Optical microscopy is widely used to analyze the properties of materials and structures, to identify and classify these structures, and to understand and control their responses to external stimuli. The extent of available applications is determined largely by the resolution offered by a particular microscopy technique. Here we present an analytic description and an experimental realization of interscale mixing microscopy, a diffraction-based imaging technique that is capable of detecting and characterizing wavelength/10 objects in far-field measurements with both coherent and incoherent broadband light. This technique is aimed at analyzing subwavelength objects based on far-field measurements of the interference created by the objects and a finite diffraction grating. A single measurement, analyzing the multiple diffraction orders, is often sufficient to determine the parameters of the object. The presented formalism opens opportunities for spectroscopy of nanoscale objects in the far field.
© 2016 Optical Society of America
The detection and visualization of subwavelength objects has numerous applications in imaging, spectroscopy, material science, biology, healthcare, and security. In conventional optical microscopy, the fundamental resolution limit is often associated with diffraction criteria by Abbe and Rayleigh [1,2]. According to these criteria, the smallest feature one can optically resolve is approximately limited to , with being vacuum wavelength and NA being the numerical aperture of the objective . In practical terms, however, the position of an isolated object can be determined with much better accuracy, and recent methods that rely on the control of a molecule’s fluorescent state have used this property to achieve imaging with nanometer resolution [4–7]. However, these methods are limited to the study of fluorescent objects. Several label-free optical imaging techniques relying on first-order diffraction and multiple measurements to access high-spatial-frequency components of the object have been demonstrated, such as structured illumination microscopy (SIM) , optical diffraction tomography (ODT) [9,10], or the far-field superlens (FSL) [11,12]. When an object with a linear optical response is imaged, the resolution of these techniques is fundamentally limited to . The imaging of unknown objects with deep subwavelength resolution is still predominately performed by relatively slow near-field scanning optical microscopy (NSOM) .
There exist several theoretical proposals to achieve fast imaging with subwavelength resolution in diffractive and tomographic setups [15,16]. In grating-assisted ODT  and interscale mixing microscopy (IMM) [18,19], a diffractive element is employed to out-couple information about the subwavelength features of an object into the far field, similarly to SIM. The final image formed with the postprocessing of information carried by multiple diffractive orders by analyzing either full electromagnetic field (ODT ) or intensity (IMM [18,19]). Although IMM somewhat resembles SIM and FSL, in contrast to the latter approaches, IMM utilizes multiple diffractive orders of the grating and, therefore, provides a resolution that is limited only by experimental noise. Here we report an experimental realization of the IMM with both coherent () and incoherent () illumination, achieving a resolution of the order of 70 nm with far-field measurements that do not require point-by-point scanning. Moreover, we present a simple analytic technique to postprocess the resulting information, often on the basis of a single diffractive measurement. The approach presented here opens the avenue for far-field microscopy and spectroscopy of nanoscale objects.
Mathematically, the process of imaging is equivalent to recovering the distribution of electromagnetic waves at the location of the object. For 1D objects in 2D (xz) space, this distribution can be described by a plane-wave spectrum parameterized by the transverse component of wavevector with wavevector-dependent amplitude (2D objects and combinations of small 3D objects can be considered similarly ):1), the information about the subwavelength features of the object is encoded in the evanescent, nonpropagating waves, which exponentially decay away from the source. As the distance to the object increases, the “signal” about the subwavelength features gets exponentially suppressed by the diffraction-limited “background” and experimental noise. Therefore, the resolution of far-field optical microscopy is directly related to the signal-to-noise ratio of the measurements .
Sparsity-based imaging  fills in the subwavelength information based on an analytic continuation of available diffraction-limited measurements of relatively sparse (isolated) objects. For densely placed objects, several techniques aim at enhancing the signal carried by evanescent-waves. NSOM collects the near-field information directly, at a cost of slow, point-by-point imaging. The concept of superoscillations [22,23], band-limited functions that oscillate faster than its fastest Fourier component, has also been instrumental in achieving subwavelength resolution in the scanning microscopy regime. Superlenses , hyperlenses [24–26], and other metamaterial-based direct imaging solutions have been used to propagate evanescent information in relatively narrow frequency ranges where (a component of) the permittivity of the metamaterial is close to its plasmon resonance (superlens) or zero (hyperlens). In addition, metamaterial solutions that provide magnification typically have rather limited field of view.
Alternatively, information about can be diffracted into the propagating waves and thus be directly measured in the far field. The use of periodic structures to achieve superresolution was first investigated a half-century ago by using multiple gratings to generate a structured light beam and then using the diffraction of this beam by the object to gain access to the evanescent part of the spectrum [27,28]. Today, diffraction-based characterization allows the analysis of the profile of highly periodic diffracting structures with a resolution of the order of single nanometer [29,30]. Structured illumination microscopy [8,31] allows the imaging of unknown objects with resolution at the expense of multiple measurements and computer postprocessing. The far-field superlens  uses near-field excitation of an object along with resonantly enhanced fields of surface plasmons to achieve resolution. Since both SIM and FSL utilize only first-order diffraction (for nonfluorescent samples; higher order SIM is possible only on fluorescent samples ), these resolution limits cannot be further improved within the existing frameworks.
ODT, a technique that aims to analyze the internal structure of transparent objects based on multiple measurements of the light scattered by the object is becoming a powerful tool for analysis of some biological samples . However, stable and reasonably fast reconstruction of ODT measurements yields resolution similar to that of SIM and is applicable primarily to analysis of low-scattering (low refractive index contrast) samples.
Several extensions of ODT to 2D tomography with subwavelength resolution have recently been proposed theoretically [15,17]. The majority of these techniques require measurements of both field amplitude and phase in order to reconstruct the image, which is accomplished by fitting measured data to simulations . While possible in theory, such fitting is typically prone to both numerical and experimental noise, especially since optical phase can rarely be measured directly [15,33]. It is likely that practical realization of such techniques will require prohibitively extensive characterization of both the sample and the scattering environment, as well as the response function of the particular optical setup . Even then, recovery of optical phase may present a great challenge.
IMM, first proposed in , relies on a diffraction element—positioned in near-field proximity to an object—to fold the evanescent information into the propagation zone (Fig. 1). Such folding essentially mixes the information about subwavelength features of the object with the information about diffraction-limited features. Since, in contrast to SIM and FSL, IMM uses multiple diffraction orders, it can in principle achieve unlimited resolution. In practice, the resolution of IMM is limited by the ability to resolve a high-order diffracted signal among the zero-order diffracted background. It has been shown, numerically, that realistic diffracting structures enable up to resolution. The maximum resolution is limited by experimental noise and by instabilities in numerical calculations, which can be somewhat mitigated with multiple measurements .
Several algorithms for recovering the information about the object based on postprocessing of the multiple measurements of the field  or intensity [18,19] profile have been developed. It was shown that when the scattering by an unperturbed diffraction grating is characterized well enough to be reproduced numerically, image recovery can then be mapped onto an optimization problem that minimizes the deviation between the predicted optical response of the test object in front of the unperturbed diffraction element and the measured signal. However, as suggested by the previous experimental studies , such characterization provides extreme constraints on experimental setup. Analysis of our experimental data is consistent with these assessments.
In this work, the problem of image recovery in IMM is addressed by the development of a new, straightforward analytic signal-processing technique that allows pinpointing of the far-field diffractive signatures of the object and isolates these signatures from the (relatively strong) background created by the diffraction element. As result, we show that only a single measurement may be sufficient to achieve, theoretically and experimentally, deep subwavelength resolution with far-field IMM measurements.
2. EXPERIMENTAL SETUP
A series of finite gratings (25 periods) were fabricated in a thin (100 nm) gold film on a glass substrate. Two different sets of samples, with periods and , each having a slit width of 125 nm, were fabricated. Each set contains three structures: (i) a reference, perfectly periodic grating, (ii) a grating with an absent slit in the center, and (iii) a grating with a slit misplaced by (Fig. 2). Apart from the scaling, the results from samples of two different periods are similar to each other. Here we present results for the set of samples with smaller period, where the smallest defect was of the order of . The samples were characterized using a 532 nm CW laser, as well as with broadband incoherent white-light illumination of 600–800 nm spectral range, putting the smallest size of the defect in the range of . For the measurements, the samples were illuminated with a objective (0.95 NA) using a quasi-plane wave obtained by focusing the light at the back focal plane of the objective (Fig. 2). The incidence angle was controlled by a pair of galvanometric mirrors conjugated to the focal plane of the illumination objective and the scattered light was detected in transmission using a objective (1.49 NA). The back focal plane of the detection objective (which corresponds to the Fourier plane) was then directly imaged onto an imaging spectrometer. Coherent measurements were performed by imaging the back focal plane directly onto the camera, while white-light measurements were performed with a 300 μm wide slit used to select only the specular transmission in the axis perpendicular to the grating, producing a far-field diffraction pattern as a function of wavelength.
3. THEORETICAL ANALYSIS
Numerical extraction of information about the unknown object was suggested as a basis for IMM in previous studies [18,19]. However, it was shown that, in practice, the implementation of such algorithms require extreme effort in equipment calibration [19,30]. Here we present a new, analytic implementation of IMM. The reference diffraction element used in this work represents a finite grating with slits with width and period . The (magnetic) field generated in the near-field proximity of such a grating, illuminated by a single, normally incident plane wave, is given by
The equations above can be used to illustrate the difference between the IMM and SIM-like diffraction techniques. In the IMM formalism, the sample is interrogated by a single plane wave, and signal processing is focused on the cancellation of all principal diffracted beams, analyzing the interference in the ringing tails of the principle orders. As a result, the information about the object can be extracted based on a single measurement. In contrast, in SIM-like techniques, the sample is interrogated by multiple beams, and the measurement aims at detecting the zero-order interference, which needs to be postprocessed and requires multiple measurements.
On the implementation level, the signal-carrying interference term can be enhanced by analyzing the product that suppresses the intensity around the diffraction maxima of the ideal grating. The spatial profile and the position of the object can be analyzed by considering the Fourier transform of the power spectrum,
The use of the Fourier transform to postprocess optical measurements is a common practice in ODT, SIM, and now IMM. However, we note again that the IMM implementation proposed here is distinct from both ODT and SIM. ODT explicitly relies on small index contrast between the sample and the environment, while SIM and IMM do not have this limitation. At the same time, IMM provides significant resolution enhancement compared to both SIM and ODT.
Figure 3 illustrates the performance of IMM, as defined by Eq. (7) for several representative objects. The operating wavelength, geometry of the grating, and data sampling rate reflect those used in our experimental setup. First, we consider the evolution of the Fourier spectrum of a grating with a single object as the object is moved across the grating. For a fixed position of the object [inset in Fig. 3(a)], the signal has two main features. The first is a maximum located at that corresponds to the period of the grating and represents the contribution from the main diffraction peak. The second feature represents the interaction of the object with the grating. Its position is directly related to the position of the object with a resolution of the order of the slit of the grating. Its intensity, in the limit , is proportional to the size of the defect [Fig. 3(b)]. When the object size approaches the size of the grating slit, the intensity dependence on the size slowly deviates from a linear relationship. The signal from the object itself [ term in Eq. (6)], is not seen in these spectra.
Note that (i) the signatures of subwavelength objects can be clearly resolved and (ii) the Fourier spectrum, constructed according to Eqs. (5) and (7), represents a direct measurement of both position and size of the object with subwavelength accuracy.
The product [see Eqs. (5) and (6)] determines the true resolution of the imaging system. The first term representing the ideal grating will have its centroid located at double the frequency of the interference term due to the square of the dominant sine term. The secondary peak is related to the position of the object, with a symmetry about . Taken together, the positions of these maxima allow a precise calibration of the diffraction element and a measurement of the position of the object with relationship to the diffracting element with great accuracy. Separate calibration (for example, using a diffraction grating with one missing slit), can be used to determine the sizes of the unknown objects as well as their positions.
The aforementioned symmetry about yields ambiguity of the sign of the object displacement from the center of the diffractive element and, thus, represents a limitation of the current implementation of IMM. This ambiguity can be avoided in a non-static arrangement in which the imaging target, as moving the object over the periodic aperture, would break the symmetry, and provides unambiguous position measurement of the object. Alternatively, the ambiguity can be avoided by restricting the operational field of view of IMM systems.
The proposed recovery procedure allows for imaging of composite objects. Thus, Fig. 3(c) clearly shows that Eq. (7) can be used to differentiate between an isolated object and two objects, and , positioned in the neighboring openings of the grating. The total scattered signal represents the combined size of the objects, while the spatial distribution of the secondary diffracted peaks reflects the locations of the parts of the composite object. Therefore, with proper calibration, it is possible to determine both the size and the locations of these objects (with exception of the position ambiguity mentioned above). One of the main conclusions of the proposed theoretical analysis is that reducing the period of the grating represents a straightforward way to improve resolution of IMM formalism.
The diffraction signal of isolated and objects and two objects within the same opening of the diffraction grating is compared in Fig. 3(d). It is seen that, with further signal processing, it may be possible to recover the composite objects located within the same opening. We defer the development of such signal processing and its experimental validation to our future work.
The diffraction patterns for the objects under consideration were experimentally measured and simulated using Eq. (7). Two different types of measurements were analyzed.
In the first set of measurements, the samples were interrogated with coherent light, for different incident angles. To compare the predictions of theoretical analysis to experimental spectra, experimental data from CCD images were translated to superimpose the data corresponding to different incident angles. The resultant spectra were Fourier-transformed to convert the angular information into the real-space domain according to Eq. (7). Figure 4(a) illustrates the result of postprocessing a single measurement. Figure 4(b) represents the statistics of nine measurements corresponding to different incident angles. Figure 4(c) shows theoretical predictions according to Eq. (7). It is seen that experimental data are in good agreement with theoretical predictions, with positions of the diffracted peaks representing positions of the objects in the middle of the grating, and the ratio of the amplitudes representing the ratio of the dimensions of the defects [Sample 1 () and Sample 2 ()]. It is clearly seen that IMM can resolve -scale defects. Interestingly, a single measurement provides enough information to resolve the defect.
In the second set of measurements, the sample was illuminated by a broadband incoherent source. Prior to translation, the contrast of raw CCD images was enhanced by subtracting the background and the images were de-skewed to compensate for misalignment of the optical setup (this eliminated a small, drift of the image). The resulting experimental data is compared to theoretical predictions in Fig. 5. Once again, it is seen that the predictions of Eq. (7) are in agreement with experimental data and that both half-period and quarter-period defects can be clearly distinguished from the ideal grating and from each other. The spectrum of the defect-free grating has a single wavelength-independent peak representing the period of the grating. The spectra of the two samples with defects contain additional features. The location of these features represents the position of the defect (central slit of the grating), while their structure describes the characteristic of the defect: the missing slit sample yields a double-peak pattern; the shifted slit yields a more complex, broader (but lower amplitude) spectral peaks. Note that, because of the mutual alignment of optical elements and CCDs, the broadband spectra provide approximately 4 times the number of data points for each wavelength as compared to a single-wavelength setup. As result, the fine structure of the IMM signal can be resolved much better in Fig. 5 than in Fig. 4.
5. DISCUSSION AND CONCLUSIONS
We have demonstrated that, by using multiple diffractive orders of a grating in near-field proximity to a nanoscale object, it is possible to recover deep subwavelength information, usually lost in the evanescent wave spectrum, with far-field measurements. A simple analytic technique capable of predicting the position and size of the opaque object in close proximity to a diffractive element has been developed and validated in the experiments. The resolution is related to recovering the parameters of the full spatial spectrum of light diffracted by the sample and grating system. Resolution of the order of has been achieved experimentally. Notably, the developed formalism has been shown to work for both coherent and incoherent excitation, opening the pathway to the spectroscopy of nanoscale objects.
In practical imaging systems, the period of the grating and its duty cycle need to be optimized as smaller period generally yields better resolution while at the same time potentially decreases signal-to-noise ratio.
The approach presented here for one-dimensional objects can be extended to the imaging and spectroscopy of two-dimensional structures with the help of 2D diffractive elements. In order to achieve reliable recovery of two-dimensional objects, future studies will need to identify the stability of such recoveries with respect to the shape of the diffractive element (circular versus rectangular versus hexagonal gratings), the Fourier/Bessel transform used, as well as the robustness of the signal recovery with respect to noise and imperfections of optical setup. These studies, although not trivial, are fundamentally similar to the approach described in this work.
Finally, we note that in realistic objects, the presented formalism will measure the combination of the geometrical size and optical transparency of the object. The spectral dependence of this combination, determined by the developed formalism, provides the direct measurement of the spectral response of the nanoscale object.
National Science Foundation (NSF) (DMR-1209761, ECCS-1102183); Engineering and Physical Sciences Research Council (EPSRC); Royal Society; Wolfson Foundation.
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