As indicated in Figure , an optical fiber is basically a two-layered structure comprised of dielectric material. It has a uniform circular cross-section along a straight longitudinal axis. The inner region is called the core and the outer region the cladding. We first analyze the electromagnetic fields present in optical fibers to derive several important propagation characteristics.
We study fields in phasor form, with 
the operating (radian)
frequency. Assume the core is a perfect dielectric characterized by 
, and define the parameter k as:
We also assume power flow is occurring along the longitudinal axis. Set this to be the z-axis and so the electric field has the form:
where 
lies in the transverse plane (has no z-component), and 
are the pola
r coordinates for the transverse plane.
Similarly, the magnetic field is given by:
The parameter 
is called the phase constant. It can be shown
that real power flow in the +z-direction occurs when is real and
positive. The phase and group velocities of the field, 
and 
,
respectively, are defined as:
Moreover, the group velocity is the speed of propagation of energy or information along the optical fiber.
Define q as:
Then Maxwell's equations for 
and 
are:
and:
The method of modal analysis seeks to represent the field as the
superposition of several special types of fields. There are several types of
modes. First, we must outright reject TEM (transverse electric and
magnetic) modes, in which both 
and 
, since no (nonzero) TEM
modes can lead to real power flow in this situation. TE (transverse
electric) modes have , while TM (transverse magnetic) modes
have . Although TE or TM modes lead to a somewhat simplified
analysis, there are several important fields which cannot be expressed as
the superposition of such modes. Therefore, we must also consider more
general hybrid modes; HE modes have 
and EH modes have 
.
The geometry of the situation leads us to seek functions periodic in 
. Therefore, we assume the -dependence has the form
where 
is an integer (possibly positive, negative or 0). Thus:
and hence:
Moreover, since this is in the core region, we impose that 
to stay finite as 
.
Thus, the solution is the 
order Bessel function of the first kind, 
.
A similar result holds for 
.
Now let us distinguish between the core 
, characterized
by 
, and the cladding 
characterized
by 
. Define:
Imposing that the core function stays finite at r=0, that the cladding
function decay to 0 at 
, and that 
for
real power flow, we have that u>0 and w>0. Thus, in particular, 
. Note, in particular, that this requires 
, which is the case in real fibers.
It may be convenient at times to express 
and 
in terms of the
respective indices of refraction 
and 
and the wavelength 
in free-space:
The longitudinal components in the core are given by:
and those in the cladding are:
In particular, we impose the same value characterize the fields in
the core and cladding. This is necessary to achieve phase match conditions
at r=a; for example, 
must be continuous at r=a. Also, 
is
the order Bessel function of the second kind. It can
be shown that 
(asymptotically) as . In fact, 
for real 
.
To summarize, to have the field in the cladding decay we must have:
The condition 
is called cutoff. Additionally, if 
is not real valued,
that is if 
, then we have no
real power flow. Hence we must also have:
with a corresponding cutoff condition 
. To summarize:
where 
is the free-space wavenumber.
The appropriate boundary conditions we must impose at r=a are:
These equations correspond to the property that the tangential component of the electric field must be continuous along a dielectric boundary. The first leads boundary condition leads to:
and the second to:
We must also impose:
These correspond to the fact that the tangential magnetic field is continuous along a dielectric boundary if there is no surface current present.
We have a total of four homogeneous equations in A,B,C,D:
Since we seek nonzero fields, we cannot have all four constants A,B,C,D equal to zero; hence:
This condition must be met for a field to be present. Define:
Then equation (1.22) becomes:
Given k, that is , the values u,w are known functions of . He
nce, (1.24) specifies as a function of frequency , and as a function of the parameter . This equation has only
discrete solutions, and in general for each there will be several
roots, denoted as:
The corresponding modes are denoted as TE,
, 
or 
, as appropriate.
Let us examine the TE and TM cases in particular. We obtain TE by setting A=B=0, and we seek nonzero C,D; and similarly TM is obtained by setting C=D=0 and we seek nonzero A,B. In each case, this req
uires a 
submatrix of M to have nonzero determinant, and in particular can be shown
to require:
Thus, there can be no -variation (there is radial symmetry) for TE
and TM modes. The equation determining for TE 
is:
Similarly, the equation determining for TM is:
If 
, we do not have TE or TM modes, and the analysis becomes very
complex. However, if 
, we can
apply an important class of approximations which lead to weakly guided
waves.
The cutoff conditions 
for lower order
modes are summarized in Table below.
Table 1.1: Cutoff Conditions for Low Order Modes
We now discuss the V-number, also called the V-parameter or the normalized frequency. The value V is defined as:
and is dimensionless. Note that the value 
is proportional
to frequency (up to a factor equal to the speed of light), and hence V is
called the normalized frequency.
The value V is related to the number of modes a fiber can support. Also define the normalized propagation constant b as:
Note that 0<b<1 corresponds to propagation.
A graph of b or 
versus V shows, for fixed V, only several
modes are possible. In particular, the HE 
mode exists
(corresponds to a value b in the range 0 to 1) for all V, down to V=0. No other mode exists until V=2.405 (this is the smallest root of 
).
Hence, below this value of V, all modes other than HE
are cutoff. For this reason, HE is called the dominant
mode.
We define the numerical aperture of a fiber to be:
so that:
The physical significance of the numerical aperture can be obtained using the ray theory approximation to wave propagation. Refer to Figure 1. We assume light propagates in the fiber like a plane wave reflecting at top and bottom core-cladding boundaries according to total internal reflection (TIR); TIR occurs when the incidence angle at a dielectric boundary exceed the critical angle, that is when:
It can be shown that the field in the cladding when TIR occurs has no real
power flow associated with it, and is called evanescent. If we imagine
launching a light wave into the fiber core from air, with an entry angle 
, then:
Thus, there is a maximum entry angle for which power will be launched in the fiber, and we define numerical aperture in terms of this maximum angle as:
The two definitions of numerical aperture coincide when 
.
Equation (1.32) implies two basic factors increase V, and hence
increase the number of modes present in light propagating in a fiber. First,
if the diameter a is increased (relative to wavelength) then V is
increased. Hence, single-mode fibers have a narrow core diameter, and
permit only one mode (the HE mode) to propagate; multi-mode fibers have a larger core diameter and permit many modes. Additionally, a
large numerical aperture enlarges V. Now, equation (1.35)
implies a larger NA corresponds to the ability of the fiber to accept a
larger beamwidth of the input light signal. LED light sources produce much
broader beams than semiconductor laser diodes, and hence LED sources must be
used with larger NA fibers, and hence multi-mode fibers.
Let us consider multi-mode fibers (so that the total number of modes M) is
large with a laser beam input. Approximate 
by 
, so
the solid acceptance angle for a light input (specified by (1.35)) is:
For a waveguide or laser with radiation at , the number of modes
per unit solid angle is approximately:
where 
is the area; the factor 2 refers to two polarized
orientations. Then:
Let us return to the weakly guided fiber approximation. Define 
as:
and assume 
. Two mode with the same value for are
said to be degenerate. We associate degenerate modes together since
they have identical propagation characteristics, although different field
distributions. In other words, we consider all linear combinations of a
class of degenerate modes to be a mode unto itself. We group the primary
lower order modes according to their degeneracies:
Numerically, 
implies:
and we get:
The positive sign leads to EH modes:
and the negative sign leads to HE modes:
To summarize:
where:
Hence, all modes with the same j and m are degenerate; for example 
and 
. This pair of degenerate modes are called
LP (linearly polarized) modes, since they can be combined to
yield fixed orientation. That is, in a complete set of modes, only one E
and one H component are significant, say the polarized along one
axis and perpendicular to it. Equivalent solutions are obtained
with the polarization reversed. These two cases can be combined with 
and 
so
four mode patterns form one 
mode.
