Single-shot extended-object super-oscillatory imaging

The pursuit of observing finer objects with optical microscopes has continued unabated over the past few centuries. The inherent wave property of light—diffraction—however, limits human observation by blurring the images. Many efforts have been devoted to overcoming the diffraction limit.¹ The scanning optical near-field imaging collects the evanescent waves that carry deep sub-diffraction details. This technique is yet hardly applicable to imaging inside cells since the scanned near-field tip would otherwise be contaminated.² An emergent non-invasive route to deeply sub-wavelength localization resolution utilizes a long dwell time to improve the sensitivity of waves interacting with sub-wavelength objects.^3,4 Several far-field optical microscopy techniques, such as patterned light illumination techniques⁵ and localization-based techniques,^6,7 significantly enhance optical resolution without the near-field limitations, but these techniques typically rely on fluorescent labeling and time-consuming fine scanning. Fluorescence microscopy is occasionally disabled since the fluorescent dye cannot be bound to inorganic objects, including silicon microchips and nanoparticles. Moreover, the diminished imaging speed caused by fine scanning hinders visualizing biological activities in real-time. Non-scanning structured illumination microscopy^8,9 and Fourier ptychographic microscopy^10,11 still require the acquisition of multiple images under active illuminations. Although these microscopy techniques suffer from fluorescent labeling, long scanning time, and active illuminations, the emerging single-shot super-oscillatory (SO) imaging technique^12–17 does not rely on these constraints.

SO phenomena¹⁸ allow for arbitrarily-high local spatial frequencies that are beyond the cutoff frequency within an imaging system, thus carrying the sub-diffraction information for super-resolution. These phenomena can be explained by the observation¹⁹ that certain functions may exhibit faster local oscillations than the fastest Fourier component of their spectrum. This is due to the fact^20,21 that in the Wigner representation, the local Fourier transform can have both positive and negative values, leading to delicate cancellations in the Fourier integration over the entire function. Put differently, a function can oscillate quickly over a confined segment of its argument in a manner that causes the high-frequency components of the function outside of the segment to cancel out.

Single-shot SO imaging leverages the engineered SO point-spread function (PSF) that has a main lobe oscillating faster than the highest Fourier component^19,22–24 to convert such sub-diffraction information into a human-readable format. This sub-diffraction main lobe enhances optical resolution through convolution with the objects.²⁵ A properly-sized region of interest (ROI), constructed by the main lobe and several low sidelobes, as shown in Fig. 1, is essential to SO imaging.^26,27 If not, the high sidelobes surrounding the ROI would overshadow the super-resolution details within the ROI. Although the issue of compact ROI can be addressed by SO illumination-based microscopy,^28–33 this imaging technique depends on the confocal imaging modality, which requires fine scanning. On the contrary, single-shot SO imaging with an extended ROI does not rely on scanning, which would significantly improve the imaging speed. The essential differences between scanning and single-shot SO imaging can be found in Sec. S6 of the supplementary material.

The structural elements of an SO PSF include the main lobe, sidelobes, ROI, and sidebands, as shown in Fig. 1. The size of the ROI, the full width at half maximum (FWHM) of the main lobe, the sidelobe intensity, and the sideband intensity construct a trade-off relationship for forming an SO PSF^12,34 (e.g., as the FWHM of the main lobe is shrunk, the sideband intensity would increase polynomially^35,36). The design of SO PSFs hence spawns various optimization algorithms,³⁷ such as particle swarm optimization,^28,38 the genetic algorithm,³⁹ and the linear programming method.⁴⁰ These objective optimization methods work well for reducing both the FWHM of the main lobe and the sideband intensity. The extreme extension of ROI (e.g., dozens of wavelengths), however, remains a challenge. Although the sidelobes within the ROI can be sharply raised to squeeze the main lobe and reduce the sideband intensity to an acceptable low level to remove the size limitation of the ROI, these elevated sidelobes adjacent to the main lobe would severely blur the images.

The extension of the ROI is always desirable for single-shot SO imaging since the elongated span of the region of low sidelobes can make imaging extended objects realizable without interference from the sidebands outside the ROI. Many explorations have been conducted recently to extend the ROI. A 0.95-NA (numerical aperture) SO spot was generated with an ROI of 15λ in radius by padding zeros within the ROI with carefully prescribed positions.⁴¹ The optical SO needles^42–45 have been demonstrated to extend the ROI to a maximum of 15λ in radius. An ROI of more than ten times the width of the Airy disk was numerically realized by a semi-analytic optimization method.³⁴ A 0.03-NA incoherent single-shot SO imaging system achieved an ROI of 400 μm in radius, which is 19λ/NA.¹⁴ This SO PSF¹⁴ is designed based on an optimization model with several preset constraints. A moon-like aperture was proposed to eliminate the sidebands outside the ROI along a pre-defined cut by breaking the circular symmetry of the SO light field.⁴⁶ Although the sidebands are completely removed along one cut, this SO spot still has limited applications to the 2D single-shot SO imaging due to the sidebands retained along the remaining cuts. A 1D SO waveform was produced without sidebands by an active “Huygens’ box.”^47,48 This Huygens’ box, however, makes it demanding to incorporate itself into an optical imaging system.

The manipulation of the ROI usually relies on a predetermined size for parameter optimization.^17,30,34 Nevertheless, as the predetermined size increases sharply, the number of undetermined parameters for the production of SO masks will accordingly grow. The consequent high-dimensional problems may render the parameter-finding task unsuccessful. Our methodology is optimization-free since we construct the radially periodic SO masks (i.e., the weights are periodic) to push the borders—sidebands—of SO PSFs away from the main lobe by only densifying the periodic weights within the spatial frequency bandwidth without efforts to optimize the weights.

In our previous work,⁴⁹ one periodic binary SO mask was utilized to provide robust persistence of the SO properties as the SO imaging system was integrated into existing microscopes. However, several major challenges remained unresolved. First, feasible ROI extension is unexplored; second, it was unclear what the tradeoffs are between extending the ROI and maintaining SO properties. In this paper, we make the following contributions: First, generalized periodic SO masks are proposed to extend the ROI, the size of which is arbitrarily adjustable. Second, we set up two imaging systems with a low NA and a high NA, respectively, to implement extended-object single-shot SO imaging. Our objects include an NBS 1952 resolution test target, a 1951 USAF resolution test target, and a Kimtech wiper made from wood fibers, all of which are significantly extended objects beyond the conventional customized objects for single-shot SO imaging.^{12,13,16,17,49} Through our comparison of the low-NA and high-NA experiments, we discovered two limitations unique to the high-NA SO imaging system: low-SNR and imperfect incoherence.

The conventional single-shot SO PSF has a constrained ROI that is typically appropriate for imaging small objects with a size of several wavelengths. As the object size increases beyond the radius of the ROI, the involved sideband interference would render the super-resolved details within the ROI unrecognizable, as exhibited in Fig. 2. This challenge cripples the scope of applications for SO imaging. Our extended-object single-shot SO imaging instead makes it realizable to image extended objects by freely extending the ROI larger than the size of the objects to be imaged, as shown in Fig. 2. The zoomed-in intensity-normalized SO image in Fig. 2 unveils super-resolved details by comparison with the diffraction-limited image.

Different from the periodic binary SO mask without ROI extension in Ref. 49, we propose in this work the generalized periodic SO masks for extendable ROIs. Such SO masks can be generated by a sequence of operations: choosing a periodic function as the radial profile, determining how many periods are included within the spatial frequency bandwidth of the imaging system, and finally discretizing the truncated periodic function to obtain the weights [see Fig. 3(a) for an example]. Since we aim for extendable ROIs, the sidebands are preferred to be adjustable. Based on the fact that the SO mask and SO PSF construct a Fourier-transform pair, an appropriate radial profile is the cosine function since it can offer such flexibility.

The creation of oscillating lobes that span the ROI between the two impulses is realized by the truncation operation, which can be expressed as G_W = G · W, where G is a cosine function, W is the window function that is usually a circular aperture within an imaging system, and G_W is the truncated function. The corresponding SO PSF is hence FT[G_W] = FT[G] ⊗ FT[W], where FT[W] provides oscillating lobes since W is a finite aperture and “⊗” denotes convolution. Figure 3(b) displays FT[G_W] in the 1D and 2D cases, respectively, where a central SO lobe can be observed in both the 1D and 2D SO functions g_2D and g_1D by comparison with the diffraction-limited ones h_2D and h_1D. The difference is that this central SO lobe in the 2D case is obviously higher than other lobes between the sidebands, which is necessary for an applicable SO PSF to imaging but is not true in the 1D case. This difference can be explained by the fact that the basis function for the 1D case is e^j2πfx, whereas that in the 2D case is J₀(2πfr) (the Bessel functions of the first kind). The latter one has a protruding main lobe.

Extending the ROI can be achieved by including more periods within the span of the window function (i.e., densifying the weights within a specific spatial frequency bandwidth). This can be performed by making the radial profile—the cosine function—oscillate faster, considering that the span of the window function is unalterable (i.e., the spatial frequency bandwidth of an imaging system is usually not changeable). An example of extended ROI (20 periods included) vs compact ROI (one period included) is exhibited in Fig. 3(c). The extension scale is approximately equal to the number of periods.

Since the generalized periodic binary SO mask (with much denser weights within the spatial bandwidth than the periodic binary SO mask in Ref. 49.) is the simplest version of our method, a complete analysis of such SO masks is hence implemented. Figure 4(a) illustrates an example of an extended SO PSF obtained by a periodic binary SO mask, where the sideband intensity is lower than 20 dB and the ROI is larger than 100λ. A detailed exploration of various characteristics of the extended SO PSF is presented in Figs. 4(b)–4(f) as the number of weights over $−k0,k0$ (k₀ = 2π/λ) increases. A short summary concluded from Fig. 4 is that: (1) the SO FWHM remains stable as the ROI is continuously extended, which is 0.7 times the diffraction-limited one; (2) the extended ROI has a radius that is directly proportional to the number of weights (mathematically, $R≈M2NAλ$⁠, where M is the number of weights); (3) the intensity of the nearest sidelobe to the main lobe varies mildly; (4) the Strehl ratio, the ratio of the peak intensity of the main lobe of the SO PSF to that of the diffraction-limited PSF, decays asymptotically; and (5) the ratio of the energy occupied by the main lobe to that of the entire SO PSF has a similar tendency to the Strehl ratio.

It should be noted that the energy ratio of the main lobe to the entire SO PSF is lower than 1% as the number of weights is more than 87, as shown in Fig. 4(f). In combination with the diffraction efficiency, it will be challenging to preserve the SO qualities against the system noise with a significantly extended ROI in an experiment.

Figure 5 depicts the experimental SO imaging system that includes the illumination system, the magnification system, and the SO imaging system. The illumination system uses a rotating diffuser to reduce the coherence of a HeNe laser. This coherence reduction can diminish laser speckles and decrease sidelobe interference (the experimental imaging comparison between coherent and coherence-reduced SO imaging can be found in Sec. S3 of the supplementary material). The magnification system forms a diffraction-limited image on an intermediate image plane where the iris resides. This iris controls the size of the diffraction-limited image to be super-resolved by the subsequent SO imaging system, where the liquid-crystal spatial light modulator (SLM) achieves Fourier frequency modulation. The pictures of the SO masks to be displayed on the SLM are depicted in Fig. 6.

Previous single-shot SO imaging could only image customized small objects with a constrained ROI. The extended SO PSFs have sidebands away from the main beam, which allows for the SO imaging of extended objects. Here we image the extended objects on an NBS 1963A resolution test target to measure the SO resolution. Our experimental results are illustrated in Fig. 7, where the Rayleigh diffraction limit is 18 cycles/mm and the SO resolution is 25 cycles/mm, or 0.72 times the diffraction limit. The calculated radii for the ROIs of the extended SO PSFs No. 1 and No. 2 are 21λ/NA and 26λ/NA, respectively. These two radii are significantly larger than the objects. We can also observe that the SNR within the ROI is low due to the scattering of the intense sidebands and the high sidelobes adjacent to the main beam, as shown in Fig. 4. The cross-sectional plot for the compact SO PSF is not displayed since its ROI is smaller than the sizes of the objects, the sidebands hence destroying the super-resolution details.

Here, we utilize the extended SO PSFs to image structural defects in a Kimtech delicate task wiper. The experimental imaging results are shown in Fig. 8, where within the areas encircled by white circles, several defects cannot be distinguished by the diffraction-limited PSF, which, however, can be discerned by the extended SO PSFs Nos. 1 and 2. Such structural defects of artificial structures made from fibers are imaged without any staining or fluorescent dye. The ROI of the compact SO PSF and the non-extended SO PSF obtained by a binary SO mask in Ref. 49 in Fig. 8 are both smaller than the objects, and they thus cannot super-resolve the sub-diffraction defects within the ROI.

We also set up a high-NA imaging system to demonstrate the extended-object SO imaging for high resolution. The diffraction-limited image is shown in Fig. 9, where the measured diffraction-limited resolution is 0.435 μm since the grating structure of Element No. 2 of Group No. 10 of the 1951 USAF resolution test target can be resolved, whereas that of Element No. 3 cannot.

The cross-section of the SO image is presented in Fig. 9 for resolution measurements, whose location is denoted by the black line. The grating structures of Elements No. 3–5 of Group No. 10 can all be resolved by the extended SO PSFs through the observations of three peaks within the black dashed-line boxes in the cross-sectional plot for the objects with an area of 28λ × 28λ, while these gratings cannot be resolved by the diffraction-limited imaging. The resolution of the SO imaging is therefore 0.308 μm, which is smaller than half the illumination wavelength (632.8 nm) and 0.7 times the measured diffraction-limited resolution.

A low SNR within the ROI in the SO images in Fig. 9, however, weakens the validity of our high-NA resolution measurements. The reason for the low SNR is that we capture these images with an exposure time of more than 1000 ms. The long exposure time makes the background noise accumulate to a high level. This capturing setting is operated due to the implementation of first-order diffraction imaging by an SLM with a low diffraction efficiency and the low energy ratio of the extended SO PSF, as shown in Fig. 4(f). The reason for the low diffraction efficiency is that the zeroth-order and the other first-order diffractions—two first-order diffractions that are symmetric about the zeroth-order diffraction, one of which is chosen to capture the images—take up most of the illumination energy. The first-order diffraction imaging is crucial for SO persistence since the designed SO mask is circular but the active area of the SLM is rectangular, on which the pixels in the redundant area must be assigned with zero-magnitude modulation for preserving weak SO qualities (see Sec. S2 in the supplementary material for how to realize the zero-magnitude modulation at the first-order diffraction and why it is difficult to do that at the zeroth-order diffraction, and see Sec. S4 for the experimental SO imaging result without zero-magnitude modulation). The improvements on the SNR issue will be discussed in Sec. IV.

CMOS camera sensors can be affected by different types of noise, which include photon noise, dark current, read noise, and ADC conversion noise.¹³ Photon noise is a type of Poisson noise that occurs due to the random arrival rate of photons at each pixel during the exposure. This type of noise is most dominant at high signal levels. Dark current, which is also Poisson noise, is caused by the spontaneous release of electrons in the electronics as a result of random thermal activity. However, dark current is usually consistent for a given temperature and can be eliminated through calibration.⁵¹ Additionally, the impact of dark current is minimal when using a CMOS camera for exposures that are under a few seconds.⁵² As a result, we focus on studying the effect of photon noise, which is the most significant Poisson noise source.

Read noise and ADC noise are both modeled as zero-mean Gaussian random variables.¹³ Read noise is generated during sensor readout and is most significant at low signal levels. ADC noise, on the other hand, occurs after sensor readout and amplification and includes noise from the amplifiers and quantization. These two types of noise are independent and additive, so they can be modeled together as a single Gaussian random variable. Therefore, photon noise is modeled as a Poisson random variable, while read noise and ADC noise are combined into a single Gaussian random variable for simulations.

Figure 10 displays the simulation results for a comprehensive noise analysis of extended-object SO imaging, where Fig. 10(a) is the SO image of a two-hole object with a center-to-center distance 0.7 times the Rayleigh diffraction limit. The comparison between noise-free diffraction-limited and SO images is plotted in Fig. 10(b), where the former is completely blurred, whereas the latter has a clear dip.

The SO image corrupted by photon noise is exhibited in Fig. 10(c), where the resolution deterioration by photon noise is slight; therefore, we focus on the additive noise (read noise and ADC noise altogether) and utilize the structural similarity index measure (SSIM) between the noise-corrupted SO image and the noise-free SO image to quantitatively measure the resolution deterioration. Figure 10(d) depicts the SSIM value, whic increases with the SNR of the additive noise. Three examples of zoomed-in SO images are presented with three different SNRs: 0, 13, and 38 dB. The corresponding SSIM values are 0, 0.5, and 1, respectively. We observe that the super-resolution imaging is completely destroyed with an SNR of 0 dB but still valid with an SNR of 13 dB. Figures 10(e1) and 10(e2) indicate that the SNR of 13 dB with an SSIM value half the maximum, where the dip is well retained in Fig. 10(e1) but becomes indistinguishable in Fig. 10(e2), can be considered the critical SNR. This occurs as the center-to-center distance of the two-hole object is decreased from 0.7 times to 0.6 times the Rayleigh diffraction limit.

Through our comparison of the low-NA and high-NA experiments, we discovered two limitations unique to the high-NA imaging system: low-SNR (due to insufficient transmission illumination energy) and imperfect incoherence (due to the spatial correlation of the coherence-reduction illumination on the object plane; details of which can be found in Sec. S5 of the supplementary material). The former impacts the persistence of SO qualities in a high-NA imaging system, while the latter prevents us from achieving higher resolution through incoherent deconvolution techniques.^13,49 However, we believe that these limitations can be mitigated by developing a feasible coherent deconvolution technique for extended-object SO imaging. Such a technique would enable us to address these issues and potentially unlock higher-quality extended-object SO imaging in high-NA systems.

Another approach to SNR and incoherence improvements is utilizing the main diffraction order, which has a high diffraction efficiency and an incoherent light source for SO imaging. Our imaging experiments at zeroth-order diffraction with a red LED in Sec. S4 of the supplementary material present an enhanced SNR and reduced speckles with an exposure time of merely tens of milliseconds. However, the resolution of extended-object SO imaging at zeroth-order diffraction without zero-magnitude modulation is imperceptibly better than the diffraction-limited one. The reason why first-order diffraction imaging is not carried out with an LED is that the dispersion would occur due to the relatively wide spectral bandwidth of an LED (compared with a HeNe laser), which would cause image overlapping. Therefore, it is more appropriate to conduct extended-object SO imaging with LEDs at zeroth-order diffraction for SNR and incoherence improvements with dedicated SO masks for precise zero-magnitude modulation.

In this paper, we propose generalized periodic SO masks for the borders of the SO PSFs—sidebands—to be freely adjustable, which enables the imaging of arbitrarily extended objects. The ROI of the extended SO PSFs can be theoretically extended without limits by means of densifying periodic weights within the spatial frequency bandwidth. A coherence reduction is introduced in the experiments by a rotating diffuser to diminish both the laser speckles and the sidelobe interference. Our low-NA measurements demonstrate the extendable ROI of the generalized binary SO masks with an SO resolution 0.72 times the Rayleigh diffraction limit and extended ROIs more than ten times the size of some typical ROIs^15–17 around 2λ/NA. We also utilize the extended SO PSFs to successfully super-resolve the sub-diffraction structural defects of a Kimtech wiper made from wood fibers. Our experimental high-NA SO imaging results reveal that a low SNR is a great challenge for high-resolution single-shot extended-object SO imaging. Such a challenge can be mitigated by implementing extended-object SO imaging at the zeroth-order diffraction with practical circular SO masks (e.g., metasurface-based SO masks) instead of rectangular SO masks to be displayed on an SLM. The dramatically extended SO patterns can overcome the ROI-related hurdles in various applications such as high-density data storage,²³ acoustic SO imaging,⁵³ super-narrow frequency conversion,⁵⁴ and temporal SO pulses.⁵⁵ 

The supplementary material presents additional experimental and simulation results, and the weights of the SO masks used in this work.

This research is funded by the Government of Canada Tri-agency New Frontiers in Research Fund (NFRF).

The authors have no conflicts to disclose.

Haitang Yang: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead). Yitian Liu: Investigation (supporting); Validation (supporting); Writing – review & editing (supporting). George V. Eleftheriades: Funding acquisition (lead); Project administration (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.