1. Introduction

Numerous applications in materials science, device characterization, security, and biology require imaging with subwavelength resolution. However, the resolution limit of conventional optical microscopy is fundamentally limited by the diffraction limit to approximately half of the vacuum wavelength (λ) [1] with exact limit being related to signal-to-noise ratio in the measurements [2]. Immersion microscopy [3], sparsity based computational microscopy [4], and several other diffraction-based techniques [5–8] discussed below, have been successful in improving the resolution of optical microscopy and metrology [9, 10]. However, resolution of realistic far-field imaging systems remains limited to approximately one-quarter of free-space wavelength. Label-free imaging with deep subwavelength resolution is typically reserved to scanning near-field microscopy (SNOM) [11] or near-field tomography [12, 13] techniques that impose significant constraints on image acquisition rates. Diffraction-imaging techniques capable of deep subwavelength resolution based on far-field field [14] and intensity [15] measurements have been recently proposed. In the latter approach, interscale mixing microscopy (IMM), an object is positioned in the near-field proximity to a diffractive elements that outcouples information about subwavelength features of the object to the far-field where it can be detected by conventional means, followed by computational reconstruction of the object. In this work we analyze numerical stability of IMM and present convenient parameterizaiton of thin objects that significantly improves noise tolerance of computational image reconstruction.

The optical radiation, scattered or emitted by the object, can be represented as a linear combination of plane waves [1]:

In such representation, the information about the features of the object with the typical feature size Δ is encoded in the plane waves having transverse wavevectors of the order of k_⊥ ∼ 2π/Δ [16]. At the same time, evolution of this information in the direction of the imaging axis (z axis in this manuscript) is given by exp(−ik_zz) with wavevector component k_z related to k_⊥ and to angular frequency of optical radiation ω = 2πc/λ via dispersion relation

It can be clearly seen that information corresponding to Δ ≪ λ corresponds to imaginary values of k_z, and thus it exponentially decays away from the object [2].

The main motivation behind improving the resolution of optical microscopy can be thus related to the detection and reconstruction of the radiation corresponding to larger values of k_⊥. For some objects, recovery of evanescent part of the spectrum can be completed based on measurements of the propagation part of the spectrum [4]. Immersion microscopy-related techniques aim to increase the effective refractive index n and thus postpone the onset of diffraction limit. SNOM uses ultra-sharp tips to scatter (diffract) the radiation from the near-field of the object to the far-field zone; the shape of the tip is optimized to predominantly diffract information about small features of the object. Structured Illumination Microscopy (SIM) [5] effectively doubles resolution by illuminating object with beams with k_⊥ ∼ nω/c = k₀ and analyzing the diffracted light. In a far-field superlens (FSL) [7], an object is first imaged directly, and is then imaged through a plasmonic resonant structure that is designed to amplify the information about subwavelength features of the object and scatter this information to the far-field by outcoupling it through first-order diffraction of the diffraction grating. In practical systems, resolution of both SIM and FSL is of the order of λ /4. In related metrology technique, scatterfield microscopy [9] the subwavelength details of the profile of relatively large diffraction gratings are recovered based on diffraction patterns.

In contrast to the above techniques, the IMM, originally proposed in [15], is capable of performing imaging of localized, wavelength-scale objects with deep subwavelength resolution. In the original proposal, the technique utilized a diffraction grating to out-couple information corresponding to multiple length-scales of the object to the far-field via multiple diffraction orders of the grating. Hence, the resultant diffraction pattern measured at far-field contains ‘mixed’ contributions from both evanescent and propagating parts of the light scattered from the object. High resolution images are then reconstructed by using a series of computational post-processing steps aimed to ‘unmix’ these contributions. While there is no fundamental limit on coupling between different parts of the spectrum, accuracy and resolution of the recovered image and the robustness of the technique are dependent on i) the ability of diffractive elements to couple the evanescent part of the spectrum into propagating modes, and ii) the efficiency of computational optimization techniques to un-couple the spectrum and cope with the noise in the measurement data [2]. Therefore, both diffractive elements and the numerical procedure can be optimized to improve image recovery.

The basic approach for IMM formulated in [15] has been successful in retriving images of light emitting subwavelenght objects placed under a metallic grating. However, it has opened several fundamental questions regarding potential stability and resolution limit of IMM. In this work we further develop the IMM in three important aspects. First, we demonstrate a five-fold increase in noise tolerance during image recovery process. Second, we demonstrate the recovery of small light-blocking objects (in contrast to light-emitting objects considered previously), a crucial step towards experimental realization of IMM. Finally, we demonstrate diffraction-based imaging with both plasmonic and non-plasmonic gratings. Highly stable recoveries of wavelength-scale objects with resolutions of the order of λ /20 are demonstrated by representing the objects as collections of sub-wavelength “pixels”. New designs of diffractive elements that improve the resolution and stability of IMM technique are proposed.

The rest of manuscript is organized as follows. In Section 2 we present details of the proposed formalism, and introduce the pixel basis. Section 3 presents analysis of imaging performance of periodic diffractive elements. In Section 4 we analyze imaging performance of chirped diffractive elements. Section 5 concludes the manuscript.

2. Mathematical foundations of Interscale Mixing Microscopy (IMM)

The schematic of the system is presented in Fig. 1. The [sufficiently thin] object is positioned in the near-field vicinity [at the distance y₀ ≪ λ] of the diffractive elements (geometry of the diffractive elements will be discussed below). The object is then excited by plane waves propagating at [multiple] incident angles θ_i. Angular distribution of light intensity away from the diffractive element is recorded for each incident angle and is used to numerically reconstruct transparency of the object.

 

Fig. 1 Schematic of IMM; Main figure: the diffraction-based high-resolution imaging setup; inset: line object represented as a set of pixels with amplitudes b_i

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In order to estimate the realistic performance of the presented technique, in this work we use “computational experiment” approach. In this approach, we use finite-element method (FEM) [17] solutions of Maxwell equations to emulate experimental measurements, by solving for electromagnetic field distribution resulting from the interaction of light generated (scattered) by a set of subwavelength objects and diffraction grating. The simulation domain in FEM is 60λ wide and 50λ tall, surrounded by perfectly matched layers. The domain, meshed with maxium mesh element size of λ /30 encompassess elements of diffraction gratings, finite-sized objects, and sources of radiation used to generate electromagnetic field. The field (intensity) calculated by FEM software is then used in lieu of experimental data.

In a separate set of calculations, rigorous-coupled-wave-analysis (RCWA) [18] solutions of Maxwell equations are used in image recoveries. The artifacts of FEM (mostly related to finite size of the simulation domain) are fundamentally different from the artifacts of RCWA (related to implicit periodicity of the geometry); therefore, the pair of numerical techniques in some sense emulates real situation when data generated in experiment and image is recovered by a computer. As an added benefit, our technique allows us to compare recoveries based on intensity- and field-measurements (the latter are routinely performed at THz and GHz frequencies) and allow us to controllably add noise to our “measurements”. In this work we restrict ourselves to recovering thin one dimensional (line-shaped) objects. Generalization of the presented approach to two-dimensional objects, although relatively straightforward (see, e.g. [14] for example of field-based imaging), is devoted to future work.

In the image reconstruction process, we begin by using RCWA to compute the grating-specific transfer function τ(k_x,θ_p), which defines the contribution of information originally encoded into plane waves with wavevector component k_⊥ = k_x, at the object plane, to the far field propagating in the direction θ_p. The transfer function is calculated for a broad spectrum of k_x values, spanning both the propagation (|k_x| ≤ k₀) and evanescent (|k_x| > k₀) regions. The far-field intensity of the light behind the diffractive element can be now related to the emission (scattering) spectrum of the object positioned in front of the element. Explicitly, consider an isolated object excited by the field propagating along the z axis. Assuming that the spectrum of the field scattered by this object is given by a(k_x), the far-field intensity I_p in the direction θ_p behind the diffractive elements due to this same object excited by the plane wave coming from direction θ_i (see Fig. 1) can be written as:

The imaging problem is mathematically equivalent to recovering the unknown amplitude spectrum a(k_x) based on measurements of I_p. Here we perform this task by minimizing the deviation between the measured intensity patterns I_meas and calculated intensity patterns I_p for multiple measurement directions θ_p and incident angles θ_i,

It is critical to use a sufficient number of k_x terms in Eq. (3) in order to adequately calculate the distribution I_p(θ_p,θ_i) and to avoid the artifacts related to spectrum discretization. However, the limited number of measurements is typically not sufficient to solve for “digitized” version of the object spectrum a(k_x). It is therefore required to develop a model that can parameterize the spectrum of the object with relatively few parameters, and use the optimization procedure [Eq. (4)] to deduce numerical values of these parameters. The original work [15] used Taylor series representation for the spectrum of the source. However, detailed analysis demonstrates that such model is overly sensitive to measurement noise, especially for recovering evanescent part of the spectrum |k_x| ≫ k₀. Representing the spectrum as linear combination of orthogonal polynomials, as well as linear combination of harmonic- and Bessel- functions yields qualitatively similar results. Here we propose to represent the object as linear combination of finite-sized pixels and use Eq. (4) to calculate the amplitude of these pixels, essentially calculating the transparency profile of the object. On the implementation level, we divide the object plane into N pixels; each pixel having width p_x and centered around x_n in the y₀ plane. Each pixel produces an electromagnetic field that is equivalent to a single slit diffraction pattern with an unknown amplitude b_n. Hence, the object spectrum can be expressed as:

Note that in principle b_n can be extracted as a complex number that represents both amplitude and phase of the pixel. The main advantage of the pixel basis expansion is that the number of unknown quantities that need to be optimized now can be reduced depending on the object size and the target resolution. Most importantly, as seen below, these results are stable and have significant noise tolerance.

Two types of objects were studied in our “numerical experiments”. The objects of the first type, “sources”, represent the luminescent objects, represented in FEM as line currents of predefined length; these objects correspond to luminescent tags or finite-sized slits in the screen. The objects of the second type “blocks”, are modeled as perfect electric conductor lines or λ /50-thick metal and dielectric blocks, excited by incoming electromagnetic beams. In all cases, objects were positioned at y₀ = λ /40, and the field (intensity) was measured along the circular arc with radius of 30λ ; both measurement and incident angles θ_p,θ_i spanned between −60° and 60°, with increments of 1° for θ_p and 20° for θ_i respectively. In all recoveries, we attempt to recover the field scattered by the objects across one-wavelength area under the grating, centered around x = 0, with target resolution of (p_x ∼ λ /20); amplitudes of 20 (λ/20-wide) pixels are recovered for luminous objects; amplitudes of 16 pixels are recovered for scattering objects.

In stability analysis, random noise −1 ≤ r_n ≤ 1 is added to “measured” intensity:

3. IMM with periodic diffractive elements

We begin by analyzing the performance of the IMM technique with simplest possible diffractive element, periodic diffraction grating with period Λ. In order for the grating to provide substantial interscale mixing, and provide efficient coupling of evanescent information into propagating waves, grating period should be of the order of free-space wavelength. Gratings with substantially larger periods couple evanescent waves only through high diffraction orders and lose efficiency. Gratings with substantially subwavelength period have Bloch vector q = 2π/Λ ≫ k₀, and thus leave “gaps” in the shifted spectrum.

Figure 2 illustrates imaging performance of periodic 0.1λ -thick diffraction plasmonic (ε = −100 −0.1i) grating with Λ = 0.7λ period, and 50% filling fraction used to recover a set of three sources with sizes λ /20, λ /10, and λ /5. It is clearly seen that presented computational imaging technique is highly tolerant to experimental noise [up to 20%]. Interestingly, the noise-related artifacts in a pixel basis do not always lead to disappearance of the smallest features of the composite source. Rather, the noise reduces the overall contrast level, and introduces parasitic sources.

 

Fig. 2 Image reconstruction using pixel basis expansion; (a) The far-field intensity pattern created by the three subwavelength sources λ /20,λ /10, and λ /5 in length [green line in panels (d)…(f)]; (b) typical distribution of angle-dependent random noise (inset) and noise-affected far-field intensity pattern corresponding to noise level of 10% (main panel); (c) transfer function of the particular diffraction grating used in computational experiments (parameters presented in the text). Panels (d), (e), and (f) represent recoveries corresponding to 5%, 10%, and 20% noise level respectively; blue lines and shaded areas represent mean and standard deviation of recoveries representing different realizations of noise; red lines represent noise-less recoveries; green lines correspond to actual configuratoin of the sources

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The performance of IMM can be significantly improved if the full field (as opposed to intensity) can be measured in experiment. Figure 3 summarizes recoveries of the same set of objects as in Fig. 2 when electromagnetic field extracted from FEM model is used in linear analog of Eq. (4) to recover the parameters of the three luminous objects. Similar to Eq. (6), random noise of the form $H_{noise}=\delta \cdot (r_n+1)\cdot \exp (\pi i{r}_n^{\prime })\cdot \max \left|H_{meas}\right|/2$, with $-1\le r_n,r_n^{\prime }\le 1$ is added to the FEM-derived “experimental” field prior to RCWA-based recoveries to analyze stability of the field-based IMM.

 

Fig. 3 Similar to Fig. 2(d)–2(f), but with recoveries based on field (as opposed to intensity) measurements; panels (a), (b), and (c) represent recoveries of the same set of objects when the noise with magnitude of 5%, 10%, and 20% (in terms of maximum value of the field amplitude) is added to the “measured” field

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To assess the ability of IMM to detect small passive objects, compound “block” objects composed from two PEC lines are incorporated into FEM model; the objects are illuminated with a gaussian beam with beam width of the order of few wavelengths in size as shown in Fig. 4(a). The measured intensity in the far-field now contains the contributions from both the input beam and the scattered light from the objects coupled through the grating. The recovery process is similar to the one described through Eqs. (3)–(6). However, to increase stability of the recoveries, it is advantageous to isolate the contributions of the Gaussian background field that often dominates the measured intensity I_p from the scattered field. To perform such procedure, I_p in Eq. (3) is now expressed as:

 

Fig. 4 Image reconstruction of scattering objects (blocks) (a) Distribution of electromagnetic field across the system; inset highlights the compex interaction between the incident field and the scattering objects; position of objects (modeled as PEC lines) indicated by black lines in inset; (b) transfer function of the grating used in recovery; Panels (c), (d) correspond to recovery of the objects that fit inside the gap of the grating (h = 0.5λ) with 2% (c) and 4% (d) random noise respectively; panels (e,f) represent recoveries of objects with compound size greater than air gap of periodic grating that are recovered using a two arm grating system with arm separation h = 0.55λ; in panels (c…f) blue lines and shaded areas represent mean and standard deviation of the distribution of recovered objects; red lines represent noiseless recoveries; black horizontal lines represent positions of the objects in the FEM setup

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As expected, presence of strong background field, along with interaction between the objects and diffractive elements, unfavorably affects the imaging performance of the proposed technique. However, despite these challenges, the proposed approach is substantially resilient to be able to recover objects of the order of λ /16 as shown in Fig. 4(c) with a noise tolerance of 2% using plasmonic (ε = −100 −0.1i) periodic diffraction grating having a period of 0.5λ, filing fraction of 50%, 0.1λ thickness. Note that since the noise level is defined as percentage of measured intensity, the noise introduced into the measurements is comparable (and even sometimes bigger than) the perturbation of far-field intensity caused by scattering of light by the objects.

During the image recovery process, specifically in the case of blocks, it is seen that the objects residing behind the air gaps of the diffraction grating are recovered more accurately than the objects placed behind the metallic parts of the grating. Therefore, to image larger objects, we propose to use a diffractive elements where the two grating arms are separated by a gap with pre-selected size h(≈λ). Such compound-grating system minimizes the multiple reflections between objects and the metallic parts of the grating and enables accurate recoveries of relatively large objects as illustrated in Fig. 4(e) and 4(f). Another advantage of the compound-grating system comes from the fact that such design introduces a natural focal point into the imaging system. Furthermore, it can be straightforwardly extended to 2D scanning microscopes, potentially resulting in highly parallel SNOMs that image ∼ λ × λ -regions with deep subwavelength resolution.

To analyze sensitivity of IMM to variations of refractive index of the scattering element, the PEC lines in above model are replaced with thin λ /50-thick plasmonic (ε = −100 −0.1i) and dielectric (ε = 12.96,ε = 2) materials. IMM is capable of recovering the high-index contrast objects, and is able to resolve two components of low-index objects. However, its performance in calculating the sizes of low-index objects is reduced as compared with high-index objects [see Fig. 5(a)].

 

Fig. 5 (a) reconstruction of the two-block system, as described in Fig. 4 with PEC lines replaced with thin (λ/50) objects with finite permittivity ε; lines of different color represent blocks of different permittivity (b,c) Image reconstruction of (b) light emitting sources and (c) light scattering objects (blocks) using dielectric gratings.

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It is important to note that IMM does not rely on resonance in plasmonic gratings. In principle, any diffracting structure can be used to trigger coupling between different parts of the wave-vector spectrum. To illutrate this point, we implemeted IMM technique to recover both light-emitting and light-scattering objects positioned behind the dielectric grating using numerical protocols described above for IMM with plasmonic gratings. The results of such recovery for lossy dielectric grating with permittivity ε = 10 − i are shown in Fig. 5(b) and 5(c). It is seen that even in the absence of metallic grating IMM still able to achieve ~λ /20 resolution, which indicates significant tolerance of IMM technique to grating-object interaction. Comparable recoveries were obtained for dielectric gratings with somewhat smaller loss [Im(ε) ~ −0.5]. However, further reduction of |Im(ε)|, accompanied by resonant coupling of incident light into leaky mode supported by the grating degrated performance of perfectly matched layers in our finite-element solver and rendered the results of our numerical experiment in the limit Im(ε) → 0 increasingly unreliable. We therefore were not able to test the performance of IMM with lossless dielectric gratings.

4. Image reconstruction using aperiodic diffractive elements

Naturally, periodic diffraction gratings are just one (and possibly the simplest) example of diffractive elements. Numerous diffractive elements, also known as meta-surfaces, have recently been proposed for controlling light refraction and for generation of unconventional optical beams [20–23]. Similar meta-surfaces can be designed to increase the mixing efficiency of the diffractive elements used here for the proposed computational imaging. In this work, we restrict ourselves to analyzing the perspectives offered by aperiodic “chirped” gratings with linearly varying periodicity in both directions form center.

Three grating designs are analyzed. Periodicity of the first grating varies between 0.7λ at the center to 0.2λ at the ends with 30 elements on each side from the center; periodicity of the second grating is unchanged at 0.7λ ; periodicity of the last grating increases from 0.7λ to 1.2λ with 30 elements on each side from the center. Image recovery with all three designs of diffractive elements is tested for noise tolerance with sparse and dense objects (Fig. 6). In contrast with fully periodic gratings that produce identical diffracting performance when the object is moved over one period under the grating, chirped gratings create a well-defined focal region in the central position of the imaging system. Chirped grating also smooth-out diffraction performance of the element, distributing the mixed information over the measurement space.

 

Fig. 6 Image reconstruction using aperiodic diffractive elements. Image recoveries of dense object (a)–(c) [λ/10 sized sources with λ/10 spacing]and sparse object (d)–(f) [λ/10 sized sources with λ /2 spacing] using three types of gratings with decreasing periodicity from center to ends (a,d), equal periodicity from center to ends (b,e) and increasing periodicity from center to ends (c,f). While all grating systems are able to recover all three objects, the system with decreasing periodicity tends to minimize the appearance of parasitic objects in the case of noisy measurements. Panel (g) quantifies the quality of image recovery via mean square deviation between recovered and original fields.

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Chirped diffraction elements can be used as tips for scanning microscopes which would be similar to NSOMs but would image wavelength-scale area at single tip location. Alternatively, chirped elements can be implemented into imaging structures where small objects are moved (flown) over the focal spot of the diffraction element. Multiple measurements of the same moving object can be used to further increase signal-to-noise ratio of the imaging system and further improve its resolution.

It is seen that a diffractive element with decreasing periodicity from center to the two sides of the arms performs better in imaging both dense and sparse objects in the presence of noise when compared to the other two types of elements tested. The grating element with decreasing periodicity is also relatively immune to the appearance of the parasitic objects, particularly in the case of recovering sparse objects from noisy measurements.

To quantify performance of a particular diffractive element, we calculated the mean square deviation between original and recovered fields:

5. Conclusion

To conclude, we presented Interscale Mixing Microscopy, a computational imaging technique based on diffraction measurements. The presented technique allows to image wavelength-scale objects with deep subwavelength resolution. Convenient parameterization of both light-emitting and light-blocking objects was developed usign a pixel-based expansion. Stability of the numerical image recoveries as a function of object size, separation, and diffractive element design was analyzed. Stable numerical recoveries of light-scattering objects pave the way for experimental realization of IMM.

Design of efficient diffracting elements for practical IMM realizations represents a complex optimization problem. As a guide, RCWA technique can be used to optimize permittivity, thickness, periodicity, chirp, and vertical profile of the diffraction grating by calculating grating transfer function τ(k_x,θ_p), and maximizing the transfer of information about subwavelength features into propagating zone. At the same time, design of experimental setup often imposes severe constraints on operating wavelength, choice of materials, dimensions (and often on grating profile) that can be manufactured, severely limiting the available optimization space.

Also, fabricated diffraction elements are often different from their design. In practical implementations of IMM diffraction elements, as well as illumination profile of the setup, and other parameters of optical response of the system need to be characterized [9, 10] and transfer function τ(k_x,θ_p) needs to be adjusted to incorporate these peculiarities of experimental setup.