Using only classical electromagnetic energy conservation laws and causality, we show that the net average power absorbed by any mechanically isolated illuminated medium in steady state must be zero, but that for linear model media it is nonzero. This contradiction implies that all media must behave inelastically. We also show in general that the average power absorbed at an incident frequency, which is equal to the total taken from an incident wave minus that scattered elastically, is also equal to the average power scattered inelastically plus that carried off mechanically, if any. Finally, we infer that while the conventional linear theory cannot predict the spectral distribution of inelastic scattering, it may be applied as always to predict the propagation, absorption, and elastic scattering of weak illumination in passive media.

Topics
Energy conservation, Classical electromagnetism, Inelastic scattering, Elastic scattering
1.
J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed., Sec. 6.7, p. 259.
2.
See, e.g., J. R. Taylor, Scattering Theory: The Quantum Theory on Non-relativistic Collisions (Wiley, New York, 1972), Chaps. 16–22.
3.
Reference 1, Sec. 6.8, p. 263, and Sec. 16.8, p. 766, contains allusions to this, as does Sec. F.8 in E. J. Konopinski, Electromagnetic Fields and Relativistic Particles (McGraw–Hill, New York, 1981).
I could find no mention of any of the ideas in this paragraph in ten other well-known electromagnetic theory textbooks.
4.
See, e.g., Ref. 1, Eqs. (6.110)–(6.112).
5.
Here, stationary means that the properties of the medium are time independent. If this were not true, then P(r,ω) would depend on several E(r,ω′) even if the response were linear.
6.
Equation (5) is equivalent to the space and time integrals of Eq. (6.124) in Ref. 1.
7.
This follows from the definition of a passive dielectric, i.e., one in which a wave cannot be amplified as it travels.
8.
See, e.g., G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, San Diego, 1995), 4th ed., Sec. 7.3.
9.
G. H.
Goedecke
, “
On electromagnetic conservation laws
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10.
Reference 1, Sec. 10.11. We use “incident power” to designate the second integral in Eq. (6), which Jackson calls “total power taken from the incident wave.”
11.
An analogous result is well known in multichannel quantum mechanical scattering. See, e.g., Ref. 2, Sec. 19d, on the existence and use of an optical potential in quantum mechanics that “reduces the computation of elastic scattering to an equivalent one-channel problem.”
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© 2001 American Association of Physics Teachers.
2001
American Association of Physics Teachers
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