ARTICLE IN PRESS
Optik 119 (2008) 86–89 www.elsevier.de/ijleo
Effects of third-order dispersion on soliton switching in ﬁber nonlinear directional couplers Yuntuan Fanga,, Jun Zhoub a
Department of Physics, Zhenjiang Watercraft College, Zhenjiang 212003, China Department of Optical Engineering, Jiangsu University, Zhenjiang 212013, China
Received 6 February 2006; accepted 23 June 2006
Abstract The split-step Fourier method is used to study the energy switching characteristics of ﬁber nonlinear directional couplers with the third-order dispersion. The effects of the third-order dispersion increases with the third-order dispersion coefﬁcient and input power and result in pulse shift and energy decreases. Adding high-order nonlinear can partly overcomes these effects. r 2006 Elsevier GmbH. All rights reserved. Keywords: Nonlinear optics; Optical soliton; Split-step Fourier transformation; Fiber nonlinear directional coupler
Since Jensen ﬁrst depicted the coupling character of ﬁber nonlinear directional couplers , it arise people’s great researching interest [2–7]. The advantage of using solitons for all-optical switching in nonlinear interferometers has been discussed by Doran and Wood . Trillo et al.  further pointed out that the switching efﬁciency double when soliton inputs were used in ﬁber nonlinear directional couplers, compared with quasi-cw pulses were used. In Trillo’s numerical calculation, the propagation of pulses in a nonlinear dual-core ﬁber directional coupler was described in terms of two linearly coupled nonlinear Schro¨dinger equations (NLSEs), in which the high-order dispersion and nonlinear effects were ignored. However, if the pulses are too narrow or the high-order dispersion and nonlinear of the material is too large, the NLSEs must include the high-order term. In this paper, we consider the effect of third-order dispersion on soliton switching in ﬁber nonlinear directional couplers. Corresponding author.
E-mail address: [email protected]
(Y. Fang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.010
The nonlinear-coupled equations including the effect of third-order dispersion for a two-core nonlinear coupler in the soliton units can be expressed as [8,9] qu 1 q2 u q3 u ¼i þ s 3 þ ijuj2 u þ iKv, 2 qz 2 qt qt
qv 1 q2 v q3 v ¼i þ s þ ijvj2 v þ iKu, (2) qz 2 qt2 qt3 where u and v are the slowly varying envelope amplitude of the modal ﬁeld in the ﬁrst and second waveguides. s ¼ b3/6 and b3 is the third-order dispersion coefﬁcient. K is the linear coupling coefﬁcient between the two waveguides. In general case, the above equations cannot be solved analytically, so the numerical method is applied. The most and widely used numerical method solving NLSE is the split-step Fourier method (SSFM) because of its simplicity, ﬂexibility, good accuracy, and relatively modest computing cost . The SSFM assumes that the propagation of the optical pulses from z to z+h is carried out in two steps, where h is a small distance. In the ﬁrst step from z to z+h/2, nonlinearlity
ARTICLE IN PRESS Y. Fang, J. Zhou / Optik 119 (2008) 86–89
acts alone, while in the second step from z+h/2 to z+h, only the linearity terms act alone. Hence, (1) can spilt into a linear and a nonlinear part. Mathematically 1 qu ¼ ijuj2 u, 2 qz
1 qu 1 q2 u q3 u ¼i þ s þ iKv. (4) 2 qz 2 qt2 qt3 In the ﬁrst step, juj2 is regarded as invariable. Eq. (3) can be exactly solved and the iterative scheme can be expressed as uðz þ h=2; tÞ ¼ uðz; tÞ expðiu2 hÞ.
Taking the Fourier transformation of (5), we have Uðo; z þ h=2Þ ¼ F ½uðz; tÞ expðijuj2 hÞ,
where U(o, z+h/2) is the Fourier transformation of u(z+h/2, t). For the second step, we take the Fourier transformation of (10) in the same way and obtain Uðo; z þ hÞ ¼ Uðo; z þ h=2Þ exp½ih=2ðo2 2so3 Þ þ iKV ðo; z þ h=2Þh.
The same process is simultaneously carried out with Eq. (2). After taking (6) into (7) and carrying out the reversal Fourier transformation, we can implement the numerical analysis on the pulse propagation in the two waveguides. We suppose that the initial conditions uðz ¼ 0; tÞ ¼ A sec hðtÞ;
vðz ¼ 0; tÞ ¼ 0.
According to , the half-beat length of the plinear ﬃﬃﬃﬃ coupler is given by L ¼ p/2K. Then, using As ¼ 2 K to obtain those values for which the peak input power equals the cw switching power. In order to verify our program, we give K ¼ 1 and A ¼ 1 for Eq. (8), and consider the case without third-order dispersion (s ¼ 0) in Eqs. (1) and (2). Fig. 1 shows the result, which is consistent with . Next, we study the energy switching characteristics that can be achieved (by the method of ) by using our solitonlike input pulses with different s, for the case of K ¼ 1/4 and L ¼ 2p. In this case, the cw switching power Ps ¼ As2 ¼ 1. The numerical results are shown in Fig. 2. As can been seen, the transmissions have same steep curves in the range of 1oPo1.5 for all
Fig. 1. Evolution of pulses in the ﬁber nonlinear directional coupler for soliton like input with A2 ¼ 1 and K ¼ 1.
Fig. 2. Averaged transmission versus input peak power P ¼ A2 for different values of s.
values of s. Then P ¼ 1 becomes the threshold power of the switching. It is clear that the third-order dispersion with small s hardly inﬂuence the averaged transmission value, especially for the range of Po1.5. Then the switching function still takes effect. However, when the value of s increases, the averaged transmission value quickly decreases with the value of P increasing in the range of P41.5. To the worst, the averaged transmission at P ¼ 3 is even less than 0.5, which basically disable the switching function. Fig. 3 shows the ﬁnal wave shapes through the directional couplers on the conditions of P ¼ 1, 1.5 and 3, corresponding to s ¼ 0, 0.1, 0.14 and 0.18. For the case of P ¼ 1 and 1.5, whatever the values of s are, the position or intensity of the pulse in two ﬁbers changes little. But when P ¼ 3, with the value of s increasing, not only the pulse intensity for u decreases, but also the position of the pulse for u results in shift in the right direction, which means that a time delay occurs when the u pulse goes through coupler. In order to solve the question that the switching function of directional couplers is disabled in the cases of the large s and P, we try to use ﬁbers adulterated by high nonlinear material. On the condition of large input power, the ﬁbers will result in highorder nonlinear, thus the Eqs. (1) and (2) must include high-order nonlinear term, which can be rewritten by qu 1 q2 u q3 u ¼i þ s þ ið1 ejuj2 Þjuj2 u þ iKv, qz 2 qt2 qt3
qv 1 q2 v q3 v ¼i þ s þ ið1 ejvj2 Þjvj2 v þ iKu, (10) qz 2 qt2 qt3 where e is the ﬁfth-order nonlinear coefﬁcient. These equations are called as the third–ﬁfth order NLSE’s . SSFM method can be also used to solve Eqs. (9) and (10) and study the energy switching characteristics. Fig. 4 shows the result for different values of e and s. From Fig. 4, we ﬁnd that the averaged transmission hardly decreases after P41.5 in spite of s ¼ 0.18, which is attributed to that the ﬁfth order nonlinear term partly overcomes the third-order dispersion. Fig. 5 depicts the
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Fig. 4. Averaged transmission versus input peak power for different values of s and e.
Fig. 5. The ﬁnal wave shape of u through the directional coupler on the condition of P ¼ 3 and s ¼ 0.18.
ﬁnal wave shape of u through the directional couplers on the condition of P ¼ 3, corresponding to s ¼ 0.18. Comparing Fig. 5 with Fig. 3, we ﬁnd the ﬁfth order nonlinear also decreases the pulse shift. However, we also notice that, after adding the ﬁfth-order nonlinear term, the curve of averaged transmission becomes more tilted than that without the ﬁfth order nonlinear term at the critical input power of the switching, which is a passive factor to the energy switching characteristics. In conclusion, we have investigated time-dependent propagation of short pulses in a nonlinear directional coupler. We focus on the effects of the third-order dispersion on the energy switching characteristics and ﬁnd that the third-order dispersion takes effect only for large third-order dispersion coefﬁcient and input power. Supported by the National Natural Science Foundation of China under Grant no. 10574058.
Fig. 3. The ﬁnal wave shapes through the directional couplers on the conditions of P ¼ 1, 1.5 and 3, corresponding to s ¼ 0.0 (solid line), s ¼ 0.1 (dotted line), s ¼ 0.14 (dashed– dotted line) and s ¼ 0.18 (dash line), respectively.
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