Grid Job Routing Algorithms - Phosphorus

Grid Job Routing Algorithmscan be observed even at low powers considering that the light is confined in a relative small area over longinteraction lengths due to the extremely low attenuation coefficient and the event of optical amplifiers. This isthe reason why nonlinear effects can not be ignored when considering light propagation in optical fibers.One manifestation of the intensity dependence of the refractive index occurs through self-phase modulation(SPM), a phenomenon that leads to spectral broadening of optical pulses travelling along a fiber. Cross-phasemodulation (XPM) is the analogue of SPM but this time the induced phase depends not only on its ownintensity, but also on the one of the other co-propagating lightwaves. Another nonlinear effect that can berelevant in optical fibers is four wave mixing (FWM). Due to this phenomenon new frequencies are generatedthat coincide with the transmission channel and induce dependent interferences which degrade the transmittedsignal.The propagation of the signal across the fibre is described by the nonlinear Schrödinger differential equation(NLSE)∂∂zAα2i2∂∂t2162( z,t) + A( z,t) + β A( z,t) − β A( z,t) − iγA( z,t) A( z,t) 0∂∂t32 23=3(16)where A(z,t) is the envelope of the transmitted signal which is assumed to vary slowly compared to the carrierwave. The equation as shown takes into account only the Kerr effect, however it can be modified to includeRaman scattering as well. This equation cannot be solved in the general case, and therefore a numericalsolution has to be used. The most common technique used for this purpose is the Split Step Fourier method,where the linear and non-linear parts of the differential equation are solved independently for a small section ofthe fibre, and their result is added together and used for the calculation of the next small step. The complicationis that the linear part is best solved in the frequency domain, whereas the non-linear part is best solved in thetime domain, requiring the calculation of a considerable number of Fourier Transforms. This is usually a timeconsuming process, particularly for large WDM systems. Therefore, analytical models for the WDM effectslike the ones presented here are of considerable interest.Self Phase Modulation (SPM)The first effect of the nonlinear refractive index that must be considered in both single and multi-channeltransmission is self phase modulation, that is, the change of the optical phase of a channel by its own intensity.The NLSE can also be expressed as [Agrawal95]2∂A1 ∂ A 12i − β + iaA + γ A A = 02 2∂z2 ∂T2(17)where A is proportional to the slowly varying amplitude of the electric field of the pulse envelope, β 2 is thesecond-order GVD, and γ is the nonlinear coefficient, defined as⎛⎜⎝2π nγ =λ Αeff2⎞⎟⎠. T=t-z/u g is the frame ofreference moving with the pulse at the group velocity, u g . It can be found from (Eq.17) that the second term isProject:PHOSPHORUSDeliverable Number: D.5.3Date of Issue: 31/06/07EC Contract No.: 034115Document Code: Phosphorus-WP5-D5.335