Radially Polarized Light for Detection and Nanolocalization of Dielectric Particles on a Planar Substrate

A fast noninvasive method based on scattering from a focused radially polarized light to detect and localize subwavelength nanoparticles on a substrate is presented. The technique relies on polarization matching in the far field between scattered and spurious reflected fields. Results show a localization uncertainty of $\approx {10}^{-4}{\lambda }^2$ is possible for a particle of area $\approx {\lambda }^2/16$. The effect of simple pupil shaping is also shown.

Figure 1

A comparison between the single dipole model (blue line) and FEM simulations (green diamond) of the difference of the total intensities in the left and right halves of the pupil for the $x$ component of the far field. FEM simulations are for a particle of diameter 100 nm on a silicon substrate ($\lambda =405\mathrm{nm}$, $\mathrm{NA}=0.9$, substrate refractive index $5.42+0.33i$, and particle refractive index 1.5). In the FEM model [18], a radially polarized beam is focused at the interface with the NA divided into ${10}^4$ incident angles. The computational domain was chosen such that the first zero of the focused electric field components fits completely in it. For $x<100\mathrm{nm}$, the variation is almost linear with a slope of $1.1\%/\mathrm{nm}$ of maximum signal, which is approximately proportional to $r_p(1+r_p)\alpha$, for given NA, $\lambda$, and ${\mathbf{A}}^{\text{inc}}$. Deviation starts to build up after $x=90\mathrm{nm}$ as the transverse components become stronger (this area, $0.16{\lambda }^2$, which is slightly less than the longitudinal spot size $0.2{\lambda }^2$ [16], is where our model remains valid). Inset: $x$-polarized field $|E_x^{\text{pupil}}|$ at the pupil. For a large shift of 100 nm, visible asymmetry is observed.

Figure 2

Experimental schematics. The modeling of the problem is illustrated in the inset. The scattered field consists of directly scattered (${\mathbf{A}}_p^{\text{dip}}$) and reflected scattered (${\mathbf{A}}_p^{\text{dip},r}$), where, from our assumption, ${\mathbf{A}}_p^{\text{dip},r}=r_p{\mathbf{A}}_p^{\text{dip}}$. The specular reflected field is ${\mathbf{A}}_p^{\text{sp}}=r_p{A}_p^{\text{inc}}\mathbf{p}$. Thus, for each $\mathbf{k}$, the total outgoing signal is ${\mathbf{A}}_p^{\text{out}}={\mathbf{A}}_p^{\text{sp}}+{\mathbf{A}}_p^{\text{dip}}+{\mathbf{A}}_p^{\text{dip},r}$.

Figure 3

Experimental LR and TB differential scanning maps of a $3\times 3\mu {\mathrm{m}}^2$ area of the sample containing one nanoparticle. A Gaussian low-pass filter has been applied to reduce noise of the data. The plots are normalized and in absolute numbers. To draw the maps, first, frames on the rightmost line parallel to $y$ are taken as a reference. Each of them is subtracted from all other frames on the same scan line parallel to $x$ to eliminate effects of internal reflections from optical elements. Then, differential signals are calculated between TB or LR halves (for example, $LR=|\underset{L}{\iint }{|E_x^{\text{pupil}}|}^2-\underset{R}{\iint }{|E_x^{\text{pupil}}|}^2|$, where $L$ and $R$, respectively, denote the left and right half of the pupil). Finally this signal is normalized by the maximum of each map. (a) LR : FULL PUPIL, (b) TB : FULL PUPIL, (c) LR : ANNULAR PUPIL, and (d) TB : ANNULAR PUPIL.

Figure 4

Top: Experimental LR profile comparison along $x$ for the annular aperture (blue solid line) and full aperture (green broken line). Bottom: Experimental LR (blue solid line) and TB (green broken line) profile of the same scan line parallel to $y$. For both figures, the experimental noise level is shown by the black arrow, approximately to scale. Actual scan points are spaced 20 nm apart.