Optical fiber transmission takes place through guided modes. The modes are determined from the Eigen values of second-order differential equations and their boundary conditions, by an analysis similar to that used for propagation in cylindrical waveguides
- The exact solution of Maxwell’s equations for a cylindrical homogeneous core dielectric waveguide involves much algebra and yields a complex result
- The cylindrical waveguide is bounded in two dimensions rather than one.
- Thus two integers, l and m, are necessary in order to specify the modes
- For the cylindrical waveguide consider TElm and TMlm modes
- These modes correspond to meridional rays traveling within the fiber.
- Hybrid modes where Ez and Hz are nonzero also occur within the cylindrical waveguide.
- These modes are designated HElm and EHlm depending upon whether the components of H or E make the larger contribution to the transverse (to the fiber axis) field.
- These fibers satisfy the weakly guiding approximation [Ref. 16] where the relative index difference Δ<<1. This corresponds to small grazing angles θ in Eq. (1).
- Δ is usually less than 0.03 (3%) for optical communications fibers
- For weakly guiding structures with dominant forward propagation, mode theory gives dominant transverse field components.
- Hence approximate solutions for the full set of HE, EH, TE and TM modes may be given by two linearly polarized components.
- These linearly polarized (LP) modes are not exact modes of the fiber except for the fundamental (lowest order) mode.
- As Δ in weakly guiding fibers is very small, then HE–EH mode pairs occur which have almost identical propagation constants.
- Such modes are said to be degenerate.
- The super-positions of these degenerating modes characterized by a common propagation constant correspond to particular LP modes regardless of their HE, EH, TE or TM field configurations.
- This linear combination of degenerate modes obtained from the exact solution produces a useful simplification in the analysis of weakly guiding fibers.
- The relationship between the traditional HE, EH, TE and TM mode designations and the LPlm mode designations is shown in Table 1. The mode subscripts l and m are related to the electric field intensity profile for a particular LP mode
- There are in general 2l field maxima around the circumference of the fiber core and m field maxima along a radius vector
Table 1 Correspondence between the lower order in linearly polarized modes and the traditional exact modes
- The subscript l in the LP notation now corresponds to HE and EH modes with labels l 1 and l − 1 respectively
- The electric field intensity profiles for the lowest three LP modes, together with the electric field distribution of their constituent exact modes, are shown in Figure 2.15.
- It may be observed from the field configurations of the exact modes that the field strength in the transverse direction (Ex or Ey) is identical for the modes which belong to the same LP mode.
- Hence the origin of the term ‘linearly polarized