Electromagnetic Metasurfaces. Christophe Caloz

(2.40b)

Subtracting 2.39a pre-multiplied by from 2.40b pre-multiplied by , and subtracting 2.40a pre-multiplied by from 2.39b pre-multiplied by yields

      (2.41a)

      (2.41b)

      Using the identity reduces these equations to

(2.42a)

(2.42b)

Subtracting 2.42a from 2.42b yields

      (2.43)

      We may now integrate this equation over the volume and apply the Gauss theorem, which gives

(2.44)

The surface integral on the left-hand side of (2.44) and the second volume integral on its right-hand side both vanish when integrated over an unbounded medium [28, 81]. This results into

(2.45)

      which represents a fundamental relation for reciprocity. We next include the bianisotropic material parameters in this relation. For this purpose, we use the bianisotropic constitutive relations (2.4), which may be explicitly rewritten as

(2.46a)

      (2.46b)

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