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CHAPTER 4
Stimulated Emission
Devices LASERS
1
• Ali Javan and his associates William Bennett Jr. and Donald Herriott
at Bell Labs were first to successfully demonstrate a continuous
wave (cw) helium-neon laser operation (1960-1962).
2
4.1
STIMULATED EMISSION AND
PHOTON AMPLIFICATION
3
Absorption, spontaneous emission
and stimulated emission
• Absorption: E1 + h  E2
• Emission: E2 E1 + h
– Spontaneous emission: The electron undergoes the downward
transition by itself spontaneously.
– Stimulated emission: The downward transition is induced by
another photon .
4
Spontaneous Emission
• The electron falls down from
level E2 to E1 and emits a
photon h = E2 – E1 in a
random direction.
• The transition is spontaneous
provided that the state with E1
is not already occupied.
5
Stimulated emission
• An incoming photon of energy
h = E2 – E1 stimulates the
whole emission process by
inducing the electron at E2 to
transit down to E1
• Emitted photon and incoming
photon
–
–
–
–
in phase
same direction
same polarization
same energy
• One incoming photon results in two outgoing
photons which are in phase.
 photon amplification
6
Population inversion
• To obtain stimulated emission, the incoming photon should
not be absorbed by another atom at E1.
• We must have the majority of the atoms at the energy level E2.
• When there are more atoms at E2 than at E1 we then have
what is called a population inversion.
• With only two levels we can never achieve population
inversion because, in the steady state, the incoming photon
flux cause as many upward excitations as downward
stimulated emissions.
7
The principle of the laser
Three energy level system
a) Atoms in the ground state are pumped up to the energy
level E3 by incoming photons of energy h13 = E3–E1.
b) Atoms at E3 rapidly decay to the metastable state at
energy level E2 by emitting photons or emitting lattice
vibrations; h 32 = E3–E2.
8
The principle of the laser
c) As the states at E2 are long-lived, they quickly become
populated and there is a population inversion between E2 and
E 1.
d) A random photon (from a spontaneous decay) of energy h 21 =
E2–E1 can initiate stimulated emission. Photons from this
stimulated emission can themselves further stimulate emissions
leading to an avalanche of stimulated emissions and coherent
photons being emitted.
 The emission from E2 to E1 is called the lasing emission.
9
LASER
• A system for photon amplification
• LASER  Light Amplification by
Stimulated Emission of Radiation
10
4.2
STIMULATED EMISSION RATE
AND EINSTEIN COEFFICIENTS
11
Einstein B12 coefficient
• N1: the number of atoms per unit volume with energy E1
• N2: the number of atoms per unit volume with energy E2
• R12: the rate of transition from E1 to E2 by absorption
 R12 = B12N1 (h)
– B12 : the Einstein coefficient for absorption.
–  (h) : the photon energy density per unit frequency
12
Einstein A21, B21 coefficients
• R21: the rate of transition from E2 to E1 by
spontaneous and stimulated emission
R12 = A21N2 + B21N2  (h)
– A21: the Einstein coefficient for spontaneous emission
– B12: the Einstein coefficient for stimulated emission
13
In thermal equilibrium
(no external excitation)
• There is no net change with time in the
population at E1 anf E2

R12 = R21
• Boltzmann statistics demands
 ( E2  E1 ) 
N2
 exp  

N1
kBT 

14
Plank’s black body radiation
distribution law
• In thermal equilibrium, radiation from the
atoms must give rise to an equilibrium
photon energy density, eq(h), that is
given by
eq (h ) 
8 h
3

 h
c exp 
 k BT

3
 
  1
 
15
B12  B21
A21 / B21  8 h / c
3
3
The ratio of stimulated to spontaneous emission
R21 (stim) B21 N 2  (h ) B21  (h )
c3



 (h )
3
R21 (spon)
A21 N 2
A21
8 h
The ratio of stimulated emission to adsorption
R21 (stim)
N2

R12 (absorp) N1
16
The ratio of stimulated emission to
adsorption
R21 (stim)
N2

•
R12 (absorp) N1
For stimulated photon emission to exceed
photon absorption, we need to achieve
population inversion, that is N2 > N1.
17
The ratio of stimulated to
spontaneous emission
R21 (stim)
c3

 (h )
•
3
R21 (spon) 8 h
For stimulated photon emission to far
exceed spontaneous emission, we must
have a large photon concentration which is
achieved by building an optical cavity to
contain the photons.
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Population inversion
• Boltzmann statistics
 ( E2  E1 ) 
N2
 exp  

N1
kBT 

• N2 > N1
T < 0 , i.e. a negative absolute temperature!
• The laser principle is based on nonthermal equilibrium.
19
4.3
OPTICAL FIBER AMPLIFIERS
20
Optical amplifier
• A light signal that is traveling along an
optical fiber over a long distance suffers
marked attenuation.
• It becomes necessary to regenerate the
light signal at certain intervals for long haul
communications over several thousand
kilometers.
• We can amplify the signal directly by using
an optical amplifier
21
Erbium doped fiber amplifier (EDFA)
• Host fiber core material:
Glass based on SiO2-GeO (or other glass
forming oxide such as Al2O3)
• Dopants in the core region:
Er3+ ions (or other rare earth ions such as
neodymium (Nd3+) ions).
• It is easily fused to a single mode long
distance fiber by a technique called
splicing.
22
Energy diagram for Er3+ in glass fiber medium
long-lived level
from a laser diode
• Er3+ ions are pumped (by a laser diode ~980 nm) from E1 to E3.
• Er3+ ions non-radiatively decay rapidly from E3 to a long-live (~10ms)
level E2 .
 Population inversion between E2 and E1.
• Signal photons at 1550 nm (0.80 eV) give rise to stimulated transitions
from E2 to E1.
23
Optical gain
•
•
•
•
•
N1 : the number of Er3+ ions at E1
N2 : the number of Er3+ ions at E2
Absorption rate (E1 to E2)  N1
Stimulated emission rate (E2 to E1)  N2
Net optical gain Gop
Gop  K ( N 2  N1 )
where K is a constant that depends on the
pumping intensity.
24
Schematic diagram of an EDFA
• The Er3+-doped fiber is pumped by feeding the light from a
laser pump diode, through a coupler, into the Er3+-doped
fiber.
• Optical isolators inserted at the entry and exit end allow
only the signals at 1550 nm to pass in one direction and
prevent the 980 nm light from propagation back or forward
into the communication system.
25
4.4
GAS LASERS: THE He-Ne
LASER
26
HeNe laser
• Lasing emission: 632.8 nm red color
• The actual stimulated emission occurs
from the Ne atoms.
• He atoms are used to excite the Ne atoms
by atomic collisions.
27
HeNe laser
• The HeNe laser consists of a gaseous mixture of Ne and
He atoms in a gas discharge tube.
• An optical cavity is formed by the end-mirrors so that
reflection of photons back into the lasing medium builds
up the photon concentration in the cavity.
28
He and Ne atoms
• Ne
– inert gas
– ground state: 1s22s22p6 (represented by 2p6)
– excited state: 2p55s1
• He
– inert gas
– ground state: 1s2
– exited state: 1s12s1
29
Excited He atom
• Dc or RF high voltage  electrical discharge  He
atoms become excited by collisions with the drifting
electrons:
He + e  He* + e
• He*
– excited He atom (1s12s1) with parallel spins
– metastable (long lasting) state
• He* (1s12s1)  He (1s2)
 l  1 no photon emission or absorption
 He* cannot spontaneously emit a photon and decay
down to the He (1s2) ground state.
 Large number of He* build up during the electrical
discharge.
30
He*-Ne collisions
• He* (1s12s1)  He (1s2) E = 20.61 eV
• Ne* (2p55s1)  Ne (1p6) E = 20.66 eV
• When He* collides with Ne, it transfers its
energy to Ne:
He* + Ne  He + Ne*
• With many He*-Ne collisions
 population inversion between (2p55s1) and
(2p53p1) states
31
The principle of HeNe laser
• A population inversion between (2p55s1) and (2p53p1) states
• A spontaneous emission from one Ne* atom falling from (2p55s1) to
(2p53p1) gives rise to an avalanche of stimulated emission
processes.
• Lasing emission:  = 632.8 nm in the red.
32
Lasing transitions in the HeNe laser
• (2p55s1)  (2p53p1)
  = 632.8 nm (red)
  = 543.5 nm (green)
• (2p55s1)  (2p54p1)
  = 3.39 m (infrared)
• To suppress lasing
emissions at the
unwanted wavelengths,
the reflecting mirrors can
be made wavelength
selective.
33
Lasing transitions in the HeNe laser
• (2p55s1)  (2p53p1)
  = 632.8 nm (red)
  = 543.5 nm (green)
• (2p55s1)  (2p54p1)
  = 3.39 m (infrared)
• To suppress lasing
emissions at the
unwanted wavelengths,
the reflecting mirrors can
be made wavelength
selective.
34
HeNe laser
He : Ne = 5 : 1
Pressure ~ torrs
• Lasing intensity  as tube length 
since more Ne atoms are used in stimulated emission.
• Lasing intensity  as tube diameter 
since Ne atoms in the (2p53s1) states can only return to
the ground state by collisions with the walls of tube.
35
EXAMPLE 4.4.1 Efficiency of the
HeNe laser
• A typical low-power 5 mW HeNe laser tube
operates at a dc voltage 2000V and carries a
current of 7 mA.
• What is the efficient of the laser?
36
EXAMPLE 4.4.1 Efficiency of the
HeNe laser
Solution
Output Light Power

• Efficiency 
Input Electrical Power
 0.036%
•
•
5 103 W
(7 103 A)(2000V)
Typical HeNe laser efficiencies < 0.1%
Laser has high concentration of coherent
photon:
5 103 W
 (0.5 103 m)2
Beam diameter = 1 mm
 6.366 kW/m2
37
EXAMPLE 4.4.2 Laser beam
divergence
• A typical He-Ne laser
– Diameter of output beam = 1 mm
– Divergence = 1 mrad
• What is the diameter of the beam at a distance
of 10 m?
38
EXAMPLE 4.4.2 Laser beam
divergence
Solution
We can assume that the laser beam
emanates like a light-cone.
divergence  2  1 mrad
   0.5 103 rad
tan   r / L
 r  L tan   (10 m) tan(0.5  103 rad)
 (10)(5 104 ) m = 5 mm
 Diameter of the beam  2 r  diameter of output beam
 10 mm +1 mm  11 mm
39
4.5
THE OUTPUT SPECTRUM OF
A GAS LASER
40
Output radiation from a gas laser
• The output radiation from a gas laser is
not actually at one single well-defined
wavelength corresponding to the lasing
transition, but covers a spectrum of
wavelengths with a central peak.
The result of the broadening of the emitted
spectrum by the Doppler effect.
41
Kinetic molecular theory
• The gas atoms are in random motion
with an average kinetic energy


1
1
3
2
2
2
2
M v  M v x  v y  v z  k BT
2
2
2
We have to consider the random motion of
atoms in the laser tube.
42
Doppler Effect
• When a gas atom is moving away from the observer
1 = 0 (1 vx /c)
• When a gas atom is moving towards the observer
2 = 0 (1+ vx /c)
where
0 : the source frequency
1, 2 : the detected frequency
vx : the relative velocity of the atom along the laser tube
(x-axis)
c : the speed of light
43
Doppler broadened linewidth
• Since the atoms are in random motion the
observer will detect a range of frequencies
due to the Doppler effect
Doppler broadened linewidth
vx
vx
2 0v x
 rms   2  1   0 (1  )  0 (1  ) 
c
c
c
kBT
where v x 
M
1
1
2
M v x  k BT
2
2
44
Maxwell velocity distribution
• The velocity of gas atoms obey the
Maxwell distribution
 M 
P(v x , v y ,v z )  

 2 k BT 
3/2
 M
2
2
2 
exp 
vx vy vz 
 2 k BT



P(vx, vy, vz) dvx dvy dvz = The probability for finding the atom
within velocity range of (vx+dvx, vy +dvy, vz+dvz)
The stimulated emission wavelength
exhibits a distribution about a central
wavelength 0 = c/0
45
Optical gain lineshape
• The variation in the optical gain with the
wavelength is called the optical gain
lineshape.
• For the Doppler broadening, this lineshape
turns out to be a Gaussian function.
•  = 2- 1 ~ 2-5 GHz
(for many gas lasers)
•  ~ 0.02 A for He-Ne
laser
46
Frequency linewidth
• When we consider the Maxwell velocity
distribution, we find that the linewidth 1/2
between the half-intensity points (full width
at half maximum FWHM) is given by
2kBT ln(2)
1/2  2 0
2
Mc
M: the mass of the lasing atom or molecule
 1/2   rms

~ 18%
 rms
47
Laser cavity modes
• Consider an optical cavity of length L with parallel end
mirrors a Fabry-Perot optical resonator or etalon
• Only standing waves with certain wavelengths can be
maintained within the optical cavity

m( )  L
2
Laser cavity modes in a gas laser
where m is the mode number
• Modes that exist along the cavity axis are called axial (or
longitudinal) modes
48
Output spectrum
• We have spikes of intensity in the output.
• There is a finite width to the individual intensity spikes
which is primarily due to nonidealities of the optical
cavity such as thermal fluctuation of L and nonideal
mirrors (R < 100 %)
– Typical He-He laser  spike width ~1 MHz
– Highly stabilized gas laser  spike width ~ 1 kHz
49
EXAMPLE 4.5.1 Doppler broadened
linewidth
• He-Ne laser
–
–
–
–
•
•
•
•
Transition for  = 632.8 nm
Gas discharge temperature ~ 127 C
Atomic mass of Ne = 20.2 (g/mol)
Laser tube L = 40 cm
Linewidth in the output wavelength spectrum?
Mode number m of the central wavelength?
Separation between two consecutive modes?
How many modes within the linewidth 1/2 of the
optical gain curve?
50
EXAMPLE 4.5.1 Doppler broadened
linewidth
Solution
• The mass of He atom
M  20.2 103 kg mol1 / 6.02 1023 mol1
 3.35 1026 kg
• The central frequency
 0  c / 0  (3 108 m s 1 ) / (632.8 109 m)
 4.74 1014 s 1
• The rms velocity vx along x
 x  kBT / M  [(1.38 1023 JK 1 )(127  273 K)/(3.35 1026 kg)]1/2
 405.8 ms-1
51
EXAMPLE 4.5.1 Doppler broadened
linewidth
Solution (cont.)
• The rms frequency linewidth
 rms
vx
vx
v0v x
  0 (1  )  0 (1  )  2
c
c
c
 2(4.74  1014 s 1 )(405.8 m s 1 ) / (3 108 m s 1 )
 1.282 GHz
• The observed FWHM frequency width
 1/2  2 0
23
2k BT ln(2)
2(1.38

10
)(127  273) ln(2)
14

2(4.748

10
)
Mc 2
(3.35 1026 )(3 108 ) 2
 1.51 GHz
 1/2   rms 1.51  1.282


 18%
 rms
1.282
52
EXAMPLE 4.5.1 Doppler broadened
linewidth
Solution (cont.)
• To get FWHM wavelength width 1/2, differentiate  = c/
d
c

 2 
d


 1/2   1/2  /   (1.51109 Hz)(632.8 109 m)/(4.74 1014 s 1 )
 2.02 1012 m  0.0020 nm
• For  = 0= 632.8 nm, the corresponding mode number m0 is
m0  2 L / 0  (2  0.4 m)/(632.8 109 m)  1.264 106
• actual m0 has to be the closest integer value to 1.264106
53
EXAMPLE 4.5.1 Doppler broadened
linewidth
Solution (cont.)
• The separation m between two consecutive modes
2L 2L
2 L 02
m  m  m1 

 2 
m m 1 m
2L
(632.8 109 m) 2
 m 
 5.006 1013 m  0.501 pm
2  0.4 m
• The number of modes within the linewidth
1/2 2.02 pm
Linewidth of spectrum
Modes 


 4.03
Separation of two modes m
0.501 pm
• We can expect at most 4 to 5 modes within the linewidth
of the output.
54
EXAMPLE 4.5.1 Doppler broadened
linewidth
Solution (cont.)
• Number of laser modes depends on how the cavity
modes intersect the optical gain curve.
55
4.6
LASER OSCILLATION
CONDITION
56
4.6 LASER OSCILLATION CONDITIONS
Energy band Diagrams
57
4.6 LASER OSCILLATION CONDITIONS
A. Optical Gain Coefficient g
58
• Consider an EM wave propagating in a medium
along the x-direction.
• If the light intensity were decreasing
P( x   x)  P( x)e x
 : absorption coefficient
• If the light intensity were
increasing
P ( x   x )  P ( x ) e  g x
g : optical gain coefficient
59
Optical gain coefficient
•
P ( x   x )  P ( x )e
 P( x)  P0e
 g x
g x
 P( x)

 P0e g x g  p( x)g
x
P
g=
P x
The gain coefficient g is defined as the
fractional change in the light power (or intensity)
per unit distance
60
Optical gain coefficient
• Optical power
P  N ph h
– Nph : the concentration of cohenernt photons
– h : photon energy
• In time  t photons travel a distance
 x  (c / n) t
– n: refractive index
• The optical gain coefficient
 N ph
P
n  N ph
g=


P x N ph x cN ph  t
61
•
 N ph
dt
 Net rate of stimulated photon emission
 N 2 B21  (h )  N1B21  (h )
Stimulated emission Absorption
 ( N 2  N1 ) B21  (h )
• We can neglect spontaneous
emission which are in random
direction and do not, on average
contribute to the directional wave.
g  g ( )
62
Lineshape function
• The emission and absorption processes would
be distributed in photon energy or frequency
over some frequency interval  (due to Doppler
broadening or broadening of the energy levels
E2 and E1).
• The optical gain will reflect this distribution
g  g ( )
• The spectral shape of the
gain curve is called the
lineshape function.
63
General optical gain coefficient
n  N ph
g
cN ph  t
 N ph
dt
 ( N 2  N1 ) B21  (h )
 (h 0 ) 
N ph h 0
Radiation energy density

per unit frequency at h0
N ph h 0
n
n
g ( 0 ) 
( N 2  N1 ) B21  (h 0 ) 
( N 2  N1 ) B21
cN ph
cN ph

B21nh 0
 General optical gain coefficient
g ( 0 )  ( N 2  N1 )
c
This equation gives the optical gain of the medium
at the center frequency 0
64
4.6 LASER OSCILLATION CONDITIONS
B. Threshold Gain gth
65
Power condition for maintaining oscillations
• Consider an optical cavity with mirrors at the ends. The
cavity contains a laser medium so that lasing emissions
build up to a steady state.
• Pi : initial optical power
• Pf : final optical power
• Under steady state condition, oscillations do not build up
and do not die out
Power condition for maintaining oscillations
Pi  Pf
•
Net round-trip optical gain Gop  Pf / Pi  1
66
• Reflections at the faces 1 and 2 reduce the
optical power by the reflectances R1, and R2 of
the faces.
• As the wave propagates, its power increases as
exp(g x)
• Losses
– Absorption by impurities and free carriers
– Scattering at defects and inhomogenities
These losses decrease the power as exp(x)
– : the attenuation or loss coefficient of the
medium
67
Threshold optical gain
• The power Pf after one round trip of path length 2L is
given by
Pf  Pi R1R2 exp[g (2 L)]exp[ (2 L)]
• For steady state oscillations Gop=Pf /Pi = 1 must be
satisfied
Pf  Pi R1R2 exp[g (2 L)]exp[ (2 L)]  Pi
 exp[g (2 L)]  exp[ (2 L)] / R1R2
1
1
 gth   
ln(
)
2 L R1R2
Threshold optical gain
• gth is the optical gain needed in the medium to achieve a
continuous wave lasing emission.
68
Threshold population inversion
1
1
)
• gth    ln(
2 L R1R2
• The necessary gth has to be obtained by suitably
pumping the medium so that N2 is sufficiently
greater than N1.
• This corresponds to a threshold population
inversion or N2 N1 = (N2 N1)th
• Form
B21nh 0
c
c
 ( N 2  N1 )th  gth
B21nh 0
g ( 0 )  ( N 2  N1 )
 [ 
Threshold population inversion
1
1
c
ln(
)]
2 L R1R2 B21nh 0
69
• Until the pump rate can bring (N2N1) to the threshold
value (N2N1)th, there would be no coherent radiation
output.
• When the pumping rate exceeds the threshold value,
then (N2N1) remains clamped at (N2N1)th because this
controls the optical gain g which must remain at gth.
• Additional pumping increases the rate of stimulated
transitions and hence increases the optical output power
P0.
70
4.6 LASER OSCILLATION CONDITIONS
C. Phase Condition and Laser Modes
71
Phase condition for laser oscillations
• Unless the total phase change after one round
trip from Ei to Ef is a multiple of 2, the wave Ef
cannot be identical to the initial wave Ei .
• We need an additional condition:
round trip  m(2 ), m  1, 2,
Phase condition for
laser oscillations
72
Longitudinal axial modes
• If k = 2/ is the free space wavevector, only those
special k value, denoted as km, that satisfy round-trip =
m(2) can exits as radiation in the cavity
round trip = nkm (2 L)  m(2 )
n
2
m
(2 L)  m(2 )
The usual mode condition
m(
m
2n
)L
• These modes are controlled by the length L of the optical
cavity along its axis and are called longitudinal axial
modes.
73
Off-axis transverse mode
• All practical laser cavity have a
finite transverse size, a size
perpendicular to the cavity
axis.
• Furthermore, not all cavities
have flat reflectors at the ends.
An off-axis mode is able to
self-replicate after one round
trip.
74
Off-axis transverse mode
• Such a mode would be non-axial.
• Its properties would be determined not
only by the off-axis round-trip distance and
but also by the transverse size of the
cavity.
• The greater the transverse size, the more
of these off-axis modes can exist.
75
Transverse modes
• A mode represents particular electric field
pattern in the cavity that can replicate itself
after one round trip.
• A mode with a certain field pattern at a
reflector can propagate to the other
reflector and back again return the same
field pattern.
• Transverse modes or transverse
electric and magnetic (TEM) modes.
76
TEM modes
• Each allowed mode corresponds to a distinct spatial field
distribution at a reflector.
• TEMp,q,m
– p: the number of nodes in the field distribution along the
transverse direction y
– q: the number of nodes in the field distribution along the
transverse direction z
– m: the number of nodes along the cavity axis x (usually m is very
large (~ 106 in gas lasers) and is not written.)
77
TEM00 mode
• The lowest order mode.
• Gaussian intensity distribution across the
beam cross section everywhere inside and
outside cavity.
• It has the lowest divergence angle.
• Many laser designs optimize on TEM00
while suppressing other modes.
• Such a design usually requires restrictions
in the transverse size of the cavity.
78
EXAMPLE 4.6.1 Threshold population
inversion for the He-Ne laser
• Show that the threshold population inversion Nth
= (N2- N1)th can be written as
Nth  gth
8 n 2 0 sp 
c2
– 0 = peak emission frequency (at peak of output
spectrum)
– n = refractive index
– sp = 1/A21 = mean time for spontaneous transition
–  = optical gain bandwidth (frequency-linewidth of the
optical gain linewidth)
79
EXAMPLE 4.6.1 Threshold population
inversion for the He-Ne laser (cont.)
• He-Ne laser
– 0 = 632.8 nm
– L = 50 cm
– R1 = 100%
– R2 = 90%
– linewidth  = 1.5 GHz
– loss coefficient   0.05 m-1
– sp = 1/A21  300 ns
–n1
• What is the threshold population inversion ?
80
EXAMPLE 4.6.1 Threshold population
inversion for the He-Ne laser
Solution
• The relationship between Einstein A, B coefficients is
A21 8 hn 3 3

B21
c3
• The threshold population inversion is then
c
N th  ( N 2  N1 )th  g th
 g th
B21nh 0
c
A21c3
nh 0
3 3
8 hn  0
8 n  0 sp 
8 n 2 02 
 g th
 g th
2
A21c
c2
2
2
81
EXAMPLE 4.6.1 Threshold population
inversion for the He-Ne laser
Solution (cont.)
• For He-Ne laser
– The emission frequency
3 108
5
0  

4.74

10
Hz
9
0 632.8 10
c
– The threshold gain
1
1
1
1
-1
gth   
ln(
)  0.05 m 
ln(
)
2 L R1R2
2(0.5 m) 1 0.9
 0.155 m 1
82
EXAMPLE 4.6.1 Threshold population
inversion for the He-Ne laser
Solution (cont.)
– The threshold population inversion
Nth  ( N 2  N1 )th  gth
8 n 2 02 sp 
c2
2
15
9
9 1
8

(1)
(4.74

10
Hz)(300

10
s)(1.5

10
s )
1
 (0.155 m )
(3 108 m/s)2
 4.4 1015 m3
This is the threshold population inversion for
Ne atoms in configuration 2p55s1 and 2p53p1
N2
N1
83
4.7
PRINCIPLE OF THE LASER
DIODE
84
Degenerately doped pn junction
• Fermi level:
– EFp in the p-side is in the VB
– EFn in the n-side is in the CB
• No bias  Fermi level is
continuous across the diode 
EFp = EFn
• Depletion region (SCL) is very
narrow.
• Built-in potential barrier eV0
prevents electron (hole)
diffusion from n+ (p+)-side to p+
(n+)-side.
85
Forward bias
• If eV = EF > Eg
The applied voltage diminishes
the built-in potential barrier to
almost zero
Electrons from n+-side and
holes from p+-side flow into the
SCL.
SCL region is no longer
depleted.
86
Population inversion
• In the SCL, there are more
electrons in the CB at
energies near Ec than
electrons in the VB near Ev.
There is a population
inversion between energies
near Ec and those near Ev
around the junction.
87
Inversion layer
• The population inversion
region is a layer along the
junction and is called the
inversion layer or the
active region.
• The region where there is
population inversion has an
optical gain because an
incoming photon is more
likely to cause stimulated
emission than being
absorbed.
88
At low temperature (T  0 K)
• The states between Ec and EFn are filled with electrons
and those between EFp and Ev are empty.
• Eg < hv (photon energy) < EFn – EFp Stimulated emission
• hv (photon energy) > EFn – EFp Absorption
89
At T > 0
• As the temperature increases
 the Fermi-Dirac function spreads the energy distribution
of electron in CB to above EFn and hole below EFp in the
VB
 a reduction in optical gain
• The optical gain depends on EFn – EFp which depends on
the applied voltage and hence on the diode current.
90
Injection pumping
• The population inversion between energies near
Ec and those near Ev is achieved by the injection
of carriers across the junction under a
sufficiently large forward bias.
• The pumping mechanism is the forward diode
and the pumping energy is supplied by external
battery.
• This type of pumping is called injection pumping.
91
Homojunction laser diode
• The pn junction uses the
same direct bandgap
semiconductor material.
• The ends of the crystal are
cleaved to be flat and optically
polished to provide reflection
and hence form an optical
cavity.
• Photons that are reflected
from the cleaved surface
stimulate more photons of the
same frequency and so on.
92
Modes in optical cavity
• The wavelength of the radiation
that can builds up in the cavity
is determined by the length L
of the cavity
m

2n
L
• Each radiation satisfying the
above relationship is essentially
a resonant frequency of the
cavity, that is a mode of the
cavity.
93
Threshold current
• Lasing radiation is only obtained when the
optical gain in the medium can overcome
the photon losses from the cavity, which
require the diode current I exceed a
threshold value
I  I th
– Ith : threshold current
94
Characteristics of a laser diode
• I < Ith : Spontaneous emission
• I > Ith : Stimulated emission
• Above Ith, the light intensity becomes coherent radiation
consisting of cavity wavelength (or modes) and
increases steeply with current.
• The number of modes in the output spectrum and their
relative strengths depend on the diode current.
95
Main problem with the
homojunction laser diode
• The threshold current density Jth is too high
for practical uses.
• Ex. for GaAs at room temperature Jth ~ 500
A mm-2
GaAs homojunction laser can only be
operated continuously at very low
temperature.
• Jth can be reduced by orders of magnitude
by using heterostructured semiconductor
laser diodes.
96
4.8
HETEROSTRUCTURE LASER
DIODES
97
Reduction of threshold current Ith
Requires
• Improving the rate of stimulated emission
• Improving the efficiency of the optical
cavity
We need both
• Carrier confinement
• Photon confinement
98
Carrier confinement
• We can confine the injected electrons and
holes to a narrow region around the
junction.
Narrowing of the active region
Less current is needed to establish the
necessary concentration of carriers for
population inversion.
99
Photon confinement
• We can build a dielectric waveguide
around the optical gain region
 To increase the photon concentration and
hence the probability of stimulated
emission.
100
Double heterostructure (DH) device
• A DH device is based on two junctions between different
semiconductor materials with different bandgaps.
– Eg (AlGaAs)  2 eV
– Eg (GaAs)  1.4 eV
• p-GaAs region : thin layer ~ 0.1 – 0.2 m
 the active layer in which lasing recombination takes
place.
101
Carrier confinement
• p-GaAs and p-AlGaAs
– heavily p-type doped
degenerate with EF in the VB
• When a sufficiently large
forward bias is applied
– Ec of n-AlGaAs moves above Ec
of p-GaAs
 A large injection of electrons in
the CB of n-AlGaAs into p-GaAs
These electrons are confined to the CB of p-GaAs since there is
a barrier Ec between p-GaAs and p-AlGaAs due to the change
in the bandgap.
102
• p-GaAs is a thin layer
The concentration of
injected electrons in the
p-GaAs layer can be
increased quickly even
with moderate increases
in forward current.
This effectively reduces the threshold current for
population inversion or optical gain.
103
Photon confinement
• Eg(AlGaAs) > Eg (GaAs)
Index n (AlGaAs) < Index
n(GaAs)
The change in the refractive
index defines an optical
dielectric waveguide that
confines the photons to the
active region of the optical
cavity.
This increase in the photon
concentration increases the
rate of stimulated emission.
104
Double heterostructure laser diode
•
•
The p and n-AlGaAs layers provide carrier and optical confinement in the vertical
direction by forming heterojunctions with p-GaAs.
The advantage of the AlGaAs/GaAs heterojunction only a small lattice mismatch
between the two crystal stuuctre and hence negligible strain induced interfacial
defects (e.g. dislocations) in the device.
105
Contacting layer
• There is an additional p-GaAs layer, call
contacting layer, next to p-AlGaAs.
• It can be seen that the electrodes are
attached to the GaAs semiconductor
materials rather than AlGaAs.
• This choice allows for better contacting
and avoids Schottky junctions which limit
the current.
106
Stripe geometry
• Stripe geometry or stripe
contact on p-GaAs
• J is greatest along the
central path 1, and
decreases away from
path1, towards 2 or 3.
• The current is confined to
flow within paths 2 and 3.
• The current density paths through the active layer where
J is greater than the threshold value Jth define the active
region where population inversion and hence optical
gain takes place.
107
Gain guided lasers
• The width of the active region, or the optical gain region,
is defined by the current density from the stripe contact.
• Optical gain is highest where the current density is
greatest.
• Such lasers are called gain guided.
108
Advantages to using a stripe geometry
• The reduced contact area also reduces
the threshold current Ith
– Typical strip widths W ~ few m
 Ith ~ tens of mA.
• The reduced emission area makes light
coupling to optical fibers easier.
109
Reducing the reflection losses from
the rear crystal facet
• n(GaAs) = 3.7
 Reflectance ~ 0.33
• Dielectric mirror at the
rear facet
Reflectance 1
Optical gain 
 Ith 
110
Buried DH laser diode
• The active layer, p-GaAs, is bound vertically and laterally
by a wider bandgap semiconductor, AlGaAs, which has
lower refractive index.
 The photons are confined to the active or optical gain
region which increases the rate of stimulated emission
and hence efficiency of the diode.
111
Index guided LDs
• The optical power is confined to the waveguide defined
by the refractive index variation, these diodes are called
index guided.
• If the buried heterostructure has the right dimension
compared with the wavelength of the radiation then only
the fundamental mode can exist in the waveguide
structure
single mode laser diode.
112
EXAMPLE 4.8.1 Modes in a laser and the
optical cavity length
• AlGaAs based heterostructure laser diode
– Optical cavity L = 200 m
– Peak radiation  = 900 nm
– Refractive index of GaAs n  3.7
• What is the mode integer m of the peak radiation
and the separation between the modes of the
cavity?
• If the optical gain has a FWHM wavelength width
1/2  6 nm, how many modes are there within
the bandwidth?
• How many modes are there if L = 20 m?
113
EXAMPLE 4.8.1 Modes in a laser and the
optical cavity length
Solution
•
m

2n
L
2(3.7)(200 106 )
m


900 109
 1644.4 or 1644
• The wavelength separation m between
2nL
modes m and m+1 is
2n L 2n L 2n L  2
m  m  m 1 

 2 
m
m 1 m
2n L
(900 109 ) 2
10


5.47

10
m  0.547nm
6
2(3.7)(200 10 )
114
EXAMPLE 4.8.1 Modes in a laser
and the optical cavity length
Solution (cont.)
• 1/2  6 nm

1/2
m
6 nm

 10.968
0.547 nm
There will be 10 modes within the bandwidth.
• If L=20 m,
2
(900 109 ) 2
9
m 


5.47

10
m  5.47nm
6
1/2
m
2n L 2(3.7)(20 10 )
6 nm

 1.0968
5.47 nm
There will be one mode that corresponds to about 900 nm.
115
EXAMPLE 4.8.1 Modes in a laser
and the optical cavity length
Solution (cont.)

• m L
2n
2(3.7)(20 106 )
m

 164.4 or 164
9

900 10
6
2nL 2(3.7)(20 10 )
 

 902.4 nm
m
164
2nL
Reducing the cavity length suppresses
higher modes.
116
4.9
ELEMENTARY LASER DIODE
CHARACTERISTICS
117
Factors of the output spectrum
of a laser diode (LD)
• The nature of the optical resonator used to
build the laser oscillations
• The optical gain curve (lineshape) of the
active medium.
118
Laser cavity
• Fabry-Perot cavity
• Length L determines the longitudinal mode separation.
• Width W and Height H determine the transverse modes
(or lateral modes).
119
• If the transverse dimensions (W and H) are
sufficiently small, only the lowest
transverse mode, TEM00 mode, will exit.
• This TEM00 mode will have longitudinal
modes whose separation depends on L.
m

2n
L
2n L 2n L 2n L  2
m  m  m 1 

 2 
m
m 1 m
2n L
120
Beam divergence
• The emerging laser beam exhibits divergence.
• This is due to diffraction of the waves at the
cavity ends.
• The smallest aperture (H ) causes the greatest
diffraction.
121
Output spectra from LDs
• The spectrum is either
multimode or single mode
depending on the optical
resonator structure and
pumping current level.
• Index guided LDs
– Low output power  multimode
– High output power single
mode
• Gain guided LDs
– tend to remain multimode even
at high diode currents.
122
Temperature dependence of the LD output
• As the temperature increases, the threshold
current increases steeply, typically as the
exponential of the absolute temperature.
123
Mode hopping
Single mode LDs
• The peak emission wavelength 0 exhibits
“jumps” at certain temperatures.
At new operating temperature, another mode
fulfills the laser oscillation condition.
124
Mode hopping
• Between mode hops, 0 increases slowly with T
due to the slight increase in n and L.
• If mode hope are undesirable, then the device
structure must be such to keep the modes
sufficiently separated.

2n L
2n
m
d m
d  2n L 




dT
dT  m 
m
m 
 L  m 
2
2n L
125
Multimode LDs
• Gain guided LDs
The output spectrum has many modes so that 0
vs. T behavior tends to follow the changes in the
bandgap rather than the cavity properties.
Eg (T )  Eg (0) 

T 
126
Slope efficiency slope
• slope 
Po
(W/A or W/mA)
I  I th
– Po : the output optical power
– I: the diode current
– Ith : the threshold current
• Slope efficiency determines the output optical power in
terms of the diode current above the threshold current.
• Slope efficiency depends on the LD structure as well as
the semiconductor packaging.
• Typically, slope  1 W/A
127
Conversion efficiency
Output of optical power
• Conversion efficiency 
Iutput of electrical power
Po

IV
• The conversion efficiency gauges the overall
efficiency of the conversion from the input of
electrical power to the output of optical power.
• In some modern LDs this maybe as high as 3040 %.
128
EXAMPLE 4.9.1 Laser output
wavelength variations
• The refractive index n of GaAs has a
temperature dependence
4
d n / dT  1.5 10 K
1
• Estimate the change in the emitted
wavelength 870 nm per degree in
temperature between mode hops.
129
EXAMPLE 4.9.1 Laser output
wavelength variations
Solution
•
m
2n L
2n
m
d m
d  2n L  2 L d n m d n




dT
dT  m  m dT
n dT
870 nm

(1.5 104 K 1 )
3.7
 0.035 nm K 1
m
 L  m 
• Index n will also depend on the optical gain of the
medium and hence its temperature dependence is
likely to be somewhat higher than the dn/dt value
we used.
130
4.10
STEADY STATE
SEMICONDUCTOR RATE
EQUATIONS
131
Active layer rate equation
• Consider a DH LD
under forward bias.
• Under steady state
operation:
Rate of electron injection into the active layer by current
 Rate of spontaneous emissions  Rate of stimulated emissions
I
n

 CnN ph
edLW  sp
Active layer rate equation
d: thickness, L : length, W: width, I: current, n : the injected electron
concentration, Nph: the coherent photon concentration in the active layer,
sp: the average time for spontaneous recombination, C: a constnt
(depends on B21)
132
Active layer rate equation
I
n
•

 CnN ph
edLW  sp
• As the current increases and provides more pumping, Nph
increases (helped by the optical cavity), and eventually
the stimulated term dominates the spontaneous term.
• The output light power P0 is proportional to Nph.
133
Rate of stimulated emissions
• Consider the coherent photon concentration Nph
in the cavity. Under steady state condition:
Rate of coherent photon loss in the cavity
 Rate of stimulated emissions
N ph
 CnN ph
 ph
ph: the average time for a photon to be lost from the
cavity due to transmission through the end-faces,
scattering and absorption in the semiconductor.
134
Threshold concentration
• nth : threshold electron concentration
• Ith: threshold current
• nth and Ith  the condition when the
stimulated emission just overcomes the
spontaneous emission and the total loss
mechanisms in ph.
• This occurs when injected n reaches nth
nth  n 
1
C ph
Threshold concentration
135
Threshold current
• When I > Ith, the output optical power
increases sharply with current
Nph = 0 when I = Ith
I
n

 CnN ph
edLw  sp
I th
nth


edLW  sp
 I th 
nth edLW
 sp
Threshold current
136
Threshold current
Ith 
nth edLW
 sp
 Ith decreases with d, L and W
 low Ith for the heterostructure and stripe
geometry lasers.
137
• Above threshold, form
I
n

 CnN ph
edLW  sp
with n clamped at nth,
I  I th
 Cnth N ph
edLW
• Using
nth 
1
C ph
and defining J = I/WL we can find
N ph 
 ph
ed
( J  J th )
Coherent photon concentration
138
Output optical power
1
( N ph )(Cacity Volume)(Photon energy)
P0  2
(1  R)
t
• t = nL/c: the time for photons crossing the laser
cavity length L
• (1R): the fraction of the radiation power that will
escape.
• (1/2)NphOnly half of the photons in the cavity
would be moving towards the output face of the
crystal
139
Laser diode equation
t  n L / c
N ph 
 ph
( J  J ph )
ed
1
( N ph )(Cacity Volume)(Photon energy)
(1  R)
P0  2
t

 1  ph
 2 ed ( J  J ph )  (dWL)(hc /  )

(1  R)

nL / c
 hc 2W ph (1  R) 
Laser diode equation
P0  
 ( J  J ph )
2en


140
141
4.11
LIGHT EMITTERS FOR OPTICAL
FIBER COMMUNICATIONS
142
Light emitters for optical
communications
• LEDs
–
–
–
–
–
simpler to drive
more economic
have a longer lifetime
provide the necessary output power
wider output spectrum
Short haul application (e.g. local networks)
• Laser diodes
– narrow linewidth
– high output power
– higher signal bandwidth capability
Long-haul and wide bandwidth communications
143
144
145
Rise time
• The speed of response of an emitter is
generally described by a rise time.
• Rise time is the time it takes for the output
optical power to rise from 10% to 90% in
response to a step current input.
• LED: Rise time  5-20 ns
• Laser diode: Rise time < 1 ns
Laser diodes are used whenever wide
bandwidths are required.
146
4.12
SINGLE FREQUENCY SOLID
STATE LASERS
147
Active layer rate equation
• Consider a DH LD
under forward bias.
• Under steady state
operation:
Rate of electron injection into the active layer by current
 Rate of spontaneous emissions  Rate of stimulated emissions
I
n

 CnN ph
edLW  sp
Active layer rate equation
d: thickness, L : length, W: width, I: current, n : the injected electron
concentration, Nph: the coherent photon concentration in the active layer,
sp: the average time for spontaneous recombination, C: a constnt
(depends on B21)
148
Active layer rate equation
I
n
•

 CnN ph
edLW  sp
• As the current increases and provides more pumping, Nph
increases (helped by the optical cavity), and eventually
the stimulated term dominates the spontaneous term.
• The output light power P0 is proportional to Nph.
149
Rate of stimulated emissions
• Consider the coherent photon concentration Nph
in the cavity. Under steady state condition:
Rate of coherent photon loss in the cavity
 Rate of stimulated emissions
N ph
 CnN ph
 ph
ph: the average time for a photon to be lost from the
cavity due to transmission through the end-faces,
scattering and absorption in the semiconductor.
150
• ph: the average time for a photon to be lost from the
cavity due to transmission through the end-faces,
scattering and absorption in the semiconductor.
• t : the total attenuation coefficient representing all the
loss mechanisms.
 ph  n / (ct )
exp(t /  ph )  exp(t (ct ) / n)  exp(t x)
151
Threshold concentration
• nth : threshold electron concentration
• Ith: threshold current
• nth and Ith  the condition when the
stimulated emission just overcomes the
spontaneous emission and the total loss
mechanisms in ph.
• This occurs when injected n reaches nth
nth  n 
1
C ph
Threshold concentration
152
Threshold current
• When I > Ith, the output optical power
increases sharply with current
Nph = 0 when I = Ith
I
n

 CnN ph
edLw  sp
I th
nth


edLW  sp
 I th 
nth edLW
 sp
Threshold current
153
Threshold current
Ith 
nth edLW
 sp
 Ith decrease with d, L and W
 low Ith for the heterostructure and stripe
geometry lasers.
154
• Above threshold, form
I
n

 CnN ph
edLW  sp
with n clamped at nth,
I  I th
 Cnth N ph
edLW
• Using
nth 
1
C ph
and defining J = I/WL we can find
N ph 
 ph
ed
( J  J th )
Coherent photon concentration
155
Output optical power
1
( N ph )(Cacity Volume)(Photon energy)
P0  2
(1  R)
t
• t = nL/c: the time for photons crossing the laser
cavity length L
• (1R): the fraction of the radiation power that will
escape.
• (1/2)NphOnly half of the photons in the cavity
would be moving towards the output face of the
crystal
156
Laser diode equation
t  n L / c
N ph 
 ph
( J  J ph )
ed
1
( N ph )(Cacity Volume)(Photon energy)
(1  R)
P0  2
t

 1  ph
 2 ed ( J  J ph )  (dWL)(hc /  )

(1  R)

nL / c
 hc 2W ph (1  R) 
Laser diode equation
P0  
 ( J  J ph )
2en


157
158
4.11
LIGHT EMITTERS FOR OPTICAL
FIBER COMMUNICATIONS
159
Light emitters for optical
communications
• LEDs
–
–
–
–
–
simpler to drive
more economic
have a longer lifetime
provide the necessary output power
wider output spectrum
Short haul application (e.g. local networks)
• Laser diodes
– narrow linewidth
– high output power
– higher signal bandwidth capability
Long-haul and wide bandwidth communications
160
161
162
Rise time
• The speed of response of an emitter is
generally described by a rise time.
• Rise time is the time it takes for the output
optical power to rise from 10% to 90% in
response to a step current input.
• LED: Rise time  5-20 ns
• Laser diode: Rise time < 1 ns
Laser diodes are used whenever wide
bandwidths are required.
163
4.12
SINGLE FREQUENCY SOLID
STATE LASERS
164
Methods of ensuring a single mode
of radiation in the laser cavity
• Using frequency selective dielectric
mirrors at the cleaved surface.
Distributed Bragg reflector (DBR) laser
• Using a corrugated guiding layer next to
the active layer.
Distributed feedback (DFB) laser
• Using two different optical cavities which
are coupled.
Cleaved-coupled-cavity (C3) laser
165
Distributed Bragg reflector (DBR)
• The DBR is a mirror that has been designed like
a reflection type diffraction grating.
• It has a periodic corrugated structure.
• Partial reflection from the corrugations interface
constructively to give a reflected wave only when
the wavelength corresponds to twice the
corrugation periodicity.
166
Distributed Bragg reflector (DBR)
• : the period of corrugation
• 2 = the optical path difference between waves A and B
• A and B interfere constructively if 2 is a multiple of the
wavelength within the medium.
• Each of these wavelength is called a Bragg wavelength
B :
B
2  q , q  1, 2,3,
n
– n : the refractive index of the corrugated material
– q : the diffraction order
• The DBR has a high reflectance around B but low
reflectance away from B
 Only that particular cavity mode, within the optical gain
curve, that is close to B can lase and exist in the output.
167
Distributed feedback (DFB) laser
• In the normal laser, the crystal faces provides
the necessary optical feedback into the cavity to
build up the photon concentration.
• In the distributed feedback (DFB) laser, there
is a corrugated layer, call the guiding layer,
next to the active layer; radiation spreads from
the active layer to the guiding layer.
168
Distributed feedback (DFB) laser
• These corrugations in the refractive index act as
optical feedback over the length of the cavity by
producing partial reflections.
Optical feedback is distributed over the cavity
length.
169
Allowed DFB lasing modes
• Traveling waves are reflected partially and
periodically as they propagate.
• The medium alters the wave-amplitudes via
optical gain.
• The allowed DFB modes:
m  B 
B2
2n L
(m  1),
2 n
where m  0,1, 2,
B 
, q  1, 2,3,
q
L  corrugation length
170
Allowed DFB lasing modes
• m  B 
•
•
•
•
B2
(m  1)
2n L
The relative threshold gain for higher modes is so large
that only the m = 0 mode effectively lases.
A perfect symmetric device has two equally spaced
modes placed around B.
In reality, either inevitable asymmetry introduced by the
fabrication process, or asymmetry introduced on purpose,
leads the only one of the modes to appear.
Typically, L >>   B >> B2 /(2nL)  0  B
171
Cleaved-coupled-cavity (C3) laser
• In the cleaved-coupled-cavity (C3) laser device, two
different laser optical cavities L and D (different lengths)
are coupled.
• The two lasers are pumped by different currents.
• Only those waves that can exist as modes in both
cavities are now allowed because the system has been
coupled.
• The wide separation between the modes results in a
single mode operation.
172
EXAMPLE 4.12.1 DFB Laser
• DFB laser
– Corrugation period  = 0.22 m
– Grating length L= 400 m
– Effective refractive index of medium n  3.5
• Assuming a first order grating, calculate
– the Bragg wavelength
– the mode wavelengths and their separation
173
EXAMPLE 4.12.1 DFB Laser
Solution
• The Bragg wavelength is given by
q
B
n
 2  B 
2n 2(0.22 m)(3.5)

 1.540 m
q
1
• The symmetric DFB laser wavelengths about B are
B2
(1.54 μm)2
m  B 
(m  1)  1.54 μm 
(m  1)
2n L
2(3.5)(400 μm)
• The m = 0 mode wavelengths are
(1.54 μm)2
0  1.54 μm 
(0  1)  1.5391 or 1.5408  m
2(3.5)(400 μm)
• The two are separated by 0.0017 m (or 1.7 nm).
• Due to asymmetry, only one mode will appear in the output and
174
for most practical purpose wavelength can be taken as B.
4.13
QUANTUM WELL DEVICES
175
Quantum well device
• A typical quantum well (QW) device has an
ultra thin (typically, d < 50 nm) narrow bandgap
semiconductor sandwiched between two wider
bandgap semiconductors.
• A QW device is a heterostructure device.
• Ex. AlGaAs/GaAs/AlGaAs
system.
176
Quantum well device
• Ec and  Ev , the discontinuities in Ec and Ev at the interfaces,
depend on the semiconductor materials and their doping.
• GaAs/AlGaAs heterostructure:
Ec  ( Eg 2  Eg1 )  60%, Ev  ( Eg 2  Eg1 )  40%
• Ec : the potential energy barrier
 The conduction electrons are confined in the x-direction but free in
the yz plane.
 Two dimensional electron gas confined in the x-direction.
177
Energy in a quantum well device
h2 ny2
h2 nz2
hn
• E  Ec  * 2  * 2  * 2
8me d
8me Dy 8me Dz
2 2
• Dy, Dz >> d  the minimum energy E1  n = 1
• E2  n = 2
• E3  n = 3
178
Density of states g(E)
• g(E) = number of quantum states per unit volume
• In the 2D electron gas, g(E) is constant at E1 until
E2 where it increases as a step and remains
constant until E3 where again it increases as a
step by the same amount and at every value of En.
• In the bulk semiconductor, g(E) E1/2
179
Single quantum well (SQW) lasers
• g(E) is finite and substantial at E1
 A large concentration of
electrons in the CB can easily
occur at E1 .
• Similarly, a large concentration
of holes in the VB can easily
occur at E1 .

• Under a forward bias:
 Electrons are injected into the GaAs layer (active layer).
The electron concentration at E1 increases rapidly with
current.
Population inversion occurs quickly without the need of
large current.
180
Advantages of the SQW lasers
• The threshold current is markedly reduced.
– SQW laser: Ith ~ 0.51 mA
– Double heterostructure laser: Ith ~ 1050 mA
• Majority of electrons are at and near E1 and holes
are at and near E1
The range of emitted photon energies are very
close to E1  E1
The linewidth in the output spectrum is substantially
narrower than that in bulk semiconductor laser.
181
Multiple quantum well (MQW) lasers
182
EXAMPLE 4.13.1 A GaAs quantum well
• GaAs quantum well
– Effective mass of electron in GaAs me* = 0.07 me
– Effective mass of hole in GaAs mh* = 0.50 me
– Thickness d = 10 nm
• Calculate
– First two electron energy levels
– First hole energy level
– The change in the emission wavelength with
respect to bulk GaAs (Eg = 1.42 eV)
183
EXAMPLE 4.13.1 A GaAs quantum well
Solution
• The electron energy levels with respect to Ec in GaAs
h2n2
En  En  Ec  * 2
8me d
(6.626 1034 J  s) 2 n 2
1 eV
2



(0.0538
eV)
n
8(0.07  9.111031 Kg)(10 10 9 m) 2 1.602 10 19 J
E1  0.0537 eV, E2  0.215 eV
• The hole energy levels below Ev
2 2
34
2 2
h
n
(6.626

10
J

s)
n
1 eV
2
En  * 2 


(0.0075
eV)
n
8mh d
8(0.5  9.111031 Kg)(10 109 m) 2 1.602 10 19 J
E1  0.0075 eV
184
EXAMPLE 4.13.1 A GaAs quantum well
Solution (cont.)
• The wavelength of emission from bulk GaAs with Eg = 1.42 eV
hc (6.626 1034 J  s)(3 108 m/s)
g 

Eg (1.42 eV)(1.60 1019 J /1 eV)
 874 109 m  874 nm
• The wavelength of emission from GaAs QW
QW
(6.626 1034 J  s)(3 108 m/s)


19
(1.42
+0.0537+0.0075
eV)(1.60

10
J /1 eV)

Eg  E1  E1
hc
 839 109 m  839 nm
• The difference is
g  QW  874  839 nm  35 nm
185
4.14
VERTICAL CAVITY SURFACE
EMITTING LASERS (VCSELs)
186
Vertical cavity surface laser
(VCSELs)
• The optical cavity axis is
along the direction of
current flow rather than
perpendicular to the current
flow as in conventional
laser diodes.
• The active region length is very short (< 0.1 m)
compared with the lateral dimensions
High reflectance end mirrors (~ 90%) are needed
because the short cavity length L reduced the
optical gain of the active layer.
187
Vertical cavity surface laser
(VCSELs)
• Dielectric mirrors (~20-30
layers)  distributed Bragg
reflector (DBR).
It has a high degree of
wavelength selective
reflectance at the required
free-space wavelength 
d1 
1
4


4n1
, d2 
2
4


4n2
 n1d1  n2 d 2 

2
188
4.15
OPTICAL LASER
AMPLIFIERS
189
Laser amplifiers
• A semiconductor structure can also be used as an
optical amplifier that amplifies light waves passing
through its active region.
• The wavelength of radiation to be amplified must fall
within the optical gain bandwidth of the laser.
• Such a device would not be a laser oscillator, emitting
lasing emission without an input, but an optical amplifier
with input and output ports for light entry.
(b)
190
Traveling wave semiconductor laser
amplifier
• The ends of the optical cavity have antireflection
(AR) coating.
 The optical cavity does not act as an efficient optical
resonator, a condition for laser-oscillations.
• Light from an optical fiber is coupled into the active
region of laser structure.
• As the radiation propagates
through the active layer,
optically guided by this layer, it
becomes amplified by the
induced stimulated emissions,
and leaves the optical cavity
with higher intensity.
191
Fabry-Perot laser amplifier
• It is operated below the threshold current for laser
oscillations.
The active region has an optical gain but not sufficient
to sustain a self-lasing output.
• Light passing through such an active region will be
amplified by stimulated emissions.
Pump current
• Optical frequencies around the
resonant frequencies will
experience higher gain than
those away from resonant
frequencies.
192
4.16
HOLOGRAPHY
193
•
Dennis Gabor (1900 - 1979), inventor of holography, is standing next to his
holographic portrait. Professor Gabor was a Hungarian born British physicist
who published his holography invention in Nature in 1948 while he as at
Thomson-Houston Co. Ltd, at a time when coherent light from lasers was
not yet available. He was subsequently a professor of applied electron
physics at Imperial College, University of London.
194
Holography
• A technique of reproducing three
dimensional optical images of an object by
using a highly coherent radiation from a
laser source.
195
Principle of holography
Reference beam
Reflected wave from the cat.
Ecat will have both amplitude
and phase variations that
represent the cat’s surface.
196
Reflected wave Ecat
• If we were looking at the cat, our eyes
would register the wavefront of the
reflected wave Ecat.
• Moving our head around we would capture
different portions of the reflected wave and
we would see the cat as a three
dimensional object.
197
• The reflected wave from the cat (Ecat) is made
to interfere with the reference wave (Eref) at the
photographic plate and give rise to a
complicated interference pattern that depends
on the magnitude and phase variation in Ecat.
• The recorded interference pattern in the
photographic film is called a hologram. It
contains all the information necessary to
reconstruct the wavefronts Ecat reflected from
the cat and hence produce a three dimensional
image.
198
Principle of holography
Reference beam
Diffracted beam
Diffracted beam
The recorded interference
pattern in the photographic film
199
Virtual and real image
• One diffracted beam is an exact replica of
the original wavefront Ecat from the cat.
• The observer sees this wavefront as if the
waves were reflected from the original cat
and registers a three dimensional image of
the cat. Virtual image
• There is a second image, called real
image, which is of lower quality.
200
• d sin   m
• We can qualitatively think of a diffracted
beam from one locality in the hologram as
being determined by the local separation d
between interference pattern produced by
Ecat, the whole diffracted beam depends
on Ecat and the diffracted beam wavefront
is an exact scaled replica of Ecat.
201
• Suppose that the photographic plate is in the xy
plane.
• The reference wave
Eref ( x, y)  U r ( x, y )e j t
U r ( x, y) : the amplitude
• The reflected wave from the cat
Ecat ( x, y )  U ( x, y )e j t
• When Eref and Ecat interference, the “brightness”
of the photographic image depends on the
intensity and hence on
I ( x, y )  Eref  Ecat  U r  U  (U r  U )(U r*  U * )
2
2
202
The pattern on the photographic plat
I ( x, y )  UU *  U rU r*  U r*U  U r U *
– UU* : the intensity of the
reflected wave
– UrUr* : the intensity of the
reference wave
– Ur*U and UrU* contain the
information on the
magnitude and phase of
U(x,y)
203
• When we illuminate the hologram, I(x,y),
with the reference beam Ur , the
transmitted light wave has a complex
magnitude Ut(x,y)
U t  U r I ( x, y )  U r [UU *  U rU r*  U r*U  U r U * ]
 U t  U r (UU *  U rU r* )  (U rU r* )U  (U rU r )U *
 U t  a  bU ( x, y )  cU * ( x, y )
where a  U r (UU *  U rU r* )
b  (U rU r* )
c  (U rU r )  U r2
a, b, and c are constants.
204
The complex magnitude of transmitted wave
• U t  a  bU ( x, y )  cU * ( x, y )
• a = Ur(UU*+UrUr*): the through beam
• bU(x,y): the diffracted wave
a scaled version of U(x,y) and represents the
original wavefront amplitude from the cat (includes
the magnitude and phase information)
wavefront reconstruction
the virtual image
• cU*(x,y): the diffracted wave
Its amplitude is the complex
conjugate of U(x,y)
the real image
the conjugate image
205