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2.3 Waveguide optics 13
For waveguides with a cross section larger than 10 µm the propagation
can be treated using ray optics. Also, a larger number of modes is allowed
simultaneously in the fibre. The waveguides treated in this thesis have
cross sectional dimensions in the 100–200 µm range, and are thereby
clearly multi-mode fibres.
2.3.1 Snell’s law
When a beam of light travels from one medium to another with different
refractive index, the light is refracted according to Snell’s law:
n1 sin(θ1) = n2 sin(θ2) (2.5)
where n1 and n2 are the refractive indices of the first and second material,
respectively, and θ1 is the incoming, and θ2 the outgoing angle with
respect to the normal of the interface between the two materials.
When moving from a dense to a less dense medium, meaning that
n1 > n2, it is readily seen that no solution exists to the equation when
θ1 exceeds a critical angle θc defined by
n2 n1 θ 2 θ1
θc = arcsin n2
n1
(2.6)
Figure 2.2: The principle of total internal
reflection. A light beam incident on
an interface between a medium of refractive
index n1 and a medium of refractive
index n2 < n1, and entering at an angle
θ1 smaller than a critical angle θc is
partly refracted and partly reflected. If
the incident angle exceeds the critical angle,
no light is refracted. Instead, total
internal reflection occurs.
As illustrated in Fig. 2.2 a beam incident at an angle θ1 < θc the
beam is partly reflected, and partly refracted at the interface. The larger
the incident angle, the larger the part of the light which is reflected. For
a beam entering at an angle θ2 > θc, no light is refracted. Instead all
light undergoes total internal reflection.