Chapter2 : Optical fiber & its types

Exa 2.1 : page 35

In [25]:
#Given data :
n1=1.40 #refractive index
delta=1.0 #relative refractive index difference in %
#Formula : n2/n1=1-delta
n2=n1*(1-delta/100) #refractive index(unitless)
print "Refractive index of cladding = %0.3f " %n2 

Refractive index of cladding = 1.386

Exa 2.2 : page 35

In [26]:
from math import asin, pi
#Given data :
n1=1.50 #refractive index
n2=1.47 #refractive index
#Formula : sin(theta_C)=n2/n1 
theta_c=asin(n2/n1)*180/pi #in degree
print "Critical Angle at core cladding interface = %0.1f Degree " %theta_c 

Critical Angle at core cladding interface = 78.5 Degree

Exa 2.3 : page 35

In [27]:
from math import sqrt
#Given data :
delta=1.0 #relative refractive index difference in %
n1=1.50 #refractive index
#Formula : NA=n1*sqrt(2*delta) 
NA=n1*sqrt(2*delta/100) 
print "Numerical Aperture of the fibre = %0.3f " %NA 

Numerical Aperture of the fibre = 0.212

Exa 2.4 : page 35

In [28]:
from math import sqrt, pi
#Given data :
delta=1 #relative refractive index difference in %
n1=1.55 #refractive index
n2=1.51 #refractive index
#Formula : NA=sqrt(n1**2-n2**2) 
NA=sqrt(n1**2-n2**2)
print "Numerical Aperture of the fibre = %0.2f " %NA 
#Formula : NA=sin(fi_o).....(max)
fi_o_max=asin(NA)*180/pi #in Degree
print "Acceptance angle = %0.1f degree " %fi_o_max 

Numerical Aperture of the fibre = 0.35 
Acceptance angle = 20.5 degree

Exa 2.5 : page 36

In [29]:
from math import asin, pi
#Given data :
NA=0.40 #Unitless
n1=1.50 #refractive index
delta=1 #relative refractive index difference in %
#Part (a) :
#Formula : NA=sin(fi_o).....(max)
fi_o_max=asin(NA)*180/pi #in Degree
print "(a) Acceptance angle = %0.1f degree " %fi_o_max 
#Part (b) :
#Formula : n2/n1=1-delta
n2=n1*(1-delta/100) #refractive index(unitless)
#Formula : sin(theta_C)=n2/n1 
theta_c=asin((n2/n1))*180/pi #in degree
print "(b) Critical Angle at core cladding interface = %0.1f Degree  " %theta_c 

(a) Acceptance angle = 23.6 degree 
(b) Critical Angle at core cladding interface = 81.9 Degree

Exa 2.6 : page 36

In [30]:
from math import sin, pi, sqrt
#Given data :
v=2*10**8 #in m/s
fi_c=60 #in degree
#Part (a)
#Formula : v=c/n 
c=3*10**8 #in m/s
n1=c/v #unitless
print "(a) Refractive index of core = %0.2f " %n1 
#Formula : sin(fi_c)=n2/n1 
n2=n1*sin(fi_c*pi/180) #unitless
print "  & Refractive index of cladding = %0.2f " %n2 
#Part (b)
NA=sqrt(n1**2-n2**2) #Unitless
print "(b) Numerical Aperture = %0.2f " %NA 

(a) Refractive index of core = 1.50 
  & Refractive index of cladding = 1.30 
(b) Numerical Aperture = 0.75

Exa 2.7 : page 37

In [31]:
from math import pi
#Given data :
d=30 #in um
a=d/2 #in um
lamda=0.80 #in um
NA=0.74 #Unitless
V=2*pi*a*NA/lamda #V number
print "V number = %0.1f " %V 

V number = 87.2

Exa 2.8 : page 37

In [32]:
#Given data :
d=60 #in um
a=d/2 #in um
delta=1 #relative refractive index difference in %
lamda=0.80 #in um
n1=1.5 #Unitless
#Part (a)
#Formula : v=2*pi*a*n1*NA/lambda 
#NA=sqrt(2*delta)
v=2*pi*a*n1*sqrt(2*delta/100)/lamda #Normalized frequency
print "(a) Normalized frequency for the fiber = %0.2f " %v 
#Part (b)
print "(b) Only the modes with cut-off v numbers below this value will propagate." 
N=v**2/2 #No. of modes supported
print "    Number of modes supported = %0.f " %round(N) 
#Note : Answer in the book is wrong.

(a) Normalized frequency for the fiber = 49.98 
(b) Only the modes with cut-off v numbers below this value will propagate.
    Number of modes supported = 1249

Exa 2.9 : page 38

In [33]:
from math import ceil
#Given data :
NA=0.16 #Unitless
d=30 #in um
a=d/2 #in um
n1=1.50 #Unitless
lamda=0.9 #in um
v=2*pi*a*NA/lamda #V number
N=v**2/2 #No. of modes propagate
print "Number of guided modes in the fibre = %0.f " %ceil(N) 

Number of guided modes in the fibre = 141

Exa 2.10 : page 38

In [34]:
#Given data :
fi_o=22 #in Degree
delta=3 #relative refractive index difference in %
#Part (a) :
#Formula : NA=sin(fi_o).....(max)
NA=sin(fi_o*pi/180) #Numerical Aperture(Unitless)
print "(a) Numerical Aperture = %0.3f " %NA 
#Part (b) :
#Formula : n2/n1=1-delta
#Let say, n2/n1=n2byn1
n2byn1=(1-delta/100) #refractive index(unitless)
#Formula : sin(fi_C)=n2/n1 
fi_c=asin(n2byn1)*180/pi #in degree
print "(b) Critical Angle at core cladding interface = %0.2f Degree " %fi_c

(a) Numerical Aperture = 0.375 
(b) Critical Angle at core cladding interface = 75.93 Degree

Exa 2.11 : page 38

In [35]:
#Given data :
delta=0.45 #relative refractive index difference in %
fi_o=0.115 #in Radian
c=3*10**8 #speed of light in m/s
#Formula : NA=sin(fi_o).....(max)
NA=sin(fi_o) #Numerical Aperture(Unitless)
#Formula : NA=n1*sqrt(2*delta)
n1=NA/sqrt(2*delta/100) #unitless
#Formula : n1=c/v 
v=c/n1 #in m/s
print "Speed of light in fibre core is %0.2e "%round(v,3),"m/s" 

Speed of light in fibre core is 2.48e+08  m/s

Exa 2.12 : page 39

In [36]:
from __future__ import division
#Given data :
n1=1.5 #Unitless
delta=1 #relative refractive index difference in %
lamda=1.3 #in um
N=1100 #No. of modes
#Formula : v=2*pi*a*n1*NA/lamda 
#NA=sqrt(2*delta)
#v=sqrt(2*N)
a=(sqrt(2*N)*lamda)/(2*pi*n1*sqrt(2*delta/100)) #Normalized frequency
print "Diameter of the fiber core = %0.1f micro meter" %(2*a)

Diameter of the fiber core = 91.5 micro meter

Exa 2.13 : page 40

In [37]:
#Given data :
n1=1.52 #unitless
fi_o=8 #in Degree
#Formula : sin(fi_o)=n1*sqrt(2*delta)
delta=(sin(fi_o*pi/180)/n1)**2/2 #Relative refractive index
print "The value of relative refractive index difference =",round(delta*100,2),"%" 

The value of relative refractive index difference = 0.42 %

Exa 2.14 : page 40

In [38]:
#Given data :
N=700 #No. of modes
d=30 #in um
a=d/2 #in um
NA=0.62 #Numerical Aperture
#Formula : v=2*sqrt(N) and v=2*pi*a*NA/lamda
lamda=2*pi*a*NA/(2*sqrt(N)) #in um
print "Wavelength of light propagating in fibre = %0.2f micro meter" %lamda 

Wavelength of light propagating in fibre = 1.10 micro meter

Exa 2.15 : page 40

In [39]:
#Given data :
n1=1.5 #unitless
alfa=2 #characteristic index profile
d=40 #in um
a=d/2 #in um
#Part (a) :
lamda=1.3 #in um
delta=1 
#Formula : v=2*pi*a*NA/lamda=2*pi*a*(n1*sqrt(2*delta))/lamda
v=2*pi*a*(n1*sqrt(2*delta/100))/lamda #Unitless
print "(a) Normalized Frequency for single mode transmission = %0.1f " %v 
#Part (b) :
#Formula : N=(alfa/alfa+2)*(v**2/2)
N=(alfa/(alfa+2))*(v**2/2) #No. of guided modes
print "(b) No. of guided modes propagating in the fibre = %d " %N 

(a) Normalized Frequency for single mode transmission = 20.5 
(b) No. of guided modes propagating in the fibre = 105

Exa 2.16 : page 41

In [40]:
#Given data :
d=60 #in um
a=d/2 #in um
NA=0.25 #Unitless
lamda=1.1 #in um
v=2*pi*a*NA/lamda #unitless
N=v**2/4 #No. of modes
print "Number of supported guided modes = %d" %N 

Number of supported guided modes = 458

Exa 2.17 :page 40

In [41]:
#Given data :
d=10 #in um
a=d/2 #in um
lamda_c=1.3 #in um
n1=1.55 #unitless
#Part (a)
#for single mode transmission cut-off wavelength is lamda_c=2*pi*a*n1*sqrt(2*delta)/2.405
delta=(lamda_c*2.405/(2*pi*a*n1))**2/2 #unitless
print "(a) Normalized refractive index difference = %0.5f %% " %delta 
#Part (b) 
#Formula : n2/n1=delta
n2=n1*(1-delta) 
print "(b) Refractive index of cladding glass = %0.3f " %n2 
#Part (c) :
fi_o=asin(n1*sqrt(2*delta))*180/pi #in degree
print "(c) Acceptance angle = %0.1f degree " %fi_o

(a) Normalized refractive index difference = 0.00206 % 
(b) Refractive index of cladding glass = 1.547 
(c) Acceptance angle = 5.7 degree

Exa 2.18 : page 42

In [42]:
#Given data :
d=7 #in um
a=d/2 #in um
n1=1.49 #unitless
delta=1 #relative refractive index difference in %
#Part (a)
#Formula : lamda_c=2*pi*a*n1*sqrt(2*delta)/2.405 
lamda_c=2*pi*a*n1*sqrt(2*delta/100)/2.405 #in um
print "(a) Shortest wavelength of the light = %0.2f micro meter " %lamda_c 
#Part (b) 
#Formula : delta=(1/2)*{2.405*lamda_c/(2*pi*a*n1)}**2
d=10 #in um
a=d/2 #in um
delta=(1/2)*(2.405*lamda_c/(2*pi*a*n1))**2 #unitless
print "(b) Maximum possible relative refractive index difference = %0.2f %% "  %(delta*100)

(a) Shortest wavelength of the light = 1.93 micro meter 
(b) Maximum possible relative refractive index difference = 0.49 %

Exa 2.19 : page 42

In [43]:
#Given data :
n1=1.49 #unitless
n2=1.48 #unitless
lamda_c=1.5 #in um
#Formula : a=2.405*lamda_c/(2*pi*sqrt(n1**2-n2**2))
a=2.405*lamda_c/(2*pi*sqrt(n1**2-n2**2)) #in um
print "Fibre core diameter = %0.2f micro meter" %(2*a)

Fibre core diameter = 6.66 micro meter

Exa 2.20 : page 43

In [44]:
#Given data :
N=742 #No. of guided modes(unitless)
n1=1.5 #unitlessnm
alfa=2 #characteristic index profile
NA=0.3 #unitless
d=70 #in um
a=d/2 #in um
alfa=2 #Graded index profile for parabolic
#Formula : N=(alfa/(alfa+2))/(v**2/2)
v=sqrt(N*((alfa+2)/alfa)*2) #Unitless
#Formula : v=2*pi*a*NA/lambda
lamda=2*pi*a*NA/v #in um
print "Wavelength of light propagating in fibre = %0.2f micro meter " %(lamda)
#Formula : lambvda_c=lambda=2*pi*a*NA/(2.405*(sqrt((alfa+2)/alfa)))
a=lamda*(2.405*(sqrt((alfa+2)/alfa)))/(2*pi*NA) #in um
print "Diameter of fibre = %0.2f micro meter " %(2*a) 
#Note : Answer in the book is not accurate.

Wavelength of light propagating in fibre = 1.21 micro meter 
Diameter of fibre = 4.37 micro meter

Exa 2.21 : page 44

In [45]:
#Given data :
n1=1.447 #unitless
n2=1.442 #unitless
lamda=1.3 #in um
d=7.2 #in um
a=d/2 #in um
#Formula : v=2*pi*a*sqrt(n1**2-n2**2)/lambda
v=2*pi*a*sqrt(n1**2-n2**2)/lamda #unitless
print "Value of v = %0.3f " %v

Value of v = 2.091

Exa 2.23 : page 44

In [46]:
#Given data :
alfa=1.9 
#characteristic index profile
#Formula : v=2.405*sqrt[(alfa+2)/alfa]
v=2.405*sqrt((alfa+2)/alfa) #unitless
print "Value of v = %0.2f " %v
#Note : Answer in the book is not accurate.

Value of v = 3.45

Exa 2.24 : page 45

In [47]:
from sympy import symbols, solve
a = symbols('a')
#Given data :
delta=1 #relative refractive index difference in %
n1=1.47 #unitless
lamda=1.5 #in um
alfa=2 #unitless
v1 = 2*pi*a*n1*sqrt(2*delta/100)/lamda  # eqn(1)
v2 = 2.405*sqrt((2+alfa)/alfa)  # eqn(2)
a = solve(v1-v2,a)  # in um
d = 2*a[0] # diameter in um
print "The diameter = %0.1f micro meter " %d

The diameter = 7.8 micro meter

Exa 2.25 : page 45

In [48]:
#Given data :
delta=1 #relative refractive index difference in %
n1=1.47 #unitless
lamda=1.5 #in um
alfa=2 #unitless
#Formula : v=2*pi*a*n1*sqrt(2*delta)/lambda
a=2.405*lamda/(2*pi*n1*sqrt(2*delta/100)) 
print "The diameter = %0.2f micro meter " %(2*a)

The diameter = 5.52 micro meter