Lecture 11: Semiconductor lasers and light

Lecture 11: Semiconductor lasers and light-emitting diodes



Electroluminescence in pn junctions
Light-emitting diodes
Semiconductor lasers
References: This lecture partially follows the materials from Photonic Devices, Jia-Ming
Liu, Chapter 13. Also from Fundamentals of Photonics, 2nd ed., Saleh &
Teich, Chapters 16-17. And from Physics of Optoelectronics, Michael A.
Parker, CRC Taylor and Francis, pp.47-78
1
Electroluminescence in pn
junctions
2
Electroluminescence

The spontaneous emission of light due to the radiative
recombination from within the diode structure is known as
electroluminescence (EL).

The term electroluminescence is used when the optical
emission results from the application of an electric field.

The light is emitted at the site of carrier recombination which
is primarily close to the junction, although recombination
may take place through the whole diode structure as carriers
diffuse away from the junction region.

The amount of radiative recombination and the emission area
within the structure is dependent upon the semiconductor
materials used and the fabrication of the device.
3
3
Forming a p+-n+ junction in thermal equilibrium
W
p+
p+ --
n+
-
x
++
+ + n+
++
++
E
Ecp
energy
Eg
EFn
Ecn
Ecp
Eg
Evp
EFp
eV0
Evp
EFp
eV0 > Eg
EFn
Ecn
Eg
Eg
Evn
position
depletion
Evn
position
4
Electron energy
Heavily doped p+-n+ junction in thermal equilibrium
Eg
eV0 > Eg
EFc
EFv
Eg
depletion
position
5
Electron energy
Heavily doped p+-n+ junction under forward bias
injection
Eg
h
e(V0-V)
EFc
h
Eg
eV > Eg
EFv
injection
Active
position
• When a forward bias nearly equals or exceeding the bandgap
voltage there is conduction. (eV > Eg)
• At high injection carrier density in such a junction there is an
active region near the depletion layer that contains
simultaneously degenerate populations of electrons and holes.
• The injection carrier may be largely electrons injected into the p-n
region because of their larger mobility.
6
Radiative recombination in a forward-biased p-n junction

The increased concentration of minority carriers in the
opposite type region in the forward-biased p-n diode of
direct-bandgap materials leads to the radiative
recombination of carriers across the bandgap.

The normally empty electron states in the conduction band of
the p-type material and the normally empty hole states in the
valence band of the n-type material are populated by injected
carriers which recombine with the majority carriers across the
bandgap.

The energy released by this electron-hole recombination is
approximately equal to the bandgap energy Eg.
7
7
Light-emitting diodes
•
•
•
•
•
•
•
Internal quantum efficiency
Extraction efficiency
External quantum efficiency
Power conversion efficiency
Responsivity
Spectral distribution
Modulation
8
LED, SOA, LD
n
p
Light-emitting diode
n
p
Semiconductor
Optical Amplifier
n
p
Laser Diode
9
Light-emitting diodes (LEDs)
• The light output of an LED is the spontaneous emission generated by
radiative recombination of electrons and holes in the active region of
the diode under forward bias.
• The semiconductor material is direct-bandgap to ensure high
quantum efficiency, often III-V semiconductors.
• An LED emits incoherent, non-directional, and unpolarized
spontaneous photons that are not amplified by stimulated emission.
• An LED does not have a threshold current. It starts emitting light as
soon as an injection current flows across the junction.
10
Electron energy
Heavily doped p-n junction under forward bias
injection
e(V0-V)
EFc
Eg
h
h
Eg
eV > Eg
EFv
injection
• The internal photon flux:  = int i/e
position
int: int. quantum efficiency
(injection electroluminescence)
11
Internal quantum efficiency
• The internal quantum efficiency int of a semiconductor material:
the ratio of the radiative electron-hole recombination coefficient to the
total (radiative and nonradiative) recombination coefficient.
• This parameter is significant because it determines the efficiency of
light generation in a semiconductor material.
• Recall that the total rate of recombination = r n p [cm-3 s-1]
• If the recombination coefficient r is split into a sum of radiative and
nonradiative parts, r = rr + rnr, the internal quantum efficiency is
int = rr / r = rr / (rr + rnr)
12
Recombination lifetimes
• The internal quantum efficiency may also be written in terms of the
recombination lifetimes as  is inversely proportional to r.
• Define the radiative and nonradiative recombination lifetimes r and nr
1/ = 1/r + 1/nr
• The internal quantum efficiency is then given by rr/r = (1/r)/(1/)
int =  / r = nr / (r + nr)
*Semiconductor optical sources require int to be large
(in typical direct bandgap materials r ≈ nr).
13
Order-of-magnitude values for recombination coefficients
and lifetimes
material
rr(cm3 s-1)
r
nr

int
Si
10-15
10 ms
100 ns
100 ns
10-5
GaAs
10-10
100 ns
100 ns
50 ns
0.5
*assuming n-type material with a carrier concentration no = 1017 cm-3 and defect centers with a
concentration 1015 cm-3 at T = 300 K
• The radiative lifetime for bulk Si is orders of magnitude longer than its
overall lifetime because of its indirect bandgap (electron momentum
mismatched). This results in a small internal quantum efficiency.
• For GaAs, the radiative transitions are sufficiently fast because of its
direct bandgap (electron momentum matched), and the internal quantum
14
efficiency is large.
Electroluminescence in the presence of carrier injection
• The internal photon flux  (photons per second), generated within a
volume V of the semiconductor, is directly proportional to the carrierpair injection rate R (electron-hole pairs/cm3-s).
• The steady-state excess-carrier concentration n = R, where 
is the total recombination lifetime (1/ = 1/r + 1/nr).
• The injection of RV carrier pairs per second therefore leads to the
generation of a photon flux  = int RV photons/s.
 = int RV = int V n/ = V n/r
Or
 = int V (i/e)/V = int i/e
15
Enhancing the internal photon flux
  n/r
=> Need higher n, shorter r
1. The excess carriers in a LED with homojunction (same materials on
the p and n sides) are neither confined nor concentrated but are spread
by carrier diffusion.
 The thickness of the active layer in a homojunction is normally
on the order of one to a few micrometers, depending on the diffusion
lengths of electrons and holes.
2. There is no waveguiding mechanism in the structure for optical
confinement. It is therefore difficult to control the spatial mode
characteristics (essential for laser diode).
16
A p+-n+ homojunction under forward bias
Eg
h
Optical
Refractive
Excess carrier
field
index
distribution
distribution profile
EFv
EFc
Eg
~ 1 - few m
Photons generated
can be absorbed
outside the active
region
carriers diffuse
x
homostructure
x
no waveguiding
x
17
Double heterostructures
• Very effective carrier and optical confinement can be simultaneously
accomplished with double heterostructures. A basic configuration can
be either P-p-N or P-n-N (the capital P, N represents wide-gap materials,
p, n represents narrow-gap materials). The middle layer is a narrow-gap
material. (e.g. Ga1-yAlyAs-GaAs-Ga1-xAlxAs)
• Almost all of the excess carriers created by current injection are
injected into the narrow-gap active layer and are confined within this
layer by the energy barriers of the heterojunctions on both sides of the
active layer.
• Because the narrow-gap active layer has a higher refractive index than
the wide-gap outer layers on both sides, an optical waveguide with the
active layer being the waveguide core is built into the double
18
heterostructure.
A P+-p-N+ double heterostructure under forward bias
(GaAlAs/GaAs/GaAlAs)
EFc
Ec
Optical
Refractive
Excess carrier
field
index
distribution
distribution profile
h
EFv
Ev
wide-gap outer
layers are
transparent to the
optical wave
~ 0.1 m
carriers confined
x
~few %
Double
heterostructure
x
waveguiding
x
19
Double heterostructures
20
LED power
e.g. The radiative and nonradiative recombination lifetimes of
the minority carriers in the active region of a LED are 60 ns and
100 ns. Determine the total carrier recombination lifetime and
the power internally generated within the device when the peak
emission wavelength is 870 nm at a driving current of 40 mA.
• The total carrier recombination lifetime is given by
 = rnr / (r +nr) = 37.5 ns
• The internal quantum efficiency
int =  / r = 0.625
=> Pint = int i/e • (1240 eV-nm / 870 nm) = 36 mW!
(However, this power level is not readily out-coupled from the device! )21
Output photon flux and efficiency
• The photon flux spontaneously generated in the junction active region
is radiated uniformly in all directions. However, the flux that emerges
from the device (output photon flux) depends on the direction of
A
emission.
B
n
c
C
l1
active region
p
e.g. Ray A at normal incidence is partially reflected. Ray B at oblique
incidence suffers more reflection. Ray C lies outside the critical angle
and thus is trapped in the structure by total internal reflection.
22
• The photon flux (optical power) traveling in the direction of ray A
(normal incidence) is attenuated by the factor
1 = exp(-l1)
where  is the absorption coefficient (cm-1) of the n-type material, and l1
is the distance from the junction to the surface of the device.
• For normal incidence, reflection at the semiconductor-air boundary
permits only a fraction of the light to escape (see Fresnel reflection in
Lecture 2)
reflectance
2 = 1 – [(n-1)2/(n+1)2] = 4n / (n+1)2
where n is the refractive index of the semiconductor material.
(For GaAs, n = 3.6, 2 = 0.68. The overall transmittance for the
photon flux (power) traveling in the direction of ray A is A=1 2)
23
• The photon flux traveling in the direction of ray B has farther to travel
suffers a larger absorption;
a larger incident angle at the semiconductor-air interface
=> a greater Fresnel reflection loss
=> B < A
• The photon flux emitted along directions lying outside a cone of
critical angle c = sin-1(1/n) (ray C) suffer total internal reflection.
 C = 0
(for n = 3.6, c = 16o)
Only rays that lie inside the cone of critical angle can escape – so
called “escape cone”
24
Escape cone

r
c
An arbitrary point source
with spherical emission
in the active junction
• The fraction of light lies within the escape cone from a point
source:
A / 4r2 = (1 – cos c)/2 ≈ 1/4n2
25
• Here we estimate the fraction of the total generated photon flux that
lies within the escape cone. The area of the circular disk cap atop this
cone is (assuming a spherical emission distribution radius r)
c
A = ∫ 2r sin r d = 2r2 (1 - cos c)
0
• The fraction of the emitted light that lies within the solid angle
subtended by this escape cone is A/4r2
=> 3 = ½ (1 – cos c) = ½ (1 – (1 – 1/n2)1/2) ≈ 1/4n2
e.g. For a material with refractive index n = 3.6, only 1.9% of the
total generated photon flux lies within the escape cone.
• The efficiency with which the internal photons can be extracted
from the LED structure is known as the extraction efficiency e.
26
Output photon flux and external quantum efficiency
• The output photon flux o is related to the internal photon flux 
o = e  = e (int i/e)
where the extraction efficiency e specifies how much of the internal
photon flux is transmitted out of the structure.
• A single quantum efficiency that accommodates both e and int
is the external quantum efficiency ext
ext ≡ e int
=> The output photon flux o = ext i/e
=> ext is simply the ratio of the output photon flux o to the injected
electron flux i/e.
27
Output optical power
• The LED output optical power Po:
Po = h o = ext h i/e
• The internal efficiency int for LEDs ranges between 50% and just
about 100%, while the extraction efficiency e can be rather low.
The external quantum efficiency ext of LEDs is thus typically low.
28
Responsivity
• The responsivity R of an LED is defined as the ratio of the emitted
optical power Po to injected current i, i.e. R = Po/i
R = Po/i = h o/i = ext h/e
• The responsivity in W/A, when o is expressed in m,
R = ext 1.24/o
• The linear dependence of the LED output power Po on the injected
current i is valid only when the current is less than a certain value
(say tens of mA on a typical LED). For larger currents, saturation
causes the proportionality to fail.
29
Output optical power Po (mW)
Optical power at the output of an LED vs. injection current
*saturation at high injection
current (“droop” --- the loss of
efficiency at high power)
Slope = responsivity R = ext 1.24/

Current i (mA)
30
Power-conversion efficiency

Another measure of performance is the power-conversion
efficiency (or wall-plug efficiency), defined as the ratio of the
emitted optical power Po to the applied electrical power.
c ≡ Po / iV = ext h/eV
where V is the voltage drop across the device

Note that c  ext because h  eV, where eV = EFc – EFv in
a degenerate (heavily doped) junction.
31
Surface-emitting and edge-emitting
• Surface-emitting diodes radiate from the face parallel to the p-n
junction plane.
(The light emitted in the opposite direction can be reflected by a
metallic contact.)
• Edge-emitting diodes radiate from the edge of the junction region.
Multimode
fiber
32
Spatial pattern of surface-emitting LEDs
• The far-field radiation pattern for light emitted into air from a planar
surface-emitting LED is given by a Lambertian distribution:
I() = Io cos 
=> The intensity decreases to half its value at  = 60o
I()
Planar
surface

Io
Lambertian spatial pattern in the
absence of a lens
• In contrast, the radiation pattern from edge-emitting LEDs (and laser
diodes) is usually quite narrow and can often be empirically described
by the function (cos )s, with s > 1.
33
MM fiber
SM fiber
surface-emitting
edge-emitting
• The coupling efficiency couple (assuming Lambertian spatial pattern):
a

couple =  I() sin  d  I() sin  d
0
= sin2 a = NA2
0
a: Fiber acceptance angle
34
Epoxy-encapsulated LED
• Transparent epoxy lenses of different shapes alter the emission pattern
in different ways (e.g. hemispherical vs. parabolic lenses)
LED chip
• Epoxy lenses can also enhance the extraction efficiency e – a lens
with a refractive index close to that of the semiconductor reduces index
mismatch, and thus optimizes the extraction of light from the
semiconductor into the epoxy. (epoxy : semiconductor ~ 1.5 : 3.5)
In practice, epoxy lenses can yield a factor of 2-3 enhancement in
light extraction.
35
Enhancing the extraction efficiency
• LED die geometry designs vs. simple planar-surface-emitting LEDs
(limited by Fresnel reflection)
• Roughen the planar surface - permitting rays beyond the critical
angle to escape via scattering
• Contact geometry designs - Top-emitting LEDs make use of currentspreading layers, which are transparent conductive semiconductor layers
(typically indium-tin-oxide (ITO)) that spread the region of light
emission beyond that surrounding the electrical contact.
• Also include the use of reflective and transparent contacts,
transparent substrates (flip-chip packaging allows light to be extracted
through the substrate), distributed Bragg reflectors, 2D photonic crystals,
36
etc.
Summary: LED efficiencies
• Internal quantum efficiency int - only a fraction of the electron-hole
recombinations are radiative in nature
• Extraction efficiency e – only a small fraction of the light generated
in the junction region can escape from the high-index medium
• External quantum efficiency ext = e int (can be measured from
the responsivity R = Po/i)
• Power-conversion (wall-plug) efficiency c – efficiency of converting
electrical power to optical power (c ≤ ext)
• Coupling efficiency couple – only a fraction of the light emitted from
the LED can be coupled (e.g. to an optical fiber)
37
Spectral distribution
• The spectral intensity Rsp() of light spontaneously emitted from a
semiconductor in quasi-equilibrium (upon injection) can be determined
as a function of the concentration of injected carriers n.
• The spectral intensity of the direct band-to-band injectionelectroluminescence has precisely the same shape as the thermalequilibrium spectral intensity, but its magnitude is increased by the
factor exp [(EFc – EFv)/kBT], which can be very large in a presence of
injection. (assuming EFc, EFv within the bandgap for this simple
enhancement factor, eV = EFc - EFv)
Eg
1.2
1.3
1.4
2kBT
1.5
1.6
1.7
h
38
Spectral intensities vs. wavelength for LEDs
*LEDs are
broadband
incoherent
sources.
0.2
0.3
0.4
0.5
Wavelength m
0.6
0.7
AlN: the largest III-nitride bandgap, emitting at 210 nm
AlGaN: mid and near UV
InGaN: violet, blue, and green
AlInGaP: yellow, orange, and red
InGaAsP: near IR (1.3 – 1.55 m)
39
Modulation
40
Direct current modulation
• An LED can be directly modulated by applying the modulation signal
to the injection current, an approach known as direct current-modulation.
• There are two factors that limit the modulation bandwidth of an LED:
the junction capacitance and the diffusion capacitance.
• Because an LED is operated under a forward bias, the diffusion
capacitance is the dominating factor for its frequency response.
• The diffusion capacitance is a function of the carrier lifetime , which
is the total carrier recombination lifetime (1/ = 1/r + 1/nr),
because it is associated with the injection and removal of carriers in the
diffusion region in response to the modulation on the injection current.
The intrinsic speed of an LED is primarily determined by the lifetime
41
of the injected carriers in the active region.
• For an LED that is biased at a DC injection current level io and is
modulated at an angular frequency  = 2f with a modulation index m,
the total time-dependent current that is injected to the LED is
i(t) = io + i1(t) = io (1 + m cos t)
• In the linear response regime under the condition that m << 1 (i.e.
small-signal modulation), the output optical power of the LED in
response to this modulation can be expressed as
P(t) = Po + P1(t) = Po [1 + |r| cos (t – )]
where Po is the constant optical output power at the bias current level
of io, |r| is the magnitude of the response to the modulation, and  is the
phase delay of the response to the modulated signal (due to the carrier
42
lifetime ).
• For an LED modulated in the linear response regime, the complex
response as a function of modulation frequency  is (following a RC
low-pass filter analysis)
r() = |r| ei = m/(1 – i)
• The frequency response and modulation bandwidth of an LED are
usually measured in terms of the electrical power spectrum of a
broadband, high-speed photodetector.
• In the linear operating regime of the detector, the detector current
is linearly proportional to the optical power of the LED.
The electrical power spectrum of the detector output is proportional
to |r|2:
R(f) = |r(f)|2 = m2/(1 + 42f22)
A 3-dB bandwidth of
f3dB = 1/2
43
Normalized modulation response
R(f)/R(0) (dB)
Normalized modulation response
0
 = 10 ns
-1
*in electronics,
f3dB ≈ 0.35/rise time
-2
=> rise time ≈ 2.2 
-3
-4
f3dB = 1/2 = 15.9 MHz
-5
0
5
10
15
Modulation frequency, f (MHz)
20
44
Modulation bandwidth
• The spontaneous carrier lifetime  is normally on the order of a
few hundred to 1 ns for an LED.
The modulation bandwidth of an LED is typically in the range
of a few megahertz to a few hundred megahertz.
• A modulation bandwidth up to 1 GHz can be obtained with a
reduction in the internal quantum efficiency (int = /r) of the LED by
reducing the carrier lifetime to the sub-nanosecond range.
• Aside from this intrinsic response speed determined by the carrier
lifetime, the modulation bandwidth of an LED can be further limited
by parasitic effects from its electrical contacts and packaging, as well
as from its driving circuitry.
45
Power-bandwidth product
A 3-dB bandwidth
f3dB = 1/2
=> int f3dB = (/r) (1/2) = 1/2r
• One can obtain a certain internal-quantum-efficiency-bandwidth
product by choosing the semiconductor with a certain radiative lifetime.
At an injection current i, the output optical power and the smallsignal modulation bandwidth of an LED have the following powerbandwidth product (i.e. a tradeoff between power and bandwidth):
Pof3dB = e int (i/e) h (1/2) = e (i/e) h (1/2r)
At a given injection level, the modulation bandwidth of an LED is
inversely proportional to its output power. A high-power LED tends to
46
have a low speed, and vice versa. (P0f3dB  i)
Semiconductor lasers
Semiconductor as a gain medium
•
Transition rates for semiconductors in quasi-equilibrium
•
Rate equations
•
Current pumping
•
Laser threshold current
•
Steady-state laser photon flux
•
Power output characteristics
•
Spatial characteristics
•
Spectral characteristics
•
Typical laser diode specifications
•
Single-mode laser diode structures
•
Wavelength-tunable laser diodes
•
Direct modulation
Ref. Physics of Optoelectronics, Michael A. Parker, CRC Taylor and Francis, pp.47-78
•
47
Semiconductor lasers
*Some useful characteristics of semiconductor lasers:
1. Capable of emitting high powers (e.g. continuous wave ~ W).
2. A relatively directional output beam (compared with LEDs) permits
high coupling efficiency (~ 50 %) into single-mode fibers.
3. A relatively narrow spectral width of the emitted light allows
operation at high bit rates (~ 10 Gb/s), as fiber dispersion becomes
less critical for such an optical source.
48
Laser diodes
• A laser diode (LD) is a semiconductor optical amplifier (SOA) that
has an optical feedback.
• A semiconductor optical amplifier is a forward-biased heavily-doped
p+-n+ junction fabricated from a direct-bandgap semiconductor material.
• The injected current is sufficiently large to provide optical gain.
• The optical feedback is usually implemented by cleaving the
semiconductor material along its crystal planes.
• The sharp refractive index difference between the crystal (~3.5) and
the surrounding air causes the cleaved surfaces to act as reflectors.
49
Laser diodes
 The semiconductor crystal therefore in general can act both as a
gain medium and as a Fabry-Perot optical resonator.
• Provided that the gain coefficient is sufficiently large, the feedback
converts the optical amplifier into an optical oscillator, i.e. a laser.
• The device is called a laser diode or a diode laser or a semiconductor
injection laser.
cleaved surface
+
p+ n+
i
-
cleaved surface
50
Turning semiconductor amplifiers into laser diodes
• In the case of semiconductor lasers, external mirrors are not required
as the two cleaved laser facets act as partially reflecting mirrors
Current injection
Active region
cleaved facets
Laser output
Gain medium
R1
R2
cavity length d
51
Semiconductor as a gain medium
• The basic principle: creation of population inversion, stimulated
emission becomes more prevalent than absorption.
• The population inversion is usually attained by electric-current
injection in some form of a p+-n+ junction diode (also possible by optical
pumping for basic research)
a forward bias voltage causes carrier pairs to be injected into the
junction region, where they recombine by means of stimulated emission.
• Here we discuss the semiconductor gain and bandwidth upon
electrical pumping scheme.
52
Absorption and stimulated emission
E
E
h
h
h
h
k
absorption
coherent
photons
k
stimulated emission
• When stimulated emission is more likely than absorption
=> net optical gain (a net increase in photon flux)
=> material can serve as a coherent optical amplifier
53
Population inversion by carrier injection
• In a semiconductor, population inversion can be obtained by means of
high carrier injection which results in simultaneously heavily populated
electrons and holes in the same spatial region.
Electron
energy
EFc
Eg
filled
EFv
54
• Incident photons with energy Eg < h < (EFc - EFv) cannot be absorbed
because the necessary conduction band states are occupied! (and the
necessary valance band states are empty)
Electron
energy
EFc
h
x
Eg
filled
EFv
55
Instead, these photons can induce downward transitions of an electron
from a filled conduction band state into an empty valence band
state. => emitting coherent photons!
Electron
energy
EFc
Amplification
by stimulated
emission!
h
filled
EFv
The condition for stimulated emission under population inversion:
EFc - EFv > h  > Eg
56
Population inversion in a forward-biased heavily doped p+-n+ junction
Electron
energy
EFc
Eg
h
Eg
EFv
active
region
(~m)
• Upon high injection carrier density in a heavily-doped p+-n+
junction there exists an active region near the depletion layer, which
contains simultaneously heavily populated electrons and holes –
population inverted!
57
Population inversion in a P+-p-N+ double heterostructure under
forward bias (e.g. GaAlAs/GaAs/GaAlAs)
EFc
Ec
h
EFv
Ev
~ 0.1 m
active
region
filled
• The thin narrow-gap active region of a double heterostructure
contains simultaneously heavily populated electrons and holes in a
confined active region – population inverted!
58
Transition rates for semiconductors in quasi-equilibrium
• Recall expressions for the rate of stimulated emission Re() and the
rate of photon absorption Ra():
Re() = B21 u() Pc(E2) [1 – Pv(E1)] ()
(m-3)
Ra() = B12 u() Pv(E1) [1 – Pc(E2)] ()
(m-3)
in the presence of an optical radiation field that has a spectral intensity
I() = (c/n) u()
B12 = B21 = c3/(8n3h3sp)
joint density of states () = ((2mr)3/2/ħ2) (h – Eg)1/2
h ≥ Eg
59
• Stimulated emission is more prevalent than absorption when:
Re() > Ra()
Pc(E2) [1 – Pv(E1)] > Pv(E1) [1 – Pc(E2)]
Pc(E2) > Pv(E1)
(possible if, e.g., E2 < EFc , E1 > EFv)
• This defines the population inversion in a semiconductor. The quasiFermi levels are determined by the pumping (injection) level (EFc – EFv
= eV > Eg, where V is the forward bias voltage).
60
Broadband optical gain
E
Optical
gain (broadband)
EFc
E2
h
Eg
E1
EFv
k
FWHM = gain
bandwidth
EFC - EFV
frequency
61
Rate equations
62
The rate equations

We use the rate equations to describe how the gain, pump,
feedback, and output coupler mechanisms affect the carrier
and photon concentration in a device.

The rate equations manifest the matter-light interaction
(emission, absorption) through the gain term.

The photon rate equation describes the effects of the output
coupler and feedback mechanism through a relaxation term
incorporating the cavity lifetime.

We will use the rate equations to determine the output optical
power vs. input current (P-I curves) and the modulation
response to a sinusoidal bias current.
63
Total carrier and photon rate

The rate equations describe the number of electron-hole pairs
(i.e. carrier pairs) and the number of photons in the cavity.

For intrinsic semiconductors or carriers created by optical
absorption or electrical pumping,
the number of pairs must be identical to the number of
electrons.
=> we discuss only the number of electrons, and
assume charge neutrality (n = p) in all portions of the active
region.
64
Electron density and photon density

Here we denote the excess electron density (number of
electrons per volume) by “n”
=> nVa represents the total number of excess electrons in the
active region of volume Va.

Also, let  be the photon density (number of photons per
volume).
=> The total number of photons in the modal volume must
be V and the total number of photons in the active region
must be Va.
65
Optical confinement factor



The active region (i.e. gain region) has volume Va, which is
smaller than the modal volume V containing the optical
energy.
The simplest model assumes that the optical power is
uniformly distributed in V and is zero outside this volume.
The optical confinement factor  specifies the fraction of the
optical mode that overlaps the gain region
 = Va/V
V
Va
66
Carrier rate equation

The carrier rate equation has the basic form (N: total number
of carriers)
dN/dt = Generation – recombination

The carrier rate equation:
dN/dt = -(stimulated emission) + (absorption) + (Pump) –
(non-radiative recombination) – (spontaneous
recombination)

This equation calculates the change in the number of carriers
“nVa” in the active region.

Absorption and pumping increase the number while emission
and recombination decrease it.
67
The pump term

The pump term describes the number of electron-hole pairs
that contribute to the photon emission process of the active
region in each second due to the bias current i.
Pump = int i/e
where int is the internal quantum efficiency.

The number of electron-hole pairs that contribute to the photon
emission process in each unit of volume (cm3) of the active
region in each second can be related to the bias current i by
Pump-current number density = int (i/e)Va-1
68
Photon rate equation


The radiative processes that decrease the total number of
carriers N must increase the total number of photons V in
the modal volume V.
A photon rate equation:
d/dt = (stimulated emission) – (absorption)
– (optical loss) + (fraction of spontaneous
emission)

The “optical loss” term accounts for the optical energy lost
from the cavity --- scatters out of the cavity sidewalls and
some passes through the mirrors (laser diode facets).

The light passing through the mirrors, although considered to
be an “optical loss,” gives a useful output.
69
Spontaneous emission into the cavity mode





The number of photons in the lasing mode increases not only
from stimulated emission but also from the spontaneous
emission.
Excess electrons and holes can spontaneously recombine and
emit photons in all directions.
The wavelength range of spontaneously emitted photons
spans over the gain bandwidth.
Some of the spontaneously emitted photons propagate in
exactly the correct direction to enter the laser mode path.
Of those photons that enter the laser mode path, a fraction of
them have exactly the right frequency to match that of the
lasing mode.
=> This small fraction of spontaneously emitted photons adds to
the photon density  of the cavity.
70
Spontaneous emission into the cavity mode
Fabry-Perot cavity mode path
Optical
gain (broadband)
r
Eg
EFC - EFV
freq
• Only small fraction of spontaneously
emitted photons propagate in exactly
the correct direction to enter the laser
mode path.
• Of those photons that enter the laser
mode path, only a fraction of them have
exactly the right frequency to match that of
the lasing mode.
71
Spontaneous emission into the cavity mode

The rate of spontaneous emission into the cavity mode can be
written as
VRsp = Vrrn2


where rr is the radiative recombination coefficient (cm3 s-1), 
is the geometrical factor that gives the fraction of the total
spontaneously emitted photons that actually couple into the
laser mode. ( typically ranges from 10-2 to 10-5)
The small fraction of spontaneous photons coupling into the
cavity with the right frequency start the lasing process.
Above threshold, however, it wastes a significant fraction of
the pump energy, thus raising the laser threshold current.
72
Optical losses





The optical loss term in the rate equation describes changes in
the photon density that are related to the optical components
of the laser cavity.
Consider the cavity as the confined structure bounded by two
mirrors (Fabry-Perot resonator) and the sidewalls.
As the photons bounce back and forth between the mirrors,
some are lost through the mirrors and some are lost
through the sidewalls.
Other loss mechanisms also influence the photon density.
e.g. free carriers can absorb light (known as free-carrier
absorption) when the light electromagnetic waves drive the
motion of the electrons and the surrounding medium damps
this motion by converting the kinetic energy into heat.
73
Optical losses
E
Sidewall
scattering
loss
Useful
output
Free-carrier
absorption
k

Free-carrier absorption: not a valence-to-conduction
resonant process; free-carriers make transition from a filled
stated to an empty state within the same band, a so-called
intraband absorption process. The excess kinetic energy of
the “hot” carriers is eventually released through
thermalization (as phonons), i.e. heat or lattice vibration.
74
Cavity optical loss



The total number of photons in the modal volume V
decreases because of these optical losses.
The number of photons lost from the cavity must depend on
the number of photons inside the cavity.
Thus, a simple differential equation expresses the dynamics
in the absence of other sources or losses of photons
V d/dt = -V / 
=> (t) = 0 exp (-t/)

=> the initial photon density in the cavity decays
exponentially
All of the optical losses contribute to an overall relaxation
time  --- the cavity lifetime (typically ~ 10-12 s ~ ps).
75
Optical loss coefficient


The cavity lifetime  describes a lumped device.
In order to include a spatial dimension, we define an optical
loss per unit length r [cm-1].
r = 1/(vg)



or
1/ = rvg
where vg represents the group velocity of the cavity light.
Thus, the optical loss (per unit length) r gives the number of
photons lost in each unit length of cavity.
We picture the optical loss r as taking place along the laser
cavity length.
Suppose vg = c/ng ~ 108 m/s and  ~ 10-12 s,
then r ~ 104 m-1 or 102 cm-1
76
Internal loss and mirror loss







Light can be lost from a cavity due to either distributed or
point-loss mechanisms.
Distributed loss refers to energy loss along the length of a
device.
e.g. distributed loss includes optical energy lost through the
sidewalls and free carrier absorption.
Often distributed loss is also termed “internal loss,” denoted
by int, because it refers to light leaving the laser structure in
a manner other than through the end mirrors.
Point-loss refers to energy loss at specific points.
e.g. the laser mirrors
However, it is convenient to describe the mirror loss as if it
were a “distributed loss” and denote it as m.
77
Mirror loss




How to picture the mirror loss m as distributed along the
length d of the cavity?
Assume that both mirrors have the same power reflectance R
(~0.34 for GaAs).
The loss per mirror must be m/2.
The rate of mirror loss can be written as
(assume R = exp (-m/2  2d))
1/m = vgm = (vg/d) ln (1/R)

The reciprocal of the cavity lifetime (total cavity decay rate)
becomes
1/ = 1/int + 1/m = vgr = vg(int + m)

The internal loss and single mirror loss are typically on the
order of 30 cm-1.
78
What does the loss coefficient value mean?

e.g. For a single mirror loss coefficient of m ~ 30 cm-1

For a typical semiconductor laser diode length d ~ 300 m
m d ~ 30 cm-1 x 300 m = 9000 x 10-4 = 0.9 ~ 1
=> exp (-md) ~ 1/e ~ 30% - 40%

Recall reflectance R = (n – 1)2/(n+1)2 for normal incidence.
For n ~ 3.5, R ~ 31%

And the mirror loss lifetime m = 1/(vgm) ~ (108 m/s x 30
cm-1)-1 ~ 3 ps
79
Cavities with a gain medium

If the cavities include a gain medium, two
things can happen:

An absorptive medium (g < 0) causes the photon
density to relax faster than .

A medium with positive material gain (g > 0)
causes the photon density to grow instead of
decay.
80
Gain





Stimulated emission and absorption affect the number of
conduction-band electrons and valence-band holes.
These processes determine the rate of change of the number
of carriers in the active region and the number of photons in
the modal volume.
A photon perturbs the energy levels of electron-hole pairs for
the semiconductor and induces radiative recombination.
The number of photons increases by one while the number of
conduction electrons decreases by the same number.
The conduction-band electrons and valence-band holes
produce “gain” in the sense that incident photons with the
proper wavelength can stimulate carrier recombination and
thereby produce more photons with the same characteristics
as the incident ones.
81
Gain




Gain in general can be defined as the ratio of the “output
number of photons” to the “input number of photons”
e.g. a “gain of two” means one input photon resulting in two
identical output photons
Gain can be <1 (output < input).
e.g. The same semiconductor can absorb photons from the
laser beam by promoting a valence electron to the conduction
band.

The stimulated emission increases the number of photons in
the laser while the absorption decreases the number.

=> the gain should describe the difference between the
emission and absorption rates.
82
Stimulated emission or absorption rates

The change in the total number of photons V in the modal
volume V due to gain and absorption must be proportional
to the number of photons present
Rstim V = V d/dt   V

Rstim represents the net number of photons produced by
stimulated emission (Rstim > 0) or absorbed (Rstim < 0) in each
unit volume in each second (cm-3 s-1).

However, only those photons in the active region (volume
Va) can stimulate additional photons as the electron-hole
pairs are confined to that region
Rstim V = V d/dt   Va
83
Temporal gain

Define the temporal gain gt (s-1)
Rstim V = V d/dt = gt  Va

Rstim = d/dt = gt  Va/V =  gt
where  = Va/V is the confinement factor.


The temporal gain gt must depend on the number of excess
conduction-band electrons n per unit volume in the active
region:
gt = gt(n)
The temporal gain gives exponential growth / decay of
photon number density in time
 = 0 exp (gtt)
84
Material gain



We define the material gain g (cm-1) in terms of the number
of photons produced in the medium in each unit of length for
each photon entering that unit length.
We can find the material gain from the temporal gain by
changing the units of gt from “per second” to those of the
material gain g, namely “per unit length.”
Change of variables
d/dt = (d/dz) (dz/dt) = (d/dz) vg
=> ddz =   (gt/vg) =  g
Again, g = g(n) depends on the number of excess conductionband electrons n per unit volume in the active region. (gt(n) =
vg g(n))
85
Material gain
ddz =  g(n)
=> (z) =  exp (g(n)z)
=> The “single-pass” gain G = (z)/0 = exp (g(n)z)



The material gain g(n) can produce either gain or absorption
(i.e. increase or decrease the number of photons in each unit
of length), depending on the value of n.
In the absence of injection, n = 0, we expect photons to be
absorbed => G < 1 and g < 0
For sufficiently large n, the material gain g becomes positive
(g > 0) and produces stimulated emission.
86
Material gain in terms of stimulated rates
Rstim V = V d/dt = V (d/dz) vg = vg g  Va
Rstim = g (Va) (vg/V
Photon # in
the active region
Photon through each unit
area per unit time (cm-2 s-1)
Photon flux in the active region
per unit area per unit time (cm-2 s-1)

To account for frequency dependence, Rstim then need to
represent the net number of photons produced by stimulated
emission (Rstim > 0) or absorbed (Rstim < 0) in each unit
volume in each second per unit frequency interval. (cm-3 s-1
Hz-1 or cm-3)
87
Gain and absorption coefficients vs. frequency

Define the gain coefficient (cm-1) in quasi-equilibrium
(recall Pc(E2) > Pv(E1), Eg < h < EFc – EFv):
g() = (h/I()) [Re() – Ra()]
= (c2/8n22sp) () [Pc(E2) – Pv(E1)]

where I()/h = vgu()/h is the photon flux per unit area (cm2).
The absorption coefficient (cm-1) in thermal equilibrium
(taking +ve sign):
() = (c2/8n22sp) () [P(E1) – P(E2)]
≈ (c2/8n22sp) () where P(E1) ~ 1, P(E2) ~ 0
** The larger the absorption coefficient in thermal equilibrium the
larger the gain coefficient when pumped! **
88
Current pumping
89
Material transparency

The semiconductor material becomes “transparent” (material
transparency) when the rate of absorption just equals the rate
of stimulated emission.
=> one incident photon produces exactly one photon in the
output.
=> the single-pass gain must be unity, i.e. G = 1.
=> The material gain upon transparency g(n0) = 0.

The transparency density n0 (number per unit volume)
represents the number of excess conduction-band electrons per
volume required to achieve transparency.
90
1.8  1018
200
1.6  1018
1.4  1018
100
1.2 1018
+ve g
0
-ve g
Peak gain coefficient gp (cm-1)
Gain coefficient g() (cm-1)
Gain coefficient g() for an InGaAsP optical amplifier
net gain
transparency
n = 1  1018 cm-3
0.90
0.92
0.94
0.96
h (eV)
1.0
1.5
2.0
n (1018 cm-3)
• Both the amplifier bandwidth and the peak value of the gain coefficient
increase with n. The bandwidth is defined at the FWHM of the gain
91
profile, also called the 3-dB gain bandwidth.
Differential gain

The peak gain coefficient curves can be approximated by a
straight line at n0 by making a Taylor expansion about the
transparency density n0 to find
gp = gp(n)  g0(n – n0)  (n/n0 – 1)
Peak gain coefficient gp

g0 = dgp/dn is typically called the differential gain (cm2). It
has a unit of cross section.
• The quantity  represents the
absorption coefficient in the
absence of injection (n = 0).
Slope = g0 ≈ /n0
gain
loss
n0
-≈ -g0n0
n
• n0 represents the injectedcarrier concentration at which
emission and absorption just
balance each other (the
transparency condition).
92
Transparency current density
• Within the linear approximation, the peak gain coefficient is
linearly related to the injected current density J (A cm-2)
gp ≈ (J/J0 – 1)
The transparency current density J0 is given by
J0 = (el/intr) n
where l is the active region thickness
• When J = 0, the peak gain coefficient gp = - becomes the
absorption coefficient.
• When J = J0, gp = 0 and the material is transparent => exhibits
neither gain nor loss.
• Net gain can be attained in a semiconductor junction only when
J > J0.
93
Injected current density
• If an electric current i is injected through an area A = wd, into an
active region Va = volume lA (where l is the active region thickness),
the steady-state carrier injection rate is i/elA = J/el per second per
unit volume, where J = i/A is the injected current density (A cm-2).
The steady-state injected carrier concentration is
given as (recombination = injection)
w
n/ = J/el
or
J = (el/intr) n
d
+
l
p+ n+
i
-
the “pump-current number density” = int i/eVa
( is the total recombination lifetime, r is the
radiative recombination lifetime, int = /r)
94
Peak gain coefficient gp (cm-1)
Peak gain coefficient as a function of current density for
the approximate linear model
*Net gain can be attained in a
semiconductor junction only when J > J0.
gain
loss
J0
Current density J (A cm-2)
J0 = (el/intr) n0
-
• Note that J0 is directly proportional to the junction thickness l
=> a lower transparency current density J0 is attained by using a
narrower active-region thickness. (another motivation for using
double heterostructures where l is ~ 0.1 m or quantum wells)
95
e.g. Gain of an InGaAsP SOA
An InGaAsP semiconductor optical amplifier operating at 300o K
has the following parameters: r = 2.5 ns, int = 0.5,
n0 = 1.25 x 1018 cm-3, and  = 600 cm-1. The junction has thickness
l = 2 m (not a double heterostructure), length d = 200 m, and
width w = 10 m.
The transparency current density J0 = 3.2 x 104 A/cm2
A slightly larger current density J = 3.5 x 104 A/cm2 provides a
peak gain coefficient gp ≈ 56 cm-1.
An amplifier gain (i.e. single-pass gain) at the peak gain G = exp(gpd)
= exp(1.12) ≈ 3
However, as the junction area A = wd = 2 x 10-5 cm2, a rather large
injection current i = JA = 700 mA (!) is required to produce this
96
current density.
The electron rate equation


Now we combine all of the individual terms into the rate
equations.
The equation for the number of conduction-band electrons in
the active region
Vadn/dt = -(stimulated emission) + (absorption) + (pump) –
(non-radiative recombination) – (spontaneous recombination)
Vadn/dt = -Vavgg + int i/e – (n/) Va


For good materials, the recombination term approximately
equals the spontaneous radiative recombination rate (n/ 
rrn2)

dn/dt = -vgg + int (i/e)Va-1 – rrn2
97
The photon rate equation

The photon rate equation
Vd/dt = +(stimulated emission) - (absorption) – (optical
loss) + (fraction of spontaneous recombination)
=> Vd/dt = Vavgg – V/ + rrn2V

Using the optical confinement factor of  = Va/V
=> d/dt = vgg – / + rrn2
98
The rate equations
Electron rate equation: dn/dt = -vgg + int (i/e)Va-1 – rrn2
Photon rate equation:




d/dt = vgg – / + rrn2
Here we use the rate equations mainly to find the output
power (also cavity power) as a function of the bias current.
We can also use them for a small-signal analysis of time
response of the laser beam to small changes in the bias
current.
Note that the laser rate equations are coupled (the electron
equation depends on , the photon equation depends on n)
and nonlinear as g is a function of n !
In general the rate equations should be generalized to a
partial differential equation that includes a spatial coordinate.
99
Power output
characteristics
100
The power-current curves





The relation between optical output power and the pump
strength provides the most fundamental information on the
operation of light-emitting devices.
The rate equations provide power vs. current curves for
semiconductor lasers and light-emitting diodes.
The power-current curves are alternatively termed P-I or L-I
curves.
The most important results can be found using
approximations to these highly nonlinear equations.
Therefore, separate approximations must be applied to the
lasing and non-lasing regimes of operation.
101
Photon density vs. pump-current number density


We solve the rate equations for the steady-state photon
density  inside the laser cavity as a function of the steadystate pump-current number density int(i/e)Va-1.
The rate equations are
dn/dt = -vgg(n) + int(i/e)Va-1 – rrn2
d/dt = vgg(n) – / + rrn2



A system attains steady state when all of the time derivatives
become zero.
We assume that the laser has been operating for a long time
compared with the time constants  (~ ps) and (~ ns).
We define the effective carrier lifetime  = 1/(rrn), assuming
large excess carrier density.
102
The steady-state equations

For sufficiently long times (>> , ), the rate equations
become the steady-state equations
0 = -vgg(n) + int(i/e)Va-1 – rrn2
0 = vgg(n) – / + rrn2

The above equations describe the steady-state photon density,
which refers to the optical power density.

The steady-state photon equation thus requires the amplitude
of the light waves to be independent of time.
(i.e. the power contained in the light waves neither grows nor
decays with time).
103
Below lasing threshold

“Below lasing threshold” means that the laser has insufficient
gain to support oscillation.

Small value of current density J
=> small values for the excess carrier density n and the
photon density 

There exists a “threshold” current density Jth for which
J > Jth produces lasing
J < Jth produces only spontaneous emission
104
Below lasing threshold

For J < Jth, the photon density  in the cavity remains
relatively small compared with that attained for lasing.
=> the stimulated emission / absorption terms vgg(n) are
negligible

The steady-state photon density equation provides
 = rrn2
for the photon density for spontaneous emission

The steady-state carrier density equation provides
rrn2 = int(i/e)Va-1
105
The photon-current relation below lasing threshold

Thus, the photon-current relation
 =  int(i/e)Va-1

This is the spontaneous emission photon density in the cavity.

The photon density is linear in the pump-current number
density int(i/e)Va-1 (or linear in the bias current i).

The factor  accounts for the geometrical factor describing
the coupling of spontaneous emission to the cavity mode.
106
Above lasing threshold

“above lasing threshold” --- the situation of sufficiently large
pump current (or pump power) to produce stimulated
emission in steady state (J > Jth).

We assume that stimulated emission provides the primary
source of cavity photons whereas the number of
spontaneously emitted photons remains relatively small.

The ratio of spontaneous to stimulated photons in the lasing
mode is further reduced by the geometrical coupling
coefficient .

Thus, above lasing threshold we neglect the term rrn2.
107
The steady-state laser equations

The steady-state laser equations become
0 = -vgg(n) + int(i/e)Va-1 – rrn2
0 = vgg(n) – /

The steady-state photon density equation can be solved for
vgg(n)
vgg(n) = 1/(

Recall that the material gain g(n) and the temporal gain gt(n)
are related vgg(n) = gt(n)
Steady-state temporal gain gt(n) = 1/(
108
Threshold carrier density

The steady-state temporal gain gt(n) is a constant 1/() and
independent of “n”.
=> this requires the steady-state n to be a constant !


The “threshold carrier density” nth represents the approximate
value of the carrier density n to produce steady-state laser
oscillation n  nth
=> The steady-state carrier density remains fixed regardless
of the magnitude of the current above lasing threshold !
Below lasing threshold, the approximation n  nth does not
hold as the device produces mostly spontaneous emission
=> the spontaneous emission term in the photon rate
equation cannot be ignored.
109
Steady-state gain equals the loss

At steady state above lasing threshold, the value of the
temporal gain therefore can be written as
gt(nth) = 1/

If we write the cavity lifetime in terms of the loss coefficients
(cm-1)
1/ = vgr

The steady-state material gain (cm-1) can be written as
g(nth) = gth = r (The gain condition !)

The gain equals the loss (and remains approximately fixed at
g(nth) for currents larger than the threshold current) when the
laser oscillates.
110
The photon-current equation

We can obtain the photon-current curve from the steady-state
carrier density equation when the laser oscillates
   (int(i/e)Va-1 – rrnth2)

Note that we have replaced the carrier density n with its
threshold value nth.

The equation describes a straight line that passes through the
threshold pump-current number density rrnth2 and has slope


The threshold pump current number density becomes larger
for materials (and laser designs) with greater tendency to
spontaneously emit (with larger rr) !
111
nth
clamped at nth
(additional carriers
recombine immediately
under the effect of
stimulated emission
and feedback)
ith
current
Photon
density 
Steady-state
carrier density n
Steady-state carrier density and photon density as
functions of injection current
ith
current
• Below threshold, the laser photon density is zero; any increase in
the pumping rate is manifested as an increase in the spontaneousemission photon flux, but there is no sustained oscillation.
• Above threshold, the steady-state internal laser photon density is
directly proportional to the initial population inversion (initial injected
carrier density), and therefore increases with the pumping rate, yet
112
the gain g(n) remains clamped at the threshold value ( g(nth)).
Gain at threshold


Above threshold, the gain does not vary much from gth = g(nth).
Recall the differential gain is the slope of the gain g(n)
g0(n) = dg(n)/dn


For lasing, the differential gain is evaluated at the threshold
density nth.
The lowest order Taylor series approximation centered on the
transparency density n0 is
g(n) ≈ g0(n – n0).
=> The gain at threshold must be
gth = g(nth) ≈ g0(nth – n0)
113
Threshold current density
• Recall that within the linear approximation, the peak gain coefficient
is linearly related to the injected current density J:
gp ≈ (J/J0 – 1)
where J0 is the transparency current density.
• Setting gp = gth = r/, the threshold injected current density Jth:
Jth ≈ [(r/ + )/] J0
The threshold current density is larger than the transparency
current density by the factor (r/ + )/ which is ~ 1 - 2 (for good
active materials with high gain in a low-loss cavity).
• The threshold injected current ith = JthA and the transparency current
114
i0 = J0A, where A is the active region cross-sectional area.
Remarks on threshold current density
• The threshold current density Jth is a key parameter in characterizing
the laser-diode performance: smaller values of Jth indicate superior
performance.
• Jth can be minimized by (Jth  J0):
maximizing the internal quantum efficiency int;
minimizing the resonator loss coefficient r,
minimizing the transparency injected-carrier concentration n0,
minimizing the active-region thickness l
(key merit of using double heterostructures and quantum wells)115
e.g. Threshold current for an InGaAsP heterostructure laser diode
Consider an InGaAsP (active layer) / InP (cladding) double
heterostructure laser diode with the material parameters:
n0 = 1.25 x 1018 cm-3,  = 600 cm-1, r = 2.5 ns, n = 3.5, int = 0.5
at T = 300o K. Assume that the dimensions of the junction are
d = 200 m, w = 10 m, and l = 0.1 m. Assume the resonator loss
coefficient r = 118 cm-1.
The transparency current density J0 = 1600 A/cm2
The threshold current density Jth = 1915 A/cm2
 The threshold current ith = 38 mA. (*Note that it is such reasonably
small threshold current that enables continuous-wave (CW) operation
of double-heterostructure laser diodes at room temperature.)
116
Evolution of the threshold current density of semiconductor lasers
10000
4.3 kA/cm2
(1968)
Impact of double heterostructures
900 A/cm2
(1970)
Jth (A/cm2)
1000
Impact of quantum wells
160 A/cm2
(1981)
100
Impact of quantum dots
A/cm2
40
(1988)
10
1965
1970
1975
1980
1985
1990
19 A/cm2
(2000)
1995
2000
2005
2010
Year
Zhores Alferov, Double heterostructure lasers: early days and future perspectives,
IEEE Journal on Selected Topics in Quantum Electronics, Vol. 6, pp. 832-840, Nov/Dec 2000
117
Power output from two cavity mirrors

Now we convert the basic concepts (photon and current
number densities) into measurable quantities like optical
power (Watts) and bias current (Amps) using simple scaling
factors.

Dimensional analysis for the power passing through both
laser mirrors (assume equal reflectivity)
Power out both mirrors = Po = Energy / sec
= (energy/photon  photons/volume  modal volume) / m
= (hc/    V vgm
118
P-I below threshold


We can find the output power from both mirrors as a function
of the bias current i for a laser operating below threshold i <
ith (LED regime).
We can find the P-I curves from the number density relation
below threshold
 =  int(i/e)Va-1)
=> Po = (hc/) ( int(i/e)Va-1) V vgm

Recall the cavity lifetime  in terms of the mirror loss m and
the internal scattering / free-carrier absorption losses int
1/ = 1/m + 1/int = vg (m + int)
119
P-I below threshold

The P-I below threshold
Po = (hc/) ( int(i/e)Va-1) V vgm
=  (int/) (hc/e) (m/(m+int)) i
=> the output power below threshold is linear in the bias
current i.

The modal coupling coefficient  causes the output power to
be of smaller magnitude than the power for the same laser
above threshold.
120
P-I above threshold


Now we find the output power from both mirrors as a
function of the bias current i for a laser operating above
threshold i > ith.
Using the photon and carrier number density relation
 =  (int(i/e)Va-1 – rrnth2)

Po/[(hc/)Vvgm] =  [(int(i/e)Va-1 – int(ith/e)Va-1)]




Po = int (hc/e) (m/(int + m)) (i – ith)
The P-I relation above threshold represents a straight line
with an intercept of ith.
The mirror loss and the internal loss determine the slope of
the line.
Smaller mirror reflectivity gives larger loss m and therefore
larger output power.
121
Power output of injection lasers
• The internal laser power above threshold:
P = int (hc/e) (i – ith) = (hc/) int (i – ith)/e
• Only part of this power can be extracted through the cavity mirrors,
and the rest is dissipated inside the laser resonator.
The output laser power if the light transmitted through both mirrors
is used (assume R = R1 = R2 => total mirror loss m = (1/d)ln(1/R))
Po = int (hc/e) (i – ith) ∙ (1/d) ln(1/R) / r
= e int (hc/e) (i – ith) = ext (hc/e) (i – ith)
extraction
efficiency (m/r)
external differential
quantum efficiency
122
External differential quantum efficiency
• The external differential quantum efficiency ext is defined as
ext = d(Po/(hc/) / d(i/e)
Output optical power Po (mW)
=> dPo/di = ext hc/e = ext 1.24/ ≡ R
(W/A)
e.g. InGaAsP/InGaAsP:
o: 1550 nm
ith: 15 mA
ext: 0.33
R: 0.26 W/A
slope R is known as the
differential responsivity
(or slope efficiency) --- we can extract
ext from measuring R
ith
Drive current i (mA)
123
e.g. Efficiencies for double-heterostructure InGaAsP laser diodes
Consider again an InGaAsP/InP double-heterostructure laser diode with
int = 0.5, m = 59 cm-1, r = 118 cm-1, and ith = 38 mA.
If the light from both output faces is used, the extraction efficiency is
e = m/r = 0.5
The external differential quantum efficiency is
ext = e int = 0.25
At o = 1300 nm, the differential responsivity of this laser is
R = dPo/di = ext 1.24/1.3 = 0.24 W/A
For i = 50 mA, i – ith = 12 mA and Po = 12  0.24 = 2.9 mW
124
Light output (power)
P-I characteristics
much steeper than LED
Coherent
emission
(Lasing)
Incoherent
emission
Threshold current ith
(typically few 10 mA’s
using double heterostructures)
Current
125
Comparison of LED and LD efficiencies and powers
• When operated below threshold, laser diodes produce spontaneous
emission and behave as light-emitting diodes.
• There is a one-to-one correspondence between the efficiencies
quantities for the LED and the LD.
• The superior performance of the laser results from the fact that the
extraction efficiency e for the LD is greater than that for the LED.
• This stems from the fact that the laser operates on the basis of
stimulated emission, which causes the laser light to be concentrated
in particular modes so that it can be more readily extracted.
A laser diode operated above threshold has a value of ext (10’s of %)
126
that is larger than the value of ext for an LED (fraction of %).
Power-conversion efficiency
• The power-conversion efficiency (wall-plug efficiency):
c ≡ Po/iV
c = ext [(i – ith)/i] (h/eV)
@ i = 2ith
=> c = (ext/2) (h/eV) < ext
• Laser diodes can exhibit power-conversion efficiencies in excess of
50%, which is well above that for other types of lasers.
• The electrical power that is not transformed into light is transformed
into heat.
• Because laser diodes generate substantial amount of heat they are
usually mounted on heat sinks, which help dissipate the heat and
stabilize the temperature.
127
Typical laser diode threshold current temperature dependence
output
power
(mW)
T = 20 30 40 50 60oC
Threshold current increases
with p-n junction temperature
ith2
ith1
x ~2 – ~3
current (mA)
Threshold current: ith  exp (T/To)
(empirical)
ith1 = ith2 exp[(T1 – T2)/T0]
(To ~ 40 – 75 K for InGaAsP)
128
More on temperature dependence of a laser diode
• As the temperature increases, the diode’s gain decreases, and so more
current is required before oscillation begins (threshold current increases
by about 1.5%/oC)
• Thermal generated minority carriers, holes in the n layer and electrons
in the p layer recombine with majority free electrons and holes in the
doped regions outside the active layer, reducing the number of charges
reaching the active layer, thereby reducing gain.
• Reducing in gain leads to an increase in threshold current.
129
Spatial characteristics
130
Spatial characteristics
• Like other lasers, oscillation in laser diodes takes the form of
transverse and longitudinal modes.
• The transverse modes are modes of the dielectric waveguide created
by the different layers of the laser diode. Recall that the spatial
distributions in the transverse direction can be described by the integer
mode indices (p, q).
• The transverse modes can be determined by using the waveguide
theory for an optical waveguide with rectangular cross section of
dimensions l and w.
• If l/o is sufficiently small, the waveguide admits only a single mode
in the transverse direction perpendicular to the junction plane.
131
Lateral modes
• However, w is usually larger than o => the waveguide will support
several modes in the direction parallel to the plane of the junction.
• Modes in the direction parallel to the junction plane are called lateral
modes. The larger the ratio w/o, the greater the number of lateral modes
possible.
l
w
• Optical-intensity (near-field) spatial distributions for the laser
waveguide modes (p, q) = (transverse, lateral) = (1, 1), (1, 2) and (1, 3)
132
Eliminating higher-order lateral modes
• Higher-order lateral modes have a wider spatial spread, thus less
confined and has r that is greater than that for lower-order modes.
some of the highest-order modes fail to oscillate; others
oscillate at a lower power than the fundamental (lowest-order) mode.
• To attain high-power single-spatial-mode operation, the number of
waveguide modes must be reduced by decreasing the dimensions of the
active-layer cross section (l and w)
a single-mode waveguide; reducing the junction area also reduces
the threshold current.
• Higher-order lateral modes may be eliminated by making use of gainguided or index-guided LD configurations.
133
Far-field radiation pattern
• A laser diode with an active layer of dimensions l and w emits coherent
light with far-field angular divergence ≈ o/l (radians) in the plane
perpendicular to the junction and ≈ o/w (radians) in the plane parallel
to the junction. The angular divergence determines the far-field
radiation pattern.
• Due to the small size of its active layer, the laser diode is characterized
by an angular divergence larger than that of most other lasers.
o/l
Elliptical
beam
o/w
e.g. for l = 2 m, w = 10 m,
and o = 800 nm, the divergence
angles are ≈ 23o and 5o.
*The highly asymmetric elliptical
distribution of laser-diode
light can make collimating it tricky!
134
Laser spectrum
135
Laser spectrum






The basic difference between a semiconductor laser and other classes of
lasers, such as fiber lasers, is that a semiconductor laser has a very short
cavity and a high optical gain.
Thus, a semiconductor laser has a larger longitudinal mode spacing and a
larger linewidth than most other lasers.
A semiconductor laser typically has a cavity length in the range of 200500 m with a corresponding longitudinal mode spacing in the range of
100-200 GHz.
Because the gain bandwidth of a semiconductor is typically in the range
of 10-20 THz, a multimode semiconductor laser easily oscillates in 10-20
longitudinal modes.
The linewidth of each longitudinal mode is typically on the order of 10
MHz, but can be as narrow as 1 MHz or as broad as 100 MHz.
The linewidth narrows, but the number of oscillating modes increases, as
the laser is injected at a current level high above its threshold.
136
Laser spectrum





In many applications, a laser oscillating in a single frequency is
desired.
There are many different approaches to making a semiconductor
laser oscillates in a single longitudinal frequency.
E.g. The use of a very short cavity, as is the case of a verticalcavity surface-emitting laser (VCSEL), and the use of a frequencyselective grating, as is the case of the distributed Bragg reflector
(DBR) laser.
For these single-frequency lasers, the linewidth is still in the typical
range of 1-100 MHz.
It is possible to obtain single-frequency output with a linewidth on
the order of 100 kHz or less by injection locking with a narrowlinewidth, single-frequency master laser source or by using a
highly frequency-selective external grating as one optical-feedback
element.
137
Spectral characteristics
• The spectral width of the semiconductor gain coefficient is relatively
wide (~10 THz) because transitions occur between two energy bands.
• Simultaneous oscillations of many longitudinal modes in such
homogeneously broadened medium is possible (by spatial hole burning).
• The semiconductor resonator length d is significantly smaller than
that of most other types of lasers.
 The frequency spacing of adjacent resonator modes  = c/2nd is
therefore relatively large. Nevertheless, many such modes can still fit
within the broad bandwidth B over which the unsaturated gain exceeds
the loss.
=> The number of possible laser modes is M  B/
138
e.g. Number of longitudinal modes in an InGaAsP laser diode
An InGaAsP crystal (n = 3.5) of length d = 400 m has resonator
modes spaced by
 = c/2nd ≈ 107 GHz
Near the central wavelength o = 1300 nm, this frequency spacing
corresponds to a free-space wavelength spacing
 = o2/2nd ≈ 0.6 nm
If the spectral width B = 1.2 THz (a wavelength width  = 7 nm),
then approximately B/  11 longitudinal modes may oscillate.
*To obtain single-mode lasing, the resonator length d would have to be
reduced so that B ≈ c/2nd, requiring a cavity of length d ≈ 36 m.
(A shortened resonator length reduces the amplifier gain exp(gpd).) 139
Growth of oscillation in an ideal homogeneously
broadened medium
go()
g()
r
g()



• Immediately following laser turn-on, all modal frequencies for which
the small-signal gain coefficient exceeds the loss coefficient begin to
grow, with the central modes growing at the highest rate. After a short
time the gain saturates so that the central modes continue to grow while
the peripheral modes, for which the loss has become greater than the gain,
140
are attenuated and eventually vanish. Only a single mode survives.
Homogeneously broadened medium
• Immediately after being turned on, all laser modes for which the
initial gain is greater than the loss begin to grow.
=> photon-flux densities 1, 2,…, M are created in the M modes.
• Modes whose frequencies lie closest to the gain peak frequency grow
most quickly and acquire the highest photon-flux densities.
• These photons interact with the medium and uniformly deplete the gain
across the gain profile by depleting the population inversion.
• The saturated gain:
M
g() = go()/[1 + ∑j/s(j)]
j=1
where s(j) is the saturation photon-flux density associated with mode141j.
Homogeneously broadened medium
• Under ideal steady-state conditions, the surviving mode has the
frequency lying closest to the gain peak and the power in this preferred
mode remains stable, while laser oscillation at all other modes vanishes.
• Semiconductors tend to be homogeneously broadened as intraband
scattering processes are very fast (~0.1 ps). [So it does not matter which
optical transitions (modes) deplete the gain, the carrier distribution within
the band quickly, within ~0.1 ps, return to quasi-equilibrium, and the whole gain
profile is uniformly depleted.] => Suggesting single-mode lasing
• In practice, however, homogeneously broadened lasers do indeed
oscillate on multiple modes because the different modes occupy
different spatial portions of the active medium.
=> When oscillation on the most central mode is established, the gain
coefficient can still exceed the loss coefficient at those locations
where the standing-wave electric field of the most central mode
vanishes.
142
Spatial hole burning
Standing wave
distribution of
lasing mode 0
Active region

go()
r
g()
Standing wave
distribution of
lasing mode 1

143
Spatial hole burning
• This phenomenon is known as spatial hole burning. It allows another
mode, whose peak fields are located near the energy nulls of the
central mode, the opportunity to lase.
permits the simultaneous oscillation of multiple longitudinal modes in
a homogeneously broadened medium such as a semiconductor.
• Spatial hole burning is particularly prevalent in short cavities in
which there are few standing-wave cycles.
=>permits the fields of different longitudinal modes, which are
distributed along the resonator axis, to overlap less, thereby allowing
partial spatial hole burning to occur.
144
Multimode spectrum of a 1550nm laser diode
3dB bandwidth
~3 nm
145
Typical laser diode
specifications
146
(AlGaAs laser diode)
147
AlGaAs laser diode
Temp.specifications
• ~4 nm linewidth
• multimode lasing
148
(InGaAsP laser diodes)
149
InGaAsP Fabry-Perot laser diodes
150
Single-mode laser diodes
151
Single-mode laser diodes
• Essential for Dense-Wavelength-Division Multiplexing
(DWDM) technology – channel spacing is only 50 GHz in the 1550 nm
window (i.e. 0.4 nm channel spacing or 64 channels within ~ 35 nm
bandwidth of the C-band)
• Single-mode laser diodes: eliminate all but one of the longitudinal
modes
• Recall the longitudinal mode spacing:  = 2 / (2nd)
 > the gain bandwidth => only the single mode within the gain
bandwidth lases
But this either imposes very narrow gain bandwidth or very small
diode size !
152
Multimode vs. singlemode laser spectra
3-dB linewidth
3-dB linewidth
153
Single longitudinal modes
• Operation on a single longitudinal mode, which produces a singlefrequency output, may be achieved by reducing the length d of the
resonator so that the frequency spacing between adjacent longitudinal
modes exceeds the spectral width of the amplifying medium.
• Better approach for attaining single-frequency operation involves
the use of distributed reflectors (Bragg gratings) in place of the cleaved
crystal surfaces that serve as lumped mirrors in the Fabry-Perot
configuration. When distributed feedback is provided, the surfaces of
the crystal are antireflection (AR) coated to minimize reflections.
e.g. Bragg gratings as frequency-selective reflectors can be placed
in the plane of the junction (Distributed Feedback lasers) or outside the
active region (Distributed Bragg Reflector lasers, Vertical Cavity
154
Surface Emitting Lasers).
Distributed-feedback (DFB) laser diodes
• The most popular techniques for WDM
p-contact
p-InP
Bragg grating provides distributed feedback
p-InGaAsP (grating)
InGaAsP MQW active region
n-InP
n-contact
AR coating
MQW: multiple quantum well
The fabricated Bragg grating selectively reflects only one wavelength.
155
The grating in DFB lasers
• The laser has a corrugated structure etched internally just above (or
below) the active region.
• The corrugation forms an optical grating that selectively reflects light
according to its wavelength.
• This grating acts as a distributed filter, allowing only one of the
cavity longitudinal modes to propagate back and forth.
• The grating interacts directly with the evanescent mode in the space
just above (or below) the active layer.
• The grating is not placed in the active layer, because etching in this
region could introduce defects that would lower the efficiency of the
laser, resulting in a higher threshold current.
156
Bragg condition
• The operating wavelength is determined from the Bragg condition
 = m (o/2neff)
 is the grating period, o/neff is the wavelength as measured in the
diode as a waveguide, and m is the integer order of the Bragg
diffraction. (usually m = 1)
neff is the effective refractive index of the lasing mode in the active
layer --- neff lies somewhere between the index of the guiding layer
(the active region of the diode) and that of the cladding layers
For double-heterostructures, the active region is the higher index
narrow-bandgap region (say n ~ 3.5), and the cladding region is
the lower-index wide-bandgap region (say n ~ 3.2).
157
DFB laser radiates only one wavelength B – a single longitudinal mode
d ~ 100 m
Antireflection
(AR)
Single
longitudinal
mode
subm
AR
Active region
DFB laser
B

For an InGaAsP DFB laser operating at B = 1.55 m,  is about 220 nm
if we use the first-order Bragg diffraction (m = 1) and neff ~ 3.2 – 3.5.
158
Power-current characteristics of DFB laser diodes
Funabashi et al.: Recent advances in DFB
lasers for ultradense WDM applications,
IEEE JSTQE, Vol. 10, March/April 2004
Different cavity lengths of 400, 600, 800, and 1200 m. The inset shows
the singlemode laser spectrum from a packaged 800-m long DFB laser
159
at a fiber-coupled power of 150 mW @ 600 mA.
DFB laser modules
Funabashi et al.: Recent advances in DFB
lasers for ultradense WDM applications,
IEEE JSTQE, Vol. 10, March/April 2004
160
DFB laser characteristics
• Narrow linewidths (typically 0.1 – 0.2 nm), attractive for long-haul
high-bandwidth transmission.
• Less temperature dependence than most conventional laser
diodes
The grating tends to stabilize the output wavelength, which varies with
temperature changes in the refractive index.  neff = m o/2
Typical temperature-induced wavelength shifts are just under 0.1 nm/oC,
a performance 3-5 times better than that of conventional laser diodes.
161
Vertical-cavity surface-emitting laser diodes
• The vertical-cavity surface-emitting laser (VCSEL) was developed
in the 1990s, several decades after the edge-emitting laser diode.
• This diode emits from its surface rather than from its side. The lasing
is perpendicular to the plane defined by the active layer.
• Instead of cleaved facets, the optical feedback is provided by Bragg
reflectors (or distributed Bragg reflectors DBRs) consisting of layers
with alternating high and low refractive indices.
• Because of the very short cavity length (thereby a short gain medium),
very high (≥ 99%) reflectivity are required, so the reflectors typically
have 20 to 40 layer pairs.
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VCSEL schematic
Circular-shaped laser beam
output vertically
Patterned or semitransparent metal electrodes
DBR (20-40
layer pairs)
p
active region
short gain region
DBR (20 – 40
layer pairs)
n
Metal electrodes
*The upper DBR is partially transmissive at the laser-output
wavelength.
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VCSEL merits

Due to the short cavity length, the longitudinal-mode spacing
is large compared with the width of the gain curve.

If the resonant wavelength is close to the gain peak, singlelongitudinal-mode operation occurs without the need for any
additional wavelength selectivity.

VCSELs have short cavity lengths, which tend to decrease
response times (i.e. short photon cavity lifetimes ). The
result is that VCSELs can be modulated at very high speeds.
(e.g. 850 nm VCSELs can be operated at well above 10 Gb/s.)

The beam pattern is circular, the spot size can be made
compatible with that of a single-mode fiber, making the
coupling from laser to fiber more efficient (compared with
the elliptical beam from an edge-emitting diode laser).
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VCSEL applications
• VCSELs operating in the visible spectrum are appropriate as sources
for plastic optical fiber (e.g. for automotive) systems.
• VCSELs are often selected as sources for short-reach datacom (LAN)
networks operating at 850 nm. Applications include the high-speed
Gigabit Ethernet.
• Longer-wavelength VCSELs (emitting in the 1300 and 1550 nm
wavelengths) can be considered for high-capacity point-to-point fiber
systems.
• Because of the geometry, monolithic (grown on the same substrate)
two-dimensional laser-diode arrays can be formed. Such arrays can be
useful in fiber optic-network interconnects and possibly in other
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communication applications (such as on-chip optical interconnects).
Wavelength-tunable laser
diodes
166
Wavelength-tunable laser diodes
• Sources that are precisely tunable to operate at specific wavelengths
(e.g. in WDM systems, where wavelengths are spaced by fractions of a
nm) --- a wavelength tunable laser diode can serve multiple WDM
channels and potentially save cost, think using 64 fixed-wavelength
diodes vs. a few wavelength-tunable laser diodes!
• e.g. A DFB laser diode can be tuned by changing the temperature or
by changing its drive current.
• The output wavelength shifts a few tenths of a nanometer per degree
Celsius because of the dependence of the material refractive index
on temperature.
• The larger the drive current, the larger the heating of the device.
Tuning is on the order of 10-2 nm/mA. e.g. a change of 10 mA
produces a variation in wavelength of only 0.1 nm (less than WDM
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channel spacing).
Wavelength-tunable semiconductor lasers
n
m
L
m
Mode selection (m)
External-cavity tunable laser
m = 2nL
=> = n/n + L/L – m/m
Larry A. Coldren et al., Tunable semiconductor lasers: a tutorial, Journal of Lightwave Technology,
Vol. 22, pp. 193 – 202, Jan. 2004
168
Key mechanisms for semiconductor laser wavelength
tuning

By differential analysis
m = 2nL
m + m = 2nL + 2nL
(m + m)/m = (2nL + 2nL)/2nL
m/m +  = n/n + L/L
=>
 = n/n + L/L - m/m
thermal
or electrical
injection
cavity
length
tuning
mode selection
filtering
169
Example: wavelength tuning by varying the refractive
index
• The tuning range  is proportional to the change in the effective
refractive index (neff), having cavity length and cavity mode fixed
/ = neff/neff
• Consider the maximum expected range of variation in the effective
index is 1%. The corresponding tuning range would then be
 = 0.01 
For  ~ 1550 nm,  ~ 15 nm (This is quite decent as it covers about
half the C-band!)
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Tunable Distributed-Bragg Reflector (DBR) laser diodes
IGain
IPhase
IBragg
Metal electrodes
p
active region
n
Metal electrodes
Gain



Phase
Bragg
A separate current controls the Bragg wavelength by changing the
temperature in the Bragg region. (need three separate electrodes!)
Heating causes a variation in the effective refractive index of the Bragg
region, changing its operating wavelength.
From the Bragg condition:  neff = m o/2
neff/neff
171
Wavelength tunable VCSELs

A tunable cantilever VCSEL. The device consists of a bottom n-DBR, a cavity
layer with an active region, and a top mirror. The top mirror, in turn, consists of
three parts: a p-DBR, an airgap, and a top n-DBR, which is freely suspended
above the laser cavity and supported via a cantilever structure. Laser drive current
is injected through the middle contact via the p-DBR. An oxide aperture is
formed in the p-DBR section above the cavity layer to provide efficient current
guiding and optical index guiding. A top tuning contact is fabricated on the top nDBR.
Connie J. Chang-Hasnain, Tunable VCSEL, IEEE Journal on Selected Topics in Quantum Electronics,
Vol. 6, pp. 978 – 987, Nov/Dec. 2000
172
Modulation characteristics
173
Laser diodes temporal response
• Laser diodes respond much faster than LEDs, primarily because the
rise time of an LED is determined by the natural spontaneous-emission
lifetime sp of the material.
• The rise time of a laser diode depends upon the stimulated-emission
lifetime.
• In a semiconductor, the spontaneous lifetime is the average time that
free charge carriers exist in the active layer before recombining
spontaneously (from injection to recombination).
• The stimulated-emission lifetime is the average time that free charge
carriers exist in the active layer before being induced to recombine by
stimulated emission.
174
Stimulated lifetime << spontaneous lifetime
• For a laser medium to have gain, the stimulated lifetime must be
shorter than the spontaneous lifetime.
• Otherwise, spontaneous recombination would occur before stimulated
emission could begin, decreasing the population inversion and inhibiting
gain and oscillation.
• The faster stimulated-emission process, which dominates
recombination in a laser diode, ensures that a laser diode responds
more quickly to changes in the injected current than a LED.
175
Typical LED rise time ~ 2 – 50 ns
Using 3-dB electrical bandwidth f3dB = 0.35/rise time
=> 3-dB bandwidth < 0.35 / (2 ns) = 175 MHz
Stimulated emission from injection lasers occurs over a much
shorter period.
Rise times: ~ 0.1 – 1 ns
3-dB bandwidth < 0.35 / (0.1 ns) = 3.5 GHz
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Direct modulation
The modulation of a laser diode can be accomplished by changing the
drive current.
This type of modulation is known as internal or direct modulation.
The intensity of the radiated power is modulated - intensity modulation.
Drawbacks of direct modulation: (1) restricted bandwidth and
(2) laser frequency drift (due to the phase modulation of the
semiconductor gain medium upon free-carrier density change).
*Note: Laser diode direct modulation is now only used for relatively low-speed
modulation (below GHz). For GHz and beyond, we typically employ external
modulation, namely, running the diode laser at steady-state (continuous-wave
operation) and modulate the laser beam with an external modulator (which has
a bandwidth on the order of ten GHz).
177
Direct modulation
• The coupled rate equations (given by the stimulated emission term
vgg(n)) => laser diode behaves like a damped oscillator (2nd-order ODE
in d2/dt2) before reaching steady-state condition
• The direct modulation frequency cannot exceed the laser diode
relaxation oscillation frequency without significant power drop.
(*Biasing above threshold is needed in order to accelerate the switching of a laser
diode from on to off.)
178
current
threshold
(gain=loss)
Under a step-like electrical input
small-signal bias
time (ns)
relaxation osc. period
Photon
density
gain clamping
condition @
steady state
time (ns)
179
How fast can we modulate a laser diode?
Low frequency (modulated
under steady-state)
averaged pulse power
@ Relaxation frequency
time (ns)
time (ns)
time (ns)
time (ns)
1st pulse power only (highest
average power)
180
Small-signal modulation behavior
> Relaxation frequency
Laser
diode
time (ns)
LED
(LED does not
have the coupled
stimulated emission
term)
time (ns)
reduced average power
Relaxation oscillation
frequency
181
Relaxation oscillation f ~ (1/2) [1/(sp )1/2] (i/ith – 1)1/2
(i ↑ f ↑; ↓ f ↑)
For sp ~ 1 ns,  ~ 2 ps for a 300 m laser
When the injection current ~ 2ith, the maximum modulation
frequency is a few GHz.
LED: f3dB ≈ 1/2sp ~ 100 MHz
LD: relaxation oscillation f ≈ 1/2(sp)1/2 ~ GHz
*For beyond GHz modulation, we usually use external modulation.
182