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Abstract
Extinction and attenuation by particles in an absorbing host have suffered a long-lasting controversy, which has impeded the physical insights on the radiative transfer in the voids dispersed composite. In this paper, we outline the existing extinction definitions, including an equivalence theorem neglecting the host absorption, the near-field analytical definition neglecting the far-field effects, and the operational way which simulates the actual detector readings. It is shown that, under the independent scattering approximation, the generalized operational definition is equivalent to a recent effective medium method according to the rigorous theory of multiple scattering. Using this generalized extinction, we show the important influences of the host absorption on the void extinction. Specifically, at the void resonance, the extinction cross sections of the small voids can be positive, zero, and even negative, which is regulated quantitively by host absorption. Considering the voids in SiC or Ag, the intriguing properties are verified through the attenuation coefficient calculated by the Maxwell-Garnett effective medium theory. In contrast, the equivalent theorem cannot describe any void resonance structures in the absorbing media. Also, the near-field definition fails to generate negative extinction and cannot thus describe the diminished total absorption by the voids. Our results might provide a better understanding of complex scattering theory in absorbing media.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light transport phenomena in the particle dispersed composite when the host is absorbing are omnipresent in nature, governing many fundamental processes in physics, biology, chemistry, and engineering. Applications include the radiation transfer in atmospheric gases [1], radiative heat transfer in biological tissues [2] and semitransparent porous media [3], optimal design of paint coatings [4] and solar cells [5], and voids tunable optical properties of metal films [6], amorphous semiconductors [7], and metamaterials [8–10].
Extinction and scattering by particles in absorbing media are of fundamental importance in determining radiative properties of the composite. To simplify the problem, a classical assumption was presented by van de Hulst [11], which was also termed as the equivalence theorem. If the complex refraction index of the absorbing host is represented as ${n_h} = n_h^{\prime} + in_h^{{\prime\prime}}$, the theorem assumes that the optical cross sections of the particles in the absorbing host are equivalent to those in its transparent counterpart (${n_h} = n_h^{\prime} + i0$). In other words, this theorem is valid only if the effects of absorption on scattering can be ignored [12]. Since the equivalence theorem is widely used in practical applications [11–18], the validity or feasibility must be carefully considered under the premise of the scattering theory in the absorbing media.
However, two urgent problems arise when one generalizes the scattering theory to the absorbing host. The conventional analytical way which integrates the energy flow over the far-field imaginary spherical surface around the particle (i) fails to link with a measurable quantity and (ii) derives the far-field extinction dependent upon the distance from the particle [19–23]. One of the solutions is to integrate the energy flow at the particle surface known as the near-field definition [24–28], which essentially neglects the host absorption outside the particle. The near-field extinction has been used to derive the attenuation coefficient of the absorbing film embedding metal clusters by Lebedev et al. [24,25]. Although Videen et al. [22] have qualitatively analyzed the practical incompatibility of the near-field definition with attenuation, there is still a lack of a quantitative study on shortcomings of the near-field definition.
The other solution uses the operational way according to the actual measurement configuration which models the readings of a forward detector in the absence and presence of the particle [23]. Therefore, the generalized operational way has clearly physical insights since the extinction cross section of the particle is thus the difference of two readings [29,30]. Using the generalized operational extinction definition, the equivalence theorem has been evaluated under the diverse circumstances, for example, plasmonic nanoparticles in polymer [31], aerosol in an atmosphere [1], pigments in a coating [4], soft particles in absorbing gases [32], and microplastic in water [12]. Still, a fundamental case of voids in absorbing host has been overlooked, which owns an important role in metamaterials and nano-plasmonics [9,10,33–36] by virtue of the void resonance. Since Galeener [7,37] has proved the necessary role of the host absorption $n_h^{{\prime\prime}}$ on the void resonance, the classical equivalence theorem should be inspected carefully for the void composite.
In this paper, we study extinction and attenuation by the dilute voids in the absorbing media under the conventional equivalence theorem, near-field definition, and generalized operational definition. In Section 2, we outline the different definitions of the particulate extinction in the absorbing host in detail. We note that, under the independent scattering approximation, the attenuation coefficient from the generalized operational definition is equivalent to that calculated from the effective dielectric function modeled by the rigorously first-principle multiple scattering theory. In Section 3, using the generalized definition, we show the important role of host absorption on extinction and attenuation by the voids. The void resonance occurs in the presence of the host absorption, which coincides with the results according to the Maxwell-Garnett theory. We identify the invalidity of the equivalence theorem for the absorption host media since neglecting the host absorption fails to predict any void resonance structures. Furthermore, the near-field definition is inaccurate to calculate the attenuation coefficient because it cannot describe the diminished effects of the voids on the host absorption. The conclusion is summarized in Section 4.
2. Particle extinction and attenuation in absorbing host
For convenience, the notations and conventions are introduced as follows. The time-harmonic term of ${e^{ - i\omega t}}$ is used in this paper. Let ${n_p}$ denote the refractive index of a dielectric particle and ${n_h} = n_h^{\prime} + in_h^{{\prime\prime}}$ is the complex refractive index of the absorbing host medium. The dielectric functions of the particle and host are represented by ${\varepsilon _p}$ and ${\varepsilon _h} = \varepsilon _h^{\prime} + i\varepsilon _h^{{\prime\prime}}$, respectively. The non-magnetic materials are considered; thus, we have ${\varepsilon _p} = n_p^2$ and ${\varepsilon _h} = n_h^2$. The wavelength and angular frequency of the fields in a vacuum are $\lambda $ and $\omega $, respectively. The wave number in the vacuum is given by ${k_0} = 2\pi /\lambda = \omega /c$ with c the vacuum light speed. The radius of the spherical particle is a. Thus, the vacuum size parameter of the particle is ${x_0} = {k_0}a$. Furthermore, the volume fraction of the particles in the host is f.
Figure 1 illustrates the two routines considering and neglecting host absorption. In both routines, the host attenuation process is common. The main difference between the two routines is that the particle scattering occurs in an absorbing host or its transparent counterpart. We need to make it clear that the absorbing host is represented by ${n_h} = n_h^{\prime} + in_h^{{\prime\prime}}$ while its transparent counterpart ignores $n_h^{{\prime\prime}}$ with ${n_h} = n_h^{\prime} + i0$. This simplification is common in practical applications of the radiative transfer and has also been inspected recently for electromagnetic energy and chirality [38]. Here, the particle scattering process in the absorbing host is briefly reviewed.
Fig. 1. Schematic diagram of the two routines calculating attenuation of the particulate slabs. (a)-(c) represent the routine in the transparent host, while (d)-(f) represent the one corresponding to the absorbing host. The incident and scattered fields in (c) will not be attenuated. Still, these fields in (f) are decayed by host absorption, which is demonstrated by the faded color and narrowed width of the arrows.
2.1 Extinction definition
It is important to note first in Figs. 1(c) and (f) that the incident fields in the absorbing host decay exponentially along the light propagation but are spatially uniform in a transparent host. As for the scattered fields, besides the inverse of the squared distance, the exponential attenuation in the absorbing host must be also considered. Fortunately, these decays have been described well by Mie expansion theory and the electromagnetic boundary conditions. Therefore, the Mie coefficients, ${a_l}$, ${b_l}$, ${c_l}$ and ${d_l}$, have included the host absorption [39,40]:
In a transparent host medium, there are two conventional ways to define the extinction cross section of a particle [41], as shown in Fig. 2. The first one is the analytical way which defines extinction by integrating the energy flow over the surface of a large imaginary sphere [40]. The second one is modeling the diminished signal of a detector by an interposed particle between the source and detector, which is termed as the operational way [30]. The analytical and operational definitions give the same results for the transparent host. When the host is transparent, the conventional extinction is [40,42,43]
Fig. 2. Two conventional ways to define the extinction cross sections of a particle. (a) Analytical way defines extinction by integrating the energy flow over the surface of a large imaginary sphere with the radius of R. The derivation is usually conducted in the far-field zone with $R \to \infty $; however, it can also be equivalently done in the near-field zone with $R \ge a$. (b) Operational way models the diminished signal of a detector by an interposed particle between the source and detector. The black dashed arrow represents the sensor reading without the particle, while the blue arrow represents the reading with the intercepting particle.
For an absorbing host medium, however, the situation is largely complex, and the discrepancy arises for the analytical way in the far-field zone. Specifically, the far-field integral in the analytical way has vague physical meanings and cannot correspond to or convert to the physically measurable quantity [21]. Furthermore, the nonvanishing integral of the incident intensity in the volume of the imaginary sphere leads to a controversy on the classical unitary relation among extinction, scattering, and absorption [20]. Even worse, the far-field integral will derive the R-dependent physical quantities including the scattering and extinction cross sections. These imaginary-sphere dependent quantities cannot represent the inherent properties of the particle itself. To eliminate the distance dependence, the different R-scaled factors have been adopted by different works, which leads to the discrepancy and ambiguity in the definition of extinction [19,45–47].
The solution to this distance-dependent problem for the conventional far-field analytical way has come from the two methods: i) the imaginary sphere is taken at the particle surface with $R = a$, which is called the near-field definition [24–28]; ii) the operational way considering the real measurement configuration is adopted [21–23,48,49].
For the method (i), the integral of electromagnetic fields at the particle surface derives the distance-independent extinction, which has thus been called the inherent, true, or actual extinction cross sections [50]. This near-field extinction has been given by the relation of [27]
The method (ii) of the operational way in the absorbing host requires the theoretical simulation on the readings of a forward detector. By the way, the importance of the operational definition in the optical measurement of a single nano-object can be found in Refs. [52–55]. This configuration corresponds to an actual extinction experiment and includes the far-field effects which are neglected in the method (i). At first, Bohren and Gilra [21] generalized the operational way to derive extinction of the spherical particle in the absorbing host using the stationary phase approximation method. And then, Mishchenko [23,48] fixed the minor formula error in Ref. [21] and gave the generalized extinction of a sphere in the absorbing host as
2.2 Attenuation coefficient
Let the incident wave propagate in the composite along a certain direction. Along the propagation distance of L, the attenuated energy removed from the incident wave follows the exponential decay law with the factor of $\exp ({ - {\alpha_{\textrm{tot}}}L} )$. The attenuation coefficient ${\alpha _{\textrm{tot}}}$ results from the summation of absorption in the host and extinction by the particles, which is given by [24]
In the absence of particles, the attenuation due to the host absorption is In the framework of independent scattering approximation, the particulate attenuation can be represented by where the uniform particle sizes are assumed and f is the volume fraction of the particles. It should be noted that ${Q_{\textrm{ext}}}$ in Eq. (14) would be replaced by $Q_{\textrm{ext}}^{\textrm{con}}$ in Eq. (5), $Q_{\textrm{ext}}^{\textrm{NF}}$ in Eq. (6), or $Q_{\textrm{ext}}^{\textrm{gen}}$ in Eq. (11), since these definitions are the research focus of this paper. In the absorbing host, the attenuation coefficient of the particles composite has been studied under the transparent host assumption [11,56], near-field definition [24,25], and generalized operational definition [1,4,32].An alternative method to describe the total attenuation in the composite materials resorts to the effective dielectric function according to the effective medium theory. In the method, the problem of the extinction discrepancy of single particles in the absorbing host is avoided. The effective attenuation in the particles composite is [57,58]
As indicated by Nolte [57], for small volume fractions in the transparent host medium, the attenuation coefficient calculated from particulate extinction agrees well with the Maxwell-Garnett results. However, caution is necessary when the host medium is absorbing since the two methods do not agree anymore, even for small volume fractions [57]. This disagreement may result from the conventional definition of extinction improperly used in the absorbing host.
Recently, using rigorous multiple scattering theory from the first principles of electromagnetic scattering, avoiding any heuristic approximation, Durant et al. [3,29] derived the effective dielectric function of the scattering random media when the host is absorbing. The morphology-dependent resonances and correlations between the particles have been included in their derivations. Within the independent scattering approximation, Durant et al. gave the effective dielectric function of the particle composite for the absorbing host as [3,29]
For convenience, the above definitions to calculate the attenuation coefficients are listed in Table 1, where the features of the definitions are concluded. Also, the symbols and equations are shown in the table to guide the results in the next section.
Table 1. Comparisons among the diverse methods calculating the extinction and attenuation coefficients.
3. Results and discussion
The particle is selected as a void and the host is chosen to be the silicon carbide (SiC) of which the optical constants are drawn in Fig. 3. In Fig. 3(a), the complex refractive index is shown. Figure 3(b) shows the complex dielectric function of SiC to clear illustrate the resonance condition. The cyan lines labeled with $- 0.5$ and $- 4$ represent the resonance positions corresponding to the void resonance with ${\varepsilon _p} = 1$ and dielectric resonance with ${\varepsilon _p} = 8$, respectively. It should be noted that the resonance condition occurs at about
Figures 4(a) and (b) show the extinction efficiencies of the void in the SiC host calculated by the far-field and near-field definitions, respectively. When host dissipation is neglected, the two definitions give the same results [66]. If host dissipation is considered, the main difference between the two definitions lies within the range of $11 – 13\; \mu \textrm{m}$. Specifically, the near-field extinction is always positive, whereas the far-field definition can give negative extinction. The phenomenon of negative extinction by the particle in the absorbing host has been reported and explained in Refs. [19,21,48,67]. However, as far as we know, this phenomenon of the dielectric particles in a dispersive and absorbing host has not been studied. Focusing on the far-field results, the small particle extinction is described well by the electric dipole contribution, which is clearly seen in Fig. 4(a). In this result, the dipole extinction of a dielectric sphere embedded in the absorbing host can be derived by the Taylor expansion of the ${a_1}$ Mie coefficient:Fig. 4. Extinction of a void in the SiC host in the spectral range of 10-14 $\mu \textrm{m}$. The far- and near-field definitions are illustrated in (a) and (b), respectively. The radius of the void is $a = 0.1\; \mu \textrm{m}$. The solid lines represent the results considering host absorption including the far-field generalized definition $Q_{\textrm{ext}}^{\textrm{gen}}$ and the near-field definition $Q_{\textrm{ext}}^{\textrm{NF}}$. The blue dashed lines represent the conventional extinction $Q_{\textrm{ext}}^{\textrm{con}}$ which neglects host absorption. The red dotted line in (a) shows the dipole contribution of the generalized extinction $Q_{\textrm{ext,}l = 1}^{\textrm{gen}}$ which is described by the quasistatic approximation.
It should be addressed that the dielectric sphere is assumed with the real and positive ${\varepsilon _p}$. The ratio of ${\varepsilon _h}/{\varepsilon _p}$ in Eq. (20) is the inverse of the conventional ratio of ${\varepsilon _p}/{\varepsilon _h} $ which is usually used in the particle scattering formulas.
To illustrate the effects of host absorption on particle extinction, we show the value of $\textrm{Im}\left[ {\frac{{{\varepsilon_h}}}{{{\varepsilon_p}}}\frac{{1 - {\varepsilon_h}/{\varepsilon_p}}}{{1 + 2{\varepsilon_h}/{\varepsilon_p}}}} \right]$ in the space of $\varepsilon _h^{\prime}/{\varepsilon _p}$ and $\varepsilon _h^{{\prime\prime}}/{\varepsilon _p}$ in Fig. 5. The resonant enhancement on extinction occurs at the resonance condition of around $\varepsilon _h^{\prime}/{\varepsilon _p} ={-} \frac{1}{2}$. However, when host absorption $\varepsilon _h^{{\prime\prime}}/{\varepsilon _p}$ increases, the extinction will cross zero and tend to be negative. At the off-resonance region, the extinction is always negative. For convenience, the zero value is indicated by the dashed line in Fig. 5, which is calculated by $\textrm{Im}\left[ {\frac{{{\varepsilon_h}}}{{{\varepsilon_p}}}\frac{{1 - {\varepsilon_h}/{\varepsilon_p}}}{{1 + 2{\varepsilon_h}/{\varepsilon_p}}}} \right] = 0$. For SiC, the void resonance occurs at two different wavelengths which are 10.4 $\mu \textrm{m}$ and 12.6 $\mu \textrm{m}$. At $\lambda = 10.4\; \mu \textrm{m}$, $\varepsilon _h^{{\prime\prime}}$ is smaller than 1 so that the resonant enhancement is achieved. However, at $\lambda = 12.6\; \mu \textrm{m}$, $\varepsilon _h^{{\prime\prime}}$ is much larger than 1 which leads to the negative resonance. These distinguishing features of the void resonances induced by the SiC absorption can be clearly seen in Fig. 4(a).
Fig. 5. Value of $Q_{\textrm{ext,}l = 1}^{\textrm{gen}}\sim \textrm{Im}\left[ {\frac{{{\varepsilon_h}}}{{{\varepsilon_p}}}\frac{{1 - {\varepsilon_h}/{\varepsilon_p}}}{{1 + 2{\varepsilon_h}/{\varepsilon_p}}}} \right]$ in Eq. (20) versus $\varepsilon _h^{\prime}/{\varepsilon _p} $ and $\varepsilon _h^{{\prime\prime}}/{\varepsilon _p}$. The dashed line highlights the positions with zero values of $Q_{\textrm{ext,}l = 1}^{\textrm{gen}}$.
The main result of Fig. 4(a) is the extinction comparison between the generalized definition in the absorbing host and the conventional one in its transparent counterpart. Intuitive results show a significant difference between the generalized and conventional definitions. On the one hand, the conventional definition neglecting host dissipation cannot describe the void resonance structures, since the dominant factor of $n_h^{{\prime\prime}}$ determining the void resonance is neglected. On the other hand, the generalized extinction cross sections between the two resonance wavelengths are negative, which cannot be either described by the conventional definition since $Q_{\textrm{ext}}^{\textrm{con}} \ge 0$.
Figure 6 illustrates the attenuation coefficients of the voids with the radius of $0.1\; \mu \textrm{m}$ in the SiC host. The volume fraction of the voids is $f = 0.02$, which is an upper threshold satisfying the independent scattering approximation verified by the experimental and numerical results [68]. The attenuation coefficients are normalized by the host attenuation ${\alpha _h}$. Figure 6(a) shows the normalized attenuation in the range of $10 – 11\; \mu \textrm{m}$ including the first void resonance. For this resonance structure, we could observe the almost overlapping curves among the generalized, near-field, and Maxwell-Garnett (MG) methods. However, the conventional method neglecting host absorption cannot describe any physical feature of the void resonance. Figure 6(b) shows the normalized attenuation $\alpha /{\alpha _h}$ in the range of $12 – 14\; \mu \textrm{m}$ where negative extinction occurs. In this range, only the generalized method can describe the diminished effects of voids on the effective absorption. Specifically, the MG result will give the value of $\alpha /{\alpha _h} \approx 0.97$ which is smaller than 1. However, the near-field and conventional definitions yield the value greater than 1. This straightly shows that the near-field and conventional definitions are not sufficient. It is imperative to note that MG theory has been largely used in the porous material and has shown good agreement with the experimental results [7,69–73].
Fig. 6. Dimensional attenuation $\alpha /{\alpha _h}$ of voids in the SiC host in the wavelength range of (a) 10-11 $\mu \textrm{m}$ and (b) 12-14 $\mu \textrm{m}$. The radius of the voids is $0.1\; \mu \textrm{m}$. The used volume fraction of the voids in calculations is $f = 0.02$. ${\alpha _h}$ represents the attenuation coefficients arose from the host, $\alpha _{\textrm{tot}}^{\textrm{gen}}$ is the total attenuation of the host and the generalized void extinction considering host absorption, $\alpha _{\textrm{tot}}^{\textrm{NF}}$ is the total attenuation corresponding to the near-field definition. $\alpha _{\textrm{tot}}^{\textrm{con}}$ means the total attenuation from the host and the void extinction neglecting host absorption, and $\alpha _{\textrm{eff}}^{\textrm{MG}}$ is calculated according to the effective medium approximation.
As a final result, Fig. 7 shows the attenuation coefficient of the voids in Ag, since this nonporous metal structure can be used to tune the plasmonic properties by changing the void content and radius [70,71,74–76]. The dielectric function of Ag is used from Ref. [77]. The void resonance in Ag is located at about $\lambda = 0.33\mu \textrm{m}$. At the resonance, the conventional extinction cannot describe the resonance. And off the resonance, the near-field extinction cannot describe the diminished effects of the voids on the host absorption. These results coincide with the case of the SiC host which is mentioned above.
Fig. 7. Dimensional attenuation $\alpha /{\alpha _{\textrm{h}}}$ of voids in the Ag host. The radius of the voids is $10\; \textrm{nm}$. The used volume fraction of the voids in calculations is $f = 0.02$. The physical meanings of different lines are the same as those in Fig. 6.
4. Conclusions
In this paper, we have studied extinction by small voids in the absorbing SiC with the generalized operational definition which matches with configurations of an actual measurement. Using this generalized definition, we visually illustrate a considerable role of host absorption on extinction by the small voids. At the void resonance with small host dissipation, the void extinction is intensively positive; when host dissipation increases, the void extinction crosses zero and tends to be negative. These results indicate that the conventional extinction definition neglecting the host absorption is not suitable, since it cannot model the void resonance. As for attenuation under the independent scattering regime, we identify an equivalence between the generalized operational extinction and the effective dielectric function from the rigorous multiple scattering theory. In contrast, the near-field analytical definition cannot describe the diminished effects of the voids on host absorption. Our work contributes to a better understanding on basic concepts of electromagnetic scattering in absorbing media and serves to the related applications of a particle composite.
Funding
National Natural Science Foundation of China (52076123); Natural Science Foundation of Shandong Province (ZR2022QE245).
Acknowledgments
S. Zhang thanks Michael Tribelsky, Kirill Koshelev, and Mario Hentschel for useful discussions about the void resonance in the absorbing host.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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