# Lecture 11: Semiconductor lasers and light

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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 = rnr / (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 / 4r2 = (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 = ∫ 2r sin r d = 2r2 (1 - cos c) 0 • The fraction of the emitted light that lies within the solid angle subtended by this escape cone is A/4r2 => 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 = 2f 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 + 42f22) 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) = 1/2r • 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 (1/2r) 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/(8n3h3sp) 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 VRsp = Vrrn2 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 = vgm = (vg/d) ln (1/R) The reciprocal of the cavity lifetime (total cavity decay rate) becomes 1/ = 1/int + 1/m = vgr = 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/(vgm) ~ (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 => ddz = (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 ddz = 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()] = (c2/8n22sp) () [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): () = (c2/8n22sp) () [P(E1) – P(E2)] ≈ (c2/8n22sp) () 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/intr) 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/intr) 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/intr) 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 Vd/dt = +(stimulated emission) - (absorption) – (optical loss) + (fraction of spontaneous recombination) => Vd/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/ = vgr 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 vgm 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 vgm 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 vgm = (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/)Vvgm] = [(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 subm 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. 162 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. 163 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). 164 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 165 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 167 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 = 2nL + 2nL (m + m)/m = (2nL + 2nL)/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!) 170 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 176 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 ≈ 1/2sp ~ 100 MHz LD: relaxation oscillation f ≈ 1/2(sp)1/2 ~ GHz *For beyond GHz modulation, we usually use external modulation. 182