Optical Fiber Communication
Lecture 2:
Nature of light and Ray theory
Dr. Ghusoon Mohsin Ali
M.Sc. in Electronics & Communication
Department of Electrical Engineering
College of Engineering
Al-Mustansiriya University
1
Laws of Reflection & Refraction
When a ray is incident on the interface between two dielectrics of different refractive
indices (e.g. glass-air), reflection and refraction occur.
Reflection law: angle
of incidence=angle of
reflection
Figure 2.5 Refraction of light
at a media interface
Snell’s law of refraction:
n1 sin 1  n2 sin 2
If n2 < n1 , then the angle of refraction is greater than the
angle of incidence and the refracted ray is said to have moved
away from the normal.
If the angle of incidence (ϕ1) is increased further, the angle
of refraction (ϕ2) also increases in accordance with the Snell’s
law and at a particular angle of incidence the angle of
refraction becomes 90o and the refracted ray grazes along the
media interface.
 This angle of incidence is called the critical angle of
incidence (θc) of medium 2 with respect to medium 1. One
should note here that critical angle is media-relative.
If (ϕ1) is increased beyond the critical angle, there exists no
refracted ray and the incident light ray is then reflected back
into the same medium. This phenomenon is called the total
internal reflection of light.
The word ‘total’ signifies that the entire light energy that
was incident on the media interface is reflected back into the
same medium. Total Internal Reflection (TIR) obeys the laws
of reflection of light.
This phenomenon shows that light energy can be made to
remain confined in the same medium.
Thus we can see that there are two basic requirements for a
TIR to occur:
1. The medium from which light is incident, must be optically
denser than the other medium. In figure 2.5 n2 < n1.
2. The angle of incidence in the denser medium must be
greater than the critical angle.
Total internal reflection, Critical angle
2
Transmitted
(refracted) light
k
t
n2
n 1 > n2
k
i
1
Incident
light
k
r
Reflected
light
 2  90
Evanescent wave
1
c
Critical angle
(a)
(b)
1   c
1   c
n1 sin 1  n2 sin 2
n2
sin  c 
n1
1   c TIR
(c)
1   c
LAUNCHING OF LIGHT INTO AN OPTICAL FIBER
Light propagates inside an optical fiber by virtue of multiple TIRs at the corecladding interface.
The refractive index of the core glass is greater than that of the cladding. This meets
the first condition for a TIR.
 All the light energy that is launched into the optical fiber through its tip does not get
guided along the fiber. Only those light rays propagate through the fiber which are
launched into the fiber at such an angle that the refracted ray inside the core of the
optical fiber is incident on the core-cladding interface at an angle greater than the
critical angle of the core with respect to the cladding.
Figure 2.8: Launching of Meridional Rays
Figure 2.8 shows planes which contain the fiber axis are called meridionalplanes and consequently the rays lying in a meridional-plane are called
meridional-rays.
There may be infinite number of planes that pass through the axis of the
fiber and consequently there are an infinite number of meridional planes.
These meridional rays which get totally internally reflected at the corecladding boundary meet again at the axis of the optical fiber as shown in the
figure 2.9 below.
Figure 2.9: Meridional
Rays meeting at the axis.
Skew rays
Another way of launching a light ray into an optical fiber is to launch it in
such a way that it does not lie in any meridional plane. These rays are
called skew rays.
Consequently, at the output, skew rays will have minimum energy at the
axis of the optical fiber and it will gradually increase towards the periphery
of the core.
Thus, on the whole, there are two ways of launching light into an optical
fiber; light can be launched either as meridional or as skew rays.
Figure 2.10: Launching of Skew Rays
Assuming that light is launched as meridional rays into the optical fiber,
The incident ray AO (shown by dotted line) is incident at an angle ϕ with the axis
of the fibre. The refracted ray for AO in the core (dotted line ON1) fails to be
incident on the core-cladding interface at angle greater or equal to the critical angle
of the core w.r.t. cladding and hence refracts out of the core and is lost to the
cladding.
Figure 2.11: Launching of Light into an Optical Fiber
The ray CO is launched into the fiber at such an angle ‘α’
that its refracted ray is incident at the core-cladding boundary
at its critical angle ‘θc’.
If any light ray is launched at an Fiber Optic angle more
than α then the refracted ray just refracts out to the cladding
because the angle of incidence of its refracted ray at the corecladding interface is less than the critical angle.
Thus the angle α is indicative of the maximum possible
angle of launching of a light ray that is accepted by the fiber.
Consequently, the angle α is called the angle of acceptance
of the fiber core.
Since the optical fiber is symmetrical about its axis, it is very
clear that all the launched rays, which make an angle α with
the axis, considered together, form a sort of a cone.
This cone is called the acceptance cone.
NUMERICAL APERTURE OF OPTICAL FIBER
An incident ray AO is incident from medium1 at the tip of the fiber making an
angle α with the axis of the fiber, which is the acceptance angle of the fibre.
The refracted ray for this incident ray in the core then is incident at the corecladding interface at the critical angle θc of the core with respect to the cladding.
The angle of refraction for critical angle of incidence is 900 and the refracted ray
thus grazes along the core-cladding boundary along BC as shown in the figure
2.12.
Figure 2.12: Launching of Light into an Optical Fiber
According Snell’s laws, the incident and the refracted rays lie in the same
meridional plane, which is the plane of the paper in this case. Applying Snell’s law at
the medium1-core interface we get:
n sin   n1 sin 
From the figure it is clear that,
(2.1), we get:

n sin   n1 cos  c
(2.1)

2
c
substituting this in equation
(2.2)
From the basic trigonometric ratios,
cos  c  1  sin  c
2
(2.3)
Applying Snell’s law at the core-cladding interface we get:
n2
sin  c 
n1
 n2 
cos  c  1   
 n1 
2
(2.4)
Substituting equation (2.4) in equation (2.2) we get:
n sin   n1 cos  c
n sin   n12  n22
Since the initial medium 1 from which the light is launched is air most of the times,
n = 1. The angle α is indicative of light accepting capability of the optical fiber.
Greater the value of α, more is the light accepted by the optical fiber. In other
words, the optical fiber acts as some kind of aperture that accepts only some
amount of the total light energy incident on it. The light accepting efficiency of this
aperture is thus indicated by sin α and hence this quantity is called as the
numerical aperture (N.A.)
N . A  sin   n12  n22
(2.5)
relative refractive index Δ
n1  n2

n1
NA  n1 2 
N . A  sin 
H.W
Numerical Aperture indicates the light collecting efficiency of an optical fiber.
More the value of N.A. better is the fiber.
 For greater values of N.A, either the refractive index of the core (n1) has to be
increased or the refractive index of the cladding (n2) has to be reduced. Since
the value of its refractive index n1 is thus fixed (approximately 1.5). The only
option thus available with us is to reduce the value of n2. But the lowest value of
1 for air because till date no material is known which has a refractive index
lower than that. If we make n2 =1, we would then get the maximum possible
N.A. for an optical fiber. But then we are basically talking about removing the
cladding because. Thus one can clearly say that from the point of view of light
accepting efficiency, the presence of a cladding is undesirable.
However, on optical fibers is not just to put light inside an optical fiber with the
best efficiency but also to propagate the light over long distances with the least
attenuation.
In other words, light launching efficiency is just one of the key characteristic
aspects of an optical fiber. There are other attributes too which have to be given
importance while determining the quality of an optical fiber. One of such
attributes of an optical fiber is its bandwidth. Large bandwidths are desirable
for high data rates of transmission.
When optical fiber is used for transmission of information, light signal launched into it
cannot be of continuous nature.
In an optical fiber light is launched in the form of optical pulses
Light energy launched into the fiber may be considered to travel in the form of
numerous rays
These rays travel different paths inside the core of an optical fiber because different
light rays are incident on the tip of the optical fiber at different angles
This causes different light rays in the acceptance cone to travel along different paths in
the core of the optical fiber and accordingly take different time intervals to travel a given
distance too, which leads to a phenomenon of pulse broadening inside the core of the
optical fiber.
Thus the pulse of light which might originally be of width T seconds now might be of
T+ΔT seconds inside the fiber core.
Figure 2.13: PulseBroadening
inside optical fiber
core
The amount of broadening is measured in terms of the increase in the pulse time
width and is denoted by ΔT. the value of ΔT is given by:
L n1 n1  n 2 
T 
c
n2
(2.6)
Where, ΔT= Pulse Broadening; c = velocity of light in free space; n1 = refractive
index of core and n2 = refractive index of the cladding.
The quantity L is the horizontal distance travelled before suffering the first total
internal reflection by the refracted ray OB which corresponds to the incident ray
AO, incident at the acceptance angle as shown in the figure.
The amount of pulse broadening is effectively the difference in time of travel
between the ray travelling along the axis and the incident ray AO.
This pulse broadening effect signifies that if a second pulse is now launched into
the fiber within the time interval T+ΔT, the two pulses will overlap and no
identifiable data would be obtained on the output.
Thus for a given length L, there would be a corresponding value of ΔT (from
equation 2.6) which would limit the rate at which light pulses can be launched into
the optical fiber.
This indirectly limits the bandwidth available on the fiber. Thus we can say that
more the pulse broadening lower the bandwidth. That is:
1
Bandwidth( BW ) 
T
(2.7)
In equation 2.6, we see that the value of ΔT is dependent on the value of L, the
difference (n1 – n2) as well as the value of n1/n2.
But reducing the value of L would signify the reduction in the length of the optical
fiber, which is not desirable.
As 1<n2<n1, the ratio, n1 / n2 is very close to 1.
Thus for low ΔT values, the only option available with us is to decrease the value
(n1–n2) or in other words, to increase the refractive index of the cladding n2.
One can now notice that a contradictory situation has been generated as to
whether the cladding should be removed for high NA or to use a cladding of large
refractive index value for higher bandwidth? The answer to this query is purely
application specific.
That means if an optical fiber is used as a sensor (say), where lowest possible light
has to be accepted, we use fiber with low n2 values.
When the optical fiber is used for data communication, fibers with high values of n2
are used. For practical communication purposes the value of (n1 – n2) is made of
the order of about 10-3 to 10-4. If the cladding is removed, the value of n2 becomes 1
and the value of the above difference becomes about 0.5. The bandwidth
corresponding to this value of n1-n2 is of the order of few Kilohertz, which is far
worse than that of a normal twisted pair of wires.
PHASE-FRONT (WAVE-FRONT) BASED STUDY OF TIR
Let us now have a study of the phenomenon of total internal reflection at the
core-cladding interface on a backdrop of the wave-fronts of the incident and the
reflected light.
The red and green colored dotted lines represent the wave fronts of the light
rays which are perpendicular to their direction of propagation.
Figure 2.14: Total Internal Reflection of Light inside a fiber core.
Thus, when two similar colored wave-fronts meet, they interfere
constructively and dissimilar colored wave-fronts interfere
destructively.
In the core, the interference between the incident and the reflected
wave-fronts constitutes a standing wave pattern of varying light
intensity with discrete maxima and minima in a direction normal to
the core-cladding interface.
Total internal reflection is also accompanied by an abrupt phase
change between the incident and the reflected rays at the corecladding boundary. This phase change depends on the angle of
incidence of the incident ray at the core-cladding boundary, the
refractive index or the core and cladding and various other
parameters.
If we refer to the electromagnetic wave theory of light, it shows that
at total internal reflection, the light intensity inside the cladding is not
completely zero.
Instead, there exist some decaying fields in the cladding, which do
not carry any power but support the total internal reflection
phenomenon
These fields are called as evanescent fields.
The importance of these evanescent fields in the TIR is that
even the slightest disturbance to these fields in the cladding
could lead to the failure of the TIR at the core-cladding
boundary accompanied by leakage of optical power to the
cladding.
This is one of the instances when the ray-model of light
becomes inadequate in explaining the phenomena exhibited
by light.
The evanescent fields are decaying fields,
Larger the value of the angle of incidence of the incident ray
at the core-cladding boundary, sharper is the decay of the
evanescent fields.
Thus there must me a sufficient thickness of cladding
provided for these evanescent fields to be accommodated so
that they decay to a negligibly small value in the cladding and
cannot be disturbed by external sources.
Figure 2.15 below shows two parallel rays that are launched into
an optical fiber and they propagate as shown.
The dotted lines represent the wave-fronts of the rays.
Hence for a sustained constructive interference, the distance
between these two phase-fronts must be multiples of 2π
 Mathematically,
d
s1  AB 
sin 
(2.7)
Figure 2.15: Propagation of Light rays in an Optical fiber
d
cos 2   sin 2  
s 2  CD  AE  AB sin( / 2  2 ) 
sin 
(2.8)
If δ is the phase change undergone in each TIR of Ray 1, interfere constructively if
phase difference is 2π x integer
For a sustained constructive interference, a phase difference must have of either 0
or integral multiples of 2π. That is, for an integer m (=0,1,2,3,…) the following
condition must be satisfied:
(2.9)
2n1
( s1  s 2 )  2  2m

2n1 d sin 
   m

(2.10)
The significance of the equation 2.10 is that only those rays, which are
incident on the tip of the fiber at angles such that their angle of refraction in
the core satisfies equation (2.10), can successfully travel along the fiber. If
we concentrate on equation (2.10), we find that since ‘m’ can take only
discrete integral values, the value of angle θ is also discrete.
This suggests that there are only some discrete launching angles within
the acceptance cone (N.A. cone) for which the rays can propagate inside
the fiber core.
Thus the condition that the launching angle of the incident ray should be
within the acceptance cone is necessary but not sufficient.
 Any ray that is not launched at these discrete angles will not propagate
inside the optical fiber.
This discretization in the values of launching angles lead to formation of
what are called as modes in an optical fiber, which are nothing but different
patterns of light intensity distribution around the axis of the core.
The Wave Model of Light
Wave-Model of light treats light as a transverse electromagnetic wave. Then the
propagation of light inside an optical fiber is explained in terms of the propagation of
an electromagnetic wave inside a bound medium like the optical fiber which is a
cylindrical dielectric waveguide.
Fig. 3.4: TEM nature of Light
Unguided electromagnetic waves in free space, can be described as a
superposition of plane waves; these can be described as TEM modes .
However in any sort of waveguide where boundary conditions are imposed by a
physical structure, a wave of a particular frequency can be described in terms of
a transverse mode (or superposition of such modes). These modes generally
follow different propagation constants.
Modes in waveguides can be further classified as follows:
Transverse electromagnetic (TEM) modes: neither electric nor magnetic field
in the direction of propagation (Ez=0 and Hz=0),..
Transverse electric (TE) modes: no electric field in the direction of
propagation. ( Ez=0),These are sometimes called H modes because there is
only a magnetic field along the direction of propagation (H is the conventional
symbol for magnetic field).
Transverse magnetic (TM) modes: no magnetic field in the direction of
propagation ( Hz=0),. These are sometimes called E modes because there is
only an electric field along the direction of propagation.
Hybrid modes: non-zero electric and magnetic fields in the direction of
propagation (Ez≠0 and Hz≠0),.
Different modes
Any ray of light which is launched within the Numerical Aperture Cone
(Acceptance cone) of an optical fiber propagates through the optical fiber core if
and only if its launching angle is such that the angle of refraction (θ) of its
refracted ray, in the core, satisfies the following phase condition given by below
equation. It should be also obvious that, whenever we talk of a single launched
ray, there exist an annular ring of similarly launched rays making the same
launching angle with the fiber axis.
2n1d sin 

   m
(m=0,1,2,3,…)
Figure 3.1: Annular rings of different modes.
(3.1)
The number of different values of ‘m’ signifies the number of different
possible launching angles which can successfully propagate in the optical
fiber core.
There may be N possible modes of propagation for which the rays
successfully travel along the fiber creating unique light intensity patterns
around the axis of the core.
The ray that is launched along the axis of the fiber propagates without any
phase condition requirement to be satisfied and corresponds to the first
mode of propagation, also called as the zero order mode ( m=0) of
propagation.
Any ray that is launched outside this cone does not propagate along the
fiber although it might correspond to a particular mode. This is shown in the
figure above by the ray AO.
The optical fiber too is selective in accepting only those rays which satisfy
the basic phase conditions and the other rays are rejected by the fiber
although they may lie within the acceptance cone of the fiber.
Thus there are only a finite number of modes that are allowed in an optical
fiber and the other modes are rejected. This leads to a further decrease in
the light accepting efficiency of fiber.
Treating light a transverse electromagnetic wave, we find that when meridional rays
propagate along the fiber, their electric and magnetic fields of all the rays superimpose
to result in electric and magnetic field distribution which may be either transverse electric
(TEx) or transverse magnetic (TMx) in nature.
The subscript ‘x’ denotes the definite number of maxima and minima in the resultant
light intensity pattern.
The propagation of skew rays, on the other hand, results in a particularly special form
of modes which are neither TE nor TM in nature and are called as Hybrid modes.
When we refer to modal propagation in dielectric waveguides, we find that unlike
metallic waveguides, there is a special set of modes that exists in a dielectric waveguide
in addition to TE and TM modes. This set of modes is called as hybrid mode. The optical
fiber is actually a cylindrical dielectric waveguide and so it can exhibit hybrid modes as
well. Rigorous analysis shows that hybrid mode is in fact the lowest order mode that can
propagate in an optical fiber. Since hybrid mode is the lowest order mode, it can be
analytically shown that the mode of the ray that propagates in the fiber along the axis is
hybrid in nature.
The different types of modes that propagate inside an optical fiber which may be
TEx, TMx, or hybrid in nature.
Figure 3.2 below shows different intensity patterns created by superposition of
the wave-fronts of all the light rays for Transverse Electric modes that propagate in
an optical fiber.
The fields that are shown in the cladding region are actually the evanescent
fields that exist in the cladding owing to the boundary condition requirement at the
core-cladding interface.
Figure 3.2: Different TE modes in an optical fiber
For very low launching angles with respect to the axis of the fiber, the intensity
pattern created is the one which is shown by TE0 in the above figure. There
exists a maximum intensity region around the axis of the core and as we move
towards the periphery of the core the fields start to decay. These fields eventually
decay down to negligibly low value in the cladding as shown in the figure.
If the launching angle is increased further, we get the intensity patterns as that
shown for TE1 and TE2 in the above figure.
The subscript of TE in fact indicates the number of destructive interferences in
the pattern where the field intensity crosses the zero level, or in other words,
creates a optically dark area.
So, for TE0 we have no dark area, for TE1 we have one dark area at the axis,
for TE2 we have two dark areas and so on.
This situation is well obvious from the above figure which shows the number of
times the field intensity pattern crosses the zero level corresponding to the
subscript of TE.
This subscript is also termed as the index of the mode. As we further increase
the launching angle with respect to the axis, more zeros are crossed and we get
the higher indices of the mode. The above discussion is also true for TM mode as
well and hence these modes are not referred to separately.