Download i THE NUMERICAL APERTURE

REF: Optics by Ajoy Ghatak
Chapter -27
Optical fiber vs Copper
Fiber has these advantages compared with metal wires
• Bandwidth – more data per second
• Longer distance
• Faster
• Lower loss
• Immunity to crosstalk
• Special applications like medical imaging and other
sensors are only possible with fiber because they
use light directly
Disadvantage
• Optical fiber is more expensive per meter than copper
• Optical fiber can not be join together as easily as copper cable.
• requires training , expensive measurement equipment
TOTAL INTERNAL REFLECTION
(a) For a ray incident on a denser medium (n2 > n1 ), the
angle of refraction is less than the angle of incidence.
(a) For a ray incident on a rarer medium (n2 < n1 ), the
angle of refraction is greater than the angle of incidence.
ϕ1
 n2 

n
 1
1  c  sin 1 
(a) The angle of incidence, for which the angle of refraction is 90o,
is known as the critical angle and is denoted by c.
the angle of refraction 2 = 90o. When the angle of incidence exceeds the critical angle
(i.e., when 1 > c), there is no refracted ray and known as total internal reflection.
THE OPTICAL FIBER
A (glass) fiber consists of a cylindrical central
core cladded by a material of slightly lower
refractive index.
Refractive index distribution for a step-index
fiber.
 n1 0<r<a
n
r>a
 n2
where n1 and n2 (< n1) represent, respectively,
the refractive indices of core and cladding and a
represents the radius of the core.
Light rays incident on the core-cladding interface at
an angle greater than the critical angle are trapped
inside the core of the fiber.
critical incident angle (ϕC)
critical propagation angle (θC)
for a ray entering the fiber, if the angle of
incidence (at the core-cladding interface) is
greater than the critical angle c, then the
ray will undergo TIR at that interface.
 n2 
  c  sin  
 n1 
1
Or  should be less than c:
(θC=90- ϕC)
 n2 
   c  cos  
 n1 
1
• all the energy in the ray is reflected back to the core.
• The ray then crosses to the other side of the core
and will meet the cladding on the other side at an
angle which again causes TIR.
• The ray is then reflected back across the previous
side of the core again.
• Through multiple TIR, ray guided from one end to
another end of fiber with out loss
Types of optical fibers:
1.
Step-index fibre
depending on refractive index:
2. Graded or variable index fiber
depending on modes:
Modes in fibre
all rays with propagation angle less than αc will propagate through the fibre.
• no. Of possible ways for a ray propagating in fibre.
• depends upon αc (n1 & n2 )
• increases as the RI difference increases.
• lower modes are having angles much less than αc .
• higher order modes travel a larger path length and hence lag behind the
lower order modes and produce distortion called intermodal distortion.
• number of modes is determined by V parameter
• V number depends on numerical aperture, source wavelength,
and diameter of core
Numerical Aperture
The numerical aperture of the fiber is closely related to the critical angle and is
used in the specification for optical fiber and the components
The quantity sin im is known as the numerical aperture (NA) of
the fiber and is a measure of light-gathering power of fiber.
• if a cone of light is incident on one end of the fiber, it will be guided
through it provided the semi angle of the cone is less than im
(Acceptance Angle)
•
The angle of acceptance is twice that given by the numerical aperture
THE NUMERICAL APERTURE
a ray incident on the entrance aperture of the fiber, making an angle i with
the axis. Assuming the outside medium to have a refractive index n0
sin i n1

sin  n0
ray has to suffer total internal reflection.
z
n2
sin  ( cos ) 
n1
n
sin i  1
n0
 n2 
1 

n
 1
2
or,
sin i 
n12  n22
no2
THE NUMERICAL APERTURE
a ray incident on the entrance aperture of the fiber, making an angle i with
the axis. Assuming the outside medium to have a refractive index n0
sin i n1

sin  n0
ray has to suffer total internal reflection.
z
n2
sin  ( cos ) 
n1
n
sin i  1
n0
 n2 
1 

n
 1
2
or,
n12  n22
no2
sin i 
If the outside medium is air, i.e., n0 = 1; and therefore the maximum value of
sin i for a ray to be guided is given by
N . A.  sin im  n  n
2
1
2
2
Numerical Aperture for graded Index Fiber
In the graded index fiber , the numerical aperture is a function of
position across the core.
(n  n )  r  x
1
2
nr  n
1
x
NA  n (r )  n
2
2
2
r
NA(r  0) 1-   for r  a
a
=0 for r>a
1 2
n
1
 
a
=
x :the refractive index profile variation
NA(r=0): numerical aperture at the centre of the fibre core.
NA(0)= n 2 (0)  n22  n12  n22
For a graded index fiber numerical aperture decrease from axial
numerical aperture NA(r=0) to zero as r increases from zero to
core radius ' a '.
Numerical Aperture for graded Index Fiber
In the graded index fiber , the numerical aperture is a function of
position across the core.
(n  n )  r  x
1
2
nr  n
1
x
NA  n (r )  n
2
2
2
r
NA(r  0) 1-   for r  a
a
=0 for r>a
1 2
n
1
 
a
=
x :the refractive index profile variation
NA(r=0): numerical aperture at the centre of the fibre core.
NA(0)= n 2 (0)  n22  n12  n22
For a graded index fiber numerical aperture decrease from axial
numerical aperture NA(r=0) to zero as r increases from zero to
core radius ' a '.
Number of Modes and Cut-off Parameters of Fibers
The number of modes
cut-off parameter (normalized cut off frequency)
d 2 2
 d d 2 2
d
n1  n2 or,
V V  n1( NA
 n)2 or, V  ( NA)
V number V 
o
o o
o
The approximate total number of modes
V2
Number of Modes (N) 
2
If the external medium around the fiber has a refractive index no then
d
V
no ( NA)
o
n12  n22
as NA=
no
Attenuation and Signal Losses in Optical Fibers
The reduction in amplitude (or power) and intensity of a signal
as it is guided through an optical fibre is called attenuation.
loss depends on the wavelength of the light and on the propagating material.
For silica glass, the shorter wavelengths are attenuated the most
The loss /decrease in signal strength along a fibre are due to
•bending losses
•scattering losses
•absorption losses (due to material or impurities)
•connector loss, splice loss, loss at terminals etc.
Attenuation and Signal Losses in Optical Fibers
•Bending losses:
Macro-bending losses
Micro-bending loss
macro-bending loss is caused by micro-bending loss is caused by
the curvature of entire fiber axis micro deformations of the fiber axis.
•Absorption losses:
•Molecules/atoms/electrons vibrates at particular frequency
•Source frequency matches with these frequency (IR region)
•Selective absorption can happen by these molecules of material
•Absorption losses increases with the impurities in material of fiber or
wapor/water/gas in the core of fiber
Scattering losses
(1) Due to Rough and irregular surface:
even at the molecular level of the glass, can reflect rays in many random directions.
(2) Due to irregular refractive index:
small change in core’s refractive index as an optical obstacle for ray which
change the direction of the original beam.
Attenuation and Signal Losses in Optical Fibers
•connector loss, splice loss, loss at terminals
• In fiber-optic systems, the losses from splices and connections
can be more than in the cable itself
– Axial or angular misalignment
– Air gaps between the fibers
– Rough surfaces at the ends of fibers
Attenuation coefficient
Attenuation losses in optical fibers are generally measured in terms of the decibel  dB  .
Due to attenuation, the power output ( Pout ) at the end of 1 km of optical fibre drops to
some fraction  say k  of the input power ( Pin ) , that is,
Pout  k Pin
After 2 km
Pout  k 2 Pin
Similarly after L km
Pout =k L Pin
or,
Pout
 kL
Pin
Taking log of both sides and then multiply by 10 gives power loss in dB as
P
Power Loss (dB) =10 log out  10log k L  L10log k   L
Pin
where  is the attenuation coefficient of the fiber in dB/km.
 Pout
10
 = log 
L
 Pin

 dB / km

 Pout 
10
To indicate loss we introduce negative sign in the expression    log 
 dB / km
L
 Pin 
Attenuation coefficient
Attenuation losses in optical fibers are generally measured in terms of the decibel  dB  .
Due to attenuation, the power output ( Pout ) at the end of 1 km of optical fibre drops to
some fraction  say k  of the input power ( Pin ) , that is,
Pout  k Pin
After 2 km
Pout  k 2 Pin
Similarly after L km
Pout =k L Pin
or,
Pout
 kL
Pin
Taking log of both sides and then multiply by 10 gives power loss in dB as
P
Power Loss (dB) =10 log out  10log k L  L10log k   L
Pin
where  is the attenuation coefficient of the fiber in dB/km.
 Pout
10
 = log 
L
 Pin

 dB / km

 Pout 
10
To indicate loss we introduce negative sign in the expression    log 
 dB / km
L
 Pin 
Attenuation coefficient
Attenuation losses in optical fibers are generally measured in terms of the decibel  dB  .
Due to attenuation, the power output ( Pout ) at the end of 1 km of optical fibre drops to
some fraction  say k  of the input power ( Pin ) , that is,
Pout  k Pin
After 2 km
Pout  k 2 Pin
Similarly after L km
Pout =k L Pin
or,
Pout
 kL
Pin
Taking log of both sides and then multiply by 10 gives power loss in dB as
P
Power Loss (dB) =10 log out  10log k L  L10log k   L
Pin
where  is the attenuation coefficient of the fiber in dB/km.
 Pout
10
 = log 
L
 Pin

 dB / km

 Pout 
10
To indicate loss we introduce negative sign in the expression    log 
 dB / km
L
 Pin 
Attenuation coefficient
Attenuation losses in optical fibers are generally measured in terms of the decibel  dB  .
Due to attenuation, the power output ( Pout ) at the end of 1 km of optical fibre drops to
some fraction  say k  of the input power ( Pin ) , that is,
Pout  k Pin
After 2 km
Pout  k 2 Pin
Similarly after L km
Pout =k L Pin
or,
Pout
 kL
Pin
Taking log of both sides and then multiply by 10 gives power loss in dB as
P
Power Loss (dB) =10 log out  10log k L  L10log k   L
Pin
where  is the attenuation coefficient of the fiber in dB/km.
 Pout
10
 = log 
L
 Pin

 dB / km

 Pout 
10
To indicate loss we introduce negative sign in the expression    log 
 dB / km
L
 Pin 
Attenuation coefficient
Attenuation losses in optical fibers are generally measured in terms of the decibel  dB  .
Due to attenuation, the power output ( Pout ) at the end of 1 km of optical fibre drops to
some fraction  say k  of the input power ( Pin ) , that is,
Pout  k Pin
After 2 km
Pout  k 2 Pin
Similarly after L km
Pout =k L Pin
Taking log of both sides and then multiply by 10 gives power loss in dB as
P
Power Loss (dB) =10 log out  10log k L  L10log k   L
Pin
where  is the attenuation coefficient of the fiber in dB/km.
 Pout
10
 = log 
L
 Pin
or,
Pout
 kL
Pin
Pout  Pin10
L
10

 dB / km

 Pout 
10
To indicate loss we introduce negative sign in the expression    log 
 dB / km
L
 Pin 
Numerical : Compute the maximum value of  ( relative
refractive index) and n2 (cladding) of a single mode
fibre of core diameter 10 m and core refractive
index 1.5. The fibre is coupled to a light source with a
of 1.3m. V cut-off for single mode propagation is
2.405. Also calculate the acceptance angle.
n1  n2
relative difference of index  
n1
=0.0022
n2 (cladding)=1.497
Acceptance angle = 5.71deg