# 8.3 Fiber Optics

By successive total internal reflections, as illustrated in Fig. 8-12(a), light can be guided through thin dielectric rods made of glass or transparent plastic known as optical fibers. Because the light is confined to traveling within the rod, the only loss in power is due to reflections at the sending and receiving ends of the fiber and absorption by the fiber material (because it is not a perfect dielectric). Optical fibers are useful for the transmission of wide-band signals as well as many imaging applications.

# Figure 8-12 Waves can be guided along optical fibers as long as the reflection angles exceed the critical angle for total internal reflection.

(a) Optical fiber

(b) Successive internal reflections

Figure 8 12 Full Alternative Text

An optical fiber usually consists of a cylindrical fiber core with an index of refraction n f, surrounded by another cylinder of lower index of refraction, n c, called the cladding (Fig. 8-12(b)). The cladding layer serves to optically isolate the fiber when a large number of fibers are packed in close proximity, thereby avoiding leakage of light from one fiber into another. To ensure total internal reflection, the incident angle θ 3 in the fiber core must be equal to or greater than the critical angle θ c for a wave in the fiber medium (with n f) incident upon the cladding medium (with n c). From Eq. (8.32a), we have

sin θ c=n c n f. (8.35)

To meet the total reflection requirement θ 3≥θ c, it is necessary that sin θ 3≥n c/n f. The angle θ 2 is the complement of angle θ 3; hence, cos θ 2=sin θ 3. The necessary condition therefore may be written as

cos θ 2≥n c n f. (8.36)

Moreover, θ 2 is related to the incidence angle on the face of the fiber, θ i, by Snell’s law:

sin θ 2=n 0 n f sin θ i, (8.37)

where n 0 is the index of refraction of the medium surrounding the fiber (n 0=1 for air and n 0=1.33 if the fiber is in water) or

cos θ 2=1−n 0 n f 2 sin 2 θ i 1/2. (8.38)

Using Eq. (8.38) on the left-hand side of Eq. (8.36) and then solving for sin θ i gives

sin θ i≤1 n 0(n f 2−n c 2)1/2. (8.39)

# 8.3.1 Maximum Acceptance Cone

The acceptance angle θ a is defined as the maximum value of θ i for which the condition of total internal reflection remains satisfied:

The angle θ a is equal to half the angle of the acceptance cone of the fiber. Any ray of light incident upon the face of the core fiber at an incidence angle within the acceptance cone can propagate down the core. This means that there can be a large number of ray paths, called modes, by which light energy can travel in the core. Rays characterized by large angles θ i travel longer paths than rays that propagate along the axis of the fiber, as illustrated by the three modes shown in Fig. 8-13. Consequently, different modes have different transit times between the two ends of the fiber. This property of optical fibers is called modal dispersion and has the undesirable effect of changing the shape of pulses used for the transmission of digital data. When a rectangular pulse of light incident upon the face of the fiber gets broken up into many modes and the different modes do not arrive at the other end of the fiber at the same time, the pulse gets distorted—both in shape and length. In the example shown in Fig. 8-13, the narrow rectangular pulses at the input side of the optical fiber are of width τ i separated by a time duration T. After propagating through the fiber core, modal dispersion causes the pulses to look more like spread-out sine waves with spread-out temporal width τ. If the output pulses spread out so much that τ>T, the output signals will smear out, making it impossible to decipher the transmitted message from the output signal. Hence, to ensure that the transmitted pulses remain distinguishable at the output side of the fiber, it is necessary that τ be shorter than T. As a safety margin, it is common practice to design the transmission system such that T≥2 τ.

# Figure 8-13 Distortion of rectangular pulses caused by modal dispersion in optical fibers.

Figure 8 13 Full Alternative Text

The spread-out width τ is equal to the time delay Δ t between the arrival of the slowest ray and the arrival of the fastest ray. The slowest ray is the one traveling the longest distance and corresponds to the ray incident upon the input face of the fiber at the acceptance angle θ a. From the geometry in Fig. 8-12(b) and Eq. (8.36), this ray corresponds to cos θ 2=n c/n f. For an optical fiber of length l, the length of the path traveled by such a ray is

l max=l cos θ 2=l n f n c, (8.41)

and its travel time in the fiber at velocity u p=c/n f is

t max=l max u p=l n f 2 c n c. (8.42)

The minimum time of travel is realized by the axial ray and is given by

t min=l u p=l c n f. (8.43)

The total time delay is therefore

τ=Δ t=t max−t min=l n f c n f−1 n c(s). (8.44)

As we stated before, to retrieve the desired information from the transmitted signals, it is advisable that T, which is the interpulse period of the input train of pulses, be no shorter than 2 τ. This in turn means that the maximum data rate (in bits per second), or equivalently the number of pulses per second, that can be transmitted through the fiber is limited to

f p=1 T=1 2 τ=c n c 2 l n f(n f−n c)(bits/s). (8.45)

# Example 8-5: Transmission Data Rate on Optical Fibers

A 1-km long optical fiber (in air) is made of a fiber core with an index of refraction of 1.52 and a cladding with an index of refraction of 1.49. Determine

1. the acceptance angle θ a and

2. the maximum usable data rate of signals that can be transmitted through the fiber.

Solution: (a) From Eq. (8.40),

sin θ a=1 n 0(n f 2−n c 2)1/2=[(1.52)2−(1.49)2]1/2=0.3,

which corresponds to θ a=17.5°.

(b) From Eq. (8.45),

f p=c n c 2 l n f(n f−n c)=3×10 8×1.49 2×10 3×1.52(1.52−1.49)=4.9 (Mb/s).

# 8.3.2 Confined Acceptance Cone

The expression for the data rate given by Eq. (8.45) pertains to the case where the light coupled to the fiber is confined to the acceptance cone defined by the acceptance angle θ a given by Eq. (8.40). If the entrance cone is larger than the acceptance cone, then some of the light will leak out of the fiber through the cladding, which not only reduces the power carried by the light but may also cause interference with signals carried by neighboring fibers.

On the other hand, if the entrance cone can be made to be smaller than the acceptance cone, the data rate can be safely increased. Let us denote θ i(max) as the maximum angle of the entrance cone. Per Eq. (8.38), θ 2(max) is related to θ i(max) by

cos θ 2(max)=1−n 0 n f 2 sin 2 θ i(max)1/2.

Repetition of the steps between Eqs. (8.41) and (8.45) leads to

f p=c 2 l n f cos θ 2(max)1−cos θ 2(max)(θ i(max)<θ a).