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Abstract
Quantitative phase imaging (QPI) is an invaluable microscopic technology for definitively imaging phase objects such as biological cells and optical fibers. Traditionally, the condenser lens in QPI produces disk illumination of the object. However, it has been realized by numerous investigators that annular illumination can produce higher-resolution images. Although this performance improvement is impressive and well documented, the evidence presented has invariably been qualitative in nature. Recently, a theoretical basis for annular illumination was presented by Bao et al. [Appl. Opt. 58, 137 (2019) [CrossRef] ]. In our current work, systematic experimental QPI measurements are made with a reference phase mask to rigorously document the performance of annular illumination. In both theory and experiment, three spatial-frequency regions are identified: low, mid, and high. The low spatial-frequency region response is very similar for disk and annular illumination, both theoretically and experimentally. Theoretically, the high spatial-frequency region response is predicted to be much better for the annular illumination compared to the disk illumination––and is experimentally confirmed. In addition, the mid-spatial-frequency region response is theoretically predicted to be less for annular illumination than for disk illumination. This theoretical degradation of the mid-spatial-frequency region is only slightly experimentally observed. This bonus, although not well understood, further elevates the performance of annular illumination over disk illumination.
© 2022 Optica Publishing Group
1. INTRODUCTION
Transparent objects like biological cells or optical fibers have historically been challenging to image. Techniques like fluorescence microscopy [1] make use of external agents or “labeling markers” like fluorescent dyes [2,3] to highlight the internal structure of the objects and enhance the contrast. Although this technique has been widely used and improved over time, the use of external fluorescent agents renders the biological samples prone to phototoxicity and photobleaching, depending on the type of agent used [4,5]. This makes it unusable on samples that are incompatible with fluorescent markers and does not allow for its use in the continuous imaging of cells over extended periods of time. On the other hand, phase-based methods use only the internal structure of cells and other objects being imaged. However, some phase imaging methods, such as phase contrast microscopy [6,7] and differential interference contrast microscopy [8,9] are useful only in qualitative imaging and cannot be used to obtain quantitative data due to their noninvertible nature. Quantitative phase imaging (QPI) techniques can be used to measure the optical path thickness (2D QPI) or determine the refractive index distribution (3D QPI) of the object.
QPI has been applied in a wide variety of biological studies [10]. One of the major areas of study is the measurement of the dry mass and dry mass density of the cells [11–14]. It is used to measure cell growth, follow cell cycles, measure the effects of drugs, as well as measure cell metabolism. QPI has been used to analyze blood coagulation processes [15], detect cancers and tumors [16,17], study optomechanical properties of cancer cells [18], and understand details like intracellular mass transport [19] and cell volume [20]. Beyond the field of clinical diagnostics and biological studies, QPI is also applicable to semiconductor research in revealing defects and features that would otherwise be difficult to observe. It has been used for imaging particles embedded in 3D microscopic structures [21], determining the number of layers in 2D materials and imaging them [22], inspecting silicon wafer defects in manufacturing applications [23,24], and imaging nanoscale electrostatic and magnetic fields in nanostructured materials [25]. QPI has also been applied to the analysis and characterization of optical fibers and fiber Bragg gratings [18,26,27].
While most microscopy techniques use disk illumination, annular illumination has also been of interest to the scientific community since early investigations by Airy [28]. In recent years, annular illumination has been applied to numerous imaging and microscopy techniques. For example, in confocal microscopy, the use of annular illumination has improved the image quality [29,30], produced narrower point spread functions, and improved resolution [31,32]. Sheppard [33] found that the use of annular illumination provided a linear response over a wider range of phase gradients with a better resolution and improved low-frequency response. Annular illumination has also been applied to numerous other fields: third-harmonic generation microscopy [34], focal modulation microscopy [35], dark-field Brillouin microscopy [36], localized surface plasmon microscopy [37,38], multiphoton microscopy [39,40], stimulated emission depletion microscopy [41], and fluorescence microscopy [42,43].
In addition to the various microscopy techniques, annular illumination has been applied to 2D and 3D QPI. In 2D QPI, the transport-of-intensity equation (TIE) is one of the frequently used approaches due to its simple requirements and deterministic nature. Zuo et al. [44] resolved the problem of lowered phase contrast and strong low-frequency artifacts in TIE imaging by using annular illumination, which resulted in improved resolution. Huang et al. [45] proposed a phase retrieval method using annular pupils and annular sector pupils, whereas Li et al. [46] identified that a thin annulus with an NA matching that of the objective is the optimal illumination pattern for TIE-based QPI. Other applications of annular illumination in 2D QPI include quantitative differential phase contrast (DPC) [47,48], color-multiplexed DPC [49,50], Fourier ptychographic microscopy [51–54], and single-shot wavelength-selective QPM [55]. Li et al. [56] also used annular illumination through a programmable LED array for phase imaging with better noise performance and higher resolution. In 3D QPI, tomography is a frequently used approach to obtain information about the sample. Li et al. [57,58] improved the intensity diffraction tomography technique by incorporating annular illumination using a programmable LED array to reconstruct 3D refractive index distributions. A novel tomographic technique used two disk illumination apertures and one annular aperture to combine multifrequency components for improved resolution and SNR [59].
Apart from the various microscopy and QPI techniques, annular illumination has also been used in photolithography systems and has led to improvements in resolution, depth of focus, and contrast [60–62]. More recently, annular illumination was used to obtain a half-pitch resolution of 8 nm [63] in EUV photolithography. Furthermore, annular illumination has also been applied to scanning electron microscopy [64], silicon wafer inspection techniques [65], X-ray imaging [66], imaging of bones and tissues [67,68], ophthalmoscopy [69,70], and optical micromachining [71].
In all the papers mentioned above, although the benefit of annular illumination was realized, quantitative comparisons to disk illumination have generally been lacking and the two illumination types have not been thoroughly evaluated within a single analytical framework. Qualitatively, it has been shown that annular illumination is better, but the underlying basis has not been rigorously established. Now, with the QPI approaches developed by Jenkins et al. [72] and later generalized by Bao et al. [73–75], an analytical framework applicable to both disk and annular illumination is available for a systematic evaluation of the use of annular illumination in QPI.
The present work is a detailed quantitative comparison of disk and annular illumination as a function of the resulting phase image spatial-frequency response. This work is based on experimental measurements made with a standard-type phase test chart rather than based on specific objects such as beads, optical fibers, lenslet arrays, phantoms, and biological cells. Further, these experimental measurements are then shown to correlate with the annular illumination phase imaging analytical predictions from [75].
2. QPI METHODOLOGY
A QPI method called weighted least squares multifilter phase imaging with partially coherent light (WLS-MFPI-PC) that was developed by Bao et al. [75] was used in this work to calculate 2D phase images from experimental data. WLS-MFPI-PC uses multiple defocused intensity patterns of the sample to reconstruct a high-contrast phase image. The method uses multiple orders of Savitzky–Golay differentiation filters (SGDF) to estimate the derivative of intensity at the focal plane for each order of SGDF. The intensity derivative is related to the unknown phase by a phase contrast transfer function (PCTF). The PCTF is obtained by integrating the generalized nonparaxial 3D phase optical transfer function (POTF) along the optical ($z$) axis, as given by Eq. 5(b) in [72]. The Fourier transform of the intensity derivative is divided by the PCTF to obtain the unknown phase. The phase estimated using each SGDF order is then combined in the frequency domain using a weighted least squares approach such that the phase information is weighted according to the PCTF magnitude and the SGDF impulse response for a given order. As a result, every SGDF order produces a phase estimation via inversion of the individual PCTF, which is then combined with phase estimation from other SGDF orders to produce a single reconstructed phase image. The magnitude of the transfer function used for inversion plays an important role in determining the reliability of the phase recovery. A high transfer function magnitude results in a higher SNR and hence a better phase recovery.
Figure 1 shows the weakly defocused phase contrast transfer function (WD-PCTF) curves for the disk (${T_{{\rm{WD}}}}$) and annular (${T_{{\rm{WA}}}}$) illumination cases. The WD-PCTF is the PCTF in the limit of infinitesimally small defocus distance (distance between defocus plane and the in-focus plane), and can be calculated using Eq. (6) in [72]. It is 2D in nature, with both axes representing spatial frequencies. The radial symmetry of the 2D WD-PCTF was used to find the average value of the function at every radius value, from zero frequency to the cutoff frequency. These average values are plotted in Fig. 1 to show the 1D variation of the WD-PCTF magnitude as a function of spatial frequency. This 1D variation is equivalent to one half of the line profile over a single row of the 2D WD-PCTF plot. The three distinct regions in the WD-PCTF curve in Fig. 1 can be identified from the figure. In the low-frequency region, the WD-PCTF for disk and annular illumination are almost the same, implying no advantage of one type over the other. In the mid-frequency region, the disk WD-PCTF has a higher value than the annular WD-PCTF, implying that the phase recovery has a higher SNR for disk illumination. In the high-frequency region, the annular WD-PCTF has a higher value than the WD-PCTF for the disk illumination case, implying that annular illumination is advantageous for higher spatial frequencies. This was predicted by Bao et al. in [75], but not verified experimentally. In this paper, we used 2D QPI experiments on a standard resolution test chart to verify this characteristic.
Fig. 1. WD-PCTF curves radially averaged from 2D data for disk (dashed red) and annular (solid blue) illumination. Three separate spatial-frequency regions are also denoted.
In the following section, the difference in performance of the two illumination types based on experimental data is compared to the trend predicted in the three regions by the analytically calculated WD-PCTF curves depicted in Fig. 1.
3. EXPERIMENTAL CONFIGURATION
2D QPI experiments were performed using the configuration shown in Fig. 2. The system was composed of an Olympus BX60 microscope, a Physik Instrumente (PI) piezoelectric actuator controlled using a digital piezo controller, and a Pixelink 5MP monochrome camera. The various components of the system are controlled using a LabVIEW program with a user interface.
The sample is first brought into focus, setting the in-focus plane as the reference plane for the system. Then the piezoelectric actuator moves the objective along the optical axis, enabling the automatic capture of defocused intensity patterns via the camera. The user can set the defocus step (distance traveled by the objective between the consecutive defocused patterns), and the total scanning range of the piezoelectric actuator. Using these two quantities, the LabVIEW program determines the number of patterns to be recorded. After all the intensity patterns are obtained, the program execution ends and the user can use the saved patterns to reconstruct a high-contrast phase image.
The configuration used for the experiments incorporates an Olympus UMPlanFl 50x objective (${{\rm NA}_{\rm{o}}} = 0.75$), an Olympus U-POC-2 condenser (C1) for disk illumination, and an Olympus U-PCD2 phase contrast condenser (C2) for annular illumination. C2 has annuli of different radii for use with corresponding phase contrast objectives in PCM. However, for our experiments, C2 was used with a conventional microscope objective. The condenser annulus that corresponds to the $40\times$ setting was used and had ${\rm NA_{\rm{c}}} = 0.331$ and ${\rm NA_{{\rm{ci}}}} = 0.293$, where ${{\rm NA}_{\rm{c}}}$ refers to the outer numerical aperture and ${{\rm NA}_{{\rm{ci}}}}$ refers to the inner numerical aperture of the annulus. To match the condenser NA for the case of disk illumination to that for the case of annular illumination, C1 was set to ${{\rm NA}_{\rm{c}}} = 0.331$, where ${{\rm NA}_{\rm{c}}}$ is the outer NA of the condenser. Naturally, the inner NA of the condenser for the case of disk illumination was zero. These NA values are depicted in Fig. 3.
Fig. 3. Schematic of (a) disk illumination and (b) annular illumination types depicting the various numerical apertures. The radii of the circles are proportional to their corresponding numerical apertures. The dotted circles represent the objective, whereas the solid circles represent the condenser apertures. The yellow region depicts the illumination region. The symbols ${\rho _{\rm{p}}}$, ${\rho _{\rm{s}}}$, and ${\rho _{{\rm{si}}}}$ represent the spatial frequencies corresponding, respectively, to the numerical apertures of the objective (pupil), outer radius of the annulus (source), and the inner radius of the annulus (source).
A custom-fabricated phase mask was used as a test object in the 2D QPI experiments. The mask is a standard-type resolution test chart etched in a silica substrate using electron beam lithography. The etch depths of the patterns on the mask were measured using atomic force microscopy. The ridges (bars) on the test chart patterns are elevated above the background. As a result, when the mask is illuminated, the light transmitted through the ridges has a phase delay relative to the background. The mask used in this work, named G02, had an etch depth of 179 nm and a refractive index of 1.4601 at a 546 nm wavelength, corresponding to a phase delay of 0.948 rad.
Fig. 4. (a) Ideal image used as a standard for comparison; Simulated phase images for (b) disk illumination and (c) annular illumination. The colorbar shows a phase variation scale ranging from ${-}{0.05}\;{\rm{rad}}$ to 0.125 rad.
4. SIMULATION RESULTS
Simulations were performed to compare the two illumination types and to determine if their performance as a function of the spatial frequency followed the trends predicted by the theory. Figure 4(a) shows the reference phase image for the ideal case used in the simulations. It contains a set of patterns the same as in the G02 mask with a uniformly zero-valued background. However, the ridges have a phase delay value of 0.1 rad instead of 0.948 rad, as in the G02 mask, to make a closer approximation to the weak object assumption used in the MFPI-PC theory [72,75]. Figures 4(b) and 4(c) illustrate the simulated phase images for the two illumination types, using the WLS-MFPI-PC method, with the same configuration parameters as the experimental configuration previously described. An SNR of 56 dB is used to approximate the noise in the experimental conditions.
The spatial-frequency response of the images was used for comparison of the phase recovery performance and was calculated using the 2D fast Fourier transform (FFT) of the images with a zero frequency component at the center. The 2D FFT for the ideal case, disk illumination case, and the annular illumination case will be referred to, respectively, as ${F_{{\rm{Ideal}}}}$, ${F_{{\rm{Disk}}}}$, and ${F_{{\rm{Annular}}}}$. The deviation of the frequency response of the simulated images from the frequency response of the ideal image is given by
where ${\delta _{{\rm{Disk}}}}$ and ${\delta _{{\rm{Annular}}}}$ refer to the 2D deviation from the ideal case. These quantities are also used to calculate the deviation for the experimental images in the following section. Figure 5 shows the ${\delta _{{\rm{Disk}}}}$ and ${\delta _{{\rm{Annular}}}}$ curves normalized with respect to ${F_{{\rm{Ideal}}}}$. All the quantities are inherently 2D. However, the plots were radially averaged, in a way similar to the previously mentioned WD-PCTF plots, to display a 1D magnitude variation as a function of the spatial frequency. In the low-frequency region, both the disk illumination (dotted) and the annular illumination (solid) curves have almost equal values of deviation, representing a similar performance in this region, as depicted in the low-frequency region in Fig. 1. In the mid-frequency region, the deviation in both annular and disk illumination is low, similar to the low-frequency region. In the high-frequency region, the deviation in the disk case is very high compared to that in the annular case, implying a strong advantage for annular illumination in the higher spatial frequencies. It should be noted that the deviation in frequency response cannot be quantitatively compared to the changes in the WD-PCTF magnitude, as there is no analytical relationship between the two quantities.Fig. 5. Normalized simulated deviations radially averaged over 2D images for simulated phase images using disk and annular illumination types.
The normalized difference of deviation ($\Delta \delta$) for simulated images is given by
Fig. 6. Normalized difference of deviation $\Delta \delta$ for simulated images and the plot of ${\log}({T_{{\rm{WA}}}}/{T_{{\rm{WD}}}})$.
Fig. 7. Experimentally reconstructed phase images for (a) disk illumination and (b) annular illumination cases. The red highlighted region shows a unique set of patterns. The patterns are magnified to show that the smallest patterns were resolved clearly. The colorbar shows the phase variation scale ranging from ${-}{0.622}\;{\rm{rad}}$ to 1.265 rad.
Figure 6 shows the $\Delta \delta$ (solid) curve radially averaged over 2D data and the log of the ratio ${T_{{\rm{WA}}}}/{T_{{\rm{WD}}}}$ (dotted). The three performance regions depicted in Fig. 1 also occur in Fig. 6. Although the mid-frequency region in Fig. 5 does not show any noticeable difference in the two illumination types, the difference becomes clear in Fig. 6 when the ratio of the two normalized deviations is depicted. Furthermore, the solid curve demonstrates that the advantage of annular illumination in the higher frequencies outweighs its disadvantage in the mid-frequency region. This is because the positive peak of $\Delta \delta$ has a much higher absolute value than its negative minimum. The cutoff frequency in both Figs. 5 and 6 is same as that in the experimental cases, and is dependent on the regularization threshold used in processing [75]. Increasing the width of the annulus results in a gradual change in the QPI performance from that of annular to disk illumination (see Appendix A).
5. EXPERIMENTAL RESULTS
A stack of defocused intensity patterns was recorded each for the disk and the annular illumination cases, and was used to reconstruct the phase images for the two illumination types. The patterns on the G02 mask were transparent when the mask was in crisp focus, and thus satisfied the weak scattering condition often observed in unstained biological samples. Figure 7 depicts the phase images reconstructed using the WLS-MFPI-PC method [75] for the disk and the annular illumination cases. The reconstructed phase images show the test chart patterns in high contrast. The magnified regions on the phase images contain a unique set of patterns on the mask, and further show the smallest patterns on the mask. The annular illumination phase image shows a higher contrast between the ridges and the background.
A. Phase Measurement
The phase variation over the row highlighted by the yellow line in the phase image for the disk and annular cases was calculated. The phase variation exhibits sharp edges and a low-frequency component in the background for both illumination types. Figure 8 shows the phase variation over the highlighted row for the annular illumination case. The low-frequency component affects the background and the ridges equally, which makes the top and bottom edges of the ridges parallel in the line profile. To calculate the phase delay in the ridges, an approximate linear fitting was used to eliminate the effect of the background variation. Since all the ridges on the mask were of equal height, the phase delay of the light through all the ridges was expected to be equal. In the disk illumination phase image, the average phase delay was measured as 0.969 rad with a standard deviation of 0.042 rad; in the annular illumination phase image, the average phase delay was 0.931 rad with a standard deviation of 0.062 rad. The measured phase values for both the illumination cases are within one standard deviation from the ideal value (0.948 rad). Thus, the WLS-MFPI-PC method is highly reliable for phase recovery and holds promise for ubiquitous applications in phase imaging.
Fig. 8. Line profile over the highlighted row (yellow in Fig. 7) in annular illumination phase image of the G02 mask. The red dashed lines depict the linear fitting used to determine the average phase delay (height in $y$ axis shown by yellow long-dashed lines) for each elevated region.
Fig. 9. (a) Ideal image used as a standard for comparison. Experimental phase images for (b) disk illumination and (c) annular illumination. This region is the intermediate inset shown in Fig. 7 and is used for the frequency response comparison. The colorbar shows a phase variation scale ranging from ${-}{0.622}\;{\rm{rad}}$ to 1.265 rad.
B. Spatial Resolution Estimation
The smallest bars on the G02 phase mask had a center-to-center distance of $1\;\unicode{x00B5}{\rm m}$. Figure 7 shows magnified images of the smallest bar patterns on the mask for disk illumination (left) and annular illumination (right). In both cases, the smallest bar patterns are resolved clearly and thus do not represent the smallest possible resolvable distance. However, the higher contrast in the annular case resolves the smallest bar patterns better than the disk case, indicating a higher resolution. By comparing these test chart images to phase test chart images published in [58], it is estimated that, for disk illumination, the just-resolved bar pattern would occur at about nine standard element steps further down on the test chart. For annular illumination, the just-resolved bar pattern would occur at about 11 standard element steps further down. Thus, the estimated resolution for disk illumination is 397 nm, and for annular illumination it is 315 nm. The Rayleigh criterion for the resolution limit of a microscope system is given by $0.61\lambda /({{\rm NA}_{\rm{c}}} + {{\rm NA}_{\rm{o}}})$. For a wavelength of 546 nm, ${{\rm NA}_{\rm{c}}}$ of 0.331, and ${{\rm NA}_{\rm{o}}}$ of 0.75, the theoretical resolution limit is 308 nm. Therefore, the estimated resolution for the annular case is closer to the theoretical limit compared to that of disk illumination, validating the advantage of using annular illumination for higher spatial resolution.
C. Systematic Evaluation
The present work includes a detailed comparison of the performance of disk and annular illumination as a function of spatial frequency, using the WLS-MFPI-PC method. To make such a comparison, the spatial-frequency responses of the experimental phase images from both illumination types were compared to the spatial frequency of an ideal image. Furthermore, the ideal response was then compared to the WD-PCTF variation over the same range of spatial frequencies to validate the analytical model and predictions presented in [75].
Figure 9(a) shows the ideal image used as a standard to compare the experimental performance of disk and annular illumination as a function of spatial frequency. The background in the ideal image is uniformly zero, whereas the bars have a value of 0.948 corresponding to the ideal phase delay of 0.948 rad. Figures 9(b) and 9(c) show the experimentally reconstructed phase images used for analysis, respectively, of the disk and annular illumination cases (same as the magnified area in Fig. 7). All three images have equal dimensions of $742 \times 742$ pixels with a pixel size of 92.2 nm.
Figure 10 shows the deviations for disk and annular illumination cases normalized with respect to the spatial-frequency response of the ideal case, ${F_{{\rm{Ideal}}}}$, and radially averaged. Here, ${F_{{\rm{Ideal}}}}$, ${F_{{\rm{Disk}}}}$, and ${F_{{\rm{Annular}}}}$ correspond to the spatial-frequency responses of the ideal and experimental images shown in Fig. 9. The two experimental curves in Fig. 10 clearly depict the three distinct regions from the analytically calculated WD-PCTF curves in Fig. 1. Furthermore, the boundaries separating the three regions also match the WD-PCTF curves from Fig. 1. In the low-frequency region ($0 {-} 0.78\,\,\unicode{x00B5}{{\rm{m}}^{- 1}}$), the solid and the dotted curve have almost equal value, implying no advantage to using one illumination type over the other. Near the zero frequency ($0 {-} 0.15\,\,\unicode{x00B5}{{\rm{m}}^{- 1}}$), the value of the transfer function is very low, which results in an unexpected peak due to the effect of noise. In the mid-frequency region ($0.78 {-} 1.57\,\,\unicode{x00B5}{{\rm{m}}^{- 1}}$), the deviation in annular case is higher compared to the disk case, which implies that there is an advantage to using disk illumination over annular type. However, in the high-frequency region ($1.57{-}1.86\,\,\unicode{x00B5} {{\rm{m}}^{- 1}}$), the deviation from ideal is very large for disk illumination compared to the deviation for the annular case, which is closer to a zero value. This shows a strong advantage of annular illumination over the disk type in the high-frequency region. A low value of normalized deviation illustrates that the response for annular case is closer to the ideal response compared to the disk case in the higher-frequency region. Since the recovery of the fine details of a sample in the phase image is dependent on the higher spatial-frequency components, the advantage of using annular illumination is significant.
Fig. 10. Normalized experimental deviations radially averaged over 2D data for experimental phase images using disk and annular illumination types. The cutoff frequency is the spatial-frequency value at which the transfer function magnitude is lower than the regularization threshold. The dotted curve represents the disk illumination case, whereas the solid curve represents the annular illumination case.
Fig. 11. Normalized difference of deviation and the ratio of annular WD-PCTF to disk WD-PCTF as a function of spatial frequency, radially averaged from the 2D data.
The $\Delta \delta$ as a function of the spatial frequency is presented in Fig. 11, along with a plot of the difference of the annular WD-PCTF to the disk WD-PCTF (from Fig. 1). Both the curves were plotted by radially averaging the 2D data. The experimentally obtained blue curve representing the normalized difference of deviation clearly highlights the three regions depicted by the theoretically calculated black curve. Furthermore, the experimental curve validates the theory by showing that the advantage of annular illumination in the higher spatial frequency compared to disk illumination is much higher than its disadvantage in the mid-frequency regions. The benefit of annular illumination is fully realized in the high-frequency regions, and the penalty experimentally observed in the mid-frequency regions is smaller than expected from the theory.
6. SUMMARY AND DISCUSSION
In the present work, the performance of disk and annular illumination types in QPI was analyzed and the analytical model and predictions presented in [75] were experimentally validated. This was done by performing 2D QPI experiments on a custom-fabricated, standard-type phase mask with disk and annular illumination using the WLS-MFPI-PC imaging technique. The spatial-frequency response of the reconstructed phase images for disk and annular illumination cases was analyzed and compared to the analytical predictions from [75]. The newly developed state-of-the-art QPI system was developed using conventional components without extensive modifications to a standard bright-field microscope or the need for custom elements. Thus, a standard microscope configuration in any lab can be used as a QPI system without making major modifications. A phase-contrast condenser was used to generate annular illumination.
The following points were concluded from the analysis of the phase images:
- 1. The phase images showed the patterns on the transparent phase mask in high contrast.
- 2. Annular illumination provided higher contrast in phase images, corresponding to a higher spatial resolution.
- 3. The phase values recovered using both the illumination types were close to the ideal values, and were accurate within one standard deviation.
- 4. Spatial resolution limit was estimated for both illumination types. The resolution limit for the annular illumination case was estimated to be close to the theoretical microscope resolution limit.
- 5. Spatial frequency responses in the experimental phase images matched the trend predicted by the WD-PCTF curves shown in Fig. 1 and in [75]. In the low-frequency region, both the illumination types performed similarly. In the mid-frequency region, the disk illumination had a slight advantage. In the high-frequency region, the annular illumination had a significant advantage.
- 6. Annular illumination was shown to be advantageous in achieving higher spatial resolution and recovery of higher spatial-frequency components.
In this work, the variation of the WD-PCTF as a function of spatial frequency was used to compare the performance of disk and annular illumination. In the previous version of this method [73,74], a single WD-PCTF was inverted to estimate the phase for all the SGDF orders. However, it was determined that the WD-PCTF can have a value close to zero in the mid-frequency region, resulting in an unreliable phase recovery. In WLS-MFPI-PC, the WD-PCTF was replaced with PCTFs corresponding to individual SGDF orders, which often have a higher absolute value when the WD-PCTF is close to zero, and thus lead to a better phase recovery performance. Nevertheless, the use of WD-PCTF in this work is valid because, for the combination of objective and condenser numerical apertures used in this work, the value of the WD-PCTF is sufficiently high in the mid-frequency region, making it a reliable standard to compare the performance of phase recovery using disk and annular illumination.
Since the analysis presented in [75] gives correct results, that analysis can be used to predict the improvement in the high spatial-frequency response for a given combination of phase contrast condenser and objective lens and thus determine if the combination would be sufficient for a particular application before using it. Furthermore, for a given objective lens, that analysis could be used to optimize the annulus dimensions for 1) the highest spatial resolution, 2) the best trade-off between high-frequency gain and mid-frequency loss, or 3) other requirements according to the specific application. In conclusion, annular illumination presents a promising, low-cost approach for high spatial-frequency imaging and the analysis of [75] can be used to accurately predict the experimental performance.
APPENDIX A: EFFECT OF ANNULUS WIDTH
The width of the annulus in annular illumination affects the imaging performance in QPI. Figure 12 shows the calculated POTFs for various annular widths. The outer numerical aperture of the condenser is kept constant at the above value of 0.331, whereas the inner numerical aperture is varied from 0 (disk illumination), 0.05, 0.10, 0.15, 0.20, and 0.293. As the inner numerical aperture of the annulus is reduced (i.e., as the thickness of the annulus is increased), the imaging performance for the annular case is expected to approach that of the disk case.
Fig. 12. POTFs corresponding to NAc = 0.331 and NAci = 0 (black-dotted), 0.05 (brown), 0.10 (green), 0.15 (yellow), 0.20 (purple), and 0.293 (blue-dashed).
Simulations were performed using the WLS-MFPI-PC method for the above-mentioned NAci values to analyze the variation in the imaging performance as a function of the annulus width. The results of the simulations match the theoretical predictions from Fig. 12 and are discussed below. Figure 13 shows the normalized deviations for the disk illumination case with NAc = 0.331 and NAci = 0, and the annular illumination case with NAc = 0.331, and NAci = 0.05, 0.10, 0.15, 0.20, and 0.293. As the width of the annulus is increased, the normalized deviation in the high-frequency region approaches the values for the disk case, matching the predictions from the POTF plots.
Fig. 13. Normalized deviations for disk and annular illumination cases with NAc = 0.331 and NAci = 0 (black-dotted), 0.05 (brown), 0.10 (green), 0.15 (yellow), 0.20 (purple), and 0.293 (blue-dashed).
Figure 14 shows the plot for the normalized difference of deviation for various NAci values. This parameter provides a direct comparison between the performance of disk illumination and annular illumination. As the width of the annulus is gradually increased, the normalized difference of the deviation approaches zero, which means that the disk and annular cases perform similarly, matching the anticipated performance (see Dataset 1, Ref. [76]).
Funding
National Science Foundation; Directorate for Engineering (DGE-1148903, ECCS-1915971).
Acknowledgment
The authors acknowledge Devin Brown for fabricating the phase mask used in this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1, Ref. [76].
REFERENCES
1. J. W. Lichtman and J.-A. Conchello, “Fluorescence microscopy,” Nat. Methods 2, 910–919 (2005). [CrossRef]
2. G. Lukinaviius, G. Y. Mitronova, S. Schnorrenberg, A. N. Butkevich, H. Barthel, V. N. Belov, and S. W. Hell, “Fluorescent dyes and probes for super-resolution microscopy of microtubules and tracheoles in living cells and tissues,” Chem. Sci. 9, 3324–3334 (2018). [CrossRef]
3. D. Hara, Y. Okamoto, S. Fujiyama, Y. Nishikawa, Y. Norimine, Y. Kazuta, T. Motoki, H. Amino, and S. Iwanaga, “An orange-fluorescent dye with long-lasting blinking for single molecule localization microscopy,” Proc. SPIE 11256, 112560G (2020). [CrossRef]
4. R. Dixit and R. Cyr, “Cell damage and reactive oxygen species production induced by fluorescence microscopy: effect on mitosis and guidelines for non-invasive fluorescence microscopy,” Plant J. 36, 280–290 (2003). [CrossRef]
5. J. W. Dobrucki, D. Feret, and A. Noatynska, “Scattering of exciting light by live cells in fluorescence confocal imaging: phototoxic effects and relevance for FRAP studies,” Biophys. J. 93, 1778–1786 (2007). [CrossRef]
6. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Phys. Scr. 9, 686–698 (1942).
7. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects part II,” Physica 9, 974–986 (1942). [CrossRef]
8. G. Nomarski and A. R. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Met. Paris 52, 121–134 (1955). [CrossRef]
9. M. Pluta, “Nomarski’s DIC microscopy: a review,” Proc. SPIE 1846, 10–25 (1994). [CrossRef]
10. T. Cacace, V. Bianco, and P. Ferraro, “Quantitative phase imaging trends in biomedical applications,” Opt. Laser Eng. 135, 106188 (2020). [CrossRef]
11. B. Kemper, M. Gotte, B. Greve, and S. Ketelhut, “Prospects and challenges of quantitative phase imaging in tumor cell biology,” Proc. SPIE 9718, 97180N (2016). [CrossRef]
12. C. Allier, L. Herve, O. Mandula, P. Blandin, Y. Usson, J. Savatier, S. Monneret, and S. Morales, “Quantitative phase imaging of adherent mammalian cells: a comparative study,” Biomed. Opt. Express 10, 2768–2783 (2019). [CrossRef]
13. S. Aknoun, J. Savatier, P. Bon, F. Galland, L. Abdeladim, B. Wattellier, and S. Monneret, “Living cell dry mass measurement using quantitative phase imaging with quadriwave lateral shearing interferometry: an accuracy and sensitivity discussion,” J. Biomed. Opt 20, 126009 (2015). [CrossRef]
14. A. T. Y. Mok, K. C. M. Lee, K. K. Y. Wong, and K. K. Tsia, “Label-free cell-cycle analysis by high-throughput quantitative phase time-stretch imaging flow cytometry,” Proc. SPIE 10505, 105050J (2018). [CrossRef]
15. F. Hideki and A. Yoshihisa, “Three-dimensional quantitative phase imaging of blood coagulation structures by optical projection tomography in flow cytometry using digital holographic microscopy,” J. Biomed. Opt. 24, 031012 (2018). [CrossRef]
16. P. Wang, R. Bista, R. Bhargava, R. E. Brand, and Y. Liu, “Spatial-domain low-coherence quantitative phase microscopy for cancer diagnosis,” Proc. SPIE 7889, 78891Q (2011). [CrossRef]
17. Z. El-Schich, A. L. Mölder, and A. G. Wingren, “Quantitative phase imaging for label-free analysis of cancer cells—Focus on digital holographic microscopy,” Appl. Sci. 8, 1027 (2018). [CrossRef]
18. G. M. Noah, Y. Bao, and T. K. Gaylord, “Characterization of fiber Bragg gratings using a 3D quantitative phase imaging approach,” in OSA Frontiers (2020), paper JW3A.63.
19. Y. Li, J. Di, W. Wu, P. Shang, and J. Zhao, “Quantitative investigation on morphology and intracellular transport dynamics of migrating cells,” Appl. Opt. 58, G162–G168 (2019). [CrossRef]
20. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005). [CrossRef]
21. J. Y. Park, H. Desta, M. C. Maloney, A. Sharikova, and A. Khmaladze, “Digital holographic phase imaging of particles embedded in microscopic structures in three dimensions,” Proc. SPIE 9870, 98700B (2016). [CrossRef]
22. A. Ghosh, J. Noble, A. Sebastian, S. Das, and Z. Liu, “Digital holography for non-invasive quantitative imaging of two-dimensional materials,” J. Appl. Phys. 127, 084901 (2020). [CrossRef]
23. R. Zhou, C. Edwards, G. Popescu, and L. Goddard, “Semiconductor defect metrology using laser-based quantitative phase imaging,” Proc. SPIE 9336, 93361I (2015). [CrossRef]
24. W. Qu, Y. Wen, Z. Wang, F. Yang, L. Huang, and C. Zuo, “Quantitative phase measurement for wafer-level optics,” Proc. SPIE 9524, 95242J (2015). [CrossRef]
25. M. R. McCartney, N. Agarwal, S. Chung, D. A. Cullen, M.-G. Han, K. He, L. Li, H. Wang, L. Zhou, and D. J. Smith, “Quantitative phase imaging of nanoscale electrostatic and magnetic fields using off-axis electron holography,” Ultramicroscopy 110, 375–382 (2010). [CrossRef]
26. Y. Bao and T. K. Gaylord, “Quantitative phase imaging of fiber Bragg gratings in multicore fibers,” Appl. Opt. 57, 10062–10071 (2018). [CrossRef]
27. G. M. Noah, Y. Bao, and T. K. Gaylord, “Cross-sectional refractive-index variations in fiber Bragg gratings measured by quantitative phase imaging,” Opt. Lett. 45, 53–56 (2020). [CrossRef]
28. G. B. Airy, “I. On the diffraction of an annular aperture,” London Edinburgh Dub. Philos. Mag. J. Sci. 18(114), 1–10 (1841). [CrossRef]
29. M. Gu, T. Tannous, and J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1996). [CrossRef]
30. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000). [CrossRef]
31. S. Lindek, C. Cremer, and E. H. K. Stelzer, “Confocal theta fluorescence microscopy with annular apertures,” Appl. Opt. 35, 126–130 (1996). [CrossRef]
32. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef]
33. C. J. R. Sheppard, “Partially coherent microscope imaging system in phase space: effect of defocus and phase reconstruction,” J. Opt. Soc. Am. A 35, 1846–1854 (2018). [CrossRef]
34. S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A 10, 125103 (2008). [CrossRef]
35. W. Gong, K. Si, N. Chen, and C. J. R. Sheppard, “Focal modulation microscopy with annular apertures: a numerical study,” J. Biophoton. 3, 476–484 (2010). [CrossRef]
36. G. Antonacci, “Dark-field Brillouin microscopy,” Opt. Lett. 42, 1432–1435 (2017). [CrossRef]
37. G. Terakado and H. Kano, “Localized surface plasmon microscope for simultaneous imaging of refractive index and fluorescence distributions: Fluorescence enhancement by annular pupil illumination,” in Biomedical Optics and 3-D Imaging, OSA Technical Digest (CD) (Optica Publishing Group, 2010), paper BSuD62.
38. G. Terakado, J. Ning, K. Watanabe, and H. Kano, “High-resolution simultaneous microscopy of refractive index and fluorescent intensity distributions by using localized surface plasmons,” Appl. Opt. 52, 3324–3328 (2013). [CrossRef]
39. J. Borglin, D. N. Wang, N. J. Durr, D. Hanstorp, A. Ben-Yakar, and M. B. Ericson, “Annular beam shaping in multiphoton microscopy to reduce out-of-focus background,” Int. J. Spectrosc. 2017, 7560141 (2017). [CrossRef]
40. A. Doi, R. Oketani, Y. Nawa, and K. Fujita, “High-resolution imaging in two-photon excitation microscopy using in situ estimations of the point spread function,” Biomed. Opt. Express 9, 202–213 (2018). [CrossRef]
41. B. Wang, J. Shi, T. Zhang, X. Xu, Y. Cao, and X. Li, “Improved lateral resolution with an annular vortex depletion beam in STED microscopy,” Opt. Lett. 42, 4885–4888 (2017). [CrossRef]
42. Y. Zhou and X. Li, “Super-resolution by annular pupil inserted scattered light fluorescence microscopes,” Opt. Rev. 24, 398–405 (2017). [CrossRef]
43. J. Hu and Y. Shen, “Annular oblique epi-illumination design for ultra-large-scale and high-resolution fluorescence microscopy,” Proc. SPIE 10815, 108150M (2018). [CrossRef]
44. C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7, 7654 (2017). [CrossRef]
45. S. Huang, F. Xi, C. Liu, and Z. Jiang, “Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation,” J. Opt. Soc. Am. A 29, 513–520 (2012). [CrossRef]
46. J. Li, Q. Chen, J. Sun, J. Zhang, X. Pan, and C. Zuo, “Optimal illumination pattern for transport-of-intensity quantitative phase microscopy,” Opt. Express 26, 27599–27614 (2018). [CrossRef]
47. Y. Fan, J. Sun, Q. Chen, J. Zhang, and C. Zuo, “Wide-field anti-aliased quantitative differential phase contrast microscopy,” Opt. Express 26, 25129–25146 (2018). [CrossRef]
48. Y. Fan, J. Sun, Q. Chen, X. Pan, L. Tian, and C. Zuo, “Optimal illumination scheme for isotropic quantitative differential phase contrast microscopy,” Photon. Res. 7, 890–904 (2019). [CrossRef]
49. Y. Fan, J. Sun, Q. Chen, X. Pan, M. Trusiak, and C. Zuo, “Single-shot isotropic quantitative phase microscopy based on color-multiplexed differential phase contrast,” APL Photon. 4, 121301 (2019). [CrossRef]
50. Y. Fan, J. Sun, Q. Chen, X. Pan, and C. Zuo, “Video-rate isotropic quantitative differential phase contrast microscopy based on color-multiplexed annular illumination,” Proc. SPIE 11186, 111860F (2019). [CrossRef]
51. A. Pan, C. Shen, B. Yao, and C. Yang, “Single-shot Fourier ptychographic microscopy via annular monochrome LED array,” in Frontiers in Optics + Laser Science APS/DLS, OSA Technical Digest (Optica Publishing Group, 2019), paper FTh3F.4.
52. J. Sun, C. Zuo, and Q. Chen, “Optimal annular illumination pattern for Fourier ptychographic microscopy based on spectrum aliasing minimization,” Proc. SPIE 11549, 115490C (2020). [CrossRef]
53. J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8, 7669 (2018). [CrossRef]
54. J. S. Sun, C. Zuo, and Q. Chen, “Video-rate quantitative phase microscopy based on Fourier ptychography with annular illuminations,” Proc. SPIE 10827, 108270M (2018). [CrossRef]
55. W. You, W. Lu, and X. Liu, “Single-shot wavelength-selective quantitative phase microscopy by partial aperture imaging and polarization-phase-division multiplexing,” Opt. Express 28, 34825–34834 (2020). [CrossRef]
56. J. Li, Q. Chen, J. Zhang, Y. Zhang, L. Lu, and C. Zuo, “Efficient quantitative phase microscopy using programmable annular LED illumination,” Biomed. Opt. Express 8, 4687–4705 (2017). [CrossRef]
57. J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Proc. SPIE 11186, 111860Y (2019). [CrossRef]
58. J. Li, A. Matlock, Y. Li, Q. Chen, L. Tian, and C. Zuo, “Resolution-enhanced intensity diffraction tomography in high numerical aperture label-free microscopy,” Photon. Res. 8, 1818–1826 (2020). [CrossRef]
59. J. Li, S. Jiasong, Z. Jialin, C. Qian, and Z. Chao, “Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations,” Proc. SPIE 10887, 108870O (2019). [CrossRef]
60. K. Kamon, T. Miyamoto, Y. Myoi, H. Nagata, M. Tanaka, and K. Horie, “Photolithography system using annular illumination,” Jpn. J. Appl. Phys. 30, 3021–3029 (1991). [CrossRef]
61. K. Tounai, H. Tanabe, H. Nozue, and K. Kasama, “Resolution improvement with annular illumination,” Proc. SPIE 1674, 753–764 (1992). [CrossRef]
62. W. N. Partlo, P. J. Tompkins, P. G. Dewa, and P. F. Michaloski, “Depth of focus and resolution enhancement of I-line and deep-UV lithography using annular illumination,” Proc. SPIE 1927, 137–157 (1993). [CrossRef]
63. C. Anderson, A. Allezy, W. L. Chao, C. Cork, W. Cork, R. Delano, J. DePonte, M. Dickinson, G. Gaines, J. Gamsby, E. Gullikson, G. Jones, S. Meyers, R. Miyakawa, P. Naulleau, S. Rekawa, F. Salmassi, B. Vollmer, D. Zehm, and W. H. Zhu, “Overview and status of the 0.5-NA micro-field exposure tool at Berkeley Lab,” Proc. SPIE 10957, 1095708 (2019). [CrossRef]
64. M. Enyama, K. Hamada, M. Fukuda, and H. Kazumi, “Method of improving image sharpness for annular-illumination scanning electron microscopes,” Jpn. J. Appl. Phys. 55, 06GD02 (2016). [CrossRef]
65. T. Miyoshi, S. Takahashi, Y. Takaya, and S. Shimada, “High sensitivity optical detection of oriented microdefects on silicon wafer surfaces using annular illumination,” CIRP Ann. Manuf. Technol. 50, 389–392 (2001). [CrossRef]
66. Y. Horikawa, “Resolution of annular-pupil optical systems,” J. Opt. Soc. Am. A 11, 1985–1992 (1994). [CrossRef]
67. E. Rascevska, K. J. Francis, and S. Manohar, “Annular illumination photoacoustic probe for needle guidance in medical interventions,” Proc. SPIE 11077, 110770L (2019). [CrossRef]
68. M. V. Schulmerich, K. A. Dooley, M. D. Morris, T. M. Vanasse, and S. A. Goldstein, “Transcutaneous fiber optic Raman spectroscopy of bone using annular illumination and a circular array of collection fibers,” J. Biomed. Opt. 11, 060502 (2006). [CrossRef]
69. Y. N. Sulai and A. Dubra, “Adaptive optics scanning ophthalmoscopy with annular pupils,” Biomed. Opt. Express 3, 1647–1661 (2012). [CrossRef]
70. B. Vohnsen and D. Rativa, “Ultrasmall spot size scanning laser ophthalmoscopy,” Biomed. Opt. Express 2, 1597–1609 (2011). [CrossRef]
71. K.-Y. Hsu, Y.-C. Tung, M.-H. Chung, and C.-K. Lee, “Design and fabrication of sub-wavelength annular apertures for femtosecond laser machining,” Proc. SPIE 9351, 93510U (2015). [CrossRef]
72. M. H. Jenkins, J. M. Long, and T. K. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, D29–D39 (2014). [CrossRef]
73. Y. Bao and T. K. Gaylord, “Quantitative phase imaging method based on an analytical nonparaxial partially coherent phase optical transfer function,” J. Opt. Soc. Am. A 33, 2125–2136 (2016). [CrossRef]
74. Y. Bao and T. K. Gaylord, “Quantitative phase imaging method based on an analytical nonparaxial partially coherent phase optical transfer function: erratum,” J. Opt. Soc. Am. A 34, 1329 (2017). [CrossRef]
75. Y. Bao, G. C. Dong, and T. K. Gaylord, “Weighted-least squares multi-filter phase imaging with partially coherent light: characteristics of annular illumination,” Appl. Opt. 58, 137–146 (2019). [CrossRef]
76. P. Kulkarni, Y. Bao, and T. K. Gaylord, “Dataset for Annular illiumination in 2D quantitative phase imaging: systematic evaluation,” figshare, 2022, https://doi.org/10.6084/m9.figshare.17710280
