Download Chapter 2 Optical fibers

Chapter 2 Optical fibers
Chapter 2 Optical fibers
Optical fiber losses have been reduced from about 1000 dB/km before 1970, to around 20 dB/km
in 1970, and 0.2 dB/km at 1.55-µm region in 1979.
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•
•
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Most optical fiber consist of a cylindrical piece of glass surrounded by an acrylic (made from
a thermoplastic resin) jacket. The glass consists of a core, of refractive index n1, and a
surrounding cladding, of refractive index n2, with n1 > n2. The index of the jacket is such that
any light reaching the cladding-jacket index is dissipated (or stripped) out of the cladding.
Light propagates down the fiber through the core at the speed, v ≅ 2 × 108 m/s, so that it takes
approximately 5 µs to travel down a length of 1 km of fiber.
The outside diameter (OD) of standard communications fiber is 125 micron (µm) while the
core diameter is 9 µm for single-mode fiber and 50 or 62.5 µm for multimode fiber. The OD of
the jacketed fiber is 250 µm.
The refractive index of the core is higher than that of the cladding, but by less than 1%.
What are the physical principles that describe light propagation in a fiber?
• Light is bound to the core by total internal reflection. The Geometrical Optics description is
valid when the core radius, a is much greater than the wavelength λ, i.e., a >> λ.
•
Wave-propagation theory (Maxwell’s Equations) must be used when a is of the same order as
λ.
What are some of the degradations affecting light propagation in an optical fiber?
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There are several mechanisms that affect the rate of attenuation of the optical signal. These
include Rayleigh scattering, micro-bending and macro-bending, modal attenuation and
several absorption mechanisms.
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Modal dispersion is a source of signal degradation that results in pulse broadening.
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Chromatic dispersion is another source of optical-signal degradation.
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Chapter 2 Optical fibers
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There are more that will be discussed when encountered.
Types of optical fibers:
Step-index fibers
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Graded-index fibers
Single-mode fibers
Multi-mode fibers
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Chapter 2 Optical fibers
Geometrical-Optics Description
Law of Refraction (Snell's Law): When a ray of light is refracted at an interface dividing two
uniform media, the transmitted ray remains within the plane of incidence and the sine of the angle
of refraction is directly proportional to the sine of the angle of incidence.
φ1 = φ '1
n1 sin φ1 = n2 sin φ2
Reflected and refracted rays for
the light incident at the
interface of two media
Critical Angle and Total Internal Reflection:
When light passes from a medium of larger refractive index
index
n1 into one of smaller refractive
n2 , the refracted ray is bent away from the normal. If the incident ray is at the critical angle
θ c , the angle of refraction is 900. The critical angle is determined from Snell's Law and is given
by
sinθ c = n2 / n1 .
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Chapter 2 Optical fibers
When the angle of incidence exceeds the critical angle, all the incident light is reflected back into
the medium from which it came. This is known as total internal reflection.
Example 1:
(a) Apply Snell’s law, we have
n1 sin( 90 0 − θ 1 ) = n2 sin( 90 0 − θ 2 )
The critical angle:
φ c = sin −1 (
n
1
) = sin −1 (
) = 40.50
n2
1.54
Step-index fibers
Based on the Snell’s Law of Refraction, we have the following relation at the fiber-air interface
n0 sin θ i = n1 sin θ r
where n0 and nr are refractive indices of the air and the fiber core.
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Chapter 2 Optical fibers
The refracted ray hit the core-cladding interface and is refracted again. To confine the light within
the fiber, the angle of incidence φ should be greater than the critical angle φ c , which is given
sin φ c = n2 / n1
n2 is the cladding index. Since such reflections occur throughout the fiber length, all rays
with φ > φ c remain confined to the fiber core. This is the basic mechanism behind light
where
confinement in optical fiber.
Numerical Aperture:
The maximum angle that the incident ray should make with the fiber axis to remain confined inside
the core:
n0 sin θ i = n1 cosφ c = (n12 − n22 )1 / 2
n0 sin θ i is known as numerical aperture (NA), which represents the light-gathering capacity of
an optical fiber.
For
n1 ≈ n2
NA = (n − n )
2
1
2 1 /2
2
= n1 (2 ∆ )
[
= (n1 + n 2 )(n1 − n 2 )
]
1/ 2

n − n2 
≈ 2n12 ( 1
)
n1


1/ 2
1/ 2
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Chapter 2 Optical fibers
∆ should be made as large as possible in order to couple maximum light into the fiber. However,
this type of fiber is not useful for high bit rate communications because of multipath dispersion or
modal dispersion.
Multipath dispersion: different rays travel along paths of different lengths, which leads to rays
disperse in time. A short pulse would broaden considerably as a result of different path lengths.
The estimation of pulse broadening:
∆T =
 L n12
n1 
L
=

∆
c  sinφc − L  c n2
It is the delay between the two rays taking the shortest and longest paths.
To be able to retrieve the information in case of dispersion, the broadening
than the allocated bit slot TB = 1 / B , that is, ∆T < TB , or ∆T ⋅ B < 1 .
∆T
should be less
So we have
∆T ⋅ B =

n1 
L
L n12
n2 c
 B =

∆B ⇒ BL < 2
c  sin φc − L 
c n2
n2 ∆
For example, an unclad fiber with n1 = 1.5 , n2 = 1 , then the BL < 0.4 (Mbps)-km. A cladded fiber
with ∆ = 2 ×10 −3 , BL < 100 (Mbps)-km. Such fiber can communicate data at 10 Mbps over
distance over 10 km, which may be suitable for Local Area Networks (LAN).
Example 2:
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Chapter 2 Optical fibers
Graded-Index Fibers
The refractive index of the core in graded-index fibers decreases gradually from its maximum
value n1 to its minimum value n2 at the core-cladding interface.
Most graded-index fibers can be modeled as
[
]
n 1 − ∆( ρ / a )α ;
n( ρ ) =  1
n1 [1 − ∆];
ρ<a
ρ ≥a
where a is the fiber core radius. α determines the index profile. A step index profile is
approached with a large α . A parabolic-index profile corresponds to α = 2 .
The multipath dispersion or intermodal dispersion is reduced with graded-index fibers. The reason
is that the path is longer for more oblique rays, but the ray velocity changes along the path because
of variations in the refractive index.
Geometrical optics can be used to show that a parabolic-index profile leads to nondispersive
pulse propagation within the paraxial approximation. The trajectory of a paraxial ray is obtained
by solving
∂ 2 ρ 1 dn
=
dz 2 n dρ
where ρ is the distance of the ray from the axis.
It can be proved that a parabolic-index fiber does not exhibit intermodal dispersion. The BL
product of graded index fibers is improved by nearly three orders of magnitude over the step-index
fibers.
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Chapter 2 Optical fibers
Example 3:
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