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FEED LINE BASICS 291

It would not be an exaggeration to say that the
actual inventors of fiber were Dutch astronomer normal
Willebrord Snellius who in 1621 discovered the
law of refraction carrying his name. Snell’s law
relays the plane wave incident and refraction
1
angle (see Figure 6.3.2) on the flat surface
2
separating two half space with different dielectric
and magnetic constants as
sin = sin
1 1 2 2
�
0 0�
1,2 = � 1,2 1,2 = 0� 1,2 =

0 1,2
(6.10) Figure 6.3.2 Snell’s law illustration
We assumed for simplification that both mediums are loss-free and 1,2 = 1 that is typical for
the most materials used in optical band. Recall that the parameter 1,2 = � 1,2 is called the
medium refractive index. Let start our discussion from the fibers that Figure 6.3.1b - d
illustrates:

b) The main idea of this fiber design is to keep the light energy inside the core using the
phenomenon of total internal reflection. The core refractive index (typically, around = 1.458
1
at wavelength of 1550 nm for pure silica) is > ( is the cladding index in Figure 6.3.1b)
2
1
2
and thus the ratio > 1. Then according to Snell’s law (6.10) sin = ( ) sin ,
⁄
⁄
2
2
1
1
2
1
i.e. sin > sin and > as it illustrates the drawing in Figure 6.3.2. As the incident
2
1
1
2
angle growths it reaches the so-called critical value when sin = ⁄ , sin = 1

1
2
2
1
and = 90°. It is clear that the refracted wave runs along the boundary between dielectrics.
2
Meanwhile, the question boils down what happens if 90° > > and ( )sin > 1.
⁄
1
1
2

At first glance, the surprising demand follows from Snell’s law that sin > 1, i.e. the sinus
2
of refraction angle must be real but greater than one. Formally, it can be fulfilled if =
2
90° + and
⁄
sin = cos() = cosh() = ( )sin ≥ 1 � (6.11)
1
1
2
2
−1
= cosh �( )sin �
⁄
1
2
1
Then the propagation of refracted plane wave can be described as (see Chapter 3)
(− 2 ) (− 2 sin 2 − 2 cos 2 )
� � ~ = (6.12)

Here sin = sin and cos = cos(90° + ) = − sin() = −sinh(). Therefore,
2
2
2
1
1
− 2 sinh() (− 1 sin 1 )
� � ~ (6.13)

Therefore, Snell’s law “tells” us that something new and unusual must happen:
1. If > ℜ( ) = 90° for any incident angle, i.e. the refracted EM wave propagates and

2
carries its energy parallel to the interface between dielectrics (x-axis in Figure 6.3.2) and
in that direction only. There is no refraction in the conventional sense.