<title>Interrelation between reciprocity relations in spatial and temporal domains for partially-coherent beams</title>

Abstract

Reciprocity inequalities (uncertainty relations) are studied for a finite light pulse with complex spatio-temporal structure and for statistical ensemble of such pulses. A possibility to create partially coherent pulses (and ensembles) with parameters close to minima of these inequalities both in temporal and spatial domains are discussed. The structure of optimal beams is analyzed from point of view of modal treatment of coherence (biorthogonal KarhunenLoéve expansion).

Purity-bounded uncertainty relations in multidimensional space—generalized purity

“…Indeed, the spectral decomposition of density matrix is closely connected to the Schmidt decomposition of non-separable states, see, e. g. [28]. Approach to uncertainty principle for entangled states can be based on mathematically analogous case of uncertainty (reciprocity) relations for pulsed partially coherent classical beam [25].…”

“…It is also necessary to note that the inequality, mathematically practically the same as uncertainty relation, but with another physical meaning, is often used for classical wave fields, e. g. in optics [8][9][10][11][12]23]. Results of the present article, as well as of preceding papers [23,24] could be used, with appropriate change of notations, for classical partially coherent fields and sources (in 1-, 2-and 3-dimensional space [25]). …”