Far-Field Super-Resolution Imaging By Nonlinear Excited Evanescent Waves

Abbe's resolution limit, one of the best-known physical limitations, poses a great challenge for any wave systems in imaging, wave transport, and dynamics. Originally formulated in linear optics, this Abbe's limit can be broken using nonlinear optical interactions. Here we extend the Abbe theory into a nonlinear regime and experimentally demonstrate a far-field, label-free, and scan-free super-resolution imaging technique based on nonlinear four-wave mixing to retrieve near-field scattered evanescent waves, achieving sub-wavelength resolution of $\lambda/15.6$. This method paves the way for application in biomedical imaging, semiconductor metrology, and photolithography.


Introduction
The spatial resolution of an imaging system is limited by the Abbe theory 1 , posing a great challenge for many areas like biomedical imaging, astronomy, and photolithography. For example, direct live images are crucial to understanding biological processes at the subcellular level, e.g. virus behaviors 2,3 ; the finest nanostructures fabricated by photolithography on semiconductor chips is also defined by the Abbe's limit. According to Abbe's theory, subwavelength image features are usually associated with near-field evanescent waves, which decay exponentially with distance similar to electron wavefunctions in quantum tunneling 4 and diffusive waves in Anderson localization 5,6 . Insufficient far-field detection of these evanescent fields in conventional optics with finite numerical apertures ultimately precludes imaging resolution better than λ/2. Like scanning tunneling electron technique in condensedmatter physics converting tunneling currents into conducting ones 7 , near-field scanning optical microscope (NSOM) can improve the resolution by converting evanescent waves into propagating ones 2 but requires near-field scanning. Under the same framework, recent advancements in metamaterials allow similar conversions using superlens 8 , hyperlens 9,10 , and surface plasmon polaritons (SPP) 11 to achieve super-resolution imaging but with demanding nanofabrications. On the other hand, X-ray and electron microscopes, even with nanometer resolution, may be potentially harmful to biomedical applications, it is still highly desirable to break the Abbe's limit and realize superresolution imaging for optical waves.
Nonlinear optics may offer an alternative way to beat the Abbe's limit, which is originally formulated for linear waves [12][13][14] . By introducing nonlinear spatial wave mixing into an imaging system, it becomes possible to retrieve those undetected waves in the far-field and reconstruct to improve the resolution 13 , effectively, generalizing the Abbe theory into nonlinear optics regime. However, evanescent waves are absent in prior works, the imaging resolutions have not reached the wavelength scale yet till this work. Meanwhile, evanescent waves can be manipulated through nonlinear wave mixings 14-16 based on surface phase-matching conditions, e.g. free-space coupling of SPP 17,18 , darkfield imaging 19 . These works show a unique way to nonlinearly couple non-propagating evanescent waves with 2 propagating ones into far-field, addressing the key issue for the aforementioned super-resolution imaging problem in the framework of nonlinear Abbe theory. Moreover, it may enable a fluorophore-label-free imaging method in contrast to the existing techniques like stimulated emission depletion (STED) microscopy 20 , stochastic optical reconstruction microscopy (STORM) 3 , structured illumination microscopy (SIM) 21,22 , where label-free imaging techniques are highly desirable not only in biomedical imaging 23 , but also in other areas like semiconductor metrology processes.
In this work, we experimentally show nonlinear wave mixings including evanescent waves in the framework of nonlinear Abbe theory and demonstrate a far-field, label-free, and scan-free super-resolution imaging scheme to resolve sub-wavelength structures on a semiconductor silicon-on-insulator (SOI) wafer. To break the Abbe's resolution limit, a nonlinear four-wave mixing (FWM) technique is implemented to excite localized near-field evanescent wave-based illumination with large spatial wave vectors, such that near-field waves containing the finest sub-wavelength imaging features can be converted into propagating ones for far-field detections, effectively enlarging the numerical apertures. Combined with an iterative Fourier ptychography method 24,25 , the reconstructed images can reach a resolution limit down to /15.6  with respect to the input probe's wavelength. Moreover, this FWM imaging scheme can also cooperate with nano-slit grating structures, which theoretically can provide additional resolution enhancement. This technique may offer a new way for critical imaging tasks sensitive to fluorescent labels like biomedical applications, semiconductor processes as well as photolithography.

Methods
According to linear Abbe's theory, a sub-wavelength object illuminated by a coherent wave with wave vector k can cause scattering waves to radiate over a wide-angle into the far-field. Meanwhile, a portion of scattered light with large wave vectors becomes evanescent and confined only to the object's surface in the near field as shown in Fig.   1(a). Such evanescent components correspond to large k portions in the Fourier space [ Fig. 1(b)], determining the finest feature of the object. How to retrieve these near-field evanescent waves in the far-field is the key to tackle the problem of Abbe's diffraction limit 26 . Linearly, evanescent waves can be scattered off by a sharp sub-wavelength tip in the case of NSOM and convert into propagating waves for far-field detection. Similarly, we can expect that nonlinear wave mixing under certain phase-matching conditions allows similar conversions between near-field evanescent waves and far-field incident waves as shown in prior works 17,18,27 . Here we purposely excite a localized evanescent wave with wave vector 3,eva k through nonlinear FWM with far-field launched pump and probe beams [ Fig. 1(a)]. Consequently, the excited evanescent waves will be scattered off by the sub-wavelength target into radiating ones at various angles. In the spatial-frequency domain, i.e. the k-space [ Fig. 1(b)], this scattering process can be described by in-plane momentum conservation 28 : where 3,0 k is the wavenumber of FWM in a vacuum, Λ represent various spatial features of the target. In reality, the imaging target contains a wide spectrum of spatial modes, extending to a wide-spreading disk in the k-space [ Fig.  1(b)]. Under a normal-incident illumination, these spatial modes collected in the far-field through an imaging lens with limited NA lie within the range of in the k-space. In contrast, if illuminated by the evanescent wave excited by FWM, the effective passband in the k-space of the same imaging system will be shifted by 3,eva k [ Fig. 1(b)], meaning that most parts of the passband lie within the evanescent regime of the target spectrum, where lie the finest sub-wavelength features of imaging object. In this manner, these evanescent waves with large wave vectors carrying sub-wavelength details can be retrieved in the far-field with a conventional imaging objective lens. Later, combining with certain numerical reconstruction methods, we shall be able to restore far-field super-resolved images with subwavelength resolution. In a similar approach, coherent total internal reflection (TIR) microscopy relied on linear TIR excited evanescent waves 29 can also enhance the imaging resolution, but limited by medium refractive indexes. The key mechanism to excite near-field evanescent waves through FWM can be understood from nonlinear phase-matching conditions at the interface. In linear optics, the laws of reflection and refraction at an interface can be directly deduced from electromagnetic boundary conditions laid out by Fresnel formalism 30 . Under the same framework, the well-known linear Snell's law can be generalized into a nonlinear regime where light waves are incident onto a nonlinear medium boundary by satisfying boundary conditions for all individual frequencies during nonlinear conversions 31 . For the case of FWMs, the resulting equations governing reflection and refraction have been developed in Ref 15,27 as well as our prior work 14 , where the nonlinear reflection law reads: 3 3 ( 1, 2,3 i = ) are wave vectors of the pump, probe, and FWM signal beams, respectively; i n are refractive indexes of the surrounding medium, i  represent their incident/output angles. Effectively, the relations of these transverse wave vectors represent in-plane momentum conservation law as shown in Fig. 2(a). Such a partial phase-matching condition only manifests itself near surfaces 19,32 and can be enhanced by thin-film structures 27 , making the wave evanescent. These generated evanescent waves are bounded at the interface, but with large in-plane wave vector, which is crucial for the aforementioned scattering imaging process.
Experimentally, we first verify Eq. (2) in a reflection configuration by synchronously launching a pump and a 5 probe beam onto a flat SOI wafer to excite near-field FWM as shown in Fig. 2 Fig. 3(d), which is the region of interest to explore our evanescent wave-based sub-wavelength imaging. Also, such nonlinear excited evanescent waves have been previously demonstrated for surface plasmon coupling 17,18 , dark field imaging 19 .
To illustrate the case of evanescent wave excitation, we perform a numerical simulation of surface FWM by finite-difference time-domain (FDTD) method. The calculated 2 || E distribution of the FWM signal shows that the signal wave is localized near the focal spot inside the top silicon layer of the SOI wafer [ Fig. 2(e)]. This is a sharp contrast to nonlinear excited surface plasmon mode near a metal-dielectric interface 11,18 , where surface plasmons can further propagate under certain phase matching. Such nonlinear excited FWM waves serve as localized evanescent light sources for sub-wavelength imaging purposes later. According to the aforementioned discussions on imaging resolution, it is essential to obtain large transverse wave vectors for excited evanescent waves. As shown in the inset of Fig. 2(e), the effective wavelength of such evanescent wave has been reduced to ~200nm , half of its free-space wavelength (~403nm ), leading to the relation of wave vectors as 3,eva 3,0 2 kk = , which is the same result according to Eq. (2) for given input angles in the simulation ( 12 ,

40
= ). Effectively, by varying the input angles of pump and probe beams, we manage to locally excite evanescent waves with variable wave vectors. Such FWM processes enable an active, flexible manner to control signal beam's wave vectors for imaging processes later.

Results
To implement an evanescent wave-based sub-wavelength imaging, we compare imaging qualities under several illumination schemes in Fig. 3. Consider a double-slit with a width of w and center-to-center distance a [ Fig.  3(j)] illuminated by an evanescent wave with wave vector 3,eva k , the reflected far-field image through an NA limited imaging system in the spatial domain can be formulated as 34 : Previously, a similar idea has been applied in total internal reflection microscopy (TIR) to improve the resolution 29 . But the key problem is to retrieve images and improve the corresponding resolution under such evanescent wave illumination. Here we restore sub-wavelength images using a computational imaging technique named Fourier ptychography (FP) 24,25 , by stacking multiple low-resolution images with pre-determined illumination angles and iteratively reconstructing a final image with much-enhanced resolution (see supplementary materials). Previously this technique has shown great successes in wide-field, high-resolution imaging 24 , and fast live-cell imaging 25 . In our case, we extend this technique into a nonlinear optics regime by including nonlinear excited . After the iteration converges, an intensity dip appears in the reconstructed image, making the two slits resolvable now [ Fig. 3(i)]. Experimentally, a pair of 90nm-width slits 7 spaced 50nm apart on SOI substrate [ Fig. 3(j)] can be successfully resolved using this technique as shown in Fig.  3(i), 3(k). The image resolution is sharply improved, in contrast, over those images formed under probe beam normal illumination [ Fig. 3(g)] and solely signal illumination at 3,eva1 Fig. 3(h)]. these results are also well confirmed by the numerical simulation. Due to the current image system's limitation, e.g. NA, the estimated image resolution is around 50nm (gap distance). To further shrink the gap distance and improve the system resolution, the wave vector 3,eva k must be increased accordingly.
At last, we implement this sub-wavelength imaging technique for a nano-slit array with 110nm slit width and 400nm period in Fig. 4. The fine features are clearly revealed [ Fig. 4(b-e)] using this technique as compared to the blurred one [ Fig. 4(a), 4(c)] under the probe normal illumination. And image contrast has also been significantly improved as shown in Fig. 4(e) because such an FP technique helps to replenish the high spatial frequency parts of images which majorly contribute to a high signal-to-noise ratio (SNR). In the meantime, localized near-field evanescent waves serve as an excellent dark-field illumination source. After filtering out the pump/probe frequencies, they clear out background noises for the high SNR. Moreover, this iterative reconstruction algorithm enables phaseretrieval ability in revealing sensitive information like depth/height 35 and is potential for topographic imaging applications in the future. Interestingly, this grating structure can also provide a platform facilitating evanescent wave excitation. The grating introduces an additional in-plane momentum | | | 2π/| D = G (first-order), inversely proportional to the spatial period D , to FWM processes. These grating modes can be involved in the nonlinear process 12,13 and affect the generated signal beam's wave vector 27 . Under this new scheme, the new phase-matching condition should include the grating momentum term as: Accordingly, Fig. 4

Discussions
In the current configuration, the resolution of our imaging technique can reach around 50nm, ~/15.6  with respect to the probe's wavelength. In the future, a UV laser source with wavelengths smaller than 300nm and the assistance of the aforementioned nano-grating structures may finally put the resolution limit down to a few tens of nanometers, showing great potential in sub-wavelength imaging. Especially, we expect this technique could be beneficial for semiconductor metrology, where UV absorption would not be an issue for semiconductor materials. As for biomedical imaging applications, our technique offers a label-free, scan-free, and far-field super-resolution capability, which is much demanded in this area. To avoid possible laser damage to the bio-imaging samples, we may consider previous similar approaches using nonlinear excited surface plasmon waves to separate laser focus and light-sensitive biosamples 11 . But compared with fixed wavelength SPP/LSP illumination, our method allows continuous varying of wave vectors, which enables nonlinear FP reconstruction for better imaging qualities. Moreover, given the nonlinear nature of FWM, we also expect our method combined with the coherent anti-stokes Raman scattering (CARS) technique 23,36,37 together may offer chemical-specified, far-field super-resolution imaging, by pairing pump/probe beams' frequencies according to molecules' vibrational energy. At last, in a reversed manner, such excited evanescent waves with large spatial wave vectors are capable of focusing light into tiny spots below the Abbe's diffraction limit in a similar way, this enables a possibility for a new type of high-resolution photolithography mechanism on silicon's surface 38 .

Conclusions
In conclusion, we experimentally realize a super-resolution imaging method based on surface nonlinear FWM excited evanescent wave illumination, which enables label-free, far-field imaging well beyond Abbe's diffraction limit. Such locally excited evanescent waves may also be beneficial for other applications beyond imaging like photolithography.