Third-Order Nonlinearity Measurement Techniques

Abstract

To measure the degenerate (single-frequency) optical nonlinearities, third-order nonlinearity measurement and their related techniques were employed. When a laser beam is induced on a nonlinear (NL) medium, a phase change is easily identified using third-order nonlinearity measurement techniques (Z-scan). When the sample material is scanned on Z-axis, the phase change is denoted by sign and magnitude, the refractive index which is directly related to the change in the index of refraction. The nonlinear absorption from the absorption coefficient is independent of the index of refraction which is a required parameter for calculating nonlinear refraction. Further, the change in transmission caused by nonlinear absorption of the subjected material is related to the change in absorption coefficient which is easily determined by the Z-scan technique. From Z-scan responses, real and imaginary parts of third-order nonlinear susceptibility (χ3) can be determined. The Z-scan technique is an interesting process that leads to optical power limiting and nonlinear optical propagation.

1. Introduction

To examine the irritation dynamics and time-resolved technique under sonorant agitation, Z-scan is an effective tool. The intensity-dependent nonlinear (NL) absorption coefficient (Δα) and magnitude of the nonlinear index (n₂) (Kerr nonlinearity) are vital key factors for photo-induced dissociation in reduced crystals, optical material, and photonic applications [1, 2]. Nonlinear absorption (NLA) and nonlinear refraction (NLR) in solid and liquid can be effectively measured using the Z-scan technique [3]. This method is relatively simple due to its single-beam technique measuring both the sign and magnitude of NLR and NLA, and its spatial beam distortions principle is purely dependent on the sign of the nonlinearity [2]. Methods like beam distortion [4], elliptical polarization [5], multi-wave mixing [6], nonlinear interferometry [7], the indexing change (Δn), and absorption change (Δα) can be done directly without curve fitting. Z-scan method is the most sensitive mechanism that directly determines the refractive index change. The electronic structure of materials is shed by the frequency dispersion study from the nonlinear susceptibility (χ) due to the internal atomic and molecular resonances. The advantage reveals that Z-scan studies become a fundamental screening for third-order nonlinearity [3, 8]. The sample is placed on the Z-axis along with the positive and negative directions, and the input laser beam was converted into polarized Gaussian beam along the waist of the motorized translation stage [9] as shown in Figure 1.

Figure 1.

Experimental arrangement of Z-scan technique.

2. Technique

The usual “closed aperture” (CA) Z-scan technique (i.e. aperture in place in the far field) for nonlinear refraction is shown in Figure 1 (sample displaying self-focusing). Sample transmittance of polarized Gaussian beam through the aperture is monitored by far field as a function of the Z position of NL in the vicinity of the linear optics focal position [10]. The scan range for the examination of the sample depends on the beam parameter and thickness of the sample L. A significant limitation is the diffraction length of the focused beam defined as a polarized Gaussian beam. A “thin” sample has a thickness of L. Even though all the information is theoretically contained within a scan range of ±Z direction, it is preferable to scan the sample for approximately ±5 Z to determine the linear transmittance [2]. If the subjected material surface is rough or imperfect, it leads to background noise in the output. A reference detector can be used to monitor the normalized transmittance. To eliminate the possible noise due to spatial beam variations, this reference arm can be further to include a lens and an aperture identical to those in the nonlinear arm [11] as shown in Figure 1.

3. Interpretation

A typical CA Z-scan output for a thin sample exhibiting nonlinear refraction peak (T_P)–the valley (T_V) is shown in Figure 2, where ΔT_P-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).

Figure 2.

Z-scan for third-order positive nonlinearity.

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 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.

Figure 3.

Z-scan graph of a thin nonlinear absorber.

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 n_I, 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 (ΔT_P-V) for a third-order nonlinear refractive process in the absence of NLA is given by Eq. (1):

$$\mathrm{\Delta }{T}_{p-v}\cong 0.4061{\left(1-S\right)}^{0.27}\left|\mathrm{\Delta }\phi \right|$$E1

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

$$S=1-exp\left[\frac{-2r_a^2}{{\omega }_a^2}\right]$$E2

Here, r_a is the radius of the aperture and w_a is the beam radius at the aperture.

$$\mathrm{\Delta }{\phi }_0={\mathit{kn}}_2{I}_0{L}_{\mathit{eff}}$$E3

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

$$\mathrm{\Delta }{\phi }_0=\frac{2\pi }{\lambda }{n}_2{I}_0{L}_{\mathit{eff}}$$E4

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

$$L_{\mathit{eff}}=\frac{1-e^{-\mathit{\alpha L}}}{\alpha }$$E5

where L is the sample length, L = L_eff (absence of linear absorption), and α is the linear absorption coefficient. The distance between peak and valley is measured in Z-axis, and ΔT_P-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):

$$\left|\mathrm{\Delta }{Z}_{\mathit{pv}}\right|\approx 1.7Z_0$$E6

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):

$$\mathrm{\Delta }{T}_{\mathit{pv}}\cong 0.21{\left(1-S\right)}^{0.27}\left|\mathrm{\Delta }{\phi }_0\right|$$E7

and

$$\left|\mathrm{\Delta }{Z}_{\mathit{pv}}\right|\approx 1.2Z$$E8

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.

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 third-order nonlinear losses with response times much less than the pulse width (i.e. two-photon absorption), and for a Gaussian temporal shape pulse, the normalized change in transmitted energy ΔT with Z is given by Eq. (9) [16]:

$$\mathrm{\Delta }T\approx -\frac{q_0}{2\sqrt{2}}\frac{1}{\left[1+Z^2/Z_0^2\right]}$$E9

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):

$$\alpha =\frac{q_0}{I_0{L}_{\mathit{eff}}}$$E10

$$\mathrm{\Delta }T=\left|1-\frac{1}{q_0}1n\left(1+q_0\right)\right|$$E11

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 n_I 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 Z-scans, 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.

Figure 4.

Open aperture (OA) curve.

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

$$\beta =\frac{2\sqrt{2\mathrm{\Delta }T}}{I_0{L}_{\mathit{eff}}}\left(m/W\right)$$E12

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, 21, 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.

Figure 5.

Closed aperture (CA) curve.

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

$$n=\frac{\mathrm{\Delta }\phi }{{\mathit{kI}}_0{L}_{\mathit{eff}}}\left(m^2/W\right)$$E13

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

$$\left|\mathrm{\Delta }\phi \right|=\frac{\mathrm{\Delta }{T}_{p-v}}{0.406{\left(1-S\right)}^{0.27}}$$E14

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]:

$$R_e{\chi }_{\left(\mathit{esu}\right)}^{\left(3\right)}=\frac{{10}^{-4}{\epsilon }_0{C}^2{n}_0^2{n}_2}{\pi }\left({\mathit{cm}}^2/W\right)$$E15

$$I_e{\chi }_{\left(\mathit{esu}\right)}^{\left(3\right)}=\frac{{10}^{-2}{\epsilon }_0{C}^2{n}_0^2\mathit{\lambda \beta }}{\pi }\left(\mathit{cm}/W\right)$$E16

where ε₀ 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):

$${\chi }^{\left(3\right)}=\sqrt{{\left(R_e{\chi }^{\left(3\right)}\right)}^2+{\left(I_m{\chi }^{\left(3\right)}\right)}^2}$$E17

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.

Acknowledgments

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, 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.

Conflict of interest

The authors declare no conflict of interest.

Compliance with ethical standards

This study was approved by the School of Arts and Science, VINAYAKA MISSION RESEARCH FOUNDATION, AV campus, Chennai.