Superenhanced three-dimensional confinement of light by compound metal-dielectric microspheres

Yulong Ku; Cuifang Kuang; Xiang Hao; Yi Xue; Haifeng Li; Xu Liu

doi:10.1364/OE.20.016981

1. Introduction

Sub-wavelength dimensional confinement of light is widely investigated because of its great potential value in various applications such as nanopatterning [1,2], laser cleaning [3,4], fluorescence correlation spectroscopy [5–7], tissue engineering [8] and optical data storage [9]. In general, this strong concentration effect can be achieved through the electromagnetic resonances induced by metallic structures such as gratings, pinholes, tip or nanoparticles [10–13]. However, losses in metals and technically challenging nanofabrication processes may be a handicap in their way to application. Strong confinement of light obtained through the localized surface plasmon of nanoparticles is a widely spread method, while the nanoscale particles are too tiny to localize accurately.

In the search for simple dielectric structures to concentrate light at the nanoscale, dielectric spheres of micrometer dimensions have been recently investigated [14–19]. The so called photonic nanojets, highly localized wave fields emerging directly behind dielectric microspheres, are widely discussed because of its light-lossless property. Numerical studies have been done on the photonic nanojets, most of them focusing on the influence caused by factors such as size of the sphere, wavelength of the illuminating beams, material’s refractive index and the refractive index of the surrounding [20–22]. And further concentrated of light have been studied through a set of bi-spheres [23]. However, three-dimensional sub-wavelength confinement could hardly be achieved due to its long extension along the optical axis. Recently, the cylindrical vector beams were introduced as the illuminate wave and its influence on property of the photonic nanojets was studied [15,16]. The result shows that the axial size of the jets can be reduced when the microspheres are illuminated by radially polarized beams. But it is also unsuitable for the applications requiring very high transversal and longitudinal resolutions.

In this letter, we introduce a technique of a compound set of metal-dielectric microspheres illuminated by focused radially polarized beams to concentrate light at three dimensional sub-wavelength scale. A micrometer silver sphere is set in a photonic nanojets region which induced by radially polarized beams. This illuminating condition is more efficient to excite the surface plasmon, which demonstrates longitudinal concentration waves of electrons formed at silver microsphere’s surface. Then they generate electromagnetically amplified local fields, the so called hotspots which can be created much smaller than a diffraction-limited size in principle. With micrometer scale of silver sphere, the challenge of nanofabrication processes can be reduced remarkably. Result shows that light can be confined to an effective volume of 0.095 (λ/n)³ with almost 6-fold improvement in peak intensity compared to simple photonic nanojets. And full width at half maximum (FWHM) could be optimized to 73nm (~0.11λ) in propagation direction and 146nm (~0.23λ) in transversal direction separately.

2. Confinement of light via compound metal-dielectric microspheres

The Lorentz-Mie theory is generally used to find the electromagnetic field around a dielectric sphere, where calculations of far-field scattering properties of spheres are more routinely performed. The total field outside the sphere is considered as the sum of the incident and scattered fields [24]. By expanding an incident plane wave of unity amplitude in spherical harmonics as:

E_{i n c} (r) = \sum_{n = 1}^{\infty} i^{n} {(2 n + 1) / [n (n + 1)]} [M_{o \ln}^{(1)} (r) - i N_{e \ln}^{(1)} (r)]

where M and N are the vector spherical harmonics. The expansion for the scattered field, valid for all points outside of the sphere, is given by:

E_{scat} (r) = \sum_{n = 1}^{\infty} i^{n} {(2 n + 1) / [n (n + 1)]} [i a_{n} N_{e \ln}^{(3)} (r) - b_{n} M_{o \ln}^{(3)} (r)]

where a_n and b_n are the scattering coefficients and the superscripts appended to M and N denote the kind of spherical Bessel function. Then, the total external electric field induced by the plane wave is given by

E_{i n c} (r)

+

E_{s c a t} (r)

.

Although such systems can be analyzed with generalized Mie theory, In the case of present, a more comprehensive study that incorporates the effects of surface plasmon requires a more computationally study. Considering that the Finite Difference Time Domain (FDTD) method based on vector electromagnetic wave theory can accurately demonstrate light propagation in media, the calculations in the present case is simulated using the FDTD Solution trial version by the Lumerical Solution, Inc.

In this simulation, the illuminating beam acts as a major point in confining light. And radial polarized illumination has attracted special interest due to its unique optical properties in the focal region. A method for efficiently obtaining essentially pure radial polarized beam directly from a laser is presented [25]. It is based on the selection and coherent summation of two orthogonally polarized TEM₀₁ modes. The scalar field distribution of them are considered as orienting along the x and y axis, respectively. In cylindrical coordinate, the field distributions of the TEM₀₁ (x) and TEM₀₁ (y) modes can be expressed by:

E_{x} (r, θ) = E_{0} \sqrt{ρ} \exp (- ρ / 2) \cos (θ)

E_{y} (r, θ) = E_{0} \sqrt{ρ} \exp (- ρ / 2) \sin (θ)

where r and θ are the cylindrical coordinates, E₀ is the magnitude of the field,

ρ = 2 r^{2} / w^{2}

with ω as the waist of the Gaussian beam. With orthogonal linear polarizations, the coherent summation of such TEM₀₁ (x) and TEM₀₁(y) modes, leads to the formation of radial polarized mode, whose vectorial field distributions have the form:

E_{r} (r, θ) = \overset{\land}{x} E_{(x)} (r, θ) + \overset{\land}{y} E_{(y)} (r, θ) = \overset{\land}{r} E_{0} \sqrt{ρ} \exp (- ρ / 2)

Radially polarized beams can be seen as p-polarized wave in all directions. Surface plasmon mode in silver sphere placed behind the dielectric microsphere is excited more efficiently by the photonic nanojets which has abundant scattered high frequency composition. Then they propagate along the surface of silver sphere, and a three dimensional sub-wavelength hotspot with high peak intensity is generated.

2.1 Three dimensional sub-wavelength confinement of light

Figure 1(a) visualizes the FDTD-computed scattered electric-field (E-field) envelope for a homogeneous 6µm diameter microsphere (n = 1.6) in solution with refractive index of 1.33, which is illuminated by linear polarized beams at λ = 635nm. This visualization is the cross section in the center of the microsphere and normalized relative to the incident wave amplitude. The black circle represents the dielectric microsphere section. A so called photonic nanojet beam which has a narrow transversal extent but a large longitudinal extent is present behind the sphere in Fig. 1(a). The three dimensional extension of the nanojet beams strongly depend on refractive index’s contrast n/n_s between the microsphere and the surrounding medium. Although the longitudinal extend can be reduced by focused radial polarized beams, the dimension of photonic jets along the transversal axis is particularly large for low index contrast. It is unsuitable for applications such as nanopatterning, requiring high transverse and longitudinal resolution in water-based solution or polymeric media. In addition, the microsphere focuses light far from the sphere and with low peak intensity. Multiple lobes of intense field will emerge between the sphere and the main beam which may spoil expected results.

Fig. 1 (a) The total electric field intensity map in linear scale for simple photonic nanojets (illuminated by linearly polarized waves at λ = 635nm). (b) The total electric field intensity map in linear scale for compound microspheres illuminated by radial polarized beams at λ = 635nm. In both cases, spheres are surrounded by solution with n = 1.33. The black circle represents the 6µm dielectric microsphere (n = 1.6) section, and the white circle represents the 1.4µm silver microsphere section. (c) and (e) display intensity distributions for dielectric microsphere along transverse axis (x) at beam waist and propagation axis (z) corresponding to (a), and so as to (d) and (f) corresponding to (b).

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Figure 1(b) visualizes the scattered E-field envelope for compound metal-dielectric microspheres of the same diameter as specified in Fig. 1(a) except that a silver microsphere ( $ε$ = −17.3995 + i2.2572) is added behind it, wherein the black circle represents the 6µm dielectric microsphere section, and the white circle represents the 1.4µm silver microsphere section. The incident wave is replaced by a focused radially polarized beam ( $w_{x y}$ = 15µm) with the same wavelength specified in Fig. 1(a). In the present condition, two spheres are touching. And Fig. 1(d) and 1(f) are the characteristic corresponding to Fig. 1(b). Figure 1(d) displays the intensity distribution for compound microspheres along a transverse axis (x) at the waist of the focused light field and Fig. 1(f) displays intensity distribution for compound microspheres along propagation axis (z). To characterize the spot, the intensity enhancement of the electromagnetic field has been displayed on two axes of interest: the propagation axis z and a transverse axis x at beam waist. The transversal and longitudinal waists are defined at $I_{\max} / e^{2}$ and are denoted $w_{x y}$ and $w_{z}$ respectively. Result shows that light is further confined by the compound spheres both longitudinally and transversally with $w_{x y}$ = 230nm and $w_{z}$ = 280nm. If the effective volume behind the sphere is defined by $π^{3 / 2} {(w_{x y} / 2)}^{2} w_{z} / 2$ [20], a remarkably small volume as 0.095 (λ/n)³ can be achieved.

To compare with the simple photonic nanojets, detailed calculations have been done to characterize the jet beams. Figure 1(c) and 1(e) displays intensity distribution for dielectric microsphere along transverse axis (x) for $z (I_{\max})$ and propagation axis (z) corresponding to Fig. 1(a).

2.2 Compare with conventional methods

Table 1 compares characteristics width of light field corresponding to single dielectric microsphere and that of compound metal-dielectric microspheres. It is remarkable that the light field is further confined by the compound spheres both transversally and longitudinally. It is apparent that the transversal modify of the beam is based on the surface plasmon modes. The maximum intensity has been enhanced and moved toward the sphere surface makes the longitudinal modification of the beam seems spectacular. It is noticeable that the effective volume is reduced to as small as 0.095 (λ/n)³, with almost 6-fold improvement in peak intensity. Although the light field of photonic nanojets can be further confined by introducing focused Gaussian beams and smaller diameter (effective volume as 0.6 (λ/n)³) [20] or radially polarized beams illuminating [18], the effective volume can’t meet the needs of special applications. It is well known that simple photonic nanojets capacity of strong concentration of light comes from the high index contrast, the transversal and longitudinal waists can be reduced by introducing higher index contrasts. However, the effective volume can’t be as small as that achieved by compound metal-dielectric microspheres. It must be pointed out that the ratio of index contrast has to be lower than 1.4 because for higher contrasts, the electromagnetic field is fully confined inside the dielectric sphere. For applications such as nanopatterning, requiring high transverse and longitudinal resolution in water-based solution or polymeric media, these methods may be unsuitable. As to the compound metal-dielectric microspheres, these problems can be avoided.

Table 1. Summary of the characteristics width corresponding to different methods

View Table

3. Discussion

A series of calculations have been done to make this phenomenon clearer. The silver sphere is replaced by spheres in different materials such as silicon, silica, gold and others. High concentrated hotspot exists only if metallic microspheres are illuminated. Thus a hypothesis is made that it is the coupling of the nanojet to surface plasmons on the silver surface that give rise to the hotspot with reduced transverse and longitudinal field localization. In this condition, it is obvious that strong concentration of light by compound metal-dielectric microspheres is due to the interplay of two contributions. The well-known surface plasmon modes provide the transversal and longitudinal confinement of light. And the photonic nanojets of dielectric microsphere provide an ideal illuminating environment. To further understand the phenomenon and optimize the focused field, some numerical experiments are undertaken. In order to highlights the nuances of focused light fields, maps are sketched with changed color-bar.

3.1 Influence of silver sphere’s size

First, the diameter of silver sphere is changed from 1µm to 1.8µm, and characteristics of the concentration field are analyzed. Figure 2(a) and 2(b) display the electric-field intensity maps, which correspond to the same set in Fig. 1(b) except for different sizes of spheres. And Fig. 2(a) represents the compound microspheres with a 6µm dielectric sphere (blue circle) and a 1µm silver sphere (white circle), the same as to Fig. 2(b) with a 1.8µm silver sphere. In the present case, the compound microspheres are touching. Compared with the Fig. 1(b), we can conclude that diameter of silver sphere is a critical point to achieve strong concentration of light field. Too smaller or bigger diameter of the silver sphere will make the confinement worse. Figure 2(a) shows that when the sphere is too small, most of the energy slides aside the sphere so that light of the photonic jet used for inducing surface plasmon modes is little. Meanwhile, radiative loss of plasmons on the surface of silver sphere becomes more serious because of the smaller radius of curvature. When the silver sphere exceeds a critical size as illustrated in Fig. 2(b), most of the light emerging from the dielectric sphere is blocked by silver sphere, another hotspot occurs in front of silver sphere which is of minor interest. And intensity of focused field is also reduced for more severe propagation losses of plasmons. In addition, the peak intensity of concentration light field on rear side of silver sphere is 3 times smaller than that in optimal case.

Fig. 2 Total electric field intensity map in linear scale for compound microspheres of different sizes illuminated by radial polarized beams at λ = 635nm. And (a) represents the compound microspheres with a 6µm dielectric sphere (blue circle) and a 1µm silver sphere (white circle), so as to (b) with a 1.8µm silver sphere (white circle).

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Figure 3(a) displays intensity distribution for compound microspheres with different diameters of silver spheres along transverse axis (x) at the waist of the focused light field. The green dash dot line represents the case of compound microspheres with 1µm diameter silver sphere. And so as to 1.2µm silver sphere in blue dot line, 1.4µm in black straight line, 1.5µm in red dash line and 1.8µm in magenta short dash line. It is obvious that peak intensity of the confined light field changes sharply as the diameter of silver sphere increases. Figure 3(b) illuminates that the transversal FWHM increases as bigger diameter of the silver sphere. This slow increasing was due to the modification of plasmons dispersion on surface of spheres with different radius of curvature [26]. In case of the fixed jets field and touching mode, an optimal size can be selected for the ideal confinement of light.

Fig. 3 (a) displays intensity distribution for compound microspheres with different silver spheres sizes along transverse axis (x). The green dash dot line represents the case of compound microspheres with 1µm diameter silver sphere. And so as to 1.2µm silver sphere in blue dot line, 1.4µm in black straight line, 1.5µm in red dash line, 1.8µm in magenta short dash line. (b) illuminates that the FWHM at the waist of focused light field increases as bigger sphere size.

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3.2 influence of silver sphere’s position

Another key point to achieve strong concentration of light is the position of the silver sphere. Figure 4 displays the total intensity distributions for compound microspheres with a similar structure specified in Fig. 2(a) except that the two spheres are separated. And illuminating condition is the same. Figure 4(a) represents the silver sphere is 0.4µm away from dielectric sphere’s surface, and so as to Fig. 4(b) with a 1µm distance. Compare with Fig. 2(a), the peak intensity increases as the distance added. If the dielectric microsphere can be seen as a high aperture lens, it is obvious that there is an optimal position to achieve ideal confinement of light with a fixed sphere size.

Fig. 4 Total electric field intensity map in linear scale for compound microspheres of different position illuminated by radial polarized beams at λ = 635nm. The blue circle represents the dielectric microsphere section, and the white circle represents the silver microsphere section. Diameter of dielectric microsphere is 6µm, and 1µm for the silver sphere. Refractive index of the dielectric microsphere is 1.6. (a) represents the silver sphere is 0.4µm away from dielectric sphere’s surface. (b) represents the silver sphere is 1µm away from dielectric sphere’s surface.

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For further understanding, a series of numerical experiments are undertaken. Figure 5(a) displays that peak intensity as a function of silver sphere’s position. It is obvious that curve of the peak intensity has an abnormal decrease when the silver sphere is about 0.4µm away from the dielectric microsphere’s surface, which is different with the case of a high aperture lens (in the ideal case of lens’s condition, the peak intensity should be increasing from rear side of lens to the focal plane without singularity). Therefore, the intensity distribution of single 6µm-diameter dielectric microsphere is under consideration. In this case, condition is the same with Fig. 4 except that silver sphere is removed. To make it clear, a comparison between intensity curve of positions and intensity distribution of photonic nanojet have been done. Figure 5(b) displays the intensity distribution of 6µm dielectric microsphere along propagation axis (z). It can be seen from the Fig. 5(b) that, the second peak intensity along propagation axis appears in the vicinity of 0.4µm after dielectric sphere’s surface. And the dimension of this peak intensity in axial direction is well corresponding to the indentation of the curve in Fig. 5(a). It is obvious that when the distance between two spheres is 0.4µm, this part of energy is reflected and induces another hotspot which is of minor interest. Consequently, the peak intensity of the confinement field is reduced.

Fig. 5 (a) Peak intensity as a function of silver sphere’s position. (b) Intensity distribution for 6µm dielectric microsphere along propagation axis (z).

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3.3 Different polarizations of incident beams

The illuminating beams act as a significant factor which determinates the property of the light field on the rear side of the microspheres. Apart from the radially polarized beams, other incident waves such as linearly polarized, circularly polarized and azimuthally polarized beams also can incite the light field emerging from rear side of compound spheres presents unique properties. In order to make out how different illumination conditions affect the properties, numerical studies have been done. Figure 6 visualize the intensity distribution maps, normalized relative to the incident wave amplitude, for compound metal-dielectric microspheres in solution with refractive index of 1.33. These xz visualizations are in the plane cutting through the center of the microspheres. And xy visualizations are in the plane cutting at 20nm away from the rear side of microspheres.

Fig. 6 Total electric field intensity map in linear scale for compound microspheres illuminated by different focused beams with 15µm waist at λ = 635nm. (a) and (b) represent the compound microspheres (6µm dielectric microsphere with n = 1.6 in black circle and 1.4µm silver sphere in white circle) illuminated by linearly polarized waves. (c) and (d) are similar with (a) (b), while illuminated by circularly polarized waves. (e) and (f) represent compound spheres are illuminated by azimuthally polarized beams. The silver sphere is 1µm in this part, and represented by black circle in (f).

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In our investigations, firstly we studied the case of linearly polarized beams illumination. It can be seen from Fig. 6(a) and 6(b) that, two hotspots appear on rear side of silver sphere along the transversal axis (x) which is the direction of the beam polarization. Only in this direction, could the incident beam be seen as p-polarization and generates the surface plasmon efficiently. Figure 6(c) and 6(d) display the condition of circularly polarized beams illumination. We have shown that a sub-wavelength doughnut shaped spot appears on the rear surface of the silver sphere. To our knowledge, a similar doughnut shaped spot can be achieved with a high NA lens or photonic nanojets illuminated by azimuthal polarized beams. Compared with the traditional methods mentioned in ref [18], the compound spheres significantly squeezes the dark spot in the intensity distribution, decreasing the peak-to-peak width by 32% from 0.41λ of the simple nanojets to only 0.28λ. What’s more, it also has a sharply profile of the intensity along transversal axis (x or y), FWHM of the each side of ring is smaller than 0.2λ. The interesting case of using azimuthally polarized beams for illumination was studied to further understand the phenomenon. Figure 6(e) and 6(f) show that, a well-known doughnut shaped spot which is similar to the depletion field in common STED-concept technique appear due to the photonic nanojets. However, it has nearly nothing to do with the silver sphere, while the light field around the silver microsphere is not affected except for a little blocked. It is due to the s-polarization in all directions in the case of this illumination, wherein the hotspot can’t be excited efficiently.

To illustrate the formations of different hotspots on rear side of silver spheres more clearly, a simplified model is used to explain it. Figure 7(a) is the model of silver sphere illuminated by linearly polarized beams. As can be seen from it, states of the polarization on both sides should be the same at a certain moment. However, if we consider the two points on surface of it P₁ and P₂, the vibration of electrons should have opposite directions. One is from the silver sphere to the media, and the other side is from the media to silver sphere. This opposite directions induces the plasmons interference on rear side of silver have a phase difference of π. In this way, two hotspots exist along the direction of beam polarization. The case of circularly polarized beams is similar to it. On the contrary, phases and states of polarization on both sides should be the same at a certain moment, as shown in Fig. 7(b) which is the model of radially polarized beams’ incidence. Vibration directions of electrons on both sides also should be the same. Finally, it induces the unique hotspot on rear side of silver sphere. These studies further demonstrate that improper illuminating conditions have no benefit to the strong confinement of light.

Fig. 7 Simplified models of silver spheres illuminated by different incident beams. (a) is corresponding to linearly polarized beams and (b) is for radially polarized beams.

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4. Conclusion

This study shows that a compound set of metal-dielectric microspheres, permits one to confine light in an effective volume as small as 0.095 (λ/n)³, together with high peak intensity when the spheres are illuminated by a focused radially polarized beam. This strong confinement arises from the surface plasmon hotspots on the rear side of the silver microsphere induced by photonic nanojets of the dielectric microsphere. Full width at half maximum (FWHM) of the focused light field could be optimized to 73nm in axial direction and 146nm in transversal direction separately. Further discussion shows that size and position of silver sphere are two key points to achieve strong concentration of light. In the case of a fixed jet field, an optimal size or position can be selected. The beam shaped in that way is suitable for applications requiring small effective volume and/or strong peak intensities, particularly in polymeric media or water-based solution. It may have potential value for the development about super-resolution microsphere microscopy for nano-imaging of biomolecules that normally live in a liquid environment.

Acknowledgments

This work is financially supported by grants from the Qianjiang Talent Project (2011R10010), the Doctoral Fund of Ministry of Education of China (20110101120061), and the Fundamental Research Funds for the Central Universities (2012FZA5004).

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	Single dielectric microsphere	Metal-dielectric microspheres	Ratio to simple dielectric sphere
	Single dielectric microsphere	Metal-dielectric microspheres	Ratio to simple dielectric sphere
Transverse waist $w_{x y}$	740nm	230nm	27%
Longitudinal waist $w_{z}$	5560nm	280nm	5%
Peak intensity	91 (a.u.)	536 (a.u.)	589%
Effective volume V	19.5 ${(λ / n)}^{3}$	0.095 ${(λ / n)}^{3}$	0.5%

Superenhanced three-dimensional confinement of light by compound metal-dielectric microspheres

Abstract

1. Introduction

2. Confinement of light via compound metal-dielectric microspheres

2.1 Three dimensional sub-wavelength confinement of light

2.2 Compare with conventional methods

3. Discussion

3.1 Influence of silver sphere’s size

3.2 influence of silver sphere’s position

3.3 Different polarizations of incident beams

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (5)

Optics Express