Soliton Propagation In Doped Optical Fibers

(Physics Department, Jilin University, Changchun, 130023)Electromagnetically induced transparency is typical quantum coherent process. Typical electromagnetically induced transparency involved in the interaction of atom and electromagnetic fields. If considering the system as the whole to probe, we can find the coupling levels of the atom have splitting and displacement via strong electromagnetic fields. In this case, the light field of the resonance with former atom levels will feel two offset resonance transition levels, which are resulted from superposition of wave functions of the former levels. Photons can be absorbed through different superimposing channels. Due to quantum coherence between these channels, the resonant incident light propagates transparently.According to the function of atom levels and coupling fields, it is divided three patterns for EIT, as shown in Fig. (1).The pulse profile may be changed when it transmits through medium of two-level atom. McCall and Hahn had found that pulse could propagate keeping profile stable without its power weakening for the pulse whose area is equal toθ=integral from n=-∞to∞(g/h)2εdt = 2nÏ€, as which called self induced transparency (SIT).Because of pulse profile invariable through medium, medium absorb no energy from the pulse.In a three-level atomic system driven by resonant microwave field, SIT happens in proper conditions. As shown in Fig.2, reverse parity levels |a> and |b>, |a> and|c> are driven by light field (ω₁) and light field (ω₂), respectively. Usingcoherent microwave field to drive same parity levels |b>and |c>, their wave functions superpose, so that there exists quantum beats in the system. When the initial phase of driving field is zero orÏ€, either area theorem or stale pulse profile are the same as that in two-level atomic system for SIT.Now we discuss soliton mechanism and propagation.In multimode optical fiber there exists models dispersion, it induces pulse broadening. In monomode optical fiber there is another kind of dispersion, which is internal model dispersion or group velocity dispersion, it also causes pulse broadening. In linear approximate, pulse may be divided into superimposing of lots harmonic waves. Each harmonic wave has phase velocity v_p =ω/β(ω), whereβ(ω)is the propagation constant. If function β(ω)includes the term of second or higher order integral powers ofω, phase velocity will change with frequency, and lead to phase shift of signal spectrums, pulse broadening. Because phase velocity of each component harmonic wave is different, propagation of whole wave packet is described by group velocity, its definition:v_g =dω/dβ.Without loss and gain, optical fiber dispersion effect may expandβwith center frequencyω₀,Omitting high-order items, we get,In the formula,We can express optical fiber dispersion using dispersion coefficient D,From the above formula, as D > 0,β₂ < 0, that means (?)v_g/(?)ω>0, groupvelocity will increase while frequency increase, which are called abnormal dispersion district. At this situation, the component of high frequencies in a pulse will transfer faster than those of low frequencies. In normal dispersion district where D < 0,β₂, > 0,(?)v_g /(?)ω< 0 , group velocity reduction withfrequency increasing, so the component of high frequencies in a pulse will transfer slower than those of low frequencies. Self-phase modulation is a kind of nonlinear effect. When pulse intensity reaches a value, there exhibits a kind of nonlinear effect between refractive index of fiber and intensity of electric field, which is Kerr effect, expressed as:n = n₀ +n₂ |E|² +…(4)Where n₀ is linear refractive of a medium, n₂ is the lowest nonlinearrefractive index. The second item in above formula adds pulses a extra displacement, that isIt shows an extra phase emerges with a variable pulse intensity, that is so-called self-phase modulation . Its offset of frequencyThat means spectrum compressed. This cause a distribution for compression in the whole frequency extant, called chirp. Pulses have lower frequencies in its front edge ((?)(Δφ)> 0) than those in the back edge ((?)(Δφ) < 0). In theabnormal dispersive district((?)v_g /(?)ω>0),ΔΦincreases (Δω>0) in thefront edge of pulse, whereasΔφincrease (Δω>0) in the back edge of pulse, which is positive chirp. Pulse speed in the front edge is slow, and fast in the back edge, which causes pulse compressed.With balance of the two effects of group velocity dispersion and self-phase modulation pulse shape will be invariable to promulgate as called solitons.A soliton can be described by nonlinear Schrodinger equation:The fundamental soliton corresponds to the above equation: u(ξ,Ï„)=sechÏ„exp(iξ/2) (8)Refer Figs. (3) . It is evident that intensity of profile is invariable along the variable distance, so soliton can transmit stably without shape changing as shown in fig. 4.For two pulse propagating in doped optical fiber, the behavior of solitons are described by the following nonlinear Schrodinger equations:The main task in this these are to solve the above equations (9) using methods of finite-difference and Split-Step Fourier. Firstly, we introduce finite-difference method. The basic step is suing the grid of discrete points stand for continuous district. Discrete variable function is approximately equal to continuous variable function in the continuous district and difference coefficients is approximate that of differential quotient, so that algebra equation sets are instead of original differential equations as shown in fig. 5.Next we introduce Split-Step Fourier Methods. It divides equations (10) into linear and nonlinear sections to solve nonlinear sections (12)by difference scheme, then inserting solutions of nonlinear sections as initial condition of nonlinear section to resolve equation (14). Fastest Fourier Transform (13) is adopted as so called Split-Step Fourier Methods.For the above method to solve NLS equations (9), specific process: Consider a general evolution equation of the formu_t =(L + N)u u(x,0) = u₀(x) (10)The solutions to equations (10) may be advanced from one time-level to the next by means of the following formula,u(x, t+Δt)≡exp[Δt(L + N)]u(x, t) (11)Then dividing the right parts of equations (11) into linear term exp(Δt*L) and nonlinear term exp(Δt*N) to solve. First space is discretization, then Time is integration. From t to t +Δt by difference scheme, we can get:Using fast Fourier Transform, we getThen By inverse Fourier Transform, we obtained:If we get solution of t +Δt, solutions to t + nΔt (n=2,3,4…) are gained . Through two kinds of ways, we can solve equations () which describe pulses represented byA₁₃(Ï„) =(?)sech(Ï„)å’ŒA₁₂(Ï„)=r(?)sech(Ï„), respectively. The coordinate r represents anormalized time for the initial pulse duration in aGalilean frame of reference, i.e.Ï„=(t-z/v_g)/T₀ where T₀ is the pulse widthand v_g is its group velocity. We introduce parameterΔto represent the intensity ratio between pump and signal at the input point, i.e.Δ= |A₁₂|²/|A₁₃|² = r². The initial conditions c1 = 0, c2 = 0, c3 = -1 so that thepopulation is in the ground state. The input power P₀ satisfies the coexistencecondition P_0(N=1)^NLS = P_0(2Ï€)^SIT. In Figs. 6 and 7, we have plotted the intensity profiles of pump and signal respectively. We observe a cloning process where a soliton as the pump field disappears, reappearing at the signal frequency. At this stage the newly created soliton propagates freely undistorted and without being absorbed. The medium has become transparent to the cloned SIT-NLS soliton.Let us raise the power of the pump to form a 4Ï€- (N = 2) soliton. The intensity profiles of pump and signal are depicted in Figs. 8 and 9 respectively. Fig. 9 shows that a N = 2 soliton is excited for a very short time due to the interaction between the nonlinearity and group velocity dispersion associated to the NLS equation. Then SIT quickly dominates and induces the pump pulse breakup into two 2Ï€solitary waves. Subsequently, two 2Ï€solitary waves the signal are simultaneously amplified with signal frequency while both split waves at the pump frequency undergo strong attenuation, demonstrating that the energy of the pump has been transferred to the signal soliton pair.An important physical aspect is observed by considering the combined SIT-NLS effect after the energy transfer process finished. At this stage, the pump is gone and the pair travels unaltered in the absence of the NLS component, as depicted in Fig. 10, By contrast, in the presence of nonlinear dispersive effect Fig. 11, the amplitude of the higher pulse displays a small oscillation whose period is approximate z₀, so that one may identify this periodicity with the characteristic NLS soliton period [Fig. 11]. Consequently the areas of both pulses oscillate around the 2Ï€value.