I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. OPTICAL SETUP DESCRIP...III. THEORYIV. EXPERIMENTAL RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESFar-field fluorescent super-resolution techniques such as stimulated emission depletion,11. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: Stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994). https://doi.org/10.1364/ol.19.000780 structured illumination microscopy,22. R. Heintzmann and T. Huser, “Super-resolution structured illumination microscopy,” Chem. Rev. 117(23), 13890–13908 (2017). https://doi.org/10.1021/acs.chemrev.7b00218 photoactivated localization microscopy,33. S. Manley, J. M. Gillette, G. H. Patterson, H. Shroff, H. F. Hess, E. Betzig, and J. Lippincott-Schwartz, “High-density mapping of single-molecule trajectories with photoactivated localization microscopy,” Nat. Methods 5(2), 155–157 (2008). https://doi.org/10.1038/nmeth.1176 and stochastic optical reconstruction microscopy44. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006). https://doi.org/10.1038/nmeth929 have become, over recent years, a standard in biomedical imaging. These methods produce images with spatial resolution reaching values way below the diffraction limit of light. The techniques mentioned above exploit sub-diffraction limited imaging of non-linear specimen responses achieved by labeling with fluorescent dyes or quantum dots. Artificial labeling is also popular for providing a high degree of specificity. However, several studies have shown that labeling changes the behavior of the studied biological specimen.5,65. U. Schnell, F. Dijk, K. A. Sjollema, and B. N. G. Giepmans, “Immunolabeling artifacts and the need for live-cell imaging,” Nat. Methods 9(2), 152–158 (2012). https://doi.org/10.1038/nmeth.18556. Z. Xiaoling, L. Yuchi, Z. Meirong, N. Xiaobing, and H. Yinguo, “Calibration of a fringe projection profilometry system using virtual phase calibrating model planes,” J. Opt. A: Pure Appl. Opt. 7(4), 192–197 (2005). https://doi.org/10.1088/1464-4258/7/4/007 Therefore, label-free imaging techniques are a more appropriate choice in many biomedical applications. No need for labeling also allows for studying artificial micro and nanostructures.7,87. P. Bouchal, P. Dvořák, J. Babocký, Z. Bouchal, F. Ligmajer, M. Hrtoň, V. Křápek, A. Faßbender, S. Linden, R. Chmelík, and T. Šikola, “High-resolution quantitative phase imaging of plasmonic metasurfaces with sensitivity down to a single nanoantenna,” Nano Lett. 19(2), 1242–1250 (2019). https://doi.org/10.1021/acs.nanolett.8b047768. T. Fordey, P. Bouchal, P. Schovánek, M. Baránek, Z. Bouchal, P. Dvořák, M. Hrtoň, K. Rovenská, F. Ligmajer, R. Chmelík, and T. Šikola, “Single-shot three-dimensional orientation imaging of nanorods using spin to orbital angular momentum conversion,” Nano Lett. 21(17), 7244–7251 (2021). https://doi.org/10.1021/acs.nanolett.1c02278 Nonetheless, breaking the diffraction limit in label-free imaging techniques is more challenging because of the missing non-linear specimen response.99. K. Wicker and R. Heintzmann, “Resolving a misconception about structured illumination,” Nat. Photonics 8(5), 342–344 (2014). https://doi.org/10.1038/nphoton.2014.88Quantitative phase imaging (QPI) has established an irreplaceable role among label-free imaging techniques thanks to its capability to quantitatively measure morphology and intrinsic specimen contrast with nanoscale sensitivity.1010. Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). https://doi.org/10.1038/s41566-018-0253-x The possible super-resolution QPI will satisfy the need for quantitative observation of previously unresolved specimen features and allow increasing the space-bandwidth product (SBP),1111. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, C. Ferreira, and Z. Zalevsky, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13(3), 470 (1996). https://doi.org/10.1364/josaa.13.000470 crucial for high-throughput studies. High SBP is important in identifying rare events, for example, in drug discovery,1212. P. Lang, K. Yeow, A. Nichols, and A. Scheer, “Cellular imaging in drug discovery,” Nat. Rev. Drug Discovery 5(4), 343–356 (2006). https://doi.org/10.1038/nrd2008 cancer-cell biology,13,1413. B. Gál, M. Veselý, J. Čolláková, M. Nekulová, V. Jůzová, R. Chmelík, and P. Veselý, “Distinctive behaviour of live biopsy-derived carcinoma cells unveiled using coherence-controlled holographic microscopy,” PLoS One 12(8), e0183399 (2017). https://doi.org/10.1371/journal.pone.018339914. O. Tolde, A. Gandalovičová, A. Křížová, P. Veselý, R. Chmelík, D. Rosel, and J. Brábek, “Quantitative phase imaging unravels new insight into dynamics of mesenchymal and amoeboid cancer cell invasion,” Sci. Rep. 8(1), 012020 (2018). https://doi.org/10.1038/s41598-018-30408-7 or stem-cell research.1515. M. R. Costa, F. Ortega, M. S. Brill, R. Beckervordersandforth, C. Petrone, T. Schroeder, M. Götz, and B. Berninger, “Continuous live imaging of adult neural stem cell division and lineage progression in vitro,” Development 138(6), 1057–1068 (2011). https://doi.org/10.1242/dev.061663 The recent development of automated data analysis and classification by artificial intelligence16,1716. L. Strbkova, D. Zicha, P. Vesely, and R. Chmelik, “Automated classification of cell morphology by coherence-controlled holographic microscopy,” J. Biomed. Opt. 22(8), 1–9 (2017). https://doi.org/10.1117/1.jbo.22.8.08600817. Y. Jo, H. Cho, S. Y. Lee, G. Choi, G. Kim, H.-s. Min, and Y. Park, “Quantitative phase imaging and artificial intelligence: A review,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–14 (2019). https://doi.org/10.1109/jstqe.2018.2859234 exaggerates this ever-increasing demand for high-resolution quantitative data. So far, the proposed approaches to QPI super-resolution are based on oblique illumination,18,1918. V. Micó, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express 16(23), 19260–19270 (2008). https://doi.org/10.1364/oe.16.01926019. M. Ďuriš, P. Bouchal, K. Rovenská, and R. Chmelík, “Coherence-encoded synthetic aperture for super-resolution quantitative phase imaging,” APL Photonics 7(4), 046105 (2022). https://doi.org/10.1063/5.0081134 structured illumination,2020. S. Chowdhury, W. J. Eldridge, A. Wax, and J. A. Izatt, “Structured illumination multimodal 3D-resolved quantitative phase and fluorescence sub-diffraction microscopy,” Biomed. Opt. Express 8(5), 2496–2518 (2017). https://doi.org/10.1364/boe.8.002496 and speckle illumination,21,2221. Y. Park, W. Choi, Z. Yaqoob, R. Dasari, K. Badizadegan, and M. S. Feld, “Speckle-field digital holographic microscopy,” Opt. Express 17(15), 12285–12292 (2009). https://doi.org/10.1364/oe.17.01228522. Y. Baek, K. Lee, and Y. Park, “High-resolution holographic microscopy exploiting speckle-correlation scattering matrix,” Phys. Rev. Appl. 10(2), 024053 (2018). https://doi.org/10.1103/physrevapplied.10.024053 which, combined with post-processing, provide synthetic images with an effectively enlarged numerical aperture (NA). These synthetic aperture methods enhance resolving power by essentially multiplexing the spatial-frequency content of the object spectrum into an unused degree of freedom in the system, sacrificing acquisition speed, quantitative information accuracy, or a field of view (FOV).Recent advances in superoscillatory hotspot creation23–2523. F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9(3), 1249–1254 (2009). https://doi.org/10.1021/nl900201424. K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, and C.-W. Qiu, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser Photonics Rev. 8(1), 152–157 (2014). https://doi.org/10.1002/lpor.20130012325. G. Chen, Z.-Q. Wen, and C.-W. Qiu, “Superoscillation: From physics to optical applications,” Light: Sci. Appl. 8(1), 56 (2019). https://doi.org/10.1038/s41377-019-0163-9 that allowed the development of novel approaches to coherent label-free super-resolution microscopy could also be adopted for QPI. However, current implementations of superoscillations also sacrifice some of the valuable microscope properties similar to the synthetic aperture methods. Band-limited fields containing superoscillations oscillate locally faster than the highest Fourier component. When carried over to optical imaging, this means that the focal spot can be made much smaller than allowed by the Abbe–Rayleigh limit. This was first investigated in 1952 by di Francia,2626. G. T. di Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9(3), 426–438 (1952). https://doi.org/10.1007/BF02903413 but only recently have these principles been applied to practical microscopy.27,2827. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). https://doi.org/10.1038/nmat328028. E. T. F. Rogers, S. Quraishe, K. S. Rogers, T. A. Newman, P. J. S. Smith, and N. I. Zheludev, “Far-field unlabeled super-resolution imaging with superoscillatory illumination,” APL Photonics 5(6), 066107 (2020). https://doi.org/10.1063/1.5144918 A superoscillatory sub-diffraction limited focal hotspot can be produced, for example, by coherently illuminating a specially designed mask of concentric annuli of varying complex transmission and widths.2727. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). https://doi.org/10.1038/nmat3280 The concentric annuli mask design can push the central hotspot radius significantly beyond the diffraction limit, but at the cost of high-intensity sidelobes,2727. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). https://doi.org/10.1038/nmat3280 which degrade the image quality in standard wide-field imaging. An alternative approach to amplitude and phase modulation is the application of light states with spatially structured polarization, such as the focusing of radially and azimuthally polarized Laguerre–Gaussian beams.29,3029. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5(2), 86–92 (2018). https://doi.org/10.1364/optica.5.00008630. X. Liu, W. Yan, Z. Nie, Y. Liang, Y. Wang, Z. Jiang, Y. Song, and X. Zhang, “Longitudinal magnetization superoscillation enabled by high-order azimuthally polarized Laguerre-Gaussian vortex modes,” Opt. Express 29(16), 26137–26149 (2021). https://doi.org/10.1364/oe.434190 The pioneering experimental research utilizing superoscillations initially demonstrated the super-resolution imaging only in a very small FOV2727. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). https://doi.org/10.1038/nmat3280 dictated by the distance of the first high-intensity sidelobe. To remove the FOV constraint, Rogers et al.2828. E. T. F. Rogers, S. Quraishe, K. S. Rogers, T. A. Newman, P. J. S. Smith, and N. I. Zheludev, “Far-field unlabeled super-resolution imaging with superoscillatory illumination,” APL Photonics 5(6), 066107 (2020). https://doi.org/10.1063/1.5144918 combined confocal detection with superoscillatory illumination. They create the super-resolution image thanks to the coherent illumination pattern with a sub-diffraction limited central hotspot and strong sidelobes. Subsequently, confocal detection eliminates the image distorting sidelobe effects at the cost of scanning the illumination pattern. Despite the great potential for resolution improvement, intensity imaging does not apply to most biological and other weakly scattering specimens and lacks quantitative information. Implementation of similar principles in QPI is thus a desirable yet challenging task due to the complexity and susceptibility of interferometric systems.In this paper, we propose a method that does not have to sacrifice any of the favorable microscope properties to achieve super-resolved QPI. To the best of our knowledge, we show for the first time that partially coherent broad-source interferometers are capable of single-shot widefield super-resolution imaging by shaping the so-called coherence gate.3131. M. Ďuriš and R. Chmelík, “Coherence gate manipulation for enhanced imaging through scattering media by non-ballistic light in partially coherent interferometric systems,” Opt. Lett. 46(18), 4486–4489 (2021). https://doi.org/10.1364/OL.432484 Our approach is based on the fact that the point spread function (PSF) of the partially coherent system is a product of the shaped coherence-gating function1919. M. Ďuriš, P. Bouchal, K. Rovenská, and R. Chmelík, “Coherence-encoded synthetic aperture for super-resolution quantitative phase imaging,” APL Photonics 7(4), 046105 (2022). https://doi.org/10.1063/5.0081134 (CGF) and the function describing the diffraction-limited image spot (Airy pattern). We shape the CGF by manipulating the illumination in the conjugated source plane similarly to the superoscillatory hotspot creation techniques. The product of the superoscillatory CGF with the Airy spot created by the objective in the object arm minimizes the sidelobes in the unbounded region while the CGF central peak delivers the super-resolving power. The minimization of sidelobes and resolution improvement co-occur in the entire field of view and allow single-shot widefield imaging. The imaging thus resembles confocal detection but with parallel filtration of all image points in the field of view. The images maintain quantitative phase information and extend the potential of superoscillations toward the QPI.We first demonstrate the effects of the superoscillatory CGF using simulated data. Then, due to the highly aberrated pupil plane of our experimental setup, we focus in the experimental part on a limiting case between the superoscillatory and super-resolution CGF. In both situations, the hotspot width is below the Rayleigh criterion. The distinction criterion between the super-resolution function and the superoscillatory one was proposed by Huang et al.2424. K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, and C.-W. Qiu, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser Photonics Rev. 8(1), 152–157 (2014). https://doi.org/10.1002/lpor.201300123 (we provide more details on the definition of the superoscillatory and super-resolution focal spot in the supplementary material). We create the CGF in this limiting case by using a simple amplitude annular mask, which proves experimentally robust. We demonstrate experimentally QPI resolution enhancement using only the limiting case, but the principle of our method is extendable to the superoscillatory focal spot region, promising higher resolution improvement. An experiment with a phase resolution target shows a resolving power improvement of 19%, and we show practical feasibility by applying the proposed method to the imaging of biological specimens.

II. OPTICAL SETUP DESCRIPTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. OPTICAL SETUP DESCRIP... <<III. THEORYIV. EXPERIMENTAL RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESThe proposed principles generally apply to various partially coherent interferometric systems. Without loss of generality, we will further describe the optical setup and theoretical framework of the used coherence-controlled holographic microscope3232. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt'ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21(12), 14747–14762 (2013). https://doi.org/10.1364/oe.21.014747 (CCHM), commercially available as the Telight Q-Phase. The optical setup (see Fig. 1) is an adaptation of the Mach–Zehnder interferometer. It consists of an object and reference arm containing two optically equivalent microscope systems. This holographic setup guarantees off-axis hologram formation in the interference plane (IP) for broad sources of an arbitrary degree of coherence. The possibility of using partially coherent sources is provided by the diffraction grating (DG; transmission phase grating with groove frequency 150 mm−1, blazed at 760 nm for the first diffraction order) implemented in the reference arm according to principles proposed by Leith and Upatnieks.3333. E. N. Leith and J. Upatnieks, “Holography with achromatic-fringe systems,” J. Opt. Soc. Am. 57(8), 975 (1967). https://doi.org/10.1364/josa.57.000975 In our system, an LED (LED Engin LZ4-00R208, peak wavelength at 660 nm, power up to 2.9 W) is used for illumination to provide a spatially broad incoherent source, and the illuminating light is made quasi-monochromatic after passing the interference filter (IF) with a central wavelength of 660 and 10 nm full width at half maximum. The source is imaged by a pair of achromatic doublets (simplified as L in Fig. 1; focal lengths 63.5 and 350 mm) through a beam splitter (BS) to the front focal planes of the condensers (C; Nikon LWD condenser lenses, 0.52 NA, with adjustable aperture stop). This plane in object and reference arms and respective condenser properties can be described according to Ref. 3434. R. Chmelik, M. Slaba, V. Kollarova, T. Slaby, M. Lostak, J. Collakova, and Z. Dostal, “The role of coherence in image formation in holographic microscopy,” Prog. Opt. 59, 267–335 (2014). https://doi.org/10.1016/b978-0-444-63379-8.00005-2 by the pupil functions PCo(Kt) and PCr(Kt), respectively, where Kt = (Kx, Ky) is the transverse wave vector of a plane wave behind condensers. The coordinates of Kt are proportional to the respective source point (pupil-plane) coordinates. For this reason, pupil properties can be characterized by a function of Kt. We use reduced wave vector notation |K| = 1/λ, where λ is the wavelength of light, and K = (Kt, Kz) = (Kx, Ky, Kz), where Kz=|K|2−|Kt|2. We modulate the condenser pupil planes to produce the sub-diffraction limited coherence gate, as explained in Sec. . The fundamental image properties also depend on the parameters of the object and reference arm objective lenses (O; Nikon Plan Fluorite Objectives, 10x/0.3 NA/16 mm WD) in combination with tube lenses (TL; Nikon, focal length 200 mm), characterized by the pupil functions POo(Kt) and POr(Kt). Stepper and piezo motors provide fine adjustment of the microscope optical components, which we use for the measurement of the coherence-gating function. The holograms are recorded in IP using an Andor Zyla 4.2 sCMOS camera.As shown in Fig. 1, we place the phase or amplitude mask in one or both of the front focal planes of the condensers. We designed the masks to shape the CGF when imaging with 10x/0.3 NA objective lenses. In simulations, we assume the phase mask is composed of concentric annuli, with the phase shift being either 0 or π radians. We also carried out simulations with the amplitude mask subsequently used in experiments. The amplitude mask is a single annulus cut by a laser cutter into a metal sheet. An inner circle of the annulus has a diameter of 16.4 mm. The outer circle diameter is about 18 mm, but more importantly, the pupil diameter in the front focal plane of the condensers is limited by the aperture stop to ∼17.3 mm (corresponding to 0.30 condenser NA).

III. THEORY

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. OPTICAL SETUP DESCRIP...III. THEORY <<IV. EXPERIMENTAL RESULTSV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESQuantitative phase information can be extracted from the measured holograms. As we work with the off-axis holographic setup, we reconstruct holograms by carrier removal in the Fourier plane.3232. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt'ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21(12), 14747–14762 (2013). https://doi.org/10.1364/oe.21.014747 In partially coherent systems, the hologram cross-correlation term depends on the transversal displacement Δq = (Δx, Δy) and relative time-delay τ of the object-scattered and reference fields. The cross-correlation function is conveniently described by a mutual coherence function3131. M. Ďuriš and R. Chmelík, “Coherence gate manipulation for enhanced imaging through scattering media by non-ballistic light in partially coherent interferometric systems,” Opt. Lett. 46(18), 4486–4489 (2021). https://doi.org/10.1364/OL.432484 (MCF) Γ(q, q − Δq, τ) of the two fields, where q = (x, y) is the position of a point in the image plane specified by the coordinates of the optically conjugated point in the object plane. The modulus and phase image for particular Δq and τ are obtained as the modulus and argument of Γ, respectively. The interferometric imaging for a given time-delay τ and transverse displacement Δq can be called a partial MCF measurement.3131. M. Ďuriš and R. Chmelík, “Coherence gate manipulation for enhanced imaging through scattering media by non-ballistic light in partially coherent interferometric systems,” Opt. Lett. 46(18), 4486–4489 (2021). https://doi.org/10.1364/OL.432484 The complete MCF is acquired by measuring and reconstructing holograms for all accessible Δq and τ. In this work, we use in experiments quasi-monochromatic illumination. Therefore, the influence of temporal coherence is minimal and manifests mainly as a speckle noise reduction. We set τ = 0 at the beginning of each experiment. The standard imaging conditions in low-coherence interferometers are when Δq = (0, 0). We use this setting for the majority of our experiments. However, as we show further, the complete MCF measurement and hence the manipulation with Δq is crucial for a measurement of the coherence-gating function. Our further analysis will stay within the limits of scalar wave approximation. More detailed mathematical derivations of the following equations are provided in the supplementary material. If we assume complete spatial source incoherence, τ = 0, and Δq as a parameter, the expression for the measured MCF, has according to Ref. 1919. M. Ďuriš, P. Bouchal, K. Rovenská, and R. Chmelík, “Coherence-encoded synthetic aperture for super-resolution quantitative phase imaging,” APL Photonics 7(4), 046105 (2022). https://doi.org/10.1063/5.0081134, the formwhere tq is a complex transmission of the specimen, the symbol ⊗ denotes convolution, and hq;Δq=poqG*q−Δq is a PSF of the imaging system, where poq=∬POoKtexp2πiKt⋅qd2Kt andGq=∬PCo*KtPCrKtPOrKtexp2πiKt⋅qd2Kt.(2)We call function Gq the coherence-gating function19,3119. M. Ďuriš, P. Bouchal, K. Rovenská, and R. Chmelík, “Coherence-encoded synthetic aperture for super-resolution quantitative phase imaging,” APL Photonics 7(4), 046105 (2022). https://doi.org/10.1063/5.008113431. M. Ďuriš and R. Chmelík, “Coherence gate manipulation for enhanced imaging through scattering media by non-ballistic light in partially coherent interferometric systems,” Opt. Lett. 46(18), 4486–4489 (2021). https://doi.org/10.1364/OL.432484 (CGF). The integration regions in poq and Gq are given by the extent of the pupil functions POoKt and PCo*KtPCrKtPOrKt, respectively. These boundaries define the band-limit of poq and Gq. The CGF provides filtering of multiply scattered light when imaging through turbid media.31,3231. M. Ďuriš and R. Chmelík, “Coherence gate manipulation for enhanced imaging through scattering media by non-ballistic light in partially coherent interferometric systems,” Opt. Lett. 46(18), 4486–4489 (2021). https://doi.org/10.1364/OL.43248432. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt'ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express 21(12), 14747–14762 (2013). https://doi.org/10.1364/oe.21.014747 Here we do not intend to use the coherence gate to mitigate unwanted scattering effects, but we unconventionally shape the coherence gate to obtain sub-diffraction limited PSF. For circular apertures, we can describe the CGF Gq and poq using the Bessel function of the first kind as Gq=2J1(μ)/(μ) and poq=2J1(ν)/(ν), where μ=2πKNACq and ν=2πKNAOq, with NAC ≤ NAO.To obtain the sub-diffraction limited resolution of QPI argΓq;Δq=(0,0), systems’s PSF hq=poqG*q must have the central peak radius below the diffraction limit. To maintain quantitative phase information in the image, the sidelobes of the PSF must also be negligible. Numerous studies24,26,27,3524. K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, and C.-W. Qiu, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser Photonics Rev. 8(1), 152–157 (2014). https://doi.org/10.1002/lpor.20130012326. G. T. di Francia, “Super-gain antennas and optical resolving power,” Il Nuovo Cimento 9(3), 426–438 (1952). https://doi.org/10.1007/BF0290341327. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). https://doi.org/10.1038/nmat328035. N. Reza and L. Hazra, “Toraldo filters with concentric unequal annuli of fixed phase by evolutionary programming,” J. Opt. Soc. Am. A 30(2), 189–195 (2013). https://doi.org/10.1364/josaa.30.000189 have shown that a superoscillatory focal spot can be created by coherently illuminating a phase or amplitude mask composed of concentric annuli of different widths and complex transmission. Superoscillations are then formed by constructive and destructive interference near the focal spot. As we use partially coherent illumination in our microscope system, it is not possible to create the superoscillatory focal spot observable in the field’s intensity by interference as proposed for coherent light. However, we can adopt the principles normally applied to coherent systems and shape the system’s PSF, the product of G*q−Δq and poq, by altering one or both of these functions. By modulating the pupil function POoKt of the object-arm objective, we can affect poq, but as Eq. (2) suggests, we have more options for Gq, because this function can be shaped by modulating one or more pupil functions PCoKt, PCrKt and POrKt of the condensers and the reference-arm objective, respectively. It is also experimentally easier to modulate the condenser pupil planes. Therefore, we will focus on shaping the CGF. However, similar results can be achieved by shaping poq, or both at the same time. Equation (2) describing CGF formation shows that Gq can be shaped similarly to coherent imaging even though the plane waves exp2πiKt⋅q superposed in Eq. (2) are mutually incoherent. The PCo*KtPCrKtPOrKt dictates whether these plane waves are constructively or destructively superposed. This allows us to use approaches designed for coherent imaging even in a system operating with partially coherent light. The expression in Eq. (2) is in fact van Cittert–Zernike theorem,3636. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). which describes the relationship between the mutual coherence function (CGF in our case) and the modulation of the pupil plane for partially coherent broad source illumination. As we can control the constructiveness of the plane wave superposition, theoretically, it should be possible to create observable superoscillations in partially coherent systems. However, not in the field’s intensity but in the mutual coherence of two fields (in our case, the CGF), hence the need for the interferometric system.For demonstration, we simulate the imaging and calculate the PSFs for three cases with different CGF shapes: first, the diffraction-limited case, when a full unmodulated condenser aperture is assumed; second, the limiting case of the superoscillation, when the amplitude mask with narrow annulus is used and the CGF is represented by the Bessel function J0(2πKNACq); and third, the case with a superoscillatory CGF produced by three-zone phase modulation. For all three cases, we assume that the poq function is the Airy pattern for NAO = NAC = 0.30, and this function is represented in Figs. 2(a)–2(c) by yellow dashed curves. The CGF Gq for the diffraction limited case is also the Airy pattern [see the red dashed curve in Fig. 2(a)]. The CCHM PSF [the product of poq and Gq] is in Figs. 2(a)–