Analysis of nonlinear optical switching in an erbium-doped fiber

Abstract

Abstruct-We present a mathematical model of the strong, resonantly enhanced nonlinear phase shift recently reported in Er-doped fibers, which relates the phase shift and signal loss to the fiber parameters and the pump and signal wavelengths. Predictions are in fair agreement with the phase shift and loss measured in an experimental Er-doped fiber switch based on this effect. A strong, nearly wavelength independent contribution to the nonlinear phase shift is observed in the switch. The model suggests that this effect is due to the same nonlinear effect arising from one or more VUV transitions in Er3+.

Dynamic Bragg gratings induced in erbium-doped fiber at phase-Modulated beams' coupling

“…Nonlinear changes in refractive index and absorption/gain coefficients in erbium-doped fibers are usually explained by the material dispersion, which is strictly, by the Kramers-Kronig Relations (KKR), connected with the change of Er absorption spectrum under the optical pumping [9]- [11]. The resonant refractive index change in the fiber core is related to the change in the real part of atomic susceptibility as , where is calculated with the use of the KKR and the data on the fiber absorption and gain spectra as [12] (1) where is the relative change in the excitedstate population (DNE is the change in the excited state popultion;…”

“…Thus, the amplitudes of the refractive index and gain gratings can be found [see (2) and (5)] as (11) The gratings' amplitudes calculated with the use of formulas (11) can be taken to estimate the modulation depth of the output signal beam [see (6)] and the grating diffraction efficiency.…”

Resonantly enhanced nonlinear optical switching in rare-earth-doped fibers

“…We assume that most of the index modulation at the signal wavelength arises from a single absorption transition, namely from the ground state to the I11/2 level (level 3). The nonlinear contribution n13 of the 1->3 transition to the refractive index change at the signal frequency V can be written as: [5] = g3 1: MT13g'(v) 16irn0r13 (1) SPIE In this expression, AN13=N1/g1 is the population difference between levels 1 and 3, where N1 and gi are the ground state population and degeneracy, respectively. The refractive index of the material is o, 13 iS the center wavelength of the l->3 transition, g and tf,3 the degeneracy and radiative lifetimeof level 3, respectively.…”

“…The z-dependence arises from pump absorption and pump absorption saturation along the fiber. The expression for n13(z) is then integrated along the length 1 of the fiber, which gives the nonlinear phase change at the signal frequency: [5] L\ -g3 1 'bs '2 (v) 0 -g1 8n02hc A?3 A 1,3g13 (2) In Eq. 2, Xp and are the pump and signal wavelengths, respectively, g is the degeneracy of the level 3, h is Planck's constant, and c is the speed of light in vacuum.…”

“…The nonlinear phase shift resulting from this effect can be described by relatively simple mathematical expressions. [5] For example, to model our Er-doped fiber switch, which was pumped at 1.48 m and probed near 920 nm, we consider a material in which pumping occurs from the ground state I15/2 (level 1) to the metastable manifold I13/2 (level 2). We assume that most of the index modulation at the signal wavelength arises from a single absorption transition, namely from the ground state to the I11/2 level (level 3).…”