GaAlAs quantum well

Optics Communications 332 (2014) 136–143

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Simultaneous effects of the intense laser field and the electric field on the nonlinear optical properties in GaAs/GaAlAs quantum well Emine Ozturk n Cumhuriyet University, Department of Physics, 58140 Sivas, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 4 April 2014 Received in revised form 26 May 2014 Accepted 1 July 2014 Available online 15 July 2014

In this study, both the linear and nonlinear intersubband optical absorption coefficients and the refractive index changes are calculated as dependent on the applied electric field (F) and the laser field intensity parameter (α0 ). Our results show that the location and the size of the linear and total absorption coefficients and refractive index changes depend on the applied external field effects. The shape of confined potential profile, the energy levels and the squared dipole moment matrix elements are changed as dependent on the F and α0 . We have found that the intense laser field and the applied electric field can be used as a way to control the linear, nonlinear and total optical properties of the quantum well structures. While for F ¼ 0 the intersubband absorption spectrum shows blue-shift up to the critical laser field value ( for about α0 o L=3), this spectrum shows red-shift for laser field values greater than this certain value. In the presence F, we obtain two critical laser field values (for about α0 o L=3 and α0 42L=3). The absorption spectra display blue-shift up to the first critical laser field values, these spectra indicate red-shift up to the second critical values and show again blue-shift for the laser field values greater than these second certain values. Therefore, the variation of the absorption coefficients and refraction index changes which can be appropriate for various optical modulators and infrared optical device applications can be smooth obtained by the alteration of electric field and intense laser field. & 2014 Elsevier B.V. All rights reserved.

Keywords: Linear and nonlinear intersubband transitions Absorption coefficient Refractive index change Intense laser field Electric field

1. Introduction Many experiments have been done on the electronic and optical properties of bulk semiconductors and low dimensional heterostructures under intense dynamic fields [1–4]. Owing to the development of modern technology the fabrication of artificial semiconductor nanostructures is now feasible [5,6]. Among these structures, the most extensively studied is the GaAs/AlxGa1  xAs quantum wells (QW) because their interesting ‘new’ intrinsic properties are of great interest for technological applications in high-performance electronic and optoelectronic devices [7–10]. Linear and nonlinear optical processes related to the intersubband transitions in low-dimensional semiconductor systems have attracted considerable attention due to the strong quantum confinement effect, leading to small energy separation between state levels, large values of dipole transition matrix elements and possibility of achieving resonance conditions. The large oscillator strengths observed in semiconductor QWs are responsible for the large detected optical transition dipole moments. These dipole moments are clear evidence that the large optical nonlinearities

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may be obtained in semiconductor QW structures. Especially, linear and nonlinear optical properties in these structures are intensely studied due to their potential for device applications in the infrared region of the electromagnetic spectrum [11–13]. The nonlinear optical properties of the low-dimensional systems are significantly dependent on the existence of some asymmetry of the confining potential [14–17]. Such an asymmetry in the potential profile can be obtained either by applying an electric field or by compositionally grading the potential profile [18–21]. It is also well known that the shape of the confining potential of the QW structures significantly affects the nonlinear optical properties. The linear and nonlinear optical absorptions based on intersubband transitions and the refractive index changes in semiconductor QW structures with different confinement potentials have been studied extensively by many authors [19,22–28]. As it is known, the effect of a high-frequency intense laser field also leads to major modifications in the shape of the confining potential of QW structures [19,29,30]. The effects of intense laser field on the confining potential and corresponding bound states play an important role in the optoelectronic device modeling. Lima et al. [31] have studied the transition from single to double QW potential induced by intense laser field in a semiconductor QW. Their numerical results show that the rapid approximation of the excited levels for GaAs/ GaAlAs QWs with the increase in the laser field intensity indicates

E. Ozturk / Optics Communications 332 (2014) 136–143

the possibility of enhancing the population inversion in the optical pumping scheme, which is interesting for the design of powerful QW lasers. Therefore, we consider that it is important to investigate the effect of such a field on both the linear and the nonlinear optical properties under the applied electric field. The intense laser field and electric field can be used as a way to control the electronic and optical properties of the quantum well structures. Thus, the effect of an intense high frequency laser field on the intersubband optical transitions in a semiconductor quantum well under the applied electric field is investigated in this work. These transitions are predicted to have a narrow bandwidth and are of practical interest for the design of tunable optical semiconductor devices.

2. Theory Within effective-mass approximation, the state energy levels and wave functions for electrons in GaAs–Ga1  xAlxAs QW grown along the z-axis in the presence applied electric field can be provided by solving the on dimensional Schrödinger equation with a proper Hamiltonian [10,17,25]. 2



ℏ2 d þ VðzÞ þeFz 2 mn dz2

! ϕðzÞ ¼ EϕðzÞ

ð1Þ

where mn is the electron effective mass, e is the electron charge, F is the strength of the electric field applied in z-direction. Looking to add the nonresonant polarized intense laser effects along the z-direction, it has been followed the Floquet method [32]. For sufficiently high frequencies, a closed-form expression for the dressed potential valid for α0 r L=2 is presented. Interestingly, for α0 4 L=2, the formation of a double-well potential is predicted. Consequently, the second term at the left hand side in Eq. (1) must be replaced by VðzÞ-hV iðz; α0 Þ where for α0 r L=2    V0 L=2 þz Θðα0  z  L=2Þarccos α0 π   L=2 z þΘðα0 þ z  L=2Þarccos α0

hV iðz; α0 Þ ¼

and for α0 4 L=2      V0 z  L=2 z þ L=2 hV iðz; α0 Þ ¼ P þ arcsin arcsin α0 α0 P  L L  α0 o z r  þ α0 2 2 here α0 ¼

!   1=2 Ι laser  e  8π 1=2 mn c Ω2

βð3Þ ðω; IÞ ¼ ω

137

 rffiffiffiffi μ I jM 21 j2 n εr 2ε0 nr c

2 44jM 21 j2 

e4 σ v ℏΓ 21 ðE21  ℏωÞ2 þ ðℏΓ 21 Þ2

o2

n o3 jM 22  M 11 j2 ðE21  ℏωÞ2  ðℏΓ 21 Þ2 þ 2ðE21 ÞðE21  ℏωÞ 5 ðE21 Þ2 þ ðℏΓ 21 Þ2

ð5Þ and the linear and the third-order nonlinear refractive index changes can be expressed as [39–40,42], respectively.  ð1Þ  Δn ðωÞ σ v e2 jM 21 j2 ðE21  ℏωÞ ¼ ð6Þ nr 2n2r ε0 ðE21  ℏωÞ2 þ ðℏΓ 21 Þ2  ð3Þ  Δn ðω; IÞ μc e4 σ v I ¼  3 jM 21 j2 nr 4nr ε0 ½ðE21  ℏωÞ2 þ ðℏΓ 21 Þ2 2 " h i 2  ðM 22  M 11 Þ ðE21  ℏωÞ ðE21 ÞðE21  ℏωÞ  ðℏΓ 21 Þ2  4ðE21  ℏωÞjM 21 j2  2 2 ðE21 Þ þ ðℏΓ 21 Þ



 ðℏΓ 21 Þ2 ½2ðE21 Þ  ℏω 

ð7Þ

here I is the optical intensity of incident electromagnetic wave (with the angular frequency ω) that excites the structure and leads to the intersubband optical transition, μ is the magnetic permeability, ω is the angular frequency of the incident photon, (E21 ¼ E2  E1 ), E1 and E2 denote the quantized energy levels for the ground and second states, respectively, εr is the real part of the permittivity, nr is the refractive index, σ v is the carrier density. In additional, ℏΓ 21 ¼ ℏ=T 21 are damping term associated with the life time of the electrons due to intersubband scattering. This quantity is related to a high extent with electron–phonon interaction, and they would need to be evaluated taking into account the particular features of the phonon modes and the electronic states in the structure. However, keeping the pace with previous works, we shall adopt a fixed value for them (taken as T 21 ¼0.14 ps in this work), which have been already reported in the literature [10,28,39,41]. The dipole moment matrix elements is defined by Z ϕnf ðzÞzϕi ðzÞdz; ði; f ¼ 1; 2Þ ð8Þ Mf i ¼ The total absorption coefficient is given by

ð2aÞ

βðω; IÞ ¼ βð1Þ ðωÞ þ βð3Þ ðω; IÞ

ð9Þ

and the total refraction index change can be written as Δnðω; IÞ Δnð1Þ ðωÞ Δnð3Þ ðω; IÞ ¼ þ U nr nr nr

ð2bÞ

ð3Þ

is the laser-dressing parameter [33], Ι laser is the average intensity of the laser, Ω is the laser frequency,V 0 is the discontinuity in the conduction band edge, L is the well width and c is the velocity of the light. Further details about dressed potential in Eq. (2) can be found in Refs. [31,34–37] After the energies and their corresponding wave functions are obtained, the first-order linear absorption coefficient βð1Þ ðωÞand the third-order nonlinear absorption coefficient βð3Þ ðω; IÞfor the intersubband transitions between two energy states can be clearly calculated as [38–41], rffiffiffiffi μ e2 σ v ℏΓ 21 βð1Þ ðωÞ ¼ ω ð4Þ jM 21 j2 εr ðE21  ℏωÞ2 þ ðℏΓ 21 Þ2

ð10Þ

3. Results and discussion We have theoretically investigated the linear and nonlinear intersubband optical properties for the (1–2) transition in GaAs– Ga1  xAlxAs QW under an applied electric field and intense laser field. In this study, for numerical calculations, we have taken mn ¼0.0665 m0 (where m0 is the free electron mass), the barrier height V0 ¼ 228 meV, the well widths L ¼180 Å, σ v ¼ 3  1016 cm  3 , nr ¼3.2, and I ¼0.4 MW/cm2. Fig. 1 demonstrates the first two electron energy levels and the associated densities of probability (solid curves) and the shape of the potential profiles (dashed curves) for the QW under intense laser radiation via the Eq. (2). Applied electric field values are F ¼0, 20 and 40 kV/cm. For F¼0, the evolution of the QW profile associated with the variation in the laser field can be seen in these figures as the transition from a single to a double QW potential. By applying the electric field the QWs are tilted and the electrons move towards the left-side of the structure, thus with increasing electric field the energy level of the ground state decreases and the electrons in this state are mostly localized in

E. Ozturk / Optics Communications 332 (2014) 136–143

350

350

300

300

250

250

200

200

E i (meV)

E i (meV)

138

150 100

100

50

50

0

0

-50

-50 -200

-100

0

z(

100

-200

200

-100

0

z(

)

350

350

300

300

E i (meV)

250

E i (meV)

150

200 150

100

200

100

200

)

250 200 150

100

100

50

50

0 -200

-100

0

z(

100

200

-200

-100

0

z( )

)

Fig. 1. For a) α0 ¼ 0, b) α0 ¼ 40 Å, c) α0 ¼ 90 Å and d) α0 ¼ 120 Å, the first two electron energy levels and the associated densities of probability (solid curves) and the potential profiles (dashed curves) under the applied electric field (The black, red and blue curves are for F¼ 0, 20 and 40 kV/cm, respectively).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

150 120

E 21 (meV)

the left-side of the well. While for F¼ 0 the electron wave function has a symmetric character, in these QWs the electron wave function shows an asymmetric character for applied electric fields. This configuration is important for the nonlinear optical response of the system. Also, the first two electron energy levels and the associated densities of probability give the significant values of the square dipole moment matrix elements. Fig. 2 shows the variation of the energy difference between the ground and second states as a function of the electric field for different laser field values (α0 ¼ 0, 40, 90 and 120 Å). The inset displays the change of the energy level of the ground (solid curves) and the second (dashed curves) state under the applied electric field. As seen from this inset, with the increasing electric field the energy levels of the ground states decrease, between F¼0 and F¼50 kV/cm these energy ranges are 8.8, 4.7, 5.6 and 23.4 meV (respectively for α0 ¼ 0, 40, 90 and 120 Å). The influence of F in the case of the ground state is clearly more pronounced for α0 ¼120 Å. We can explain this behavior as follows; for α0 ¼0, the well width is great and the ground state energy level is small at the bottom of the quantum well (in comparison to α0 ¼ 40 and 90 Å). This energy level is affected from the changing potential profile by applied electric field. By increasing the laser field (up to α0 r L=2) the effective well width decreases, the energy level increases and the wave functions of this state tends to localize in the upper part of the potential well with wider width. While for α0 rL=2the ground state energy level changes slightly with F, the variation of this energy is evident for α0 4 L=2. For α0 ¼120 Å, the QW structure has the property of a symmetric double wells and is shallower. Thus, this structure changes significantly with increasing electric field.

90 60

B

30

A

C

D

0

0

10

20

30

40

50

F (kV/cm) Fig. 2. The energy difference between the ground and second states as a function of the electric field for different laser field values. The inset shows the change of the energy level of the ground (solid curves) and the second (dashed curves) state under the applied electric field. (Notations A, B, C, D are associated with α0 ¼ 0, 40, 90, 120 Å, respectively.).

For α0 ¼ 120 Å, the energy range of the ground state with increasing F are different from α0 ¼ 0, 40 and 90 Å. While by increasing F the energy levels of the second states are nearly constant for α0 ¼ 0 and 40 Å, these levels are decrease for α0 ¼ 90 and 120 Å. In the presence of the applied electric field the potential profile turns to a

E. Ozturk / Optics Communications 332 (2014) 136–143

triangular shape and QW bottom is shifted downwards. This effect dominates over the effective reduction of the well width and barrier height due to the potential profile deformation. Also, the energy differences of the first two states are nearly constant for α0 ¼ 40 and 120 Å, since with increasing electric field both E1 and E2 are changed at the same rate. Whereas for α0 ¼ 0, the variation

D

2

M 21 (x10 ) (cm )

30

-14

20

A

2

C

10

0

B

0

10

20

30 F (kV/cm)

40

50

Fig. 3. The square dipole matrix element as a function of the electric field for different laser field values. (Notations A, B, C and D are associated with α0 ¼ 0, 40, 90, 120 Å, respectively.).

of E2  E1 increases with increasing F, because the electric field pushes the electrons towards the descending potential barrier, and while E1 decreases, E2 increases lightly. In contrast to behavior for α0 ¼ 0, for α0 ¼ 90 Å, the change of E2 E1 decreases with increasing F, and while E1 decreases lightly, E2 decreases clearly, thus two state energies get to close to each other. By considering the variation of the energy difference, it should point out that by applying electric field we can obtain a red or blue-shift in the intersubband optical transitions. Fig. 3 presents the dependence of the square dipole matrix element on the electric field for various laser parameters. By increasing the electric field the potential wells are tilted, the geometric confinement of the electrons changes thus it can penetrate into the left potential barriers easily. This penetration modifies the subband dispersion relations and causes a variation in the overlap function between the ground and second subband. For α0 ¼ 0 the square dipole matrix element decreases with F, because in the electric field the ground-state wave function moves to the left hand of the well while the first-excited-state wave function shifts to the opposite direction, as shown in Fig. 1a. Thus the overlap between E1 and E2 state wave functions decreases with F. Also, the change of the square dipole matrix element by the applied electric field are nearly constant for α0 ¼ 40 Å while this change increases for α0 ¼ 90 Å due to the overlap between the first two state wave functions. For α0 ¼120 Å, the variation of the square dipole matrix element with increasing F are different from α0 ¼0, 40 and 90 Å since the applied electric field change the probability of finding of the electrons in right or left hand of the double QW.

4000

4000 F=0 F=20 kV/cm F=40 kV/cm

2000

3000 Absorption coefficients ( cm-1 )

Absorption coefficients ( cm -1)

3000

1000 0 -1000 -2000

2000 1000 0 -1000 -2000 -3000

-3000 -4000 0.01

139

0.02

0.03

0.04

0.05

-4000 0.04

0.06

0.05

0.06

0.07

Photon energy ( eV )

Photon energy ( eV )

4000 3000 Absorption coefficients ( cm -1)

Absorption coefficients ( cm -1)

3000 2000 1000 0 -1000 -2000

2000 1000 0 -1000 -2000 -3000

-3000 -4000 0.03

0.04

0.05

Photon energy ( eV )

0.06

0.07

-4000 0.00

0.01

0.02

0.03

0.04

0.05

Photon energy ( eV )

Fig. 4. For a) α0 ¼ 0, b) α0 ¼ 40 Å, c) α0 ¼ 90 Å and d) α0 ¼ 120 Å, the linear (dashed curves), nonlinear (dotted curves) and total (solid curves) absorption coefficients as a function of the photon energy under the applied electric field. (The black, red and blue curves are for F ¼0, 20 and 40 kV/cm, respectively.)(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E. Ozturk / Optics Communications 332 (2014) 136–143

Respectively, for α0 ¼0, 40, 90 and 120 Å in Fig. 4 (a, b, c, and d), we plot the linear, nonlinear and total absorption coefficients as a function of the photon energy, without and with applied electric field. There is no shift at the resonant peak position with incident optical intensity. The total absorption coefficient βðω; IÞ reduces by the negative βð3Þ ðω; IÞ contribution in the presence incident optical intensity. Therefore, the contribution of both the linear and the nonlinear terms should be considered in the calculation of βðω; IÞ near the resonance frequencyðE2 E1 ffi ℏωÞ. These resonance frequency values and the resonant peak values of the linear absorption coefficient are given in Table 1 for α0 ¼ 0, 40, 90 and 120 Å, without and with applied electric field. Also, the squared intrasubband matrix element,jM 22  M 11 j2 , is offered in Table 1 for different α0 values under the applied electric field. The intrasubband matrix elements are zero for zero electric field due to the odd symmetry of the integrand. As seen from Fig. 4 (a, b, and d), because the

differences between the energy levels increase, the absorption spectra show blue-shift with F for α0 ¼ 0, 40 and 120 Å, respectively. This shift is very small for α0 ¼ 40 Å (see Fig. 4b). The resonant peak of total absorption coefficient can be bleached at sufficiently specific incident optical intensities. This is due to the negative contribution of the third-order nonlinear term. From Fig. 4d, we can conclude that the total optical absorption coefficients branch distinctly with F. Also, for α0 ¼ 90 Å the absorption

70 60 50

E21 (meV)

140

Table 1 For different laser field values the squared intrasubband matrix element, the resonance frequency values and the resonant peak values of the linear absorption coefficient under the applied electric field.

α0 ðÅÞ

F¼ 0

F¼ 20

F¼ 40

F¼ 0

F¼ 20

F¼40

F¼ 0

F ¼20

F¼ 40

0 40 90 120

0 0 0 0

3.34 0.16 8.62 4.55

9.52 0.45 22.71 0.26

5.72 8.28 7.95 3.42

6.00 8.34 7.49 3.87

6.77 8.51 6.29 4.14

3265 3345 3221 3330

3179 3341 3046 3272

3001 3330 2786 3345

20

βð1Þ ðωÞðcm  1 Þ

0

0.06

0.06

0.04

0.04

0.02

0.02

0.00

-0.02

-0.04

-0.04

-0.06

-0.06 0.04

0.05

0.06

0.04

0.05

0.06

0.06

0.04

0.04

0.02

0.02

0.00

-0.02

-0.04

-0.04

-0.06

-0.06 0.05

90

( ) 0

120

150

0.06

0.07

0.00

-0.02

0.04

60

Photon energy ( eV )

n /n r

n /n r

Photon energy ( eV )

0.03

30

0.00

-0.02

0.03

0

Fig. 6. The energy difference between the first two states as a function of the intense laser field for different electric field values.

n /n r

n /n r

ωð1013Þ ðs  1 Þ

0.02

30

10

jM 22  M 11 j2 ð10  14 cm2 Þ

0.01

40

0.06

Photon ener gy ( eV )

0.07

0.00

0.01

0.02

0.03

0.04

0.05

Photon energy ( eV )

Fig. 5. For a) α0 ¼ 0, b) α0 ¼ 40 Å, c) α0 ¼ 90 Å and d) α0 ¼ 120 Å, the linear (dashed curves), nonlinear (dotted curves) and total (solid curves) refractive index changes as a function of the photon energy under the applied electric field. (The black, red and blue curves are for F ¼0, 20 and 40 kV/cm, respectively.)(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E. Ozturk / Optics Communications 332 (2014) 136–143

4000

Absorption coefficients ( cm -1)

3000

F

E

A D

0.12

C

B

0.08

2000 1000

E A

-0.04

-1000

-0.08

-2000

-0.12 0.02

0.04

0.00

0.06

D

0.02

4000

0.04

0.06

0.08 E

B

A

3000

E

0.06

C

D

A

0.04

D

F

2000

B

C

0.02

F

n /n r

Absorption coefficients ( cm -1)

C

Photon energy ( eV )

Photon energy ( eV )

b

B

0.00

0

0.00

F

0.04

n /n r

a

141

0.00 -0.02

1000

-0.04 -0.06

0

-0.08 0.00

0.02

0.04

0.00

0.06

0.02

Photon energy ( eV )

4000

0.06

0.08 B

E

3000

0.06

A

F D

E F

0.04

C

D

A

B C

0.02 2000

n /n r

Absorption coefficients ( cm -1)

0.04 Photon energy ( eV )

0.00 -0.02

1000

-0.04 -0.06

0

-0.08 0.00

0.02

0.04

0.06

Photon energy ( eV )

0.00

0.02

0.04

0.06

Photon energy ( eV )

Fig. 7. For a) F¼ 0, b) F¼ 20 kV/cm, c) F¼ 40 kV/cm, the linear (dashed curves) and total (solid curves) absorption coefficients versus the photon energy for different intense laser field values. (Notations A, B, C, D, E and F are associated with α0 ¼ 0, 40, 70, 90, 120 and 140 Å, respectively.).

Fig. 8. For a) F ¼0, b) F¼ 20 kV/cm, c) F¼ 40 kV/cm, the linear (dashed curves) and total (solid curves) refractive index changes versus the photon energy for different intense laser field values. (Notations A, B, C, D, E and F are associated with α0 ¼ 0, 40, 70, 90, 120 and 140 Å, respectively.).

spectra show red-shift with F (see Fig. 4c), since the difference between the energy levels decreases. In our calculations, the magnitude and position of the absorption coefficient depend on the electric field strength and the intense laser field value.

Fig. 5 shows the linear, nonlinear and total refractive index changes versus the photon energy for different intense laser field values, without and with electric field. This figure clearly shows that the total refractive index change is related to the electric field

142

E. Ozturk / Optics Communications 332 (2014) 136–143

60 50 40 30 20 10 0

0

30

60

90

120

150

Fig. 9. The square dipole matrix element as a function of the intense laser field for different electric field values.

and intense laser field. As the electric field increases, the total refractive index changes vary in the magnitude and also shift toward higher or lower energies. The main reason for this resonance shift is the variation between energy intervals of two different electronic states which an optical transition occurs. All refractive index changes are changed in energy and magnitude with changing electric field and intense laser field. Fig. 6 demonstrates the relation of the energy difference between states versus the laser field amplitude. As seen from this figure, the state energy difference changes abruptly whenever the laser-field amplitude reaches the certain values. These critical amplitude values affect both the linear and nonlinear intersubband transitions. This situation is related with the modification in the QW potential profile geometry induced by the laser effect. Respectively, for F ¼0, 20 and 40 kV/cm in Fig. 7 (a, b, and c), we plot the linear and total absorption coefficients as a function of the photon energy for different intense laser field values. While for F¼0 the intersubband absorption spectrum shows blue-shift up to the critical laser field value (α0 o 67 Å), this spectrum shows redshift for laser field values greater than this certain value. Due to the asymmetric character of the electron wave function in the presence F, the particle is mostly confined on the left-hand side of the structure. Thus, the laser field and the applied electric field sensitivity of the electronic structure is quite different compared with results of zero electric field. For F¼20 and 40 kV/cm, we obtain two critical laser field values (for about α0 o L=3 andα0 42L=3). Respectively, for F ¼20 and 40 kV/cm the absorption spectra display blue-shift up to the first critical laser field values (α0 o65 and α0 o60 Å), these spectra indicate red-shift up to the second critical values (α0 o130 and α0 o 120 Å) and show again blue-shift for the laser field values greater than these second certain values. Such a dependence of the critical laser-field amplitude on the applied external field strengths can be very useful for diverse potential device applications. As seen from Fig. 7, the linear and total intersubband absorption coefficients are changed in energy and size with increasing laser-field value over a wide range of the laser-field amplitude. We can explain this behavior as follows; by increasing the laser field the effective well width decreases, which in turn increases the energy difference between the first and second state (up to the first critical laser field values) and afterwards begins to decrease when the laser field is further increased. By further increasing the laser field, the second state is pushed up to the top of the well so that is no longer localized in the narrow lower part of the well. With increasing α0 value the state wave functions tends to localize in the upper part

of the potential well with wider width, thus energy levels get to close to each other. When the applied electric field is further increased, the electronic structure is dramatically changed and a third triangular potential is created between the left-side of the well and the left barrier of the structure. By increasing applied electric field and intense laser field, the energy difference changes and the resonant peak of the linear and the total optical absorption coefficients gives different values for different electric fields and intense laser fields. (In order to consider this behavior more clearly see Fig. 1.) For different intense laser field values, Fig. 8 defines the linear and total refractive index changes as a function of the photon energy for F¼0, 20 and 40 kV/cm. We conclude that, the variation of refractive index changes as dependent on both the laser field and the electric field is similar to the variation of the absorption coefficients. The total refractive index change is reduced by the negative nonlinear refractive index change contribution. We calculated jM 21 j2 which a useful quantity is in the experimental works. Fig. 9 indicates the change of the square dipole matrix element with the intense laser field for different electric fields. Our results revealed that the jM 21 j2 values depend on the applied electric field and intense laser field. The explanation should be connected with the variation of both the transition energy, E2  E1, and the overlap between E1 and E2 state wave functions. The electron distribution becomes asymmetric due to the electric field polarization for both the electronic states. The α0 effect on the square dipole matrix element is essentially associated with the behavior of the electron wave functions in the laser radiation.

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