Three-dimensional wide-field pump-probe structured illumination microscopy.

We propose a new structured illumination scheme for achieving depth resolved wide-field pump-probe microscopy with sub-diffraction limit resolution. By acquiring coherent pump-probe images using a set of 3D structured light illumination patterns, a 3D super-resolution pump-probe image can be reconstructed. We derive the theoretical framework to describe the coherent image formation and reconstruction scheme for this structured illumination pump-probe imaging system and carry out numerical simulations to investigate its imaging performance. The results demonstrate a lateral resolution improvement by a factor of three and providing 0.5 µm level axial optical sectioning.


Introduction
The development of high-sensitivity pump-probe microscopy has enabled molecular imaging and characterization in the fields of materials science, biomedicine, and art conservation [1]. Many pump-probe microscopy modalities are based on higher order optical processes, such as transient absorption (TA) and stimulated Raman scattering (SRS), require high peak energy excitation. As a result, typical implementation of pump-probe microscopy involves tight focusing of combined "pump" and "probe" laser beams and scanning the localized focal spot to map out the whole sample volume. Using this well-established approach, SRS and TA based pump-probe microscopy has demonstrated high speed imaging with sub micrometerscale spatial resolution in visualizing the fine structures of cells and tissues with molecular specificity [2,3]. However, the optical resolution of microscopy is limited by the wave-like characteristic of the light, more specifically diffraction [4]. Many interesting biological and chemical structures are smaller than this diffraction limit. There are many recent advances in overcoming this diffraction limited resolution. These efforts have been mostly focused on fluorescent imaging utilizing unique properties of fluorescence processes. One group of these technologies utilize photoswitchable fluorophores and the centroid of the individual emitter is localized based on stochastic nature of fluorophore activation including photoactivated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM) [5][6][7]. Another group of super-resolution technologies uses reversible saturable fluorophores and patterned excitation to decrease the fluorescence emission volume for resolution enhancement. Reversible saturable optical fluorescence transitions (RESOLFT), Stimulated emission depletion (STED), and ground-state depletion (GSD) microscopies are representative technologies in this group [8][9][10]. Structured illumination microscopy (SIM) is a super-resolution method that also relies on patterned excitation, but it relaxes constraints on fluorophores since it is not based on manipulating the population in different states [11][12][13][14][15]. SIM illuminates the sample with spatially structured light to shift the high frequency information into the passband of the imaging system. While significant progresses have been made in fluorescence imaging, the advances in achieving super-resolution in pump-probe microscopy are more limited. For sub diffraction resolution, Wang et al. applied RESOLFT to TA based pump-probe microscopy to image graphene and graphite nanoplatelets [16] and Silva et al. adopted STED concept to SRS based pump-probe microscopy to image diamond [17]. These approaches can only be applied to the sample that can withstand high enough flux of laser beam to generate 'saturated transient absorption' for TA and 'decoherence' for SRS. Lu et al. developed a super-resolution imaging method based on spatiotemporal modulation and it can readily be adopted to the pump-probe microscopy with general samples [18]. However, the imaging speeds of these methods are limited because of its point scanning configuration. As an alternative method to point scanning, there have been publications to implement wide-field pump-probe microscopy based on coherent anti-Stokes Raman scattering (CARS) [19][20][21]. People have also applied SIM to pump-probe microscopy to increase the lateral resolution in two dimensions in various configurations including point scanning, line scanning and wide-field [22][23][24][25]. In this article, we extend these previous works to three dimensions (3D) in widefield configuration and propose a scheme that is compatible with pump-probe modalities where signal wavelength is not shifted from those of input beams (for example, signal wavelength is anti-Stokes shifted for CARS, but it is not for SRS). We provide the theoretical framework to implement 3D SIM pump-probe microscopy in the wide-field configuration. To validate the proposed method, numerical simulations are carried out on a planar and a nonplanar resolution targets, and 3D HeLa cell data from the online public source [26][27][28]. The results demonstrate that our scheme for 3D SIM pump-probe can provide three-dimensional imaging with nearly three times higher lateral resolution than that of conventional pumpprobe imaging.

Configuration of a wide-field pump-probe structured illumination microscope (ppSIM)
We consider a wide-field pump-probe microscope where a sample is illuminated with structured pump beams in Fig. 1(a). Pump-probe signal is detected as a small change of probe beam and the signal is a function of pump beam intensity [1]. The pump and probe beams are arranged in a collinear geometry and focused at the back aperture of the microscope objective resulting in a uniform illumination of the objective field of view for the probe beam and a structured illumination for the pump beam. The transmitted signal is imaged by a second objective. A sharp long-pass filter (not shown in Fig. 1(a)) separates the pump beam from the probe beam. For lock-in detection of the probe signal, a light chopper is used to gate the pump beam at 50% duty cycle. This modulation transfers from the pump beam onto the probe beam by the nonlinear light-matter interactions and the probe beam passing through the sample object is detected by a camera. The frame rate of the camera will be set at four times higher frequency than the chopper frequency to allow easy extract of the lock-in signal based on homodyne detection. The chopper modulation and the camera sampling are synchronized by phase-locked loop and successive waveforms are synchronously averaged to suppress nonharmonic noise. The pump-probe signal which is directly proportional to the molecular concentration is related to the AC/DC ratio as shown in Fig. 1(b) [29]. To detect electromagnetic phase and amplitude of the probe beam, which are needed for image reconstruction, phase-shifting holography technique is adopted [30]. The probe beam is divided into three paths. One beam passes through the sample object, that is being illuminated by the structured pump beam, Probe 1 in Fig. 1(a). A second beam, Probe 2 in Fig. 1(a), is delayed by the piezoelectric transducer mirror for phase shifting and split into two. A third probe beam, Probe 3 in Fig. 1(a), freely propagates and it is interfered with a part of Probe 2 on the photo-diode surface to calibrate the motion of the delay mirror. For this calibration, we record the photo-diode signal changing the position of the delay mirror until we get a full period of the sinusoidal interference signal. If the accuracy and the repeatability of the phase shift measurement is not good enough, we may replace the photo-diode with a CMOS array or a CCD camera and analyze the fringe pattern itself between Probe 2 and Probe 3 with noncollinear configuration [30,31]. The other part of Probe 2 acts as a reference wave for Probe 1 and the phase-shifted interference patterns between two beams are recorded by the camera. Fig. 1. Wide-field pump-probe structured illumination microscope (ppSIM) design (a) Schematic diagram of an optical setup (b) Synchronous sampling of probe beam signal at four points per sinusoidal wave form in each pixel of the camera (directly applicable to 50% duty ratio on-off modulation).
We represent the first probe beam (object wave) complex amplitude on the camera plane as ( ) where the initial reference phase is assumed to be zero. The amplitude ( ) , A x y of the object wave may be derived by blocking the reference wave [30] or directly calculate from the previous four measurements of the interference intensity. To retrieve a 3D object information with a 2D detector, we shift the object in the axial direction and get a 3D data stack. We will provide an explanation on the details about getting the 3D data stack and the 3D coherent transfer function in a 4f system in the Appendix A. The resulting spatial frequency components of the illumination intensity for the ppSIM with (b) a single grating period and (c) grating period scanning. (d-e) The transfer function for (d) the conventional wide-field microscopy, (e) the single grating period ppSIM, and (f) grating period scanning ppSIM in (1) 3D, (2) mn plane, and (3) ms plane. The color in 3D transfer function represents the position in s axis, not a weighting.

Theoretical framework
In this article, we will discuss the pump-probe microscopy of which light-matter interaction by the pump beam excitation is linearly proportional to the intensity of the pump beam. For example, the exact relationship between the probe beam signal and the pump beam intensity is highly non-linear for stimulated Raman scattering (SRS), but it can be linearized when the pump beam intensity is relatively low [32]. Because of this intensity dependency, the pump beam illumination is incoherent whereas the scattering of the probe beam is coherent. This is similar to the previous work about SIM coherent anti-Stokes Raman scattering (CARS) microscopy in two dimensions [23]. We develop a theoretical framework to reconstruct a 3D super-resolved image in the wide-field structured illumination pump-probe microscopy based on coherent image theory [23, 33] considering incoherent pump beam excitation.
Assuming unit magnification without loss of generality, the image field in a coherent imaging system with a uniform probe beam can be described as where ( ) , , x y z is a spatial coordinate, ( ) After Fourier transforming the image field, the image spatial frequency distribution can be described as where ( ) , , m n s is a spatial frequency coordinate corresponding to the spatial coordinate ( ) , , x y z and H is called as a coherent transfer function (CTF). We denote Fouriertransformed entity by changing from lowercase to uppercase font in this article. The resolution of a microscope is limited by the "support" or "observable region" of the CTF, the region of three-dimensional reciprocal space (spatial Fourier space) where the CTF has nonzero values in Figs. 2(d)-2(f). Our pump-probe SIM has a larger effective CTF support in Figs. 2(e1)-2(e3) and Figs. 2(f1)-2(f3) than the normal observable region with uniform illumination in Figs. 2(d1)-2(d3) by shifting higher-resolution information into the CTF support using structured illumination in Figs. 2(a)-2(c), also known as spatial frequency mixing [14].
For our SIM pump-probe microscopy, the sample is illuminated by five pump beams of which one beam propagates parallel to the optical axis and four beams cross the sample at equal incidence angles θ in Fig. 2(a). For simplicity, only the case of S-pol in xz plane is considered. All five waves have the same wavelength and polarization parallel to y axis. In Spol geometry, the five plane waves can be described as where j e  denotes the field vector of each plane wave, the amplitude of each incident plane wave is assumed to be unity for simplicity, and ŷ and ẑ are unit vectors in each y and z directions. The pump beam wave vector is 2 / k π λ = assuming air as a propagation medium and λ is the wavelength of the pump beam. The phase of each wave is denoted as j φ . Since the temporal dependence is irrelevant in terms of average intensity finally measured by the camera, it is neglected here. The structured pump beam intensity for S-pol interference pattern can be calculated as By Fourier transforming this pump beam intensity, , we obtain the pump beam spectrum in the spatial-frequency domain as , , , where sin sin 2 mn k k θ θ π λ ≡ = and cos cos 1 2 Interference among the five pump beams produces a three-dimensional excitation intensity pattern that contains seventeen Fourier components at each difference frequency between the two illumination wave vectors in Figs. 2(a) and 2(b). Substituting the previous result into Eq. (4), and mathematically expanding, we get the corresponding image field in the Fourier space as Equation (8) can be written as a linear combination of 17 spatial frequency shifted object information passing through the imaging system's passband as, where j w refers to the multiplied terms in front of shifted object information j O and system's transfer function H in Eq. (8) and it is a function of the phases of structurally illuminated pump beams except the first one. This equation shows that the measured data gains new information which is not observable with a conventional uniform illumination microscope: the spatial frequency translation of the original object O moves new information into the support of H .
Although an image data captured by the pump-probe SIM system contains more spatial frequency information compared to the conventional wide-field pump-probe microscope, high frequency spatial information is translated and mixed with other components in the raw data. To restore the data, these information components must be separated and shifted back to the original position in the spatial Fourier space. For this image reconstruction, at least 17 raw image data ( ) , we are able to use its inverse matrix ( ) to un-mix multiplexed SI components by carrying out an , a poor choice will compromise final image signal-to-noise (S/N) level because each position of the sample will get different dose of light [34]. The ideal choice of structured illumination phases will be further discussed in Appendix B.
We explained how the wide-field pump-probe SIM can provide enhanced-resolution object information through shifting object information using a structured illumination with the conventional CTF. The effective CTF of the pump-probe SIM is the convolution of the seventeen-dot illumination structure of Fig. 2(b) with the conventional CTF support in Figs. 2(d1)-2(d3), resulting in the region shown in Figs. 2(e1)-2(e3). This region fills in the missing information in 3D which gives us a maximum factor of three lateral resolution extension and at the same time provides limited axial resolution from partially filled 3D Fourier space. The procedure can be repeated with additional illumination patterns to more densely fill the Fourier space. In this article, we suggest scanning grating period with a configurable grating device like a spatial light modulator or a digital mirror device. Figure  2(c) shows the illumination structure using sixteen grating periods. While the CTF for a single grating period coarsely fills the Fourier space and the space between CTF surfaces is empty in Figs. 2(e1)-2(e3), this empty space is effectively filled in when multiple grating periods are used in Figs. 2(f1)-2(f3).

Methods for numerical simulation
We performed numerical simulations to validate the theoretical framework to obtain superresolution 3D resolved pump-probe microscopy with structured illumination using computational Fourier optics method [35]. To implement Fourier optical image formation with a computer, discrete Fourier transform (DFT) is needed and fast Fourier transform (FFT) algorithm is used. After the original object information is Fourier transformed, we synthesize the optical field captured by the proposed setup using Eq. (8) with 17 independent phase shift because there are 17 independent object information copies mixed for the captured image. Subsequently, we construct a weighting matrix for the 17 object information copies. We have 17 captured images and 17 unknown object information copies, so that we can solve the equation to get an answer by matrix inversion. The next step is re-distribute the object information copies to the right position in the spatial Fourier space because the object information copies were shifted from the original position in Eq. (8). Stitching the re-shifted object information copies into one and inverse Fourier transforming it gives us a reconstructed SIM image. To make the validation more meaningful in the practical context, simulation parameters were taken from a typical case of stimulated Raman scattering (SRS) experiments. We assumed a nonlinear sample providing a pump-probe contrast similar to SRS at the Raman shift of 753 cm −1 (corresponding to the pyrrole breathing mode ν 15 of cytochrome c) that is resonantly excited by a 755 nm pump laser and a 800 nm Stokes laser [36]. For simplicity, we used the same numerical aperture for both pump beam (755 nm) excitation and probe beam (800 nm) detection in the transmission stimulated Raman gain geometry: we used low NA (0.68) to provide a general explanation about our pump-probe structured illumination microscopy (ppSIM) with artificial samples and compared the performance between the low NA (0.68) and the high NA (0.9) for an actual biological sample data. (logarithmic scale) are displayed for amplitude with their axes normalized with the lateral cutoff frequency of the conventional microscope. An isolated three bar pattern simulation results comparing the conventional microscope and ppSIM with grating period scanning in (d, e) lateral and (f, g) axial dimensions: (d, f) cross-sectional profiles in space and (e, g) field amplitude modulations according to the spatial frequency. For the conventional wide-field imaging, the three bar patterns inside the dashed red rectangle in (a1) appear blurred and they are barely resolvable where the period of element 1 (the most coarse set) is close to the coherent Abbe diffraction limit λ/NA (1.177 µm in this study). For the ppSIM imaging, on the other hand, three bars inside the dashed rectangles in (b1, c1) are clearly distinguished with more sharply defined edges.

Pump-Probe Structured Illumination Imaging of a planar target: Calibration Chart
To verify the theory, we show a simulated pump-probe structured illumination reconstruction of 1951 USAF test chart images as our pump-probe active sample in Fig. 3. The 3D test target is built to represent a sample distribution over the volume of 40 µm × 40 µm × 40 µm (in 512 × 512 × 256 image pixels) and total 10 different grating period (1.11 µm ~11.10 µm) is used. First, we compare the imaging result of a whole USAF target image which fits in the computational field of view in lateral dimension and has a single pixel thickness [28]. The lateral resolution improvement for the pump-probe SIM image in Figs. 3(b1) and 3(c1) can be clearly seen overall when compared with the conventional wide-field image in Fig. 3(a1). We also note that the image field in xz plane from the single pixel layer object is not localized near the sample location and generates extended interference pattern along axial dimension for the conventional wide-field microscope in Fig. 3(a2) and for the ppSIM with a single grating period in Fig. 3(b2), whereas the corresponding grating period scanning ppSIM field in xz dimension shows localized sharp peaks around the sample plane and more moderate interference pattern in Fig. 3(c2).
These observation also can be confirmed with Fourier domain field amplitude in both lateral and axial dimensions: the support of the transfer function for grating period scanning pump-probe SIM in Figs. 3(c3) and 3(c4) is increased in both lateral and axial dimensions compared with that for conventional wide-field microscopy in Figs. 3(a3) and 3(a4). We also notice that the grating period scanning ppSIM in Figs Fig. 2(d1) and the bandwidth in axial dimension is very narrow in Fig. 3(a4). On the other hand, the bandwidth of our pump-probe SIM is extended to ~500 lpmm (lines per mm) in the axial dimension in Fig. 3(c4) corresponding to the localized imaged field in Fig. 3(c2). For a more quantitative analysis about resolution in space, we synthesize isolated three bar patterns which follows the MIL-STD-150A standard about USAF 1951 resolution target [37] and compare the image performance between the conventional wide-field microscope and the ppSIM with grating period scanning. We adapted the procedure in the article about coherent SIM in scattering mode [33] to measure their corresponding field amplitude modulations, defined as max min max min Here, max A and min A are the maxima and minima, respectively, of the image field amplitude of test bars. Figures 3(d) and 3(f) show the field amplitude intensity profiles taken from the images of the lateral and the axial bar patterns. We plot the simulated field amplitude modulations observed in the conventional wide-field microscopy and the grating period scanning pump-probe SIM with respect to lateral and axial bar frequencies in in Figs. 3(e) and 3(g). In the lateral case, excellent agreement with the theoretical expectations are observed in Fig. 3(e). However, axial amplitude modulation plot shows lower contrast, fluctuation, and decreased bandwidth compared to that of axial transfer function from ~500 lpmm to ~150 lpmm in Fig. 3(g). This can be explained in the context of a general characteristic of bandpass filter especially related with coherent imaging. The measured optical field is a complex valued function and it can interfere with another field constructively or destructively according to the phase. For simplicity of explanation, we can approximate the axial transfer function of ppSIM as a shifted "rect" function,  To further demonstrate the 3D sectioning capability of grating period scanning pump-probe SIM, we image a model of the three letters of the MIT logo where individual letter is located in a different axial position (separated by 8.3 µm). The same simulation condition as in the previous section was used with the volume of 40 µm × 40 µm × 40 µm (in 512 × 512 × 256 image pixels) and total 10 different grating periods (1.11 µm ~11.10 µm). The width of the bar shape in x direction is 3.438 µm and each bar is separated by 2.5 µm in both x and y directions. Each letter intentionally overlaps its neighbor about 1/3 of the bar width (1.172 µm) in x direction to validate our method with a "real 3D" sample. Figure 4(a) shows the overall configuration of the original sample object in 3D. Comparing Figs. 4(b) and 4(c), it is clear that grating period scanning ppSIM has depth-sectioning 3D imaging capability that the conventional wide-field microscope doesn't have. The xy planes in Figs. 4(b1) and 4(c1) are located in the middle of the letter 'i' which shows distinct contrast over other letters for our ppSIM in Fig. 4(c1), whereas all three letters are there with similar contrast for the conventional microscope in Fig. 4(b1). In addition, we can also observe that the edge of the letter 'i' for the ppSIM in Fig. 4(c1) is sharper than that for the conventional microscope in Fig. 4(b1) because of the lateral resolution enhancement. The images of bar patterns located in different z locations are totally elongated over the whole axial dimension for the conventional wide-field microscope in Fig. 4(b2). On the other hand, the corresponding images of each bar patterns are clearly localized to the original positions for our ppSIM in Fig. 4(c2).  The original HEC1 protein data in Figs. 5(a1) and 5(a2) show isolated dot-like sharp features with low background, while the original DNA data in Figs. 5(e1) and 5(e2) have relatively bigger island-like features with wide diffuse background. The conventional widefield microscope loses sharp features of the original data and blurred severely, but several big chunks' locations seem to be at least correlated to the actual object position laterally in Figs. 5(b1) and 5(f1). However, the axial object information is completely distorted in the xz crosssectional view in Figs. 5(b2) and 5(f2). Comparing the images of HEC1 protein and DNA of HeLa cells in Fig. 5, we can observe the imaging performance of our grating scanning pumpprobe SIM is different according to the sample itself and NA of the objective lens. For the HEC1 protein, the imaging result of low NA pump-probe SIM and that of high NA pumpprobe SIM look similar and both cases successfully reconstruct original sample information in both lateral and axial dimensions in Figs. 5(c1), 5(c2), 5(d1), and 5(d2). On the other hand, for the DNA image stack, the low NA and the high NA pump-probe SIM results are quite different. The high NA case succeeds in reconstructing the sample information in both lateral and axial dimensions in Figs. 5(h1) and 5(h2). In contrast, for the low NA case, individual island-like features turned into many smaller dot-like features in xy plane in Fig. 5(g1) and all features lost contrast with distortion in xz plane in Fig. 5(g2).

Pump-Probe Structured Illumination Imaging of Biomolecules in HeLa Cells
This difference in imaging performance among conventional, low NA ppSIM, and high NA ppSIM comes from the difference in the frequency supports or the transfer functions in the Fourier space. The bandwidth in axial direction for the conventional microscope in Fig. 5 (f3) is so thin that it loses the axial information completely in Fig. 5(f2). We can consider our grating period scanning ppSIM transfer function in the axial direction as a bandpass filter in Figs. 5(g3) and 5(h3). The low NA ppSIM distorts the wide and blunt objects more severely than the narrow and sharp objects because it blocks the low frequency information and passes the high frequency information for the axial dimension in Fig. 5(g3). The high NA ppSIM transfer function for the axial dimension passes both the low and the high frequency information in Fig. 5(h3) so that the sample information is conserved well in Fig. 5(h2).

Discussion
We have shown numerical simulation results that validate the theoretical framework proposed to obtain 3D imaging capability through structured illumination microscopy (SIM) of pumpprobe active samples. It is worthwhile now to compare our work with existing pump-probe SIM publications. Previous works about pump-probe SIM are based on intensity measurement for 2D coherent imaging [22][23][24][25]. Let us call the reconstruction method in those works as the intensity SIM framework and the one that we are suggesting in this paper as the field SIM framework. The image intensity measured in a coherent system with a uniform probe beam is given by the nonlinear relation where r is the spatial position vector, ( ) where ω is the spatial frequency vector and H is a coherent transfer function (CTF) corresponding to the system PSF. Following the intensity SIM framework in the literature, we first consider the following hypothetically extended transfer function where j ω is the spatial frequency vector and F N is a number of copies of the original transfer function. We write a corresponding "enhanced-resolution" image to this extended transfer function and a uniform illumination of the pump beam as To reconstruct these enhanced-resolution components, the intensity SIM framework considers a structured illumination pattern through the original aperture, given by The intensity SIM framework is based on the mathematical similarity found between the two Fourier images, one with hypothetically extended CTF,

ET
H ω which is exactly the same as the one given by the reconstruction method that we suggest in this article (optical field based). Let us assume that the number of images necessary for the four-point lock-in process to calculate the AC/DC ratio in Fig. 1(b) are similar between both the intensity and field SIM frameworks and we will omit them when we compare the total number of images to acquire to reconstruct a single 3D resolved image for both frameworks. The synthetic CTF we suggested to get 3D resolution consists of ( ) times less number of pump beams and 3~5 times less number of images to reconstruct a single 3D image with the same synthetic CTF than the intensity SIM framework. This difference comes from the fact that object information is linearly related to the optical field while its relation with the light intensity is nonlinear for the coherent imaging [38,39].
In addition to the difference in the number of beams and the number of images to be acquired between two SIM frameworks, we also noticed that there is a difference in easiness of artifact removal. In Fig. 2, shifted copies of the original transfer function overlap one another and may generate an artifact pattern in the final reconstructed image. We observed a periodic line pattern (z direction) in the reconstruction result of Fig. 5(h) and could readily remove it by averaging or putting a proper weighting on the overlapped Fourier regions. This artifact removal process may be possible for the intensity SIM framework with non-linear deconvolution, but it is not as straight forward as the field SIM framework. As a result, we can more easily provide a higher fidelity reconstructed image with less artifact where the regions within the spectral domain passband are equally weighted while previous work reconstructed an image where some Fourier frequencies are over-weighted. Fully comparing these two approaches would be an interesting topic, but it is beyond the scope of the current paper.
There are many prior works on using a point-scanning configuration with a lock-in amplifier to get 3D resolution with pump-probe contrast [1,2,16,32]. The fastest pointscanning pump-probe microscope generates 512 × 512 pixels at a rate of 25 frames/s [2]. For a 512 × 512 × 256 pixel 3D image, the frame rate of the point-scanning system is about 0.1 frames/s. Our system needs 4 (four-points lock-in) × 4 (phase and amplitude for optical field) × 17 (un-mixing) × 16 (different grating periods) = 4352 images for one acquisition. We can easily find a camera with >2000 frames/s for a 512 x 512 pixel field of view in the market (for example, CP80-3-M-540 from Optronis). As a result, the final estimated frame rate of our method is about 0.5 frames/s (about 5 times faster than the point-scanning configuration) assuming that the camera can process all available probe photons without saturation. Now let us consider laser power utilization to be more realistic. If the imaging system is photon shot noise limited, the amount of photons that the system can handle determines the imaging speed with a given sensitivity. Point scanning pump-probe microscopes utilizes only a fraction of available laser power (<50mW for SRS, <2mW for TA) because of sample damage in biological specimen. However, point scanning pump-probe microscopes has inherent 3D imaging capability and a single scanning over the sample volume is enough to acquire the 3D object information [2,3,32]. Our grating period scanning ppSIM approach needs to take 17 different 3D data set for one grating period and repeat 10~16 times for different grating periods to reconstruct a single 3D resolved object data, but wide-field configuration makes it possible to fully utilize the available laser power with three times higher lateral resolution. In conclusion, our method has a potential to provide higher imaging speed than the point scanning pump-probe microscope with sufficient laser power.
The grating period scanning approach advocated in this paper is important as it allows much better filling of Fourier space than the conventional wide-field microscopy and previous 2D ppSIM approaches. As a result, our approach provides not only 3D imaging capability, but also 2D image with higher fidelity without non-linear image processing.

Conclusion
We have proposed a theoretical framework to implement three-dimensional (3D) wide-field super-resolution pump-probe microscopy utilizing structured illumination microscopy technique. In the wide-field pump-probe imaging configuration, we have added a phaseshifting holography to measure the optical field of the probe beam scattered by the sample and the pump beam. It is important to note that quantitative phase and amplitude imaging of the electric field is required for the reconstruction in these coherent image conditions. The structured illumination pump beam serves as a key element that encodes the missing information in both lateral and axial dimensions into the conventional imaging passband. A rigorous 3D Fourier domain framework has been established on how to extract and reconstruct the 3D contents into an image with an enhanced resolution in the context of coherent image formation (probe beam) with an intensity dependent structured excitation (pump beam). To check the validity of the proposed method, we have simulated a pumpprobe structured illumination microscope with three different sets of samples: a computersynthesized resolution test target, the MIT logo, and actual biomolecules in HeLa cells. The reconstructed ppSIM image has been examined and compared with the diffraction-limited coherent wide-field image to evaluate the resolution and depth sectioning performances. The results have clearly demonstrated the potential of our method to enable pump-probe microscopy to achieve more than 3 times better lateral resolution and 3D imaging capability over the conventional wide-field pump-probe system. In addition, we found the drawbacks of our framework with a low NA objective related with an offset of the transfer function in axial direction and using high NA objective could solve the problems. Future work might include the experimental implementation of the actual microscope and the investigation to achieve the 3D imaging capability instead of grating period scanning, and the development of simpler way to read the optical field without using phase-shifting holography. The proposed scheme is expected to provide a new way to add super-resolution and depth sectioning capability to wide-field pump-probe optical microscopy and make it a more useful tool for biological and material science.

Appendix A
Here, we mathematically derive the 3D coherent transfer function in the 4f system which usually consists of two positive lenses with the input plane located one focal length (f1) in front of the first lens and the output plane located one focal length (f2) after the second lens. There is a derivation for the 3D coherent transfer function for the single lens imaging system in the literature [39] and we will extend its methodology for the 4f system. However, we will use the terminology including the phasor direction for electric field and the various transforms from a different publication [38].
We will use , , x y z for spatial coordinates in the Cartesian coordinate system, spatial frequencies , , m n s in the , , x y z directions for the Fourier spectrum. Radial and angular coordinates in the polar coordinate system are defined as: x y − plane) and the observation plane (the 2 2 x y − plane).
Using scalar diffraction theory and the paraxial approximation in Fig. 6, the light field in the 2 2 x y − plane (z distance far away from the 1 1 x y − ) can be calculated as which is the Fresnel diffraction integral [38]. With the same method, the complex transmittance of a thin lens can be presented as Fig. 7. 4f optical imaging system with a thin object.
We apply the previous diffraction integral to the 4f optical imaging system with two lenses in Fig. 7. There is a thin object placed in a plane at a distance 1 z from the back focal plane of the first lens. Let us assume that we have an area type detector that can measure the light field and it is placed in the front focal plane of the second lens. There is a pupil mask with transmittance ( ) pupil plane). The pupil mask determines the maximum spatial frequency that the optical system can pass and finally the spatial resolution of the optical system. The field in the detector plane ( 4 4 x y − plane) can be presented as a convolution integral like with a new function ( ) , , h x y z and demagnification factor M as ( ) Let us now apply the result to an object with a finite thickness in Fig. 8. In this case, the object function is a 3D function ( ) , , o x y z . For each of the vertical sections in the thick object at a given position 1 z , its image in the image plane is given by Eq. (21). The total field in the image plane is the superposition of the contributions from the images of all sections. The superposition principle holds only if secondary diffraction in a thick object is neglected and if the object is semi-transparent. This assumption is called the first Born approximation. Under this approximation, the image of an object with finite thickness, i.e. the image of a 3D object, is the integration of Eq. (21) with respect to 1 z . The final image field in the image plane is To retrieve a 3D object information with a 2D detector, we need to scan the object in the axial direction in Fig. 9 and the light field in the image plane with a shifted object is represented as: for each position of the sample to get the equal dose of light so that the final image signal-to-noise (S/N) level is uniform over the field of view. Let us start with one-dimensional two beam SIM in Fig. 11(a) to grab an idea for the ideal phase set. In S-pol geometry, the two plane waves can be described as This can be easily understood using Euler's formula to convert each ( ) cos term into a matching complex number just like the phasor notation and complex number addition in twodimensional space in Fig. 11(b). Three complex numbers are equally distributed on the circle whose center is located in the origin. The sum of these three complex numbers with any arbitrary angular offset from Fig. 11(b) configuration is 0 because of the symmetry and this result doesn't change with an arbitrary number of complex numbers in Fig. 11(c). We can describe this result for our purpose as