Object Motion with Structured Optical Illumination as a Basis for Far-Subwavelength Resolution

An imaging method based on object motion with structured light illumination and far-field measurement data that results in far-subwavelength image information is proposed. Simulations with realistic noisy data show that this approach will lead to the ability to distinguish object features on the nanometer scale using visible light, without the need for fluorophores. The principle is that far-field measurements with controlled motion in a spatially varying incident field add information about nanometer-scale dimensions and material properties.

Figure 1

(a) The square $6\lambda$ simulation geometry consists of two square scatterers (${\epsilon }_r=1.5$) scanned in $0.1\lambda$ steps in 2D. The range of motion is represented by the blue square ($1\lambda \times 1\lambda$) at the center of the free-space background. The red dotted lines denote the locations of the detectors ($0.5\lambda$ from the PML boundaries and about $2\lambda$ from the objects at the closest points). (b) The central scanned region is drawn to explicitly show the dimension and arrangement of the scatterers. These figures are not drawn to scale.

Figure 2

Time-averaged Poynting vector in the $\widehat{\mathbf{x}}$ direction, measured at $s_2$ for $D=0.01\lambda$, $0.02\lambda$, and $0.03\lambda$, for the two ${\epsilon }_r=1.5$ objects fixed at the center of the domain (0,0). The spatial coordinates refer to the geometry in Fig. 1. The PW case is shown in (a) and (b), and the SI case is shown in (c) and (d). To clearly show the error bars, corresponding to $\mathrm{SNR}=40\mathrm{dB}$, the data plotted in (b) and (d) are for $S_x-S_{x,D=0.02\lambda }$.

Figure 3

Numerical values of $g(\mathrm{\Delta }x,\mathrm{\Delta }y;D)$ [see Eq. (5)] for the reference separation $D_0=0.02\lambda$ when the (two ${\epsilon }_r=1.5$) scatterers are scanned along the (a) bottom and (b) left boundaries [the central region in Fig. 1] for the PW case, and (c) and (d) for the SI situation, all with a $\mathrm{SNR}=40\mathrm{dB}$ (producing the error bars, which are determined numerically with 100 Poynting vector data sets).

Figure 4

Cost functions for a simulated experiment with $D^{*}=0.02\lambda$ and ${\epsilon }_r^{*}=1.5$. (a) and (b) are the decimal logarithm of the cost function for the PW and SI cases, respectively. (c) and (d) are plots of the costs for the correct value of ${\epsilon }_r$ for the PW and SI cases, respectively. $D$ is given in terms of wavelength. The error bars give the standard deviation (estimated by performing the measurement 100 times with random data) assuming a $\mathrm{SNR}=40\mathrm{dB}$ in the Poynting vector data.

Figure 5

Cost functions for a simulated experiment with $D^{*}=0$ (a single object) and ${\epsilon }_r^{*}=2.5$. The meaning of all subfigures is the same as Fig. 4.

Figure 6

Cost functions for a simulated experiment using structured illumination with the dielectric constants of the two scatters in Fig. 1, with $D=0.05\lambda$ and ${\epsilon }_{r1}$ and ${\epsilon }_{r2}$ as unknowns. (a) The discrete cost function, showing a minimum at the correct values ${\epsilon }_{r1}^{*}=1.4$ and ${\epsilon }_{r2}^{*}=1.6$. (b) Cost as a function of ${\epsilon }_{r1}$ when ${\epsilon }_{r2}={\epsilon }_{r2}^{*}$ with error bars due to detector noise. (c) Cost as a function of ${\epsilon }_{r2}$ when ${\epsilon }_{r1}={\epsilon }_{r1}^{*}$ with the error bars.