Chapter 2 Optical Fiber Waveguide in Signal Transmission 2.1 Electromagnetic Theory for Optical Waveguide 2.2 Modes in Optical Waveguides 2.3 The Analysis of Optical Waves Propagation in the Step-Index Fiber 2.4 The Analysis of Nonlinear Optical Effects

2.1 Electromagnetic Theory for Optical Waveguide The basis for the EM wave propagation is provided by Maxwell's equations shown as following: (2-1) (2.2) • = 0 (no free charges) (2.3) • = 0 (no free poles) (2.4)

2.1 Electromagnetic Theory for Optical Waveguide The four field vectors are related by the relation : ( 2.4) ( 2.5) where e is the dielectric permittivity and m is the magnetic permeability of the medium.

2.1 Electromagnetic Theory for Optical Waveguide Substituting for and and taking the curl of Eqs. (2.1) and (2.2) gives (2.6) (2.7) By using the divergence of Eqs. (2.3) and (2.4), we can obtain the following wave equations: (2.8) (2.9) where ▽ is the Laplacian operator .

2.1 Electromagnetic Theory for Optical Waveguide For Cartesian and cylindrical coordinate systems, the above wave equations hold for each component of the field vector as shown in the following: (2.10) where y is a component of the vector field or and np is the phase velocity in the dielectric medium.

2.1 Electromagnetic Theory for Optical Waveguide It can be expressed as the following (2.11) where mr and er are the relative permeability and permittivity for the dielectric medium, and mo and eo are the permeability and permittivity of free space, respectively. The velocity of optical waves in the free space c is therefore shown as the following: (2.12)

2.1 Electromagnetic Theory for Optical Waveguide Considering planar waveguides, described by Cartesian coordinates (x, y, z), or optical fibers, described by cylindrical coordinates (r, f, z) , the Laplacian operator takes the following form: (2.13) (2.14)

2.1 Electromagnetic Theory for Optical Waveguide The basic solution of the wave equation is a sinusoidal wave, the most important form of which is a uniform plane wave given by: (2.15) where w is the angular frequency of the field and is the propagation vector which gives the direction of propagation and the rate of change of phase with distance. When l is the optical wavelength in the free space, the magnitude of the propagation vector or the vacuum propagation constant is given by: k = 2p/l (2.16)

2.2 Modes in Optical Waveguides The simplest planar waveguide may assume it consists of a dielectric slab with refractive index nf, sandwiched between two regions of lower refractive index nc and ns, as shown in Figure 2.1 Figure 2.1 A plane wave propagating in the three layer planar optical waveguide.

2.2 Modes in Optical Waveguides The wave equation in Cartesian coordinates for the TE wave propagating in a lossless three layer planar optical waveguide can be expressed as following: (2.17) The solutions are with the form: (2.18)

2.2 Modes in Optical Waveguides The transverse electric fields in each layer are shown as: Ec(x) = Ac exp(-qcx), in the cladding (2.19) Ef(x) = Af cos[Qf(x-xf)], in the guilding film (2.20) Es(x) = As exp(qsx), in the substrate (2.21) where (2.22) (2.23) By matching the boundary conditions, and choosing qc = qs = q (nc = ns = n2), and Qf = Q (nf = n1), we obtain : (2.24)

2.2 Modes in Optical Waveguides Figure 2.2 shows examples of the lowest three TE modes for m = 0, 1, and 2 in the three layer planar optical waveguide. Figure 2.2 The lowest three modes (m = 0, 1, and 2) in the three layer planar optical waveguide.

2.2 Modes in Optical Waveguides For the cylindrical core waveguide under the weak guidance conditions outlined above, the scalar wave equation can be written in the form: (2.25) where y is the field of E or H, n1 is the refractive index of the fiber core, k is the propagation constant for light in the free space, and r and f are cylindrical coordinates. The propagation constants of the guided modes b lie in the range (2.26)

2.2 Modes in Optical Waveguides Solutions of the wave equation for the cylindrical fiber are separable, having the form: (2.27) where E(r) in this case represents the dominant transverse electric field component.

2.2 Modes in Optical Waveguides Introducing the solutions given by Eq. (2.27) into Eq. (2.25) results in a differential equation of the form: (2.28) For a step index fiber with a constant refractive index core, Eq. (2.28) is a Bessel’s differential equation and the solutions are cylinder functions. The electric field can be expressed as: for R < 1 (core) for R > 1 (cladding)

2.2 Modes in Optical Waveguides where G is the amplitude coefficient and R = r/a is the normalized radial coordinate when a is the radius of the fiber core. U and W, which are respectively the eigenvalues in the core and the cladding, are defined as: (2.30) (2.31)

2.2 Modes in Optical Waveguides The sum of the squares of U and V defines a very useful quantity which is usually referred to as the normalized frequency V shown as the following: (2.32) The normalized frequency V may be expressed in terms of the numerical aperture NA and the relative refractive index difference D respectively as: (2.33) (2.34)

2.2 Modes in Optical Waveguides It is also possible to define the normalized propagation constant b for a fiber in terms of the parameters of Eq. (2.32) so that: (2.35) Using the Bessel function relations, an eigenvalue equation for the LP modes may be written as following: (2.36)

2.2 Modes in Optical Waveguides The lowest modes obtained in step index optical fibers are shown in Figure 2.3. Figure 2.3 The lowest modes against V for step index optical fibers.

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber The electromagnetic fields propagating in optical fibers can be governed by the wave equation. Taking the advantage of the cylindrical symmetry, the wave equation can be written in the cylindrical coordinate as shown in the following: (2.37) (2.37) The the refractive index n can be expressed as the form: (2.38)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber Equation (2.37) is easily solved by using the method of separation of variables and writing Ez as following: By substituting Eq. (2.39) into Eq. (2.37), we obtain the three ordinary differential equations: (2.40) (2.41) (2.42)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber Its general solution in the core and cladding regions is given by (2.43) = (2.44) where A, A’, C, and C’are constants, and Jm, Ym, Km, and Im are different kinds of Bessel functions. The parameters κ and γ are defined by (2.45) (2.46)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber The general solution of Eq. (2.43) is thus of the form = (2.48) Hz can be obtained by using the same method. The solution is the same but with different constants B and D: (2.49) = (2.50)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber Another components Er, Ef, Hr, and Hf can be expressed in terms of Hz and Ez by using Maxwell's equations. The result is : (2.51) (2.52) (2.53) (2.54)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber The eigenvalue equation is shown as the following: (2.55) This can be understood by noting that the optical field of guided modes decays exponentially inside the cladding layer, since (2.56)

2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber A parameter that plays an important role in determining the cutoff condition is defined as the following: (2.57) This parameter V is called the normalized frequency.

Propagation in the Step-Index Fiber   2.3 Analysis of Optical Waves Propagation in the Step-Index Fiber The eigenvalue equation for these two modes can be obtained by setting m = 0 and is given as the following: (2.58) (2.59) The cutoff condition for both modes is simply given as: Jo(V) = 0 (2.60) The smallest value of V of Jo(V) = 0 is 2.405. A fiber designed such that V < 2.405 supports only the fundamental HE11 mode. This is the so-called single mode optical fibers. That is the optical fiber operated in single mode.

2.4 Analysis of Nonlinear Optical Effects The nonlinear optical effects for electromagnetic fields and optical fibers are no exception,even though silica is intrinsically not a highly nonlinear material. This section discusses briefly the important nonlinear effects in optical fibers. Raman scattering and Brillouin scattering are two examples of the nonlinear scattering effects. The main difference between the two is that optical phonons participate in Raman scattering, whereas acoustic phonons participate in Brillouin scattering.

2.4 Analysis of Nonlinear Optical Effects At low power levels the scattering cross sections are sufficiently small that power loss is negligible At high power levels the nonlinear phenomena of stimulated Roman scattering (SRS) and stimulated Brillouin scattering (SBS) can lead to considerable fiber loss. A fundamental difference is that SBS in single-mode fibers occurs only in the backward direction, whereas SRS dominates in the forward direction. The threshold power level for both SRS and SBS can be estimated by considering how the scattered-light intensity grows from noise.

2.4 Analysis of Nonlinear Optical Effects For SRS the threshold power Pth, defined as the incident power at which half of the power is lost to SRS at the output end of a fiber of length Leff, is estimated from (2.61) where gR is the peak value of the Raman gain. Aeff is the effective mode cross section, often referred to the effective core area.

2.4 Analysis of Nonlinear Optical Effects Leff is the effective interaction length defined by: For SBS, the threshold power can expressed as following: (2.62) If we replace Aeff by pw2, where w is the spot size, then Pth is given by (2.63) (2.64)

2.4 Analysis of Nonlinear Optical Effects Both SRS and SBS can be used to advantage in the design of optical communication systems, since they can amplify an optical field by transferring energy to it from a pump field whose wavelength is suitably chosen. The refractive index of silica becomes necessary to include the nonlinear contribution at high powers by taking into account shown as the following: where n1 and n2 are the core and cladding indices and n2i is the nonlinear-index coefficient. . (2.65)

2.4 Analysis of Nonlinear Optical Effects If we use first-order perturbation theory to obtain the fiber modes, we find that the propagation constant becomes power dependent and can be written as (2.66) where The effect of nonlinear refraction is to produce a nonlinear phase shift given by (2.67) where accounts for the fiber loss.

2.4 Analysis of Nonlinear Optical Effects By replacing fNL << 1 by for long fibers, this condition becomes (2.68) Self-phase modulation (SPM) leads to considerable spectral broadening of pulses propagating inside the optical fiber. The intensity dependence of the refractive index can also lead to another nonlinear phenomenon known as cross-phase modulation (XPM) The phase shift for the j-th channel can be written as (2.69) where M is the total number of channels and Pj is the channel power.

2.4 Analysis of Nonlinear Optical Effects If we assume equal channel powers, the phase shift in the worst case is given as following: ( 2.70) The XPM can be a major power-limiting factor. It is considered in the context of multi-channel communication systems. The four-wave mixing (FWM) is also another nonlinear phenomenon for silica fibers. The frequency combination of the form is the most troublesome for multi-channel communication systems

2.4 Analysis of Nonlinear Optical Effects Such an energy transfer not only results in the power loss for a specific channel but also leads to inter-channel crosstalk that degrades system performance. Recently, their significance for future optical fiber soliton communication systems has been appreciated comparatively for the development of Erbium-Doped Fiber Amplifiers.