- Email: [email protected]
Modeling Optical Low-threshold Exciton Nonlinearity in Dielectric Nanocomposites
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 86 (2017) 24 – 31
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Modeling optical low-threshold exciton nonlinearity in dielectric nanocomposites D.V. Storozhenkoa*, V.P. Dzyubaa, Y.N. Kulchina,b, A.V. Amosova a
Institute of Automation and Control Processes, FEB Russian Academy of Sciences, 5 Radio Street, Vladivostok, 690041, Russia b Far Easern Federal University, 8 Suhanova Street, Vladivostok, 690950, Russia
Abstract We report on calculations of exciton nonlinearity in dielectric nanocomposites. The effect of various parameters on the spectrum of nonlinear increment to the refractive index, such as size and form factor of the nanoparticles shown. Numerical simulations of the optical response of dielectric nanoparticles Al2O3 presented. ©2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Keywords: dielectric nanoparticles; low-threshhold nonlinearity; photoluminescence; exitons
1. Introduction Unusual optical properties of dielectric nanocomposites actively investigated in the last decade. Special attention focused to features that are weak or absent in bulk dielectrics. For example, detected of additional environmental impact properties (Ganeev and Usmanov(2007)), an external field (Kecherenko andNalbanyan(2012)), the size and nature of the particle shape (Dzuba et. al.(2008)) on the transmission spectra and luminescence. Also found in(Dzuba et. al. (2010), Ganeev et. al.(2008)), the appearance of the nonlinear optical response of some dielectric nanocomposite materials in the range of intensity of the order of less than 1 kW/cm2, i.e. insufficient to run the multiphoton processes, photoionization or other non-linear processes. In (Milichko et al. (2013),Dneprovskii et
* Corresponding author. Tel.: +79242402258; E-mail address:[email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.010
25
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
al.(2010), Kupchak et al. (2008), He et al.(2004), Dneprovskii et al.(2013)) the authors, discussing the nature of the nonlinearity of this type conclude that the reason for increased impact defective or exciton levels in the dielectric particles on the overall transmission spectrum. In (Milichko et al. (2013)) invited the conditions conducive to emergence of resistant exciton states in a substance consisting of dielectric nanoparticles embedded in a transparent dielectric matrix liquid with linear optical properties in the optical range. Published in (Dzuba V. et al.(2013)) results of experimental studies of such substances have shown that at intensity of radiation about 150-250 W/cm2 the nonlinear addition to the refractive index near the resonant frequency of the absorption band was nn=10-4÷10-5. This is consistent with the proposed in [14] the theoretical model of such non-linearity. However, for a good prediction of the optical properties of nanocomposite dielectric materials necessary to investigate the influence of the size, shape and concentration of nanoparticles, it is possible with the help of numerical calculations with partial use of empirical data when setting the simulation parameters. 2. Description theory of exciton nonlinearity The basis of the model used in the study is the excitation of resonant two-tier system (Shen(1984)), in which a nonlinear optical response of the medium is proportional to the complex susceptibility per unit volume of the optical path. Considering that in the range of intensity up to 1000 W/cm2 total susceptibility dielectric substance F is the sum of the linear and nonlinear susceptibilities:
F
F R
F 0 F R
(1)
Np 2 ΔUng (Z, I ) !
'Ung (Z, I )
Z Z0 i* 2 Z Z0 * 2
ª º I / Is » 'U0 «1 2 « Z Z0 *2 1 I / I s » ¬ ¼
(2)
(3)
Where F 0 is linear part of the susceptibility of the dielectric nanoparticles, F R – the resonant nonlinear correction to the susceptibility of the dielectric nanoparticles, ߱Ͳ – resonance frequency, Γ – the half-width of the absorption line, – Planck's constant with a bar, N – number of charge carriers in the bulk of the optical path, ȟߩ݊݃ – the difference between the populations of the energy levels of states ȁ݊ and ȁ݃ , ൌ൏ ݊ȁ݁ ή ݖݎȁ݃ – the projection of the total electric dipole moment of the transition of the electron with charge ݁ nanoparticles out of ൏ ݊ȁ in state ȁ݃ on direction of polarization of the external optical radiation, ȟߩͲ – the equilibrium difference between the thermal of the population in the absence of an external field, I s – the level of saturation of a two-tier system in which the excited state has moved half of the charge carriers. Note that a priori unknown dipole moment ൌ൏ ݊ȁ݁ ή ݖݎȁ݃ , which does not know the value of the vector projection rz. Therefore, the introduction of orientation factor A( I ) can decide this problem:
p
p0 A( I )
(4)
We believe that the maximum value of the dipole moment p0 a e is proportional to the size of the nanoparticles. The dependence of orientation factor from intensity I given in the form of a power model:
A( I ) 1 e I / D
(5)
26
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
The nanoparticles shape is taken into account in the model by a factor of orientation of the polarization vector of the particle along the electric field vector, which depends on the intensity of polarized radiation. The coefficient α specifies the sensitivity to radiation intensity, and it is determined by the shape of the nanoparticles, but rather the difference between the moduli of its basis vectors. In other words, the value of this coefficient is proportional to elongation of nanoparticles that the experimentally observed in previous studies. The exact calculation and analytical representation of this factor through the distribution of sizes and shapes of nanoparticles in the bulk metamaterial with a liquid matrix – an open question. Therefore, in the described model, this factor is considered to be a predetermined constant, and D o 0 at sphericity nanoparticles form with an isotropic dielectric tensor. 3. Description of setting model parameters After algebraic manipulation of expressions (2-4) rewrite the expression additives exciton resonance nonlinear susceptibility as follows:
F R
Z Z0 i* NA(I) p0 2 'U0 2 ! Z Z0 * 2 1 I / I s
(6)
Right fraction of the expression (6) is the Lorentz model from saturation spectroscopy, which at Is=const the real part is an odd function, and the imaginary – even. Knowing that the susceptibility F is related to the dielectric constant, and with the refractive index, we note that the real part of the expression (6) defines a nonlinear addition to the refractive index and the imaginary part is the correction to the absorption coefficient of the substance. We write the expression to relate with susceptibility F R complex refractive index of the dielectric nanoparticles in solution:
n (Z, ,) n0 nn (Z, I)
n0
2SF R (Z, I) n0
(7)
Next to the numerical modeling is necessary to list all input parameters and define them a range of values. x
'U0 [0,1] – the difference between the equilibrium thermal of the population obeys the Boltzmann distribution, however in this case the model is considered a fixed temperature of 300 K, so we can assume that ȟߩͲ ൌ ܿݐݏ݊
x
a [5,100]nm – specify the size of the dielectric nanoparticles we studied the range of sizes of nanoparticles.
x
N [1012 ,1015 ]cm3 – the number of charge carriers per unit volume of optical path. The model assumed to be equal to the number of nanoparticles and is calculated by volume concentration of the substance and scope of the nanoparticles with a given size a. We studied the case of a bulk density of about f 0.3%
x
p0 a e – the maximum dipole moment of the nanoparticles, where a - the size of the nanoparticles, and the charge of an electron is specified in the GHS.
x
Z0 , * – the resonant frequency and the half-width of the absorption band can be set from experimental data. For convenience in the model, it displayed in the dimension corresponding to wavelengths λ 0 и Г λ in nm.
x
n0 – Refractive index of the bulk material of the dielectric nanoparticles on optical frequency reference is given.
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
x
D [1,100] cm2/W– factor elongation of nanoparticles with an isotropic dielectric permittivity tensor and the spherical shape of the nanoparticles D o 0 . Increasing values of the coefficient means that one of its basis vectors at times exceeds the other.
I s [20, 200] , W/cm2– saturation threshold at which half of the photoinduced excited free charge carriers. Sets of experimental data. In the analysis of the expression (6), you may notice that the susceptibility F R is proportional the value N to and that p0 2 , in turn, sets by the volume concentration f and the size of the nanoparticles a. Moreover, at f const when the N particle size increases proportionately a3 , and p0 2 increased proportionally a 2 . This allows us to conclude that with the same volume concentration of particle size reduction leads to an increase in the optical nonlinear response. As you can see from the expressions (2-4) of the nonlinear susceptibility F R varies with frequency and intensity. The behavior of the nonlinear susceptibility is convenient to consider the product of two competing processes: the orientation factor A( I ) and the difference in population levels ȟߩ݊݃ , which graphs are shown in Fig. 1. x
27
28
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
Fig. 1.Diagrams of two processes: ȟߩ݊݃ and
A( I ) . Dependences from intensity with fixed λ. In (a,b,c) α=0.25, and α=6.75 in (d,e,f).
From the analysis of patterns can be seen that formed at the intersection of function extremum shifts with wavelength. It is also demonstrated by the impact of the coefficient α on the nature of the formation of the resulting behavior. For small values of α, which corresponds to a spherical particle shape, orientation factor value A(I) within the absorption band tends to 1, giving contribution to the nonlinear susceptibility F R . Including these processes needs to select the most suitable working portion in use exciton nonlinearity of dielectric nanocomposites in practice, for example in optical transistors (Jain et al. (1976)). 4. Results of modeling Through the nonlinear susceptibility and known model for calculating the refractive index of the two-component medium, such as the Maxwell-Garnett, you can build transmission spectra, absorption and scattering of substances. Expressions for calculating the cross sections of scattering and absorption materials can be found in work (Dzyuba et al. (2011)). However, due to the superiority of the contribution of the linear component of the nonlinear several orders of magnitude it is advisable to depict the individual spectra corresponding to only the nonlinear addition to the complex refractive index. To reproduce the experimental data (Milichko et al.(2013)) with particles of Al2O3 embedded in a transparent immersion oil the model were based on the following parameters: ȟߩͲ ൌ ͲǤͻ, a 45nm , n0 = 1.65 λ0 =437 nm, Г λ=27.4 nm
f 0.3% I s 50 W/cm2 D 3.5 cm2/W Fig. 2 are graphs of complex nonlinear addition nn (Z, ,) . Cases when fixing the wavelength (Fig. 2 a, b, c) or intensity (Fig. 2 d, e, f). These diagrams are in good agreement with the experimental data (Milichko et al.(2013)).
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
Fig. 2.Diagrams of nonlinear addition with fixed λ (a,b,c) and fixed intensity (d,e,f). Real and imaginary parts shown separately.
Calculation of the spectrum of wavelengths show that the change in the intensity of the imaginary part of the nonlinear increment decreases faster than the real part. When passing through the resonance frequency λ 0 real part of
29
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D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
the nonlinear correction nn (Z, ,) changes the sign. With the change in the intensity (Fig. 2 d, e, f) the widening strips of real and imaginary parts nn (Z, ,) with a decreasing amplitude values. Also in the Fig.2 (a b c) we can see that the distance from the resonant frequency shifts in schedules intensity extremum real and imaginary parts. Also, in Fig.2 (and b) can be seen that with an increase in the intensity of the nonlinear response first increases and then decreases. Real and imaginary parts of the nonlinear correction nn (Z, ,) is convenient to estimate a two-dimensional maps of the distribution of the intensity and the wavelength presented in Figure 3:
Fig. 3.Maps of nonlinear addition
nn (Z, ,) for Al2O3 nanoparticles with λ0=437 nm, and Г λ=27.4 nm.
From the analysis of the images clearly visible allocation of the real part of the refractive index at various wavelengths and intensities. With a relatively small half-width of the absorption band, the area of non-linear change of the actual supplement covers a large optical range. This means that the presence of the resonant absorption bands in the violet portion of the spectrum has exciton nature can be used at a considerable distance up to 200 nm from it in red. 5. Conclusion It demonstrated the behavior of nonlinear optical response of exciton intensity. A distinctive feature is that the smooth increase of the intensity of radiation response first increases and then decreases. This behavior of the exciton nonlinearity in dielectric nanocomposites distinguishes it from the known quadratic or cubic nonlinearity substance. The magnitude of the nonlinear correction in the refractive index to 10 -4 is at a relatively low radiation intensity (up to 1000 W/cm2) is of practical interest for use in optical devices, for example, in optic transistors. The model allows the analysis of numerical calculations of the exciton optical nonlinearity at different frequencies and monitor the behavior of complex additives nn (Z, ,) with different particle sizes, and other parameters. For example, a numerical simulation found that the magnitude of the nonlinear optical response of F R proportional to the size of nanoparticles and their number per unit volume equal to N. When the volume concentration of nanoparticles with a decrease in their size increases the value of F R . So to increase the nonlinear optical response of nanoparticles is recommended to use a smaller size. It was found that the contribution to the nonlinear optical response of orientation factor A(I) increases with elongation of nanoparticles. To the greatest extent it appears near the resonance frequency, and a distancing from her contribution to the nonlinear optical response decreases. Also elongation of nanoparticles influences the behavior of the nonlinear optical response intensity. When increasing the ratio of size of the parties particles
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
orientation factor has a smooth change in the intensity. On the contrary, the behavior of the nonlinear optical response can serve as an indicator of spherical particles with an isotropic polarizability tensor. Acknowledgements The work was partially supported by the grants of the Complex Fundamental Research Programs of Far Eastern Branch of the Russian Academy of Sciences ( № 0262-2015-0094, № 0262-2015-0059). References Dneprovskii V. , Smirnov A., Kozlova M., 2013. Self-diffraction of laser beams in the case of resonant excitation of excitons in colloidal CdSe/ZnS quantum dots, Proc. of SPIE, Vol. 8772, p. 877209-1 – 877209-7. Dneprovskii V. S., Zhukov E. A., Kozlova M.V., et. a l, 2010.The saturation of absorption and processes when the self-resonant excitation of the main exciton transition in colloidal quantum dots, Solid State Physics 52 (9) pp. 1809-1814. Dzuba V., Kulchin Y., Milichko V., 2012. Effect of the shape of a nano-object on quantum-size states. Journal of nanoparticle research 14(11), 1208 Dzuba V., Kulchin Y., Milichko V., 2012. Photonics of Heterogeneous Dielectric Nanostructures, Nanocomposites - New Trends and Developments, Dr. Farzad Ebrahimi (Ed.), ISBN: 978-953-51-0762-0, InTech, DOI: 10.5772/50212. Dzuba V.P., Krasnok A.E., Kulchin Y.N., 2010. Nonlinear refractive index of dielectric nanocomposites in weak opticals fields, Technical Physics letters 36(11) pp. 973-977. Dzyuba, V.P., Krasnok, A.E., Kulchin, J.N., Dzyuba, I.V., 2011. A Model of Nonlinear Optical Transmittance for Insulator Nanocomposites. Semiconductors 45 (3), 295-301. Ganeev R. A. and UsmanovT., 2007. “Nonlinear-optical parameters of various media,”IOP Quantum Electron37(7), 605–622. Ganeev, R. A., Suzuki, M., Baba, M., Ichihara, M., Kuroda, H., 2008. Low- and high-order nonlinear optical properties of BaTiO3 and SrTiO3 nanoparticles. Journal of Optical Society of America B, 25(3), 325-333. He J., Ji, W., Ma, G. H., Tang, S. H., Elim, H. I., Sun, W. X., Zhang, Z. H., & Chin, W. S., 2004. Excitonic nonlinear absorption in CdSnanocrystals studied using Z-scan technique. Journal of Applied Physics, 95(11), 6381-6386. Jain, K. (1976). «Optical transistor». Appl. Phys. Lett. 28 (12): 719. Kecherenko M.G., Nalbanyan V.M., 2012. Modification of the spectrum electric dipole polarizability of a cluster of two conducting spherical nanoparticles in an external magnetic field. Herald OGU 1 pp. 141-149. Kupchak I.M., Kruchenko Yu.V., Korbutyak D.V., et al., 2008. Exciton states and photoluminescence of Si and Ge nanocrystals in Al2O3 matrix. Semiconductors 2008, 42 (10) pp. 1213-1218. Milichko V., Dzuba V., Kulchin Y., 2013. Abnormal optical nonlinear dielectric nanodispersions, Quantum Electronics, 43 (6) pp.567-573. Milichko V., Dzuba V., Kulchin Y., 2013. Unusual nonlinear optical properties of SiO 2 nanocomposite in weak optical fields, Appl Phys A (2013) 111:319–322. Milichko V.A., et al.,2013.Photo-induced electric polarizability of Fe3O4 nanoparticles in weak optical fields.Nanoscale Research Letters 2013 8:317. ShenY.R., John Wiley and Sons , 1984. The principles of nonlinear optics.
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ScienceDirect Physics Procedia 86 (2017) 24 – 31
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Modeling optical low-threshold exciton nonlinearity in dielectric nanocomposites D.V. Storozhenkoa*, V.P. Dzyubaa, Y.N. Kulchina,b, A.V. Amosova a
Institute of Automation and Control Processes, FEB Russian Academy of Sciences, 5 Radio Street, Vladivostok, 690041, Russia b Far Easern Federal University, 8 Suhanova Street, Vladivostok, 690950, Russia
Abstract We report on calculations of exciton nonlinearity in dielectric nanocomposites. The effect of various parameters on the spectrum of nonlinear increment to the refractive index, such as size and form factor of the nanoparticles shown. Numerical simulations of the optical response of dielectric nanoparticles Al2O3 presented. ©2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Keywords: dielectric nanoparticles; low-threshhold nonlinearity; photoluminescence; exitons
1. Introduction Unusual optical properties of dielectric nanocomposites actively investigated in the last decade. Special attention focused to features that are weak or absent in bulk dielectrics. For example, detected of additional environmental impact properties (Ganeev and Usmanov(2007)), an external field (Kecherenko andNalbanyan(2012)), the size and nature of the particle shape (Dzuba et. al.(2008)) on the transmission spectra and luminescence. Also found in(Dzuba et. al. (2010), Ganeev et. al.(2008)), the appearance of the nonlinear optical response of some dielectric nanocomposite materials in the range of intensity of the order of less than 1 kW/cm2, i.e. insufficient to run the multiphoton processes, photoionization or other non-linear processes. In (Milichko et al. (2013),Dneprovskii et
* Corresponding author. Tel.: +79242402258; E-mail address:[email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.010
25
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
al.(2010), Kupchak et al. (2008), He et al.(2004), Dneprovskii et al.(2013)) the authors, discussing the nature of the nonlinearity of this type conclude that the reason for increased impact defective or exciton levels in the dielectric particles on the overall transmission spectrum. In (Milichko et al. (2013)) invited the conditions conducive to emergence of resistant exciton states in a substance consisting of dielectric nanoparticles embedded in a transparent dielectric matrix liquid with linear optical properties in the optical range. Published in (Dzuba V. et al.(2013)) results of experimental studies of such substances have shown that at intensity of radiation about 150-250 W/cm2 the nonlinear addition to the refractive index near the resonant frequency of the absorption band was nn=10-4÷10-5. This is consistent with the proposed in [14] the theoretical model of such non-linearity. However, for a good prediction of the optical properties of nanocomposite dielectric materials necessary to investigate the influence of the size, shape and concentration of nanoparticles, it is possible with the help of numerical calculations with partial use of empirical data when setting the simulation parameters. 2. Description theory of exciton nonlinearity The basis of the model used in the study is the excitation of resonant two-tier system (Shen(1984)), in which a nonlinear optical response of the medium is proportional to the complex susceptibility per unit volume of the optical path. Considering that in the range of intensity up to 1000 W/cm2 total susceptibility dielectric substance F is the sum of the linear and nonlinear susceptibilities:
F
F R
F 0 F R
(1)
Np 2 ΔUng (Z, I ) !
'Ung (Z, I )
Z Z0 i* 2 Z Z0 * 2
ª º I / Is » 'U0 «1 2 « Z Z0 *2 1 I / I s » ¬ ¼
(2)
(3)
Where F 0 is linear part of the susceptibility of the dielectric nanoparticles, F R – the resonant nonlinear correction to the susceptibility of the dielectric nanoparticles, ߱Ͳ – resonance frequency, Γ – the half-width of the absorption line, – Planck's constant with a bar, N – number of charge carriers in the bulk of the optical path, ȟߩ݊݃ – the difference between the populations of the energy levels of states ȁ݊ and ȁ݃ , ൌ൏ ݊ȁ݁ ή ݖݎȁ݃ – the projection of the total electric dipole moment of the transition of the electron with charge ݁ nanoparticles out of ൏ ݊ȁ in state ȁ݃ on direction of polarization of the external optical radiation, ȟߩͲ – the equilibrium difference between the thermal of the population in the absence of an external field, I s – the level of saturation of a two-tier system in which the excited state has moved half of the charge carriers. Note that a priori unknown dipole moment ൌ൏ ݊ȁ݁ ή ݖݎȁ݃ , which does not know the value of the vector projection rz. Therefore, the introduction of orientation factor A( I ) can decide this problem:
p
p0 A( I )
(4)
We believe that the maximum value of the dipole moment p0 a e is proportional to the size of the nanoparticles. The dependence of orientation factor from intensity I given in the form of a power model:
A( I ) 1 e I / D
(5)
26
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
The nanoparticles shape is taken into account in the model by a factor of orientation of the polarization vector of the particle along the electric field vector, which depends on the intensity of polarized radiation. The coefficient α specifies the sensitivity to radiation intensity, and it is determined by the shape of the nanoparticles, but rather the difference between the moduli of its basis vectors. In other words, the value of this coefficient is proportional to elongation of nanoparticles that the experimentally observed in previous studies. The exact calculation and analytical representation of this factor through the distribution of sizes and shapes of nanoparticles in the bulk metamaterial with a liquid matrix – an open question. Therefore, in the described model, this factor is considered to be a predetermined constant, and D o 0 at sphericity nanoparticles form with an isotropic dielectric tensor. 3. Description of setting model parameters After algebraic manipulation of expressions (2-4) rewrite the expression additives exciton resonance nonlinear susceptibility as follows:
F R
Z Z0 i* NA(I) p0 2 'U0 2 ! Z Z0 * 2 1 I / I s
(6)
Right fraction of the expression (6) is the Lorentz model from saturation spectroscopy, which at Is=const the real part is an odd function, and the imaginary – even. Knowing that the susceptibility F is related to the dielectric constant, and with the refractive index, we note that the real part of the expression (6) defines a nonlinear addition to the refractive index and the imaginary part is the correction to the absorption coefficient of the substance. We write the expression to relate with susceptibility F R complex refractive index of the dielectric nanoparticles in solution:
n (Z, ,) n0 nn (Z, I)
n0
2SF R (Z, I) n0
(7)
Next to the numerical modeling is necessary to list all input parameters and define them a range of values. x
'U0 [0,1] – the difference between the equilibrium thermal of the population obeys the Boltzmann distribution, however in this case the model is considered a fixed temperature of 300 K, so we can assume that ȟߩͲ ൌ ܿݐݏ݊
x
a [5,100]nm – specify the size of the dielectric nanoparticles we studied the range of sizes of nanoparticles.
x
N [1012 ,1015 ]cm3 – the number of charge carriers per unit volume of optical path. The model assumed to be equal to the number of nanoparticles and is calculated by volume concentration of the substance and scope of the nanoparticles with a given size a. We studied the case of a bulk density of about f 0.3%
x
p0 a e – the maximum dipole moment of the nanoparticles, where a - the size of the nanoparticles, and the charge of an electron is specified in the GHS.
x
Z0 , * – the resonant frequency and the half-width of the absorption band can be set from experimental data. For convenience in the model, it displayed in the dimension corresponding to wavelengths λ 0 и Г λ in nm.
x
n0 – Refractive index of the bulk material of the dielectric nanoparticles on optical frequency reference is given.
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
x
D [1,100] cm2/W– factor elongation of nanoparticles with an isotropic dielectric permittivity tensor and the spherical shape of the nanoparticles D o 0 . Increasing values of the coefficient means that one of its basis vectors at times exceeds the other.
I s [20, 200] , W/cm2– saturation threshold at which half of the photoinduced excited free charge carriers. Sets of experimental data. In the analysis of the expression (6), you may notice that the susceptibility F R is proportional the value N to and that p0 2 , in turn, sets by the volume concentration f and the size of the nanoparticles a. Moreover, at f const when the N particle size increases proportionately a3 , and p0 2 increased proportionally a 2 . This allows us to conclude that with the same volume concentration of particle size reduction leads to an increase in the optical nonlinear response. As you can see from the expressions (2-4) of the nonlinear susceptibility F R varies with frequency and intensity. The behavior of the nonlinear susceptibility is convenient to consider the product of two competing processes: the orientation factor A( I ) and the difference in population levels ȟߩ݊݃ , which graphs are shown in Fig. 1. x
27
28
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
Fig. 1.Diagrams of two processes: ȟߩ݊݃ and
A( I ) . Dependences from intensity with fixed λ. In (a,b,c) α=0.25, and α=6.75 in (d,e,f).
From the analysis of patterns can be seen that formed at the intersection of function extremum shifts with wavelength. It is also demonstrated by the impact of the coefficient α on the nature of the formation of the resulting behavior. For small values of α, which corresponds to a spherical particle shape, orientation factor value A(I) within the absorption band tends to 1, giving contribution to the nonlinear susceptibility F R . Including these processes needs to select the most suitable working portion in use exciton nonlinearity of dielectric nanocomposites in practice, for example in optical transistors (Jain et al. (1976)). 4. Results of modeling Through the nonlinear susceptibility and known model for calculating the refractive index of the two-component medium, such as the Maxwell-Garnett, you can build transmission spectra, absorption and scattering of substances. Expressions for calculating the cross sections of scattering and absorption materials can be found in work (Dzyuba et al. (2011)). However, due to the superiority of the contribution of the linear component of the nonlinear several orders of magnitude it is advisable to depict the individual spectra corresponding to only the nonlinear addition to the complex refractive index. To reproduce the experimental data (Milichko et al.(2013)) with particles of Al2O3 embedded in a transparent immersion oil the model were based on the following parameters: ȟߩͲ ൌ ͲǤͻ, a 45nm , n0 = 1.65 λ0 =437 nm, Г λ=27.4 nm
f 0.3% I s 50 W/cm2 D 3.5 cm2/W Fig. 2 are graphs of complex nonlinear addition nn (Z, ,) . Cases when fixing the wavelength (Fig. 2 a, b, c) or intensity (Fig. 2 d, e, f). These diagrams are in good agreement with the experimental data (Milichko et al.(2013)).
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
Fig. 2.Diagrams of nonlinear addition with fixed λ (a,b,c) and fixed intensity (d,e,f). Real and imaginary parts shown separately.
Calculation of the spectrum of wavelengths show that the change in the intensity of the imaginary part of the nonlinear increment decreases faster than the real part. When passing through the resonance frequency λ 0 real part of
29
30
D.V. Storozhenko et al. / Physics Procedia 86 (2017) 24 – 31
the nonlinear correction nn (Z, ,) changes the sign. With the change in the intensity (Fig. 2 d, e, f) the widening strips of real and imaginary parts nn (Z, ,) with a decreasing amplitude values. Also in the Fig.2 (a b c) we can see that the distance from the resonant frequency shifts in schedules intensity extremum real and imaginary parts. Also, in Fig.2 (and b) can be seen that with an increase in the intensity of the nonlinear response first increases and then decreases. Real and imaginary parts of the nonlinear correction nn (Z, ,) is convenient to estimate a two-dimensional maps of the distribution of the intensity and the wavelength presented in Figure 3:
Fig. 3.Maps of nonlinear addition
nn (Z, ,) for Al2O3 nanoparticles with λ0=437 nm, and Г λ=27.4 nm.
From the analysis of the images clearly visible allocation of the real part of the refractive index at various wavelengths and intensities. With a relatively small half-width of the absorption band, the area of non-linear change of the actual supplement covers a large optical range. This means that the presence of the resonant absorption bands in the violet portion of the spectrum has exciton nature can be used at a considerable distance up to 200 nm from it in red. 5. Conclusion It demonstrated the behavior of nonlinear optical response of exciton intensity. A distinctive feature is that the smooth increase of the intensity of radiation response first increases and then decreases. This behavior of the exciton nonlinearity in dielectric nanocomposites distinguishes it from the known quadratic or cubic nonlinearity substance. The magnitude of the nonlinear correction in the refractive index to 10 -4 is at a relatively low radiation intensity (up to 1000 W/cm2) is of practical interest for use in optical devices, for example, in optic transistors. The model allows the analysis of numerical calculations of the exciton optical nonlinearity at different frequencies and monitor the behavior of complex additives nn (Z, ,) with different particle sizes, and other parameters. For example, a numerical simulation found that the magnitude of the nonlinear optical response of F R proportional to the size of nanoparticles and their number per unit volume equal to N. When the volume concentration of nanoparticles with a decrease in their size increases the value of F R . So to increase the nonlinear optical response of nanoparticles is recommended to use a smaller size. It was found that the contribution to the nonlinear optical response of orientation factor A(I) increases with elongation of nanoparticles. To the greatest extent it appears near the resonance frequency, and a distancing from her contribution to the nonlinear optical response decreases. Also elongation of nanoparticles influences the behavior of the nonlinear optical response intensity. When increasing the ratio of size of the parties particles
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orientation factor has a smooth change in the intensity. On the contrary, the behavior of the nonlinear optical response can serve as an indicator of spherical particles with an isotropic polarizability tensor. Acknowledgements The work was partially supported by the grants of the Complex Fundamental Research Programs of Far Eastern Branch of the Russian Academy of Sciences ( № 0262-2015-0094, № 0262-2015-0059). References Dneprovskii V. , Smirnov A., Kozlova M., 2013. Self-diffraction of laser beams in the case of resonant excitation of excitons in colloidal CdSe/ZnS quantum dots, Proc. of SPIE, Vol. 8772, p. 877209-1 – 877209-7. Dneprovskii V. S., Zhukov E. A., Kozlova M.V., et. a l, 2010.The saturation of absorption and processes when the self-resonant excitation of the main exciton transition in colloidal quantum dots, Solid State Physics 52 (9) pp. 1809-1814. Dzuba V., Kulchin Y., Milichko V., 2012. Effect of the shape of a nano-object on quantum-size states. Journal of nanoparticle research 14(11), 1208 Dzuba V., Kulchin Y., Milichko V., 2012. Photonics of Heterogeneous Dielectric Nanostructures, Nanocomposites - New Trends and Developments, Dr. Farzad Ebrahimi (Ed.), ISBN: 978-953-51-0762-0, InTech, DOI: 10.5772/50212. Dzuba V.P., Krasnok A.E., Kulchin Y.N., 2010. Nonlinear refractive index of dielectric nanocomposites in weak opticals fields, Technical Physics letters 36(11) pp. 973-977. Dzyuba, V.P., Krasnok, A.E., Kulchin, J.N., Dzyuba, I.V., 2011. A Model of Nonlinear Optical Transmittance for Insulator Nanocomposites. Semiconductors 45 (3), 295-301. Ganeev R. A. and UsmanovT., 2007. “Nonlinear-optical parameters of various media,”IOP Quantum Electron37(7), 605–622. Ganeev, R. A., Suzuki, M., Baba, M., Ichihara, M., Kuroda, H., 2008. Low- and high-order nonlinear optical properties of BaTiO3 and SrTiO3 nanoparticles. Journal of Optical Society of America B, 25(3), 325-333. He J., Ji, W., Ma, G. H., Tang, S. H., Elim, H. I., Sun, W. X., Zhang, Z. H., & Chin, W. S., 2004. Excitonic nonlinear absorption in CdSnanocrystals studied using Z-scan technique. Journal of Applied Physics, 95(11), 6381-6386. Jain, K. (1976). «Optical transistor». Appl. Phys. Lett. 28 (12): 719. Kecherenko M.G., Nalbanyan V.M., 2012. Modification of the spectrum electric dipole polarizability of a cluster of two conducting spherical nanoparticles in an external magnetic field. Herald OGU 1 pp. 141-149. Kupchak I.M., Kruchenko Yu.V., Korbutyak D.V., et al., 2008. Exciton states and photoluminescence of Si and Ge nanocrystals in Al2O3 matrix. Semiconductors 2008, 42 (10) pp. 1213-1218. Milichko V., Dzuba V., Kulchin Y., 2013. Abnormal optical nonlinear dielectric nanodispersions, Quantum Electronics, 43 (6) pp.567-573. Milichko V., Dzuba V., Kulchin Y., 2013. Unusual nonlinear optical properties of SiO 2 nanocomposite in weak optical fields, Appl Phys A (2013) 111:319–322. Milichko V.A., et al.,2013.Photo-induced electric polarizability of Fe3O4 nanoparticles in weak optical fields.Nanoscale Research Letters 2013 8:317. ShenY.R., John Wiley and Sons , 1984. The principles of nonlinear optics.
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