Interferometric Spectroscopy of Scattered Light Can Quantify the Statistics of Subdiffractional Refractive-Index Fluctuations

Despite major importance in physics, biology, and other sciences, the optical sensing of nanoscale structures in the far zone remains an open problem due to the fundamental diffraction limit of resolution. We establish that the expected value of spectral variance (${\tilde{\Sigma }}^2$) of a far-field, diffraction-limited microscope image can quantify the refractive-index fluctuations of a label-free, weakly scattering sample at subdiffraction length scales. We report the general expression of $\tilde{\Sigma }$ for an arbitrary refractive-index distribution. For an exponential refractive-index spatial correlation, we obtain a closed-form solution of $\tilde{\Sigma }$ that is in excellent agreement with three-dimensional finite-difference time-domain solutions of Maxwell’s equations. Sensing complex inhomogeneous media at the nanoscale can benefit fields from material science to medical diagnostics.

Figure 1

Sample: RI of the middle layer is random, and RIs of the top and bottom layers are constant; RI as a function of depth is shown in gray. The coherent sum of $U^{(r)}$ and $U^{(s)}$ is detected. Reflection from the bottom of the substrate (glass slide) is negligible, as its thickness (1 mm) is much larger than the microscope’s depth of field (for most setups, $0.5–15\mu \mathrm{m}$).

Figure 2

Spatial-frequency space with $k_z$ axis antiparallel to ${\mathbf{k}}_{\mathbf{i}}$. (a) Cross section of $T_{\Delta {\mathbf{k}}_{\mathbf{s}}}$, $T_{k\mathrm{NA}}$, and their interception $T_{3\mathrm{D}}$; (b) PSD of the RI fluctuation (blue) and $T_{3\mathrm{D}}$ (gray) when $l_c\ll L$ and (c) $l_c\lesssim L$.

Figure 3

Illustration of $\tilde{\Sigma }$ dependence on $l_c$ predicted by the quadrature-form [Eq. (5)] and the closed-form [Eq. (7)] analytical expressions for $\tilde{\Sigma }$ (circles and solid lines, respectively) and by FDTD (solid lines with error bars representing standard deviation between 20 realizations of each statistical condition), calculated for (a) $L=0.5\mu \mathrm{m}$, $\Delta k=4.9\mu {\mathrm{m}}^{-1}$, $k_c=16.8\mu {\mathrm{m}}^{-1}$, (b) $L=1.5\mu \mathrm{m}$, $\Delta k=4.9\mu {\mathrm{m}}^{-1}$, $k_c=16.8\mu {\mathrm{m}}^{-1}$, and (c) $L=2.0\mu \mathrm{m}$. $\Delta k=11.9\mu {\mathrm{m}}^{-1}$, $k_c=18.1\mu {\mathrm{m}}^{-1}$ (wave number values inside the sample). Data are shown normalized by ${\Gamma }_{01}^2$, the image intensity in the absence of RI fluctuations inside the sample.

Figure 4

$40\times$ magnification, 0.6 NA microscope images of samples with $L=2\mu \mathrm{m}$ were synthesized by FDTD. Bright-field images of samples with (a) $l_c=20\mathrm{nm}$ and (b) $l_c=50\mathrm{nm}$; $\Sigma (x^{\prime },y^{\prime })/{\Gamma }_{01}^2$ obtained from the wavelength-resolved image of (c) the sample with $l_c=20\mathrm{nm}$ and (d) $l_c=50\mathrm{nm}$; (e) RI of the two samples as a function of $z$ along central voxels ($x_0$, $y_0$), and (f) image spectra of the corresponding pixels ($x_0^{\prime }$, $y_0^{\prime }$).