Optical microscopy is one of the most widely used imaging techniques in science and industry with various fields of application. Due to the wave properties of light and resulting diffraction effects, conventional optical microscopes operating in the far field are subject to the fundamental lateral resolution limit. According to Abbe, the minimum resolvable period length $l_{x,\min }$ of a periodic measurement object is given by $l_{x,\min }=\lambda /(2\mathrm{NA})$ with the light wavelength $\lambda$ and the numerical aperture (NA) of the microscope objective lens.1

To overcome this fundamental resolution limit, several optical measurement techniques have been developed based on near- and far-field detection of light fields. For an overview of the methods to overcome or bypass the resolution limit, we refer to Huszka and Gijs.2

One of the most promising and simple methods to enhance the lateral resolution is placing microelements such as solid immersion lenses (SILs),3^,4 microspheres,5^,6 or microcylinders7^,8 in the measurement object’s near field. Due to its simplicity and resolution improvement using a setup still operating in the far field and hence remaining fast and contactless, microsphere and microcylinder-assisted microscopy (MAM) is part of various experimental and theoretical studies published over the last years. Due to the broad variety of papers addressing MAM, a detailed overview is out of the scope of this paper. Therefore, we just list a few and refer to Wang and Luk’yanchuk,9 Darafsheh,10 or Wu and Hong11 for more information. Generally, it should be mentioned that microspheres are usually applied in experimental studies due to the resolution enhancement in both lateral directions, whereas microcylinders are more often considered in theoretical studies for computational reasons. Furthermore, MAM can be combined with other techniques such as digital holography,12^,13 confocal,14^–16 interference,12^,17^–19 or fluorescence microscopy20^,21 to enhance the lateral or axial resolution.

Despite many investigations, the major reason for the obtained resolution enhancement is not clearly clarified yet. However, evanescent waves (EWs),22^,23 photonic nanojets (PNs),5^,24 whispering-gallery modes (WGMs),25^,26 and super-resonances27 are effects frequently mentioned in the context of MAM. A detailed overview of optical phenomena occurring in microelements is provided by Minin et al.28 Other recent studies come to the conclusion that the resolution enhancement mainly results from a local increase of the effective NA.10^,29^–31 This result is in agreement with the observation that the point spread function of a microelement is a more accurate indicator for resolution compared with the PN.32

To analyze the imaging properties of microelements and study the influences of several phenomena potentially contributing to resolution enhancement, various numerical models are developed.30^,33^–36 However, full modeling of MAM including conical illumination and detection, considering coherence effects as well as a rigorous treatment of the scattering process, has been missing until recently.9 In recent studies, we present such a full modeling of microcylinder-assisted interference,30 confocal,37 and conventional38 microscopy based on the theory published elsewhere.39^–41 Using the models, a quantification of the resolution enhancement based on the imaging of gratings is outlined.30^,38 With this, we found that the local NA increase due to the microelement is the most likely reason for the resolution enhancement in the case of phase object imaging, and no impact of WGMs has been observed. Furthermore, the enhanced resolution is independent of the NA of the objective lens of the microscope. This observation is in good agreement with measurement results obtained by Duocastella et al.29

Nonetheless, the results obtained for imaging phase objects can not be generalized, as the imaging of objects modulating the amplitude of the electric field referred to as amplitude objects, where other types of EWs occur, plays an important role in conventional MAM. Furthermore, MAM can operate in transmission as well as reflection mode.9 Therefore, another recent study deals with modeling microscopic imaging of amplitude objects in reflection and transmission mode. As a result, an additional resolution improvement is demonstrated for amplitude imaging in transmission mode, which is supposed to be mainly attributed to EWs.38 However, all results shown in Ref. 38 are obtained for wavelengths exciting WGMs. This study extends Ref. 38 through a more detailed investigation of WGMs as one of the effects frequently named in the context of super-resolution in MAM.

This section briefly introduces the microscopic setup and the modeling used in this study. Figure 1(a) displays a schematic representation of an exemplary microcylinder-assisted microscope in reflection mode. The illumination path is shown in red, and the imaging path is shown in blue. A diffuser illuminated by an LED represents a homogeneous, spatially extended light source, which is imaged in the back focal plane of the microscope objective, yielding spatially incoherent illumination. Later in this paper, we consider solely a vertically incident plane wave for coherent illumination corresponding to a point light source instead of the diffuser, leading to a reduced illumination NA and thus a reduced computation time. However, the resolution enhancement achieved using a microcylinder is independent of the illumination NA, as shown elsewhere.30

The microcylinder (radius $r=2.5\mu \mathrm{m}$, refractive index $n=1.5$) is placed directly above an amplitude grating consisting of glass ($n=1.5$) and aluminum with a lateral extent of $l_x/2$, respectively, where $l_x$ is the period length of the amplitude grating. For comparability, the parameters of the microcylinder are similar to a previous publication.30 These are generally adapted from real experiments,5^,16^,42^,43 where the refractive indices are approximately $n\approx 1.5$ and the radii are in the range of 2 to $10\mu \mathrm{m}$. However, analyses using varying parameters are shown exemplarily in the Appendix.

The light–surface interaction of the incident light with the measurement object is simulated based on the finite element method (FEM) using the open-source software NGSolve.44Figure 1(b) shows the geometry of the FEM setup, where the microcylinder surrounded by air is placed on an amplitude grating of finite or infinite thickness. The top and the bottom boundaries of the geometry are extended by so-called perfectly matched layers (PMLs), which damp the scattered field exponentially to zero without reflections at interfaces to artificially constitute the case of an infinitely extended space. Laterally, the geometry is assumed to be periodic with period length $L_x=13.2\mu \mathrm{m}$ considered in the modeling by quasi-periodic boundary conditions. In this context, it should be noted that the period length of the amplitude grating is labeled $l_x$, whereas $L_x$ refers to the period length of the whole simulation setup including several periods of the grating and the microcylinder. For more information and explanation on the modeling, we refer to previous studies.30^,39 The incident wave, which is i.a. considered by appropriate boundary conditions for the electric field ${\mathbf{E}}_{\text{top}}$ at the top boundary and ${\mathbf{E}}_{\text{bottom}}$ at the bottom boundary, can be chosen to propagate from the top boundary to the bottom boundary or vice versa depending on the microscope configuration.

Like all numerical models, the presented FEM modeling is limited in its applicability. One major limitation is the assumed periodicity of the setup. However, as explained in a previous study,30 varying the period length $L_x$ between 10 and $20\mu \mathrm{m}$ does not significantly influence simulated microscope images, and the obtained resolution limits remain unchanged. Furthermore, in many experiments, several microspheres are placed on the object’s surface close to each other still achieving improved resolution,45^–47 justifying this assumption. Nonetheless, if required, FEM simulations can be performed similarly for nonperiodic objects simply by extending the left and right boundaries by PMLs as, e.g., done elsewhere,48 where the FEM model used in this paper has also been validated. Another limiting factor is the discretized area, which is mainly restricted due to computational resources. In addition, it should be mentioned that the cylinder and the grating are assumed to be of ideal shape and materials, which can be described by the given refractive indices. Compared with realistic measurement setups, systems considered in this paper are assumed to be free of aberrations and perfectly aligned. However, if required, these effects could be considered in the model by appropriate pupil functions as explained by Pahl et al.41 As a final remark, it should be noted that due to the sensitivity of scattered fields in microsphere and microcylinder-assisted setups to slight changes in the system, a quantitative comparison between experimental and simulation results is extremely challenging and out of the scope of this paper.

In this study, we compare the achievable resolution enhancement for three different imaging configurations sketched in Fig. 1(c). In the first case (left figure), which corresponds to the setup shown in Fig. 1(a), the microscope operates in the reflection mode. Hence, the microcylinder affects both the illumination and the imaging paths. The other two configurations correspond to the transmission mode, where the microcylinder is solely used for either imaging (middle figure) or illumination (right figure). Therefore, we analyze the influence of the microcylinder on the illumination and the imaging process separately.

To optically resolve small structures, short light wavelengths in the range of 400 to 490 nm are used. In this range, the wavelengths are selected to excite a WGM. These specific wavelengths are found as described by Foreman et al.49 Using spectrally broader light sources such as LEDs, it is expected that at least a few wavelengths of exciting WGMs are contained by the spectrum.23Figure 2 presents extracts of simulated intensities for vertically incident plane waves computed for TE [Figs. 2(a)–2(c)], wavelength $\lambda =448.975\mathrm{nm}$) and TM [Figs. 2(d)–2(f)], $\lambda =442.951\mathrm{nm}$) polarizations in reflection [Figs. 2(a), 2(b), 2(d), 2(e)] and transmission [Figs. 2(c), 2(f)] modes. For TE polarization, the refractive index of aluminum is given by $n=0.389+4.287\mathrm{i}$, for TM polarization, $n=0.379+4.228\mathrm{i}$.50 In all figures, the intensity of the total field, which equals the sum of the incident and scattered fields, is displayed.

To analyze the influence of the thickness $d$ of the amplitude object [see Fig. 1(b)], the simulations in reflection mode are performed for a grating of infinite [Figs. 2(a), 2(d)] and finite thicknesses of $d=100\mathrm{nm}$ [Figs. 2(b), 2(e)]. Comparing Figs. 2(a) and 2(b) as well as Figs. 2(d) and 2(e), the fields especially inside the cylinder differ depending on the thickness of the grating. Hence, internal reflections from the bottom of the grating and eventually EWs occurring at the interface between the grating and air seem to have an impact. Thus, we compare results for both infinite and finite thicknesses in this study. Furthermore, placing the cylinder below the grating instead of above strongly affects the fields because, in this case, the cylinder no longer influences the illumination of the grating, but the imaging of the transmitted field as shown by comparison of Figs. 2(b) and 2(e) to Figs. 2(c) and 2(f).

To calculate realistic image stacks obtained by a conventional microscope while the grating with the microcylinder on it moves through the focus of the objective lens, near-field calculations as exemplarily depicted in Fig. 2 are repeated for a number of discrete incident angles forming a cone of wave vectors, where the maximum angle of incidence is limited by the NA of the objective as described for interference microscopy by Pahl et al.39 In the following, we first show simulated microscope images depending on the axial scanning position also referred to as image stacks and outline a method to quantify the lateral resolution (Sec. 3.1). Afterward, resolution analyses are performed, and the impact of different imaging configurations and physical mechanisms on the achieved resolution is investigated (Sec. 3.2).

Figures 3(a) and 3(b) show simulated intensity values obtained for $\mathrm{NA}=0.9$ and monochromatic TM polarized light of $\lambda =440\mathrm{nm}$. The results are calculated for a period $l_x=300\mathrm{nm}$ of a grating of infinite thickness in reflection mode. The results are shown in the $xz$-plane, where $x$ represents the lateral coordinate and $z$ the axial coordinate. From Figs. 3(a)–3(b), the grating is laterally shifted by half a period (180 deg), yielding the complementary grating, as sketched in Fig. 1(d). As shown in the figures, the shift of the grating leads to intensity changes in the focal region of the cylinder (see horizontal lines). Note that the focal region of the field imaged by the cylinder is axially shifted compared with the virtual image plane. The focal region used for analysis in this study is simply approximated by an area around the plane, where the focal plane is assumed to be located. The exact focal plane can be analyzed, e.g., by time reversal studies.17^,19 Furthermore, it should be mentioned that, due to the high NA of the system, the grating can be resolved without a microcylinder as well. However, by varying the grating period, this simulation strategy can be used to analyze the resolution limit.

Figure 3(c) displays the difference between the intensities shown in Figs. 3(a) and 3(b). As a result, the grating is clearly visible in the focal area of the microcylinder (see horizontal lines). Cross sections of Figs. 3(a)–3(c) extracted from the axial position marked by the horizontal lines of the same color are shown in Fig. 3(d). Note that the intensities extracted from Figs. 3(a) and 3(b) are reduced by a constant offset for better clarity. Obviously, a phase shift of 180 deg appears between the red and green curves, which can be assigned to the complementation of the grating according to Figs. 3(a) and 3(b). The difference $\mathrm{\Delta }I$ between the intensities obtained for a grating and the complemented one shows an enhanced modulation depth. If the grating is not resolved by the microcylinder-assisted microscope setup, $\mathrm{\Delta }I$ is expected to become 0. Hence, the standard deviation $\mathrm{std}(\mathrm{\Delta }I)$ of $\mathrm{\Delta }I$ can be used as an indicator of whether the grating is resolved or not.

In the following, $\mathrm{std}(\mathrm{\Delta }I)$ is calculated for different microscope setups explained in Sec. 2 depending on the period length $l_x$ of the grating. Note that we consider only vertically incident plane wave illumination in the following resolution studies to reduce the computational burden. In a previous study,30 we demonstrated that the resolution enhancement is independent of the NA of the objective lens and solely depends on the imaging properties of the microcylinder. The NA of the imaging objective lens is assumed to be 0.55. The selected illumination wavelengths are similar to those used in Fig. 2 to excite WGMs. Note that the whole area of $5\mu \mathrm{m}$ of the cylinder is considered in the std calculation. A more detailed analysis of the optimum field of view of a microsphere of a certain radius is, e.g., provided by Darafsheh.10

The procedure described here is based on a previous publication30 and oriented toward the Abbe limit. Thus, a more detailed explanation and a description of existing resolution limits in the context of MAM can be found in this previous publication.30

Resolution analyses are performed for the different imaging configurations sketched in Fig. 1(c). Hence, this section first shows results for a microscope setup operating in the reflection mode and then in the transmission mode. Finally, the impact of WGMs on the resolution achieved for all three configurations is analyzed in more detail.

Figure 3(e) displays simulated std values obtained for TM polarized light as well as TE polarized light depending on the period length $l_x$ of the grating. Furthermore, for TM polarized light, an infinitely (inf) thick grating as well as a grating of finite (fin) thickness $d=100\mathrm{nm}$ is considered. All std values are normalized by the value obtained from an infinitely thick grating of $l_x=300\mathrm{nm}$ and TM polarization. All three curves show the expected behavior and generally decrease with decreasing period length. This is in agreement with previous results obtained for interference microscopy.30 Comparing the results of different polarization states, the resolution enhancement seems to be better for TM polarization. This agrees with experimental observations published by Darafsheh et al.,46 where a better contrast is achieved using TM polarized light in microsphere-assisted microscopy. Thus, in the following, we mainly focus on TM polarization. In all three cases, the resolution is significantly increased compared with the lateral resolution $l_{x,\min }\approx 400\mathrm{nm}$ according to the Abbe limit assuming $\mathrm{NA}=0.55$ for illumination and detection and air as the surrounding medium. Furthermore, the resolution limit for TM polarized light appears to be below 180 nm. This result is in good agreement with the results obtained for interference microscopy30 and can be explained by a local enhancement of the NA combined with a limited field of view through the cylinder. For an explanation of the effect of an improved resolution if only a few periods of the grating are in the field of view, we refer to the result reported by Lehmann et al.51 Generally, the std values obtained for an infinitely thick grating and a grating of finite thickness differ. Hence, internal reflections appearing inside the grating at the interface between the grating and air seem to affect the results. However, the resolution enhancement appears to be independent of the thickness.

Figure 3(f) shows std values depending on the period length $l_x$ if the microscope operates in transmission mode, where the microcylinder can be arranged in the illumination (ill) and imaging (im) path. For both cases, the illumination is assumed to be TM polarized. For comparison, a result is shown, in which the cylinder consists of air corresponding to the absence of the microcylinder (no cyl). As expected, without a microcylinder, the grating is not resolved for the values considered of $l_x$ and, hence, std is 0. In addition, a high-resolution confocal microscope with $\mathrm{NA}=0.95$ without a microcylinder has been simulated. For this purpose, the simulation model described elsewhere40 is applied to transmitting amplitude gratings, and analog to the MAM case, std values of simulated image stacks are calculated and shown in Fig. 3(f). The std values show that the grating is still resolved up to a period length of $l_x=240\mathrm{nm}$ and is no longer resolved at $l_x=200\mathrm{nm}$. This matches the value of $l_{x,\min }\approx 233\mathrm{nm}$ predicted according to Abbe and underlines that the metric introduced in this paper is consistent with the Abbe theory.

However, for both transmission cases, the resolution appears to improve compared with the reflection mode and below $l_x=140\mathrm{nm}$. Therefore, in the case of transmission, the focused illumination as well as the imaging of EWs seem to affect the resolution enhancement. This result is opposed to expectations concluded from simulation results obtained in previous studies for interferometric measurements of phase objects.30 To gain a deeper understanding of the mechanisms that lead to resolution enhancement, Figs. 3(g) and 3(h) show simulated near fields obtained for $l_x=150\mathrm{nm}$. Comparing Figs. 3(g) and 3(h), which differ only by the complementation of the grating above the cylinder, large differences in the fields obtained inside the cylinder appear. In both cases, a WGM is visible at the boundary of the cylinder. It should be noted that both intensities, which are calculated by the absolute square of the electric field neglecting other constant factors, are obtained for the unity amplitude of the incident electric field. Hence, the field is significantly enhanced inside the cylinder in both figures. However, the shape and characteristics of the WGMs differ significantly. Therefore, the fields inside the cylinder and the shape of the excited WGM strongly depend on the phase of the grating above it. This result is an indicator for the conversion of EWs to propagating waves by microspheres and microcylinders as discussed in more detail by Zhou et al.26 and Boudoukha et al.23

In comparison to the reflection mode, the effect of EWs is expected to be enhanced for transmission because, on the one hand, the background field is expected to be significantly weakened and, on the other hand, further EWs occur at the interface between glass and air. Nonetheless, even in the reflective case, slight contributions of the grating appear for periods smaller than 180 nm. However, these contributions are strongly reduced compared with longer period lengths. This probably follows from higher amplitudes of the reflected and diffracted field components compared with the transmitted field.

The resolution enhancement reached by the cylinder in the illumination path seems to be similar to that obtained for the cylinder in the imaging path. This can be caused by the illumination of a small spot comprising only a few periods combined with the influence of EWs occurring at the interface between the cylinder and the grating.

To analyze the role of WGMs in the context of resolution, Figs. 4(a)–4(c) display the standard deviation depending on the period length $l_x$ for two different wavelengths, one wavelength that excites WGMs and the other not. The results for reflection [Fig. 4(a)] and transmission modes are shown. In the case of the transmission mode, the cylinder is again placed in either the imaging path [Fig. 4(b)] or the illumination path [Fig. 4(c)]. All curves are normalized by the value of the standard deviation obtained for TM polarized light from an infinitely thick grating with $l_x=300\mathrm{nm}$. The curves in Figs. 4(a)–4(c) correspond to the curves of the same color in Figs. 3(e) and 3(f). The black curves are obtained for $\lambda =450\mathrm{nm}$, where no WGMs appear for the given cylinder parameters. Especially for both transmission cases, the resolution is significantly lower compared with the resolution achieved with WGM appearance. Note that a slight difference is to be expected due to the differences in wavelength, but the apparent difference is significantly larger. Hence, the resolution enhancement in Fig. 3(f) is attributed to the occurrence of WGMs.

Figures 4(d) and 4(e) depict simulated near fields for vertically incident light of $\lambda =450\mathrm{nm}$ with the microcylinder placed below the grating [Fig. 4(d)] and the complementary grating [Fig. 4(e)]. In accordance with the electric fields in Figs. 3(g) and 3(h), the period length of the gratings is $l_x=150\mathrm{nm}$. However, in Fig. 4(b), the grating is not resolved for $\lambda =450\mathrm{nm}$. Hence, as expected, the fields shown in Figs. 4(d) and 4(e) are equal, and no WGM is apparent. This is an additional indicator for the assumption that WGMs excited by EWs affect the resolution improvement.

The results shown so far demonstrate that the grating periodicity can be detected for small period lengths, if WGMs are excited. However, evidence that the grating structure can be obtained from the results has not been provided yet. Figure 4(f) displays the difference $\mathrm{\Delta }I$ assuming a grating with period length $l_x=150\mathrm{nm}$ obtained in transmission mode with the microcylinder placed in the imaging path. Both illumination and imaging NAs are 0.55. From Fig. 4(f), the periodicity of the grating can be seen in the focal plane of the microcylinder. Therefore, the grating seems to be resolved, which indicates the required evidence. Similar to the results simulated for interference microscopy,30 the grating is significantly magnified, and the magnification is higher compared with Fig. 3(c) due to the smaller NA, although the period length is halved from Figs. 3(c) to 4(f).

To further underline the assumption that WGMs excited by EWs contribute to the resolution enhancement and to investigate the sensitivity of the simulated setup to parameter changes, additional simulations are performed, and the results are shown in the Appendix. The major findings of this investigation are summarized as follows.

Figure 5 shows simulations considering a distance $\delta$ between the grating and the cylinder for TM-polarized light and a wavelength of $\lambda =442.951\mathrm{nm}$. To examine the influence of the grating on the WGM, the intensity difference $\mathrm{\Delta }I$ between the grating ($l_x=150\mu \mathrm{m}$) and the complementary grating is shown in the images. As the distance [10 nm, Fig. 5(a), 50 nm, Fig. 5(b), 100 nm, Fig. 5(c)] between grating and cylinder increases, the intensity of this difference decreases. At a distance of 300 nm [Fig. 5(d)], no more difference can be recognized. If the two intensity distributions for the corresponding gratings are plotted individually [Fig. 5(a), 5(b)], a WGM can still be identified. This could be an indication that EWs propagating along the grating are coupling into the cylinder and have an influence on the WGM. A deeper insight into the convergence of EWs into propagating waves and a mathematical description is given by Ben-Aryeh et al.,52 Yang et al.,53 and Zhou et al.26

Figure 6 shows a slight shift in wavelength and different angles of incidence for the most recent setup. Moving further away from the wavelength that excites a WGM, the difference in the intensities obtained from the grating and its complement decreases. As an example, this is shown for the wavelengths 442.8 nm [Fig. 6(a)], 442.7 nm [Fig. 6(b)], and 442.5 nm [Fig. 6(c)]. At different angles of incidence [0 deg, Fig. 6(d), 30 deg, Fig. 6(e), 56.31 deg, Fig. 6(f)], the excited WGMs change in direction and intensity, but the intensities are not equal to zero at all angles. This indicates that the grating can be resolved at all investigated angles of incidence. In addition, the choice of cylinder (based on variants of the refractive index and radius) and its influence on the improvement in resolution were investigated. Figure 7 shows the curve for the refractive index $n=1.5$, the radius of $r=2.5\mu \mathrm{m}$, and the corresponding wavelength of $\lambda =442.951\mathrm{nm}$, which has already been shown in Fig. 3(b). In comparison, a second curve is shown, where the radius [Figs. 7(a), 7(b)] and the refractive index [Fig. 7(c), 7(d)] have been changed. The illumination wavelength was adjusted for the different setups so that it continues to excite a WGM. This illustration gives the impression that a smaller radius gives better results and the larger radius gives worse results in terms of resolution improvement. When comparing the changed refractive indices, it appears that a lower refractive index leads to similar results, whereas a higher refractive index leads to poorer results. This shows that the improved resolution depends on the selected parameters of the microcylinder. In this regard, it should be noted that the system investigated in this study is complex and requires further studies.

In sum, the resolution enhancement of MAM obtained using amplitude gratings is more significant than expected from previous studies30 concluded from simulations of interference microscopy for phase gratings. Especially in the case of transmission mode, the resolution improvement is larger and more visible due to the attenuated background field. The conversion of EWs to propagating waves seems to be the most likely reason. This conversion is induced by whispering gallery modes, which are excited by the EWs. In the case of the phase objects studied previously,30 no appreciable influence of EWs is expected, and no additional resolution enhancement is achieved by exciting WGMs. Instead, the local NA enhancement and a limited field of view turned out as the main reasons for resolution enhancement. However, depending on the profile to be analyzed, the resolution is moreover improved by the conversion of evanescent to propagating waves. This result accurately reflects the diversity of results and resolution capabilities achieved in the literature.

We present an FEM-based simulation model of microcylinder-assisted microscopy and show a way to analyze the resolution capabilities considering the imaging of gratings, which affect the amplitude of either reflected or transmitted light waves. Compared with previous studies, where phase gratings are imaged by interference microscopy, microcylinder-assisted microscopic imaging of amplitude gratings is studied. The resolution enhancement obtained by these simulations is compared for setups operating in reflection and transmission modes. In the case of transmission, we further distinguish between placing the microcylinder in the illumination path or the imaging path of the microscope. The results of this investigation can be summarized as follows.

1. •In both cases, reflection and transmission, the resolution is significantly enhanced compared with the fundamental resolution limitation, which appears without microelement.
2. •In previous studies, a local increase of the NA by the microelement combined with a limited field of view, which is only a portion of the cross-section of the microelement, is identified as the major reason leading to an enhanced resolution of phase objects. However, the resolution obtained in this study is further improved.
3. •The conversion of EWs to propagating waves is identified as the most likely reason for this additional resolution improvement, which in particular occurs in transmission mode imaging. In the case of reflection, the improvement, especially for extremely small period lengths, suffers from a dominant background field.
4. •As the additional resolution enhancement is not apparent for light wavelengths, which do not excite WGMs, the resolution improvement is attributed to WGMs, converting evanescent to propagating waves.

Compared with previous investigations, the resolution improvement through microelements can be attributed to different reasons and strongly depends on the measurement configuration and the object to be measured. This theoretical finding reflects the diversity of resolution capabilities achieved in various experimental studies. Therefore, the results give an interesting insight into the complexity of effects leading to resolution improvement in MAM.

More detailed parameter studies and quantitative comparisons with measurement results are planned for future studies. In addition, an extension of the model to 3D scattering objects and, thus, microspheres instead of microcylinders is of great interest.

To underline the impact of EWs on the obtained resolution and further demonstrate the sensitivity of the setup on measurement parameters such as the light wavelength, or the radius and refractive index of the microcylinder, additional simulation results with varying parameters are shown as follows. Note that this is not a comprehensive parameter study. Instead, some results are shown, which give rise to future investigations.

If EWs play a role for the excited WGMs and the associated resolution enhancement, the impact is expected to reduce with increasing distance between microcylinder and grating due to the decay of the EWs. Figures 3(g) and 3(h) show that the shape of the excited WGM within the microcylinder is significantly influenced by the lateral position of the grating above. Therefore, Figs. 5(a)–5(d) display the difference $\mathrm{\Delta }I$ between intensities obtained from a grating and its complement for a gap $\delta$ between grating and microcylinder of 10 nm [Fig. 5(a)], 50 nm [Fig. 5(b)], 100 nm [Fig. 5(c)], and 300 nm [Fig. 5(d)]. All other parameters remain unchanged from Figs. 3(g), 3(h). As expected, the difference in the intensities decreases with increasing distance and completely vanishes for $\delta =300\mathrm{nm}$. Figures 5(e) and 5(f) present the corresponding intensities for the grating [Fig. 5(e)] and its complement [Fig. 5(f)] with $\delta =300\mathrm{nm}$. Obviously, for both positions of the grating, the same WGM is excited from the transmitted field. However, as no difference between both intensities is visible, no information on the phase of the grating is contained in the WGM. As a consequence, EWs obviously contribute to the obtained super-resolution effect.

The excitation of a WGM, which is required for the obtained super-resolution, is quite sensitive to the light wavelength. Figures 6(a)–6(c) display the intensity differences $\mathrm{\Delta }I$ between the intensities obtained from a grating and its complement, for slightly different wavelengths of $\lambda =442.8\mathrm{nm}$ [Fig. 6(a)], $\lambda =442.7\mathrm{nm}$ [Fig. 6(b)], and $\lambda =442.5\mathrm{nm}$ [Fig. 6(c)]. All of the results are obtained for a grating of period length $l_x=150\mathrm{nm}$, and hence, the difference in the intensities indicates if the grating is still resolved due to the microcylinder or not. For comparison, Fig. 6(d) shows $\mathrm{\Delta }I$ for $\lambda =442.951\mathrm{nm}$ used for the studies in the main text and thus corresponds to the difference between the results shown in Figs. 3(g) and 3(h). Changing the wavelength by $\mathrm{\Delta }\lambda =0.15\mathrm{nm}$ [difference between Figs. 6(a) and 6(d)] does not seem to reduce the amplitude of WGM. However, if the wavelength is further reduced, the WGM is significantly reduced for $\lambda =442.7\mathrm{nm}$ and disappears for $\lambda =442.5\mathrm{nm}$. Hence, the robustness of the results to variations in the light wavelength seems to be in the subnanometer range.

So far, the resolution analyses have been performed in light of normal incidence to reduce the simulation time. To analyze the impact of the incident angle ${\theta }_{\mathrm{inc}}$, which is defined with respect to the $z$-axis (for a sketch with definition, see Ref. 41), Figs. 6(e) and 6(f) display intensity differences for ${\theta }_{\mathrm{inc}}=30\mathrm{deg}$ [Fig. 6(e)] and the Brewster angle of glass ${\theta }_{\mathrm{inc}}=56.31\mathrm{deg}$ [Fig. 6(f)]. Independent of the angle of incidence, the differences in the excited WGM are visible, and hence, it seems that the grating can be resolved.

As microspheres and microcylinders of varying size and refractive index are used in literature (for a review, we refer to Refs. 9–11), resolution analyses shown in Sec. 3 are repeated with different properties of the microcylinder. Figure 7 displays std values following the procedure described in Sec. 3 for a microscope in transmission mode with the microcylinder of $r=2\mu \mathrm{m}$, $n=1.5$ [Fig. 7(a)], $r=3.5\mu \mathrm{m}$, $n=1.5$ [Fig. 7(b)], $r=2.5\mu \mathrm{m}$, $n=1.4$ [Fig. 7(c)], and $r=2.5\mu \mathrm{m}$, $n=1.6$ [Fig. 7(d)] placed in the imaging path. Note that the blue curve, marked as original in all four plots, corresponds to the blue curve shown in Figs. 3(f) and 4(b). The illumination wavelength is adjusted appropriately to excite a WGM. The results indicate that the achieved resolution is degraded using larger microcylinders or those of higher refractive index. For $n=1.4$, the resolution seems to be slightly deteriorated compared to $n=1.5$ as well. With respect to Fig. 7(a), the resolution seems to be similar for both sizes. As a consequence, the resolution shows a significant dependency on the parameters of the microcylinder, giving cause for more detailed studies in the future. Finally, it should be mentioned that the presented parameter studies are related to a specific example and require much more investigation to be generalizable.

Felix Rosenthal worked in the field of measurement technology at the University of Kassel, Germany, after completing his master’s degree in physics. He has been a research assistant since 2023 and is working on his dissertation. His research focuses on the determination of the 3D transfer function of optical instruments and the processing of data.

Tobias Pahl received his master’s degree in physics in 2018 from the University of Münster, Germany. He has been working as a research assistant and a PhD candidate in the Measurement Technology Group of the Department of Electrical Engineering and Computer Science at the University of Kassel since 2019. His main research interests are interference, focus-variation, and confocal microscopes with high lateral resolution and their modeling.

Lucie Hüser studied electrical engineering at the University of Kassel. She has been working as a research assistant in the Measurement Technology Group of the Department of Electrical Engineering and Computer Science at the University of Kassel since 2018. Her main research areas are interference microscopes with high numerical apertures and near-field support in interference microscopy.

Michael Diehl began studying physics at the University of Marburg, Germany, and completed his studies at the University of Göttingen, Germany, with a doctorate in 1994. After a postdoc in the X-ray Microscopy Research Center at the University of Göttingen, he moved to various industrial companies from 1995 to 2019, managing projects and product developments in the fields of traffic engineering, software development, and optical metrology. Since 2019, he has been working at the Measurement Technology Section, Faculty of Electrical Engineering and Computer Science, University of Kassel, Germany.

Tim Eckhardt studied electrical engineering at the University of Kassel and graduated with a master’s degree in 2022. He then gained one year of professional experience as a development engineer before becoming a research assistant in the Department of Metrology at the University of Kassel in November 2023. As part of his doctorate, he is working on the development and optimization of optical point sensors, in particular through the use of micro-optical lenses such as microspheres and the high-frequency modulation of interference signals. His scientific focus lies in the field of interferometry.

Sebastian Hagemeier has been working as a research assistant in the Department of Metrology at the University of Kassel, Germany, since October 2014. He completed his doctorate degree in engineering sciences in February 2022. His scientific work includes the comparison of the transfer behavior of various surface topography sensors in a multisensor measuring system. He is also working on the development and investigation of a fiber-coupled confocal-interferometric distance sensor for precise high-speed profilometry.

Peter Lehmann studied physics at the University of Karlsruhe, Germany. He received his PhD in engineering at the University of Bremen, Germany in 1994 and received his habilitation degree in 2002. From 2001 to 2008, he coordinated research projects related to optical metrology in an industrial company. Since 2008, he has been a full professor and holds the chair in measurement technology at the Faculty of Electrical Engineering and Computer Science, University of Kassel, Germany.