Abstract

Two different versions of an optical theorem for a scattering body embedded inside a lossy background medium are derived in this paper. The corresponding fundamental upper bounds on absorption are then obtained in closed form by elementary optimization techniques. The first version is formulated in terms of polarization currents (or equivalent currents) inside the scatterer and generalizes previous results given for a lossless medium. The corresponding bound is referred to here as a variational bound and is valid for an arbitrary geometry with a given material property. The second version is formulated in terms of the T-matrix parameters of an arbitrary linear scatterer circumscribed by a spherical volume and gives a new fundamental upper bound on the total absorption of an inclusion with an arbitrary material property (including general bianisotropic materials). The two bounds are fundamentally different as they are based on different assumptions regarding the structure and the material property. Numerical examples including homogeneous and layered (core-shell) spheres are given to demonstrate that the two bounds provide complimentary information in a given scattering problem.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Fundamental limits on the scattering and absorption in resonant electromagnetic structures have been considered in various formulations and applications such as with small dipole scatterers [1], antennas [2,3], absorbers [4], high-impedance surfaces [5], passive metamaterials [6,7], and absorption and scattering [810]. Notably, these problems are almost always formulated for a lossless background medium such as vacuum. In most cases there are very good reasons for doing this, at least when the background losses are sufficiently small. In fact, it turns out that the presence of the lossy background medium not only implies an obstructive complication of the analytical derivations, it also nullifies the validity of many powerful theorems and constraints, see e.g., [1121] with references. In essence, the problem is that any optical theorem for a scattering body embedded in a lossy medium will depend on the geometry of the scatterer. A vivid illustration of this is the optical theorem and the associated upper bounds on dipole scattering and absorption of a small (dipole) scatterer in a lossless medium which are based solely on the polarizability of the scatterer [1], and which no longer is valid for a lossy background [12,18,20]. Hence, for a lossy background medium the associated optical theorems must be modified, and it can be expected that any analytical results regarding the optimal absorption will become dependent on the geometry of the scatterer, see e.g., [11,12]. As e.g., the scattering cross-section presented by a scatterer under optimal absorption conditions, may no longer be equal to its absorption cross section, as it is expected for a lossless background medium [1,22].

There is a number of application areas where the surrounding losses clearly cannot be neglected. This includes typically medical applications such as localized electrophoretic heating of a bio-targeted and electrically charged gold nanoparticle suspension as a radiotherapeutic hyperthermia-based method to treat cancer, cf., [2327]. Corresponding applications in the optical domain are concerned with light in biological tissue [28], and the use of gold nanoparticles for plasmonic photothermal therapy [29]. Surface-enhanced biological sensing with molecular monolayer spectroscopy is another related application, see e.g., [30]. In photonic applications and in plasmonics, dielectric substrates based on polymeric media are usually considered to be lossless at optical frequencies [31,32]. However, some of the substances that are used can also show significant losses such as with z-doped PMMA materials [33]. Another important medical application is concerned with implantable antennas that are used in telemetry applications as a part of communication link of health-monitoring and health-care systems [34,35]. The presence of losses in the background medium affects the reliability of such links, especially the performance of in-body antennas. In this application, however, the aim is to reduce the amount of power absorbed by a human body, and this can be achieved by encapsulation of the implant by a biocompatible insulator [36]. Finally, we mention the terrestrial gaseous atmosphere which has a diversity of rotational-vibrational absorption bands ranging from microwave to optical frequencies [37,38]. Important applications include antennas and short range communications at 60 GHz [3943] (absorption bands of oxygen) as well as the study of radiative transfer in the presence of aerosols and cloud particles in the atmosphere [21,37].

In this paper we present two different versions of an optical theorem and the associated absorption bounds for a scattering body embedded inside a lossy background medium. These versions are based on the interior and the exterior fields, i.e., the equivalent (polarization) currents inside the scatterer and the T-matrix parameters of the scatterer, respectively. The two versions of the optical theorem express the same power balance, and yet they are fundamentally different (and hence complimentary), since they are based on different assumptions regarding the properties of the scatterer. The first version is an extension to lossy media regarding the absorption bounds given in [8], and is valid for an arbitrary geometry with a given material property. Even though the basic optimization technique is the same as in [8], it is demonstrated how the seemingly trivial extension to lossy media insidiously requires a careful analysis where it is not sufficient just to replace a real-valued wave number for a complex-valued one. In particular, the new optical theorem shows that the extinct power must be expressed as an affine form in the equivalent currents implying subtle changes in the final form of the fundamental bound on absorption. The second version is a refinement of the fundamental bounds on multipole absorption given in [11], and is valid for a spherical geometry with an arbitrary material property. In particular, we formulate an optical theorem and derive the associated absorption bound for the total fields including all the electric and magnetic multipoles. It is proved that the bound is valid not only for a rotationally invariant sphere as in [11], but also for general heterogeneous bianisotropic materials. We prove also that the new bound, which is given by a multipole summation formula, is convergent whenever there are non-zero losses in the exterior domain. In this way, the results also provide a new way to determine the number of useful multipoles in a given scattering problem, i.e., as a function of the electrical size of the scatterer as well as of the losses in the exterior domain. Through the numerical examples, we show that the new fundamental bounds give complementary information on the absorption of scattering objects in lossy media. The derived bounds are applicable for arbitrary objects made of arbitrary materials, which gives a possibility to find such an electrically small structure that has an absorption peak close to the fundamental bounds.

The rest of the paper is organized as follows: In Section 2 is given the optical theorem based on the interior fields and in Section 3 the corresponding bounds on absorption by variational calculus. In Section 4, we consider the optical theorem and the associated bounds based on the exterior fields using the T-matrix formalism. In Section 5 is illustrated the numerical examples, and the paper is summarized in Section 6. Finally, in Appendix A. is shown the derivation of the maximal absorbed power based on calculus of variations, and in Appendix B. is put the most important definitions and formulas that are used regarding the spherical vector wave expansion.

2. Optical theorem based on the interior fields

2.1 Notation and conventions

The electric and magnetic field intensities $\boldsymbol {E}$ and $\boldsymbol {H}$ are given in SI-units [44] and the time convention for time harmonic fields (phasors) is given by $\textrm{e} ^{-\textrm{i} \omega t}$, where $\omega$ is the angular frequency and $t$ the time. Let $\mu _0$, $\epsilon _0$, $\eta _0$ and $\mathrm {c}_0$ denote the permeability, the permittivity, the wave impedance and the speed of light in vacuum, respectively, and where $\eta _0=\sqrt {\mu _0/\epsilon _0}$ and $\mathrm {c}_0=1/\sqrt {\mu _0\epsilon _0}$. The wave number of vacuum is given by $k_0=\omega \sqrt {\mu _0\epsilon _0}$, and hence $\omega \mu _0=k_0\eta _0$ and $\omega \epsilon _0=k_0\eta _0^{-1}$. The real and imaginary parts and the complex conjugate of a complex number $\zeta$ are denoted by $ {\mathrm {Re}}\left \{\zeta \right \}$, $ {\mathrm {Im}}\left \{\zeta \right \}$ and $\zeta ^{*}$, respectively. For dyadics, the notation $(\cdot )^{\dagger }$ denotes the Hermitian transpose.

2.2 Extinction, scattering and absorption

Consider a scattering problem consisting of a scattering body $V$ bounded by the surface $\partial V$ and which is embedded in an infinite homogeneous and isotropic background medium having relative permeability $\mu _{\mathrm {b}}$ and relative permittivity $\epsilon _{\mathrm {b}}$, see Fig. 1. The background medium is passive and in general lossy, with complex-valued parameters satisfying $ {\mathrm {Im}}\{\mu _{\mathrm {b}}\}\geq 0$ and $ {\mathrm {Im}}\{\epsilon _{\mathrm {b}}\}\geq 0$. The scatterer $V$ consists of a linear material bounded by a finite open set with volume denoted by the same letter $V$, and which will contain the contrast sources. Hence, both the incident (i) and scattered (s) fields satisfy the following source-free Maxwell’s equations for the exterior region $\mathbb {R}^{3}\setminus V$

$$\left\{\begin{array}{l} \nabla\times\boldsymbol{E}_{\mathrm{\{i,\;s\}}}=\textrm{i} k_0\eta_0\mu_{\mathrm{b}}\boldsymbol{H}_{\mathrm{\{i,\;s\}}}, \\ \nabla\times\boldsymbol{H}_{\mathrm{\{i,\;s\}}}={-}\textrm{i} k_0\eta_0^{{-}1}\epsilon_{\mathrm{b}}\boldsymbol{E}_{\mathrm{\{i,\;s\}}}, \end{array}\right.$$
and the total fields are denoted $\boldsymbol {E}=\boldsymbol {E}_{\mathrm {i}}+\boldsymbol {E}_{\mathrm {s}}$ and $\boldsymbol {H}=\boldsymbol {H}_{\mathrm {i}}+\boldsymbol {H}_{\mathrm {s}}$. It is noted that the incident fields satisfy Eq. (1) in the whole of $\mathbb {R}^{3}$.

 figure: Fig. 1.

Fig. 1. Problem setup. Here, $\epsilon _{\mathrm {b}}$ and $\mu _{\mathrm {b}}$ denote the relative permittivity and permeability of the passive background medium, respectively, and $\hat {\boldsymbol {n}}$ the outward unit vector.

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The interior scattering medium is characterized by the following constitutive relations for a general bianisotropic linear material

$$\left\{\begin{array}{l} \boldsymbol{D}=\epsilon_0\boldsymbol{\epsilon}\cdot\boldsymbol{E}+\frac{1}{\mathrm{c}_0}\boldsymbol{\chi}_{\mathrm{em}}\cdot\boldsymbol{H}, \\ \boldsymbol{B}=\frac{1}{\mathrm{c}_0}\boldsymbol{\chi}_{\mathrm{me}}\cdot\boldsymbol{E} +\mu_0\boldsymbol{\mu}\cdot\boldsymbol{H}, \end{array}\right.$$
where $\boldsymbol {B}$ is the magnetic flux density and $\boldsymbol {D}$ the electric flux density and where the relative permittivity and permeability dyadics are $\boldsymbol {\epsilon }=\epsilon _{\mathrm {b}}\boldsymbol {I}+\boldsymbol {\chi }_{\mathrm {ee}}$ and $\boldsymbol {\mu }=\mu _{\mathrm {b}}\boldsymbol {I}+\boldsymbol {\chi }_{\mathrm {mm}}$, respectively, and where $\boldsymbol {\chi }_{\mathrm {ee}}$, $\boldsymbol {\chi }_{\mathrm {mm}}$, $\boldsymbol {\chi }_{\mathrm {em}}$, and $\boldsymbol {\chi }_{\mathrm {me}}$ are dimensionless susceptibility dyadics. By following the standard volume equivalence principles [45], the Maxwell’s equations for the interior region $V$
$$\left\{\begin{array}{l} \nabla\times\boldsymbol{E}=\textrm{i} k_0\boldsymbol{\chi}_{\mathrm{me}}\cdot\boldsymbol{E} +\textrm{i} k_0\eta_0\boldsymbol{\mu}\cdot\boldsymbol{H}, \\ \nabla\times\boldsymbol{H}={-}\textrm{i} k_0\eta_0^{{-}1}\boldsymbol{\epsilon}\cdot\boldsymbol{E}-\textrm{i} k_0\boldsymbol{\chi}_{\mathrm{em}}\cdot\boldsymbol{H}, \end{array}\right.$$
can now be reformulated in terms of the background medium as
$$\left\{\begin{array}{l} \nabla\times\boldsymbol{E}=\textrm{i} k_0\eta_0\mu_{\mathrm{b}}\boldsymbol{H}-\boldsymbol{J}_{\mathrm{m}}, \\ \nabla\times\boldsymbol{H}={-}\textrm{i} k_0\eta_0^{{-}1}\epsilon_{\mathrm{b}}\boldsymbol{E}+\boldsymbol{J}_{\mathrm{e}}, \end{array}\right.$$
which is just Eq. (3) rewritten based on the equivalent electric and magnetic contrast currents
$$\left\{\begin{array}{l} \boldsymbol{J}_{\mathrm{e}}={-}\textrm{i} k_0\eta_0^{{-}1}\boldsymbol{\chi}_{\mathrm{ee}}\cdot\boldsymbol{E}-\textrm{i} k_0\boldsymbol{\chi}_{\mathrm{em}}\cdot\boldsymbol{H}, \\ \boldsymbol{J}_{\mathrm{m}}={-}\textrm{i} k_0\boldsymbol{\chi}_{\mathrm{me}}\cdot\boldsymbol{E}-\textrm{i} k_0\eta_0\boldsymbol{\chi}_{\mathrm{mm}}\cdot\boldsymbol{H}. \end{array}\right.$$
The power balance at the surface $\partial V$ just outside $V$ (where $\boldsymbol {E}=\boldsymbol {E}_{\mathrm {i}}+\boldsymbol {E}_{\mathrm {s}}$) is obtained by using the corresponding Poynting’s vectors and can be expressed as
$$P_{\mathrm{a}}={-}P_{\mathrm{s}}+P_{\mathrm{t}}+P_{\mathrm{i}},$$
where $P_{\mathrm {a}}$, $P_{\mathrm {s}}$, $P_{\mathrm {t}}$ and $P_{\mathrm {i}}$ are the absorbed, scattered, extinct (total) and the absorbed incident powers, respectively, defined by
$$P_{\mathrm{a}} ={-}\frac{1}{2}{\mathrm{Re}}\left\{ \int_{\partial V}\boldsymbol{E}\times\boldsymbol{H}^{*} \cdot \hat{\boldsymbol{n}}\thinspace\textrm{d} S \right\},$$
$$P_{\mathrm{s}} = \frac{1}{2}{\mathrm{Re}}\left\{ \int_{\partial V}\boldsymbol{E}_{\mathrm{s}}\times\boldsymbol{H}_{\mathrm{s}}^{*} \cdot \hat{\boldsymbol{n}}\thinspace\textrm{d} S \right\},$$
$$P_{\mathrm{t}} ={-}\frac{1}{2}{\mathrm{Re}}\left\{ \int_{\partial V}\left(\boldsymbol{E}_{\mathrm{i}}\times\boldsymbol{H}_{\mathrm{s}}^{*} + \boldsymbol{E}_{\mathrm{s}}\times\boldsymbol{H}_{\mathrm{i}}^{*}\right) \cdot \hat{\boldsymbol{n}}\thinspace\textrm{d} S \right\},$$
$$P_{\mathrm{i}} ={-}\frac{1}{2}{\mathrm{Re}}\left\{ \int_{\partial V}\boldsymbol{E}_{\mathrm{i}}\times\boldsymbol{H}_{\mathrm{i}}^{*} \cdot \hat{\boldsymbol{n}}\thinspace\textrm{d} S \right\},$$
and where the surface integrals are defined with an outward unit normal $\hat {\boldsymbol {n}}$, see also [46, Eq. (3.19)]. Based on the Poyntings theorem (the divergence theorem) $\int _{\partial V}\boldsymbol {E}\times \boldsymbol {H}^{*} \cdot \hat {\boldsymbol {n}}\textrm{d} S=\int _{V} \left (\boldsymbol {H}^{*}\cdot \nabla \times \boldsymbol {E}-\boldsymbol {E}\cdot \nabla \times \boldsymbol {H}^{*}\right ) \textrm{d} v$, and by employing the following identities on $\partial V$
$$\left\{\begin{array}{l} \hat{\boldsymbol{n}}\times(\boldsymbol{E}_{\mathrm{i}}+\boldsymbol{E}_{\mathrm{s}})=\hat{\boldsymbol{n}}\times\boldsymbol{E}, \\ \hat{\boldsymbol{n}}\times(\boldsymbol{H}_{\mathrm{i}}+\boldsymbol{H}_{\mathrm{s}})=\hat{\boldsymbol{n}}\times\boldsymbol{H}, \end{array}\right.$$
the vector identity $\hat {\boldsymbol {n}}\cdot \boldsymbol {X}\times \boldsymbol {Y}=\hat {\boldsymbol {n}}\times \boldsymbol {X}\cdot \boldsymbol {Y}$ and the Maxwell’s equations given by Eq. (3), it is possible to show that Eq. (6) gives an optical theorem for the lossy background where
$$P_{\mathrm{a}} = \frac{k_0}{2\eta_0}{\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}^{*}\cdot \boldsymbol{M}_{\mathrm{a}}\cdot\boldsymbol{F}\thinspace\textrm{d} v\right\},$$
$$P_{\mathrm{t}} = \frac{k_0}{2\eta_0}{\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot \boldsymbol{M}_{\mathrm{t}}\cdot\boldsymbol{F}\thinspace\textrm{d} v\right\} -2P_{\mathrm{i}},$$
$$P_{\mathrm{i}} = \frac{k_0}{2\eta_0}{\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot \boldsymbol{M}_{\mathrm{b}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v\right\},$$
and which are based solely on the interior fields. Here, the field quantities are defined as
$$\boldsymbol{F}=\left(\begin{array}{c} \boldsymbol{E} \\ \eta_0\boldsymbol{H} \end{array}\right), \quad \boldsymbol{F}_{\mathrm{i}}=\left(\begin{array}{c} \boldsymbol{E}_{\mathrm{i}} \\ \eta_0\boldsymbol{H}_{\mathrm{i}} \end{array}\right),$$
and the material dyadics are given by
$$\boldsymbol{M}_{\mathrm{a}}=\left(\begin{array}{cc} \boldsymbol{\epsilon} & \boldsymbol{\chi}_{\mathrm{em}} \\ \boldsymbol{\chi}_{\mathrm{me}} & \boldsymbol{\mu} \end{array}\right) =\boldsymbol{\chi}+\boldsymbol{M}_{\mathrm{b}}$$
where
$$\boldsymbol{\chi}=\left(\begin{array}{cc} \boldsymbol{\chi}_{\mathrm{ee}} & \boldsymbol{\chi}_{\mathrm{em}} \\ \boldsymbol{\chi}_{\mathrm{me}} & \boldsymbol{\chi}_{\mathrm{mm}} \end{array}\right), \quad \boldsymbol{M}_{\mathrm{b}}=\left(\begin{array}{cc} \epsilon_{\mathrm{b}}\boldsymbol{I} & \boldsymbol{0} \\ \boldsymbol{0} & \mu_{\mathrm{b}}\boldsymbol{I} \end{array}\right),$$
and
$$\boldsymbol{M}_{\mathrm{t}}= \left(\begin{array}{cc} \boldsymbol{\epsilon}-\epsilon_{\mathrm{b}}^{*}\boldsymbol{I} & \boldsymbol{\chi}_{\mathrm{em}} \\ \boldsymbol{\chi}_{\mathrm{me}} & \boldsymbol{\mu}-\mu_{\mathrm{b}}^{*}\boldsymbol{I} \end{array}\right) =\boldsymbol{\chi}+\textrm{i} 2{\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{b}}\}.$$
It is noted that Eqs. (12)–(18) generalize previous expressions which have been given for a lossless exterior medium [8, Eqs. (4)–(7) on p. 3338] where $P_{\mathrm {i}}=0$, and both $\boldsymbol {M}_{\mathrm {a}}$ and $\boldsymbol {M}_{\mathrm {t}}$ given above can be replaced by the susceptibility dyadic $\boldsymbol {\chi }$.

It is observed that $P_{\mathrm {a}}$ is represented by a positive definite (strictly convex) quadratic form and $P_{\mathrm {t}}$ by an affine form in the field quantities. Note in particular the additional power balancing term $-2P_{\mathrm {i}}$ that is present in Eq. (13). Finally, it is noted that in the present formulation it is sufficient to derive three terms as in Eqs. (12)–(14) since the fourth term $P_{\mathrm {s}}$ will then be given by the optical theorem Eq. (6).

3. Fundamental bounds on absorption by variational calculus

The fundamental bounds on absorption derived in [8] are generalized below for the case with a lossy background medium. The derivation is based on the optical theorem expressed in Eq. (6) together with Eqs. (12)–(18) above.

3.1 General bianisotropic media

The optimization problem of interest is given by

$$ \begin{array}{llll} & \mathrm{maximize} & P_{\mathrm{a}} \\ & \mathrm{subject\ to} & P_{\mathrm{s}} \geq 0, \end{array} $$
where the optimization is with respect to the interior fields $\boldsymbol {F}$ of the structure, and where the scattered power $P_{\mathrm {s}}=-P_{\mathrm {a}}+P_{\mathrm {t}}+P_{\mathrm {i}}$ is used as the non-negative constraint. In this case, the constraint is organized as $-P_{\mathrm {a}}+P_{\mathrm {t}}+2P_{\mathrm {i}}-P_{\mathrm {i}}\geq 0$ where $P_{\mathrm {a}}$ is the positive definite quadratic form expressed in Eq. (12), $P_{\mathrm {t}}+2P_{\mathrm {i}}$ is the linear form given by Eq. (13) and $P_{\mathrm {i}}$ is given by Eq. (14). This is a convex maximization problem having a unique solution at the boundary of the feasible region (active constraint), cf., [47].

By using the method of Lagrange multipliers [48] and variational calculus, it can be shown that the optimal bound on absorbed power $P_{\mathrm {a}}^{\mathrm {opt}}$ is given by

$$P_{\mathrm{a}}^{\mathrm {opt}}=\frac{k_0\alpha^{2}}{8\eta_0}\int_{V} \boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{t}}\cdot\left({\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace \textrm{d} v,$$
see the detailed derivation of the result Eq. (62) in Appendix A.1. The parameter $\alpha$ is found by inserting the stationary solution Eq. (60) into the active constraint in Eq. (19), yielding the quadratic equation
$$\alpha^{2}+2\alpha=q,$$
where
$$q=\frac{\displaystyle 4\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot{\mathrm{Im}}\left\{\boldsymbol{M}_{\mathrm{b}}\right\}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v} {\displaystyle\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{t}}\cdot\left({\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v}.$$
The denominator in Eq. (22) is convex in $\boldsymbol {M}_{\mathrm {a}}$ for $ {\mathrm {Im}}\{\boldsymbol {M}_{\mathrm {a}}\}>0$, and thus by its minimization, it can be shown that Eq. (22) is maximal for $\boldsymbol {M}_{\mathrm {a}}=\boldsymbol {M}_{\mathrm {b}}$ implying that $q\leq 1$, see the proof in Appendix A.2. The maximizing root of Eq. (21) is hence given by
$$\alpha={-}1-\sqrt{1-q}.$$
The expression in Eq. (20) together with Eqs. (16)–(18), Eq. (22) and Eq. (23) generalizes the previous result in [8] which has been given for a lossless exterior medium. In particular, by considering a lossless exterior medium with e.g., $\mu _{\mathrm {b}}=\epsilon _{\mathrm {b}}=1$, it is seen that $q=0$ (which implies that $0\leq q\leq 1$), $\alpha =-2$, $\lambda =2$, $\boldsymbol {M}_{\mathrm {t}}=\boldsymbol {\chi }$ and $ {\mathrm {Im}}\{\boldsymbol {M}_{\mathrm {a}}\}= {\mathrm {Im}}\{\boldsymbol {\chi }\}$, so that Eq. (20) reproduces the corresponding result in [8, Eq. (23b) on p. 3342]. Optimization of the scattered power can be treated similarly.

3.2 Piecewise homogeneous and isotropic dielectric structures

Important special cases are with the optimal absorption of piecewise homogeneous and isotropic dielectric structures in a lossy surrounding dielectric medium. In this case, the problem only involves the electric losses and we can simplify the notation by writing $\boldsymbol {F}=\boldsymbol {E}$, $\boldsymbol {F}_{\mathrm {i}}=\boldsymbol {E}_{\mathrm {i}}$ and $\boldsymbol {\chi }_i=\boldsymbol {\chi }_{\mathrm {ee},\;i}=(\epsilon _i-\epsilon _{\mathrm {b}})\boldsymbol {I}$, where $i=1,\ldots ,N$ is related to the corresponding homogeneous component of the composed scatterer. Note that for $N=1$, the problem simplifies to the homogeneous structure. The associated material dyadics are given by

$$\boldsymbol{M}_{\mathrm{a},\;i}=\epsilon_i\boldsymbol{I}, \quad \boldsymbol{M}_{\mathrm{b}}=\epsilon_{\mathrm{b}}\boldsymbol{I}, \quad \boldsymbol{M}_{\mathrm{t},\;i}=(\epsilon_i-\epsilon_{\mathrm{b}}^{*})\boldsymbol{I},$$
and the expression in Eq. (20) becomes
$$P_{\mathrm{a}}^{\mathrm {var}}=\frac{k_0\alpha^{2}}{8\eta_0}\sum_{i=1}^{N}\frac{\left|\epsilon_i-\epsilon_{\mathrm{b}}^{*}\right|^{2}}{{\mathrm{Im}}\{\epsilon_i\}}\int_{V_i}\left|\boldsymbol{E}_{\mathrm{i}}(\boldsymbol r)\right|^{2}\textrm{d} v,$$
for the total volume of scattering body $V=\sum _i V_i$, $i=1,\ldots ,N$, and where $\alpha$ is given by Eq. (23), and $q$ is obtained from Eq. (22) as
$$q=\displaystyle\cfrac{4{\mathrm{Im}}\{\epsilon_{\mathrm{b}}\}\displaystyle\int_{V}\left|\boldsymbol{E}_{\mathrm{i}}(\boldsymbol r)\right|^{2}\textrm{d} v}{\displaystyle\sum_{i=1}^{N}\cfrac{\left|\epsilon_i-\epsilon_{\mathrm{b}}^{*}\right|^{2}}{{\mathrm{Im}}\{\epsilon_i\}}\displaystyle\int_{V_i}\left|\boldsymbol{E}_{\mathrm{i}}(\boldsymbol r)\right|^{2}\textrm{d} v}.$$
Assume now that the scatterer is an $N$-layered sphere $V_a$ of total radius $a$, $N\geq 1$, and the incident field is a plane wave $\boldsymbol {E}_{\mathrm {i}}(\boldsymbol r)=\boldsymbol {E}_0\textrm{e} ^{\mathrm {i} k_{\mathrm {b}}\hat {\boldsymbol {k}}\cdot \boldsymbol {r}}$ with vector amplitude $\boldsymbol {E}_0$, propagation direction $\hat {\boldsymbol {k}}$ and where $k_{\mathrm {b}}=k_0\sqrt {\epsilon _{\mathrm {b}}}$ is the wave number of the background medium. By expanding the plane wave in regular spherical vector waves as expressed in Eq. (70), it can readily be shown that
$$\int_{V_a}\left|\boldsymbol{E}_{\mathrm{i}}(\boldsymbol r)\right|^{2}\textrm{d} v = \left| \boldsymbol{E}_0 \right|^{2} 2\pi \sum_{\tau=1}^{2}\sum_{l=1}^{\infty}(2l+1)W_{\tau l}(k_{\mathrm{b}},\;a),$$
with $W_{\tau l}(k_{\mathrm {b}},\;a)$ defined as
$$W_{\tau l}(k,\;a)=\int_{V_a}\left|\textbf{v}_{\tau ml}(k\boldsymbol{r})\right|^{2}\textrm{d} v,$$
for $\tau =1,2$ and $l\geq 1$ and where $\textrm{d} v=r^{2}\textrm{d} \Omega \textrm{d} r$, and where we have employed the orthogonality relationships [26, Eqs. (A.18) and (A.19)], as well as [11, Eqs. (5) and (6)]. It is observed that for electrically small objects of size $k_0a<1$, the influence of the background medium can be appropriately neglected, and thus the incident field can be assumed to have a constant amplitude $\boldsymbol {E}_{\mathrm {i}}(\boldsymbol r)=\boldsymbol {E}_0$. Hence, the relationship in Eq. (27) simplifies as
$$\int_{V_a}\left|\boldsymbol{E}_{\mathrm{i}}(\boldsymbol r)\right|^{2}\textrm{d} v= \left| \boldsymbol{E}_0 \right|^{2} V_{\mathrm{a}},$$
where $V_a=\sum _i^{N}V_i=4\pi a^{3}/3$ is the volume of the layered sphere. Note that the expression in Eq. (29) is always valid for the case with a lossless surrounding medium, where $k_{\mathrm {b}}$ is real-valued ($k_{\mathrm {b}}=k_{\mathrm {b}}^{*}$).

The variational upper bound on the absorption cross section $\sigma _{\mathrm {a}}^{\mathrm {var}}$ is obtained by normalizing with the intensity of the plane wave at the origin $\boldsymbol {r}=\boldsymbol {0}$, i.e.,

$$I_{\mathrm{i}}=\left| \boldsymbol{E}_0 \right|^{2}{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}/2\eta_0,$$
giving
$$\sigma_{\mathrm{a}}^{\mathrm {var}}=\frac{k_0}{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}\frac{\alpha^{2}}{4}\sum_{i=1}^{N}\frac{\left|\epsilon_i-\epsilon_{\mathrm{b}}^{*}\right|^{2}}{{\mathrm{Im}}\{\epsilon_i\}} \int_{V_i}\left|\textrm{e}^{\mathrm{i} k_{\mathrm{b}}\hat{\boldsymbol{k}}\cdot\boldsymbol{r}} \right|^{2}\textrm{d} v.$$
To give an explicit formula for Eq. (31), it is more convenient to express the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {var}}=\sigma _{\mathrm {a}}^{\mathrm {var}}/\pi a^{2}$ for the $N$-layered sphere as
$$Q_{\mathrm{a}}^{\mathrm {var}}=\frac{k_0a}{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}\frac{\alpha^{2}}{4}\sum_{i=1}^{N}\frac{\left|\epsilon_i-\epsilon_{\mathrm{b}}^{*}\right|^{2}}{{\mathrm{Im}}\{\epsilon_i\}} \frac{1}{\pi a^{3}}\int_{V_i}\left|\textrm{e}^{\mathrm{i} k_{\mathrm{b}}\hat{\boldsymbol{k}}\cdot\boldsymbol{r}} \right|^{2}\textrm{d} v,$$
where
$$\frac{1}{\pi a^{3}}\int_{V_i}\left|\textrm{e}^{\mathrm{i} k_{\mathrm{b}}\hat{\boldsymbol{k}}\cdot\boldsymbol{r}} \right|^{2}\textrm{d} v =2\sum_{\tau=1}^{2}\sum_{l=1}^{\infty}\frac{(2l+1)}{a^{3}}\left[W_{\tau l}(k_{\mathrm{b}},\;a_i)-W_{\tau l}(k_{\mathrm{b}},\;a_{i-1})\right],$$
and where $a_i$ is the radius of each subsphere for $i=1,\ldots ,N$, $a_N=a$ and $a_0=0$. Here,
$$W_{1 l}(k_{\mathrm{b}},\;a_i)=a_i^{2}\frac{{\mathrm{Im}}\left\{ k_{\mathrm{b}}\mathrm{j}_{l+1}( k_{\mathrm{b}}a_i)\mathrm{j}_{l}^{*}( k_{\mathrm{b}}a_i)\right\}}{{\mathrm{Im}}\{k_{\mathrm{b}}^{2}\}},$$
and
$$W_{2 l}(k_{\mathrm{b}},\;a_i)=\frac{(l+1)W_{1,\;l-1}(k_{\mathrm{b}},\;a_i)+lW_{1,\;l+1}(k_{\mathrm{b}},\;a_i)}{(2l+1)},$$
are readily obtained from [26, Eqs. (A.20) and (A.21)]. Note that for $i=1$, the last term in Eq. (33) vanishes because of $W_{\tau l}(k_{\mathrm {b}},\;a_0)=0$ for $a_0=0$, see Eqs. (34) and (35), respectively.

In the case of a homogeneous ($N=1$) sphere in a lossless medium where $ {\mathrm {Im}}\{\epsilon _{\mathrm {b}}\}=0$, we have $q=0$, $\alpha =-2$ and the integral in Eq. (29) so that the bound in Eq. (25) simplifies to

$$P_{\mathrm{a}}^{\mathrm {var}}=\frac{k_0}{2\eta_0}\frac{\left|\epsilon-\epsilon_{\mathrm{b}}\right|^{2}}{{\mathrm{Im}}\{\epsilon\}}\left| \boldsymbol{E}_0 \right|^{2} V_a,$$
and which reproduces the corresponding result in [8, Eq. (32b) on p. 3345].

4. Optical theorem based on the exterior fields

4.1 Notation and conventions

The definition of the spherical vector waves [44,46,4952] is summarized in Appendix B. The field expansions based on the spherical vector waves [44,46,4952] are presented in Appendix B. In particular, the regular spherical Bessel functions, the Neumann functions, the spherical Hankel functions of the first kind and the corresponding Riccati-Bessel functions [52] are denoted $\mathrm {j}_l(z)$, $\mathrm {y}_l(z)$, $\mathrm {h}_l^{(1)}(z)=\mathrm {j}_l(z)+\textrm{i} \mathrm {y}_l(z)$, $\psi _l(z)=z\mathrm {j}_l(z)$ and $\xi _l(z)=z\mathrm {h}_l^{(1)}(z)$, respectively, all of order $l$.

4.2 Optical theorem and physical bounds for a spherical region in a lossy medium

We consider the physical bounds on absorption that can be derived from the optical theorem when it is formulated in terms of the multipole coefficients of a scattering problem. In particular, the scatterer is here embedded in a spherical region surrounded by a lossy medium, as shown in Fig. 1. Hence, the scatterer may consist of a general bianisotropic linear material and is bounded by a spherical surface of radius $a$. The surrounding medium is an infinite homogeneous and isotropic dielectric free space having relative permittivity $\epsilon _{\mathrm {b}}$ and wave number $k_{\mathrm {b}}=k_0\sqrt {\epsilon _{\mathrm {b}}}$. For simplicity, it is assumed that the background is non-magnetic (a magnetic background with relative permeability $\mu _{\mathrm {b}}\neq 1$ can straightforwardly be added to the analysis if required). The background is furthermore assumed to be passive, and possibly lossy, so that $ {\mathrm {Im}}\{\epsilon _{\mathrm {b}}\}\geq 0$, and with permittivity $\epsilon _{\mathrm {b}}$ that does not reside at the negative part of the real axis, which corresponds to the branch cut of the square root.

The optical theorem is once again given by the power balance Eq. (6) with the absorbed, scattered, extinct (total) and the absorbed incident powers defined by Eq. (7) – (10), respectively. Let $a_{\tau ml}^{\mathrm i}$ and $f_{\tau ml}$ denote the multipole coefficients of the incident (regular) and the scattered (outgoing) spherical vector waves, respectively, as defined in Eq. (70). Based on the orthogonality of the spherical vector waves on the spherical surface $\partial V_a$ as given by [11, Eqs. (A.31) and (A.32)], it can be shown that

$$P_{\mathrm{s}} = \frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\sum_{\tau,\;m,\;l}A_{\tau l}\left|f_{\tau ml}\right|^{2},$$
$$P_{\mathrm{t}} = \frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\sum_{\tau,\;m,\;l}2{\mathrm{Re}}\{B_{\tau l}a_{\tau ml}^{\mathrm i*}f_{\tau ml}\},$$
$$P_{\mathrm{i}} = \frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\sum_{\tau,\;m,\;l}C_{\tau l}\left|a_{\tau ml}^{\mathrm i}\right|^{2},$$
where
$$A_{\tau l} = \frac{1}{{\mathrm{Re}}\{k_{\mathrm{b}}\}}\left\{ \begin{array}{ll} -{\mathrm{Im}}\{k_{\mathrm{b}}^{*}\xi_l\xi_l^{\prime *}\}& \tau=1, \\ {\mathrm{Im}}\{k_{\mathrm{b}}^{*}\xi_l^{\prime}\xi_l^{*}\}& \tau=2, \end{array} \right.$$
$$B_{\tau l} = \frac{1}{\textrm{i} 2{\mathrm{Re}}\{k_{\mathrm{b}}\}}\left\{ \begin{array}{ll} k_{\mathrm{b}}^{*}\xi_l\psi_l^{\prime *}-k_{\mathrm{b}}\psi_l^{*}\xi_l^{\prime} & \tau=1, \\ -k_{\mathrm{b}}^{*}\xi_l^{\prime}\psi_l^{*}+k_{\mathrm{b}}\psi_l^{\prime *}\xi_l & \tau=2, \end{array} \right.$$
$$C_{\tau l} = \frac{1}{{\mathrm{Re}}\{k_{\mathrm{b}}\}}\left\{ \begin{array}{ll} {\mathrm{Im}}\{k_{\mathrm{b}}^{*}\psi_l\psi_l^{\prime *}\}& \tau=1, \\ -{\mathrm{Im}}\{k_{\mathrm{b}}^{*}\psi_l^{\prime}\psi_l^{*}\}& \tau=2, \end{array} \right.$$
for $\tau =1,2$ and $l=1,\ldots ,\infty$, and where the arguments of the Riccati-Bessel functions are $z=k_{\mathrm {b}}a$, see also [11, Eqs. (8)–(11)] and [16, Eqs. (6) and (7)]. By applying Poynting’s theorem to the scattered and the absorbed incident powers defined by Eqs. (8) and (10), it follows that $P_{\mathrm {s}}\geq 0$ and $P_{\mathrm {i}}\geq 0$ for a passive background medium, and hence that $A_{\tau l}> 0$ and $C_{\tau l}\geq 0$. Note that $B_{\tau l}$ is a complex-valued constant. For a lossless medium with $ {\mathrm {Im}}\{k_{\mathrm {b}}\}=0$, we can employ the Wronskian of the Riccati-Bessel functions $\psi _l\xi _l^{\prime }-\psi _l^{\prime }\xi _l=\textrm{i}$ and use $\xi _l^{*}=2\psi _l-\xi _l$ to show that the coefficients defined in Eqs. (40)–(42) become $A_{\tau l}=1$, $B_{\tau l}=-1/2$ and $C_{\tau l}=0$ in agreement with e.g., [52, Eq. (7.18)], see also [11, Eqs. (9)–(12)].

4.2.1 Optimal absorption of an arbitrary linear scatterer circumscribed by a sphere

Consider the contribution to the absorbed power from a single partial wave with fixed multi-index $(\tau ,\;m,\;l)$,

$$P_{\mathrm{a},\tau ml}=\frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\left[{-}A_{\tau l}\left|f_{\tau ml}\right|^{2} +2{\mathrm{Re}}\{B_{\tau l}a_{\tau ml}^{\mathrm i*}f_{\tau ml}\}+C_{\tau l}\left|a_{\tau ml}^{\mathrm i}\right|^{2}\right],$$
where we have employed the optical theorem given by Eq. (6) as well as Eqs. (37)–(39). Let the scattering coefficients $f_{\tau ml}$ be given by the T-matrix [52, Eq. (7.34)] for an arbitrary linear scatterer inside the spherical surface $\partial V_a$, so that
$$f_{n}=\sum_{n^{\prime}} T_{n,\;n^{\prime}}a_{n^{\prime}}^{\mathrm i},$$
and where we have introduced the multi-index notation $n=(\tau ,\;m,\;l)$. It is observed that the expression in Eq. (43) is a concave function of the complex-valued variables $T_{n,\;n^{\prime }}$ with respect to the primed index $n^{\prime }$. Differentiating Eq. (43) with respect to $T_{n,\;n^{\prime }}$ (for fixed $n$) gives the condition for stationarity
$$A_{\tau l}a_{n^{\prime}}^{\mathrm i} \sum_{n^{\prime\prime}}a_{n^{\prime\prime}}^{\mathrm{i}*} T_{n,\;n^{\prime\prime}}^{*}=B_{\tau l}a_{n}^{\mathrm{i}*}a_{n^{\prime}}^{\mathrm i},$$
which is an infinite-dimensional linear system of equations in the double-primed indices of $T_{n,\;n^{\prime \prime }}$. The corresponding system matrix $a_{n^{\prime }}^{\mathrm i}a_{n^{\prime \prime }}^{\mathrm {i}*}$ is of rank one and is in general unbounded. Assuming that this matrix is either bounded, or is truncated to some finite dimension, the corresponding matrix norm is given by
$$g=\sum_{\tau,\;m,\;l}\left| a_{\tau ml}^{\mathrm i}\right|^{2},$$
and the unique minimum norm (pseudo-inverse) T-matrix solution to Eq. (45) is given by
$$T_{n,\;n^{\prime}}= \frac{B_{\tau l}^{*}}{A_{\tau l}g}a_{n}^{\mathrm{i}}a_{n^{\prime}}^{\mathrm{i}*}.$$
By inserting Eq. (44) into Eq. (43) and completing the squares using Eq. (47), it can be shown that
$$\begin{aligned} P_{\mathrm{a},\tau ml}&=\frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\Bigg\{{-}A_{\tau l}\left| \sum_{\tau^{\prime} m^{\prime} l^{\prime}}\left(T_{\tau ml,\tau^{\prime} m^{\prime} l^{\prime}}- \frac{B_{\tau l}^{*}a_{\tau ml}^{\mathrm i}a_{\tau^{\prime} m^{\prime} l^{\prime}}^{\mathrm{i}*}}{A_{\tau l}g}\right) a_{\tau^{\prime} m^{\prime} l^{\prime}}^{\mathrm i} \right|^{2} \\ & \left. +\left(\frac{\left| B_{\tau l} \right|^{2}}{A_{\tau l}}+C_{\tau l}\right)\left| a_{\tau ml}^{\mathrm i}\right|^{2}\right\}. \end{aligned}$$
Due to the concavity of this expression ($A_{\tau l}>0$) it follows that the last term, which is independent of $T_{\tau ml,\tau ^{\prime } m^{\prime } l^{\prime }}$, gives the optimal absorption. Summing over the $\tau ml$-indices, the optimal absorption is hence obtained as
$$P_{\mathrm{a}}^{\mathrm {opt}}=\frac{{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}}{2\left|k_{\mathrm{b}}\right|^{2}\eta_0}\sum_{\tau,\;m,\;l} \left(\frac{\left| B_{\tau l} \right|^{2}}{A_{\tau l}}+C_{\tau l}\right)\left| a_{\tau ml}^{\mathrm i}\right|^{2}.$$
It is emphasized that the infinite dimensional matrix equation in Eq. (45) in general is related to an unbounded operator where the series in Eq. (46) does not converge (the corresponding matrix norm does not exist). However, this is merely a mathematical subtlety that does not pose any real problem here. Hence, considering that the T-matrix in Eq. (44) can be truncated to a finite size $L$ with $l,\;l^{\prime }\leq L$, the bound in Eq. (50) can be interpreted as the optimal absorption with respect to all incident and scattered fields up to multipole order $L$, as $L\rightarrow \infty$. Note in particular that the individual terms appearing in Eq. (50) do not depend on the truncation order, and it is only the interpretation of the partial sums that depend on $L$.

For an incident plane wave where $\boldsymbol {E}_{\mathrm {i}}(\boldsymbol r)=\boldsymbol {E}_0\textrm{e}^{\mathrm {i} k_{\mathrm {b}}\hat {\boldsymbol {k}}\cdot \boldsymbol {r}}$, the multipole coefficients $a_{\tau ml}^{\mathrm i}$ are given by [11, Eq. (5)], and the optimal bound becomes

$$P_{\mathrm{a}}^{\mathrm {opt}}=\frac{\pi{\mathrm{Re}}\{\sqrt{\epsilon_{\mathrm{b}}}\}\left| \boldsymbol{E}_0\right|^{2}}{\left|k_{\mathrm{b}}\right|^{2}\eta_0}\sum_{\tau=1}^{2}\sum_{l=1}^{\infty}(2l+1) \left(\frac{\left| B_{\tau l} \right|^{2}}{A_{\tau l}}+C_{\tau l}\right),$$
where we have made use of the sum identities [11, Eq. (6)] for the vector spherical harmonics. The corresponding optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ is obtained by normalizing with the intensity $I_{\mathrm {i}}$ of the plane wave at the origin $\boldsymbol {r}=\boldsymbol {0}$ given by Eq. (30), as well as with the geometrical area cross section of the sphere $\pi a^{2}$, giving
$$Q_{\mathrm{a}}^{\mathrm {opt}}=\frac{2}{\left|k_{\mathrm{b}}a\right|^{2}}\sum_{\tau=1}^{2}\sum_{l=1}^{\infty}(2l+1) \left(\frac{\left| B_{\tau l} \right|^{2}}{A_{\tau l}}+C_{\tau l}\right).$$
In the next section we will show that Eq. (52) converges whenever there are losses in the exterior medium and $ {\mathrm {Im}}\{k_{\mathrm {b}}\}> 0$. In Fig. 2 is illustrated the convergence of the expression given by Eq. (52) by plotting the partial sums against the number of included multipoles $L$. The calculations are for electrical sizes $k_0a\in \{0.1,1,10\}$ and with background losses $\epsilon _{\mathrm {b}}^{\prime \prime }\in \{10^{-9},10^{-3},10^{-1}\}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$. Clearly, with increasing external losses there are fewer modes that can contribute to the absorption inside the sphere, which is due to the interaction of the reactive near-fields of the higher order modes with the lossy exterior domain.

 figure: Fig. 2.

Fig. 2. Optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ of a sphere in a lossy medium, plotted as a function of the number of included multipoles $L$.

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In the lossless case, when $ {\mathrm {Im}}\{k_{\mathrm {b}}\}=0$, the truncated partial sums of Eq. (52) can be calculated as

$$Q_{\mathrm{a}}^{\mathrm{opt},L}=\frac{2}{(k_{\mathrm{b}}a)^{2}}\sum_{\tau=1}^{2}\sum_{l=1}^{L} \frac{2l+1}{4}=\frac{1}{(k_{\mathrm{b}}a)^{2}}L(L+2),$$
where $L$ is the truncated maximal multipole order, which can be determined by the method proposed in [53, p. 1508]. Interestingly, the obtained result in Eq. (53) is similar as the expression for maximum gain derived by Harrington in [54, Eq. (11) on p. 221].

4.2.2 Proof of convergence

To prove that Eq. (52) converges for a lossy medium where $ {\mathrm {Im}}\{k_{\mathrm {b}}\}> 0$, we consider the following power series expansions of the regular Riccati-Bessel functions

$$\psi_l(z)=\sum_{k=0}^{\infty} \alpha_{kl}z^{l+1+2k},$$
and the singular Riccati-Hankel functions
$$\xi_l(z)=\textrm{i}\sum_{k=0}^{l} \beta_{kl}z^{{-}l+2k}+\alpha_{0l}z^{l+1}+{\cal O}\{z^{l+2}\},$$
where $\alpha _{kl}=(-1/2)^{k}/k!(2l+2k+1)!!$ and $\beta _{kl}=-(1/2)^{k}(2l-2k-1)!!/k!$ cf., [55, Eqs. (10.53.1) and (10.53.2)] and where ${\cal O}\{\cdot \}$ denotes the big ordo defined in [56, p. 4]. By inserting Eqs. (54) and (55) into Eqs. (40)–(42) and retaining only the most dominating terms for fixed $z=k_{\mathrm {b}}a$ and increasing $l$, it is found that
$$A_{\tau l} \sim \left\{ \begin{array}{ll} \displaystyle \frac{2\beta_{0l}\beta_{1l}\sin 2\theta}{|z|^{2l-1}\cos\theta} & \tau=1, \\[0.3cm] \displaystyle \frac{l\beta_{0l}^{2}\sin 2\theta}{|z|^{2l+1}\cos\theta} & \tau=2, \end{array} \right.$$
$$B_{\tau l} \sim \left\{ \begin{array}{ll} \displaystyle -\frac{1}{2}\frac{\textrm{e}^{-\textrm{i}\theta(2l+1)}}{\cos\theta} & \tau=1, \\[0.3cm] \displaystyle -\frac{1}{2}\frac{l\textrm{e}^{-\textrm{i}\theta(2l+3)}+(l+1)\textrm{e}^{-\textrm{i}\theta(2l-1)}}{(2l+1)\cos\theta} & \tau=2, \end{array} \right.$$
$$C_{\tau l} \sim \left\{ \begin{array}{ll} \displaystyle \frac{-2\alpha_{0l}\alpha_{1l}|z|^{2l+3}\sin 2\theta}{\cos\theta} & \tau=1, \\[0.3cm] \displaystyle \frac{(l+1)\alpha_{0l}^{2}|z|^{2l+1}\sin 2\theta}{\cos\theta} & \tau=2, \end{array} \right.$$
and where $z=|z|\textrm{e}^{\textrm{i} \theta }$. Note that $\alpha _{0l}=1/(2l+1)!!$, $\alpha _{1l}=-(1/2)/(2l+3)!!$, $\beta _{0l}=-(2l-1)!!$ and $\beta _{1l}=-(1/2)(2l-3)!!$. Convergence of Eq. (52) can now be established by considering the factorial increase of the $A_{\tau l}$ coefficients, the boundedness of the $B_{\tau l}$ coefficients and the factorial decrease of the $C_{\tau l}$ coefficients for large $l$. In the lossless case when $\theta =0$, we have $A_{\tau l}=1$, $B_{\tau l}=-1/2$ and $C_{\tau l}=0$ and Eq. (52) is divergent as demonstrated in Eq. (53).

5. Numerical examples

In this section, we illustrate the theory that has been developed in Sections 3 and 4 in comparison with the normalized absorption cross sections of spherical objects embedded in a lossy medium. As objects of study, homogeneous and layered (core-shell) spheres are selected. The dielectric background medium is characterized by permittivity $\epsilon _{\mathrm {b}}=\epsilon _{\mathrm {b}}^{\prime }+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, where the choice of $\epsilon _{\mathrm {b}}^{\prime \prime }$ is based on the skin depth of human skin $\alpha =2k_0\epsilon _{\mathrm {b}}^{\prime \prime }$, $\alpha ^{-1}\in (10^{-4},10^{-2})$ cm, see [28, Table 3.2 on p. 49]. Note that the real part of the background permittivity does not play the key role in comparisons presented below, and thus we consistently choose $\epsilon _{\mathrm {b}}^{\prime }=1$ despite that the refractive index of human tissue $n\approx 1.33$ [28, Table 3.8 on p. 63]. In addition to this investigation, the absorption of spherical objects embedded in an almost lossless medium with relative permittivity $\epsilon _{\mathrm {b}}=1+\textrm{i} 10^{-9}$ is also considered.

In Fig. 3 is shown a comparison of the optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ given by Eq. (52), the absorption of a homogeneous sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$ (full Mie solution) obtained by normalization of Eq. (43) with the intensity $I_{\mathrm {i}}$ of the plane wave at the origin given by Eq. (30) and the geometrical area cross section $\pi a^{2}$, and the corresponding variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$ for a homogeneous object given by Eq. (32). The background relative permittivity is $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$ with various levels of background loss: $\epsilon _{\mathrm {b}}^{\prime \prime }\in \{10^{-9},10^{-3},10^{-1}\}$. The calculations of $Q_{\mathrm {a}}^{\mathrm {Au}}$ and $Q_{\mathrm {a}}^{\mathrm {var}}$ are for two different radii of gold spheres (20 nm and 89 nm) and the same photon energy range 1–5 eV (corresponds to the wavelength range 248–1240 nm) according to the Brendel-Bormann (BB) model fitted to experimental data as in [57, the dielectric model in Eq. (11) with parameter values from Table 1 and Table 3]. The sphere of radius $a=89$ nm has been tuned to optimal electric-dipole absorption for a lossless background as in [11, Fig. 4]. It is noted that the variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$ depends very weakly on the background loss for these parameter ranges, and the bound is therefore plotted only for $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$ (the plots for $\epsilon _{\mathrm {b}}^{\prime \prime }\in \{10^{-3},10^{-1}\}$ almost coincide). As can be seen in this plot, the two bounds $Q_{\mathrm {a}}^{\mathrm {opt}}$ and $Q_{\mathrm {a}}^{\mathrm {var}}$, which are derived under different assumptions (arbitrary structure of linear bianisotropic materials inside the sphere vs arbitrary structure of gold inside the sphere), give complementary information about the upper bounds on absorption. At the same time, the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {Au}}$ is not tight with respect to the optimal bounds as shown in Fig. 3, despite it is known that its electric-dipole contribution approaches the bound for electric-dipole absorption when the radius of object is 89 nm, see [11, Fig. 4].

 figure: Fig. 3.

Fig. 3. Comparison between the optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the absorption of a homogeneous sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$, and the corresponding variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$; all plotted as functions of the electrical size $k_0a$. The plots are for various levels of background loss $\epsilon _{\mathrm {b}}^{\prime \prime }$, and the calculations of $Q_{\mathrm {a}}^{\mathrm {var}}$ and $Q_{\mathrm {a}}^{\mathrm {Au}}$ are for two different radii of the sphere $a_1=20$ nm (to the left) and $a_2=89$ nm (to the right).

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 figure: Fig. 4.

Fig. 4. Comparison of the various upper bounds $Q_{\mathrm {a}}^{\mathrm {opt}}$, $Q_{\mathrm {a},2}^{\mathrm {opt}}$ (the electric multipole contribution), $Q_{\mathrm {a},21}^{\mathrm {opt}}$ (the electric-dipole upper bound), $Q_{\mathrm {a}}^{\mathrm {var}}$ and the absorption of a sphere $Q_{\mathrm {a}}$ tuned to optimal electric (plasmonic) dipole resonance at $k_0a=0.1$ for: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024+\textrm{i} 0.0040$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024 + \textrm{i} 0.0020$.

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Now, we would like to find such spherical objects which are resonant at small electrical size, but at the same time are of a reasonable physical size and have a resonance absorption peak close to the optimal absorption bound. To fit these requirements, one way is to “tune" a homogeneous sphere to the resonance at the desirable electrical size. An alternative approach is to consider a layered sphere constructed of a dielectric core and coated with a metallic shell.

 figure: Fig. 5.

Fig. 5. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the total absorption of a sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$ (ratio $r/d=0$), and the total absorption of a multilayered sphere made of silicon (core of radius $r$) and gold (shell of thickness $d)Q_{\mathrm {a}}^{\mathrm {Si,Au}}$; all are plotted as functions of the electrical size $k_0a$. The plots are made for a fixed level of a background loss $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$ and a fixed total radius of spheres $a=89$ nm.

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In Fig. 4(a) is shown a comparison of the various upper bounds on absorption and the absorption of a sphere tuned to optimal electric (plasmonic) dipole resonance. Here, $Q_{\mathrm {a}}$ denotes the full Mie solution for a homogeneous sphere with a (hypothetical) fixed value of permittivity $\epsilon =-2\epsilon _{\mathrm {b}}^{*}-\frac {12}{5}\epsilon _{\mathrm {b}}^{*2}(0.1)^{2} +\textrm{i} 2\epsilon _{\mathrm {b}}^{*2}\sqrt {\epsilon _{\mathrm {b}}^{*}}(0.1)^{3}$ which has been tuned to optimal electric-dipole resonance at $k_0a=0.1$, cf., [11, Eq. (55)]. The optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ is given by Eq. (52), $Q_{\mathrm {a},2}^{\mathrm {opt}}$ and $Q_{\mathrm {a},21}^{\mathrm {opt}}$ denote the corresponding electric multipole contribution ($\tau =2,~l=1,2,3,\ldots$) and the normalized electric-dipole absorption cross section ($\tau =2,~l=1$), respectively. The corresponding variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$ is given by Eq. (32). All calculations have been made for $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$. As can be seen in Fig. 2, for this combination of electrical size and background loss, it is (almost) sufficient to consider the dipole ($l=1$) contribution at resonance, which explains why the Mie solution $Q_{\mathrm {a}}$ in Fig. 4(a) is (almost) tight with the upper bounds $Q_{\mathrm {a},21}^{\mathrm {opt}}$ and $Q_{\mathrm {a},2}^{\mathrm {opt}}$, respectively. In Fig. 4(b) is shown the same calculations, except that here the background loss is given by $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$. Again, as can be seen in Fig. 2, with such small background losses the optimal $Q_{\mathrm {a}}^{\mathrm {opt}}$ is based on at least multipole orders up to $L=3$, which explains why the Mie solution $Q_{\mathrm {a}}$ in Fig. 4(b) is not tight with the corresponding upper bound $Q_{\mathrm {a},2}^{\mathrm {opt}}$. Interestingly that at the same time, the Mie solution can approach the upper electric dipole bound $Q_{\mathrm {a},21}^{\mathrm {opt}}$ when such an amount of losses in the background takes place.

Figure 5 depicts a comparison of the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ and its electric-dipole component $Q_{\mathrm {a,21}}^{\mathrm {opt}}$ with the total absorption $Q_{\mathrm {a}}$ of spherical objects of the total radius $a=89$ nm. In this plot, the following objects have been considered: a homogeneous sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$ (special case with ratio $r/d=0$) and three designs of a layered sphere, where the core is made of silicon, and it is coated by gold. The absorption of the layered sphere is based on the normalization of Eq. (43) similarly as in the previous examples, but here, the scattering coefficients $f_{\tau ml}$ defined in Eq. (44) are expressed in terms of the transition matrices $t_{\tau l}^{(i)}$ for layered spherical objects, see [52, p. 437]. The designs have been considered for three different ratios between the radius of core $r$ and the thickness of shell $d$: $r/d\in \{2,3,5\}$. Note that $r$ and the total radius $a$ coincide with $r_1$ and $r_2$ ($n=2$), respectively, introduced in [52, Fig. 8.21 on p. 436]. The dielectric properties of silicon are represented by Drude-Lorentz model that fits the measurement data and valid in the photon range $1-6$ eV, see [58, the dielectric model in Eq. (4) with parameter values in Table 1]. The permittivity of background is $\epsilon _{\mathrm {b}}=1+\textrm{i} 10^{-1}$. As can be seen from Fig. 5, by replacing a part of the metallic sphere with silicon, it is possible to obtain a plasmonic resonance at smaller electrical sizes $k_0a$. This is a magnetic dipole resonance, which is inherent in dielectric materials [32]. It should be noted that by increasing the ratio between the radius of the silicon core and the thickness of the gold shell, the composed sphere becomes resonant at smaller electrical sizes: e.g., for $r/d=5$, the layered sphere is resonant at $k_0a\approx 0.49$, while the sphere with $r/d=2$ and the gold sphere are resonant at $k_0a\approx 0.64$ and $k_0a\approx 1.13$, respectively. Hence, by increasing the ratio between the dielectric and metallic sizes, the resonance of the composed object will move towards the resonance of the dielectric sphere that occurs when the wavelength inside the sphere approximately equals to its diameter [32,59].

In Figs. 6(a)–(c) is shown a comparison of the various upper bounds on absorption, and the absorption of a 2-layered core-shell sphere of different designs for three different levels of losses in the background: $\epsilon _{\mathrm {b}}^{\prime \prime }\in \{10^{-1},10^{-3},10^{-9}\}$, where $\epsilon _{\mathrm {b}} = 1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$. The considered designs are with cores of radii $r\in \{30,60,70\}$ nm and the corresponding thicknesses of shells $d\in \{10,20,10\}$ nm. The core is made of germanium (Ge), which is characterized by Drude-Lorentz model [58, the dielectric model in Eq. (4) with parameter values in Table 1] valid in the photon energy range $0.5-6.0$ eV. The dielectric core is coated by gold (Au) which is characterized by the BB model [57]. The corresponding variational upper bound for the 2-layered sphere $Q_{\mathrm {a}}^{\mathrm {var}}$ has been obtained by Eq. (32). As can be seen from these figures, the composed structure based on Ge and Au is resonant at small electrical sizes $k_0a$, but the magnitude of these resonances is not tight to none of the multipole upper bounds, even when the amount of the background losses is high, see Fig. 6(a) with results for $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$. By comparison of the results on the upper bounds for absorption $Q_{\mathrm {a}}^{\mathrm {opt}}$ obtained by Eq. (52) and $Q_{\mathrm {a}}^{\mathrm {var}}$ for different amounts of losses in the background in Figs. 6(a)–(c), it should be noted that these bounds provide a complementary information on absorption, and thus this conclusion is valid both for homogeneous and layered spherical objects. It can be concluded that for backgrounds with strong losses, the multipole bound $Q_{\mathrm {a}}^{\mathrm {opt}}$ brings more information on absorption limitations, while the variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$ is more tight for smaller objects that are embedded in low loss surrounding media, which complements the results obtained by $Q_{\mathrm {a}}^{\mathrm {opt}}$, see e.g., $Q_{\mathrm {a}}^{\mathrm {var}}(r,\;d)$ for $r=30$ nm and $d=10$ nm in Fig. 6(c).

6. Summary and conclusions

In this paper, two fundamental multipole bounds on absorption of scattering objects embedded in a lossy surrounding medium have been derived. The derivation of these bounds have been made under two fundamentally different assumptions: based on equivalent currents inside the scatterer, and with respect to the external fields using the T-matrix parameters, respectively. The first bound depends on the material properties of scatterer as well as on its shape, while the second bound is applicable to spherical objects made of an arbitrary material. Through the numerical examples, it has been illustrated that the derived bounds can complement each other, depending on the amount of losses in the surrounding medium.

 figure: Fig. 6.

Fig. 6. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, variational bound on the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {var}}$, and total absorption of a multilayered sphere made of germanium (core of radius $r$) and gold (shell of thickness $d)\;Q_{\mathrm {a}}^{\mathrm {Ge,Au}}$. All the results are plotted as functions of the electrical size $k_0a$ for different levels of losses in the background medium with relative permittivity $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$; c) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$.

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A. Derivations based on calculus of variation

A.1 Optimal power absorption

Consider the Lagrangian functional for the optimization problem formulated in Eq. (19) which is given by

$$\begin{aligned} {\cal L}(\boldsymbol{F},\lambda)&=(1-\lambda){\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}^{*}\cdot\boldsymbol{M}_{\mathrm{a}}\cdot\boldsymbol{F}\thinspace\textrm{d} v\right\} +\lambda{\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{t}}\cdot\boldsymbol{F}\thinspace\textrm{d} v\right\} \\ & -\lambda{\mathrm{Im}}\left\{\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{b}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v\right\}, \end{aligned}$$
where $\lambda$ is the Lagrange multiplier. Taking the first variation of Eq. (59) yields
$$\delta\!{\cal L}(\boldsymbol{F},\lambda)={\mathrm{Im}}\left\{\int_{V}\delta\!\boldsymbol{F}^{*}\cdot \Big[(1-\lambda)\left(\boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{a}}^{{\dagger}}\right)\cdot\boldsymbol{F} -\lambda\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}} \Big]\thinspace\textrm{d} v \right\},$$
where $(\cdot )^{\dagger }$ denotes the Hermitian transpose. Hence, a stationary solution with $\delta \!{\cal L}(\boldsymbol {F},\lambda )=0$ is given by
$$\boldsymbol{F}=\frac{\alpha}{2\textrm{i}}\left({\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}},$$
where $\alpha =\lambda /(1-\lambda )$, and
$${\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\}=\frac{\boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{a}}^{{\dagger}}}{2\textrm{i}}.$$
Inserting the solution to Eq. (61) into Eq. (12) gives the optimal absorption
$$P_{\mathrm{a}}^{\mathrm {opt}}=\frac{k_0\alpha^{2}}{8\eta_0}\int_{V} \boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{t}}\cdot\left({\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v.$$

A.2 Maximization of parameter $q$

Consider the function $f(\boldsymbol {M}_{\mathrm {a}})$

$$f(\boldsymbol{M}_{\mathrm{a}})=\int_V \boldsymbol{F}_{\mathrm{i}}^{*}\cdot\boldsymbol{M}_{\mathrm{t}}\cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\boldsymbol{M}_{\mathrm{t}}^{{\dagger}}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v,$$
which represents the denominator of parameter $q$ defined in Eq. (22), and where $\boldsymbol {M}_{\mathrm {t}} = \boldsymbol {M}_{\mathrm {a}} - \boldsymbol {M}_{\mathrm {b}}^{\dagger }$ is based on definitions in Eqs. (16)–(18). The function in Eq. (64) is convex in $\boldsymbol {M}_{\mathrm {a}}$ for $ {\mathrm {Im}}\{\boldsymbol {M}_{\mathrm {a}}\}>0$, and hence in order to maximize $q$ in Eq. (22), $f(\boldsymbol {M}_{\mathrm {a}})$ has to be minimized. The first variation of Eq. (64) is given by
$$\begin{aligned} \delta\!f(\boldsymbol{M}_{\mathrm{a}}) &= \int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\delta\!\boldsymbol{M}_{\mathrm{a}}\cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1}\cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right)^{\dagger}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v \\ & - \int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right) \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1}\cdot{\mathrm{Im}}\{\delta\!\boldsymbol{M}_{\mathrm{a}}\} \\& \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1}\cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right)^{\dagger}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v \\ & +\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right) \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1}\cdot\delta\!\boldsymbol{M}_{\mathrm{a}}^{\dagger}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v, \end{aligned}$$
and where we have employed the relation
$$\delta\!\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1} ={-}\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot{\mathrm{Im}}\{\delta\!\boldsymbol{M}_{\mathrm{a}}\}\cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}.$$
The stationarity condition $\delta \!f(\boldsymbol {M}_{\mathrm {a}})=0$ for all $\delta \!\boldsymbol {M}_{\mathrm {a}}$ and all $\boldsymbol {F}_{\mathrm {i}}$ gives the simplified condition
$$\begin{aligned} &{\mathrm{Re}}\left\{ \delta\!\boldsymbol{M}_{\mathrm{a}}\cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1}\cdot\left( \boldsymbol{M}_{\mathrm{a}}- \boldsymbol{M}_{\mathrm{b}}^{\dagger} \right)^{\dagger} \right. \\ & -\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right) \left. \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1} \cdot\frac{\delta\!\boldsymbol{M}_{\mathrm{a}}}{2\textrm{i}} \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{-1} \cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right)^{\dagger}\right\}=\boldsymbol{0}, \end{aligned}$$
and which can be reorganized as
$${\mathrm{Re}}\left\{\left[\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{{\dagger}} \right)\cdot\frac{\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}}{2\textrm{i}} -\boldsymbol{I}\right] \cdot\delta\!\boldsymbol{M}_{\mathrm{a}}\cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\} \right)^{{-}1}\cdot\left( \boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{{\dagger}} \right)^{{\dagger}} \right\}=\boldsymbol{0}.$$

Assuming that both $ {\mathrm {Im}}\{\boldsymbol {M}_{\mathrm {a}}\}>0$ and $ {\mathrm {Im}}\{\boldsymbol {M}_{\mathrm {b}}\}>0$ and hence that $\boldsymbol {M}_{\mathrm {a}}\neq \boldsymbol {M}_{\mathrm {b}}^{\dagger }$, the optimal solution is obtained by requiring that the first line within parenthesis above vanishes, yielding

$${\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{a}}\}=\frac{\boldsymbol{M}_{\mathrm{a}}-\boldsymbol{M}_{\mathrm{b}}^{{\dagger}}}{2\textrm{i}},$$
or equivalently that $\boldsymbol {M}_{\mathrm {a}}=\boldsymbol {M}_{\mathrm {b}}$. This solution implies that the minimum of $f(\boldsymbol {M}_{\mathrm {a}})$ in Eq. (64) is given by
$$ \begin{aligned} f(\boldsymbol{M}_{\mathrm{a}})\left|_{\boldsymbol{M}_{\mathrm{a}} =\boldsymbol{M}_{\mathrm{b}}}\right. & = \int_{V} \boldsymbol{F}_{\mathrm{i}}^{*}\cdot\left( \boldsymbol{M}_{\mathrm{b}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right) \cdot\left( {\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{b}}\} \right)^{-1} \cdot\left( \boldsymbol{M}_{\mathrm{b}}-\boldsymbol{M}_{\mathrm{b}}^{\dagger} \right)^{\dagger}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v \\ & = 4\int_{V}\boldsymbol{F}_{\mathrm{i}}^{*}\cdot{\mathrm{Im}}\{\boldsymbol{M}_{\mathrm{b}}\}\cdot\boldsymbol{F}_{\mathrm{i}}\thinspace\textrm{d} v. \end{aligned}$$
By substitution of the result of Eq. (70) into the denominator of Eq. (22), it is finally seen that $q$ reaches its maximum at $1$, and hence $q\leq 1$.

B. Spherical vector wave field expansion

Consider a source-free, homogeneous and isotropic medium with relative permittivity and permeability $\epsilon$ and $\mu$, respectively, wave number $k=k_0\sqrt {\mu \epsilon }$ and wave impedance $\eta _0\eta$ where $\eta =\sqrt {\mu /\epsilon }$.

The electromagnetic field can then be expanded in spherical vector waves as

$$\left\{\begin{array}{l} \boldsymbol{E}(\boldsymbol{r})=\displaystyle\sum_{\tau,\;m,\;l}a_{\tau ml}\textbf{v}_{\tau ml}(k\boldsymbol{r})+f_{\tau ml}\textbf{u}_{\tau ml}(k\boldsymbol{r}), \\ \boldsymbol{H}(\boldsymbol{r})=\displaystyle\frac{1}{\textrm{i}\eta_0\eta}\sum_{\tau,\;m,\;l}a_{\tau ml}\textbf{v}_{\bar{\tau} ml}(k\boldsymbol{r})+f_{\tau ml}\textbf{u}_{\bar{\tau} ml}(k\boldsymbol{r}), \end{array}\right.$$
where $\textbf {v}_{\tau ml}(k\boldsymbol {r})$ and $\textbf {u}_{\tau ml}(k\boldsymbol {r})$ are the regular and the outgoing spherical vector waves, respectively, and $a_{\tau ml}$ and $f_{\tau ml}$ the corresponding multipole coefficients, see e.g., [44,46,4952]. Here, $l=1,2,\ldots ,$ is the multipole order, $m=-l,\;\ldots\;,\;l$, the azimuthal index and $\tau =1,2$, where $\tau =1$ indicates a transverse electric (TE) magnetic multipole and $\tau =2$ a transverse magnetic (TM) electric multipole, and $\bar {\tau }$ denotes the dual index, i.e., $\bar {1}=2$ and $\bar {2}=1$.

Funding

Swedish Foundation for Strategic Research (AM13-0011).

Disclosures

The authors declare no conflicts of interest.

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References

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  1. S. Tretyakov, “Maximizing absorption and scattering by dipole particles,” Plasmonics 9(4), 935–944 (2014).
    [Crossref]
  2. M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. London, Ser. A 463(2086), 2589–2607 (2007).
    [Crossref]
  3. C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A: Math. Theor. 40(36), 11165–11182 (2007).
    [Crossref]
  4. K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48(8), 1230–1234 (2000).
    [Crossref]
  5. M. Gustafsson and D. Sjöberg, “Physical bounds and sum rules for high-impedance surfaces,” IEEE Trans. Antennas Propag. 59(6), 2196–2204 (2011).
    [Crossref]
  6. J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D: Appl. Phys. 39(6), 1226–1229 (2006).
    [Crossref]
  7. C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on metamaterials: Restrictions on scattering and absorption over a frequency interval,” J. Phys. D: Appl. Phys. 40(22), 7146–7151 (2007).
    [Crossref]
  8. O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. Express 24(4), 3329–3364 (2016).
    [Crossref]
  9. H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, “Fundamental limits to near-field optical response over any bandwidth,” Phys. Rev. X 9(1), 011043 (2019).
    [Crossref]
  10. S. Molesky, W. Jin, P. S. Venkataram, and A. W. Rodriguez, “Bounds on absorption and thermal radiation for arbitrary objects,” arXiv preprint arXiv:1907.04418 (2019).
  11. S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
    [Crossref]
  12. S. Nordebo, M. Mirmoosa, and S. Tretyakov, “On the quasistatic optimal plasmonic resonances in lossy media,” J. Appl. Phys. 125(10), 103105 (2019).
    [Crossref]
  13. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
    [Crossref]
  14. A. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
    [Crossref]
  15. Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
    [Crossref]
  16. I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. A 18(6), 1275–1278 (2001).
    [Crossref]
  17. I. W. Sudiarta and P. Chỳlek, “Mie scattering by a spherical particle in an absorbing medium,” Appl. Opt. 41(18), 3545–3546 (2002).
    [Crossref]
  18. J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. A 23(11), 2784–2796 (2006).
    [Crossref]
  19. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24(9), 2943–2952 (2007).
    [Crossref]
  20. M. I. Mishchenko, G. Videen, and P. Yang, “Extinction by a homogeneous spherical particle in an absorbing medium,” Opt. Lett. 42(23), 4873–4876 (2017).
    [Crossref]
  21. M. I. Mishchenko and J. M. Dlugach, “Multiple scattering of polarized light by particles in an absorbing medium,” Appl. Opt. 58(18), 4871–4877 (2019).
    [Crossref]
  22. R. Dezert, P. Richetti, and A. Baron, “Complete multipolar description of reflection and transmission across a metasurface for perfect absorption of light,” Opt. Express 27(19), 26317–26330 (2019).
    [Crossref]
  23. S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radio-frequency electric field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
    [Crossref]
  24. E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
    [Crossref]
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2019 (6)

H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, “Fundamental limits to near-field optical response over any bandwidth,” Phys. Rev. X 9(1), 011043 (2019).
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S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
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S. Nordebo, M. Mirmoosa, and S. Tretyakov, “On the quasistatic optimal plasmonic resonances in lossy media,” J. Appl. Phys. 125(10), 103105 (2019).
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M. I. Mishchenko and J. M. Dlugach, “Multiple scattering of polarized light by particles in an absorbing medium,” Appl. Opt. 58(18), 4871–4877 (2019).
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R. Dezert, P. Richetti, and A. Baron, “Complete multipolar description of reflection and transmission across a metasurface for perfect absorption of light,” Opt. Express 27(19), 26317–26330 (2019).
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A. K. Skrivervik, M. Bosiljevac, and Z. Sipus, “Fundamental limits for implanted antennas: Maximum power density reaching free space,” IEEE Trans. Antennas Propag. 67(8), 4978–4988 (2019).
[Crossref]

2017 (4)

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

M. I. Mishchenko, G. Videen, and P. Yang, “Extinction by a homogeneous spherical particle in an absorbing medium,” Opt. Lett. 42(23), 4873–4876 (2017).
[Crossref]

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

2016 (1)

2015 (1)

W. Al-Taay, S. F. Oboudi, E. Yousif, M. A. Nabi, R. M. Yusop, and D. Derawi, “Fabrication and characterization of nickel chloride doped PMMA films,” Adv. Mater. Sci. Eng. 2015, 1–5 (2015).
[Crossref]

2014 (2)

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

S. Tretyakov, “Maximizing absorption and scattering by dipole particles,” Plasmonics 9(4), 935–944 (2014).
[Crossref]

2012 (6)

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radio-frequency electric field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
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E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
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A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20(18), 20599–20604 (2012).
[Crossref]

F. Merli, L. Bolomey, F. Gorostidi, B. Fuchs, J.-F. Zurcher, Y. Barrandon, E. Meurville, J. R. Mosig, and A. K. Skrivervik, “Example of data telemetry for biomedical applications: An in vivo experiment,” IEEE Antennas Wirel. Propag. Lett. 11, 1650–1654 (2012).
[Crossref]

J. Wang, H. Zhang, T. Lv, and T. A. Gulliver, “Capacity of 60 GHz wireless communication systems over fading channels,” J. Networks 7(1), 203–209 (2012).
[Crossref]

A. E. Miroshnichenko, B. Luk’yanchuk, S. A. Maier, and Y. S. Kivshar, “Optically induced interaction of magnetic moments in hybrid metamaterials,” ACS Nano 6(1), 837–842 (2012).
[Crossref]

2011 (2)

F. Merli, B. Fuchs, J. R. Mosig, and A. K. Skrivervik, “The effect of insulating layers on the performance of implanted antennas,” IEEE Trans. Antennas Propag. 59(1), 21–31 (2011).
[Crossref]

M. Gustafsson and D. Sjöberg, “Physical bounds and sum rules for high-impedance surfaces,” IEEE Trans. Antennas Propag. 59(6), 2196–2204 (2011).
[Crossref]

2008 (1)

X. Huang, P. K. Jain, I. H. El-Sayed, and M. A. El-Sayed, “Plasmonic photothermal therapy (PPTT) using gold nanoparticles,” Lasers Med. Sci. 23(3), 217–228 (2008).
[Crossref]

2007 (5)

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. A 24(9), 2943–2952 (2007).
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C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on metamaterials: Restrictions on scattering and absorption over a frequency interval,” J. Phys. D: Appl. Phys. 40(22), 7146–7151 (2007).
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M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. London, Ser. A 463(2086), 2589–2607 (2007).
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C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A: Math. Theor. 40(36), 11165–11182 (2007).
[Crossref]

C. Park and T. S. Rappaport, “Short-range wireless communications for next-generation networks: UWB, 60 GHz millimeter-wave WPAN, and ZigBee,” IEEE Wirel. Commun. 14(4), 70–78 (2007).
[Crossref]

2006 (2)

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D: Appl. Phys. 39(6), 1226–1229 (2006).
[Crossref]

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. A 23(11), 2784–2796 (2006).
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2002 (1)

2001 (2)

2000 (1)

K. N. Rozanov, “Ultimate thickness to bandwidth ratio of radar absorbers,” IEEE Trans. Antennas Propag. 48(8), 1230–1234 (2000).
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1999 (1)

A. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

1998 (1)

1985 (1)

N. D. Hawkins, R. Steele, D. C. Rickard, and C. R. Shepherd, “Path loss characteristics of 60 GHz transmissions,” Elect. Lett. 21(22), 1054–1055 (1985).
[Crossref]

1980 (1)

1979 (1)

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

1971 (1)

R. Progelhof, J. Franey, and T. W. Haas, “Absorption coefficient of unpigmented poly(methyl methacrylate), polystyrene, polycarbonate and poly(4-methylpentene-1) sheets,” J. Appl. Polym. Sci. 15(7), 1803–1807 (1971).
[Crossref]

1963 (1)

M. L. Meeks and A. E. Lilley, “The microwave spectrum of oxygen in the Earth’s atmosphere,” J. Geophys. Res. 68(6), 1683–1703 (1963).
[Crossref]

1958 (1)

R. Harrington, “On the gain and beamwidth of directional antennas,” IEEE Trans. Antennas Propag. 6(3), 219–225 (1958).
[Crossref]

1947 (1)

J. H. V. Vleck, “The absorption of microwaves by oxygen,” Phys. Rev. 71(7), 413–424 (1947).
[Crossref]

Ackerson, B. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Ackerson, C. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Adams, R. A.

R. A. Adams, Calculus: a complete course (Addison-Wesley, 1995), 3rd ed.

Al-Taay, W.

W. Al-Taay, S. F. Oboudi, E. Yousif, M. A. Nabi, R. M. Yusop, and D. Derawi, “Fabrication and characterization of nickel chloride doped PMMA films,” Adv. Mater. Sci. Eng. 2015, 1–5 (2015).
[Crossref]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 2001), 5th ed.

Auwera, J. V.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Barbe, A.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Baron, A.

Barrandon, Y.

F. Merli, L. Bolomey, F. Gorostidi, B. Fuchs, J.-F. Zurcher, Y. Barrandon, E. Meurville, J. R. Mosig, and A. K. Skrivervik, “Example of data telemetry for biomedical applications: An in vivo experiment,” IEEE Antennas Wirel. Propag. Lett. 11, 1650–1654 (2012).
[Crossref]

Bayford, R.

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

Belov, P. A.

Bernath, P. F.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Birk, M.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. H. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, J. J. H. T. Furtenbacher, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

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

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Bolomey, L.

F. Merli, L. Bolomey, F. Gorostidi, B. Fuchs, J.-F. Zurcher, Y. Barrandon, E. Meurville, J. R. Mosig, and A. K. Skrivervik, “Example of data telemetry for biomedical applications: An in vivo experiment,” IEEE Antennas Wirel. Propag. Lett. 11, 1650–1654 (2012).
[Crossref]

Bosiljevac, M.

A. K. Skrivervik, M. Bosiljevac, and Z. Sipus, “Fundamental limits for implanted antennas: Maximum power density reaching free space,” IEEE Trans. Antennas Propag. 67(8), 4978–4988 (2019).
[Crossref]

Boström, A.

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Phys. Rev. (1)

J. H. V. Vleck, “The absorption of microwaves by oxygen,” Phys. Rev. 71(7), 413–424 (1947).
[Crossref]

Phys. Rev. B (1)

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Phys. Rev. X (1)

H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, “Fundamental limits to near-field optical response over any bandwidth,” Phys. Rev. X 9(1), 011043 (2019).
[Crossref]

Plasmonics (1)

S. Tretyakov, “Maximizing absorption and scattering by dipole particles,” Plasmonics 9(4), 935–944 (2014).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. London, Ser. A 463(2086), 2589–2607 (2007).
[Crossref]

Other (16)

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S. A. Maier, Plasmonics: Fundamentals and Applications (Springer-Verlag, 2007).

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H. S. Sehmi, W. Langbein, and E. A. Muljarov, “Optimizing the Drude-Lorentz model for material permittivity: Examples for semiconductors,” in 2017 Progress In Electromagnetics Research Symposium-Spring (PIERS) (IEEE, 2017), pp. 994–1000.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

F. W. J. Olver, Asymptotics and special functions (A K Peters, Ltd, 1997).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.

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

D. G. Luenberger, Optimization by vector space methods (John Wiley & Sons, 1969).

R. A. Adams, Calculus: a complete course (Addison-Wesley, 1995), 3rd ed.

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A. Boström, G. Kristensson, and S. Ström, “Transformation properties of plane, spherical and cylindrical scalar and vector wave functions,” in Field Representations and Introduction to Scattering, V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds. (Elsevier Science Publishers, 1991), pp. 165–210, Acoustic, Electromagnetic and Elastic Wave Scattering, chap. 4.

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Figures (6)

Fig. 1.
Fig. 1. Problem setup. Here, $\epsilon _{\mathrm {b}}$ and $\mu _{\mathrm {b}}$ denote the relative permittivity and permeability of the passive background medium, respectively, and $\hat {\boldsymbol {n}}$ the outward unit vector.
Fig. 2.
Fig. 2. Optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$ of a sphere in a lossy medium, plotted as a function of the number of included multipoles $L$.
Fig. 3.
Fig. 3. Comparison between the optimal normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the absorption of a homogeneous sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$, and the corresponding variational bound $Q_{\mathrm {a}}^{\mathrm {var}}$; all plotted as functions of the electrical size $k_0a$. The plots are for various levels of background loss $\epsilon _{\mathrm {b}}^{\prime \prime }$, and the calculations of $Q_{\mathrm {a}}^{\mathrm {var}}$ and $Q_{\mathrm {a}}^{\mathrm {Au}}$ are for two different radii of the sphere $a_1=20$ nm (to the left) and $a_2=89$ nm (to the right).
Fig. 4.
Fig. 4. Comparison of the various upper bounds $Q_{\mathrm {a}}^{\mathrm {opt}}$, $Q_{\mathrm {a},2}^{\mathrm {opt}}$ (the electric multipole contribution), $Q_{\mathrm {a},21}^{\mathrm {opt}}$ (the electric-dipole upper bound), $Q_{\mathrm {a}}^{\mathrm {var}}$ and the absorption of a sphere $Q_{\mathrm {a}}$ tuned to optimal electric (plasmonic) dipole resonance at $k_0a=0.1$ for: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024+\textrm{i} 0.0040$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$ where $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$, and the permittivity of the sphere is $\epsilon =-2.024 + \textrm{i} 0.0020$.
Fig. 5.
Fig. 5. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, the total absorption of a sphere made of gold $Q_{\mathrm {a}}^{\mathrm {Au}}$ (ratio $r/d=0$), and the total absorption of a multilayered sphere made of silicon (core of radius $r$) and gold (shell of thickness $d)Q_{\mathrm {a}}^{\mathrm {Si,Au}}$; all are plotted as functions of the electrical size $k_0a$. The plots are made for a fixed level of a background loss $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$ and a fixed total radius of spheres $a=89$ nm.
Fig. 6.
Fig. 6. Comparison between the optimal total normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {opt}}$, variational bound on the normalized absorption cross section $Q_{\mathrm {a}}^{\mathrm {var}}$, and total absorption of a multilayered sphere made of germanium (core of radius $r$) and gold (shell of thickness $d)\;Q_{\mathrm {a}}^{\mathrm {Ge,Au}}$. All the results are plotted as functions of the electrical size $k_0a$ for different levels of losses in the background medium with relative permittivity $\epsilon _{\mathrm {b}}=1+\textrm{i} \epsilon _{\mathrm {b}}^{\prime \prime }$: a) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-1}$; b) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-3}$; c) $\epsilon _{\mathrm {b}}^{\prime \prime }=10^{-9}$.

Equations (70)

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{ × E { i , s } = i k 0 η 0 μ b H { i , s } , × H { i , s } = i k 0 η 0 1 ϵ b E { i , s } ,
{ D = ϵ 0 ϵ E + 1 c 0 χ e m H , B = 1 c 0 χ m e E + μ 0 μ H ,
{ × E = i k 0 χ m e E + i k 0 η 0 μ H , × H = i k 0 η 0 1 ϵ E i k 0 χ e m H ,
{ × E = i k 0 η 0 μ b H J m , × H = i k 0 η 0 1 ϵ b E + J e ,
{ J e = i k 0 η 0 1 χ e e E i k 0 χ e m H , J m = i k 0 χ m e E i k 0 η 0 χ m m H .
P a = P s + P t + P i ,
P a = 1 2 R e { V E × H n ^ d S } ,
P s = 1 2 R e { V E s × H s n ^ d S } ,
P t = 1 2 R e { V ( E i × H s + E s × H i ) n ^ d S } ,
P i = 1 2 R e { V E i × H i n ^ d S } ,
{ n ^ × ( E i + E s ) = n ^ × E , n ^ × ( H i + H s ) = n ^ × H ,
P a = k 0 2 η 0 I m { V F M a F d v } ,
P t = k 0 2 η 0 I m { V F i M t F d v } 2 P i ,
P i = k 0 2 η 0 I m { V F i M b F i d v } ,
F = ( E η 0 H ) , F i = ( E i η 0 H i ) ,
M a = ( ϵ χ e m χ m e μ ) = χ + M b
χ = ( χ e e χ e m χ m e χ m m ) , M b = ( ϵ b I 0 0 μ b I ) ,
M t = ( ϵ ϵ b I χ e m χ m e μ μ b I ) = χ + i 2 I m { M b } .
m a x i m i z e P a s u b j e c t   t o P s 0 ,
P a o p t = k 0 α 2 8 η 0 V F i M t ( I m { M a } ) 1 M t F i d v ,
α 2 + 2 α = q ,
q = 4 V F i I m { M b } F i d v V F i M t ( I m { M a } ) 1 M t F i d v .
α = 1 1 q .
M a , i = ϵ i I , M b = ϵ b I , M t , i = ( ϵ i ϵ b ) I ,
P a v a r = k 0 α 2 8 η 0 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | E i ( r ) | 2 d v ,
q = 4 I m { ϵ b } V | E i ( r ) | 2 d v i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | E i ( r ) | 2 d v .
V a | E i ( r ) | 2 d v = | E 0 | 2 2 π τ = 1 2 l = 1 ( 2 l + 1 ) W τ l ( k b , a ) ,
W τ l ( k , a ) = V a | v τ m l ( k r ) | 2 d v ,
V a | E i ( r ) | 2 d v = | E 0 | 2 V a ,
I i = | E 0 | 2 R e { ϵ b } / 2 η 0 ,
σ a v a r = k 0 R e { ϵ b } α 2 4 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } V i | e i k b k ^ r | 2 d v .
Q a v a r = k 0 a R e { ϵ b } α 2 4 i = 1 N | ϵ i ϵ b | 2 I m { ϵ i } 1 π a 3 V i | e i k b k ^ r | 2 d v ,
1 π a 3 V i | e i k b k ^ r | 2 d v = 2 τ = 1 2 l = 1 ( 2 l + 1 ) a 3 [ W τ l ( k b , a i ) W τ l ( k b , a i 1 ) ] ,
W 1 l ( k b , a i ) = a i 2 I m { k b j l + 1 ( k b a i ) j l ( k b a i ) } I m { k b 2 } ,
W 2 l ( k b , a i ) = ( l + 1 ) W 1 , l 1 ( k b , a i ) + l W 1 , l + 1 ( k b , a i ) ( 2 l + 1 ) ,
P a v a r = k 0 2 η 0 | ϵ ϵ b | 2 I m { ϵ } | E 0 | 2 V a ,
P s = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l A τ l | f τ m l | 2 ,
P t = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l 2 R e { B τ l a τ m l i f τ m l } ,
P i = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l C τ l | a τ m l i | 2 ,
A τ l = 1 R e { k b } { I m { k b ξ l ξ l } τ = 1 , I m { k b ξ l ξ l } τ = 2 ,
B τ l = 1 i 2 R e { k b } { k b ξ l ψ l k b ψ l ξ l τ = 1 , k b ξ l ψ l + k b ψ l ξ l τ = 2 ,
C τ l = 1 R e { k b } { I m { k b ψ l ψ l } τ = 1 , I m { k b ψ l ψ l } τ = 2 ,
P a , τ m l = R e { ϵ b } 2 | k b | 2 η 0 [ A τ l | f τ m l | 2 + 2 R e { B τ l a τ m l i f τ m l } + C τ l | a τ m l i | 2 ] ,
f n = n T n , n a n i ,
A τ l a n i n a n i T n , n = B τ l a n i a n i ,
g = τ , m , l | a τ m l i | 2 ,
T n , n = B τ l A τ l g a n i a n i .
P a , τ m l = R e { ϵ b } 2 | k b | 2 η 0 { A τ l | τ m l ( T τ m l , τ m l B τ l a τ m l i a τ m l i A τ l g ) a τ m l i | 2 + ( | B τ l | 2 A τ l + C τ l ) | a τ m l i | 2 } .
P a o p t = R e { ϵ b } 2 | k b | 2 η 0 τ , m , l ( | B τ l | 2 A τ l + C τ l ) | a τ m l i | 2 .
P a o p t = π R e { ϵ b } | E 0 | 2 | k b | 2 η 0 τ = 1 2 l = 1 ( 2 l + 1 ) ( | B τ l | 2 A τ l + C τ l ) ,
Q a o p t = 2 | k b a | 2 τ = 1 2 l = 1 ( 2 l + 1 ) ( | B τ l | 2 A τ l + C τ l ) .
Q a o p t , L = 2 ( k b a ) 2 τ = 1 2 l = 1 L 2 l + 1 4 = 1 ( k b a ) 2 L ( L + 2 ) ,
ψ l ( z ) = k = 0 α k l z l + 1 + 2 k ,
ξ l ( z ) = i k = 0 l β k l z l + 2 k + α 0 l z l + 1 + O { z l + 2 } ,
A τ l { 2 β 0 l β 1 l sin 2 θ | z | 2 l 1 cos θ τ = 1 , l β 0 l 2 sin 2 θ | z | 2 l + 1 cos θ τ = 2 ,
B τ l { 1 2 e i θ ( 2 l + 1 ) cos θ τ = 1 , 1 2 l e i θ ( 2 l + 3 ) + ( l + 1 ) e i θ ( 2 l 1 ) ( 2 l + 1 ) cos θ τ = 2 ,
C τ l { 2 α 0 l α 1 l | z | 2 l + 3 sin 2 θ cos θ τ = 1 , ( l + 1 ) α 0 l 2 | z | 2 l + 1 sin 2 θ cos θ τ = 2 ,
L ( F , λ ) = ( 1 λ ) I m { V F M a F d v } + λ I m { V F i M t F d v } λ I m { V F i M b F i d v } ,
δ L ( F , λ ) = I m { V δ F [ ( 1 λ ) ( M a M a ) F λ M t F i ] d v } ,
F = α 2 i ( I m { M a } ) 1 M t F i ,
I m { M a } = M a M a 2 i .
P a o p t = k 0 α 2 8 η 0 V F i M t ( I m { M a } ) 1 M t F i d v .
f ( M a ) = V F i M t ( I m { M a } ) 1 M t F i d v ,
δ f ( M a ) = V F i δ M a ( I m { M a } ) 1 ( M a M b ) F i d v V F i ( M a M b ) ( I m { M a } ) 1 I m { δ M a } ( I m { M a } ) 1 ( M a M b ) F i d v + V F i ( M a M b ) ( I m { M a } ) 1 δ M a F i d v ,
δ ( I m { M a } ) 1 = ( I m { M a } ) 1 I m { δ M a } ( I m { M a } ) 1 .
R e { δ M a ( I m { M a } ) 1 ( M a M b ) ( M a M b ) ( I m { M a } ) 1 δ M a 2 i ( I m { M a } ) 1 ( M a M b ) } = 0 ,
R e { [ ( M a M b ) ( I m { M a } ) 1 2 i I ] δ M a ( I m { M a } ) 1 ( M a M b ) } = 0 .
I m { M a } = M a M b 2 i ,
f ( M a ) | M a = M b = V F i ( M b M b ) ( I m { M b } ) 1 ( M b M b ) F i d v = 4 V F i I m { M b } F i d v .
{ E ( r ) = τ , m , l a τ m l v τ m l ( k r ) + f τ m l u τ m l ( k r ) , H ( r ) = 1 i η 0 η τ , m , l a τ m l v τ ¯ m l ( k r ) + f τ m l u τ ¯ m l ( k r ) ,

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