Enormous enhancements of the Kerr nonlinearity at C-band telecommunication wavelength in an Er3+-doped YAG crystal

Physica B 442 (2014) 60–65

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Enormous enhancements of the Kerr nonlinearity at C-band telecommunication wavelength in an Er3 þ -doped YAG crystal Hamid Reza Hamedi n Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, LT-01108 Vilnius, Lithuania

art ic l e i nf o

a b s t r a c t

Article history: Received 3 February 2014 Received in revised form 20 February 2014 Accepted 24 February 2014 Available online 4 March 2014

A novel solid configuration is proposed to achieve a giant Kerr nonlinearity with reduced absorption under conditions of slow light levels. It is shown that an enhanced Kerr nonlinearity accompanied with negligible absorption can be obtained just through the proper tuning of intensity of coherent driving field at C-band telecommunication wavelength which is practical for communication applications. Moreover, the impact of incoherent pump field as well as frequency detuning of coherent field on manipulating the linear and nonlinear optical properties of the yttrium–aluminum-garnet (YAG) crystal medium is discussed. The presented results may be of interest to researchers in the field of all-optical signal processing and solid-state quantum information science. & 2014 Elsevier B.V. All rights reserved.

Keywords: Kerr nonlinearity Er3 þ -doped yttrium–aluminum-garnet crystal Coherent and incoherent fields

1. Introduction Nonlinear optical properties of an atomic medium can be substantially modified by applying external fields. Quantum coherence and quantum interference are the basic mechanisms for modifying the response of the atomic medium to the applied fields. The effect of electromagnetically induced transparency (EIT) [1] has led to many interesting nonlinear optical phenomena such as Kerr nonlinearity [2] and optical bistability [3–6]. Due to its potential applications in many areas of optics, such as quantum computing, quantum communication, and fast optical switching, the Kerr nonlinearity has recently been developed in multi-level gas systems [7–12]. The refractive part of third order susceptibility which is known as Kerr nonlinearity plays a crucial role in nonlinear optics. Enhanced Kerr nonlinearity with reduced linear absorption has potential applications in quantum nondemolition measurements [9], quantum teleportation [10], cross-phase modulation [11], and self-phase modulation for the generation of optical solitons [12]. It is shown that a large Kerr nonlinearity with reduced linear and nonlinear absorption causes the nonlinear optics to be studied at low light power [13–15]. In fact, it is desirable to have large third-order nonlinear susceptibilities under conditions of low light power and high sensitivities [16,17], since they can be used for realization of single-photon nonlinear devices. This requires that the linear susceptibilities should be as small as possible for all pump and absorption losses. In recent years, both experimentally and theoretically [18–21], tremendous interest has been aroused in the study of large third-order nonlinear susceptibility with reduced

n

Tel.: þ 370 67218909. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.physb.2014.02.045 0921-4526 & 2014 Elsevier B.V. All rights reserved.

absorption in gases media. It is desirable that one investigate the giant Kerr nonlinearity in a solid-state medium. In our recent work, a scheme for enhancement of Kerr nonlinearity with vanishing absorption in a three-level ladder configuration n-doped semiconductor quantum well was proposed [22]. It is shown that the Kerr nonlinearity can be controlled and even enhanced by the intensity of coupling fields. In recent years, many kinds of nonlinear quantum optical phenomena based on the quantum interference and quantum coherence have been studied in the rare-earth-metal ion-doped crystals [23–25], such as Optical Switches [25], Light Storage [26], Large Refractive Index Without Absorption [27] and Four-Wave Mixing [28,29]. The reason for this is that the phenomena in these crystals have many potentially important applications in optoelectronics and solid-state quantum information science. Otherwise, these crystals have many inherent characteristics that the atomic systems do not have, such as long optical coherence times [30], and optical controllability of the ionic states. In this paper, we intend to study the refractive part of the third order susceptibility of an Er3 þ -doped YAG crystal. It is found that a giant Kerr nonlinearity with reduced absorption can be obtained under conditions of slow light levels. The effect of an incoherent pumping field on linear and nonlinear susceptibilities is then discussed. The four-level scheme we study in this work is mainly according to Ref. [31], but our work is very different from that work. First of all, the incoherent field is used to control the Kerr nonlinearity, and the quantum interference effect is created by coherent and incoherent fields. In other words, the underlying mechanism is very different from those of the conventional schemes. Thus, it may provide some new possibilities for technological applications in solid-state quantum information science. Second, this configuration has many advantages which the other

H.R. Hamedi / Physica B 442 (2014) 60–65

quantum coherent media do not have: (a) rare-earth-doped crystals have properties similar to those of atomic vapors but with the advantage of no atomic diffusion; (b) yttrium–aluminum garnet is an excellent optical host material compared with others which are sensitive to moisture; and (c) the Er3 þ -doped YAG crystal is an efficient active medium for lasers operating in the middle infrared band, coinciding with the telecommunications band [27]. Third, our study is much more practical than its gaseous counterpart due to its flexible design and the widely adjustable parameters. The paper is organized as follows. In Section 2 the model and equation are presented. The results are discussed in Section 3. Finally, some simple conclusions are given in Section 4.

ρ_ 33 ¼  ðγ 31 þ γ 32 Þρ33 þ γ 43 ρ44 þ Rðρ11  ρ33 Þ ρ_ 21 ¼  iΩp ðρ22  ρ11 Þ þiΩc ρ41 þ ðiΔp  Γ 21 Þρ21 ρ_ 31 ¼  iΩp ρ32  Γ 31 ρ31 ρ_ 41 ¼  iΩp ρ42 þ iΩc ρ21 þ ½iðΔp þ Δc Þ  Γ 41 ρ41 ρ_ 32 ¼  iΩp ρ31  iΩc ρ34  ðiΔp þ Γ 32 Þρ32 ρ_ 42 ¼  iΩc ðρ44  ρ22 Þ  iΩp ρ41 þ ðiΔc  Γ 42 Þρ42 ρ_ 43 ¼ iΩc ρ23 þ ½iðΔp þ Δc Þ  Γ 43 ρ43 ρ11 þ ρ22 þ ρ33 þ ρ44 ¼ 1

61

ð2Þ

where

Γ 21 ¼ ðγ 21 þ r þ γ dph 21 Þ=2 Γ 31 ¼ ðγ 31 þ γ 32 þ 2r þ γ dph 31 Þ=2 Γ 41 ¼ ðγ 41 þ γ 42 þ γ 43 þ r þ γ dph 41 Þ=2 Γ 32 ¼ ðγ 31 þ γ 32 þ γ 21 þ r þ γ dph 32 Þ=2

2. Theoretical model and equations Consider a four-level Er3 þ ionic system in an Er3 þ -doped YAG crystal as depicted in Fig. 1. The experimental system for this scheme can be realized by Er3 þ -doped YAG crystal with j4 I15=2 〉, j4 I13=2 〉, j4 I11=2 〉; and j4 I9=2 〉 denoting the j1〉; j2〉 j3〉 and j4〉 state levels, respectively. The states j1〉; j2〉 and j3〉 comprise a standard V-type three levels EIT subsystem, while the states j1〉, j2〉, and j4〉 form a usual ladder-type three-level EIT subsystem [27]. A coherent field with frequency ωc couples levels j2〉 and j4〉. The ground level j1〉 is coupled to level j2〉 by a weak probe field with frequency ωp via a fast decay from level j3〉, and level j2〉 is populated by an incoherent pump process interacting with the transition j1〉-j3〉. If we take the level j1〉 as the energy origin and choose H 0 ¼ ðωc þ ωp Þj4〉〈4jþ ωp j2〉〈2j, in the interaction picture and under the rotating-wave approximation, the interaction Hamiltonian of the system is given as follows: H I ¼  ðΔc þ Δp Þj4〉〈4j  Δp j2〉〈2j  ðΩc j4〉〈2j þ Ωp j2〉〈1j þ H:C:Þ

Γ 43 ¼ ðγ 41 þ γ 42 þ γ 43 þ γ 31 þ γ 32 þ r þ γ dph 43 Þ=2:

ð3Þ

Here, Γ mn ðm anÞ are the total decay rates which are added phenomenologically in the above density-matrix equations and are composed of a population spontaneous emission contribution as well as a dephasing contribution. The first contribution γ mn ðm a nÞ designates the population spontaneous damping from jm〉 tojn〉, while the other contribution γ dph mn ðm a nÞ, determined by electron–electron, interface roughness, and phonon scattering processes [32], is the dephasing decay rate of the quantum coherence of the |m〉2|n〉 transition. In contrast to many atomic schemes, the γ dph mn is the dominant contribution to the total decay rate Γ mn , which is the major obstacle in the observation of coherent effects such as EIT in solid material media [27].

ð1Þ

where the detuning parameters are defined as Δp ¼ ωp  ω21 and Δc ¼ ωc  ω42 . The corresponding Rabi-frequencies of coupling and probe fields are defined as Ωc ¼ ℘42 Ec =2ℏ and Ωp ¼ ℘21 Ep =2ℏ where ℘42 denotes the reduced electric dipole moment of j4〉-j2〉 transition, while ℘21 signifies the electric dipole moment associated with j2〉-j1〉. In addition, Ec and Ep are the amplitudes of the coupling and probe fields; ω21 and ω42 are resonant frequencies associated with the corresponding optical transitions j2〉-j1〉 and j4〉-j2〉, respectively. The density matrix equation of the motion in dipole and rotating wave approximations for this system can be written as

ρ_ 11 ¼ iΩp ðρ21  ρ12 Þ þ γ 21 ρ22 þ γ 31 ρ33 þ γ 41 ρ44 þ Rðρ33  ρ11 Þ ρ_ 22 ¼ iΩp ðρ12  ρ21 Þ þ iΩc ðρ42  ρ24 Þ  γ 21 ρ22 þ γ 32 ρ33 þ γ 42 ρ44

Γ 42 ¼ ðγ 41 þ γ 42 þ γ 43 þ γ 21 þ γ dph 42 Þ=2

3. Results and discussion In this section, we investigate the Kerr nonlinearity of the Er3 þ doped YAG crystal medium. First, we derive an analytical expression for the first- and third-order susceptibilities. We will show that the medium can be used as an optical medium to achieve a large Kerr nonlinearity with negligible probe absorption. This may be important for achieving large nonlinearity under low power conditions. Let the probe field be weak compared to the coupling field. To solve the density matrix equations of motion, in the steady state, given by Eq. (2), we should expand the density matrix ð1Þ ð2Þ ð3Þ elements as ρij ¼ ρð0Þ ij þ ρij þ ρij þ ρij þ ⋯. The zeroth order soluð0Þ tion of ρð0Þ 11 will be identity, i.e. ρ11 ¼ 1, while the other elements are set to be zero. The linear and nonlinear responses of the

Fig. 1. A schematic diagram of a four-level Er3 þ ionic system in an Er3 þ - doped YAG crystal sample.

62

H.R. Hamedi / Physica B 442 (2014) 60–65

medium to the applied fields are determined, respectively, by linear and nonlinear susceptibility χ ð1Þ and χ ð3Þ :

Thus, the linear and nonlinear susceptibility χ ð1Þ and χ ð3Þ can be defined as follows [33,34]:

ð1Þ ρ21 ¼

 Ωp ½ðΔp þ Δc Þ þ iΓ 41  w

4ðaÞ

χ ð1Þ ¼

ð1Þ ρ41 ¼

 iΩc ρð1Þ iðΔp þ Δc Þ  Γ 41 21

4ðbÞ

χ ð3Þ ¼

o 1n ð2Þ iΩp ðρð1Þ  ρð1Þ Þ þ γ 21 ρð2Þ þ ðγ 31 þ rÞρð2Þ þ γ 41 ρ44 21 12 22 33 r

4ðcÞ

ð2Þ ρ11 ¼

ð2Þ ρ22 ¼

1 n

γ 21

ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ iΩp ðρð1Þ 12  ρ21 Þ þ iΩc ðρ42  ρ24 Þ þ γ 32 ρ33 þ γ 42 ρ44

o

4ðdÞ

ð2Þ ρ33 ¼

o 1n  ðγ 31 þ γ 32 Þρð2Þ þ γ 43 ρð2Þ þ r ρð1Þ 33 44 11 r

4ðeÞ

ð2Þ ρ44 ¼

 iΩc ðρð2Þ  ρð2Þ Þ γ 41 þ γ 42 þ γ 43 42 24

4ðfÞ

ð2Þ ρ42 ¼

n o 1 ð2Þ ð1Þ iΩc ðρ44  ρð2Þ 22 Þ þ iΩp ρ41 iΔc  Γ 42

4ðgÞ

ð3Þ ρ21 ¼

Ωp w

ð2Þ ð2Þ ½  ½ðΔp þ Δc Þ þ iΓ 41 ðρð2Þ 22  ρ11 Þ þ Ωc ρ42 

4ðhÞ

where w ¼ Ωc þ Γ 21 ð  iΔc þ Γ 41 Þ  Δp ðΔc þ iΓ 41 þ Γ 21 Þ  Δp 2

2

ð5Þ

2N γ 221 ð1Þ ρ ε0 ℏΩp 21 2N γ 421

ð3Þ 3 21

3ε0 ℏ3 Ωp

ρ

6ðaÞ

6ðbÞ

where N is the number density of the Er3 þ ions in the sample. The real and imaginary parts of χ ð1Þ correspond, respectively, to the linear dispersion and absorption. Moreover, the Kerr nonlinearity is proportional to the refraction part of the third order susceptibility χ ð3Þ , while the imaginary part of χ ð3Þ determines the nonlinear absorption. We can get population spontaneous rates γ mn of the Er3 þ ions in Er3 þ -doped YAG crystals containing 0:52 at% concentrations of Er3 þ ions at room temperature [27,35]. So it is reasonable that we can choose the parameters as γ 21 ¼ 239:1 s  1 ; γ 31 ¼ 0:8γ 21 ; γ 41 ¼ 0:86γ 21 ; γ 32 ¼ 10γ 21 ; γ 42 ¼ 0:29γ 21 and γ 43 ¼ 0:04γ 21 in the following. Based on Refs. [27,30], it is reasonable for us to estimate the dph dph dph dph dephasing decay rate as γ dph 21 ¼ γ 31 ¼ γ 41 ¼ γ 32 ¼ γ 42 ¼ dph γ 43 ¼ 15γ 21 : In this paper, all the parameters have been scaled by γ 21 ¼ 239:1 s  1 : Now, we are interested in the effect of the coherent and incoherent fields on linear and nonlinear susceptibilities. We assume γ 21 ¼ γ . Moreover, we choose ð2N γ 221 =ε0 ℏΩp Þ  1 and ð2N γ 221 =ε0 ℏΩp Þ  1. In the first step, the impact of the coherent

Fig. 2. Linear and nonlinear susceptibility versus probe field wavelength. (a, c) Linear absorption (dash line), and linear dispersion (solid line). (b, d) Nonlinear absorption dph dph dph (dash line), and Kerr nonlinearity (solid line). The selected parameters are γ 21 ¼ γ; γ 31 ¼ 0:8γ; γ 41 ¼ 0:86γ; γ 32 ¼ 10γ; γ 42 ¼ 0:29γ; γ 43 ¼ 0:04γ; γ dph 21 ¼ γ 31 ¼ γ 41 ¼ γ 32 ¼ dph γ dph 42 ¼ γ 43 ¼ 15γ 21 : The other parameters are r ¼ 0,(a, b) Ωc ¼ 8γ and (c, d) Ωc ¼3γ.

H.R. Hamedi / Physica B 442 (2014) 60–65

field on linear and nonlinear susceptibilities is discussed. Variations of linear and nonlinear susceptibility versus probe field wavelength are shown in Fig. 2. It is obvious that when Ωc ¼ 0:8γ and for r ¼ 0; the weak Kerr nonlinearity is accompanied with a strong linear and nonlinear absorption. In addition, it can be seen from Fig. 2(a) that the slope of linear dispersion is negative which shows superluminal light propagation. Based on our knowledge, it is ideal to achieve large Kerr nonlinearities with reduced

Fig. 3. Transmission coefficient versus probe field wavelength. The selected parameters are the same as in Fig. 2.

63

absorption under the condition of slow light levels. Thus, the obtained result is undesirable for nonlinear optical application. To overcome this situation, we increase the intensity of the coherent coupling field. An investigation on Fig. 2(c) shows that for Ωc ¼ 3γ , the linear dispersion changes from negative to positive at λ ¼ 1550 nm, that is corresponding to subluminal light propagation. In addition, the refractive part of third order susceptibility i.e. Kerr nonlinearity enhances dramatically around λ ¼ 1550 nm λ ¼ 1550 nm and reaches approximately 1  10  11 . In addition, the corresponding linear and nonlinear absorptions become negligible and EIT window appears. Physically, without the incoherent pump, the states j1〉; j2〉 and j4〉 form a usual ladder type threelevel EIT system. In such case, the strong coupling laser field produces a pair of dressed states j þ 〉 andpjffiffiffi〉that are superposition of states j2〉 and j4〉, i.e. j 7 〉 ¼ ð1= 2Þðj2〉 7j4〉Þ(Δc ¼ 0) [36]. Then probe absorption from level j1〉to the dressed states j7 〉can be understood by Fig. 1(b). In principle, there are two pathways for probe absorption, which create two absorption peaks as can be seen from Fig. 2(c, d). Therefore, applying strong coupling field between levels j2〉 and j4〉 will sharply suppress the absorption and enhance the Kerr nonlinearity of the Er3 þ doped YAG crystal medium. Thus, simply an inappropriate situation with large absorption can be changed to an applicable case for nonlinear application just by proper tuning of Rabi-frequency Ωc . In order to test the validity of the analysis described above, we also display the corresponding transmission coefficient versus probe field wavelength in Fig. 3. In the slowly-varying-envelope approximation, the Rabi frequency Ωp of the probe laser field obeys the

Fig. 4. Real and imaginary parts of the linear susceptibility (a, b), and real and imaginary parts of the nonlinear susceptibility (c, d) versus probe field wavelength for the different rates of incoherent pumping fields. The selected parameters are Ωc ¼ 3γ and the other parameters are the same as in Fig. 2.

64

H.R. Hamedi / Physica B 442 (2014) 60–65

following Maxwell's wave equation in the steady state limit: ∂Ep N ωp ℘21 ¼i ρ21 ∂z 2cε0

ð7Þ

Here c is the light speed in free space. In the linear regime, for in given Ωp at the atomic medium input, we can easily arrive at the out steady-state probe field Ωp at the medium output as    ρ21 in Ωout ð8Þ p ¼ Ωp exp  α Im

Ωp

Fig. 5. Transmission coefficient versus probe field wavelength. The selected parameters are Ωc ¼ 3γ and the other parameters are the same as in Fig. 2.

where α ¼ N ωp ℘21 =2cε0 . Thus, the normalized transmission coefficient of the probe laser field for the transition j1〉-j2〉 can be written as T ¼ Ωp =Ωp out

in

ð9Þ

Fig. 3(a) shows the transmission properties of the probe field propagating through the YAG crystal medium. It is obvious that by increasing the intensity of coupling field Ωc to 3γ the transmission coefficient is enhanced at the line center of the probe transmission spectrum and reaches approximately 1. It shows a good agreement with the probe absorption spectra given in Fig. 2. To investigate how the rate of incoherent pump field affects the linear and nonlinear susceptibilities of the YAG crystal medium, we plot the linear and nonlinear absorption and dispersion versus probe field wavelength in the presence of coherent coupling field Ωc in Fig. 4. We found that when Ωc ¼ 3γ , the slope of linear dispersion converts from positive to negative corresponding to the switch from subluminal to superluminal light propagation (Fig. 4 (a)). Simultaneously, as we go from r ¼ 0 to r ¼ 5γ (Fig. 4(c)), the maximal Kerr nonlinearity will be decreased significantly around λ ¼ 1550  10  6 m. In this case, it is clear from Fig. 4(b, d) that the linear and nonlinear absorptions decrease dramatically. Physically, due to the fast decay from state j3〉 to the metastable state j2〉, increasing of the incoherent pumping rate r results in enhancement of population difference between levels j2〉 andj1〉. In other words, for the stronger incoherent pump fields, more population will be pumped to levelj2〉. More population on level j2〉 and less population on level j1〉 can lead to reduction of the probe absorption and eventually, causes Kerr nonlinearity coefficient to be decreased. Thus, by applying an increasingly incoherent

Fig. 6. Linear and nonlinear absorption (a, b) as well as Kerr nonlinearity (c) versus probe field wavelength. The selected parameters are Ωc ¼ 10γ, r ¼5γ and the other parameters are the same as in Fig. 2.

H.R. Hamedi / Physica B 442 (2014) 60–65

65

on linear and nonlinear susceptibility is discussed. It is theoretically illustrated that an enhanced Kerr nonlinearity with reduced absorption can be achieved by increasing the intensity of coherent field at long wavelength λ ¼ 1550 nm coinciding with C-band telecommunication wavelength. In addition, the effect of an incoherent pump field on Kerr nonlinearity behavior of the solid medium is investigated. The controllable Kerr nonlinearity in such a crystal medium can enable more efficient all-optical switches and logic-gate devices for optical computing quantum information processing.

Acknowledgments The presented work has been supported by the Lithuanian Research Council (No. VP1-3.1-ŠMM-01-V-03-001).

Fig. 7. Transmission coefficient versus probe field wavelength. The selected parameters are Ωc ¼ 10γ, r ¼ 5γ and the other parameters are the same as in Fig. 2.

pumping field between the states j1〉 and j3〉, we can modify the absorption for the weak probe field on the transition j2〉 to j1〉 and affect the Kerr nonlinearity of the Er3 þ -doped YAG crystal medium, which will have destructive and constructive effects on Kerr nonlinearity, dispersion and absorption of the medium. The effect of incoherent pumping field on transmission coefficient of the medium is shown in Fig. 5. It is found that the transmissioncoefficient is very sensitive to the rate of incoherent pump field. We observe that in the case r ¼ 0, a low transmission appears around long wavelength λ ¼ 1:55 μm, while increasing r to r ¼ 5γ causes a high transmission. These results are in a good agreement with our previous obtained results. Now, we consider how the frequency detuning of coherent field affects the Kerr nonlinearity as well as linear and nonlinear absorption of the YAG crystal medium. The linear and nonlinear absorption spectra as a function of probe field wavelength can be observed in Fig. 6(a) and (b). It is found that when we switch from the on-resonance control field Δc ¼ 0 to off-resonance case (Δc ¼ 10γ ), one of two symmetric absorption peaks becomes higher, while the other one decreases. Moreover, the whole spectral profiles of linear and nonlinear absorption shift toward the left. Simultaneously, it can be seen from Fig. 6(c) that the maximal Kerr nonlinearity is enhanced inside the shifted reduced absorption window (solid line). Thus, another approach to achieve large Kerr nonlinearity accompanied with negligible absorption is presented just via proper tuning of frequency detuning of coherent field. The corresponding transmission coefficient is displayed in Fig. 7. Explicitly, switching from an on-resonance control field to an off-resonance case, the whole transmission spectrum moves to the left. This outcome supports the accuracy of the previous results.

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]

4. Conclusion

[33]

In our solid configuration, the effect of intensity and frequency detuning of coherent driving field as well as incoherent pump field

[34] [35] [36]

Y. Wu, X. Yang, Phys. Rev. A 71 (2005) 053806. Ying Wu, L. Deng, Phys. Rev. Lett. 93 (2004) 143904. Dong-chao Cheng, Cheng-pu Liu, Shang-qing Gong, Opt. Commun. 263 (2006) 111. J. Li, X. Lu, J. Luo, Q. Huang, Phys. Rev. A 74 (2006) 035801. Jia-Hua Li, Xin-You Lü, Jing-Min Luo, Qiu-Jun Huang, Phys. Rev. A 74 (2006) 035801. Zhiping Wang, Opt. Commun. 282 (2009) 4745. I.L. Chuang, Y. Yamamoto, Phys. Rev. Lett. 76 (1996) 4281. J.C. Howell, A. Lamas-Linares, D. Bouwmeester, Phys. Rev. Lett. 88 (2002) 030401. Y.F. Xiao, K. Sahin, V. Gaddam, C. Hua, N. Imoto, Lan. Yang, Opt. Express 16 (2008) 26. J. Howell, J. Yeazell, Phys. Rev. A 62 (2000) 032311. H. Schmidt, A. Imamoglu, Opt. Lett. 21 (1996) 1936. Vladimir Tikhonenko, Jason Christou, Barry Luther- Davies, Phys. Rev. Lett. 76 (1996) 2698. Y. Wu, X. Yang, Appl. Phys. Lett. 91 (2007) 094104. Y. Wu, L. Deng, Opt. Lett. 29 (2004) 2064. Seyyed Hossein Asadpour, Hamid Reza Hamedi, Mostafa Sahrai, J. Lumin. 132 (2012) 2188. Wen-juan Jiang, Xiang’an Yan, Jian-ping Song, Huai-bin Zheng, Cshuanxing Wu, Bao-yin Yin, Yanpeng Zhang, Opt. Commun. 282 (2009) 101. P.R. Hemmer, D.P. Katz, J. Donoghue, M. Cronin- Golomb, M.S. Shahriar, P. Kumar, Opt. Lett. 20 (1995) 982. Yanfeng Bai, Wenxing Yang, Xiaoqiang Yu, Opt. Commun. 283 (2010) 5062. Sun Hui, Niu Yue-Ping, Gong Shang-Qing, Chin. Phys. 16(02) (2007) 429. Hui Sun, Shangqing Gong, Yueping Niu, Shiqi Jin, R1uxin Li, Zhizhan Xu, Phys. Rev. B 74 (2006) 155314. Yueping Niu, Shangqing Gong, Phys. Rev. A 73 (2006) 053811. Seyyed Hossein Asadpour, Hamid Reza Hamedi, Opt. Quant. Electron. 45 (2013) 11. B.S. Ham, P.R. Hemmer, M.S. Shahriar, Opt. Commun. 144 (1997) 227. B.S. Ham, P.R. Hemmer, Phys. Rev. Lett. 84 (2000) 4080. B.S. Ham, Appl. Phys. Lett. 78 (2001) 3382. J.J. Longdell, E. Fraval, M.J. Sellars, N.B. Manson, Phys. Rev. Lett. 95 (2005) 063601. H.F. Zhang, J.H. Wu, X.M. Su, J.Y. Gao, Phys. Rev. A 66 (2002) 053816. Y. Wu, M.G. Payne, E.W. Hagley, L. Deng, Opt. Lett. 29 (2004) 2294. Y. Wu, X. Yang, Phys. Rev. B 76 (2007) 054425. Y. Sun, C.W. Thiel, R.L. Cone, R.W. Equall, R.L. Hutcheson, J. Lumin. 98 (2002) 281. J.H. Li, Phys. Rev. B 75 (2007) 155329 ([6 pages]). G.B. Serapiglia, E. Paspalakis, C. Sirtori, K.L. Vodopyanov, C.C. Phillips, Phys. Rev. Lett. 84 (2000) 1019. M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK, 1997. R.W. Boyd, Nonlinear Optics, Academic, San Diego, 1992. Q.Y. Wang, S.Y. Zhang, Y.Q. Jia, J. Alloys Compd. 202 (1993) 1. A. Imamoglu, S.E. Harris, Opt. Lett. 14 (1989) 1344.