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Linear and nonlinear intersubband optical absorption and refractive index change in asymmetrical semi-exponential quantum wells Guanghui Liu, Kangxian Guo ⇑, Qingjie Wu Department of Physics, College of Physics and Electronic Engineering, Guangzhou University, Guangzhou 510006, PR China

a r t i c l e

i n f o

Article history: Received 6 March 2012 Received in revised form 28 April 2012 Accepted 30 April 2012 Available online 17 May 2012 Keywords: Quantum well Nonlinear optical properties

a b s t r a c t Linear and nonlinear intersubband optical absorption and refractive index change in asymmetrical semi-exponential quantum wells are theoretically investigated within the framework of the compact–density–matrix approach and iterative method. The wave functions are obtained by using the effective mass approximation. The energy levels are obtained by numerical method. It is found that the optical absorption coefﬁcients and refractive index changes are strongly affected not only by r and U 0 , but also by the incident optical intensity. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In the past few years, linear and nonlinear optical properties related to intersubband transitions in semiconducting material have been a subject of great interest. Such properties have become the important physical foundation for many optoelectronic devices such as high-speed electro-optical modulators, far infrared photodetectors, semiconductor optical ampliﬁers and so on. Especially with the rapid advances in nanofabrication techniques such as molecular beam epitaxy and metal–organic chemical vapor deposition, the research scope has already been widely expanded in this area. In recent years, much attention are focused on low-dimensional quantum systems such as quantum well, quantum wires and quantum dots [1–12]. It is well known that due to the existence of a strong quantum conﬁnement effect, the nonlinear effects in low-dimensional quantum systems can be enhanced more dramatically over those in bulk materials, which is crucial for the evolution of the emerging nanoelectronics. In this paper, we focus attention on the quantum wells.

⇑ Corresponding author. E-mail address: [email protected] (K. Guo). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.04.023

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Low-dimensional quantum wells have different shapes and sizes such as double or single triangular, parabolic or semi-parabolic, inﬁnite square well and so on. The inﬂuences of changing quantum wells parameters on the quantum states of carriers are often investigated. Besides, the application of either an electric or a magnetic ﬁeld, and also of external perturbation like hydrostatic pressure or temperature, changes the quantum states of carriers conﬁned in a quantum well. Therefore, the linear and nonlinear optical properties of quantum well are studied by many researchers. Some of these researchers do a lot of work on linear and nonlinear intersubband optical absorption and refractive index change. For instance, Chen et al. investigated the linear and nonlinear intersubband optical absorption in double triangular quantum wells under applying an electric ﬁeld, and found that the linear and the nonlinear optical absorption coefﬁcients have complex relationships with right-well width [13]. Besides, he also studied the linear and nonlinear intersubband refractive index changes in asymmetric coupled quantum wells, with the result that the structure parameters, such as the barrier width and the right-well width, have great inﬂuences on the total refractive index changes [14]. Keshavarz et al. reported the linear and nonlinear intersubband optical absorption in symmetric double-parabolic quantum wells with a suitable numerical method, and also revealed that the structure parameters such as the barrier and the well width really affect the optical characteristics of there structures [15]. In addition, Yesilgul discussed the linear and nonlinear intersubband optical absorption coefﬁcients and refractive index changes in symmetric double semi-V-shaped quantum wells [16], and he together with others also analyzed the linear and nonlinear intersubband optical absorption coefﬁcients and refractive index changes in a V-shaped quantum well under the applied electric and magnetic ﬁelds [17]. However, the linear and nonlinear intersubband optical absorption and refractive index change in asymmetrical semi-exponential quantum wells (ASEQW) has not been investigated. In this paper, a detailed study will be given about the current problem. This paper is organized as follows. In Section 2, we obtain the eigenfunctions and the energy eigenvalues. The analytical expressions for the linear and nonlinear optical absorption coefﬁcients and refractive index changes are obtained by the compact–density–matrix approach and iterative method. In Section 3, the numerical results and discussions are performed. Finally, a brief conclusion is made in Section 4. 2. Theory 2.1. Energy eigenvalues and eigenfunctions of an electron conﬁned in asymmetrical semi-exponential quantum wells. Let us consider an electron conﬁned in ASEQW. Within the framework of effective mass approximation, the Hamiltonian of the system is given by 2

h @2 @2 @2 6H ¼ þ þ 2m @x2 @y2 @z2

! þ VðzÞ;

ð1Þ

with

( UðzÞ ¼

U 0 ez=r 1 z P 0

ð2Þ

z < 0;

1

where z represents the growth direction of the quantum well, h is Planck constant, m is the effective mass of the conduction band, and both U 0 and r are positive parameters, respectively. The time independent Schrödinger equation for the electron is given by

Hwn;k ðrÞ ¼ en;k wn;k ðrÞ:

ð3Þ

The eigenfunctions wn;k ðrÞ and the energy eigenlvalues ikk rk

wn;k ðrÞ ¼ /n ðzÞuc ðrÞe

;

en;k are given by ð4Þ

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185

with 2

en;k ¼ En þ

h 2 k ; 2m k

ð5Þ

where kk and rk are the wave vector and coordinate in the x–y plane. uc ðrÞ is the periodic part of the Bloch function in the conduction band at k ¼ 0. /n ðzÞ and En are the solutions of one-dimensional Schrödinger equation

Hz /n ðzÞ ¼ En /n ðzÞ;

ð6Þ

where Hz is the z part of the Hamiltonian H, and it is given by 2

Hz ¼

@2 h þ UðzÞ: 2m @z2

ð7Þ

For Eq. (6), let us set the following quantities:

a2 ¼

8m U 0 r2 h

2

;b ¼

8m ðEn þ U 0 Þr2 h

2

;

n ¼ aez=2r ;

ð8Þ

therefore, the Schrödinger equation can be rewritten as 2

n2 where

d /n ðnÞ

þn

2

dn

d/n ðnÞ ðt2 þ n2 Þ/n ðnÞ ¼ 0; dn

ð9Þ

pﬃﬃﬃ

t ¼ i b. The equation is a modiﬁed Bessel equation [18]. Its solution is /n ðnÞ ¼ AK t ðnÞ þ BIt ðnÞ;

where A and B are arbitrary constants. Due to the fact that It ðnÞ !

ð10Þ p1ﬃﬃﬃﬃﬃﬃ en 2pn

diverges exponentially for

n ! 1, and the wavefunctions must satisfy bound state condition, we must set B = 0. Therefore, Eq. (10) reduces to

/n ðzÞ ¼ AK ipﬃﬃb ðaez=2r Þ:

ð11Þ

The normalized coefﬁcient A can be obtained by normalized condition, and the energy eigenvalues En can be numerically solved by the standard continuous condition K ipﬃﬃb ðaÞ ¼ 0. 2.2. Linear and nonlinear intersubband optical absorption coefﬁcients and refractive index changes In this section ,we will give a brief derivation of the linear and third-order nonlinear optical absorption coefﬁcients and refractive index changes by the compact density matrix method and the iterative procedure. Suppose our system is excited by an electromagnetic ﬁeld . The electric ﬁeld vector of the electromagnetic ﬁeld is

EðtÞ ¼ E0 cosðxtÞ ¼ e E expðixtÞ þ e E expðixtÞ;

ð12Þ

where x is the frequency of the external incident ﬁeld with a polarization vector normal to the ASEQW. Then the evolution of the density matrix operator q is given by the time-dependent Schrödinger equation [19]:

@ qij 1 ¼ ½H0 ezEðtÞ; qij Cij ðq qð0Þ Þij ; ih @t

ð13Þ

where H0 is the Hamiltonian of the system in the absence of the electromagnetic ﬁeld ~ EðtÞ; qð0Þ is the unperturbed density matrix, and Cij is the phenomenological relaxation rate, caused by the electron– phonon, electron–electron and other collision processes. Here we select Cij ¼ C0 ¼ 1=T 0 when i – j for simplicity. Eq. (13) can be solved by using iterative method [19],

qðtÞ ¼

X

qðnÞ ðtÞ;

n

ð14Þ

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with ðnþ1Þ

@ qij

@t

¼

o 1 1n ðnþ1Þ ½ez; qðnÞ ij EðtÞ: ½H0 ; qðnþ1Þ ij Cij qij ih ih

ð15Þ

The electronic polarization can be expanded as Eq. (14). Neglecting the terms which contribute little to our calculations, we can get a concise expression of the electronic polarization, 3 3ixt e ixt þ e0 vð3Þ e PðtÞ ¼ e0 vð1Þ þ c:c; x Ee x E e ð1Þ

ð16Þ ð3Þ

where e0 is the vacuum permittivity, vx and vx are the linear and third-order nonlinear susceptibility coefﬁcients, respectively. The electronic polarization of the nth order electronic polarization is given as

PðnÞ ðtÞ ¼

1 TrðqðnÞ ezÞ; V

ð17Þ

where V is the volume of interaction and Tr denotes the trace or summation over the diagonal elements of the matrix qðnÞ ez. With the compact density matrix approach and the iterative procedure, the analytical expressions of the linear and the third-order nonlinear susceptibilities for a two-level quantum system are given as follows [4,13,15]:

rt jM21 j2 ; E21 hx i h C0

e0 vð1Þ ðxÞ ¼

ð18Þ

and

"

#

r jM j2 jEj2 4jM 21 j2 ðM22 M 11 Þ2 ; e0 v ðxÞ ¼ t 21 E21 hx ihC0 ðE21 hxÞ2 þ ðhCÞ2 ðE21 ihC0 ÞðE21 hx ihC0 ð3Þ

ð19Þ

where rt is the electron density, and E21 ¼ E2 E1 is the energy interval of two different electronic states. Mij ¼ jhwi jezjwj ijði; j ¼ 0; 1Þ is the dipole transition matrix element. The absorption coefﬁcients and refractive index changes are given by

DnðxÞ ¼ Re nr

vðxÞ 2n2r

ð20Þ

;

and

rﬃﬃﬃﬃﬃ

l Imðe0 vðxÞÞ; eR

aðxÞ ¼ x

ð21Þ

where nr is the refractive index, l is the permeability, and eR is the real part of the permittivity, respectively. From Eqs. (18)–(21), the linear and the third-order nonlinear optical absorption coefﬁcients can be written as

rﬃﬃﬃﬃﬃ

að1Þ ðxÞ ¼ x

l jM 21 j2 rt hC0 ; R ðE21 hxÞ2 þ ðhC0 Þ2 rﬃﬃﬃﬃﬃ

að3Þ ðx; IÞ ¼ x

ð22Þ

l I jM 221 j2 rt hC0 eR 2e0 nr c ½ðE21 hxÞ2 þ ðhC0 Þ2 2

h i9 8 2 < jM 22 M11 j2 3E221 4E21 hx þ h ðx2 C20 Þ = 4jM21 j2 ; : ; E2 þ ðhC0 Þ2

ð23Þ

21

where hx is the incident photon energy, and I ¼ 2e0 nr cj e Ej2 is the incident optical intensity. Therefore, the total optical absorption coefﬁcients can be written as

aðxÞ ¼ að1Þ ðxÞ þ að3Þ ðxÞ:

ð24Þ

G. Liu et al. / Superlattices and Microstructures 52 (2012) 183–192

187

The linear and the third-order nonlinear refractive index changes can be expressed as

Mnð1Þ ðxÞ rt jM 21 j2 E21 hx ¼ ; nr 2n2r e0 ðE21 hxÞ2 þ ðhC0 Þ2 ( Mnð3Þ ðx; IÞ rt jM21 j2 lcI ðM22 M 11 Þ2 ¼ 4ðE21 hxÞjM 21 j2 þ 2 2 2 3 nr 4nr e0 ½ðE21 hxÞ þ ðhC0 Þ ðE21 Þ2 þ ðhC0 Þ2 n o ð hC0 Þ2 ð2E21 h xÞ ðE21 hxÞ½E21 ðE21 hxÞ ðhC0 Þ2 g :

ð25Þ

ð26Þ

The total change of refractive index can be written as

Mnðx; IÞ Mnð1Þ ðxÞ Mnð3Þ ðx; IÞ ¼ þ nr nr nr

ð27Þ

3. Results and discussion In this section, we will discuss the optical absorption coefﬁcients and refractive index changes in ASEQW. The parameters used in our calculations are as follows [20]: m ¼ 0:067m0 (where m0 is the electron mass), e0 ¼ 8:85 1012 Fm1 ; rt ¼ 5:0 1022 m3 , nr ¼ 3:2, and C0 ¼ 1=0:14ps. The linear að1Þ ðxÞ, the third-order nonlinear að3Þ ðx; IÞ and the total aðx; IÞ absorption coefﬁcients as a function of the incident photon energy, with U 0 ¼ 40 meV; I ¼ 0:3 MW=cm2 , and four different values of r are shown in Fig. 1. From the ﬁgure, it can be clearly seen that for each r, the að1Þ ðxÞ; að3Þ ðx; IÞ and aðx; IÞ, as a function of the incident photon energy hx has an prominent peak, respectively, and the prominent peaks have the same position, due to the one-photon resonance enhancement. Besides, the linear term is of opposite sign to the nonlinear term, and the linear term makes larger contribution to the total term than the nonlinear term. In addition, a very important feature of the ﬁgure is that with the increase of r, the að1Þ ðxÞ; að3Þ ðx; IÞ and aðx; IÞ move toward the lower energy regions . The physical reason for this feature is that as r increases, the quantum conﬁnement of the electron becomes weak, which results in the decrease of the energy interval E21 . From the ﬁgure, it also should

Fig. 1. The linear að1Þ ðxÞ, the third-order nonlinear að3Þ ðx; IÞ and the total aðx; IÞ absorption coefﬁcients as a function of the incident photon energy, with U 0 ¼ 40 meV; I ¼ 0:3 MW=cm2 , and four different values of r.

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G. Liu et al. / Superlattices and Microstructures 52 (2012) 183–192

be noted that the resonant peaks of the linear term is not a monotonous function of r but a concave function of r. This is due to the fact that with the increase of r, the quantum conﬁnement is weakened, which leads to the increase of the overlap between different electronic states and the decrease of the energy interval E21 . According to Eq. (22), when the photon energy is equal to the energy interval E21 , the dipole transition matrix element jM21 j2 and the energy interval E21 are the two main factors inﬂuencing the resonant peaks of the linear term. Therefore, at initial stage, the resonant peaks of the linear term are mainly decided by E21 , but with an increment in r, the quick increase of jM 21 j2 makes the leading role played by E21 become weak. At last, jM 21 j2 becomes the main factor deciding the resonant peaks of the linear term, which can be seen from C in the ﬁgure . Moreover, we also can clearly see that the increase of r makes the resonant peaks of the total aðx; IÞ become small. For D in the ﬁgure, the resonant peak of the total aðx; IÞ splits up into two separate peaks known as bleaching effect. The bleaching effect will be discussed in Fig. 5. As for að3Þ ðx; IÞ, the resonant peaks shows a complex relationship with r. This is because the resonant peak of C is smaller than that of B. It is difﬁcult for us to understand the above feature. From Eq. (23), when hx ¼ E21 , although að3Þ ðx; IÞ is proportional to jM21 j4 , the nonzero difference between M 22 and M 11 together with hx ¼ E21 has still some inﬂuences on að3Þ ðx; IÞ. In Fig. 2, we present the að1Þ ðxÞ; að3Þ ðx; IÞ and aðx; IÞ as a function of the incident photon energy for four different values of U 0 , with r ¼ 10 nm; I ¼ 0:3 MW=cm2 . By observing Fig. 2, it is noticed that with the enhancement of U 0 , the resonant peaks of að1Þ ðxÞ; að3Þ ðx; IÞ and aðx; IÞ exhibit a blue shift. This is due to the fact that the separation of the electron states get strengthened with increasing b. From the ﬁgure, it is obviously observed that the resonant peaks of the nonlinear term become larger with increasing U 0 . The reason for the phenomenon is that with the increase of U 0 ; M 421 has become the main factor deciding the nonlinear term. In addition, the resonant peaks of the linear term ﬁrstly becomes small, and next becomes larger, and again becomes small at last. This feature can be explained as follows. First, the resonant peaks of the að1Þ ðxÞ are not only dependent on the transition matrix element becoming small, but also on the energy difference becoming large. Secondly, From A to C the decrease of M221 makes more contribution to the að1Þ ðxÞ than E21 , but from C to D the increase of E21 is the main factor. At last, from D to E the decrease of M221 becomes important again. Moreover, the resonant peaks of the total aðx; IÞ about A, B and C remains almost constant. However, from C to D an obvious increment in the resonant peaks of aðx; IÞ occurs.

Fig. 2. The linear að1Þ ðxÞ, the third-order nonlinear að3Þ ðx; IÞ and the total aðx; IÞ absorption coefﬁcients as a function of the incident photon energy for four different values of U 0 , with r ¼ 10 nm; I ¼ 0:3 MW=cm2 .

G. Liu et al. / Superlattices and Microstructures 52 (2012) 183–192

189

Fig. 3. The linear Dnð1Þ =nr , the third-order nonlinear Dnð3Þ =nr and the total Dn=nr refractive index changes as a function of the incident photon energy, with U 0 ¼ 40 meV; I ¼ 1 MW=cm2 and three different values of r.

Fig. 3 depicts the linear Dnð1Þ =nr , the third-order nonlinear Dnð3Þ =nr and the total Dn=nr refractive index changes as a function of the incident photon energy, with U 0 ¼ 40 meV; I ¼ 1 MW=cm2 and three different values of r. From the ﬁgure, we can see that the linear refractive change is the opposite in sign of the nonlinear refractive index change. Therefore, the total refractive index change will be reduced by the nonlinear contribution. Besides, it is also noted that with the increase of r, a red shift appears. It is attributed to the fact that the weaker the quantum conﬁnement is, the smaller the energy interval E21 is. Moreover, we also see that with the increase of r, the maximum value of Dnð1Þ =nr ; Dnð3Þ =nr and Dn=nr obviously increases. It is easy to explain the feature. According to Eq. (25), the linear term depends on the product of ðE21 h xÞ and M 221 , and the horizontal ordinate of the maximum value of the linear term is close to E21 . Therefore, the difference between E21 and the horizontal ordinate is so small. There is no doubt that M221 decides the variation of the maximum value of the linear term. As for the nonlinear term, M 421 make itself become more important comparing to the difference between E21 and the horizontal ordinate . Especially, the inﬂuences of ðM 22 M 11 Þ2 on the nonlinear term are weakened by very small difference between E21 and the abscissa. Therefore, the importance of M421 becomes the ﬁrst. In addition, we also ﬁnd that when ðE21 ¼ hxÞ, for r ¼ 30 nm, the intersection of the linear Dnð1Þ =nr , the nonlinear Dnð3Þ =nr and the total Dn=nr is obvious nonzero whereas for r ¼ 5; 10 nm, the intersection is almost zero. In Fig. 4, we present the linear Dnð1Þ =nr , the third-order nonlinear Dnð3Þ =nr and the total Dn=nr refractive index changes as a function of the incident photon energy, with r ¼ 40 nm; I ¼ 0:8 MW=cm2 , and four different values of U 0 . From the ﬁgure, we can see that with the enhancement of U 0 , the maximum values of Dnð1Þ =nr ; Dnð3Þ =nr and Dn=nr shift to the higher energy regions. The physical origin is in agreement with Fig. 2. Besides, It can be clearly seen that with the increase of U 0 , the maximum value of Dnð1Þ =nr ; Dnð3Þ =nr and Dn=nr become small. The reason for the feature lies in the fact that increasing U 0 diminishes the penetration of the wave function into the barrier material, reduces the extended area of the wave function, and in effect decreases the magnitude of dipole moment matrix element M 21 . Therefore, a huge reduction in M 221 and M 421 exceeds a modest increase of the difference between the horizontal ordinate of the maximum value of the three refractive index changes and E21 , which leads to the result above. In Fig. 5, we plot the total optical absorption coefﬁcient aðx; IÞ as a function of incident photon energy hx for four different values of the incident optical intensity I with r ¼ 10 nm and U 0 ¼ 40 meV.

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Fig. 4. The linear Dnð1Þ =nr , the third-order nonlinear Dnð3Þ =nr and the total Dn=nr refractive index changes as a function of the incident photon energy, with r ¼ 40 nm; I ¼ 0:8 MW=cm2 , and four different values of U 0 .

Fig. 5. The total optical absorption coefﬁcient aðx; IÞ as a function of incident photon energy hx for four different values of the incident optical intensity I with r ¼ 10 nm and U 0 ¼ 40 meV.

From this ﬁgure, it is clearly found that the peaks of the total absorption coefﬁcient aðx; IÞ decrease prominently with the increase of I. In addition, it is obvious that the resonant peak of absorption coefﬁcient can be bleached at higher intensities. For instance, we can obviously see the bleaching effect when I is equal to 0:9 MW=cm2 and 1:2 MW=cm2 , respectively. Besides, the stronger the bleaching effect is, the more prominently the resonant peak is split up into two peaks. The reason for the feature is that the large the incident photon intensity is ,the more the nonlinear term makes contribution to the total absorption coefﬁcient, especially to the resonant peak. Therefore, we should do more research on

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191

Fig. 6. The total refractive index changes Dn=nr ðx; IÞ as a function of incident photon energy hx for ﬁve different values of the incident optical intensity I with r ¼ 10 nm and U 0 ¼ 40 meV.

nonlinear absorption with a strong incident optical intensity I. It should be pointed out that the period of oscillation of the incident optical intensity should be much greater the transit time of the electron so that the electron is too fast to see the potential-well distortion induced by the incident optical intensity. If is not only the frequency of the laser high, but also the incident optical intensity is very strong, we should consider laser dressed potential energy. we also make emphasis on this point in the following discussion about the refractive index changes. In Fig. 6, we plot the total refractive index changes Dn=nr ðx; IÞ as a function of incident photon energy h x for ﬁve different values of the incident optical intensity I with r ¼ 10 nm and U 0 ¼ 40 meV. From this ﬁgure, it can be clearly seen that when I is not too large, the maximum value of the total refractive index changes reduces with the increase of the incident optical intensity whereas when I is large enough, the maximum value of the total refractive index changes will increase with the increase of the incident optical intensity. For instance, In the ﬁgure, A, B and C explain the former whereas D and E explain the latter. Besides, we should realize that with the increase of I, the horizontal ordinate of the maximum value will change from the left of the point of intersection F to the right of that . As is known to us, the stronger the optical intensity is, the more intense the interaction between the photon and the medium will be, and the more obvious the nonlinear effects will be. But the linear effect is not related to the incident optical intensity, as shown in Eq. (25). Therefore, if we want to get obvious nonlinear optics effect about the refractive index change, we should import larger incident optical intensity. Of course, if I is very large, the structure of material may be broken. 4. Conclusion In this paper, we have studied theoretically linear and nonlinear intersubband optical absorption and refractive index change in asymmetrical semi-exponential quantum wells in detail. The calculations mainly focus on the dependence of optical absorption coefﬁcients and refractive index change on r; V 0 and the incident optical intensity I. The calculated results show that with the increase of r, the resonant peaks of the three absorption coefﬁcients and the maximum values of the three refractive index changes exhibit a red shift except that with the increase of U 0 , a blue shift is exhibited. The results of our paper also reveal that the resonant peaks of the three absorption coefﬁcients and the maximum values of the three refractive index are strongly affected by r; V 0 . Besides, increasing the

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incident optical intensity I make the bleaching effect more obvious for the total optical absorption, and make the maximum values of the total refractive index change ﬁrst become small and next become large. Finally, we hope that the results above can make a positive contribution to our scientiﬁc research in nonlinear optics ﬁeld. Acknowledgments This Work is supported by the National Natural Science Foundation of China (under Grant No. 61178003), Guangdong Provincial Department of Science and Technology (under Grant No. 2011B0 10400006), and the Science and Information Technology Bureau of Guangzhou (under Grant No. 11C62010688). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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