Optical reciprocity theorem

Finite models. Because of their mathematical simplicity, theoretical models of the types described above are those most frequently analyzed in detail in the reactor literature. However, the self-adjoint nature of the differential operators postulated is not typical of real reactor problems neither is the symmetry of the associated integral kernels. This is because, physically, neutron life-histories are not reversible in heterogeneous reactors, even statistically the optical reciprocity theorem [4, p. 82] is not vaUd when slowing down is considered. 

It is interesting to note that Eq. (1.377) and Eq. (1.379) coincide exactly with the square of Eq. (1.199) and Eq. (1.201), respectively, which describe the field-enhancements of a dielectric sphere in the quasi-static limit. This is a manifestation of the so called optical reciprocity theorem and it can be shown that it holds for arbitrary geometry [46] (see also Sec. 5.3.3). 

The estimate of the enhancement factor due to the first term in Eq. (5.45) is often performed by invoking the optical reciprocity theorem (ORT) [75]. It states that the interaction of a dipole fi placed in position 0 with an EM field 62 created by a dipole fi2 in position M... 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

For optical waveguides, we take A to be the infinite cross-section A. The line integral is then over the circle r = oo, where r is the cylindrical radius, and, if either or both of the barred and unbarred fields represent bound modes, then F vanishes as r- 00, since bound-mode field amplitudes fall off exponentially. Thus we drop the line integral in Eq. (31-4) and obtain the conjugated form of the reciprocity theorem... 

A large number of measurements by KrishnaN confirmed this relation and therewith the reciprocity theorem One can deduce qualitatively the siae and shape of the particles from the values of Pu, Pv and ph If Pu deviates from 0 this means that the particles are either large or geometrically or optically anisotropic If pv deviates from 0 then the particles are geometrically or optically anisotropic If ph deviates from 1 then they arc large ... 

However, the theorem of reciprocity is a wave optical argument that does not consider intensities where we easily find the differences. For example, if one thinks of a STEM as an inverted HRTEM one would not detect any intensity in an image since it is an inherent property of a point detector to collect no intensity. On the other extreme side, the ability to form an intense and focused probe is a valuable ability that boosts local spectroscopy. Obviously, the best choice of tools cannot be a matter of exclusion but must relate to the problem at hand that needs solving . 

---