Huge light scattering from active anisotropic spherical particles

Xiaofeng Fan; Zexiang Shen; Boris Luk’yanchuk

doi:10.1364/OE.18.024868

1. Introduction

Controlling light energy into nanometer scale is one of the rapidly growing fields of material physics and nanotechnology[1, 2, 3] which hosts a lot of important research directions with potential applications such as high-resolution optical imaging[4, 5], small-scale sensing techniques[7, 8], and numerous biomedical applications[9, 10, 11, 12]. Presently, designing the metal/dielectric interface with the use of surface plasmon polaritons is considered to be an effective approach of manipulating light on nano scale. With the isolated small metal particle, the local electromagnetic field can be enhanced by the surface plasmons localized on the surface and thus be utilized to enhance Raman scattering[13, 14], fluorescence of single molecule[15, 16] and transfer resonantly the energy of exciton[17], and so on. With designing the proper periodic pattern on metal materials or the interface of active medium, dielectric and metal, the different plasmon modes can be hybridized and tune the resonance frequency for guiding electromagnetic wave on nanosized structure[18, 19, 20, 21] and lasing of semiconductor nano particle or fluorescence molecules in dielectric media with assistance of plasmons[22, 23, 24, 25, 26, 27, 28, 29]. Therefore, the anisotropic materials or designed metamaterials with anisotropy have been a subject of great interest, since the merging of plasmonics and these materials may open up a new perspective to control the electromagnetic wave, such as the achievement of negative index metamaterials in the optical frequencies[30].

Prodan et al. have studied the plasmon hybridization of complex nanoshell structures with a metallic shell and a dielectric core[31, 19]. The interaction of bare plasmon modes of individual surface is demonstrated. Furthermore, Bergman et al. predicted that the surface plasmon can be amplified by stimulated emission of radiation and thus result in lasing with the generator of coherent surface plasmons[27]. Recently, it was experimentally demonstrated that the spaser is realized on a conjugate structure with a metallic core and a dye-doped dielectric shell which is as an active medium to overcome the inherent loss of surface plasmon inside metal core[28]. In this paper, we study the fancy effect of light scattering on the small particles with radial anisotropic permittivity. Some progress has been made in this direction about spherical anisotropic structures[32, 33, 34, 35, 36, 37, 38, 39]. The method of moments, coupled-dipole methods, second-harmonic generation approach and expanded Mie theory have been developed to expand the field expressions[32, 33, 34, 36]. Some new effects of light scattering by anisotropic materials, such as the additional increase in field enhancement near surface plasmon resonance frequencies induced by anisotropy, have been analyzed[39]. Here, with modified Mie theory, we will analyze the space of parameters including the transverse permittivity ε_t, the longitudinal permittivity ε_r, the particle size a and wavelength of incident wave λ. We consider the anisotropy of permittivity ε results in the spherical particles have different properties in different directions, such as metallic property and light-active dielectric property. The huge light scattering is found in special region of parameter space after inspecting the extinction, scattering, absorption and radar cross section. This fantastic effect is suspected to be due to the coupling between active medium and plasmons and be possible to result in spaser.

2. Theoretical model

Different analytical methods have been developed to deal with the light scattering with medium with different structures[40, 41]. For planar multilayered structures, the 3D Fourier transform technique was usually used to relate the space and spectral domains to the analysis of the waves and fields[42, 43]. For the boundary-value problems and periodic structures, the Green’s functional technique as a kernel is used to solve the integral equation[44]. For the spherical and cylindrical structures, the Lorenz-Mie approach is a powerful method of separation of variables to expand angularly the electromagnetic field[45, 46, 47, 48, 49]. Here, we chose to modify the Mie theory to deal with the particle with radial anisotropy[38, 40].

We assume that the plane wave with the electric field polarized along the x axis is scattered by the particle immersed in homogeneous medium. Considering the scattering about monochromatic wave, the time dependence e^−iωt part can be suppressed and the electric and magnetic vectors satisfy the time-free Maxwell’s equations:

\nabla \vec{H} = - i k_{0} ɛ \cdot \vec{E} and \nabla \vec{E} = {ik}_{0} μ \cdot \vec{H},

where k₀ is the wave vector of light in vacuum. The uniaxial anisotropy of particle is defined by the constitutive tensors of the permittivity and permeability as:

ɛ = (\begin{matrix} ɛ_{r} & 0 & 0 \\ 0 & ɛ_{t} & 0 \\ 0 & 0 & ɛ_{t} \end{matrix}) and μ = (\begin{matrix} μ_{r} & 0 & 0 \\ 0 & μ_{t} & 0 \\ 0 & 0 & μ_{t} \end{matrix})

where the coordinate system is spherical coordinate, and ε_r, ε_t, μ_r and μ_t are the longitudinal permittivity, transverse permittivity, longitudinal permeability and transverse permeability, respectively. In general case, these four values ε_r, ε_t, μ_r and μ_t will be complex numbers, i.e. ε_r = Re[ε_r] + i Im[ε_r], ε_t = Re[ε_t] + i Im[ε_t], etc. As usual, the electric and magnetic vectors can be expressed by the Debye’s scaler potentials Π_TE and Π_TM. Thus, the Maxwell vector equations are transferred as the scaler equations about the magnetic Π_TE and electric Π_TM potentials, which are expressed as:

\frac{ɛ_{r}}{ɛ_{t}} \frac{\partial^{2} Π_{TM}}{\partial r^{2}} + \frac{1}{r^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial Π_{TM}}{\partial θ}) + \frac{1}{r^{2} {sin}^{2} θ} \frac{\partial^{2} Π_{TM}}{\partial φ^{2}} + k_{0}^{2} ɛ_{r} μ_{t} Π_{TM} = 0,

\frac{μ_{r}}{μ_{r}} \frac{\partial^{2} Π_{TE}}{\partial r^{2}} + \frac{1}{r^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial Π_{TE}}{\partial θ}) + \frac{1}{r^{2} {sin}^{2} θ} \frac{\partial^{2} Π_{TE}}{\partial φ^{2}} + k_{0}^{2} ɛ_{t} μ_{r} Π_{TE} = 0,

for the scattering by particle with anisotropic permittivity ε and permeability μ. Solving these equations with corresponding boundary conditions, the scattering amplitudes

B_{l}^{e}

(electric) and

B_{l}^{m}

(magnetic) can be found to be as:

B_{l}^{e} = i^{l + 1} \frac{2 l + 1}{l (l + 1)} b_{l}^{e} and B_{l}^{m} = i^{l + 1} \frac{2 l + 1}{l (l + 1)} b_{l}^{m},

with,

b_{l}^{e} = \frac{\sqrt{ɛ_{t}} Φ_{l}^{'} (k_{0} a) Φ_{v_{1}} (k_{t} a) - \sqrt{μ_{t}} Φ_{l} (k_{0} a) Φ_{v_{1}}^{'} (k_{t} a)}{\sqrt{ɛ_{t}} ξ_{l}^{'} (k_{0} a) Φ_{v_{1}} (k_{t} a) - \sqrt{μ_{t}} ξ_{l} (k_{0} a) Φ_{v_{1}}^{'} (k_{t} a)},

b_{l}^{m} = \frac{\sqrt{ɛ_{t}} Φ_{l} (k_{0} a) Φ_{v_{2}}^{'} (k_{t} a) - \sqrt{μ_{t}} Φ_{l}^{'} (k_{0} a) Φ_{v_{2}} (k_{t} a)}{\sqrt{ɛ_{t}} ξ_{l} (k_{0} a) Φ_{v_{2}}^{'} (k_{t} a) - \sqrt{μ_{t}} ξ_{l}^{'} (k_{0} a) Φ_{v_{2}} (k_{t} a)},

where

k_{t} = k_{0} \sqrt{ɛ_{t} μ_{t}}

is the wave vector in anisotropic spheres. the functions Φ_l(x) and ξ_l(x) are given by

Φ_{l} (x) = \sqrt{\frac{π x}{2}} J_{l + \frac{1}{2}} (x),

ξ_{l} (x) = \sqrt{\frac{π x}{2}} (J_{l + \frac{1}{2}} (x) + i N_{l + \frac{1}{2}} (x))

where J_l(x) and N_l(x) are usual Bessel function and Neumann function. The order v₁ and v₂ of the spherical Bessel function Φ_v₁ and Φ_v₂ are:

v_{1} = {[l (l + 1) \frac{ɛ_{t}}{ɛ_{r}} + \frac{1}{4}]}^{1 / 2} - \frac{1}{2},

and

v_{2} = {[l (l + 1) \frac{μ_{t}}{μ_{r}} + \frac{1}{4}]}^{1 / 2} - \frac{1}{2} .

With the solution about Π_TE and Π_TM, we can analyze the distribution of electromagnetic fields around the particle. The lost of total energy from incident wave and the energy flux due to backscattering from the particle can be also analyzed. With optical cross-section theorem, the forward scattering amplitude can be evaluated by extinction, scattering and absorption cross sections. The backward scattering amplitude can be evaluated by radar cross section. For the convenience of discussion, the dimensionless cross sections Q is introduced by the formula Q = σ_sc/σ_geom, where σ_sc is the optical cross-section of the particle and σ_geom = πa² is the geometrical cross section with the radius a. By the scattering amplitudes $b_{l}^{e}$ and $b_{l}^{m}$ , the dimensionless extinction, scattering and backscattering cross section can be expressed as:

\begin{array}{r} Q_{ext} = \frac{2}{k_{0}^{2} a^{2}} \sum_{l = 1}^{\infty} (2 l + 1) Re (b_{l}^{e} + b_{l}^{m}), \\ Q_{sca} = \frac{2}{k_{0}^{2} a^{2}} \sum_{l = 1}^{\infty} (2 l + 1) [{| b_{l}^{e} |}^{2} + {| b_{l}^{m} |}^{2}], \\ Q_{rbs} = \frac{1}{k_{0} a} Re {| \sum_{l = 1}^{\infty} {(- 1)}^{l} (2 l + 1) (b_{l}^{e} - b_{l}^{m}) |}^{2} . \end{array}

With the extinction and scattering cross section, the dimensionless absorption cross section can be defined by the formula, Q_abs = Q_ext – Q_sca. For discussing the light scattering due to surface plasmons, the scattering amplitudes in equ.(6)and(7) can be expressed in a more convenient form:

b_{l}^{e} = \frac{F_{b}^{e} (l)}{F_{b}^{e} (l) + i G_{b}^{e} (l)} and b_{l}^{m} = \frac{F_{b}^{m} (l)}{F_{b}^{m} (l) + i G_{b}^{m} (l)},

with,

\begin{array}{l} F_{b}^{e} = \sqrt{ɛ_{t}} Φ_{l}^{'} (k_{0} a) Φ_{v 1} (k_{t} a) - \sqrt{μ_{t}} Φ_{l} (k_{0} a) Φ_{v 1}^{'} (k_{t} a), \\ G_{b}^{e} = \sqrt{ɛ_{t}} χ_{l}^{'} (k_{0} a) Φ_{v 1} (k_{t} a) - \sqrt{μ_{t}} χ_{l} (k_{0} a) Φ_{v 1}^{'} (k_{t} a), \\ F_{b}^{m} = \sqrt{ɛ_{t}} Φ_{l} (k_{0} a) Φ_{v 2}^{'} (k_{t} a) - \sqrt{μ_{t}} Φ_{l}^{'} (k_{0} a) Φ_{v 2} (k_{t} a), \\ G_{b}^{m} = \sqrt{ɛ_{t}} χ_{l} (k_{0} a) Φ_{v 2}^{'} (k_{t} a) - \sqrt{μ_{t}} χ_{l}^{'} (k_{0} a) Φ_{v 2} (k_{t} a), \end{array}

where

χ_{l} (x) = \sqrt{\frac{π x}{2}} N_{l + \frac{1}{2}} (x)

.

3. Results and Discussion

The special properties of surface plasmon are expected to exist in the anisotropic materials. Since the electron collective movements are shown clearly in metal, let us consider firstly the strange effect in small isotropic metallic sphere. The relative dielectric permittivity in the Drude model is described as:

ɛ_{D} = 1 - \frac{ω_{p}^{2}}{ω^{2} + i γ ω} with ω_{p} = {(\frac{{ne}^{2}}{ɛ_{0} m_{0}})}^{1 / 2},

where ω_p, γ, n, e m₀ and ε₀ are the plasma frequency, frequency of electron collisions, concentration of electron, charge of electron, mass of electron and vacuum permittivity, respectively. Obviously, the value is complex, i.e. ε_D = Re[ε_D] + i Im[ε_D]. Here, ω_p exhibits the properties of bulk plasmons. With the decrease of size, the new plasmon modes, surface plasmons will be possible to be excited. The nondimensional quantity

q = a \sqrt{ɛ_{m}} k_{0}

, can be as size parameter to analyze the scattering effects with the radius of the spherical particle a and the dielectric permittivities of media ε_m. At small q, the electric dipole scattering plays the dominant role. the scattering from the magnetic amplitudes

b_{l}^{m}

can be also neglected. Now we consider the case which is far from the resonances. The amplitude

b_{1}^{e}

of electric dipole can be approximately expressed as

(- 2 i / 3) \frac{ɛ_{D} - 1}{ɛ_{D} + 2} q^{3}

. This results in classical extinction efficiency

Q_{scat}^{R} (≃ \frac{8}{3} {| \frac{ɛ_{D} - 1}{ɛ_{D} + 2} |}^{2} q^{4})

of Rayleigh scattering. It should be noticed that the

Q_{scat}^{R}

has a singularity at ε_D = −2. Therefore, with neglecting the frequency of electron collisions (γ = 0), it is obtained that the resonance frequency of dipole surface plasmon

ω_{s p} = ω_{p} / \sqrt{3}

. By introducing normalized frequency ω_R = ω/ω_sp and normalized collision frequency γ_R = γ/ω_sp, the Drude dielectric permittivity can be rewritten as:

ɛ_{D} = 1 - \frac{3}{ω_{R}^{2} + γ_{R}^{2}} + i \frac{γ_{R}}{ω_{R}} \frac{3}{ω_{R}^{2} + γ_{R}^{2}} .

For the cases of weak dissipation (γ_R ∈ [10⁻¹, 10⁻³]), the imaginary part of dielectric permittivity Im[ε] will be less than 0.3. In the process of deducing Rayleigh formula, the

F_{b}^{e} (l)

in denominator part of

b_{l}^{e}

is ignored, relative to

G_{b}^{e} (l)

. However, for the case which is near the resonances (

G_{b}^{e} (l) \to 0

), the

F_{b}^{e} (l)

will can not be ignored. Actually,

F_{b}^{e} (l)

corresponds to the radiative damping as shown in Refs.[50, 51]. With the consideration of radiative damping, the singularity of Rayleigh extinction efficiency will be disappeared. Meanwhile, a series of surface localized electromagnetic modes with the resonance frequencies

ω_{R} (l) (\sim \sqrt{3 l / (2 l + 1)})

appear in the formula for small particle. This means the surface plasmons will be possible to be excited when the condition (Re[ε_D] < −1) is satisfied for the real part of dielectric permittivity Re[ε_D] of the particle.

Under the nondissipative limit (Im[ε_D] = 0), the resonance extinction cross section from small particle increases with increase in the order of the resonance modes, as shown in Ref[50]. However, in the real metal materials, the different dissipative mechanisms, such as the electron-electron collision, electron-phonon coupling and electron-defect interaction, will result in the damping of the collective moving of electrons with Im[ε_D] > 0. With the weak dissipative damping, the resonance frequencies are not changed obviously, whereas the extinction cross section of each resonance modes will decrease quickly. As shown in Fig. 1, the decrease of the cross sections of higher order modes is quicker than that of dipole resonance cross sections.

Fig. 1 The maximal value of Q_ext for each resonance mode as a function of the dissipative damping Im[ε_D] for the size q = 1 (A) and q = 0.5 (B). The Q_ext as a function of both frequency ω and Im[ε_D] for dipole and quadrupole modes of the particle with size q = 1 shown in the inset of (A).

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For the particle size q ≤ 1 with Im[ε_D] ≥ 0.07 at each resonance frequency, the dipole resonance becomes greater than that of the higher order modes, such as quadruple. For an example, the dipole resonance of the particle with size q = 0.5 becomes the dominant resonance mode when the image part of permittivity ε_D is larger than 0.02. The quick decrease of cross section of high order resonance modes may be attributed to their relative small characteristic widths. As we known, the natural width of the arbitrary resonance is given by the following expression[50]:

γ_{l} = \frac{q^{2 l + 1} (l + 1)}{{[l (2 l - 1)!!]}^{2} {(d ɛ_{D} / d ω)}^{l}}

where the derivative (dε_D/dω)^l is taken at the corresponding resonance frequency ω = ω_l. There is an extremely sharp decrease in γ_l with the increase of l. At the same time, the characteristic width γ_l will increase and the different resonance modes become to incorporate, following the increase of size q. As shown in the inset of Fig. 1A, the incorporation between dipole and quadruple resonances induces that the quadruple resonance still holds an important rule in scattering process even at Im[ε_D] = 0.3 for the size q = 1. Whatever, we can find that the dipole resonance mode should be considered to be as an important role for the application of surface plasmons.

Since the small size is more effective for the surface plasmons as found in Fig. 1A and B, we will detect the giant resonance cross section in the size rang 0 < q < 1 for the small metal sphere. As the result of Rayleigh formula, the dipole scattering section increases quickly following ε_D reaches asymptotically the singular value −2 (in Fig. 2A) under the nondissipative limit, while the size q arrives at the limit value 0 from Mie theory as shown in Fig. 4A. With the dissipative damping, the resonance frequency of the maximal resonance cross section will have a large change, though the resonance frequency is just with an inconspicuous change by following the increase of Im[ε_D] at fixed size q. As Fig. 2A shown, the maximal resonance frequency has a red-shift with the decrease of the maximal cross section as a function of Im[ε_D]. This means the maximal resonance scattering should be appeared on a particle with a limited small size and not on a particle with q → 0.

Fig. 2 The maximal value of Q_ext as a function of Re[ε_D] with different Im[ε_D] (A), the maximal or minimal values of Q_sca and Q_abs as a function of Im[ε_D] with Re[ε_D] = −2.2 (B), maximal value of Q_ext as a function of Im[ε_r] with different Re[ε_r] and the ratio ε_t/ε_r (C and D).

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Fig. 4 Log[Q_ext] as a function of permittivity ε_D and size q under the nondissipative limit (Im[ε_D] = 0) (A), the maximal value of Log[Q_ext] as a function of Re[ε_D] and Im[ε_D] (B), the maximal value of Log[Q_sca] as a function of Im[ε_r] and the ratio |ε_t|/|ε_r| for Re[ε_r] = −2.5 and ε_t = (−2.5 + i Im[ε_r])|ε_t|/|ε_r|(C), Log[Q_sca] as a function of the ratio |ε_t|/|ε_r| and size q for ε_r = −2.5 −0.1i and ε_t = (−2.5 −0.1i)|ε_t|/|ε_r|(D).

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Near plasmon resonance frequencies, the anisotropy of permittivity can lead to an additional increase in field enhancement has been shown. In Fig. 3A, the maximal scattering section Q_{max ext} as a function of both ε_r and ε_t/ε_r is demonstrated under the nondissipative limit (Im[ε_r] = 0 and Im[ε_t] = 0). It can be found that the maximal value of Q_{max ext} will exist at the relative small ratio of ε_t to ε_r with the decrease of ε_r. Since the scattering efficiencies are inversely proportional to q², the change of Q_{max ext} following the change of the ratio ε_t/ε_r may be due to the shift in the resonance positions. As Fig. 3B shown, the maximal value of Q_ext is at the small size q with the decrease of the ratio ε_t/ε_r. Furthermore, the Q_ext also increases with the decease of q and increase of ε_r at fixed ratio ε_t/ε_r = 0.75 shown in Fig. 3C. Therefore, the enhancement of resonance scattering efficiencies can be attributed mostly to the decrease of the size q.

Fig. 3 Log[Q_{max ext}] as a function of both ε_r and ε_t/ε_r (A), Q_ext as a function of q and ε_t/ε_r with fixed longitudinal permittivity ε_r = −2.5 (B) and Q_ext as a function of q and ε_r with fixed ratio ε_t/ε_r = 0.75 (C), with the nondissipative limit (Im[ε_r] = 0 and Im[ε_t] = 0).

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As we shown, the resonance scatting section decreases quickly with the enhancement of the dissipative damping. If an assumed active mechanism with the negative value for Im[ε_D] is introduced, whether the scattering efficiency can increase need to be checked further. In Fig. 2B, the resonance cross sections of scattering (Q_{max sca}) and absorption (Q_{min abs} or Q_{max abs}) as the functions of Im[ε_D] with the fixed Re[ε_D] (= −2.2) are analyzed. we find that Q_{max sca} has a maximal value at the point with Im[ε_D] = −0.05. At the same time, there is a minimal value for Q_{min abs}. It is well known that the absorption is attributed to the dissipative damping. With the decrease of Im[ε_D], the efficiency of absorption decreases, whereas the resonance scattering cross section increases. Thus, there is a maximal value for Q_abs at special point of Im[ε_D]. At the limit of small size parameter, the maximal value of Q_abs is given by $Q_{abs \max}^{l} = \frac{1}{q^{2}} (l + 1 / 2)$ [52]. Then at the nondissipative limit, the absorbtion cross section (Q_abs) will become zero. The negative Q_sca at the negative Im[ε_D] means the particle can emit light with some active mechanism. Therefore, the minimal value of Q_{min abs} at Im[ε_D] = −0.05 means there is light emitting with special resonance mechanism. In Fig. 4B, the Q_{max sca} as the function of both Re[ε_D] and Im[ε_D] is shown. It is found that the maximal resonance scattering cross section exists with the more negative value of Im[ε_D] for the smaller Re[ε_D]. In Fig. 2 C, D and Fig. 4 C, we demonstrate that the anisotropy of permittivity with different negative image part may result in the more large resonance cross sections of scattering and emitting. In Fig. 4D, we found that the giant resonance cross section is located at the region of special particle size.

In practice, the active metal is impossible to exist, since the overlap of valance band and conduction band at the Fermi level will result in an impotent radiation transition for any visible light and the energy gain from the model of absorbtion-emission can not be realized. However, if the dielectric or insulate, such as silica, is introduced in the system, the light absorbtion-emission model can be used to be as the mechanism of energy gain. Therefore, the active mechanism can be introduced by the anisotropic permittivity with ε_r or ε_t for the active medium. Then it is expected that the light gain can be coupled with the collective moving of electrons from metal. The giant resonance cross section for light scattering and adsorption (or emission) will be possible to be obtained.

For the metal particle, the giant scattering resonance cross section due to the surface plasmons is localized at the small size with q < 1. For dielectric particle, there isn’t the ability to trap the electromagnetic field which results in the scattering cross section is small in the nano size for visible light. Since the surface plasmons mechanism is more effective for small size, we will limit the size of the anisotropic spherical particle in the region of 0 < q ≤ 2. As Fig. 5A shown, the maximal value of Q_sca is as a function of Re[ε_t] and Im[ε_t] with fixed longitudinal permittivity ε_r = 2.5 – 0.05i. We can found that there is a region with Re[ε_t] ∼ −1.8 for giant scattering cross section duo to the resonance. It is known that the resonance from the electric dipole mode needs ε ≤ −2 for metal particle. Therefore, it is possible that there is a relation for both ε_r and ε_t in the region of resonance. It is also found that Im[ε_t] is less than 0.05 for the region of resonance. It seems that the energy gain from the active dielectric part (ε_r) must be larger than the dissipative damping from the metal part (ε_t) for the giant resonance cross section.

Fig. 5 the resonance cross section Log[Q_sca] as a function of transverse permittivity (Re[ε_t] and Re[ε_t]) with the fixed longitudinal permittivity ε_r = 2.5 – 0.05i (A), the maximal value of scattering amplitude | $b_{1}^{e}$ | as a function of Re[ε_t] and Im[ε_t] with the fixed ε_r = 2.5 – 0.05i (B).

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Is it possible that the giant resonance cross section is from the contribution of different scattering modes? After the analysis of scattering cross section, it is found that the electronic dipole mode is the major contribution for the special anisotropic particle with the size 0 < q ≤ 2. In Fig. 5B, the absolute value of the scattering amplitude $b_{1}^{e}$ from electric dipole mode as a function of Re[ε_t] and Im[ε_t] with fixed longitudinal permittivity ε_r = 2.5 – 0.05i is shown. It demonstrates the major contribution of electronic dipole mode. Therefore, it is considered that the electronic dipole mode will be possible to be the effective major mode to couple with the active medium for the particle with size 0 < q ≤ 2. Firstly, the longitudinal permittivity ε_r as the active medium is considered to be contributed to the the giant scattering cross section. In Fig. 6A, | $b_{1 \max}^{e}$ | as a function of Re[ε_r] and Re[ε_t] with fixed image parts Im[ε_r] = 0.05 and Im[ε_t] = −0.008 is shown. Obviously, there is a quasi-linear region for the resonance cross section. By fitting the extremum of | $b_{1 \max}^{e}$ | about Re[ε_r] and Re[ε_t], we find there is a quasi-linear relation (Re[ε_t] = − 2.62 + 0.608Re[ε_r] − 0.110Re[ε_r]² + 0.008Re[ε_r]³) for resonance region. Secondly, the transverse permittivity ε_t is considered to be as the active medium for the giant scattering cross section. In Fig. 6B, | $b_{1 \max}^{e}$ | as a function of Re[ε_r] and Re[ε_t] with fixed image parts Im[ε_r] = 0.001 and Im[ε_t] = −0.02 is shown. It is found that there is also a linear relation about Re[ε_r] and Re[ε_t] for the resonance region. By fitting the value of | $b_{1 \max}^{e}$ |, the linear relation, Re[ε_t] = 0.389 – 0.154Re[ε_r], is found.

Fig. 6 the maximal value of scattering amplitude | $b_{1}^{e}$ | as a function of Re[ε_r] and Re[ε_t] with fixed Im[ε_r](= −0.05) and Im[ε_t](= 0.008) for ε_r as active medium (A), with fixed Im[ε_r](= 0.001) and Im[ε_t](= −0.02) for ε_t as active medium (B).

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With the relation about the real parts of both ε_r and ε_t, we can detect the dependence of the size about the resonance cross section in the resonance region in detail. From Fig. 7A and B, it is found that the size for resonance cross section changes with the real part of permittivity. Furthermore, by the relation of Re[ε_r] and Re[ε_t], the size is more dependent to the permittivity with the dielectric property. Then we can explore the dependence of resonance cross section to the dissipative damping from metal and the gain energy of active dielectric. Considering the weak dissipative limit, we just analyze the Im[ε_t] or Im[ε_r] in the region [0, 0.3] for the damping. As shown in Fig. 7C and D, there is the special relation bewteen Im[ε_t] and Im[ε_r] for the resonance process. It gives us an amazed conclusion that it is not the case that larger energy gain results in a stronger coupling to surface plasomons and then a larger resonance cross section. In additional, with the special ratio of Im[ε_t] and Im[ε_r], there are the giant resonance cross sections, such as for backscattering shown in Fig. 8. The giant cross sections should be attributed to the the resonance of electric dipole mode with the assistance of gain energy from the active ε_r or ε_t and this maybe result in spaser.

Fig. 7 the scattering amplitude $Log [| b_{1}^{e} |]$ as a function of Re[ε_r] and q with fixed Im[ε_r](= −0.05) and Im[ε_t](= 0.008) (Re[ε_t] is chosen by the formula Re[ε_t] = −2.62 + 0.608Re[ε_r] − 0.110Re[ε_r]² + 0.008Re[ε_r]³)(A), the scattering amplitude $Log [| b_{1}^{e} |]$ as a function of Re[ε_r] and q with fixed Im[ε_r](= 0.001) and Im[ε_t](= −0.02) (Re[ε_t] is chosen by the formula Re[ε_t] = 0.389 – 0.154Re[ε_r])(B), the maximal value of scattering amplitude $Log [| b_{1}^{e} |]$ as a function of f(Im[ε_r]) and Im[ε_t] with ε_r = 3 – f(Im[ε_t])Im[ε_t] i and Re[ε_t] = −1.5604 (C) and the maximal value of scattering amplitude $Log [| b_{1}^{e} |]$ as a function of f(Im[ε_r]) and Im[ε_r] with Re[ε_r] = −12 and ε_t = 2.237 – f(Im[ε_r])Im[ε_r]i (D).

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Fig. 8 High scattering efficiencies with huge radar backscattering cross section. Q_rbs as the function of size q for ε_r with the property of energy-gain (A and B) and for ε_t with the property of energy-gain (C and D).

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4. Conclusion

With the expanded Mie theory, the Maxwell’s equations are solved by the Debye’s scalar potentials and the electromagnetic fields are expressed in terms of Bessel functions with Legendre functions. Thus, the light scattering of spherical structure with anisotropic permittivity can be analyzed effectively. For metal particle, small dissipative damping will result in the rapid decrease of resonance scattering cross section. Furthermore, the resonance cross section of higher order scattering mode deceases more quickly than that of dipole mode. This explains that why the electric dipole approximation is an effective method for small metal particle, and it also implies that the dipole mode may be an effective way to control light energy into nanometer scale.

With designing the transverse permittivity ε_t and the longitudinal permittivity ε_r, we can introduce an active mechanism to compensate the damping dissipation of plasmons and even enhance the resonance of plasmons. With the analysis of extinction, scattering, absorption and radar cross section, the electric anisotropy effects on scattering efficiency are studied systematically for small partial. Following the change of the transverse permittivity ε_t, the longitudinal permittivity ε_r and the particle size q, the huge scattering cross sections are found at the regions with special parameters. The huge cross sections are considered to be due to the coupling between the surface plasmon with electric dipole mode and the active medium. This coupling results in the strong light emitting and scattering.

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Huge light scattering from active anisotropic spherical particles

Abstract

1. Introduction

2. Theoretical model

3. Results and Discussion

4. Conclusion

References and links

Cited By

Figures (8)

Equations (17)

Optics Express