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Numerical simulation of solitons switching and propagating in asymmetric directional couplers
Optics Communications 285 (2012) 118–123
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Numerical simulation of solitons switching and propagating in asymmetric directional couplers Qiliang Li ⁎, Aixin Zhang, Xiaofeng Hua Institute of Communication and Information System, Hangzhou Dianzi University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 23 July 2011 Received in revised form 4 September 2011 Accepted 5 September 2011 Available online 21 September 2011 Keywords: Asymmetric directional couplers Switching threshold power Coupling length Solitons pulse evolution
a b s t r a c t A numerical investigation, based on the use of split step Fourier transformation algorithm, of all-optical solitons switching in asymmetric directional couplers is presented. The numerical algorithm is described in details. The analysis highlights the influence of the different effective mode area, the phase- and group–velocity mismatch, the different dispersion between two cores on the switching and propagation of short pulses. The investigation indicates that the phase velocity mismatch and the different effective mode area can reduce the coupling length while the different group velocity and the different dispersion between two cores do not change the coupling length. We have also found that the increase of effective mode area ratio can lead to an increase of the switching threshold power but improve significantly the switching steepness, the increase of the phase velocity mismatch can cause a decrease of the switching threshold power but degrade the switching steepness, the increase of the ratio of dispersion can result in a decrease of the switching threshold power and vary the switching steepness, the increase of group velocity mismatch can give rise to an increase of the switching threshold power but improve obviously the switching steepness. Furthermore, the group velocity mismatch can induce solitons pulse to walk off or stretch in the asymmetric directional coupler. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nonlinear directional couplers (NLDCs), which are made of two or multi guiding channels exhibiting an intensity-dependence refractive, have potentials for all-optical signal processing and optical computing [1–2], many interesting applications of these devices have stimulated a great deal of researches since it was theoretically studied by Jensen [3] and experimental fundamentals of phenomenon of the optical selfswitching was presented by Maier [4]. There are two important operating modes in NLDCs. In the first mode, at low power level, the NLDC behaves as a linear device. By evanescent coupling, the channels switch between cores with a period that depends on their wavelengths. For a coupling length, when a pulse enters an input port, the pulse emerges from the output port of cross channel. In the second mode, high power detunes the coupler, and induces changes in the refractive index with n = n0 + n2I, in which n0 is the linear refractive index at low power, n2 is the Kerr nonlinear coefficient, I is the light intensity. Thus, the high power can prevent signal pulses from switching between two cores in couplers. Therefore the device operates as a power controlled switch and can be used for demultiplexing and routing of a stream of pulses [5–6]. Generally, two cores of couplers are assumed to be symmetrical. However, in practice, the couplers with fully identical cores are
⁎ Corresponding author. Tel.: +86 571 86919123. E-mail address: [email protected] (Q. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.003
seldom existent. Furthermore, compared with symmetric directional couplers, the asymmetric directional couplers with dissimilar cores can be easily made and may even exhibit many interesting switching performances. In fact, as already evidenced [7–9], a NLDC has better switching performance in the presence of different nonlinear cores, and coupler with non-identical cores can have better performances than identical cores. To our knowledge, corresponding to the continuous wave, the solitons switching has much more advantages because the solitons can keep their shapes during propagation in fiber. Continued interests in solitons switching and in asymmetric directional couplers are driven because of the excellent switching performance. Many theories and experiments have proved that solitons can exist and propagate in asymmetric dual-core optical fiber. Kaup et al. proved that the solitons can exist in asymmetric dualcore optical fibers in a 1997 numerical simulation [10]. Later, they still found that the bright solitons may exist in the normal dispersion region [11]. Mak et al. studied spatial solitons and their stability in a pair of asymmetric linearly coupled waveguides with intrinsic quadratic nonlinearity in 1998 [12]. Zafrany et al. studied that the solitons in a linearly coupled system with separated dispersion and nonlinearity [13]. Much works have been done about solitons switching in asymmetric directional couplers [14–15]. Of course, the asymmetric dual-core directional couplers have proved to be widely used for all-optical communications [16–17]. These applications of the asymmetric directional couplers in all-optical communications motivate researchers to further research different dynamics features about the asymmetric directional couplers.
Q. Li et al. / Optics Communications 285 (2012) 118–123
As we have mentioned above, in practice, the identical cores are hardly realized, herein we introduce three ways which yield the asymmetry between the two cores artificially. In the first of these, the dissimilarity of diameters between the two cores, which can not only generate the phase velocity mismatch but also cause the different nonlinear coefficient, makes the coupler asymmetric [10]. In the second approach, using the birefringent fibers, which can make the phase velocity mismatched and the nonlinear coefficient unchangeable [18], can also realize the asymmetry. In the third method, using the different dispersion fibers can change the group velocity dispersion to achieve the asymmetry. In this paper, we extend the analysis of switching performance with introduction of the asymmetry in the directional couplers by coupled mode theory. The paper of He et al. [2] studied the influence of the group velocity mismatch and the different effect mode area on the switching performance. In our model of dual-core directional couplers, the asymmetry mainly includes the different effect mode area and different dispersion, also includes the phase- and group velocity mismatch. We also use split-step Fourier transformation (SSFT) [19] to investigate the switching performance and the solitons pulse evolution during propagation. The paper is organized as follows. In Section 2, we introduce the theory model. In Section 3, we use the split-step Fourier method to solve the coupled equations. Section 4 devotes the computer simulation to investigating the switching performance in the asymmetric directional couplers of different asymmetrical parameters. The conclusion is drawn in Section 5.
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where we suppose that the first core is of anomalous dispersion (β21 b 0) and introduce the asymmetry parameters [20]: dp ¼ δa LD dg ¼ ðβ12 β11 ÞLD =T0 d2 ¼ β22 =β21 dn ¼ γ2 =γ1 k ¼ κLD In physics, dp and dg are phase- and group velocity mismatch, respectively. The ratio d2 and dn are the difference of dispersion and effective mode area, respectively. 3. Numerical algorithm To solve CNLSEs of Eqs. (2a) and (2b) and understand the behavior of solitons switching in asymmetric directional coupler, we use a technique based on the split step Fourier method. The technique was usually used in several problems of optical solitons. Firstly, considering the dispersion, coupling, phase- and group velocity mismatch from ξ to ξ + h/2 and from ξ + h/2 to ξ + h, we have 2
∂u1 i ∂ u1 þ idp u1 þ iku2 ¼ 2 ∂τ2 ∂ξ
ð3aÞ
∂u2 ∂u i ∂2 u1 −idp u2 þ iku1 : ¼ −dg 1 þ d2 ∂ξ ∂τ 2 ∂τ2
ð3bÞ
The nonlinear effect of the whole h is considered at the midsegment located at ξ + h/2, so we get
2. Theoretical model
∂u1 2 ¼ iju1 j u1 ∂ξ
ð4aÞ
Ultrashort solitons pulse propagation in an asymmetric directional coupler array, with nearest-neighbor coupling, can be described by coupled nonlinear Schrödinger equations (CNLSEs) [20]:
∂u2 2 ¼ idn ju2 j u2 : ∂ξ
ð4bÞ
∂A1 ∂A i ∂2 A 2 2 þ β11 1 þ β21 21 ¼ iκ12 A2 þ iδa A1 þ i γ1 jA1 j þ C12 jA2 j A1 2 ∂z ∂t ∂t ð1aÞ ∂A2 ∂A i ∂2 A 2 2 þ β12 2 þ β22 22 ¼ iκ21 A1 iδa A2 þ i γ2 jA2 j þ C21 jA1 j A2 2 ∂z ∂t ∂t ð1bÞ where vgm ≡ 1/β1m, m = 1, 2 is the group velocity and β2m is the group-velocity dispersion (GVD) in the mth core , δa ¼ 12 ðβ01 β02 Þ is a measure of asymmetry between the two cores, γ1, γ2 are the nonlinearity parameters of core 1 and core 2, respectively. κ12 and κ21 are the coupling parameters, C12 and C21 are the cross-phase modulation parameters. In practice, the cross-phase modulation (XPM) is weak, so the XPM can be neglected. Here we set the coupling coefficients in the two cores to be consistent. LD = T02/|β21| is the dispersion length. Performing the following transformation: z t τ ¼ t=T0 ; ξ ¼ þ vg1 τ; u1 ¼ ðγ1 LD Þ1=2 A1 ; u2 ¼ ðγ1 LD Þ1=2 A2 ; LD LD
Performing the Fourier transformation for Eqs. (3), we obtain ∂ u˜ 1 ¼ ∂ξ
i 2 ω þ idp u˜ 1 þ ik u˜ 2 2
ð5aÞ
∂ u˜ 2 i 2 ¼ ik u˜ 1 þ idg ω d2 ω idp u˜ 2 : 2 ∂ξ
ð5bÞ
Where ũ1 and ũ2 is the Fourier transformation of u1 and u2. Solving Eqs. (5a) and (5b), we can suppose that initial condition u1(0, ω) = u10, u2(0, ω) = 0 and get u˜ 1 ξ þ h2 ; ω , u˜ 2 ξ þ h2 ; ω at ξ + h/2: "
2
u˜ 1 ðξ þ h=2; ωÞ ¼ expðik1 h=2ÞV cosðk2 h=2Þ u1 ðξ; ωÞ
ω =2 þ dp k1 u˜ 2 ðξ; ωÞ k
#
2
þik2 =k sinðk2 h=2Þ u˜ 2 ðξ; ωÞg þ
ω =2 þ dp k1 u˜ 2 ðξ þ h=2; ωÞ; k
ð6aÞ u˜ 2 ðξ þ h=2; ωÞ ¼ expðik1 h=2ÞV cosðk2 h=2Þ u˜ 2 ðξ; ωÞ ; " # 2 ω =2 þ dp k1 k þi sinðk2 h=2Þ u˜ 1 ðξ; ωÞ u2 ðξ; ωÞ g k2 k2
ð6bÞ
We can obtain normalized CNLSEs as follows: where k1 ¼ ∂u1 i ∂2 u1 2 iku2 idp u1 iju1 j u1 ¼ 0 ∂ξ 2 ∂τ2
ð2aÞ
∂u2 ∂u i ∂2 u2 2 iku1 þ idp u2 idn ju2 j u2 ¼ 0 þ dg 2 d2 ∂ξ ∂τ 2 ∂τ2
ð2bÞ
2
ð1þd2 Þω2 þ2dg ω , 4 4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ 14 A þ B þ C , here 3
A ¼ ð1 d2 Þ ω þ 4ðd2 1Þdg ω ; 2 B ¼ 4 dg 2dp þ 2dp d2 dp ω ; 2 2 C ¼ 16dp dg ω þ 16 dp þ k :
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Q. Li et al. / Optics Communications 285 (2012) 118–123
Secondly, considering the nonlinear effect of the whole h at the mid-segment located at ξ + h/2, we obtain
h i h h e1 ξ þ ; ω · exp ih ju1 ðξ; τÞj2 þ ju1 ðξ þ h; τÞj2 =2 ; u1NL ξ þ ; τ ¼ IFFT u 2 2
ð7aÞ
h i h h 2 2 ∼ u2NL ξ þ ; τ ¼ IFFT u2 ξ þ ; ω · exp idn h ju1 ðξ; τÞj þ ju1 ðξ þ h; τÞj =2 : 2 2
ð7bÞ Thirdly, we only consider the dispersion, coupling, phase velocity mismatch and group velocity mismatch from ξ + h/2 to ξ + h, we have
k h k h h u˜ 1 ðξ þ h; ωÞ ¼ exp i 1 ⋅V cos 2 ðFFT u1NL ξ þ ; τ 2 2 2
ð8aÞ
1 2
ω þ dp k1 h 2 FFT u1NL ξ þ ; τ Þ 2 k
k k h h þi 2 sin 2 FFT u1NL ξ þ ; τ g 2 2 k
þ
1 2 ω þ dp k1 ∼ 2 u2 ðξ þ h; ωÞ; k
k h k h h u˜ 2 ðξ þ h; ωÞ ¼ exp i 1 ⋅ cos 2 FFT u2NL ξ þ ; τ 2 2 2
k h k h þi sin 2 FFT u1NL ξ þ ; τ 2 k2 2
V V
ð8bÞ
ω2 =2 þ dP k1 h FFT u2NL ξ þ ; τ : 2 k2
tt
u1 ðξ þ h; τÞ ¼ IFFT ½u˜ 1 ðξ þ h; ωÞ;
ð9aÞ
u2 ðξ þ h; τÞ ¼ IFFT ½u˜ 2 ðξ þ h; ωÞ:
ð9bÞ
4. Results and discussion In this section, to compute the transmission coefficients which characterize switching- and propagation-performance of solitons in an asymmetric directional coupler, the following initial conditions, which correspond to the edge excitation of coupler, are considered: u1 ðξ ¼ 0; τÞ ¼ P sechðτÞ;
ð10aÞ
u2 ðξ ¼ 0; τÞ ¼ 0:
ð10bÞ
Here P is the input peak power. We can ignore the initial chirp which is small in ideal solitons communication systems. The pulse is fully input in first core, so we define the first core as bar channel, and the second core as cross channel. Here we define the transmission coefficient of core i (the i core fiber) as Ti: ∞
∫ jui ðξL ; τÞj2 dτ −∞ ∞
∫ ju1 ðξL ; τÞj2 dτ −∞
;
4.1. Periodic characteristics of propagation in two cores Fig. 1(a)–(h) shows the curves of energy transfer coefficient T with the change of coupling transmission distance ξ in the case of lowpower input (P = 1). As shown in Fig. 1(a) and (b), one can find that the coupling length remains unchanged while the group velocities of the two cores are mismatched. In Fig. 1(c) and (d), one can notice that the increase of phase velocity mismatch dp leads to shortening the coupling length. And the result indicates that the change of phase velocity mismatch causes coupling length to modify, furthermore, the power can not be fully coupled into the second core at the length of LC. But, it is more interesting that the total energy can be transferred into the first core at the length of 2LC. Thus, the coupling length is not only determined by coupling coefficient but also affected by phase velocity mismatch. In Fig. 1(e) and (f), one can infer that when the two cores have the same sign of dispersion, the coupling length is almost the same as the symmetric coupler. In Fig. 1(g) and (h), one can notices that the increase of the nonlinear coefficient ratio leads to a decrease of the coupling length. 4.2. Switching characteristics
Finally, at ξ + h, we obtain
Ti ¼
Where ξL is transmission distance at which the power is fully coupled from the first core into the second core. To convert the solitons units back to real units, the investigation is restricted our general considerations to the real communication fiber, which parameters are chosen as λ=1.55 μm, β21 =−20 ps2/km, the coupling coefficient of κ=0.8 cm−1, a fiber effective area of Aeff =20 μm2, and the nonlinearity coefficient of γ=10 W−1/km. The coupling length Lc =π/2κ≈1.96 cm, and when the solitons pulse width is usually T0 =50 fs, the dispersion length LD =T02/|β21|=1.25 cm. As we know, when the dispersion length is comparable to the coupling length, the solitons switching can occur. According to the parameters of the first core, which are given above, those of the second core are properly modified.
ð11Þ
In Fig. 2(a)–(d), one has a plot of the transmission coefficients of the bar state of the coupler as a function of the input power. Fig. 2(a) shows the transmission characteristics of the bar channel with the diverse ratio of effective mode area, here dg = 0, dp = 0, d2 = 1, k = 1, the calculation is performed at the coupling length LC = π/2. One can notice that the decrease of effective mode area of the cross core (or the increase of the ratio of nonlinear coefficient), when the effective mode area of the first core unchanged and the power is injected in the first core, leads to the increase of the switching threshold power, but improves the switching steepness. In Fig. 2(b), one has the transmission characteristics of the bar channel with the different phase velocity mismatches, we set dg = 0, dn = 1, d2 = 1, k = 1, when dp ≠ 0, from Fig. 1(c) and (d), we know that the coupling length changes with the introduction of the phase velocity mismatch. Thenew coupling length can be estimated by using the forpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mula LC ¼ π= 2 k2 þ δp 2 . At the new coupling length LC, one can find that whether dp N 0 or dp b 0, the threshold power decreases with the increasing of dp, but the switching steepness degrades. So the threshold power can be decreased by decreasing the phase velocity of the cross core. Fig. 2(c) represents the transmission performances of the bar channel with the diverse ratios of dispersion, we take parameters dg = 0, dn = 1, dp = 0, k = 1. At the coupling length LC = π/2, it shows that the increase of the dispersion ratio d2 leads to a decrease of the switching threshold power and degrades the switching steepness. In Fig. 2(d), one has a curve of the transmission performances of the bar channel as a function of input power for different group velocity mismatches. Here d2 = 0, dn = 1, dp = 0, k = 1. We know from Fig. 1(a) and (b) that the coupling length remains unchanged with
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Fig. 1. Transmission curves of energy transfer coefficient being with the change of coupling transmission distance in a coupler. (a) dg = 0, dn = 1, dp = 0, d2 = 1, k = 1;(b) dg = 0.5, dn = 1, dp = 0, d2 = 1, k = 1;(c) dp = 0.5, dg = 0, dn = 1, d2 = 1, k = 1; (d) dp = − 0.5, dg = 0, dn = 1, d2 = 1, k = 1;(e) d2 = − 1.5, dg = 0, dn = 1, dp = 0, k = 1; (f) d2 = 1.5, dg = 0, dn = 1, dp = 0, k = 1; (g) dn = 0.4,d2 = − 1.5, dg = 0, dp = 0, k = 1; and (h) dn = 3, d2 = − 1.5, dg = 0, dp = 0, k = 1.
the changing of group velocity. At the length LC = π/2, as displayed in Fig. 2(d), one can notice that the switching threshold power increases with increasing of dg, and the switching steepness degrades. 4.3. The solitons evolution during propagation The numerical simulation of solitons propagation in an asymmetric dual-core nonlinear coupler is shown in Figs. 3 and 4.
Fig. 3(a1) and (a2) has a plot of solitons evolution in the bar channel and cross channel with different efficient mode area, one can find that the pulse peak power get lower in two cores when the efficient mode area of the cross channel is half of that of the bar channel (dn = 0.5), the solitons pulse is significantly broadened during propagation, however, as show in Fig. 3(b1) and (b2), the pulse can keep the high peak power when the efficient mode area of the cross channel is twice of that of the bar channel (dn = 2).
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1
1
0.8
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dp=0
dn=0.5
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dn=0.2
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dn=1 0.2 0 0
dp=-0.5
0.2
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P0 (Lc=pi/(2))
P0 (Lc=pi/(2k))
(a)
(b) 1
0.8
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T
1
d2=0.5
0.4
dg=0
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dg=1.5
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0
2
4
6
0
0
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P0 (Lc=pi/(2))
P0 (Lc=pi/(2))
(c)
(d)
6
Fig. 2. The switching characteristic of the bar channel from numerical method. (a) dg = 0, dp = 0, d2 = 1, k = 1; (b) dg = 0, dn = 1, d2 = 1, k = 1; (c) dg = 0, dn = 1, dp = 0, k = 1; and (d) d2 = 1, dn = 1, dp = 0, k = 1.
Fig. 4(a) shows that the group velocity mismatch leads to the solitons pulse walk-off. Fig. 4(b) shows that the pulse significantly broadens in the cross channel in the presence of normal-dispersion in the cross channel but in the presence of anomalous-dispersion in
the bar channel, but the pulse seldom broadens in the bar channel. In addition, the dispersion difference of the two cores results in the decrease of the peak power. Fig. 4(c), (d) and (e) show that the pulse hardly spreads with phase velocity mismatch. As shown in Fig 4. (e), when the enlargement of phase velocity mismatch to a certain extent causes, most of energy remains in the bar channel, and no energy is transferred into the cross channel. 5. Conclusion
(a1)
(a2)
(b1)
(b2)
Fig. 3. Solitons evolution in the bar channel and cross channel. (a1) and (a2) dn = 0.5, dg = 0, d2 = 1, dp = 0, k = 1; (b1) and (b2) dn = 2, dg = 0, d2 = 1, dp = 0, k = 1.
In conclusions, we concern about the impact of the different effective mode area, the phase- and group –velocity mismatch, the different dispersion between two cores on the switching and propagation of short pulses in an asymmetric NLDC. A general method, based on SSFT, is used to solve CNLSEs which include the evanescent coupling between neighboring waveguides. The investigation indicates that the phase velocity mismatch and the different effective mode area can shorten the coupling length, while the different group velocity and the different dispersion between two cores do not change the coupling length. The theory of coupled-mode in asymmetric directional couplers developed in this work also reveals that the four different asymmetrical parameters can influence the switching and propagatingperformances. We have pointed out that the increase of effective mode area ratio leads to an increase of the switching threshold power but improves significantly the switching steepness, the increasing of the phase velocity mismatch causes a decrease of the switching threshold power but degrades the switching steepness , the increase of the ratio of dispersion results in a decrease of the switching threshold power and varies the switching steepness, the increase of group velocity mismatch leads to an increase of
Q. Li et al. / Optics Communications 285 (2012) 118–123
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the switching threshold power but improves the switching steepness. Furthermore, in the asymmetric directional coupler the group velocity mismatch can induce solitons pulse to walk off or stretch. Acknowledgment
(a1)
(a2)
We are grateful to reviewers for helpful comments. This work was supported by the National Science Foundation of China under grant 10904028 and the Natural Science Foundation of Zhejiang Province under grant Y1110078. References [1] [2] [3] [4] [5] [6]
(b1)
(b2)
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(c1)
(c2)
[17] [18] [19] [20]
(d1)
(d2)
(e1)
(e2)
Fig. 4. Evolution of the solitons amplitudes in the bar channel and cross channel. (a) the asymmetry is only shown in the group velocity (dg = 0.5, dn = 1, dp = 0, d2 = 1, k = 1); (b) the two cores have opposite dispersion(d2 = − 1,dg = 0, dn = 1, dp = 0, k = 1);(c),(d),(e)are about the effect of different phase velocity mismatch on pulse evolution, (c)dp = − 0.5, dg = 0, dn = 1, d2 = 1,k = 1; (d)dp = 0.5, dg = 0, dn = 1,d2 = 1,k = 1;(e)dp = 1, dg = 0, dn = 1, d2 = 1,k = 1.
Amarendra K. Sarma, Optics Communication 284 (2011) 186. Xiujun He, Kang Xie, Anping Xiang, Optik 122 (2011) 1222. S.M. Jensen, IEEE Journal of Quantum Electronics 18 (1982) 1580. A.A. Maier, Physics-Uspekhi 38 (1995) 991. Qiliang Li, Yuyong Xie, Yinfang Zhu, Sheng Qian, Optics Communication 281 (2008) 2811. Qiliang Li, Yuyong Xie, Yinfang Zhu, Yongming Qi, Zhijing Zhao, IEEE Journal of Lightwave Technology 27 (2009) 2933. M. Bertolotti, M. Monaco, C. Sibilia, Optics Communication 116 (1995) 405. J. Atai, Y. Chen, Journal of Applied Physics 72 (1992) 24. J. Atai, Y. Chen, IEEE Journal of Quantum Electronics 29 (1993) 242. D.J. Kaup, T.I. Lakoba, B.A. Malomed, Journal of the Optical Society of America B 14 (1997) 1199. D.J. Kaup, B.A. Malomed, Journal of the Optical Society of America B 15 (1998) 2838. A. Zafrany, B.A. Malomed, I.M. Merhasin, Chaos 15 (2005) 037108. William C.K. Mak, B.A. Malomed, P.L. Chu, Optics Communication 154 (1998) 145. K.Z. No´brega, M.G. da Silva, A.S.B. Sombra, Optics Communication 173 (2000) 413. M. Liu, P. Shum, Optics Express 11 (2003) 116. J.S. de Almeida, J.W.M. Menezes, M.G. Da Silva, A.S.B. Sombra, W.B. Fraga, J.C. Sales, A.C. Ferreira, S.P. Marciano, A.F.G.F. Filho, Fiber and Integrated Optics 26 (2007) 217. W.B. Fraga, J.W.M. Menezes, M.G. da Silva, et al., Optics Communication 262 (2006) 32. J. Atai, B.A. Malomed, Optics Communication 221 (2003) 55. P.M. Ramos, C.R. Paiva, IEEE Journal of Quantum Electronics 35 (1999) 983. G.P. Agrawal, Nonlinear Fiber Optics, 4th ed. Academic Press, Boston, 2007.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Numerical simulation of solitons switching and propagating in asymmetric directional couplers Qiliang Li ⁎, Aixin Zhang, Xiaofeng Hua Institute of Communication and Information System, Hangzhou Dianzi University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 23 July 2011 Received in revised form 4 September 2011 Accepted 5 September 2011 Available online 21 September 2011 Keywords: Asymmetric directional couplers Switching threshold power Coupling length Solitons pulse evolution
a b s t r a c t A numerical investigation, based on the use of split step Fourier transformation algorithm, of all-optical solitons switching in asymmetric directional couplers is presented. The numerical algorithm is described in details. The analysis highlights the influence of the different effective mode area, the phase- and group–velocity mismatch, the different dispersion between two cores on the switching and propagation of short pulses. The investigation indicates that the phase velocity mismatch and the different effective mode area can reduce the coupling length while the different group velocity and the different dispersion between two cores do not change the coupling length. We have also found that the increase of effective mode area ratio can lead to an increase of the switching threshold power but improve significantly the switching steepness, the increase of the phase velocity mismatch can cause a decrease of the switching threshold power but degrade the switching steepness, the increase of the ratio of dispersion can result in a decrease of the switching threshold power and vary the switching steepness, the increase of group velocity mismatch can give rise to an increase of the switching threshold power but improve obviously the switching steepness. Furthermore, the group velocity mismatch can induce solitons pulse to walk off or stretch in the asymmetric directional coupler. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nonlinear directional couplers (NLDCs), which are made of two or multi guiding channels exhibiting an intensity-dependence refractive, have potentials for all-optical signal processing and optical computing [1–2], many interesting applications of these devices have stimulated a great deal of researches since it was theoretically studied by Jensen [3] and experimental fundamentals of phenomenon of the optical selfswitching was presented by Maier [4]. There are two important operating modes in NLDCs. In the first mode, at low power level, the NLDC behaves as a linear device. By evanescent coupling, the channels switch between cores with a period that depends on their wavelengths. For a coupling length, when a pulse enters an input port, the pulse emerges from the output port of cross channel. In the second mode, high power detunes the coupler, and induces changes in the refractive index with n = n0 + n2I, in which n0 is the linear refractive index at low power, n2 is the Kerr nonlinear coefficient, I is the light intensity. Thus, the high power can prevent signal pulses from switching between two cores in couplers. Therefore the device operates as a power controlled switch and can be used for demultiplexing and routing of a stream of pulses [5–6]. Generally, two cores of couplers are assumed to be symmetrical. However, in practice, the couplers with fully identical cores are
⁎ Corresponding author. Tel.: +86 571 86919123. E-mail address: [email protected] (Q. Li). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.003
seldom existent. Furthermore, compared with symmetric directional couplers, the asymmetric directional couplers with dissimilar cores can be easily made and may even exhibit many interesting switching performances. In fact, as already evidenced [7–9], a NLDC has better switching performance in the presence of different nonlinear cores, and coupler with non-identical cores can have better performances than identical cores. To our knowledge, corresponding to the continuous wave, the solitons switching has much more advantages because the solitons can keep their shapes during propagation in fiber. Continued interests in solitons switching and in asymmetric directional couplers are driven because of the excellent switching performance. Many theories and experiments have proved that solitons can exist and propagate in asymmetric dual-core optical fiber. Kaup et al. proved that the solitons can exist in asymmetric dualcore optical fibers in a 1997 numerical simulation [10]. Later, they still found that the bright solitons may exist in the normal dispersion region [11]. Mak et al. studied spatial solitons and their stability in a pair of asymmetric linearly coupled waveguides with intrinsic quadratic nonlinearity in 1998 [12]. Zafrany et al. studied that the solitons in a linearly coupled system with separated dispersion and nonlinearity [13]. Much works have been done about solitons switching in asymmetric directional couplers [14–15]. Of course, the asymmetric dual-core directional couplers have proved to be widely used for all-optical communications [16–17]. These applications of the asymmetric directional couplers in all-optical communications motivate researchers to further research different dynamics features about the asymmetric directional couplers.
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As we have mentioned above, in practice, the identical cores are hardly realized, herein we introduce three ways which yield the asymmetry between the two cores artificially. In the first of these, the dissimilarity of diameters between the two cores, which can not only generate the phase velocity mismatch but also cause the different nonlinear coefficient, makes the coupler asymmetric [10]. In the second approach, using the birefringent fibers, which can make the phase velocity mismatched and the nonlinear coefficient unchangeable [18], can also realize the asymmetry. In the third method, using the different dispersion fibers can change the group velocity dispersion to achieve the asymmetry. In this paper, we extend the analysis of switching performance with introduction of the asymmetry in the directional couplers by coupled mode theory. The paper of He et al. [2] studied the influence of the group velocity mismatch and the different effect mode area on the switching performance. In our model of dual-core directional couplers, the asymmetry mainly includes the different effect mode area and different dispersion, also includes the phase- and group velocity mismatch. We also use split-step Fourier transformation (SSFT) [19] to investigate the switching performance and the solitons pulse evolution during propagation. The paper is organized as follows. In Section 2, we introduce the theory model. In Section 3, we use the split-step Fourier method to solve the coupled equations. Section 4 devotes the computer simulation to investigating the switching performance in the asymmetric directional couplers of different asymmetrical parameters. The conclusion is drawn in Section 5.
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where we suppose that the first core is of anomalous dispersion (β21 b 0) and introduce the asymmetry parameters [20]: dp ¼ δa LD dg ¼ ðβ12 β11 ÞLD =T0 d2 ¼ β22 =β21 dn ¼ γ2 =γ1 k ¼ κLD In physics, dp and dg are phase- and group velocity mismatch, respectively. The ratio d2 and dn are the difference of dispersion and effective mode area, respectively. 3. Numerical algorithm To solve CNLSEs of Eqs. (2a) and (2b) and understand the behavior of solitons switching in asymmetric directional coupler, we use a technique based on the split step Fourier method. The technique was usually used in several problems of optical solitons. Firstly, considering the dispersion, coupling, phase- and group velocity mismatch from ξ to ξ + h/2 and from ξ + h/2 to ξ + h, we have 2
∂u1 i ∂ u1 þ idp u1 þ iku2 ¼ 2 ∂τ2 ∂ξ
ð3aÞ
∂u2 ∂u i ∂2 u1 −idp u2 þ iku1 : ¼ −dg 1 þ d2 ∂ξ ∂τ 2 ∂τ2
ð3bÞ
The nonlinear effect of the whole h is considered at the midsegment located at ξ + h/2, so we get
2. Theoretical model
∂u1 2 ¼ iju1 j u1 ∂ξ
ð4aÞ
Ultrashort solitons pulse propagation in an asymmetric directional coupler array, with nearest-neighbor coupling, can be described by coupled nonlinear Schrödinger equations (CNLSEs) [20]:
∂u2 2 ¼ idn ju2 j u2 : ∂ξ
ð4bÞ
∂A1 ∂A i ∂2 A 2 2 þ β11 1 þ β21 21 ¼ iκ12 A2 þ iδa A1 þ i γ1 jA1 j þ C12 jA2 j A1 2 ∂z ∂t ∂t ð1aÞ ∂A2 ∂A i ∂2 A 2 2 þ β12 2 þ β22 22 ¼ iκ21 A1 iδa A2 þ i γ2 jA2 j þ C21 jA1 j A2 2 ∂z ∂t ∂t ð1bÞ where vgm ≡ 1/β1m, m = 1, 2 is the group velocity and β2m is the group-velocity dispersion (GVD) in the mth core , δa ¼ 12 ðβ01 β02 Þ is a measure of asymmetry between the two cores, γ1, γ2 are the nonlinearity parameters of core 1 and core 2, respectively. κ12 and κ21 are the coupling parameters, C12 and C21 are the cross-phase modulation parameters. In practice, the cross-phase modulation (XPM) is weak, so the XPM can be neglected. Here we set the coupling coefficients in the two cores to be consistent. LD = T02/|β21| is the dispersion length. Performing the following transformation: z t τ ¼ t=T0 ; ξ ¼ þ vg1 τ; u1 ¼ ðγ1 LD Þ1=2 A1 ; u2 ¼ ðγ1 LD Þ1=2 A2 ; LD LD
Performing the Fourier transformation for Eqs. (3), we obtain ∂ u˜ 1 ¼ ∂ξ
i 2 ω þ idp u˜ 1 þ ik u˜ 2 2
ð5aÞ
∂ u˜ 2 i 2 ¼ ik u˜ 1 þ idg ω d2 ω idp u˜ 2 : 2 ∂ξ
ð5bÞ
Where ũ1 and ũ2 is the Fourier transformation of u1 and u2. Solving Eqs. (5a) and (5b), we can suppose that initial condition u1(0, ω) = u10, u2(0, ω) = 0 and get u˜ 1 ξ þ h2 ; ω , u˜ 2 ξ þ h2 ; ω at ξ + h/2: "
2
u˜ 1 ðξ þ h=2; ωÞ ¼ expðik1 h=2ÞV cosðk2 h=2Þ u1 ðξ; ωÞ
ω =2 þ dp k1 u˜ 2 ðξ; ωÞ k
#
2
þik2 =k sinðk2 h=2Þ u˜ 2 ðξ; ωÞg þ
ω =2 þ dp k1 u˜ 2 ðξ þ h=2; ωÞ; k
ð6aÞ u˜ 2 ðξ þ h=2; ωÞ ¼ expðik1 h=2ÞV cosðk2 h=2Þ u˜ 2 ðξ; ωÞ ; " # 2 ω =2 þ dp k1 k þi sinðk2 h=2Þ u˜ 1 ðξ; ωÞ u2 ðξ; ωÞ g k2 k2
ð6bÞ
We can obtain normalized CNLSEs as follows: where k1 ¼ ∂u1 i ∂2 u1 2 iku2 idp u1 iju1 j u1 ¼ 0 ∂ξ 2 ∂τ2
ð2aÞ
∂u2 ∂u i ∂2 u2 2 iku1 þ idp u2 idn ju2 j u2 ¼ 0 þ dg 2 d2 ∂ξ ∂τ 2 ∂τ2
ð2bÞ
2
ð1þd2 Þω2 þ2dg ω , 4 4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ 14 A þ B þ C , here 3
A ¼ ð1 d2 Þ ω þ 4ðd2 1Þdg ω ; 2 B ¼ 4 dg 2dp þ 2dp d2 dp ω ; 2 2 C ¼ 16dp dg ω þ 16 dp þ k :
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Secondly, considering the nonlinear effect of the whole h at the mid-segment located at ξ + h/2, we obtain
h i h h e1 ξ þ ; ω · exp ih ju1 ðξ; τÞj2 þ ju1 ðξ þ h; τÞj2 =2 ; u1NL ξ þ ; τ ¼ IFFT u 2 2
ð7aÞ
h i h h 2 2 ∼ u2NL ξ þ ; τ ¼ IFFT u2 ξ þ ; ω · exp idn h ju1 ðξ; τÞj þ ju1 ðξ þ h; τÞj =2 : 2 2
ð7bÞ Thirdly, we only consider the dispersion, coupling, phase velocity mismatch and group velocity mismatch from ξ + h/2 to ξ + h, we have
k h k h h u˜ 1 ðξ þ h; ωÞ ¼ exp i 1 ⋅V cos 2 ðFFT u1NL ξ þ ; τ 2 2 2
ð8aÞ
1 2
ω þ dp k1 h 2 FFT u1NL ξ þ ; τ Þ 2 k
k k h h þi 2 sin 2 FFT u1NL ξ þ ; τ g 2 2 k
þ
1 2 ω þ dp k1 ∼ 2 u2 ðξ þ h; ωÞ; k
k h k h h u˜ 2 ðξ þ h; ωÞ ¼ exp i 1 ⋅ cos 2 FFT u2NL ξ þ ; τ 2 2 2
k h k h þi sin 2 FFT u1NL ξ þ ; τ 2 k2 2
V V
ð8bÞ
ω2 =2 þ dP k1 h FFT u2NL ξ þ ; τ : 2 k2
tt
u1 ðξ þ h; τÞ ¼ IFFT ½u˜ 1 ðξ þ h; ωÞ;
ð9aÞ
u2 ðξ þ h; τÞ ¼ IFFT ½u˜ 2 ðξ þ h; ωÞ:
ð9bÞ
4. Results and discussion In this section, to compute the transmission coefficients which characterize switching- and propagation-performance of solitons in an asymmetric directional coupler, the following initial conditions, which correspond to the edge excitation of coupler, are considered: u1 ðξ ¼ 0; τÞ ¼ P sechðτÞ;
ð10aÞ
u2 ðξ ¼ 0; τÞ ¼ 0:
ð10bÞ
Here P is the input peak power. We can ignore the initial chirp which is small in ideal solitons communication systems. The pulse is fully input in first core, so we define the first core as bar channel, and the second core as cross channel. Here we define the transmission coefficient of core i (the i core fiber) as Ti: ∞
∫ jui ðξL ; τÞj2 dτ −∞ ∞
∫ ju1 ðξL ; τÞj2 dτ −∞
;
4.1. Periodic characteristics of propagation in two cores Fig. 1(a)–(h) shows the curves of energy transfer coefficient T with the change of coupling transmission distance ξ in the case of lowpower input (P = 1). As shown in Fig. 1(a) and (b), one can find that the coupling length remains unchanged while the group velocities of the two cores are mismatched. In Fig. 1(c) and (d), one can notice that the increase of phase velocity mismatch dp leads to shortening the coupling length. And the result indicates that the change of phase velocity mismatch causes coupling length to modify, furthermore, the power can not be fully coupled into the second core at the length of LC. But, it is more interesting that the total energy can be transferred into the first core at the length of 2LC. Thus, the coupling length is not only determined by coupling coefficient but also affected by phase velocity mismatch. In Fig. 1(e) and (f), one can infer that when the two cores have the same sign of dispersion, the coupling length is almost the same as the symmetric coupler. In Fig. 1(g) and (h), one can notices that the increase of the nonlinear coefficient ratio leads to a decrease of the coupling length. 4.2. Switching characteristics
Finally, at ξ + h, we obtain
Ti ¼
Where ξL is transmission distance at which the power is fully coupled from the first core into the second core. To convert the solitons units back to real units, the investigation is restricted our general considerations to the real communication fiber, which parameters are chosen as λ=1.55 μm, β21 =−20 ps2/km, the coupling coefficient of κ=0.8 cm−1, a fiber effective area of Aeff =20 μm2, and the nonlinearity coefficient of γ=10 W−1/km. The coupling length Lc =π/2κ≈1.96 cm, and when the solitons pulse width is usually T0 =50 fs, the dispersion length LD =T02/|β21|=1.25 cm. As we know, when the dispersion length is comparable to the coupling length, the solitons switching can occur. According to the parameters of the first core, which are given above, those of the second core are properly modified.
ð11Þ
In Fig. 2(a)–(d), one has a plot of the transmission coefficients of the bar state of the coupler as a function of the input power. Fig. 2(a) shows the transmission characteristics of the bar channel with the diverse ratio of effective mode area, here dg = 0, dp = 0, d2 = 1, k = 1, the calculation is performed at the coupling length LC = π/2. One can notice that the decrease of effective mode area of the cross core (or the increase of the ratio of nonlinear coefficient), when the effective mode area of the first core unchanged and the power is injected in the first core, leads to the increase of the switching threshold power, but improves the switching steepness. In Fig. 2(b), one has the transmission characteristics of the bar channel with the different phase velocity mismatches, we set dg = 0, dn = 1, d2 = 1, k = 1, when dp ≠ 0, from Fig. 1(c) and (d), we know that the coupling length changes with the introduction of the phase velocity mismatch. Thenew coupling length can be estimated by using the forpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mula LC ¼ π= 2 k2 þ δp 2 . At the new coupling length LC, one can find that whether dp N 0 or dp b 0, the threshold power decreases with the increasing of dp, but the switching steepness degrades. So the threshold power can be decreased by decreasing the phase velocity of the cross core. Fig. 2(c) represents the transmission performances of the bar channel with the diverse ratios of dispersion, we take parameters dg = 0, dn = 1, dp = 0, k = 1. At the coupling length LC = π/2, it shows that the increase of the dispersion ratio d2 leads to a decrease of the switching threshold power and degrades the switching steepness. In Fig. 2(d), one has a curve of the transmission performances of the bar channel as a function of input power for different group velocity mismatches. Here d2 = 0, dn = 1, dp = 0, k = 1. We know from Fig. 1(a) and (b) that the coupling length remains unchanged with
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(g)
(h)
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4
Fig. 1. Transmission curves of energy transfer coefficient being with the change of coupling transmission distance in a coupler. (a) dg = 0, dn = 1, dp = 0, d2 = 1, k = 1;(b) dg = 0.5, dn = 1, dp = 0, d2 = 1, k = 1;(c) dp = 0.5, dg = 0, dn = 1, d2 = 1, k = 1; (d) dp = − 0.5, dg = 0, dn = 1, d2 = 1, k = 1;(e) d2 = − 1.5, dg = 0, dn = 1, dp = 0, k = 1; (f) d2 = 1.5, dg = 0, dn = 1, dp = 0, k = 1; (g) dn = 0.4,d2 = − 1.5, dg = 0, dp = 0, k = 1; and (h) dn = 3, d2 = − 1.5, dg = 0, dp = 0, k = 1.
the changing of group velocity. At the length LC = π/2, as displayed in Fig. 2(d), one can notice that the switching threshold power increases with increasing of dg, and the switching steepness degrades. 4.3. The solitons evolution during propagation The numerical simulation of solitons propagation in an asymmetric dual-core nonlinear coupler is shown in Figs. 3 and 4.
Fig. 3(a1) and (a2) has a plot of solitons evolution in the bar channel and cross channel with different efficient mode area, one can find that the pulse peak power get lower in two cores when the efficient mode area of the cross channel is half of that of the bar channel (dn = 0.5), the solitons pulse is significantly broadened during propagation, however, as show in Fig. 3(b1) and (b2), the pulse can keep the high peak power when the efficient mode area of the cross channel is twice of that of the bar channel (dn = 2).
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1
1
0.8
0.8
0.6
0.6
dp=0
dn=0.5
0.4
dp=0.3
T
T
dn=0.2
dp=0.5 0.4
dp=-0.3
dn=1 0.2 0 0
dp=-0.5
0.2
dn=2
0 2
4
6
0
2
4
6
P0 (Lc=pi/(2))
P0 (Lc=pi/(2k))
(a)
(b) 1
0.8
0.8
0.6
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T
T
1
d2=0.5
0.4
dg=0
0.4
dg=0.8 d2=1 0.2
dg=1.5
0.2 d2=2
0
0
2
4
6
0
0
2
4
P0 (Lc=pi/(2))
P0 (Lc=pi/(2))
(c)
(d)
6
Fig. 2. The switching characteristic of the bar channel from numerical method. (a) dg = 0, dp = 0, d2 = 1, k = 1; (b) dg = 0, dn = 1, d2 = 1, k = 1; (c) dg = 0, dn = 1, dp = 0, k = 1; and (d) d2 = 1, dn = 1, dp = 0, k = 1.
Fig. 4(a) shows that the group velocity mismatch leads to the solitons pulse walk-off. Fig. 4(b) shows that the pulse significantly broadens in the cross channel in the presence of normal-dispersion in the cross channel but in the presence of anomalous-dispersion in
the bar channel, but the pulse seldom broadens in the bar channel. In addition, the dispersion difference of the two cores results in the decrease of the peak power. Fig. 4(c), (d) and (e) show that the pulse hardly spreads with phase velocity mismatch. As shown in Fig 4. (e), when the enlargement of phase velocity mismatch to a certain extent causes, most of energy remains in the bar channel, and no energy is transferred into the cross channel. 5. Conclusion
(a1)
(a2)
(b1)
(b2)
Fig. 3. Solitons evolution in the bar channel and cross channel. (a1) and (a2) dn = 0.5, dg = 0, d2 = 1, dp = 0, k = 1; (b1) and (b2) dn = 2, dg = 0, d2 = 1, dp = 0, k = 1.
In conclusions, we concern about the impact of the different effective mode area, the phase- and group –velocity mismatch, the different dispersion between two cores on the switching and propagation of short pulses in an asymmetric NLDC. A general method, based on SSFT, is used to solve CNLSEs which include the evanescent coupling between neighboring waveguides. The investigation indicates that the phase velocity mismatch and the different effective mode area can shorten the coupling length, while the different group velocity and the different dispersion between two cores do not change the coupling length. The theory of coupled-mode in asymmetric directional couplers developed in this work also reveals that the four different asymmetrical parameters can influence the switching and propagatingperformances. We have pointed out that the increase of effective mode area ratio leads to an increase of the switching threshold power but improves significantly the switching steepness, the increasing of the phase velocity mismatch causes a decrease of the switching threshold power but degrades the switching steepness , the increase of the ratio of dispersion results in a decrease of the switching threshold power and varies the switching steepness, the increase of group velocity mismatch leads to an increase of
Q. Li et al. / Optics Communications 285 (2012) 118–123
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the switching threshold power but improves the switching steepness. Furthermore, in the asymmetric directional coupler the group velocity mismatch can induce solitons pulse to walk off or stretch. Acknowledgment
(a1)
(a2)
We are grateful to reviewers for helpful comments. This work was supported by the National Science Foundation of China under grant 10904028 and the Natural Science Foundation of Zhejiang Province under grant Y1110078. References [1] [2] [3] [4] [5] [6]
(b1)
(b2)
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(c1)
(c2)
[17] [18] [19] [20]
(d1)
(d2)
(e1)
(e2)
Fig. 4. Evolution of the solitons amplitudes in the bar channel and cross channel. (a) the asymmetry is only shown in the group velocity (dg = 0.5, dn = 1, dp = 0, d2 = 1, k = 1); (b) the two cores have opposite dispersion(d2 = − 1,dg = 0, dn = 1, dp = 0, k = 1);(c),(d),(e)are about the effect of different phase velocity mismatch on pulse evolution, (c)dp = − 0.5, dg = 0, dn = 1, d2 = 1,k = 1; (d)dp = 0.5, dg = 0, dn = 1,d2 = 1,k = 1;(e)dp = 1, dg = 0, dn = 1, d2 = 1,k = 1.
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