Effect of intense laser field on the nonlinear optical susceptibilities in an asymmetric single quantum well

Superlattices and Microstructures 61 (2013) 124–133

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Effect of intense laser field on the nonlinear optical susceptibilities in an asymmetric single quantum well B.K. Panda a,⇑, S. Panda b a b

Department of Physics, Ravenshaw University, Cuttack 753 003, India Department of Mathematics, Ravenshaw University, Cuttack 753 003, India

a r t i c l e

i n f o

Article history: Received 27 December 2012 Received in revised form 1 May 2013 Accepted 2 June 2013 Available online 14 June 2013 Keywords: Laser radiation effects Excitons Eigenvalues and eigen functions Fourier series Optoelectronic devices Infrared detectors

a b s t r a c t In the present paper, the effect of a high-frequency laser field on a single asymmetric rectangular quantum well is studied in the Fourier series method. The effect of laser field on the confining potential of the quantum well is taken through the laser dressing parameter which depends upon the strength of intensity and the magnitude of frequency of the laser field. The step type confining potential of the rectangular asymmetric well is modified by the laser dressing parameter to a smooth function. In this quantum well, linear, third-order nonlinear, nonlinear optical rectification and second harmonic generation susceptibilities under another laser source with low frequency are calculated using the density matrix method. The linewidths for susceptibilities are estimated taking the electron scattering with the longitudinal optic phonon. Compared to laser undressed well, the nonlinear optical properties in the laser dressed well are blue shifted and enhanced due to larger energy separations and oscillator strengths, respectively. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In the last two decades, there has been considerable interest in the nonlinear optical properties of the semiconductor quantum well (QW) systems as a result of important advances in both the epitaxial growth of the semiconductor heterojunctions and the laser technologies. Using the molecular beam epitaxy method, the QWs with a variety form of confining potentials have been grown in the ⇑ Corresponding author. Tel.: +91 674 2421764; fax: +91 674 2302047. E-mail address: [email protected] (B.K. Panda), . 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.06.002

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laboratory. With the availability of intense CO2 THz laser source in recent years, strongly laser-driven semiconductor heterostructures have received attention [1]. The effect of laser field on the semiconductor QW systems is taken through the renormalization of electron confining potential [2]. With a second low-energy laser source, the optical properties in a QW are usually studied. The carrier confinement in a QW results in several nondegenerate energy levels with narrow intersubband energy separations. The optical properties can be enhanced dramatically in the QW systems compared to bulk semiconductors. This happens as the oscillator strengths in the semiconductor QW structures are larger than those in the bulk semiconductors. Moreover, infrared absorption and emission of photons are possible due to narrow intersubband separations [3]. Based on the intersubband transitions, a number of device applications such as far-infrared photodetectors [4], electro-optical modulators [5] and infrared lasers [6] have been proposed and realized. The linear and nonlinear intersubband absorption coefficients within the conduction band of a laser dressed square [7], graded [8], inverse V-shaped [9] and compositional asymmetric [10] AlGaAs/ GaAs quantum wells have been studied with and without an applied electric field. The absorption coefficients are found to get enhanced in the laser dressed QWs due to the renormalization of confining potentials. When the intensity of the laser source is high, the absorption spectra in an asymmetric QW or in a biased QW are bleached due to strong nonlinear absorption coefficient. The calculation of second-order susceptibilities requires an asymmetric QW which can be achieved either by using the sophisticated material growth technology or by applying external electric field to the symmetric QW. The structural asymmetry can be grown by the molecular beam epitaxy method through compositional asymmetric QW [11] or asymmetrically coupled quantum wells [12]. The asymmetry in the structurally symmetric QW can be introduced by an applied dc electric field [13,14]. The susceptibility for nonlinear optical rectification (NOR) has been studied in a biased rectangular QW [15–17], an asymmetric double triangular QW under applied electric field [18] and a step QW [19]. The susceptibility for second harmonic generation (SHG) has been calculated in a symmetric QW under applied electric field [20,21] and an asymmetric single QW under hydrostatic pressure [22]. It has been found that the nonlinearities in absorption coefficient, NOR and SHG are much higher in a compositional asymmetric well than in a biased symmetric QW. To the best of our knowledge, the second-order optical nonlinearities for NOR and SHG have not been calculated in a laser dressed single asymmetric rectangular quantum well (ARQW) where it is expected that the nonlinearities in optical properties will be further enhanced due to modification of the confining potential of the QW. In the present work, the effect of THz laser field on the confining potential of a single ARQW is studied. The energies, envelope functions and intersubband linewidths are calculated in the Fourier series method. The susceptibilities for linear, third-order nonlinear, NOR and SHG due to a low-frequency laser radiation source are calculated in the density matrix approach. The results obtained with laser dressed confining potentials are compared with those obtained in the QW under zero optical field. 2. Theoretical methods A monochromatic laser radiation field with THz frequency X and vector potential Aðz; tÞ is incident along the growth direction (z-direction) of the QW. In the dipole approximation, Aðz; tÞ  AðtÞ ¼ ez A0 cos Xt, where ez is the unit vector of polarization and A0 is the amplitude of the vector potential. By applying the time-dependent transition z ! z þ aðtÞ, the effective mass equation in the momentum gauge describing the interaction dynamics in the laboratory frame of reference is transformed as [23–25]

"

# 2  @ h 1 @ @/ðz; tÞ  þ V½z þ aðtÞ /ðz; tÞ ¼ ih ; @t 2 @z m ðzÞ @z

ð1Þ

where V½z þ aðtÞ is the laser driven dressed confining potential. The classical quiver motion of an electron in the laser field is given by

aðtÞ ¼ ez a0 sin Xt

a0 ¼

eA0 ; m ð0ÞcX

ð2Þ

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where a0 is the laser dressing parameter, c is the velocity of light and m ð0Þ is the effective mass of electron in the well region. The time-dependent translation on the position-dependent effective mass is neglected. The potential V½z þ aðtÞ is a periodic function in time for a given z. Therefore, V½z þ aðtÞ can be expanded in a Fourier series as [2] 1 X 1 X

V½z þ aðtÞ ¼

ð1Þm VðkÞJ m ð2pka0 =LÞ expði2pkz=LÞ expðimXtÞ;

ð3Þ

k¼1m¼1

where J m is the Bessel function of order m, L is the length of QW including both well and barrier widths, VðkÞ is kth inverse Fourier component of VðzÞ,

VðkÞ ¼

1 L

Z

L 2

VðzÞ expði2pkz=LÞdz:

ð4Þ

2L

Using the Floquet approach, the time-dependent effective mass Eq. (1) is reduced to time-independent effective mass equation in the high-frequency limit Xs  1, where s is the transit time of electron in the well [26]. We can thus write

"

# 2  @ h 1 @  þ V d ðz; a0 Þ Wn ðzÞ ¼ En Wn ðzÞ; 2 @z m ðzÞ @z

ð5Þ

where En and Wn ðzÞ are nth level energy and envelope function, respectively and V d ðz; a0 Þ is the laser dressed confining potential. Using the standard procedure V d ðz; a0 Þ is obtained by taking the timeaverage of the potential V½z þ aðtÞ which is valid when the laser is tuned far from any resonance. The laser dressed confining potential is thus obtained as

V d ðz; a0 Þ ¼

X

Z

2p

2p

X

V½z þ aðtÞdt:

ð6Þ

0

Substituting Eq. (3) in Eq. (6), we find the dressed potential as

V d ðz; a0 Þ ¼

1 X

V d ðk; a0 Þ expði2pkz=LÞ;

ð7Þ

k¼1

where kth component of the inverse Fourier transform of the laser-dressed potential is given as

V d ðk; a0 Þ ¼ VðkÞJ 0 ð2pka0 =LÞ:

ð8Þ

Here J 0 ðxÞ is the Bessel function of 0th order. The effective mass Eq. (5) is solved in the Fourier series method [27,28]. In this method the envelope function Wn ðzÞ is expanded in a Fourier series as

rffiffiffi 1 1X Wn ðzÞ ¼ C n ðkÞ expði2pkz=LÞ; L k¼1

ð9Þ

where C n ðkÞ is the kth Fourier component. The inverse of the position-dependent effective mass ð1=m Þ can be expanded in a Fourier series with Fourier coefficients mðkÞ as,

1 m ðzÞ

¼

1 X

mðkÞ expði2pkz=LÞ:

ð10Þ

k¼1

Substituting Eqs. (7), (9) and (10) in Eq. (5), the effective mass equation is expressed as

"   # 2 2 X h  2p mðk  lÞkl þ V d ðk  l; a0 Þ C n ðlÞ ¼ En C n ðkÞ: 2 L l

ð11Þ

Both En and C n ðkÞ are obtained by diagonalizing the matrix HC n ¼ En C n . The QW is incident upon by a second optical field with low frequency x and polarization normal to the QW. The electronic polarization up to third-order in a QW under the optical field with the electric

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field vector along the growth direction of QW EðtÞ ¼ ez ½e expðixtÞ þ e expðixtÞ can be expanded as [29]

Pel ðtÞ ¼ ½0 vð1Þ ðxÞe expðixtÞ þ 0 vð2Þ ð2xÞe2 expð2ixtÞ þ 0 vð3Þ ðxÞe jej2 expðixtÞ þ 0 vð3Þ ð3xÞe3 expð3ixtÞ þ c:c: þ 0 vð2Þ ð0Þe2 ;

ð12Þ

where 0 is the vacuum permittivity, v ðxÞ, v ð0Þ; v ð2xÞ; v ð3xÞ and v ðxÞ are linear, NOR, SHG, third harmonic generation (THG) and third-order nonlinear susceptibilities, respectively. Using the density matrix method, the expressions for linear, third-order nonlinear and second-order susceptibilities in a QW containing two levels are given as [13,30,31] ð1Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

qs jl21 j2 ; E21  hx  iC21 " # 2 qs jl21 j I 4jl21 j2 jl22  l11 j2 ð3Þ ; 0 v ðxÞ ¼   E21  hx  iC21 ðE21  hxÞ2 þ C221 ðE21  iC21 ÞðE21  h  x  iC21 Þ

0 vð1Þ ðxÞ ¼

ð2Þ

v ð2xÞ ¼

qs

0

# jl21 j2 ðl22  l11 Þ ; ð2hx  E21  iC21 Þð hx  E21  iC21 Þ

ð13Þ

"

ð15Þ

where qs is the electron density, the energy separation E21 ¼ E2  E1 , the linewidth C21 ¼ C1 þ C2 where Ci is the linewidth of ith level and the laser intensity I ¼ jej2 . The THG is neglected here as it does not contribute in a two-level system. The dipole matrix element between i and j levels is defined as

lij ¼ e

Z

L 2

2L

dzWi ðzÞzWj ðzÞ:

Using Eq. (9) in Eq. (16),

lij ¼ e

Z

L 2

2L

ð16Þ

lij is derived as

Wi ðzÞWj ðzÞdz

ð17Þ

where j1 ðxÞ is the first-order spherical Bessel function. The density of electrons at finite temperature T is given as

qs ¼

m kB T 2

ph L

log

  1 þ exp½ðEF  E1 Þ=kB T ; 1 þ exp½ðEF  E2 Þ=kB T

ð18Þ

where EF is the Fermi energy and it is obtained from the charge neutrality condition. The linewidth Ci for the ith level arises as a result of the electron scattering with different phonon modes. The electron scattering with the longitudinal optic (LO) phonon mode is the most dominating among all phonon modes. The momentum-dependent linewidth of an electron is calculated from the self-energy of an electron interacting with the LO phonon as Ci ðkt Þ ¼ 2ImR½kt ; Ei ðkt Þ. The one-phonon self-energy of the electron in the second-order perturbation theory is given as [32] Ri ½kt ;Ei ðkt Þ ¼

 XXX jMðqt ; qz Þj2 jGji ðqz Þj2 j–i qt

qz

 nB ½hxLO  þ nF ½Ei ðkt þ qt Þ nB ½hxLO  þ 1  nF ½Ei ðkt þ qt Þ þ ; Ei ðkt Þ  Ej ðkt þ qt Þ  hxLO  id Ei ðkt Þ  Ej ðkt þ qt Þ þ hxLO þ id ð19Þ

where nB and nF are the distribution functions for boson and fermion, respectively, xLO is the LO phonon frequency and the energy of ith electron including the transverse kinetic energy is given as 2 2 Ei ðkt Þ ¼ Ei þ  h kt =2m ð0Þ. The electron-phonon interaction in the Fröhlich form is defined as [32]

jMðqt ; qz Þj2 ¼

2pe2 hxLO q2t þ q2z



1



1

1 0

 ;

ð20Þ

where 0 and 1 are static and high-frequency dielectric constants, respectively. The structure factor is defined as

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Gji ðqz Þ ¼

Z

L 2

2L

Wj ðzÞeiqz z Wi ðzÞdz:

ð21Þ

Substituting Eq. (9) in Eq. (21), we find

Gji ðqz Þ ¼

Z

L 2

2L

Wj ðzÞeiqz z Wi ðzÞdz

ð22Þ

where j0 ðxÞ is the spherical Bessel function for 0th order. At low temperature nF ½Ei ðkt þ qt Þ is negligibly small,

Ci ðkt Þ ¼ 2p

2 XX X  jMðqt ; qz Þj2 jGji ðqz Þj2 nB ½ hxLO dðEj ðkt þ qt Þ  Ei ðkt Þ  hxLO Þ j–i qt

qz

 þfnB ½hxLO  þ 1gdðEj ðkt þ qt Þ  Ei ðkt Þ þ hxLO Þ :

ð23Þ

After lengthy but straightforward algebra [33], we find m ð0ÞxLO e2 Ci ðkt Þ ¼ h



1



 2 Z 1 X

1 0

j–i

2 1

0

3

nB ðhxLO Þ nB ðhxLO Þ þ 1 7 dqz jGji ðqz Þj 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; 2 2 2 2 q4z þ 2q2z ð2kt  Dij Þ þ ðDij Þ q4z þ 2q2z ð2kt  Dþij Þ þ ðDþij Þ 26

ð24Þ

Dij

2

where ¼ 2m ð0Þ½Ej  Ei   hxLO = h . The linewidth for ith level described above refers to one electron at momentum kt . The mean linewidth at temperature T is obtained by averaging over all electrons having different kt by the Fermi-Dirac distribution function as 

R

Ci ¼

dEt nF ½Ei ðkt ; kB TÞCi ðkt Þ R ; dEt nF ½Ei ðkt ; kB TÞ

ð25Þ 2 2

where the transverse kinetic energy Et ¼  h kt =2m ð0Þ 3. Application to asymmetric rectangular quantum well In this work a single ARQW consisting of ALxl Ga1xl As=GaAs=Alxr Ga1xr As heterostructures, where xl and xr are Al concentrations in the left and right barrier regions, respectively is taken for calculating the electronic structure. The well width W = 140 Å and the Al concentrations xl = 0.4 and xr = 0.2. The asymmetric single QW is described as

8 > < V 0 ðxl Þ z 6 W=2 jzj < W=2 VðzÞ ¼ 0 > : V 0 ðxr Þ z P W=2:

ð26Þ

The barrier height V 0 ðxÞ is taken as V 0 ðxÞ ¼ 0:6ð1:155x þ 0:37x2 Þ eV. The kth component of the inverse Fourier transformation of VðzÞ is obtained by substituting Eq. (26) in Eq. (4) as

VðkÞ ¼

   1 W W ðV 0 ðxl Þ þ V 0 ðxr ÞÞ j0 ðpkÞ  j0 pk 2 L L    i W W ; þ ðV 0 ðxl Þ  V 0 ðxr ÞÞ y0 ðpkÞ  y0 pk 2 L L

ð27Þ

where j0 ðxÞ and y0 ðxÞ are the spherical Bessel function and the Legendre duplication formula of 0th order, respectively. The laser driven confining potentials V d ðz; a0 Þ for various values of a0 are shown in Fig. 1. The radiation field induces a variation in the laser dressed potential from a step type to a smooth function. The effective well width at the bottom of the dressed well decreases from W to W  2a0 , while it turns out as W þ 2a0 at the top of the well. The effective well width at the half of the barrier heights in both sides of the well, however, does not change. As a result of this the effective well width increases with increasing a0 [34].

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The kth inverse Fourier component of 1=m ðzÞ in the QW may be derived as

mðkÞ ¼

      1 1 1 W W 1 W W j þ j j ð p kÞ  p k p k þ 0 0 0 2 m ðxl Þ m ðxr Þ L L m ð0Þ L L     i 1 1 W W y0 ðpkÞ  y0 pk ;  þ 2 m ðxl Þ m ðxr Þ L L

ð28Þ

where m ðxÞ ¼ ð0:067 þ 0:083xÞm0 with m0 being the electron rest mass. For a symmetric well ðxl ¼ xr Þ, the imaginary part vanishes and it reproduces the analytic results for VðkÞ and mðkÞ previously given by us [28]. The energy levels are calculated from Eq. (11) by diagonalizing 601  601 matrix. The barrier height at the right side of the well is 147.48 meV. There are two bound states in the well. The bound state energies E1 ; E2 and their separations E21 calculated at different a0 are shown in Table 1. Both the bound state energies increase with a0 since the effective well width of the laser driven QW increases with a0 . The optical Stark shift for the excited state is larger than that of the ground state. The Fermi energies are calculated at different a0 taking the electron density 1  1016 cm3 are are shown in Table 1. With this value of electron density, the Fermi energy is located just above the ground state energy. The linewidths for n = 1 and 2 corresponding to the electron scattering with LO phonon are calculated using 0 ¼ 13:1; 1 ¼ 10:89; xLO ¼ 36:25 meV and T ¼ 100 K. The linewidths C1 ; C2 and C21 as a function of a0 are shown in Table 1. Both C1 and C2 decrease with increasing a0 . It is further found that C1 is significantly higher than C2 at lower a0 , but at higher a0 the difference between their values decreases. It is well known that the linewidth due to electric field induced tunneling for n ¼ 2 level is higher than that of n ¼ 1 level. The scattering process gives just the opposite behavior for linewidths compared to tunneling process. The dipole matrix elements l11 ; l22 and l21 as a function of a0 are shown in Fig. 2. While l11 and l22 increase with a0 ; l21 decreases with it. Compared to l11 ; l22 increases rapidly with a0 . At a0 ¼ 40 Å, l22 is higher than l21 . The asymmetry in the well introduces asymmetry in the ground and first excited state envelope functions. When the laser dressing parameter is increased, the lower part of the well is narrowed while the upper part is widened. As a result of this further asymmetry is introduced in the well. Compared to the ground state envelope function, the envelope function corresponding to the first excited is more anisotropic. Therefore, l22 increases rapidly with a0 compared to

400

Vd (z,α0) (meV)

300

200

100

0 -300

-200

-100

0

100

200

300

z (Å) Fig. 1. Laser dressed asymmetric rectangular confining potential under laser dressing parameter ða0 Þ. Black, red, green, blue and brown lines represent the dressed confining potentials corresponding to a0 ¼ 0; 10; 20; 30 and 40 Å, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Table 1 Calculated energy levels, linewidths and Fermi energy in an asymmetric quantum well. E1 ðmeVÞ

E2 ðmeVÞ

E21 ðmeVÞ

C1 ðmeVÞ

C2 ðmeVÞ

C21 ðmeVÞ

EF ðmeVÞ

18.246 19.510 22.378 26.716 32.835

72.132 76.037 84.219 94.263 103.53

53.886 56.527 61.841 67.547 70.695

3.589 3.201 2.631 2.150 1.759

1.945 1.818 1.589 1.345 1.128

5.534 5.019 4.220 3.495 2.887

21.159 22.434 25.316 29.661 35.783

Dipole moments (10

-29

mC)

60 50 40 30 20 10 0

0

10

20

30

40

Laser Dressing Parameter (α0)(Å) Fig. 2. Dipole matrix elements l11 (black line), l22 (red line) and l21 (blue line) as a function of laser dressing parameter a0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

l11 . On the other hand, the parity of the excited state is different from that of the ground state so that their overlap decreases with increase in a0 and thus the strength of l21 decreases with increasing a0 . The absorption coefficient is calculated from the susceptibilities in Eqs. (13) and (14) as

aðx; IÞ ¼ x

rffiffiffiffiffi

l Im½0 vð1Þ ðxÞ þ Im½0 vð3Þ ðx; IÞ ; R

ð29Þ

where l is the permeability of the system and the real part of the permittivity R ¼ n2r 0 with nr ¼ 3:2 as the refractive index. In our calculation the laser intensity I = 0.2 MW/cm2. The first and second terms in Eq. (29) are the linear and nonlinear absorption coefficients, respectively. The linear, nonlinear and total absorption coefficients as a function of photon energy  hx corresponding to a0 ¼ 0; 20 and 40 Å are shown in Fig. 3. As can be seen from this figure, the large contribution due to the linear absorption coefficient is opposite in sign of the nonlinear absorption coefficient. The absorption coefficients are shifted to higher photon energies and the peaks of the absorption coefficients increase with increasing a0 . It is found that the að3Þ ðx; IÞ is large at a0 ¼ 40 Å for which aðx; IÞ is bleached. This behavior can be explained in terms of the calculated linewidths C21 ; l11 ; l22 and l21 . The increase in ðl22  l11 Þ at 40 Å increases vð3Þ ðx; IÞ. The change in refractive index Dn=nr is related to the susceptibility as

Dn 1 ¼ 2 Re½vð1Þ ðxÞ þ Re½vð3Þ Þx; IÞ ; nr 2nr 0

ð30Þ

where the first and second terms correspond to linear and third-order nonlinear changes in refractive indices, respectively. The linear Dnð1Þ =nr , the third-order nonlinear Dnð3Þ =nr and the total Dn=nr

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131

Absorption Coefficients (cm-1)

8000 6000 4000 2000 0 -2000 -4000 -6000

0

20

40

60

80

100

120

Photon Energy (meV) Fig. 3. Linear (thin solid line), nonlinear (dashed solid line) and total (thick solid line) absorption coefficients as a function of photon energy  hx for a0 = 0 (black line), 20 Å (blue line) and 40 Å (brown line) in a single ARQW shown in Fig.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

changes in refractive indices are plotted as a function of the incident photon energy  hx in Fig. 4. There are two peaks at E21 þ C21 and E21  C21 in each curve. The linear and nonlinear changes in refractive indices are of opposite sign. Therefore the total change in refractive index is reduced. The calculation of change in refractive index using only the linear term is not accurate enough. The inclusion of nonlinear term is essential in the calculation of total change in refractive index. The change in refractive index shifts towards the high energy direction as a0 increases. The blue-shift in Dn=nr results from the higher energy interval E21 in the dressed well. Moreover, the magnitude of Dn=nr increases with increase in a0 due to increase in certain dipole moments. The NOR susceptibility vð2Þ ð0Þ calculated at various a0 are shown as a function of  hx in Fig. 5. The maximum value of vð2Þ ð0Þ occurs due to the one-photon resonance enhancement at  hx ¼ E21 . The peak of NOR susceptibility shifts to higher photon energy and the intensity of NOR increases with higher a0 . The shifting of NOR susceptibility to higher photon energy for increased a0 results from

Change in Refractive Index (Δn/nr)

0.2

0.1

0

-0.1

-0.2

0

20

40

60

80

100

120

Photon Energy (meV) Fig. 4. Linear (thin solid line), nonlinear (dashed solid line) and total (thick solid line) changes in refractive indices as a function of photon energy  hx for a0 ¼ 0 (black line), 20 Å (blue line) and 40 Å (brown line) in a single ARQW shown in Fig.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Optical Rectification (m/V)

8e-11

6e-11

4e-11

2e-11

0 20

40

60

80

100

120

Photon Energy (meV) Fig. 5. Optical rectification susceptibility vð2Þ ð0Þ as a function of photon energy for different laser dressing parameters in a laser driven asymmetric QW shown in Fig.1. The lines for different colors are for different a0 as described in Fig.1.

Second Harmonic Generations (m/V)

4e-12

3e-12

2e-12

1e-12

0

0

40

80

120

Photon Energy (meV) Fig. 6. Second harmonic generation susceptibility jvð2Þ ð2xÞj as a function of photon energy  hx for different laser dressing parameters in the laser dressed asymmetric QW shown in Fig.1. The lines for different colors are for different a0 as described in Fig.1.

higher separation energies E21 . The increase in the peak value can be easily accounted by higher jl21 j2 ðl22  l11 Þ for increased a0 values. The calculated SGH as a function of photon energy  hx at various a0 are shown in Fig. 6. There are two peaks in SHG [22]. The first and the second peaks are found at 2 hx ¼ E21 and  hx ¼ E21 , respectively. The effect of a0 causes a blue-shift in the photon energy corresponding to the peak value of SHG susceptibility since a0 causes an increment in E21 . Moreover, the peak value of SHG susceptibility increases with higher a0 as a result of higher jl21 j2 ðl22  l11 Þ value. These observations are the same as in NOR studies. 4. Conclusions In the present work, the effect of radiation field on a single ARQW is studied. The radiation field increases the effective well width, but does not change the barrier height and effective mass. The

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Fourier series method is adopted for calculating the bound state energies and the corresponding envelope functions. The linewidths calculated using the electron scattering with LO phonon decreases with increasing laser dressing parameter. The nonlinear effects on the absorption coefficients, NOR and SHG become stronger when the laser dressing parameter increases. For xl ¼ xr , the nonlinearity in absorption coefficient decreases and both OR and SHG vanishes. We hope that the present work will further motivate theoretical and experimental investigations of the second-order nonlinear optical properties in the single ARQW structures. Acknowledgements Authors acknowledge the computational facilities provided by Institute of Physics, Bhubaneswar. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

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