**3. Interpretation**

A typical CA Z-scan output for a thin sample exhibiting nonlinear refraction peak (TP)–the valley (TV) is shown in **Figure 2**, where ΔTP-V is the difference between transmittance peak and valley transmittance. A self-defocusing nonlinearity results in a peak followed by a valley in the normalized transmittance as the sample is moved away from the lens (i.e. increase in the Z direction).

The normalization is performed in such a way that the transmittance is unity for the sample far from the "focus where the nonlinearity is negligible [12]. The negative lensing in the sample placed before the focus moves the focal position further from

**Figure 1.** *Experimental arrangement of Z-scan technique.*

*Third-Order Nonlinearity Measurement Techniques DOI: http://dx.doi.org/10.5772/intechopen.106506*

**Figure 2.** *Z-scan for third-order positive nonlinearity.*

**Figure 3.** *Z-scan graph of a thin nonlinear absorber.*

the sample placed increasing the aperture transmittance. The experiment with T = 1 is referred to as an "open aperture" (OA) Z-scan and allows direct measurement of nonlinear absorption in the sample as shown in **Figure 3**.

#### **4. Nonlinear refraction**

Nonlinear refraction (NLR) in the absence of nonlinear absorption (NLA) is determined by using a finite aperture to calculate the transmittance of the NL medium in a far field as a function of the position (Z) of the sample. The illumination of the sample is done by the self-focused polarized Gaussian beam that attains different incident field

strengths at different positions of Z [3]. Obtained values can be convenient to plot T, the transmittance normalized to the linear transmittance of the system. The nonlinearity can be estimated from the difference between the maximum (peak) and minimum (valley) values of the normalized transmittance (ΔT). For a thin optical Kerr medium where the refractive index varies linearly with irradiance nonlinear refractive index coefficient nI, ΔT is proportional to the nonlinear phase distortion (shift) on the axis with the sample at the waist [13, 14]. The empirically determined relation between the induced phase distortion (Δϕ0) and normalized transmittance (ΔTP-V) for a thirdorder nonlinear refractive process in the absence of NLA is given by Eq. (1):

$$
\Delta T\_{p-v} \cong \mathbf{0}.406 \mathbf{1} (\mathbf{1} - \mathbf{S})^{0.27} |\Delta \rho| \tag{1}
$$

where S is the linear transmittance of the aperture in the absence of the sample and Δϕ is the axis phase shift.

$$S = 1 - \exp\left[\frac{-2r\_a^2}{\alpha\_a^2}\right] \tag{2}$$

Here, ra is the radius of the aperture and wa is the beam radius at the aperture.

$$
\Delta \rho\_0 = k n\_2 I\_0 L\_{\text{eff}} \tag{3}
$$

Here, Leff is an effective sample length and k = 2π/λ, so,

$$
\Delta\rho\_0 = \frac{2\pi}{\lambda} n\_2 I\_0 L\_{\text{eff}} \tag{4}
$$

Here, λ is the wavelength, n2 is the nonlinear index of refraction, and I0 is the axial irradiance at the waist.

$$L\_{\rm eff} = \frac{1 - e^{-aL}}{a} \tag{5}$$

where L is the sample length, L = Leff (absence of linear absorption), and α is the linear absorption coefficient. The distance between peak and valley is measured in Zaxis, and ΔTP-V is a direct measure of the diffraction length of the incident beam for a given order nonlinear response. In a standard Z-scan (i.e. using a Gaussian laser beam and a far-field aperture), Eq. (6) gives the relation for third-order nonlinearity (Z):

$$|\Delta Z\_{pv}| \approx \mathbf{1.7Z\_0} \tag{6}$$

This gives the focal spot size of the beam for diffraction-limited optics independent of the irradiance for small nonlinearities.

In principle, the Z-scan can be used to measure very small spot sizes using a very thin sample. Nonlinear absorption measurements are usually done by removing the aperture to collect the maximum intensity from the sample.

#### **5. Higher-order nonlinearities**

The nonlinear optical effects give index changes proportional to the irradiance (Δn). For a fifth-order (χ(5)), NLR becomes the dominant mechanism in

semiconductors when Δn is induced by two-photon generated free carriers. This type of nonlinearity is derived from simple relations that accurately characterize the Z-scan data [15]. A Gaussian beam and far-field aperture is given by Eq. (7):

$$
\Delta T\_{p^{w}} \cong \mathbf{0}.2\mathbf{1}(\mathbf{1}-\mathbf{S})^{0.27}|\Delta\rho\_{0}|\tag{7}
$$

and

$$|\Delta Z\_{pv}| \approx 1.2Z\tag{8}$$

For this case, the data analysis becomes very complicated due to the simultaneous process of χ(3)and χ(5) using several Z-scans at different irradiances [15]. This procedure makes use of simple relations of Eqs. (1) and (3) to estimate the nonlinear coefficients associated with both χ(3)and χ(5) processes.
