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**Chapter 4 Optical Sources**

4.2 LIGHT-EMITTING DIODES (LEDs) LED Structures Quantum Efficiency and LED Power Modulation of LED 4.3 LASER DIODES Modes and Threshold Conditions 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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**4.2 Light-Emitting Diodes (LEDs)**

Characteristics of LEDs: Low speed ( < 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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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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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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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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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 Schematic of an edge-emitting double-hetero-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 ( mm), the contact stripes for the edge emitter are 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 half- power width of q|| = 120o. In the plane perpendicular to the junction, the half-power beam width q= has been made as small as 25-35o 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 ) (4-9)

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**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) (4-11)

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**4.2.3 Quantum Efficiency and LED Power**

LEDs having double-heterojunction structures can have quantum efficiencies of %. 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 / ( ) 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) _ = 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) (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) (4-16) It follows that the optical power emitted from the LED is P = hextPint = Pint/n(n+1) (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 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 Frequency response of an optical source showing the electrical and optical 3-dB bandwidth points.

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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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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 Ratiooptic = 10.log[P(w)/P(0)]**

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/ (4-20) or I(w)/I(0) = 1/2½ = 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)] (4-21)

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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 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 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 This cavity is approximately mm long, mm wide, and 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 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 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)] (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] = (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 Ggth = 33 cm-1 for a laser 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 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 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 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. High-quality 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 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) = (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

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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. (4-41) with g(l) = 0.5g(0) yields s = 1.70 nm.

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Single-Mode Lasers Three types of laser configurations using frequency-selective reflector (the corrugated phase grating) are shown in Fig 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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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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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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Single-Mode Lasers Figure Three types of laser structures using built-in frequency-selective resonator gratings: (a) Distributed-feedback (DFB) laser, (b) Distributed-Bragg- reflector (DBR) laser, (c) Distributed-reflector (DR) laser.

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Single-Mode Lasers Figure Output spectrum symmetrically distributed around in an idealized distributed-feedback (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/ (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

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**4.3.7 Modulation of Laser Diodes**

Figure Example of the relaxation-oscillation 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 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 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 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 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 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 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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