Simultaneous Measurement of Multiple Parameters of a Subwavelength Structure Based on the Weak Value Formalism

A mathematical extension of the weak value formalism to the simultaneous measurement of multiple parameters is presented in the context of an optical focused vector beam scatterometry experiment. In this example, preselection and postselection are achieved via spatially varying polarization control, which can be tailored to optimize the sensitivity to parameter variations. Initial experiments for the two-parameter case demonstrate that this method can be used to measure physical parameters with resolutions at least 1000 times smaller than the wavelength of illumination.

Figure 1

(a) Schematic of a focused vector beam scatterometry experiment in which preselection and postselection are achieved via spatially varying polarization control. $\mathrm{NPBS}=\text{nonpolarizing}$ beam splitter. (b) The structure’s scattering matrix $\mathbb{R}({\mathbf{u}}^{\prime },\mathbf{u})$ provides the coupling between incident and reflected plane waves with directions ${\mathbf{u}}^{\prime }$ and $\mathbf{u}$.

Figure 2

Experimentally measured (top row) and optimized simulated (bottom row) input polarizations and output intensity distributions for a two-parameter measurement of critical dimension and sample orientation. (a) and (d) Incident polarization and normalized intensity after transmission through the beam splitter. An annular apodization profile was used to prevent unwanted backreflections on axis. The transmitted polarization $\mathbf{P}$ of the output analyzer is shown in the upper left corner. Right- and left-handed polarization states are represented by green and red ellipses, respectively. (b) and (e) Output intensity distributions for 49 experimental or simulated measurements, normalized to the same scale as the input intensity. The axis labels indicate the parameter values associated with the intensity distributions shown in each row and column. (c) and (f) Differences between each intensity distribution and the mean distribution over the parameter range.

Figure 3

Estimated parameters from each of the 49 images shown in Fig. 2. Error bars connect each estimate to the associated true parameter values. The ellipses represent the minimum standard deviation error expected from a measurement of 7500 photons.

Figure 4

Retrieved parameters from output intensities containing 1000 photons, simulated for the input polarization and analyzer shown in Fig. 2. Data are shown for 10 000 trials over 49 true parameter values (each shown in a different color). The ellipses represent the expected standard deviation errors for each true parameter value.