CHAPTER 15. LASER AND FIBER OPTICS
The laser is essentially an optical amplifier. The word laser is an acronym that
stands for light amplification by the stimulated emission of radiation.
15.1 Einstein’s theory
Consider atomic transition between two states, E1 (ground state) and E2 (excited
states). Stimulated absorption occurs whenever radiation containing photons of energy
h= E2E1 is incident on the matter. The spontaneous emission takes place whenever
atoms are in an excited state. No external radiation is required to initiate the emission.
The emitted photon has an energy of h= E2E1 but is emitted in a random direction. By
contrast, stimulated emission requires the presence of external radiation. When an
incident photon of resonant energy h= E2E1 passes by an atom in excite state, it
“stimulate” the atom to drop to the lower ground state. In the process, the atom releases a
photon of the same energy, direction, phase, and polarization as that of the photon passing
by. The net effect is two identical photons in the place of one, or an increase in the
intensity of the incident “beam”. It is precisely this process of stimulated emission that
makes possible the amplification of light in lasers.
Fig.15.1 Radiative processes that affect the
number of atoms at energy E1 and E2.
Spontaneous emission (A21):
N
 dN 2 
  A21 N 2   2



 dt  spont
(15.1)
where N2 is the population of level E2, A21 is the radiative rate, and  is the spontaneous
radiative lifetime.
Stimulated emission (B21):
 dN 2 

   B21 N 2   
 dt  se
(15.2)
where () is the photon density as a function of frequency.
Absorption (B12):
 dN1 

   B12 N1   
 dt  abs
(15.3)
Absorption is also a stimulated process since it depends on the strength of the photon field.
In effect, stimulated absorption and stimulated emission are inverse processes. The A and
B coefficients are constants characteristic of the atom.
Einstein has made the following assumptions:
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(a). Thermodynamic equilibrium at arbitrary temperature T exists between the radiation
field and atom.
(b). The radiation field () has the spectral distribution characteristic of a blackbody at
temperature T.
8h 3
1
   
3
h kT
c
e
1
(c). The atom population densities are distributed according to the Boltzman distribution
at that temperature.
 E 
N1  N 0 exp   1 
 kT 
 E 
N 2  N 0 exp   2 
 kT 
and
(d). Population densities are constant in time.
Finally, we have,
A21 8h 3
B21

(15.4)
c3
and
B12  B21
(15.5)
Eq.(15.5) is valid for the case of nondegenerate energy states. By comparing Eqs.(15.2)
and (15.3), if N2 is greater than N1 and a radiation field interacts with the atoms, stimulated
emission exceeds absorption and photons will be added to the field. This leads to an
increase in (), an amplification. This is the condition of population inversion since
under normal equilibrium condition N2 is less than N1.
According to Eq.(15.4), when the frequency is higher, A21 becomes larger than B21.
Since A21 is related to the spontaneous emission and does not contribute to the photon
amplification, lasers of short wavelength radiation (UV or X-ray, for example) would be
more difficult to build and operate.
For the successful operation of a laser, two ideas are important: stimulated emission
and population inversion.
15.2 Essential Elements of a Laser
The laser device is an optical oscillator that emits an intense, highly collimated
beam of coherent radiation. The device consists of three basic elements: a pump, an
amplifying medium, and an optical cavity or resonator.
The pump is an external energy source that produces a population inversion in the
laser medium. Pumps can be optical, electrical, chemical, or thermal in nature, so long as
they provide energy that can be coupled into the laser medium to excite the atoms and
create the required population inversion.
Many lasers are named after the type of laser medium used, for example, heliumneon (He-Ne), carbon dioxide (CO2), and neodymium:yttrium aluminum garnet
(Nd:YAG). The laser medium, which may be a gas, liquid, or solid, determines the
wavelength of the laser radiation.
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Fig.15.2 Basic elements of a laser. (a)
Integral device with output beam.
(b) Pump. (c) Resonator. (d) Laser
medium.
Resonator is an optical “feedback device” that directs photons back and forth
through the laser (amplifying) medium. In its basic form, it consists of a pair of carefully
aligned plane or curved mirrors centered along the optical axis of the laser system. One of
the mirrors is chosen with a reflectivity as close to 100% as possible. The other is selected
with a reflectivity somewhat less than 100% to allow part of the internally reflecting beam
to escape and become the useful laser output beam. The fundamental mode that appears in
the output laser beam is the TEM00 mode. The transverse variation in the irradiance of this
mode is Gaussian in shape, with peak irradiance at the center and exponentially decreasing
irradiance toward the edges.
15.3 Simplified Description of Laser Operation
The four-step process is illustrated in Fig.15.3. In step 1, a large number of atoms
are excited from the ground state E0 to several excited states, collectively labeled E3. In
step 2, these atoms decay in a short time (usually radiationless) to a special level (upper
laser level) E2. This level has a long lifetime with a typical value of 10 -3s, while most
excited levels may have lifetimes in the order of 10 -8s. Since level E2 is metastable, the
population of this level easily piles up to a value much larger than that of level E1 (lower
laser level), which is an ordinary level that decays to the ground states quite rapidly (step
4). The population N1 cannot build to a large value. The required population inversion for
the light amplification is thus been built. When a photon of resonant energy h=E2E1
passes by any of the N2 atoms in the upper laser level, stimulated emission can occur (step
3).
In summary, the laser process depends on the followings:
 A population inversion
 Seed photons
 An optical cavity
 Laser beam through the output coupler mirror
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Fig.15.3 Operation mechanism for a
general four-level laser.
15.4 Characteristics of Laser Beam
The light emitted by a laser is monochromatic. The emission determined by a
single pair of energy of an identical set of atoms levels is called fluorescence. The emitted
light has a wavelength spread  about a center wavelength 0, where 0=c/v0 and v0=(E2E1)/h.  is often called linewidth. When the linewidth is measured at the half maximum
level of the lineshape plot, it is called the FWHM linewidth, that is, “full width at half
maximum”. The FWHM linewidth for an ordinary discharge lamp is about 0.1nm, that for
a cadmium low-pressure lamp is about 0.0013nm, while that for a He-Ne laser is only
about 10-8nm.
The optical property that most distinguishes the laser from other light sources is
coherence. The laser is regarded as the first truly coherent light source. Other light
sources, such as the sun or a gas discharge lamp, are at best only partially coherent.
Coherence is a measure of the degree of phase correlation that exists in the radiation field
of a light source at different locations and different times. It is often described in term of a
temporal coherence, which is a measure of the degree of monochromaticity of the light,
and a spatial coherence, which is a measure of the uniformity of phase across the optical
wavefront. In the stimulated emission, each photon added to the stimulating radiation has
a phase, polarization, energy and direction identical to that of the amplified light wave in
the laser cavity. Thus the laser light created and emitted is both temporally and spatially
coherent.
For typical lasers, both the spatial coherence and temporal coherence of laser light
are far superior to that for light from other sources. The coherence time tc of a laser is a
measure of the average time interval over which one can continue to predict the correct
phase of the laser beam at a given point in space. The coherence length Lc is related to the
coherence time by equation Lc=ctc, where c is the speed of light. Thus the coherence
length is the average length of light beam along which the phase of the wave remains
unchanged. For the He-Ne laser, the typical coherence time is of the order of milliseconds
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(compared with about 10-11s for light from a sodium discharge lamp), and the coherence
length for the same laser is thousands of kilometers (compared with fractions of a
centimeter for the sodium lamp).
The astonishing degree of directionality of a laser is due to the geometrical design
of the laser cavity and to the monochromatic and coherent nature of light generated in the
cavity. The beam-spread angle is given by the relationship,
1.27

(15.6)
D
where  is the wavelength of the laser beam and D is the diameter of the laser beam at its
beam waist.
Fig.15.4 External and internal laser beam for a given cavity. Diffraction or beam spread appears to be
caused by an effective aperture of diameter D, located at the beam waist. The beam waist is
determined by the design of the laser cavity and depends on the radii of curvature of the two
mirrors and the distance between the mirrors.
In Fig.15.4, the wavefronts that pass through the effective aperture or beam waist
are plane waves, but the irradiance of the laser light is not uniform across the plane. For
the lowest-order transverse mode, the TEM00 mode or the Gaussian beam, the irradiance
of the beam decreases exponentially toward the edges of the beam in accordance with the
Gaussian form e 8 y D , where y measures the transverse beam direction and D is the beam
width at a given position along the beam.
2
2
Lasers are classified in many ways. Sometimes they are grouped according to the
state of matter represented by the laser medium: gas, liquid, or solid. Sometimes they are
classified according to how they are pumped: flashlamp, electrical discharge, chemical
actions, and so on. Other classifications divide them according to the nature of their
output [pulsed or continuous (cw)] and according to their spectral region of emission
(infrared, visible, or ultraviolet).
The wavelength for several common lasers is:
He-Ne (gas):
632.8 nm
Ruby (solid):
694.3 nm
CO2 (gas):
10.6 m
Nd:YAG (solid):
1.064 m
Ar (gas):
488 nm or 514.5 nm
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15.5 Optical Fibers
An optical fiber is a cylindrical dielectric waveguide made of low-loss materials
such as silica glass. It has a central core in which the light is guided, embedded in an outer
cladding of slightly lower refractive index (Fig.15.5). Light rays incident on the corecladding boundary at angles greater than the critical angle undergo total internal reflection
and are guided through the core without refraction. Rays of greater inclination to the fiber
axis lose part of their power into the cladding at each reflection and are not guided.
Fig.15.5 An optical fiber.
Applications: Light pipes: transporting of light, imaging, and communications. The
advantages of fiber-optics communication systems over the conventional two-wire, coaxial
cable or microwave waveguide systems are high information-carrying capacity, immunity
to electromagnetic interference, and freedom from signal leakage.
15.6 Optics of Propagation:
Suppose the fiber itself has a refractive index n1, the encasing medium (called
cladding) has a index n2, and the end faces are exposed to a medium of index n0 (Fig.15.6).
Fig.15.6 (a) Propagation
of light rays through an
optical fiber. Ray B
defines the maximum
input cone of rays that
satisfying total internal
reflection at the walls
of the fiber.
(b)
Propagation
of
a
typical
light
ray
through an optical
fiber.
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The critical angle c must satisfy,
sin  c 
n2
n1
(15.7)
At the input surface, the maximum half-angle m of a cone of rays must obey the Snell’s
law,
n0 sin  m  n1 sin  m
(15.8)
Since  c   m  90  , it comes that the numerical aperture is,
N . A.  n0 sin  m  n1 cos  c  n12  n22
(15.9)
The skip distance between two successive reflections of a ray of light propagating in the
fiber is given by,
Ls  d cot  
(15.10)
where d is the fiber diameter. Relating ’ to the entrance angle  by Snell’s law, we can
obtain,
2
 n1 
  1
Ls  d 
n
sin

 0

(15.11)
Surface scratches or irregularities, as well as surface dust, moisture, or grease,
become sources of loss that rapidly diminish light energy. The cladding is used to protect
the optical quality of the fiber. Cladding materials need not to be highly transparent, but
must be compatible with the fiber core in terms of expansion coefficients, for example.
The cladding of a fiber actually increases the critical angle for internal reflection and
reduces the numerical aperture of the fiber. The cladding of a fiber can also prevent the
cross talk between fibers in communication applications.
The optic fiber cores, which are homogeneous in composition, characterized by a
single index of refraction n1 are called step-index fibers. If the refractive index changes
continuously from the core axis as a function of radius, the fibers are called graded-index
fibers.
15.7 Allowed Modes
Not every ray that enters an optical fiber within its acceptance cone can propagate
successfully through the fiber. Only certain ray directions or modes are allowed.
As shown in Fig.15.7, noticing that the ray represents plane waves moving up and
down in the waveguide, it is evident that such waves overlap and interfere with one
another. Points A and C lie on a common wavefront of such waves. If the net phase
change that develops between A and C is some multiple of 2, then the interfering
wavefront experience constructive interference and corresponding ray directions are
allowed. The optical path difference is,
  n1  AB  BC   2n1d cos   m
(15.12)
The maximum number of allowed mode is when =c, the critical angle,
2dn1 cos  c 2d
m

n12  n22
(15.13)


90
Fig.15.7 Section of a slab waveguide showing a
successfully propagating ray or one of the
possible modes. The geometry is used to
determine the condition for constructive
interference.
The above analysis is for slab waveguide, in the case of a cylindrical fiber, the
maximum number of modes is,
2
1  d

mmax  
N . A.
(15.14)
2 

The number of possible modes increases with the ratio d/. Larger diameter fibers are
multimode fibers. If d/ is small enough to make mmax<2, the fiber allows only the axial
mode to propagate. This is the monomode (or single mode) fiber. A careful analysis
indicates that single-mode performance results when
d


2.4
 N . A.
(15.15)
15.8 Attenuation
The intensity of light propagating through a fiber attenuates due to extrinsic and
intrinsic losses. Among the extrinsic losses are inhomogeneities and geometric effects.
Other extrinsic losses occur as light is coupled into and out of the fiber. At the fiber input
there are losses due to the restrictions of numerical aperture, as well as losses due to
inevitable reflections at the interface, the so-called Fresnel losses. Losses can include
mismatch of coupled fiber ends, involving core diameter and lateral and angular alignment.
Separation and numerical aperture incompatibility are also possible to cause losses.
Intrinsic losses are due to absorption, both by the core material and by residual
impurities, and by Rayleigh scattering from microscopic inhomogeneities, dimensionally
smaller than the optical wavelength.
Absorption losses over a length L of fiber can be described by the usual exponential
law for light irradiance,
I  I 0 e L
(15.16)
where  is an attenuation or absorption coefficient for the fiber, a function of wavelength.
The defining equation for the absorption coefficient in decibels (db) is,
P 
 db  10 log 10  1 
(15.17)
 P2 
where P1 and P2 refer to power levels of the light at two different cross sections 1 and 2.
15.9 Distortion
The signal waveform can be distorted due to several mechanisms: modal distortion,
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material dispersion, and waveguide dispersion.
severity.
They are in an order of decreasing
Modal distortion occurs because propagating rays (fiber modes) travel at different
times, broadening the square wave as shown in Fig.15.8. The shortest distance L from A
to B is taken by the axial ray; the longest distance L’ from A to B is taken by the steepest
propagating ray that reflects at the critical angle c.
Fig.15.8
Schematic of modal distortion.
Shown are the extreme paths: an axial ray
and one propagating at the critical angle.
The modal distortion can be expressed as a temporal pulse spread per unit length,

  n  n
    1  1  1
(15.18)
 L  c  n2

Modal distortion can be lessened by reducing the number of propagating modes.
The best solution is to use a monomode fiber, with only one propagating mode. The next
best is to use a graded index (GRIN) fiber. A GRIN fiber is produced with a refractive
index that decreases gradually from the core axis as a function of radius. In general, the
variation of refractive index with radius can be expressed as,

r
nr   n1 1  2   ,
a
where  
0r a
(15.19)
n12  n22 n1  n2

. The parameter  is chosen to minimize modal distortion.
2n12
n1
Even if modal distortion is absent, some pulse broadening still occurs because the
refractive index is a function of wavelength. The monochromatic the light, the less the
distortion due to material dispersion. Fig.15.9 illustrates material dispersion by showing
the progress of two square pulses (initially coincident) in a fiber at wavelengths 1 and 2
with corresponding refractive indices n1 and n2. The material dispersion can be expressed
as,
 d 2n
 
   
  M
(15.20)
c d2
 L
where M is a property of the core material and  is the spectral width of the input wave.
Pulse broadening due to material dispersion is much smaller than that due to modal
distortion.
The waveguide dispersion is a geometrical effect that depends on waveguide
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parameters. It is a small effect that becomes important only when the other pulsebroadening effects have been essentially eliminated. The waveguide dispersion results
from the dependence of the field distribution in the fiber on the ratio between the core
radius and the wavelength. If this ratio is altered, the relative portions of optical power in
the core and cladding are different, the group velocity of the mode is altered.
The variation of the index of refraction causes not only a spread in the group
velocity of the component waves but also a variation in the zigzag pattern. Dispersion due
to the spread of the index of refraction is called material dispersion. Dispersion due to the
variation in the zigzag pattern is called waveguide dispersion.
Fig.15.9 A square wave input arrives at the
fiber end at different times, depending
on wavelength.
Fig.15.10
Schematic for waveguide
dispersion.
The zigzag pattern is
changed.
References:
1.
A.H.Tunnacliffe and J.G.Hirst, Optics, The Association of British Dispensing
Opticians, 1996.
2.
F.L.Pedrotti and L.S.Pedrotti, Introduction to Optics, Prentice Hall, New Jersey,
1993.
3.
D.Halliday, R.Resnick and J.Walker, Fundamentals of Physics, 5th edition, John
Willey & Sons, Inc., Singapore, 1997.
93