- Email: [email protected]
Generation of high-power parabolic pulses in quantum dot waveguide amplifiers
Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of high-power parabolic pulses in quantum dot waveguide amplifiers
T
⁎
Jun-Rong He , Siliu Xu, Li Xue School of Electronic and Information Engineering, HuBei University of Science and Technology, Xianning 437100, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Self-similar techniques Quantum dot waveguide amplifiers High-power parabolic pulses
The self-similarity techniques are used to study pulse propagation in a quantum dot waveguide amplifier with an arbitrary longitudinal gain profile. Generation of high-power parabolic pulses are analyzed under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. In particular, the periodic distributed gain enables us to obtain a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp. Finally, a possible experimental protocol is proposed to generate the high-power parabolic pulses in realistic waveguides.
1. Introduction In the past decades, extensive research works have been carried out to find the exact analytical and asymptotic similaritons in gain amplifier systems due to their potential applications in nonlinearity and dispersion management systems [1–3]. These optical similaritons possess many attractive features that make them potentially useful for various applications in fiber-optic telecommunications and photonics, since they can maintain their overall shapes but allow their amplitudes and widths changing with the modulation of the system's parameters such as dispersion, nonlinearity, gain, and inhomogeneity [4,5]. Fiber optics and waveguide optics are used in most of the important applications. It is common knowledge that the so-called self-similar solution was used in early studies as a qualitative test for the self-focusing theory and ultra-short pulses generation [6–9]. Generally speaking, optical similaritons can be divided into two categories. The first category is the exact optical similaritons, which are mainly described by the exact solitary-wave solutions, including the bright and dark soliton solutions [10–14], the quasisoliton solutions [15], the nonlinear Bloch waves [16], and the solitons on the continuous-wave background [17]. The second category is the asymptotic optical similaritons, which are mainly described by the parabolic, Hermite-Gaussian, and hybrid functions [18–23] and exist in a wide range of gain amplifier when the strict balance of system parameters is broken. Among these optical similaritons, the asymptotic parabolic similariton has attracted more attentions due to: (i) they can be easily generated from arbitrary input optical waves; (ii) they have strict linear chirps, which are important to the effective compression of optical waves; (iii) their stabilities are guaranteed even with the high power input, and so on. These advantages usually does not belong to the exact optical similaritons. It should be noted that the previously studies are usually focused on optical pulses (beams) propagating in graded-index waveguide amplifiers with cubic Kerr nonlinearity, i.e., the refractive index is n(z, x) = n0 + n2I(z, x), where z is the propagation distance, x is the spatial coordinate, and I is the pulse intensity. Here the first term describes the linear part of the refractive index and the last term represents the Kerr-type nonlinearity with n2 being the cubic coefficient (positive n2 for self-focusing nonlinearity and negative n2 for self-defocusing nonlinearity). When the intensity of the optical pulse exceeds a certain value, a higher-order nonlinear
⁎
Corresponding author. E-mail address: [email protected] (J.-R. He).
https://doi.org/10.1016/j.ijleo.2019.01.103 Received 29 January 2019; Accepted 29 January 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
J.-R. He, et al.
effect such as the quintic nonlinearity should be taken into account [24], which leads to the refractive index as
n (z , x ) = n 0 + n2 I (z , x ) + n4 I (z , x )2 ,
(1)
where n4 is the quintic coefficient of the nonlinearity which may assume positive or negative values. In experiment, the cubic-quintic nonlinearities can be obtained by doping a fiber with two appropriate semiconductor materials [26,27]. On the other hand, recent study suggests that the quintic nonlinearity could be realized in a centrosymmetric nonlinear medium doped with resonant impurities in the limit of a large light carrier frequency detuning from the impurity resonance [28]. The resonant impurities could be rare-earth element atoms, erbium-doped glasses, or semiconductors doped with quantum dots (QDs). The authors demonstrate that stable exact bright spatial similaritons can be supported by homogeneous conservative optical media with quintic nonlinearities. However, there is no work on this model for obtaining asymptotic self-similar solutions, which have important applications in many fields of nonlinear science, especially in nonlinear optical-fiber and waveguide amplifiers. Moreover, in the realistic optics fiber and waveguides, there will always be some factors influencing the propagation of pulses. These effects may lead the pulses to be amplification (or attenuation) during the propagation. Thus, a model including the gain (loss) effects in optics fiber and waveguides is reasonable. For these reasons, in this work we study the generation of parabolic pulses in quantum dot waveguide amplifiers. The paper is organized as follows. In Section 2, we provide the theoretical model to describe the nonlinear wave propagation in a quantum dot waveguide amplifier. In Section 3, we study the generation of high-power parabolic pulses under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. In particular, the periodic distributed gain enables us to obtain a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp, which have important applications in nonlinear optical-fiber and waveguide amplifiers. In Section 4, we give a possible experimental protocol which may generate the reported self-similar parabolic pulses in semiconductor quantum dot waveguide amplifiers. In Section 5, the main results of the paper are summarized. 2. The model Under the paraxial and the slowly varying envelope approximations, the nonlinear wave equation governing pulse propagation in a quantum dot waveguide amplifier given by Eq. (1) can be written as
i
Nd eg ω2 ig(z ) ∂u 1 ∂ 2u k n k n + + σ∞ + 0 2 |u|2 u + 0 4 |u|4 u = u, 2 ∂z 2k 0 ∂x 2 2ϵ 0 k 0 c 2 n0 n0
(2)
where g(z) is the gain (loss) coefficient. The third term on the left of the equation above represents the non-linear polarization induced by QDs, where N is the dopant density, deg is the dipole matrix element between the excited state and the ground state, and σ∞ is the stable value of the atomic dipole moment. In the cw limit and assuming the light carrier frequency are far enough from the QDs, σ∞ is possible to developed into
σ∞ ≈
2dge u ℏ∆
(1 −
4γ⊥ |dge |2 |u|2 γ∥
ℏ2∆2
+
16γ⊥2 |dge |4 |u|4 γ∥2 ℏ4∆4
), (3)
where γ⊥(γ∥) is a transverse (longitudinal) decay rate of the atomic dipole moment (inversion), and Δ is a detuning of the incident light from atomic impurity resonance satisfying Δ ≫ γ⊥. If one chooses ∆ =
3
4γ⊥ | dge |4 N γ∥ ℏ3 ϵ0 n 0 | n2 |
, the QD-generated nonlinearity and the
third-order nonlinearity described by the nonlinear interaction of light with the bulk medium in Eq. (2) offset each other, causing an effective renormalized quintic nonlinearity with the coefficient
n4eff = n4 + n2 3
4γ⊥ ϵ 20 n02 n22 , γ∥ |dge |2 N 2
(4)
which indicates that n4eff may be positive or negative, depending on the sign of n2 and n4. Introducing the normalized variables U = (k 0 |n4eff |LD / n 0)1/4u, G (Z ) = g (z ) LD , X = x / w0 and Z = z/LD with LD = k 0 w02 being the diffraction length, then Eq. (2) can be rewritten in a dimensionless form
i
∂U 1 ∂ 2U iG + + α|U |4 U = U, ∂Z 2 ∂X 2 2
(5)
where α = |n4eff|/n4eff = ± 1 corresponds to self-focusing (+) and self-defocusing (−) quintic nonlinearity of the waveguide, and G(Z) is a function of the normalized distance Z. When the relative strength of the diffraction term is much less than that of the quintic nonlinearity (the initial power is large enough), the form of asymptotic solution can be found. In the following, we consider the case α = −1 (Our direct numerical simulations show that the asymptotic pulses cannot be generated in waveguide with the quintic focusing nonlinearity). In this situation, the cubic nonlinearity coefficient n2 should be self-defocusing; the quintic nonlinearity coefficient n4 may assume positive or negative values because it can be neglected in such a medium [28]. The self-similar solution is sought in the form 1 1 U (Z , X ) = exp[ E (Z )][W (Z )]− 2 Θ (θ)exp[iΦ (Z , X )], 2
(6) 1107
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J.-R. He, et al.
Fig. 1. Attenuation rate as a function of distance Z expressed by condition (9) for different gain profiles. The parameters are G0 = Ze = Zp = 1.
The explicit dynamics of the field profile depend on the phase evolution which is found to be given by
Φ (Z , X ) = B (Z ) +
(X − XC )2 dW (Z ) , 2W (Z ) dZ
(7)
Z
where E ≡ ∫0 G (Z′)dZ′ and θ = (X − XC)/W(Z) with W(Z) being the width of the solution, XC denotes the pulse position, and B(Z) is a phase offset. Substituting Eqs. (6) and (7) into Eq. (5), we obtain the following differential equation for the function Θ:
−
1 ∂ 2Θ W2 dB (Z ) W3 d2W (Z ) 2 exp[−2E (Z )] 2 + Θ+ θ Θ + Θ5 = 0, 2 ∂θ exp[2E (Z )] dZ 2 exp[2E (Z )] dZ 2
(8)
To obtain high-power parabolic pulses, we consider the situation when the relative strength of the diffraction term is much less than that of the quintic nonlinearity, i.e., the first term in the left hand side of Eq. (8) can be neglected. This requires that
1 exp[−2E (Z )] ≪ 1, 2
(9)
Next, we will study three different gain profiles to simulate the amplification behaviors for (a) exponential change with G(Z) = G0 exp(Z/Ze), (b) constant gain with G(Z) = G0, and (c) periodic change with G(Z) = 1 + ϵ cos(Z/Zp), ϵ ∈ (−1, 1). Fig. 1 shows the attenuation rates of the above three gain profiles as a function of distance Z. It is seen that the exponential gain attenuates fastest, followed by constant gain. The attenuation rate of periodic gain depends on the sign of ϵ: when ϵ is negative, its attenuation rate is faster than constant gain; when ϵ is positive, its attenuation rate is slower than constant gain. This property determines how fast the input pulse evolves into a parabolic one. 3. Numerical simulations 3.1. System with exponential gain We consider that the gain profile is G(Z) = G0 exp(Z/Z0). The control of gain is possible in experiments [20]. Direct numerical simulations are shown in Fig. 2. Figs. 2(a) and (b) show the self-similar evolution of the parabolic pulse in the waveguide amplifier with the Gaussian input pulse and the hyperbolic secant input pulse, respectively. It is found that the pulse has evolved toward a parabolic solution in the amplifier after about 1.2 propagation distance. Moreover, the simulations show low-amplitude wings on the parabolic pulse whose amplitude decreases with increasing propagation distance, vanishing as Z→ ∞. Another significant property is that the input pulses with different shapes and width but of same initial power converge to the same parabolic pulse, see Fig. 2(c) and (d). 3.2. System with periodic gain We consider that the gain profile is G(Z) = 1 + ϵ cos(Z/Zp), described by a periodic modulated function. Direct numerical 1108
Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
J.-R. He, et al.
Fig. 2. Evolutions of the pulse in quantum dot waveguide amplifier with exponential gain, starting with (a) the Gaussian input pulse and (b) the hyperbolic secant input pulse. From the top to bottom, the propagation distance is Z = 1.8, 1.6, 1.4, and 1.2, respectively. (c) and (d) Evolutions of pulse width and pulse peak amplitude as functions of propagation distance for Gaussian pulses. The direct numerical simulations for Eq. (5) have the same initial power Pin = 1 but different full width half maximum (FWHM). The parameters are G0 = Z0 = 1.
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Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
J.-R. He, et al.
Fig. 3. Evolutions of the pulse in quantum dot waveguide amplifier with periodic gain, starting with the Gaussian input pulse for (a) ϵ = −0.5 and (b)ϵ = 0.5. From the top to bottom, the propagation distance is Z = 4.4, 4.0, 3.6, 3.2, and 2.8, respectively. The direct numerical simulations for Eq. (5) have the same initial power Pin = 1 for Zp = 1.
simulations are shown in Fig. 3. Fig. 3(a) and (b) show the self-similar evolution of the parabolic pulse in the waveguide amplifier with the Gaussian input pulse for ϵ = −0.5 and ϵ = 0.5, respectively. It is observed that the pulse has evolved toward a parabolic solution in the amplifier and shows low-amplitude wing with the increase of propagation distance. In addition, we find that the central region of Fig. 3(b) is wider than that of Fig. 3(a). This may be explained as: the slower the attenuation rate expressed by condition (9) is, the larger the pulse broadening is, and the larger the frequency chirp is, which results in the maximum spectral width. This means that it is the preferred profile to obtain the minimum pulse duration after compression. When ϵ = 0, we have G(Z) = 1, corresponding to a constant gain. In this case, Eq. (8) can be solved analytically, which has been discussed in Ref. [29]. Our work is different: (i) the model in this paper is derived from in quantum dot waveguide amplifiers, while the model in Ref. [29] appears in quintic nonlinear media. (ii) In our paper the gain profile is a function of propagation distance Z while in Ref. [29] it is considered to be a constant. Therefore, our results are more general. In fact, we can see that the results obtained here are more interesting and experimental when the gain varies with the propagation distance. For instance, when ϵ > 0, we can obtain a fast amplification of high power pulses in parabolic state, which has the largest broadening and frequency chirp, thus generating the largest spectral width. (iii) We have considered the gain varies periodically with propagation distance Z, which has not been reported for obtaining parabolic pulses to our knowledge. 3.3. Initial chirped pulses In the case of linearly chirped Gaussian pulses, the input function can be written as 2
1 + ic ⎛ X ⎞ U (0, X ) = exp ⎡ ⎢− 2 ⎝ X0 ⎠ ⎣ ⎜
⎟
⎤, ⎥ ⎦
(10)
where c is a chirp parameter. It is common to refer to the chirp as being positive or negative, depending on whether c is positive or negative. Employing the Fourier transformation, we find that the spectral half-width (at 1/e-intensity point) of the input Gaussian pulse is ∆Ω = 1 + c 2 / X0 . Clearly, spectral width of a pulse is enhanced by a factor of 1 + c 2 in the presence of linear chirp. In the absence of frequency chirp (c = 0), the spectral width satisfies the relation ΔΩX0 = 1. Fig. 4 shows the intensity and spectrum power of the output in quantum dot waveguide amplifier at Z = 5 with a linearly chirped Gaussian input pulse. It is found that a chirped Gaussian pulse for c > 0 broadens monotonically at a rate faster than that of the chirp-free pulse (c = 0). The reason is related to the fact that the diffraction-induced chirp and input chirp have the same sign, resulting in a superposition effect. When c < 0, the contribution of the diffraction-induced chirp is oppositive to that of the input chirp. In this case, the chirped Gaussian pulse also broadens at a rate faster than that of the chirp-free pulse after a proper propagation distance, but with a slower of broadening speed than that of the positive chirp. 1110
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J.-R. He, et al.
Fig. 4. (a) The intensity of the output in quantum dot waveguide amplifier at Z = 5, starting with the linearly chirped Gaussian input pulse. (b) The output spectrum power as a function of spatial frequency corresponding to (a).
4. Possible experimental generation We now give a suitable experimental protocol to generate the obtained high-power parabolic pulses in realistic waveguides. We consider a 1 μm thick planar waveguide (the input pulse is confined in the y direction), such as ZnSe [30], doped with CdSe QDs for N ≈ 1012 cm−3, which are grown by molecular beam epitaxy using a thermal activation procedure [31]. A typical value of the dipole matrix element is |dge| ≈ 10−28 Cm at a transition wavelength in the middle of λ≈500 nm [32] and γ⊥ = 2.5γ∥ [28]. The refractive index in Eq. (1) for ZnSe can be graded with n0 ≈ 2.7 and n2 ≈ −7 × 10−14 cm2/W near 500 nm [14,33], which lead to n4eff ≈ −10−21 cm4/W2 and Δ ≈ 1012 s−1. On the other hand, a typical exciton lifetime of CdSe QD is roughly 100 ps [34], translating into γ⊥ ≈ 1010 s−1, such that the system is well within the confines of a purely dispersive large-detuning regime Δ ≫ γ⊥. For the transverse scale typically of tens of microns order of magnitude, at z ≈ 7 cm (corresponding to five propagation distance units), one can obtain that the peak intensity is about 300 MW/cm2 and the required input power Pin∼10 W. Such input power levels can be realized under quasi-cw conditions (such as Nd:YAG laser) for which our cw theory remains applicable. 5. Conclusions In conclusion, we have shown that the high-power parabolic pulses can be generated in a quantum dot waveguide amplifier with an arbitrary longitudinal gain profile for defocusing nonlinearity. Analysis of high-power parabolic pulses were implemented under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. The results showed that the periodic distributed gain can support a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp. We have also presented a possible experimental protocol to generate the high-power parabolic pulses for the longitudinal distribution of gain in realistic waveguides. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 11847103 and the Field Grade Project of HuBei University of Science and Technology under Grant No. 2016-19XB006. References [1] G.P. Agrawal, Nonlinear Fiber Optics, 4th ed., Academic, New York, 2007. [2] J.D. Moores, Opt. Lett. 21 (1996) 555.
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Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
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[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [26] [27] [28] [29] [30] [31] [32] [33] [34]
S. Chen, J.M. Dudley, Phys. Rev. Lett. 102 (2009) 233903. B.S. Azimov, M.M. Sagatov, A.P. Sukhorukov, Sov. J. Quantum Electron. 21 (1991) 785. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, J. Mod. Optics 57 (2010) 1456. V.I. Talanov, Radiophys. Quant. Electron. 9 (1966) 260. V.I. Talanov, JETP Lett. 11 (1970) 199. J.H. Marburger, Prog. Quant. Electr. 4 (1975) 35. E.G. Lariontsev, V.N. Serkin, Sov. J. Quantum Electron. 5 (1975) 796. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102. V.N. Serkin, A. Hasegawa, Phys. Rev. Lett. 85 (2000) 4502. V.N. Serkin, T.L. Belyaeva, JETP Lett. 74 (2001) 573. V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. Lett. 90 (2003) 113902. S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901. S. Kumar, A. Hasegawa, Opt. Lett. 22 (1997) 372. Y. Zhang, B. Wu, Phys. Rev. Lett. 102 (2009) 093905. J.F. Wang, L. Li, Z.H. Li, G.S. Zhou, D. Mihalache, B.A. Malomed, Opt. Commun. 263 (2006) 328. M.E. Fermann, V.I. Kruglov, B.C. Thomsen, J.M. Dudley, J.D. Harvey, Phys. Rev. Lett. 84 (2000) 6010. V.I. Kruglov, A.C. Peacock, J.M. Dudley, J.D. Harvey, Opt. Lett. 25 (2000) 1753. V.I. Kruglov, A.C. Peacock, J.D. Harvey, J.M. Dudley, J. Opt. Soc. Am. B 19 (2002) 461. J.M. Dudley, C. Finot, D.J. Richardson, G. Millot, Nat. Phys. 3 (2007) 597. C. Finot, G. Millot, Opt. Express 13 (2005) 5825. S. Chen, L. Yi, D.S. Guo, P. Lu, Phys. Rev. E 72 (2005) 016622. G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, UK, 1996. J. Herrmann, Opt. Commun. 87 (1992) 161. C. De Angelis, IEEE J. Quantum Electron. 30 (1994) 818. S.A. Ponomarenko, S. Haghgoo, Phys. Rev. A 81 (2010) 051801(R). H.Q. Jin, Braz. J. Phys. 45 (2015) 439. S.R. Skinner, G.R. Allan, D.R. Andersen, A.L. Smirl, IEEE J. Quantum Electron. 27 (9) (1991) 2211. M. Rabe, M. Lowisch, F. Henneberger, J. Cryst. Growth 184/185 (1998) 248. K. Watanabe, H. Nakano, A. Honold, Y. Yamamoto, Phys. Rev. Lett. 62 (1989) 2257. G.R. Allan, S.R. Skinner, D.R. Andersen, A.L. Smirl, Opt. Lett. 16 (1991) 156. T. Flissikowski, A. Hundt, M. Lowisch, M. Rabe, F. Henneberger, Phys. Rev. Lett. 86 (2001) 3172.
1112
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of high-power parabolic pulses in quantum dot waveguide amplifiers
T
⁎
Jun-Rong He , Siliu Xu, Li Xue School of Electronic and Information Engineering, HuBei University of Science and Technology, Xianning 437100, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Self-similar techniques Quantum dot waveguide amplifiers High-power parabolic pulses
The self-similarity techniques are used to study pulse propagation in a quantum dot waveguide amplifier with an arbitrary longitudinal gain profile. Generation of high-power parabolic pulses are analyzed under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. In particular, the periodic distributed gain enables us to obtain a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp. Finally, a possible experimental protocol is proposed to generate the high-power parabolic pulses in realistic waveguides.
1. Introduction In the past decades, extensive research works have been carried out to find the exact analytical and asymptotic similaritons in gain amplifier systems due to their potential applications in nonlinearity and dispersion management systems [1–3]. These optical similaritons possess many attractive features that make them potentially useful for various applications in fiber-optic telecommunications and photonics, since they can maintain their overall shapes but allow their amplitudes and widths changing with the modulation of the system's parameters such as dispersion, nonlinearity, gain, and inhomogeneity [4,5]. Fiber optics and waveguide optics are used in most of the important applications. It is common knowledge that the so-called self-similar solution was used in early studies as a qualitative test for the self-focusing theory and ultra-short pulses generation [6–9]. Generally speaking, optical similaritons can be divided into two categories. The first category is the exact optical similaritons, which are mainly described by the exact solitary-wave solutions, including the bright and dark soliton solutions [10–14], the quasisoliton solutions [15], the nonlinear Bloch waves [16], and the solitons on the continuous-wave background [17]. The second category is the asymptotic optical similaritons, which are mainly described by the parabolic, Hermite-Gaussian, and hybrid functions [18–23] and exist in a wide range of gain amplifier when the strict balance of system parameters is broken. Among these optical similaritons, the asymptotic parabolic similariton has attracted more attentions due to: (i) they can be easily generated from arbitrary input optical waves; (ii) they have strict linear chirps, which are important to the effective compression of optical waves; (iii) their stabilities are guaranteed even with the high power input, and so on. These advantages usually does not belong to the exact optical similaritons. It should be noted that the previously studies are usually focused on optical pulses (beams) propagating in graded-index waveguide amplifiers with cubic Kerr nonlinearity, i.e., the refractive index is n(z, x) = n0 + n2I(z, x), where z is the propagation distance, x is the spatial coordinate, and I is the pulse intensity. Here the first term describes the linear part of the refractive index and the last term represents the Kerr-type nonlinearity with n2 being the cubic coefficient (positive n2 for self-focusing nonlinearity and negative n2 for self-defocusing nonlinearity). When the intensity of the optical pulse exceeds a certain value, a higher-order nonlinear
⁎
Corresponding author. E-mail address: [email protected] (J.-R. He).
https://doi.org/10.1016/j.ijleo.2019.01.103 Received 29 January 2019; Accepted 29 January 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 182 (2019) 1106–1112
J.-R. He, et al.
effect such as the quintic nonlinearity should be taken into account [24], which leads to the refractive index as
n (z , x ) = n 0 + n2 I (z , x ) + n4 I (z , x )2 ,
(1)
where n4 is the quintic coefficient of the nonlinearity which may assume positive or negative values. In experiment, the cubic-quintic nonlinearities can be obtained by doping a fiber with two appropriate semiconductor materials [26,27]. On the other hand, recent study suggests that the quintic nonlinearity could be realized in a centrosymmetric nonlinear medium doped with resonant impurities in the limit of a large light carrier frequency detuning from the impurity resonance [28]. The resonant impurities could be rare-earth element atoms, erbium-doped glasses, or semiconductors doped with quantum dots (QDs). The authors demonstrate that stable exact bright spatial similaritons can be supported by homogeneous conservative optical media with quintic nonlinearities. However, there is no work on this model for obtaining asymptotic self-similar solutions, which have important applications in many fields of nonlinear science, especially in nonlinear optical-fiber and waveguide amplifiers. Moreover, in the realistic optics fiber and waveguides, there will always be some factors influencing the propagation of pulses. These effects may lead the pulses to be amplification (or attenuation) during the propagation. Thus, a model including the gain (loss) effects in optics fiber and waveguides is reasonable. For these reasons, in this work we study the generation of parabolic pulses in quantum dot waveguide amplifiers. The paper is organized as follows. In Section 2, we provide the theoretical model to describe the nonlinear wave propagation in a quantum dot waveguide amplifier. In Section 3, we study the generation of high-power parabolic pulses under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. In particular, the periodic distributed gain enables us to obtain a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp, which have important applications in nonlinear optical-fiber and waveguide amplifiers. In Section 4, we give a possible experimental protocol which may generate the reported self-similar parabolic pulses in semiconductor quantum dot waveguide amplifiers. In Section 5, the main results of the paper are summarized. 2. The model Under the paraxial and the slowly varying envelope approximations, the nonlinear wave equation governing pulse propagation in a quantum dot waveguide amplifier given by Eq. (1) can be written as
i
Nd eg ω2 ig(z ) ∂u 1 ∂ 2u k n k n + + σ∞ + 0 2 |u|2 u + 0 4 |u|4 u = u, 2 ∂z 2k 0 ∂x 2 2ϵ 0 k 0 c 2 n0 n0
(2)
where g(z) is the gain (loss) coefficient. The third term on the left of the equation above represents the non-linear polarization induced by QDs, where N is the dopant density, deg is the dipole matrix element between the excited state and the ground state, and σ∞ is the stable value of the atomic dipole moment. In the cw limit and assuming the light carrier frequency are far enough from the QDs, σ∞ is possible to developed into
σ∞ ≈
2dge u ℏ∆
(1 −
4γ⊥ |dge |2 |u|2 γ∥
ℏ2∆2
+
16γ⊥2 |dge |4 |u|4 γ∥2 ℏ4∆4
), (3)
where γ⊥(γ∥) is a transverse (longitudinal) decay rate of the atomic dipole moment (inversion), and Δ is a detuning of the incident light from atomic impurity resonance satisfying Δ ≫ γ⊥. If one chooses ∆ =
3
4γ⊥ | dge |4 N γ∥ ℏ3 ϵ0 n 0 | n2 |
, the QD-generated nonlinearity and the
third-order nonlinearity described by the nonlinear interaction of light with the bulk medium in Eq. (2) offset each other, causing an effective renormalized quintic nonlinearity with the coefficient
n4eff = n4 + n2 3
4γ⊥ ϵ 20 n02 n22 , γ∥ |dge |2 N 2
(4)
which indicates that n4eff may be positive or negative, depending on the sign of n2 and n4. Introducing the normalized variables U = (k 0 |n4eff |LD / n 0)1/4u, G (Z ) = g (z ) LD , X = x / w0 and Z = z/LD with LD = k 0 w02 being the diffraction length, then Eq. (2) can be rewritten in a dimensionless form
i
∂U 1 ∂ 2U iG + + α|U |4 U = U, ∂Z 2 ∂X 2 2
(5)
where α = |n4eff|/n4eff = ± 1 corresponds to self-focusing (+) and self-defocusing (−) quintic nonlinearity of the waveguide, and G(Z) is a function of the normalized distance Z. When the relative strength of the diffraction term is much less than that of the quintic nonlinearity (the initial power is large enough), the form of asymptotic solution can be found. In the following, we consider the case α = −1 (Our direct numerical simulations show that the asymptotic pulses cannot be generated in waveguide with the quintic focusing nonlinearity). In this situation, the cubic nonlinearity coefficient n2 should be self-defocusing; the quintic nonlinearity coefficient n4 may assume positive or negative values because it can be neglected in such a medium [28]. The self-similar solution is sought in the form 1 1 U (Z , X ) = exp[ E (Z )][W (Z )]− 2 Θ (θ)exp[iΦ (Z , X )], 2
(6) 1107
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Fig. 1. Attenuation rate as a function of distance Z expressed by condition (9) for different gain profiles. The parameters are G0 = Ze = Zp = 1.
The explicit dynamics of the field profile depend on the phase evolution which is found to be given by
Φ (Z , X ) = B (Z ) +
(X − XC )2 dW (Z ) , 2W (Z ) dZ
(7)
Z
where E ≡ ∫0 G (Z′)dZ′ and θ = (X − XC)/W(Z) with W(Z) being the width of the solution, XC denotes the pulse position, and B(Z) is a phase offset. Substituting Eqs. (6) and (7) into Eq. (5), we obtain the following differential equation for the function Θ:
−
1 ∂ 2Θ W2 dB (Z ) W3 d2W (Z ) 2 exp[−2E (Z )] 2 + Θ+ θ Θ + Θ5 = 0, 2 ∂θ exp[2E (Z )] dZ 2 exp[2E (Z )] dZ 2
(8)
To obtain high-power parabolic pulses, we consider the situation when the relative strength of the diffraction term is much less than that of the quintic nonlinearity, i.e., the first term in the left hand side of Eq. (8) can be neglected. This requires that
1 exp[−2E (Z )] ≪ 1, 2
(9)
Next, we will study three different gain profiles to simulate the amplification behaviors for (a) exponential change with G(Z) = G0 exp(Z/Ze), (b) constant gain with G(Z) = G0, and (c) periodic change with G(Z) = 1 + ϵ cos(Z/Zp), ϵ ∈ (−1, 1). Fig. 1 shows the attenuation rates of the above three gain profiles as a function of distance Z. It is seen that the exponential gain attenuates fastest, followed by constant gain. The attenuation rate of periodic gain depends on the sign of ϵ: when ϵ is negative, its attenuation rate is faster than constant gain; when ϵ is positive, its attenuation rate is slower than constant gain. This property determines how fast the input pulse evolves into a parabolic one. 3. Numerical simulations 3.1. System with exponential gain We consider that the gain profile is G(Z) = G0 exp(Z/Z0). The control of gain is possible in experiments [20]. Direct numerical simulations are shown in Fig. 2. Figs. 2(a) and (b) show the self-similar evolution of the parabolic pulse in the waveguide amplifier with the Gaussian input pulse and the hyperbolic secant input pulse, respectively. It is found that the pulse has evolved toward a parabolic solution in the amplifier after about 1.2 propagation distance. Moreover, the simulations show low-amplitude wings on the parabolic pulse whose amplitude decreases with increasing propagation distance, vanishing as Z→ ∞. Another significant property is that the input pulses with different shapes and width but of same initial power converge to the same parabolic pulse, see Fig. 2(c) and (d). 3.2. System with periodic gain We consider that the gain profile is G(Z) = 1 + ϵ cos(Z/Zp), described by a periodic modulated function. Direct numerical 1108
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Fig. 2. Evolutions of the pulse in quantum dot waveguide amplifier with exponential gain, starting with (a) the Gaussian input pulse and (b) the hyperbolic secant input pulse. From the top to bottom, the propagation distance is Z = 1.8, 1.6, 1.4, and 1.2, respectively. (c) and (d) Evolutions of pulse width and pulse peak amplitude as functions of propagation distance for Gaussian pulses. The direct numerical simulations for Eq. (5) have the same initial power Pin = 1 but different full width half maximum (FWHM). The parameters are G0 = Z0 = 1.
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Fig. 3. Evolutions of the pulse in quantum dot waveguide amplifier with periodic gain, starting with the Gaussian input pulse for (a) ϵ = −0.5 and (b)ϵ = 0.5. From the top to bottom, the propagation distance is Z = 4.4, 4.0, 3.6, 3.2, and 2.8, respectively. The direct numerical simulations for Eq. (5) have the same initial power Pin = 1 for Zp = 1.
simulations are shown in Fig. 3. Fig. 3(a) and (b) show the self-similar evolution of the parabolic pulse in the waveguide amplifier with the Gaussian input pulse for ϵ = −0.5 and ϵ = 0.5, respectively. It is observed that the pulse has evolved toward a parabolic solution in the amplifier and shows low-amplitude wing with the increase of propagation distance. In addition, we find that the central region of Fig. 3(b) is wider than that of Fig. 3(a). This may be explained as: the slower the attenuation rate expressed by condition (9) is, the larger the pulse broadening is, and the larger the frequency chirp is, which results in the maximum spectral width. This means that it is the preferred profile to obtain the minimum pulse duration after compression. When ϵ = 0, we have G(Z) = 1, corresponding to a constant gain. In this case, Eq. (8) can be solved analytically, which has been discussed in Ref. [29]. Our work is different: (i) the model in this paper is derived from in quantum dot waveguide amplifiers, while the model in Ref. [29] appears in quintic nonlinear media. (ii) In our paper the gain profile is a function of propagation distance Z while in Ref. [29] it is considered to be a constant. Therefore, our results are more general. In fact, we can see that the results obtained here are more interesting and experimental when the gain varies with the propagation distance. For instance, when ϵ > 0, we can obtain a fast amplification of high power pulses in parabolic state, which has the largest broadening and frequency chirp, thus generating the largest spectral width. (iii) We have considered the gain varies periodically with propagation distance Z, which has not been reported for obtaining parabolic pulses to our knowledge. 3.3. Initial chirped pulses In the case of linearly chirped Gaussian pulses, the input function can be written as 2
1 + ic ⎛ X ⎞ U (0, X ) = exp ⎡ ⎢− 2 ⎝ X0 ⎠ ⎣ ⎜
⎟
⎤, ⎥ ⎦
(10)
where c is a chirp parameter. It is common to refer to the chirp as being positive or negative, depending on whether c is positive or negative. Employing the Fourier transformation, we find that the spectral half-width (at 1/e-intensity point) of the input Gaussian pulse is ∆Ω = 1 + c 2 / X0 . Clearly, spectral width of a pulse is enhanced by a factor of 1 + c 2 in the presence of linear chirp. In the absence of frequency chirp (c = 0), the spectral width satisfies the relation ΔΩX0 = 1. Fig. 4 shows the intensity and spectrum power of the output in quantum dot waveguide amplifier at Z = 5 with a linearly chirped Gaussian input pulse. It is found that a chirped Gaussian pulse for c > 0 broadens monotonically at a rate faster than that of the chirp-free pulse (c = 0). The reason is related to the fact that the diffraction-induced chirp and input chirp have the same sign, resulting in a superposition effect. When c < 0, the contribution of the diffraction-induced chirp is oppositive to that of the input chirp. In this case, the chirped Gaussian pulse also broadens at a rate faster than that of the chirp-free pulse after a proper propagation distance, but with a slower of broadening speed than that of the positive chirp. 1110
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Fig. 4. (a) The intensity of the output in quantum dot waveguide amplifier at Z = 5, starting with the linearly chirped Gaussian input pulse. (b) The output spectrum power as a function of spatial frequency corresponding to (a).
4. Possible experimental generation We now give a suitable experimental protocol to generate the obtained high-power parabolic pulses in realistic waveguides. We consider a 1 μm thick planar waveguide (the input pulse is confined in the y direction), such as ZnSe [30], doped with CdSe QDs for N ≈ 1012 cm−3, which are grown by molecular beam epitaxy using a thermal activation procedure [31]. A typical value of the dipole matrix element is |dge| ≈ 10−28 Cm at a transition wavelength in the middle of λ≈500 nm [32] and γ⊥ = 2.5γ∥ [28]. The refractive index in Eq. (1) for ZnSe can be graded with n0 ≈ 2.7 and n2 ≈ −7 × 10−14 cm2/W near 500 nm [14,33], which lead to n4eff ≈ −10−21 cm4/W2 and Δ ≈ 1012 s−1. On the other hand, a typical exciton lifetime of CdSe QD is roughly 100 ps [34], translating into γ⊥ ≈ 1010 s−1, such that the system is well within the confines of a purely dispersive large-detuning regime Δ ≫ γ⊥. For the transverse scale typically of tens of microns order of magnitude, at z ≈ 7 cm (corresponding to five propagation distance units), one can obtain that the peak intensity is about 300 MW/cm2 and the required input power Pin∼10 W. Such input power levels can be realized under quasi-cw conditions (such as Nd:YAG laser) for which our cw theory remains applicable. 5. Conclusions In conclusion, we have shown that the high-power parabolic pulses can be generated in a quantum dot waveguide amplifier with an arbitrary longitudinal gain profile for defocusing nonlinearity. Analysis of high-power parabolic pulses were implemented under different gain profiles by considering initial chirp-free and chirped pulses via numerical simulations. The results showed that the periodic distributed gain can support a fast amplification of high power pulses in parabolic state with the largest broadening and frequency chirp. We have also presented a possible experimental protocol to generate the high-power parabolic pulses for the longitudinal distribution of gain in realistic waveguides. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 11847103 and the Field Grade Project of HuBei University of Science and Technology under Grant No. 2016-19XB006. References [1] G.P. Agrawal, Nonlinear Fiber Optics, 4th ed., Academic, New York, 2007. [2] J.D. Moores, Opt. Lett. 21 (1996) 555.
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