1. INTRODUCTION

Imaging devices with resolutions beyond diffraction limit have always been challenging to realize but are highly anticipated in the fields of biomedical optics, material science, and nanofabrication [1–4]. Information regarding subwavelength features of an object usually resides in the evanescent waves that decay in the vicinity of the object [5]. Conventional microscopy based on ray optics and dielectric lenses fails to efficiently refract the evanescent waves. Consequently, the information about subwavelength features is discarded, limiting the ideal spatial resolution to ${\lambda }_0/2$. So far, the most reliable technique for high-resolution imaging is near-field scanning optical microscopy [6] (NSOM). NSOM relies on collecting the evanescent information before it decays in the near field of an object. It operates by collecting a point-by-point (pixel-by-pixel) image of the object using a sharp metallic tip of subwavelength dimensions placed in the near field of the object. The image resolution does not have any fundamental limit but depends on the radius of the tip, its distance from the object, and experimental noise. The technique has been successful in providing resolutions of the order of tens of nanometers. However, it is relatively slow paced due to the requirement of mechanical movements with subwavelength steps, making it unsuitable for real-time scenarios.

The requirement of high-resolution imaging devices and the hurdle of scanning is not only limited to visible frequencies. There are some biomedical scenarios where cancers, such as colon cancer [4,7,8] and skin cancer [9], are observed to be detectable only through terahertz radiation. Improving image resolutions at such frequencies is also of immense importance [10,11].

On the other hand, two-dimensional materials, such as grapheme [12,13], transition metal dichalcogenides (TMDCs) [14–16], and boron nitrides [17], have been shown to possess metallic properties in the terahertz frequency regime with extensive ability to tune the resonating responses over a broad range of frequencies. The ability to engineer the electronic and optical properties of materials actively using external stimuli has boosted many applications in the field of electro-optics [16,18]. In addition, extreme effort has been made to push the plasmonic resonance of two-dimensional heterostructures to the visible frequency regime [19,20] and also to induce gate-bias tunability to optical properties of existing plasmonic materials at visible and near-IR frequencies [21,22].

In this paper, utilizing the tunable resonance property of graphene, we present the alternate wide-field microscopy technique of graphene-based near-field optical microscopy (GNOM), which is free of mechanical movements based on the principles of diffraction-based microscopy [23–33]. In contrast to a moving tip, as in NSOM or other scanning-based imaging devices, we dynamically vary the optical properties (spatial surface conductivity profile) of a planar digitized graphene sheet placed in the near field of an object whose subwavelength features need to be resolved. The scattered light is collected at a far-field distance using conventional microscopy methods. However, as is the case of most diffraction-based imaging techniques, the measurements do not represent a final image of the object and computational postprocessing is required to obtain a high-resolution image. The scanning of optical properties of the sheet, hence the measurements, can be performed dramatically faster in time by varying the electrostatic bias using programmable electronics. Using computational experiments, we show that image resolution of the order of ${\lambda }_0/16$ (comparable to NSOM) can be achieved using GNOM. Even though we use properties of a graphene sheet at terahertz frequencies for proof of principle studies, the technique can be adapted to any tunable material at any frequency range.

Diffraction-based imaging techniques, such as structured illumination microscopy (SIM) [24], optical diffraction tomography (ODT) [25,27,33], scanning far-field superlens (FSL) [23,26] and interscale mixing microscopy (IMM) [28–30], have come into focus recently, where conventional wide-field microscopy measurements are postprocessed with numerical algorithms to resolve subwavelength features of an object beyond the diffraction limit. SIM relies on Moiré fringes where objects are illuminated with patterned light formed by interference of beams from multiple angles. More images are collected by shifting the fringe pattern, and the data are postprocessed in the Fourier domain to deconvolute the object data from the pattern data. The maximum resolution is limited to ${\lambda }_0/4$, corresponding to the periodicity of the fringe pattern. ODT relies on measurements of amplitude and phase of the scattered field from the object by illuminating from multiple directions, where the postprocessing aims to retrieve the refractive index profile of the object, assuming it as a scattering potential. The maximum resolution is also limited to ${\lambda }_0/4$ [34]. Even though both techniques are fast paced in comparison to NSOM in terms of measurements, the resolution in these techniques is fundamentally limited to the order of ${\lambda }_0/4$ or less since both cases do not capture the information of the evanescent spectrum decaying in the near field.

The latter two techniques (FSL and IMM) utilize a diffraction grating in the near field of the object to transfer the evanescent information fading in the vicinity of the object to the far field. From the principles of diffraction, a diffraction grating reflects and transmits an incident plane wave or evanescent wave with tangential spatial frequency component $k_{x0}$ into a number of diffraction orders, each characterized by a wave vector component defined by the Bragg-periodic condition

FSL relies on the design of the grating where evanescent waves of the object around the surface plasmon polariton (SPP) resonance wave vector [23] of the grating are folded to the propagation regime using only $m=\pm 1$ diffraction orders. This simplifies the postprocessing of the data; however, the resolutions are limited to the surface plasmon resonance of the grating material ($\sim {\lambda }_0/4$).

On the other hand, IMM is designed with no such (resonance) restrictions and includes multiple diffraction orders where the grating can be designed either with metallic or dielectric material [28]. Far-field intensity measurements from multiple incident angles are considered to numerically unfold the evanescent spectrum of the object from multiple diffraction orders. Recently, image resolution of the order of ${\lambda }_0/10$ has been demonstrated experimentally using IMM combined with a postprocessing method based on scalar diffraction theory [29]. The resolution limit of the IMM technique does not have any fundamental barrier apart from its dependence on the distance between the diffraction element and the object, and experimental noise. IMM is fast paced in terms of measurement in comparison to pixel-by-pixel scanning. However, it is relatively slow due to the requirement of multiple incident angular measurements to accurately unfold the evanescent spectrum. The multiple incident angle measurements with the rotation of source or object are typically prone to alignment errors [29,35]. Also, the near-field interaction is limited to one diffraction element impelling the limited number of multiple incident angle measurements the responsibility to transfer nonredundant near-field information of the object to the far field. In addition, the resolution is also hampered by the material loss of the diffraction gratings.

GNOM is free from the above imperfections with the ability to actively vary the diffraction properties of graphene without disturbing the system [36–39]. Optical properties of graphene are usually described by the real and imaginary parts of its surface conductivity, whose magnitudes are determined by the doping concentration of electrons injected using an electrostatic bias [40–42]. In the terahertz regime, the conductivity usually follows the Drude model and is linearly proportional to the square root of the surface doping concentration ($n_s$). For a given doping density, the conductivity of a graphene sheet at terahertz wavelengths can be expressed as [43]

A schematic of the setup of GNOM is shown in Fig. 1(a). A planar digitized graphene sheet is attached to a substrate and is placed on top (in close proximity) of an object that needs to be imaged with the graphene side down. The digitization of the sheet can be performed by considering graphene strips each of width $d$ placed side-by-side on a substrate (in the $x–y$-plane) with each strip attached to electrodes along the longitudinal direction ($y$ axis). By controlling the voltage of each strip, various patterns of the surface conductivity profile with a step size of $d$ can be programmed digitally [46]. For example, a surface grating of period $6d$ and duty cycle $3d$ can be created by periodically turning on a constant voltage for three consecutive strips and by turning off the next three strips. From the current fabrication technologies, strip width $d$ can be made as small as 50 nm [47]. For terahertz wavelengths (${\lambda }_0\approx 10\mathrm{\mu m}$), the strip width is around ${\lambda }_0/200$, allowing an almost smooth variation of the profile with respect to the wavelength.

 

Fig. 1. (a) Schematic setup of GNOM. Objects are placed on a substrate $(\epsilon =2.25)$. Graphene sheet slices with electrodes are placed at a distance $h$ on top of the object. Red and black colors of the graphene slices represent different values of surface conductivity. The inset is the schematic for transmission function calculation. (b), (c) The relative variation in field amplitude at angles of 0° and 30°, respectively, for five different objects numbered 1–5 shown as insets. Object $1\to \text{total size}(5{\lambda }_0)$, Object $2\to \text{total size}(3{\lambda }_0)$, Object $3\to \text{total size}(2{\lambda }_0)$, Object $4\to \text{total size}(2{\lambda }_0)$ with subwavelength features (${\lambda }_0/8$, ${\lambda }_0/4$, ${\lambda }_0/2$, ${\lambda }_0/8$, ${\lambda }_0/4$), and Object $5\to \text{total size}(2{\lambda }_0)$ with subwavelength feature sizes (${\lambda }_0/16$, ${\lambda }_0/8$, ${\lambda }_0/4$, ${\lambda }_0/2$, ${\lambda }_0/16$, ${\lambda }_0/4$). The gaps between the subwavelength features are of relative size. For objects with the smallest subwavelength features (more evanescent spectrum) the variation in relative amplitude is high and irregular (nonredundant) in pattern. $|H|$ and $|H_0|$ represent the amplitude of the field with and without a grating, respectively.

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Light emitted from the object in the form of both propagation and evanescent waves are allowed to interact with various engineered conductivity profiles discussed in later sections. Each profile couples (“mixes”) a part of the evanescent spectrum to the propagation regime that eventually gets transmitted to the far fields. The transmitted light through each profile is measured at a far-field distance through the substrate of graphene. Since the interaction of each evanescent wave with each conductivity profile is unique, each transmitted signal inherently contains nonredundant information about the object’s evanescent spectrum.

To demonstrate the uniqueness in the interaction of evanescent waves with each conductivity profile, we calculated (using COMSOL, RF Module) [48] the relative variation in the amplitude of the transmitted signal with and without the conductivity profile for a set of objects comprised of different subwavelength features. The object that we consider here is a transparent light-emitting source with an overall lateral dimension of wavelength order but is chopped into subwavelength pieces and gaps, as shown in the inset of Figs. 1(b) and 1(c) (numbers 4 and 5). An efficient imaging system is expected to recover the dimensions and locations of both pieces and gaps. In COMSOL, such sources can be defined as surface currents of required lengths by using the current density boundary condition. The surface currents, designed to radiate transverse-magnetic- (TM) polarized light, are then placed below a surface grating of unit cell length $\mathrm{\Lambda }$ made of two components of equal length ratios. The conductivities of the two components are defined as ${\sigma }_1=0.0025+0.2387i(E_f=1\mathrm{eV})$ and ${\sigma }_2={10}^{-4}(E_f=0\mathrm{eV})$, respectively (a small real part for ${\sigma }_2$ is assumed to represent interband transitions at zero doping concentration [49–56]). The light emitted from the sources and transmitted through the grating is first calculated in the near-field zone along a planar boundary in the transmitted region and then a Fourier transform is applied to compute far-field strengths. The relative variation in the far-field amplitudes with and without the grating as a function of periodicity of the grating at 0° and 30° transmitted angles are presented in Figs. 1(b) and 1(c), respectively. Since objects with smaller feature sizes contribute more to the evanescent spectrum, the magnitude of the relative variation of the far field for objects with no subwavelength features (numbers 1–3 in the insets) is less when compared to objects with subwavelength features (numbers 4 and 5 in the insets). Out of the three objects with the same overall size (numbers 3–5), the one with the smallest subwavelength features (number 5) gets maximum variation. The irregularity in the pattern as a function of periodicity represents the uniqueness in the interaction of each grating with each object. The similarity in the relative variation of the fields at both angles [in Figs. 1(b) and 1(c)] demonstrates that the object–grating interaction weakly depends on the measuring direction. In later sections, by defining a suitable transfer function for each grating, we demonstrate recovery of the locations and sizes of pieces and gaps of an object from such far-field measurements.

Note that even though the transparent nature of the utilized virtual object is an ideal assumption, diffraction-based imaging techniques such as IMM started with similar theoretical assumptions have been successful in experimental demonstrations [28–30].

2. OPTICAL TRANSFER FUNCTION OF GRAPHENE GRATINGS

The image reconstruction process is equivalent to calculating the electromagnetic field of the object at the object plane. The Fourier spectrum of the magnetic field of the object in the object plane can be expressed as

Postprocessing of intensity measurements using the GNOM technique requires calculation of the transmission coefficient (or coupling coefficient) of each incident plane wave or evanescent wave, represented by $k_x$, into its diffraction orders. In order to do so, consider a graphene sheet with a one-dimensional surface conductivity profile given by $\sigma (x)$ between two media (region I and region II) with permittivity ${\epsilon }_{\mathrm{I}}$ and ${\epsilon }_{\mathrm{II}}$, respectively, as shown in the inset of Fig. 1(a). Assuming TM-polarized waves with $\overrightarrow{E}=\{E_x,0,E_z\}$ and $\overrightarrow{H}=\{0,H_y,0\}$, solving Maxwell’s equations in Gaussian units and Cartesian coordinates, the tangential electric and magnetic field components can be expressed as

The matrix $T^{+}$ represents the coupling coefficients (or “mixing” coefficients) of incident plane or evanescent waves with wave vectors represented by $k_{xm}$ to transmitted plane or evanescent waves with the same $k_{xm}$, which can serve as the optical transfer function. However, note that the wave vector spectrum $k_{xm}$ is discrete with spacing between them defined by the periodicity of the grating $k_{xm+1}-k_{xm}=2\pi /\mathrm{\Lambda }$. For gratings with periodicities less than or of the order of the wavelength, the discretized $k_{xm}$ spectrum does not represent the object spectrum accurately. Instead, one can define a super cell structure with a large but finite number of repeating unit cells, where $\mathrm{\Lambda }$ now represents the periodicity of the super cell. Since $\mathrm{\Lambda }\to \infty$, the $k_{xm}$ set is dense enough to represent the object spectrum. In such case, the matrix $T^{+}$ itself represents the transfer function ($T$) of the grating. However, the numerical decomposition in Eq. (3) requires a large number of slices for accuracy making it impractical to invert the matrices [in Eq. (9)] of such high order.

Alternately, the matrix $T^{+}$ can be computed for each $k_{x0}$ [in Eq. (1)] spanning from $-\pi /\mathrm{\Lambda }$ to $-\pi /\mathrm{\Lambda }$, ($k_{x0}\to k_{x0;l}\in [-\pi /\mathrm{\Lambda },-\pi /\mathrm{\Lambda })$) to fill the gaps in the $k_{xm}$ spectrum. The optical transfer function $T$ can be obtained by dividing it into number of block matrices as

Also, it has been noted that due to the formation of grain boundaries, the surface conductivity of graphene and other two-dimensional materials are not generally uniformly controllable over a wide spatial range [58]. Note that the above equations are also valid in such cases when $\sigma (x)$ is not periodic. However, precalibration of the graphene sheets to identify and calculate the possible surface conductivity profiles is needed. For such nonuniform cases, the inversion of large matrices is inevitable. In addition, the above formulation is also applicable for thin gratings or metasurfaces (of thickness $d_0\ll {\lambda }_0$) made of bulk materials. Equation (9) can be computed by converting the bulk permittivity pattern of the materials $({\epsilon }_r(x))$ to surface conductivity pattern $(\sigma (x))$ using $\sigma (x)=-i({\epsilon }_r(x)-1)k_0{d}_0$ [59].

3. IMAGE RECONSTRUCTION WITH BINARY GRAPHENE GRATINGS

The transfer of evanescent waves to the propagation regime is usually accompanied by the distribution of light into multiple diffraction orders that eventually decay in the near field of the grating. In order to get more information from the object, here we utilize far-field measurements taken by varying the periodicity of the graphene grating. As discussed before, in practice such variation can be programmed using electronics for graphene without disturbing the system. It has been shown in Figs. 1(b) and 1(c) that different subwavelength features of the object interact differently with each grating and consequently transfer nonredundant information regarding the subwavelength features of the object to the far field. Here we present the numerical image recovery procedure for GNOM utilizing only one far-field measurement in the normal direction through the graphene gratings as a function of its periodicity.

In order to evaluate the performance of GNOM in reconstructing subwavelength features of an object, we performed computational experiments with calculations from COMSOL, considered as “measurements.” A test object is designed with a total length of $2{\lambda }_0$ composed of subwavelength features of dimensions ranging from ${\lambda }_0/16$ to ${\lambda }_0/2$. The field pattern of the object is shown as the dashed line in Fig. 2(c). The object is placed in the near field of a grating at a distance of 100 nm radiating transverse-magnetic-polarized waves that pass through the grating, as shown in the schematic in Fig. 1(a). Far-field “measurements” are taken using the far-field node in COMSOL by defining the periodicity of the grating as $\mathrm{\Lambda }={\lambda }_0/f$, where $f$ varies from 1 to 16 in 300 steps. The overall grating size is fixed to $10{\lambda }_0$ in all cases. The diffraction gratings are composed of binary units with $E_f=0\mathrm{eV}$ for one of the components. To compensate for the loss of phase information, intensity measurements are taken at two Fermi energies of graphene $E_f=1\mathrm{eV}$ and $E_f=0.9\mathrm{eV}$ for the other component. The intensity “measurements” in the normal direction of the grating as a function of periodicity for the test object are shown in Fig. 2(a). Note that $E_f=0\mathrm{eV}$ may represent zero conductivity according to Eq. (2). However, a small nonzero real part ($\sigma ={10}^{-4}$) is assumed here to represent the interband conductivity of graphene when the doping concentration is zero [49–56].

 

Fig. 2. Image reconstruction using binary graphene gratings. (a) Transmitted far-field intensity “measurements” of the test object (calculated using COMSOL) in the normal direction with grating periodicity reconfigured from $\mathrm{\Lambda }={\lambda }_0$ to $\mathrm{\Lambda }={\lambda }_0/16$ in 300 steps, ${\lambda }_0=10\mathrm{\mu m}$. The unit cell has two equal size components with ${\sigma }_1\propto E_f$ and ${\sigma }_2={10}^{-4}$. The object-to-grating distance is $h=100\mathrm{nm}$. (b) The corresponding transmission function matrix ($T^G$) with rows as the rows of the transmission function of each grating in the normal direction calculated using Eq. (10), where color represents the magnitude of the matrix element. Upper and lower sections represent gratings with the two Fermi levels of graphene. (c) The test object (dotted line) and the reconstructed image (solid line) using Eq. (13). The overall size of the object is $2{\lambda }_0$, and the subwavelength feature sizes are (${\lambda }_0/16$, ${\lambda }_0/8$, ${\lambda }_0/4$, ${\lambda }_0/2$, ${\lambda }_0/4$) with gaps of relative size between them. The reconstructed image resolves the gaps and features of the object.

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To begin the process of image reconstruction from the far-field measurements, the transfer function $T$ $(=T^{(g)})$ is calculated for a grating with each periodicity using Eq. (10), where $g$ is an index to represent each grating. Since only one far-field measurement per grating is considered in recovery, only one row (central row for normal direction) of each $T^{(g)}$ is required in postprocessing. A composite matrix ($T^G$) is then defined whose rows are filled with a central row of each $T^{(g)}$, i.e., $T_{gm}^G=T_{0m}^{(g)}$. The composite transmission matrix $T^G$ relates the amplitude spectrum of the object to transmitted far fields by all gratings simultaneously. A section of the elements of matrix $T^G$ is shown in Fig. 2(b). The number of nonzero elements in each row indicates the number of diffraction orders coupled into the propagation regime and the magnitude shows the strength of the coupling.

The amplitude spectrum of the object is expanded in the pixel basis, as described in [28], where the object plane is divided into a number of pixels. The problem of image reconstruction is now reduced to the problem of finding the field amplitudes of each pixel $(h_p)$ centered at a coordinate $x_p$ with width $P$, where $P={\lambda }_0/16$ in this example. Using the fact that the Fourier transform of each pixel is a sinc function, the transfer function $(T^F)$ relating the amplitude of the field $(h_p)$ at each pixel $(x_p)$ to the transmitted far-field amplitude $(a_g)$ by each grating can be defined as

4. IMAGE RECONSTRUCTION WITH OPTIMIZED GRAPHENE GRATINGS

The Imaging performance of GNOM relies on (i) efficiency of the diffraction element to couple (“mix”) the evanescent waves to the propagation regime and (ii) the accuracy of the image reconstruction mechanism to uncouple (“unmix”) the contributions of evanescent and propagation parts originated from the object. However, mixing too many diffraction orders into one far-field measurement complicates the unmixing algorithm, leading to the parasitic artifacts seen in Fig. 2(c). In order to keep the unmixing algorithm accurate and efficient, the diffraction element itself can be designed to share the unmixing process. Using the genetic algorithm [62–64], here we designed a number of optimized unit cell patterns with a fixed periodicity $\mathrm{\Lambda }$, where each pattern enhances the coupling of a specific part of the evanescent spectrum of the object to the propagation regime suppressing the others. The design aim of each unit cell is to maximize the coupling of an evanescent wave with wave vector $k_{xa}$ to its corresponding $m$th diffraction order in the propagation regime $k_{xb}=k_{xa}+m2\pi /\mathrm{\Lambda }$ ($|k_{xb}|<k_0$) suppressing others. The design principle is also close to the concept of a metasurface [65–69], where an incident plane wave with wave vector $k_{xa}$ is coupled to a plane wave with wave vector $k_{xb}$ at an interface with periodic phase gradient $d\varphi /dx=k_{xa}-k_{xb}$. However, instead of forcing the phase gradient by engineering each unit of the unit cell, we utilized the genetic algorithm to optimize the unit cell pattern.

The periodicity of the unit cell $\mathrm{\Lambda }$ is fixed equal to the free-space wavelength ($\mathrm{\Lambda }={\lambda }_0$) and then made into 32 pieces of equal length. The free parameter of each unit of the unit cell is only the Fermi level of the graphene $E_f$, which relates to the surface conductivity by Eq. (2). The magnitude of $E_f$ is limited to be in the range of $0\to 2\mathrm{eV}$ [44,45], which corresponds to the feasible range in practice. It can be verified that for gratings with periodicity of the unit cell $\mathrm{\Lambda }={\lambda }_0$, for any chosen $k_{xa}$ there always exists an integer $m$ such that it is the $m$th and $m+1$th diffraction orders fall in the propagation regime. To limit the measurement cone to less than 30°, we consider only one out of two diffraction orders that satisfies the condition $k_{xb}\le k_0/2$ as the output direction. The objective function of the optimization process is defined using the transmission function $(T^{+})$ in Eq. (9). The far-field transmission into the wave with wave vector $k_{xb}$ is given by one row of the matrix represented as $T_{bn}^{+}$ (the subscript $b$ stands for wave vector $k_{xb}$). To maximize the coupling of the wave with wave vector $k_{xa}$ to the wave with wave vector $k_{xb}$ and suppress others, it is enough to maximize the element $T_{ba}^{+}$ in the row $T_{bn}^{+}$ while suppressing others. Hence, the objective function for optimization can be defined as

 

Fig. 3. Image reconstruction using gratings with optimized unit cells. (a) Fermi level, consequently the surface conductivity, profile of unit cells optimized to maximize the coupling of each wave vector $(k_{xa})$ to the corresponding diffraction order in the propagation regime ($\mathrm{\Lambda }={\lambda }_0$, ${\lambda }_0=10\mathrm{\mu m}$). Each unit cell has 32 elements. A color bar is displayed for clarity. (b) Line plots of the same quantity for three representative wave vectors. (c) The elements of composite transmission matrix $T^G$, where each row is the row of the transmission function calculated using Eq. (10) for corresponding transmission direction of each optimized grating. (d) The real part of the magnetic field of the object (black dashed line) from COMSOL and the corresponding image (solid and dotted lines) recovered analytically using Eqs. (3) and (15) with no numerical postprocessing. The inset shows the corresponding amplitude spectrum of the object and image. The electric and magnetic fields of the image are computed using Eq. (4) as $H=H_z$ and $E=\sqrt{E_x^2+E_z^2}$.

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Even though the optimization maximizes the element $T_{ba}^{+}$ by suppressing others, the direct coupling of propagation wave to propagation wave $T_{bb}^{+}$ is inevitable due to limited range of $E_f$ and fixed geometrical parameters. To perform the imaging recovery process, the composite transmission function of all gratings $T^G$ can be defined as a sparse matrix, where each row is filled with elements of the transmission function of each optimized grating carefully aligned to the corresponding $k_{xm}$ values. The elements of the matrix $T^G$ are presented in Fig. 3(c). It can be observed that in comparison to the matrix of regular periodic gratings in Fig. 2(b), the number of dominant elements is minimized per each row. The dominant elements along the diagonal represent the maximized element of each row and the transmission band around the central column represents the direct coupling of propagation to propagation orders. It is evident that the transmission measurements taken through each of these gratings consist of dominant contributions only from the propagation part of the object and a specific evanescent wave vector region minimizing the efforts of postprocessing algorithms. Note that there is only one transmission measurement required using each grating similar to the regular periodic case. However, the required measuring angle varies with each grating since $k_{xb}$ (corresponds to measuring angle) depends on $k_{xa}$.

The ideal capability of image recovery using optimized gratings can be utilized if both amplitude and phase of the transmitted field can be measured along the required angle. Assuming $A_{\text{meas}}^{(g)}(k_{xb})$ is the measured complex amplitude of the field at a far-field distance in the corresponding $k_{xb}$ direction with a grating $(g)$ and assuming only dominant elements of the transmission function contribute to the far field, an equation for the amplitude spectrum of the object can be written as

Far-field transmission “measurements” through optimized gratings are taken using COMSOL for the same test object. The amplitude spectrum is calculated from the field measurement of each grating using Eq. (15). The image of the object is then recovered from the calculated amplitude spectrum using Eq. (3) and is presented in Fig. 3(d). The image clearly reconstructs all the subwavelength features of the object to the order of ${\lambda }_0/16$ without any complex postprocessing methods. The calculated amplitude spectrum is shown in the inset of Fig. 3(d) in comparison to the actual amplitude spectrum. Relatively large discrepancies in the amplitudes are observed in the range of $0.5\le |k_x/k_0|\le 3$, where the contribution from the diagonal elements of the $T^G$ matrix is weak. These discrepancies resulted in oscillations of the order of ${\lambda }_0$ in the image of the object. Note that here the recovery is not based on the pixel basis. Hence, the field of the object can be expressed as a continuous function in space.

The postprocessing of the data can also be performed using the same pixel-basis method described in Eqs. (11) and (13), which can handle intensity measurements. It will be shown in the next section that optimized gratings provide better imaging performance (with fewer artifacts) and high noise tolerance in comparison to periodic gratings.

5. NOISE TOLERANCE

In this section, we test and compare noise tolerance of the imaging process using regular periodic gratings and gratings with optimized unit cells. We performed the image reconstruction process [using Eqs. (11) and (13)] by adding multiple realizations of random noise patterns to the measured data proportional to its peak intensity. The calculated mean and standard deviation of field amplitudes of each pixel are plotted in Fig. 4(a), representing the imaging performance of regular periodic gratings at different noise levels. Clearly, the parasitic artifacts in the objects increase in amplitude, dissolving the sharpness of the edges, specifically in larger objects. At 20% noise, the recovery almost fails in distinguishing the central objects. Figure 4(b) represents image recoveries using gratings with optimized unit cells. Significant improvement in the imaging performance with minimum artifacts is observed for low noise levels in comparison to regular periodic gratings. Both the subwavelength features and features of wavelength order are clearly distinguished, even at higher noise levels. The error bar (standard deviation) in all cases is almost proportional to the noise level.

 

Fig. 4. Noise tolerance of the image reconstruction. Image reconstruction using (a) binary graphene gratings and (b) gratings with optimized unit cells by adding multiple realizations of random noise to the measured far-field intensity. The dotted line represents the object; the solid line and the shaded region represent the mean image and the standard deviation of all noise patterns, respectively. The addition of noise to the “measured” intensity is performed as $I_{\text{meas};\text{noise}}={|H_{\text{meas}}+Nr\max (|H_{\text{meas}}|)|}^2$, where $I$ represents intensity, $H$ represents complex field, $N$ is the noise percent level, and $r$ is a random number between $-1$ and 1. In comparison to (a), (b) has less artifacts surrounding the object and reconstructs all the features of the object at noise levels as high as 20%.

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6. DISCUSSION

The imaging performance of the proposed GNOM technique depends on the key parameters of the system (i) the smallest achievable width or spacing between graphene strips, (ii) the minimum distance possible between the objects and the gratings, and (iii) the tunability range of the surface conductivity of the gratings via external stimuli. Even though the transmission function calculation in Eq. (10) and the pixel-basis approach described in Eq. (11) do not limit the resolution of the image, the reliable final resolution of the technique is determined by the above [particularly (i) and (iii)]. If one aims on resolving an object with resolution ${\lambda }_0/f$ ($f>1$), the maximum width of the smallest strip should be necessarily equal or less than ${\lambda }_0/f$. The maximum distance between the object and the gratings should not exceed ${\lambda }_0/f$ (since evanescent components decay faster than ${\lambda }_0/f$). While the above two are the necessary requirements, the tunability range of the surface conductivity determines the coupling efficiency of the evanescent waves to propagating waves. Note that the imaging coupling (imaging performance) weakly depends on the actual magnitude conductivity but strongly depends on the range of tunability of the conductivity. For example, one can design optimized grating profiles similar to Fig. 3(a), restricting the range of the Fermi level between $-1$ and 1 eV to obtain qualitatively similar imaging performance.

One of the known drawbacks of GNOM (or some diffraction-based techniques) is its strategy in handling imperfect measurements. In spatial scanning techniques, such as NSOM, one imperfect measurement usually affects the quality of only one pixel of the image. In contrast, the computational postprocessing of GNOM may distribute one imperfect measurement into a number of pixels or completely compensate it with the help of other accurate measurements. This property is evident from Fig. 4 where image recovery using data with higher noise levels does not necessarily affect the subwavelength features of the objects but rather increases the overall intensity level of the object (decreases the image contrast). Another known drawback is its inability to image only a specific section of the object where NSOM fits best.

In conclusion, we presented a new graphene-based high-resolution mechanical-scanning-free optical microscopy technique (GNOM) based on the principles of diffraction imaging. In contrast to mechanical scanning of a sharp tip (as in NSOM), we utilize active electronic scanning of diffraction properties of atomic thin graphene gratings placed in the near field of an object. We provided engineered designs of spatial patterns and optical properties of the gratings that can efficiently and systematically out-couple the evanescent information of the object to far fields mixed with diffraction orders. By defining a transmission function of graphene interface gratings, we then provided computational postprocessing methods to “unmix” the evanescent information and consequently recovered the detailed spatial field profile of the object. Using computational experiments, we demonstrated that the technique is capable of imaging objects with an overall size of $2{\lambda }_0$ and a pixel resolution of ${\lambda }_0/16$. Numerical optimization based on the genetic algorithm is then used to design an optimum set of diffraction gratings that can potentially eliminate the requirement of postprocessing or at least minimize the numerical artifacts in the image. Noise tolerance and known drawbacks are discussed.

The proposed technique has the capability to transform into a fully static high-resolution imaging setup without the requirement of any linear or angular movements of objects, sources, or tips. The variation of optical properties of graphene gratings using electrostatic bias can be performed dramatically faster by making the technique more suitable for real-time imaging scenarios. Considering the extensive ongoing research on active tunable plasmonic materials at visible and near-IR frequencies [19–22], the proposed technique has more avenues to improve and extend to a broad range of frequencies.