A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently.

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Energy transfer between Tm³⁺ and Tb³⁺ dependent on the power density of pump laser was investigated in NaYF₄: Tb³⁺,Tm³⁺,Yb³⁺ microcrystals. Under the excitation of a 976-nm near-infrared laser at various power densities, Tb³⁺-Tm³⁺-Yb³⁺ doped samples exhibited intense visible emissions with tunable color between green and blue. The ratio of blue and green emission were determined by energy transfer between Tm³⁺ and Tb³⁺. When the power density of pump laser was low, the energy transfer process from Tm³⁺ (³F₄) to Tb³⁺ (⁷F₀) occurred efficiently. Upconversion processes in Tm³⁺ were inhibited, only visible emissions from Tb³⁺ with green color were observed. When the power density increased, energy transfer from the ³F₄ (Tm³⁺) to ⁷F₀ level (Tb³⁺) was restrained and population on high energy levels of Tm³⁺ was increased. Contribution of upconversion emissions from Tm³⁺ gradually became dominant. The emission color was tuned from green to blue with increasing the power density. Energy transfer processes between low-lying levels of activators, such as Tm³⁺ will greatly reduce the population on certain levels for further high-order upconversion processes. The Tb³⁺-Tm³⁺-Yb³⁺ doped phosphors are promising materials for detecting the condition of power density of the invisible near-infrared laser.

This paper describes the radius of convergence of a self-excited feedback interconnection of two convergent Fliess operators. For either locally convergent or globally convergent operators, a fundamental and achievable positive lower bound on the radius of convergence for the composite system is computed.

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