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Photonic nanojet super-resolution in immersed ordered assembly of dielectric microspheres
Journal of Quantitative Spectroscopy & Radiative Transfer 200 (2017) 32–37
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Photonic nanojet super-resolution in immersed ordered assembly of dielectric microspheres Y.E. Geints∗, A.A. Zemlyanov Zuev Institute of Atmospheric Optics SB RAS, 1 Zuev Square, Tomsk 634021, Russia
a r t i c l e
i n f o
Article history: Received 28 February 2017 Revised 27 March 2017 Accepted 2 June 2017 Available online 6 June 2017 Keywords: Photonic nanojet Microsphere Microassembly of particles Immersion
a b s t r a c t Specific spatially-localized optical field structure, which is often referred to as a photonic nanojet (PNJ), is formed in the near-field scattering area of non-absorbing dielectric micron-sized particle exposed to an optical radiation. By virtue of the finite-difference time-domain technique we numerically simulate the two-dimensional array of PNJs created by an ordered single-layer microassembly of glass microspheres immersed in a transparent polymer matrix. The behavior of the main PNJ parameters (length, diameter, and intensity) is analyzed subject to the immersion depth of the microparticles and cooperative interference effects of the neighboring microspheres. We show that depending on microassembly configuration, the PNJ quality can be significantly improved; in particular, the PNJ spatial resolution better than λ/5 can be achieved. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Rapid advances in optical technologies open new prospects in ultrahigh-resolution microscopy, nano-optics, nanomaterial design, and precision diagnostics of dispersed medium [1–4]. Most of these problems require high-localized optical fluxes created near material objects of different physical nature. To date, several approaches are developed to produce ultrahigh localized optical fields on nanometer scales, such as the excitation of surface plasmon-polariton resonances in metal nanoparticles [5], nanoantennas [6] or optical tips [7]. However, all these techniques use the metal objects as a light focusing tool, which strongly absorb optical radiation and can produce unwanted heating of the samples under study. In this context, non-absorbing dielectric microobjects of various geometric shapes, physical properties, and inner composition are currently the focus of researcher attention from the whole world [8–14]. Such mesoscale microobjects, i.e., the objects with the characteristic spatial scale on the order of a light wavelength, are fairly promising in obtaining light localization within extremely small area. Diffraction of electromagnetic radiation at microparticles produces near their surfaces the highly-localized regions of enhanced intensity, the so-called photonic nanojets (PNJ) [9,15]. The physical nature of these nanojets formation is related to the aberrational character of optical near-field focusing at a transparent wavelength-sized particle [16]. Under these conditions the for∗
Corresponding author. E-mail address: [email protected] (Y.E. Geints).
http://dx.doi.org/10.1016/j.jqsrt.2017.06.001 0022-4073/© 2017 Elsevier Ltd. All rights reserved.
mation of a focal region with the spatial super-resolution (up to subdiffraction in size), high intensity, and increased length is possible due to the constructive interference between the scattered and incident fields in the particle shadow. To date, the control over PNJ dimensional and amplitude parameters is the most important challenge for the improving jet spatial resolution, working range and intensity. The most obvious strategy of PNJ manipulation is the variation of the size, shape, optical properties, and the structure of the mother microparticle which creates the jet [17–20]. Here, the impressive results are achieved and, in particular, several research groups reported on the possibilities of producing extra-long [21], extra-narrow [22] or super-intense PNJs [23]. A different resolution improving strategy of optical imaging devices is a technique successfully applied in the sphere-based immersion microscopy, when an object of interest is viewed through a microspherical lens deposited in immersion medium (usually liquid) being optically denser than the air [24,25]. According to the principles of physical optics, this provides for an increase in the spatial resolution of the image although decreases object magnification and contrast. It turns out that in this technique, not only the optical properties of the immersion layer play an important role, but also its arrangement relative to the microparticle is important. As was shown for glass microspheres [26–30], the semi-immersed particles often ensure a better resolution than the full immersed or non-immersed spheres. It is important to understand that the spatial resolution of a PNJ is not the same as the spatial resolution of an image that is obtained by means of this PNJ. Hereafter, the term “PNJ spatial
Y.E. Geints, A.A. Zemlyanov / Journal of Quantitative Spectroscopy & Radiative Transfer 200 (2017) 32–37
resolution” connotes exactly the transversal spatial size (usually FWHM) of a PNJ. Notice that the most studies devoted to the control over the photonic jets usually consider single isolated particles in a free environment. The optical field distribution near an isolated particle depends only on the characteristics of incident light and the microphysical parameters of the particle. In practice, the manipulation of localized optical fluxes induced by individual isolated microparticles is a challenging task because of the technical problems related to the spatial fixation of microparticles on the object surface. In view of this, the formation of a PNJ cluster via the deposition of an ordered array of microparticles (microassembly) on a transparent substrate, e.g., on a flexible polymer film [22], seems currently the most promising way to solve this problem. This secures the microparticles in the matrix, which then can be moved to the desired spatial position of the object for PNJs production. A conventional way of a PNJ cluster creation is the use of a particle self-assembly that is formed during particle self-arrangement on an object surface or on a special substrate. Generally, this process is related to the precipitation and evaporation of microparticles from a colloidal solution deposited on a substrate, with subsequent formation of a single- or multilayer matrix of closely or loosely packed particles [25,31]. Such silicone films (PDMA, PMMA) with embedded glass microspheres [25] are used, e.g., in the ultrahigh resolution microscopy for subwavelength and even subdiffraction light focusing [22]. Sphere-containing silicone films are unique because the embedded microparticles can be fixed in any spatial configuration, but not only in the form of closely packed particles. Then a flexible polymer matrix can be carried to any surface and optically irradiated. Therefore, the study of cooperative interference effects during formation of PNG from ensembles of differently arranged particles becomes urgent. It is evident that the use of a group of microparticles, instead of an individual particle, embedded into a fixing matrix or placed on a substrate can significantly influence the parameters of individual nanojets due to the interference of optical fields during the radiation diffraction at neighboring microparticles and interaction between the scattered field and the substrate. This problem was studied previously in several works [32–35]. The effect of absorbing substrate on the optical field scattered at the ensemble of seven microspheres was studied theoretically in [33], where it was found that both the substrate and neighboring particles reduce up to several times the optical field intensity in the focal regions in comparison with an isolated sphere in vacuum. Elongation of a PNJ and the shift of its maximum toward the surface of parent spheres arranged into a hexagonal cluster were observed in [34]. Recently [35], we considered the near-field scattering region of a single-layer cluster of identical microspheres embedded into a transparent matrix. We found that the collective interference effects occurring during PNJ array formation manifest themselves mainly in the pulsations of photonic jet intensity and its length when sphere cluster period is varied. In this work, we continue studying the near-field diffraction patterns upon PNJ formation by a single-layer assembly of identical microspheres embedded in the matrix with refractive index contrast. To this end, we numerically solve the set of Maxwell differential equations by virtue of the finite-difference time-domain (FDTD) method. Spatial and amplitude parameters of the produced PNJs are examined versus the inter-particle gap, particle size, and optical contrast of the immersion matrix. In contrast to our previous study [35], in this work we focused on the effect of the immersion depth of a particle on PNJ parameters. We show that depending on the microassembly configuration, PNJ quality can be significantly improved; in particular, one can achieve the sub-diffraction jet resolution (< λ/5). Worth noting, that similar effect was previously reported in [26] for a half-immersed isolated microsphere
33
Fig. 1. (a) Microassembly of spheres immersed in the film; (b) optical field intensity enhancement B in a vicinity of glass spheres with D = 2λ immersed in PDMA film. The arrows show light incidence direction.
exposed to an optical radiation. However, in a microsphere assembly these super-resolution PNJs may not be observed because of strong coupling of the optical fields if the gap between individual spheres becomes much shorter than the incident radiation wavelength. 2. Numerical simulation technique We considered the classical spheres self-assembly in the form of a hexagonal single-layer cluster of spherical particles (Fig. 1(a)). Identical spheres with the diameter D are deposited on a plane grid with the lattice period d and constitute close (d = D) or loosely (d > D) spheres packing. Particle assembly is immersed in a transparent dielectric matrix (film). The immersion depth f can vary within the limits 0 ≤ f ≤ D. The near-field diffraction region of the microsphere assembly was simulated in the 3D domain bounded by perfectly matched layers, using commercial FDTD software package (Lumerical Solutions Inc.), which implements the numerical integration of Maxwell equations for the electromagnetic field components. The accuracy of numerical solution was improved by the adaptive mesh grid, which was denser in the regions of sharp permittivity gradients (particles rim). The total number of grid nodes (Yee cells) was about 108 , in which case spatial and temporal grid steps were 2 nm and 0.06 fs, respectively. A dielectric permittivity was specified inside the calculation domain according to the geometry of spheres arrangement. The spheres were assumed to be made of optical glass with the refractive index ns and no absorption in the optical wavelength range. The assemblies of spheres are immersed in a non-absorbing silicone (PDMA) film with the refractive index nm = 1.46 surrounded by air (n0 = 1). To minimize edge effects, a three-line cluster of nine spheres is used to simulate the hexagonal geometry; the PNJ parameters are studied at the cluster center. At the initial instant a monochromatic optical wave with the wavelength λ = 0.532 μm is switched-on at the lower boundary of the domain. The spatial form of the wave is a plane linear polarized wave with the electric field vector directed, for definiteness, along the x-axis. Optical wave propagates through an assembly of particles in the positive z-direction and forms time-dependent diffraction pattern in the near-field, which is then averaged over
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Fig. 2. Optical intensity profiles near isolated microspheres (D = 2λ) (а) in air and (b) immersed in polymer film (f/D = 75%).
the time interval of 2 ps. This time is sufficient to account for all transition processes in spheres and matrix. The resulted spatial distributions of the relative field intensity B(r ) = |E(r )|2 /E02 (E0 is the incident wave amplitude) is analyzed in order to reveal localized regions of enhanced intensity (PNJs) and determine their dimensional and amplitude characteristics. 3. Simulation results and discussion Fig. 1(b) exemplifies the photonic nanojets formed by an assembly of semi-immersed in the polymer film spherical particles with ns = 1.6 and D = 2λ. Spatial distribution of the relative optical intensity B in the xz-plane is shown on the gray scale. The formation of compact high-intensity PNJs is clearly seen near the top shadow surfaces of the spheres. As will be shown below, jet parameters generally depend on the spatial arrangement of particles, their optical contrast and immersion depth into the film. For a quantitative characterization of a PNJ, we use two size parameters, namely, the length L and the width R (shown in Fig. 1(b)). These parameters are calculated as the full-width at halfmaximum values (FWHM). The PNJ magnitude is characterized by the (relative) peak intensity Bmax of the optical field. First, it is instructive to demonstrate how the immersion of a sphere inside the optically contrasted film affects the near-field spatial structure of a PNJ produced. Fig. 2(a) and (b) show the intensity distribution for a spherical particle placed either on top of the film or semi-immersed inside it. It is evident that the shape, intensity, and size of the PNJ differ and depend on the environments. Thus, for a non-immersed microparticle (totally in air), the intensity maximum is located inside the sphere near its shadow hemisphere (Fig. 2(a)). Here, the PNJ is formed through the field leakage from the internal focus toward the exterior that leads to extremely short (several fractions of the wavelength) but highintensity jet creation. If a sphere is semi-immersed in the film with the refractive index higher than that of air (Fig. 2(b)), the spatial structure of the PNJ changes dramatically. In this case, the PNJ is visually detached from the particle rim and possesses lower field amplitude but markedly longer extent (L ∼ 2λ). This is the direct consequence of optical contrast decrease for a wave entering the particle from the immersion layer. In its turn, lower optical contrast ns /nm decreases the numerical aperture of the spherical particle-lens that
Fig. 3. (a) PNJ intensity Bmax , width Rx , and (b) PNJ length L versus intersphere gap g in differently immersed microassemblies.
elongates its focal waist similarly as jet formation in gradient-index spheres [19,36]. Consider now the parameters of PNJs produced by different microassembly configurations. Fig. 3(a) and (b) show the effect of the assembly period d on the key PNJ parameters. For convenience, hereafter we introduce the intersphere gap parameter g that is defined in terms of particle diameter, g = (d − D). So, g = 0 corresponds to the close packed spheres, whereas g ≈ D describes the loosely packed assembly. In these figures, differently filled points depict sphere microassemblies with different immersion depths f expressed in per cents of sphere diameter. Interestingly, PNJs demonstrate high variability of their parameters depending on the intersphere gap. This is caused by the cooperative interference effects [35] of the neighboring particles, which are less pronounced for PNJ amplitude while jet length and width can differ by several times even at small changes of the gap. In general, the assemblies with fully immersed microspheres (f/D = 100%) produce the PNJs with the parameters more stable against variations in g. Furthermore, in this case closely packed microspheres form PNJs with close intensities and widths but much smaller lengths than loosely packed particles. In contrast, the assemblies of half- and non-immersed spheres are very sensitive to the variations of the cluster period. Here, several particle configurations can produce PNJ with extremal characteristics. Indeed, the analysis of Fig. 3(a) shows that PNJ peak intensity becomes higher (Bmax ≈ 50 atf = 0) at moderate values of the intersphere gap (g ≈ 0.4) than in the cases of close packing (Bmax = 21)
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Fig. 5. Transverse intensity profiles B(x, y) near half-immersed spheres (f = D/3) at different distances z from particle surface.
Fig. 4. (a) PNJ intensity and width as functions of the immersion depth for g = 0 (close packing) and g = 0.25D (loose packing); (b) near-field intensity profile for different immersion depths f (g = 0.25D).
Fig. 6. Geometrical rays traces in (a) fully- and (b) semi-immersed spherical particle. Rays propagate from left to right. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
or clearly loose packing (Bmax = 39). If the gap between half immersed particles equals to 1/4 or 3/4 of their diameter then the produced PNJ is extremely narrow. In this situation, PNJ size along the x-axis exhibits super-resolution, Rx = 0.182λ ≈ λ/5.3, because this more than twofold exceeds the Abbe diffraction limit for the conventional far-field optics focusing ( ∼ λ/2). Fig. 4(a) shows the effect of microsphere immersion depth on the parameters of PNJ array. Apparently, the subdiffraction PNJ is attainable only from semi-immersed (25% < f/D ≤ 50%) loosely packed particles, Besides, PNJ intensity is also higher in this case. This fact can be of practical interest, e.g., for sphere-assisted superresolution microscopy [37], where a transparent microsphere is used as a final focusing tool with subdiffraction texturing of an object. According to our calculations, an attempt to improve the resolution of such optical system by means of spheres immersion in optical denser medium will even worsen the spatial resolution until the microspheres are fully immersed. It is worth to emphasize that the desired resolution improvement could be attainable only in the case of semi-immersed microspheres. The longitudinal structure of the PNJs from a microassembly of loosely packed spheres is shown in Fig. 4(b) for several depths of particle immersion. It is clear that PNJ intensity profile has two maxima, which are z-separated by about 1.5λ. As the immersion depth decreases, the intensity of the maxima increases and they approach the mother particle surface. When the particles are halfimmersed (f < D/2) then the first (main) PNJ maximum almost coincides with the microsphere surface. Secondary PNJ intensity
maximum continues moving toward the particle as f decreases, which results in total PNJ length shortening (Fig. 3(b)). The lateral field structure of these maxima also differs. Consider Fig. 5(a) and (b) which show the relative optical intensity distribution B(x, y) in two cross-sections downstream the photonic flux. Spatial positions of these cross-sections are chosen so that they contain the first and second field maxima. The distance z between the sections and the particle surface is indicated in the figures. From Fig. 5(b) it is clear that far from the particle surface PNJ exhibits transverse ellipticity which is due to the linear polarization of optical wave incident on sphere assembly. PNJ in the second field maximum has the width Ry along y-axis (parallel to the polarization plane) which is 1.5 times larger than the analogous scale along x-axis, Rx = 190 nm ≈ λ/3.32. Intensity profile of the main PNJ maximum (Fig. 5(a)) has the shape of a distorted ellipse that indicates the presence of strong aberrations upon wave focusing. The size of the nanojet takes the minimum, Rx = 95 nm, close to the particle surface that is less than 1/5 of laser wavelength and is far below the diffraction limit. Worth noticing, that this PNJ structure is typical only for the case of semi-immersed spheres and is not observed for fully- or non-immersed particles. Strong wave aberrations upon light focusing by a semiimmersed sphere can be readily understood from the geometrical optics. Consider Fig. 6(a) and (b), which show geometrical ray traces passing through a sphere being fully- and semi-embedded in an optically dense block. Ray traces are rendered by COMSOL Multiphysics software package. As seen, for geometrical rays, the outer
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lations for minimal PNJ width Rx are plotted in Fig. 7(a) and (b) as the color 2D-maps with intersphere gap g and immersion depth f/D as the dependent variables. From these figures it follows that both microassemblies demonstrate the ranges of their parameters where generated PNJs are extremely narrow (R ≈ λ/5). If bigger particles with the diameter D = 4λ (Fig. 7(a)) are used in the assembly, then the superresolution PNJ occurs only for half-immersed loosely packed particles at the lattice period, d > 1.1 D. The assembly of optically denser microspheres (Fig. 7(b)) in most cases produces the PNJs with the subdiffraction resolution in terms of Abbe diffraction limit ( ∼ λ/2). Here, the narrowest photonic nanojets are formed by semi- or fully-immersed microspheres at any intersphere gap excluding only the close packing of particles. 4. Conclusions We considered the spatial structure of the near-field scattering of an optical wave at ordered hexagonal microassemblies of non-absorbing dielectric microspheres immersed in a transparent matrix. By virtue of the finite-difference time-domain technique, the evolution of dimensional and amplitude parameters of produced photonic nanojets array is studied by varying assembly immersion depth. The main conclusion of our research is that the reported earlier PNJ super-resolution ( ∼ λ/5) from individual half-immersed microsphere [26,30] can be also observed in sphere microassemblies with different geometry and optical properties. Moreover, this effect is apparent not only for half-immersed spheres but also for semi- and in some cases for fully-immersed particles, depending on the microassembly configuration. Principal condition for ultra-narrow PNJ formation is loose packing of spheres when the cooperative interference within the microassembly becomes weak. References
Fig. 7. PNJ width from sphere assemblies with (a) D = 4λ, ns = 1.6; (b) D = 2λ, ns = 2.1. Digits are presented for better reading.
surface of the fully-immersed particle comprises the boundary of constant optical contrast, ns /nm . This causes geometrically similar refraction of peripheral and central rays on the sphere and results in the formation of one external focus, which is markedly stretched along z-axis due to the spherical aberrations. In contrast, for a semi-immersed sphere, several parts of its illuminated face (shown by bold red lines in Fig. 6(b)) possess higher optical contrast ns /n0 than the remaining surface due to lower air refractivity. This causes stronger rays refractions in these regions and produces the additional zone of rays focusing located close to particle rim. This additional focus is sharper, thus having smaller focal waist. As a result, in this semi-embedded sphere configuration the longitudinal PNJ shape becomes clearly bimodal (see Fig. 4(b)). The robustness of a PNJ with spatial super-resolution is of importance, in view of possible structural and optical parameters of microsphere assembly variations. To investigate this point, we calculated PNJ key parameters produced by sphere microassemblies either with doubled diameter or with higher refractive index, ns = 2.1. The latter case simulates a cluster of spheres made, e.g., of high density barium titanate glass (BTG). The results of our calcu-
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Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Photonic nanojet super-resolution in immersed ordered assembly of dielectric microspheres Y.E. Geints∗, A.A. Zemlyanov Zuev Institute of Atmospheric Optics SB RAS, 1 Zuev Square, Tomsk 634021, Russia
a r t i c l e
i n f o
Article history: Received 28 February 2017 Revised 27 March 2017 Accepted 2 June 2017 Available online 6 June 2017 Keywords: Photonic nanojet Microsphere Microassembly of particles Immersion
a b s t r a c t Specific spatially-localized optical field structure, which is often referred to as a photonic nanojet (PNJ), is formed in the near-field scattering area of non-absorbing dielectric micron-sized particle exposed to an optical radiation. By virtue of the finite-difference time-domain technique we numerically simulate the two-dimensional array of PNJs created by an ordered single-layer microassembly of glass microspheres immersed in a transparent polymer matrix. The behavior of the main PNJ parameters (length, diameter, and intensity) is analyzed subject to the immersion depth of the microparticles and cooperative interference effects of the neighboring microspheres. We show that depending on microassembly configuration, the PNJ quality can be significantly improved; in particular, the PNJ spatial resolution better than λ/5 can be achieved. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Rapid advances in optical technologies open new prospects in ultrahigh-resolution microscopy, nano-optics, nanomaterial design, and precision diagnostics of dispersed medium [1–4]. Most of these problems require high-localized optical fluxes created near material objects of different physical nature. To date, several approaches are developed to produce ultrahigh localized optical fields on nanometer scales, such as the excitation of surface plasmon-polariton resonances in metal nanoparticles [5], nanoantennas [6] or optical tips [7]. However, all these techniques use the metal objects as a light focusing tool, which strongly absorb optical radiation and can produce unwanted heating of the samples under study. In this context, non-absorbing dielectric microobjects of various geometric shapes, physical properties, and inner composition are currently the focus of researcher attention from the whole world [8–14]. Such mesoscale microobjects, i.e., the objects with the characteristic spatial scale on the order of a light wavelength, are fairly promising in obtaining light localization within extremely small area. Diffraction of electromagnetic radiation at microparticles produces near their surfaces the highly-localized regions of enhanced intensity, the so-called photonic nanojets (PNJ) [9,15]. The physical nature of these nanojets formation is related to the aberrational character of optical near-field focusing at a transparent wavelength-sized particle [16]. Under these conditions the for∗
Corresponding author. E-mail address: [email protected] (Y.E. Geints).
http://dx.doi.org/10.1016/j.jqsrt.2017.06.001 0022-4073/© 2017 Elsevier Ltd. All rights reserved.
mation of a focal region with the spatial super-resolution (up to subdiffraction in size), high intensity, and increased length is possible due to the constructive interference between the scattered and incident fields in the particle shadow. To date, the control over PNJ dimensional and amplitude parameters is the most important challenge for the improving jet spatial resolution, working range and intensity. The most obvious strategy of PNJ manipulation is the variation of the size, shape, optical properties, and the structure of the mother microparticle which creates the jet [17–20]. Here, the impressive results are achieved and, in particular, several research groups reported on the possibilities of producing extra-long [21], extra-narrow [22] or super-intense PNJs [23]. A different resolution improving strategy of optical imaging devices is a technique successfully applied in the sphere-based immersion microscopy, when an object of interest is viewed through a microspherical lens deposited in immersion medium (usually liquid) being optically denser than the air [24,25]. According to the principles of physical optics, this provides for an increase in the spatial resolution of the image although decreases object magnification and contrast. It turns out that in this technique, not only the optical properties of the immersion layer play an important role, but also its arrangement relative to the microparticle is important. As was shown for glass microspheres [26–30], the semi-immersed particles often ensure a better resolution than the full immersed or non-immersed spheres. It is important to understand that the spatial resolution of a PNJ is not the same as the spatial resolution of an image that is obtained by means of this PNJ. Hereafter, the term “PNJ spatial
Y.E. Geints, A.A. Zemlyanov / Journal of Quantitative Spectroscopy & Radiative Transfer 200 (2017) 32–37
resolution” connotes exactly the transversal spatial size (usually FWHM) of a PNJ. Notice that the most studies devoted to the control over the photonic jets usually consider single isolated particles in a free environment. The optical field distribution near an isolated particle depends only on the characteristics of incident light and the microphysical parameters of the particle. In practice, the manipulation of localized optical fluxes induced by individual isolated microparticles is a challenging task because of the technical problems related to the spatial fixation of microparticles on the object surface. In view of this, the formation of a PNJ cluster via the deposition of an ordered array of microparticles (microassembly) on a transparent substrate, e.g., on a flexible polymer film [22], seems currently the most promising way to solve this problem. This secures the microparticles in the matrix, which then can be moved to the desired spatial position of the object for PNJs production. A conventional way of a PNJ cluster creation is the use of a particle self-assembly that is formed during particle self-arrangement on an object surface or on a special substrate. Generally, this process is related to the precipitation and evaporation of microparticles from a colloidal solution deposited on a substrate, with subsequent formation of a single- or multilayer matrix of closely or loosely packed particles [25,31]. Such silicone films (PDMA, PMMA) with embedded glass microspheres [25] are used, e.g., in the ultrahigh resolution microscopy for subwavelength and even subdiffraction light focusing [22]. Sphere-containing silicone films are unique because the embedded microparticles can be fixed in any spatial configuration, but not only in the form of closely packed particles. Then a flexible polymer matrix can be carried to any surface and optically irradiated. Therefore, the study of cooperative interference effects during formation of PNG from ensembles of differently arranged particles becomes urgent. It is evident that the use of a group of microparticles, instead of an individual particle, embedded into a fixing matrix or placed on a substrate can significantly influence the parameters of individual nanojets due to the interference of optical fields during the radiation diffraction at neighboring microparticles and interaction between the scattered field and the substrate. This problem was studied previously in several works [32–35]. The effect of absorbing substrate on the optical field scattered at the ensemble of seven microspheres was studied theoretically in [33], where it was found that both the substrate and neighboring particles reduce up to several times the optical field intensity in the focal regions in comparison with an isolated sphere in vacuum. Elongation of a PNJ and the shift of its maximum toward the surface of parent spheres arranged into a hexagonal cluster were observed in [34]. Recently [35], we considered the near-field scattering region of a single-layer cluster of identical microspheres embedded into a transparent matrix. We found that the collective interference effects occurring during PNJ array formation manifest themselves mainly in the pulsations of photonic jet intensity and its length when sphere cluster period is varied. In this work, we continue studying the near-field diffraction patterns upon PNJ formation by a single-layer assembly of identical microspheres embedded in the matrix with refractive index contrast. To this end, we numerically solve the set of Maxwell differential equations by virtue of the finite-difference time-domain (FDTD) method. Spatial and amplitude parameters of the produced PNJs are examined versus the inter-particle gap, particle size, and optical contrast of the immersion matrix. In contrast to our previous study [35], in this work we focused on the effect of the immersion depth of a particle on PNJ parameters. We show that depending on the microassembly configuration, PNJ quality can be significantly improved; in particular, one can achieve the sub-diffraction jet resolution (< λ/5). Worth noting, that similar effect was previously reported in [26] for a half-immersed isolated microsphere
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Fig. 1. (a) Microassembly of spheres immersed in the film; (b) optical field intensity enhancement B in a vicinity of glass spheres with D = 2λ immersed in PDMA film. The arrows show light incidence direction.
exposed to an optical radiation. However, in a microsphere assembly these super-resolution PNJs may not be observed because of strong coupling of the optical fields if the gap between individual spheres becomes much shorter than the incident radiation wavelength. 2. Numerical simulation technique We considered the classical spheres self-assembly in the form of a hexagonal single-layer cluster of spherical particles (Fig. 1(a)). Identical spheres with the diameter D are deposited on a plane grid with the lattice period d and constitute close (d = D) or loosely (d > D) spheres packing. Particle assembly is immersed in a transparent dielectric matrix (film). The immersion depth f can vary within the limits 0 ≤ f ≤ D. The near-field diffraction region of the microsphere assembly was simulated in the 3D domain bounded by perfectly matched layers, using commercial FDTD software package (Lumerical Solutions Inc.), which implements the numerical integration of Maxwell equations for the electromagnetic field components. The accuracy of numerical solution was improved by the adaptive mesh grid, which was denser in the regions of sharp permittivity gradients (particles rim). The total number of grid nodes (Yee cells) was about 108 , in which case spatial and temporal grid steps were 2 nm and 0.06 fs, respectively. A dielectric permittivity was specified inside the calculation domain according to the geometry of spheres arrangement. The spheres were assumed to be made of optical glass with the refractive index ns and no absorption in the optical wavelength range. The assemblies of spheres are immersed in a non-absorbing silicone (PDMA) film with the refractive index nm = 1.46 surrounded by air (n0 = 1). To minimize edge effects, a three-line cluster of nine spheres is used to simulate the hexagonal geometry; the PNJ parameters are studied at the cluster center. At the initial instant a monochromatic optical wave with the wavelength λ = 0.532 μm is switched-on at the lower boundary of the domain. The spatial form of the wave is a plane linear polarized wave with the electric field vector directed, for definiteness, along the x-axis. Optical wave propagates through an assembly of particles in the positive z-direction and forms time-dependent diffraction pattern in the near-field, which is then averaged over
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Fig. 2. Optical intensity profiles near isolated microspheres (D = 2λ) (а) in air and (b) immersed in polymer film (f/D = 75%).
the time interval of 2 ps. This time is sufficient to account for all transition processes in spheres and matrix. The resulted spatial distributions of the relative field intensity B(r ) = |E(r )|2 /E02 (E0 is the incident wave amplitude) is analyzed in order to reveal localized regions of enhanced intensity (PNJs) and determine their dimensional and amplitude characteristics. 3. Simulation results and discussion Fig. 1(b) exemplifies the photonic nanojets formed by an assembly of semi-immersed in the polymer film spherical particles with ns = 1.6 and D = 2λ. Spatial distribution of the relative optical intensity B in the xz-plane is shown on the gray scale. The formation of compact high-intensity PNJs is clearly seen near the top shadow surfaces of the spheres. As will be shown below, jet parameters generally depend on the spatial arrangement of particles, their optical contrast and immersion depth into the film. For a quantitative characterization of a PNJ, we use two size parameters, namely, the length L and the width R (shown in Fig. 1(b)). These parameters are calculated as the full-width at halfmaximum values (FWHM). The PNJ magnitude is characterized by the (relative) peak intensity Bmax of the optical field. First, it is instructive to demonstrate how the immersion of a sphere inside the optically contrasted film affects the near-field spatial structure of a PNJ produced. Fig. 2(a) and (b) show the intensity distribution for a spherical particle placed either on top of the film or semi-immersed inside it. It is evident that the shape, intensity, and size of the PNJ differ and depend on the environments. Thus, for a non-immersed microparticle (totally in air), the intensity maximum is located inside the sphere near its shadow hemisphere (Fig. 2(a)). Here, the PNJ is formed through the field leakage from the internal focus toward the exterior that leads to extremely short (several fractions of the wavelength) but highintensity jet creation. If a sphere is semi-immersed in the film with the refractive index higher than that of air (Fig. 2(b)), the spatial structure of the PNJ changes dramatically. In this case, the PNJ is visually detached from the particle rim and possesses lower field amplitude but markedly longer extent (L ∼ 2λ). This is the direct consequence of optical contrast decrease for a wave entering the particle from the immersion layer. In its turn, lower optical contrast ns /nm decreases the numerical aperture of the spherical particle-lens that
Fig. 3. (a) PNJ intensity Bmax , width Rx , and (b) PNJ length L versus intersphere gap g in differently immersed microassemblies.
elongates its focal waist similarly as jet formation in gradient-index spheres [19,36]. Consider now the parameters of PNJs produced by different microassembly configurations. Fig. 3(a) and (b) show the effect of the assembly period d on the key PNJ parameters. For convenience, hereafter we introduce the intersphere gap parameter g that is defined in terms of particle diameter, g = (d − D). So, g = 0 corresponds to the close packed spheres, whereas g ≈ D describes the loosely packed assembly. In these figures, differently filled points depict sphere microassemblies with different immersion depths f expressed in per cents of sphere diameter. Interestingly, PNJs demonstrate high variability of their parameters depending on the intersphere gap. This is caused by the cooperative interference effects [35] of the neighboring particles, which are less pronounced for PNJ amplitude while jet length and width can differ by several times even at small changes of the gap. In general, the assemblies with fully immersed microspheres (f/D = 100%) produce the PNJs with the parameters more stable against variations in g. Furthermore, in this case closely packed microspheres form PNJs with close intensities and widths but much smaller lengths than loosely packed particles. In contrast, the assemblies of half- and non-immersed spheres are very sensitive to the variations of the cluster period. Here, several particle configurations can produce PNJ with extremal characteristics. Indeed, the analysis of Fig. 3(a) shows that PNJ peak intensity becomes higher (Bmax ≈ 50 atf = 0) at moderate values of the intersphere gap (g ≈ 0.4) than in the cases of close packing (Bmax = 21)
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Fig. 5. Transverse intensity profiles B(x, y) near half-immersed spheres (f = D/3) at different distances z from particle surface.
Fig. 4. (a) PNJ intensity and width as functions of the immersion depth for g = 0 (close packing) and g = 0.25D (loose packing); (b) near-field intensity profile for different immersion depths f (g = 0.25D).
Fig. 6. Geometrical rays traces in (a) fully- and (b) semi-immersed spherical particle. Rays propagate from left to right. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
or clearly loose packing (Bmax = 39). If the gap between half immersed particles equals to 1/4 or 3/4 of their diameter then the produced PNJ is extremely narrow. In this situation, PNJ size along the x-axis exhibits super-resolution, Rx = 0.182λ ≈ λ/5.3, because this more than twofold exceeds the Abbe diffraction limit for the conventional far-field optics focusing ( ∼ λ/2). Fig. 4(a) shows the effect of microsphere immersion depth on the parameters of PNJ array. Apparently, the subdiffraction PNJ is attainable only from semi-immersed (25% < f/D ≤ 50%) loosely packed particles, Besides, PNJ intensity is also higher in this case. This fact can be of practical interest, e.g., for sphere-assisted superresolution microscopy [37], where a transparent microsphere is used as a final focusing tool with subdiffraction texturing of an object. According to our calculations, an attempt to improve the resolution of such optical system by means of spheres immersion in optical denser medium will even worsen the spatial resolution until the microspheres are fully immersed. It is worth to emphasize that the desired resolution improvement could be attainable only in the case of semi-immersed microspheres. The longitudinal structure of the PNJs from a microassembly of loosely packed spheres is shown in Fig. 4(b) for several depths of particle immersion. It is clear that PNJ intensity profile has two maxima, which are z-separated by about 1.5λ. As the immersion depth decreases, the intensity of the maxima increases and they approach the mother particle surface. When the particles are halfimmersed (f < D/2) then the first (main) PNJ maximum almost coincides with the microsphere surface. Secondary PNJ intensity
maximum continues moving toward the particle as f decreases, which results in total PNJ length shortening (Fig. 3(b)). The lateral field structure of these maxima also differs. Consider Fig. 5(a) and (b) which show the relative optical intensity distribution B(x, y) in two cross-sections downstream the photonic flux. Spatial positions of these cross-sections are chosen so that they contain the first and second field maxima. The distance z between the sections and the particle surface is indicated in the figures. From Fig. 5(b) it is clear that far from the particle surface PNJ exhibits transverse ellipticity which is due to the linear polarization of optical wave incident on sphere assembly. PNJ in the second field maximum has the width Ry along y-axis (parallel to the polarization plane) which is 1.5 times larger than the analogous scale along x-axis, Rx = 190 nm ≈ λ/3.32. Intensity profile of the main PNJ maximum (Fig. 5(a)) has the shape of a distorted ellipse that indicates the presence of strong aberrations upon wave focusing. The size of the nanojet takes the minimum, Rx = 95 nm, close to the particle surface that is less than 1/5 of laser wavelength and is far below the diffraction limit. Worth noticing, that this PNJ structure is typical only for the case of semi-immersed spheres and is not observed for fully- or non-immersed particles. Strong wave aberrations upon light focusing by a semiimmersed sphere can be readily understood from the geometrical optics. Consider Fig. 6(a) and (b), which show geometrical ray traces passing through a sphere being fully- and semi-embedded in an optically dense block. Ray traces are rendered by COMSOL Multiphysics software package. As seen, for geometrical rays, the outer
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lations for minimal PNJ width Rx are plotted in Fig. 7(a) and (b) as the color 2D-maps with intersphere gap g and immersion depth f/D as the dependent variables. From these figures it follows that both microassemblies demonstrate the ranges of their parameters where generated PNJs are extremely narrow (R ≈ λ/5). If bigger particles with the diameter D = 4λ (Fig. 7(a)) are used in the assembly, then the superresolution PNJ occurs only for half-immersed loosely packed particles at the lattice period, d > 1.1 D. The assembly of optically denser microspheres (Fig. 7(b)) in most cases produces the PNJs with the subdiffraction resolution in terms of Abbe diffraction limit ( ∼ λ/2). Here, the narrowest photonic nanojets are formed by semi- or fully-immersed microspheres at any intersphere gap excluding only the close packing of particles. 4. Conclusions We considered the spatial structure of the near-field scattering of an optical wave at ordered hexagonal microassemblies of non-absorbing dielectric microspheres immersed in a transparent matrix. By virtue of the finite-difference time-domain technique, the evolution of dimensional and amplitude parameters of produced photonic nanojets array is studied by varying assembly immersion depth. The main conclusion of our research is that the reported earlier PNJ super-resolution ( ∼ λ/5) from individual half-immersed microsphere [26,30] can be also observed in sphere microassemblies with different geometry and optical properties. Moreover, this effect is apparent not only for half-immersed spheres but also for semi- and in some cases for fully-immersed particles, depending on the microassembly configuration. Principal condition for ultra-narrow PNJ formation is loose packing of spheres when the cooperative interference within the microassembly becomes weak. References
Fig. 7. PNJ width from sphere assemblies with (a) D = 4λ, ns = 1.6; (b) D = 2λ, ns = 2.1. Digits are presented for better reading.
surface of the fully-immersed particle comprises the boundary of constant optical contrast, ns /nm . This causes geometrically similar refraction of peripheral and central rays on the sphere and results in the formation of one external focus, which is markedly stretched along z-axis due to the spherical aberrations. In contrast, for a semi-immersed sphere, several parts of its illuminated face (shown by bold red lines in Fig. 6(b)) possess higher optical contrast ns /n0 than the remaining surface due to lower air refractivity. This causes stronger rays refractions in these regions and produces the additional zone of rays focusing located close to particle rim. This additional focus is sharper, thus having smaller focal waist. As a result, in this semi-embedded sphere configuration the longitudinal PNJ shape becomes clearly bimodal (see Fig. 4(b)). The robustness of a PNJ with spatial super-resolution is of importance, in view of possible structural and optical parameters of microsphere assembly variations. To investigate this point, we calculated PNJ key parameters produced by sphere microassemblies either with doubled diameter or with higher refractive index, ns = 2.1. The latter case simulates a cluster of spheres made, e.g., of high density barium titanate glass (BTG). The results of our calcu-
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