An entropy-controlled objective chip for reflective confocal microscopy with subdiffraction-limit resolution

Planar diffractive lenses (PDLs) with optimized but disordered structures can focus light beyond the diffraction limit. However, these disordered structures have inevitably destroyed wide-field imaging capability, limiting their applications in microscopy. Here, we introduce information entropy S to evaluate the disorder of an objective chip by using the probability of its structural deviation from standard Fresnel zone plates. Inspired by the theory of entropy change, we predict an equilibrium point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{0}=0.5$$\end{document}S0=0.5 to balance wide-field imaging (theoretically evaluated by the Strehl ratio) and subdiffraction-limit focusing. To verify this, a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${NA}=0.9$$\end{document}NA=0.9 objective chip with a record-long focal length of 1 mm is designed with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=0.535$$\end{document}S=0.535, which is the nearest to the equilibrium point among all reported PDLs. Consequently, our fabricated chip can focus light with subdiffraction-limit size of 0.44 λ and image fine details with spatial frequencies up to 4000 lp/mm experimentally. These unprecedented performances enable ultracompact reflective confocal microscopy for superresolution imaging.


Introduction
Since the first microscope was invented in 1595 by a Dutch father-son team Hans and Zacharias Janssen 1 , optical objectives have developed rapidly with improved performance in aberration correction, field of view, magnification, and numerical aperture, except its imaging resolution being limited by diffraction of light 2 .Based on refractive optics 3 , traditional objectives need multiple elements with carefully designed curvatures and air intervals for better imaging, thus leading to a bulky volume.The situation of objectives has been sustained for over 400 years until subwavelength-thick metalenses were reported with ultracompact volume in 2016 4 .However, highaspect-ratio dielectric nanobricks in the metalenses exhibit high efficiency under normal incidence only with a small tolerance of tilting angle 5 , resulting in large spatial-frequency details of objects being rejected by the metalenses operating in an imaging mode 6,7 .Consequently, the imaging resolutions of the metalenses and their related scanning confocal microscopies (SCMs) are diffraction-limited to 0.51λ/NA (λ is the wavelength of light and NA is the numerical aperture of the metalens) 8 .
Since only the focusing properties of these planar lenses are designed without considering the capability of direct wide-field imaging, superoscillation and supercritical lenses are usually used as optical probes in an SCM 7 , where an additional refraction-based objective is mandatorily required to collect the transmitted light through the objects.This leads to the fact that all these planar-lensbased SCMs must operate in transmission mode and are only valid for objects sitting on a transparent substrate that introduces spherical aberration, which requires collection objectives with coverslip collars for correction 2 .Without involving these issues, reflective SCMs are, therefore, more popular for noninvasive and in vivo imaging of various specimens 19 .Despite the strong requirements from applications, it is still difficult to demonstrate reflective planar-lens-based SCMs with better resolving power than commercial SCMs, due to the lack of high-performance planar objectives that possess dual functionalities of focusing and imaging beyond the diffraction limit.
The challenges to desigining such a planar objective are twofold.First, designing a planar objective for subdiffraction-limit focusing generally leads to an irregularly distributed phase or amplitude 18 .However, such structural disorder is not preferred in its imaging counterpart, where the analytical phase or amplitude is preferred for the constructive reconstruction of objects 7 .Such a dilemma is a fundamental barrier to demonstrating planar objectives with subdiffraction-limit resolution in both focusing and imaging.Second, the diameter of the planar objectives should be sufficiently large to suppress the diffraction effect of collected light by the planar objectives (when working in the imaging mode) for better image formation.Correspondingly, the number of fine structures in the planar objectives is extremely large because of the short wavelength in the visible spectrum, thereby increasing the technical difficulty in the design and fabrication of planar objectives.
To overcome these challenges, we propose a disorder-controlled objective chip that functionally integrates a binary-phase Fresnel zone plate (FZP) and a weakly perturbed few-ring phase mask into a single ultrathin element.By introducing the concept of information entropy, we theoretically predict that an objective chip with its entropy at S0=0.5 can maintain the imaging and superfocusing properties simultaneously.Using deep-ultraviolet (DUV) lithography, the fabricated objective chip experimentally exhibits a focal spot of 0.44λ (below the Rayleigh criterion of 0.51λ/NA=0.57λ)without strong sidebands and the capability of imaging fine objects with spatial frequency of 4000 lp/mm.Benefiting from this, an ultracompact reflective SCM is built with an imaging resolution (center-to-center) of 200 nm at λ=405 nm and a record-long working distance of 1 mm, superior to the state-of-the-art SCMs.

Disorder of objective chip
Since standard FZPs have analytical complex transmission, i.e.,   =      (where   and   are the amplitude and phase modulation, respectively), they can realize wide-field but diffraction-limit imaging.For an objective chip, its complex transmission  ℎ =      deviates irregularly from that of an FZP, thus creating undesired disorder for imaging purposes.
Assuming that the deviations are ∆ =  ℎ −   for amplitude and ∆ =  ℎ −   for phase, we can rewrite the transmission of the objective chip as  ℎ = (  + ∆) (  +∆) =      (1 + ∆   ⁄ ) ∆ =   •  ∆ , where the deviated transmission  ∆ = (1 + ∆   ⁄ ) ∆ introduces optical aberration in imaging but offers more degrees of freedom for subdiffraction-limit focusing.Thus, for an arbitrary objective chip, its complex transmission contains a standard imaging part   (i.e., FZP) and an additional aberration part  ∆ .In the design of an objective chip, the item  ∆ is fundamentally important in building the connection between imaging and focusing.
Considering that  ∆ is distributed spatially in an irregular way, we first investigate its statistical property of such deviation by defining a dimensionless parameter-deviation probability where ∆= ∆ with a maximum value of ∆  = 1 for the amplitude deviation, ∆= ∆ with a maximum value of ∆  =  for the phase deviation, the objective chip is divided into minimum diffraction subunits (i.e., zones in the FZP) with their boundaries of   = √ + () 2 4 ⁄ (m is the index of the m-th zone, λ is the wavelength and f is the focal length of the lens), M is the total number of zones contained in the FZP, and the modulus is used to ensure the non-negative probability.Because the standard FZPs have pure amplitude (i.e.,   = 0 in   for binaryamplitude FZPs) or phase (i.e.,   is a constant in   for binary-phase FZPs) modulation, we only need to calculate ΔA or Δφ in Eq. ( 1) for most objective chips.According to its definition, the deviation probability p1 ranges from 0 to 1, leaving a probability p2=1-p1 for the unchanged part.
It behaves like an information channel with binary values, where the entropy is usually used to evaluate the disorder of this information system 20 .
Similarly, based on our defined probability p1 and p2, we can also calculate the disorder of an objective chip by using the information entropy S. When p1=0 or p1=1, the corresponding S equals zero, which means high certainty without any disorder.It agrees with the real cases that both objective chips with p1=0 and p1=1 refer to standard FZPs, where  ∆ = 1.When p1=0.5, the entropy S=1, which implies the highest disorder because half of the zones are reversed randomly.Although high disorder offers large degrees of freedom for optical super-focusing, it also destroys the imaging capability of the objective chip due to optical aberration dominated by its random  ∆ .Thus, we infer that the entropy S is symmetric about p1=0.5, where the peak is located.At the side of 0≤ p1 ≤0.5, the entropy S increases monotonically from 0 to 1. High certainty at S=0 is helpful in widefield imaging but high disorder at S=1 is required to realize super-focusing.
To obtain a good balance between imaging and super-focusing, we introduce a thermodynamic analog that the change of entropy for an isolated system originates from outside work or heat transfer 21 .In our case, a virtual work W is assumed to govern the change ΔS of entropy by following a straightforward relationship  ∝ .The change of entropy from S=0 to S0 requires the virtual work W1, while the change from S0 to S=1 requires virtual work W2.We suggest that, when both virtual works are equal, i.e., W1=W2, good balance between imaging and super-focusing is achieved with an equality  1 =  2 (i.e.,  0 − 0 = 1 −  0 ), leading to the equilibrium point S0=0.5.
Interestingly, at this equilibrium point, its relative deviation probability p1 equals 0.11, which is smaller than the middle point p1=0.25 of the interested range 0≤ p1 ≤0.5.This implies that the entropy S is more sensitive to the intrinsic disorder of the objective chip than the deviation probability p1, thereby indicating the rationality of the proposed equilibrium point S0=0.5.

Strehl ratio and focal size of the objective chip
To quantitatively investigate the imaging and focusing properties of an objective chip with different disorders, a binary-phase objective chip (Fig. 1a) with a focal length f=1mm and NA=0.9 (implying its diameter of 4.13 mm) is exemplified here to enhance the optical efficiency in both focusing and imaging.Compared with its corresponding binary-phase FZP, this proposed objective chip has only the phase deviation of Δφ because of ΔA=0, leaving  ∆ =  ∆ .This implies that any binary-phase objective chip can be taken as a combination of an analytical FZP and an additional few-ring phase (i.e., ∆) mask, as sketched in Fig. 1b.Since the phase  ∆ in the few-ring mask introduces optical aberration, we evaluate its influence on imaging quality by using the relative Strehl ratio (SR) 3

𝑆𝑅 =
ℎ (0,0,=) where the incident electric field E0 is taken as unity in this work, the wavenumber k=2π/λ, R 2 =(u- x) 2 +(v-y) 2 +z 2 , r 2 =x 2 +y 2 , tanφ=y/x, x and y are the Cartesian coordinates at the initial plane of the objective chip, u=v=0 and z=f are used to obtain the on-axis intensity at the focal plane, and   is fixed once the FZP is given.In Eq. ( 2), the on-axis intensity of the FZP, having the same modulation (phase or amplitude) type as that of the designed objective chip, is used as the denominator to avoid the influence of the focusing efficiency of the FZP with different modulation types.Therefore, Eq. ( 2) defines a relative Strehl ratio, which is more useful in evaluating the optical aberration of imaging systems.
For a binary-phase objective chip with its deviation probability p1 (only 0≤ p1 ≤0.5 is considered in the following because the entropy S is symmetric about p1=0.5), its relative Strehl ratio can be approximated as  = (1 − 2 Similarly, its focal spot is also controlled by the entropy S or p1.Although it is difficult to derive the spot size analytically, we obtain its upper (rmax) and lower (rmin) boundaries numerically (see Supplementary Section 1), as illustrated in Fig. 1c.For the larger entropy S, the spot size is valued in the wider range, which offers more opportunities to realize super-focusing.Hence, a large entropy S with high disorder is required in super-focusing.These results reveal that imaging and super-focusing have completely opposite requirement for the entropy S of an objective chip, which doubly confirms that the entropy S0=0.5 leads to good balance.

Design of objective chip
Since the FZP contained in the objective chip is given with analytically described structures, we only need to optimize its deviation part  ∆ =  ∆ , which refers to a few-ring phase mask (see Fig. 1b).Considering that the ideal equilibrium point S0=0.5 has a small deviation probability, only 5 rings are used to avoid introducing high disorder.In our design, all radii   (n=1, 2, …, 5) of these 5 rings are chosen from the radii of zones in the FZP, which means that no new finer structure is created when combining the FZP and this 5-ring phase mask.Such a design strategy has twofold significance.First, it greatly enhances the speed of optimization.Because all the structural details of the objective chip are given by the radii rm of the FZP, we can calculate the electric field of each zone ahead of optimization and then store it as a database for quick reading during the design, thus avoiding repeated calculation.For example, to design our objective chip with 6387 zones, it takes only 6 mins to run 500 iterations by using the particle swarm algorithm in a personal computer, enhancing the speed by a factor of ~3.8×10 5 (see the calculation details in the Methods).Second, it allows large-scale and low-cost fabrication of our objective chip because its smallest feature size is theoretically fixed to   = lim →∞ (  −  −1 ) = /2.

Fabrication of objective chip
The possibility of employing semiconductor processes to fabricate planar flat optics will ultimately allow mass production of flat optics at a low cost and push this technique for wide market adoption.As discussed above, our design strategy makes low-cost and fast DUV lithography feasible for fabricating a 4-mm-diameter objective chip.Under the conditions of λ=405 nm, f=1 mm and NA=0.9 in this work, Δrmin is only 225 nm, which is within the capability of commercial DUV lithography (with a critical dimension of 200 nm).This key feature will greatly facilitate the future manufacturing of our objective chips by reducing the cost compared with other flat lenses with critical dimensions smaller than 200 nm that would require much more costly 12-inch immersion lithography 27 .We note that E-beam lithography is able to write the pattern at high resolution; however, it is not the technique for large-scale fabrication due to its speed limit and the inevitable stitching error (with a small writing square of several tens of micrometers).The fabrication details of our objective chip are provided in the Methods.
The inset of Fig. 1a shows the microscopic image of our fabricated objective chip, where different colors arise from optical scattering of daylight.To reveal the fine details, we show scanning-electron-microscopy (SEM) images of the objective chip in Fig. 2a, where both simulated and experimental widths from the center to the outermost boundary are also provided with a maximum deviation of <75 nm.The possible reason comes from insufficient exposure time of photoresist under DUV radiation, which can be solved by increasing the exposure time.Using a profilometer, we characterize the groove depth of ~530 nm (see the insert in Fig. 2a) that yields a phase modulation of ~1.23π, which is larger than the ideal value of π due to overetching issues.
Nevertheless, we emphasize that such a deviation of 0.23π in phase modulation leads to a theoretical decrement of only ~2.6% in the focusing efficiency of the objective chip, see the simulation details in the Methods.

Subdiffraction-limit focusing by objective chip
To verify the focusing capability of the objective chip, we first measure the diffraction field near the focus of the objective chip under the illumination of collimated circularly polarized light by using a 0.95-NA objective lens, see the experimental details in Supplementary Section 3. Figure 2b shows the x-z and y-z cross-section of the measured light intensity, which reveals a record-long focal length of 1 mm (compared with the previous planar diffractive lenses 9,14,15,22,28,29 ).The axial depth of focus (DOF) is extended from the simulated ~500 nm to the experimental ~5500 nm, which is caused by the fabrication error of the groove width in the objective chip (see Fig. 2a).Such a long DOF offers good tolerance to sample alignment in an SCM 13 .Fortunately, the focusing spot size of the objective chip is less influenced by the weak phase perturbation from the imperfect zones, as confirmed by the good agreement between the simulated and experimental line-scanning intensity profiles (Fig. 2c).To quantitatively evaluate the focusing effect, we present the experimental FWHM of the focal spot size along the propagation of light in Fig. 2d, indicating the varying FWHM from 170 nm to 210 nm (tightly close to the simulated 180 nm).Compared with the Rayleigh criterion of 0.51λ/NA, these achieved focal spots confirm the subdiffraction-limit focusing capability of the proposed objective chip.Due to its supercritical feature with lateral FWHMs above the superoscillation criterion 26 , the focusing spots have no strong sideband, as observed in Supplementary Video 1, which dynamically records the focusing process near the focal plane.
The focusing efficiency, defined as the ratio of the focused power (experimentally filtered by a 150 μm-diameter pinhole at the focal plane) to the total power incident on the objective chip, is measured to be 12.3% (see the measurement details in Supplementary Section 4), which is lower than its theoretical efficiency of 18.7% (see its calculation in the Methods), as shown in Fig. 2e.
This discrepancy in efficiency is attributed to incomplete constructive interference of light diffracted from two neighboring zones because of the insufficient etching widths.Although our achieved efficiency is not as high as those of traditional objectives and metalenses, it still exhibits significant enhancement in comparison with those of amplitude-type planar diffractive lenses 9,14,15,22,29 .

Direct wide-field imaging by objective chip
To investigate its wide-field imaging, a knife-edge object (see its microscopic image at the leftbottom of Fig. 2f) is located at a distance of z=1.2f from the objective chip 30 .After illuminating the knife-edge object, the transmitted light is collected by our objective chip.According to the imaging formula of a lens, we can roughly estimate its imaging distance of 6f at the other side of the objective chip, thereby exhibiting an imaging magnification of 5×.The recorded image of the knife-edge object is shown at the bottom-right panel of Fig. 2f, which reveals a well-defined boundary at the edge.A dynamic video that records its imaging process by tuning the axial position of such a knifeedge object is provided in Supplementary Video 2, which is captured in a homemade measurement setup (see Supplementary Fig. 8 and Supplementary Section 5).
The modulation transfer function (MTF) of this objective chip is characterized by using the linescanning intensity across the edge in the image.To decrease the experimental error, we employ a mean of the line-scanning intensity (with its spatial dimension scaled down by a factor of its magnification M=5) at the red-rectangle region in the image of Fig. 2f to recover the line spread function (LSF) of this imaging configuration.After fitting the line-scanning intensity with an error function (taken as a convolution between a Gaussian function and a jump function), we carry out the deconvolution of the fitted error function, yielding the retrieved LSF (Fig. 2g).Using a Fourier transformation of the retrieved LSF, we finally obtain its MTF (Fig. 2h), which indicates a cut-off frequency of 4000 lp/mm.This implies an imaging resolution of 250 nm, which corresponds to an effective NA of 0.83 (evaluated by using 0.51λ/NAeff=250 nm) for imaging.Compared with the previous metalens with a cut-off spatial frequency of 2000 lp/mm 4 , our objective chip achieves a twofold enhancement in resolving power when operating in the mode of direct wide-field imaging.
These experimental results have confirmed that our objective chip has sufficient imaging ability to collect high spatial frequencies from fine details of objects, which is superior to all previous superoscillation 9,10,12,22,28,29 and supercritical lenses [13][14][15] .

Objective-chip-based reflective SCM
A high-resolution reflective SCM (Fig. 3a, see its working principle in the Methods) has been built successfully due to both features (i.e., subdiffraction-limit focusing and direct wide-field imaging) of our objective chip.First, the enhanced focusing efficiency allows more light to illuminate the object in a reflective mode, which underpins subsequent collection and detection of reflected light.Figure 3b shows the experimental signals detected by the photomultiplier tube (PMT) and charge-coupled device (CCD) when the nano-objects are moved longitudinally near the focal plane of z=1 mm.It reveals that the PMT signal reaches its maximum for the in-focus (i.e., Δz=0) nano-objects and decreases gradually with the increment of the out-of-focus distance |Δz|, which is doubly checked by the CCD images (see the inserts in Fig. 3b and Supplementary Video 3).The FWHM of the PMT measured intensity profile is ~5 μm, which agrees with the experimental DOF of 5.5 μm (see Figs. 2b and 2d).This result confirms that our objective chip can efficiently focus the incident beam and collect the reflected light.Second, a powerful imaging ability with an effective NA of 0.83 is required to enhance the practical imaging resolution of an SCM.Theoretically, we have already shown that the resolution of an SCM is less influenced by the NA of the collection objective 7 , which, however, is valid only for infinitesimal point objects.For real objects with finite sizes ranging from tens to hundreds of nanometers, the NA of the collection objective should be larger than 0.7 for a better resolution in an SCM, see the simulated proofs in Supplementary Section 6. Due to these two features of our objective lens mentioned above, the subdiffraction-limit focusing and the high-resolution imaging enable a reflective SCM.More experimental details about the scanning imaging are provided in the Methods.
To test its resolution, we provide the imaging results of 50 nm-width and 2-length double slits with center-to-center (CTC) distances ranging from 190 nm to 270 nm.As shown in Fig. 3c, these slits are etched on a 140 nm-thick chromium film on a quartz substrate.Using a 0.9 NA objective for a fair comparison, the coherent bright-field microscope cannot resolve these double slits (Fig. 3d) while conventional reflective SCM can only resolve the double slits with CTC distances larger than 240 nm (Fig. 3e).In contrast, our objective-chip-based reflective CSM has an enhanced resolution so that double slits with a CTC distance of 200 nm can be distinguished with a valley of intensity in the image (Fig. 3f), where the distortion is caused by mechanical variation of the sample.The qualitative comparisons among their line-scanning intensity profiles (Fig. 3g) doubly verify an imaging resolution of 200 nm achieved by our SCM.In addition, all these experimental results regarding scanning images are confirmed by our simulations (see Supplementary Section 7 and Supplementary Fig. 10) with the theory of SCM 7,31,32 .
Complex nano-objects can also be imaged with a high resolution by using our SCM.Figure 4a shows the SEM image of a dolphin (composed of 50 nm-width curves) with a total size of 8 μm×8 μm. Due to their limited resolutions, both coherent BF microscopy and traditional SCM can map only rough contours of the dolphin but lose fine details, such as the eye and tail (Figs.4b and 4c).
In comparison, our SCM can clearly resolve all these fine details (Fig. 4d) with a narrower line width (Fig. 4e).Furthermore, two lines with a CTC distance of 225 nm (see the dashed-red lines at the lower rows in Figs.4b-4d) in the tail can be distinguished only by using our SCM.The low contrast of intensity in the image for our SCM comes from the relatively lower focusing efficiency of the objective chip in comparison with the traditional objective.However, it has little influence on the resolution and clarity of the image, as observed in Fig. 4d.

Discussion
Among all the planar-lens-based SCMs, our current SCM has the advantages of eliminating bulky objectives, a millimeter-level working distance, reflection-mode operation, working for both transparent and non-transparent substrates, and a competitive resolution of 0.49λ, as shown in Supplementary Table 1.For commercial objectives, pursuing a high NA and long working distance simultaneously leads to an increment in the diameters of optical elements and the accompanying optical aberrations that need large-scale nonspherical surfaces for correction 3 , thereby yielding extremely high costs in both the fabrication and design of elements.This issue does not exist in our objective chip, where the smallest feature of λ/2 will not change with increasing NA and focal length, so the same fabrication tools and design methods reported in this work can also be used to develop more advanced objective chips with even larger NAs and longer focal lengths.
Our objective chip has a fabrication cost of ~$42 dollars (estimated by the total price of 300 chips in an 8-inch quartz wafer, see Supplementary Fig. 11), which is ~100 times cheaper than the price of commercial objectives.This objective chip has a volume of 4 mm×4 mm×0.5 mm, which indicates a shrinking factor of 4300 (~3.6 orders of magnitude) compared with traditional objectives (ZEISS, EC Epiplan-Neufluar 100×, NA 0.9, M27).
Developing this reflective configuration makes a truly important step to push the technology of planar-lens-based SCMs towards practical applications, because many samples have opaque substrates that are incompatible with all the previous planar-lens-based SCMs.Note that the objects used in this work are made in a high-reflectivity metal film, which helps to enhance the imaging contrast.If the difference between the optical reflection of the object and its surrounding background is not obvious, one should increase the optical efficiency of the objective chip and the sensitivity of the optical detector.The focusing efficiency of the objective chip can be enhanced further if multilevel phase elements 33 or high-efficiency dielectric metasurfaces 4,5,[34][35][36][37][38][39][40][41][42][43] are used.The detector can also be updated to the single-photon level for a better recording of collected photons by our objective chip, enabling the characterization of less-reflective biological tissues and cells even in a living body.
In summary, we have proposed information entropy to evaluate the disorder of an optimized planar lens.The suggested equilibrium point S0=0.5 is used to guide the quick design of a 1-mm focal-length, high-NA and low-cost objective chip with efficiency-enhanced subdiffraction-limit focusing and direct wide-field imaging.These advantages open the way to demonstrate compact and high-resolution reflective SCM with planar lenses, which will greatly benefit from optical to biomedical imaging.

Methods
Design and optimization details.To avoid the creation of additional finer structures when combining the FZP and the few-ring mask, each radius ρn (n=0, 1, 2, … , N) in the N-ring mask is valued within the radii (i.e., rm) of belts in the FZP.To highlight these selected radii in the zone plates, we label them    =   , which means that the n th ring of the few-ring mask has the same radius as the Mn th belt of the zone plate.Although it decreases the degree of freedom to design the few-ring mask, significant benefits are achieved in simplifying the optimization and fabricating the sample.
Benefiting from this design strategy, we can express the electric field of light focused by the objective chip as where Am is the electric field of light diffracting from the m th belt in the FZP, the wavenumber k=2π/λ, R 2 =(x-v) 2 +(y-u) 2 +z 2 , r 2 =x 2 +y 2 , tanφ=y/x, x and y are the spatial coordinates at the plane of the objective chip, x, y and z stand for the spatial position of interest.Considering the nonparaxial propagation of light in such a high-NA objective chip, we calculate Am by using the rigorous Rayleigh-Sommerfeld diffraction integral without any approximation 7,44 .Although thousands of belts are included in this objective chip, only N-1 variables (i.e., M1, M2, …, MN-1 because M0=0 and MN=6391) are unknown in Eq. ( 3) because all rm are given.Since the phase jump of π occurs at the Mn th belt (n=1, 2, …, N-1), both the Mn th and (Mn+1) th belts are combined into one, leading to the belt number of M-(N-1) in the final objective chip.
Since no new belt appears in this strategy, the electric field Am can be calculated ahead of optimization and then stored in a database, thus enhancing the design speed.The particle swarm algorithm 45 is used to optimize the N-1 parameters, see the details in Supplementary Section 2. In our design, N=5 is employed with 4 unknown parameters, which can be determined with the values of M1=283, M2=850, M3=1046 and M4=1258 by carrying out 500 iterations in ~6 minutes in a personal computer (Intel Core i5-7500 CPU 3.40 GHz, 32G RAM).In each iteration, the optimization algorithm contains 20 populations, each of which stands for one design of objective chip.If our design strategy with the pre-calculated database is not used, we can roughly estimate its time cost of 3.8×10 4 (=3.8×20×500)hours to finish the design by running 500 iterations, because it will take ~3.8 hours to calculate the focal field of a single objective chip by numerically integrating all the zones with Rayleigh-Sommerfeld diffraction under the same computation environment.Thus, our design strategy accelerates the optimization by a factor of 3.8×10 5 .In our designed objective chip, the phase-reversed zones contain two parts: 1) from m=284 to m=850 and 2) from m=1047 to m=1258, resulting in p1=(567+213)/6391=0.122 and p2=1-p1=0.878.According to the definition of information entropy, we have S=0.535, which is tightly close to the equilibrium point S0=0.5.

Fabrication details.
The designed objective chip is fabricated through deep-ultraviolet (DUV, Nikon S204) exposure process.The quartz substrate is first deposited with 200 nm-thick aluminum using a physical vapour deposition (PVD) system (AMAT Endura).Then, a 300 nm-thick positive photoresist (UV135) is coated and baked.Subsequently, the photoresist is patterned using the DUV lithography.After development, the aluminum film without photoresist is etched sufficiently by an inductively coupled plasma (ICP) etching system (LAM 9600).Thus, the patterns of the objective chip are transferred into the aluminum film after removing the residual photoresist.Next, using the aluminum film as masking layer, the quartz substrate is etched for a designed thickness by an inductively coupled plasma-reactive ion etching (ICP-RIE) system (Oxford, Plasma Pro System100 ICP380).Finally, the aluminum hard mask is removed by Tetramethylammonium Hydroxide (TMAH, 2.5%) solvent, yielding the expected phase-type objective chip.
Theoretical efficiency of the objective chip.To obtain the theoretical efficiency of the objective chip, we first calculate the electric field   of the m th belt in the zone plate at the focal plane (ignoring the influence from the width error of etched belts).For the sake of convenient simulation, we calculate the one-dimensional field along the radial direction within the range of λ (starting from r=0), where the focused light is concentrated.To evaluate the experimental error, we update Eq. ( 3) by considering an actual phase difference of ∆ as   | 2 .Finally, the theoretical focusing efficiency of the objective chip can be evaluated as where   =  2 (1/) denotes the optical efficiency of multilevel phase elements, and L is the number of phase levels.For the binary-phase FZP, we have   = 40.5%.Note that, Eq. ( 6) depends on the geometric parameters Mn and Δφ, which allows us to conveniently investigate optical efficiency of the objective chip.For example, once the geometric parameters are fixed in this work, we can simulate the theoretical focusing efficiency for different Δφ values (see Fig. 2e), which reveals a peak efficiency of 21.3% at Δφ=π.For the experimental Δφ=1.23π, its corresponding theoretical efficiency is 18.7% with a deviation of 2.6% from the peak efficiency.

Work principle of objective-chip-based reflective SCM.
A schematic diagram of the objectivechip-based reflective SCM is given in Fig. 3a.The confocal configuration consists of our objective chip and two tube lenses (TL1 and TL2), where their focal planes are conjugated with that of the objective chip.The objective chip is illuminated by a collimated light beam with a wavelength λ=405 nm.
To increase the signal-to-noise ratio of the entire system, we suppress the light reflected from the back-surface (i.e., the bare-quartz side without any structure) of the objective chip by utilizing two orthogonal linear polarizers (LP1 and LP2) and a quarter-waveplate (QWP) thin film, as sketched in Fig. 3a.Because both LP1 and LP2 have orthogonal transmission directions, the reflected light from the back-surface of the objective chip is blocked efficiently, leaving a very weak background with a four-lobe pattern (see the insert in Fig. 3b).It is important to note that such a four-lobe pattern has a dark center, where the signal light reflected by the objects is located, thus leading to spatial separation between the noise and signal.The 60 μm-thick QWP thin film (with an angle of 45 degrees between its fast axis and the transmission directions of both LPs) is adhered to the structure side of the objective chip to obtain circular polarization (CP), which enables us to realize a circularly symmetric focal spot for isotropic scanning of the image.Moreover, the reflected CP signal light from nano-objects passes through the QWP thin film again and is converted into linear polarization with its direction aligned to the transmission direction of LP2.Therefore, the second linear polarizer (LP2) can block the noise light reflected from the back surface of the objective chip and transmit the signal light efficiently, thus increasing the signal-to-noise ratio of the PMT signals.
After tuning the signal, we coarsely move the objective chip mounted on the electric stage toward the nano-objects.When the nano-objects are close to the focal plane, the PMT signals behave like that shown in Fig. 3b, having a peak when the nano-objects are in focus.At the same time, the recorded pattern at the CCD becomes the smallest.Due to the conjugation relationship between the objects and the PMT, we can adjust the PMT to collect an optical signal at the focal plane of TL2 filtered by a 10 μm-diameter pinhole.Then, we finely move the nano-objects mounted on the 3D piezo stage to the focal plane by observing the signal collected by the PMT.When the PMT signal reaches its maximum (see Fig. 3b), we assume that the nano-objects are at the focal plane of the objective chip.Finally, the nano-objects can be scanned at the focal plane, and the signal collected by the PMT can be recorded simultaneously to complete the scanning image.
Experimental details about the scanning imaging.The 3D piezo stage (PI-545.3R8S)and the controller (PI-E727) are integrated into a single device with a scanning resolution of ~1 nm.The PMT has the module of the WiTec 3000R series.We utilize the LabVIEW language to control the movement of the 3D piezo stage and read the signal collected by the PMT with a DAQ card (NI USB-6000, 12-bit, sampling rate: 10 Ks/s) simultaneously.A 10  -diameter fiber (Thorlab M64L01 10 0.1 NA) is employed here as the pinhole.
In the experiment testing the imaging resolution, the scanning range of double slits is 3 μm×1 μm with 100×100 pixels, which takes ~15 minutes to finish one image.The scanning speed can be updated further by using a high-speed stage and digital-analog converter.The 2.4 μm×0.6 μm range of scanning images in Figs.3(c)-(f) is employed to fully cover the objects.In the experiment for imaging complex nano-objects, the scanning range is 9 μm×9 μm with 150×150 sampling points,   To evaluate its resolving power, the spatial coordinate y is scaled down by its magnification of 5.
The experimental ESF is fitted by an error function, the deviation of which outputs the line spread function (LSF).(h) Retrieved modulation transfer function (MTF, solid-circle curve) of the objective chip by using the Fourier transform of the achieved LSF in (g).The diffraction limit (dashed curve) is also provided for a better comparison.

. 15 Section 1. Mathematical basis for Strehl ratio and focal size of an objective chip
To reveal the relationship between the information entropy S and optical properties of an objective chip, we investigate its Strehl ratio and focal size under different deviation probability p1.
For a given p1, the relative Strehl ratio and focal size change because the locations of the deviated zones in the objective chip are different.It means that the relative Strehl ratio and focal size have a certain range with the fixed minimum and maximum values, which can be determined mathematically by using diffraction properties of each zone.
First, we derive the minimum and maximum Strehl ratios.Because the binary-phase objective chip is reported here with the modulation phase of 0 and π, we can directly use its complex modulation of 1 and -1, respectively.In our design strategy, the objective chip is functionally divided into a binary-phase FZP and an N-ring phase mask.The phase of π in the N-ring phase mask realizes the reversal (from 1 to -1, or from -1 to 1) of the complex modulation.Assuming that the odd and even rings have the phase of 0 and π respectively, it means that the electric fields contributed by the zones in the even rings are removed from those of the standard FZP and then are used to interfere constructively with those of the zones in the odd rings.For the objective chip containing a N-ring phase mask, Eq. (3) describing the total electric fields of our objective chip can be rewritten as where  denotes a set of the indices of all the phase-reversed zones (i.e., contained in the even rings of the N-ring phase mask).For our design strategy used in this work, we can obtain the number of the set  by using M•p1, where M is the total number of zones in the corresponding FZP and p1 is the deviation probability (see Eq. ( 1) in the main text) of the phase-reversed zones.Considering its university, Eq. ( S1) is valid for all binary-phase planar lenses.where EFZP(0, 0, f) is in phase with a0=(-1) m Am(0, 0, f).Meanwhile, for different m, the item a0 is also in phase with each other and nearly a constant with a slow variation from a0min=0.87 to a0max=2, which can be numerically calculated by using the rigorous Rayleigh-Sommerfeld diffraction integral.
Due to their feature of slow variation, the on-axis intensity  0 from all the zones with the zone indices  ∈  are assumed to be identical.Thus, ∑  0 ∈ can be approximated by  0  1 .By applying these assumptions, Eq. ( S2) can be approximated as where   = √  and 0.87≤a0≤2.Eq. ( S3) reveals the direct link between SR and p1 (which To verify the analytical SRmin and SRmax, we have simulated the range of SR by using the proposed objective chip with a 5-ring phase mask.The limited N=5 of the phase mask allows us to go through all possible solutions quickly without any optimization because all the Am can be calculated ahead.By controlling the number and position of the phase-reversed zones in the 5-few ring phase mask, the simulations are implemented within the range of 0 ≤  1 ≤ 0.35 with an interval of 0.01, which is enough here because low disorder  1 is important to develop the objective chip with good balance between imaging and super-focusing.The simulated SRmin and SRmax are shown in Supplementary Fig. S1a, exhibiting good agreement with their analytical values.
Their slight deviations come from the approximations made during its derivation.Therefore, these results have confirmed that the analytical SRmin and SRmax give a good prediction for the range of Strehl ratio.
Second, the focal size of an objective chip with different p1 can also be predicted by using Eq.
(S1).As shown in Eq. ( S1), the electric field of the objective chip is taken as the coherent superposition of the electric fields from all the zones.Diffraction behavior of each zone is important in predicting the focal size of the objective chip.Because the width of each zone is small, the diffraction field from each zone is mainly determined by its focusing angle between the outmost boundary of each zone and optical axis.When the focusing angle is large, the relative focal spot size of diffraction field from one zone is small; vice versa.For an objective chip, its maximum focal spot is achieved when the contribution from the outermost zones is small, where the electric field at the focal plane is where the phases of the outermost zones are reversed with a deviation probability p1.From Eq. (S4), we numerically predict the maximum focal size, which depends on only p1.In contrast, when the phases of the inner zones are reversed, the minimum focal size can be predicted by using the electric field at the focal plane and simulated rmax agree with each other for the interested range of 0 ≤  1 ≤ 0.35.However, for the case of rmin, the discrepancy between the prediction and simulation increases with the increment of p1, which is caused by the large error of the approximation in Eq. (S5).In fact, for a large p1, the disorder in the objective chip is higher than the predicted one in Eq. (S5), so that the minimum focal spot has more choices with a larger range than that predicted by Eq. (S5).Despite this, our prediction in Eq. (S5) shows the same decreasing tendency when p1 increases, thereby confirming the validity of the predicted focal sizes.

Section 2. Optimization of the objective chip
According to the design strategy described in the main text, we implement the optimization of the objective chip with four key steps, as discussed below.
where m=0, 1, 2,…, M, the wavelength λ=405 nm, the focal length f =1mm.In our design, the total number M of belts in this BPFZP is M=6391 and the radius of BPFZP is ~2.0648mm, which yields a numerical aperture (NA) of 0.9, as shown in Supplementary Fig. 2. The binary phase is employed to enhance optical efficiency of the objective chip.

Calculating the focal field of each belt in the BPFZP
Benefiting from our design strategy, no new ring is created during our optimization because all the radii can be described by Eq. (S6).It means that the focal field of light from each zone can be calculated ahead of optimization and then stored in a database, so that we directly revisit the relative focal field during the optimization.Thus, the time cost will be significantly shorten.Since we need to evaluate the lateral focal size and the longitudinal depth of focus, both focal fields along the radial (i.e., r) and longitudinal (i.e., z) direction are calculated ahead, where the positons of interest are: 1) the lateral positions 0≤r≤λ at the focal plane z=1000 μm; 2) the longitudinal positions 950 μm≤z≤1050 μm at the on-axis position r=0.By using Rayleigh-Sommerfeld diffraction theory without any approximation for high accuracy, we calculate the focal fields of each belt along the lateral and longitudinal positions, and store them in two matrices (i.e., Ar and Az, see Supplementary Figs. 3a and 3b) respectively.According to the electric field stored in the 1 th , 3000 th and 6391 th column of Ar and Az, the normalized intensity of diffraction field at two target positions of corresponding belts of BPFZP are exemplified in Supplementary Figs.3c and 3d, respectively.To show its convenience, the focal fields at two target positions for a BPFZP can be calculated as

Optimizing the 5-ring phase mask
To optimize the detailed structures of the 5-ring phase mask, we use the well-matured particle swarm optimization (PSO) algorithms that have been used frequently to design various lenses, especially for super-oscillation and super-critical lenses.Considering the limited number of the fewring mask, the optimization will be implemented with its standard version of the PSO algorithm.•  0 , (S8) where n is the ring number of the few-ring phase mask, NA0=0.9 is used in this work.From Eq. (S8), one can induce that NA5=NA0, which means that the outer boundary of the 5-th ring in the phase mask refers to the maximum radius of 2.046 mm.The five parameters in each particle are related with the difference of NA between two neighboring rings.In this definition, we have built the oneto-one relationship between the particle parameters and the structures of the objective chip.In addition, such a definition will offer full degree of freedom to go through all the possible solutions because each xi can be valued within 0<xi<ꝏ.The normalization factor of 1 ∑   5 =1 ⁄ is quite helpful to make the maximum value of NA0, thus enabling each particle to yield a physically meaningful objective chip.Thus, the universal properties of design an objective chip are maintained in our definition, which is an important step to implement this optimization.In one iteration, each xi is updated with the PSO algorithm (as described in the dashed rectangle of Supplementary Fig. 4).Based on the optimized xi, we derive the NA parameters in Eq. ( S8), from which we find each ).
To avoid the creation of additional finer structures when combining the zone plate and 5-ring mask, the optimized   is approximated by the closet rm, which is labelled as    =   .Thus, we can find all the   , hereby fixing the geometric structures of the few-ring phase mask in each iteration.
According to the updated   and the well-built database, we can calculate the electric fields that changes from 0.5 to 0.9.From these simulated results, one can see that, when NA≤0.6, the scanning images will be distorted seriously with a bad imaging resolution.In comparison, the imaging resolution is high if NA≥0.7, which suggests the smallest NA of 0.7 to achieve highresolution imaging in a SCM.Therefore, the NA of collection objective is also critical for highquality imaging of real objects in SCM.Moreover, the resolution reduces with the increment of slit width, because the contrast of image is getting lower.This result is consistent with the fact that the fixed 200nm-CTC-distance double slits will not be resolved when the double silt merges into one slit with the continuing increment of slit width.Supplementary Fig. 10 shows the simulated images that reveal the different resolution in various microscopies.As observed in our experiment, the coherent bright-field microscopy cannot resolve any double slit, suggesting its low resolving power.But, the traditional SCM by using refraction-based objective can only resolve the double slits with the CTC distance large than 240 nm, which agrees with our experimental results in Fig. 3 of main text.
To show the advantages of our proposed SCMs, we give a detailed comparison with the previously reported SCMs by presenting their various parameters.One can find that we have proposed the first planar-lens-based reflective SCM and the ultra-long working distance at the level of millimeter.More importantly, we can use industrial DUV lithography to fabricate all these objective chips in a mass-product and low-cost way, which pushes this objective chip towards practical applications.Supplementary Fig. 11 shows the images of our fabricated objective chips, which validates the feasibility for industrial mass production.

Supplementary
) where M0=0, M1=283, M2=850, M3=1046, M4=1258, and M5=6391, ∆ = 2  ( − 1) ℎ is the phase difference between etched and unetched zones in the objective chip at a wavelength of λ=405 nm, the refraction index of the quartz substrate is n=1.47 and the experimental etching depth  ℎ =530 nm (referring to ∆ = 1.23).To further evaluate the energy flux in the circular area with a radius of λ at the focal plane of the objective chip, we employ the expression:  ℎ () = | ℎ ()| 2 .Similarly, we can acquire the energy flux of the standard binary-phase zone plate in the same area with   = ∫ ∫   ( which takes ~34 minutes.Only the 8 μm×8 μm range is shown in Figs.4(b)-(d) to highlight more details of the images.

Figure 1 .
Figures and Captions

Figure 2 .
Figure 2. Focusing and imaging properties of our objective chip.(a) Simulated (curves) and experimental (dots) widths of belts at the different regions of our objective chip.Inset: SEM images of the different regions by addressing the corresponding zone numbers.The etched depth (left-

Figure 3 .
Figure 3. Reflective scanning confocal microscopy based on our objective chip.(a) Sketch of the optical setup of the objective-chip-based reflective SCM.LP: linear polarizer; BS: beam splitter; QWP: quarter wave plate; L: lens; PH: pinhole; PMT: photomultiplier tube.(b) Detected signals (PMT, curve) and images (CCD, inserts) when the nano-objects are scanned with the out-of-focus distance Δz=z-f.In the CCD images, the dashed circles denote the position of the focused signal light.In the PMT signals, the nonzero background (~0.1) is caused by the incompletely suppressed light (i.e., the four-lobe patterns) reflected from the back surface of the objective chip.(c-f) Double slits (c, SEM) and their images by using coherent bright-field microscopy (d), traditional RSCM (e) and objective-chip-based RSCM (f).The CTC distances of double slits Ⅰ to Ⅵ are 190 nm, 200 nm, 220 nm, 240 nm, 250 nm and 270 nm, respectively.The height and width of each slit are 2 μm and 50 nm respectively.Scale bar: 300 nm.(g) Line-scanning intensity profiles of images by using different microscopies.

Figure 4 .
Figure 4. Imaging complex nano-objects by using different microscopies.(a) SEM image of a "dolphin" object composed of 50 nm-width curves.(b-d) Images (upper row) by using coherent bright-field microscopy (b), traditional RSCM (c) and objective-chip-based RSCM (d).Their zoomed-in images with more details are shown in the lower row.(e) The line-scanning profiles of the imaged dolphins (along the colored lines of the upper low in b-d) by using coherent bright-field microscopy (green circles), traditional RSCM (triangles) and objective-chip-based RSCM (red squares).These line-scanning data are fitted by using a Gaussian shape to guide the eyes.
ℎ (, , ) =   (, , ) − 2 .(S4) and (S5), we can calculate the maximum and minimum focal spots, as shown in the solid lines in Supplementary Fig.S1b.Because both Eqs.(S4) and (S5) have no limitation about the number N, we use our proposed objective chip with a 5-ring phase mask to verify the predicted minimum (rmin) and maximum (rmax) focal size.Similarly, we go through all the possible solutions by changing the number and position of the phase-reversed zone in the second and fourth rings of the few-ring phase mask, which can be implemented together with the above calculation of the Strehl ratio.The simulated rmin and rmax are provided in Supplementary Fig.1b.Both the predicted

2. 1 .
Determining the radius of each belt in a standard zone plate Supplementary Fig. 2. Sketch of standard zone plate with 6391 belts.The imaging parameters are λ=405 nm, f=250 μm, which indicates a radius of 2 mm.Considering the non-paraxial feature of this objective chip, a rigorous formula of standard binary phase Fresnel zone plate (BPFZP) should be used to calculate the radius rm of the m th belt with ) where   () is the radial-position electric field of light diffracting from the m th belt in the zone plate and saved in the m th column of Ar,   () is the longitudinal-position electric field of light diffracting from the m th belt in the zone plate and stored in the m th column of Az.According to this definition,   ()=  ()=0 when m=0.Based on these database, we can calculate any focal field of the objective chip only if the phase of each zone is given.Supplementary Fig. 3. Data preparation for optimization.The Ar (a) and Az (b) database show the way of saving the pre-calculated data in our optimization.The normalized line intensity at the focal plane (c) and on the optical axis (d) of the 1 th , 3000 th and 6391 th belt of BPFZP.

For a 5
-ring mask, the dimension of particle D is 5.In our algorithm, the size or population of the particle is 20.The details and flowchart of PSO are shown in Supplementary Fig. 4. Since the electric fields (Ar and Az) have been calculated in advance, the calculation of fitness of each particle in iteration can be finished quickly, as shown below.Supplementary Fig. 4. The detailed flowchart of our built PSO.To correlate the particle parameters [ 1 ,  2 ,  3 ,  4 ,  5 ] with the unknown structures of the designed objective chip, we define the relative NA at the each boundary of the 5-ring phase mask by using

Supplementary Fig. 9 .
Simulated images of double slits by using collection objectives with different NAs.(a) Double slits with the center-to-center distances of 200 nm and the silt widths ranging from 10nm to 100nm.The length of slits are 2 μm.(b) Simulated scanning images under different NAs (0.5-0.9) and slit widths (10nm-100nm).(c) The line-scanning profiles of these slit images for the collection objectives with different NAs.NA=0.9 for a fair comparison.For our objective-chip-based RSCM, the PSF of condenser lens is the simulated electric field of our objective chip and the PSF of collector lens is Airy spot  1 ()  with a measured NA of 0.83.