Dual-arm Z-scan measurement of the third-order nonlinearity in the silver island films

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Dual-arm Z-scan measurement of the third-order nonlinearity in the

silver island films

Danil Amosov

First name Last name Название, кол-во страниц University Of Eastern Finand

MSc Thesis May 2018

Department of Physics and Mathematics University of Eastern Finland

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Danil Amosov Dual-arm Z-scan measurement of the third- order nonlinearity in the silver island films, 38 pages

Master’s Degree Programme in Photonics

Supervisors: Prof. Yuri Svirko

Dr. Viatcheslav Vanyukov Prof. Elena Kolobkova

Abstract

In this Thesis, we studied the nonlinear optical properties of the silver island films using dual-arm z-scan measurements. Change the z-scan setting in dual- arm scan mode allows us to increase the accuracy of the measurements and obtain both nonlinear refractive index and nonlinear absorption coefficient in the same experiment. We compare refractive indices of the films annealed at a different temperature and demonstrate that the lower optical losses, the higher the optical nonlinearity of the films.

Keywords: Light and matter interaction; glass metall nanocomposite; z-scan;

nonlinear refraction; nonlinear absorption, nonlinear susceptibility, silver island film

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CHAPTER 1 Introduction

In this Thesis, we report on the investigation of the nonlinear optical properties of silver island nano-films deposited on the glass surface. The measurements of the third-order nonlinearity were performed by using a z-scan technique extended to measure the nonlinear refraction and the nonlinear absorption in one single scan. In order to avoid a contribution of the glass substrate in the measured value of the nonlinear refractive index and the nonlinear absorption coefficient of metal nanoparticles we studied the dependence of the transmittance on the glass thickness and obtained the data when the signal from the glass matrix is negligibly small compared to the nonlinear optical response arising from metal nanoparticles.

1.1 Nonlinear effects in optics

Nonlinear optics studies optical phenomena that occur due to the modifica- tion of the properties of a matter by light. Optical nonlinearity is conventionally described in terms of the dependence of the dipole moment per unit volume, or polarization P(t), on the electric field strength:

𝑃 = 𝑃⁽¹⁾+ 𝑃⁽²⁾+ 𝑃⁽³⁾ + ⋯ = 𝜀₀𝜒⁽¹⁾𝐸 + 𝜀₀𝜒⁽²⁾𝐸² + 𝜀₀𝜒⁽³⁾𝐸³+ ⋯ (1.1) where the linear optical component is described in terms of the linear susceptibility ⁽¹⁾. The second and the third terms in this series describe the second- and third-order nonlinear phenomena, respectively. Correspondingly, ⁽²⁾ and ⁽³⁾ are the second- and third-order nonlinear susceptibilities. Since the polarization and the electric field are vectors, the linear susceptibility is the second rank tensor while the second-order susceptibility is the third-rank tensor, and the third- rank susceptibility is the fourth-rank tensor and so on.

The second order nonlinear processes are prohibited in a media with inver- sion symmetry (e.g. gases). The third-order nonlinear processes can occur both for centrosymmetric and non-centrosymmetric medium.

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Since the nonlinear constitutive equation that relates to the polarization and the electric field is an expansion in power series in the electric field strength, one may introduce the dimensionless expansion parameter (E/E^at),

𝑃 = 𝑃⁽¹⁾+ 𝑃⁽²⁾+ 𝑃⁽³⁾ + ⋯ = 𝜀₀𝜒⁽¹⁾[1 + 𝐸

𝐸_𝑎𝑡+ ( 𝐸 𝐸_𝑎𝑡)

2

+ . . ] 𝐸 (1.2) where Eat is an internal field. The nonlinearity becomes important as soon as the electric field strength in the light wave becomes comparable with the internal filed.

In the case of the electronic mechanism of the optical nonlinearity, which will be considered in this Thesis, E^at= e/4πε⁰a ² where a is the Bohr radius. In hydrogen atom a = 0.5 A, i.e. E^at = 6 x 10¹¹ V/m. Therefore,

𝜒⁽²⁾≈ 𝜒⁽¹⁾

𝐸_𝑎𝑡 ≈ 10⁻¹²𝑚 𝑉 𝜒⁽³⁾≈ 𝜒⁽¹⁾

(𝐸_𝑎𝑡)² ≈ 10⁻²⁴𝑚² 𝑉²

(1.3)

In practice, the magnitude of nonlinear susceptibility depends on the origin of the nonlinear response and varies in a wide range. For example, ex- citons in semiconductors have Bohr radius up to 1000A, i.e. the third-order nonlinear susceptibility may be as high as 10^-15 m²/V².

It is worth noting that concept of the nonlinear susceptibility, i.e. expansion of the polarization in series of the electric field strength can be intro- duced only at E < E^at. In terms of the light intensity, this corresponds to the following

𝐼 < 𝐼_𝑎𝑡 =1

2𝑐𝜀₀𝐸_𝑎𝑡² ≈ 10²¹𝑊/𝑚² (1.4) The nonlinear optical susceptibilities depend on the frequencies of the light waves involved and can have real and imaginary parts. The real part of the third-order susceptibility is usually associated with propagation effects (e.g.

self-focusing), while it’s imaginary part is responsible for in intensity- dependent absorption of light waves.

When the light wave at frequency ω propagates in the medium, the electric field can be presented in the following form:

𝐸(𝑡) = 𝐸(𝜔) exp{−𝑖𝜔𝑡} + 𝐸^∗(𝜔) exp{−𝑖𝜔𝑡} , (1.5)

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the third-order nonlinear polarization at the same frequency is

𝑃⁽³⁾(𝜔) = 3𝜒⁽³⁾(𝜔 = 𝜔 + 𝜔 − 𝜔)|𝐸(𝜔)|²𝐸(𝜔) (1.6) Correspondingly, the total polarization at a frequency of the incident wave yields

𝑝(𝜔) = 𝜒⁽¹⁾𝐸(𝜔) + 3𝜒⁽³⁾(𝜔 = 𝜔 + 𝜔 − 𝜔)|𝐸(𝜔)|²𝐸(𝜔) (1.7) This allows us to describe the wave propagation in terms of the effective susceptibility 𝜒_eff = 𝜒⁽¹⁾+ 3𝜒⁽³⁾|𝐸(𝜔)|² and effective refractive index 𝑛² = 1 + 𝜒_𝑒𝑓𝑓.

1.2 Anharmonic oscillator model of the optical nonlinearity The electronic mechanism of the optical nonlinearity can be described in terms of the anharmonic oscillator model. In the framework of this model, the displacement of the bound electron from its equilibrium position in due to the oscillating electric field can be described by the following equation:

𝑑²𝑥

𝑑𝑡² + 𝜔₀²𝑥 = ( 𝑒

𝑚_𝑒) 𝐸(𝑡) + 𝛽𝑥²+ 𝛿𝑥³ (1.8) where 𝜔₀² is the eigen frequency, β and δ are anharmonicity coefficients. In order to estimate anharmonicity parameters one may expect that linear and nonlinear in x terms in the right-hand-side of equation (1.7) are comparable when the displacement of the electron from the equilibrium position is of the order of the equilibrium electron orbit radius 𝑟_𝑎, i.e. the nonlinear force βx² has a value of the same order as the linear force 𝛽𝑟_𝑎² = 𝑒𝐸_𝑎𝑡/𝑚 ≈ 𝜔₀²𝑟_𝑎. Therefore, β/δ ≈ ω⁰²/β≈ r^а, that is the anharmonic oscillator model allows us to present the polarization of the medium in series of the ratio of the incident electric field and atomic field:

|𝑝^(𝑛+1)

𝑝^(𝑛) | ≈ 𝑒|𝐸|

𝑚_𝑒𝜔₀²𝑟_𝑎 ≈ |𝐸|

|𝐸_𝑎𝑚| (1.9)

The amplitude of the intensity of the light vector of the wave should be compared with the intensity of the intra-atomic field, Eat = 6 x 10¹¹ V/m. There- fore, even for intensities of the order of 10¹⁴ W / m², which occur in the focus of a Q-switched laser, nonlinearity can be regarded as a small perturbation:

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|𝑝^(𝑛+1)|

|𝑝^(𝑛)| ≈ |𝐸|

|𝐸_𝑎𝑚|≈ 3 ∙ 10⁻³ (1.10)

It should be noted that this ratio increases at resonance, i.e. when the frequency of the external electric field is closed to the eigen frequency of the oscillator 𝜔₀. However, even small nonlinear effects can be detected due to the high sensitivity of optical indicators.

It is worth noting that oscillator models are widely used in nonlinear optics. They are effective for modelling linear and nonlinear optical response allowing one to describe the generation of higher harmonics, sum- and difference frequency generation, bistability and other phenomena [1].

1.3 The third-order susceptibility tensor

The linear or nonlinear response of the medium to the external light field is characterized by the relative magnitude of the applied electric field strength with respect to the intra-atomic fields. The intensity of the intraatomic fields E^at is ~ 10¹⁰-10¹¹ V/m. For non-laser sources, the electric filed strength does not ex- ceeds 10³ V/m, i.e. E << Eat. In this case, the response of the medium is linear.

However for ultrashort laser pulses with a large peak intensity, the E/Eat is not small and optical response of the medium becomes nonlinear.

In optical glasses (as in crystals with a center of symmetry) the second- order nonlinear susceptibility χ⁽²⁾ is zero [1,2], and the optical properties are determined by the third-order susceptibility tensor(1.11):

𝜒_𝑖𝑘(𝐸) = 𝜒_𝑖𝑘⁽¹⁾+ ∑ ∑ 𝜒_{𝑖𝑘𝑗𝑚}⁽³⁾ 𝐸_𝑗𝐸_𝑚

3

𝑚=1 3

𝑗=1

(1.11)

The third-order nonlinearity manifests itself as the appearance of a wave at the frequency ω⁴ = ω¹ ± ω² ± ω³when light waves at frequencies ω¹ ,ω² and ω³ entering the medium. In the four-frequency interaction, the components of the tensor χ⁽³⁾ depend on the frequencies of all four waves. That is why the interaction of intense light waves with the medium with third-order nonlinearity is often referred to as four-wave mixing.

Figure 1 shows possible nonlinear optical phenomena when a single intense light wave at frequency ω propagates in the medium with the third-order nonlinearity. These are third harmonic generation and frequency-degenerate

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fourwave mixing. The later manifests itself as self-action phenomena, which are studed in this Thesis.

Figure 1 – The nonlinear processes in the medium with the third-order nonlinearity. (a) The third harmonic generation, (b-c) the nonlinear refraction and the nonlinear absorption [2].

1.4 Self-action of light waves

Self-action is one of the most important classes of the nonlinear optical phenomena. Self-action is associated with the intensity-dependent change in the refractive index or the absorption coefficient of the medium when a light wave propagates in it. One of the most important self-action phenomena is self- focusing, which essentially affect the propagation of the intense light beam in various medium. In particular, the avalanche increase in the intensity of the light field in self-focusing causes in many cases an optical breakdown of the medium.

In the isotropic medium, the dependence of the refraction index and absorption coefficient on the intensity of the linearly polarized light is described by the following equations:

𝑛 = 𝑛₀ + 𝑛₂𝐼,

𝛼 = 𝛼₀+ 𝛼₂𝐼 (1.12)

where 𝑛₀ (𝑛₂) and 𝛼₀ (𝛼₂) are linear (nonlinear) refraction index and absorption coefficient, and I is the light intensity. This approximation is valid when third- order nonlinearity dominates the nonlinear response. Correspondingly, for the linearly polarized beam, 𝑛₂ and 𝛼₂ are determined by the real and imaginary parts of the third-order susceptibility 𝜒⁽³⁾ of the material:

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4𝑛₀² 𝛼₂ = 3𝜋 𝑅𝑒{𝜒⁽³⁾}𝑍₀

𝜆𝑛₀²

(1.13)

where 𝜆 is the light wavelength, 𝑍₀ = √𝜇₀/𝜀₀ = 377 Ω is the vacuum imped- ance. The values of nonlinear refraction index and nonlinear absorption coefficient are determined by the mechanism of the nonlinear response of the medium.

The character of the evolution of the light beam depends on the sign of the refractive index n2 (Figure 2). In transparent media, as a rule, n2 > 0, and n2 < 0 more often occurs for media with absorption.

Figure 2 – The change in the width of the light beam in linear (1) and nonlinear (2-4) media [3].

In a medium with n2 > 0, the region where the field amplitude is larger becomes optically denser, and it is to this region that light rays are collected.

When the threshold value of the power is exceeded, the refractive index of the nonlinear medium at the center of the beam increases, and gradually decreases to the periphery of the beam. As a result, the optical medium becomes similar to a positive gradient (nonlinear) lens and transforms the initially flat wave front of the light wave into a converging one (Figure 3) [4].

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Figure 3 – Transformation of a light wave into a convergent self-focusing.

In a linear medium, the beam cross-section increases with the distance due to diffraction (Figure 2, curve 1). When a light beam with a Gaussian amplitude profile [3].

𝐸(𝑟) = 𝐸₀𝑒𝑥𝑝 (−𝑟²

𝑎²) (1.14)

experiences self-focusing, the peripheral rays deviate to the beam axis and con- verge at a nonlinear focus (Figure 2, curve 4). That is a spontaneous compres- sion of an aperture-limited beam of light in a medium with a positive nonlinear refractive index [5] takes place. Self-focusing may be accompanied by an increase in the radiation power density in the cross-section and the formation of filamentary waveguide channels in such a medium. In very special conditions, self-focusing may compensate diffraction allowing the beam propagating without distortion for a long distance (soliton propagation, curve 3 in Figure 2).

In self-focusing regime, the beam width is described by the formula (1.15):

𝑎(𝑧) = 𝑎₀√(1 −𝑧

𝑅)² + (1 − 𝑊 𝑊_𝑘𝑝)𝑧²

𝑙_𝑑² (1.15)

where R is the radius of curvature of the spherical wave front, d = 2πa0/λ is the diffraction length of the beam. Mechanism of the self-focusing is a change in the intensity-dependent refractive index, which is determined by the distribution of the intensity of the light wave. As a result, the peripheral rays of the beam tend to a zone with a greater optical density. It is worth noting that self-focusing may compete with diffraction divergence. When these processes compensate each other, the self-channeling and soliton propagation may occurs [3]

In the case of n²< 0, the optical medium plays the role of a negative (di- vergent) lens, and the light beam defocuses (Figure 2, curve 2). The defocusing is often observed due to the thermal mechanism of the optical nonlinearity.

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Specifically, when an intense beam propagates in a slightly absorbing medium, the medium temperature increases, Δ𝑇 ∝ 𝐼, and therefore the refractive index changes proportionally to ^𝜕𝑛_𝜕𝑇𝐼, i.e. 𝑛₂ ∝^𝜕𝑛_𝜕𝑇. Since usually ^𝜕𝑛_𝜕𝑇 < 0, thermal nonlinearity usually leads to defocusing.

1.5 The effect of self-induced refraction

In the plane wave approximation for the cubic nonlinear material χ^{(2 )}= 0 and χ⁽³⁾≠0, the nonlinear polarization excited by the monochromatic wave 𝐸 = 𝐸₀cos (𝜔𝑡 − 𝑘𝑧) can be written in the scalar form (1.16):

𝑃 = 𝜀₀(𝜒⁽¹⁾𝐸 + 𝜒⁽³⁾𝐸³)

= 𝜀₀(𝜒⁽¹⁾+ 3 4⁄ 𝜒⁽³⁾𝐸₀²)𝐸₀cos(𝜔𝑡 − 𝑘𝑧) +1

4𝜒⁽³⁾𝐸₀³cos(3(𝜔𝑡 − 𝑘𝑧)) = 𝑃^𝜔 + 𝑃^3𝜔

(1.16)

The amplitude of the polarization at frequency ω is P(ω) = (χ⁽¹⁾+3/4 χ⁽³⁾E⁰²)E⁰. The presence of the 3/4χ⁽³⁾E⁰²additive to the coefficient of linear susceptibility χ⁽¹⁾ shows that the refractive index in a nonlinear medium is not constant, and depends on the amplitude of the light wave field. Dielectric permea- bility in a cubic-nonlinear mediumcan be described as follows (1.17)

𝜀_{𝑛𝑜𝑛𝑙𝑖𝑛} = 1 + 𝜒⁽¹⁾+3

4𝜒⁽³⁾𝐸₀² = 𝜀 + 𝜀₂𝐸₀² (1.17) composed of the linear permittivity ɛ = 1 + χ⁽¹⁾ and a nonlinear addition proportional to the square of the amplitude, with ɛ2 = 3χ⁽³⁾/4. One can notice, that the linear refractive index is related to the real part of ɛ, and the linear (single- photon) absorption coefficient is related to the imaginary part of ɛ. Similarly, the real part of the quantity ɛ2×E20 in (1.17) is related to the nonlinear refractive index n2, and the imaginary part to the nonlinear absorption coefficient α2 [6].

Under the condition of ɛ2 << ɛ, the refractive index in a nonlinear medium can be expressed as follows (1.18):

𝑛_{𝑛𝑜𝑛𝑙𝑖𝑛} = √𝜀𝑛𝑜𝑛𝑙𝑖𝑛 = 𝑛₀+ 3

8𝑛₀𝜒⁽³⁾𝐸₀² (1.18)

where n0 = √𝜀 is the linear refractive index. Since the intensity of the light beam is related to the electric field strength by the relation (1.19)

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2𝑍₀ (1.19)

the expression (1.18) can be written in the following form (1.20):

𝑛_{𝑛𝑜𝑛𝑙𝑖𝑛} = 𝑛₀+ 𝑛₂𝐼 (1.20)

where

𝑛₂ = 3𝑍₀

4𝑛₀²𝜒⁽³⁾ (1.21)

Thus, the propagation of an intense light wave in a nonlinear cubic medium leads to a change in the refractive index proportional to the square of the wave amplitude or the intensity of the laser radiation. The change in the refractive index of the medium under the action of the radiation propagating in it is called the self-induced refraction effect, which can be described in terms of the slowly varying amplitude approach [3].

In the framework of this approach we assume that the complex amplitude of the propagating wave slowly changes in the medium:

𝐸(𝑡, 𝑧) = 𝐴(𝑧) exp(−𝑖𝜔𝑡 + 𝑖𝑘𝑧) + 𝑐. 𝑐. (1.22) A(z) satisfies the following equations [4]:

𝑑𝐴

𝑑𝑧 = 𝑖𝜔²

2𝑘𝑐²𝑛₂|𝐴|²𝐴 (1.23)

Since all frequencies are the same, the phase-matching condition is automatically fulfilled. Therefore, the effects associated with self-action always accumulate when the beam propagates. By presenting the complex amplitude in the form

𝐴₁ = 𝑎₁𝑒^𝑖𝜑¹ (1.24)

where a1 and φ1 are real amplitude and phase, which satisfy the following equations:

𝑑𝑎₁

𝑑𝑧 = 0 (1.25)

𝑑𝜑₁

𝑑𝑧 = 𝜔² 2𝑘𝑐²𝑛₂𝑎₁²

(1.26)

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Equation (1.26) describes a nonlinear phase shift arising when a plane light wave propagates in a cubic-nonlinear medium. The self-focusing phenom- enon is a special case of self-induced refraction and is due to the fact that the refractive index of the medium changes in a strong light field of a laser beam bounded in space. If the sign of the nonlinear change in the refractive index is such that it increases in the region occupied by the beam, this region becomes optically denser, and the peripheral rays deviate toward the center of the beam.

Figure 4 shows the phase fronts and the path of the rays in a light beam propagating in a medium with a nonlinear refractive index: n = n0 + n2I, n2 > 0. In the figure, the arrows show the path of the rays, the dashed lines the surface of the constant phase, the solid bold line the beam intensity profile.

Figure 4 – The phase fronts and ray paths in a light beam with self- focusing [3].

Optical glasses have small values of n² for typical compositions of multi- component glasses and fused quartz n² = 10^-16 cm²/W (Figure 5). In particular, for fused quartz (SiO²), n²= 1.6×10^-16 cm²/W. However, there are optical glasses with significantly higher third-order nonlinearity, for which the values of n² are 2-3 orders of magnitude larger than those of fused quartz. These include, for example, chalcogenide glass, known for its variety of composition. Figure 5 shows data for sulfide (As-S) and selenide (As-Se) chalcogenide glasses.

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Figure 5 – The refractive index diagram of optical glasses.

In order to obtain changes in the refractive index measured in the experiment, a high intensity of laser radiation is required. Thus, at an intensity of 1 GW / cm²in quartz glass, the change in the refractive index n²I ≈ 1.6×10^-7 and in chalcogenide glass based on arsenic selenide ~ 10^-4. However, it is worth noting that the breakdown intensity depends on the composition of the glass. In the continuous mode of laser generation, the intensities of hundreds of kW/cm² already lead to the destruction of the material. As is known from [5], the material destruction threshold increases with decreasing pulse duration. Therefore, to observe the nonlinear effects it is necessary to use a pulsed radiation.

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CHAPTER 2 Theory of the Z-scan method

Z-scan technique described in details by Sheikh-Bahae et.al in [7], is well known and is used to study the nonlinear optical properties of various materials including solids and dispersions. The method makes it possible not only to determine the magnitude and the sign of the nonlinear refractive index n2, but also the two-photon absorption coefficient, which are essential parameters of any material.

The z-scan method is based on the self-focusing effect of a converging Gaussian beam in a sample with nonlinear optical properties. The method for determining the nonlinear optical constants of the media is based on moving the sample along the optical axis alongside the focal point of the lens used to form a convergent Gaussian beam and measuring the energy of the light passing through the sample.

In the simplified schematic diagram of z-scan technique shown in Fig- ure 6, a linearly polarized Gaussian beam propagating in z-direction is focused by a focusing lens L. The sample S (can be solid and suspended) moves along the z-direction, and the intensity of the transmitted radiation is recorded at the terminal aperture of the photodetector D2 in the far field, as a function of the sample position relative to the focal plane z = 0. The reference detector D1 is used to avoid the fluctuations in the incident light energy.

Figure 6 – Schematic diagram of z-scan technique: D¹, D² - photodetec- tors, BS – beam splitter, L – focusing lens, S-sample.

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In the absence of nonlinear light scattering [8], the open aperture z-scan technique allows measuring the two-photon absorption coefficient of the media.

Figure 7 a shows a schematic diagram of the z-scan technique with an open aperture (aperture in front of the detector D² is removed) configuration. When the sample is moved along the focused light beam along the z-axis (from –z to +z), the transmitted through the sample light energy is measured as a function of the sample position relative to the focal point of the focused laser beam (Figure 7 a). For the Gaussian beam and in well-aligned experimental conditions this dependence is symmetrical with respect to z = 0 (Figure 7 b). Depending on the nonlinear mechanism initiating the nonlinear extinction of the incident light energy, the transmittance curve at z = 0 can show minimum (as in Figure 7 b), or maximum (in case of saturable absorption [9]). In Figure 7 b Tref is the transmission of the sample at a low incident light intensity (in the absence of nonlinear effects), and Tdet is the transmission of the sample at a high intensity of the incident radiation.

Figure 7 – a – schematic diagram of open aperture z-scan configuration; b – the sample transmission measured as a function of the sample position relative to the focal point z = 0.

In order to determine the nonlinear refractive index of the media, the closed aperture z-scan technique is used in the absence of nonlinear light scattering. Since the spot size in the plane of the detector D2 changes due to the self- focusing effect, the placement of the aperture in front of the detector D2 provides the measurement of the magnitude and the sign of the nonlinear refractive index. Figures 8 a, b show the difference in the propagation of the light beam for the two different sample positions relative to z = 0 (z > 0 (a), and z < 0 (b)) in a closed aperture z-scan for the media with a positive nonlinear refractive index (n²> 0). This difference in a light propagation results in the corresponding transmittance curve shown in Figure 8 c. In the case where the sample with a

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positive nonlinear refractive index n² is far from the focus of the lens, the intensity of the radiation passing through the sample is small and, as the thickness of the sample is not great, the transmission varies little as it moves. As soon as the sample is closer to the focus, the intensity in the beam becomes sufficient for the appearance of the self-focusing (n²> 0) or the self-defocusing (n²< 0) effects in the sample.

Figure 8 – a and b are the schematic diagrams of closed-aperture z-scan at different positions of the sample relative to the focus of the lens; c- the sample transmission as a function of the sample position for the sample with a positive nonlinear refractive index n2 > 0.

The beam power transmitted through the aperture with radius r can be calculated by integrating the field strength E(r, t) a within the aperture (2.1):

𝑃_𝑟(∆Ф₀) = 𝑐𝜀₀𝑛₀𝜋 ∫ |𝐸^𝑟 _{𝑎(𝑟,𝑡)}|²𝑟𝑑𝑟

0 (2.1)

where c is the speed of light in vacuum, ɛ0 is the electric constant, and n⁰ is the refractive index.

The phase change due to the self-action of the beam is determined as follows (2.2):

∆Ф₀(𝑡) = 𝑘∆𝑛₀(𝑡)𝐿_𝑒𝑓𝑓 (2.2)

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where 𝐿_𝑒𝑓𝑓 = ( 1 − 𝑒𝑥𝑝^{(−𝛼𝐿)})/𝛼 , L is the length of the sample, α coefficient of linear absorption. ∆𝑛₀(𝑡) = 𝑛₂𝐼(𝑡, 𝑟 = 0) is the change of the refractive index.

The transmission of the aperture is determined by the formula (2.3):

𝑇(𝑧) = ∫ 𝑃_−∞^∞ _𝑇(∆Ф₀(𝑡)𝑑𝑡

𝑆 ∫ 𝑃_−∞^∞ _𝐼(𝑡)𝑑𝑡 (2.3)

where PI is the initial beam power incident on the sample, S = 1 − exp (−2 𝑟² ⁄ 𝑤² ) is the linear transmission of the aperture, 𝑤 is the beam radius at the aperture in the linear propagation mode (in the absence of nonlinearity), r is the aperture radius. In the open aperture configuration (S = 1), transmission of the pulses with a Gaussian envelope is described by the following equation:

𝑇(𝑧, 𝑆 = 1) = 1

√𝜋𝑞0(𝑧, 0)∫ 𝐼𝑛[𝐼 + 𝑞^∞ ₀(𝑧, 0)𝑒^−𝑡²]

−∞

𝑑𝑡 (2.4)

where, 𝑞₀(𝑧, 𝑡) = 𝛽𝐼₀(𝑡)𝐿_𝑒𝑓𝑓/(1 +^𝑧_𝑧²

02) , z0 is the diffraction length of a Gaussian beam, β is the two-photon absorption coefficient, I⁰(t) = I(t, r = 0) is the instanta- neous intensity of the pulse at the beam axis.

If |q0|< 1, the transmission can be represented as a series (2.5):

𝑇(𝑧, 𝑆 = 1) = ∑ [−𝑞₀(𝑧, 0)]^𝑚 (𝑚 + 1)³^⁄²

∞

𝑚=0

(2.5)

If in this expression only the first two terms are taken, then in the far zone one can obtain a simplified expression for T with an open diaphragm (2.6):

𝑇_{𝑜𝑝𝑒𝑛}(𝑧) = 1 − 𝛽𝐼₀𝐿

2√2(1 + 𝑥²) (2.6)

For a closed aperture, one can obtain similar expression:

𝑇_{𝑐𝑙𝑜𝑠𝑒𝑑}(𝑧) ≅ 1 +4𝑥(∆𝜑₀)(1 − 𝑆)^0.25

(1 + 𝑥²)(9 + 𝑥²) (2.7) where x = z/z⁰, and z⁰ is the Rayleigh length.

The incident beam induced changes in the refraction index and/or absorption coefficient in the irradiated area of the sample leading to a phase shift, which is proportional to the intensity. Shifting the sample with respect to focal point changes the intensity distribution across the beam resulting in the wave

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front distortion. As a result in the far zone, the amount of energy carried by the central part of the beam changes depending on the sample position. A typical dependence of the transmittance as a function z is presented in Figure 9. In the thin sample approximation, for Gaussian beam the energy difference ∆T between the peak and the valley of the transmittance curve is given by the following equation [10]:

∆T = 0.406(1-S)^0.25|∆Ф^о| (2.8) where ∆T is the transmittance energy difference between the peak and the valley, and ∆Ф⁰ = 2πLn²I/λ is the nonlinear phase shift at z = 0.

Figure 9 – A typical result of a thin sample in z-scan measurement in the closed aperture configuration [11].

Figure 10 – The example of the graph obtained in the z-scan experiment, where solid line represent material with a positive n2 and the dashed line represent material with negative n2 [12].

If the material has a positive refractive index, a valley is situated at z < 0 and pick at z > 0 (solid line in Figure 10), while for material with negative re-

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fractive index starts with a peak followed by a valley (dotted line in Figure 10).

The peak represents the position where the self-focusing of the beam occurs, i.e.

sample causes the beam to collimate and increase the light energy in the vicinity of the beam axis. The valley represents the defocusing when the sample causes the beam to diverge, reducing the energy in the central part of the beam [12].

It is worth noting that if the absorption coefficient of the medium does not depend on intensity, the total transmitted energy remains the same. This allows one to measure nonlinear absorption coefficient by performing open aperture z-scan measurements when all transmitted photons are collected by a detector. Combining open and closed aperture z-scan measurements can be used to obtain both nonlinear refraction index and nonlinear absorption coefficient. The open aperture is used to determine the nonlinear absorption, while the close aperture scan is used to determine the nonlinear refraction [13].

This technique can be used to determine the optical nonlinearity of different materials such as organic, carbon, glasses and dielectrics with a response time of the nonlinearity spanning from milliseconds to femtoseconds. However to obtain accurate and reliable results from this technique one needs to know the thickness of the sample, the distance from the sample to the aperture, the power of the laser beam, the wave front distortions produced by the lens and well as many others. One way to minimize the uncertainty effects is to use a reference sample with known nonlinearity, however finding such a sample is not a simple task. Also, if the z- scan measurements are used to determine both the nonlinear refraction index and absorption coefficient, their ratio should be the same in the material under study and in the reference sample [14].

One of the advantages of using the z-scan is that it’s only required a single laser beam and that the nonlinearity in refraction and absorption can be determined with a minimum amount of data analysis compared with others methods. As well as by using this technique we can gain more information not only about the nonlinearity of the material but also information about how to maximize the geometry of the optical power, for example, it can identify the ultimate sample position and thickness that allow the best results in the optical limiting applications [14].

It is important to note that in most cases, nonlinear refraction does not occur by itself, but usually in combination with nonlinear absorption. This means that the data from the Z-scan will contain both nonlinear refraction and nonlinear absorption. To extract a nonlinear refractive index, one needs to per-

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form a closed aperture z-scan to measure the overall transmittance of the sample [13]. The measured transmission coefficient is then independent of the nonlinear refraction and depends only on the nonlinear absorption. In the following paragraphs, it will be shown that data from such a z-scan with a remote aperture, during construction, forms a valley symmetrical around the focus. This z- scan of the open diaphragm is used to determine the nonlinear absorption coefficient. A nonlinear refractive index can be obtained by dividing the data obtained from the z-scan using an aperture in situ with the data obtained from the open-aperture z-scan [14]. The data of these two scans, as well as the result of the division, can be seen in Figure 11.

Figure 11 – A) a closed aperture z-scan, B) an open aperture z-scan, and C) the result of dividing the closed aperture z-scan by the open aperture z-scan.

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CHAPTER 3 Z-scan measurements with silver island films

3.1 Fabrication of the silver island films

Figure 12 shows schematically the process of film deposition and the growth of the structure of the nano-island silver film.

Figure 12 - Steps for the preparation of nanoisland films [15].

The soda-lime glass is immersed in the melt AgxNa1-x NO3 (x = 0,01-0,15) in which ion exchange occurs between the glass and the solution and a nanostructural structure is formed on the surface of the glass. The melt temperature is 325°C, the dipping time varies from 5 minutes to an hour. Further, (II) after glass annealing (the heat treatment temperature can also be varied) in a hydrogen atmosphere the sample is aged at a temperature of 75-350°C, from 30

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seconds to 3 hours. During this the silver film on the nano system is increased.

The next step is figure 13 (III) is thermal polishing of ion-exchange glass using a profiled anode at a temperature of 200-350°C, a constant current of 500-600V, and a time of 30 seconds to 30 minutes. This stage allows regulating the distribution of silver ions on the surface. It is shown in Figure 13 (IV) that after the annealing of the polished ion-exchangeable glass, the silver nanostructures of the structure can grow only where the anode electrode did not contact the surface of the sample during polishing. At the same time, hydrogen that penetrat- ed the glass makes silver ions less; so silver forms nanoparticles, in the process of diffusion and self-organization, both on the glass surface and in the volume of the glass.

Annealing of the ion exchange glasses was carried out in a reducing hydrogen medium at 200-400°C for duration of 10 minutes. During this stage, hydrogen (which is formed from the decomposition of water vapor on the surface of the glass) diffuses into the near-surface region of glass samples and repeated- ly forces silver ions of neutral atoms to replace silver in bonds with unreceptive oxygen (3.1):

Si − O − Ag⁺+1

2H₂ →≡ Si − O − H⁺+ Ag⁰ (3.1) A decrease in temperature leads to the formation of the glass matrix of a solid solution of neutral silver, which, due to the low solubility of atomic silver in the glass, tends to decompose. This decomposition promotes the formation of silver clusters and nanoparticles on the surface of glass samples, namely self- assembled nanolayer films, which are a strong absorber for silver atoms. These nanolayer s films demonstrated applicability in surface-enhanced Raman scattering [15].

3.2 Optical characterization of the fabricated samples

In the nonlinear optical experiments, we studied silver island films embedded in the glass matrix. We studied the samples with different heat treatment temperature. For all the samples the heat treatment time was 10 minutes. The thickness of the sublayer is 0.5 mm, such a small thickness is necessary in order to minimize measurement errors, to maximally remove the effects from the substrate. Figures 14 and 15 show the photos of the samples. As one can see the first two samples (Figure 13) at a temperature of 200 and 250°C

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are almost transparent whilst the second pair of 300 and 450°C is much darker (Figure 14).

Figure 13 – Glass metal nanocomposite samples fabricated at a temperature of 200 and 250°C.

Figure 14 – Glass metal nanocomposite samples fabricated at a temperature of 300 and 400°C.

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Figure 15 – The absorption spectra of studied silver nanoisland films.

Figure 15 shows the absorption spectra of silver island films embedded in a glass. One can see that all the samples have an absorption peak at 400 nm.

It can be noted that heat-treated samples at 300 and 400°C have significantly higher linear absorption. At a wavelength of 800 nm at which z-scan was performed, linear absorption of samples treated at 200 and 250°C is practically ab- sent.

3.3 Dual-Arm Z-Scan measurements

Dual arm z-scan technique we used in the experiments allows one to measure samples in several times faster with an increased accuracy. In this technique, measurements of nonlinear absorption and a nonlinear index of refraction occur at the same time and are measured at the same point. The installation on a conventional z-scan to a dual-arm z-scan was performed in the course of this studies. The modified setup allows one to get the division of the closed and open-aperture z-scan data obtained in the course of a single scan.

The division data have been automatically recorded on a PC providing a con- venient way immediately to analyze the obtained data and making the needed alignment in experimental geometry.

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3.4 Experimental setup

To create a dual arm scheme, a beam splitter was required, which would share the laser beam after passing through the sample. This can not be a classic 50/50 beam splitter. This is because there are internal reflections in it and they degrade the quality of the measurement because of interference [16]. For the first working scheme, a prism of 10 mm thick was chosen as the beam splitter (Figure 16.)

Figure 16 - The photo of the experimental setup showing the first iteration of a beamsplitter used to divide the beam one for closed-aperture arm and the second one for the open-aperture arm.

In this setup, the laser beams were limited by the thickness of the prism, the internal reflections in the prism itself did not give a good result in measurements with low energy. In order to overcome this, we used as the beam splitter a normal glass with a thickness of 15 mm. This allowed us to take the inner reflected beam as well as to choose the incident beam diameter without limitations. This also allowed to improve the quality of measurements, as well as to simplify the calibration of the installation before use (Figure17).

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Figure 17 – The final iteration of a glass beam splitter used for the measurements.

For the measurements of the energy of the incident and transmitted light pulses we used ThorLabs detectors shown in Figure 18.

Figure 18 – ThorLabs photodetector used in the experiments that prior to the measurements was checked on the linearity of the signal detection.

In the experiments, we employed an Integra-C (Figure 19) compact and versatile amplifier to generate femtosecond laser pulses at the wavelength of 800nm. The device includes a built-in laser pump, gain (s) and a subframe/compressor that amplifies the energy of the initial pulse to several mJ units. Seeds can be provided with a fiber-optic generator or an external oscillator, which are built into the box. The amplifier system can be tuned to deliver

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picosecond (1-3 ps), femtosecond (80-130 fs), or ultra-short (< 40 fs) output pulses. In our experiments the pulse duration was 100 fs. For the most accurate experiments, the laser provides constant performance. A good contrast both before and after pulses are provided by the construction of Pockel cells, which eliminates the error in the results of the experiment. Active thermal stabilization provides stable working energy in a wide temperature range, facilitating con- trol of the laser installation.

Figure 19 – Quantronix Integra-C laser system providing 800nm femtosecond laser pulses employed in the experiments.

The built dual-arm z-scan experimental setup is shown in Figure 20.

Polarizers were used to smoothly adjust the radiation incident on the sample.

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Figure 20 – Dual-arm z-scan setup employed in the measurements of the nonlinear refractive and nonlinear absorption coefficients of silver nanoisland films.

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CHAPTER 4 Results and Discussion

4.1 Measurement results

For the trial measurements, the samples with the silver films on a 1,5 mm thick glass matrix were taken. During the test, it was found that the thickness of the substrate (glass) of 1,5 mm is too large and its contribution to the measured response is too great. Specifically, it was difficult to evaluate whether the silver film had nonlinear properties, or not from the results obtained with a 1.5mm glass. In order to reduce the contribution arising from the glass itself, we changed the glass matrix to the one with a thickness of 0.5 mm. This made it possible to significantly reduce the influence of the substrate on the data obtained.

Since we used the samples with the different processing temperature, the density of silver particles was different, and accordingly, the transmission coefficient was also different. In the course of the measurements, it was found that only the samples processed at 200°C and 250°C were suitable for the z-scan experiments. The other two samples had very low transmittance and very high sensitivity to laser radiation. Even with the lowest energy that was achieved at the experimental conditions, the samples received irreversible damage.

Therefore in the measurements of nonlinear refraction and nonlinear absorption the samples of heat-treated at 200 and 250°C were used, and the series of measurements at different laser radiation energies were made.

At first the initial energy, at which no nonlinear effects were observed in the reference sample (glass without coating), was found that is the scanning dependence was linear at low incident light intensities. This energy has been found at the range of 1nJ. Secondly at the same energy a series of scans of samples with silver films were carried out. The results are shown in Figure 21.

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Figure 21 – The dependence of the normalized transmittance on the sample position z at the energy of the incident pulse of 1,1nJ.

As one can see from the graph, the samples with silver coating exhibit non-linear effect when the sample passes through a focal point. This can tell us that the coated samples exhibit a nonlinear optical effect.

In order to determine how much the coating affects the nonlinear optical properties, further experiments with an increase in the incident on the sample energy were made. Thus it was possible to find out how strong the properties can have glass coated with a silver film in comparison with ordinary glass. The energy increased from 1nJ up to 1.7nJ. The results are shown in Figures 22-24.

At incident pulse energy of 1,3nJ (Figure 22) one can see that the sample curves are very similar, both in shape and in peak values. Accordingly, for this simulation, the samples have similar nonlinear characteristics.

By increasing the energy to 1.5nJ, the nonlinear effects in the samples were amplified. The difference between samples with silver film and glass becomes more pronounced. The below graph (Figure 23) is clearly demonstrating the effect of the silver coating.

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Figure 22 – The dependence of the normalized transmittance on the sample position z at the energy of the incident pulse of 1,3nJ.

Figure 23 – Dependence of the normalized transmittance on the sample position z at the energy of the incident pulse 1,5nJ.

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Then the energy was increased further. The largest value of energy, effective for obtaining nonlinear properties, was 1.7nJ (Figure 24). It should be noted that the curves for the sample 250°C are not symmetrical, and it can be assumed that this energy is too high for a given coating, and the sample does not with- stand such energy.

Figure 24 – Dependence of the normalized transmittance on the sample position z at the energy of the incident pulse 1,7nJ.

A further increase in energy does not give a strong effect. The coating gets irreversible damage in the experiments and the nonlinear effect in the glass itself (substrate) become comparable with that observed on the samples.

4.2 Discussion

From the measured dependence of the normalized transmission on the sample position (Figures 21-24), one can obtain the nonlinear refractive index of the samples using theory presented in Chapter 3. In order to calculate the nonlinearity of the silver island film, the measurements of the peak-valley differences (maximum and minimum) and characteristics of the laser beam are required [17].

From the obtained data, it can be seen that 200°C and 250°C samples are slightly different in their nonlinear properties. It should be noted that at low

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energies the effect on sample 250°C is slightly higher, and with increasing energy, the 200°C sample becomes more optically active. The remaining samples with a higher processing temperature have a significantly higher absorption index and a significantly greater sensitivity to light. Nonlinear properties on them were not detected since even with the minimum possible energy received at the installation, the samples were damaged. This property of high absorption of these samples can be used and useful for some other purpose.

To calculate the refractive index, we used the formulas obtained earlier from the equations [18,19]:

∆𝑇 = 0.408(1 − 𝑆)^0.267|∆Ф₀|

∆Ф₀ = (2𝜋 𝜆⁄ )𝑛₂𝐼𝐿_𝑒𝑓𝑓

(4.1) (4.2) The results of the calculations can be seen in Table (1)

Sample annealing temperature

Laser pulse Energy

1,1nJ 1.3nJ 1.5nJ 1.7nJ

200°C 3.37×10^-12 2.47×10^-12 3.4×10^-12 3.39×10^-12 250°C 2.82×10^-12 1.93×10^-12 2.14×10^-12 1.7×10^-12 Table 1. – Nonlinear refractive index of the silver island films measured in the experiment.

For the 200°C sample, the averaged refractive index was defined as (3,16

± 0.46)×10^-12, and for the at250°C sample it was (2,15 ± 0.56)×10^-12. Such a result can be explained by the fact that transmittance of the sample fabricated at 250°C is lower than that in the sample fabricated at 200°C. The higher the temperature, the thicker the film and the bigger the nano-islands. Therefore, one may expect that the number of gaps between nano-islands in the irradiated area of the film is smaller. Since the enhancement of the local electric field is associated with these gaps, reduction of the gaps number results in the lower local filed and hence the optical nonlinearity of the film. It is worth noting that the increase of the optical losses results in the lower intensity of the transmitted light, i.e. to the decrease of the signal-to-noise ratio. As a result, the experimental error for the measurements with sample fabricated at 250°C is bigger.

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CHAPTER 5 Conclusions

In this work, we studied nonlinear optical properties of silver nanoisland films. In order to take into account both nonlinear absorption and nonlinear transmission, we developed the dual-arm z-scan setup, which allows one to perform simultaneous measurements in the open and closed z-scan configura- tions. We demonstrated that nonlinear refraction coefficients of the silver island films fabricated at temperatures of 200°C and 250°C are comparable, however, the thinner film possesses the stronger nonlinearity. Specifically, nonlinear refractive indices for these samples calculated for these samples are (3.16 ± 0.46)10^-12 for 200°C film and (2.15 ± 0.56)10^-12 for 250°C film. Such differences as a result between samples can be a consequence of the fact that the sample has a different thickness and density of the nanoislands on the glass surface and, therefore, the different number of the inter-island gaps, which are associated with the enhanced local field. It is worth noting that samples fabricated at higher annealing temperatures have been found to be unsuitable for the for z-scan measurements because of high optical losses

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REFERENCES

1. Chapple P.B., Staromlynska J., Hermann J.A., et al. J. Nonlinear Opt.

Phys. Mater., 251 (1997).

2. Boyd, R.W. ”Nonlinear Optics”, Academic Press, London. (1992) 3. Sukhorukov A.P. “Diffraction of light beams in nonlinear media” // So- rosovsky educational journal 1996. No. 5. P. 85-92.

4. Bahaa E. A. Saleh, Malvin Carl Teich, Fundamentals of Photonics (John Wiley, New York, 1991).

5. A. Zakery, S.R. Elliott. Optical Nonlinearities of chalcogenide glasses and their Appications. Springer-Verlag Berlin Heidelberg, 2007. 201 p.

6. Vivien L., Riehl D., Lançon P., Hache F., Anglaret E. Pulse duration and wavelength effects on the optical limiting behavior of carbon nanotube suspensions // Optics Letters. 2001. V. 26, № 4. P. 223-225.

7. Sheik-Bahae M., Hutchings D. C., Hagan D. J., Van Stryland E. W. Dis- persion of Bound Electronic Nonlinear Refraction in Solids // IEEE Journal of Quantum Electronics 1991.V.27. No. 6. P. 1296-1309.

8.Viatcheslav Vanyukov " Effects of Nonlinear Light Scattering on Opti- cal Limiting in Nano carbon Suspensions", Publications of University of eastern Finland , Dissertation in Frosty and national Science Number 182, 2015.

9. Viatcheslav Vanyukov, Gennady Mikheev, Tatyana Mogileva, Alexey Puzyr, Vladimir Bondar, Dmitry Lyashenko, Andrey Chuvilin, "Saturable absorption in detonation nanodiamond dispersions," Journal of Nanophotonics 11(3), 2017

10. Josset S., Muller O., Schmidlin L., Pichot V., Spitzer D. Nonlinear optical properties of detonation nanodiamond in the near infrared: Effects of con- centration and size distribution // Diam. Relat. Mater. 2013. V. 32. P. 66-71.

11. Olivier T, Billard F and Akhouayri H. (2004). Nanosecond Z-scan measurements of the nonlinear refractive index of fused silica, Optics Express, Vol. 12, No. 7, pp. 1377-1382.

12 M. Sheik-Bahae, A.A. Said, T.-H. Wei, D.J. Hagan, E.W.Van Stryland, IEEE J. Quantum Electron. 1990 760.

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36

13. Sheik-Bahae M., Said A.A., Wei T.-H., Hagan D.J., Stryland E.W. Sen- sitive measurement of optical nonlinearities using a single beam // IEEE J.

Quantum Electron. 1990. V. 26, № 4. P. 760-769.

14. M. Sheik-Bahae, A.A. Said, E.W. Van Stryland, Opt. Lett.14 1995 955.

15. S. Chervinskii, I. Reduto, A. Kamenskii, I. S. Mukhin, and A. A.

Lipovskii, "2D-patterning of self-assembled silver nanoisland films," Faraday Discussions, 10.1039/C5FD00129C vol. 186, no. 0, pp. 107-121, 2016.

16. R. Y. Krivenkov, G. M. Mikheev, R. G. Zonov, T. N. Mogileva, V. M.

Styapshin, “Automated complex for the investigation of the optical limiting and nonlinear light scattering by the z-scan technique”, CPM, 17:3 (2015), 471–481

17. Rhee B.K., Byun J.S. and Van Stryland E.W. (1996). Z-scan using cir- cularly symmetric beams, Journal of the Optical Society of America, Vol. 13, No.

12, pp. 2720-2723.

18. E.W. Van Stryland, S.-B. Mansoor, "Z-scan measurements of optical nonlinearities," in Characterization Techniques and Tabulations for Organic Nonlinear Materials (1998), pp. 655–692.

19 Stryland E.W., Sheik-Bahae M. Z-scan measurements of optical nonlinearities // Charact. Tech. Tabul. Org. Nonlinear Mater. 1998. № 3. P. 655-692.

Dual-arm Z-scan measurement of the third-order nonlinearity in the silver island films