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Gain-induced soliton switching in fiber nonlinear directional coupler
Optik 123 (2012) 2247–2249
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Gain-induced soliton switching in fiber nonlinear directional coupler Xiujun He a,b,∗ , Kang Xie a , Huajun Yang a a b
School of Optoelectronic Information, University of Electronics Science & Technology of China, Chengdu 610054, China Department of Optoelectronic Technology, Chengdu University of Information Technology of China, Chengdu 610225, China
a r t i c l e
i n f o
Article history: Received 31 May 2011 Accepted 27 October 2011
Keywords: Gain-induced Fiber nonlinear coupler Switching characteristics Output coupling ratio
a b s t r a c t With study for switching characteristics and output coupling ratio of fiber nonlinear directional couplers (NLDC), we found that the influence elements on the switching characteristics and output coupling ratio had gain of core 1 and 2, nonlinear coefficient of those, the input power and width of input soliton. We also found numerically that switching efficiencies were improved by controlling gain of core 1, different output coupling ratio would be obtained by controlling gain of core 1 or 2. And we can change the gain by changing pump power of the optical fiber amplifiers. From this, we also made fiber coupler which output coupling ratio was changeable. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction All optical switching devices are attracting considerable interest as fast as switching components for high-bit-rate system in the future. Optical fiber coupler has been studied for their potential applications to ultra fast all optical switching process, such as optical switch [1–4]. In a nonlinear coupler, constructed from a Kerr-model medium depending nonlinear refractive index on the laser intensity, is given by expression n = n0 + n2 I, where n0 is refractive index at low intensity and n2 is Kerr nonlinear coefficient [2]. Jensen showed that varying input light in the nonlinear coupler could result in pulse switching between two cores. Accordingly he foresaw possible use of a nonlinear directional coupler as an optical switch. Previous studies of soliton switching in dual core optical fibers have shown excellent switching characteristics, with efficiencies around 96% for a wide range of input energies [5]. By comparing to switching behavior of fundamental, second order, and quasi-solitons, it was observed that fundamental soliton has the most suitable features for optical switching [5]. Indeed, it has been shown that pulse breakup may be avoided whilst input signal is a soliton [6–8]. Since then, soliton switching in nonlinear fiber couplers has been receiving for considerable attention. In this paper, we will present a theoretical analysis of switching solitons in fiber nonlinear directional coupler (NLDC). We can improve switching characteristics of NLDC and obtain different output coupling ratios of NLDC by gain induction. After computer
simulation and by changing gain of core 1 and core 2, this showed their excellent switching characteristics and also obtained different output coupling ratios of NLDC. We can add optical fiber amplifiers in NLDC. So we can change the gain by changing pump power of the optical fiber amplifiers. From this, we can make fiber coupler which output coupling ratio is changeable, i.e. variable fiber coupler. 2. Theory Optical soliton propagating in NLDC shall be described by a pair of coupled nonlinear Schrödinger equation in normalized parameter as follows: i
1 ∂2 A1 ∂A1 + + g31 |A1 |2 A1 + kA2 = ig1 A1 2 ∂ 2 ∂
(1)
i
1 ∂2 A2 ∂A2 + g32 |A2 |2 A2 + kA1 = ig2 A2 + 2 ∂ 2 ∂
(2)
where A1 and A2 are modal field amplitudes in soliton unit with core 1 and 2. Here and are normalized length and time in soliton unit with =z/LD and =t/T0 . Here LD = T02 /|ˇ2 |, where ˇ2 is group velocity dispersion of core 1 and core 2. LD and T0 are dispersion length and input pulse width, respectively. g31 = LD /Lnl1 , g32 = LD /Lnl2 , where Lnl1 = 1/( 1 P0 ), Lnl2 = 1/( 2 P0 ), where 1 and 2 are nonlinear coefficient of core 1 and core 2, respectively. g1 and g2 are the gain of core 1 and 2. k is normalized coupling coefficient and is related to normalized coupling length LC by k = /2LC . 3. Results and discussion
∗ Corresponding author at: School of Optoelectronic Information, University of Electronics Science & Technology of China, Chengdu 610054, China. E-mail address: xiujun [email protected] (X. He). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.11.003
As the set of coupled Eqs. (1) and (2) are not analytically solvable, we solve them numerically by fast Fourier transform method for
2248
X. He et al. / Optik 123 (2012) 2247–2249
1
1 (a)
Transmission T
0.8
(b)
0.8
Transmission T
(c)
0.6 (a) (b) (c) (d) (e)
0.4 0.2 0
(d)
0.6
(e)
0.4
0.2
0
5
10
15
20
0
Normalized input power P 0
0
5
10
15
20
Normalized input power P0 Fig. 1. Transmission coefficient as a function of input power for (a) g1 = 0.4, (b) g1 = 0.6, (c) g1 = 0.8, (d) g1 = 1, and (e) g1 = 1.2 with k = 1, r = 1, g2 = 0, L = /2, g31 = 1, g32 = 1.
linear dispersive part and by fourth-order Runge–Kutta method, with automatic control of step size for given accuracy of results for nonlinear part. Initial pulse at input core is given by: u1 (0, ) =
P0 sec h(r)
(3)
u2 (0, ) = 0
(4)
where P0 and r represent soliton power, width inversion, respectively. We can define transmission Ti as a function of the pulse energies:
+∞
Ti =
+∞ −∞
−∞
|ui (L , )|2 d
(5)
|u1 (L , )|2 + |u2 (L , )|2 d
with i = 1, 2 and fiber coupler with length of L . Applying for numerical calculation based on Eqs. (1) and (2), we shall demonstrate several important properties of gain-induced soliton switching in fiber nonlinear directional coupler. 3.1. Influence of gain of core 1 g1 and gain of core 2 on switching dynamics In order to observe influence of g1 and g2 on switching characteristics and output coupling ratio, we have integrated in system of Eqs. (1) and (2). The result was depicted in Figs. 1 and 2. This
Fig. 3. Transmission coefficient as a function of input power for (a) g31 = 0.4, (b) g31 = 0.6, (c) g31 = 0.8, (d) g31 = 1, and (e) g31 = 1.2 with k = 1, r = 1, g1 = 0, g2 = 0, L = /2, g32 = 1.
shows that, when gain of core 1 g1 became much bigger, output coupling ratio (T2 /T1 ) was much bigger; with same output coupling ratio, minimum input power was much less. Herein, with increase in g1 , excellent switching characteristics and less switching threshold was also obtained. However, with increase in gain of core 2 g2 , output coupling ratio was also lowered. With same output coupling ratio, minimum input power was basically same. So we can improve switching performance by changing gain of core 1 g1 , and make fiber coupler whose output coupling ratio varied by changing gain of core 1 and core 2. We can change the gain by changing pump power of the fiber coupler with optical fiber amplifiers. According this, a changeable fiber coupler can be made. 3.2. Influence of nonlinear coefficient of core 1 g31 and nonlinear coefficient of core 2 on switching dynamics Viewing from Fig. 3, we found that, when nonlinear coefficient of core 1 g31 was varied, with same output coupling ratio, bigger g31 was lower input power required. Therefore, if expecting to obtain same output coupling ratio, we can choice a little bigger nonlinear coefficient of core 1 g31 . From Fig. 4, we found that, compared with g31 , nonlinear coefficient of core 2 g32 have less influence in output coupling ratio. But, with same output coupling ratio, much bigger
1
1 (a)
0.8
(b)
Transmission T
Transmission T
0.8
(c) (d)
0.6
(e) 0.4
(a) (b)
0.4
(c) (d)
0.2
0.2
0
0.6
0 0
5
10
15
20
Normalized input power P0 Fig. 2. Transmission coefficient as a function of input power for (a) g2 = 0.4, (b) g2 = 0.6, (c) g2 = 0.8, (d) g2 = 1, and (e) g2 = 1.2 with k = 1, r = 1, g1 = 0, L = /2, g31 = 1, g32 = 1.
(e)
0
5
10
15
20
Normalized input power P0 Fig. 4. Transmission coefficient as a function of input power for (a) g32 = 0.4, (b) g32 = 0.6, (c) g32 = 0.8, (d) g32 = 1, and (e) g32 = 1.2 with k = 1, r = 1, g1 = 0, g2 = 0, L = /2, g31 = 1.
X. He et al. / Optik 123 (2012) 2247–2249
3.3. Influence of coupled coefficient k and input soliton width inversion r on switching dynamics
1
Transmission T
0.8
0.6 (a)
0.4
(b) (c)
0.2
(d) (e)
0
0
5
10
15
20
Normalized input power P0 Fig. 5. Transmission coefficient as a function of input power for (a) k = 0.4, (b) k = 0.6, (c) k = 0.8, (d) k = 1, and (e) k = 1.2 with r = 1, g1 = 0, g2 = 0, L = /2, g31 = 1, g32 = 1.
0.8
Transmission T
In Fig. 5, the value of coupled coefficient k has influence on output coupling ratio. We can change the coupled coefficient k by changing the core-to-core distances of coupler. The result shown that, the bigger the value of coupled coefficient k is, the lower the output coupling ratio is. So we shall also change output coupling ratio by altering k. Herein, with the same output coupling ratio, required minimum input power becomes almost equal. In Fig. 6, the value of soliton width inversion r has influence in output coupling ratio and corresponding minimum input powers. This shown that, the bigger A is (i.e. less soliton width is), the lower the output coupling ratio is, the bigger minimum input power under same output coupling ratio. 4. Conclusion In this work, we have carried out a detailed numerical study for gain-induced soliton switching in fiber nonlinear directional coupler (NLDC). This has been clearly shown that we improved switching characteristics of NLDC and obtained different output coupling ratio of NLDC by gain induction. By computer simulation and by changing gain of core 1 and core 2, This also shown excellent switching characteristics and obtained different output coupling ratio of NLDC. From this, we shall make fiber coupler which output coupling ratio is changable, i.e. variable fiber coupler.
1
0.6 (a) (b)
0.4
References
(c) (d)
0.2
(e) 0
2249
0
5
10
15
20
Normalized input power P0 Fig. 6. Transmission coefficient as a function of input power for (a) r = 0.4, (b) r = 0.6, (c) r = 0.8, (d) r = 1, and (e) r = 1.2 with k = 1, g1 = 0, g2 = 0, L = /2, g31 = 1, g32 = 1.
g32 is also required for bigger input power. For all, with the same output coupling ratio, lower input power would be obtained by increasing in g31 or decrease in g32 , but influence of g31 is much more.
[1] S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegman, Soliton switching in fiber nonlinear directional coupler, Opt. Lett. 13 (1988) 672–674. [2] M. Romagnoli, S. Trillo, S. WabnitZ, Soliton switching in nonlinear coupler, Opt. Quantum. Electron. 24 (1992), Sl237–S1267. [3] I. Uzunov, R. Muschall, M. Golles, Yu. Kivshar, B. Malomed, F. Lederer, Pulse switching in nonlinear fiber directional coupler, Phys. Rev. E 51 (1995) 2527–2537. [4] G. Agrawal, Nonlinear Fiber Optics, Academic Press, London, 1995. [5] P.L. Chu, B.A. Malomed, G.D. Peng, Soliton switching and propagation in nonlinear fiber couplers: analytical results, J. Opt. Soc. Am. B 10 (1993) 1379–1385. [6] P. Belanger, C. Pare, Soliton switching and energy coupling in two-mode fibers: analytical results, Phys. Rev. A41 (1990) 5254–5256. [7] F. Abdullaev, R. Abrarov, S. Darmanyan, Dynamics of solitons in coupled optical fiber, Opt. Lett. 14 (1989) 131–133. [8] J.M. Fang, M.J. Potasek, Optical switching in the presence of two-photon absorption for a nonlinear directional coupler, Phys. Rev. A45 (1992) 5305.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Gain-induced soliton switching in fiber nonlinear directional coupler Xiujun He a,b,∗ , Kang Xie a , Huajun Yang a a b
School of Optoelectronic Information, University of Electronics Science & Technology of China, Chengdu 610054, China Department of Optoelectronic Technology, Chengdu University of Information Technology of China, Chengdu 610225, China
a r t i c l e
i n f o
Article history: Received 31 May 2011 Accepted 27 October 2011
Keywords: Gain-induced Fiber nonlinear coupler Switching characteristics Output coupling ratio
a b s t r a c t With study for switching characteristics and output coupling ratio of fiber nonlinear directional couplers (NLDC), we found that the influence elements on the switching characteristics and output coupling ratio had gain of core 1 and 2, nonlinear coefficient of those, the input power and width of input soliton. We also found numerically that switching efficiencies were improved by controlling gain of core 1, different output coupling ratio would be obtained by controlling gain of core 1 or 2. And we can change the gain by changing pump power of the optical fiber amplifiers. From this, we also made fiber coupler which output coupling ratio was changeable. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction All optical switching devices are attracting considerable interest as fast as switching components for high-bit-rate system in the future. Optical fiber coupler has been studied for their potential applications to ultra fast all optical switching process, such as optical switch [1–4]. In a nonlinear coupler, constructed from a Kerr-model medium depending nonlinear refractive index on the laser intensity, is given by expression n = n0 + n2 I, where n0 is refractive index at low intensity and n2 is Kerr nonlinear coefficient [2]. Jensen showed that varying input light in the nonlinear coupler could result in pulse switching between two cores. Accordingly he foresaw possible use of a nonlinear directional coupler as an optical switch. Previous studies of soliton switching in dual core optical fibers have shown excellent switching characteristics, with efficiencies around 96% for a wide range of input energies [5]. By comparing to switching behavior of fundamental, second order, and quasi-solitons, it was observed that fundamental soliton has the most suitable features for optical switching [5]. Indeed, it has been shown that pulse breakup may be avoided whilst input signal is a soliton [6–8]. Since then, soliton switching in nonlinear fiber couplers has been receiving for considerable attention. In this paper, we will present a theoretical analysis of switching solitons in fiber nonlinear directional coupler (NLDC). We can improve switching characteristics of NLDC and obtain different output coupling ratios of NLDC by gain induction. After computer
simulation and by changing gain of core 1 and core 2, this showed their excellent switching characteristics and also obtained different output coupling ratios of NLDC. We can add optical fiber amplifiers in NLDC. So we can change the gain by changing pump power of the optical fiber amplifiers. From this, we can make fiber coupler which output coupling ratio is changeable, i.e. variable fiber coupler. 2. Theory Optical soliton propagating in NLDC shall be described by a pair of coupled nonlinear Schrödinger equation in normalized parameter as follows: i
1 ∂2 A1 ∂A1 + + g31 |A1 |2 A1 + kA2 = ig1 A1 2 ∂ 2 ∂
(1)
i
1 ∂2 A2 ∂A2 + g32 |A2 |2 A2 + kA1 = ig2 A2 + 2 ∂ 2 ∂
(2)
where A1 and A2 are modal field amplitudes in soliton unit with core 1 and 2. Here and are normalized length and time in soliton unit with =z/LD and =t/T0 . Here LD = T02 /|ˇ2 |, where ˇ2 is group velocity dispersion of core 1 and core 2. LD and T0 are dispersion length and input pulse width, respectively. g31 = LD /Lnl1 , g32 = LD /Lnl2 , where Lnl1 = 1/( 1 P0 ), Lnl2 = 1/( 2 P0 ), where 1 and 2 are nonlinear coefficient of core 1 and core 2, respectively. g1 and g2 are the gain of core 1 and 2. k is normalized coupling coefficient and is related to normalized coupling length LC by k = /2LC . 3. Results and discussion
∗ Corresponding author at: School of Optoelectronic Information, University of Electronics Science & Technology of China, Chengdu 610054, China. E-mail address: xiujun [email protected] (X. He). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.11.003
As the set of coupled Eqs. (1) and (2) are not analytically solvable, we solve them numerically by fast Fourier transform method for
2248
X. He et al. / Optik 123 (2012) 2247–2249
1
1 (a)
Transmission T
0.8
(b)
0.8
Transmission T
(c)
0.6 (a) (b) (c) (d) (e)
0.4 0.2 0
(d)
0.6
(e)
0.4
0.2
0
5
10
15
20
0
Normalized input power P 0
0
5
10
15
20
Normalized input power P0 Fig. 1. Transmission coefficient as a function of input power for (a) g1 = 0.4, (b) g1 = 0.6, (c) g1 = 0.8, (d) g1 = 1, and (e) g1 = 1.2 with k = 1, r = 1, g2 = 0, L = /2, g31 = 1, g32 = 1.
linear dispersive part and by fourth-order Runge–Kutta method, with automatic control of step size for given accuracy of results for nonlinear part. Initial pulse at input core is given by: u1 (0, ) =
P0 sec h(r)
(3)
u2 (0, ) = 0
(4)
where P0 and r represent soliton power, width inversion, respectively. We can define transmission Ti as a function of the pulse energies:
+∞
Ti =
+∞ −∞
−∞
|ui (L , )|2 d
(5)
|u1 (L , )|2 + |u2 (L , )|2 d
with i = 1, 2 and fiber coupler with length of L . Applying for numerical calculation based on Eqs. (1) and (2), we shall demonstrate several important properties of gain-induced soliton switching in fiber nonlinear directional coupler. 3.1. Influence of gain of core 1 g1 and gain of core 2 on switching dynamics In order to observe influence of g1 and g2 on switching characteristics and output coupling ratio, we have integrated in system of Eqs. (1) and (2). The result was depicted in Figs. 1 and 2. This
Fig. 3. Transmission coefficient as a function of input power for (a) g31 = 0.4, (b) g31 = 0.6, (c) g31 = 0.8, (d) g31 = 1, and (e) g31 = 1.2 with k = 1, r = 1, g1 = 0, g2 = 0, L = /2, g32 = 1.
shows that, when gain of core 1 g1 became much bigger, output coupling ratio (T2 /T1 ) was much bigger; with same output coupling ratio, minimum input power was much less. Herein, with increase in g1 , excellent switching characteristics and less switching threshold was also obtained. However, with increase in gain of core 2 g2 , output coupling ratio was also lowered. With same output coupling ratio, minimum input power was basically same. So we can improve switching performance by changing gain of core 1 g1 , and make fiber coupler whose output coupling ratio varied by changing gain of core 1 and core 2. We can change the gain by changing pump power of the fiber coupler with optical fiber amplifiers. According this, a changeable fiber coupler can be made. 3.2. Influence of nonlinear coefficient of core 1 g31 and nonlinear coefficient of core 2 on switching dynamics Viewing from Fig. 3, we found that, when nonlinear coefficient of core 1 g31 was varied, with same output coupling ratio, bigger g31 was lower input power required. Therefore, if expecting to obtain same output coupling ratio, we can choice a little bigger nonlinear coefficient of core 1 g31 . From Fig. 4, we found that, compared with g31 , nonlinear coefficient of core 2 g32 have less influence in output coupling ratio. But, with same output coupling ratio, much bigger
1
1 (a)
0.8
(b)
Transmission T
Transmission T
0.8
(c) (d)
0.6
(e) 0.4
(a) (b)
0.4
(c) (d)
0.2
0.2
0
0.6
0 0
5
10
15
20
Normalized input power P0 Fig. 2. Transmission coefficient as a function of input power for (a) g2 = 0.4, (b) g2 = 0.6, (c) g2 = 0.8, (d) g2 = 1, and (e) g2 = 1.2 with k = 1, r = 1, g1 = 0, L = /2, g31 = 1, g32 = 1.
(e)
0
5
10
15
20
Normalized input power P0 Fig. 4. Transmission coefficient as a function of input power for (a) g32 = 0.4, (b) g32 = 0.6, (c) g32 = 0.8, (d) g32 = 1, and (e) g32 = 1.2 with k = 1, r = 1, g1 = 0, g2 = 0, L = /2, g31 = 1.
X. He et al. / Optik 123 (2012) 2247–2249
3.3. Influence of coupled coefficient k and input soliton width inversion r on switching dynamics
1
Transmission T
0.8
0.6 (a)
0.4
(b) (c)
0.2
(d) (e)
0
0
5
10
15
20
Normalized input power P0 Fig. 5. Transmission coefficient as a function of input power for (a) k = 0.4, (b) k = 0.6, (c) k = 0.8, (d) k = 1, and (e) k = 1.2 with r = 1, g1 = 0, g2 = 0, L = /2, g31 = 1, g32 = 1.
0.8
Transmission T
In Fig. 5, the value of coupled coefficient k has influence on output coupling ratio. We can change the coupled coefficient k by changing the core-to-core distances of coupler. The result shown that, the bigger the value of coupled coefficient k is, the lower the output coupling ratio is. So we shall also change output coupling ratio by altering k. Herein, with the same output coupling ratio, required minimum input power becomes almost equal. In Fig. 6, the value of soliton width inversion r has influence in output coupling ratio and corresponding minimum input powers. This shown that, the bigger A is (i.e. less soliton width is), the lower the output coupling ratio is, the bigger minimum input power under same output coupling ratio. 4. Conclusion In this work, we have carried out a detailed numerical study for gain-induced soliton switching in fiber nonlinear directional coupler (NLDC). This has been clearly shown that we improved switching characteristics of NLDC and obtained different output coupling ratio of NLDC by gain induction. By computer simulation and by changing gain of core 1 and core 2, This also shown excellent switching characteristics and obtained different output coupling ratio of NLDC. From this, we shall make fiber coupler which output coupling ratio is changable, i.e. variable fiber coupler.
1
0.6 (a) (b)
0.4
References
(c) (d)
0.2
(e) 0
2249
0
5
10
15
20
Normalized input power P0 Fig. 6. Transmission coefficient as a function of input power for (a) r = 0.4, (b) r = 0.6, (c) r = 0.8, (d) r = 1, and (e) r = 1.2 with k = 1, g1 = 0, g2 = 0, L = /2, g31 = 1, g32 = 1.
g32 is also required for bigger input power. For all, with the same output coupling ratio, lower input power would be obtained by increasing in g31 or decrease in g32 , but influence of g31 is much more.
[1] S. Trillo, S. Wabnitz, E.M. Wright, G.I. Stegman, Soliton switching in fiber nonlinear directional coupler, Opt. Lett. 13 (1988) 672–674. [2] M. Romagnoli, S. Trillo, S. WabnitZ, Soliton switching in nonlinear coupler, Opt. Quantum. Electron. 24 (1992), Sl237–S1267. [3] I. Uzunov, R. Muschall, M. Golles, Yu. Kivshar, B. Malomed, F. Lederer, Pulse switching in nonlinear fiber directional coupler, Phys. Rev. E 51 (1995) 2527–2537. [4] G. Agrawal, Nonlinear Fiber Optics, Academic Press, London, 1995. [5] P.L. Chu, B.A. Malomed, G.D. Peng, Soliton switching and propagation in nonlinear fiber couplers: analytical results, J. Opt. Soc. Am. B 10 (1993) 1379–1385. [6] P. Belanger, C. Pare, Soliton switching and energy coupling in two-mode fibers: analytical results, Phys. Rev. A41 (1990) 5254–5256. [7] F. Abdullaev, R. Abrarov, S. Darmanyan, Dynamics of solitons in coupled optical fiber, Opt. Lett. 14 (1989) 131–133. [8] J.M. Fang, M.J. Potasek, Optical switching in the presence of two-photon absorption for a nonlinear directional coupler, Phys. Rev. A45 (1992) 5305.