Basic principles of light propagation

Ray Theory (Geometrical Optics)

Snell’s law
The angles of the rays are measured with respect to the normal. This is a line
drawn at right angles to the boundary line between the two refractive indices.
The angles of the incoming and outgoing rays are called the angles of
incidence and refraction respectively. These terms are illustrated in Figure 1

Figure 1: The names of the parts

Law of reflection
θ I =θ r

Law of refraction (Snell’s law)

The angle increases as it crosses from the higher refractive index material to
the one with the lower refractive index.
Snell’s law states the relationship between the refractive indices of the
materials and the sine of the angles.
n1 sin θ1=n 2 sin θ2
Where: n1 and n2 are the refractive indices of the two materials, and sin θ1 and
sin θ2 are the angles of incidence and refraction respectively.

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Example

Calculate the angle shown as θ2 in Figure 2

The first material has a refractive index of 1.51 and the angle of incidence is
38° and the second material has a refractive index of 1.46.

Figure2
Solution
Starting with Snell’s law:
n1 sin θ1=n 2 sin θ2
°
1.51 sin 38 =1.46 sin θ 2
°
1.51 sin 38
sin θ2=
1.46

sin θ2=0.6367

θ2=arcsin 0.6367
°
θ2=39.55

Critical angle
The angle of the ray increases as it enters the material having a lower
refractive index.
As the angle of incidence in the first material is increased, there will come a
time when, eventually, the angle of refraction reaches 90° and the light is
refracted along the boundary between the two materials. The angle of
incidence which results in this effect is called the critical angle. We can

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calculate the value of the critical angle by assuming the angle of refraction to
be 90° and transposing Snell’s law:
°
n1 sin θ1=n 2 sin 90
−1
θc =sin (n2 /n 1)

If n1 >n 2 , then we can have

Figure 3: Critical angle

Example

A light ray is traveling in a transparent material of refractive index 1.51 and

approaches a second material of refractive index 1.46. Calculate the critical
angle.

Solution

Using the formula for the critical angle

−1
θc =sin (n2 /n 1)
−1
θc =sin (1.46 /1.51)
°
θc =75.2

Total internal reflection

The critical angle is well named as its value is indeed critical to the operation of
optic fibers. At angles of incidence less than the critical angle, the ray is
refracted.

However, if the light approaches the boundary at an angle greater than the
critical angle, the light is actually reflected from the boundary region back into

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the first material. This effect is called total internal reflection (TIR). Figure 4
shows these effects.

Any ray launched at an angle greater than the critical angle will be propagated
along the optic fiber.

Figure 4 Total internal reflection

At angles of incidence θ I > θc, the light is totally reflected back into the incidence higher refractive index
medium. This is known as total internal reflection (T. I. R)

Figure 5: Light can bounce its way along the fiber

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 The choice of frequency

Electromagnetic waves

Radio waves and light are electromagnetic waves. The rate at

which they alternate in polarity is called their frequency (f ) and is
measured in Hertz (Hz), where 1 Hz 1 cycle per second.

The speed of the electromagnetic wave (v) in free space is

approximately 3∗108 m/s. The term ‘m/s ’ means meters per second.
The distance traveled
during each cycle, called the wavelength ( λ ), can be calculated by
the relationship:
speed of light
wavelength=
frequency
In symbols, this is:
v
λ=
f

Electromagnetic spectrum

In the early days of radio transmission when the information

transmitted was
mostly restricted to the Morse code and speech, low frequencies
(long waves)
were used. The range of frequencies able to be transmitted, called
the bandwidth, was very low. This inevitably restricted us to low
speed data transmission and poor quality transmission (Figure
3.1)

In fiber optics, we find it more convenient to use the wavelength

of the light instead of the frequency.

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 Figure 3.1 Fiber optics use visible and infrared light

Need for Wider Bandwidth: Over time, increasing bandwidth was required to transmit more
complex information at higher speeds. This was achieved by increasing the frequency of the radio
signals, as higher frequencies provide greater usable bandwidth.

Television Development: The need for wider bandwidth during television's advent led to the same
approach—using higher frequencies.

Standard Approach: For about 60 years, the solution to bandwidth challenges was consistent:
higher frequency for greater bandwidth.

Fiber Optics Revolution:

Fiber optics introduced the potential for significantly higher transmission rates using light instead of
radio signals.

Early experiments explored the visible spectrum for optimal light frequencies.

Challenges with High Frequencies in Fiber Optics:

 Higher frequencies caused exponentially higher transmission losses, increasing by the fourth
power (e.g., doubling frequency increased losses by 24 (16 times)).
 These losses made high-frequency transmission impractical for long distances.

Solution with Infrared:

 Fiber optic systems leverage the inherently high bandwidth of light, allowing for the use of
lower frequencies to minimize losses and extend transmission range.
 The infrared spectrum, with its wide range of wavelengths, became the standard for fiber optic
communications.
 Visible light is typically reserved for short-range transmission (e.g., plastic fibers) or diagnostic
purposes due to its visibility.

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Windows

Infrared Light for Communication:

Infrared wavelengths are used for most fiber optic communication systems. However, some
wavelengths are easier and less expensive to manufacture for light sources and photodetectors.

Undesirable Wavelengths:

Example: 1380 nm has high losses due to water content (hydroxyl ions) in the glass, which absorb
energy at this wavelength.
Glass manufacturing must minimize water content to as low as 1 part in 10⁹ to reduce losses.

Standard Wavelengths (Windows):

It makes commercial sense to agree on standard wavelengths to ensure that equipment from
different manufacturers is compatible. These standard wavelengths are called windows and we
optimize the performance of fibers and light sources so that they perform at their best within one of
these windows

Common Windows:

850 nm Window:

 Higher losses; used for short-range communication (e.g., local area networks up to 10 km).
 Advantages: Lower cost, easier installation.

1300 nm and 1550 nm Windows:

 Much lower losses, making them ideal for long-distance communication.

 Increasingly used in local area and campus networks due to higher bandwidth when paired with
lasers.

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 Propagation of light along the fiber

The contamination on the surface changes the refractive index of the material
surrounding the glass. Previously, it was air which has a refractive index of 1 and
it is now grease which, in common with all other materials, has a refractive
index greater than 1.
This will locally increase the critical angle and some of the light will now find
itself approaching the surface at an angle less than the new critical angle. It will
then be able to escape.

Dirt, grease, rain, in fact any contamination will allow leakage of the light.

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 Figure 4.2 Any contamination causes a power loss

Solution
Cover the fiber with another layer of glass as shown in Figure 4.3.

Figure 4.3 How to keep the core clean

The original glass, called the core, now has a new layer, the cladding, added
around the outside during manufacture. The core and the cladding form a single
solid fiber of glass.

To protect the optic fiber from surface scratches, we add a layer of soft plastic to the
outside of the cladding. This extra layer is called the primary buffer (sometimes
called the primary coating or just the buffer) and is present only to provide
mechanical protection and has nothing to do with light transmission.

Figure 4.9 The completed optic fiber

Three important points:

 The optic fiber is solid, there is no hole through the middle

 The buffer and the jacket are only for mechanical protection

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 The light is transmitted through the core but to a small extent it travels in the
cladding and so the optical clarity of the cladding is still important

Figure 4.11 Light enters cladding during reflection

We can now see that an opaque cladding would prevent the ray from being propagated along the
fiber since the light would not be able to pass through the cladding.

Getting the light into the fiber

The ray shown has bounced its way along the fiber at the critical angle. It
leaves the fiber as the one on the outermost edge (Figure 4.13).

Assume the same values that we have used previously.

core refractive index 1.5
cladding refractive index is 1.48

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This gives an approximate critical angle (as in Figure 4.14) of:

As the angle is measured from the normal, the angle between the cladding boundary
and the ray is actually 90–80.6° = 9.4° as shown in Figure 4.15.

The refractive index of the air is 1 (near enough), so we can calculate the angle
at which the ray leaves the fiber by applying Snell’s law.
n1 sin θ1=n 2 sin θ2
n1 = refractive index of the core 1.5
sin θ1 = sine of the angle of incidence (9.4°)
n2 = refractive index of air 1.0
sin θ2 = sine of the angle of refraction.

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We have now calculated the angle at which the light spreads out as 14.18°: not a very
large angle but typical for a glass fiber (Figure 4.16).

Incidentally, plastic fibers have a greater angle, around 27°.

Since light direction is reversible, this 14.18° is also the angle at which light can
approach the core and be propagated along the fiber (Figure 4.17).

The optic fiber is circular and therefore this angle is applicable in two dimensions and
would look like a cone as shown in Figure 4.18.

Cone of acceptance
The cone of acceptance is the angle within which the light is accepted
into the core and is able to travel along the fiber.

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The angle is measured from the axis of the cone so the total angle of
convergence is actually twice the stated value.

Numerical aperture (NA)

The numerical aperture of a fiber is a figure which represents its

light gathering
capability. We have just seen that the acceptance angle also
determines how much light is able to enter the fiber and so we must
expect an easy relationship between the numerical aperture and the
cone of acceptance as they are both essentially measurements of
the same thing.
The formula for the numerical aperture is based on the refractive
indices of the core and the cladding.

Names given to different rays

We have seen that rays approaching from within the cone of
acceptance are successfully propagated along the fiber. The position
and the angle at which the ray strikes the core will determine the
exact path taken by the ray. There are three possibilities, called the
skew, meridional and the axial ray as shown in Figure 4.31.

The skew ray never passes through the center of the core. Instead
it reflects off
the core/cladding interface and bounces around the outside of the
core. It moves
forward in a shape of a spiral staircase built from straight sections.
The meridional ray enters the core and passes through its center.
Thereafter,
assuming the surfaces of the core are parallel, it will always be
reflected to pass
through the center.

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The axial ray is a particular ray that just happens to travel straight
through the
center of the core.

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