Figure 1

Overview of the relevant parameters in the model: the sum frequency and source waves have $k$ vectors ${\mathbf{k}}_0$, ${\mathbf{k}}_1$, and ${\mathbf{k}}_2$ in order of decreasing frequency. The angle between the propagation direction of the lowest frequency wave and the positive $z$ axis is $\alpha$, the opening angle between source waves $\beta$. Local fields are expressed in spherical coordinates $\left(r^{\prime },{\theta }^{\prime },{\varphi }^{\prime }\right)$, which are defined with respect to the $y$ axis. The sum-frequency scattering pattern is parametrized using the scattering angle $\theta$.Reuse & Permissions

Figure 2

(Color) Illustration of the nonlinear polarization built up by combining source waves of ${\lambda }_1=800\text{nm}$ and ${\lambda }_2=3447\text{nm}$. The radial components of the local electric fields ${\mathbf{E}}_1$ and ${\mathbf{E}}_2$, represented by the left and middle spheres, couple according to the relation $P_{\perp }={\chi }_{\perp \perp \perp }^{\left(2\right)}{E}_{1,\perp }{E}_{2,\perp }$, represented in the right sphere. The electric fields, as well as the polarization vary in phase (represented by color) and amplitude (represented by the saturation of the color). The inner product of the polarization with the sum-frequency wave yields the sum-frequency amplitude. For both source waves, a unity index of refraction was taken, though the procedure also applies for nonunity values. The rightmost image shows a legend for the colors used: blue and red correspond to positive and negative real components, while green corresponds to the amplitude of imaginary components.Reuse & Permissions

Figure 3

Comparison of results obtained with numerical Mie simulation to previous analytical expressions: (a) Scattering pattern of sum-frequency generation by source beams at an opening angle $\beta$ of $15°$ for which ${\lambda }_1=800\text{nm}$ and ${\lambda }_2=3447\text{nm}$ at the surface of a particle of radius $R=500\text{nm}$, for which ${\chi }_{\perp \perp \perp }^{\left(2\right)}=1$. Numerical results (solid curve) correspond exactly to analytical results (dashed curve; offset for clarity). (b) Scattering pattern corresponding to the small particle limit $\left(R=10\text{nm}\right)$. A second-harmonic scattering pattern was simulated using a source wavelength of 800 nm and ${\chi }_{\perp \perp \perp }^{\left(2\right)}=1$ (solid curve), which corresponds exactly to analytical results in the small particle limit (dashed curve; offset for clarity). In both cases, the polarization combination was $ppp$.Reuse & Permissions

Figure 4

(Color) Simulations of nonlinear scattering intensities for the $ppp$ polarization as a function of scattering angle (horizontal axis) and particle size (vertical axis), using a noncollinear scattering setup consisting of source waves of wavelength ${\lambda }_1=800\text{nm}$ and ${\lambda }_2=2708\text{nm}$ under an angle at $15°$ with respect to each other. The topmost image shows scattering intensities in the case of a spherical particle of unity index of refraction $\left[\left(\Delta n_{0,1,2}\approx 0\right)\right]$, the bottom image those of spherical water droplets, with $n_0=1.33-1.3\cdot {10}^{-8}i$, $n_1=1.33-1.25\cdot {10}^{-7}i$, and $n_2=1.19-0.019i$. In both cases, the particle surfaces were chosen to be isotropic with ${\mathit{\chi }}^{\left(2\right)}$ components taken from literature: ${\chi }_{\perp \perp \perp }^{\left(2\right)}=1$, ${\chi }_{\Vert \Vert \perp }^{\left(2\right)}=0.50+0.017i$, and ${\chi }_{\Vert \perp \Vert }^{\left(2\right)}={\chi }_{\perp \Vert \Vert }^{\left(2\right)}=0.089-0.011i$. For clarity, each scattering pattern has been normalized to its peak intensity.Reuse & Permissions

Figure 5

Comparison of size-dependent integrated scattering intensities (solid black curves) as well intensities integrated over the forward (gray curves; $-90$ to $90°$) and backward (dashed black curves; 90 to $-90°$) semicircle. The scattering geometry as well as the wavelength used were identical to those used in Fig. 4. All intensities have been divided by $R^4$ and normalized with respect to the resulting maximum value of the total scattered intensity. The insets show the same curves without dividing by $R^4$. In the case of water droplets, the curve has been offset for clarity. All graphs show values for the $ppp$ polarization combination. The left graph represents the RGD limit. Here, the dominant backscattering contributions are due to initial appearance of first- and second-order scattering maxima, which appear at low radii and whose position shifts forward as the particle radius increases. At larger radii, the dominant scattering direction is forward. The graph on the right represents the intensities for water droplets. In the limit of small particles, the intensities follow a similar dependence on radius, though at larger radii a backscatter peak appears along with size-dependent oscillations in overall intensity due to shape resonances. At larger radii, the scattered energy is divided between forward and backward scatterings of which the backward contribution eventually dominates.Reuse & Permissions