The forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability dyadics. It is useful for deriving bounds on the interaction between scatterers and electromagnetic fields, antenna bandwidth and directivity and energy transmission through sub-wavelength apertures. The sum rule is valid for linearly polarized plane waves impinging on linear, passive and time translational invariant scattering objects in free space. Here, a time-domain approach is used to clarify the derivation and the used assumptions. The time-domain forward scattered field defines an impulse response. Energy conservation shows that this impulse response is the kernel of a passive convolution operator, which implies that the Fourier transform of the impulse response is a Herglotz function. The forward scattering sum rule is finally constructed from integral identities for Herglotz functions.
The forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability of the scatterer defined by its electric and magnetic polarizability dyadics. This sum rule was introduced for spheroidal dielectrics in Purcell (1969) and generalized to arbitrary heterogeneous objects in Sohl et al. (2007a). Related dispersion relations are considered by several authors, see e.g. Newton (2002), Nussenzveig (1972) and Panofsky & Phillips (1962). In addition to the insight the identity provides in scattering theory, it offers physical bounds on the interaction between arbitrary obstacles and electromagnetic fields (Sohl et al. 2007a). For example, it is useful in the analysis of metamaterials, where it shows that an increased interaction is traded against bandwidth (Sohl et al. 2007b, 2008). It has also been used to demonstrate how size and shape affect performance in antenna theory (Gustafsson et al. 2007, 2009; Sohl & Gustafsson 2008; Derneryd et al. 2009) and for extra-ordinary transmission of energy through sub-wavelength apertures (Gustafsson 2009).
Here, a time-domain derivation of the forward scattering sum rule is presented. The derivation is based on transient scattering of an incident plane wave in the form of a (Dirac) delta pulse. The scattered field of an ordinary incident pulse with finite power density is retrieved from convolution between the impulse response and the pulse shape. Moreover, the corresponding time-harmonic forward scattered field is obtained from a multiplication with the transfer function, i.e. the Fourier transform of the impulse response. These properties are related to the impulse response and transfer function that are commonly used in linear system theory (Zemanian 1996). The sum rule relates the extincted (i.e. the scattered and absorbed) power of an object integrated over all wavelengths with the polarizability of the object times the incident power flux. The corresponding physical bounds show that the bandwidth times the extinction is bounded by the polarizability. Moreover, the polarizability dyadic of an object is bounded by the high-contrast polarizability dyadic that only depends on the geometry of the object (Sohl et al. 2007a). This demonstrates that large extinction cross sections of sub-wavelength objects are only possible over limited frequency intervals.
The time-domain analysis shows that, under the assumption of passivity, the impulse response is the kernel of a passive convolution operator (Zemanian 1996). The corresponding transfer function is then a positive real function (Guillemin 1949; Zemanian 1996) or a Herglotz (Nussenzveig 1972) or Nevanlinna or Pick (Donoghue 1969) function. This choice depends on the considered right or upper complex half plane that is often induced by the time conventions ejωt and e−iωt, respectively, where i2=−1 and j=−i. Here, Herglotz functions are considered (Nussenzveig 1972). They are defined as holomorphic mappings from the upper complex half plane into itself. Integral identities for Herglotz functions are used to derive the forward scattering sum rule (Fano 1950; Gustafsson et al. 2007; Sohl et al. 2007a; Bernland et al. 2010). It is straightforward to transform the derivation to positive real functions (Zemanian 1996) using the time convention ejωt.
This paper is organized as follows. In §2, the forward scattering impulse response is introduced and the extincted energy of an incident time-domain plane wave is derived. The corresponding results in the frequency domain are analysed in §3. Sum rules, related physical bounds and numerical examples are given in §4. Section 5 contains the conclusions.
2. Time domain forward scattering
Introduce a coordinate system such that the finite scattering object is confined to the region z1≤z≤0 leaving z≥0 as free space, see figure 1a. Consider a transient incident plane wave in the form of a (Dirac) delta distribution, δ, propagating in the z-direction, i.e. the incident electric and magnetic field intensities are 2.1 respectively, where , c0 denotes the speed of light in free space, and η0 is the intrinsic impedance of free space. The spatial and temporal support (the support is the region where the field is non-zero) of the incident field is given by the plane defined by t=z/c0, see figure 1b. The (total) field is equal to the incident field until the pulse reaches the object, i.e. for t<z1/c0. The support of the field is more complex for larger times owing to the interaction between the incident field and the object. In general, it requires a solution of the Maxwell equations with an accurate model of the constitutive relations to determine the field and its support, see figures 2a and 3. However, for the purpose of deriving the forward scattering sum rule, it is sufficient to determine a region that contains the support of the field, see figure 1b. This region is solely based on the fundamental property that the wavefront velocity is bounded by the speed of light in free space and that the incident wave does not reach the object until the time t=z1/c0.
Before specific material models of the scattering object and the Maxwell equations are considered, it is important to review the underlying physics that the constitutive relations should fulfil. A fundamental property is that the propagation of an electromagnetic wave is limited by the speed of light in free space. Note that this is a bound on the wavefront velocity and not the phase velocity or group velocity that are commonly used for time harmonic fields. The wavefront velocity quantifies the speed that the wavefront propagates with in the direction orthogonal to itself. This is related to the Huygens principle, which states that each point of an advancing wavefront is the centre of a new disturbance (Evans 1998).
Mathematically, it is necessary to consider a set of constitutive relations that guarantee these properties when used together with the Maxwell equations. Here, the analysis is restricted to linear, time translational invariant, continuous, passive, isotropic and non-magnetic material models for simplicity. This gives the temporally dispersive constitutive relations of the form 2.2 and B=μ0H, where ϵ0 and μ0 denote the intrinsic permittivity and permeability of free space, respectively, is the instantaneous response, and χt the electric susceptibility kernel. To guarantee a well-posed solution of the Maxwell equations, it is also assumed that |∂χt/∂t| is integrable in t. The instantaneous response models the direct response of the medium whereas the susceptibility kernel models the temporal dispersion. One can argue that for materials as the medium cannot react instantaneously to the electromagnetic field (Jackson 1975). However, here the general case with an arbitrary is considered. This is motivated from a modelling point of view where is the permittivity for sufficiently high frequencies (or equivalently fast temporal responses) compared with the frequency range of interest (Gustafsson 2003).
The wave-front velocity in the medium is now given by . This is easily understood for the plane-wave solution in a homogeneous medium. General inhomogeneous objects require further analysis, where the general properties follow from the analysis of symmetric hyperbolic systems (Kreiss & Lorenz 1989; Evans 1998) together with the observation that the principal part of the Maxwell equations determine the propagation of the wavefront. This reduces the analysis to the case of non-dispersive material models, i.e. χt=0 in equation (2.2).
Decompose the (total) field into incident and scattered fields as 2.3 see also the numerical example in figure 2. The surface integral representation (A6) of the scattered field is used to determine the co-polarized scattered far field in the forward direction as , where 2.4 It is realized that the field of an incident δ-pulse does not decay according to the inverse square law. However, the far field of an incident square integrable pulse is determined by convolution of the pulse shape with the impulse response as outlined in appendix A.
The surface representation (2.4) is used to show that ht is causal, i.e. ht(t)=0 for t<0. Here, it is convenient to consider the scattered field on the surface defined by z=0 with , which simplifies the surface integral representation (2.4) into 2.5 where . It is seen that ht is the component of the scattered field that is co-polarized with the incident field at z=0. The properties of ht(t) can hence be obtained from the corresponding properties of the scattered field. It is obvious that ht(t) is quiescent for t<0 as the same holds for the scattered field as well as for the total field.
The impulse response in the forward direction ht(t) is now used to express the extincted energy, cf. the time-domain optical theorem (de Hoop 1984; Karlsson 2000). A compactly supported smooth pulse shape, E(t), is considered to ensure that the extincted energy is well defined. The total and scattered fields induced by the incident field, 2.6 is represented as a convolution between the impulse response, E(δ)s, and the incident pulse shape, i.e. 2.7 and similarly for Hs. The extincted energy, Wext, is the sum of the absorbed and scattered energies. The scattered energy at time T with respect to the region V is defined as the energy of the scattered field outside V , i.e. 2.8 where the equality follows from conservation of energy in the form of the Poynting’s theorem (Jackson 1975; van Bladel 2007). Similarly, the absorbed energy is defined as the difference between the total energy and the incident energy in V 2.9 Note that the absorbed and scattered energies at a time T depend on the considered region V . Now the extincted energy can be written as an integral of the energy flux through the surface ∂V , i.e. 2.10
This form is used in the time-domain version of the optical theorem (Karlsson 2000). It is observed that Wext(T) is independent of T for large times as Ei and Hi in equation (2.10) are quiescent after the incident pulse has passed the region V . In particular, this implies that is independent of the considered region V . The sign follows directly from the non-negative signs of equations (2.8) and (2.9), where the passivity of the constitutive relations (2.2) and the finite support of E(t) are used.
Use equation (2.6) and the relation (2.7) of the scattered field to rewrite the extincted energy as 2.11 where the surface is chosen as the surface z=0 and ht is identified from the surface integral representation of the scattered field in the forward direction (2.5). This relation is valid for all compactly supported smooth functions E(t), and classifies the impulse response, ht(t), as a (tempered) distribution of positive type (Reed & Simon 1975). Passivity and the finite speed of propagation is now used to show that Wext(T)≥0 for all E(t). The extincted energy (2.10) is first rewritten as a volume integral over the exterior region, i.e. 2.12 Using equation (2.12) with as the region z≥0, it follows that Wext(T)≥0. This characterizes ht(t) as the kernel of a passive convolution operator (Zemanian 1996) that has the general representation (Zemanian 1996) 2.13 where B≥0, θ(t)=0 for t<0, θ(t)=1 for t>0, and dβ(ξ)/(1+ξ2) is a finite measure, i.e. .
The constant B in equation (2.13) is further restricted. The finite speed of propagation restricts the surface integral (2.5) to the projection of the geometrical cross section on the surface z=0 for short times, i.e. as t→0. It also creates a shadow zone, denoted by in figure 3b, where E(δ)(t,r)=η0H(δ)(t,r)=0, behind impenetrable objects or objects with , causing and as t→0 in this region. This gives a short-time behaviour of the form ht(t)≈2Aδ(t) as t→0 for these types of objects, where denotes the area of , cf. the extinction paradox. This shows that B=0 in equation (2.13).
3. Fourier domain forward scattering
The derivation of the forward scattering sum rule is based on integral identities in the Fourier domain Sohl et al. (2007a). A Fourier transform of the impulse response, ht, defines h(k) as 3.1 where k=ω/c0 with Imk>0 denotes the wavenumber and the multiplication with the imaginary unit, i, is used to rotate the range to the upper complex half plane. The Fourier (Laplace) transform of ht(t) defined by equation (2.13) classifies h(k) as a (symmetric) Herglotz function (Donoghue 1969; Nussenzveig 1972; Bernland et al. 2010), which can be represented by the integral 3.2 where Imk>0 and (Nussenzveig 1972; Zemanian 1996). The cross symmetry h(k)=−h*(−k*) used in equation (3.2), where a star denotes the complex conjugate, follows from the real valued ht. It is observed that dβ(ξ)=Imh(ξ)dξ/π if Imh(ξ) is sufficiently regular (Donoghue 1969) offering a simple relation with the Hilbert transform and the Kramers–Kronig relations (Jackson 1975; King 2009). A Herglotz function is holomorphic in Imk>0 and its imaginary part is non-negative, Imh(k)≥0, i.e. h(k) maps the upper half plane into itself. The high-frequency limit is h(k)/k→B≥0 as , where is a shorthand notation for limits such that for some α>0.
The Plancherel formula expresses the extincted energy (2.11) as 3.3 where σext(k)=Imh(k) is the extinction cross section. The extinction cross section of an object is also related to the forward scattering of the object via the optical theorem (Nussenzveig 1972; Newton 2002), i.e. σext(k)=Imh(k), where and denotes the far-field amplitude. The function h(k) can hence be considered as a holomorphic continuation of σext(k) into Imk>0.
The surface integral representation of the electromagnetic field expresses the far field in the scattered field on the surface z=0, cf. equation (2.5). Written in the function h(k) it reads 3.4 where the incident wave is the unit-amplitude time-harmonic plane wave and and denote the Fourier transforms of E(δ)s and H(δ)s, respectively.
The low-frequency asymptotic expansion of the far field is well known and e.g. analysed in Kleinman & Senior (1986). Here, a simplified derivation based on the volume integral representation (A2) is presented, see Kleinman & Senior (1986) for a more general derivation. It is valid for the constitutive relations (2.2) without a static conductivity. The low-frequency asymptotic expansions and as k→0 together with the induced current inserted into the volume integral representation (A2) are used to get 3.5 where γe denotes the electric polarizability dyadic (Kleinman & Senior 1986; van Bladel 2007).
The high-frequency asymptotic is more involved. Obviously, the high-frequency response depends on the corresponding high-frequency limit of the constitutive relations. As discussed above, it is sometimes argued that the constitutive relations reduce to their free space values in this limit. This would simplify the analysis; however, here the general constitutive relations (2.2) are considered. The representation (3.2) shows that it is sufficient to consider the high-frequency asymptote on the imaginary axis k′=0 as , where k=k′+ik′′. The high-frequency asymptotic is hence related to the short-time behaviour of ht(t) since 3.6 for any τ1>0. The second integral decays exponentially with k′′, so the high-frequency asymptotic is given by the first integral with an arbitrary small τ1. The short-time response of equation (2.5) shows that ht(t)≈2Aδ(t) as t→0 for impenetrable objects and hence h(k)→2Ai as , where A denotes the cross-section area of the shadow zone , see figure 3.
4. Sum rules and physical bounds
Integrate h(k) over a line in the upper complex half plane and use the low-frequency asymptote (3.5) to get the sum rule (Sohl et al. 2007a; Bernland et al. 2010) 4.1 where, if necessary, the integral in the centre is a generalized integral defined as the limit of the integral to the left. It is often convenient to use the wavelength λ=2π/k to express the identity as the integrated extinction (Sohl et al. 2007a) 4.2 where the symbol σext is reused to denote the extinction cross section as a function of the wavelength. This shows that the extinction cross section integrated over all wavelengths is proportional to the polarizability .
It is known that γe is monotone in the material parameters (Jones 1985; Sjöberg 2009). That is γe1≤γe2 if ϵ1(r)≤ϵ2(r) for all points r in the object, where the inequality means γe1≤γe2 if for all . It is hence convenient to introduce the high-contrast polarizability dyadic such that for all objects confined to the same volume (Sohl et al. 2007a). The high-contrast polarizability dyadics are illustrated for a square patch and a square split ring resonator in figure 4a. Note that is larger for the patch than for the split ring as the patch can be constructed from the split ring by adding material. The variational principle together with the sum rule (4.2) show that the integrated extinction is monotone in the material parameters.
Bound the integral (4.2) as 4.3 where Λ=[λ1,λ2] and |Λ|=λ2−λ1 to get a simple bound on the bandwidth times the extinction cross section (Sohl et al. 2007a). The bounds are illustrated by the shaded boxes in figure 4 that are constructed to have similar total areas as the areas under the corresponding curves. An interpretation of the bound is that it is not possible to design a scatterer with σext(λ) that does not intersect the box.
A second example is provided by spherical scatterers composed of aluminium (Al), silver (Ag), gold (Au) and copper (Cu), using the permittivity models in Ung & Sheng (2007). The extinction cross sections for spheres with radius a=50 nm are depicted in figure 5 as functions of the wavelength λ. It is observed that σext is large compared with the cross-section area πa2 at some resonance wavelengths and that σext is small for λ>1 μm and σext(λ)→0 as λ→0. The polarizabilities of the spheres are as they have a static conductivity. This is also confirmed by numerical integration of the sum rule (4.2). The areas under the curves defined by σext(λ) are hence given by the radii of the spheres. The dispersion characteristics of the materials distribute the area to different wavelengths. The large values of σext(λ) for silver around λ≈0.4 μm must hence be compensated by reduced values at other wavelengths as seen in the figure. A comparison with the bound (4.3) shows that 4.4 if λ1=0 is used. This gives a rough estimate of the average distribution of σext, e.g. σext/πa2=3 gives λ2=2/3 μm.
The polarizabilities of hollow and solid metallic spheres are identical owing to the static conductivity of metals (Kleinman & Senior 1986). This implies that the integrated extinctions (4.2) are identical for hollow and solid spheres composed of metallic materials. The extinction cross sections of hollow silver spheres with outer radius a=50 nm and inner radii 0.5a and 0.9a are depicted in figure 6. Note that the increased values of σext(λ) around 0.7 μm are compensated by low values at the shorter wavelengths.
A time-domain approach is used to derive the forward scattering sum rule. The time-domain analysis highlights the used assumptions such as causality and passivity. The use of causality is clearly observed in the definition of the forward scattering impulse response. It follows from the fact that the wave-front speed cannot exceed the speed of light in free space. Passivity enters through conservation of energy, which states that the extincted energy is non-negative. This also shows that the impulse response is the kernel of a passive convolution operator, and, hence, constructs a Herglotz function in the Fourier domain (Zemanian 1996).
The forward scattering sum rule has previously been used to derive bounds on scattering and absorption of metamaterials (Sohl et al. 2007b, 2008), antenna bandwidth and directivity (Gustafsson et al. 2007, 2009), and extraordinary transmission of power through sub-wavelength apertures (Gustafsson 2009). Here, the sum rule and bounds are exemplified for resonant perfectly conducting structures and spheres composed of various metals. It is observed that the sum rule offers simple estimates of the overall behaviour of the extinction cross section, σext(λ), in a way that the dispersion characteristics of the metals determine the resonances but the (long-wavelength) polarizability dyadic determines the total area under the curve σext(λ). This is particularly important in e.g. cloaking of objects in free space (Alù & Engheta 2008), where it is noted that the cloaking material increases the polarizability as it adds material (Jones 1985; Sjöberg 2009) and hence the area under σext(λ). Reduction of σext(λ) at some desired wavelengths must then be compensated by increased values of σext(λ) at other wavelengths.
Appendix A. Integral representation of the scattered field
Here, some classical integral representations are summarized, see e.g. van Bladel (2007). The electric field is A1 in the far-field region. The volume integral representation of the far field from a current distribution is A2 which is used in the derivation of the low-frequency asymptotic expansions (3.5).
The surface integral representation is used in the definition of the impulse response, ht in equation (2.4) and the transfer function , in equation (3.4). It is given by A3 which gives the function () A4 in the Fourier domain. The corresponding time-domain case is obtained by an inverse Fourier transform, i.e. A5 that gives A6 The time-domain co-polarized case in the forward direction is A7 where A8
- Received December 31, 2009.
- Accepted April 26, 2010.
- © 2010 The Royal Society