Optical dispersions through intracellular inhomogeneities

The transport of intensity equation (TIE) exhibits a noninterferometric correlation between the intensity and phase variations of intermediate fields (e.g., light and electrons) in biological imaging. Previous TIE formulations have generally assumed free-space propagation of monochromatic, coherent field functions crossing phase distributions along a longitudinal direction. In this study, we modify the TIE with fractal (or self-similar) organization models based on intracellular refractive index turbulence. We then implement TIE simulations over a broad range of fractal dimensions and wavelengths. Simulation results show how the intensity propagation through the spatial fluctuation of intracellular refractive index interconnects fractal dimensionality with intensity dispersion (or transmissivity) within the picometer to micrometer wavelength range. Additionally, we provide a spatial autocorrelation of phase derivatives, which allows for the direct measurement and reconstruction of intracellular fractal profiles from optical and electron microscopy imaging.

Figure 1

For a given wavelength $\lambda$, the initial intensity distribution of a complex plane wave $I(r_{\perp },z_{\mathrm{initial}})$ propagates through the intracellular refractive index $n(r_{\perp },z)$, exhibiting the final intensity distribution $I(r_{\perp },z_{\mathrm{final}})$.

Figure 2

Fractal model of intracellular refractive index turbulence. (a) The WM covariance as a function of distance between two different pixel positions $\rho$. Each colored line denotes one of the four representative models of fractal media: medium I (blue), medium II (green), medium III (magenta), and medium IV (orange). (b) An example of the mean differential scattering cross section per unit volume plotted in spherical coordinates ($l_c=30$ nm). An incident wave function polarized in the vertical plane can propagate from left to right along the longitudinal direction. The dimple is located at the origin.

Figure 3

The simulation results for thicker cell sample ($\sim 10 \mu \mathrm{m}$). (a) Final intensity distribution for medium I ($D_f=3.25$). Scale bar: 1.00 $\mu \mathrm{m}$. (b) Final intensity distribution for medium III ($D_f=4.00$). (c) Histogram comparison between initial and final intensity distributions for medium I ($D_f=3.25$). Blue and red colored areas represent histograms for initial and final intensity distributions, respectively. (d) Comparison of intensity histograms for medium III ($D_f=4.00$). (e) RMS variations arising from intensity propagation shown as a function of the $z$ axis. Each colored line denotes one of the four different fractal models: medium I (blue), medium II (green), medium III (magenta), and medium IV (orange). (f) Averaged fractional errors shown as a function of the $z$ axis.

Figure 4

Optical depth for sample thickness ($L=110$ nm) represented over a wide range of effective wave number $k{n}_0$ values, from visible light (green) to electron beam (blue). Color lines denote the four different fractal media. If $\tau \gg 1$, then the sample is optically thick. If $\tau \ll 1$, then the sample is optically thin. The dashed line represents $\tau =1$.