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Abstract
Improving the lateral resolution is a key focus of the research on optical measuring systems to expand the fields of application for optical metrology. By means of microspheres put on an object in a microscope, and therefore used as a near-field support, it has already been shown that a superresolution of structures below Abbe’s diffraction limit is possible. The following investigations give more detailed theoretical and experimental insight into the physical mechanisms responsible for the transition of near-field information to the far field. In particular, the effects of microspheres as near-field support on the behavior of phase-evaluating interference microscopes close to the optical resolution limit are studied experimentally as well as with numerical simulations. Special attention is drawn to measured data taken with a Linnik microscope of high numerical aperture. Finally, the measurement results of grating structures with a period below Abbe’s diffraction limit are presented.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Optical measuring systems have a decisive advantage in surface profilometry because of their ability to measure contactless, fast, and sometimes label-free. Therefore, the application of these systems in research and industrial applications such as quality control is often the method of choice. With interference microscopy, for example, an axial resolution in the single-digit nanometer range can be achieved. In lateral resolution, however, optical imaging systems are always bound by diffraction effects to the resolution limit depending on the illumination wavelength and the numerical aperture (NA) of the imaging system. For the smallest resolvable lateral distance $ {d_{\min}} $ it applies
In recent research, the use of microspheres as near-field enhancing support has been proven to enable optical microscopy below the resolution limit. This approach uses the fact that microspheres create a strong focus of light, which gives access to near-field information. This was presented in Ref. [1] as a potential technique for super-resolution microscopy and is described as photonic nanojet. Wang et al. [2,3] have shown that a resolution of 50 nm related to the structure width is possible. In Ref. [4], a resolution of 100 nm with respect to structure widths is obtained. A resolution down to $ \lambda /7 $ is achieved in Ref. [5]. In Ref. [6], the connection of the microsphere approach to the Mie theory is investigated, and Ref. [7] shows the classical imaging process related to super-resolution with microspheres. Darafsheh et al. [8] and Allen et al. [9,10] deal with the incorporation of microspheres in substrates for better handling. In Ref. [11], the influence of the refractive index of the surrounding medium is investigated. An application concerning the imaging of adenoviruses with dimensions on the order of 75 nm is shown in Ref. [12]. In Refs. [13,14], the position of the virtual image plane in the imaging process through the microsphere is described in detail.
The properties of photonic nanojets have been studied in numerous publications. In Refs. [15,16], the theoretical basics and possible applications are discussed. Yang et al. [17] show the influence of the photonic nanojet waist on the resolution. Rockstuhl et al. describe possibilities for photonic nanojet engineering [18,19]. The role of coherence is explored in Ref. [20]. Phase anomalies are discussed in Ref. [21] and Ref. [22] deals with the imaging properties of a microsphere when illuminated with an optical needle. The use of biological materials to produce a photonic nanojet is shown in Refs. [23,24]. In Ref. [25], the lighting configuration and related imaging behavior is discussed.
Furthermore, microspheres were used in combination with other measuring systems to improve the resolution. Duocastella et al. [26] show the combination of a microsphere with an AFM cantilever. As shown in Ref. [27], structures down to 25 nm width could be resolved using a combination of microspheres with confocal microscopy. Upputuri et al. [28] combined Raman scattering microscopy with microspheres to achieve a resolution enhancement.
Further investigations make use of microspheres for near-field support in interferometry. This type of investigation was reported by Wang et al. [29] and Montgomery et al. [30–32] by means of a white-light Linnik interferometer. In Refs. [33,34], a Mirau interferometer is used. Leong-Hoi et al. [35] also use a Linnik configuration to measure nanomaterials. The microsphere-assisted microscopy provides a measuring method that enables super-resolution, which opens up a broad field of application for optical profilometry. Until now, as documented in literature, the use of microspheres provides improved lateral resolution capabilities in 3D measurement. However, in most cases, the same lateral resolution capabilities could also be achieved using a microscope objective of higher NA combined with a light source with a lower center wavelength. Our approach is to use high NA objectives and to study if the lateral resolution can be further improved.
The results presented below aim at the application of microspheres in interferometric systems with high NAs. A Linnik interferometer with $100\times$ magnifying lenses and a NA of 0.9 is used to acquire an image stack of interference patterns employing a depth scan. Furthermore, the behavior of the phase at different evaluation wavelengths of the interferometric data in microsphere-assisted interferometry is studied. When using a microsphere, a shift occurs in the evaluation wavelength at which phase information about the topography can be obtained. The relationship between the evaluation wavelength and the measured structure width is discussed in Ref. [36].
To gain a deeper understanding of the behavior of the phase when using microspheres, numerical simulations of the electromagnetic field distribution in the near field were performed. Particular emphasis was put on a complete representation of the imaging process. To investigate the phase behavior in the virtual image plane created by the microsphere, a time reversal simulation was performed.
2. EXPERIMENTAL SETUP
For the measurement results presented in this paper, a Linnik interferometer with near-field microsphere support was used. The Linnik interferometer is equipped with two microscope objectives, each with a NA of 0.9 and $100\times$ magnification (Olympus MPLFLN-BDP). For image acquisition, a scientific CMOS camera (Hamamatsu, ORCA-flash 2.8) is used and the light source is a royal blue LED (LUXEON Rebel Color Line). The used $ {{\rm SiO}_2} $-microspheres with a diameter of 5 µm are fabricated by micromod Partikeltechnologie GmbH. The spheres were put on the measurement object in a water suspension. After the water evaporated, the measurement was carried out in a dry surrounding. The depth scan was performed using a piezo stage (Physik Instrumente P-620.ZCD). Figure 1(a) shows a schematic representation of the experimental setup. The classical structure of a Linnik interferometer has been extended by an additional mirror in the reference arm. This allows microspheres to be put even in the reference arm [37]. In Fig. 1(b), an enlarged section of the specimen is shown to illustrate the position of the virtual image plane. Figure 1(c) shows a photograph of the experimental setup. Compared to the schematic setup presented in Fig. 1(a), the illumination path is redirected by 90° for practical reasons using a 45° mirror.
Fig. 1. (a) Experimental setup consisting of a Linnik interferometer, lighting and image acquisition units, and a piezo scanner; (b) enlarged section showing the microsphere on the specimen’s surface and the virtual image plane; and (c) photograph of the experimental setup.
3. EVALUATION WAVELENGTH IN PHASE EVALUATION ALGORITHMS
The evaluation of interference signals acquired with broadband light sources involves the use of various algorithms, which consider both the envelope of the interferogram and the phase data of the interference. By considering phase data in the evaluation, higher accuracy and robustness can be achieved [38]. Therefore, our topic of interest is the further investigation of the phase evaluation while using near-field assistance with the application of microspheres put on the measurement object in combination with high NA microscope objectives. Investigations on the behavior of phase analysis at high NAs are presented in Ref. [36]. Especially in measurements of structures near the resolution limit, these considerations play an important role.
In the following section, there will be a discussion of the role of the evaluation wavelength on the results of phase evaluation when using microscope objectives of high NA. Subsequently, these investigations are considered for use in near-field assisted interference microscopy and the conclusion that can be drawn to super-resolution. Furthermore, for a more detailed insight into the imaging mechanisms, numerical simulations of the field distributions in the near field were performed. In particular, observation of the phase in the virtual image plane is of special interest.
A. Phase Evaluation and High Numerical Apertures
When acquiring interferometric measurement data using short-coherent light sources, it is assumed at low NAs that the appropriate evaluation wavelength for the phase evaluation of the interference signals equals the center wavelength of the illumination source used. However, as shown in Refs. [36,39], it is no longer the case when using microscope objectives with higher NAs or profile heights of the measurement object greater than $ \lambda /4 $. In investigations of the resolution limit, grating structures of different period lengths are often used as measuring objects to maintain a reproducibility of the conditions. The incident angles, which are still accessible at high NAs in combination with diffraction at structures with lateral dimensions in the order of the lateral resolution limit, result in a shift of the spectral components present in the interferogram to higher wavelengths. This leads to a characteristic dependence of the measured surface topography on the evaluation wavelength, as it will be presented in the following investigation.
In Fig. 2, both simulated and experimentally determined spectra of interferograms for different NAs are plotted. The LED used to perform the measurements shown in Fig. 2(b) is also a royal blue LED (the same LED as the one used for the experimental data presented in this paper).
Fig. 2. (a) Simulated and (b) measured spectra of interferograms for different numerical apertures. (Reprinted by permission from Springer Nature: [P. Lehmann, S. Tereschenko, B. Allendorf, S. Hagemeier, and L. Hüser, J. Eur. Opt. Soc. 15,5 (2019) [36]).
The spectra are calculated from interferograms obtained during the depth scan from a single camera pixel using a plane mirror as a measurement object by using a discrete Fourier transform and rescaling of the spatial frequency axis with regard to wavelengths instead of wave numbers. Thus, the measured interference signals are directly connected to the spectral components of the interferogram. In particular for high NAs, a clear deviation of the spectrum of the interferogram compared to the spectrum of the light source used appears. This shift leads to enlarged fringe spacing and thus to a systematic error of the height values obtained from the measurement data, which is also commonly known and studied as the NA effect [40–42]. The dependency between this shift and the evaluation wavelength of the phase evaluation was investigated in Ref. [36], and it could be shown that the lateral resolution limit of the measurement is essentially related to the selected evaluation wavelength. This connection will be discussed in the following chapter in detail.
B. Dependence of Phase Evaluation on Microspheres
Analyzing the phase of interferometric measurement data, which were generated by use of the near-field support of a microsphere, the influence of the spheres on the measurement results at different evaluation wavelengths was studied. As shown in Ref. [36], the most appropriate evaluation wavelength belonging to a periodic structure can be calculated using the angle of its first-order diffraction maximum. It leads to a measured surface topography that is closest to the nominal surface structure. In Fig. 3, the modulation depth of a measured rectangular grating structure (period length 300 nm) is plotted over the used evaluation wavelength. The shift of the maximum modulation depth to lower wavelengths can be clearly observed by comparison with the result obtained without a microsphere. With an AFM, a structure height of 85 nm was measured (results of AFM measurements are shown in chapter A). This contradicts the manufacturer’s data [43], which indicate a maximum structure height of 30 nm, but both the optical and the tactile measurements agree. Therefore, the measured height is taken as the correct height.
Fig. 3. Measured structure height (modulation depth) of the 300 nm grating is plotted with respect to the evaluation wavelength $ {\lambda _{{\rm eval}}} $, both with and without a microsphere. The black dashed line shows the structure height of 85 nm measured with the AFM.
The 300 nm period length structure under investigation can still be resolved with the interferometric system without near-field support, so this result is not yet considered as super-resolution. In further measurements, structures down to 230 nm in period length were successfully measured. The consideration of the evaluation wavelength in the phase evaluation for this structure, however, provides information about the transmission mechanisms of the applied microspheres. Therefore, it will be discussed in more detail below.
In Ref. [36], for a certain angle of incidence $ {\theta _e} $, the following relation between the illumination wavelength $ \lambda $ and the evaluation wavelength $ {\lambda _{{\rm eval}}} $ is given as
For the angle $ {\theta _{{\rm diff}}} $ belonging to the first diffraction order, the relationship
holds, where $ \Lambda $ is the period length of the considered grating structure. For $ \Lambda = 300\;{\rm nm} $ and an illumination wavelength $ \lambda = 460\;{\rm nm} $, the equation results in an angle of $ {\theta _{{\rm diff}}} = {50^ \circ } $ for the first diffraction order, which is quite close to the angle $ {\theta _e} = {56^ \circ } $ following from $ {\lambda _{{\rm eval}}} = 820\;{\rm nm} $ and $ \lambda = 460\;{\rm nm} $. Due to the numerical aperture of 0.9, these angles are well within the angular range covered by the objective lenses. The slight deviation may be due to the spectral bandwidth of the illumination source. The illumination wavelength is included in the formulas only as a discrete value, while in reality the nonuniform illumination of the objective’s pupil and the dependence of the incident irradiance on the radial position within the pupil play a role in the imaging process.When considering the measured modulation depth with the help of a microsphere, a clear shift of the maximum modulation depth toward lower evaluation wavelengths can be observed. In addition, plausible results were also obtained for the depth of the structure with near-field support. The visible peak in the evaluated modulation depth at $ {\lambda _{{\rm eval}}} = 640\;{\rm nm} $ in Fig. 3 would correspond to an angle $ {\theta _e} = {44^ \circ } $ without a sphere. This leads to the conclusion that image information in the spatial frequency domain is shifted by the microsphere toward the optical axis and thus the diffraction orders can be captured with an imaging system with a lower NA. This phenomenon can be considered as one of the relevant mechanisms to enable the capability for super-resolution.
C. Time Reversal Simulations for Phase Analysis
To find a theoretical explanatory model for the imaging processes in near-field assisted microscopy, numerical simulations have already been used in some publications to study the electromagnetic field distributions in the near field. In particular, considering the behavior of photonic nanojets, numerous results have been published. In general, however, the studies only partly show the imaging process. Publications such as [1] show the field distribution resulting from a wave impinging on the sphere, while in other studies it is assumed that the structure can be considered as a point light source and thus the back propagation through the sphere can be investigated [33].
During the simulative validation of the presented results of interferometric phase analysis, the entire imaging process through the sphere will be considered to be able to assess the behavior of the phase of the propagating wavefront. The phase information obtained by the objective is essential for an interferometric phase evaluation and therefore is of particular interest. In Ref. [33], simulations were presented that include the phase of interference signals. Two point light sources were placed as object points under the sphere and the resulting field distribution was calculated. Subsequently, a simulation for time reversal propagation without a sphere was carried out to determine the position of the virtual image plane. This approach will now be followed up in the simulations shown below to get a more detailed insight into the phase behavior in near-field assisted interference microscopy. The numerical simulations were performed with finite difference time domain (FDTD) methods.
To get a more complete view of the imaging mechanisms by means of microspheres, the simulation setup was arranged as realistically as possible. The microsphere was placed in the center of the simulation field and the source was a plane wave above the sphere. A grating structure was placed under the sphere. In the simulation, therefore, the entire beam path through the sphere on the grating and the back reflection through the sphere was examined. The resulting field was acquired above the sphere and stored for the next step. To investigate more precisely how the field distribution is influenced by a microsphere, in the second step the approach from [33] was followed to investigate the back propagation of the fields by means of time reversal calculations.
Since the virtual image plane is shifted by microsphere interference microscopy, the position of this plane had to be determined first to execute the phase observations at the correct position. For this purpose, a point light source was placed under the sphere and then in time reversal, the waist of the field distribution as the position of the virtual image plane was determined.
To validate that it is possible to study the behavior of the phase by considering the time reversal field, the time reversal field distribution was first considered for the simulation case of a grating without a sphere. The result is shown in Fig. 4(a). For consideration of the phase, a profile cut was made in the virtual image plane (marked by the dashed white line in the figure). The phase profile in this section is shown in Fig. 5(a). The structure of the grating is clearly visible, which shows that this method is suitable for making further investigations of the phase behavior.
Fig. 4. Magnitude of time reversal field distributions for a grating (600 nm period): (a) without a microsphere and (b) with a microsphere. The position of the virtual image plane is marked with the white dashed line.
Fig. 5. Phase profile in the virtual image plane for (a) a grating structure (600 nm period) without microsphere; (b) phase profile assuming a microsphere (diameter 5 µm, $ {{\rm SiO}_2} $) on a plane surface; (c) phase profile assuming a microsphere on a grating structure; and (d) the grating structure in (c) was inverted.
To have a reference of how the phase is affected only by the microsphere, a simulation scenario was created to examine this particular case. For this purpose, the microsphere was placed on a plane surface without a grating structure. The behavior of the phase profile in the time reversal simulation can be seen in Fig. 5(b).
Finally, the phase behavior for the simulation case of a grating with a microsphere was investigated. The time reversal field distribution for this case can be seen in Fig. 4(b), and the phase in the virtual image plane is shown in Fig. 5(c). To distinguish the grating structure from the phase distortion of the microsphere, a second simulation was carried out in which the grating was inverted. The phase profile for this case is shown in Fig. 5(d).
It can be seen from Fig. 5(b) that there is a region of about 2 µm around the central axis, in which the phase is not strongly distorted by the insertion of the microsphere. This range is somewhat larger in the experimental measurements, which can be explained by the fact that in the simulation setup only vertically incident rays are considered.
Considering the simulation of the grating structure below, the microsphere [Figs. 5(c) and 5(d)], the enlarged structure of the grating placed under the microsphere can be seen. The period in Fig. 5(c) is approximately 1, 2 µm corresponding to a magnification factor of 2 introduced by the microsphere. The phase shift of 180° between the grating structures used for the two simulations is also clearly visible. The results show that the phase profile in the virtual image plane is not only a superposition of the phases from the individual simulations of grating and sphere [Figs. 5(a) and 5(b)], but also the phase profile in the near field is significantly influenced by the microsphere. The enlargement of the grating structures in the phase profile can provide a possible theoretical explanation model for the super-resolution. Since the magnification already takes place in the near field and directly influences the phase, it becomes possible to transmit the information into the far field, where the objective is placed. Also, the shift of the evaluation wavelength described in chapter B can be explained with this result. Fine structures are already enlarged in the near field and thus they are made accessible in the far field.
4. EXPERIMENTAL RESULTS
The measurement object used throughout this section is the Nanoscale Linewidth/Pitch Standard fabricated by Supracon AG [43]. It covers fine structures with period lengths of 160 nm to 4 µm. Due to the fine gradations in the range from 200 to 400 nm, it is possible to carry out detailed experimental investigations in the range of the resolution limit of the measurement setup used. To validate the optically measured structure heights, the fine structures were additionally measured with an AFM. These results and the results of near-field assisted interference microscopy will be presented.
The measurable area with the microspheres used throughout these experiments ($ {{\rm SiO}_2} $, 5 µm diameter) covers an area of about 2 µm in diameter. For the presented measurements on gratings with period lengths of 230 nm and 300 nm, accurate measurement data can be obtained for a range of about 5 to 6 periods.
It should be noted at this point that the measurement standard consists of $ {{\rm SiO}_2} $ coated with silicon. Since the top layer of the structure is pure silicon, but the bottom layer in the grooves is made from $ {{\rm SiO}_2} $, two different materials are to be considered. These materials have different refractive indices and absorption coefficients. Thus, the Fresnel coefficients were used to investigate the resulting phase shift for all angles of incidence occurring at an NA of 0.9. Neglecting absorption results in a phase shift of 0° for p-polarized light and 180° for the s-polarization for both materials.
It has been found that the deviation from these values, including the absorption coefficient for $ {{\rm SiO}_2} $, is nearly zero. For silicon, it is not more than 1° for s-polarization, and no more than 2° for p-polarization. This results in a maximum phase shift compared to the reflection on the $ {{\rm SiO}_2} $ surface of 4° maximum, which would correspond to a maximum systematic error of 4.8 nm with respect to the optical path length difference. However, the measurements presented below show no significant relevance of this error.
A. Results of AFM Measurements
Since the results of the optical measurements differ from the manufacturer’s information, reference measurements by established measurement methods were necessary for validation purposes. Therefore, the measurement object used was additionally measured by an AFM. The results of the measurements regarding the structures of interest are shown in Fig. 6.
Fig. 6. AFM results for 300 nm period length grating structure on the measurement object in (a) and (b); 230 nm period length grating structure in (c) and (d).
For the structure with a period length of 300 nm, a height of 85 nm was determined. For the 230 nm structure, the measured height is 55 nm.
B. Results Obtained at Different Evaluation Wavelengths
To illustrate the effects of near-field support, interferometric surface measurements of various grating structures with and without microspheres were performed using the described Pitch Standard by Supracon. In addition to the modulation depth of the 300 nm structure shown in Fig. 3, the corresponding profile sections for the evaluation wavelengths 640 nm and 820 nm with and without a microsphere are shown in Fig. 7.
Fig. 7. Profile sections of the phase analysis of the 300 nm structure on the measurement object. (a) and (c) were with microsphere; and (b) and (d) measured without microsphere. The evaluation wavelength $ {\lambda _{\rm eval}} $ is 640 nm for (a) and (b) and 820 nm for (c) and (d).
In Fig. 8, the topographic measurement results through the microsphere for the two different evaluation wavelengths are shown. These wavelengths correspond to the maximum structure depth of the measured surface profiles corresponding to either a measurement with or without a microsphere, which is shown in Fig. 3. Note that in Figs. 7 and 9 the scale of the abscissa is not changed even if a microsphere is inserted. Thus, in Fig. 7(a) the period of the measured structure is 300 nm, although it seems to be 400 nm according to the scaling of the x axis.
Fig. 8. Measured topography of the grating structure with 300 nm period length. The evaluation wavelengths are (a) 640 nm and (b) 820 nm. The black dashed lines indicate the position of the profile cuts presented in Figs. 7(a) and 7(c).
Fig. 9. (a) Profile section and (b) surface topography of the 230 nm structure. The interferometric measurement data were obtained by means of phase evaluation. The evaluation wavelength was 650 nm.
For interferometric measurements without a microsphere, the influence of the evaluation wavelength on the measurement result becomes very clear, as already demonstrated and examined in Ref. [36]. However, for measurements with a microsphere, it can be observed that even at a lower evaluation wavelength, which is closer to the effective wavelength of the system used, the structure can be resolved. For higher evaluation wavelengths, phase information is no longer available. The irregularities in Fig. 7(c) can be attributed to the low signal quality of the interferograms at the evaluation wavelength.
C. Super-resolution Measurements
By means of microsphere-assisted near-field support it was also possible to measure topographies down to a period of 230 nm corresponding to a width of the individually measurable structures of 115 nm, as it is demonstrated in Fig. 9. These structures can no longer be resolved by the system without microspheres, since the Abbe limit for the resolution of a grating period results in 256 nm.
In the results presented in Fig. 9, the phase of the interferometric measurement data is obtained from the interference signals using an evaluation wavelength of 650 nm. The measured structure depth is in agreement with the AFM measurement results presented in subsection A. In the surface topography shown in Fig. 9(b), phase jumps are visible in the peripheral areas of the sphere. This is due to the algorithms used and the quality of the measured data. In the edge regions of the sphere, it is not possible to obtain qualitatively good measurement signals, since the incident light waves are primarily reflected at the strongly tilted edge region of the sphere. As a consequence, there are no accessible interferometric measurement data in this area.
5. CONCLUSION
In the presented simulations and the corresponding measurement results, the super-resolving capabilities of microspheres, which act as near-field support, was confirmed and further investigated. Particular attention is paid to the evaluation of the interferometric data and the influence that microspheres have on the imaging processes at this point. It has been shown that a shift of the image information in the spatial frequency domain of the optical system shifts toward the optical axis. The results obtained by near-field assisted interference microscopy could be validated by AFM results.
The analysis of the evaluation wavelength used for the phase evaluation of the interferometric data shows a clear influence of the microspheres. The microspheres manipulate the high-frequency near-field information and make it accessible in the far field captured by the imaging system. This allows interferometric measurements of structures below Abbe’s diffraction limit. The evaluation wavelengths used for the phase evaluation of near-field supported measurement data are closer to the illumination wavelength than the wavelengths that are required to obtain similar results from interferometric measurement data without using a microsphere. This could be one of the decisive explanatory effects for the working principle of microspheres in context with interference microscopy. To confirm the influence of the microspheres on the phase profiles in the near field, numerical simulations of the electromagnetic field distributions were performed. These simulations showed that structures visible in the electromagnetic field’s phase distribution the virtual image plane are already enlarged by microspheres and thus made accessible in the far field.
Finally, measurement results are shown that confirm the super-resolution capabilities of microspheres. Grating structures with periods slightly below Abbe’s diffraction limit could be successfully measured using a Linnik interferometer combined with $ {{\rm SiO}_2} $ microspheres for near-field support.
In further investigations, the effects of the microspheres on the phase profiles at different evaluation wavelengths will be discussed in more detail. In addition, the simulations of the near field are to be expanded. Different angles of incidence as well as an insight to the transition process of the phase information from the virtual image to the imaging system should be included.
Funding
Deutsche Forschungsgemeinschaft (LE 992/15-1).
Disclosures
The authors declare no conflicts of interest.
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