58
2 Optical Fiber Structures and Light Guiding Principles
Thus only waves that have those angles θ that satisfy the condition in Eq. (2.26)
will propagate in the dielectric slab waveguide.
2.4 Modes in Circular Waveguides
To attain a detailed understanding of the optical power propagation mechanism in a
fiber, it is necessary to solve Maxwell’s equations subject to their cylindrical boundary
conditions of the electric and magnetic fields at the interface between the core and
the cladding of the fiber. This has been done in extensive detail in a number of works
[6, 10, 14–18]. Because a complete mathematical treatment is beyond the scope of
this book, only the results of the analysis of Maxwell’s equations is given here.
When solving Maxwell’s equations for hollow metallic waveguides, only transverse electric (TE) modes and transverse magnetic (TM) modes are involved.
However, in optical fibers the core-cladding boundary conditions lead to a coupling
between the electric and magnetic field components. This gives rise to hybrid electromagnetic modes, which makes optical waveguide analysis more complex than
metallic waveguide analysis. The hybrid modes are designated as EH or HE modes,
depending on whether the transverse electric field (the E field) or the transverse
magnetic field (the H field), respectively, is larger for that mode. The two lowestorder modes are called HE 11 and TE 01 , where the subscripts refer to possible modes
of propagation of the optical field.
2.4.1 Basic Modal Concepts
Before progressing with a discussion of mode theory in circular optical fibers, this
section will qualitatively examine the appearance of modal fields in the planar dielectric slab waveguide shown in Fig. 2.19. The core of this waveguide is a dielectric
slab of index n 1 that is sandwiched between two dielectric layers that have refractive
indices n 2 < n 1 . These surrounding layers are called the cladding. This represents
the simplest form of an optical waveguide and can serve as a model to gain an
understanding of wave propagation in optical fibers. In fact, a cross-sectional view
of the slab waveguide looks the same as the cross-sectional view of an optical fiber
cut along its axis. Figure 2.19 shows the field patterns of several of the lower-order
transverse electric (TE) modes (which are solutions of Maxwell’s equations for the
slab waveguide). The order of a mode is equal to the number of field zeros across the
guide (the locations where the field is zero at the center of the guide or at the axis of
the fiber). The order of the mode is also related to the angle that the ray congruence
corresponding to this mode makes with the plane of the waveguide (or the axis of
a fiber); that is, the steeper the angle, the higher the order of the mode. The plots
show that the electric fields of the guided modes are not completely confined to the
central dielectric slab (i.e., they do not go to zero at the guide-cladding interface),
2 Optical Fiber Structures and Light Guiding Principles
Thus only waves that have those angles θ that satisfy the condition in Eq. (2.26)
will propagate in the dielectric slab waveguide.
2.4 Modes in Circular Waveguides
To attain a detailed understanding of the optical power propagation mechanism in a
fiber, it is necessary to solve Maxwell’s equations subject to their cylindrical boundary
conditions of the electric and magnetic fields at the interface between the core and
the cladding of the fiber. This has been done in extensive detail in a number of works
[6, 10, 14–18]. Because a complete mathematical treatment is beyond the scope of
this book, only the results of the analysis of Maxwell’s equations is given here.
When solving Maxwell’s equations for hollow metallic waveguides, only transverse electric (TE) modes and transverse magnetic (TM) modes are involved.
However, in optical fibers the core-cladding boundary conditions lead to a coupling
between the electric and magnetic field components. This gives rise to hybrid electromagnetic modes, which makes optical waveguide analysis more complex than
metallic waveguide analysis. The hybrid modes are designated as EH or HE modes,
depending on whether the transverse electric field (the E field) or the transverse
magnetic field (the H field), respectively, is larger for that mode. The two lowestorder modes are called HE 11 and TE 01 , where the subscripts refer to possible modes
of propagation of the optical field.
2.4.1 Basic Modal Concepts
Before progressing with a discussion of mode theory in circular optical fibers, this
section will qualitatively examine the appearance of modal fields in the planar dielectric slab waveguide shown in Fig. 2.19. The core of this waveguide is a dielectric
slab of index n 1 that is sandwiched between two dielectric layers that have refractive
indices n 2 < n 1 . These surrounding layers are called the cladding. This represents
the simplest form of an optical waveguide and can serve as a model to gain an
understanding of wave propagation in optical fibers. In fact, a cross-sectional view
of the slab waveguide looks the same as the cross-sectional view of an optical fiber
cut along its axis. Figure 2.19 shows the field patterns of several of the lower-order
transverse electric (TE) modes (which are solutions of Maxwell’s equations for the
slab waveguide). The order of a mode is equal to the number of field zeros across the
guide (the locations where the field is zero at the center of the guide or at the axis of
the fiber). The order of the mode is also related to the angle that the ray congruence
corresponding to this mode makes with the plane of the waveguide (or the axis of
a fiber); that is, the steeper the angle, the higher the order of the mode. The plots
show that the electric fields of the guided modes are not completely confined to the
central dielectric slab (i.e., they do not go to zero at the guide-cladding interface),