Anomalous forward scattering of dielectric gain nanoparticles

Ya-Ming Xie; Wei Tan; Zhi-Guo Wang

doi:10.1364/OE.23.002091

1. Introduction

The study of electromagnetic scattering by small particles has a long history [1,2] and is of great interest to various fields [3–6], such as astrophysics, biophysics, material science and so on. At present, the demand of future light-on-chip integration for efficient manipulation of light radiation on nanoscale has attracted renewed interest in the research of light scattering of nanoparticles, which can serve as substitutions for invalid conventional optical elements [7].

An early formal study of the scattering problem for a spherical particle was carried out by Mie more than a century ago [1], which can explain many optical phenomena in our daily life [2], for example, the blue color of the sky. Nowadays, with the appearance of metamaterials whose optical parameters can be artificially designed, some anomalous but very fascinating electromagnetic scattering behaviors have been observed [8,9]. In the field of anomalous electromagnetic scattering, the pioneering work can be traced to Kerker et al., who systematically analyzed the light scattering of small magnetodielectric spheres [10], predicting that a particle with diameter much smaller than incident wavelength would show zero forward scattering if its relative permittivity $ε_{p}$ and permeability $μ_{p}$ satisfied the relation $ε_{p} = (4 - μ_{p}) / (2 μ_{p} + 1)$ . García-Cámara et al. [11] have offered an interesting discussion on the validity of Kerker’s condition as a function of the distance to particle, and it is shown that there exists a smooth evolution from dipolar scattering behavior to directional scattering behavior as the distance increasing [11]. Alú et al. [12] have indicated that there exists an inconsistency between Kerker’s zero-forward-scattering condition and the optical theorem since radiation damping is completely ignored during the derivation of this condition, and it can be resolved by considering power balance in the quasi-static analysis. García-Cámara et al. [13] further proposed the corrected expression of Kerker’s zero-forward-scattering condition that includes radiative correction and satisfies the optical theorem. It is noticed that in order to achieve zero forward scattering, gain must be introduced in the magnetic material [13]. In addition, the expression of zero-forward-scattering condition for spherical particles with radial anisotropy has also been derived [14]. Mehta et al. [15,16] have reported their observation of zero forward scattering by magnetic spheres according to Kerker’s criteria, however, it has been questioned by several researchers [12,17,18]. Thus to date the experimental confirmation of Kerker’s zero-forward-scattering condition is still absence.

According to Mie [1] theory, the key ingredient for achieving zero forward scattering in small nanoparticles involves complete destructive interference between the electric and magnetic dipolar responses to incident wave. For metamaterials and metallic media, large dissipation is an inevitable problem, which has turned researchers’ attention to dielectric materials with high permittivity [7,19–22]. In the past few years, researchers have shown theoretically [19,20] and experimentally [21] that dielectric nanoparticles with moderate relative permittivity present strong magnetic and electric dipolar resonances simultaneously in the visible, as well as near-infrared frequencies. For such dielectric nanoparticles, the spectral proximity between the electric and magnetic dipolar responses allows coherent effect between them, causing phenomena like angle-suppressed scattering [20,22]. Recently, Geffrin et al. have given an unambiguous experimental verification of totally suppressed backscattering and partially suppressed forward scattering for a subwavelength silicon sphere [7]. However, restricted by the dielectric permittivity of silicon material [12,22], only a minimum not zero forward scattering can be achieved, thus realizing zero forward scattering in nonmagnetic dielectric spherical particles remains as an open question.

For usual lossy and lossless dielectric spherical particles, the real parts of Mie dipolar expansion coefficients ( $a_{1}$ and $b_{1}$ ) are always positive [12,20,22], which causes that the electric and magnetic dipolar responses cannot offset completely at forward direction [7,20,22]. By introducing gain [23–25] in material, it is noted that forward scattering intensity can get significant increased with specific material permittivity under certain incident wavelength. More interestingly, through appending appropriate gain, the electric and magnetic dipolar responses can achieve total destructive interference in forward direction at specific incident wavelength. When the quadrupole and other higher order Mie coefficients are negligible, zero forward scattering can be observed and backscattering will get enhanced. In contrast, if the quadrupole and dipolar Mie coefficients are of the same order of magnitude, coherent effect between them will affect the directionality of light scattering of the spherical particle.

This paper is organized as follows. In Section 2, we review the general theory of light scattering by a nonmagnetic dielectric spherical particle and give a possible solution to the realization of zero forward scattering by adding gain in material. In Section 3, we explore the anomalous forward scattering of a dielectric gain spherical nanoparticle, including minimum and maximum. Conclusions are made in Section 4.

2. Theoretical model

Supposing an isotropic homogeneous nonmagnetic dielectric spherical particle with radius $a$ and relative permittivity $ε_{p} = m_{p}^{2}$ is immersed in a non-absorbing homogenous medium with real relative permittivity $ε = m_{s}^{2}$ , the scattered radiant intensity at far field can be expressed as [2]

I_{1} = \frac{λ^{2}}{4 π^{2} m_{s}^{2} r^{2}} \sin^{2} φ {| S_{1} (\cos θ) |}^{2}, (TE for φ {=90}^{\circ})

I_{2} = \frac{λ^{2}}{4 π^{2} m_{s}^{2} r^{2}} \cos^{2} φ {| S_{2} (\cos θ) |}^{2}, (TM for φ {=0}^{\circ})

where

r

is the distance to the observer,

φ

is the angle between the incident electric vector and the scattering plane,

θ

is the angle between the incident and scattering directions, and

λ

is the incident wavelength in vacuum.

S_{1}

and

S_{2}

are the corresponding amplitude functions [2].

I_{1}

and

I_{2}

correspond to the polarized components of incident electric vector perpendicular and parallel to the scattering plane, respectively.

For forward direction, the total scattering intensity can be obtained as

I (θ = 0^{\circ}) = \frac{λ^{2}}{16 π^{2} m_{s}^{2} r^{2}} {| \sum_{n = 1}^{\infty} (2 n + 1) (a_{n} + b_{n}) |}^{2} .

Here the

φ

dependence is eliminated, implying that the forward scattering is azimuthally symmetric. It is obvious that if

\sum_{n = 1}^{\infty} (2 n + 1) (a_{n} + b_{n}) = 0

gets satisfied, the forward scattering is reduced to zero, regardless of particle size. Apparently,

a_{n} = - b_{n}

is the simplest way to fulfill this expression.

Following the form of Bohern [2], the Mie expansion coefficients ${a_{n}}$ and ${b_{n}}$ can be expressed as

a_{n} = \frac{ψ_{n} (m x) ψ_{n}^{'} (x) - \frac{1}{m} ψ_{n} (x) ψ_{n}^{'} (m x)}{ψ_{n} (m x) ξ_{n}^{'} (x) - \frac{1}{m} ξ_{n} (x) ψ_{n}^{'} (m x)},

b_{n} = \frac{ψ_{n} (m x) ψ_{n}^{'} (x) - m ψ_{n} (x) ψ_{n}^{'} (m x)}{ψ_{n} (m x) ξ_{n}^{'} (x) - m ξ_{n} (x) ψ_{n}^{'} (m x)} .

Here the radial functions

ψ_{n}

and

ξ_{n}

are the Ricatti Bessel and Hankel functions of arguments

x = 2 π m_{s} a / λ

and

m x = 2 π m_{p} a / λ

. Note that as

m

approaches unity, both

a_{n}

and

b_{n}

vanish, hence the condition

a_{n} = - b_{n}

is satisfied. This is as it should be: when the particle disappears, so does the scattered field. However, if the particle has different material with surrounding medium,

a_{n} = - b_{n}

will be a rare harsh condition and it can hardly be fulfilled in practice.

Under long wavelength limitation ( $a / λ ≪ 1$ ), the first two terms of Mie coefficients, $a_{1}$ and $b_{1}$ , are a good approximation of light scattering, which means all coefficients higher than $a_{1}$ and $b_{1}$ can be neglected. Thus the forward scattering intensity can be approximatively reduced to

I (θ = 0^{\circ}) = \frac{9 λ^{2}}{16 π^{2} m_{s}^{2} r^{2}} {| a_{1} + b_{1} |}^{2} .

Obviously, with the existence of opposite dipolar electric and magnetic responses (

a_{1} = - b_{1}

), forward scattering intensity can be reduced to zero, and simultaneously backscattering intensity will get enhanced since the backscattering is proportional to

| a_{1} - b_{1} |^{2}

. The cancelation of forward scattering and enhancement of backscattering originate from the destructive and constructive interferences between the magnetic and electric dipolar responses, respectively. Nevertheless, it can be derived from Eqs. (4) and (5) that the real parts of

a_{1}

and

b_{1}

are always positive for lossy and lossless materials. Thus in order to achieve zero-forward-scattering condition

a_{1} = - b_{1}

, gain needs to be introduced in material [13].

It has been shown that in the absence of lasing and in the limit of weak electric fields, the propagation of an electromagnetic wave in gain medium can be described by the Maxwell equations, using a field-independent dielectric permittivity with a negative imaginary part $Im (ε) < 0$ [26–28]. Due to the simplicity of this approach, it is often used for analyzing experiments with metamaterial structures coupled to gain media [29,30]. Thus in the article we also adopt this theoretical model to describe gain material.

3. Results and discussion

Figure 1(a) shows the 3D plot of forward scattering intensity of a spherical particle with radius $a = 100 nm$ and real part of permittivity $Re (ε_{p}) = 16$ as a function of incident wavelength $λ$ and imaginary part of permittivity $Im (ε_{p})$ , and Fig. 1(b) provides the corresponding 2D pseudocolor plot. Due to the introduction of gain, forward scattering intensity experiences a remarkable enhancement at points C and D, and the corresponding parameters at these two maximum points are C ( $λ = 832 nm$ , $Im (ε_{p}) = - 1.1$ ) and D ( $λ = 590 nm$ , $Im (ε_{p}) = - 2.2$ ). More interestingly, forward scattering intensity is reduced to nearly zero at points A and B, implying that here the electric and magnetic dipolar responses of the spherical particle can totally offset at forward direction. The corresponding parameters at these two minimum points are A ( $λ = 768 nm$ , $Im (ε_{p}) = - 2.8$ ) and B ( $λ = 517 nm$ , $Im (ε_{p}) = - 0.8$ ). In the parameter range considered in Fig. 1(b), no minimum point is found in the parameter region $Im (ε_{p}) \geq 0$ , which confirms that the zero-forward-scattering condition cannot be met in lossy and lossless media. With the broadening of incident wavelength range, it is noticed that more extreme points will appear for incident wavelength less than 500 nm but no extreme point will exist for incident wavelength larger than 1000 nm. As for another parameter $Im (ε_{p})$ , no more extra extreme point can be discovered when its scope gets enlarged.

Fig. 1 (a) 3D plot of forward scattering intensity (logarithmic scale) considering only dipolar Mie coefficients as a function of incident wavelength $λ$ and imaginary part of permittivity $Im (ε_{p})$ for a spherical particle with radius $a = 100 nm$ and real part of permittivity $Re (ε_{p}) = 16$ in vacuum. (b) The corresponding 2D pseudocolor plot of Fig. 1(a).

Download Full Size | PDF

With the development of technology, it is noticed that different kinds of gain materials have been used in experiments [24,25,31] and several significant breakthroughs have been obtained, such as plasmon lasers at deep subwavelength scale [24] and observation of parity-time symmetry in optics [25]. To our knowledge, PbS quantum dots (Qdots) doped glass [31] is a candidate material which has a chance to meet the optical properties required at these extreme points in Figs. 1(a) and 1(b). It is shown that in the 500–1500 nm spectral range, the real part of refractive index of PbS Qdots-doped glass experiences a slow change in the vicinity of 4, while the imaginary part has a continuous change ranging from 0 to –4 [31]. Besides, the refractive index of PbS Qdots-doped glass can be adjusted through altering the Qdot diameter or the volume fraction of Qdot in glass matrix [31]. Therefore, it is feasible to produce the specific dielectric gain nanoparticles required above in experiments (for example, refractive index $m_{p} = \sqrt{ε_{p}} \approx 4.0 - 0.35 i$ at 768 nm), which makes our design practical.

In order to explain the different physical mechanisms behind these four extreme points in Figs. 1(a) and 1(b), we provide the corresponding electric and magnetic Mie coefficients to the third order ( $n = 3$ ) in Table 1. As shown in Table 1, the electric and magnetic dipolar terms can totally cancel out at two minimum points A and B, but the behaviors of their quadrupole terms are quite different. For minimum point A, the norms of quadrupole terms are an order of magnitude smaller than those of its dipolar terms and the induced electric and magnetic quadrupoles oscillate in-phase at forward direction. While for minimum point B, the dipolar and quadrupole terms are of the same order of magnitude and the electric- and magnetic-induced quadrupoles oscillate out-phase at forward direction. These different behaviors of quadrupole terms between points A and B will cause diversity in their scattering diagrams as shown below. As for maximum point C, the absolute value of magnetic dipolar term is much larger than other terms, which implies that here the scattering properties of the spherical particle is dominated by its magnetic dipolar response, and in a similar way the dominant term at maximum point D is its electric dipolar response.

Table 1. Numerical Results of the First Six Electric ( $a_{1}$ , $a_{2}$ and $a_{3}$ ) and Magnetic ( $b_{1}$ , $b_{2}$ and $b_{3}$ ) Mie Coefficients at Four Extreme Points Shown in Figs. 1(a) and 1(b)

View Table

In the following section, we will demonstrate how the different behaviors of Mie coefficients of these four extreme points affect the far-field scattering diagram. Figure 2(a) shows the far-field scattering diagram of a spherical particle with radius $a = 100 nm$ at minimum point A, where the permittivity of particle is $ε_{p} = 16 - 2.8 i$ and the corresponding incident wavelength is $λ = 768 nm$ . The left half part of Fig. 2(a) gives the scattering intensity calculated by all Mie coefficients, while the right half part presents the scattering intensity considering only dipolar terms. Both TE and TM polarizations are provided in Fig. 2(a), and the arrow denotes the propagation direction of incident wave.

Fig. 2 Scattering properties at minimum point A. (a) Scattering diagram for a spherical particle with radius $a = 100 nm$ and permittivity $ε_{p} = 16 - 2.8 i$ at incident wavelength $λ = 768 nm$ under both TE and TM polarizations. The left half part of graph shows the scattering diagram considering all Mie coefficients while the right half part presents the scattering diagram considering only dipolar coefficients. (b) Distribution of the scattered electric field intensity (logarithmic scale) around the spherical particle.

Download Full Size | PDF

From Fig. 1(a) and Table 1 it is verified that the electric and magnetic dipolar terms can completely offset in forward direction at minimum point A, thus in the right half part of Fig. 2(a) the forward scattering intensity is reduced to zero. From Eqs. (1) and (2), it is known that the scattering intensities under TE and TM polarizations are proportional to $| S_{1} |^{2}$ and $| S_{2} |^{2}$ , respectively. When considering only dipolar terms, zero-forward-scattering condition $a_{1} = - b_{1}$ will cause $| S_{1} |^{2} = | S_{2} |^{2}$ , hence two different polarizations will get identical scattering pattern as shown in the right half part of Fig. 2(a). Simultaneously, the electric and magnetic dipolar responses will emerge a complete coherent enhancement at backward direction, thus in the right half part of Fig. 2(a) the scattering intensity reaches its maximum in this direction.

Although forward scattering intensity will get increased when higher order ( $n \geq 2$ ) Mie coefficients are considered, it is still much smaller than the scattering intensity in other direction, since the absolute values of quadrupole and other higher order coefficients are far less than those of dipolar terms as shown in Table 1 and the dipolar terms can only totally offset in forward direction. Hence in the left half part of Fig. 2(a) a nearly zero forward scattering is observed and the scattering diagrams in the left and right half parts are basically the same. Nevertheless, the inclusion of quadrupole and other higher order Mie coefficients will cause a little difference in the scattering diagram under different polarizations, which can be seen clearly in the left half part of Fig. 2(a).

For illustrative purposes, the simulated distribution of scattered electric field intensity around the particle is plotted in Fig. 2(b), in order to show how the scattered wave evolves from the near- to far-field and how they are correlated. The propagation direction of incident wave is + x direction and the electric vector polarizes along z axis. It can be seen that electromagnetic scattering is greatly suppressed in forward direction while the modulus of electric field is relatively large in backward direction, which are consistent with the discussions of scattering diagrams shown in Fig. 2(a). The electric field intensity reaches maximum near the surface of the particle along the polarization direction of incident electric vector, which corresponds to the electric field distribution characteristic produced by the induced electric dipole moment. This observed evolution of scattered electric field intensity with increasing distance $r$ has also been reported in Ref [11].

Figure 3(a) plots the far-field scattering diagram of a spherical particle with radius $a = 100 nm$ at minimum point B, here the incident wavelength is $λ = 517 nm$ and the permittivity of material is $ε_{p} = 16 - 0.8 i$ . The scattering intensity in the left half part is calculated with all Mie coefficients, while in the right half part only dipolar coefficients are included. Both TE and TM polarizations are provided in Fig. 3(a) and the arrow represents incident wave direction.

Fig. 3 Scattering properties at minimum point B. (a) Scattering diagram for a spherical particle with radius $a = 100 nm$ and permittivity $ε_{p} = 16 - 0.8 i$ at incident wavelength $λ = 517 nm$ under both TE and TM polarizations. The left half part of graph gives the scattering diagram considering all Mie coefficients while the right half part presents the scattering diagram considering only dipolar coefficients. Inset: Enlarged area for demonstrating scattering minimum at a non-zero angle. (b) Distribution of the scattered electric field intensity (logarithmic scale) around the spherical particle.

Download Full Size | PDF

In the right half part of Fig. 3(a), one can observe zero forward scattering, maximum backscattering and the same scattering diagram under TE and TM polarizations. All these phenomena are due to the zero-forward-scattering condition satisfied at minimum point B, just like the situation at minimum point A.

It has been noticed from Table 1 that at minimum point B the quadrupole terms are of the same order of magnitude with dipolar terms, which implies that the inclusion of quadrupole coefficients in the calculation of scattering intensity will cause obvious difference comparing with the calculation under long wavelength approximation, as observed in Fig. 3(a). Because the dipolar terms can completely offset and the quadrupole terms can partly cancel each other out at forward direction, the scattering intensity may still remain as a minimal value in this direction, which is found in the scattering diagram under TE polarization in the left half part of Fig. 3(a). More interestingly, the interference between dipolar and quadrupole terms has a chance to shift the scattering minimum away from forward direction, which can be seen clearly from the inset of Fig. 3(a). Besides, the scattering maximum will also deviate from backward direction due to the coherence between dipolar and quadrupole coefficients, which may have potential applications in steering incident light into specific direction at nanoscale. Finally, an obvious difference between the scattering diagrams under TE and TM polarizations is observed in the left half part of Fig. 3(a) due to the consideration of higher order ( $n \geq 2$ ) Mie coefficients.

Figure 3(b) provides the simulated distribution of scattered electric field intensity around the particle. The propagation direction of incident light is + x direction and the polarization orientation of incident electric vector is along z axis. It is clear that the electric filed intensity decays very rapidly in forward direction while the modulus of electric field is relatively large in backward direction, which are consistent with the analyses of scattering diagrams shown in Fig. 3(a). Along the surface of the particle there exist four areas where the scattered electric field intensity shows a relative maximum, which corresponds to the electric field distribution characteristic caused by the induced electric quadrupole moment.

Figure 4 gives the scattering diagrams and the corresponding scattered electromagnetic field distributions at two maximum points C and D. Figure 4(a) shows the far-field scattering diagram (including all Mie coefficients) of a spherical particle with radius $a = 100 nm$ and permittivity $ε_{p} = 16 - 1.1 i$ at incident wavelength $λ = 832 nm$ , two different polarizations TE and TM are offered and the arrow represents the propagation direction of incident wave. Figure 4(b) provides the simulated distribution of scattered magnetic field intensity around the particle (the incident electric vector polarizes along z axis). From Table 1 it is known that the absolute value of magnetic dipolar term $b_{1}$ is far larger than other terms, which indicates that the excited particle can be seen as a magnetic dipole at maximum point C. Thus in Fig. 4(b) the typical magnetic field intensity distribution caused by a magnetic dipole is observed, the modulus of magnetic field reaches maximum at the center of the particle and rapidly reduces with increasing distance $r$ .

Fig. 4 Scattering properties at two maximum points C and D. (a) Scattering diagram considering all Mie coefficients under both TE and TM polarizations and (b) nearby distribution of the scattered magnetic field intensity around a spherical particle with radius $a = 100 nm$ and permittivity $ε_{p} = 16 - 1.1 i$ at incident wavelength $λ = 832 nm$ . (c) Scattering diagram considering all Mie coefficients under both TE and TM polarizations and (d) nearby distribution of the scattered electric field intensity around a spherical particle with radius $a = 100 nm$ and permittivity $ε_{p} = 16 - 2.2 i$ at incident wavelength $λ = 590 nm$ .

Download Full Size | PDF

Figures 4(c) and 4(d) give the far-field scattering diagram and the simulated distribution of scattered electric field intensity of a spherical particle at incident wavelength $λ = 590 nm$ , here the parameters of the particle are $a = 100 nm$ and $ε_{p} = 16 - 2.2 i$ . From Table 1 it is verified that at maximum point D the dominant term is the electric dipolar coefficient $a_{1}$ , which means that here the excited particle can be treated as an electric dipole. In Fig. 4(d) it is seen that the electric field intensity gets relative maximum near the surface of the particle along the polarization direction of incident electric vector, which is just the typical electric field intensity distribution induced by an electric dipole.

4. Conclusion

In summary, the light scattering of an isotropic homogenous nonmagnetic dielectric spherical particle with different gains has been analyzed. Due to the introduction of gain, in general the scattering intensity of spherical particles will get increased. It is shown that under certain incident wavelength, with specific material permittivity the forward scattering intensity of a spherical particle can be enhanced dramatically, which is attributed to the excitation of a magnetic or electric dipolar resonance. In addition to enhancement effect, it is verified that by adding proper gain in dielectric spherical particle, zero-forward-scattering condition can be fulfilled at certain incident wavelength. When the quadrupole and higher order terms are negligible, zero forward scattering can be observed and the total interference enhancement between electric and magnetic dipolar responses at backward direction will cause the scattering intensity reaching maximum. However, if the quadrupole terms have the same order of magnitude with dipolar terms, coherent effect between dipolar and quadrupole responses will lead to the scattering minimum and maximum shift away from forward and backward directions, respectively. The results in this work may shed light on the manipulation of the directionality of light scattering at nanoscale.

Acknowledgments

We acknowledge financial support from the National Natural Science Foundation of China grant no. 11174222 and no. 91230203, and the National Basic Research Program (973) of China grant no. 2011CB922203.

References and links

1. G. Mie, “Contribution to the optics of turbid media, particularly colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908). [CrossRef]

2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

3. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

4. L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, “Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance,” Proc. Natl. Acad. Sci. U.S.A. 100(23), 13549–13554 (2003). [CrossRef] [PubMed]

5. P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed, “Noble metals on the nanoscale: optical and photothermal properties and some applications in imaging, sensing, biology, and medicine,” Acc. Chem. Res. 41(12), 1578–1586 (2008). [CrossRef] [PubMed]

6. S. Derom, A. Berthelot, A. Pillonnet, O. Benamara, A. M. Jurdyc, C. Girard, and G. Colas des Francs, “Metal enhanced fluorescence in rare earth doped plasmonic core-shell nanoparticles,” Nanotechnology 24(49), 495704 (2013). [CrossRef] [PubMed]

7. J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012). [CrossRef] [PubMed]

8. B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A 25(2), 327–334 (2008). [CrossRef] [PubMed]

9. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010). [CrossRef] [PubMed]

10. M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]

11. B. García-Cámara, J. M. Saiz, F. González, and F. Moreno, “Distance limit of the directionality conditions for the scattering of nanoparticles,” Metamaterials (Amst.) 4(1), 15–23 (2010). [CrossRef]

12. A. Alú and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem,” J. Nanophoton. 4(1), 041590 (2010). [CrossRef]

13. B. García-Cámara, R. A. de la Osa, J. M. Saiz, F. González, and F. Moreno, “Directionality in scattering by nanoparticles: Kerker’s null-scattering conditions revisited,” Opt. Lett. 36(5), 728–730 (2011). [CrossRef] [PubMed]

14. Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Controlling light scattering and polarization by spherical particles with radial anisotropy,” Opt. Express 21(7), 8091–8100 (2013). [CrossRef] [PubMed]

15. R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96(12), 127402 (2006). [CrossRef] [PubMed]

16. H. Bhatt, R. Patel, and R. V. Mehta, “Magnetically induced Mie resonance in a magnetic sphere suspended in a ferrofluid,” J. Opt. Soc. Am. A 27(4), 873–877 (2010). [CrossRef] [PubMed]

17. B. García-Cámara, F. Moreno, F. González, and J. M. Saiz, “Comment on ‘experimental evidence of zero forward scattering by magnetic spheres’,” Phys. Rev. Lett. 98(17), 179701 (2007). [CrossRef] [PubMed]

18. H. Ramachandran and N. Kumar, “Comment on ‘experimental evidence of zero forward scattering by magnetic spheres’,” Phys. Rev. Lett. 100(22), 229703 (2008). [CrossRef] [PubMed]

19. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19(6), 4815–4826 (2011). [CrossRef] [PubMed]

20. R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton. 5(1), 053512 (2011). [CrossRef]

21. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012). [CrossRef] [PubMed]

22. M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28(1), 54–60 (2011). [CrossRef] [PubMed]

23. R. Frank and A. Lubatsch, “Scalar wave propagation in random amplifying media: Influence of localization effects on length and time scales and threshold behavior,” Phys. Rev. A 84(1), 013814 (2011). [CrossRef]

24. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef] [PubMed]

25. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

26. B. Crosignani, P. DiPorto, and S. Solimeno, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

27. X. Hu, J. Cao, M. Li, Z. Ye, M. Miyawaki, and K. M. Ho, “Modeling of three-dimensional photonic crystal lasers in a frequency domain: A scattering matrix solution,” Phys. Rev. B 77(20), 205104 (2008). [CrossRef]

28. A. V. Dorofeenko, A. A. Zyablovsky, A. A. Pukhov, A. A. Lisyansky, and A. P. Vinogradov, “Light propagation in composite materials with gain layers,” Phys. Usp. 55(11), 1080–1097 (2012). [CrossRef]

29. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010). [CrossRef] [PubMed]

30. N. Meinzer, M. Konig, M. Ruther, S. Linden, G. Khitrova, H. M. Gibbs, K. Busch, and M. Wegener, “Distance-dependence of the coupling between split-ring resonators and single-quantum-well gain,” Appl. Phys. Lett. 99(11), 111104 (2011). [CrossRef]

31. I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2(5), 496–500 (2012). [CrossRef]

	$n = 1$ (dipole)	$n = 2$ (quadrupole)	$n = 3$ (octupole)
$a_{n}$ (A)	+ 1.497 × 10⁻¹−4.183 × 10⁻¹i	−2.456 × 10⁻⁴−1.081 × 10⁻²i	−4.585 × 10⁻⁶−1.754 × 10⁻⁴i
$b_{n}$ (A)	−1.497 × 10⁻¹ + 4.183 × 10⁻¹i	−1.571 × 10⁻³−4.052 × 10⁻³i	−8.346 × 10⁻⁶−3.188 × 10⁻⁵i
$a_{n}$ (B)	−5.986 × 10⁻²−2.730 × 10⁻¹i	+ 4.888 × 10⁻³−1.030 × 10⁻¹i	−2.489 × 10⁻⁵−2.777 × 10⁻³i
$b_{n}$ (B)	+ 5.986 × 10⁻² + 2.730 × 10⁻¹i	−1.112 × 10⁻² + 9.978 × 10⁻²i	−4.020 × 10⁻⁴−2.307 × 10⁻³i
$a_{n}$ (C)	+ 9.297 × 10⁻²−3.064 × 10⁻¹i	−3.930 × 10⁻⁵−7.135 × 10⁻³i	−1.036 × 10⁻⁶−9.984 × 10⁻⁵i
$b_{n}$ (C)	−4.013 × 10⁺¹−6.847 × 10⁻¹i	−2.707 × 10⁻⁴−2.139 × 10⁻³i	−1.445 × 10⁻⁶−1.495 × 10⁻⁵i
$a_{n}$ (D)	−2.289 × 10⁺¹−5.833 × 10⁺¹i	−3.529 × 10⁻⁴−4.411 × 10⁻²i	−2.638 × 10⁻⁵−1.097 × 10⁻³i
$b_{n}$ (D)	+ 3.403 × 10⁻² + 2.821 × 10⁻¹i	−8.001 × 10⁻²−2.713 × 10⁻²i	−1.405 × 10⁻⁴−4.555 × 10⁻⁴i

Anomalous forward scattering of dielectric gain nanoparticles

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (6)

Optics Express