Page 59 - Optical Communications Essentials

Optical Fibers

Optical Fibers 49

and be lost in the cladding. Now suppose the medium outside of the fiber is air for
which n 1.00. By applying Snell’s law to the air-fiber interface boundary, the condi-
tion for the critical angle can be related to the maximum entrance angle θ 0,max through
the relationship
sin θ 0,max n 1 sin θ c n 1 n 2 1/2
2
2
where θ c π 2 φ c . Thus those rays having entrance angles θ 0 less than θ 0,max will be
totally internally reflected at the core-cladding interface.

Example
1. Suppose the core index n 1 1.480 and the cladding index n 2 1.465. Then the crit-
ical angle is φ c arcsin(1.465/1.480) 82°, so that θ c π 2 φ c 8°.
2. With this critical angle, the maximum entrance angle is
θ 0,max arcsin (n 1 sin θ c ) arcsin (1.480 sin 8°) 11.9°

4.2. Optical Fiber Modes
Although it is not directly obvious from the ray picture shown in Fig. 4.2, only
a finite set of rays at certain discrete angles greater than or equal to the criti-
cal angle φ c is capable of propagating along a fiber. These angles are related to
a set of electromagnetic wave patterns or field distributions called modes that
can propagate along a fiber. When the fiber core diameter is on the order of 8 to
10µm, which is only a few times the value of the wavelength, then only the one
single fundamental ray that travels straight along the axis is allowed to propa-
gate in a fiber. Such a fiber is referred to as a single-mode fiber. The operational
characteristics of single-mode fibers cannot be explained by a ray picture, but
instead need to be analyzed in terms of the fundamental mode by using the elec-
tromagnetic wave theory. Fibers with larger core diameters (e.g., greater than
or equal to 50µm) support many propagating rays or modes and are known as
multimode fibers. A number of performance characteristics of multimode fibers
can be explained by ray theory whereas other attributes (such as the optical cou-
pling concept presented in Chap. 8) need to be described by wave theory.
Figure 4.3 shows the field patterns of the three lowest-order transverse elec-
tric (TE) modes as seen in a cross-sectional view of an optical fiber. They are the
TE 0 , TE 1 , and TE 2 modes and illustrate three of many possible power distribu-
tion patterns in the fiber core. The subscript refers to the order of the mode,
which is equal to the number of zero crossings within the guide. In single-mode
fibers only the lowest-order or fundamental mode (TE 0 ) will be guided along the
2
fiber core. Its 1/e width is called the mode field diameter.
As the plots in Fig. 4.3 show, the power distributions are not confined com-
pletely to the core, but instead extend partially into the cladding. The fields
vary harmonically within the core guiding region of index n 1 and decay
exponentially outside of this region (in the cladding). For low-order modes the

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