Electromagnetic Metasurfaces. Christophe Caloz

Electromagnetic Metasurfaces

1 EndFraction plus StartFraction StartAbsoluteValue upper U 2 Superscript left-parenthesis minus right-parenthesis Baseline EndAbsoluteValue squared Over eta 2 EndFraction equals StartFraction StartAbsoluteValue upper U 1 Superscript left-parenthesis minus right-parenthesis Baseline EndAbsoluteValue squared Over eta 1 EndFraction plus StartFraction StartAbsoluteValue upper U 2 Superscript left-parenthesis plus right-parenthesis Baseline EndAbsoluteValue squared Over eta 2 EndFraction comma"/>

where eta 1 and eta 2 are the impedances of medium 1 and 2, respectively, and are assumed to be real quantities. Substituting (2.78) into (2.79), and considering that StartAbsoluteValue upper U EndAbsoluteValue squared equals upper U upper U Superscript asterisk , yields

(2.80)

Since this equality holds for any field value, all the terms in brackets must vanish. We have thus

(2.81) StartAbsoluteValue upper S 11 EndAbsoluteValue squared plus StartAbsoluteValue upper S 21 EndAbsoluteValue squared StartFraction eta 1 Over eta 2 EndFraction equals 1 and StartAbsoluteValue upper S 22 EndAbsoluteValue squared plus StartAbsoluteValue upper S 12 EndAbsoluteValue squared StartFraction eta 2 Over eta 1 EndFraction equals 1 comma

which indicates how the incident power distributes in terms of the reflected and transmitted power, and

(2.82) StartFraction upper S 11 upper S 12 Superscript asterisk Baseline Over eta 1 EndFraction plus StartFraction upper S 22 Superscript asterisk Baseline upper S 21 Over eta 2 EndFraction equals 0 and StartFraction upper S 11 Superscript asterisk Baseline upper S 12 Over eta 1 EndFraction plus StartFraction upper S 22 upper S 21 Superscript asterisk Baseline Over eta 2 EndFraction equals 0 comma

which implies that the phases of the scattered waves are related to each other.

For a reciprocal slab with the same media at both sides, i.e. eta 1 equals eta 2 , these expressions reduce to

(2.83) StartAbsoluteValue upper S 11 EndAbsoluteValue squared plus StartAbsoluteValue upper S 21 EndAbsoluteValue squared equals 1 and StartAbsoluteValue upper S 22 EndAbsoluteValue squared plus StartAbsoluteValue upper S 12 EndAbsoluteValue squared equals 1 comma

and

(2.84) StartFraction t Superscript asterisk Baseline Over t EndFraction equals minus StartFraction upper S 22 Superscript asterisk Baseline Over upper S 11 EndFraction and StartAbsoluteValue upper S 11 EndAbsoluteValue squared equals StartAbsoluteValue upper S 22 EndAbsoluteValue squared comma

where t equals upper S 12 equals upper S 21 by reciprocity. Interestingly, if the slab is gainless, lossless, reciprocal, and isotropic, as for instance a thin beam splitting blade, then upper S 11 equals upper S 22 equals r and (Eq. 2.84) tells us that the argument of must necessarily be in quadrature with respect to .

2.7 Classification of Bianisotropic Media

It is sometimes convenient to classify bianisotropic media in terms of their tensorial constitutive parameters. For this purpose, we split the susceptibility tensors chi overbar overbar Subscript em and into a tensor, nu overbar overbar , related to nonreciprocity, and a tensor, kappa overbar overbar , related to chirality [148], as

(2.85) chi overbar overbar Subscript em Baseline equals nu overbar overbar minus j kappa overbar overbar and chi overbar overbar Subscript me Baseline equals left-parenthesis nu overbar overbar plus j kappa overbar overbar right-parenthesis Superscript normal upper T Baseline period

Table 2.3 classifies the reciprocal ( nu overbar overbar equals 0 and kappa overbar overbar not-equals 0 ) and nonreciprocal ( nu overbar overbar not-equals 0 and kappa overbar overbar equals 0 ) bianisotropic media depending on whether the tensors or are diagonal or not, and symmetric or antisymmetric, respectively. This classification relies on the fact that an arbitrary tensor upper M overbar overbar , which may represent either (for a reciprocal medium) or nu overbar overbar (for a nonreciprocal

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