Stimulated Emission Devices:
LASERS
Kasap Chapter 4
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Contents
4.1 Stimulated Emission and Photon Amplification
4.2 Stimulated Emission Rate and Einstein Coefficients
4.3 Optical Fiber Amplifiers
4.4 Gas Laser
4.5 The Output Spectrum of Gas Laser
4.6 Laser Oscillation Conditions
4.7 Principle of the Laser Diode
4.8 Heterostructure Laser Diodes
4.9 Elementary Laser Diode Characteristics
4.10 Steady State Semiconductor Rate Equation
4.11. Light Emitters for Optical Fiber Communications
4.12 Single Frequency Solid State Lasers
4.13 Quantum Well Devices
4.14 Vertical Cavity Surface Emitting Lasers (VCSELs)
4.15 Optical Laser Amplifiers
4.16 Holography
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Spontaneous emission:
random direction
E1 is empty
As if e- is oscillating in freq. v
Stimulated emission: induced
Two photons are
in phase, same direction
same polarization, same energy
Incoming e- couples to the e- in E2
Basis for photon amplification
Optical Cavity feedback
Re-absorbed? => population inversion
high optical intensity
not in two level systems
4.1 Simulated Emission and
Photon Amplification
(in steady state: R12= R21 )
E2
hυ
hυ
E2
E2
h υ In
hυ
Out
hυ
E1
Fig. 4.1 (a) Absorption
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E1
E1
(b) Spontaneous emission (c) Stimulated emission
Absorption, spontaneous (random photon) emission and stimulated
emission.
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Three level system
LASER: Light Amplification by Stimulated Emission of Radiation
e.g. ruby laser: chromium ions Cr3+ in Al2O3 crystal (See p.6)
Feedback: silvered mirror and partially silvered mirror
Pumping
(optical)
(Note: usually, more efficient in four level systems )
E3
E3
hυ32
hυ 13
E3
E2
E2
Metastable
state (long-
E2
E1
(a)
•
E2
IN
OUT
hυ 21
lived state)
E1
E3
E1
(b )
(c)
http://www.llnl.gov/nif/library/aboutlasers/how.html
hυ 2
E1
Coherent photons
(d)
Fig. 4.2
The principle of the LASER. (a) Atoms in the ground state are pumped up to the energy level E3 by
incoming photons of energy hυ13 = E3 –E1 . (b) Atoms at E3 rapidly decay to the metastable state at
energy level E2 by emitting photons or emitting lattice vibrations; hυ32 = E3 –E2 . (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 hυ21 = E2 –E1 can initiate stimulated
emission. Photons from this stimulated emission can themselves further stimulate emissions leading to 5an
avalanche
stimulated emissions and coherent photons being emitted.
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Ni : atoms / unit volume at Ei
Upward transition rate
R12 = B12 N1 ρ (hν )
4.2 Stimulated Emission Rate
and Einstein Coefficients
Theodore Harold Maiman was born in 1927
in Los Angeles, son of an electrical
engineer. He studied engineering physics at
Colorado University, while repairing
electrical appliances to pay for college, and
then obtained a Ph.D. from Stanford.
Theodore Maiman constructed this first
laser in 1960 while working at Hughes
Research Laboratories (T.H. Maiman,
"Stimulated optical radiation in ruby lasers",
Nature, 187, 493, 1960). There is a vertical
chromium ion doped ruby rod in the center
of a helical xenon flash tube. The ruby rod
has mirrored ends. The xenon flash provides
optical pumping of the chromium ions in
the ruby rod. The output is a pulse of red
laser light. (Courtesy of HRL Laboratories,
LLC, Malibu, California.)
6
B12 B21, A21 : Einstein Coefficients
ρ ( hν ) : photon energy density / freq.
= # photons / volume at hv
Downward transition rate
R21 = A21 N 2 + B21 N 2 ρ (hν )
spontaneous
stimulated
Thermal equilibrium: no change with time in the populations at E1 and E2
R12 = R21
A useful LASER medium must have a higher efficiency of
stimulated emission compared with spontaneous emission
and absorption.
⎡ (E − E1 ) ⎤
N2
= exp ⎢− 2
⎥
N1
k BT ⎦
⎣
(only) in thermal equilibrium, by
8πhν 3
=> ρ eq (hν ) =
Plank’s black body radiation
⎡ ⎛ hν ⎞ ⎤
distribution law
⎟⎟ − 1⎥
c 3 ⎢exp⎜⎜
=> B12 = B21
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Boltzmann statistics
A21 8πhν 3
=
B21
c3
⎣
⎝ k BT ⎠
⎦
(ρ is much larger in laser operation)
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A21 8πhν 3
=
B21
c3
B12 = B21
Ratio of stimulated to spontaneous emission
R21 ( stim) B21 N 2 ρ (hν ) B21 ρ (hν )
=
=
R21 ( spon)
A21 N 2
A21
=
c3
ρ ( hν )
8π hν 3
4.3 Optical Fiber Amplifiers
Ratio of stimulated emission to absorption
R21 ( stim)
N
= 2
R12 (absorp ) N1
Long haul communication suffers attenuation.
⇒Regenerate signal by Optical-Electrical-Optical
Pros:
Cons:
⇒Amplify optical signal directly => optical amplifier
Pros:
Cons:
Optical Cavity provides
1. feedback 2. high optical intensity
R21(stim) > R12 (absorp) =>
Stim. emission >> spon. emission => large photon concentration
achieved by optical cavity
EDFA: erbium (Er3+) doped fiber amplifier
Other dopants: e.g. Nd3+
Splicing: fibers are fused together
Population inversion => depart from thermal equilibrium (negative abs. temp.)
The laser principle is based on non-thermal equilibrium.
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Er 3+ -doped
fiber (10 - 20 m)
Optically pumped (usu. by LD)
Optical gain:
1240 (eV ⋅ nm)
E = hv =
λ (nm)
Gop = K ( N 2 − N1 )
K: dep. on pumping intensity
Energy of the Er3 + ion
in the glass fiber
Pump 0.80 eV
E2: long-lived ~ 10ms
E2
1550 nm
0
Optical
isolator
λ = 1550 nm
Signal out
λ = 1550 nm
Termination
A simplified schematic illustration of an EDFA (optical amplifier). The
erbium-ion doped fiber is pumped by feeding the light from a laser pump
diode, through a coupler, into the erbium ion doped fiber.
?1999 S.O. Kasap,
E1
Energy diagram for the Er3+ ion in the glass fiber medium and light amplification
by stimulated emission from E 2 to E1. Dashed arrows indicate radiationless
transitions (energy emission by lattice vibrations)
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Splice
Optoelectronics
(Prentice Hall)
1550 nm
Out
In
Fig. 4.3
Splice
Fig. 4.4
Non-radiative decay
980 nm
Wavelength-selective
coupler
Pump laser diode
λ = 980 nm
E ′3
E3
1.54 eV
1.27 eV
Optical
isolator
Signal in
E’3 at 810 nm is less efficient.
Photodetector: monitor the pumping level
Closely spaced collection of several levels:
1525-1565 nm (40nm) used in WDM system
gain flattening to overcome non-uniform gain
Pumping at 1480 nm is also used
Gain efficiency: 8-10 dB/mW
e.g. 103 gain by a few mW
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12
•
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).
4.4 Gas LASERS: The He-Ne
LASER
He-Ne laser @632.8nm (red) from Ne atoms, He used to excite Ne
Ne: 1s22s22p6 or 2p6
He: 1s2
excited Ne: 2p55s1
Excited He: 1s12s1
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Flat mirror (Reflectivity = 0.999)
Concave mirror (Reflectivity = 0.985)
Very thin tube
He-Ne gas mixture
He
1
Laser beam
20.61 eV
Ne
1
(1s 2s )
Collisions
(2p5 5s1 )
20.66 eV
632.8 nm
Lasing emission
(2p5 3p1 )
Fast spontaneous decay
Fig. 4.5
Current regulated HV power supply
~600 nm
A schematic illustration of the He-Ne laser
(2p5 3s1 )
Electron impact
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Collisions with the walls
Using dc or RF high voltage:
He atoms to become excited by collisions with drifting electrons
He + e − → He* + e −
Excited He*: 1s12s1 parallel spin, metastable,
not allowed to simply decay back to ground state,
Excited He collides with a Ne atom
He* + Ne → He + Ne*
=> population inversion of Ne
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0
Fig. 4.6
15
(1s 2 )
(2p6 )
Ground states
The principle of operation of the He-Ne laser. He-Ne laser energy levels
(for 632.8 nm emission).
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16
He
(1s12s1)
(1s12s1)
Ne
(2p55s1)
20.6 eV
Collisions
Ne(2p53p1): 10 closely spaced energy levels
(2p54p1)
red
543.5 nm
1523 nm
1152 nm
Lasing emissions
3.39 μm
(2p54s1)632.8 nm
19.8 eV
Ne(2p55s1): 4 closely spaced energy levels
green
Ne(2p53s1) is a metastable level,
to ground state by colliding with the walls of laser tube
=> narrow tube is better
1118 nm
Fast spontaneous decay
~600 nm
(2p53s1)
(metastable, requiring spin flip to
return to 2p6)
Typ. He-Ne laser
He: Ne = 5:1 several torrs
longer and narrower tube is better
99.9% flat reflecting mirror, and 99% concave reflecting mirror, convergent lens
diameter: 0.5-1 mm
divergence: 1 mrad few milliwatts
polarized light by Brewster angle
Laser radiation
Collisions with the walls
0
2
(1s )
θ
(2p6)
Laser tube
Ground states
Various lasing transitions in the He-Ne laser
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IR: 3.39 μm
prevented by freq. selective mirrors
Also energy level -- Ne(2p54p1)
(2p53p1)
Electron impact
Red: 632.8 nm
Green: 543 nm
Gaussian beam
Δr
L
2θ =
4λ
π ( 2 wo )
θ
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?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Example 4.4.2 Laser beam divergence
Typical He-Ne laser:
output beam diameter = 1 mm
divergence = 1 mrad
What is the diameter of the beam at a distance of 10m?
Example 4.4.1 Efficiency of the HeNe laser
Typical low-power He-Ne laser tube:
power = 5mW
operating dc voltage = 2000V
current = 7 mA
What is the efficiency?
We assume the laser beam is a cone, as the below figure,
with angle 2θ =1 mrad.
As the light is propagating, the spot size will be bigger,
output power
5 ×10-3
Efficiency =
=
=0.036%
input power 7 × 10-3 × 2000
Δr
,
L
⎛1
⎞
so Δr = (10m)tan ⎜ 10−3 rad ⎟ = 5mm
⎝2
⎠
so the diameter of the spot is 11mm.
with the relationship tanθ =
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θ
Laser output
Δr
L
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Finite width due to nonidealities of cavities
acoustic and thermal fluctuations of L
nonideal end mirror (R<100%)
4.5 The Output Spectrum of a Gas Laser
Relative intensity
Optical Gain
Average kinetic energy of molecules: (3/2)kBT
Doppler effect:
⎛
moving
v1 = v0 ⎜1 −
away
⎝
vx ⎞
⎟
c ⎠
Doppler
broadening
(a)
λο
⎛ v ⎞
moving
v2 = v0 ⎜1 + x ⎟
toward
c ⎠
⎝
(c)
λ
λ
δλm
axial (longitudinal) modes
Doppler broadened linewidth: Δv approx. v2-v1
Allowed Oscillations (Cavity Modes)
Optical gain lineshape around λ0= c/v0
(b)
m(λ/2) = L
Δv = 2-5GHz for many gas lasers, He-Ne laser ~0.02 Å
Fig. 4.8
Full width at half maximum
linewidth (FWHM)
Δv1/ 2 = 2v0
2k BT ln(2)
Mc 2
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cavity mode
m: mode number
⎛λ⎞
m⎜ ⎟ = L
⎝2⎠
(b)
λ
Mirror
Stationary EM oscillations
Mirror
(a) Optical gain vs. wavelength characteristics (called the optical gain curve) of the
lasing medium. (b) Allowed modes and their wavelengths due to stationary EM waves
within the optical cavity. (c) The output spectrum (relative intensity vs. wavelength) is
determined by satisfying (a) and (b) simultaneously, assuming no cavity losses.22
Example 4.5.1 Doppler broadened linewidth
axial (or longitudinal) modes
Gaussian beam
Example 4.5.1
(a)
δλm
L
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Typ. freq. width of an individual spike in He-Ne laser is ~1MHz ( low ~1kHz)
Optical gain between
FWHM points
Typ. ~1Mhz for He-Ne
Stabilized gas laser as low
as 1kHz
δλm
Cavity modes
5 modes
4 modes
Fig. 4.9
Number of laser modes
depends on how the
cavity modes intersect
the optical gain curve.
In this case we are
looking at modes
within the linewidth
Δλ1/2.
λ
901 37500 光電導論?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Calculate the Doppler broadened linewidths in frequency and wavelength for the
He-Ne laser transition for λ = 632.8nm if the gas discharge temperature is about
127°C. The atomic mass of Ne is 20.2 g/mol. The laser tube length is 40 cm.
Find: (a) the linewidth in the output wavelength spectrum
(b) the mode number m of the central wavelength
(c) the separation between two consecutive modes
(d) number of modes within the lindwidth Δλ1/2
Optical gain between
FWHM points
(a)
(b)
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δλm
Cavity modes
5 modes
4 modes
λ
Number of laser modes
depends on how the
cavity modes intersect
the optical gain curve.
In this case we are
looking at modes
within the linewidth
Δλ1/2.
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4.6 LASER Oscillation Conditions
Solution of Example 4.5.1
Δν 1 2 = 2ν 0
2k BT ln 2
2(1.38 × 10−23 )(400)(ln 2)
14
2(4.748
10
)
=
×
= 1.51GHz.
(3.35 × 10−26 )(3 ×108 ) 2
Mc 2
λ
dλ
c
From
=− 2 =−
⇒ Δλ1 2 ≈ Δν 1 2 − λ = 2.02 × 10−12 m.
ν
dν
ν
ν
And from m(λ / 2) = L, we know
δλm = λm − λm +1 =
hυ
E1
δλm =
x
5 modes case
λ
2
0
= (632.8 × 10−9 m) 2 /(2 × 0.4m) = 5.006 × 10−13 m
2L
The number of modes in the FWHM of optical gain lineshape is:
δλm
≈ 4.03 ⇒ There are 4 or 5 modes.
4 modes case
A. Optical Gain Coefficient g
(Fig. 4.10)
g=
δ N ph
δP
n δ N ph
=
=
Pδ x N phδ x cN ph δ t
= Net rate of stimulated photon emission
= N 2 B21 ρ (hν ) − N1 B21 ρ (hν )
Radiation energy density
per unit freq. at hv0
ρ (hν 0 ) ≈
B12 = B21
(stimulated – absorption)
(considering directional wave
=> spon. emission is neglected)
N ph hν 0
Δν
General optical gain coefficient at v0
g ( ν0 ) ≈ ( N 2 − N1 )
B12 = B21
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= (N 2 − N1 )B21 ρ (hν )
(optical gain coefficient)
= Net rate of stimulated photon emission
= (N 2 − N1 )B21 ρ (hν )
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
dt
P proportional to Nph (concentration of coherent photons)
= N 2 B21 ρ (hν ) − N1 B21 ρ (hν )
(b)
(a) A laser medium with an optical gain (b) The optical gain curve of the medium. The
dashed line is the approximate derivation in the text.
dN ph
exp(-αx) α: absorption coefficient
dt
υ
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4.6 LASER Oscillation Conditions
dN ph
g( υ )
υο
(a)
25
δ N ph
n δ N ph
δP
=
=
Pδ x N phδ x cN ph δ t
Δυ
P+ δP
P
δx
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exp(gx) g: optical gain coefficient
Fig. 4.10
δλm
0
g=
g( υ o)
2
0
Δλ1 2
Optical Gain
Laser medium
6
2L 2L
2L λ
−
≈
=
m m + 1 m2 2L
lineshape function:
spectral shape of the gain curve
E2
δλm
m0 = 2 L / λ0 = (2 × 0.4m) /(632.8 × 10 m) = 1.264 × 10 .
−9
A. Optical Gain Coefficient g
B21nhν0
cΔν
A21 8πhν 3
=
B21
c3
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Pf = PR
i 1 R2 exp ⎡
⎣ g ( 2 L ) ⎤⎦ exp ⎡⎣ −γ ( 2 L ) ⎤⎦
B. Optical Threshold Gain gth
Pf
Gop =
Pi
Reflecting
surface
=1
Pf
Ef
2
Pi
gth = γ +
Losses: R1, R2,
absorption (e.g. by impurities, free carriers)
scattering (defects and inhomogenities)
others
Reflecting
surface
Ei
Steady state EM oscillations
R2
Cavity axis
1
x
Gop =
Pf
Optical cavity resonator
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Δφround- trip = m(2π )
nkm ( 2 L ) ≅ m ( 2π )
⎛λ ⎞
m⎜ m ⎟ = L
⎝ 2n ⎠
Threshold pump rate
longitudinal (axial) modes
TEM00
Each transverse mode with a given p,q has a set of longitudinal modes.
Usu. m is very large ~ 106 in gas lasers
31
30
TEM10
TEM00
TEM10
(a)
Spherical
mirror
plane waves are assumed
Gaussian beams are the solutions
Off-axis modes can exist and replicate themselves
=> transverse modes or transverse electric and magnetic (TEM) modes
Note: N 2 , N1: fx of pumping
( N 2 − N1 ) ∝ pumping
Simplified description of a laser oscillator. (N2 − N1) and
coherent output power (Po) vs. pump rate under continuous
wave steady state operation.
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Optical cavity
A mode with a certain field pattern at a reflector can propagate to the other
reflector and back again and return the same field pattern.
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Threshold population
inversion
Transverse modes depend on: optical cavity dimensions, reflector sizes,..
Cartesian (rectangular) or polar (circular) symmetry about the cavity axis
(Brewster angle)
TEMpqm
highly desirable TEM00:
lowest mode, radially symmetric, lowest divergence
Phase condition for laser oscillations
Ideally, infinitely wide mirrors
Practically, finite size mirrors
Po = Lasing output power
(N2 − N1)th
Fig. 4.12
C. Phase condition and Laser modes
Approx. laser
cavity modes
=1
N2 − N1
Pump rate
901 37500 光電導論 ?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Neglect φ changes
at mirrors
Pi
B21nhν0
cΔν
Threshold population inversion
cΔν
( N 2 − N1 )th ≈ gth
B21nhν0
(N2 − N1) and Po
Pf = PR
i 1 R2 exp ⎡
⎣ g ( 2 L ) ⎤⎦ exp ⎡⎣ −γ ( 2 L ) ⎤⎦
Ef = Ei in Fig. 4.11
1 ⎛ 1 ⎞
ln ⎜
⎟
2 L ⎝ R1 R2 ⎠
R1
L
Fig. 4.11
g ( ν0 ) ≈ ( N 2 − N1 )
Threshold optical gain
Steady state conditions, no optical power loss in the round trip
=> net round-trip optical gain Gop = 1.
(b)
Wave fronts
TEM11
TEM01
(c)
TEM11
TEM01
(d)
Fig. 4.13
Laser Modes (a) An off-axis transverse mode is able to self-replicate after one round
trip. (b) Wavefronts in a self-replicating wave (c) Four low order transverse cavity
modes and their fields. (d) Intensity patterns in the modes of (c).
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32
Example 4.6.1 Threshold population inversion for the He-Ne laser
Show that the threshold population inversion ΔNth = (N2-N1)th :
ΔN th ≈ gth
8π n v τ Δv
2 2
0 sp
2
Solution of Example 4.6.1
( N 2 − N1 )th ≈ gth
c
cΔν
B21nhν0
ν0 = peak emission frequency
n = refractive index
τsp = 1/A21 = mean time for spontaneous transition
Δν = optical gain bandwidth (frequency linewidth of optical gain lineshape)
where
,
cΔν
B21nhν0
8π n ν 0τ sp Δν
c Δν
= g th
A21c 3
c2
nhν 0
3 3
8π hn ν 0
2
632.8 × 10 −9 m
from the given condition, g th = γ +
ΔNth ≈ g th
8π n2ν 02τ sp Δν
c
= 4.4 × 10 m
15
2
2
= 4.74 × 1014 Hz,
1 ⎛ 1 ⎞
−1
ln ⎜
⎟ = 0.155m
2L ⎝ R1R2 ⎠
= (0.155m −1 )
8π (4.74 × 1014 s −1 )(12 )(300 × 10 −9 s )(1.5 × 109 s −1 )
(3 × 108 ms −1 )2
−3
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4.7 Principle of the Laser Diode
Junction
n+
ΔEF= eV > Eg
Ec
Eg
eVo
Ho les in V B
Electro ns
c3
∴ ΔNth = (N2 − N1 )th ≈ g th
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Ev
EF p
3
−1
8
And ν 0 = 3 × 10 ms
Ex: He-Ne laser:
operation wavelength = 623.8 nm
tube length L = 50 cm tube diameter = 1.5 mm
mirror reflectances: 100% and 90%
linewidth Δν = 1.5 GHz
loss coefficient γ ≈ 0.05 m-1
spontaneous decay time constant τsp = 1/A21 ≈ 300 ns
n≈1
What is the threshold population inversion?
p+
A21 / B21 = 8π hν
( N 2 − N1 )th ≈ gth
Electro ns in C B
Ec
EF n
Ec
n+
p+
Eg
In v ers io n
reg io n
EF n
Ec
eV
E. Degenerate and Non-degenerate Semiconductors
n << N c and p << N v
Non-degenerate semiconductor:
The electron statistics ~the Boltzmann statistics (4)
Pauli exclusion principle can be neglected.
n > N c or p > N v
Degenerate semiconductor:
The electron statistics ~the Fermi-Dirac statistics (1)
Pauli exclusion principle becomes important.
More metal-like than semiconductor-like.
E
EF p
(a)
Ev
Impurities
forming a band
(b)
degenerate doping
n > Nc, p>Nv
V
The energy band diagram of a degenerately doped p-n with no bias. (b) Band
diagram with a sufficiently large forward bias to cause population inversion and
35
hence stimulated emission.
Fig. 4.14
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CB
EFn
Ec
CB
g(E)
Ev
Bands overlap with CB or VB.
EFn is within CB or above Ec
EFp is within VB or below Ev
(a)
Ec
Ev
EFp
VB
(a) Degenerate n-type semiconductor. Large number of donors form a
band that overlaps the CB. (b) Degenerate p-type semiconductor.
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O t l t
i
(P
ti
H ll)
(b)
36
B. Semiconductor Statistics
g ( E ) ∝ ( E − Ec )
Density of States (DOS)
f (E) =
Probability of finding eFermi-Dirac function
c
2
1
⎛ E − EF ⎞
1 + exp ⎜
⎟
⎝ k BT ⎠
(1)
ΔE F = eV
Electric work input or output per e-
f (E F ) =
However there may be no state.
1
2
Electron
concentration in CB
(E
1
n=∫
Ec + x
Ec
[
N c = 2 2πm k T / h
901 37500 光電導論
Mass Action Law
Junction
Eg
Ev
EF p
Ho les in V B
Electro ns
Electro ns in C B
EF n
Ec
Eg
Ev
In v ers io n
reg io n
EF n
Ec
eV
38
Fig. 4.14(a) degenerate doping
No applied voltage, EFp = EFn
- Diminishes the built-in potential barrier
Î SCL is no longer depleted
- In SCL, more electrons in the conduction band at energies near Ec
than electrons in the valence band near Ev
- Called inversion layer or active layer
- Incoming photons stimulate direct recombinations
More stimulated emission than absorption => optical gain
V
The energy band diagram of a degenerately doped p-n with no bias. (b) Band
diagram with a sufficiently large forward bias to cause population inversion and
39
hence stimulated emission.
Fig. 4.14
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S.O. Kasap, Optoelectronics (Prentice Hall)
(7)
Fig. 4.15(a) Population inversion
- occurs between energies near Ec and those near Ev around the junction
(b)
degenerate doping
n > Nc, p>Nv
k BT
) = ni2
Fig. 4.14(b) applied voltage V, ΔEF = eV > Eg
n+
EF p
(a)
Eg
(6)
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p+
Ec
3/ 2
37
ΔEF= eV > Eg
eVo
]
n=p=ni (intrinsic concentration)
]
n+
np = N c N v exp(−
(4)
4.7 Principle of the Laser Diode
p+
[
⎛N ⎞
1
1
E g − kT ln ⎜ c ⎟
2
2
⎝ Nv ⎠
1
Typically, N c ~ N v and E Fi ≈ Ev + E g
2
(3)
2 3/ 2
Ec
N v = 2 2πmh* k BT / h 2
E Fi = Ev +
g CB ( E ) f ( E ) dE
*
e B
Effective DOS at the
VB edge
(5)
Intrinsic semiconductor (pure crystal), n=p
⎡ E − EF ⎤
n = N c exp ⎢ − c
⎥
k BT ⎦
⎣
Effective DOS at the
CB edge
⎡ ( E − Ev ) ⎤
p ≈ N v exp ⎢ − F
⎥
k BT ⎦
⎣
* EF determines the electron and hole concentration.
(2)
⎛ (E − E f )
⎞
− E f ) >> k BT → f (E ) ≈ exp⎜ −
⎟ Boltzmann Statistics
k
T
B ⎠
⎝
Non-degenerate
Semiconductor
(# of e-<<# of states)
Hole concentration
in CB
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Eg < hv < EFn - EFp
Stimulated emission
hv >EFn - EFp
Absorption
40
Optical Cavity: feedback
Injection pumping: pumping by the forward diode current
high optical intensity
⎛λ ⎞
m⎜ m ⎟ = L
⎝ 2n ⎠
Energy
EF n
Ec
Optical gain
CB
EF n − EF p
L
Eg
Holes in VB
= Empty states
VB
Resonant frequency
Mode
hυ
0
Ev
EF p
Cleaved surface mirror
Electrons
in CB
eV
Current
p+
At T > 0
GaAs
L
At T = 0
Optical absorption
n+
GaAs
Density of states
(b)
(a)
Fig. 4.15
(a) The density of states and energy distribution of electrons and holes in
the conduction and valence bands respectively at T ≈ 0 in the SCL
under forward bias such that E Fn − E Fp > E g . Holes in the VB are empty
states. (b) Gain vs. photon energy.
Fig. 4.16
Optical P ower
Active region
(stimulated emission region)
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?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Laser
Optical P ower
I > Ith => coherent
LED
Stimulated
emission
0
λ
4.8 Heterostructure Laser Diodes
Laser
Optical P ower
Spontaneous
emission
λ
I
Ith
λ
Transparency current: stimulated emission = absorption
Threshold current Ith: optical gain in the medium can overcome
the photon losses from the cavity
(g=gth)
(typ. 32% reflecting
for n=3.6)
Typical output optical power vs. diode current (I) characteristics and the corresponding
output spectrum of a laser diode.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Problems of homojunction LD:
threshold current density Jth is too high for practical use
~500 A/mm2 for GaAs, can only be operated at very low temperature
Jth can be reduced by orders of magnitude by using heterojunction LD
901 37500 光電導論
Electrode
A schematic illustration of a GaAs homojunction laser
diode. The cleaved surfaces act as reflecting mirrors.
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?1999 S.O.
Kasap, Optoelectronics (Prentice Hall)
Optical Power
Electrode
Reduction of the threshold current by improving:
A. Rate of stimulated emission
Î carrier confinement => establish required e-h concentration for pop. inv.
B. Efficiency of the optical cavity
Î photon confinement by waveguide
Both can be achieved by heterostructured devices
LEDs vs LDs: stimulated emission > spontaneous emission (require good cavity)
Fig. 4.18
p-GaAs and p-AlGaAs are degenerately doped
43
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Fig. 4.18
n
AlGaAs
(a)
p
p
GaAs
AlGaAs
(a) A double
heterostructure diode has
two junctions which are
between two different
bandgap semiconductors
(GaAs and AlGaAs).
(~0.1 μm)
Electrons in CB
Ec
ΔEc
Ec
1.4 eV
2 eV
2 eV
Ev
(b)
Ev
Holes in VB
Refractive
index
Photon
density
(c) Higher bandgap
materials have a
lower refractive
index
Δn ~ 5%
Active
region
(c)
(b) Simplified energy
band diagram under a
large forward bias.
Lasing recombination
takes place in the pGaAs layer, the
active layer
p-GaAs: 870-900 nm (dep. on doping, can also be AlyGa1-yAs for controlled wavelength)
Stripe contact: define current density
Cleaved reflecting surface
Gain guided
W
=> reduce Ith, better coupling to fiber
W: few μm, Ith: tens mA
(avoid Schottky junction)
Oxide insulator
p-GaAs (Contacting layer)
p-Al xGa1-xAs (Confining layer)
p-GaAs (Active layer)
n-Al xGa1-xAs (Confining layer)
n-GaAs (Substrate)
small lattice mismatch
negligible strain induced
interfacial defects
Elliptical
less nonradiative recomb. laser
beam
(d) AlGaAs layers
provide lateral optical
confinement.
(d)
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?1999
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1 ⎛ 1 ⎞
ln ⎜
⎟
2 L ⎝ R1 R2 ⎠
R=0.33 @ n=3.7 (GaAs)
Stripe geometry => poor optical confinement in lateral direction
Buried double heterostructure
Index guided: better lateral optical confinement
can be single mode by small V number
Oxide insulation
p+-AlGaAs (Contacting layer)
p-AlGaAs (Confining layer)
n-AlGaAs
p-GaAs (Active layer)
n-AlGaAs (Confining layer)
2
1
Current
paths
Substrate
3
Substrate
Electrode
Cleaved reflecting surface
Active region where J > Jt h.
(Emission region)
Fig. 4.19
Schematic illustration of the the structure of a double heterojunction stripe
contact laser diode
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Efficiency can be further improved by higher reflection in the facet => dielectric mirror
=> reduce threshold current Ith
gth = γ +
L
Stripe electrode
Example 4.8.1 Modes in a laser and the optical cavity length
AlGaAs based heterostructure laser diode
optical cavity length = 200 μm
peak radiation: 870 nm
refractive index of GaAs = 3.7
FWHM wavelength width (gain vs. wavelength) = 6 nm
Electrode
(a) What is the mode integer m of the peak radiation?
(b) What is the separation between the modes of the cavity?
(c) How many modes are there within the bandwidth?
(d) How many modes are there if cavity length = 20 μm?
n-GaAs (Substrate)
λ~900nm GaAs/AlGaAs
λ~1.3, 1.55μm InGaGaP (InP substrate)
Schematic illustration of the cross sectional structure of a buried
heterostructure laser diode.
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Finite width due to nonidealities of cavities
acoustic and thermal fluctuations of L
nonideal end mirror (R<100%)
(a)
Solution of Example 4.8.1
Relative intensity
Optical Gain
The relationship between the free-space wavelength of cavity mode
and the length of the cavity is: m
∴m =
2nL
λ
2n
λο
= 1644.4 ≈ 1644
λ
2nL 2nL 2nL λ 2
−
≈
=
2nL
m
m + 1 m2
(900 × 10 −9 ) 2
= 0.547 nm
When L = 200 μ m, δλm =
2(3.7)(200 × 10 −6 )
When L = 20 μ m, δλm = 5.47 nm
And δλm =
There are
Doppler
broadening
(a)
=L
Δλ1 2
(c)
λ
λ
δλm
axial (longitudinal) modes
Allowed Oscillations (Cavity Modes)
m(λ/2) = L
L
(b)
δλm
Fig. 4.8
δλm modes in the optical gain linewidth,
when L =200 μ m, there are 10 modes in the linewidth,
when L =20 μ m, there is only one mode in the linewidth.
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λ
Mirror
Stationary EM oscillations
Mirror
(a) Optical gain vs. wavelength characteristics (called the optical gain curve) of the
lasing medium. (b) Allowed modes and their wavelengths due to stationary EM waves
within the optical cavity. (c) The output spectrum (relative intensity vs. wavelength) is
determined by satisfying (a) and (b) simultaneously, assuming no cavity losses.50
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37500
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Kasap, Optoelectronics (Prentice Hall)
4.9 Elementary Laser Diode
Characteristics
alternating n1 & n2 with
thicknesses in quarter wavelengths
Dielectric mirror
Fabry-Perot cavity
Output Spectrum dep. on
A. Optical Resonator
L: longitudinal mode separation
W, H: lateral modes
small W, H => single mode, TEM00
divergence: smaller aperture => larger diffraction
(e.g. H in Fig. 4.21)
B. Optical Gain Curve
Length, L
Height, H
Diffraction
limited laser
beam
Fig. 4.21 The laser cavity definitions and the output laser beam
characteristics.
?1999 S.O. Kasap, Optoelectronics(Prentice Hall)
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Width W
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Relative optical power
P o = 5 mW
Temp.
Index guided
may become single mode at high I
Gain guided
still multimode even at high I
(no lateral optical confinement)
Refractive Index
cavity length
Avoid mode hops
sufficiently separated modes to reduce the chance of mode hopping
thermoelectric (TE) cooler to control device temp.
Single mode
788
P o = 3 mW
Red shift:
temperature-induced gain shifting
due to heating
(a)
778
786
λo
(nm)
784
782
776
782
8
Slope efficiency
P0
I − I th
Conversion efficiency
=
0
0
20
40
60
80
I (mA)
40
50
P/(I-Ith)
Conversion efficiency: output optical power / input electric power
<1 W/A, typ.
4
Fig. 4.23
30
Case temperature (° C)
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ηslope =
2
30
50 20
40
Case temperature (° C)
53
Slope efficiency
6
30
50 20
40
Case temperature (° C)
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Threshold current increases exponentially with the absolute temperature.
50 °C
20
Peak wavelength vs. case temperature characteristics. (a) Mode hops in the output
spectrum of a single mode LD. (b) Restricted mode hops and none over the temperature
range of interest (20 - 40 °C). (c) Output spectrum from a multimode LD.
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S.O. Kasap, Optoelectronics (Prentice Hall)
0 °C
25 °C
Mode hopping
Fig. 4.24
Fig. 4.22
Output spectra of lasing emission from an index guided LD.
At sufficiently high diode currents corresponding to high
optical power, the operation becomes single mode. (Note:
Relative power scale applies to each spectrum individually and
not between spectra)
10
Multimode
(c)
778
P o = 1 mW
λ (nm)
P o (mW)
Single mode
(b)
780
780
center wavelength
output optical power
input electrical power
may be as high as 30-40%
Output optical power vs. diode current as three different temperatures. The
threshold current shifts to higher temperatures.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Example 4.9.1 Laser output wavelength variations
GaAs: n = refractive index
dn/dT = 1.5 x 10-4 K-1
emitted wavelength = 870 nm
Estimate the change in the emitted wavelength per degree
change in the temperature between mode hops
Consider one specific wavelength λm , m(
λm
2n
d λm
d ⎡ 2 ⎤ 2 L dn λm dn
then
=
nL ≈
=
,
dT
dT ⎢⎣ m ⎥⎦ m dT
n dT
d λm 870nm
≈
1.5 ×10−4 K −1 = 0.035nmK −1.
dT
3.7
)=L
Note: in reality, n depends on the optical gain of the cavity medium Î dλ/dT ↑
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4.10 Steady State Semiconductor
Rate Equation
From
Consider double heterojunction (DH) laser diode (LD) (Fig. 4.19)
Rate of electron inject by current (neglect non-radiative recombination)
= Rate of spontaneous emission + Rate of stimulated emission
n: injected electron concentration
I
n
=
+ CnN ph (1)
C: constant dep. on B21
edLW τ sp
Nph: coherent photon encouraged
Active layer
by the optical cavity (mode)
τ ph
= CnN ph
At threshold nth =
Rate of the coherent photon loss = Rate of stimulated emission
τph: due to trans. thru end-faces, scattering, absorption
= CnN ph
1
Cτ ph
Scattering, absorption, out-coupling
Stimulated emission just balanced by (all loss + spontaneous emission)
Right before lasing, coherent photon by the cavity: Nph = 0
I = I th =
nth edLW
τ sp
from
I
n
=
+ CnN ph
edLW τ sp
∴ dLW Î Ith
Steady state:
τ ph
N ph
(steady state)
Under steady state operation
N ph
Threshold: simulated emission just overcomes
the spontaneous emission + total loss mechanisms in τph
Smaller Ith: heterostructure
and stripe geometry
Cleaved reflecting surface
W
L
Stripe electrode
Oxide insulator
p-GaAs (Contacting layer)
p-Al xGa1-xAs (Confining layer)
p-GaAs (Active layer)
n-Al xGa1-xAs (Confining layer)
n-GaAs (Substrate)
Elliptical
laser
beam
2
1
Current
paths
Substrate
3
Cleaved reflecting surface
Active region where J > Jth.
(Emission region)
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Above threshold
O/P optical
power
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I − I th
= Cnth N ph
edLW
J=I/WL => N ph =
τ ph
ed
(J − J th )
⎛1
⎞
⎜ N ph ⎟(Cavity Volume)(Photon energy )
2
⎝
⎠
(1 − R )
P0 =
Δt
Δt=nL/c
n Po
4.11 Light Emitters for Optical
Fiber Communications
Long haul: LD
narrow linewidth and high output power
Threshold population
inversion
n
Schematic illustration of the the structure of a double heterojunction stripe
contact laser diode
Short haul: LED
larger Δλ
simpler to drive, economic, longer lifetime
usually with multimode graded index fibers
⎡ hc 2τ phW (1 − R ) ⎤
⎥ ( J − J th )
2enλ
⎢⎣
⎥⎦
=> Laser diode equation P0 = ⎢
nth
Substrate
Electrode
Comparison, see Fig. 4.26 and Table 4.1
P o = Lasing output power ∝ Nph
I
Fig. 4.25
Ith
Nph: coherent photon concentratoin
Simplified and idealized description of a semiconductor laser
diode based on rate equations. Injected electron concentration
n and coherent radiation output power Po vs. diode current I.
901 37500 光電導論
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Optoelectronics (Prentice Hall)
A laser diode pigtailed to a fiber. Two of the leads are for
a back-facet photodetector to allow the monitoring of the
laser output power (Courtesy of Alcatel)
59
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rise time: light output 10% to 90%
LD has very short τr Î used when wide bandwidth (bit rate) is required
Laser diode
One mode by suppressing unwanted mode through design, Δλ ~ 0.01~0.1nm
Δλ < 0.01 nm in single frequency LD
Light power
Laser diode
Long haul
10 mW
LED
5 mW
Short haul
Current
0
50 mA
Fig. 4.26
100 mA
Typical optical power output vs. forward current
for a LED and a laser diode.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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4.12 Single Frequency Solid State Lasers
Distributed Bragg reflector (DBR) laser
q
λB
n
= 2Λ
λB: Bragg wavelength
q: diffraction order
Corrugated
dielectric structure
2nL
( m + 1)
Ideally, perfectly symmetric (b)
Practically, asymmetry induced on purpose (c)
Λ
q( λ B /2n) = Λ
63
Commercially available 1.55μm Δλ~0.1nm
L>>Λ λm~λB
Λ
Corrugated grating
(b)
Fig. 4.27
(a) Distributed Bragg reflection (DBR) laser principle. (b) Partially reflected waves
at the corrugations can only constitute a reflected wave when the wavelength
satisfies the Bragg condition. Reflected waves A and B interfere constructive when
q( λB/2n) = Λ.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
901 37500 光電導論
Right and left traveling waves are coupled
set up a standing wave if they are coherently coupled
(round-trip phase change =2π)
λB2
m=0 dominates
because relative threshold gain of higher modes is higher
A
B
Active layer
Distributed feedback (DFB) laser
λm = λB ±
Distributed Bragg
reflector
(a)
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Ideal lasing emission
Optical power
Guiding layer
Active layer
0.1 nm
(a)
(b)
λB
λ
λ (nm)
(c)
Fig. 4.28
(a) Distributed feedback (DFB) laser structure. (b) Ideal lasing emission output. (c)
Typical output spectrum from a DFB laser.
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Kasap, Optoelectronics (Prentice Hall)
Example 4.12.1 DFB Laser
DFB laser has: corrugation period Λ = 0.22 μm
grating length = 400 μm
effective refractive index of the medium = 3.5
For a first order grating, calculate:
(a) Bragg wavelength
(b) mode wavelength
(c) mode separation
Cleaved-coupled-cavity (C3) laser
Two lasers are pumped by different currents
Only those waves that can exist as modes in both cavities are allowed
Wide separation between the modes => single mode operation
Cavity Modes
In L
Active
layer
L
D
(a)
Fig. 4.29
The Bragg wavelength is :
λ
In D
λB =
λ
In both
L and D
λm = λB ±
(b)
Cleaved-coupled-cavity
λB
(1.54 μ m) 2
(0 + 1)
2nL
2(3.5)(400 μ m)
∴ The wavelength of the mode with m = 0 is:
λ
(C3)
2Λn 2(0.22 μ m)(3.5)
=
= 1.540 μ m
1
q
laser
(m + 1) = 1.54 ±
λ0 = 1.5392 μ m or 1.5408μ m.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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4.13 Quantum Well Devices
ΔEc, ΔEv dep. on materials
E = Ec +
Fig. 4.30
Ultra thin, typ. < 50nm, narrow bandgap semiconductor devices
Lattice match is required
E
Dy
h 2 n y2
h2n2
h 2 nz2
+
+
8me*d 2 8me* D y2 8me* Dz2
Step density of states
a large concentration of electrons (holes) can easily occur at E1 ( E’1 )
population inversion occurs quickly without a large current
Advantages
Ith is markedly reduced (eg. 0.5-1mA in SQW, 10-50mA in DH laser)
linewidth is substantially narrower because e- (h+) near E1 (E’1)67
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d
Ec
d << Dx, Dy
One dimensional PE well in x-direction and as if free in yz-plane
E = Ec +
h 2 n y2
h2n2
h 2 nz2
+
+
8me*d 2 8me* D y2 8me* Dz2
D z AlGaAs
AlGaAs
Δ² E c n = 2
E
d
n=1
Eg2
z
Eg1
y
x
Ev
QW
QW
Δ² E v
Bulk
g(E)
Density of states
GaAs
Fig. 4.30 (a)
Bulk
E3
E2
E1
(b)
(c)
A quantum well (QW) device. (a) Schematic illustration of a quantum well (QW) structure in which a
thin layer of GaAs is sandwiched between two wider bandgap semiconductors (AlGaAs). (b) The
conduction electrons in the GaAs layer are confined (by Ec) in the x-direction to a small length d so
that their energy is quantized. (c) The density of states of a two-dimensional QW. The density of states
is constant at each quantized energy level.
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S.O. Kasap,
Optoelectronics (Prentice Hall)
Multiple quantum well (MQW) laser
E
Ec
Larger volume
MQW spectrum < bulk spectrum, but
MQW still not necessarily single mode
=> MQW +DFB
E1
h υ = E 1 ? E′ 1
Active layer
E′ 1
Ev
E
SQW: power not large
Enhancement by MQW
Barrier layer
Ec
Fig. 4.31
In single quantum well (SQW) lasers electrons are
injected by the forward current into the thin GaAs
layer which serves as the active layer. Population
inversion between E1 and E ′1 is reached even with a
small forward current which results in stimulated
emissions.
Ev
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Optoelectronics
Fig. 4.32
A multiple quantum well (MQW) structure.
Electrons are injected by the forward current
into active layers which are quantum wells.
A 1550 nm MQW-DFB InGaAsP
laser diode pigtail-coupled to a
fiber. (Courtesy of Alcatel.)70
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?1999 S.O.
Kasap, Optoelectronics (Prentice Hall)
(Prentice Hall)
Example 4.13.1 A GaAs quantum well
Electron effective mass: 0.07me (me: electron mass in vacuum)
Hole effective mass: 0.5me
Quantum well thickness = 10 nm
Find: (a) first two electron energy levels
(b) hole energy (below Ev)
(c) change in the emission wavelength w.r.t. bulk GaAs
(energy gap of bulk GaAs = 1.42 eV)
Solution of Example 4.13.1
ε n = En − Ec =
h2n2
, where n is the quantum number,
8me*d 2
from the given conditions, we get ε1 = 0.0537eV , ε 2 = 0.215eV .
And the energy of holes is ε n' =
h2n2
,
8mh* d 2
from the given conditions, we get ε 1' = 0.0075eV .
Now we calculate the emission wavelength of bulk GaAs,
from Eg =1.42eV, we get λg =
hc
= 874nm.
Eg
Now for GaAs quantum well,
λQW =
hc
= 839nm,
Eg + ε1 + ε 1'
The emission wavelength shifts 35nm.
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4.14 Vertical Cavity Surface
Emitting Lasers (VCSELs)
See Fig. 4.33
Dielectric mirrors
n1d1 + n2 d 2 =
Contact
n1d1 + n2 d 2 =
λ/4n 2
λ/4n 1
1
λ
2
Dielectric mirror
Active layer (MQW)
1
λ
2
Dielectric mirror
(DBR structure)
High reflectance end mirrors are needed due to short cavity length
20-30 or so layers to obtain the required reflectance (99%)
Substrate
Active layer: very thin <0.1μm, likely MQW
e.g. 980 nm InGaAs in GaAs substrate
Contact
(pumping EDFA)
Circular cross-section
Height: several microns => single mode longitudinally (maybe not laterally)
Surface emission
Fig. 4.33 A simplified schematic illustration of a vertical cavity
Maybe not laterally, dep. on lateral size In practice, single lateral mode
Diameter < 8 μm spectrum width Δλ~0.5 nm
surface emitting laser (VCSEL).
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Microlaser: cavity dimension in the microns range
Matrix emitter: arrayed microlaser, broad area surface emittiner laser sourse
optical inter connect, optical computing
higher optical power than conventional laser, few watts
4.15 Optical Laser Amplifiers
Amplified by induced stimulated emission
Pumped to achieve optical gain
(i.e. population inversion)
Random spontaneous emission
=> noise => filtered out
Typ. ~20dB, dep. on the AR coating
Pump current
Active region
AR = Antireflection
coating
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Under threshold current
Amplified by stim. Emission
Multiple reflection
Higher gain, less stable
Signal out
Signal in
An 850 nm VCSEL diode.
(Courtesy of Honeywell.)
SEM (scanning electron microscope) of the first low-threshold VCSELs
developed at Bell Laboratories in 1989. The largest device area is 5 µm
75
in diameter (Courtesy of Dr. Axel Scherer Caltech.)
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?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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AR
(a) Traveling wave amplifier
Partial mirror
Partial mirror
(a) Fabry-Perot amplifier
Fig. 4.34 Simplified schematic illustrations of two types of laser amplifiers
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?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
76
d (sin θ m ± sin θ i ) = mλ ;
4.16 Holography
First-order
Zero-order
A technique of reproducing three dimensional (3D) optical image of an object
by using a highly coherent radiation from a laser source
Fig. 4.35
Ecat: both amplitude and phase variations representing the cat’s surface
Reflected Ecat interferes with Eref at photographic plate
Interference pattern depends on the magnitude and phase variations in Ecat.
Hologram: recorded interference patter in the photographic film
Incident
light wave
Thru beam: real image
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(b) Reflection grating
(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A
reflection grating. An incident light beam results in various "diffracted" beams. The
zero-order diffracted beam is the normal reflected beam with an angle of reflection equal
to the angle of incidence.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
78
Ecat ( x, y ) = U (x, y )e jωt
Observer
Ecat (x,y)
Eref (x,y)
Laser
beam
Laser
beam
Cat
Eref (x,y)
2
Through beam
I ( x, y ) = UU * + U rU r* + U r*U + U rU *
Illuminating the hologram with reference beam
(
U t ∝ U r I ( x, y ) = U r UU * + U rU r* + U r*U + U rU *
Hologram
i.e.
Real cat imag
Virtual cat image
)
Pattern on the hologram
Ecat (x,y)
(a)
(
I ( x, y ) = Eref + Ecat = U r + U = (U r + U ) U r* + U *
2
(b)
A highly simplified illustration of holography. (a) A laser beam is made to interfere with
the diffracted beam from the subject to produce a hologram. (b) Shining the laser beam
through the hologram generates a real and a virtual image.
(
) (
)
)
U t ∝ U r I ( x, y ) = U r UU * + U rU r* + U rU r* U + U r2U *
U t ∝ a + bU ( x, y ) + cU ( x, y )
*
a = Ur(UU*+UrUr*) constant: through beam
bU: scaled version of U: virtual image
cU*: complex conjugate of U: real image, conjugate image
Eye can not detect phase shift. Observer always sees the positive image.
?1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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m = 0 Zero-order
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Mirror
Fig. 4.35
m = 1 First-order
m = -1 First-orde
Eref ( x, y ) = U r ( x, y )e jωt
Photographic plate
(8)
First-order
(a) Transmission grating
(Bragg condition d sinθ = mλ )
Diffracted beam: virtual image
Incident
light wave
m = 0, ± 1, ± 2,...
Holography is a method of wavefront reconstruction.
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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. (From M.D.E.C. Photo Lab, Courtesy AIP Emilio Segrè
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Visual Archives, AIP.)
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