Nonlinear resonant change in refractive index of a doped saturable medium at short-pulse excitation

Abstract

A novel method is reported for calculating the nonlinear resonant change in refractive index of a doped saturable medium at the short-pulse excitation. The derived system of equations for energy density and phase of a short pulse propagating in the doped medium allows one to address a rather general case of the resonant index non-linearity caused by the dope's population perturbation (bleaching) effect. As an example, the energy-dependent changes in the refractive index and transmission coefficient of a Cr 4+ :YAG crystal under the ns-range (wavelength 1.06 µm) excitation are modeled. It is known [1] that propagation of a short pulse through a resonantly amplifying (or absorbing) medium with a "slow" relaxation of the excited state is addressed by the Avizonis-Grotbeck's equation:where ε = E p s σ GSA hυ is the ratio of incident pulse energy density E p /s (where E p and s are the pulse energy and laser beam geometrical cross-section) and saturating energy density hυ/σ GSA (where hυ is the energy quanta at the resonant wavelength and σ GSA is the ground-state absorption (GSA) of resonantly-absorbing dope centers embedded in the material), α 0 = σ GSA N 0 is the weak-signal resonant absorption coefficient (where N 0 is the concentration of dope centers), γ is the residual (non-saturable) loss coefficient, and z is the coordinate inside the sample. Eq. (1) holds if the coefficient γ relates (i) to all the "passive" losses (owing to scattering or presence of non-resonant impurities) and (ii) to the loss stemming from the excited-state absorption (ESA) [2] in the system of dope centers (if such a source of loss exists, see Fig. 1). Notice that "slow" relaxation of the excited state means that decay time τ rel of the latter is essentially longer than pulse-width τ p .Solving Eq. (1), one is able to calculate nonlinear (energy-dependent) transmission coefficient of the medium as the ratio of normalized input (ε in ) and output (ε out ) pulse energy densities: T (ε in ) = ε out /ε in . Meanwhile, to the best of our knowledge there is not known an analogous equation concerning the resonant nonlinear change ∆n res (ε in ) in refractive index n of the medium that should accompany the transmission coefficient nonlinearity. Despite there have been published a few re-

Nonlinear Change in Refractive Index and Transmission Coefficient of ZnSe:Fe<sup>2+</sup> at Long-Pulse 2.94-μm Excitation

“…Following the approach [27], the equation for phase difference Δφ sat , experienced by phase φ of the pulse travelling in the sample (which is complementary to the resonant-absorption saturation process addressed by Equation (1) via Kramers-Kronig relations), can be written as:…”

“…Accordingly, the average Fe 2+ concentrations n 0 , SSA-values α 0 , and passive losses γ were determined; see Table 1. The saturation energy density was fixed in calculations (E s =0.08 J/cm 2 ) as defined by the value of the GSA cross-section [21] in analogy to [27] [28]. The most reliable values of these parameters were found to be δ = 0.03 and η = −0.4 (at λ 0 = 2.94 µm).…”

Passively Q-switched mode-locking in a diode-pumped c-cut Nd:LuVO4laser with Cr4+:YAG

“…Cr 4+ :YAG absorber has high thermal conductivity, low saturable intensity, fine chemical and physical properties [15,16], consequently, it has been extensively employed in Q-switched lasers [17,18]. Further studies show that there is excited state absorption (ESA) in Cr 4+ :YAG [19], so it is also considered to realize mode-locking when the ESA is strongly saturated by strong intracavity intensity.…”