On mathematical and physical principles of transformations of the coherent radar backscatter matrix

The congruential rule advanced by Graves for polarization basis transformation of the radar backscatter matrix is now often misinterpreted as an example of consimilarity transformation. However, consimilarity transformations imply a physically unrealistic antilinear time-reversal operation. This is just one of the approaches found in literature to the description of transformations where the role of conjugation has been misunderstood. In this paper, the different approaches are examined in particular in respect to the role of conjugation. In order to justify and correctly derive the congruential rule for polarization basis transformation and properly place the role of conjugation, the origin of the problem is traced back to the derivation of the antenna height from the transmitted field. In fact, careful consideration of the role played by the Green’s dyadic operator relating the antenna height to the transmitted field shows that, under general unitary basis transformation, it is not justified to assume a scalar relationship between them. Invariance of the voltage equation shows that antenna states and wave states must in fact lie in dual spaces, a distinction not captured in conventional Jones vector formalism. Introducing spinor formalism, and with the use of an alternate spin frame for the transmitted field a mathematically consistent implementation of the directional wave formalism is obtained. Examples are given comparing the wider generality of the congruential rule in both active and passive transformations with the consimilarity rule.

Asymmetric oxidation of vinyl- and ethynyl terthiophene ligands in triruthenium complexes

A series of ruthenium(II) complexes [{RuCl(CO)(PMe3)3(–CHvCH–)}nX], 1a–1c (1a: n = 3, X = 3,3’’- dimethyl-2,2’:3’,2’’-terthiophene; 1b: n = 2, X = 2,2’-bithiophene; 1c: n = 2, X = 2,3-bis(3-methylthiophen- 2-yl)benzothiophene) and [{Cp*(dppe)2Ru(–CuC–)}3X], 1d (X = 3,3’’-dimethyl-2,2’:3’,2’’- terthiophene), were prepared and characterized by 1H, 13C and 31P NMR. Their redox, spectroscopic and bonding properties were studied with a range of spectro-electrochemical methods in combination with density functional theory calculations. The first two anodic steps observed for 1a and 1d are largely localized on the lateral frameworks of the molecular triangle, the direct conjugation between them being precluded due to the photostable open form of the dithienyl ethene moiety. The third anodic step is then mainly localized on the centerpiece of the triangular structure, affecting both bithiophene laterals. The experimental IR and UV-vis-NIR spectroelectrochemical data and, largely, also DFT calculations account for this explanation, being further supported by direct comparison with the anodic behavior of reference diruthenium complexes 1b and 1c.