In this study samples are probed by high intensities in order to observe a measurable EFISH signal from the Si/SiO2 interfaces. The optical properties of silicon can change drastically under intense laser irradiation therefore equations for low intensity irradiation cannot fully describe the transmittance and absorbance of light in silicon. Under intense fs laser irradiation (> 1 GW/cm2), the probability of a material absorbing more than one photon before relaxing to ground state can be greatly enhanced [47]. If multiphoton absorption at high intensities in silicon is strong, the transmitted fundamental beam can be depleted in the Si leading to low beam transmission.
During multiphoton absorption electron-hole pairs in silicon are generated by the absorption of two or more photons, the probability of an electron to absorb at least two photons is increased at high intensities. In silicon multiphoton absorption such as two or three photon
absorption is strong in the mid- or far-infrared wavelength regions where two or three photons are simultaneously absorbed through virtual intermediate states in the indirect energy gap of 1.1 eV [55]. Multiphoton absorption can be greatly enhanced by the presence of intermediate resonances. At a fundamental wavelength of 800 nm (1.55 eV) three photon absorption can occur through resonance two photon absorption via the silicon direct bandgap of 3.1 eV.
The change in the nonlinear absorption coefficient is related to the imaginary part of the effective third order susceptibility tensor χ(3). The functional form of χ(3) depends on the symmetry and orientation of the crystal [56, 57]. Single photon, two, three and four photon absorption are proportional to the magnitude of the imaginary part of first, third, fifth and seventh order susceptibility tensors respectively. Since χ(n+2) ≪ χ(n), higher order multi- photon absorption coefficients are small to measure therefore high intensities are required to observe any transmission change.
The attenuation of the laser beam caused by two photon absorption can be represented by the differential equation [46, 58]
dI
dz = −αI − βI
2 (3.38)
where α is the linear absorption coefficient and β is the two photon absorption coefficient.
For a three photon absorption process, the intensity attenuation is given by
dI
dz = −αI − γI
3 (3.39)
where γ is the three photon absorption coefficient.
The solution to equation 3.38 is given by
I(z) = αIoe −αz α + β(1 − e−αz)I
o
, (3.40)
in the limit that β goes to zero equation 3.40 reduces to Beer-Lambert’s law [58] of single photon absorption.
processes equations 3.38 and 3.39 can be generalised to
dI
dz = −αI − βI 2
− γI3− τI4− ... (3.41)
in which τ is the four photon absorption coefficient.
The investigation of multiphoton absorption in particular two photon absorption at different wavelengths for different materials has been reported in literature using the common z-scan technique [59, 60, 61, 62, 63, 64, 65, 66, 67]. Z-scan has been widely adopted as a simple single beam technique to obtain β and n2 (nonlinear refractive index) with intensity variation achieved by scanning a sample through the focal region of a Gaussian beam [61]. The z-scan method provides a sensitive and straight-forward method for the determination of the sign and the values of the real and imaginary parts of χ(3). The simplicity of both the experimental setup and the data analysis has allowed the z-scan method to become widely used by many research groups [68]. Measurements of β and n2 are performed using the closed and open aperture z-scan technique respectively.
The incident laser is focussed on the sample and measure the transmitted light as the sample is scanned through the laser focus in the z-direction in an open aperture z-scan technique. According to Dinu et al. [69], for a Gaussian beam the transmitted light for open aperture technique, is given by Topen(z) = 1 − 1 2√2 βIoLeff 1 + (z/zo)2 , (3.42)
where z is the longitudinal scan distance from the focal point with an on-axis intensity of Io (inside the sample) and zois the confocal beam parameter; Leff = α−1(1−e−αL) is the effective optical path length, α is the linear absorption coefficient and L is the sample thickness.
In a closed aperture z-scan a circular aperture with transmissivity S < 1 is placed behind the sample, and the transmission is recorded as a function of z position of the sample. The measured transmitted intensity is sensitive to small changes caused by nonlinear effects on the sample such as self focusing and self defocussing [70]. For small absorptive and refractive
−6 −4 −2 0 2 4 6 12 13 14 15 z (mm) Transmission (%) 1 GW/cm2 10 GW/cm2 20 GW/cm2 40 GW/cm2 60 GW/cm2 100 GW/cm2
Figure 3.5: The plot of change in transmitted light as the Si membrane is scanned through the focus for different peak intensities in a typical open aperture z-scan experiment. The plots are according to equation 3.42.
changes the transmissivity is given by [68, 69]
Tclosed(z) = 1 −λ8π√ 2 z/zo(1 − S)0.25Leffn2Io (1 + (z/zo)2)(9 + (z/zo)2) − 1 2√2 LeffβIo(3 − (z/zo)2) (1 + (z/zo)2)(9 + (z/zo)2) . (3.43)
Figure 3.5 shows a plot of equation 3.42 at different peak incident intensities using typical parameters such as thickness and linear absorption coefficient of the Si membrane used in this study. The two photon absorption (β) coefficient was assumed constant at different peak intensities based on the value obtained from [70] at 800 nm. The simulation shows that the transmitted signal changes as the sample is scanned through the focus. The highest incident intensity on the sample correspond to z = 0 in which a minimum in transmission change is well pronounced. At this value of z there is high intensity at focus and the probability of absorbing two or more photons is enhanced. More incident light is absorbed leading to less light being transmitted by the sample. Far from z = 0, all graphs show the same transmission value independent of peak incident intensities. In this case the sample is far from focus and no multiphoton absorption occur therefore single photon absorption process dominates in this low intensity regime.