Tunable Anomalous Scattering and Negative Asymmetry Parameter in a Gain-Functionalized Low Refractive Index Sphere.

Usually, low refractive index passive spheres exhibit strong forward scattering and a positive asymmetry parameter due to weak interference between the electric and magnetic scattering channels. In this work, we investigate, analytically and numerically, the forward scattering of light by a gain-functionalized low refractive index dielectric sphere. It is shown that by tuning the optical gain one can optimize the interference, which provides a novel paradigm to achieve the zero forward scattering and negative asymmetry parameter even for a low refractive index sphere. As a result, a low-density collection of such identical back scatterers provides an anomalous regime, where the scattering mean free path and extinction mean free path are greater than the transport mean free path. Furthermore, we also provide the numerical guideline to achieve the larger extinction mean free path without achieving preferential back-scattering.

Introduction

The scattering of light by an individual subwavelength spherical particle has remained a subject of great interest with fundamental importance for the understanding of light–matter interaction with countless applications.[1−4] One of the most desired scientific goals in the optical scattering problem is to dictate the scattered field distribution around the particle.[5,6] The optimal control on the direction of scattered momentum is a matter of interest in light-emitting and light-guiding devices[7,8] and optical manipulation.[9−12] The direction of scattered light by a spherical particle can accurately be determined in terms of material properties and the length scale involved in the course of interaction[13,14] using the Mie scattering theory.[15]

With this perspective, the direction of the scattered light by a spherical particle can be explained using the asymmetry parameter (g), which is the average cosine of all scattering angles and is given as g ≡ ⟨cos θ⟩,[13,14] where θ is the angle between the incident wavevector to the scattered wavevector. For instance, a scatterer with g = 1 exhibits zero back-scattering, and g = −1 leads to zero forward scattering, which are the so-called first and second Kerker’s conditions,[16] respectively. According to Kerker’s criterion, the extremum values of the asymmetry parameter, g = ±1, can be achieved for magnetic spheres. In contrast, the electromagnetic scattering by a usual nonmagnetic dielectric sphere is elongated in the forward direction, and hence the asymmetry parameter is expected to be 1 > g > 0. However, as a spacial case, g = 0 can be achieved for a dipole-like scattering. However, by considering a large refractive index sphere and composite sphere, one can achieve zero backward scattering with g ≈ 1, which occurs due to induced magnetic dipoles owing to the rotation of displacement currents inside the sphere.[17−20] On the contrary, the second Kerker’s condition hardly occurs in ordinary nonmagnetic scatterers due to the fundamental restriction imposed by the optical theorem. To be more specific, the observation of the second Kerker’s condition not only is challenging but also is considered to be impossible to observe for the passive sphere, as required by causality.[21]

In the past two decades, some works have been reported demonstrating the negative asymmetry parameter by a magnetic sphere,[22] a sphere with large refractive index,[19,20,23,24] a plasmonic passive sphere,[25] and an active nanoparticle with a large refractive index.[26−28] In addition, the negative asymmetry parameter is also achieved by applying an external magnetic field[29] and in a cluster made of short-range correlated identical scatterers.[30] Nevertheless, to the best of our knowledge, no study has been proposed presenting a negative or turntable asymmetry parameter in a low refractive index sphere, especially in the visible frequency range.[31]

The negative asymmetry parameter plays a key role in many applications, where the switching of the scattered light direction is not only required but also should be achieved with optimal control. For instance, the structural information on a random medium consisting of a dilute dispersion of identical spheres can be extracted by means of unusual characteristic parameters, such as extinction mean free path (EMFP), scattering mean free path (SMFP), and transport mean free path (TMFP), denoted by , , and , respectively. The EMFP describes the attenuation of the averaged field inside the medium, which is equal to the SMFP for a lossless medium. Moreover, the EMFP and SMFP can be defined in terms of the TMFP and the average single particle scattering properties such as extinction efficiency, Qext, scattering efficiency, Qs, and the asymmetry parameter.[32]

With this perspective, the negative asymmetry parameter plays a crucial role, and it may lead to an unusual multiple light-scattering regime, where the becomes larger than the . This achievement has important applications in many characterization and imaging methods such as the fabrication-tolerant broad-band operation of random spectrometers, where and are needed to be larger than .[33−35] A usual low refractive index sphere exhibits large forward scattering and positive g, which imposes severe limitations to achieve large values of and .

In this work, we present an alternative strategy to achieve negative g, along with tunable EMFP and SMFP, by employing the optical gain in the visible frequency domain. We show that the gain-functionalized multipolar particle exhibits a preferential back-scattering which can be tailored experimentally, for instance, by tuning the power of the pump laser that governs the optical gain level. Furthermore, using a realistic gain model, we numerically (by Comsol multiphysics simulation) demonstrate that by varying the gain level inside the sphere, one can achieve zero forward scattering, even for a sphere of remarkably low refractive index compared to the ones reported so far.[31] As a result, for a low-density collection of such back-scattering identical scatterers, an anomalous regime occurs, where the EMFP becomes larger than the TMFP, that is, . In addition, we also propose an alternative strategy for tunable EMFP and show that by varying the optical gain level one can achieve the larger EMFP without relaying the negative g.

Theoretical Model

Let us consider an electromagnetic plane wave with vacuum wavelength λ0 impinging on an isotropic nonmagnetic sphere of radius a, immersed in a non-absorbing medium of refractive index n. The Mie scattering theory provides the exact scattering solution to quantify the energy received from the incident field, energy scattered, and energy absorbed by the sphere in terms of extinction cross section Cext, scattering cross section Cs, and absorption cross section Ca, respectively. As per convention, we normalize these cross sections by a geometric cross section of the sphere πa2, which leads us to define normalized extinction and scattering and absorption efficiencies as[14]andrespectively, where the index denotes the order spherical harmonic, and x = ka is the size parameter in the surrounding medium with k = 2π/λ, where λ = λ0/n. In addition, and are the Mie scattering coefficients that can be calculated by applying the subsidiary boundary conditions at the sphere surface and given as[14,36]where and are Riccati-Bessel functions and m and μs are the relative refractive index and relative permeability of the sphere, respectively. In the far-field domain, the scattered radiant intensity can be defined aswhere I1 and I2 are the transverse electric (ϕ = 90°) and transverse magnetic (ϕ = 0°) components of the scattered intensity, respectively. In addition, θ and ϕ are the polar and the azimuthal angles, respectively. Furthermoreandare the components of the scattering matrix,[14] where π(cos θ) and τ(cos θ) describe the angular scattering patterns of the spherical harmonics.[14] Finally, the scattering efficiency in the forward direction (θ = 0) is given as

It can be seen in eq that the second Kerker’s condition can be achieved by solving , which presents multiple solutions for the Mie sphere. However, in the long wavelength region (λ ≫ a) where only dipolar terms (a1, b1) are excited, Qf ≃ 0 can be achieved by providing a1 ≃ −b1. This occurs when the condition is satisfied, where ϵs is the relative permittivity of the sphere.[16,21,23,36] Otherwise for a nonmagnetic sphere the second Kerker’s condition is difficult to achieve, as required by the optical theorem to preserve the causality.[21] It can be explained analytically by solving eq , where Qf = 0 occurs only if and vanish simultaneously. However, for a passive sphere, real parts of and are obliged to be positive to preserve the passivity condition. Thus, sum of nonzero (positive) real scattering amplitudes can never be zero. These limitations can be circumvented by employing the optical gain in the sphere that would allow one to achieve negative or . Thus, introduction of the optical gain is a necessary requirement to realize zero forward scattering.

Modeling of the Gain Media

The optical gain media can be designed by embedding the gain elements (e.g., fluorescent dyes molecules or quantum dots) inside a host medium of permittivity ϵs. When the gain elements are excited by means of pump frequency, they emit photons due to their de-excitations, as shown in Figure a. As a result, close to the resonance frequency of the gain elements, the composite material shows a negative imaginary refractive index that characterizes the active nature of the composite particle under exp– monochromatic time-harmonic convention. For our purpose, we consider solvatochromic dye LDS 798 molecules[37,38] as a four-level atomic system, and each level has an occupation number density N, where i = 0, 1, 2, 3, as sketched in Figure a. The total number of dye molecules per unit volume of the host medium are represent by Ndye that can be given as ∑3N = Ndye. As the incident pump laser excites the dye molecules from N0 to N3 during the de-excitation, they make a transition from the ith level to the jth level (ith> jth) with lifetime τ, where τ21 ≫ {τ32, τ10}. Thus, frequency emitted due to the transition between the level 2 to level 1 would play a key role in the design of the gain media that we call probing frequency, as shown in Figure a. Finally, the effective permittivity of such a heterogeneous composite medium can be expressed by the following dispersion relation:[39−43]where ω = 2πc/λ, c is the speed of light, Δ = 2/τ, τ is the relaxation time constant associated with energy relaxation processes of the dye molecules,[37,41] μd is the dipole moment transition amplitude of the dye molecules, ωa is the emission frequency, ℏ is the reduced Plank constant, and ϵ0 represents the permittivity of free space. In addition, Ñ is the population inversion which is counted in terms of the number of dye molecules in the excited state. The population inversion Ñ can be controlled through the action of the external pump laser: for no pump, all dye molecules are in the ground state N0 and hence Ñ = 0; on the other hand, Ñ = 1 can be achieved for a larger pump power that sets the all dye molecules in the excited state. By varying the pumping power, one can tune the gain level which allows the inferences between the electric and magnetic scattering amplitudes to be tailored. It is worth noting that the probing frequency ω is substantially different than the pump frequency (as sketched in Figure a); therefore, scattering contribution from the pump frequency can be blocked, thereby, we are not considering it in the Mie scattering calculations. Thus, the radiation pattern around the scatterer can be defined using the Mie scattering theory in terms of the asymmetry parameter:[14]The multiple scattering theory of random media made of dilute concentrations of identical scatterers provides a connection between the to the and in terms of single particle scattering cross sections and asymmetry parameter. In this assumption, the average separation between the particles is considered to be significantly larger than the particle size without positional correlation. The relation between and of light in a such dilute medium is defined as[31,32] and can be defined in terms of transport cross section Ct and scattering cross sections Cs as[31,32]where ρ is the density of scatterers in the medium. As we are considering an active sphere and the absorption cross section of the sphere would be nonzero, therefore, it is worthwhile to discuss the absorption mean free path which can be defined as .[44,45] In this case, the EMFP should be different than the SMFP that can be expressed as .[32,44,45] For the active media, it is more convenient to investigate the TMFP in terms of the EMFP and single particle scattering properties using the following relation:[25,31]It is clear that for a losses medium Ca = 0 and .

Figure 1

(a) Schematic illustration of problem: a dielectric sphere is doped with dye molecules. An incident beam and laser pump are illuminating the sphere, where ki, ks, and ke are representing the wave vectors of the incident, scattered, and emitted light, respectively. In the inset, a four-level energy scheme is presented, where the laser pump with pump frequency higher than the probing frequency inverts the population of dye molecules. In this way, the excited dye molecules emit photons of frequency ω. (b) Forward scattering efficiency Qf as a function of incident wavelength by a passive sphere Ñ = 0 (dashed) and an active sphere Ñ = 0.61 (solid). (c) Scattered field distribution around the particle for fixed Ñ = 0.61 and λ = 785 nm. (d) Qf as a function of population inversion for fixed wavelength at λ = 785 nm. In all calculations, the radius of the dielectric sphere and permittivity is fixed at a = 155 nm and ϵs = 6.7, respectively.

Results and Discussion

In the following discussion, we consider gain-functionalized composite sphere composed of silicon carbide (SiC) as a host sphere of radius a = 155 nm, relative permittivity ϵs = 6.7 and doped with solvatochromic dye LDS 798 molecules[37] with a dye concentration Ndye as depicted in Figure a. The characteristic parameters of the dye molecules (appearing in eq ) are given as ωa = 2.42 × 1015 Hz, μd = 1.33 × 10–29 Cm, τ = 1 × 10–14 s, and Ndye = 1 × 1018 cm–3.[37,38,40,41] The effective refractive index m of the composite sphere is defined using eq as . Since m is a complex number which governs the optical properties of sphere, it behaves like a passive sphere if Im(m) ≥ 0 and active sphere otherwise. In the case of active sphere, the optical gain level can be tuned by varying the population inversion by means of an external pump laser, as depicted in Figure a, that governs the population inversion by exciting and de-exciting the dye molecules.

In Figure b, we demonstrate the forward scattering efficiency as a function of wavelength by a SiC sphere (dashed line) and an active sphere (black line) for fixed population inversion at Ñ = 0.61. It shows that the passive sphere presents large forward scattering for all wavelengths, and it is the ultimate fact that a sphere with a relatively lower refractive index (m < 3)[17,31] can never realize zero forward scattering due to limitations imposed by the optical theorem.[21,36]

Here, this limitation is circumvented by shining the pump laser to activate the dye molecules inside the SiC sphere and puts them into an excited state. In this scenario, we have additional photons (emitted photons) to meet the condition for causality during the suppression of forward scattering. As shown by the black line, the forward scattering is completely suppressed at a wavelength of λ = 785 nm.

In Figure c, we numerically, using Comsol, illustrate the angular distribution of the scattered field intensity at a fixed incident wavelength of 785 nm and Ñ = 0.61. It is clearly shown that the scattered field is elongated in the backward direction with zero footprint in the forward direction. In Figure d, we calculate Qf as a function of population inversion, which shows that the required population inversion is not very sharp, and hence, one can experimentally control it by tuning the pump laser power.

It is important to emphasis that, here, Qf = 0 is achieved in the presence of optical gain which optimizes the interference between the scattering coefficients. Particularly, the photons emitted by dye molecules amplify the field around the sphere, which is counted as negative absorption. Therefore, in this scenario, the real parts of the scattering amplitudes are expected to be negative, as given in Table , where the first six, three electric a1, a2, a3 and three magnetic b1, b2, b3, coefficients are given for a passive sphere and an active sphere. The forward scattering efficiency Qf, in this case can be simplified asFor the passive sphere, all coefficients have positive real parts, and hence, a large Qf is observed, whereas in the presence of photon emission, some of them appear to be negative; therefore, eq gives Qf = 0.

Table 1

Mie Scattering Coefficients of a Dye Functionalized Sphere with Radius a = 155 nm, Permittivity ϵs = 6.7 for Two Population Inversions (Ñ = 0 and Ñ = 0.61; Incident Wavelength Is Fixed at 785 nm)

Ñ = 0
a1	0.66140–0.47320i	b1	0.77886 + 0.41501i
a2	0.00503–0.07081i	b2	0.00070–0.02649i
a3	≃0–0.00250i	b3	≃0–0.00046i

Figure a demonstrates the asymmetry parameter as a function of wavelength for a passive sphere (dashed line) and for an active sphere (black line). It shows that the asymmetry parameter is positive g ≃ 0.2 for the passive sphere due to preferential forward scattering, as shown in Figure b,d, and for active sphere g ≃ −1/2, which occurs due to the preferential back-scattering, as shown in Figure b–d. In Figure b,c, we numerically calculate the angular distribution of the scattered field for fixed population inversion Ñ = 0 and Ñ = 0.61, respectively. These three-dimensional scattering sketches illustrate that scattering is preferentially in the forward direction for the passive sphere and in the backward direction for the active sphere.

Figure 2

(a) Asymmetry parameter as a function of wavelength for a passive sphere, Ñ = 0 (dashed line), and active sphere, Ñ = 0.61 (solid line). (b) Corresponding scattered field intensity around the sphere for fixed population inversion at (b) Ñ = 0 and (c) Ñ = 0.61, where wavelength is 785 nm.

Until now, we have achieved the negative asymmetry parameter with zero forward scattering for a refractive index sphere relatively lower than that reported in ref (31). Consequently, it is easy to perceive that for all wavelengths corresponding to negative g, this should also lead to larger SMFP, that is, . However, the EMFP also depends on Qext that might have negative values at a large gain level. Thus, only g cannot provide an exact description of as a comparison to .

To this end, in Figure a, we plot the asymmetry parameter as a function of wavelength and population inversion, where the area enclosed by the dotted line corresponds to negative g, and the sphere exhibits zero forward scattering close to g = −1/2, as we have demonstrated in Figures and 2. Figure b,c illustrates and , respectively, as a function of wavelength and population inversion. It has been shown that the sphere with g > 0 and g < 0 leads to and , respectively, as one can inspect from eq . Thus, the area enclosed by the dotted line leads to a parametric space where SMFP is larger than TMFP. On the contrary, is not necessarily less than 1 for all regions where the asymmetry parameter is negative. For instance, close to the transition length, λa = 777 nm, the low population inversion Ñ < 0.21 provides weak photon emission, and we measure . As the population inversion is increased, g becomes negative and is observed. However, we note that for certain wavelength and population inversion, where the emitted photons overcome the incident photons, in this scenario, the major scattering contribution is coming from the emission ones, and hence, according to eq , extinction efficiency becomes negative. For a better understanding of this fact, let us consider point A(800,0.61) in Figure c, where g < 0, Qext < 0, and |Qext – Qsg| < |Qext|, which manifests as (further details are presented in Figure ). Thus, the introduction of optical gain provides an additional degree of freedom to tune the EMFP without strictly depending on g and a large index sphere.

Figure 3

(a) Asymmetry parameter, (b) , and (c) are plotted as a function of incident wavelength and population inversion, where the dotted line indicates (a) g = 0 and (b) . Here, radius and permittivity of the host sphere is a = 155 nm and ϵs = 6.7.

Figure 5

(a) Asymmetry parameter and (b) as a function of incident wavelength and population inversion. (c) Extinction efficiency (black line), denominator of eq D = (Qext – gQs) (dotted line) and absorption efficiency (dashed line) (d) as a function of permittivity for fixed population inversion Ñ = 0.5 as marked by black lines in (b). Here, the radius of the sphere is taken at a = 155 nm and ϵs = 2.

To gain further insight into the phenomenon, in Figure a, we present an asymmetry parameter as a function of wavelength and permittivity of the host dielectric sphere at fixed Ñ = 0.7. It is clearly shown that the gain-functionalized sphere not only exhibits negative asymmetry parameter for lower permittivity sphere ϵs ≃ 2 but also preferentially scatters light in backward directions at ϵs ≃ 5, where g = −1/2. Here, the area enclosed by the dotted line indicates negative g, and in this region, should occur. In Figure b, we calculate as a function of wavelength and permittivity of the host sphere. It is clearly shown that for an active sphere, g and do not follow the fundamental rule as they do for a passive sphere.[25] To demonstrate the role of g on , we consider two lines in the parameter space of Figure a,b, and the corresponding g and are plotted in Figure c,d, respectively, versus permittivity of the host sphere. It is demonstrated that the sphere with ϵs = 2 exhibits g > 0 and EMFP larger than TMFP (i.e., ) as marked by a vertical dotted line. On the other hand, at ϵs = 3.5, where g < 0 but EMFP is smaller than TMFP, , and at ϵs = 7, g < 0 and , as indicated by the vertical dotted lines in Figure c,d. It is worth concluding that one can achieve tunable EMFP by employing the optical gain without strictly relying on negative g and a larger refractive index sphere. In this scenario, gain level and Qext play a crucial role in the tuning of mean free paths.

Figure 4

(a) Asymmetry parameter (the dotted line indicates g = 0) and (b) as a function of incident wavelength and permittivity of the sphere. We draw a purple line-cut in (a), black line-cut in (b) at wavelength λ = 790 nm along the permittivity axis. The corresponding asymmetry parameter and are plotted in (c) and (d), respectively, as a function of permittivity of the sphere for fixed wavelength λ = 790 nm. In this calculation, the radius of the sphere is taken at a = 155 nm and Ñ = 0.7.

For deeper insights into this concept, now we are considering a sphere made of a low refractive index of 1.41 to unveil the role of gain on EMFP under the condition where g is positive. In Figure a, we calculate the asymmetry parameter as a function of wavelength and population inversion. The sphere has a low refractive index, and it presents a g > 0. In this case, it is immediately determined that the TMFP is large than the SMFP or . On the other hand, strongly depends on Ñ and wavelength, as shown in Figure b, where three distinct regions, A, , B, , and, C, , are found as indicated in Figure b. To explore physical phenomena behind the scene, we draw a line at Ñ = 0.5, which crosses through all of the regions, and we calculate the corresponding quantities such as Qext (black), denominator of eq , D = (Qext – Qsg) (dotted), and Qa (dashed) are plotted in Figure c. In the first region A, where the sphere has a smaller gain level, the extinction efficiency and D are positive quantities, as shown in Figure c. Thus, it is found that , as shown in Figure d. In the second region B, gain level increases and the particle undergoes Qext > 0, and the electromagnetic field amplification governs the denominator of eq , such that it becomes negative. As a result, is measured and shown in Figure d. Finally, in region C, the sphere has achieved higher gain level such that the emitted photons overcome the incident photons and the extinction efficiency and D become negative, as shown in Figure c. As a result, in region C, is measured and shown in Figure b,d. In a nutshell, it has been demonstrated that by applying the appropriate gain level, one can achieve the anomalous region where EMFP is larger than the TMFP without achieving the negative asymmetry parameter.

Conclusions

In conclusion, we have studied the electromagnetic scattering by a gain-functionalized low refractive index sphere. By providing a realistic optical gain model, we have, analytically and numerically (Comsol), shown that the implication of optical gain can optimize the destructive interference between the electric and magnetic scattering amplitudes. It has been demonstrated that optimal control on the optical gain level can not only achieve the negative asymmetry parameter but also completely offset the optical scattering in the forward direction by a strikingly low refractive index sphere than that reported to date. As a result, an anomalous regime occurs where the EMFP and SMFP are larger than the TMFP. Another novel finding of the presented work is the optimal control on EMFP, where we have shown that by tuning the experimentally controllable parameter (i.e., population inversion), one can achieve the large EMFP without necessarily achieving the negative asymmetry parameter. Altogether, we hope that our findings could provide a route to analyze the structural information on complex media and in the design of materials with specific transport properties.