Christophe Caloz - Electromagnetic Metasurfaces

Using the identity reduces these equations to

( 2.42a)

( 2.42b)

Subtracting 2.42afrom 2.42byields

(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)

where represents a time-odd external bias. Under phase conjugation (or time reversal), these constitutive relations become [28]

( 2.47a)

(2.47b)

Substituting ( 2.46) and ( 2.47) into ( 2.45) transforms the integrand of ( 2.45) into

( 2.48)

Expanding the bracketed terms in ( 2.48), using the identity , where the superscript denotes the transpose operation, and grouping the identical parameter terms, we obtain

( 2.49)

The nullity of the volume integral of ( 2.49) ( Eq. (2.45)) for any field implies then the general 6bianisotropic Lorentz reciprocity conditions

(2.50a)

(2.50b)

In the absence of external bias, i.e. , these conditions reduce to

( 2.51)

which may be expressed in terms of susceptibilities as

(2.52)

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