[1] The time domain version of the optical theorem is discussed. The theorem is derived from the optical theorem in the frequency domain by the Parseval's relation. It expresses the sum of the scattered and absorbed energies in terms of the scattered farfield in the forward direction. From the theorem, causality, and reciprocity, a number of results concerning the scattered and absorbed energies from a plane pulse that is scattered from bounded objects are obtained. Some of these results are verified by numerical calculations.

[2] Consider a plane wave pulse with a length of half a nanosecond that is scattered from the three perfectly conducting objects in Figure 1. It is a very hard numerical problem to obtain the scattered field with good accuracy since the geometry of the objects are complicated and since the multiple scattering effects cannot be neglected. Nevertheless, the scattered energy can be obtained within ten minutes on a 300 MHz PC or Mac, with a simple numerical program based on Mie scattering. The key to this numerical simplification is the optical theorem in the time domain. From that theorem one can prove that the scattered energy for the case in Figure 1 is three times the energy that is scattered from one perfectly conducting sphere of radius one meter.

[3] The optical theorem in the frequency domain is well studied [cf. Newton, 1976]. The theorem relates the total scattering cross-section to the scattering amplitude in the forward direction. Less attention has been paid to the time domain version of the optical theorem [cf. de Hoop, 1984; Karlsson, 2000]. It relates the sum of the scattered and absorbed energies to the farfield amplitude in the forward direction. The paper [Karlsson, 2000] presented results that can be extracted from the time domain optical theorem. In the current paper some of these results are reviewed and some new results concerning reciprocity and polarization are presented. Also, new concepts such as the domain of dependence of a scattering object, equivalent and independent scattering objects, and independent pulses are introduced.

2. Geometry and Incident Wave

[4] A plane wave impinges on a bounded region where one or more scattering objects are situated (cf. Figure 1). Outside the scattering region it is assumed to be vacuum.

[5] The incident plane wave propagates in the positive z-direction. The corresponding electric field is given by

where the vector E^{i}(z, t) lies in the xy-plane since the wave is transverse. It is assumed that the incident pulse has finite length, which implies that E_{0}(t) is zero outside some time interval t_{0} < t < t_{1}. The leading edge of the pulse then arrives at z = 0 when t = t_{0} and the trailing edge when t = t_{1}. It is possible to have t_{1} = ∞ if E_{0}(t) goes to zero when t → ∞.

3. Scattered Field

[6] The scattered field from an incident plane pulse can be expressed in terms of the induced charge and current densities in the scattering region by the Jefimenko's equation [cf. Jefimenko, 1989]. Thus

where ,

is the retarded time, and where the dot denotes time derivative. The far zone is defined as the region where the scattered field can be approximated by

where F is the farfield amplitude and r = |r|. The continuity equation implies that

where ∇′ = (d/dx′, d/dy′, d/dz′). From Jefimenko's equation, the continuity equation, and Gauss law it is seen that

[7] When the scattered farfield amplitude is integrated in time one obtains

where it has been assumed that there are no currents in the object before, and after a long time after, the pulse impinges. Thus the following general result holds for the farfield amplitude:

[8]The time mean value of the farfield amplitude is zero, in all spatial directions, regardless of the time-dependence of the incident plane wave.

[9] This result is utilized in the definition of independent incident pulses below.

4. Optical Theorem

[10] The optical theorem can be derived in the frequency domain [cf. Newton, 1976; Bohren and Huffman, 1983; Jackson, 1975], and in the time domain [cf. de Hoop, 1984; Karlsson, 2000]. It is also possible to use the Parseval's relation to transform the theorem from the time domain to the frequency domain, and vice versa. In this section the frequency domain theorem is presented and the time domain version is derived from it by the Parseval's relation.

4.1. Optical Theorem in the Frequency Domain

[11] Let and be the Fourier transforms of the incident field and the farfield, respectively. Thus

The sum of the scattered power, P_{s}, and the absorbed power, P_{a}, is given by

where s(ω) is the sum of the scattered and absorbed complex powers. The frequency domain optical theorem implies that

where k is the wave number, η_{0} is the wave impedance in vacuum, and where the asterisk denotes complex conjugate. Thus it is sufficient to know the farfield amplitude in the forward direction to obtain the sum of the scattered and absorbed powers.

4.2. Optical Theorem in the Time Domain

[12] In the time domain the scattered energy is given by

where dΩ = sinθ dθdϕ and where the surface integration is over the unit sphere. The absorbed energy is given by

where the surface S is a closed surface that circumscribes the scattering objects, is the outward directed unit normal vector to that surface, and E, H are the total electric and magnetic fields.

[13] Let f(t) and g(t) be two square integrable functions with Fourier transforms and . The Parseval's relation then reads

Since s(ω) is the spectrum of the sum of the scattered and the absorbed powers, it is seen that the sum of the scattered energy, W_{s}, and absorbed energy, W_{a}, is given by

The Parseval's relation is applicable and gives

This is the time domain version of the optical theorem. Many of the results presented in this paper are derived from the following observation:

[14]The sum of the scattered and absorbed energies is uniquely determined by the farfield amplitude in the forward direction during the time interval [t_{0}, t_{1}].

5. Implications

[15] There are a number of fundamental results that can be derived from the time domain version of the optical theorem. Some of these were presented by Karlsson [2000] and they are here complemented by new results.

5.1. Scattering From Several Objects

[16] Consider a case where an incident plane pulse of length t_{1} − t_{0} is scattered from a region with N objects. The objects are defined as energy independent for that incident pulse if

where W_{i} is the sum of the scattered and absorbed energies for object i with the other objects not present. This is the case if the multiply scattered waves are delayed at least a time t_{1} − t_{0} relative the wave front of the incident field. An example of scattering from independent objects is depicted in Figure 1. In that case the total scattered energy is the sum of the three energies for each of the three objects with the other two not present.

5.2. Domain of Dependence and Equivalent Objects

[17] The part of a scattering object that for a given pulse can contribute to the sum of the scattered and absorbed energies defines the domain of dependence for the energy for that pulse. For a nondispersive scattering object the domain is given by all points for which the shortest travel time for any ray that passes through that point is delayed less than the length of the incident pulse, i.e. less than t_{1} − t_{0}, compared to the wave front of the incident field. Two objects that have identical domain of dependence are equivalent in the sense that they have the same sum of the scattered and absorbed energies. As an example the three perfectly conducting objects in Figure 1 are equivalent for an incident pulse with pulse length .

[18]Figure 2 shows two objects that are equivalent for the incident pulse in Figure 4. The permittivity in the shaded regions is ε_{r} = 16, and everywhere else there is vacuum. The equivalence is checked numerically in Figure 3 where the integral

is given as function of t_{c} for each of the two objects when the incident pulse given in Figure 4 impinges. When t_{c} > t_{1} ≈ 8 ns the values of the integrals are seen to be constant. According to equation (15) the values of the integrals are then equal to the scattered energy. The farfield amplitude for the two objects is given in Figure 5. It is seen that the farfield amplitudes for the two objects deviate after the incident pulse has passed but the scattered energies are the same.

5.3. Independent Pulses

[19] Assume an incident field that consists of two pulses as

The field is scattered from one or several objects and gives rise to a farfield amplitude . The pulses are defined as energy independent if

Here W_{1} is the sum of the scattered and absorbed energies if only the first pulse is present and W_{2} is the sum of the scattered and absorbed energies if only the second pulse is present.

[20] If the first pulse, E_{1}(t − z/c_{0}), has support in the interval t_{0} < t < t_{1} and the second pulse, E_{2}(t − z/c_{0}), has support in the interval t_{2} < t < t_{3}, where t_{2} > t_{1}, then a sufficient condition for the pulses to be energy independent is that the farfield amplitude in the forward direction of the first pulse has died out before the second pulse arrives. In that case causality ensures that

where t_{p} < t_{2} is the time after which F_{1}(, t) is approximately zero. The last integral is zero due to equation (7) and thus

There are of course cases when two pulses are independent even if the farfield amplitude of one incident field overlaps with the other incident field. A simple example is two linearly polarized incident waves with polarization along and , respectively, that impinges on an object that is symmetric with respect to the xz- and yz-planes.

5.4. Scattering of a Discontinuous Pulse

[21] A case when the optical theorem is invaluable to the calculation of the scattered energy is when the incident pulse has a discontinuity at the trailing edge. Assume that one can calculate the scattered farfield F(, t) from a pulse

where E_{0}(t) is a continuous function with support for t_{0} < t < t_{1}. Then consider the discontinuous incident pulse E^{i}_{disc}(z, t) = E_{disc}(t − z/c_{0}), where

that gives rise to the farfield amplitude F_{disc}(, t). Causality ensures that for times t < t_{c} the farfield F(, t) is identical to the farfield F_{disc}(, t). Thus the sum of the scattered and absorbed energies for the discontinuous pulse E_{disc}^{i}(z, t) is obtained from the scattered farfield F(, t).

[22] The procedure to calculate the sum of the scattered and absorbed energies for the discontinuous incident wave E_{disc}^{i} is as follows. First calculate the scattered farfield amplitude F(, t) for the continuous incident wave E^{i}. Calculate the integral in equation (14) with F(, t) as the farfield amplitude and E_{disc}^{i} as incident field. The obtained value is the sum of the scattered and absorbed energies for the discontinuous incident wave. As an example, consider scattering from one of the objects in Figure 2. Let the incident field be discontinuous at t = t_{c} = 6 ns such that it is equal to the field in Figure 4 for t < 6 ns and zero for t ≥ 6 ns. The scattered energy for the discontinuous pulse is then given by Figure 3 as the value for t = t_{c} = 6 ns, i.e. approximately 9.5 · 10^{−12} Nm.

[24] In the time domain a medium is defined to be reciprocal at a point r in a region V if and only if

holds for all times t, for all electromagnetic fields {E^{a}, B^{a}} and {E^{b}, B^{b}}, and for every closed surface S_{r} ⊂ V around the point r. The medium in a volume V is reciprocal in V if and only if it is reciprocal at all points in V.

[25] Here H is the magnetic field and B is the magnetic flux density. The operator ⊗ is defined by

Let the fields {E^{b}, B^{b}} be the total fields from an incident plane wave pulse, traveling in the positive z-direction, e.g.,

and let {E^{b}, B^{b}} be the total fields when the incident plane wave pulse travels in the negative z-direction, i.e. when

If the scattering object is reciprocal it follows that

for all times. Here F^{a}(, t) and F^{b}(−, t) are the farfield amplitudes in the forward direction of the field E^{a} and the field E^{b}, respectively. The derivation of equation (28) is given in the Appendix A. The sum of the scattered and absorbed energies is given by equation (18). From equation (28) it is then seen that

where W^{a} and W^{b} are the sum of the scattered and absorbed energies for the incident pulses E^{ia} and E^{ib}, respectively. Thus the incident waves E^{ia}(z, t) and E^{ib}(z, t) give the same sum of the scattered and absorbed energies. This is illustrated in Figure 6.

5.6. Polarization

[26] Consider a linearly polarized plane wave that is scattered from an object that is bounded in space and is made of linear materials. If the incident wave is polarized in the direction ( cos ϕ + sin ϕ), i.e.

the sum of the scattered and absorbed energies is denoted W(ϕ). Notice that W(ϕ) = W(π + ϕ) since the object's material is linear. It is assumed that E_{0}(t) has support for t_{0} < t < t_{1}, where it is possible to have t_{1} = ∞. The optical theorem implies that if W(ϕ) is known for 0 ≤ ϕ ≤ π/2 then W(ϕ) is known for all ϕ. To see this, consider the incident wave with a polarization perpendicular to E_{1}^{i}, i.e.,

The sum of the scattered and absorbed energies for this wave is W(ϕ + π/2). Then introduce a scattering matrix that relates the farfield amplitude to the incident plane wave. In the case of an incident wave propagating in the z-direction the farfield amplitude in the forward direction, F(, t) = F_{x}(, t) + F_{y}(, t), is given by

where E^{i}(z, t) = E_{x}^{i}(z, t) + E_{y}^{i}(z, t) is the incident field. The asterisk * denotes convolution in time. The scattering matrix S(t) is the impulse response and is independent of the incident field. In the case of the incident wave in equation (30) it is seen that the sum of the scattered and absorbed energies is given by

Thus

which is independent of the angle ϕ. It follows that if W(ϕ) is known for 0 ≤ ϕ ≤ π/2 it is known for all ϕ.

6. Conclusions

[27] A number of time domain results concerning the scattered and absorbed energies for an incident plane pulse can be derived from the time domain optical theorem. In that sense it contains more physics than its frequency domain counterpart. In the paper the concepts of independent scattering objects and equivalent objects are introduced, and it is shown that these concepts are valuable to the calculation of scattered energies. Thus, for a group of independent scattering objects one may ignore the multiple scattering in the calculation of the scattered energy. Furthermore, the scattered energy from a complicated scattering object can be obtained from the scattered farfield amplitude from a simpler equivalent object. In a current project it is examined if the optical theorem in combination with the finite difference time domain method can be used to calculate the scattered energy in an efficient way.

Appendix A:: A Reciprocity Theorem

[28] Consider a scattering object made of a reciprocal material. In that case the equation (24) holds for all times t, for all electromagnetic fields {E^{a}, B^{a}} and {E^{b}, B^{b}}, and for every closed surface S_{r} ⊂ V around the point r.

[29] Let the two fields E^{a}(r, t), H^{a}(r, t) and E^{b}(r, t), H^{b}(r, t) be the total electric and magnetic fields from the two incident fields

Here E_{0}(t) is zero except for a finite time interval that for simplicity is chosen to be 0 < t < t_{1} and the origin is located inside the scattering object. The scattered electric and magnetic fields are in the far zone given by

Since the scattered field has finite energy the farfield amplitudes have to go to zero when t − r/c_{0} gets large. Thus F^{a}(, t) and F^{b}(, t) are zero for t < 0, due to causality, and negligible for t > T, for some T. Consider the surface S to be a sphere, denoted S_{R}, with radius R and with center at the origin. The radius R is large enough so that the surface of the sphere is in the far zone. The identity (24) implies

where

It is first proven that I_{1} = I_{3} = 0. Do the substitution θ′ = π − θ and ϕ′ = 2π − ϕ in the surface integral of I_{1}. After some manipulations it follows that

and hence I_{1} = 0. To see that I_{3} = 0 one observes that on the surface S_{R}

where ⊙ denotes the convolution of a scalar product (cf. equation (25)). It follows that I_{3} = 0. The remaining integral I_{2} is given by

where

Since F(, t) is approximately zero for all times except 0 < t < T, the integrals reduce to

Now E_{0}(t) is zero except when 0 < t < t_{1}. Consider the time interval 0 < t < T and choose the radius of the sphere large enough to satisfy R ≫ c_{0}T. In that case the integrand in K_{1}(, z, t) is nonzero only for and −R < z < −R + c_{0}T and the integrand in K_{2}(, z, t) is nonzero only for and R − c_{0}T < z < R. After a substitution θ′ = π − θ in the part of I_{2}(t) that contains K_{1}(, z, t) it follows that for sufficiently small T/R the integral I_{2}(t) is given by

Since I_{1}(t) = I_{3}(t) = 0 it follows that I_{2}(t) = 0 for all times, and in particular for 0 < t < T. From equation (44) it follows that this can only be fulfilled if

for all times t.

[30] In addition to the result in equation (45) there is a reciprocity result also for the case when the incident fields are given by

In that case it is straightforward to see that I_{1} = I_{3} = 0 and that

Since I_{2}(t) = 0 for all times it follows that for all times