1. Introduction

Diminutive micron-sized dielectric particles of various spatial forms (spheres, cylinders, ellipses, cones, pyramids, cuboids, etc.) are actively investigated in diffraction optics, nanophotonics, and laser physics in the last two decades (see reviews [1–4] and references therein). Scientific interest to such mesoscale objects with effective dimensions comparable to the illuminating radiation wavelength is conditioned by their unique property of modifying the spatial structure of the incident optical wave and creating a concentrated light flux of subwavelength width and enhanced intensity in the near-field scattering region. This specific and as a rule elongated area of external microparticle focus in the direction of radiation incidence [5,6] was termed as a “photonic nanojet” (PNJ) by Z. Chen et al. in 2004 [7]. These specific features of photonic nanojets cause wide opportunities for their practical application in medicine, biology, nanosensorics, optical microtechnology, optical data storage systems, optical capture and trapping of micro- and nano objects, etc. [4,8–13]. Importantly, the necessary condition for PNJ formation is maintaining the refractive index contrast of mother particle relative to the environment (relative refractive index) less than two [1,4].

Particularly, for optical manipulation of particles with characteristic sizes of the order of units and tens of nanometers (e.g., some viruses, protein molecules, submicron fraction of dust), the optical traps (OT) based on PNJ effect demonstrate certain advantages over the traditional optical tweezers which usually relay on a tightly focused laser beam [14]. The reason is that the optical forces acting on a particle subjected to the electromagnetic field are proportional to particle radius in cubic degree [15]. Therefore, when working in the nanoscale [16], the subwavelength electromagnetic field localization is already required to hold a trapped nanoparticle in the desired position. This is impossible in the conventional high-NA objective-based focusing schemes due to the fundamental diffraction limitation of the far-field focus waist ∼λ/2n₀ (here, λ is radiation wavelength, n₀ is environment refraction index).

Worthwhile noting, other types of micro-photonic devices are reported in the literature for optical capturing and manipulation of nanoparticles and atoms based on flat waveguides [17,18], microresonators [19–21], and plasma circuits [22]. However, the subdiffraction focusing in these OTs is achieved in evanescent fields which decay very quickly when moving away off media interface. Thus, the OTs of this type usually have extremely short working distances that becomes ineffective, e.g., for analyzing the flows of low-concentration nanoparticles in a liquid [23].

Usually, the PNJ-based traps use mesoscale dielectric particles that generate photonic fluxes with subwavelength localization and enhanced optical intensity near the shadow side of particle [1,13,24]. Meanwhile, typical PNJ possesses transverse dimensions below the diffraction limit [1,4,25,26]. In addition, due to sharp field amplitude gradient in a PNJ, the optical forces acting on a bead are much stronger than in the beam focus produced by a convex objective in free space [13,27]. This allows the PNJ traps to be used for capturing nanoparticles of any nature, including metallic ones. Obviously, this is challenging in conventional OTs, because the optical radiation pressure pushing target object out of the trap is much higher for metallic particles than for dielectrics. Thus, for optical manipulation of metallic nanoparticles, the confocal optical trapping technique with counter-propagating focused beams [28] or an optical mirror trap technique for dielectric particles [29] is applied

For practical purposes the OTs based on standing waves are of particular interest because the axial scattering force components are balanced from opposing directions and the essential gradient force component dominates the trap site that exceeds the axial gradient force of single-beam tweezers. Besides, the optical field localization is realized here both in transversal and longitudinal planes, achieving half of optical wavelength [30]. As applied to the photonic jet effect, two types of PNJ-based traps are usually considered: i) OT originated from twin PNJs interference produced by two particles counter-illuminated by two different radiation sources [31] and ii) OT with a dielectric microparticle placed near a reflecting mirror and exposed to a single optical source [32].

Single-beam reflection OT exploits the so-called “reflection mode” PNJ [33,34] emerging when the optical wave is directed to a metal or dielectric flat mirror with a transparent dielectric microparticle located near it. In this case, the incident beam is focused by the microparticle and specularly reflected back along the original path (in the case of normal incidence), thus forming symmetrical, counter-propagating optical fields modulated by a standing wave. It should be noted that in contrast to the twin-PNJs configuration, in specular-reflection OT precision spatial alignment, power balancing and phase matching of optical sources are less important. Meanwhile, this relatively new and promising OT type is rather poorly investigated.

In this paper we partially fill this gap and discuss in detail the physical principles and basic properties of a specular-reflection PNJ, which hereafter is termed as the “s-PNJ” to distinguish from the classical “direct” PNJ. In our study, s-PNJ is simulated by double focusing of a plane optical wave with a glass microcylinder positioned near a flat metal mirror. Using COMSOL Multiphysics FEM-solver to the Maxwell equations along with the geometrical ray trajectories analysis, we show that unlike traditional schemes of PNJ generation with single or counter-propagating optical beams, the s-PNJ retains high spatial localization and intensity enhancement even using microparticles with refractive index (RI) contrast n exceeding some critical value n^* when the classical PNJ is no longer formed. This possibility to overcome the fundamental RI limitation of ball-lens focusing is caused by the specific vortex circulation of the optical energy within the s-PNJ emerging from the interference of counter-propagating waves and contributing to the extreme optical field concentration. Worthwhile noting, the critical contrast of the refractive index depends on the particle size [1] and in case of large particles with radii R >> λ, this value equals to two in the approximation of the geometric optics, whereas at the mesoscales usually n*< 2. Additionally, we calculate the optical forces acting on a metallic nanosphere placed in s-PNJ and propose the concept of a multi-position OT integrated into a microfluidic tweezer. During our simulations it comes out that the s-PNJ based trap demonstrates multiply enhanced stiffness to random Brownian motion of trapped nanoparticle compared to the traditional PNJ trapping schemes.

2. Numerical simulation details

The photonic structure under study consists of a dielectric microcylinder with the diameter D and refractive index n₁ placed in water (RI, n₀= 1.33). Specifically, in the calculations the cylinder diameter is fixed, D = 6 µm, whereas its refractive index may vary. Hereinafter, the nonabsorbing optical materials are considered. A cylinder is illuminated by a plane optical wave (or a loosely focused laser beam) with the wavelength λ. In the classical case, a PNJ is generated near the shadow surface of the cylindrical particle (Fig. 1(a)). In the situation of s-PNJ formation (Fig. 1(b)), a thin metal plate is additionally placed near one side of the cylinder playing a role of a specularly reflective mirror for optical wave illuminating the microcylinder from the other side. This flat mirror can be not necessarily metallic but a dielectric one with high refractive index as well, such as, e.g., a silicon [35]. As a result of optical wave diffraction on a microcylinder and subsequent reflection of the scattered field from a mirror, in front of the particle a specular-reflection PNJ is formed modulated by a standing wave. Here, for simplicity only the normal optical wave incidence on the flat mirror is considered. The case with oblique cylinder illumination is the subject of the future work.

 

Fig. 1. Schematic view of (a) classical PNJ and (b) s-PNJ formed near a dielectric mesoscale microcylinder (n₁) illuminated by a plane optical wave (λ).

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During the simulations, the numerical solution to the Helmholtz vectorial equation for the components of the electromagnetic field ${\textbf E}({\textbf r} )$ in the stationary conditions is carried out:

To eliminate the effect of wave diffraction at the input port aperture, the electromagnetic field distribution is simulated in two steps. First, the calculation of the background field is performed without accounting for light transforming objects (particle, mirror). During this study step, on the input and output boundaries of the modeling structure the conditions of perfect field absorption (PML) are applied, and the Floquet-Bloch periodicity conditions are set on the other two boundaries. On the second study step, a microcylinder and a mirror at which the background field is scattered are “switched on”. The accuracy of the numerical solution of Eq. (1) is provided by adaptive mesh whose nodes get denser in the areas of sharp dielectric permeability gradients. The maximum step of the spatial grid is about 5 nm inside the metal mirror and λ/20 in the remaining domains.

3. Physical basis of s-PNJ localization

We start with the comparison of intensity localization degree in PNJs produced under different types of a dielectric microcylinder irradiation by a monochromatic plane wave. Figures 2(a)-(f) show the spatial distributions of relative optical intensity ${|E |^2}$ normalized to the initial value near 6-µm glass cylinder with the refractive index n₁ = 1.45 (RI contrast $n \equiv {{{n_1}} / {{n_0}}}$ = 1.09) for three different PNJ generation geometries.

 

Fig. 2. (a-c) 2D-distributions and (d-f) longitudinal profiles of relative optical intensity ${|E |^2}$ for three PNJ-generation schemes by exposing 6-µm glass microcylinder in water by a plane wave: (a, d) single “classical” PNJ, (b, e) twin PNJ, and (c, f) s-PNJ.

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The first profile group shown in Figs. 2(a) and 2(d) refers to PNJ formation within the “classical” geometry when the jet flux appears in the near-field shadow region of a single microparticle and is characterized by a smooth hill-shaped intensity profile [7]. Here, the transverse and longitudinal scales of the intensity change are of the order of one and several optical wavelengths, respectively.

The second PNJ configuration corresponds to the colliding PNJs originated from a pair of identical microparticles (Figs. 2(b) and 2(e)), similarly to the illuminating geometry presented in [31]. In this case, the paired photonic jets (twin PNJ) are formed through the diffraction of two counterpropagating and mutually coherent optical waves in the same spatial region between the particles. As a result of twin PNJs interference within the interparticle gap, a specific field structure is established constituting a standing wave with maxima period of ${\mathrm{\lambda} / {2{n_0}}}$ [30,37]. The amplitude of intensity maxima significantly depends on the mutual position of particles and usually only slightly exceeds the intensity in the classical PNJ.

The last case considered concerns to the PNJ formation in specular-reflection mode as proposed in [32–34] and shown in Figs. 2(c) and 2(f). In this case behind the microcylinder illuminated by an optical source, a metal plate is mounted having the thickness of 70 nm and playing a role of a return mirror. The plane optical wave is incident on dielectric microcylinder and is focused on a metal mirror. After specular reflection by the mirror, the spatially transformed optical wave is focused again by the same particle, but in opposite direction, near the illuminated surface of the cylinder. This reflected PNJ interferes with the incident radiation that leads to the formation of specific localized region with enhanced intensity modulated by a standing wave [32] which is referred here as the specular-reflection PNJ.

Unlike a twin PNJ configuration, s-PNJ is not the product of two real focused optical beams, but rather it is formed by one real and one virtual photonic flux created as an image of primary “direct” PNJ on the mirror. Because of this, the s-PNJ as a self-interference optical structure is always perfectly phase matched and retains maximal intensity [38].

Figure 3(a) shows the spatial structure of optical energy flows rendered through the calculations of spatial distribution of Pointing vector ${\textbf S} = ({{c / {8\mathrm{\pi} }}} )Re [{{\textbf E} \times {{\textbf H}^\ast }} ]$ (where E and H are electric and magnetic fields, respectively, c is the speed of light) in the region of s-PNJ formation. As seen, this case is characterized by the presence of counterpropagating energy fluxes leading to a complex structure of the optical field. Indeed, along the particle rim the energy flows are directed towards the incident wave propagation (bold red arrows), whereas in the central region where the s-PNJ arises, the reverse energy flow is observed (bold blue arrows). Such omnidirectional energy fluxes generate optical vortices in the intermediate boundary regions of the flow encounters, as noted in the figure by concentric ellipses. The vortices are observed both along the boundary zone of the PNJ and inside the microcylinder and contribute to additional concentration of optical radiation in the focusing region [39], similarly to the formation of deeply penetrating jets in the Langmuir turbulence model of ocean [40].

 

Fig. 3. Vector maps of optical energy fluxes in (a) s-PNJ (n = 1.09) and (b) twin PNJ.

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Interestingly, in case of the interference of two counter-directed PNJs as shown in Fig. 3(b), such organized large-scale circulation of optical energy in vortices is not observed. Instead, one can observe the formation a kind of stagnation zone for energy flows (denoted as “stagnation” in the figure) accompanied by a lot of small-scale vortices in the collision area of optical flows with different directions. This energy flow dynamics resembles the optical turbulence structure [41] caused by stochastic fluctuations of atmospheric air density, when the dissipation of the internal medium energy occurs via vortex motion splitting towards smaller spatial scales.

As known, the main PNJ parameters (length, width, intensity, etc.) are sensitive to the variation of microparticle size, spatial shape and RI contrast with the environment [1,2,26]. Obviously, s-PNJ should also demonstrate similar dependence of its characteristics. In Fig. 4(a) the peak intensity $E_{\max }^2 = {\max _{xy}}\{{{{|E |}^2}} \}$ of the considered PNJ types is presented in the dependence on the microcylinder RI contrast n. Figure 4(b) gives similar behavior of the PNJ transverse width w (FWHM) measured at half-maximum intensity.

 

Fig. 4. (a) Maximal intensity $E_{\max }^2$ and (b) lateral width w of PNJs under different generation schemes by changing microcylinder RI contrast n; (c, d) 2D intensity distribution ${|{E({x,y} )} |^2}$ of s-PNJ (c) and classical PNJ (d) from particle with over-limit contrast n = 2.03.

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From these figures it follows that for the chosen parameters of cylindrical microparticle and environment, the highest near-field intensity in a s-PNJ is realized with low RI contrasting particle in the range of values $n \approx $1.09 ± 0.04. A classical PNJ and a twin PNJ show the best optical field concentration at higher RI contrast of the microcylinder and water, namely $n \approx $1.61 and 1.17, respectively. Meanwhile, the width of s-PNJ reveals high variability from $w \approx $λ/2 to ∼ 5λ/2 when the parameter n changes unlike the classical PNJ which always retains its width close to λ/2. This is the consequence of the changes in angular divergence of reflected optical wave after focusing by the particle when varying its refractive index n₁ that can improve or, on the contrary, worsen the light energy concentration upon the interference with the incoming wave.

It is important to realize that the increase of particle refractive index unambiguously leads to PNJ shortening and its proximity towards the microparticle surface [1].

As an example, in Fig. 4(c) the distribution of normalized intensity ${|{E({x,y} )} |^2}$ near the microparticle with high RI (n₁ = 2.7) is presented in the situation of a classical PNJ generation. Note, in this case the microcylinder is characterized by the over-limit RI contrast with respect to the environment, n = 2.03 > n*, where n^*= 1.61 for selected wavelength and cylindrical particle size, and the region of high optical intensity is almost completely concentrated inside the shadow hemicylinder volume. As expected, a classical spindle-shaped PNJ is not formed, but instead it appears as an exponentially decaying intense “tail” near the geometric shadow of the particle. In this configuration, the position f of the external field intensity maximum, defined as the distance from the particle surface to the principal intensity maximum of PNJ, is only ∼ λ/5, and the PNJ length does not exceed the half-wavelength.

With the same parameters of the particle and optical radiation but in the configuration of specular reflection shown in Fig. 4(d), the s-PNJ is observed far from the particle surface at the position f which is practically equal to two laser wavelengths. Besides, the s-PNJ produced by an over-limit contrast particle possesses the total length of about 4λ when maintaining subwavelength transverse dimension (see, Fig. 4(b)). Notably, so far, the free-propagating PNJ with high-RI particles are reported only for truncated microcylinders or microspheres [42–44].

It should be emphasized that in the standard PNJ-based optical traps, an attempt to increase the optical trapping forces through the deeper optical field concentration in PNJ by using dielectric microparticles with higher refractive indices n₁ is inefficient because of trap working distance shortening. Indeed, starting from a certain value of RI contrast, the optical trapping of a test microobject can only be possible if this object is located very close to or at the surface of the mother microparticle. On the contrary, the optical trapping assisted by a s-PNJ effect is free of this limitation, and in this OT both low and high-contrast microparticles are able to provide wide working area of optical capture at high transverse localization of intensity.

To understand the physical causes of this unique property of s-PNJ for light concentration in over-limit contrast domain that is unusual for the PNJ optics, we consider a geometric optical treatment to the light scattering problem. It is clear that the geometric optics on meso-wavelength spatial scales (∼λ) is strictly speaking not applicable. However, as will be seen below, such a simplified qualitative analysis can shed a light on the origin of the phenomenon in question.

Within the framework of geometric-optical approximation, a photonic jet is a spatial region bounded by the outer caustics formed during focusing of the optical rays by a particle, i.e., a PNJ is a result of ray concentration due to multiple refractions on optically contrasting boundaries of physical media. The geometric rays tracing in the studied geometry is carried out by means of the COMSOL Multiphysics “Ray Optics” module. The ray tracing results are shown in Figs. 5(a) and 5(b) in the same cases of PNJ generation as in Figs. 4(c) and 4(d).

 

Fig. 5. Geometric ray tracing analysis of (a) classical PNJ and (b) s-PNJ by the high-contrast microcylinder with n > 2. Rays are colored depending on their optical path value. Internal and external ray foci formation are shown.

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As seen, the classical PNJ from the microcylinder with n >2 shown in Fig. 5(a) is formed by a divergent beam of geometric rays leaving the focal area formed inside the particle due to strong refraction at the interface. As a result, the length of focal caustic is small and the PNJ is located near the shadow surface of the microcylinder.

At the initial stage of s-PNJ formation, the ray focus is also located inside the particle as in the previous case with a classical jet, as demonstrated in Fig. 5(b). However, the presence of a mirror near the rear surface of the particle completely reconstruct the further ray traces. Indeed, a narrow and highly diverging beam of geometric rays is released from the internal focus, then it experiences specular reflection from the mirror and returns to the rear particle surface. After the refraction on the material interface with high RI, this ray beam sharply reduces its angular divergence. Further refraction of this weakly diverging beam by the illuminated particle edge leads to its rather loose focusing and formation of a long PNJ far away from the microcylinder. The increase or decrease in particle RI contrast will move the internal ray focus relative to the mirror and change the location of the external focal area.

4. Optical trapping with a s-PNJ

It is important to assess the trapping potential of the s-PNJ. To this end, we calculate the optical forces acting on a test nanoparticle placed in specular-reflection jet area. As such test object we use a gold bead with the radius R_b = 50 nm (λ/30) and refractive index n_b = 0.53-i·10.8 at λ = 1550 nm [45]. The bead is moved on two coordinate axes to build the functional dependence of the net mechanical force of the optical field.

The optical pressure forces acting on a particle are determined through the calculations of Maxwell stress tensor M(x,y) and Pointing vector S(x,y) which accounts for the net energy flux in the particle volume. In isotropic medium, the following expression for the net optical force F is used by the time averaging of the electromagnetic fields [46]:

Formally, Eq. (2) contains the sum of two forces: ${\textbf F} = {{\textbf F}_s} + {{\textbf F}_\nabla }$, where F_s is the scattering force caused by the light pressure on the particle, and ${{\textbf F}_\nabla }$ stays for the gradient optical force associated with the Lorentz force acting on an elementary charge in electric field. The scattering optical force acts in the direction of optical wave propagation, while the gradient force is directed opposite to the surface-averaged gradient of the electric field. For Rayleigh particles (R_b << λ), when the dipole approximation of particle polarizability can be used, the analytical expressions for these optical forces are known in the following form [48]:

As follows from Eq. (2), in the optical traps based on the standing wave phenomenon, i.e., OTs on the twin PNJ and a s-PNJ, the optical force caused by electromagnetic wave scattering on a particle should be substantially decreased because the counter propagating optical fluxes effectively reduce the longitudinal Pointing component S_y [15]. Obviously, under these conditions the role of the gradient optical force ${{\textbf F}_\nabla }$ increases. This will be facilitated by additional increase of intensity in the antinodes of the standing wave trap relative to the classical PNJ-based OT (see, Figs. 1(d)–1(f)).

The net optical forces acting along two coordinate axes on the nanometer Au-sphere in OTs of various types formed by the microcylinder with RI contrast n = 1.09 are shown in Figs. 6(a) and 6(b). In the simulations, the optical force is calculated according to Eq. (2) and provided by a continuous wave (CW) laser beam with the power 1 mW focused into the spot with the dimensions 10 µm ×10 µm. A test nanoparticle is immersed in water and for each type of considered OTs it is placed in the area of maximum optical intensity, where its position may vary within certain limits to obtain the required force profiles. Note a CW-laser illumination used in our simulations assumes all temperature gradients near a gold bead arising due to light absorption are already relaxed. Thus, in contrast to a pulsed laser exposure [51], we do not account for the photothermal effect, which may mediate the scattering cross-section of a metal particle.

 

Fig. 6. Net optical forces acting on a 50 nm Au-nanosphere in (a) lateral ${F_x}$ and (b) longitudinal ${F_y}$ directions; green line is plotted for a 20 nm spherule.

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It is clear from Fig. 6(a), that regular PNJ-based and twin-PNJ optical traps demonstrate close values of transverse optical force modulus $|{{F_x}} |$ being of order of 10 fN. Interestingly, regardless of higher intensity achieved in OT with the counter propagating nanojets, the transverse optical force in this trap is lower. Obviously, this is due to weaker field localization in transverse direction in the standing wave traps and a subsequent decrease in the gradient optical force F_x. At the same time, the s-PNJ optical trap proposed here shows a significantly higher transverse trapping potential with a maximum optical restoring force $|{{F_x}} |{\approx} $ 28 fN, which indicates both large gradients of field amplitude and a multiple increased intensity on the trap axis.

The longitudinal capture forces are shown in Fig. 6(b). It turned out, that a 50 nm gold particle is too large for optical trapping by the classical PNJ. One can see from this figure (black curve) that the longitudinal optical force in this case is negative over the whole jet length. This is due to strong imbalance in favor of the scattering force ${{\textbf F}_S}$ that tends to pull out the nanoparticle from the OT as the light beam propagates. However, both traps exploiting the standing wave effect (twin PNJ and s-PNJ) due to the obvious dominance of the gradient optical force ${{\textbf F}_\nabla }$ show a feasible capturing of a 50 nm gold bead in several regions positioned within the standing wave antinodes. Moreover, the optical trapping produced by the s-PNJ appears to be much stronger in this direction.

As follows from Eqs. (3)–(4), which represent the optical forces in the approximation of Rayleigh particles, the ratio $|{{{{{\textbf F}_s}} / {{{\textbf F}_\nabla }}}} |$ is proportional to the trapped particle volume V_b. Thus, by decreasing the test sphere radius, e.g., to 20 nm, it becomes possible to realize optical capture with classical PNJ (green curve in Fig. 6(b)). However, the optical trapping of such ultra-small objects may be challenging because of the increased influence of chaotic motion of a nanobead in the Brownian force field [52].

Indeed, since it is assumed that the trapped particle is in a dense medium (water), it will experience multiple random shifts from the stability point under the action of Brownian motion. Following [48], one can estimate the effective stiffness of a s-PNJ based on the smallness of Boltzmann exponent: $\exp ({ - {{{W_{op}}} / {{k_B}T}}} )< < 1$, where ${W_{op}} = \int\limits_{{\mathrm{\xi} _1}}^{{\mathrm{\xi} _0}} {Fd\mathrm{\xi} } $ is the work of optical forces to return a particle from the current coordinate ξ₁ to the position ξ₀ of a stable trapping, which it has leaved due to stochastic thermal oscillations of medium with characteristic energy k_BT (k_B is Boltzmann constant, and T is average temperature).

At normal conditions, one can use ${k_B}T \approx $ 4·10⁻²¹ J, which gives the threshold value of the optical trapping potential ${W_{op}} \ge $ 10⁻²⁰ J. Therefore, if we take the effective width of a PNJ of the order of λ/2, the effective amplitude of optical forces ${F_e} = |{{{{W_{op}}} / {({{\mathrm{\xi}_1} - {\mathrm{\xi}_0}} )}}} |$ should be not less than ${F_e}\sim $ 50 fN. Recalling Fig. 6(b), one can see that this requirement at laser power of 1 mW meets only the s-PNJ based OT, where the peak longitudinal trapping force can reach 100 fN. Other types of optical traps considered which use the PNJ effect can secure stable optical trapping only for multiply increased optical power, which in some cases may be undesirable, e.g., when manipulating the biological cells [53].

5. Concept of a multiposition optical s-PNJ trap

Based on the above analysis of optical capture forces, in Figs. 7(a)–7(c) we present a possible engineering design of a multiposition s-PNJ optical trap being a structural component of an optical tweezer.

 

Fig. 7. (a, b) Schematics of an on-chip miltiposition s-PNJ optical trap integrated in a microfluidic tweezer; (a) Perspective and (b) side views; (c) Squared optical field amplitude distribution ${|E |^2}$ OT cross-section and in microfluidic channel with trapped multiple Au NPs (zoomed picture).

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The proposed trap is fabricated as an integrated on-chip microfluidic device consisting of a glass substrate (RI n₀) with an empty channel drilled near one of its edges. The channel shape can be arbitrary, e.g., rectangular as in the figure, because it has weak effect on the distribution of optical trapping forces. On top of the substrate, right above the channel, a focusing dielectric microcylinder is mounted. It can be made as a tapered glass fiber with the refractive index providing proper RI contrast value of, e.g., n ∼ 1.1. Above the cylinder there is a metal foil acting as a flat reflecting mirror. The entire structure is fixed with transparent glue having refractive index close to n₀ and illuminated downwards by a laser beam (here, with a telecommunication wavelength λ = 1550 nm).

Analyte containing the target nanoparticles (NPs) is fed through the channel in the form of a liquid suspension. NPs experience the optical field forces in the s-PNJ and occupy OT potential wells. As a result of the global transport in a liquid flow, NPs line up in the bundles throughout the trap in the form of the “beamlets”, which can then be further processed and analyzed.

In real conditions, upon manufacturing of such optical tweezers various deformations and bends of the reflecting plane mirror may appear causing small-scale changes to the direction of illumination of the microchannel and cylindrical mesoscale lens and impairing the normal operation of the microfluidics. To estimate the influence of mirror surface roughness on the quality of s-PNJ we conducted a series of numerical simulations with non-flat reflecting mirror. The maximum intensity attained in s-PNJ is shown in Fig. 8 as a function of mirror quality.

 

Fig. 8. Effect of reflecting mirror roughness σ on the maximum s-PNJ intensity $E_{\max }^2$ in water channel of the microfluidic tweezer shown in Fig. 8(a). Inset: transverse mirror surface profile $h(x )$ at different perturbation amplitude.

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During the simulations, an ideal flat mirror of the microfluidic tweezers shown in Fig. 7(a) was replaced by a reflecting metal plate with a rough specular-reflecting surface. The surface roughness degree could change so that the thickness h of the mirror became a function of the transverse coordinate x: $h(x )= {h_0} + \sigma \tilde{\mathrm{\chi} }(x )$, where h₀ is the thickness of the smooth mirror substrate, $\tilde{\mathrm{\chi} }(x )$ is a normally distributed random number in the range -1…1, and σ denotes the amplitude of height perturbations. Such surface corrugations can be caused, e.g., by multiple lengthwise surface scratches imposed to a metal foil.

As the example, two extreme cases of rough reflecting surfaces constructed in the simulations are presented in Fig. 8 by the pictograms and the insert to this figure shows the corresponding transverse profiles $h(x )$. Each point of the calculated dependence $E_{\max }^2(\sigma )$ is obtained by averaging ten independent model runs with fixed perturbation amplitude σ and randomly generated mirror surface.

As seen, the decrease in mirror surface quality, i.e. its smoothness, leads to worsening of s-PNJ spatial localization and is expressed in decrease in its intensity. Thus, at fivefold growth of the perturbations amplitude of flat mirror from σ =λ/50 to λ/10, a ∼15% decrease of s-PNJ intensity is observed, which at further growth of the σ-parameter relaxes to the level $E_{\max }^2 \approx $ 9. Worthwhile, this value nevertheless remains above the similar intensity value in the configuration of the twin-PNJ ($E_{\max }^2 \approx $ 4) and classical PNJ ($E_{\max }^2 \approx $ 5).

6. Conclusion

We consider a specific type of optical near-field subwavelength spatial localization originated from the constructive interference of direct and backward propagated optical waves focused by a transparent dielectric microparticle located near a flat reflecting mirror. This specular-reflection photonic nanojet (s-PNJ) is formed in the shadow region of a micron-sized dielectric cylinder. A detailed analysis of optical near-field structure of the s-PNJ is carried out which shows that upon radiation specular reflection from a mirror and subsequent focusing by the dielectric microcylinder, large-scale optical vortices with counter-circulating optical energy are formed in the shadow particle region. These eddy flows promote additional concentration of optical field within the photonic jet and impart the s-PNJ based trap several unique features in contrast to the previously known optical traps using the classical PNJ effect. Particularly, the optical trap proposed is characterized by significantly stronger stability to chaotic Brownian displacements of captured metallic nanoparticle along all coordinate axes. Moreover, for high refractive index contrast microparticles (n > n*), the s-PNJ optical trap does not reduce, but increases its working zone and optical capture force.

We propose a prospective design of an optical trap based on the s-PNJ effect. Due to relatively simple technical implementation, the proposed optical tweezer can find practical application as an on-chip microfluidic device for optical sorting and lining up of target atoms or nanoobjects [54–57]. Besides, the s-PNJ based trap allows combining multi-position optical manipulation with a microfluidic channel, e.g., to secure fixing nanoobjects in a given position for their better visualization, removing [57], or to localization the target nanoobjects in a desired region to improve the limit of their detection by means of optical instrumentation [58]. This, in turn, makes it possible the continuous replacement of analytes and consequently high productivity and automatic processing of research data.