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Keywords:

  • optical;
  • theorem

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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.

image

Figure 1. Scattering of a plane incident pulse with length equation image from three perfectly conducting objects. The two lower objects have cavities shaped like circular cones. The optical theorem implies that for the given incident pulse the three objects are equivalent and independent wrt energy. The scattered energy is then equal to 3Wsphere, where Wsphere is the scattered energy for the uppermost sphere with the other two objects not present. The reason for this is that in the forward direction the multiple scattered waves and the waves scattered from the cavities are delayed more than the length of the pulse t1t0 relative the wave front.

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[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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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

  • equation image

where the vector Ei(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 E0(t) is zero outside some time interval t0 < t < t1. The leading edge of the pulse then arrives at z = 0 when t = t0 and the trailing edge when t = t1. It is possible to have t1 = ∞ if E0(t) goes to zero when t [RIGHTWARDS ARROW] ∞.

3. Scattered Field

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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

  • equation image

where equation image,

  • equation image

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

  • equation image

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

  • equation image

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

  • equation image

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

  • equation image

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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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 equation image and equation image be the Fourier transforms of the incident field and the farfield, respectively. Thus

  • equation image

The sum of the scattered power, Ps, and the absorbed power, Pa, is given by

  • equation image

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

  • equation image

where k is the wave number, η0 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

  • equation image

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

  • equation image

where the surface S is a closed surface that circumscribes the scattering objects, equation image 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 equation image and equation image. The Parseval's relation then reads

  • equation image

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, Ws, and absorbed energy, Wa, is given by

  • equation image

The Parseval's relation is applicable and gives

  • equation image

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 [t0, t1].

5. Implications

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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 t1t0 is scattered from a region with N objects. The objects are defined as energy independent for that incident pulse if

  • equation image

where Wi 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 t1t0 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 t1t0, 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 equation image.

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

  • equation image

is given as function of tc for each of the two objects when the incident pulse given in Figure 4 impinges. When tc > t1 ≈ 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.

image

Figure 2. Two spheres that are equivalent for an incident pulse of length less than 6 ns. The permittivity in the shaded regions is εr = 16 and everywhere else there is vacuum. Any ray that passes through the vacuum region of the left sphere is delayed at least 6 ns and cannot contribute to the scattered energy.

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image

Figure 3. The integral in equation (17) as a function of the time tc for the spheres in Figure 2. The incident wave is given in Figure 4. The curve for the solid sphere (solid line) and for the hollow sphere (dotted line) are indistinguishable from each other. The curves have been obtained by simulation using Mie scattering and Fourier transformation. The final value of Ws is in perfect agreement with the value obtained by numerical integration of the integral in equation (11).

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image

Figure 4. The function E0(t) = (Tt)/t0 exp(−(tT)2/t02), where T = 5 ns and t0 = 1 ns, as a function of t. The corresponding incident wave Ei(z, t) = equation imageE0(tz/c0) is used for the Figures 3 and 5.

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image

Figure 5. The farfield amplitudes in the forward direction, F(equation image, t) = equation image · F(equation image, t), for the two spheres given in Figure 2 and with the incident pulse in Figure 4. The solid line is for the solid sphere and the dashed for the hollow sphere. The curves were obtained by simulation using Mie scattering and Fourier transformation.

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5.3. Independent Pulses

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

  • equation image

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

  • equation image

Here W1 is the sum of the scattered and absorbed energies if only the first pulse is present and W2 is the sum of the scattered and absorbed energies if only the second pulse is present.

[20] If the first pulse, E1(tz/c0), has support in the interval t0 < t < t1 and the second pulse, E2(tz/c0), has support in the interval t2 < t < t3, where t2 > t1, 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

  • equation image

where tp < t2 is the time after which F1(equation image, t) is approximately zero. The last integral is zero due to equation (7) and thus

  • equation image

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 equation image and equation image, 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(equation image, t) from a pulse

  • equation image

where E0(t) is a continuous function with support for t0 < t < t1. Then consider the discontinuous incident pulse Eidisc(z, t) = equation imageEdisc(tz/c0), where

  • equation image

that gives rise to the farfield amplitude Fdisc(equation image, t). Causality ensures that for times t < tc the farfield F(equation image, t) is identical to the farfield Fdisc(equation image, t). Thus the sum of the scattered and absorbed energies for the discontinuous pulse Edisci(z, t) is obtained from the scattered farfield F(equation image, t).

[22] The procedure to calculate the sum of the scattered and absorbed energies for the discontinuous incident wave Edisci is as follows. First calculate the scattered farfield amplitude F(equation image, t) for the continuous incident wave Ei. Calculate the integral in equation (14) with F(equation image, t) as the farfield amplitude and Edisci 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 = tc = 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 = tc = 6 ns, i.e. approximately 9.5 · 10−12 Nm.

5.5. Reciprocity

[23] The following definition of a reciprocal medium is provided by Karlsson and Kristensson [1992]:

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

  • equation image

holds for all times t, for all electromagnetic fields {Ea, Ba} and {Eb, Bb}, and for every closed surface SrV 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

  • equation image

Let the fields {Eb, Bb} be the total fields from an incident plane wave pulse, traveling in the positive z-direction, e.g.,

  • equation image

and let {Eb, Bb} be the total fields when the incident plane wave pulse travels in the negative z-direction, i.e. when

  • equation image

If the scattering object is reciprocal it follows that

  • equation image

for all times. Here Fa(equation image, t) and Fb(−equation image, t) are the farfield amplitudes in the forward direction of the field Ea and the field Eb, 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

  • equation image

where Wa and Wb are the sum of the scattered and absorbed energies for the incident pulses Eia and Eib, respectively. Thus the incident waves Eia(z, t) and Eib(z, t) give the same sum of the scattered and absorbed energies. This is illustrated in Figure 6.

image

Figure 6. Due to reciprocity the sum of the absorbed and scattered energies are the same in a) and b). The scattering object is made of a reciprocal material but is otherwise arbitrary.

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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 (equation image cos ϕ + equation image sin ϕ), i.e.

  • equation image

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 E0(t) has support for t0 < t < t1, where it is possible to have t1 = ∞. 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 E1i, i.e.,

  • equation image

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(equation image, t) = equation imageFx(equation image, t) + equation imageFy(equation image, t), is given by

  • equation image

where Ei(z, t) = equation imageExi(z, t) + equation imageEyi(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

  • equation image

Thus

  • equation image

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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References

[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 {Ea, Ba} and {Eb, Bb}, and for every closed surface SrV around the point r.

[29] Let the two fields Ea(r, t), Ha(r, t) and Eb(r, t), Hb(r, t) be the total electric and magnetic fields from the two incident fields

  • equation image

Here E0(t) is zero except for a finite time interval that for simplicity is chosen to be 0 < t < t1 and the origin is located inside the scattering object. The scattered electric and magnetic fields are in the far zone given by

  • equation image

Since the scattered field has finite energy the farfield amplitudes have to go to zero when tr/c0 gets large. Thus Fa(equation image, t) and Fb(equation image, 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 SR, 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

  • equation image

where

  • equation image

It is first proven that I1 = I3 = 0. Do the substitution θ′ = π − θ and ϕ′ = 2π − ϕ in the surface integral of I1. After some manipulations it follows that

  • equation image

and hence I1 = 0. To see that I3 = 0 one observes that on the surface SR

  • equation image

where ⊙ denotes the convolution of a scalar product (cf. equation (25)). It follows that I3 = 0. The remaining integral I2 is given by

  • equation image

where

  • equation image

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

  • equation image

Now E0(t) is zero except when 0 < t < t1. Consider the time interval 0 < t < T and choose the radius of the sphere large enough to satisfy Rc0T. In that case the integrand in K1(equation image, z, t) is nonzero only for equation image and −R < z < −R + c0T and the integrand in K2(equation image, z, t) is nonzero only for equation image and Rc0T < z < R. After a substitution θ′ = π − θ in the part of I2(t) that contains K1(equation image, z, t) it follows that for sufficiently small T/R the integral I2(t) is given by

  • equation image

Since I1(t) = I3(t) = 0 it follows that I2(t) = 0 for all times, and in particular for 0 < t < T. From equation (44) it follows that this can only be fulfilled if

  • equation image

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

  • equation image

In that case it is straightforward to see that I1 = I3 = 0 and that

  • equation image

Since I2(t) = 0 for all times it follows that for all times

  • equation image

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry and Incident Wave
  5. 3. Scattered Field
  6. 4. Optical Theorem
  7. 5. Implications
  8. 6. Conclusions
  9. Appendix A:: A Reciprocity Theorem
  10. References
  • Bohren, C. F., and D. R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley, New York, 1983.
  • de Hoop, A. T., A time domain energy theorem for scattering of plane electromagnetic waves, Radio Sci., 19, 11791184, 1984.
  • Jackson, J. D., Classical Electrodynamics, 2nd ed., John Wiley, New York, 1975.
  • Jefimenko, O. D., Electricity and Magnetism, Electret Sci., Star City, 1989.
  • Karlsson, A., On the time domain version of the optical theorem, Am. J. Phys., 68(4), 344349, 2000.
  • Karlsson, A., and G. Kristensson, Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain, J. Electromagn. Waves Appl., 6(5/6), 537551, 1992.
  • Newton, R., Optical theorem and beyond, Am. J. Phys., 44, 639642, 1976.