_{1} M_{1} J_{2} M_{2} J M}^{(1,0,-1)} = \sqrt{\frac{3}{2 \pi}} (2 J_{1}+1) C_{J_{1} M_{1} J_{2} M_{2}}^{J M} \cdot \left[ \sqrt{(J_{2})(2 J_{2}-1)}\left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J+1 & 1 \end{array} \right\} C_{(J_{1})(0)(J_{2}-1)(0)}^{(J+1)0} \sqrt{\frac{J}{2J+1}} \right. \\ && {\kern15pc} \left. -\sqrt{(J_{2}+1)(2 J_{2}+3)}~ \left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J+1 & 1 \end{array} \right\} C^{(J+1) 0}_{(J_{1})(0)(J_{2}+1)(0)} \sqrt{\frac{J}{2J+1}} \right. \\&& {\kern15pc} \left.+\sqrt{(J_{2})(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J-1 & 1 \end{array} \right\} C^{(J-1) 0}_{(J_{1})(0)(J_{2}-1)(0)} \sqrt{\frac{J+1}{2J+1}} \right. \\&& {\kern15pc} \left.-\sqrt{(J_{2}+1)(2 J_{2}+3)}~ \left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J-1 & 1 \end{array} \right\} C_{(J_{1})(0)(J_{2}+1)(0)}^{(J-1) 0} \sqrt{\frac{J+1}{2J+1}} \right] \end{array} $$

$$\begin{array}{@{}rcl@{}}&&C_{J_{1} M_{1} J_{2} M_{2} J M}^{(1,1,-1)} = \sqrt{\frac{3}{2 \pi}} C_{J_{1} M_{1} J_{2} M_{2}}^{J M} \cdot \left[ \sqrt{(J_{1}+1)(J_{2})(2 J_{1}-1)(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J+1 & 1 \end{array} \right\} C_{(J_{1}-1)(0)(J_{2}-1)(0)}^{(J+1) 0} \sqrt{\frac{J}{2J+1}} \right.\\ && {\kern12pc} - \sqrt{(J_{1}+1)(J_{2}+1)(2 J_{1}-1)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J+1 & 1 \end{array} \right\} C_{(J_{1}-1)(0)(J_{2}+1)(0)}^{(J+1) 0} \sqrt{\frac{J}{2J+1}} \\ && {\kern12pc} + \sqrt{(J_{1})(J_{2})(2 J_{1}+3)(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J+1 & 1 \end{array} \right\} C^{(J+1) 0}_{(J_{1}+1)(0)(J_{2}-1)(0)} \sqrt{\frac{J}{2J+1}} \\ && {\kern12pc} - \sqrt{(J_{1})(J_{2}+1)(2 J_{1}+3)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J+1 & 1 \end{array} \right\} C^{(J+1) 0}_{(J_{1}+1)(0)(J_{2}+1)(0)} \sqrt{\frac{J}{2J+1}} \\ && {\kern12pc} + \sqrt{(J_{1}+1)(J_{2})(2 J_{1}-1)(2 J_{2}-1)} \left\{\begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J-1 & 1 \end{array}\right\} C^{(J-1) 0}_{(J_{1}-1)(0)(J_{2}-1)(0)} \sqrt{\frac{J+1}{2J+1}} \\ && {\kern12pc} - \sqrt{(J_{1}+1)(J_{2}+1)(2 J_{1}-1)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J-1 & 1 \end{array} \right\} C^{(J-1) 0}_{(J_{1}-1)(0)(J_{2}+1)(0)} \sqrt{\frac{J+1}{2J+1}} \\ &&{\kern12pc} + \sqrt{(J_{1})(J_{2})(2 J_{1}+3)(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J-1 & 1 \end{array} \right\} C^{(J-1) 0}_{(J_{1}+1)(0)(J_{2}-1)(0)} \sqrt{\frac{J+1}{2J+1}} \\ &&{\kern12pc} - \sqrt{(J_{1})(J_{2}+1)(2 J_{1}+3)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J-1 & 1 \end{array} \right\} \left. C^{(J-1) 0}_{(J_{1}+1)(0)(J_{2}+1)(0)} \sqrt{\frac{J+1}{2J+1}} \right] \end{array} $$

$$\begin{array}{@{}rcl@{}}C_{J_{1} M_{1} J_{2} M_{2} J M}^{(0,0,-1)}= \sqrt{\frac{3}{2 \pi}} (2 J_{1}+1) C_{J_{1} M_{1} J_{2} M_{2}}^{J M} &\cdot& \left[ \sqrt{(J_{2})(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J & 1 \end{array} \right\} C_{(J_{1})(0)(J_{2}-1)(0)}^{J 0}\right.\\ &&{\kern12pt}\left.-\sqrt{(J_{2}+1)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1} & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J & 1 \end{array} \right\} C_{(J_{1})(0)(J_{2}+1)(0)}^{J 0}\right] \end{array} $$

$$\begin{array}{@{}rcl@{}} C^{(0,1,-1)}_{J_{1} M_{1} J_{2} M_{2} J M} = \sqrt{\frac{3}{2 \pi}} C^{J M}_{J_{1} M_{1} J_{2} M_{2}} &\cdot& \left[ \sqrt{(J_{1}+1)(J_{2})(2 J_{1}-1)(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J & 1 \end{array} \right\} C^{J 0}_{(J_{1}-1)(0)(J_{2}-1)(0)} \right. \\ &&{\kern12pt}-\sqrt{(J_{1}+1)(J_{2}+1)(2 J_{1}-1)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}-1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J & 1 \end{array} \right\} C^{J 0}_{(J_{1}-1)(0)(J_{2}+1)(0)}\\ &&{\kern12pt}+\sqrt{(J_{1})(J_{2})(2 J_{1}+3)(2 J_{2}-1)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}-1 & 1 \\ J & J & 1 \end{array}\right\} C^{J 0}_{(J_{1}+1)(0)(J_{2}-1)(0)}\\ &&{\kern12pt}\left.-\sqrt{(J_{1})(J_{2}+1)(2 J_{1}+3)(2 J_{2}+3)} \left\{ \begin{array}{lll} J_{1} & J_{1}+1 & 1 \\ J_{2} & J_{2}+1 & 1 \\ J & J & 1 \end{array} \right\} C^{J 0}_{(J_{1}+1)(0)(J_{2}+1)(0)} \right] \end{array} $$

$$\begin{array}{@{}rcl@{}} W^{(-1,-1)}_{J_{1} M_{1} J_{2} M_{2} J M}&=& {\sqrt {\frac{{{J_{1}} }}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} }}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} - 1,{J_{1}},{M_{1}},{J_{2}} - 1,{J_{2}},{M_{2}}}}\\ &+&{\sqrt {\frac{{{J_{1}} + 1}}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} + 1}}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} + 1,{J_{1}},{M_{1}},{J_{2}} + 1,{J_{2}},{M_{2}}}}\\ &-&{\sqrt {\frac{{{J_{1}} }}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} + 1}}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} - 1,{J_{1}},{M_{1}},{J_{2}} + 1,{J_{2}},{M_{2}}}}\\ &-&{\sqrt {\frac{{{J_{1}} + 1}}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} }}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} + 1,{J_{1}},{M_{1}},{J_{2}} - 1,{J_{2}},{M_{2}}}} \end{array} $$

$$\begin{array}{@{}rcl@{}} W^{(1,1)}_{J_{1} M_{1} J_{2} M_{2} J M}&=& {\sqrt {\frac{{{J_{1}} + 1}}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} + 1}}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} - 1,{J_{1}},{M_{1}},{J_{2}} - 1,{J_{2}},{M_{2}}}} \\ &+&{\sqrt {\frac{{{J_{1}} }}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} }}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} + 1,{J_{1}},{M_{1}},{J_{2}} + 1,{J_{2}},{M_{2}}}} \\ &+&{\sqrt {\frac{{{J_{1}} + 1}}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} }}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} - 1,{J_{1}},{M_{1}},{J_{2}} + 1,{J_{2}},{M_{2}}}} \\ &+&{\sqrt {\frac{{{J_{1}} }}{{2{J_{1}} + 1}}} \sqrt {\frac{{{J_{2}} + 1}}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}} + 1,{J_{1}},{M_{1}},{J_{2}} - 1,{J_{2}},{M_{2}}}} \end{array} $$

$$W^{(0,0)}_{J_{1} M_{1} J_{2} M_{2} J M}= W_{JM}^{{J_{1}},{J_{1}},{M_{1}},{J_{2}},{J_{2}},{M_{2}}} $$

$$\begin{array}{@{}rcl@{}} W^{(1,0)}_{J_{1} M_{1} J_{2} M_{2} J M}&=& {\sqrt {\frac{{{J_{1}} + 1}}{{2{J_{1}} + 1}}} W_{JM}^{{J_{1} - 1},{J_{1}},{M_{1}},{J_{2}},{J_{2}},{M_{2}}}}\\ &+&{\sqrt {\frac{{{J_{1}} }}{{2{J_{1}} + 1}}} W_{JM}^{{J_{1} + 1},{J_{1}},{M_{1}},{J_{2}},{J_{2}},{M_{2}}}} \end{array} $$

$$\begin{array}{@{}rcl@{}} W^{(0,1)}_{J_{1} M_{1} J_{2} M_{2} J M}&=& {\sqrt {\frac{{{J_{2}} + 1}}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}},{J_{1}},{M_{1}},{J_{2}} - 1,{J_{2}},{M_{2}}}}\\ &+&{\sqrt {\frac{{{J_{2}} }}{{2{J_{2}} + 1}}} W_{JM}^{{J_{1}},{J_{1}},{M_{1}},{J_{2}} + 1,{J_{2}},{M_{2}}}} \end{array} $$

$$\begin{array}{@{}rcl@{}} W_{LM}^{L1,J1,M1,L2,J2,M2} = {( - 1)^{{J_{2}} + {L_{1}} + L}} C_{{L_{1}}0{L_{2}}0}^{L0}\,C_{{J_{1}}{M_{1}}{J_{2}}{M_{2}}}^{LM} \cdot\\ \sqrt {\frac{{(2{J_{1}} + 1)(2{J_{2}} + 1)(2{L_{1}} + 1)(2{L_{2}} + 1)}}{{4\pi (2L + 1)}}} \left\{ \begin{array}{lll} {{L_{1}}}&{{L_{2}}}&L\\ {{J_{2}}}&{{J_{1}}}&1 \end{array} \right\} \end{array} $$

and where $C_{{J_{1}}{M_{1}}{J_{2}}{M_{2}}}^{JM}$ is the Clebsch–Gordan coefficient (Chap. 8 in Ref. [33]) and the quantities in braces are Wigner 6j and 9j symbols (Chaps. 9 and 10 in Ref. [33]).

G Validation

We validated the proposed analytical solution derived in the framework of the Mie theory by means of an independent approach based on a surface integral method. The implementation details of this latter method are given in Ref. [36]. We calculated with both methods the SH power per unit solid angle ${{dP^{( {2\omega } )}( {{\hat {\bf {K}}}} )}}/{{d\Omega }}$ radiated by each of the three SH sources of interest. In Fig. 11, an illustrative case is reported of a nanoparticle with R = 100 nm and an incident wavelength of λ = 520 nm. The two solutions coincide within a tolerance of about 3 %.

Fig. 11

SH radiation diagrams as a function of θ angle, at ϕ = 0^∘ (blue) and ϕ = 90^∘ (red), for a nanosphere of size R = 100 nm and a pump wavelength λ = 520 nm. Panel (a) is relative to the bulk current density, panel (b) to the surface electric current density, and panel (c) to the surface magnetic current density. Continuous lines are the analytical solution in the framework of the Mie theory and circles are the solution computed by means of a surface integral method