The saturable absorption of visible radiation with intensities less than 0.14 kW/cm2 in the composite and the negative change in the refractive index were due to the presence of Fe3O4 nanoparticles since pure MMAS showed only the relatively weak reverse saturable absorption of yellow radiation. Therefore, the experimental data ΔT(I) and ΔTpv(I) obtained for the composite could be used to calculate the values of Δα(I) and Δn(I) for Fe3O4 nanoparticle arrays (Equation 1), and these values are listed in Figure .
Figure 6 The values of changes in the absorption coefficient, refractive index, and polarizability of Fe3O4 nanoparticles. (a) The dependency of changes in the absorption coefficients Δα of pure MMAS (circle) and Fe3O4 nanoparticle arrays (square (more ...)
Because the observed dependence of Δn
on the radiation intensity I
(Figure b) for Fe3
nanoparticle arrays could be considered a linear function, it can be assumed that Δn
was caused by the thermal effect of the radiation. We estimated the contribution of this effect to the changes of the composite refractive index using the equation [43
where chc was the MMAS heat capacity (0.7 J/g·K), ρd was the MMAS density (1.3 g/cm3), dn/dT was the MMAS thermo-optic coefficient (−10−5 K−1), and ΔE was the energy absorbed by the composite per unit volume per second. The thermal effect of cw low-intensity radiation on the change in the refractive index (red dashed lines in Figure b) was relatively small (not more than 20% for blue radiation and 8% for yellow radiation).
Generally, the possibility of a nonthermal optical response of the composite due to external optical radiation is associated with the polarization of Fe3
nanoparticles in the external field E
. Nanoparticle polarization occurs at the spatial separation of positive and negative charges, i.e., at the electron transition to higher allowed energy states (quantum number l ≠ 0). These transitions should be accompanied by the absorption of external radiation. In our case, we observed the absorption of radiation with wavelengths of 380 to 650 nm (Figure ). This absorption band consisted of three maxima (380, 480, and 650 nm), indicating the broadened quantum-size states for the electrons in Fe3
nanoparticles. Because the bandgap of magnetite is rather small (approximately 0.2 eV) [20
], the conduction and valence bands of the nanoparticles should be coupled due to quantum-size effect [44
]. Therefore, the transitions of Fe3
nanoparticle electrons to higher energy states by the action of photons with energies of 2.3 eV (λ
= 561 nm) and 2.6 eV (λ
= 442 nm) can be considered intraband transitions. In turn, these transitions result in changes in the refractive index of the media as follows [45
where e was the electron charge, c was the speed of light, ϵ0 was the electric constant, me was the electron mass, and Ne was the concentration of excited electrons, which depends on the number of photons in the beam or the radiation intensity I.
Using Equation 4 to approximate the experimentally observed behavior of Δn(I) (Figure b, blue dashed lines), we estimated that the concentration of optically excited electrons in Fe3O4 nanoparticles was approximately 1023 m−3, being the radiation intensity of less than 0.14 kW/cm2.
The amplitude of the nanoparticle polarization is determined by |E|
of the external field and the nanoparticle susceptibility (χ
) or polarizability (α
) measured in cubic angstrom. In turn, the change in the refractive index induced by the radiation is associated with the change in nanoparticle polarizability Δα
) by classical relations [48
]. Therefore, we could calculate the values of Δα
) for Fe3
nanoparticle using the experimental values of Δn
) and the following equations (SI):
where ϵ was the real part of the dielectric constant, the composite refractive index n(I) = n0 + Δn(I), and n0 was the refractive index of pure MMAS (approximately 1.5). The extinction coefficient k = αλ / 4π was significantly less than n(I) and could be ignored; χ was the nanoparticle susceptibility, and N was the nanoparticle concentration (approximately 2.3 × 1019 m−3). Therefore, the values of Δα (Å3) for Fe3O4 nanoparticle were calculated using the formula Δα (Å3) ≈ 2n × Δn(I) × 1030 / N and are presented in Figure b.
The obtained values for the changes in nanoparticle polarizability are orders of magnitude greater than those for semiconductor nanoparticles and molecules [30
] in extremely weak optical fields. In addition, the average nanoparticle volume was approximately 2.2 × 106
, and the maximum value of Δα
) was 9 × 106
. Thus, we can conclude that the nanoparticle polarization should be formed by several optical intraband transitions of nanoparticle electrons in weak optical fields.