Fig. 1. (a) Schematic representation of an exemplary microcylinder-assisted microscope in reflection mode with LED illumination, diffuser (D), condenser lens (CL), beam splitter cube (BSC), tube lens (TL), camera (Cam), microscope objective (MO), and microcylinder placed on a sample (S). The spatially incoherent Köhler illumination is sketched in red for an exemplary point located on the optical axis, and the imaging path is shown in blue. (b) Geometry of the FEM setup including the microcylinder of radius $r$ placed on an amplitude grating of period length $l_x$ and thickness $d$. The whole geometry is assumed to be periodic with period length $L_x$, and hence, quasi-periodic boundary conditions are considered for the left and right boundaries. To avoid reflections, the simulation area is extended by absorbing layers called PML on the top and the bottom of the geometry. The incident wave can be chosen to propagate from ${\mathbf{E}}_{\text{top}}$ to ${\mathbf{E}}_{\text{bottom}}$ or vice versa to consider microscope setups in reflection mode as well as transmission mode. In the case of an axially infinitely extended measurement object, the grating ends in the PML, whereby the PML is implemented as a grating as well. The far field is calculated based on the scattered field obtained at the red dotted line closely above the cylinder or closely below the measurement object, approximately where $L_x$ is marked in the figure. (c) Three different imaging configurations where the red arrows specify the direction of the illumination. The left figure shows the setup in reflection mode corresponding to the setup sketched in panel (a), where the cylinder affects both the illumination and the imaging paths. The middle and right figures display the cases of transmission mode with the cylinder in the imaging (middle) path and the illumination (right) path. (d) Sketch of an amplitude grating and the corresponding complementary grating considered measurement objects for resolution analysis.

Fig. 2. Extracts of simulated intensities obtained for a plane incident wave of unity amplitude with TE (a)–(c) and TM (d)–(f) polarizations. The illumination wavelength, which is chosen to excite a WGM, is $\lambda =448.975\mathrm{nm}$ in the case of TE polarization and $\lambda =442.951\mathrm{nm}$ for TM polarization. In all figures, the illuminating plane wave propagates from top to bottom, and the intensity of the total field given by the sum of the incident and scattered fields is shown. The object is a rectangular amplitude grating of infinite thickness placed below the cylinder (a), (d), an amplitude grating of thickness $d=100\mathrm{nm}$ placed below the cylinder (b), (e), and above the cylinder (c), (f), respectively. All amplitude gratings have a period length of $l_x=300\mathrm{nm}$ and consist of glass and aluminum. The total geometry is assumed to be periodic with a period length of $L_x=13.2\mu \mathrm{m}$. The yellow lines indicate the boundaries of the cylinder and the amplitude grating.

Fig. 3. (a), (b) Simulated intensities obtained from a grating (a) and its complement (b) with $l_x=300\mathrm{nm}$ imaged by a microscope objective of $\mathrm{NA}=0.9$ used for illumination and imaging. The illuminating light is assumed to be TM polarized with $\lambda =440\mathrm{nm}$. The difference between the intensities according to (a) and (b) is displayed in panel (c). Cross sections along the colored lines marked in (a)–(c) are plotted in panel (d). The intensities obtained along the red (a) and green (b) lines are reduced by an offset. (e) The standard deviation of intensities for a grating and its complementary grating depending on the period length $l_x$ obtained for vertically incident light of different polarizations in reflection mode assuming a detection NA of 0.55. The grating is considered infinitely thick (inf) or of finite thickness $d=100\mathrm{nm}$ (fin). (f) Standard deviation depending on $l_x$ obtained in transmission mode for TM polarization, where the microcylinder is considered for illumination (ill) and imaging (im). The detection NA is assumed to be 0.55 as well. For comparison, the standard (std) values without a cylinder (no cyl) for the same parameters and obtained by simulation of a high-resolution confocal microscope (conf) of $\mathrm{NA}=0.95$ are also shown. All std values, despite the confocal result, are normalized by the std value obtained for TM-polarized light from an infinitely thick grating with $l_x=300\mathrm{nm}$. The confocal curve is normalized to the maximum value of the curve obtained with a microcylinder for imaging. (g), (h) Simulated near fields for $l_x=150\mathrm{nm}$ for original (g) and complementary (h) gratings in transmission mode with $d=100\mathrm{nm}$ and TM polarized light. The wavelengths for TE and TM polarizations are chosen to be similar to those of Fig. 2 for all subfigures of the bottom row.

Fig. 4. (a)–(c) Standard deviation of intensities for the field distribution obtained from a grating and the corresponding complementary grating depending on the period length $l_x$ for vertically incident, TM-polarized light of $\lambda =442.951\mathrm{nm}$ and $\lambda =450\mathrm{nm}$, assuming an NA of 0.55 for detection. The results are obtained in reflection mode (a) and transmission mode path (b), (c) with the microcylinder placed in the imaging (b) or illumination path (c). The colored curves obtained with $\lambda =442.951\mathrm{nm}$ correspond to the green curve in Fig. 3(e) and to the blue and red curves in Fig. 3(f). (d), (e) Simulated near fields for a grating (d) and its complementary (e) in transmission mode with $l_x=150\mathrm{nm}$, $d=100\mathrm{nm}$, and TM polarized light of $\lambda =450\mathrm{nm}$. (f) Difference of the intensities from the grating and its complementary, simulated for $l_x=150\mathrm{nm}$, and TM polarized light of $\lambda =442.951\mathrm{nm}$ in transmission mode, where the microcylinder is placed in the imaging path. The illumination NA as well as imaging NA amounts to 0.55.

Fig. 5. Near fields simulated for a transmission grating with the same parameters used for Figs. 4(g) and 4(h). Panels (a)–(d) show the difference $\mathrm{\Delta }I$ between the intensities from a grating and its complement for a distance $\delta$ between the grating and the microcylinder of 10 nm (a), 50 nm (b), 100 nm (c), and 300 nm (d). For $\delta =300\mathrm{nm}$, the intensities of the fields from the grating and its complement are displayed in panels (e) and (f), respectively.

Fig. 6. Near fields simulated for a transmission grating with the same parameters used for Figs. 4(g) and 4(h). Panels (a)–(c) show the difference $\mathrm{\Delta }I$ between the intensities from a grating and its complement for the light wavelengths $\lambda =442.8\mathrm{nm}$ (a), $\lambda =442.7\mathrm{nm}$ (b), and $\lambda =442.5\mathrm{nm}$ (c). Panels (d)–(f) display the difference $\mathrm{\Delta }I$ between the intensities from a grating and its complement for the light wavelength $\lambda =442.951\mathrm{nm}$ with the incident angles ${\theta }_{\mathrm{inc}}=0$ (d), ${\theta }_{\mathrm{inc}}=30\mathrm{deg}$ (e), and ${\theta }_{\mathrm{inc}}=56.31\mathrm{deg}$ (f), which correspond to the Brewster angle of glass.

Fig. 7. Standard deviation of intensities for the field distribution obtained from a grating and the corresponding complementary grating depending on the period length $l_x$ for vertically incident, TM-polarized light assuming an NA of 0.55 for detection in transmission mode with the microcylinder placed in the imaging path. The results are obtained for two different radii of the microcylinder of $r=2\mu \mathrm{m}$ (a) and $r=3.5\mu \mathrm{m}$ (b), with corresponding light wavelengths of $\lambda =415.13\mathrm{nm}$ and $\lambda =437.15\mathrm{nm}$, respectively, and $n=1.5$. Furthermore, the refractive index of the microcylinder is varied to $n=1.4$ (c) and $n=1.6$ (d) for $r=2.5\mu \mathrm{m}$ and $\lambda =408.40$ (c) or 432.06 nm (d), respectively. For comparison, the blue curve ($n=1.5$, $r=2.5\mu \mathrm{m}$) displayed in Figs. 3(f) or 4(b) is shown in all four cases and marked as original. All curves are normalized to the std value of the original curve for $l_x=300\mathrm{nm}$.