§¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 59 INFLUENCE OF THIRD-ORDER DISPERSION ON SOLITON'S TRANSMISSION IN OPTICAL FIBERS Dinh Xuan Khoa (a) , Bui Dinh Thuan (a) Abstract. In this article, we use split-step Fourrier method to investigate the influence of loss and third-order dispersion on a transmission soliton in optical fibers. 1. Introduction Soliton in optical fibers is formed by the balance between phase self- modulation and dispersion caused by group velocity. The low-loss transmission of soliton in optical fibers is described by non-linear Schrodinger equation. The loss and dispersion are main factors which diminish transmission distance and durable of soliton. Normally, we consider only second-order dispersion factor. However, when pulse's width is small, higher-order dispersion factors are not negligible. Besides, in optical fibers, material dispersion equals zero at the wavelength of 1300 nm. In this case, the change of pulse's form and its durable depend on high-order dispersion. In order to investigate transmission of soliton in optical fibers, we can use back-scattering and perturbation methods. But if the loss is considered, back- scattering method does not give correct solution. In this article, we use Split-Step Fourrier algorithm to investigate the transmission of light pulse in non-linear dispersion media, thenceforth investigate the influence of high-order dispersion on soliton's transmission. 2. Transmission equation Pulse's transmission in optical fiber is described by equation [1,3] ( ) ∂ ∂ − ∂ ∂ +− ∂ ∂ + ∂ ∂ + ∂ ∂ −−= ∂ ∂ t A ATAA t i AAi t A t A i t A A z A R 2 2 0 2 3 3 3 2 2 2 1 622 ω γ β β β α (1) where A=A(z,t) is a complex envelop function of optical field. This function varies slowly with time and z position along optical fiber; α is loss factor of optical fiber; β 1 , β 2 and β 3 are first, second and third-order dispersion factors, respectively; γ is non- linear factor of optical fiber, γ/ω 0 term describes self-steeping effect, γT R term describes Raman scattering effect. NhËn bµi ngµy 19/9/2006. Söa ch÷a xong 18/12/2006. §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 60 Let us consider an optical pulse with the width about some ps. In this case, self-steeping and Raman scattering effects can be negligible. Applying the transformation t'=t-z/v g , (v g is group velocity) equation (1) is re-written as follows: AAi t A t Ai A z A 2 3 3 3 2 2 2 ' 6 1 ' 2 2 γββ α − ∂ ∂ + ∂ ∂ +−= ∂ ∂ . (2) In general cases, when second-order dispersion factor β 2 is negative and higher-order dispersion is negligible, the two specific distances of dispersion and non-linear effects are approximately equal, 2 2 0 2 β t L D = , where t 0 is the width of input pulse [2]. If the loss of optical fiber is small, we can receive pulses whose form does not change during transmission process. These pulses are solitons. So that we can normalize the equation (2) to obtain ( ) ( ) ( ) ( ) ( ) ( ) τξτξ τ τξ τ τξ τξ ξ τξ ,, ,, 2 1 , , 2 3 3 2 2 uui u B u iu u + ∂ ∂ + ∂ ∂ +Γ−= ∂ ∂ (3) where u(ξ,τ) is complex normalized amplitude of soliton pulse, ξ is normalized distance along transmission direction, τ is normalized time and Γ is normalized loss factor. Specific expressions of these quantities are as follows ( ) ( ) τξτξ ,, 0 UAA = , ( ) 10,0 =U , ( ) 0 0,0 AA = 2 0 2 2t z L z D β ξ == , ( ) 00 / ' t vzt t t g − == τ 2 2 0 2 β α t =Γ 02 3 6 t β β =Β . In case second-order dispersion factor β 2 equals zero, we can not use above- mentioned normalization. In this case, transmission distance z and absorption factor are normalized as 3 0 3 t z β ξ = 3 3 0 2 ' β α t =Γ and equation (2) is re-written ( ) ( ) ( ) ( ) ( ) τξτξ τ τξ τξ ξ τξ ,, , 6 1 ,' , 2 3 3 uui u u u + ∂ ∂ +Γ−= ∂ ∂ . (4) §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 61 Equations (3) and (4) contain absorption factor so that it is difficult to use back-scattering and perturbation methods. Here, we use Split-Step Fourier method [1]to investigate the transmission in light pulse. The method is described as follows: Considering differential equation with the form ( ) ( ) ( ) ( ) tzAANtzAL z tzA i ,. ˆ , ˆ , += ∂ ∂ . (5) L ˆ is an operator containing time derivative, N ˆ is a non-linear operator and is a function of A(z,t). Solution of equation (5) has the form ( ) ( ) [ ] ( ) tzAzANzLitzzA , ˆˆ exp, ∆+∆−=∆+ . (6) Operators L ˆ and N ˆ are non-commutative, formula (6) can not be used directly, so that we have to use approximate methods. Depending on the correctness of methods, formula (6) is normally re-written in the form [4]. ( ) ( ) ( ) zAL z zAL z NzL z zzA ∆ ∆ ∆ ∆ =∆+ ˆ . 2 exp ˆ . 2 exp ˆ .exp ˆ . 2 exp . 3. Results Using Split-Step Fourrier method, we obtain the following results: Fig1. 3D graph of the transmission of soliton with Γ =0, B= 0 §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 62 In Fig. 1, third-order dispersion and fiber loss are not considered. Not the same as in the case of linear pulse transmission, in this case, power and form of pulse does not change when it transmits along optical fiber. This is due to the cancellation between the phase self-modulation and group velocity dispersion. This case rarely happens. Fig. 2 and Fig. 3 are 3D and 2D graph of the transmission of soliton in optical fibers when the loss and third-order dispersion are considered. In this case, second-order dispersion factor is different from zero but is not so big in comparison with third-order dispersion factor (β 3 = 0,1 ps 3 /km; α=0.2 dB/km). Due to third-order dispersion, pulse's shape is deformed during its transmission. Peak power is moved to the positive direction of the axis due to the change of v g ; pulse's width is widened; vibration tail is formed at sides after transmitting along optical fibers. If the initial pulse has small width, the value of B factor is large, the influence of third-order dispersion is considerable. Fig2. 3D graph of the transmission of soliton with Γ =1.1510 -6 , B= 0.83 Fig3. 2D graph of transmission of soliton with Γ =1.1510 -6 , B= 0.83 ( ) 2 , τξ U τ ξ ( ) 2 , τξ U τ §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 63 In case the second-order dispersion factor equals zero, third-order dispersion factor must be considered. Fig. 4 and Fig. 5 correspond to the case β 2 =0 (at dispersion wavelength equals zero) and β 3 = 0.1 ps 3 /km. The amplitude of pulse decreases quickly, pulse amplitude of vibration tail which is formed at the back side of pulse decreases along optical fiber. The maximum of amplitude moves to the positive direction of the axis. Numerical calculation shows that the deformation of pulse caused by third-order dispersion at dispersion wavelength equals zero can limit the efficiency of optical fibers information system. 4. Conclusion In this article, Split-Step Fourrier method is used to investigate the influence of loss and third-order dispersion on pulse's transmission in optical fibers. It shows that the deformation of pulse caused by third-order dispersion at dispersion wavelength equals zero can limit the efficiency of optical fibers information system. ( ) 2 , τξ U τ ξ Fig4. 3D graph of the transmission of soliton with β 2 =0, Γ ‘=0.3 ( ) 2 , τξ U τ Fig5. 2D graph of the transmission of soliton with β 2 =0, Γ ‘=0.3 Đại học Vinh Tạp chí khoa học, tập XXXVI, số 2A-2007 64 References [1] Cao Long Van, Đinh Xuan Khoa, M. Trippenback, Introduction to Nonlinear Optics, Vinh, 2003. [2] Dinh Xuan Khoa and Bui Dinh Thuan Soliton study of schrodinger equation by hirota method, Vol 15, number 2, June 2005. [3] P. N. Butcher and D. Cotter, The elements of nonlinear optics, Cambridge University Press, New York, 1990. [4] A. Hasegawa and Y. Kodama, Solitons in optical communication, Oxford University Press, New York, 1995. [5] G. P. Agrawal and M. J. Potasek, Nonlinear pulse distortion in single mode optical fibers at the zero-dispersion wavelength, Phys Rev. vol33, no3. pp.1765 1776, 1986. Tóm tắt ảnh hởng của tán sắc bậc ba lên soliton lan truyền trong sợi quang Trong bài này sử dụng phơng pháp split-step Fourrier, chúng tôi đã khảo sát ảnh hởng của mất mát và tán sắc bậc 3 lên soliton lan truyền trong sợi quang. (a) Khoa Vật Lý, trờng Đại học Vinh. . hởng của tán sắc bậc ba lên soliton lan truyền trong sợi quang Trong bài này sử dụng phơng pháp split-step Fourrier, chúng tôi đã khảo sát ảnh hởng của mất mát và tán sắc bậc 3 lên soliton lan. Fourrier, chúng tôi đã khảo sát ảnh hởng của mất mát và tán sắc bậc 3 lên soliton lan truyền trong sợi quang. (a) Khoa Vật Lý, trờng Đại học Vinh. . 64 References [1] Cao Long Van, Đinh Xuan Khoa, M. Trippenback, Introduction to Nonlinear Optics, Vinh, 2003. [2] Dinh Xuan Khoa and Bui Dinh Thuan Soliton study of schrodinger equation by hirota