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Chapter 4 Optical Sources
 4.2 LIGHT-EMITTING DIODES (LEDs)
 4.2.1 LED Structures
 4.2.3 Quantum Efficiency and LED Power
 4.2.4 Modulation of LED
 4.3 LASER DIODES
 4.3.1 Modes and Threshold Conditions





4.3.2
4.3.3
4.3.4
4.3.6
4.3.7
Laser Diode Rate Equations
External Quantum Efficiency
Resonant Frequencies
Single-Mode Lasers
Modulation of Laser Diodes
 4.4 LIGHT SOURCE LINEARITY
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光纖通訊實驗室 黃振發教授 編撰
4.2 Light-Emitting Diodes (LEDs)
Characteristics of LEDs:
 Low speed ( < 100-200 Mb/s) data rates;
 Easy to couple with multimode fiber;
 Medium optical power in tens of microwatts;
 Require less complex drive circuitry
 No thermal or optical stabilization circuits needed;
 Fabricated less expensively with higher yields.
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光纖通訊實驗室 黃振發教授 編撰
4.2.1 LED Structures
 Characteristics Radiance (or brightness) of LED is
a measure of the optical power radiated into a unit
solid angle per unit area of the emitting surface.
 Emission response time is the time delay between
the application of a current pulse and the onset of
optical emission.
 Time delay is the factor limiting the bandwidth
with which the source can be modulated directly
by varying the injected current.
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光纖通訊實驗室 黃振發教授 編撰
4.2.1 LED Structures
 Quantum efficiency is related to the fraction of
injected electron-hole pairs that recombine
radiatively.
 To achieve high radiance and high quantum
efficiency, the LED’s double-hetero-junction
structure, as shown in Figures 4-9 and 4-10,
provides a means of confining the charge carriers
and the stimulated optical emission to the active
region of the pn junction.
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光纖通訊實驗室 黃振發教授 編撰
4.2.1 LED Structures
 Carrier confinement is to achieve a high level of
radiative recombination in the active region of the
device, which yields a high quantum efficiency.
 Band-gap differences of adjacent layers confine the
charge carriers.
 Optical confinement is for preventing absorption of
the emitted radiation by the material surrounding
the pn junction.
 Index differences of adjoining layers confine the
optical field to the central active layer.
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4.2.1 LED Structures
 Surface Emitter :
Figure 4-9. Schematic of a high-radiance surface-emitting
LED. The active region is limited to a circular section
that has an area compatible with the fiber-core end face.
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4.2.1 LED Structures
 A well is etched through the substrate of the device,
into which a fiber is cemented to accept the emitted
light.
 The circular active area is nominally 50 mm in
diameter and up to 2.5 mm thick.
 The emission pattern is essentially isotropic with a
120o half-power beam width.
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4.2.1 LED Structures
 In the isotropic Lambertian pattern, the emitter
source is equally bright when viewed from any
direction.
 The power diminishes as cosq, where q is the angle
between the viewing direction and the normal to
the surface.
 The power is down to 50% of its peak when q = 60o,
the total half-power beam width is 120o.
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4.2.1 LED Structures
 Edge Emitter :
Figure 4-10. Schematic of an edge-emitting doublehetero-junction LED. The output beam is lambertian
in plane of the pn of junction (q||=120o) and highly
directional perpendicular to the pn junction
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4.2.1 LED Structures
 Consists of an active junction region, and two
guiding layers. The guiding layers have a refractive
index lower than the active region but higher than
the surrounding material.
 To match the typical fiber-core diameters (50-100
mm), the contact stripes for the edge emitter are 5070 mm wide. Lengths of the active regions usually
range from 100 to 150 mm.
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4.2.1 LED Structures
 In the plane parallel to the junction, the emitted
beam is Lambertian (varying as cosq) with a halfpower width of q|| = 120o.
 In the plane perpendicular to the junction, the halfpower beam width q= has been made as small as 2535o by a proper choice of the waveguide thickness.
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4.2.3 Quantum Efficiency and LED Power
 Excess of electrons and holes in p- and n-type
material (referred to as minority carriers) is created
in semiconductor light source by carrier injection at
the device contacts.
 The excess carrier density decays exponentially with
time according to the relation
n = no exp(-t/t)
(4-6)
where no is the initial injected excess electron
density and the time constant t is the carrier
lifetime.
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4.2.3 Quantum Efficiency and LED Power
 The total rate at which carriers are generated is the
sum of the externally supplied and the thermally
generated rates.
 Externally supplied rate is given by J/qd, where J is
the current density, q is the electron charge, and d is
the thickness of the recombination region. Thermal
generation rate is given by n/t.
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4.2.3 Quantum Efficiency and LED Power
 The rate equation for carrier recombination in an
LED can be written as
dn / dt = (J/qd) - (n/t)
(4-7)
 Equilibrium condition is found by setting Eq. (4-7)
equal to zero, yielding the steady-state electron
density in the active region
n = Jt / qd
(4-8)
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4.2.3 Quantum Efficiency and LED Power
 Internal quantum efficiency in the active region is
the fraction of the electron-hole pairs that
recombine radiatively.
 If the radiative recombination rate is Rr and the
nonradiative recombination rate is Rnr, then the
internal quantum efficiency hint is the ratio of the
radiative recombination rate to the total
recombination rate:
hint = Rr / (Rr + Rnr )
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(4-9)
4.2.3 Quantum Efficiency and LED Power
 For exponential decay of excess carriers, the
radiative recombination lifetime is tr =n/Rr and the
nonradiative recombination lifetime is tnr = n/Rnr.
 Thus, the internal quantum efficiency can be
expressed as
hint = 1/[1+(tr/tnr)] = t/tr
(4-10)
where the bulk-recombination lifetime t is
(1/t) = (1/tr) + (1/tnr)
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(4-11)
4.2.3 Quantum Efficiency and LED Power
 LEDs having double-heterojunction structures can
have quantum efficiencies of 60-80 %. This high
efficiency is achieved because the thin active regions
of LEDs mitigate the self-absorption effects, which
reduces the nonradiative recombination rate.
 For current I injected into LED, the recombination
rate is
Rr + Rnr = I/q
(4-12)
Substituting Eq. (4-12) into Eq. (4-9) then yields the
photon-generating rate Rr = hint(I/q).
 Since each photon has an energy hn, the optical
power generated internally to the LED is
Pint = (hintI/q).hn
(4-13)
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4.2.3 Quantum Efficiency and LED Power
 Example 4-5:
A double-heterojunction InGaAsP LED emitting at
a peak wavelength of 1310-nm has radiative and
nonradiative recombination times of 30 and 100-ns,
respectively. The drive current is 40-mA. From Eq.
(4-11), the bulk recombination lifetime is
t = tr.tnr/(tr + tnr)
= 30 x 100 / (30 + 100) ns
= 23.1 ns
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4.2.3 Quantum Efficiency and LED Power
 Using Eq. (4-10), the internal quantum efficiency is
hint = t / tr = 23.1/30 = 0.77
Substituting this into Eq. (4-13) yields an internal
power level of
Pint = hint.(hcI/ql)
_
(6.6256x10-34J.s)(3x108m/s)(0.040A)
= 0.77 x ---------------------------------------------(1.602x10-19C)(1.31x10-6m)
= 2.92 mW
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4.2.3 Quantum Efficiency and LED Power
 External quantum efficiency hext is the ratio of the
photons emitted from the LED to the number of
internally generated photons.
 As shown in Fig. 4-15, only that fraction of light
falling within a cone defined by the critical angle
fc = p/2 - qc = sin-1(n2/n1) will cross the interface.
 Here, n1 is the refractive index of the semiconductor
material and n2 is the refractive index of the outside
material.
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4.2.3 Quantum Efficiency and LED Power
 External The external quantum efficiency can be
calculated from the expression
(4-14)
where T(f) is the Fresnel transmissivity.
 T(f) can be simplified with the expression for
normal incidence
T(0) = 4n1n2 / (n1+n2)2 .
(4-15)
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4.2.3 Quantum Efficiency and LED Power
 For the outside medium being air (n2 = 1.0) and
letting n1 = n, we have T(0) = 4n/(n+1)2.
 The external quantum efficiency is then
approximately given by
hext = 1/n(n+1)2.
(4-16)
 It follows that the optical power emitted from the
LED is
P = hextPint
= Pint/n(n+1)2.
(4-17)
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4.2.3 Quantum Efficiency and LED Power
 Example 4-6:
Assuming a typical value of n = 3.5 for the refractive
index of an LED material, then from Eq. (4-16) we
obtain hext = 1.41%.
 This shows that only a small fraction of the
internally generated optical power is emitted from
the device.
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4.2.3 Quantum Efficiency and LED Power
Figure 4-15. Only light falling a cone defined
by the critical angel will be emitted from an
optical source.
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4.2.3 Quantum Efficiency and LED Power
Figure 4-16. Frequency response of an optical
source showing the electrical and optical 3-dB
bandwidth points.
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4.2.4 Modulation of LED
 The frequency response of an LED is determined by
1). The doping level in the active region,
2).The injected carrier lifetime ti in the
recombination region,
3). The parasitic capacitance of the LED.
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4.2.4 Modulation of LED
 If the drive current is modulated at a frequency w, the
optical output power of the device will vary as
P(w) = Po[1 + (wti)2]-1/2 ,
(4-18)
where Po is the power emitted at zero modulation frequency.
 Since P(w) = I2(w)R, the ratio of the output electrical power
at the frequency w to the power at zero modulation is
Ratioelec = 10.log[P(w)/P(0)]
= 10.log [I2(w)/I2(0)]
(4-19)
where I(w) is the electrical current in the detection circuitry.
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4.2.4 Modulation of LED
 The electrical 3-dB point occurs at that frequency
point where the detected electrical power P(w) =
P(0)/2. This happens when
I2(w) / I2(0) = 1/2
(4-20)
or I(w)/I(0) = 1/2½ = 0.707.
 The optical 3-dB bandwidth of an LED can be
determined from
Ratiooptic = 10.log[P(w)/P(0)]
= 10.log [I(w)/I(0)]
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(4-21)
4.2.4 Modulation of LED
 The optical 3-dB point occurs at that frequency
where the ratio of the currents is equal to 1/2.
 As shown in Fig. 4-16, this corresponds to an
electrical power attenuation of 6 dB.
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4.3 LASER DIODES
 Laser action is the result of three key processes:
1). photon absorption,
2). spontaneous emission, and
3). stimulated emission.
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4.3 LASER DIODES
 Spontaneous Emissions
 When a photon of energy hn12 impinges on the system,
an electron in state E1 can absorb the photon energy
and be excited to state E2, as shown in Fig. 4-17a.
 The electron will shortly return to the ground state,
thereby emitting a photon of energy hn12 = E2 – E1.
 The spontaneous emissions are isotropic and of
random phase, and thus appear as a narrowband
gaussian output.
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4.3 LASER DIODES
 Stimulated Emissions
 As shown in Fig. 4-17c, if a photon of energy hn12
impinges on the system while the electron is still in its
excited state, the electron is immediately stimulated
to drop to the ground state and give off a photon of
energy hn12.
 The emitted photon in the stimulated emission is in
phase with the incident photon. Stimulated emission
will exceed absorption if the population of the excited
states is greater than that of the ground state. This
condition is known as population inversion.
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4.3 LASER DIODES
Figure 4-17. The three key transition processes involved in
laser action. The open circle represents the initial state of
the electron and the filled circle represents the final state.
Incidect photons are shown on the left of each diagram and
emitted photons are shown on the right.
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4.3 LASER DIODES
Figure 4-18. Fabry-Perot resonator cavity for a laser diode.
The rear facet can be coated with a dielectric reflector to
reduce optical loss in the cavity. The light beam emerging
from the laser forms a vertical ellipse, even though the lasing
spot at the active-area facet is a horizontal ellipse.
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4.3.1 Modes and Threshold Conditions





Characteristics of LDs:
Suitable for systems of bandwidth > 200-MHz;
Typically have response times less than 1-ns;
Having optical bandwidths of 2-nm or less;
Capable of coupling several tens of milliwatts of
luminescent power;
 Can couple with optical fibers with small cores
and small mode-field diameters.
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4.3.1 Modes and Threshold Conditions
 The radiation in the laser diode is generated within a
Fabry-Perot resonator cavity, as shown in Fig. 4-18.
 This cavity is approximately 250-500 mm long, 5-15
mm wide, and 0.1-0.2 mm thick.
 These dimensions are commonly referred to as the
longitudinal, lateral, and transverse dimensions of the
cavity, respectively.
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4.3.1 Modes and Threshold Conditions
 In the LD Fabry-Perot resonator, a pair of flat,
partially reflecting mirrors are directed to enclose
the cavity.
 The laser cavity can have many resonant
frequencies.
 The device will emit light at those resonant
frequencies for which the gain is sufficient to
overcome the losses.
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4.3.1 Modes and Threshold Conditions
 Typical distributed-feedback (DFB) laser
configuration is shown in Fig. 4-19. The lasing
action is obtained from Bragg reflectors or
distributed-feedback corrugations, which are
incorporated into the multilayer structure along the
length of the diode.
 The optical radiation within the resonance cavity of
a laser diode sets up a pattern of electric and
magnetic field lines called the modes of the cavity.
These can be separated into two independent sets of
transverse electric (TE) and transverse magnetic
(TM) modes.
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4.3.1 Modes and Threshold Conditions
 The longitudinal modes are related to the length L of
the cavity and determine the principal structure of
the frequency spectrum of the emitted optical
radiation.
 Since L is much larger than the lasing wavelength of
~1 mm, many longitudinal modes can exist.
 Lateral modes lie in the plane of the pn junction.
These modes depend on the side wall preparation
and the width of the cavity, and determine the
shape of the lateral profile of the laser beam.
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4.3.1 Modes and Threshold Conditions
 Transverse modes are associated with the
electromagnetic field and beam profile in the
direction perpendicular to the plane of the pn
junction.
 These modes largely determine such laser
characteristics as the radiation pattern and the
threshold current density.
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4.3.1 Modes and Threshold Conditions
 To determine the lasing conditions and the resonant
frequencies, we express the EM wave propagating
in the longitudinal direction in terms of the electric
field phasor
E(z,t) = I(z).exp[j(wt - bk)]
(4-22)
where I(z) is the optical field intensity, w is the
optical radian frequency, and b is the propagation
constant.
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4.3.1 Modes and Threshold Conditions
 The stimulated emission rate into a given mode is
proportional to the intensity of the radiation in that
mode.
 The radiation intensity at a photon energy hn varies
exponentially with the distance z that it traverses
along the lasing cavity according to the relationship
I(z) = I(0).exp{[Gg(hn) – ~a(hn)]z}
(4-23)
 where g is the gain coefficient in the Fabry-Perot
cavity, ~a is the effective absorption coefficient of
the material in the optical path,
 and G is the optical-field confinement factor -- the
fraction of optical power in the active layer.
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4.3.1 Modes and Threshold Conditions
Figure 4-19. Structure of a distributed-feedback
(DFB) laser diode.
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4.3.1 Modes and Threshold Conditions
 Lasing occurs when the gain of guided modes
exceed the optical loss during one roundtrip
through the cavity.
 During the roundtrip z = 2L, only the fractions R1
and R2 of the optical radiation are reflected from
the laser ends.
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4.3.1 Modes and Threshold Conditions
 R1 and R2 are the Fresnel reflection coefficients given
by
R = [(n1-n2)/(n1+n2)]2
(4-24)
for the optical reflection at an interface between
materials having refractive indices n1 and n2.
 From this lasing condition, Eq. (4-23) becomes
I(2L) = I(0)R1R2.exp{2L[Gg(hn) – ~a(hn)]} (4-25)
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4.3.1 Modes and Threshold Conditions
 At the lasing threshold, a steady-state oscillation
takes place, and the magnitude and phase of the
returned wave must be equal to those of the original
wave:
I(2L) = I(0) and exp[-j2bL] = 1
(4-26)
 Equ. (4-26) gives information concerning the
resonant frequencies of the Fabry-Perot cavity.
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4.3.1 Modes and Threshold Conditions
 The condition to just reach the lasing threshold is
the point at which the optical gain is equal to the
total loss at in the cavity.
 From Eq. (4-26), this condition is
Ggth = at
= ~a + (1/2L).ln(1/R1R2)
= ~a + aend
(4-28)
where aend is the mirror loss in the lasing cavity.
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4.3.1 Modes and Threshold Conditions
 For lasing to occur, we must have the gain g > gth.
This means that the pumping source that maintains
the population inversion must be sufficiently strong
to support or exceed all the energy-consuming
mechanisms within the lasing cavity.
 Example 4-7:
For GaAs, R1 = R2 = 0.32 for uncoated facets (i.e.,
32% of the radiation is reflected at a facet) and
~a = 10cm-1. This yields Gg = 33 cm-1 for a laser
th
diode of length L = 500 mm.
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4.3.1 Modes and Threshold Conditions
 The relationship between optical output power and
diode drive current is presented in Fig. 4-20.
 At low diode currents, only spontaneous radiation
is emitted. Both the spectral range and the lateral
beam width of this emission are broad like that of
an LED.
 A dramatic and sharply defined increase in the
power output occurs at the lasing threshold. As this
transition point is approached, the spectral range
and the beam width both narrow with increasing
drive current.
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4.3.1 Modes and Threshold Conditions
 The final spectral width of ~1 nm and the fully
narrowed lateral beam width of nominally 5-10° are
reached just past the threshold point.
 The threshold current Ith is defined by extrapolation
of the lasing region of the L-I curve, as shown in Fig.
4-20.
 At high power outputs, the slope of the curve
decreases because of junction heating.
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4.3.1 Modes and Threshold Conditions
 For laser structures that have strong carrier
confinement, the threshold current density for
stimulated emission Jth can to a good approximation
be related to the lasing-threshold optical gain by
gth = bJth
(4-29)
where b is a constant that depends on the specific
device construction.
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4.3.1 Modes and Threshold Conditions
Figure 4-20. Relationship between optical output power
and laser diode drive current. Below the lasing threshold,
the optical output is a spontaneous LED-type emission.
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4.3.2 Laser Diode Rate Equations
For a pn junction with a carrier-confinement region
of depth d :
 The rate equation governs the number of photons F
is given by
dF/dt = CnF + Rsp – F/tph
(4-30)
= stimulated emission
+ spontaneous emission + photon loss.
 The rate equation governs the number of electrons n
is given by
dn/dt = J/qd - n/tsp - CnF
(4-31)
= injection + spontaneous emission
+ stimulated emission .
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4.3.2 Laser Diode Rate Equations
 Here, C is a coefficient describing the strength of
the optical absorption and emission interactions;
 Rsp is the rate of spontaneous emission into the
lasing mode,
 tph is the photon lifetime,
 tsp is the spontaneous-recombination lifetime,
 and J is the injection-current density.
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4.3.3 External Quantum Efficiency
 External differential quantum efficiency hext is
defined as the number of photons emitted per
radiative electron-hole pair recombination above
threshold:
hext = hi (gth – a) / gth
(4-37)
Here, gth is the fixed gain coefficient and hi is the
internal quantum efficiency.
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4.3.3 External Quantum Efficiency
 Experimentally, hext is given by
(4-38)
where Eg is the band-gap energy in electron volts, dP
is the incremental change in the emitted optical
power for an incremental change dI in the drive
current, and l is the emission wavelength.
 For standard LDs, external differential quantum
efficiencies of 15-20% per facet are typical. Highquality devices have differential quantum
efficiencies of 30-40%.
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4.3.4 Resonant Frequencies
 The condition in Eq. (4-27) holds when
2bL = 2pm
(4-39)
where m is an integer.
 Using b = 2pn/l for the propagation constant from
Eq. (2-46), we have
m = L/(l/2n) = (2Ln/c)n
(4-40)
where c = nl.
 The cavity resonates when an integer number of
half-wavelengths spans the region between the
mirrors.
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4.3.4 Resonant Frequencies
 The relationship between gain and frequency can be
assumed to have the Gaussian form
g(l) = g(0)exp[-(l-lo)2/2s2]
(4-41)
 where lo is the wavelength at the center of the
spectrum, s is the spectral width of the gain, and
the maximum gain g(0) is proportional to the
population inversion.
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4.3.4 Resonant Frequencies
Figure 4-21. Typical spectrum from a gain-guided
GaAlAs/GaAs laser diode.
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4.3.4 Resonant Frequencies
 To find the frequency spacing, consider successive
modes of frequencies nm-1 and nm. From Eq. (4-40),
we have
m-1=(2Ln/c)nm-1 and m=(2Ln/c)nm
(4-43)
 Subtracting these two equations yields
(2Ln/c)(nm - nm-1) = (2Ln/c)(Dn) = 1
(4-44)
from which we have the frequency and wavelength
spacings
Dn = c / 2Ln and Dl = l2 / 2Ln .
(4-46)
 Given Eqs. (4-41) and (4-46), the output spectrum of
a multimode laser follows the plot given in Fig. 4-21.
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4.3.4 Resonant Frequencies
 Example 4-8:
A GaAs laser operating at 850-nm has a 500-mm
length and a refractive index n=3.7.
What are the frequency and wavelength spacings? If,
at the half-power point, l - lo = 2 nm, what is the
spectral width s of the gain?
 Solution:
From Eq. (4-45) we have Dn = 81-GHz, and from
Eq. (4-46) we find that Dl = 0.2 nm. Using Eq. (441) with g(l) = 0.5g(0) yields s = 1.70 nm.
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4.3.6 Single-Mode Lasers
 Three types of laser configurations using frequencyselective reflector (the corrugated phase grating) are
shown in Fig. 4-28.
 The coupling between the counter-propagating
traveling waves is at a maximum for wavelengths
close to the Bragg wavelength lB :
lB = 2neL/k ,
(4-47)
 where L is the period of the corrugations, ne is the
effective refractive index of the mode, and k is the
order of the grating.
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4.3.6 Single-Mode Lasers
 In a DFB laser (Figs. 4-28a & 4-29), the longitudinal
modes are spaced symmetrically around lB at
wavelengths
(4-48)
where m is the mode order and Le is the effective
grating length.
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4.3.6 Single-Mode Lasers
 For the DBR laser, the gratings are located at the
ends of the normal active layer to replace the
cleaved end mirrors used in the Fabry-Perot optical
resonator (Fig. 4-28b).
 The DR laser consists of active and passive
distributed reflectors (Fig. 4-28c). The structure
improves the lasing properties of conventional DFB
and DBR lasers, and has a high efficiency and high
output capability.
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4.3.6 Single-Mode Lasers
Figure 4-28. Three types of
laser structures using builtin frequency-selective
resonator gratings:
(a) Distributed-feedback
(DFB) laser,
(b) Distributed-Braggreflector (DBR) laser,
(c) Distributed-reflector
(DR) laser.
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4.3.6 Single-Mode Lasers
Figure 4-29. Output spectrum symmetrically
distributed around in an idealized distributedfeedback (DFB) laser diode.
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4.3.7 Modulation of Laser Diodes
Modulation of LDs can be realized by:
 Direct Modulation –
varying the laser drive current with the
information stream to produce a varying optical
output power.
 External Modulation –
needed for high-speed systems (> 2.5 Gb/s) to
minimize undesirable nonlinear effects such a
chirping.
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4.3.7 Modulation of Laser Diodes
 Limitation on LDs Modulation Rate:
 The spontaneous lifetime tsp is a function of the
semiconductor band structure and the carrier
concentration.
 The stimulated carrier lifetime tst depends on the
optical density in the lasing cavity and is on the
order of 10 ps.
 The photon lifetime tph is the average time that the
photon resides in the lasing cavity before being lost
either by absorption or by emission through the
facets.
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4.3.7 Modulation of Laser Diodes
 If the laser is completely turned off after each pulse,
the spontaneous carrier lifetime will limit the
modulation rate.
 At the onsets of a current pulse Ip, a period of time
td given by
td = t ln{Ip / [Ip+(IB - Ith)]}
(4-50)
is needed to achieve the population inversion to
produce a gain to overcome the optical losses in the
lasing cavity.
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4.3.7 Modulation of Laser Diodes
 In Eq. (4-50) the parameter IB is the bias current and
t is the average carrier lifetime in the combination
region when the total current I = Ip + IB is close to Ith.
 The delay time can be eliminated by dc-biasing the
diode at the lasing threshold current.
 Pulse modulation is carried out by modulating the
laser in the operating region above threshold.
 In this region, the carrier lifetime is shortened to the
stimulated emission lifetime, so that high modulation
rates are possible .
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4.3.7 Modulation of Laser Diodes
 When using a directly modulated laser diode for
high-speed transmission systems, the modulation
frequency can be no larger than the frequency of
the relaxation oscillations of the laser field.
 The relaxation oscillation depends on both the
spontaneous lifetime and the photon lifetime.
 For a linear dependence of the optical gain on
carrier density, the relaxation oscillation occurs
approximately at
f = (1/2p).[1/(tsptph)1/2].[I/Ith - 1]1/2
(4-51)
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4.3.7 Modulation of Laser Diodes
 Since tsp is ~1 ns and tph is ~2 ps for a 300-mm-long
laser, then when the injection current I is about
twice the threshold current Ith, the maximum
modulation frequency is a few GHz.
 Example of a laser having relaxation-oscillation
peak at 3-GHz is shown in Fig. 4-30.
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4.3.7 Modulation of Laser Diodes
Figure 4-30. Example of the relaxationoscillation of a laser diode.
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4.4 LIGHT SOURCE LINEARITY
 In Fig. 4-35, the electric analog signal s(t) is used to
modulate directly an optical source about a bias
point IB. With no signal input, the optical power
output is Pt.
 When the signal s(t) is applied, the optical output
power P(t) is
P(t) = Pt[1 + ms(t)]
(4-53)
 Here, m is the modulation index defined as
m = DI / IB’
(4-54)
where IB’ = IB for LEDs and IB’ = IB – Ith for laser
diodes. The parameter DI is the variation in current
about the bias point.
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4.4 LIGHT SOURCE LINEARITY
 To prevent distortions in the output signal, the
modulation must be confined to the linear region of
the L-I curve.
 If DI > IB’ (i.e., m > 100 %), the lower portion of the
signal gets cut off and severe distortion will result.
 Typical m values for analog applications range from
0.25 to 0.50.
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4.4 LIGHT SOURCE LINEARITY
Figure 4-35. Bias point and AM range for LEDs and
laser diodes.
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4.4 LIGHT SOURCE LINEARITY
 If the signal input to a nonlinear device is a simple
cosine wave x(t) = Acoswt, the output will be
y(t) = Ao +A1coswt +A2cos2wt +A3cos3wt
(4-55)
 The output signal consists of a component at the
input frequency w plus spurious components at zero
frequency, at the 2nd harmonic frequency 2w, at the
3rd harmonic frequency 3w, and so on.
 The above effect is known as harmonic distortion.
The amount of nth-order distortion is given by
nth-order harmonic distortion = 20 log(An/A1)
(4-56)
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4.4 LIGHT SOURCE LINEARITY
 To determine intermodulation distortion (IMD), the
modulating signal of a nonlinear device is taken to
be x(t) = A1coswt + A2cos2wt.
 The output signal will be of the form
y(t) = Sm,n Bmn.cos(mw1 + nw2)
(4-57)
where m, n = 0, ±1, ±2, ±3, ...
 This signal includes all the harmonics of w1 and w2
plus w2 + w1, w2 + 2w1, and so on.
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4.4 LIGHT SOURCE LINEARITY
 The sum of the absolute values of the coefficients m
and n determines the order of the IMD.
 The 2nd-order IM products are at w1 + w2 with
amplitude B11,
 the 3rd-order IM products are at w1 + 2w2 and 2w1 +
w2 with amplitudes B12 and B21, and so on.
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4.4 LIGHT SOURCE LINEARITY
 In gain-guided laser diodes, there can be
nonlinearities for optical power output versus diode
current, as is illustrated in Fig. 4-36.
 The "kinks" are a result of inhomogeneities in the
active region of the device and also arise from power
switching between the dominant lateral modes in the
laser.
 Power saturation can occur at high output levels
because of active-layer heating.
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4.4 LIGHT SOURCE LINEARITY
Figure 4-36. Example of a kink and power saturation
for optical output power versus drive current of a
laser diode.
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4.4 LIGHT SOURCE LINEARITY
 Total harmonic distortions in GaAlAs LEDs and
laser diodes tend to be in the range of 30-40 dB
below the output at the fundamental modulation
frequency for modulation index around 0.5.
 The 2nd and 3rd-order harmonic distortions are
shown in Fig. 4-37 for a GaAlAs LED.
 The harmonic distortions decrease with increasing
bias current but become large at higher modulation
frequencies.
 The IMD curves follow the same characteristics as
those in Fig. 4-37, but are 5-8 dB worse.
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4.4 LIGHT SOURCE LINEARITY
Figure 4-37. Second- and third-order harmonic distortions in a
GaAlAs LED for several modulation frequencies. The distortion
is given in terms of the power as the n-th harmonic relative to
power at the modulation frequency f1.
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