Figure 1

Illustration of a second-order vibrational sum frequency scattering experiment. The incoming beams generate a nonlinear optical polarization at the interface of a particle. The polarization is the source of the scattered sum frequency light. The magnitude (represented by the scattering lobes) and spectral shape of the scattered field are determined by the molecular properties of the interface, which are contained in the second-order nonlinear susceptibility ${\mathbf{\chi }}^{\left(2\right)}$. The angular distribution reflects both the value of the second-order susceptibility and the radius of the particle and the scattering geometry. The criterium used in the Rayleigh-Gans-Debye formalism is depicted in the upper left corner. $n_1$ and $n_2$ are corresponding refractive indices.Reuse & Permissions

Figure 2

(Color online) The scattering geometry for two incoming $\mathbf{k}$ vectors, depicting the three different axis systems used in the calculations. ${\mathbf{k}}_0$ is the wave vector of the scattered sum frequency field, ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ are the incoming beams, and $\mathbf{q}$ is the scattering vector, defined as ${\mathbf{k}}_0-{\mathbf{k}}_0^0$ (the difference between the scattered SFG beam and the sum of the incoming beams). The $(x,z)$ plane is defined by ${\mathbf{k}}_0^0$ and ${\mathbf{k}}_1$. $\theta$, the scattering angle, is the angle between ${\mathbf{k}}_0^0$ and ${\mathbf{k}}_0$, and $\varphi$ is the angle between the $(x,z)$ plane and ${\mathbf{k}}_0$, as projected onto the $(x,y)$ plane. For a certain scattering angle, $\mathbf{q}$ describes a circle in the $(x,y)$ plane (shown in the inset). In the inset, ${\mathbf{k}}_0^0$ is pointing out of the paper.Reuse & Permissions

Figure 3

Illustration of the transformation of the elements of local surface susceptibility into the elements of the effective particle susceptibility. The elements of ${\mathbf{\Gamma }}^{\left(2\right)}$ that consist of an odd number of transverse components vanish.Reuse & Permissions

Figure 4

(Color online) The four $\Lambda$ functions that determine the angular distribution of second-order surface SFG scattering. In the left panel, the real part of $\Lambda (0,0)$ and $\Lambda (0,2)$ [pattern caused by chiral components of ${\overline{\mathbf{\chi }}}^{\left(2\right)}$] is depicted and in the right panel, the imaginary part of $\Lambda (1,0)$ and $\Lambda (1,2)$ [caused by nonchiral components of ${\overline{\mathbf{\chi }}}^{\left(2\right)}$] is plotted as a function of $qR$. $A$ and $B$ refer to the angles marked in Fig. 6.Reuse & Permissions

Figure 5

(Color online) The calculated scattering pattern for all eight polarization directions. The patterns are projections onto an outgoing hemisphere, so that each value of scattering angle $\theta$ is represented by a circle (as indicated in the $ppp$ pattern). $\theta$ $\left(\varphi \right)$ ranges from $[-90,90]$ $\left([0,360]\right)\mathrm{deg}$. The intensities are plotted on a logarithmic scale for clarity. The particle radius $\left(R\right)$ is $500\mathrm{nm}$; the angle between the incoming infrared (with ${\lambda }_2=3400\mathrm{nm}$) and visible (with ${\lambda }_1=800\mathrm{nm}$) beams $\left(\beta \right)$ is 15°. The relative magnitudes of the components of ${\overline{\mathbf{\chi }}}^{\left(2\right)}$ are set such that ${\overline{\chi }}_{r^{\prime }{r}^{\prime }{r}^{\prime }}^{\left(2\right)}=1$, ${\overline{\chi }}_{{\theta }^{\prime }{r}^{\prime }{\theta }^{\prime }}^{\left(2\right)}={\overline{\chi }}_{r^{\prime }{\theta }^{\prime }{\theta }^{\prime }}^{\left(2\right)}=0$, ${\overline{\chi }}_{{\theta }^{\prime }{\theta }^{\prime }{r}^{\prime }}^{\left(2\right)}=-1$, ${\overline{\chi }}_{r^{\prime }{\theta }^{\prime }{\varphi }^{\prime }}^{\left(2\right)}=0$, ${\overline{\chi }}_{{\varphi }^{\prime }{r}^{\prime }{\theta }^{\prime }}^{\left(2\right)}=0.1$, and ${\overline{\chi }}_{{\theta }^{\prime }{\varphi }^{\prime }{r}^{\prime }}^{\left(2\right)}=0.1$.Reuse & Permissions

Figure 6

The calculated in-plane scattering patterns for nonchiral components of the scattered electrical field (top) and the chiral components (bottom) of the scattered electrical field. The relative values of ${\overline{\mathbf{\chi }}}^{\left(2\right)}$ are set such that ${\overline{\chi }}_{r^{\prime }{r}^{\prime }{r}^{\prime }}^{\left(2\right)}=1$, ${\overline{\chi }}_{{\theta }^{\prime }{r}^{\prime }{\theta }^{\prime }}^{\left(2\right)}={\overline{\chi }}_{r^{\prime }{\theta }^{\prime }{\theta }^{\prime }}^{\left(2\right)}=0$, ${\overline{\chi }}_{{\theta }^{\prime }{\theta }^{\prime }{r}^{\prime }}^{\left(2\right)}=-1$, ${\overline{\chi }}_{r^{\prime }{\theta }^{\prime }{\varphi }^{\prime }}^{\left(2\right)}=0$, ${\overline{\chi }}_{{\varphi }^{\prime }{r}^{\prime }{\theta }^{\prime }}^{\left(2\right)}=0.1$, and ${\overline{\chi }}_{{\theta }^{\prime }{\varphi }^{\prime }{r}^{\prime }}^{\left(2\right)}=0.1$ for the four nonzero in-plane patterns. At ${\theta }_A$, $\Lambda (0,2)=0$ and at ${\theta }_B$, $\Lambda (1,2)=0$.Reuse & Permissions