1. Introduction

Over the last decade, coherent anti-Stokes Raman scattering (CARS) microscopy [1] has become one of most widely used chemical imaging tools in biology and medicine, for its high sensitivity to molecular vibrations offering intrinsic chemical contrast without exogenous labeling agents introduced to biological samples. A typical implementation of CARS microscopy involves tight focusing of collinearly combined CARS excitation laser beams and scanning the highly localized focal spot in the sample to map out anti-Stokes signals point by point for constructing an image. This well-established technique has demonstrated a real-time performance and a submicrometer-scale spatial resolution in visualizing the fine structures of cells and tissues with vibrational selectivity [2,3]. There has been, however, a growing demand in CARS microscopy for pushing its spatial resolution down further to reveal the biologically relevant structures in more detail for better understanding of their functions and interactions, as is of importance as well with any microscopic techniques pursuing modern applications. A critical factor that has been known to fundamentally limit the optical far-field resolution is diffraction [4]. The fact that the physical size of many fine objects of biological interest are at or below the diffraction limit, has thus prompted a lot of efforts to break the diffraction barrier of resolution in optical microscopy.

A wide range of methods for achieving super-resolution have been explored, especially in fluorescence imaging. To name a few, stimulated emission depletion (STED) microscopy [5,6], photoactivated localization microscopy (PALM) [7], stochastic optical reconstruction microscopy (STORM) [8], and structured illumination microscopy (SIM) using fluorescence saturation [9,10] have demonstrated remarkable success allowing for unprecedented far-field resolution below tens of nanometers. The working principles of these techniques, however, heavily rely on the properties of specially designed fluorophores and can no longer be applied for imaging objects that are not either autofluorescent or fluorescence-tagged. Realizing the subdiffraction-limited resolution in label-free CARS imaging therefore requires quite a different alternative than the existing methods.

So far, several approaches to subdiffraction-limited CARS microscopy have been proposed. Beeker et al. theoretically proposed a scheme to reduce the effective spot size of CARS excitation to nanometer scale [11] while its experimental validation is not yet demonstrated unfortunately. Another theoretical investigation by Raghunathan and Potma [12] showed that a multiplicative focal volume engineering can narrow the point spread function (PSF) of the nonlinear imaging system, leading to a resolution enhancement of better than a factor of 1.5. Motivated by this work, researchers at NIST experimentally demonstrated CARS imaging resolution approaching for 130 nm, approximately twice the diffraction limit, for an isolated nanowire structure by combining the use of a Toraldo phase filter and the multiplicative nature of multiphoton PSFs [13]. In such approaches for PSF narrowing in CARS microscopy, however, a question still remains to be addressed on its validity and practical resolution limit with extended microscopic samples comprising complicated features where the spatial coherence of illumination comes into play to affect the image formation significantly. An alternative route to super-resolved CARS microscopy [14] has been also suggested which incorporates the SIM principle [15–17] with a widefield CARS imaging scheme [18] using a non-collinear phase-matching configuration. In this study, a one dimensional (1D) laterally-patterned excitation of CARS satisfying the phase-matching condition was considered and the resulting coherent image formed by 1D widefield anti-Stokes signals was theoretically modeled. A computational method was then derived for reconstructing a CARS image with super-resolution, based on the framework of coherent SIM [17] which is an essential extension from the conventional incoherent SIM [15] in order to deal with the sample illumination and emission that are both spatially coherent in CARS imaging. A numerical simulation of super-resolved CARS imaging with such patterned excitations was performed to verify a factor of three improvement in resolution over the standard widefield CARS imaging [18]. The resolution improvement factor greater than twice, the maximum obtainable value with simple fluorescence SIM, can be attributed to the nonlinear nature of CARS excitation process. Unfortunately, this super-resolved CARS method is only valid for imaging 1D samples and might be prone to a corruption of reconstructed images suffered by unwanted artifacts and deteriorated resolution when applied to real samples containing two dimensional (2D) fine structures.

In this article, we present an extensive scheme for implementing a widefield CARS microscopy with lateral resolution surpassing the diffraction limit in two dimensions. A square lattice-patterned field of CARS excitation, tailored to satisfy the phase matching condition, is a key element we employ for aliasing inaccessible spatial frequency contents of a nonlinear sample into the classical imaging passband of the system from beyond the diffraction limit. In contrast with the previous work done in the spatial domain [14], we instead derive a Fourier-domain formulation describing the coherent anti-Stokes image formation with such a 2D-patterned nonlinear excitation and provide a framework to demultiplex the high-frequency contents from a set of multiple images acquired with spatially-shifted excitation patterns. We then show a computational step of processing this extracted high-resolution information to rebuild a single CARS image with a spatial-frequency support exceeding the classical diffraction limit, which is quite analogous to the recent framework developed for 2D implementation of coherent SIM [19]. Our extension from one dimension to two dimensions is expected to allow more directions for resolution enhancement as well as to avoid the potential pitfall of the 1D spatial-domain frameworks [14] which could not even serve as the equivalent of a 2D framework with structured illumination in one direction. Confirming the validity of the proposed method, numerical simulations are carried out to produce a reconstructed CARS image with synthesized resolution target and evaluate its imaging performance for various test patterns. The results indicate that our scheme for super-resolved CARS microscopy is capable of chemical imaging with nearly three times as high 2D resolution as is possible in the standard CARS imaging.

2. Principle of 2D super-resolution CARS microscopy with structured illumination

In a nut shell, achieving a far-field super-resolution for CARS microscopy proposed in this study merges the principle of structured illumination (SI) technique and the coherent nonlinear nature of sample excitation/emission in CARS. Rendering CARS microscopy compatible with a typical SI geometry, we employ a widefield CARS imaging configuration [18] as shown in Fig. 1. A pair of degenerate pump/probe beams, impinging non-collinearly on the sample plane, is spatially overlapped with an axially propagating Stokes beam delivered from the opposite side. When the phase-matching condition (illustrated in the inset of Fig. 1) is fulfilled with the CARS excitation fields, widefield anti-Stokes signals can be efficiently generated in the finite area on the sample. Here, the oblique pump/probe beams interfere to form a laterally structured standing-wave field, which in turn leads to a spatial modulation in the nonlinear excitation permitting anti-Stokes signals to be imaged. Such a structured excitation field, in brief, acts to shift the sample’s high-frequency spectrum being aliased into the detectable passband, enabling an image reconstruction to improve the resolution as suggested in the SIM framework.

 

Fig. 1 Schematic diagram of a wide-field CARS microscopy with nonlinear structured illumination. The inset on the left shows the non-collinear phase-matching geometry of the wave vectors associated with CARS excitation.

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Due to the coherent nature of the anti-Stokes signal, however, the 2D super-resolution SIM scheme for CARS varies substantially from that applicable to fluorescence imaging. Not only should the basic imaging theory be extended to take coherence into account but also the formation of structured excitation field needs to be modified. In conventional SIM [15], a sample is typically illuminated with dense stripes of a sinusoidal “intensity” pattern to encode the sample’s undetectable high-frequency information into the observable image spectrum with smaller frequency. The multiplexed high-frequency content can be extracted by processing a series of images taken with translated illumination patterns of different phases, which then allows reconstructing an image with resolution improved in a direction perpendicular to the line pattern. Reconstructing a 2D super-resolved image with isotropic enhancement requires iterative measurements simply with rotating the illumination pattern to several orientations to yield more information to fill out 2D frequency space. Based on the superposition principle, this strategy is valid with a linear optical system that deals with incoherently scattering objects such as fluorophores. For coherent imaging, on the other hand, this is not the case because the detected image intensity is intrinsically nonlinear with both the amplitude and intensity of the object [20] while a linear relationship holds only between the object and image “field.” As pointed out previously [19], an attempt for 2D resolution enhancement in coherent SIM, should rather introduce a structured illumination containing multiple oblique beams of different frequency shifts, at “two or more” orientations, to “simultaneously” interfere at the sample. In the present study, we employ a square lattice-patterned field of CARS excitation that is able to support “two orthogonal pairs” of frequency shifting components working at the same time.

Interestingly, the nonlinearity in CARS further gives rise to a couple of significant differences, compared with a coherent SIM. One is associated with the choice of frequency shifts provided by the oblique illumination beams. They cannot be chosen arbitrarily, as large as possible to maximize the resolution-enhancement gain as is typical of linear SIM. Instead, the phase-matching condition to be met for CARS, determines the angle of obliquely incident beams, which depends on the target species under investigation (more specifically, on the molecular vibrational frequency as well as on the wavelengths used for CARS excitation). This imposes a relevant physical constraint on the super-resolution gain achievable in the present method. The other is obviously the nonlinearity in sample excitation, being quadratic to amplitude of the illuminating pump/probe field. Apparently, the effective structured illumination is therefore described by the squared field of the pump/probe beam, leading to a doubling of the frequency shift that would result from a linear sample response. As a consequence, this could allow a potential for “tripling” the frequency support of the widefield CARS imaging system, when the actual structured field frequencies are pushed to the very edge of the detection passband. Given below is a more detailed description on the proposed scheme for 2D super-resolved CARS microscopy with structured illumination, followed by a theoretical framework on how to acquire and process coherent nonlinear SIM images for reconstructing a resolution-enhanced CARS image.

2.1 Configuration of a widefield CARS microscope with nonlinear structured illumination

As illustrated in Fig. 1, we consider a widefield CARS microscope where a nonlinear sample is illuminated with 2D structured excitation beams to generate CARS signals that are imaged by a sensitive camera. Recalling that CARS is a four-wave mixing process [1], one needs to involve three excitation fields at the pump (${\omega }_p$), Stokes (${\omega }_s$), and probe (${\omega }_{prb}$) frequencies, interacting simultaneously with the sample specified by a third-order nonlinear susceptibility ${\chi }^{(3)}$. In practice, the probe light is usually derived from the same source as the pump and it is henceforth assumed to degenerate in frequency with the pump, i.e. ${\omega }_{prb}={\omega }_p$. When the beating between the pump and Stokes fields, at difference frequency${\omega }_p-{\omega }_s$, is tuned resonant with the vibrational frequency ${\Omega }_{\text{mol}}$ of a target molecule, the third probe light incident on such molecules can be scattered off to strongly generate a coherent anti-Stokes signal at frequency ${\omega }_{AS}=2{\omega }_p-{\omega }_s$ which yields a chemical contrast specific to the target molecule. An additional condition to be met, yet more onerous than this energy conservation, is the phase-matching condition that dictates momentum conservation with the fields associated in CARS; specifically, wave vectors of the excitation and emission fields must satisfy a vectorial relation given by $k_{AS}=2k_p-k_s$, as delineated in the inset of Fig. 1.

The phase matching in the widefield CARS geometry can be accomplished by projecting two pump beams onto the sample from below, at high angle but directed oppositely (at $+{\theta }_{\text{PM}}$ and $-{\theta }_{\text{PM}}$ from the surface normal of the sample, respectively) to cross each other over a finite area. The Stokes beam $E_s^{}(r;z)=\exp [-i{k}_{s}z]$, on the other hand, impinges on the sample perpendicularly from above (in the $-z$ direction), to overlap with the pair of pump beams at the same region in the sample. The three excitation beams, co-localized at the sample, are assumed to be plane waves of which wave vectors are all lying on an incident plane normal to the sample plane. When phase-matched, the anti-Stokes signal emanating from the beam-overlapping area in the sample plane propagates anti-parallel to the incident Stokes beam (in the $+z$ direction), allowing its intensity distribution to be imaged with a camera. Given the vibrational frequency ${\Omega }_{\text{mol}}$ of a target molecule, the phase-matching angle $\pm {\theta }_{\text{PM}}$ for the incident pump beams with frequency ${\omega }_p$ can be calculated readily by

A pair of phase-matched pump beams then constitutes a standing wave in the sample plane $(r;z=0)$ in a direction (say, $x$axis) as seen in Fig. 2(a), characterized by two lateral components counter-propagating with a wave number $k_{p\parallel }=k_p\sin {\theta }_{\text{PM}}$ where $k_p=\sqrt{k_{p\parallel }^2+k_{pz}^2}$. Such a standing-wave field $E_{px}^{}(r;z)=\left\{\exp \left[i(k_{px}x+{\varphi }_{px})\right]+\exp \left[-i(k_{px}x+{\varphi }_{px})\right]\right\}\exp \left[i{k}_{pz}z\right]$ would act, in a linear SIM, to shift the sample’s spectrum by $+k_{px}=(k_{p\parallel },0)$ and $-k_{px}=(-k_{p\parallel },0)$ in 2D spatial-frequency space. When superimposed by another standing-wave field $E_{py}^{}(r;z)=\left\{\exp \left[i(k_{py}y+{\varphi }_{py})\right]+\exp \left[-i(k_{py}y+{\varphi }_{py})\right]\right\}\exp \left[i{k}_{pz}z\right]$ in Fig. 2(b) of the same wave number but to the orthogonal orientation, the laterally structured field would extend in two dimension as depicted by Fig. 2(c) to permit additional frequency-shifting components at $+k_{py}=(0,k_{p\parallel })$ and $-k_{py}=(0,-k_{p\parallel })$, resulting in four regions of the sample’s spectrum being diffracted into the system’s coherent transfer function (CTF). Throughout the paper, we henceforth use a nomenclature in which $k$ is referred to as “spatial frequency” when the entity represents the same physical relevance as $f=k/2\pi$.

 

Fig. 2 Structured illumination field of CARS pump beams and its resulting nonlinear excitation field distribution in the proposed method. The associated coherent transfer functions (CTFs) that can be extended from the conventional CTF are displayed.

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Of note, the light paths in our proposed setup for structured excitation and detection do not share the same physical aperture of spatial-frequency support, giving us a freedom to choose the numerical aperture $N{A}_{DET}$ for imaging the widefield signal while the spatial frequency of standing-wave field $k_{p\parallel }$ is constrained by the phase-matching condition. Hence, the greatest gain in resolution can be achieved when the cutoff frequency $k_c=k_{AS0}N{A}_{DET}$ for imaging coherent anti-Stokes signals is adjusted to match with the frequency shift $k_{p\parallel }=k_{p0}n\sin {\theta }_{\text{PM}}$ caused by the structured field, in which $k_{AS0}$ and $k_{p0}$refer to wave numbers of the anti-Stokes and pump, respectively, and $n$ is the refractive index of sample immersion medium. By choosing a detection objective with $N{A}_{DET}=k_{p0}n\sin {\theta }_{\text{PM}}/k_{AS0}$, the square lattice-patterned field could effectively extend the observable region of a coherent sample’s spectrum beyond the classical CTF, as illustrated by Fig. 2(d) in the case of a linear sample. In CARS imaging, on the other hand, the coherent signal generated from a sample is proportional to the nonlinear excitation amplitude given by $E_{EX}(r)=E_p^2(r)E_s^{*}(r)$, not linear with the structured pump field itself. This leads to a modification to the effective structured illumination field in coherent SIM, which would in turn alter the resultant imaging behavior. Considering the physics of phase matching condition, we here note that the nonlinear excitation field is quadratic to each pair of pump fields at orthogonal orientations separately, $E_{px}^2(r)$ and $E_{py}^2(r)$, and not to the actual pump field, ${[E_{px}^{}(r)+E_{py}^{}(r)]}^2$ as a whole. This is because the mixing term therein $E_{px}^{}(r)E_{py}^{}(r)$ does not satisfy the phase-matching condition and vanishes in turn. As a result, the nonlinear structured excitation field takes the form of

The CARS pump-field pattern required in our scheme can be conveniently created by using a pixel-addressable spatial light modulator (SLM) programmed with a periodic 2D grid pattern [16]. By illuminating the SLM with a collimated pump laser beam, two sets of diffracted beams at orthogonal orientations are generated, which are then directed to a 4-f microscope system. The diffracted beams are reduced in beam size appropriately and recombined to interfere at the sample. In order to produce 2D sinusoidal fields as shown in Fig. 2(c), a Fourier-plane pupil mask can be used to permit only the $\pm 1$ diffraction orders in both orthogonal sets of diffracted pump beams as well as to reject the 0-th order. During the acquisition of a series of raw images to extract super-resolution information, one needs to manipulate the SLM’s 2D grid pattern to have a particular set of different phases $\left\{{\varphi }_{px},{\varphi }_{py}\right\}$of pattern translation in both directions.

2.2 Theoretical framework

We here develop a theoretical framework for reconstructing a 2D super-resolved image in the widefield CARS microscopy with nonlinear structured excitation. We set out its basis by treating the CARS image formation in our proposed method within the scope of coherent imaging theory [20,21]. In a coherent illumination/detection system, the in-focus image intensity $d(r)$ of an ideally thin amplitude object $s(r)$ under an illumination field $E(r)$ can be measured with a 2D-resolving detector, as given by the equation

We now consider a widefield CARS microscopy in which the CARS polarization $u_{CARS}(r)={\chi }^{(3)}N(r)E_p^2(r)E_s^{*}(r)$ is taken to be the object disturbance term $u_o(r)=s(r)\cdot E(r)$ in Eq. (3), where ${\chi }^{(3)}$is the third-order susceptibility of the target species for CARS detection, $N(r)$is the density distribution of the target species lying on the sample plane, and $E_p^{}(r)$ and $E_s^{}(r)$denote the pump and Stokes fields, respectively. Physically, the object distribution is specified as $s(x,y)={\chi }^{(3)}N(x,y)$ and the effective illumination field $E(r)$is replaced by the nonlinear excitation amplitude$E_{EX}(r)=E_p^2(r)E_s^{*}(r)$. Reminding the square lattice-patterned pump field introduced in this study and disregarding the Stokes field term $E_s^{*}(r)$ with no spatial variation, we express the 2D nonlinear structured excitation for CARS as

Next, we turn to calculate the coherent image formation in a virtual super-resolution system with an “extended passband” that consists of a set of multiple frequency-shifted CTFs, equivalent to the frequency-aliasing of a sample achievable with structured excitation. By intuition, such a hypothetic imaging passband $C_{EP}(f)$ can be synthesized as

In order to reconstruct a super-resolved CARS image, we exploit the mathematical similarity found between the two Fourier images, one with structured illumination and the other with hypothetically extended CTF. As guided by the approach used in the previous study [19], we know that the SI components $\{T_n(f)|n=0,1,\mathrm{...12}\}$ in Eq. (8) are in one-to-one correspondence with the EP components $\{V_n(f)|n=0,1,\mathrm{...12}\}$worked out in Eq. (10), and possess exactly the same analytic form as those except for their locations in the spatial frequency space. Without presenting the rigorous mathematical proofs, we here point out only the useful findings. The SI and EP components, for any arbitrary $\gamma \in n$, take general forms, $T_{\gamma }(f)=C(f)S(f+f_{\alpha })\ast C(f)S(f+f_{\beta })$and $V_{\gamma }(f)=C(f-f_{\alpha })S(f)\ast C(f-f_{\beta })S(f)$, respectively, in which $f_{\alpha }$and $f_{\beta }$are independent frequency-shift vectors$\in \left\{0,+f_{sx},-f_{sx},+f_{sy},-f_{sy}\right\}$. Interestingly, the similarity and symmetry existing in the two forms leads to a general relation

To obtain individual SI components $\{T_n(f)|n=0,1,\mathrm{...12}\}$ to be processed for the image reconstruction, a minimum of 13 SI image acquisitions ${D_{SI}(f)|}_{{\Psi }_m}$are necessary at different phases (translations) $\{{\Psi }_m\equiv {({\varphi }_{sx},{\varphi }_{sy})}_m|m=0,1,\mathrm{...12}\}$ of the nonlinear structured excitation. Using the linear algebra given by Eq. (8), we construct a $13\times 13$matrix $M_{({\Psi }_m,n)}^{}$ that describes the spectral mixing of SI components into the Fourier image, ${D_{SI}(f)|}_{{\Psi }_m}\equiv M_{({\Psi }_m,n)}^{}{T}_n(f)$. Given a set of $\{{\Psi }_m\equiv {({\varphi }_{sx},{\varphi }_{sy})}_m|m=0,1,\mathrm{...12}\}$ appropriately chosen to render $\det \left[M_{({\Psi }_m,n)}^{}\right]\ne 0$ valid, we are then able to use its inverse matrix $M_{({\Psi }_m,n)}^{-1}$ to unmix multiplexed SI components by carrying out an operation $T_n(f)\equiv M_{({\Psi }_m,n)}^{-1}{D_{SI}(f)|}_{{\Psi }_m}$with measured 2D raw image data.

3. Methods for numerical simulation

We performed numerical simulations to validate the theoretical framework we derived to obtain super-resolution in CARS microscopy with 2D structured illumination. To let the investigation give practical implications in terms of concrete values, simulation parameters were taken from a typical case of CARS microscopy experiments. We assumed a nonlinear sample providing a CARS contrast at the Raman shift of $\text{3045}{\text{cm}}^{-1}$ (corresponding to the aromatic CH stretching vibration of polystyrene) that is resonantly excited by an 803-nm pump laser and a 1064-nm Stokes laser and imaged by detecting the anti-Stokes signal generated at the wavelength ${\lambda }_{AS}$ of 645 nm. Satisfying the phase-matching condition (${\theta }_{\text{PM}}={75.8}^{\circ }$) for the sample immersed in water ($n=1.33$), the 2D-patterned pump field was formed with spatial period $f_{p\parallel }^{-1}$ of 623 nm. The numerical aperture $N{A}_{DET}$ for wide-field imaging at the anti-Stokes wavelength was set to 1.04 (for $f_{p\parallel }=N{A}_{DET}/{\lambda }_{AS}$), allowing the maximum gain of resolution improvement achievable with the present method. The system described above, would have a diffraction-limited resolution ${\delta }_{DL}={\lambda }_{AS}/2N{A}_{DET}$ of 311 nm theoretically.

3.1 Computer-synthesized resolution target

In the simulation to generate a set of raw SI-CARS images and demonstrate a reconstruction of an image with resolution enhancement, a computer-synthesized resolution target shown in Fig. 3 was used for evaluating the imaging performance of our proposed method.

 

Fig. 3 Computer-synthesized resolution test target used for the evaluation of resolving power of the proposed super-resolution CARS microscopy. The black-and-white contrast represents a binary sample density such that the sample is present only within white regions whereas its complete absence depicted in black. Colored lines are not the target’s geometrical features but merely indicate the lines along which intensity profiles will be taken from the simulated images. The yellow circle on the sector star pattern is drawn for the line along which the radial bars have a cycle period equal to the diffraction limit (indicated also by the gap sandwiched by blue arrows at the bottom right).

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The 2D test target was built to represent a sample distribution over the area of 18.5$\mu \text{m}$$\times$18.5$\mu \text{m}$ (in 1024 $\times$1024 image pixels), assuming the aforementioned molecular species plated in several different types of patterns. They include groups of bar patterns, isolated single lines at different orientations, and a sector star pattern, and. Here, black-and-white (B&W) contrast was used to indicate that the molecular species is present only in the white region with a uniform density.

The bar patterns of different cycle periods were grouped together by 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, we formed several elements with the cycle period of 180 nm, 216 nm, 252 nm, 360 nm, and 540 nm, which are labeled with numbers as 10, 12, 14, 20, and 30, respectively. Extra bar patterns with the cycle period of 720 nm and 1080 nm were added to the Group HV.

The isolated single lines at different orientations were created above the individual groups of bar patterns with the same orientation. To be used for evaluating the line spread function (LSF), these lines were delineated as thin as possible (with a single-pixel thickness corresponding to 18 nm).

The sector star (also called Siemens star) pattern, which is useful for determining the image resolution by noting how close to the center of the pattern one is able to resolve adjacent bars, was drawn to have 32 bars over 360° with the largest cycle period of 905 nm (corresponding to 2.9 times the diffraction limit ${\delta }_{DL}$) at its circumference. By examining the image of this pattern, we can evaluate the orientation-dependent resolution down to 1/6 of the diffraction limit ${\delta }_{DL}$, which is limited by the pixel resolution of the resolution target.

3.2 Numerical calculation procedures

First, we generated the 2D nonlinear structured excitation $E_{EX}(r)$ as given by Fig. 2(e), directly from the square lattice-patterned pump field $E_p^{}(r)=E_{px}^{}(r)+E_{py}^{}(r)$, following the design explained above for their parameters. The nonlinear structured excitation of the test target sample $s(r)$ in Fig. 3 was calculated by $s(r)\cdot E_{EX}(r)$ in the space domain, and its Fourier transform $F\left\{s(r)\cdot E_{EX}(r)\right\}$ implying $S(f)\otimes {\tilde{E}}_{EX}(f)$ is then used to calculate

4. Results and discussion

Using the resolution target in Fig. 3 as a sample and the imaging parameters given in the previous section, we obtained a super-resolution (SI) CARS image with structured excitation and compared its resolution with that of a conventional widefield (BF) CARS image that would be acquired with the same condition but no structured excitation. In Fig. 4, the resolution improvement in the SI-CARS image (on the right) can be clearly seen overall when compared with the BF-CARS image (on the left).

 

Fig. 4 Coherent images (top) and the associated spatial-frequency spectra (bottom) for (a,a’) the conventional widefield (BF) CARS and (b,b’) the super-resolution (SI) CARS, numerically simulated using the same resolution test target shown in Fig. 3. The dashed circle (yellow) on the area of the sector star target is drawn to indicate the location on which the radial bars have a cycle period equal to the diffraction limit${\delta }_{DL}$( = 311 nm). The radial bars at the circumferences of the five circles with increasing radii (green lines), have cycle periods of 0.58, 0.70, 0.81, 1.16, and 1.74 times the diffraction limit, respectively, which are equal to those of the line bar elements 10, 12, 14, 20, and 30. The image spectra (logarithmic scale) are displayed in false color with their horizontal and vertical axes normalized with the intensity cut-off frequency (the inverse of diffraction-limited resolution).

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At a glance, the sector star pattern in the BF-CARS image appears blurred in most of the area and the features can only be resolvable on its periphery (outside the circumference of the largest green circle drawn in Fig. 4(a)) where the cycle periods of adjacent radial bars are much larger than the diffraction limit ${\delta }_{DL}$(given by 311 nm in this study) by approximately 1.67 when we assume the minimum required visibility for resolving features to be 15%, according to the Rayleigh’s criterion. In the SI-CARS imaging, on the other hand, the same features are more sharply defined at their edges and the smaller features close to the center (reaching the imaginary circle equivalent with the smallest green circle drawn in Fig. 4(a)) can also be resolved clearly even below the diffraction limit, down to 0.62$\times {\delta }_{DL}$. Compared with the theoretical resolution boundary (indicated in Figs. 4(a) and 4(b) by the yellow dashed circles) given by the diffraction limit, the inward extension of a resolvable region with the SI-CARS scheme apparently demonstrates its super-resolution capability.

The actual resolution obtained in the simulation, however, might seem to have a sizable discrepancy with the theory. We here note that the discrepancy does not imply a flaw in our simulation but a characteristic of coherent imaging. It has been well recognized that a coherent system might be poor in its resolution compared with that of an incoherent system, depending on the nature of spatial frequency distribution possessed by objects under investigation [21]. Therefore, one needs to evaluate the resolution performance of SI-CARS imaging in a relative manner, compared with the actual resolving power of a coherent BF system of the same parameters.

The resolution improvement with the SI-CARS is also evident from its “extended” image spectrum shown in Fig. 4(b’) compared with that of the BF-CARS system in Fig. 4(a’). The footprint of the SI-CARS image spectrum is shown to have its outermost boundary, in the same orthogonal directions as the 2D structured field wave vectors, at three times the “intensity” cut-off frequency of the conventional one given by $1/{\delta }_{DL}=2N{A}_{DET}/{\lambda }_{AS}$. This shows obviously that the SI-CARS system has captured the sample’s high-resolution information beyond the diffraction limit where a conventional system would not be able to reach. Nevertheless, the extended 2D frequency space in the SI-CARS scheme was found to be anisotropic, as anticipated from its effective CTF implementation in Eq. (9) having appreciable missing regions within an ideal circular passband with $3f_c$. The high-frequency contents were shown to be rather sparse in the directions between the two orthogonal axes, though seemingly less predominant than in the footprints of the hypothetical CTF constructed as Fig. 2(f), which can be attributed to the fact that the spectrum of an intensity image is given by the autocorrelation of that of an amplitude image. Such a lack of high-frequency contents would result in the orientation dependence of the SI-CARS resolution as well as a reduced improvement factor less than three as expected theoretically.

The orientation-dependent imaging performance was examined in more detail in Fig. 5, showing the line intensity profiles taken from the images of the vertical bar elements of Group HV and the + 45°-oriented bar elements of Group DG, respectively. For the vertical bar patterns in Figs. 5(a)–5(e), the finest feature as small as 180 nm of cycle period was discernible in the SI-CARS image as in Fig. 5(a) while the smallest one exhibiting three separable peaks in the BF-CARS image was the element with cycle period of 540 nm as in Fig. 5(e). The SI-CARS scheme also allowed the next finest feature with cycle period of 216 nm ( = 0.70$\times {\delta }_{DL}$) in Fig. 5(b) to be resolved with a visibility of 67%, much better than the 17% visibility with the 540 nm ( = 1.74$\times {\delta }_{DL}$) feature in the coherent BF imaging. It can be claimed, at this point, that the SI-CARS scheme is able to outperform the BF-CARS system in resolution by a factor of at least 2.5. We further analyzed the sector star target images in both systems, more thoroughly with the Rayleigh’s resolution criterion suggesting the 15% visibility between objects, to conclude that the resolution in the SI-CARS scheme can actually provide a maximum resolution gain of 2.68 over the conventional one. The argument, however, holds only for a certain range of pattern orientations, as illustrated in Fig. 4(b) by the image blurring of the sector star target as well as of the slanted bar targets in Group HX and Group DG. Particularly with the + 45°-oriented bar patterns, the SI-CARS scheme failed to improve the image resolution at all, even slightly worse than the BF-CARS images overall as depicted in Figs. 5(a’)–5(e’). Notwithstanding the failure at $\pm {45}^{\circ }$, in contrast with the super-resolution at $0^{\circ }$ and ${90}^{\circ }$, the resolution improvement was shown to work moderately for the pattern orientations in between.

 

Fig. 5 Line intensity distributions of the images of bar patterns in the super-resolving SI-CARS (red lines) compared to the conventional BF-CARS (blue lines). The line intensity profiles are taken from the images of (a-e) the vertical bars in Group HV and (a’-e’) the bars with their line normal oriented at + 45° in Group DG. The bar-pattern elements under investigation vary with their cycle periods ${\Lambda }_{LS}$of (a,a’) 180 nm, (b,b’) 216 nm, (c,c’) 252 nm, (d,d’) 360 nm, and (e,e’) 540 nm. For comparison, the profiles of the original bar patterns on the test target are displayed together (dotted gray lines). In each line plot, the distance is normalized to the diffraction limit (${\delta }_{DL}$ = 311 nm) of resolution and close-up images of the corresponding bar-pattern element (with a field-of-view of 4 times the pattern’s cycle period) are added to the right for SI-CARS (upper) and BF-CARS (lower).

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So far, the resolution improvement in the SI-CARS scheme has been treated in a relative manner, and its absolute performance achievable was shown to be 0.62$\times {\delta }_{DL}$ ( = 194 nm for detection NA of 1.04 at the wavelength of 645 nm) for particular target patterns chosen in the present study. Regarding the ultimate resolution of the SI-CARS, however, it would be helpful to determine its point spread function (PSF), for coherent imaging itself in general exhibits somewhat complicated aspects in terms of resolving object features that might differ by sample to sample [21].

We thus simulated the orientation-dependent PSF of the SI-CARS imaging system in Fig. 6 for comparison with that of the coherent BF system with the same NA. The PSFs in the directions of $0^{\circ }$, $\pm {45}^{\circ }$, ${90}^{\circ }$, and $\pm {60}^{\circ }$were actually evaluated from the spatial spreading of the image intensity distributions taken for delta-function (single-pixel) line objects oriented at those angles, respectively. Here, the width from the central peak to first minimum was used as a common measure of resolution, as adopted in Rayleigh’s criterion. The line spread functions (LSFs) of the coherent BF system all overlapped one another, appearing as a single isotropic PSF curve (dotted line) with almost exactly the same resolution of 0.99$\times {\delta }_{DL}$(310 nm) as the diffraction limit. The SI-CARS system, in contrast, resulted in the orientation-dependent PSFs with absolute resolutions of 0.36$\times {\delta }_{DL}$(114 nm), 1.05$\times {\delta }_{DL}$(326 nm), and 0.68$\times {\delta }_{DL}$(211 nm) at $0^{\circ }$/${90}^{\circ }$, $\pm {45}^{\circ }$, and $\pm {60}^{\circ }$, respectively. This explicitly shows a potential of the proposed SI-CARS scheme to achieve a maximum of 2.7-fold enhancement in resolution with respect to the diffraction limit as well, provided that objects have appropriate amplitude structures along the directions in which the 2D structured illumination are formed. At the same time, it was found that the resolution anisotropy would impose a major limitation on the present SI-CARS scheme.

 

Fig. 6 Simulation of the point spread function (PSF) of the super-resolving SI-CARS imaging system compared to the conventional BF-CARS imaging system of the same NA. The orientation-dependent PSFs are evaluated actually from the spatial spreading of image intensity profiles taken for delta-function-like line objects oriented at 0°, 90°, + 45°, –45°, + 60°, and –60°. The PSF of the coherent BF system is isotropic to all orientations (dotted line). The curves undistinguishable due to overlap between one another are displayed in the same color. Here, the distance is normalized to the diffraction limit of resolution.

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A straightforward way to get around this problem is to isotropically fill out the missing frequency region with more than two orthogonal sets of standing wave fields as done in the present scheme. In practice, however, implementing more orientations into the structured illumination can only be accomplished at the price of an increased complexity involved with the mathematical formulation and a bulky image processing. Another possibility is to abandon the equality condition$N{A}_{DET}={\lambda }_{AS}{f}_{p\parallel }$ and employ a higher NA objective ($N{A}_{MAX}=\eta N{A}_{DET}$) instead for detecting the CARS signals, which would permit a better overlap between the shifted CTFs being used to constitute a synthetic passband $C_{EP}(f)$, alleviating the problem of drastic loss in frequency contents. A higher resolution overall (i.e.$2+\eta >3$) could be accompanied by an improved isotropy though the enhancement factor over the coherent BF system would become less (i.e.$2{\eta }^{-1}+1<3$). We here emphasize that the attempt would not add any complexity to the 2D frequency-domain framework in the present study but only need changes in parameters. This is not the case, on the other hand, for the 1D spatial-domain approaches worked out previously [14,17], in which a significant modification to the mathematical formulation would be required to render the attempt possible, let alone the existing pitfall that such a 1D framework can never be the equivalent of a 2D framework with structured illumination in one direction.

5. Conclusion

We have proposed a theoretical framework to implement two-dimensional (2D) super-resolution in CARS microscopy via nonlinear structured excitation. In the widefield CARS imaging configuration, we have tailored a square lattice-patterned pump field, satisfying the phase-matching condition for CARS, to serve as a key element that encodes a nonlinear sample’s high-frequency information into the classical imaging passband. A 2D Fourier-domain formulation has been derived to describe the coherent image formed with the nonlinear structured excitation and a rigorous framework has been established on how to extract and reconstruct the super-resolution contents into an image with an enhanced resolution. To check the validity of the proposed method, we have simulated a super-resolved CARS imaging with a computer-synthesized resolution test target. The reconstructed CARS image has been examined and compared with the diffraction-limited coherent bright-field image, for evaluating the resolution performance with various test patterns. The results have clearly demonstrated the potential of our method to enable CARS microscopy to achieve a maximum of 2.7 times better spatial resolution over the conventional widefield CARS system. Our extensive framework (from 1D to 2D) has been shown to allow a substantial benefit of resolution enhancement in two dimensions, successfully addressing the flaw in the previous 1D spatial-domain framework which would fail to describe imaging of a 2D-extended sample. Future work might include investigations to achieve the isotropic 2D super-resolution as well as to its best absolute spatial resolution by optimizing the system parameters. The proposed scheme is expected to provide a new way to add super-resolution capability to nonlinear optical microscopy tools to become more useful for biological research.