1 Introduction
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 sus-ceptibility χ(1) shows that the refractive index in a nonlinear medium is not con-stant, and depends on the amplitude of the light wave field. Dielectric permea-bility in a cubic-nonlinear mediumcan be described as follows (1.17)
𝜀𝑛𝑜𝑛𝑙𝑖𝑛 = 1 + 𝜒(1)+3
4𝜒(3)𝐸02 = 𝜀 + 𝜀2𝐸02 (1.17) composed of the linear permittivity ɛ = 1 + χ(1) and a nonlinear addition propor-tional 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):
𝑛𝑛𝑜𝑛𝑙𝑖𝑛 = √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)
11 𝐼 =𝑛0𝐸02
2𝑍0 (1.19)
the expression (1.18) can be written in the following form (1.20):
𝑛𝑛𝑜𝑛𝑙𝑖𝑛 = 𝑛0+ 𝑛2𝐼 (1.20)
where
𝑛2 = 3𝑍0
4𝑛02𝜒(3) (1.21)
Thus, the propagation of an intense light wave in a nonlinear cubic me-dium 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 refrac-tive 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 ampli-tude 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 au-tomatically fulfilled. Therefore, the effects associated with self-action always accumulate when the beam propagates. By presenting the complex amplitude in the form
𝐴1 = 𝑎1𝑒𝑖𝜑1 (1.24)
where a1 and φ1 are real amplitude and phase, which satisfy the following equa-tions:
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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 propa-gating 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 exper-iment, a high intensity of laser radiation is required. Thus, at an intensity of 1 GW / cm2 in 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 not-ing 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 mate-rial 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 materi-als 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 pass-ing 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 ap-erture (apap-erture 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 transmis-sion 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 inci-dent 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 scat-tering. 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 pro-vides the measurement of the magnitude and the sign of the nonlinear refrac-tive 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 inten-sity 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 sam-ple transmission as a function of the samsam-ple position for the samsam-ple with a posi-tive nonlinear refracposi-tive 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):
𝑃𝑟(∆Ф0) = 𝑐𝜀0𝑛0𝜋 ∫ |𝐸𝑟 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 fol-lows (2.2):
∆Ф0(𝑡) = 𝑘∆𝑛0(𝑡)𝐿𝑒𝑓𝑓 (2.2)
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where 𝐿𝑒𝑓𝑓 = ( 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 aper-ture in the linear propagation mode (in the absence of nonlinearity), r is the ap-erture radius. In the open apap-erture configuration (S = 1), transmission of the pulses with a Gaussian envelope is described by the following equation:
𝑇(𝑧, 𝑆 = 1) = 1 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): zone one can obtain a simplified expression for T with an open diaphragm (2.6):
𝑇𝑜𝑝𝑒𝑛(𝑧) = 1 − 𝛽𝐼0𝐿
2√2(1 + 𝑥2) (2.6)
For a closed aperture, one can obtain similar expression:
𝑇𝑐𝑙𝑜𝑠𝑒𝑑(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 ab-sorption 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 be-tween the peak and the valley of the transmittance curve is given by the follow-ing equation [10]:
∆T = 0.406(1-S)0.25|∆Фо| (2.8) where ∆T is the transmittance energy difference between the peak and the val-ley, 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 repre-sent 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-19
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 ap-erture 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 coeffi-cient. 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 dif-ferent 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 sin-gle laser beam and that the nonlinearity in refraction and absorption can be de-termined 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-20
form a closed aperture z-scan to measure the overall transmittance of the sam-ple [13]. The measured transmission coefficient is then independent of the non-linear refraction and depends only on the nonnon-linear absorption. In the following paragraphs, it will be shown that data from such a z-scan with a remote aper-ture, during construction, forms a valley symmetrical around the focus. This z-scan of the open diaphragm is used to determine the nonlinear absorption coef-ficient. A nonlinear refractive index can be obtained by dividing the data ob-tained from the z-scan using an aperture in situ with the data obob-tained 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 temper-ature 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 distri-bution 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 sur-face 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 drogen medium at 200-400°C for duration of 10 minutes. During this stage, hy-drogen (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 scat-tering [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 nec-essary in order to minimize measurement errors, to maximally remove the ef-fects 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 per-formed, 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 re-fraction occur at the same time and are measured at the same point. The instal-lation 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
Figure 16 - The photo of the experimental setup showing the first