Multipoint control of optical bistability in a defect slab doped with single-layer of graphene nanostructure

Optik 127 (2016) 8857–8863

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Multipoint control of optical bistability in a defect slab doped with single-layer of graphene nanostructure Gh. Solookinejad, M. Panahi, E. Ahmadi Sangachin, Seyyed Hossein Asadpour ∗ Department of Physics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

a r t i c l e

i n f o

Article history: Received 28 January 2016 Received in revised form 26 May 2016 Accepted 28 June 2016 PACS: 42.50,-p 42.65.pc Keywords: Optical bistability Group velocity Graphene nanostructure

a b s t r a c t In this study, we investigated the optical bistability (OB) properties of transmitted weak probe light from a defect dielectric medium doped by four-level graphene nanostructure. By using the transform matrix and density matrix methods, we explore the features of transmitted light versus incident light for studying the OB behaviors in a defect slab. Our numerical results show that the threshold of OB in multipoint of incident light can be easily manipulated by some controllable parameters such as Rabi-frequencies of applied fields and optical thickness of the slab. Moreover, we find that the group velocity of transmitted pulse from the defect slab can be adjusted by tuning the optical thickness of the slab. We hope that our study may be suitable for the future all-optical based devices in Nano sizes. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction Controlling light by light due to its potential application in all-optical sensors, all-optical memories and all-optical transistor have been discussed by many research groups in quantum and nonlinear optics [1–12]. One the most interesting phenomena which has potential application for developing all-optical systems are optical bistability and multistability. The experimental and theoretical investigation of OB in two, three and four-level atomic systems in an optical ring cavity has been discussed by many research groups [13–22]. For example, Joshi et al. experimentally studied the optical bistable behavior in an optical ring cavity filled with a collection of three-level rubidium atoms, interacting with two collinearly propagating laser beams. Sheng et al. [14] experimentally observed OM in an optical ring cavity containing three level -type Doppler-broadened rubidium atoms. The realization of OB and OM in semiconductor quantum wells (SQWs) and quantum dots (SQDs) have also been reported by many groups [23–30]. For example, Joshi and Xiao [23] studied the OB behavior in a unidirectional ring cavity containing three-level Ladder type SQWs. In fact, SQWs and SQDs are candidate as a two-dimensional (2D) electron gas. They have properties similar to atomic systems such as the discrete levels, but with the advantages of high nonlinear optical coefficients and large electric dipole moments, due to the small effective electron mass. They are interested due to wide applications for building the future all-optical systems and devices [31–40]. Graphene because of its potential usages in nanoelectronics and condense matter physics due to its unusual electronic properties and unique optical properties has been interested recently. Graphene is a single-atom thick allotrope of carbon with unusual two-dimensional (2D) Dirac like electronic excitations [41–43], which can be controlled effectively by external electromagnetic fields [44]. The tunability of the charge carrier density and conductivity by the bias voltage make graphene

∗ Corresponding author. E-mail address: [email protected] (S.H. Asadpour). http://dx.doi.org/10.1016/j.ijleo.2016.06.104 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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Fig. 1. (a) LLs near the K point superimposed on the electronic energy dispersion without a magnetic √ field E = ±vF |p|. The magnetic field condenses the original states in the Dirac cone into discrete energies. The LLs in graphene are unequally spaced: ∝ B. (b) Energy level diagram and optical transitions in graphene interacting with two continuous-wave control fields 1 and 2 and a weak pulsed probe field p. The states |1, |2, |3 and |4 correspond to the LLs with energy quantum numbers n = −2, −1, 0, 1, respectively. Graphene monolayer is a one-atom-thick monolayer of carbon atoms arranged in a hexagonal lattice, which we will treat as a perfect two-dimensional (2D) crystal structure in the x-y plane.

operates in both THz and optical frequency ranges. Belyanin and his coworkers [45–48], studied more linear and nonlinear optical properties of graphene system under strong magnetic field. Experimental and theoretical Investigations of optical solitons have also been analyzed in graphene [49–57]. Investigation of matched infrared solitons pairs based on FWM in a monolayer of graphene system has been presented in Ref. [52]. In this work, we will analyze the OB features of transmitted light from defect slab medium doped with a single-layer of graphene nanostructure by using the transform matrix and density matrix methods. We will demonstrate the impact of some controllable parameters such as optical thickness and Rabi-frequencies of applied fields on OB evolution of weak probe light through defect slab. 2. Model and equation 2.1. Pulse propagation in a dielectric medium By using the transfer-matrix method, the reflection and transmission coefficients of the incident mono-chromatic wave with frequency ωp in a dielectric slab can be expressed as [58]:  √ −(i/2)(1/ ε(ωp ) − ε) sin(kd) r(ωp ) = (1) ,   cos(kd) − (i/2)(1/ ε(ωp ) + ε(ωp )) sin(kd) t(ωp ) =



cos(kd) − (i/2)(1/

1

ε(ωp ) +



,

(2)

ε(ωp )) sin(kd)

For the defect dielectric slab, the dielectric function can be divided into two parts; ε(ωp ) = εb + (ωp )

(3)

where εb = 1 is the background dielectric constant and (ωp ) is the susceptibility produced by the doped four-level graphene system. From Eqs. (1)–(3), we can realize that the reflection and transmission coefficients depend on the thickness of the slab and susceptibility of the doped graphene system. In a resonance condition, thethickness of the slab  the   is employed √ √ as d = 4 εb 0 /2m , whereas, for the off-resonance condition, it is considered as d = 4 εb 0 /(2m + 1) . Here, m is an integer number, and in the following numerical calculations is chosen as m = 300. Note that the other values also are available. To investigating the OB behavior of the slab, one can use the relation between transmitted intensity Ut and the incident intensity Uin which is defined as follow: Uin = Ut /T

(4)

In the next section, we explain the level structure of single-layer graphene system under strong magnetic field which we use as a defect layer in slab medium. 2.2. Single-layer of graphene system The level structure of the doped graphene system under a strong magnetic field is presented in Fig. 1. A monolayer of graphene system is a one-atom-thick monolayer of carbon atoms arranged in a hexagonal lattice. The chosen transitions between Landau levels (LLs) can be dipole allowed when we use the peculiar selection rules for electrons in graphene i.e., |n| = ±1 (n is the energy quantum number). Some nonlinear optical features of this system have been discussed in Refs. [49–52,56,57]. By using an external magnetic field in the range 0.01-10T, one can anticipate that the

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optical transitions between the adjacent LLs fall into the infrared to THz region according to ωc  36 B(Tesla) meV . The effective-mass Hamiltonian for a monolayer graphene in the presence of magnetic field Bˆz can be given [52]:

⎛

ˆ x − i ˆy 

0

0

0

0

0

0

0

0

ˆ x − i ˆy 

0

⎜ ⎜ ˆ x + iˆ y

ˆ 0 = vF ⎜ H ⎜

⎝

0

⎞

⎟ ⎟ ⎟ ⎟ ˆ x + i ˆy ⎠  0

(5)

0 o

where vF = 3 0 /(2a) ≈ 106 m/s ( 0 ∼2.8 eV and a = 1.42 A are the nearest-neighbor hopping energy and C C spacing) is a ˆ = p ˆ + eA/cdenotes ˆ is the electron momentum the generalized momentum operator, p band parameter (Fermi velocity),  operator, e is the electron charge, and A is the vector potential, which is equal to (0, Bx) for a static magnetic field. Here, we assume that the monolayer of graphene system interacts with a bichromatic electric fields consisted of the probe field and opt = ic E/ω the continuous-wave (CW) control field. Therefore, the vector potential of optical fields A , (E = E p + E 23 )can be ˆ in the added to the vector potential of the magnetic field in the generalized momentum operator    Hamiltonian. The probe field and CW control field can be written as E p = e p+ Ep+ exp(−iωp t + ik p .r ) and E 23 = e 2+ E2+ + e 2− E2− exp(−iω2 t + ik 2 .r). Here e + and e − are the unit vectors of the right-hand circular (RHC) and left-hand circular (LHC) polarize basis, respectively. The j

j

RHC (LHC) polarized fields E2+ (E2− ) with frequencies ω2 is applied to transitions|2 → |3 (|3 → |4). Moreover, the interLandau-level transition |1 → |4 interact with a RHC polarized probe field with amplitude and carrier frequency E3 andωp , respectively. The resulting interaction Hamiltonian can be expressed in the following form: ˆ int = vF . H

e Aopt c

(6)

The density matrix of Dirac electrons in graphene coupled to the infrared laser fields by utilizing the Liouville’s equation ∂ ˆ ˆ int , ] ˆ ) ˆ ) = − i [H ˆ + R( ˆ can be obtained. Here R( ˆ indicates incoherent relaxation which may originate from disorder, ∂t

interaction with phonons, carrier-carrier interactions. As a result, the density-matrix equations of motion for four-level graphene system can be written as: ˙ 11 = i˝1 41 − i˝1 14 ˙ 22 = − 2 22 + i˝2 32 − i˝2 23 , ˙ 33 = − 3 33 + i˝2 23 + i˝3 43 − i˝2 32 − i˝3 34 , ˙ 44 = − 4 44 + i˝1 14 + i˝3 34 − i˝1 41 − i˝3 43 , 2 21 + i˝2 31 − i˝1 24 , 2 3 ) 31 + i˝2 21 + i˝3 43 − i˝1 34 , = −(iω2 + 2 4 ) 41 + i˝1 ( 11 − 44 ) + i˝3 31 , = −(iω1 + 2 3 + 2 = −(iω2 + ) 32 + i˝2 ( 22 − 33 ) + i˝3 42 , 2 4 + 2 = −(iω1 + ) 42 + i˝1 12 + i˝3 32 − i˝2 43 , 2 4 + 3 = −(iω1 − iω2 + ) 43 + i˝1 13 2

˙ 21 = − ˙ 31 ˙ 41 ˙ 32 ˙ 42 ˙ 43

(7)

+i˝3 ( 33 − 44 ) − i˝2 42 where ω1 = (εn=1 − εn=−2 )/ − ω1 = (εn=1 − εn=0 )/ − ω3 and ω2 = (εn=0 − εn=−1 )/ − ω2 represent the corresponding √ frequency detunings, and εn = sgn(n)ω c |n|is the energy of the Landau level for electrons near the Dirac point, with n = √ 0, ±1, ±2, ..., ωc = 2vF /lc , and lc = c/eB implies the magnetic length. 41 .e 1 )E1 /(2), ˝2 = ( 32 .e 2 )E2 /(2), and ˝3 = ( 43 .e 3 )E3 /(2) are the corresponding one-half Rabi-frequencies ˝1 = ( mn = m| |n with

|n = e. m|r |n = εnie m| v being the dipole matrix element for the relevant optical transition. j (j = F −εm 2, 3, 4), corresponds to the decay rate of the state |j. We use the above Eq. (8) to obtain the solution for 41 , therefore the susceptibility can be written as: =

2N| 41 |2 , εr ?˝1 41

Here, N and εr being the 2D electron concentration of graphene and the substrate dielectric constant, respectively.

(8)

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Fig. 2. The transmitted intensity of probe field Ut versus the incident intensity Uin for ˝2− = 0.6 (solid), ˝2− = 1.1 (dashed) and ˝2− = 2 (dotted). The

selected parameters are = 3 × 1013 S −1 , 4 = 3 = , 2 = 0.05 , m = 200, and p = 2 = 3 = 0.

Fig. 3. The transmition coefficient versus probe detuning for different values of LHC field. The selected parameters are same as Fig. 2.

3. Results and discussion We now turn on the numerical results of Eq. (4) via Eqs. (2) and (8). It is noted that the carrier frequency √ of the probe field can be estimated approximately the same amount of the transition frequency ω41 = (εn=1 − εn=−2 )/ = ( 2 + 1)ωc which is on the order of ω41 ∼1.86 × 1014 S −1 for graphene at the magnetic field of the value B = 1T [46–49]. While, when the magnetic field reaches up to 5T the transition frequency is estimated to be ω41 ∼3.87 × 1014 S −1 . This is confirmed that the bistable curve is located within the infrared region (p = 0 ≈ 5 m). According to the numerical estimate based on Refs. [46] and [55]. We can take a reasonable value for the decay rate 3 = 3 × 1013 S −1 , and assuming 4 = 3 , 2 = 0.05 3 these values depend on the sample quality and the substrate used in the experiment [46–49]. Besides, the √ dipole moment between the 14 |∼evF /ε4 − ε1 ∝ 1/ B. The electron concentration transitions |1 ↔ |2 in the graphene has a magnitude of the order | is N  5 × 1012 cm−2 and the substrate dielectric constant is εr  4.5 [56,57]. Dependence of transmitted intensity of probe field Ut versus the incident intensity Uin for different values of LHC field and in the absence of RHC field is demonstrated in Fig. 2. It can be seen that for ˝2− = 0.6 (solid line) the threshold of optical bistability appears in the three points of the incident light system and by enhancing the Rabi frequency ˝2− to 1.1 (dashed line) and 2 (dotted line), the threshold of optical bistability reaches to two and one point with low threshold intensity. Physically, the transmission properties of incident light to defect slab can be modified by controlling the Rabi-frequency of applied field. So, it can be seen that the variational Rabi frequency of applied field lead to reduction in threshold of OB. In Fig. 3, we display the transmission coefficient of incident light versus detuning of probe light for different values of LHC coupling field. It is realized for ˝2− = 0.6 (solid line), the transmission curve has a dip and its value is near to 0.9. However, by enhancing the value of LHC field to 1.1 (dashed line), the value decreases and also there is a dip in transmission curve. By enhancing the LHC field to 2 (dotted line), the value of transmission coefficient reaches to 0.8 and transmitted curve corresponds to a peak. According to Agarwal’s reciprocity theorem, in a lossless slab, each peak (dip) in reflection or transmission shows subluminal (superluminal) light propagation. In superluminal light propagation, the group velocity of light pulse can exceed the velocity of light in vacuum

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Fig. 4. The transmitted intensity of probe field Ut versus the incident intensity Uin for ˝2− = 0.02 (solid), ˝2− = 0.1 (dashed) and ˝2− = 0.2 (dotted).

The selected parameters are = 3 × 1013 S −1 , 4 = 3 = , 2 = 0.05 , m = 300, ˝2+ = 1.8 , andp = 2 = 3 = 0.

Fig. 5. The transmition coefficient versus probe detuning for different values of LHC field. The selected parameters are same as Fig. 4.

and even becomes negative. Therefore, in Fig. 3, we can find that the switching from superluminal to subluminal light propagation or vice versa can be possible by adjusting the intensity of LHC field. In Fig. 4 we show transmitted intensity of probe field Ut versus the incident intensity Uin for different RHC Rabifrequencies, i.e. ˝2+ = 0.5 (solid), ˝2+ = 1.5 (dashed) and ˝2+ = 3.5 (dotted). It can be realized that in the presence of a weak RHC field in the transition|3 → |2, the OB appears in the one point of the incident light. However, by increasing the Rabi-frequencies of the RHC field (˝2+ = 1.5 (dashed) and ˝2+ = 3.5 (dotted)), the threshold points of OB increases. This is very interesting result for controlling the OB at different values of incident light only by adjusting the intensity of RHC field. By applying a weak perturbing field a high-resolution peak appears in the optical spectra due to the presence of interacting dark resonances. Furthermore by decreasing the Rabi frequency of perturbing field the width and amplitude of the peak are reduced and the optical bistability behavior is obtained with reduced threshold. This is to say that the behaviors of OB can be manipulated via adjusting the Rabi-frequency of weak perturbing or strong coupling field. In Fig. 5, we plot the transmission properties of incident light through defect slab, when we change the intensity of RHC field. It can be seen that there is a peak in the transmission curve for a weak intensity of RHC field (solid line) which corresponds to the subluminal light propagation. By enhancing the Rabi frequency of RHC field to 1.5 (dashed line), the corresponding peak converts to a dip which corresponds to the superluminal light propagation. And finally, for ˝2+ = 3.5 (dotted line), the dip again converts to a peak with high coefficient. In this case, the group velocity of the transmitted light corresponds to the subluminal light propagation. Finally, we are interested discuss the impact of thickness of dielectric slab on OB and transmission properties of weak probe light. The dependence of transmitted intensity of probe field Ut versus the incident intensity Uin on the thickness of the slab (a) and transmission coefficient versus probe detuning (b) are displayed in Fig. 6. From part (a) of Fig. 6, one can find that, decreasing in parameter m leads to changing in OB properties of transmitted pulse. For m = 250 (solid line), the threshold of OB appears in two points of incident light, while for m = 200 (dashed line) and m = 150 (dotted line) the threshold appears in one point of incident light. However, in the case of m = 150, the threshold of OB is less than m = 200 one. In part (b) of Fig. 6, we show the transmission coefficient versus probe detuning. One can easily

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Fig. 6. The transmitted intensity of probe field Ut versus the incident intensity Uin (a) and (b) transmission coefficient versus probe detuning for m = 250 (solid), m = 200 (dashed) and m = 150 (dotted). The selected parameters are ˝2− = 2 , ˝2+ = 1.5 and others are same as Fig. 2.

find that the transmission curve has a dip for all values of parameter m. Here, only the value of transmission is differ for various quantities of parameter m. Moreover, according to the Agarwal’s theorem, the group velocity of transmitted light is in superluminal condition for whole of chosen parameter m. 4. Conclusion In summary, we discuss the optical bistability and transmission coefficient properties of weak probe light in a defect slab doped by four-level graphene system. We show that the controlling of OB and group velocity of transmitted light can be possible by changing the Rabi-frequencies of LHC and RHC coupling fields and thickness of the slab, respectively. We realize that the intensity threshold of OB can be adjusted at different points for incident light. Moreover, we find the proper conditions for switching between subluminal and superluminal light propagation in a defect slab. Our numerical results may be suitable for the future experimental works for developing the Nonophotonics technologies. 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]

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