Electromagnetic Metasurfaces. Christophe Caloz

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2.5 Poynting Theorem

      This section derives the time-average bianisotropic Poynting theorem [16, 81, 140], which provides the general conditions for gain and loss in terms of susceptibility tensors.

      The time-domain Maxwell–Faraday and Maxwell–Ampère equations, assuming the presence of electric current sources, , and magnetic current sources, , are, respectively, given as

(2.53a)

(2.53b)

      with the bianisotropic constitutive relations (2.4) defined by

(2.54a)

      (2.54b)

Pre-multiplying (2.53a) by and (2.53b) by and subtracting the two resulting equations yields

(2.55)

The left-hand side of (2.55) may be simplified using the vectorial identity

      (2.56)

      where the cross product corresponds to the Poynting vector . This transforms (2.55) into

(2.57)

      We shall now simplify the last two terms of this relation to provide the final form of the bianisotropic Poynting theorem. We show the derivations only for , but similar developments can be made for . From , we have that

      (2.58)

Splitting the two terms of the right-hand side into two equal parts transforms this relation into

      (2.59)

      Manipulating the terms in the right-hand side of this new relation, adding the extra null term , and using the chain rule leads to

      (2.60) Скачать книгу