# Chapter 23: Lasers, Fibre Optics and Holography¶

## Example 1, Page 662¶

In [2]:
#Variable declaration
y=630*10**(-9)#y=emitted wavelength in meters
c=3*10**8#c=velocity of light in free space in m/s

#Calculations&Results
print "The frequency of the emitted radiation is %.2e Hz"%v
h=6.62*10**(-34)#h=Planck's constant
P=1*10**(-3)#P=output power of gas laser(given)
n=P/(h*v)
print "The number of photons emitted per second is= %.2e s^-1"%n

The frequency of the emitted radiation is 4.76e+14 Hz
The number of photons emitted per second is= 3.17e+15 s^-1


## Example 2, Page 663¶

In [3]:
#Variable declaration
V=500#V=bandwidth of a He-Ne laser in Hz

#Calculations&Results
t=1./V#t=coherence time
print "The coherence time is = %.f ms"%(t*(10**3))
c=3*10**8#c=velocity of light in m/s
Lc=c/V#Lc=longitudinal coherence length
print "The longitudinal coherence length is=%.f km"%(Lc/1000)

The coherence time is = 2 ms
The longitudinal coherence length is=600 km


## Example 3, Page 663¶

In [4]:
#Variable declaration
#To obtain interference fringes of good visibility the path difference for the central fringe must be an integral multiple of each of the 2 wavelengths.
#2*d=(n1*y1)=(n2*y2)where y1 & y2 are 2 wave-lengths and d=path difference and n1 and n2 are 2 integers
#(2*d)*((1/y2)-(1/y1))=(n2-n1)=m where m is another integer
#Now m=(-2*d*Y)/(y^2)=(2*d*V)/(v*y)=(2*d*V)/c=(2*d)/Lc
Lc=600.#Lc=coherence length in km

#Calculations
d=(Lc/2)#d=minimum difference between the 2 arms of the Michelson interferometer

print "The minimum difference between the two arms of the Michelson interferometer is=%.f km"%d

The minimum difference between the two arms of the Michelson interferometer is=300 km


## Example 4, Page 663¶

In [8]:
import math

#Variable declaration
h=6.62*10**(-34)#h=Planck's constant
v=3*10**8#v=velocity of light(as normal optical source is mentioned) in m/s
kB=1.38*10**-23#kB=Boltzmann's constant
T=1000#T=temperature in Kelvin
w=6000#w=wavelength in Armstrong

#Calculations&Results
R=(math.exp((h*v)/(w*(10**-10)*kB*T)))-1#R=the ratio of the number of spontaneous to stimulated transitions
print "R=%.1e"%R
if (R>1):
print "As the ratio of the number of spontaneous to stimulated transitions (R) is >> 1 the emission is predominantly",\
"\ndue to spontaneous transitions and is thus incoherent"

R=2.6e+10
As the ratio of the number of spontaneous to stimulated transitions (R) is >> 1 the emission is predominantly
due to spontaneous transitions and is thus incoherent


## Example 5, Page 663¶

In [10]:
#Variable declaration
u=8./(10**14)#u=(V/v)=the short term frequency stability of a He-Ne gas laser
#v=c/y where c=velocity of light in vacuum  and y=wavelength
c=3*10**8#c=velocity of light in m/s
y=1153*10**(-9)#y=emitted wavelength in meters

#Calculations&Results
V=(u*c)/y
t=1./V#t=coherence time
print "The coherence time is = %.f ms"%(t*(10**3))
Lc=c/V#Lc=coherence length
print "The coherence length is=%.2e m"%Lc

The coherence time is = 48 ms
The coherence length is=1.44e+07 m


## Example 6, Page 664¶

In [11]:
#Variable declaration
#y0=vacuum wavelength for the frequency v
#c=(v*y0)
#The deviation in the wavelength is Y0=(c*V)/(v**2)
#Y0=((y0**2)*V)/c
#V being spread in frequency over the linewidth.
#V=(1/tc)
c=3*(10**8)#c=velocity of light in m/s
tc=10**(-8)#tc=coherence time in seconds
y0=650*(10**(-9))#y0=vacuum wavelength in m

#Calculations&Results
Y0=(y0**2)/(c*tc)
print "Line width is =%.1e nm"%(Y0/(10**-9))#Y0 is converted in terms of nm
Lc=c*tc#Lc=coherence length
print "The coherence length Lc is=%.f m"%Lc

Line width is =1.4e-04 nm
The coherence length Lc is=3 m


## Example 7, Page 664¶

In [13]:
import math

#Variable declaration
f=10#f=focal length in cm

#Calculations&Results
D=f*o#D=diameter of the image
print "The image radius is = %.1e cm"%r
a=math.pi*(r**2)#a=cross sectional area of the image in cm**2
P=10*10**-3#P=power in Watts
PD=P/a#PD=power density
print "Power density is = %.1e W/cm**2"%PD
y=6000*10**-8#y=wavelength in cm
d=y/o#d=coherent width
print "The lateral coherent width is = %.1f cm"%d

The image radius is = 2.5e-04 cm
Power density is = 5.1e+04 W/cm**2
The lateral coherent width is = 1.2 cm


## Example 8, Page 664¶

In [14]:
#Variable declaration
h=6.62*10**-34#h=Planck's constant
c=3*10**8#c=velocity of light in vacuum in m/s
y=632.8*10**-9#y=emitted wavelength in m

#Calculations
E=(h*c)/y#E=emitted photon energy in Joules
e=15.2*10**-19#e=energy of 2p level in Joules
P=E+e#P=Pumping energy required for transition from 3s to 2p level in a He-Ne laser

#Result
print "The desired pumping energy is = %.1f eV"%(P/(1.6*10**-19))

The desired pumping energy is = 11.5 eV


## Example 9, Page 665¶

In [15]:
import math

#Variable declaration
h=6.62*10**-34#h=Planck's constant
v=2.4*10**15#v=frequency of emitted radiation in Hz
c=3*10**8#c=velocity of light in vacuum in m/s

#Calculations
A21=1/(1.66*10**-8)#A21=mean spontaneous life time
B21=((c**3)*A21)/(8*math.pi*h*(v**3))#B21=probability of stimulated emission

#Result
print "The desired probability is = %.2e m**3/(J.s**2)"%B21

The desired probability is = 7.07e+18 m**3/(J.s**2)


## Example 10, Page 665¶

In [18]:
import math

#Variable declaration
u1=1.55#u1=refractive index of the core of the fibre

#Calculations&Results
oa=math.asin(math.sqrt((u1**2)-(u2**2)))#oa=acceptance angle
print "The acceptance angle is = %.f degrees"%math.degrees(oa)
NA=math.sin(oa)#NA=numerical aperture
print "NA=%.4f"%NA
oc=math.asin(u2/u1)#oc=critical angle
print "Critical angle=%.f degrees"%math.degrees(oc)
d=50*10**-6#d=core diameter in meters
x=d*math.tan(oc)#x=axial distance traversed by the ray between two consecutive reflections
n=1/x#n=number of reflections per metre
print "The number of reflections per metre is = %.f"%n

The acceptance angle is = 23 degrees
NA=0.3905
Critical angle=75 degrees
The number of reflections per metre is = 5207


## Example 11, Page 665¶

In [21]:
import math

#Variable declaration
ro = 0.2  #mm
u1 = 1.52

#Calculations
n_ro = 1.52-(2*ro**2)
NA = math.sqrt(u1**2-n_ro**2)
theta_A = math.degrees(math.asin(NA))

#Results
print "Numerical Arperture = %.4f"%NA
print "Acceptance angle = %.2f degrees"%theta_A

Numerical Arperture = 0.4866
Acceptance angle = 29.12 degrees