Biologically enabled sub-diffractive focusing - OSA Publishing

Biologically enabled sub-diffractive focusing E. De Tommasi,1,∗ A. C. De Luca,2 L. Lavanga,1 P. Dardano,1 M. De Stefano,3 L. De Stefano,1 C. Langella,4 I. Rendina,1 K. Dholakia,5 and M. Mazilu5 1 National

Research Council, Institute for Microelectronics and Microsystems, Department of Naples, Via P. Castellino 111, I-80131, Naples , Italy 2 National Research Council, Institute of Protein Biochemistry, Via P. Castellino 111, I-80131, Naples , Italy 3 Second University of Naples, Department of Environmental, Biological and Pharmaceutical Sciences and Technologies, Via Vivaldi 43, I-81100 Caserta, Italy 4 Second University of Naples, Department of Architecture and Industrial Design ”Luigi Vanvitelli”, Via San Lorenzo, Abazia di San Lorenzo, I-81031 Aversa (CE), Italy 5 SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, KY16 9SS, St Andrews, UK ∗ [email protected]

Abstract: Evolution shows that photonic structures are a constituent part of many animals and flora. These elements produce structural color and are useful in predator-prey interactions between animals and in the exploitation of light for photosynthetic organisms. In particular, diatoms have evolved patterned hydrated silica external valves able to confine light with extraordinary efficiency. Their evolution was probably guided by the necessity to survive in harsh conditions of sunlight deprivation. Here, we exploit such diatom valves, in conjunction with structured illumination, to realize a biological super-resolving lens to achieve sub-diffractive focusing in the far field. More precisely, we consider a single diatom valve of Arachnoidiscus genus which shows symmetries and fine features. By characterizing and using the transmission properties of this valve using the optical eigenmode technique, we are able to confine light to a tiny spot with unprecedented precision in terms of resolution limit ratio, corresponding in this case to 0.21λ /NA. © 2014 Optical Society of America OCIS codes: (100.6640) Superresolution; (160.1435) Biomaterials; (230.4110) Modulators.

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#217831 - $15.00 USD Received 28 Jul 2014; revised 19 Sep 2014; accepted 21 Sep 2014; published 27 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027214 | OPTICS EXPRESS 27214

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1.

Introduction

Over billions of years, nature has evolved to create flora and fauna exhibiting photonic structures as an inherent part of their organisms and function. Their functions range from structural colours in predator-prey schemes [1], to anti-reflecting coatings covering the eyes of moths and mosquitos [2], to microlens systems in dorsal arms of brittlestars [3], and to circular polarising selective coatings on beetles cuticles [4]. Many of these biological systems have inspired advanced photonics research relating to the form of butterflies wings [5], bird feathers and biomimetic material fabrication [6]. Among all these species, the first to explicitly deserve the term of “living photonic crystals” are diatoms [7], which are ubiquitous monocellular algae living in all waters [8–10]. It has been estimated that 20-25% of the global primary production and related oxygen amounts are due to photosynthetic activity of phytoplancton [11–13], most of which is composed of diatoms themselves. Whilst there is still an absence of an accurate estimate of the overall number of species [9], to date approximately 105 of them have been classified according to dimension (from few micrometers to few millimeters) and shape. This huge variety of species shares a common trait: the presence of two external, micro- and nano-porous valves made of amorphous hydrated silica [8, 14]. These valves are characterised by a hierarchical, complex and quasi-ordered pattern of pores whose symmetry and dimensions depend upon the species in question [15]. A single valve may comprise multiple plates, every plate being characterised by a specific thickness, lattice constant and pore diameter [8, 14, 15]. The evolutionary hypothesis suggests that these nanopores, slits and, in general, the ultrastructure of diatom valves not only ensure a protected interaction with external environment, but play a significative role in the efficient exploitation of light for photosynthesis [8, 10, 16]. Recently, the ability of single valves of Coscinodiscus wailesii diatoms to collect transmitted light in micro-spots (FWHM'8µm) has been experimentally demonstrated, for both coherent and non-coherent light sources and at different wavelengths of the visible spectrum [17–19]. Furthermore, the diffractive contribution of the valve pores, which leads to light confinement effect, has been clearly distinguished from that of a silica disk with the same dimensions of the valve by means of digital holography [20]. Numerical simulations [18], digital holography extrapolations [20] and direct measurements [19] confirm that this effect persists in an aqueous and/or cytoplasmatic environment. In these cases the spots take place even closer to the valve. Surpassing the diffraction limit in the far-field promises the realisation of super resolution imaging. Recently, a significant effort has been invested in order to focus light below the diffraction limit, but remaining in the linear regime exploiting particularly the concept of super oscillations and orthogonality. Most approaches rely on either the precise nano-fabrication of specific structures that allow the beam shaping to achieve a sub-diffraction focal spot [21, 22] or the use of adaptive computer controlled beam shapes [23–25]. In the nano-fabricated approach, the design of a super-oscillatory lens is based on nature-inspired swarm optimization and nanofabricated using focused ion beam milling or electron beam lithography. In the second case, researchers have used the optical eigenmode (OEi) decomposition to squeeze the focal spot generated by a standard microscope objective. Both methods are limited by poor coupling efficiency and large intensity sidebands outside the region of interest, which is a generic problem when trying to locally achieve sub-diffracting focal spots in the far-field. Here, we demonstrate that by combining the OEi technique with the natural optical properties of a single diatom valve of Arachnoidiscus genus it is possible to squeeze light further below the diffraction limit than either of these approaches, yet retain a relatively high efficiency. The underlying reason can be understood as the interplay between the OEi decomposition accessing the optical degrees of freedom (ODOF) of the system [26] and the valve fine nanostructure enabling a particularly efficient access to these ODOF. This novel form of far-field sub-diffractive focusing thus harnesses the potential of nature in the absence of sophisticated nano-fabrication and opens


up a wealth of new applications including advanced imaging, novel solar cell designs [27, 28] or high efficiency detectors. Further, the internal optical scattering properties of the diatom together with structured illumination provided by the OEi approach can be used as a design starting point to deliver nano-fabricated superlenses that achieve even smaller focal spots with high efficiencies. 2. 2.1.

Materials and methods Sample origin, preparation, and characterization

Single valves from Arachnoidiscus genus have been obtained from the AM671 sample of the Hustedt collection [29]. The sample dates from 1945 and was taken in Port Townsend, Washington, USA. The particular species under investigation is still classified as undetermined. General informations about history and geographical distribution of the genus and frustule structure for the known species can be found in [30]. Despite its occurrence, mainly on seaweeds of tropical coasts around the Pacific [8], very little is known about the living cells belonging to this genus. Single drops of buffer suspension containing the cleaned valves, which are characterized by a mean diameter of about 210 microns, were deposited and dried onto a quartz slide for optical characterization and onto a silicon chip for Scanning Electron Microscopy (SEM) characterization. SEM images were performed by a Field Emission Scanning Electron Microscope (FESEM, Carl Zeiss NTS GmbH 1500 Raith), with 5kV accelerating voltage and 30 µm wide aperture, exploiting secondary emission mode, which allows to show the peculiar morphology of the valves. Starting from SEM images, 3D CAD models of diatom frustules have been elaborated by means of Rhinoceros 4.0 (Robert Mc Neel & Associates) and successively rendered by Cinema 4d software (Maxon). Figure 1 shows SEM images of a single valve of Arachnoidiscus with some details at different magnifications. Frustules of Arachnoidiscus genus are heterovalvar, i.e. containing two different valves, and petri-dish shaped (see Fig. 1(g)). One valve is characterised by a planar central area (see Fig. 1(a)) and the second one by a centre ringed with elongated radial slits (not shown). In Fig. 1(c) a single pore occluded by volae (delicated flap-like outgrowths of porous silica) is visible. In general, the valve presents a clear ultrastructure characterised by progressively reduced porous features, according to position with respect to the plate with dimensions ranging from micrometers to tens of nanometers. The internal side of the valve (see Fig. 1(d)) hosts a system of costae radiating from a flange around a central ring. 2.2.

Theoretical background

The optical eigenmode (OEi) decomposition is based on defining electromagnetic fields that are additive in intensity when their associated fields are linearly superposed. For the experimental determination of the OEi, we use a set of probe fields E j with j = 1...N as an initial Hilbert basis with respect to which we define these OEi. In order to do this, we consider the intensity m0 (E) of the beam over a given region of interest (ROI) defined as: m(0) (E) =

Z ROI

EE ∗ dxdy =< E|E >ROI

(1)

where the surface integral is evaluated over the ROI. Equation 1 can be written in a general quadratic form: (0) m(0) (E) = ∑ a∗j M jk ak =< E|M (0) |E >ROI (2) ij


Fig. 1. SEM images of a single valve of an Arachnoidiscus diatom at different magnifications. a, Outer plate and b,c, details; d, inner plate and e,f, details. The presence of ultrastructure, with pores of different dimensions over different scales, is clearly visible. g, Rendering of an Arachnoidiscus frustule achieved through the construction of a 3D CAD model, derived from SEM images. The petri-dish shape of the whole frustule can be seen. #217831 - $15.00 USD Received 28 Jul 2014; revised 19 Sep 2014; accepted 21 Sep 2014; published 27 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027214 | OPTICS EXPRESS 27218

(0)

where the elements M jk are constructed by combination of the probe fields E j and Ek for (0)

j, k = 1...N such that M jk =< E j |Ek >ROI . Each probe beam corresponds to a beam that delivers a constant intensity in the ROI with different wavefront slope. The slopes are chosen such that the set of probe beams are orthogonal to each other. In practical terms, this means that on a spatial light modulator (SLM), we encode the Fourier transform of these beams corresponding to a correctly translated sin(x) sin(y)/(xy) (0) holographic mask. The optical eigenmodes OEi are defined by the eigenvectors of matrix M jk and can be expressed as: M (0) |E` >ROI = λ` |E` >ROI (3) where |E` >ROI is an optical eigenmode and λ` is the associated eigenvalue. If the initial probe basis is large enough and oversampled the optical degrees of freedom of the system, then the OEi do not depend on the number of probes (N), which, in our case, equals 169. In order to determine the smallest possible squeezed spot, we can represent the field in the focal plane of the valve as |F >= ∑Nj=1 b j |E j >ROI , with the normalization condition ∑ b∗j b j = 1. The measure of the spot size can thus be defined as the second order momentum of the intensity: (2) (4) m(2) (F) =< F|(x2 + y2 )|F >ROI = ∑ b∗j M jk bk jk (2)

(2)

where M jk =< E j |(x2 + y2 )|Ek >ROI . The eigenvector of M jk having the smallest eigenvalue corresponds to the superposition of |Ek > that delivers the smallest achievable spot in the considered ROI. Finally, the b j coefficients are converted back to the initial superposition of probe beams described in terms of a j coefficients and their associated masks. Experimentally, we encode on the SLM the phase and amplitude of the mask resulting from this superposition. This results in the generation of the OEi squeezed spot in the detection plane. 2.3.

Experimental set-up

In Fig. 2 the experimental set-up is shown in detail. Coherent radiation at 532 nm, coming from a diode laser (Laser Quantum, Opus; TEM00 mode with a >98% fit with a Gaussian profile in both x and y directions; M2 ∼ 1.1) is expanded by a telescope in order to fill the aperture of a SLM (Holoeye, LC-R 2500, resolution = 1024x768 pixels, pixel pitch = 19 µm, direct addressing rate = 72 Hz). A λ /2 waveplate is used to set the polarization which maximizes the efficiency of the SLM. The diffracted beams reflected by the SLM are collected by the Fourier lens L3. The SLM is located in the front focal plane of the Fourier lens, while an iris placed at the back focal pane of the lens selects the first-order diffracted beam. The selected beam is thus imaged in the focal plane of the objective OB1 (Olympus, UPlanFL N, 10X, NA=0.3), where the sample is initially positioned. In order to test the effect of circular polarization on light squeezing, a λ /4 waveplate is placed before OB1. The sample consists of sparse single Arachnoidiscus valves deposited onto a quartz slide, mounted on a micrometric xyz translational stage. The light transmitted by a single valve is collected by objective OB2 (Olympus, UPlanSApo, 40X, NA=0.95) and then imaged by lens L5 onto a CCD camera (Photon Focus, model MV1-D1312-40-GB-12). Measurements have been performed at different position along the direction of propagation of light (z direction) moving the glass slide by steps of 10 µm and for different light polarizations. At every single position the combination of probe beams which minimizes the second order momentum of the transmitted spot is recorded, together with the point spread function of the system.


Fig. 2. Optical set-up. Abbreviations, L: lens; M: mirror; SLM: Spatial Light Modulator; I: iris; OBi : microscope objective mounted on xyz micrometric translator; CCD: chargedcoupled-device camera. Upper left side inset: Example of four probing beams illuminating the central flange of the valve. Hue represents the experimentally recovered phase and luminosity the recovered amplitude. Lower left side inset: Composite SEM and optical image showing the relationship between sample and typical optical probe.

3. 3.1.

Results Numerical simulations

In Fig. 3(a) a simplified two-dimensional distribution of pores resembling the pore pattern and the costae system of the central region of the inner valve of an Arachnoidiscus diatom is reproduced; this is the actual region that will be invested by the incident laser beam in the experimental phase. Under illumination, every single pore can be envisaged, in virtue of Huygens principle, as a source of secondary waves whose intensity is defined proportional to the area of the single pore and phase by the phase of the incident light field. Expanding the definition of the diffraction limited Airy disk, we define the diffraction limited spot as the spot created after the valve under plane-wave illumination. Here, we consider a wavelength of 532 nm and we simulate the creation of this spot by superimposing all the contribution coming from each pore of the distribution. Figure 3(b) shows the spot transmitted at a distance of 6 µm from the pore position plane. We performed further modeling of the optical probing and the valve’s response using a set of orthogonal beams illuminating a fixed rectangular area on the diatom. For each probing beam, we calculate the field pattern in the detection plane. Based on these fields, the OEi procedure [31, 32] (described in Materials and Methods section) is used to numerically compute the generalised point spread function (gPSF), defined as the the spot obtained by the combination of probe beams which gives rise to constructive interference behind the valve. If the scattering


Fig. 3. Numerical simulations. a, Typical pore pattern resembling the pores and costae distribution of the central region of the inner valve of an Arachnoidiscus diatom; this region is the actual one invested by the incident laser beam. b, diffraction-limited spot, at a wavelength of 532 nm, evaluated at a distance of 6 µm along the optical axis; c, colour-coded phase pattern minimizing the spot size at the same distance; d, corresponding squeezed spot (sidebands have been removed for reasons of clarity, displaying only the region of interest).; e, variation of the full-width half-maximum (FWHM) of the diffraction limited spot (light green curve), of the gPSF (gray curve) and of the squeezed spot for unpolarized incident field (dark green) and of the squeezed spot for linear (blue curve) and circular (red) polarized incident fields; f, density of optical degrees of freedom as a function of distance for unpolarized (dark green curve), linear (blue curve) and circular (red curve) polarized incident fields. In d, only the region of interest is shown. Inserts in b,d, correspond to zoomed-in intensity cross sections. a-d refer to the case of unpolarized incident fields.


Fig. 4. Numerical simulations. Squeezed spot, at a wavelength of 532 nm, evaluated at a distance of 10 µm along the optical axis. a, Linear polarized incident fields; b, Circular polarized incident fields. The white circle illustrates the ROI inside which the spot size is calculated.

object would be a circular aperture, then the gPSF would correspond to the standard Airy disk point spread function (PSF). The gPSF has been used in previous studies to achieve aberration correction and focussing after random scattering media [33–35]. Comparing the size of the focal spot achieved using the gPSF to the one defined by the PSF it is possible to observe an apparent sample geometry dependent sub-diffractive spot. We achieve further focussing power through the squeezed spot, i.e. the spot corresponding to the combination of beams that minimise the spot size (Eq.4), which by its nature delivers focal spots smaller then the gPSF. Figure 3(c) shows the colour-coded phase pattern which gives rise to the smallest squeezed spot (z=6 µm), shown on Fig. 3(d). The noticeable shrinking of the spot size is accompanied by the appearance of sidebands [23], however, these sidebands are masked as we only display the region of interest. Figure 3(e), represents the full-width half-maximum (FWHM) of these different spots as a function of distance from the valve plane. This result demonstrates the potential of λ sub-diffraction (FWHM< 2NA ) and even sub-wavelength (FWHM< λ2 ) light squeezing in this case. In Fig. 3(f), we study the density of optical degrees of freedom (ODOF) as defined by the ratio between the number of OEis that significantly contribute to the spot intensity and the area of the region of interest. We observe that the minimum achievable spot size for the squeezed spot coincides with a maximum in the density of ODOF. In turn this illustrates how the rich fine structure of the diatom is crucial to this form of sub-diffractive focusing. Indeed, numerical simulations show similar results for a wide variate of diatom pore configurations. Further, Figs. 2(e) and 2(f) present the effect of the polarisation state of the incident light onto the spot size and the density of the ODOF. We observe that circular polarised light provides a smaller spot corresponding to an increased density of ODOF. The effect of the polarisation on the squeezed focal spot is highlighted in Figure 4 showing the different symmetries in the spot intensity for linear and circular polarised light. 3.2.

OEi probing and beam squeezing

Figure 2 shows the optical set-up used for OEi probing and subsequent implementation of the beam squeezing illumination. In a first step, we illuminate the valve using an expanded Gaussian beam on the back-aperture of the focussing objective (OB1 ) corresponding to a focussed beam incident onto the valve. We considered both linearly and circularly polarised fields for which we acquire the transmission images for varying positions of the valve moved along the optical illumination path. The initial position (z=0 µm) of the sample is defined as the focal plane of OB1 (see Fig. 5(a)). This plane is imaged onto the detection camera (CCD) while #217831 - $15.00 USD Received 28 Jul 2014; revised 19 Sep 2014; accepted 21 Sep 2014; published 27 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027214 | OPTICS EXPRESS 27222

the sample is scanned from z=0 µm to z=150 µm in steps of 10±5 µm. We observe that the minimal transversal dimension for the transmitted spot is obtained at z=30 µm for linear and circular polarization delivering a spot with a FWHM of about 880 nm. Using the SLM to probe and detect the experimental OEi of the system, we create beams corresponding to the gPSF and to the optimal squeezed spot. Figures 5(b)–(d) shows these beams in the case of the empty substrate or in the presence of the valve, all for the case of circular polarisation. We observe that the smallest spot-size is achieved in the presence of the valve at z=10 µm using the spot-size optimised OEi (FWHM=670 nm) for the linear polarization case (not shown here). This can be further improved by using circular polarization allowing the spot-size to decrease to FWHM=570 nm for z=20 µm (see Fig. 5(d)). We also remark that when squeezing the spot using the OEi method, we observe the appearance of sidebands outside the region of interest corresponding to a decrease of the Strehl ratio [24] accompanying all linear sub-diffraction focussing methods (see Fig. 5(d)). Figure 6(a) shows a comparison between the FWHM of the transmitted spots for three different illuminations: the OEi squeezed spot for the glass slide alone, the valve illuminated by the focussed beam and finally by the OEi squeezed spot. At a sample position of z=30 µm, the valve, under standard focussed beam illumination, is able to confine light to a spot that is narrower than the squeezed spot of the glass slide. As predicted by our simulations, further improvements are obtained by combining OEi squeezed spot illumination and the natural focussing properties of the diatom valve. The sub-diffractive nature of the squeezed spot can be highlighted by defining the squeezing ratio σ = ww0 , where w is the FWHM of the transmitted spot and w0 is the mean FWHM of the reference gPSF for the glass slide alone. Figure 6(b) shows the behaviour of squeezing ratio as a function of sample position and polarization. The minimum value of σ = 0.42 ± 0.02 for the squeezing ratio is obtained at z=20 µm for circularly polarization. This ratio compares well in performance to other far-field sub-diffraction focussing beams reported in literature (see Table 1). To understand the interplay between the diatom and OEi illumination, we determine the density of the ODOF of our system both with and without the diatom. Figures 6(c) and 6(d) show this density for linear and circular polarization and we observe, as modelled previously, that optimal squeezing is achieved in the positions of maximal density of ODOF. The higher values of experimental diffraction-limited spot and squeezed spot FWHMs respect to those who came from numerical simulations are due to the simplified 2D model we used to reproduce the diatom valve, which does not take into account its ultrastructure and irregularities but is still a good starting point in order to asses the feasibility of the subsequent measurements. Nevertheless, as stated above, the real diatom valve is characterized by a sufficiently high density of ODOF which guarantees unprecedented performances in terms of squeezing ratio respect to previous attempts reported in literature. 4.

Discussion

We have reported an original biologically-based approach for the sub-diffractive confinement of laser light in the far-field regime. The proposed approach relies on the structured illumination of a single valve of Arachnoidiscus diatom by a superposition of fields corresponding to the smallest spot-size OEi. In short, by combining the natural focusing properties of the valve and the flexible and mathematically rigorous approach of OEi, we are able to generate sub-diffractive focal spots at several positions along the optical axis of the system. Our sub-diffraction focal spots compare favourably against other far-field super focussing approaches such as the OEi based pupil filters [23–25, 36], random lenses based on materials with high indices of refraction [37] or specially engineered and micro-fabricated binary mask super-lenses [21, 22]. Table 1 summarises this comparison taking into account the numerical aperture and index of refrac-


Fig. 5. Transmitted intensity distributions acquired for different sample positions along the optical axis. The incident fields are circularly polarised. a, Focused beam illumination of the diatom valve; b, OEi based gPSF of the substrate alone (glass slide); c, OEi based gPSF for the diatom valve; d, OEi squeezed spot after transmission through the diatom valve.


Fig. 6. Focal spot-size variations as a function of diatom position for different configurations. a, Comparison between the spot-size measurements after the diatom valve under focused beam (blue), OEi squeezed spot after the valve (red, w) and after the glass slide alone (green). The incident fields are circularly polarised. b, Comparison, in terms of squeezing ratio σ = w/w0 between linear and circular polarization of OEi squeezed beams after the valve. Experimentally determined density of optical degrees of freedoms (ODOF) of the system in presence of the valve (red) and of the glass slide only (blue): c, linear polarized incident fields and d, circular polarized incident fields.


tion used in each case. Table 1. Comparison between different far-field focussing super-lenses.

Method Random diffuser [37] Airy disk (classical resolution limit) Photoporation OEi beam [25] Bessel beam Binary masks [22] OEi confocal imaging [36] OEi sub-diffraction [24] Binary masks [21] OEi sub-wavelength [23] OEi and Diatoms (this paper)

Focal spot (FWHM) 0.59λ /NA 0.51λ /NA 0.36λ /NA 0.34λ /NA 0.34λ /NA 0.31λ /NA 0.26λ /NA 0.26λ /NA 0.23λ /NA 0.21λ /NA

Squeezing ratio 116% 100% 71% 67% 67% 61% 51% 51% 45% 42%

More precisely, we have theoretically assessed the possible creation of sub-diffractive focal spots of the light when coherently illuminating the diatom valve with OEi fields and showed that the combined action, valve/OEi can deliver sub-diffractive spots at different distances from the valve plane (Fig. 2). Figure 5 offers an experimental proof of these numerical predictions. For reference comparison, we define the gPSF of the glass substrate alone as the combination of its OEi that decompose a Dirac function. Physically, this corresponds to the ’in-situ’ aberration correction of the optical system [35]. Figure 5 shows that the light transmitted by the valve under focussed beam illumination confines light to a spot narrower than the reference gPSF (see Fig. 5(a) and 5(b)). This is due, as confirmed by the numerical simulations, to the constructive superposition of the diffractive contributions coming from the single pores of the valve. The combination of natural focusing ability of the valve with the OEi method enable to further squeeze the light as shown in Fig. 5(d). The squeezing of the light under the diffraction limit is unfortunately accompanied by a super-oscillating behaviour of field with the emergence of sidebands containing a significant amount of the optical energy. In our case, the emergence of sidebands does not drastically reduce the efficiency of the spot in term of intensity: the contrast ratio, defined as (I f − Is )/(I f + Is ), where I f is the peak intensity of the central spot and Is the peak intensity of the sidebands, is, on average, higher than 50%. We also analyzed the effect of the polarization of the incident field, theoretically (see Fig. 3) and experimentally (see Fig. 6). The structure of the Arachnoidiscus valve, its pore distribution and its radial symmetry (rotational invariance for angles equal to the ones between two adjacent costae, see Fig. 1) suggest that a circular polarized field would better match the features of the valve itself, improving the focal spot squeezing, and ensuring a more isotropic distribution of the transmitted field. This effect is demonstrated by the simulation shown in Fig. 3. Experimentally, the presence of a quarter-wave plate before the input objective that varies the polarization of the field from linear to circular, enable to improve the light squeezing, as clearly shown in Fig. 6(b), with an improved squeezing ratio by 15%. In order to compare our performances with the ones obtained with other approaches, we extend the review seen in Ref. [25] in Table 1, where the FWHM of the focal spot is expressed in terms of the squeezing ratio. A value of 42% of the squeezing ratio has never been reached and reported in literature, thus, our approach delivers an improved performance in terms of light squeezing over all other studies to date. Please notice that this value, combined with the proper NA, might lead to a noticeable sub-wavelength light confinement [23]. These improvements can be explained by considering the density of the ODOF of the system as reported in Fig. 6(c) and 6(d). Indeed, an increase in the NA increases the density of the #217831 - $15.00 USD Received 28 Jul 2014; revised 19 Sep 2014; accepted 21 Sep 2014; published 27 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027214 | OPTICS EXPRESS 27226

ODOF that can be accessed by an objective, thus lowering the squeezing ratio for an OEi subdiffraction beam [25]. An ensemble evaluation of Fig. 6 reveals that the ODOF density of the system in presence of the valve is higher than the one with the substrate glass slide alone at all positions where the valve squeezes the light better than the glass slide. In particular the smallest squeezed spot corresponds to the largest density of ODOF. Similar conclusions can be drawn for both polarizations. Furthermore the improvement in light squeezing provided by the circular polarization with respect to the linear one is reflected by an increment in the ODOF density (see Figs. 6(c) and 6(d)). These effects cannot be explained by an effective change of the NA in the imaging plane as the setup maintains the NA when the valve is moved in the z-direction. These considerations enable us to conclude that the presence of the valve, with its ultrastructure, pores of different dimensions and geometries leads to an increase of the number of ODOF that can be efficiently accessed through the OEi illumination, thus noticeably improving the squeezing ratio. Further squeezing ratio improvements might be achievable by considering complex polarisation states of the incident light field [38]. 5.

Conclusion

Our approach exploits the natural super-focussing lens behavior of a single diatom valve. The structured illumination and the application of the OEi method to this sample allows to squeeze light under the diffraction limit to an unprecedented level (0.21λ /NA) when compared to other far-field sub-diffraction linear techniques reported in the literature [21]. Furthermore, it achieves relative high efficiency, potentially allowing for practical applications of this focal spot. Additionally, the structure of the diatom can be used to inspire nano-fabricated structures with improved focussing powers. The presented results suggest an interest in further studies of different species of diatoms or bio-inspired engineered pore pattern in combination with the OEi structured illumination approach. From a biological point of view, we may exploit the OEi and the focusing properties of the diatoms for imaging with sub-diffractive resolution. Acknowledgments EDT, LL, MDS, CL, LDS are supported by a FIR project RBFR12WAPY. ACDL is supported by an AIRC Start-up Grant 11454 and a FIR project RBFR12WAPY. KD thanks the UK EPSRC for funding. The authors would like to thank Dr. Antonia Auletta from the Department of Architecture and Industrial Design ”Luigi Vanvitelli” of the Second University of Naples for her skillful assistance in the realization of the 3D CAD models of the diatom frustules.
