Investigations on the electromagnetic scattering of nanometer-scale particles are significant in a variety of research fields, such as sensing , ,  and optical antennas , , . Among these researches, the abilities to control and use the scattering of electromagnetic fields by nanometer-scale particles play a vital role in numerous applications, and the research interests in these fields have been increased rapidly at the present times , . The studies on the scattering features of the small spherical particles have been investigated theoretically and experimentally for a long time, and the Mie theory is the general formal way to study the electromagnetic scattering of the spherical particles . Recently, technological success in the fabrication of the nanostructured artificial material, called metamaterial , , can produce some anomalous but amazing scattering effects, such as electromagnetic cloaking ,  and superscattering , . In this field, the pioneering work can go back to Kerker who showed unusual electromagnetic scattering properties by magnetic spheres , in which the forward scattering of the subwavelength particles could be almost suppressed if the relative permeability μ and permittivity ε satisfied certain conditions, i.e. ε=(4−m)/(2m+1) and μ≠1. Alu et al. have shown an apparent inconsistency between zero-forward scattering condition and the optical theorem using an improved quasi-static analysis consistent with power balance considerations . García-Cámara et al. further presented the amended expressions for zero-forward scattering conditions that satisfied the optical theorem . At the last part of this Letter, it is mentioned that, to satisfy the proposed zero-forward scattering condition, scatters must have negative values of the absorption cross-section, which are known as active objects. Besides, the zero-forward scattering condition of spherical particles with radially anisotropic permittivity and permeability has been derived within the quasi-static limit .
Among these demonstrations of efficient manipulation of light radiation, zero-forward scattering can be obtained from the total destructive interference between the electric and magnetic dipolar resonances. In general, the first-order dipolar responses are the easiest to be excited; thus, to a certain degree, it will dominantly determine the scattering pattern. However, the scattering features for small particles may be not only determined by the interference between the electric and magnetic dipoles in some other situations , . It has recently been shown that higher-order electric and magnetic responses are also important for the manipulation of scattering direction . Moreover, for usual lossy and lossless material, the real parts of Mie expansion coefficients an and bn are always positive , . Unfortunately, more recently, it has been theoretically demonstrated that, by embedding a metallic nanoparticle into a dielectric nanosphere, only a minimum but not zero-forward scattering can be achieved due to the restrictions imposed within the Mie theory .
In this paper, we have investigated the scattering properties of a single core-shell spherical nanoparticle consisting of a gold core and dielectric shell layer with a incident plane wave. The core-shell spherical nanoparticle can be tuned to support both electric and magnetic modes simultaneously. First, we analyzed the scattering properties of the core-shell nanoparticle and systematically investigated the condition of zero-forward scattering intensity based on the Mie theory. In contrast to the second Kerker condition, a novel mechanism is adopted to explain zero-forward scattering, which is based on complex interference between electric (dipolar and quadrupolar) and magnetic (dipolar and quadrupolar) responses and satisfies the condition of 3(a1+b1)+5(a2+b2)=0. Then, using a proper gain doped into the dielectric shell, the forward scattering intensity can get a significant enhancement. More interestingly, by appropriately adding gain in the dielectric shell, the zero-forward scattering intensity can also be achieved at certain incident wavelengths, which reveals that the electric and magnetic responses can be counteracted totally at the forward direction. Moreover, we demonstrated that the core-shell nanoparticle with suitable gain in shell is able to obtain anomalously weak scattering or superscattering. In addition, if the absolute values of dipolar and quadrupolar terms have the same order of magnitudes, the electromagnetic scattering directionalities can also be modulated due to the coherent effect between the dipolar and quadrupolar terms.
2 Theoretical development
We first considered the electromagnetic scattering of the core-shell spherical nanoparticles in the vacuum, and the core-shell spherical nanoparticle consists of a gold core and the gain material-doped dielectric shell layer. In our calculations, the real part of permittivity for dielectric shell is set as 16 and the permittivity of the metallic core can be taken from . A schematic view of the coated nanoparticle and the associated coordinate system are shown in Figure 1. The coated sphere has an inner radius r1 with a size parameter of x=kr1 (k is the wave number in the ambient medium) and an outer radius r2 with a size parameter of y=kr2. The relative permittivities of the core and shell are εc=m12 and εs=m22, respectively (m1 and m2 are the refractive indexes of the inner-medium and outer-medium, respectively). We assumed that the incident plane wave propagates along the +z-direction and the electric field is polarized along the x-direction. The scattering properties for the core-shell nanoparticle can be solved analytically based on the Mie theory and the scattering efficiency Qsca defined as the scattering cross-section divided by the cross-section of the nanoparticle is given by (1)
where an and bn are the electric and magnetic Mie coefficients of coated spheres, which can be expressed as (2)(3)(4)
Here, m=m2/m1 is the relative refractive index and the functions ψn, ξn, and χn are the related Riccati-Bessel functions. The function Dn is given by Dn=ψn′/ψn=χn′/χn+1/(ψnχn). The scattered radiant intensity (SI) in the far field can be decomposed as two polarized components :(5)(6)(7)
where I1 and I2 are the two polarized components of scattered intensity. In addition, λ is the incident wavelength, θ is the polar angle, and ϕ is the azimuthal angle. The expressions for the scattering amplitudes S1 and S2 can be expressed as (8)(9)
Here, the functions π(cosθ) and τ(cosθ) describe the angular scattering patterns of the spherical harmonics. From above, the total scattering intensity in the forward direction can be obtained as(10)
According to Eq. (10), to obtain zero-forward scattering, it is required that the electric (an) and magnetic (bn) terms of Mie coefficients satisfy the equation In general, for a small high-permittivity particle within dipole limit, the first two terms of Mie coefficients a1 and b1 dominantly contribute to the scattering field, whereas the other higher terms can be normally neglected. Therefore, the condition of a1=−b1 is expected to be the simplest case to realize zero-forward scattering. It is natural to extend the case of a1=−b1 to higher-order modes, i.e. an=−bn. Nevertheless, if the particle has a different material with the surrounding medium, an=−bn will be a quite harsh condition to realize the zero-forward scattering condition. Although the above condition can hardly be fulfilled, the problem of zero-forward scattering can be solved from a different angle. Here, we just considered the dipolar (a1, b1) and quadrupolar (a2, b2) terms, so the scattered fields can be well described using these four terms. Under this approximation, the forward scattering intensity can be simplified as(11)
From a mathematical view, both electric and magnetic terms have different amplitudes and satisfy the condition of 3(a1+b1)+5(a2+b2)=0, and the forward scattering intensity can also be reduced to zero. However, the real parts of an and bn are always positive for lossy and lossless particles, which can be derived from Eqs. (2) to (4). Thus, to ensure the establishment of the above conditions, the introduced gain material into the nanoparticles is necessary.
With increasing development of technology, different kinds of optical gain material have been used in the experiment and some significant inventions have been found, such as plasmonic lasers , , , . The propagation of electromagnetic wave in the gain-doped medium in the situation of plasmonic lasers or weak electric fields can be solved by Maxwell equations using a field-independent dielectric permittivity, which has a negative imaginary part, i.e. Im(ε)<0 . This theoretical model is usually used to characterize the metamaterial coupled to the gain medium in the experiments due to its simplicity. In this paper, we also used this approach to model the gain medium. The usual gain medium consists of dye molecules , quantum dots , rare earth , and semiconductors .
3 Results and discussion
In Figure 2A, based on Eq. (10), we show the forward scattering intensity in three-dimensional scale for a core-shell spherical particle with an inner radius r1=0.07 μm and an outer radius r2=0.1 μm as a function of the incident wavelength λ and the imaginary part of permittivity for dielectric shell Im(εs). Furthermore, Figure 2B gives the corresponding two-dimensional pseudocolor plot. In Figure 2A and B, there are two points marked as D [λ=0.659 μm and Im(εs)=−2.35] and E [λ=0.508 μm and Im(εs)=−0.85], where the forward scattering intensity obtains an obvious enhancement due to the introduction of gain material into the shell. More interestingly, within the calculated wavelength range, there are also some special points marked as A [λ=0.856 μm and Im(εs)=−0.85], B [λ=0.583 μm and Im(εs)=−5.9], and C [λ=0.507 μm and Im(εs)=−2.15], where zero-forward scattering can be obtained. The obtained zero-forward scattering means that all electric and magnetic responses of the core-shell spherical particles can be cancelled out totally in the forward direction. It is clearly found that all of the zero-forward scattering intensity occurs in the range of Im(εs)<0 and there is no minimum point located at Im(εs)≥0, which indicates that the zero-forward scattering intensity can never be achieved in lossy and lossless particles but can be achieved by the introduction of gain medium.
From the above-simulated results, some anomalous but amazing forward scattering effects can be obtained by doping gain medium into the shell. Then, the total scattering efficiency for core-shell spherical particles with an inner radius r1=0.07 μm and an outer radius r2=0.1 μm, as a function of the incident wavelength λ and the imaginary part of permittivity for dielectric shell Im(εs), can also be obtained based on Eq. (1), which has been shown in three- and two-dimensional scales in Figure 2C and D. It is found that the total scattering efficiency at maximum points D and E can also be enhanced strongly, which can be called as superscattering . Besides, for minimum points B and C, the scattering efficiency is still relatively large. It is worth noting that the scattering efficiency at minimum point A can be reduced significantly, which can be called anomalously weak scattering associated with the cloaking phenomenon , . Moreover, around minimum point A, the total scattering efficiency is also suppressed. In particular, for such a choice of r1=0.07 μm, r2=0.1 μm, and εs=16−0.85i, we can obtain superscattering at λ=0.508 μm (E) and anomalously weak scattering at λ=0.856 μm (A).
Here, we have also calculated the corresponding electric and magnetic Mie expansion coefficients from the first order to the third order at the above five extreme points as shown in Table 1. First, we concentrated our attention on minimum points A to C. It is clearly shown that their absolute values of octupolar terms are several orders of magnitude smaller than their dipolar and quadrupolar terms. For minimum point A, the absolute values of quadrupolar terms are also an order of magnitude smaller than its dipolar terms. It is noted that all absolute values of multipolar terms at minimum point A are relatively small. Thus, the scattering efficiency is significantly reduced at minimum point A . Besides, for minimum point B, the absolute values of quadrupolar terms have a close order of magnitude to their dipolar terms, and for minimum point C, the absolute values of dipolar terms are of the same order of magnitude with their quadrupolar terms. Here, it is worth mentioning that all absolute values of dipolar and quadrupolar terms at minimum points B and C are much larger than those at minimum point A. Therefore, the scattering efficiencies at minimum points B and C are relatively large. Although the detailed information cannot be demonstrated intuitively in Table 1, this table preferably provides the corresponding electric and magnetic multipolar contributions to the scattering responses. Thus, we will also try to explain zero-forward scattering intensity by the corresponding curves intuitively. Next, we will divert the attention to maximum points D and E. For maximum point D, the absolute value of the magnetic dipole term is much larger than other ones, which means that the forward scattering effects of the core-shell spherical nanoparticle are determined by the magnetic dipole response. Similarly, the dominating term for maximum point E is its magnetic quadrupolar response.
Based on the above results and analysis, now we will turn to investigate these five extreme points to obtain the intuitively concrete information. First, for point A, we just considered the dipole terms, but the higher-order terms (n≥2) can be negligible from the above analysis. Figure 3A shows the real and imaginary parts of the Mie term (a1+b1) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity εs=16−0.85i) as a function of the incident wavelength. At the location labeled by a green star, where both the real and imaginary parts of the Mie term (a1+b1) are simultaneously equal to zero, the forward scattering can reach zero nearly because of the second Kerker condition at the wavelength of 0.856 μm (corresponding to minimum point A as displayed in Figure 2).
To characterize the scattering properties of the designed core-shell nanoparticle, the corresponding far-field scattering patterns at minimum point A have been simulated for both scattering TE and TM components as shown in Figure 3B, and the arrow denotes the direction of the incident wave, from which it can also be clearly observed that the forward scattering intensity is reduced to zero due to the completely destructive interference between the electric and magnetic dipolar terms. On just considering the dipolar terms, zero-forward scattering condition a1=−b1 will cause I1=I2, which can be deduced from Eq. (1). Therefore, as shown in Figure 3B, two different scattering components (TE and TM) have nearly identical scattering patterns, although there is a little difference due to the inclusion of quadrupole and other higher-order terms but much smaller than the dipole terms. Figure 3C shows the x-component of electric field distributions (Ex) around the nanoparticle at minimum point A, which demonstrates the relationship between near- and far-field. Figure 3D shows the electric field enhancement factor distributions |E|/|E0|, where |E| and |E0| represent the amplitude of the electric field near the nanoparticles and the incident electric field, respectively. From this we can observe that the electromagnetic scattering is suppressed in the forward direction, whereas the values of the electric field are relatively large in the backward direction, which are in accordance with the far scattering patterns. Furthermore, in Figure 3C, we can observe that the electromagnetic plane waves propagate through the core-shell nanoparticle with the original wave front nearly. This is because the scattered radiant intensity in the far field and scattering efficiency are reduced significantly, which can also be verified in Figures 3B and 2.
As demonstrated in the above section, we just considered the first four terms of Mie expansion coefficients at minimum point B. The real and imaginary parts of the Mie term 3(a1+b1)+5(a2+b2) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity εs=16−5.9i) are plotted as a function of the incident wavelength as demonstrated in Figure 4A. It is clearly seen that both the real and imaginary parts of the Mie term 3(a1+b1)+5(a2+b2) are simultaneously equal to nearly zero at the wavelength of 0.583 μm as marked by the green star (corresponding to minimum point B as displayed in Figure 2). The condition of 3(a1+b1)+5(a2+b2)=0 under the dipole-quadrupole approximation can also cause zero-forward scattering intensity, as demonstrated above.
Furthermore, the corresponding far-field scattering patterns at minimum point B, including both scattering TE and TM components, are shown in Figure 4B. The arrow in Figure 4B denotes the incident wave direction. The forward scattering intensity is reduced to zero just like the situation at minimum point A. However, the difference between the TE and TM components are larger than that at minimum point A, because the absolute values of the quadrupolar terms have closer value to those of dipolar terms. The electric field distributions around the nanoparticle at minimum point B, consisting of the x-component of electric field distributions Ex and the electric field enhancement factor distributions |E|/|E0|, are displayed in Figure 4C and D, respectively. It can be seen that the electric field is decreased very rapidly in the forward direction and the values of the electric field are relatively large in the backward direction, which agree well with the far scattering patterns. Furthermore, it is found in Figure 4C that the electromagnetic plane waves propagating through the core-shell nanoparticle have been affected at backward half-plane. This is because the scattered radiant intensity in the far field and scattering efficiency are relatively large, as in Figures 4B and 2.
At minimum point C, from the above analysis, we know that the octupole and higher-order terms can be negligible. Figure 5A shows the real and imaginary parts of the Mie term 3(a1+b1)+5(a2+b2) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity εs=16−2.15i) as a function of the incident wavelength. At the location marked by the green star, both the real and imaginary parts of the Mie term 3(a1+b1)+5(a2+b2) are simultaneously equal to nearly zero at the wavelength of 0.507 μm (corresponding to minimum point C displayed in Figure 2). It should be noted that zero-forward scattering intensity can be achieved at the wavelength of 0.507 μm, where the condition of 3(a1+b1)+5(a2+b2)=0 under dipole-quadrupole approximation is satisfied.
The corresponding far-field scattering patterns at minimum point C, including both scattering TE and TM components, are further displayed in Figure 5B. The arrow in Figure 5B denotes the incident wave direction. It is found that the scattering in the forward direction is almost completely suppressed. Here, we can find that there exist other possible directions (away from the forward and backward directions), where the scattering TE component reaches a local minimum or a local maximum, thus leading to extra side scattering lobes. It is due to the complex interference between the electric and magnetic responses (dipolar and quadrupolar terms), which have the same order of magnitudes. The above analyses at minimum points B and C clearly confirm the possibility of zero-forward scattering by employing the complex interference between dipolar and quadrupolar terms without the particular requirement of satisfying the second Kerker condition. The electric field distributions around the nanoparticle at minimum point C, consisting of the x-component of electric field distributions Ex and the electric field enhancement factor distributions |E|/|E0|, are displayed in Figure 5C and D, respectively. It can be seen that the electric field is suppressed in the forward direction and the values of the electric field are relatively large in the backward direction, which are in accord with the far-field scattering patterns. Moreover, it is seen in Figure 5C that the electromagnetic plane waves propagating through the core-shell nanoparticle have been affected at backward half-plane, just like the situation at minimum point B. This is also because the scattered radiant intensity in the far field and scattering efficiency are relatively large, which can be verified in Figures 5B and 2.
Corresponding far-field scattering patterns and electric field distributions at maximum points D and E, respectively. The far-field scattering patterns for a core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity εs=16−2.35i) at the incident wavelength of 0.659 μm are shown (A) consisting of both the scattering TE and TM components. The arrow denotes the incident wave direction. (B and C) Near electric field distributions around the particle at maximum point D, consisting of the x-component of electric field distributions Ex, and the electric field enhancement factor distributions |E|/|E0|. It should be noted that the electric field shows a super-enhancement from (B and C) at maximum point D, which is associated with laser. Moreover, the electromagnetic plane waves propagating through the core-shell nanoparticle have been affected strongly, as shown in (C). This is due to the excitation of the strong magnetic dipolar mode. In addition, it can be observed that the electric field enhancement map inside the particle shows a circular distribution corresponding to the magnetic dipole oriented along the direction normal to x-z plane , which is in accord with the far scattering patterns. It can be also verified that the strong magnetic dipolar term for maximum point D can be excited, as in Table 1.
The corresponding far-field scattering patterns and electric field distributions for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity εs=16−0.85i) at the incident wavelength of 0.508 μm (at maximum point E) have been demonstrated further in Figure 6D–F. It is worth mentioning that the electric field can be enhanced strongly in Figure 6E and F at maximum point E, which is associated with laser. Moreover, the electromagnetic plane waves propagating through the core-shell nanoparticle have been affected strongly, as seen in Figure 6F. This is due to the excitation of strong magnetic quadrupolar mode. In addition, the far-field scattering pattern of the TE component presents a symmetrical four-lobe shape and that of the TM component shows symmetrical two-lobe shape, which is a typical distribution of magnetic quadrupole . The electric field enhancement factor distributions display the typical two-lobe distribution corresponding to magnetic quadrupole oriented along the y-axis, which is in accord with the corresponding far scattering pattern (scattering TM component). This behavior of maximum point E has verified that the strong magnetic quadrupolar mode can be excited, as can be observed in Table 1.
In conclusion, we use the Mie theory to investigate the scattering characteristics of a core-shell spherical nanoparticle with a gold core of radius r1=0.07 μm and different gains in the dielectric shell layer of radius r2=0.01 μm, which can be tuned to support both electric and magnetic responses simultaneously. Generally, the scattering intensity of spherical nanoparticles will be enhanced by the introduction of gain. It is seen that the forward scattering intensity of a core-shell spherical nanoparticle can be enhanced remarkably at certain incident wavelengths using the appropriate gain-doped dielectric shell, which is attributed to the excitations of magnetic dipolar and quadrupolar modes. More importantly, we have confirmed the possibility to produce zero-forward scattering intensity by adding proper gain in the dielectric shell. With the appropriate gain-doped dielectric shell, the second Kerker condition or the condition of 3(a1+b1)+5(a2+b2)=0 under dipole-quadrupole approximation can also be fulfilled at certain incident wavelengths. In addition, if the absolute values of dipolar and quadrupolar terms have the same order of magnitude, the local scattering minimum and maximum can be produced away from the forward and backward directions. Moreover, anomalously weak scattering or superscattering can also be obtained for the core-shell nanoparticle with suitable gain in shell. In particular, for such a choice of suitable gain in shell, zero-forward scattering and anomalously weak scattering can be realized at the same wavelength. Meanwhile, super-forward scattering and superscattering can also be achieved simultaneously at another wavelength. The above-mentioned features make the proposed core-shell spherical particles to have great potential applications in manipulating light at nanoscale, such as cloaking, plasmonic lasers, and optical antennas.
The authors gratefully acknowledge the financial supports for this work from the National Natural Science Foundation of China (Grant Nos. 61575060, 61501159, and 11505043) and the Fundamental Research Funds for the Central Universities (Grant No. 2015HGCH0010).
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