**6. Nonlinear absorption**

The NLA can be determined using a two-parameter fit to a closed aperture Z-scan, and it is more accurate to determine in an open aperture of Z -scan. For small thirdorder nonlinear losses with response times much less than the pulse width (i.e. twophoton absorption), and for a Gaussian temporal shape pulse, the normalized change in transmitted energy ΔT with Z is given by Eq. (9) [16]:

$$
\Delta T \approx -\frac{q\_0}{2\sqrt{2}} \frac{1}{\left[1 + Z^2/Z\_0^2\right]} \tag{9}
$$

The Lorentzian distribution of the illuminance with Z position for a focused polarized Gaussian beam is shown in **Figure 3**. If the response time is extended than the pulse width, the factor 2√2 is replaced by 2, which is independent of the temporal pulse shape. This gives the excited state of absorption cross section (σ).

#### **7. Thin nonlinear medium**

For an absorption coefficient that varies linearly with irradiance, the coefficient of nonlinear absorption α can be calculated using Eq. (11):

$$a = \frac{q\_0}{I\_0 L\_{\text{eff}}} \tag{10}$$

$$
\Delta T = \left| 1 - \frac{1}{q\_0} \mathbf{1} n \left( \mathbf{1} + q\_0 \right) \right| \tag{11}
$$

Here, I0 is the incident beam intensity, and Eq. (3) can be used for all orders of I0. The Z-scan experiment is a simple method due to its single-beam technique. The Z-scan technique yields both the sign and magnitude of the nonlinearity from nI and α. In addition, it has an advantage process of a closer similarity between Z-scan and optical power limiter geometries [17]. A comprehensive Z-scan study not only gives important information on the NL properties of the sample but also yields necessary

information regarding optimization of the optical power limiting geometry such as the optimum sample position and optimum sample thickness. This makes Z-scan an ideal tool for assessing materials for optical power-limiting applications.

### **8. Modeling studies**

The Z-scan technique is a far-field measurement method. The term "far field" is defined as a distance of 10 Rayleigh lengths between the beam waist and the aperture, and there is an 18% change in the transmittance. An eliminating error in ΔT is only 1% because both the peak and valley are calculated in the Z direction [3].

The poor aperture alignment was effectively determined by modeling studies for the subjected sample Z-scan profile. For best results, the sample should be wedged and alignment of the aperture focused on words of the center of the sample to obtain a maximum transmittance through the sample. Modeling studies also launch the effects which ensue if the sample has a lens-like profile or if it shows surface roughness or scratches. Implementing the technique of subtracting low-power from high-power Zscans, as suggested by Sheik-Bahae et al. [2] works well for extracting the NL result for imperfect surfaces.

The refracted beam from the sample gives rise to the third-order nonlinearity, and it is intensity-dependent convergence or divergence (self-focusing/defocusing). Sample position Z causes intensity variation due to the transformation of phase distortion converted to amplitude distortion of the transmitted beam [3]. As the position of the Z varies, if there is an increase in transmittance in the pre-focal stage and followed by a decrease in the post-focal region (peak-valley) Z-scan symbolizes negative NLR, whereas a valley-peak represents a positive nonlinearity. Removing the aperture leads to collecting total intensity on the detector which results in the Z-scan for flat response (purely refractive nonlinearity) due to the sensitivity of NLR depending on the aperture. The multi-photon absorption leads to valley enhancement quashes the peak saturation of absorption provides a reverse saturable absorption (RSA) effect in CA of Z-scan [18]. Significantly, this technique enables the measurement of the sign of nonlinearity by directly eliminating magnitude (real and imaginary parts). For optical signal processing applications, the sign of nonlinearity plays a key factor. It cannot be directly derived using any other techniques [19].

#### **9. Z-scan measurement technique**

Third-order NL parameters of the subject material were determined by the Z-scan technique. A wavelength of 532 nm (Gaussian laser beam) was converged using a converging lens of focal length 10 cm in the Z position of the subject sample with a thickness of 1 mm. The NL parameters such as absorption coefficient (β), refractive index (n), and susceptibility (χ(3)) were calculated using their equations. The subjected sample was displaced in the Z direction to obtain intensity-dependent absorption. The subjected sample information was recorded using a high-sensitive photo-detector [6]. A sample is drawn in between the obtained values, and a curve fit was done as shown in **Figure 4**.

The nonlinear absorption coefficient (β) was calculated using the following equation:

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

**Figure 4.** *Open aperture (OA) curve.*

$$\beta = \frac{2\sqrt{2\Delta T}}{I\_0 L\_{\text{eff}}} (m/W) \tag{12}$$

Keeping the aperture in front of the detector Z-scan was performed to get the sign and magnitude closed aperture. The NLA and NLR were essentially important in closed aperture transmittance [14, 20–22]. The ratio between CA and OA gives the real NLR. A sample plot of the closed aperture (CA) curve is depicted in **Figure 5**.

To calculate the third-order nonlinear refractive index (n), Eq. (13) was used.

$$n = \frac{\Delta \rho}{k I\_0 L\_{\text{eff}}} \left( m^2 / \mathcal{W} \right) \tag{13}$$

**Figure 5.** *Closed aperture (CA) curve.*

The on-axis phase shift at the focus was indicated by Δφ; it has been found using the Eq. (14):

$$|\Delta\rho| = \frac{\Delta T\_{p-v}}{0.406(1-\mathbb{S})^{0.27}}\tag{14}$$

The obtained values of n2 and β were used to calculate the complex equations containing the real and imaginary elements of third-order nonlinear optical susceptibility (χ(3)), which can be calculated using the Eqs. (15) and (16) [21]:

$$R\_{\epsilon} \chi\_{(\text{eu})}^{(3)} = \frac{\mathbf{1} \mathbf{0}^{-4} \epsilon\_0 C^2 n\_0^2 n\_2}{\pi} \left( c m^2 / W \right) \tag{15}$$

$$I\_{\epsilon} \chi\_{(em)}^{(3)} = \frac{10^{-2} \epsilon\_0 C^2 n\_0^2 \lambda \theta}{\pi} (cm/W) \tag{16}$$

where ε<sup>0</sup> indicates the permeability of free space, C is the velocity of light, and the n0 is the linear refractive index of the sample. The magnitude of third-order nonlinear susceptibility (χ(3)) for the subjected sample was estimated using the Eq. (17):

$$\chi^{(3)} = \sqrt{\left(R\_{\epsilon}\chi^{(3)}\right)^{2} + \left(I\_{m}\chi^{(3)}\right)^{2}}\tag{17}$$

The NL of the subjected material purely depends on the concentration of the solvent because its intermolecular interaction Eq. (17) provides a better nonlinearity.

#### **10. Conclusion**

Several methods and techniques are available to determine the third-order nonlinearity measurement techniques; among them, Z-scan is the simplest and most efficient technique. The Z-scan technique directly measures the physical processes behind the nonlinear response of a given material using a single wavelength. Nonlinear absorption and refraction invariably coexist because they are obtained from the same physical parameters. They are related by dispersion relations identical to the usual Kramers-Kronig relation that connects linear absorption to linear index, or equivalently, leads to real and imaginary parts of linear susceptibility. And also it has both a completely computed technique for determining standards and a relative measurement method. The Z-scan signal is a function of irradiance and shapes for sample position. The Z-scan technique has a great prospect to solve highly scattering problems and surface the way to characterize the NLO properties of biological and optical polished samples. It can give useful information on the order of nonlinearity as well as its sign and magnitude.

#### **Acknowledgements**

I am very much grateful to my institution VINAYAKA MISSION'S RESEARCH FOUNDATION, Salem, for providing an opportunity to carry out the research work in the Department of Physics, School of Arts and Science, Av campus, Payinoor,

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

Chennai. My heartfelt thanks to Late Dr. SM. Ravikumar, for supporting me throughout my research. I thank IIT Madras for the charazation analysis. I extend my gratitude to thank Dr. Samuel, Dr. Annie, Ms. Ivy, and Ms. Ezhil for their valuable suggestion.
