The structured illumination for qDIC microscopy (Figure 1B) can be expressed as A m ξ = 2cos (2r/Λ+δ _m ) with m = 1, 2, … , 5, and ξ indicates the orientation indices of x and y, respectively. r is the spatial position vector, Λ is the stripe’s period, and δ _m is the phase shift. Passing through the sample, the object wave under A m ξ is diffracted and relayed by the telescope system MO₂-L₅ to the image plane. After diffraction of a distance of Δz, the diffraction patterns are recorded by the CCD₁ camera. The waves along the ±1st orders of A m ξ at the camera plane are U 1 ξ and U + 1 ξ . These two waves interfere with each other at the camera plane and the intensity distribution of the interferogram captured by the camera can be written as: I m ξ ( r ) = | U + 1 ξ ( r ) | 2 + | U − 1 ξ ( r ) | 2 + U + 1 ξ ( r ) U − 1 ξ ∗ ( r ) + U − 1 ξ ( r ) U + 1 ξ ∗ ( r ) = α 0 ξ + α 1 ξ ⁡ exp ( i 2 δ m ) + α 2 ξ ⁡ exp ( - i 2 δ m ) (1) where, α 0 ξ equals to | U + 1 ξ ( r ) | 2 + | U − 1 ξ ( r ) | 2 , α 1 ξ equals to [ I + 1 ξ ( r ) I − 1 ξ ( r ) ] 1 / 2 e x p [ i ( φ d i f f ξ + 2 π r / Λ ) ] , and α 2 ξ equals to [ I + 1 ξ ( r ) I − 1 ξ ( r ) ] 1 / 2 e x p [ − i ( φ d i f f ξ + 2 π r / Λ ) ] . Here,   φ d i f f ξ denotes the phase difference between   U 1 ξ and U − 1 ξ which is generated by the lateral shearing. And δ _m = 2 (m-1)π/5 (m = 1, 2, … , 5.) is the phase shift induced by laterally translating the fringe on DMD. Despite three-step phase-shifting is enough to solve these three terms   α 0 ξ , α 1 ξ , and   α 2 ξ , we use the five-step phase shifting to achieve a better reconstruction immune to phase shift error and environmental instability [33]. Accordingly, Eq. 1 can be rewritten as: ( 1 exp ( i 2 δ 1 ) exp ( - i 2 δ 1 ) 1 exp ( i 2 δ 2 ) exp ( - i 2 δ 2 ) 1 exp ( i 2 δ 3 ) exp ( - i 2 δ 3 ) 1 exp ( i 2 δ 4 ) exp ( - i 2 δ 4 ) 1 exp ( i 2 δ 5 ) exp ( - i 2 δ 5 ) ) ⋅ ( α 0 ξ α 1 ξ α 2 ξ ) = ( I 1 ξ ( r ) I 2 ξ ( r ) I 3 ξ ( r ) I 4 ξ ( r ) I 5 ξ ( r ) ) (2) Here   α 0 ξ , α 1 ξ , and   α 2 ξ can be solved in a least-square manner via multiplying both sides of Eq. 2 with the transpose of the coefficient matrix. Therefore, α 0 ξ , α 1 ξ , and α 2 ξ along the ξ-direction can be achieved, respectively. In order to compensate for the linear phase terms in α 1 ξ and α 2 ξ induced by the oblique illumination of the ±1st orders of the structured illuminations, an accurate calibration was performed. Actually, the calibration was performed with the same procedure but in the absence of any samples, and we can get new terms   α b 0 ξ , α b 1 ξ , and α b 2 ξ . Eventually, the pure phase difference of the sample can be obtained by φ d i f f ξ = A n g { α 1 ξ / α b 1 ξ } , where Ang{·} denotes the argument retrieval operator. Similarly, the phase difference φ d i f f ξ along the x- and y-axes can be obtained by rotating the fringe 90^o with the same calculation procedure. Ultimately, the phase distribution φ(r) can be obtained by integrating φ d i f f x and φ d i f f y [34]: φ ( r ) = I F T { − i [ v x F T { φ diff x } + v y F T { φ diff y } ] 2 π ( v x ) 2 + ( v y ) 2 } (3) where v ^x and v ^y are the coordinates in the Frequency domain, respectively. FT{·} and IFT{·} represent the Fourier transform and inverse Fourier transform, respectively.

Using the same stripes projection method as in phase imaging, a resolution enhancement in non-fluorescent imaging can be realized by using coherent structured illumination. The key to enhancing the spatial resolution in non-fluorescent/scattering imaging is the synthetic-aperture effect, as shown in Figure 1C, which can bring unobservable high-frequency information into the low-frequency supporting area through oblique illumination. Different from the phase imaging, the scattering imaging here records the diffraction patterns in an in-focus manner. Mathematically, the intensity images in the CCD₁ plane can be written as I m ξ ( r ) = | h c ( r ) ⊗ [ O ( r ) ⋅ A m ξ ] | 2 , where h _c (r) is the coherent point spread function of the system, ⊗ is the convolution operator, O(r) is the object transmittance function, and A m ξ is the structured illumination filed in the sample plane. After a Fourier transform, we can obtain the spectrum distribution of I m ξ ( r ) : I ˜ m ξ ( v ) = [ G 0 ξ ( v ) + exp ( i 2 δ m ) ⋅ G + 1 ξ ( v − v 0 ξ ) + exp ( − i 2 δ m ) ⋅ G − 1 ( v + v 0 ξ ) ] (4) where I ˜ m ξ ( v ) is the Fourier transform of I m ξ ( r ) , v is the spatial frequency vector, and G m ξ (m = 0, ±1) represent the spectral components along x- and y-directions. Notably, G + 1 ξ ( v − v 0 ξ ) and G − 1 ( v + v 0 ξ ) have the high-frequency spectrum surpassing the supporting pupil of the imaging system, which was downshifted by the oblique illumination and therefore passing through the imaging system. To acquire these three spectral components, five-step phase-shifting was performed with the phase shifts δ _m = 2 (m-1)π/5. After similar mathematical operations as described in Quantitative Differential Phase-Contrast (qDIC) Microscopy With Structured Illumination different spectra along with the 0th, and ±1st orders of the structured illuminations can be solved, and an extended-spectrum can be synthesized with: G S I ( v ) = 1 2 ⋅ [ G 0 x ( v ) + G 0 y ( v ) ] + G 1 x ( v − v 0 x ) + G − 1 x ( v + v 0 x ) + G 1 y ( v − v 0 y ) + G − 1 y ( v + v 0 y ) (5) Afterward, the resolution-enhanced image can be achieved using an inverse Fourier transform on Eq. 5 and multiplying weight factor. It is worth noting that in such a coherent imaging system the phase distribution of different G m ξ (m = 0, ±1) should be compensated e.g., with the experiment without any samples, making sure that there are no additional phase shifts between different terms in Eq. 5.

In the first experiment, a confirmatory experiment is carried out to demonstrate qDIC for live samples imaging without fluorescent labeling. For this purpose, live mouse adipose stem cells were used as phase samples. The magnification and numerical aperture of the imaging system MO₂-L₅ are 10× and 0.32, respectively. Two groups of binary patterns with the orientation along the x- and y-directions were loaded on the DMD, of which the period was set as ten pixels and the modulation depth is 1. The x- and y- orientated patterns were shifted by five times and each time had a phase shift δ _m = 2 (m-1)π/5 (m = 1, 2, … , 5.). The generated diffraction patterns were recorded by the CCD₁ camera, and shown in Figures 2A,B respectively. Using the reconstruction method described with Eqs. 1, 2, both the amplitude and phase derivatives of the sample are obtained. Figures 2C,D show the phase gradients along the x- and y- orientations, respectively. Despite the contours of the cells become visible in Figures 2C,D, the result is a mixture of amplitude and phase-gradient in a nonlinear manner. The final phase distribution of the sample is then obtained by integrating the phase derivatives along the x- and y- orientations, as shown in Figure 2F. Compared to the amplitude image of the sample shown in Figure 2E, where the structures of the cells are nearly invisible, the phase image (Figure 2F) visualizes clearly the structures of the cells, and notably in a quantitative manner. The comparison reveals that qDIC can not only visualize the transparent samples with high contrast but also provide us the quantitative information on the optical path difference (OPD) of the sample.

In the second experiment, resolution-enhanced nonfluorescent/scattering imaging using coherent structured illumination was proved by imaging SiO₂ beads (diameter: 500-nm). Binary fringe patterns were loaded on DMD to generate sinusoidal fringe stripes at the sample plane after being filtered in the Fourier plane. The period of the binary patterns was set as five pixels, and after the de-magnification, the period of the sinusoidal fringe stripes was 0.95 μm at the sample plane. As explained in Super-Resolution Scattering Imaging With Coherent Structured Illumination, the binary fringe patterns were shifted by five times (yielding the phase shifts δ _m = 2 (m-1)π/5, m = 1, 2, … , 5) along the x- and y-direction, and the generated diffraction patterns are recorded in sequence by CCD₁ located at the image plane. The super-resolution reconstruction is then realized with the method elaborated in Super-Resolution Scattering Imaging With Coherent Structured Illumination. The reconstructed image of SiO₂ beads is shown in Figures 3A,B shows the conventional wide-field image obtained using a perpendicular plane-wave illumination. It is clear that the coherent structured illumination provides a high-resolution image on the SiO₂ beads.

The numerical aperture (NA _detect = 0.32) of the detection objective MO₂ limits the lateral resolution to δ _plan = 1.44 μm for the conventional imaging using a perpendicular plane wave illumination. When the structured illumination is used, the illumination angle of the ±1st diffraction orders of the fringe stripes is θ _illum = 0.30 rad, and thus the theoretical lateral resolution can be estimated with δ _str = 0.82λ/(sin (θ _illum)+NA _detect) = 0.75 μm [35]. For a quantitative evaluation of the lateral resolution, ten random SiO₂ beads were randomly chosen, and the intensity distributions along the line crossing the center of each bead are analyzed, as shown in Figure 3C. The statistics on the ten beads tells that the averaged full widths at half maximum (FWHM) under these two illuminations are 1.86 ± 0.20 μm for perpendicular uniform illumination while 1.10 ± 0.10 μm for coherent structured illumination. When considering the non-negligible size (the diameter d = 500 nm) of the SiO₂ beads, the final resolution can be calculated by δ r e s = [ ( δ p l a n / s t r ) 2 − d 2 ] 1 / 2 , yielding the lateral resolution δ r e s = 1.80 ± 0.20 μm and 0.98 ± 0.10 μm for the conventional wide-field imaging mode and coherent structured illumination mode, respectively. It should be noted that there is a mismatch between the measured and theoretical resolutions due to the system’s aberration and other uncertain factors.

In the third experiment, the dual-modality (non-fluorescent scattering/fluorescent) imaging capability of the proposed SIM apparatus was demonstrated with lily anther as the sample. The coherent SIM image was shown in Figure 4B, and the fluorescence image was captured by the camera CCD₂ after being filtered by a color filter (600/50 nm, central wavelength/full-width at half maximum) and is shown in Figure 4C. Compared with the wide-field image in Figure 4A obtained by using a uniform illumination, the SIM image in Figure 4B shows more detailed structures and clearer background. Moreover, the fluorescent image in Figure 4C shows clear pollen structures (having autofluorescence) in the context of a clean background. It is also distinct that the SIM image (transmission) and the fluorescent image (reflection) have the opposite contrast for the same sample.