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 *r*_{1}=0.07 μm and an outer radius *r*_{2}=0.1 μm as a function of the incident wavelength *λ* and the imaginary part of permittivity for dielectric shell Im(*ε*_{s}). 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(*ε*_{s})=−2.35] and E [*λ*=0.508 μm and Im(*ε*_{s})=−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(*ε*_{s})=−0.85], B [*λ*=0.583 μm and Im(*ε*_{s})=−5.9], and C [*λ*=0.507 μm and Im(*ε*_{s})=−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(*ε*_{s})<0 and there is no minimum point located at Im(*ε*_{s})≥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.

Figure 2: (A) Forward scattering intensity (logarithmic scale) in three-dimensional scale for core-shell spherical particles with an inner radius of 0.07 μm and an outer radius of 0.1 μm as a function of the incident wavelength *λ* and the imaginary part of permittivity for dielectric shell Im(*ε*_{s}). (B) Corresponding forward scattering spectra in two-dimensional pseudocolor view. (C) Total scattering intensity (logarithmic scale) in three-dimensional scale for the above core-shell spherical nanoparticles as a function of the incident wavelength *λ* and the imaginary part of permittivity for dielectric shell Im(*ε*_{s}). (D) Corresponding total scattering spectral in two-dimensional pseudocolor view.

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 *r*_{1}=0.07 μm and an outer radius *r*_{2}=0.1 μm, as a function of the incident wavelength *λ* and the imaginary part of permittivity for dielectric shell Im(*ε*_{s}), 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 [13]. 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 [12], [13]. Moreover, around minimum point A, the total scattering efficiency is also suppressed. In particular, for such a choice of *r*_{1}=0.07 μm, *r*_{2}=0.1 μm, and *ε*_{s}=16−0.85*i*, 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 . 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 [13]. 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 , 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.

Table 1: Calculated results of the first six electric (a_{1}, a_{2}, and a_{3}) and magnetic (b_{1}, b_{2}, and b_{3}) Mie expansion coefficients at the extreme points displayed in Figure 2.

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 (*a*_{1}+*b*_{1}) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity *ε*_{s}=16−0.85*i*) 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 (*a*_{1}+*b*_{1}) 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).

Figure 3: (A) Real and imaginary parts of the Mie term (*a*_{1}+*b*_{1}) for the used core-shell spherical nanoparticle as a function of the incident wavelength. The green star locates at the wavelength of 0.856 μm. (B) Corresponding far-field scattering patterns at minimum point A consisting of both scattering TE (blue line) and TM (red line) components. The corresponding electric field distributions around the nanoparticle are shown as the *x*-component of electric field distributions *Ex* (C) and the electric field enhancement factor distributions |*E*|/|*E*_{0}| (D).

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 *a*_{1}=−*b*_{1} will cause *I*_{1}=*I*_{2}, 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*|/|*E*_{0}|, where |*E*| and |*E*_{0}| 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2}) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity *ε*_{s}=16−5.9*i*) 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2}) 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2})=0 under the dipole-quadrupole approximation can also cause zero-forward scattering intensity, as demonstrated above.

Figure 4: (A) Real and imaginary parts of the Mie term 3(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2})=0 for the used core-shell spherical nanoparticle as a function of the incident wavelength. The green star locates at the wavelength of 0.583 μm. (B) Corresponding far-field scattering patterns at minimum point B consisting of both scattering TE (blue line) and TM (red line) components. The corresponding electric field distributions around the nanoparticle are shown as the *x*-component of electric field distributions *Ex* (C) and the electric field enhancement factor distributions |*E*|/|*E*_{0}| (D).

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*|/|*E*_{0}|, 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2}) for the used core-shell spherical nanoparticle (inner radius=0.07 μm, outer radius=0.1 μm, and permittivity *ε*_{s}=16−2.15*i*) 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2}) 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(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2})=0 under dipole-quadrupole approximation is satisfied.

Figure 5: (A) Real and imaginary parts of the Mie term 3(*a*_{1}+*b*_{1})+5(*a*_{2}+*b*_{2})=0 for the used core-shell spherical nanoparticle as a function of the incident wavelength. The green star locates at the wavelength of 0.507 μm. (B) Corresponding far-field scattering patterns at minimum point C consisting of both scattering TE (blue line) and TM (red line) components. The corresponding electric field distributions around the nanoparticle are shown as the *x*-component of electric field distributions *Ex* (C) and the electric field enhancement factor distributions |*E*|/|*E*_{0}| (D).

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*|/|*E*_{0}|, 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 *ε*_{s}=16−2.35*i*) 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*|/|*E*_{0}|. 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 [20], 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 .

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 *ε*_{s}=16−0.85*i*) 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 [20]. 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 .

Figure 6: The corresponding far-field scattering patterns at maximum point D (A) and maximum point E (D) consist of both scattering TE (blue line) and TM (red line) components. The corresponding electric field distributions around the nanoparticle are shown as the electric field enhancement factor distributions |*E*|/|*E*_{0}| at maximum point D (E) in (B) [(E)] and *x*-component of electric field distributions *Ex* at maximum point D (E) in (C) [(F)].

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