## Abstract

We present a method to improve the isotropy of spatial resolution in a structured illumination microscopy (SIM) implemented for imaging non-fluorescent samples. To alleviate the problem of anisotropic resolution involved with the previous scheme of coherent SIM that employs the two orthogonal standing-wave illumination, referred to as the orthogonal SIM, we introduce a hexagonal-lattice illumination that incorporates three standing-wave fields simultaneously superimposed at the orientations equally divided in the lateral plane. A theoretical formulation is worked out rigorously for the coherent image formation with such a simultaneous multiple-beam illumination and an explicit Fourier-domain framework is derived for reconstructing an image with enhanced resolution. Using a computer-synthesized resolution target as a 2D coherent sample, we perform numerical simulations to examine the imaging characteristics of our three-angle SIM compared with the orthogonal SIM. The investigation on the 2D resolving power with the various test patterns of different periods and orientations reveal that the orientation-dependent undulation of lateral resolution can be reduced from 27% to 8% by using the three-angle SIM while the best resolution (0.54 times the resolution limit of conventional coherent imaging) in the directions of structured illumination is slightly deteriorated by 4.6% from that of the orthogonal SIM.

© 2014 Optical Society of America

## 1. Introduction

Optical microscopy has been widely used in the biological sciences for its ability to visualize the fine interiors of specimens and biomolecules [1]. Diffraction of light, however, has long been imposing a critical limitation on the spatial resolution attainable with conventional optical microscopy, which often precludes its use in the observation of many biologically relevant structures at scales beyond the diffraction limit [2]. This has prompted challenging attempts to surpass the optical diffraction limit, leading eventually to a number of super-resolution techniques, such as stimulated emission depletion (STED) microscopy [3,4], photoactivated localization microscopy (PALM) [5], stochastic optical reconstruction microscopy (STORM) [6], structured illumination microscopy (SIM) [7–11], etc. Unprecedented far-field resolutions down to tens of nanometers, have been successfully demonstrated while these approaches might be flawed in that the specimens neither autofluorescent nor fluorescently tagged can no longer be imaged. It would be thus desirable to have an alternative solution to achieve super-resolution that is also applicable to biologically relevant samples for which proper fluorescence tagging may be difficult or an exogenous labeling agent should not perturb their biochemical interactions and viability.

Among the aforementioned techniques, the SIM is unique in its principle to provide a way to improve resolution without relying on fluorophores. The basic concept of SIM is to illuminate a sample with dense stripes of a sinusoidal light pattern to encode the sample’s undetectable high-frequency information into the observable passband (the diffraction-limited frequency support of the system). The high-frequency contents can be extracted from a series of images with different phases of the illumination pattern, which then allows, after appropriate back-shifting of the high-frequency contents in frequency space, reconstruction of an image with resolution improved in a direction perpendicular to the illumination pattern. With a linear sample, the maximum enhancement in resolution is given by a factor of two [7–9]. Even greater gains, approximately up to a factor of five, have been demonstrated to be attainable by exploiting the nonlinearity of a sample such as saturation [10,11]. In order to achieve quasi-isotropic resolution in the lateral direction, it is necessary to rotate the illumination pattern to several orientations in the acquisition of raw SIM image data. Most of the SIM, however, have thus far been implemented for imaging incoherently scattering objects based on fluorescence [7–15].

Although the basic idea of coherent SIM has been explored previously [16–19], it was not until recently that a substantial extension to the conventional SIM was made by Chowdhury *et al*. [20], rendering non-fluorescent, coherently scattering objects better resolvable in two dimensions (2D). A notable difficulty in realizing the 2D coherent analogue to the conventional SIM arises from the fact that the relationship between the sample’s “field” and detected image “intensity” is no longer linear, which hampers the use of a rotating sinusoidal illumination to enlarge the 2D frequency support as in the incoherent SIM [7–9]. It was shown to be rather imperative that a structured illumination field consists of multiple standing-wave components being able to cause the sample’s frequency shifts at two or more orientations “simultaneously.” With the recognition of this hallmark of 2D coherent SIM, the researchers derived a Fourier-domain formulation to model the coherent imaging with structured illumination comprising two orthogonal sets of standing-wave fields, and successfully worked out a robust framework on how to demodulate and process the high-frequency information to reconstruct a coherent image with a 2D-extended passband exceeding the diffraction limit. In the former work using the illumination pattern built with two standing waves at 0° and 90° orientations, however, anisotropic improvement of resolution is anticipated as the extension of the effective detection band is smaller in other directions, particularly for 45° and 135°.

In this paper, we present a modified framework to improve the isotropy of lateral resolution in the 2D coherent SIM. A hexagonal lattice-patterned illumination incorporating three standing-wave fields, instead of two orthogonal components considered in the previous demonstration [20], is employed to fill out the 2D image’s frequency space more isotropically. Seemingly straightforward, the attempt to increase the number of orientations in the structured illumination, would require in fact a more complicated mathematical formulation as well as extra computational burden for image processing. Here, we derive a rigorous formulation modeling the 2D coherent image with such simultaneous multiple-beam illuminations, which results in a Fourier image spectrum containing a number of cross-correlation terms between the multiplexed sample spectra with different 2D frequency shifts. An explicit framework is then established on how to extract and process the mixed high-resolution information for image reconstruction. To investigate the validity of our proposed method, we numerically simulate a coherent SIM imaging with the three-angle structured illumination applied to a computer-generated resolution test target. The lateral-resolution performance is evaluated by examining the reconstructed images of various test patterns on the target, compared with those in the 2-angle structured illumination scheme. The results clearly demonstrate the benefit of our effort to improve isotropy of the 2D resolution in coherent SIM over the existing scheme.

## 2. Principle of coherent SIM with isotropic 2D resolution

#### 2.1 Isotropic extension of the coherent passband by three-angle structured illumination

In the implementation of a 2D coherent SIM, it is essential to generate a structured illumination (SI) with multiple standing-wave fields that are simultaneously superimposed in the sample plane (spanned by$r$) where each field component $\mathrm{cos}({k}_{s(i)}\cdot r+{\varphi}_{s(i)})$ with the lateral wavevector of ${k}_{s(i)}=2\pi {f}_{s(i)}$ causes frequency shift of the sample’s spectrum by $\pm {f}_{s(i)}$. For an imaging system specified by the coherent transfer function (CTF) $C(f)$ with frequency cutoff at $\left|{f}_{c}\right|={f}_{c}$, this is analogous to acquiring high-frequency spectra of the sample through the frequency-shifted passbands$C(f\pm {f}_{s(i)})$, except for their multiplexing into the observable passband. Using an appropriate set of simultaneous standing-wave fields at different orientations, is therefore equivalent to the effective 2D extension of CTF as illustrated in Fig. 1, provided that the multiplexed information can be demodulated and shifted back to its original location.

It is apparent from the SI-extended CTFs shown in Fig. 1 that the 2D isotropy of passband extension can be improved by adding more orientations of standing-wave fields into the SI. For a set of two orthogonal standing-wave components with spatial frequencies of ${f}_{sx}=(\pm {f}_{c},0)$ and ${f}_{sy}=(0,\pm {f}_{c})$, respectively, the SI-extended CTF shown in Fig. 1(a) has twice as high cutoff frequency in the SI directions as that of a diffraction-limited system while its extension is smaller for other directions, particularly 45° and 135°. With the SI consisting of three-angle components as illustrated in Fig. 1(b), on the other hand, such anisotropy can be considerably alleviated by filling out the frequency space more isotropically. We here note that the resultant CTF, however, does not allow a direct assessment for the enhancement of resolution and its isotropy in the three-angle SI scheme because the CTF itself is not a proper transfer function to describe the “intensity” image in a coherent system. Care must be taken when evaluating the actual resolution of a coherent SIM, which instead needs to be done by examining the real-space intensity image of a specific amplitude sample.

Specifically, the quasi-isotropic scheme for coherent SIM we suggest in this study, employs three-angle SI field components simultaneously superimposed as

Regarding the aspect of experimental realization [14], the three-angle SI pattern explained above can be produced readily by using a 2D spatial light modulator (SLM) programmed with a hexagonal grid pattern. By illuminating the SLM with a collimated laser beam, three sets of diffracted beams at three equally-divided orientations are generated, which are then directed to a microscope system. The diffracted beams are reduced in beam size appropriately and recombined to interfere at the sample. In order to obtain DC-free sinusoidal field components as shown in Figs. 2(a)-2(c), a Fourier-plane pupil mask can be used to permit only the $\pm 1$ diffraction orders in the three sets of diffracted beams and reject the 0-th order as well.

In the course of acquiring a number of raw SIM images, the SLM’s pattern needs to be manipulated with a particularly chosen set of different three-angle SI phases$\Psi \equiv ({\varphi}_{s1},{\varphi}_{s2},{\varphi}_{s3})$. In Fig. 2(e), the 19 grid points indicated on the hexagonal mesh of unit-phase space exemplifies the phase vectors $\{{\Psi}_{m}\equiv {({\varphi}_{s1},{\varphi}_{s2},{\varphi}_{s3})}_{m}|m=0,1,\mathrm{...18}\}$ by which the three-angle SI pattern is required to be translated laterally to yield the 19 independent SIM images, allowing a linear algebra to solve for a complete set of image frequency contents involved with the extended-passband coherent imaging as characterized by Eq. (2). The mathematical details, associated with the three-angle SI imaging and its image reconstruction process, are described in the following.

#### 2.2 Mathematical formulation

Based on the coherent imaging theory [20,21], we rigorously derive a framework to reconstruct a 2D super-resolved image in the three-angle coherent SI scheme proposed in this study. The in-focus image intensity $d(r)$ of an amplitude object $s(r)$ illuminated by a coherent field $E(r)$, is described by the nonlinear equation

where ${u}_{i}(r)$ denotes the image amplitude, $s(r)$ is the 2D object’s response (transmittance or reflectance) function to the light amplitude $E(r)$, and ${h}_{c}(r)$ is the coherent point spread function (PSF) determined by the system’s limiting aperture. Here, the symbol $\otimes $ designates the convolution operator and $r=(x,y)$ is the vector specifying the spatial position on the sample/image plane, disregarding the system magnification for simplicity. In the spatial-frequency domain$f=({f}_{x},{f}_{y})$, the image intensity $d(r)$ can be expressed as its equivalent Fourier imageFor the hexagonal-lattice illumination with three-angle SI field components, we now find the corresponding SI Fourier image to be

The three SI components are assumed preferably to have spatial frequencies of the same magnitude $\left|{f}_{s1}\right|=\left|{f}_{s2}\right|=\left|{f}_{s3}\right|={f}_{s}\le {f}_{c}$ with their orientations at$\text{\hspace{0.17em}}\{{\theta}_{1},{\theta}_{2},{\theta}_{3}\}=\{{0}^{\circ},{120}^{\circ},{240}^{\circ}\}$, respectively. However, our formulation will still be valid for any three SI frequencies $\{{f}_{s(i)}=({f}_{s(i)}\mathrm{cos}{\theta}_{i},{f}_{s(i)}\mathrm{sin}{\theta}_{i})|i=1,2,3\}$ provided that $\left|{f}_{s(i)}\right|\le {f}_{c}$ and$\text{\hspace{0.17em}}{\theta}_{i}$’s are distinct. Using the abbreviations that refer to the SI-induced frequency shifts of the sample’s original spectrum as$\text{\hspace{0.17em}}S(f-{f}_{s1})\equiv {S}_{1}^{-}$,$\text{\hspace{0.17em}}S(f+{f}_{s1})\equiv {S}_{1}^{+}$, $\text{\hspace{0.17em}}S(f-{f}_{s2})\equiv {S}_{2}^{-}$, $\text{\hspace{0.17em}}S(f+{f}_{s2})\equiv {S}_{2}^{+}$, $\text{\hspace{0.17em}}S(f-{f}_{s3})\equiv {S}_{3}^{-}$, $\text{\hspace{0.17em}}S(f+{f}_{s3})\equiv {S}_{3}^{+}$, Eq. (6) further expands to

Next, we turn to calculate an enhanced-resolution image equivalent to what would be obtained with the “extended passband” ${C}_{EP}(f)$ in Eq. (2), hypothetically synthesized with the six frequency-shifted CTFs corresponding to the three-angle SI. Replacing the conventional passband $C(f)$ in Eq. (5) with the extended one ${C}_{EP}(f)$ given by Eq. (2), we obtain the Fourier image with extended passband

Based on the mathematical proofs given by the previous study [20], there exists one-to-one correspondence between the SI components $\{{T}_{n}(f)|n=0,1,\mathrm{...18}\}$ in Eq. (7) and the EP components $\{{V}_{n}(f)|n=0,1,\mathrm{...18}\}$worked out in Eq. (8). Each SI term in the measured Fourier image$\text{\hspace{0.17em}}{D}_{SI}(f)$, taking the general form of ${T}_{\gamma}(f)=[C(f)S(f+{f}_{\alpha})]\text{[}C(f)S(f+{f}_{\beta})]$ for any arbitrary$\gamma \in n$, is identical to the EP term ${V}_{\gamma}(f)=[C(f-{f}_{\alpha})S(f)]\text{[}C(f-{f}_{\beta})S(f)]$ comprising the hypothetical Fourier image$\text{\hspace{0.17em}}{D}_{EP}(f)$, except for the location in the frequency space. Explicitly, quite a useful relation ${T}_{\gamma}(f)={V}_{\gamma}(f+[{f}_{\alpha}-{f}_{\beta}])$ holds, for ${f}_{\alpha}$and ${f}_{\beta}$ being the independent frequency-shift vectors$\in \left\{0,+{f}_{s1},-{f}_{s1},+{f}_{s2},-{f}_{s2},+{f}_{s3},-{f}_{s3}\right\}$. Provided that the individual SI terms$\{{T}_{n}(f)|n=0,1,\mathrm{...18}\}$ can be isolated from the raw data of${D}_{SI}(f)$, it is straightforward to reconstruct a super-resolved image spectrum, by combining all these terms linearly after appropriate shifting of each SI term, in such a way that

where the frequency shifts $[{f}_{\alpha}-{f}_{\beta}]{|}_{n}$ necessary for the 19 SI terms in the present framework can be listed in series asSolving for 19 individual SI components $\{{T}_{n}(f)|n=0,1,\mathrm{...18}\}$ to be processed for the image reconstruction, requires a minimum of 19 raw image acquisitions ${{D}_{SI}(f)|}_{{\Psi}_{m}}$with different phases ${\Psi}_{m}\equiv {({\varphi}_{s1},{\varphi}_{s2},{\varphi}_{s3})}_{m}$ of the 3-angle SI pattern translation. From Eq. (7), a system of linear equations can be constructed to describe the spectral mixing of SI components in the Fourier image as ${{D}_{SI}(f)|}_{{\Psi}_{m}}\equiv {M}_{({\Psi}_{m},n)}^{}{T}_{n}(f)$ using a $19\times 19$matrix ${M}_{({\Psi}_{m},n)}^{}$. Given an appropriately chosen set of ${\Psi}_{m}\equiv {({\varphi}_{s1},{\varphi}_{s2},{\varphi}_{s3})}_{m}$ to render ${M}_{({\Psi}_{m},n)}^{}$ nonsingular, we are then able to unmix the multiplexed SI components from the measured raw image data by performing the inverse-matrix calculation for ${T}_{n}(f)\equiv {M}_{({\Psi}_{m},n)}^{-1}{{D}_{SI}(f)|}_{{\Psi}_{m}}$. A possible set of the SI pattern phases$\{{\Psi}_{m}\equiv {({\varphi}_{s1},{\varphi}_{s2},{\varphi}_{s3})}_{m}|m=0,1,\mathrm{...18}\}$ can be exemplified by the zero-shift vector of${\Psi}_{0}\equiv (0,0,0)$ and the 18 other vectors of $({\varphi}_{s(1)},{\varphi}_{s(2)},{\varphi}_{s(3)})$ constituted by choosing the pattern phase for each SI orientation particularly from${\varphi}_{s(i)}\in \left\{{\phi}_{1},{\phi}_{2},{\phi}_{3}\right\}=\left\{0,2\pi /3,4\pi /3\right\}$, ${\varphi}_{s(j)}\in \left\{{\phi}_{2},{\phi}_{3}\right\}=\left\{2\pi /3,4\pi /3\right\}$, ${\varphi}_{s(k)}\in \left\{{\phi}_{1}\right\}=\left\{0\right\}$ in which the indices $(i,j,k)$ to be the permutation of $(1,2,3)$. With such phase vectors, the 3-angle SI pattern is hence translated to the 19 grid points on the hexagonal unit-cell, as delineated in Fig. 2(e).

It is worth noting that the effective passband of the 3-angle SIM we present in this study is not an ideal extension of the conventional CTF, i.e. not only non-circular but also non- uniform in its distribution due to the overlap of 6 individual CTFs as illustrated in Fig. 1(b). This will over-weight the low spatial-frequency regions to deteriorate the resolution improvement eventually. It is certainly desirable to remove such over-weighting in the extended passband for better resolution as well as for reducing image artifacts with low-frequency features. In the incoherent SIM, a computational algorithm using a generalized Wiener filter has been demonstrated [15] to get around this problem in combining the multiple SI spectral components. A similar task for the coherent SIM, however, can be possible but much complicated to implement for the Fourier intensity spectrum of a coherent image is highly nonlinear with the object amplitude distribution, which results in SI spectral components including a number of cross-correlations between the individual over-weighted CTFs that are mixed together in the Fourier domain. Despite the over-weighting unremoved from the passband, it still holds that the coherent SIM is able to improve image resolution by capturing the sample’s high-frequency information that a conventional BF system would fail to access. High-resolution information obtainable with our 3-angle SIM seems more isotropic on the whole than that of the orthogonal SIM whereas the unwanted overlap of individual CTFs is more prevalent in our 3-angle SI scheme. In this study, actual resolution performances of the orthogonal and 3-angle SIM are compared in detail without removing the over-weighting aspect involved with the SI-extended CTFs in the both schemes.

## 3. Numerical simulation

In order to investigate the validity of our theoretical framework for 3-angle SIM and compare its 2D resolution isotropy with that of the orthogonal SIM scheme [20], we carried out numerical simulations to produce a set of raw coherent SIM image data and reconstruct an image with enhanced resolution. An imaging system was assumed to share the same limiting aperture of 0.9 NA for the illumination and detection of a coherent light at the 532 nm wavelength, setting the classical diffraction limit of resolution ${\delta}_{DL}=\lambda /2NA$ at 296 nm.

In the simulation, we used a computer-generated resolution test target shown in Fig. 3(a) for evaluating the imaging performance with various test patterns including groups of bar patterns, isolated single lines at different orientations, and a sector star pattern. The 2D test target was synthesized to represent a coherent sample over the area of 26$\mu \text{m}$$\times $26$\mu \text{m}$ (in 1024 $\times $1024 image pixels), with its black-and-white (B&W) contrast indicating the presence of thin, uniform-amplitude objects only in the white regions. The bar patterns of different cycle periods were grouped together according to their line normal orientations, named Group HV (at ${0}^{\circ}$and ${90}^{\circ}$), Group DG (at $+{45}^{\circ}$and $-{45}^{\circ}$), and Group HX (at $+{60}^{\circ}$and $-{60}^{\circ}$). In each group, several elements were formed with cycle periods ${\Lambda}_{LS}$ of 254 nm, 305 nm, 356 nm, 508 nm, and 762 nm, labeled with numbers as 10, 12, 14, 20, and 30, respectively. Extra bar patterns with cycle periods of 1016 nm and 1525 nm were added to the Group HV. The sector star (also called Siemens star) pattern was built to have 32 bars over 360° with the largest cycle period of 1278 nm (corresponding to 4.3 times the diffraction limit${\delta}_{DL}$) at its circumference.

Using the resolution target as a sample, we first calculated the coherent bright-field (BF) image as shown in Fig. 3(b), which served as a reference to be compared with the reconstructed images from the two coherent SIM methods under consideration. In the coherent BF image, the sector star pattern can be useful to determine the image resolution by noting how close to the center of the pattern one is able to resolve adjacent bars. According to the Rayleigh’s resolution criterion (i.e. the minimum required visibility of 15.2% or, equivalently, the intensity dip of 26.6% between the peaks for resolving any adjacent features), the sector star pattern appears discernible only in the region outside the circumference of the green circle overlaid in Fig. 3(b), on which the radial bars have a cycle period of 497 nm, corresponding to 1.67 times the diffraction limit ${\delta}_{DL}$.

Here, the coherent BF resolution seems to exhibit appreciable discrepancy with the theory which exactly agrees with the resolution of the incoherent BF image shown in Fig. 3(c) on the other hand. We would like to emphasize, however, that the discrepancy does not imply a flaw in our calculation but it is rather attributed to the intriguing characteristic of coherent imaging. It has long been recognized that a coherent system can be poor in image resolution, compared with that of an incoherent system [22]. Since the optical transfer function (OTF) of a diffraction-limited incoherent system extends to a frequency${f}_{d}={\delta}_{DL}^{-1}$ that is twice the “amplitude” cutoff frequency ${f}_{c}=NA/\lambda $ of the CTF of a coherent system, it is tempting to conjecture that coherent imaging will yield twice worse resolution than incoherent imaging. An attempt to compare the resolution of the two imaging systems, however, is far more complex than the transfer functions would suggest simply in terms of cutoff frequency. The difficulty arises from the fact that any direct comparison must be carried out with the same observable quantity, image intensity, while the CTF of a coherent system determines only the image amplitude and no well-defined transfer function exists for the image intensity in coherent imaging. As the coherent image’s intensity spectrum is the auto-correlation of its amplitude spectrum, the resulting intensity image could appear with spatial frequencies that can even exceed the amplitude cutoff frequency${f}_{c}$, extending to the limit of${f}_{d}=2{f}_{c}$. Therefore, the actual resolution ${\delta}_{CBF}$ in coherent BF imaging to discriminate the adjacent coherent features may vary as ${\delta}_{DL}\le {\delta}_{CBF}\le 2{\delta}_{DL}$, depending strongly on the specific nature of both the intensity and phase distribution across the object though [22]. In similar context, a coherent SIM that extends its effective passband beyond the conventional cutoff frequency${f}_{c}$, indeed allows reconstruction of an image with resolution overcoming the diffraction limit ${\delta}_{DL}$ even though the sample’s “amplitude information” itself cannot be captured from beyond Abbe’s diffraction limit in the “conventional sense” pertinent to incoherent imaging [20]. Throughout the paper, we henceforth refer to the improved resolution equivalent to what would be obtained with coherent imaging with an extended passband exceeding the conventional amplitude cutoff at ${f}_{c}$ as “super-resolution.” We evaluate the resolution performance of coherent SIM in a relative manner, comparing with the actual resolving power of a coherent BF system characterized as ${\delta}_{CBF}$ = 1.67$\times {\delta}_{DL}$.

Following the procedure described in the previous section, we next simulated a reconstruction of a coherent SIM image. A set of raw SI image data were generated in sequence for the same test target in Fig. 3(a) and Fourier-transformed, which were then used to solve for SI spectral components. The 19 extracted SI components for the 3-angle SIM are displayed in Fig. 4, each revealing the frequency-shifted high-resolution information that is in one-to-one correspondence with the extended-passband (EP) components representing the auto-correlation and 18 more cross-correlation terms given in Eq. (2). After appropriate shifting in frequency space, SI components were linearly combined to reconstruct an enhanced-resolution image. A similar procedure was performed to simulate the orthogonal SIM image for comparison.

The reconstructed images of the orthogonal SIM and the 3-angle SIM are shown in Fig. 5. Compared to the coherent BF image in Fig. 3(b), resolution improvements in the both coherent SIM images are obvious in Figs. 5(a) and 5(b), where the same sector star patterns appear more sharply and render the features closer to the center (inside the green circles indicating the actual limit of resolution ${\delta}_{CBF}$ = 1.67$\times {\delta}_{DL}$ for the coherent BF imaging) clearly visible. In the direction around the SI orientations of each coherent SIM, one can resolve the features with separation nearly a half of the coherent BF resolution, even below the diffraction limit. In the direction midway between the SI orientations, however, some degree of degradations in spatial resolution can be noted from the non-circular boundaries to which the features can be discernible. With the orthogonal SIM, the shape of such boundary appears square-like as illustrated by the yellow line in Fig. 5(a), indicating that the resolution gets worse toward the directions of 45° and 135°. The 3-angle SIM, on the other hand, is found to exhibit a hexagonal boundary of resolution limit as depicted in Fig. 5(b), implying smaller resolution degradations (at 30°, 90°, and 150°) than those of the orthogonal SI scheme (at 45° and 135°) if the best resolutions in the SI directions are identical in the both SI schemes. The aspects of anisotropic “super-resolution” are also evident from Figs. 5(c) and 5(d), which display the Fourier spectra of the reconstructed images in Figs. 5(a) and 5(b), respectively. Compared to the coherent BF image spectrum in Fig. 5(e), the footprints of the both SIM image spectra have extended boundaries at nearly twice the “intensity” cutoff frequency ${f}_{d}=1/{\delta}_{DL}=2NA/\lambda $ of the coherent BF system along the directions of the SI wavevectors. This confirms that the both SIMs are capable of detecting the sample’s high-resolution information from beyond the frequency support of the conventional coherent imaging. The 2D frequency limits of the SIM images are, however, shown to be anisotropic, indicating the lack of high-frequency contents in the directions midway between the SI wavevectors. Obviously, the orientation-dependent decrease in the maximum frequency of the image content is less appreciable with the 3-angle SIM than with the orthogonal SIM, which is in a good qualitative agreement with the aspects of anisotropic SIM resolution revealed in Figs. 5(a) and 5(b).

At this point, it would be useful to examine the auto-correlation of the effective CTF for the coherent SIM, providing an insight into the anisotropic resolution improvement involved with the two different SIM schemes. Although a CTF is the well-defined transfer function to describe a coherent image’s “amplitude” distribution in frequency space, its “intensity” spectrum, as given by Eq. (4), is more like the auto-correlation of the CTF. In Fig. 6, the auto-correlated effective CTFs (ATFs) of the two coherent SIMs are compared, illustrating the two systems’ frequency response to a point amplitude object, equivalent to the maximum possible frequency contents that can be supported by the reconstructed SIM images. One can see that the footprints of the SIM image spectra we obtained in Figs. 5(c) and 5(d) quite resemble these ATFs shown in Figs. 6(a) and 6(b) for the orthogonal and 3-angle SIMs, respectively.

A heuristic attempt was made to estimate the anisotropy of super-resolution in the two schemes, simply based on the undulation of orientation-dependent frequency cutoff in these characteristic functions. For the orthogonal SIM, the ratio of the minimum (at 45° and 135°) and maximum (at 0° and 90°) cutoff frequencies was found to be 0.86, corresponding to 14% undulation in the resolution enhancement factor. The 3-angle SIM, on the other hand, was found to exhibit a smaller undulation of 6.7% in its resolution enhancement between the best (at 0°, 60°, and 120°) and the worst (at 30°, 90°, and 150°) orientations. In terms of spatial resolution, the undulations are alternatively quantified as 16% and 7.1% for the orthogonal and 3-angle SIMs, respectively. Such a difference in the resolution anisotropy can be expected also from the intensity spread functions (ISFs) for a point amplitude object, shown in Figs. 6(a’) and 6(b’) for the orthogonal and 3-angle SIMs, respectively. For the both SIMs, the overall intensity spreading of a point object is reduced to approximately a half the coherent BF one shown in Fig. 6(c’). The anisotropy of ISF with the 3-angle SIM is apparently less predominant than with the orthogonal SIM, by noting that the central peak appears nearly circular and its side lobes in the directions between the SI wavevectors are suppressed substantially in the 3-angle SIM. The above findings show roughly that our 3-angle SI scheme could allow advantage over the orthogonal SIM in reducing the anisotropy of enhanced resolution. However, we need to emphasize that the semi-quantitative argument made here only helps gain insight on the 2D resolution characteristics of the two coherent SIMs and should not be generalized to evaluate their actual performance because the ACF and the ISF do not mathematically represent a proper transfer function and a point spread function, respectively, for the intensity image of a particular amplitude object in coherent imaging.

The actual 2D resolution of the coherent SIM was evaluated rigorously by investigating the line intensity distributions of the bar-pattern images with different cycle periods and orientations. In Figs. 7(a)-7(e), we show the magnified images of the bar-pattern elements 10, 12, 14, and 20 (oriented at 0°, 45°, 60°, and 90° for each), taken from Figs. 5(a) and 5(b) for the orthogonal SIM and the 3-angle SIM, respectively. The corresponding line intensity profiles of these images were extracted and grouped in Figs. 7(f’)-7(i’) for the orthogonal SIM and Figs. 7(f”)-7(i”) for the 3-angle SIM.

Before going into the precise assessment of 2D resolution, one can get an overview on the orientation-dependent resolving power of the two coherent SIMs. From the images of the vertical bars in Fig. 7(b), the both coherent SIMs are nearly twice better in resolution (around 0.86${\delta}_{DL}$) than the coherent BF imaging with its rough resolution (around 1.72${\delta}_{DL}$) revealed in Fig. 7(a). For the bar orientation at 45°, however, the finest pattern resolvable with the orthogonal SIM has a cycle period ${\Lambda}_{LS}$of 1.20${\delta}_{DL}$while the 3-angle SIM can resolve the pattern down to 1.03${\delta}_{DL}$. With the bars orientated at 60° in Fig. 7(d), the orthogonal SIM renders the feature with a cycle period ${\Lambda}_{LS}$ of 1.03${\delta}_{DL}$discernible whereas the 3-angle SIM has the same resolution (around 0.86${\delta}_{DL}$) as the best one achievable at 0°-orientation. In Fig. 7(e) for the bar orientation at 90°, the 3-angle SIM exhibits its worst resolution around 1.03${\delta}_{DL}$, yet better than the worst resolution (around 1.20${\delta}_{DL}$) of the orthogonal SIM at the 45°-orientation as shown in Fig. 7(c).

To accurately determine the spatial resolution of the coherent SIMs, we took the minimum required visibility of 15.2% in the line intensity profile of the image as a consistent measure for resolving any two adjacent features, as adopted in Rayleigh’s criterion. Rather than creating a number of bar patterns with the different cycle periods ${\Lambda}_{LS}$ and finding the smallest feature that exhibits a visibility greater than 15.2%, we instead fine adjusted the physical size of the entire resolution target shown in Fig. 3(a), effectively varying the cycle period ${\Lambda}_{LS}$of the bar-pattern elements in small increments, and evaluate the cycle period ${\Lambda}_{LS}$ of a designated bar-pattern element as the spatial resolution when the visibility of its line intensity profile matches the Rayleigh’s criterion. We repeated this procedure of assessing spatial resolution for the two coherent SIMs, separately at the SI orientation and the direction midway between the SI wavevectors. By doing so, we readily found that the orthogonal SIM has its best resolution (along the SI orientations at 0° and 90°) of ${\delta}_{SI-2(B)}=0.87\text{\hspace{0.05em}}{\delta}_{DL}$ and the worst resolution (at 45° and 135°, midway between the SI orientations) of ${\delta}_{SI-2(W)}=1.11\text{\hspace{0.05em}}{\delta}_{DL}$. On the other hand, the 3-angle SIM was found to allow the best resolution (along the SI orientations at 0°, 60°, and 120°) of ${\delta}_{SI-3(B)}=0.91\text{\hspace{0.05em}}{\delta}_{DL}$ and the worst resolution (at 30°, 90°, and 150°, midway between the SI orientations) of ${\delta}_{SI-3(W)}=0.98\text{\hspace{0.05em}}{\delta}_{DL}$. As a reference, the coherent BF resolution was also characterized to be ${\delta}_{CBF}=1.67\text{\hspace{0.05em}}{\delta}_{DL}$. When one defines the orientation-dependent undulation of SIM resolution by ${\Delta}_{\delta}=({\delta}_{SI(W)}-{\delta}_{SI(B)})/{\delta}_{SI(B)}$, the results indicate that the resolution undulation of ${\Delta}_{\delta (SI-2)}$ = 27.6% in the orthogonal SIM can be reduced to ${\Delta}_{\delta (SI-3)}$ = 7.7% by using the 3-angle SIM. It is worth noting that despite the decrease in the resolution undulation with the 3-angle SIM, the best resolution (${\delta}_{SI-3(B)}$) achievable in the 3-angle SIM gets slightly worse than that (${\delta}_{SI-2(B)}$) of the orthogonal SIM by 4.6%. The maximum degraded resolution of ${\delta}_{SI-3(W)}$in the 3-angle SIM is, however, still 13% better than the worst resolution ${\delta}_{SI-2(W)}$ in the orthogonal SIM. The deterioration of the 3-angle SIM resolution along the SI orientations could be attributed to its extended passband with the greater over-weightings of low-frequency contents than in the orthogonal SIM, due to the increased number of CTFs involved. Notwithstanding the small loss in resolution caused along the SI orientations, it was shown that the more overlap of the CTFs in the 3-angle SIM can indeed take advantage of isotropic filling of high-frequency space to improve the actual resolution overall. Of course, there is certainly room for improving the resolution in the present 3-angle SIM when the over-weightings in the extended passband can be removed appropriately.

In terms of the resolution enhancement factor${\eta}_{SI}={({\delta}_{SI}/{\delta}_{CBF})}^{-1}$, the orientation-dependent undulations ${\Delta}_{\eta}=({\eta}_{SI(B)}-{\eta}_{SI(W)})/{\eta}_{SI(B)}$ are expressed as ${\Delta}_{\eta (SI-2)}$ = 21.6% for the orthogonal SIM and ${\Delta}_{\eta (SI-3)}$ = 7.1% for the 3-angle SIM. Interestingly, the results are in fairly good agreement with the semi-quantitative estimates we made based on the auto-correlated CTFs given in Fig. 6.

Finally, the orientation-dependent imaging performance was examined in more detail by the intensity profile of a periodic pattern with continuously varying orientations. Figure 8 compares the orthogonal SIM and the 3-angle SIM for the line profiles taken along the circular arcs of the sector star images in Figs. 5(a) and 5(b). With the radial bars having cycle period of 2.58${\delta}_{DL}$, greater than the resolution limit (1.67${\delta}_{DL}$) of the coherent BF imaging, the both SIM schemes allow excellent image contrasts as shown in Fig. 8(a). Nonetheless the image intensity in the orthogonal SIM was found to undulate considerably with the orientation of the radial bars, which can be also noticed obviously from Fig. 5(a) by the brighter radial bars in the diagonal directions (45° and 135°) than at the SI orientations (0° and 90°). This can be attributed to the fact that the orthogonal SIM’s extended passband consists of four individual frequency-shifted CTFs to overlap in the low-frequency regime, creating the four over-weighted areas elongated in the diagonal directions as illustrated in Fig. 1(a). Therefore the line patterns with low spatial-frequency that are aligned at non-SI orientations tend to get intensified in the reconstructed image while their resolution enhancement is lower than that at the SI orientations. Moreover, the locally over-weighted passband is thought to produce appreciably high side lobes in the coherent spread function (CSF) of the orthogonal SIM, similar in shape with the ISF illustrated in Fig. 6(a’). The constructive interference between such side lobe components formed by adjacent amplitude objects seems to be accounting for the satellite image peaks that occur between the radial bars around $\pm {45}^{\circ}$. In the 3-angle SIM, on the other hand, such image imperfections were found to become less manifested. For the patterns with a cycle period reaching the resolution limit (1.67${\delta}_{DL}$) of the coherent BF imaging, it is obvious in Fig. 8(b) that the both SIMs permit perfect image visibilities in the entire directions because the object’s feature size is much greater than the enhanced resolution of the both SIMs.

The finer patterns with a period of 1.20${\delta}_{DL}$, well below the coherent BF resolution limit, could still be resolvable in the both SIMs as shown in Fig. 8(c). The image visibility, however, significantly decreased with the orthogonal SIM in the direction of$\pm {45}^{\circ}$. The advantage of the 3-angle SIM over the orthogonal SIM becomes clearer in Fig. 8(d), where the even smaller patterns with a period of 1.03${\delta}_{DL}$ can be well resolved in the entire directions with fairly high contrast in the 3-angle SIM whereas the orthogonal SIM fails to image the patterns in the direction around $\pm {45}^{\circ}$owing to the artifact of missing intensity peaks and their phase shifts. The results in Figs. 8(c) and 8(d) apparently show that the anisotropic degradation of resolution in the 3-angle SIM is smaller than that of the orthogonal SIM. For the pattern with a period as small as 0.86${\delta}_{DL}$, nearly a half the coherent BF resolution limit of 1.67${\delta}_{DL}$, the super-resolved angular range in which the image features remain discernible was found to be quite limiting for the both schemes of coherent SIM, as seen from Fig. 8(e). In accordance with our previous finding that the 3-angle SIM’s resolution is ${\delta}_{SI-3(B)}=0.91\text{\hspace{0.05em}}{\delta}_{DL}$ at best and no better than the orthogonal SIM’s maximum resolution of ${\delta}_{SI-2(B)}=0.87\text{\hspace{0.05em}}{\delta}_{DL}$, the imaging performance of the 3-angle SIM with the object features below its resolution limit is apparently lower than that of the orthogonal SIM.

## 4. Conclusion

We have presented a method for improving the isotropy of 2D super-resolution in the coherent structured illumination microscopy (SIM). Instead of an orthogonal standing-wave illumination previously used in a coherent SIM [20], a hexagonal-lattice illumination incorporating simultaneous three standing-wave fields has been introduced in this study. A mathematical formulation has been worked out to describe the coherent image formation in the 3-angle SIM and an explicit Fourier-domain framework has been derived for reconstructing an image with enhanced resolution. We have carried out numerical simulations to test the feasibility of our 3-angle SIM framework, comparing its lateral resolution with that of the previous scheme for orthogonal SIM. The investigation on the 2D imaging performance with the various test patterns of different periods and orientations has shown that the orientation-dependent degradation in the lateral resolution can be reduced from 27.6% to 7.7% (by a factor of approximately 3.6) in the 3-angle SIM, with its super-resolution along the directions of the best enhancement remains 0.54 times the resolution limit of coherent bright-field imaging, while slightly worse than that of the orthogonal SIM by 4.6%. Interestingly, the 3-angle SIM has been found to exhibit additional advantage over the previous scheme in that the “image intensity” undulation depending on the pattern orientation, being predominant in the orthogonal SIM, can be significantly alleviated by the 3-angle SIM where the 2D frequency passband is extended more isotropically than in the orthogonal SIM. The isotropic implementation of coherent SIM proposed in this study, is expected to benefit a diversity of imaging applications that require high resolution beyond the diffraction limit without relying on the fluorescence contrast of samples.

## Acknowledgments

This work was supported by the grants from Green Nano Technology Development Program and Bio-signal Analysis Technology Innovation Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT and Future Planning, Republic of Korea.

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