Imaging of sub-wavelength structures radiating coherently near microspheres Free

In the past few years, the imaging of nanoscale objects by dielectric microspheres has emerged as a surprisingly simple way of overcoming the classical diffraction limit of about λ/2, where λ is the wavelength in free space. In this method, a dielectric microsphere is placed in contact with the investigated object and its virtual image is observed using a conventional microscope, see Fig. 1(a). Microsphere imaging of various nanostructures^1–5 and biological objects^6–8 demonstrated a clear resolution advantage over conventional microscopy. Unlike the recently advanced super-resolution fluorescence techniques, where dye-molecules act as incoherent point-sources,⁹ the microsphere-assisted imaging is often applied to nanoscale structures, which can support electromagnetic modes.

An open problem of microsphere imaging is the quantification of its resolution. Mostly visual analysis of experimental images resulted in a broad range of super-resolution claims spanning the range from λ/6 to λ/17.^1,3,5–8 A more rigorous resolution analysis of metal dimers and bowties based on the convolution of their shape and the point spread function led to a resolution value ∼λ/7.^10–12 This value is still higher than the classical upper limit λ/(2 n) ≈ λ/4 in the presence of the high-index (n ≈ 2.0) sphere.

Attempts to explain the nature of the super-resolution in microsphere imaging have been made. It was proposed that the super-resolution is related to the stronger than usual focusing of light by spheres with diameters on the order of several wavelengths.¹ However, the width of photonic jets is slightly less than λ/2 in small (R ≲ 2λ, where R is the sphere radius) microspheres only.^13,14 The enhancement of focusing, therefore, seems to be insufficient to explain the super-resolution which is typically observed in larger spheres. Furthermore, resolution is ultimately related to the re-radiation of light by the object, rather than to its illumination. The modeling of a dipole radiating near a microsphere using the Mie expansion doubted the super-focusing as a plausible explanation.¹⁵ High-Q whispering gallery mode (WGM) resonances in the microsphere can slightly improve the resolution,¹⁶ but their selective excitation is unlikely in the experiments with broad band illumination. A finite-element study of imaging a dipole radiating near a microsphere resulted in the 0.24λ–0.3λ resolution range.¹⁷ The transformation of the evanescent waves to propagating waves was also discussed⁶ as a possible super-resolution mechanism.

In this work, we consider the imaging of closely spaced nanoscale components that form hybrid electromagnetic states (modes), such as in coupled metallic particles (dimers, bowties^18,19) or cavities.²⁰ An excited mode radiates like an antenna²¹ and the resultant image may not directly correspond to the geometry of the object. This also makes inapplicable the common assumption of independent (incoherent) emission by different object's points. Since a system of interacting particles forms several states, the images may differ depending on specific excitation conditions. We compare coherent and incoherent imaging with and without microspheres based on the idea that in general cases the object can be modelled as a current distribution, similar to that in an antenna. We show that the excitation of the anti-symmetric mode leads to the resolution of the sub-wavelength structure, which is not resolved in the case of symmetric mode or incoherent imaging. The selective excitation can also be used for precise localization of coupled particles.

The two-dimensional (2D) model of imaging is shown in Fig. 1(b), where the cylinder plays the role of the sphere. The imaged object is a current source located below the cylinder with refractive index n_c and radius R. The refractive index of the background is n_b. The lens (microscope objective) collects the generated fields in the far-field region and focuses them to form an image on the detector. The current is assumed to be in the x-y plane and can form charges on the boundaries. The magnetic field has only one component H_z(x, y) along z. All complex fields and currents have e^−iωt time dependence.

The field H_z(x, y) for ρ > ρ₀, where ρ₀ is the distance from the origin to the current distribution, can be represented using the standard expansion in the cylindrical coordinates ${ρ,φ}$⁠. For large distances ρ → ∞, the asymptotic form of the Hankel functions $Hn(1)$ gives the far field $f(φ)$ distribution,

where k_b = n_bω/c. The coefficients F_n in Eq. (1) are found by expanding the current into the angular functions and applying the boundary conditions.²² This accounts for all field components (evanescent and propagating) produced by the current source.

The image created by the lens is represented by the intensity distribution formed by the far fields that are formally backpropagated to the virtual image plane. In our 2D geometry, we use the intensity of the magnetic, rather than of the electric, field. The image is defined as the intensity distribution along y at a fixed position x of the virtual image plane. The plane-wave component of the far field $H̃z(h)=f(φ)/(2πkb cos φ)$⁠, where $h=kb sin φ$ is the y-projection of its wavevector, gives

and $h2+g2=kb2$⁠. The integration limits in (2) assume the maximal numerical aperture NA = n_b. The magnetic field H_z(x, y) created by the unit-magnitude point-like current at {x₀, y₀} oriented along $φ$

defines the response function R of the system

calculated using Eqs. (1) and (2). The absence of ρ-oriented current simplifies the analytical solution. The image intensity for an arbitrary distribution of coherent current $Jφ(x,y)$ or incoherent sources $Gφ(x,y)$ becomes

We assume that the current is distributed along an arc of length L = ρ₀θ of a circle with radius ρ₀ subtending an angle θ. The current can be written as

We consider antisymmetric and symmetric functions $ja,s(φ)$ with respect to the center at the angle π

where $ϕ=φ−π$ and $|ϕ|<θ/2; jφa,s(φ)=0$ for $|ϕ|>θ/2$⁠. Since θ is very small in the examples below, the current is almost along the y axis. The distance between the current extrema is L/2. For the incoherent case, we take

where g₀ defines the intensity of the incoherent source.

Current distributions (8) are chosen based on their analogy with the hybrid modes formed by two interacting resonant particles, see Fig. 2. The charges oscillate so that the currents in each particle are either in phase (symmetric mode) or out of phase (antisymmetric mode).

We consider the cylinder with n_c = 1.4 located in free space n_b = 1. We compare the images obtained with the cylinder (n_c = 1.4) and without it (n_c = 1). In the actual simulations of the case n_c = 1, we take n_c = 1.005. The size of the cylinder is always kR = 20.1 (R/λ = 3.2). At this value, there is no excitation of WGMs.^23,24

We start with the case of point-like source (3), see Fig. 3. The source is just below the cylinder at kx₀ = −21.1 and has a small displacement ky₀ = π/2 from the optical x axis. In the absence of the cylinder, Fig. 3(b), the image plane corresponds to the actual location of the object. In the presence of the cylinder, Fig. 3(a), the virtual image can be obtained in a rather wide range of x significantly below the actual location of the object. At x/λ = −7.5, the virtual image is displaced by y/λ ≈ 0.516 and broadened. The position x/λ = −7.5 is close to the maximum of the image intensity along the radial line passing the object. The calculated magnification M and the full width at half maximum (FWHM) Δ with and without the cylinder are extracted from the data in Fig. 3 and summarized in Table I. The cylinder magnifies the object and broadens it proportionately. Magnification increases if the image is taken at a lower virtual image plane but this broadens the image further. Thus, the cylinder does not increase the resolution.

We now investigate the images produced by coherent (8) and incoherent (9) sources with L/λ = 1/2, 1, see Fig. 4 for n_c = 1.4 and Fig. 5 for n_c = 1.0. Comparing frames (a1, a2) with (b1, b2) in Fig. 4, we see significant differences. Frames (a1, a2) show clearly two peaks while there is only one peak in (b1, b2). The minimum between the peaks in (a1, a2) is formed due to the destructive interference of radiation from the two particles. For Fig. 4(a1), the distance between the peaks of $jφa(φ)$ is λ/4 which is beyond the diffraction limit λ/(2n_c) = λ/2.8. Incoherent source with the same size produces only one peak, see Fig. 4(c1). Without the cylinder, Fig. 5, the excitation of the antisymmetric mode, frames (a1, a2), also provides the information about the structure of the object. However, the image is observable only at very small intervals of x near the object's plane. Such small focusing depth complicates experimental observation of this effect without using contact microlenses.

Figure 6 shows the image intensity at x/λ = −7.5 for different types and sizes of the sources. The unitless numbers used for the intensity allow one to analyze not only the spatial distributions but also to compare the intensities. Although the image of the antisymmetric mode in Fig. 6(a) has two peaks, similar to the real physical object, the image does not change with the size of the object. The two peaks are formed as a result of interference and are not related to the object's physical dimensions in the limit L ≪ λ. The intensity decreases very rapidly with decreasing size L. The decrease of the image intensity is not so rapid for the symmetric mode or incoherent source, see Figs. 6(b) and 6(c). For L/λ ≈ 1 in the case of incoherent source, see Fig. 6(c), there is an indication of the object structure in agreement with the Abbe's two-point resolution limit.

In conclusion, it has been demonstrated that the optical images of a sub-wavelength object depend strongly on the excitation of its electromagnetic modes. The modes were modeled using spatial current distributions as in two coupled particles. It was shown that the image of the antisymmetric mode gives two peaks that resemble the structure of the object. The image does not change with object size in the strongly sub-wavelength limit but the intensity of the image decreases rapidly with decreasing size. Although such images do not accurately reproduce the shape of the objects, they can render their precise localization and affect the quantification of resolution in the recent experiments with nanoplasmonic structures. Experimentally, such imaging can be realized by using a narrow-band illumination tuned in resonance with the antisymmetric mode. The antisymmetric currents can also be created, although less efficiently, using incoherent band illumination and off-resonant excitation. The zero-intensity of the central point can lead to the overestimation of resolution if the results are interpreted within the framework of incoherent imaging. In contrast, imaging at the wavelength of the symmetric mode significantly reduces the resolution compared to incoherent illumination. A dielectric microsphere in contact with the object forms the magnified image in a wide range of the focal plane positions. Furthermore, the microsphere may facilitate the excitation of the modes due to light focusing on the subwavelength scale. We believe that the coherent effects play some role in the super-resolution of nanoplasmonic structures observed in the recent experiments.

The work at the UNN was supported in part by the Ministry of Education and Science of the Russian Federation through Agreement No. 02.B.49.21.0003. The work at UNCC was supported by U.S. Army Research Office through Dr. J. T. Prater under Contract No. W911NF-09-1-0450 and by Center for Metamaterials, an NSF I/U CRC, Award No. 1068050.