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Lecture 8: Laser amplifiers Optical transitions Optical absorption and amplification Population inversion Coherent optical amplifiers: gain, nonlinearity, noise References: This lecture follows the materials from Photonic Devices, Jia-Ming Liu, Chapter 10. Also from Fundamentals of Photonics, 2nd ed., Saleh & Teich, Chapter 14. 1 Intro The word laser is an acronym for light amplification by stimulated emission of radiation. However, the term laser generally refers to a laser oscillator, which generates laser light without an input light wave. A device that amplifies a laser beam by stimulated emission is called a laser amplifier. Laser light is generally highly collimated with a very small divergence and highly coherent in time and space. It also has a relatively narrow spectral linewidth and a high intensity in comparison with light generated from ordinary sources (e.g. light-emitting diodes) Due to the process of stimulated emission, an optical wave amplified by a laser amplifier preserves most of the characteristics including the frequency spectrum, the coherence, the polarization, the divergence and the direction of propagation of the input wave. Here, we discuss the characteristics of laser amplifiers. We will discuss laser oscillators in Lecture 9. 2 Optical transitions 3 Optical transitions Optical absorption and emission occur through the interaction of optical radiation with electrons in a material system that defines the energy levels of the electrons. Depending on the properties of a given material, electrons that interact with optical radiation can be either those bound to individual atoms or those residing in the energyband structures of a material such as a semiconductor. The absorption or emission of a photon by an electron is associated with a resonant transition of the electron between a lower energy level |1> of energy E1 and an upper energy level |2> of energy E2. 4 Photon-matter interaction processes There are three fundamental processes electrons make transitions between two energy levels upon a photon of energy. E = h12 = E2 – E1 • A two-level system is a model system that only contains two energy levels with which the photon interacts. 5 Three basic photon-matter interaction processes Absorption – when the quantum energy h equals the energy difference between the two energy levels (a resonant condition); the atom gains a quantum of energy Stimulated emission – the emission of a photon is triggered by the arrival of another, resonant photon Spontaneous emission – when an atom emits a photon, losing a quantum of energy in the process Einstein in 1917 first pointed out that stimulated emission is essential in the overall balance between emission and absorption, about reaching thermal equilibrium for a system of atoms. (Einstein Relations) Stimulated emission was demonstrated in 1953 in the microwave frequency by Basov, Prokhorov and Townes (Nobel 1964) 6 Three basic photon-matter interaction processes A photon emitted by stimulated emission has the same frequency, phase, polarization and propagation direction as the optical radiation that induces the process. Spontaneously emitted photons are random in phase and polarization and are emitted in all directions, though their frequencies are still dictated by the separation between the two energy levels, subject to a degree of uncertainty determined by the linewidth of the transition. Therefore, stimulated emission results in the amplification of an optical signal, whereas spontaneous emission adds noise to an optical signal. Absorption leads to the attenuation of an optical signal. 7 Spontaneous emission An electron spontaneously falls from a higher energy level E2 to a lower one E1, the emitted photon has frequency = (E2 – E1) / h |2> |1> This photon is emitted in a random direction with arbitrary polarization. The probability of such a spontaneous jump is given quantitatively by the Einstein coefficient for spontaneous emission (known as the “Einstein A coefficient”) defined as A21 = “probability” per second of a spontaneous jump from level |2> to level |1>. 8 8 Probability per second of a spontaneous emission For example, if there are N2 population per unit volume in level |2> then N2A21 per second make jumps to level |1>. The total rate at which jumps are made between the two levels is dN2/dt = -N2A21 A negative sign because the population of level 2 is decreasing Generally an electron can make jumps to more than one lower level, unless it is in the first (lowest) excited level. 9 9 Natural lifetime The population of level |2> falls exponentially with time as electrons leave by spontaneous emission. N2 = N20 exp(-A21t) The time in which the population falls to 1/e of its initial value is called the natural lifetime of level |2>, 2 = 1/A21 The magnitude of this lifetime is determined by the actual probabilities of jumps from level |2> by spontaneous emission. 10 10 Spectral lineshape The spectral characteristic of a resonant transition is therefore never infinitely sharp. Any allowed resonant transition between two energy levels has a finite relaxation time constant because at least the upper level has a finite lifetime due to spontaneous emission. From Quantum Mechanics, the finite spectral width of a resonant transition is dictated by the uncertainty principle of quantum mechanics. Intuitively, any response that has a finite relaxation time in the time domain must have a finite spectral width in the frequency domain. (recall the impulse response discussed in Lecture 2) We will see that the rate of the induced transitions between two energy levels in a given system is directly proportional to the spontaneous emission rate from the upper to the lower level. 11 Spectral lineshape For each particular resonant transition between two energy ^ levels, there is a characteristic lineshape function g() of finite linewidth that characterizes the optical processes associated with the transition. The lineshape function is generally normalized as 0 0 ĝ( )d ĝ( )d 1 ĝ( ) 2 ĝ( ) Area = 1 12 12 Homogeneous broadening If all of the atoms in a material that participate in a resonant interaction associated with the energy levels |1> and |2> are indistinguishable, their responses to an electromagnetic field are characterized by the same resonance frequency 21 and the same relaxation constant 21. In such a homogeneous system, the physical mechanisms that contribute to the linewidth of the transition affect all atoms equally. Spectral broadening due to such mechanisms is called homogeneous broadening. Previously (in Lect. 2), we discussed that such homogeneously broadened systems can be described as the damped response characterized by a single resonance frequency and a single relaxation constant. 13 Homogeneous broadening In the interaction of a material with an optical field, the absorption and emission of optical energy are characterized by the imaginary part ” of the susceptibility of the material. Therefore, the spectral characteristics of optical absorption and emission due to a resonant transition in a homogeneously broadened medium are described by the Lorentzian lineshape function of ”(). (recall from Lect 2) Using the normalization condition, we find that the resonant transition between |1> and |2> has the following normalized Lorentzian lineshape function: h ĝ( ) 2 [( 21 )2 ( h / 2)2 ] where h is the FWHM of the lineshape 14 Inhomogeneous broadening However, in many practical situations, the simple picture that gives rise to Lorentzian lineshape is not adequate. For example, because of the Doppler effect, gas atoms with different velocities have different effective resonance frequencies even if they are otherwise identical. In solids the slightly different environments in which the resonant atoms find themselves, such as random dislocations, impurities and strain fields, also give rise to different effective resonance frequencies for differently located but otherwise identical atoms. Thus, in many cases the actual emission line must be thought of as a superposition of a large number of Lorentzian lines, each with homogeneous width k and each with a distinct center frequency k. Ref: Optical resonance and two-level atoms, L. Allen and J. H. Eberly, pp. 7-10 15 Spectral lineshape Spread in frequencies Gaussian lineshape The origin of inhomogeneous broadening. The individual Lorentzian emission lines associated with different atomic dipoles are oscillating at multiple distinct frequencies. If a dielectric material is made up of those atoms with such individual lines, its emission line will be the sum of the curves. When the individual lines are densely spaced over a frequency range large compared with their own individual widths, the total lineshape is termed inhomogeneously broadened. Ref: Optical resonance and two-level atoms, L. Allen and J. H. Eberly, pp. 7-10 16 Transition rates The transition rate of a resonant optical process measures the probability per unit time for the process to occur. The transition rate of an induced process is a function of the spectral distribution of the optical radiation and the spectral characteristics of the resonant transition. The spectral distribution of an optical field is characterized by its spectral energy density u() – the energy density of the optical radiation per unit frequency interval at the optical frequency . The total energy density of the radiation u u( )d 0 The spectral intensity distribution I() = (c/n) u(), n is the refractive index of the medium The total intensity I I( )d 0 17 Spectral energy density The energy density of a radiation field u() (joules per unit volume per unit frequency interval) can be simply related to the intensity of a plane electromagnetic wave. If the intensity of the wave is I() (watts per unit area per frequency interval) u() c = I() where c is the velocity of light in free space (in the medium of refractive index n, u() c/n = I()) V Length c in a second A 18 18 Transition rates For the upward transition from |1> to |2> associated with absorption in the frequency range between and +d is W12 ( )d B12 u( )ĝ( )d For the downward transition from |2> to |1> associated with stimulated emission W21 ( )d B21u( )ĝ( )d (s-1) (s-1) The spontaneous emission rate is independent of the energy density of the radiation and is solely determined by the transition lineshape function Wsp ( )d A21ĝ( )d (s-1) The A and B constants are the Einstein A and B coefficients. 19 Radiative processes connecting two energy levels in thermal equilibrium Population N2 Spontaneous emission Population N1 E2 Stimulated emission absorption h E1 Einstein (1917) demonstrated that the rates of the three transition processes of absorption (B12), stimulated emission (B21) and spontaneous emission (A21) are related. 20 20 Transition rates The total induced transition rates W12 W21 W 12 ( )d B12 u( )ĝ( )d 0 0 W 21 ( )d B21 u( )ĝ( )d 0 0 The total spontaneous emission rate is Wsp W sp ( )d A21 N2 0 u() |2> W12 = B12u() W21 = B21u() Wsp = A21 |1> N1 21 Transition rates The induced and the spontaneous transition rates for a given system are directly proportional to one another. The relationship can be obtained by considering the interaction of blackbody radiation with an ensemble of identical atomic systems in thermal equilibrium. The spectral energy density of the blackbody radiation at a temperature T (known as thermal radiation or blackbody radiation) is given by Planck’s formula: 1 8 n 3h 3 u( ) c3 e h /kBT 1 where kB is the Boltzmann constant, kBT is the thermal energy (kBT = 26 meV @ T = 300 K) 22 Blackbody radiation A system under thermal equilibrium produces a radiation energy density u() (J Hz-1m-3) which is identical to blackbody radiation. A blackbody absorbs 100% all the radiation falling on it, irrespective of the radiation frequency. If the inside of this body is in thermal equilibrium it must radiate as much energy as it absorbs and the emission from the body is therefore characteristic of the equilibrium temperature T inside the body => this type of radiation is often called “thermal” radiation or blackbody radiation Thermal radiation 23 23 Planck’s law of blackbody radiation Planck showed that the radiation energy density for a blackbody radiating within a frequency range to +d is given by u = (8nh3/c3) [exp(h/kBT) – 1]-1 = (8n2/c3) h [exp(h/kBT) – 1]-1 Photon energy Photon density of states in a medium Of refractive index n (number of photon modes per volume per frequency interval) Photon probability of occupancy (average number of photons in each mode according to Bose‐Einstein distribution) 24 24 Planck’s law of blackbody radiation Radiation energy density u() (JHz‐1m‐3) 180 160 140 1500 K 120 100 80 60 1000 K 40 20 0 0 5E+13 1E+14 1.5E+14 2E+14 2.5E+14 3E+14 3.5E+14 Frequency (Hz) 25 25 Transition rates If N2 and N1 are the population densities per unit volume of the atoms in levels |2> and |1>, the number of atoms per unit volume making the downward transition per unit time accompanied by the emission of radiation in a frequency range from to +d N 2 (W21 ( ) Wsp ( ))d The number of atoms per unit volume making the upward transition per unit time N1W12 ( )d 26 Transition rates In thermal equilibrium, both the blackbody radiation spectral density and the atomic population density in each energy level should reach a steady state N 2 (W21 ( ) Wsp ( )) N1W12 ( ) This is the principle of detailed balance in thermal equilibrium. The steady-state population distribution in thermal equilibrium: W12 ( ) B12 u( ) N2 N1 W21 ( ) Wsp ( ) B21u( ) A21 27 Transition rates In thermal equilibrium at temperature T, the population ratio of the atoms in the upper and the lower levels follows the Boltzmann distribution. N 2 g2 exp(h / kBT ) N1 g1 where g2 and g1 are the degeneracy factors* of these energy levels, and the energy density A21 / B21 u( ) g1B12 exp(h / kBT ) 1 g2 B21 *In an atomic or molecular system, a given energy level usually consists of a number of degenerate quantummechanical states, which have the same energy. 28 Boltzmann distribution Energy E E2 N2 << N1 in thermal equilibrium exp (-h/kBT) 1 eV [For T = 300 K, kBT = 26 meV] E1 N2 (=1) N1 (=5 x 1016) Population N N2 << N1 in thermal equilibrium 29 Transition rates Identify u() with Planck’s formula: A21 8 n 3h 3 B21 c3 g1B12 g2 B21 The spontaneous radiative lifetime of the atoms in the level |2> associated with the radiative spontaneous transition from |2> to |1> is 1 1 sp Wsp A21 30 Transition rates Therefore, the spectral dependence of the spontaneous emission rate Wsp ( ) 1 sp ĝ( ) The transition rates of both of the induced processes of absorption and stimulated emission are directly proportional to the spontaneous emission rate. c3 c2 W21 ( ) u( )ĝ( ) I( )ĝ( ) 3 3 2 3 8 n h sp 8 n h sp g2 W12 ( ) W21 ( ) g1 31 Transition cross section We often express the transition probability of an atom in its interaction with optical radiation at a frequency in terms of the transition cross section, () [m2, cm2]. I( ) W21 ( ) 21 ( ) h I( ) 12 ( ) W12 ( ) h hphoton-flux density) The emission cross section c2 e ( ) 21 ( ) ĝ( ) 2 2 8 n sp The absorption cross section g2 g2 a ( ) 12 ( ) 21 ( ) e ( ) g1 g1 32 Transition cross section For the ideal Lorentzian lineshape in a homogeneously broadened medium h ĝ( ) 2 [( 21 )2 ( h / 2)2 ] The peak value of the lineshape occurs at the center of the spectrum and is a function of linewidth h only. 2 gˆ ( 21 ) h Thus, the peak value of the emission cross section at the center wavelength of the spectrum e 2 4 2 n 2 h sp 33 Characteristics of some laser materials Gain medium Wavelength (m) System Peak cross section e (m2) Spontaneous linewidth (gain bandwidth) sp 2 HeNe 0.6328 I, 4 3.0x10-17 1.5 GHz 300 ns 30 ns Ruby (Cr3+:Al2O3) 0.6943 H, 3 1.25-2.5 x 10-24 330 GHz 3 ms 3 ms Nd:YAG 1.064 H, 4 2-10 x 10-23 150 GHz 515 s 240 s Nd:glass 1.054 I, 4 4.0 x 10-24 6 THz 330 s 330 s Er:fiber 1.53 H/I, 3 6.0 x 10-25 5 THz 10 ms 10 ms Ti:sapphire 0.66-1.1 H,Q2 3.4x10-23 100 THz 3.9 s 3.2 s Semiconductor 0.37-1.65 H/I, Q2 1-5 x 10-20 10-20 THz ~1 ns ~1 ns 34 H: homogeneously broadened; I: inhomogeneously broadened 34 Optical absorption and amplification 35 Optical absorption and amplification For a monochromatic optical field at frequency and intensity I() = I(’-) W21 = (I/h) e() and W12 = (I/h) a() The net power (time-averaged) that is transferred from the optical field to the material is the difference between that absorbed by the material and that emitted due to stimulated emission: Wp = hW12N1 – hW21N2 = [N1a() – N2e()]I Wp > 0 => net power absorption from the optical field Wp < 0 => net power flows from the medium to the optical field 36 Optical absorption and amplification absorption coefficient [m-1, cm-1] () = N1a() – N2e() = (N1 – (g1/g2)N2) a() gain coefficient [m-1, cm-1] () = N2e() – N1a() = (N2 – (g2/g1)N1) e() () > 0 and () < 0 if N1 > (g1/g2) N2 () > 0 and () < 0 if N2 > (g2/g1) N1 A material absorbs optical energy in its normal state of thermal equilibrium when the lower energy level is more populated than the upper energy level. A material must be in a nonequilibrium state of population inversion with the upper energy level more populated than the lower energy level in order to provide a net optical gain to the optical field. 37 Optical absorption and amplification For simplicity, in some later discussion we can assume the degeneracy of levels 1 and 2 are equal, i.e. g1 = g2 absorption coefficient [m-1, cm-1] () = N1a() – N2e() = (N1 – N2) () gain coefficient [m-1, cm-1] () = N2e() – N1a() = (N2 – N1) () () > 0 and () < 0 if N1 > N2 () > 0 and () < 0 if N2 > N1 And e() = a() = () 38 Resonant optical susceptibility For resonant interaction of an isotropic medium with a monochromatic plane optical field at a frequency = 2, we have E(t) Eeit E * ei t Pr es (t) 0 ( res ( )Eeit *res ( )E * eit ) where Pres is the polarization contributed by the resonant transitions and res is the resonant susceptibility. The time-averaged power density absorbed by the medium is P Wp E t 2 0 "res ( ) | E | 2 t nc "res ( ) I 39 Resonant optical susceptibility Relate the time-averaged power density absorbed by the medium to the population relation Wp "res ( ) I [ N 1 a ( ) N 2 e ( )] I nc The imaginary part of the susceptibility contributed by the resonant transitions between energy levels |1> and |2> is nc "res ( ) [N1 a ( ) N 2 e ( )] The real part ’res() can then be found through the KramersKronig relations (recall from Lect. 2) Recall from Lect. 2 that a medium has an optical loss if ” > 0, and it has an optical gain if ” < 0. It is also clear that there is a net power loss from the optical field to the medium if ”res > 0, but there is a net power gain for the optical field if ”res < 0. 40 Resonant optical susceptibility The medium has an absorption coefficient given by ( ) nc "res ( ) in the case of normal population distribution when ”res>0, whereas it has a gain coefficient given by ( ) nc "res ( ) In the case of population inversion when ”res<0 Note that the material susceptibility characterizes the response of a material to the excitation of an electromagnetic field. Therefore, the resonant susceptibility res accounts for only the contributions from the induced processes of absorption and stimulated emission, but not that from the process of spontaneous emission. 41 Resonant optical susceptibility When the phase information of the optical wave is of no interest, we can find the evolution of the intensity of the optical wave as it propagates through the medium. dI/dz = -I (-ve sign represents attenuation) in the case of optical attenuation when ”res > 0, and dI/dz = I in the case of optical amplification when ”res < 0 42 Population inversion 43 Population inversion and optical gain Population inversion is the basic condition for the presence of an optical gain. In the normal state of any system in thermal equilibrium, a low-energy state is always more populated than a high-energy state – no population inversion Population inversion in a system can only be accomplished through a process called pumping – actively exciting the atoms in a low-energy state to a high-energy state. Population inversion is a nonequilibrium state that cannot be sustained without active pumping. To maintain a constant optical gain we need continuous pumping to keep the population inversion at a constant level. Many different pumping techniques depending on the gain media: optical excitation, current injection, electric discharge, chemical reaction, and excitation with ion beams 44 Population inversion A nonequilibrium distribution showing population inversion Energy E E2 E1 N1 N2 Population N 45 Population inversion and optical gain The use of a particular pumping technique depends on the properties of the gain medium being pumped. The lasers and optical amplifiers are often made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er:glass fiber, or direct-gap semiconductors, such as GaAs and InP. For dielectric media, the most commonly used pumping technique is optical pumping either with incoherent light sources, such as flashlamps and light-emitting diodes, or with coherent light sources from other lasers. Semiconductor gain media can also be optically pumped, but they are usually pumped with electric current injection. 46 Rate equations The net rate of change of population density in a given energy level is described by a rate equation. Here we only write the rate equations for the upper laser level |2> and the lower laser level |1>. In the presence of a monochromatic, coherent optical wave of intensity I at a frequency , dN2/dt = R2 – N2/2 – (I/h) (N2e – N1a) dN1/dt = R1 – N1/1 + N2/21 + (I/h) (N2e – N1a) where R2 and R1 are the total rates of pumping into energy levels |2> and |1>, and 2 and 1 are the fluorescence lifetimes (total lifetimes) of levels |2> and |1>. The rate of population decay, including radiative and nonradiative spontaneous relaxation from |2> to |1> is 1/21. 47 Rate equations Because it is possible for the population in level |2> to relax to other energy levels, the total population decay rate of level |2> is 1/2 1/21. 2 21 sp In an optical gain medium, level |2> is known as the upper laser level and level |1> is known as the lower laser level. The fluorescence lifetime 2 of the upper laser level is an important parameter that determines the effectiveness of a gain medium. In general, the upper laser level has to be a metastable state with a relatively large 2 for a gain medium to be useful. 48 Population inversion Population inversion in a medium is generally defined as N2 > (g2/g1) N1 (N2 > N1 for g1 = g2) However, this condition does not guarantee an optical gain at a particular optical frequency when the population in each level, |1> or |2>, is distributed unevenly among its sublevels. A better condition for population inversion to guarantee an optical gain at a given frequency N2e() – N1a() > 0 The pumping requirement for the condition to be satisfied depends on the properties of a medium. For atomic and molecular media, there are three different basic systems. Each has a different pumping requirement to reach effective population inversion for an optical gain. The pumping requirement can be found by solving the coupled rate equations. 49 Two-level systems |2> hp h pump |1> When the only energy levels involved in the pumping and the relaxation processes are the upper and the lower laser levels |2> and |1>, the system can be considered as a two-level system. (i.e. p = ) Level |1> is the ground state with 1 = ∞, and level |2> relaxes only to level |1> so that 21 = 2. The total population density is Nt = N1 + N2. 50 Two-level systems No matter how a true two-level system is pumped, it is not possible to achieve population inversion for an optical gain in the steady state. The optical pump for a two-level system has to be in resonance with the transition between the two levels – inducing both downward and upward transitions. |2> hp hp pump |1> While a pumping mechanism excites atoms from the lower energy level to the upper energy level, the same pump also stimulates atoms in the upper energy level to relax to the lower energy level. 51 Two-level systems While a pumping mechanism excites atoms from the lower laser level to the upper laser level, the same pump also stimulates atoms in the upper laser level to relax to the lower laser level. R2 = -R1 = W12pN1 - W21pN2, where W12p and W21p are the pumping rates from 1 to 2 and from 2 to 1. Under these conditions, dN2/dt and dN1/dt are equivalent to each other (N1 + N2 = Nt = constant). The upward (W12p) and downward (W21p) pumping rates are not independent of each other but are directly proportional to each other because both are associated with the interaction of the same pump source with a given set of energy levels. 52 Two-level systems Take the upward pumping rate W12p = Wp and the downward pumping rate to be W21P= p Wp, where p is a constant that depends on the detailed characteristics of the two-level atomic system and the pump source. In the steady state when dN2/dt = dN1/dt = 0, N2e – N1a = [Wp2(e-pa)-a]Nt [1+(1+p)Wp2 + (I2/h)(e+a)]-1 For optical pumping p = ep/ap = e(p)/a(p), where ap and ep are the absorption and emission cross sections at the pump wavelength. 53 Two-level systems In a true two-level system, the energy levels |2> and |1> can each be degenerate with degeneracies g2 and g1, but the population densities in both levels are evenly distributed among the respective degenerate states. In this situation, p = ep/ap = g1/g2 = e/a N2e – N1a = -aNt [1+(e+a)(I/hWp/a)2]-1 < 0 No matter how a true two-level system is pumped, it is clearly not possible to attain population inversion for an optical gain in the steady state. 54 Two-level systems Intuitively, the pump for a two-level system has to be in resonance with the transition between the two levels, thus inducing downward transitions and upward transitions. In the steady state, the two-level system would reach thermal equilibrium with the pump at a finite temperature T, resulting in a Boltzmann population distribution N2/N1 = (g2/g1) exp(-h/kBT) without population inversion. 55 Quasi-two-level systems |2> hp h pump |1> However, many laser gain media including laser dyes, semiconductor gain media, and some solid-state gain media, are often pumped as a quasi-two-level system. An energy level is split into a band of closely spaced, but not exactly degenerate, sublevels with its population density unevenly distributed among these sublevels. A system is a quasi-two-level system if either or both of the two levels involved are split in such a manner. 56 Quasi-two-level systems By pumping such a quasi-two-level system properly, it is possible to reach the needed population inversion in the steady state for an optical gain at a particular laser frequency . Now the ratio p = ep/ap at the pump frequency p can be made different from the ratio e/a at the laser frequency due to the uneven population distribution among the sublevels within an energy level. The pumping requirements for a steady-state optical gain from a quasi-two-level system (see p.53) p = ep/ap < e/a; Wp > (1/2) a/(e – pa) Because the absorption spectrum is generally shifted to the short-wavelength side of the emission spectrum, these conditions can be satisfied by pumping sufficiently strongly at a higher transition energy than the photon energy corresponding 57 to the peak of the emission spectrum. Three-level systems |3> hp pump Nonradiative relaxation |2> h |1> Population inversion in steady state is possible for a threelevel system. The lower laser level |1> is the ground state (or is very close to the ground state, within an energy separation of << kBT from the ground state, s.t. it is normally populated). The atoms are pumped to an energy level |3> above the upper laser level |2>. 58 Population inversion in three-level systems Over a period the population in the metastable state N2 increases above those in the ground state N1. =>The population inversion is obtained between levels |2> and |1>. Drawback: the three-level system generally requires very high pump powers because the terminal state of the stimulated transition is the ground state. More than half the ground state atoms must be pumped into the metastable state to attain population inversion. 59 Three-level systems An effective three-level system satisfies the following conditions: Population relaxation from level |3> to level |2> is very fast and efficient, ideally 2 >> 32 ≈ 3 s. t. the atoms excited by the pump quickly end up in level |2> Level |3> lies sufficiently high above level |2> with E32 >> kBT s. t. the population in level |2> cannot be thermally excited back to level |3> The lower laser level |1> is the ground state, or its population relaxes very slowly if it is not the ground state. Under these conditions R2 ≈ WpN1, R1 ≈ -WpN1, and N1+N2 ≈ Nt 1 ≈ ∞ and 21 ≈ 2 60 Three-level systems The parameter WP is the effective pumping rate for exciting an atom in the ground state to eventually reach the upper laser level. It is proportional to the power of the pump. In the steady state with a constant pump, Wp is a constant and dN2/dt = dN1/dt = 0 N2e – N1a = (Wp2e-a)Nt [1+Wp2 + (I2/h)(e+a)]-1 The pumping condition for a constant optical gain under steady-state population inversion Wp > a/2e This condition sets the minimum pumping requirement for effective population inversion to reach an optical gain in a three-level system. Note that almost all of the population initially resides in the lower laser level |1>. To attain population inversion, the pump has to be strong enough to depopulate sufficient population density from level |1>, while the system has to be able to keep it in level |2>. In the case when a = e (i.e. g1 = g2), no population inversion occurs before at least one-half of the total population is transferred from level |1> to level |2>. 61 Erbium-doped silica fibers Er3+:silica fiber amplifier is a three-level system. 3 32 pump 2 1.55 m 1 • Pumping at 980 nm using InGaAs laser diodes; a mixture of homogeneous/inhomogeneous broadening; ~ 5.3 THz • The laser transition can also be directly pumped at 1.48 m by light from InGaAsP laser diodes – like a quasi-two-level scheme 62 62 Four-level systems Nonradiative relaxation hp |3> |2> h pump |1> Nonradiative relaxation |0> A four-level system is more efficient than a three-level system. The lower laser level |1> lies sufficiently high above the ground level |0>, with E10 >> kBT. Thus, in thermal equilibrium, the population in |1> is negligibly small compared with that in |0>. Pumping takes place from level |0> to level |3>. 63 Four-level systems Levels |3> and |2> need to satisfy the same conditions as in a three-level system. The population in level |1> relaxes very quickly back to the ground level, ideally 1 ≈ 10 << 2, s. t. level |1> remains relatively unpopulated in comparison with level |2> when the system is pumped. Under these conditions, N1 ≈ 0; R2 ≈ Wp(Nt – N2) where the effective pumping rate Wp is proportional to the pump power. In the steady state when Wp is held constant, by taking dN2/dt = 0, (ignoring dN1/dt because N1 ≈ 0) N2e – N1a ≈ N2e = (Wp2e)Nt [1 + Wp2 + (I2/h)e]-1 => No minimum pumping requirement for an ideal four-level system because level |1> is initially empty. A practical four-level system is much more efficient than a three-level system. 64 Neodymium-doped glass Nd3+:glass amplifier is a four-level system. 3 32 2 1.053 m pump 0 1 1 65 65 Neodymium-doped glass Level 1 is 0.24 eV above the ground state. This is substantially larger than the thermal energy 0.026 eV at room temperature, so that the thermal population of the lower laser is negligible. Level 3 is a collection of four absorption bands, centered at 805, 745, 585, and 520 nm. The excited ions decay rapidly from level 3 to level 2 and then remain in level 2 for a substantial time sp = 330 s. 1 is very short (~ 300 ps) The 2→1 transition is inhomogeneously broadened because of the amorphous nature of the glass, which presents a different environment at each ionic location. This material therefore has a large spontaneous linewidth (gain bandwidth) ≈ 6 THz 66 66 Coherent optical amplifiers Gain, nonlinearity, noise Coherent optical amplifiers A coherent optical amplifier is a device that increases the amplitude of an optical field while maintaining its phase. If the optical field at the input to such an amplifier is monochromatic, the output will also be monochromatic with the same frequency. The output amplitude is increased relative to the input while the phase remains unchanged or is shifted by a fixed amount. In contrast, an incoherent optical amplifier increases the intensity of an optical wave without preserving its phase. Coherent optical amplifiers are important, for example, in the amplification of weak optical pulses that have traveled through a long length of optical fiber, and as a basis to understanding laser oscillators. 68 Coherent light amplification As seen earlier, stimulated emission allows a photon in a given mode to induce an atom whose electron is in an upper energy level to undergo a transition to a lower energy level and, in the process, to emit a clone photon into the same mode as the initial photon. A clone photon has the same frequency, direction and polarization as the initial photon. These two photons in turn serve to stimulate the emission of two additional photons, and so on, while preserving these properties. The result is coherent light amplification. Because stimulated emission occurs only when the photon energy is nearly equal to the transition energy difference, the process is restricted to a band of frequencies determined by the transition linewidth. 69 Laser amplification vs. electronic amplifiers Laser amplification differs in a number of respects from electronic amplification. Electronic amplifiers rely on devices in which small changes in an injected electric current or applied voltage result in large changes in the rate of flow of charge carriers (electrons and holes in a semiconductor field-effect transistor). Tuned electronic amplifiers make use of resonant circuits (e.g. a capacitor and an inductor) to limit the gain of the amplifier to the band of frequencies of interest. In contrast, atomic, molecular, and solid-state laser amplifiers rely on differences in their allowed energy levels to provide the principal frequency selection. These entities act as natural resonators that select the frequency of operation and bandwidth of the device. 70 Population inversion Light transmitted through matter in thermal equilibrium is attenuated. This is because absorption by the large population of atoms in the lower energy level is more prevalent than stimulated emission by the smaller population of atoms in the upper level. An essential ingredient for attaining laser amplification is the presence of a greater number of atoms in the upper energy level than in the lower level. This is a nonequilibrium situation. Attaining such a population inversion requires a source of power to excite (pump) the atoms to the higher energy level. 71 Ideal coherent amplifiers An ideal coherent amplifier is a linear system that increases the amplitude of the input signal by a fixed factor, the amplifier gain. A sinusoidal input leads to a sinusoidal output at the same frequency, but with larger amplitude. The gain of the ideal amplifier is constant for all frequencies within the amplifier spectral bandwidth. The amplifier may impart to the input signal a phase shift that varies linearly with frequency (corresponding to a time delay at the output with respect to the input). gain phase Output amp. Input amplitude 72 Real coherent amplifiers Real coherent amplifiers deliver a gain and phase shift that are frequency dependent. The gain and phase shift determine the amplifier’s transfer function. For a sufficiently large input amplitude, real amplifiers generally exhibit saturation, a form of nonlinear behavior in which the output amplitude does not increase in proportion to the input amplitude. Saturation introduces harmonic components into the output, provided that the amplifier bandwidth is sufficiently broad to pass them. Real amplifiers also introduce noise, s.t. a random fluctuating component is present at the output, regardless of the input. An amplifier may therefore be characterized by the following features: Gain Bandwidth Phase shift Power source Nonlinearity and gain saturation Noise 73 Theory of laser amplification A monochromatic optical plane wave traveling in the z direction with frequency , electric field E ( z ) Re[ E ( z ) exp(i 2t )] Intensity Photon-flux density (photons per second per unit area) I ( z ) | E ( z ) |2 ( z ) I ( z ) / h Consider the atomic medium (gain or active medium) with two relevant energy levels whose energy difference nearly matches the photon energy h. The numbers of atoms per unit volume in the lower and upper energy levels are denoted N1 and N2. (assume g1 = g2) 74 Gain and bandwidth The wave is amplified with a gain coefficient () (per unit length) and undergoes a phase shift () (per unit length). () > 0 corresponds to amplification, () < 0 corresponds to attenuation. Recall that the probability density (s-1) that an unexcited atom absorbs a single photon is Wi ( ) where the transition cross section at the frequency ( ) c2 8n sp 2 2 gˆ ( ) Here we assume the probability density for stimulated emission is the same as that for absorption. (a() = e() = ()) 75 Gain coefficient The average density of absorbed photons (number of photons per unit time per unit volume) is N1Wi. Similarly, the average density of clone photons generated as a result of stimulated emission is N2Wi. The net number of photons gained per second per unit volume is therefore NWi, where N = N2 – N1 is the population density difference. N is referred to as the population difference. If N > 0, a population inversion exists, in which case the medium acts as an amplifier and the photon-flux density can increase. If N < 0, the medium acts as an attenuator and the photonflux density decreases. If N = 0, the medium is transparent. 76 Gain coefficient As the incident photons travel in the z direction, the stimulated-emission photons also travel in this direction. An external pump providing a population inversion N > 0 then causes the photon-flux density (z) to increase with z. Because emitted photons stimulate further emissions, the growth at any position z is proportional to the population at that position. (z) thus increases exponentially. 77 Gain coefficient Consider the incremental number of photons per unit area per unit time, d(z), is the number of photons gained per unit time per unit volume, NWi, multiplied by the thickness dz d NWi dz In the form of a differential equation d ( ) ( z ) dz where the gain coefficient ( ) N ( ) N c2 8n sp 2 2 gˆ ( ) 78 Gain coefficient The coefficient () represents the net gain in the photon-flux density per unit length of the medium. The photon-flux density therefore is given as ( z ) (0) exp[ ( ) z ] The optical intensity I(z) = h(z) I ( z ) I (0) exp[ ( ) z ] Thus, () also represents the gain in the intensity per unit length of the medium. 79 Gain For an interaction region of total length d, the overall gain of the laser amplifier G() is defined as the ratio of the photonflux density at the output to the photon-flux density at the input, G ( ) (d ) / (0) G ( ) exp[ ( )d ] Note that in the absence of a population inversion, N is negative (N2 < N1) and so is the gain coefficient. The medium will then attenuate light traveling in the z direction. A medium in thermal equilibrium cannot provide laser amplification. 80 Gain bandwidth The dependence of the gain coefficient () on the frequency of the incident light is contained in its proportionality to the lineshape function g(). The latter is a function of width centered about the atomic resonance frequency 0 = (E2 – E1)/h. The laser amplifier is therefore a resonant device, with a resonance frequency and bandwidth determined by the lineshape function of the atomic transition. This is because stimulated emission and absorption are governed by the atomic transition. The linewidth in frequency (Hz) and in wavelength (nm) are related by = |(c/)| = (c/2) = (2/c) 81 Gain bandwidth If the lineshape function is Lorentzian, the gain coefficient is then also Lorentzian with the same width ( / 2) 2 ( ) ( 0 ) ( 0 ) 2 ( / 2) 2 where the peak gain coefficient at the central frequency 0 ( 0 ) N c2 4 2 n 2 2 sp 82 Phase‐shift coefficient The laser amplification process also introduces a phase shift. When the lineshape is Lorentzian with linewidth g() = (/2) / [( – 0)2 +(/2)2] The amplifier phase shift per unit length turns out to be () = [( – 0)/] () This phase shift is in addition to that introduced by the medium hosting the laser atoms. 83 Gain coefficient and phase-shift coefficient for a laser amplifier with a Lorentzian lineshape function gain coefficient () Phase-shift coefficient () (Compare these with the ideal coherent amplifier gain and phase responses on p. 72)84 Rate equations R2 2 1/21 = 1/sp + 1/nr sp R1 1 1 nr 1/2 = 1/21 + 1/20 20 1/nr: non-radiative decay rate Steady-state populations of levels 1 and 2 can be maintained only if the energy levels above level 2 are continuously excited by pumping and ultimately populate level 2. Pumping serves to populate level 2 at rate R2 and depopulate level 1 at rate R1 (per unit volume per second) => levels 1 and 2 can attain non-zero steady-state populations. 85 85 Rate equations in the absence of amplifier radiation The rates of increase of the population densities of levels 2 and 1 arising from pumping and decay are dN2/dt = R2 – N2/2 dN1/dt = -R1 – N1/1 + N2/21 Steady-state population difference in the absence of amplifier radiation (dN1/dt = dN2/dt = 0) N0 = N2 – N1 = R22(1-1/21) + R11 A large gain coefficient requires a large population difference (0() = N0e()) 86 86 Rate equations in the absence of amplifier radiation To increase population difference N0 Increase pumping and de-pumping rate (R2 and R1) Long 2, but sp must be sufficiently short so as to make the radiative transition rate large Short 1 (if R1 < (2/21)R2) The physical picture: the upper level should be pumped strongly and decay slowly so that it retains its population. The lower level should be de-pumped strongly so that it quickly disposes of its population. Ideally, 21 ≈ sp << 20 so that 2 ≈ sp, and 1 << sp 87 87 Rate equations in the presence of amplifier radiation The presence of radiation near the resonance frequency 0 enables transitions between levels 2 and 1 to occur via stimulated emission and absorption. These processes are characterized by the probability density Wi = () where = I/h (assuming g1 = g2 and thus e() = a()) 2 R2 Wi-1 1 R1 sp 1 nr 20 88 88 Rate equations in the presence of amplifier radiation dN2/dt = R2 – N2/2 – N2Wi + N1Wi dN1/dt = -R1 – N1/1 + N2/21 + N2Wi - N1Wi The population density of level 2 is decreased by stimulated emission from level 2 to level 1 and increased by absorption from level 1 to level 2. Under steady-state conditions (dN1/dt = dN2/dt = 0), the population difference in the presence of amplifier radiation (assuming g1 = g2) N = N2 – N1 = N0/(1 + sWi) The characteristic time s (saturation time constant) is always positive (2 ≤ 21) is given by s = 2 + 1(1 – 2/21) 89 89 Population difference N Depletion of the steady-state population difference N0 N0/2 0 0.01 0.1 1 10 sWi If the radiation is sufficiently weak so that sWi << 1 (the small signal approximation), we may take N ≈ N0 As the amplifier radiation becomes stronger, Wi increases and ultimately the population difference N 0 (transparency). This arises because stimulated emission and absorption dominate the interaction when Wi is very large and they have equal probability densities. Even very strong radiation cannot convert a negative population difference into a positive one, nor vice versa. The quantity s plays the role of a saturation time constant, i.e. when Wi = 1/s, 90 90 N is reduced by a factor of 2 from its value when Wi = 0. Four-level pumping Pump R Laser Wi-1 Rapid decay Rapid decay Short-lived |3> Long-lived |2> Short-lived |1> Ground state |0> Here we assume that the rate of pumping into level 3, and out of level 0, are the same. 91 91 Four-level pumping An external source of energy pumps atoms from level 0 to level 3 at a rate R. If the decay from level 3 to level 2 is sufficiently fast, it may be taken to be instantaneous, in which case pumping to level 3 is equivalent to pumping level 2 at the rate R2 = R. However, in this case, atoms are neither pumped into nor out of level 1, s.t. R1 = 0. Thus, in the absence of amplifier radiation (Wi = = 0), the steady-state population difference is given by (see p.86) 1 N 0 R 2 1 21 92 Four-level pumping In most four-level systems, the nonradiative decay component in the 2 to 1 transition is negligible (sp << nr) and 20 >> sp >> 1, s.t. N 0 R sp s sp And therefore R sp N 1 spWi We have assumed that the pumping rate R is independent of the population difference N = N2 – N1. 93 93 Four-level pumping This is not always the case because the population densities of the ground state and level 3, Ng and N3, are related to N1 and N2 by N g N1 N 2 N 3 N a where the total atomic density in the system, Na, is a constant. If the pumping involves a transition between the ground state and level 3 with transition probability W, then R (N g N 3 )W If levels 1 and 3 are short-lived, then N1 ≈ N3 ≈ 0, Ng + N2 ≈ Na s.t. Ng ≈ Na - N2 ≈ Na - N 94 94 Four-level pumping Under these conditions, the pumping rate can be approximated as R ( N a N )W which reveals that the pumping rate is a linearly decreasing function of the population difference N and is thus not independent of it. This arises because the population inversion established between levels 2 and 1 reduces the number of atoms available to be pumped. We obtain sp N aW N 1 spW spWi 95 95 Four-level pumping The population difference can be written in the general form N0 N 1 sWi sp N aW N0 1 spW sp s 1 spW 96 96 Three-level pumping Pump R Laser Wi-1 Rapid decay Short-lived |3> Long-lived |2> Ground state |1> Here we assume that the rate of pumping into level 3 is the same as the rate of pumping out of level 1. 97 97 Three-level pumping Under rapid 3 to 2 decay, the three-level system (assumed R is independent of N) 1 R1 R2 R In steady state, both the rate equations provide the same result 0 R 2 21 N2 21 N 2Wi N1Wi As 32 is very short, level 3 retains a negligible steady-state population. All of the atoms that are raised to it immediately decay to level 2. N1 N 2 N a 98 98 Three-level pumping The population difference N can be cast in the form: N0 N 1 sWi Where N 0 2R 21 N a s 2 21 When nonradiative decay from level 2 to level 1 is negligible (sp << nr), 21 may be replaced by sp N 0 2 R sp N a s 2 sp 99 99 Three-level pumping Attaining a population inversion N0 > 0 in the three-level system requires a pumping rate R > Na/2sp. A substantial pump power density given by E3Na/2sp. The large population in the ground state (which is the lowest laser level) is an inherent obstacle to attaining a population inversion in a three-level system that is avoided in a fourlevel system (in which level 1 is normally empty as 1 is short). The saturation time constant s ≈ sp for the four-level pumping scheme is half that for the three-level scheme. 100 100 Three-level pumping The dependence of the pumping rate R on the population difference N can be included in the analysis of the three-level system by writing R (N1 N 3 )W N3 ≈ 0, N1 = (Na-N)/2, 1 R (N a N )W 2 Substituting this in the principal equation 2R sp N a N 1 2 spWi 101 101 Three-level pumping We can write the population difference in the usual form N0 N 1 sWi N a ( spW 1) N0 1 spW 2 sp s 1 spW As in the four-level scheme, N0 and s saturates as the pumping transition probability W increases. 102 102 Saturated gain in homogeneously broadened media The gain coefficient () of a laser medium depends on the population difference N. N is governed by the pumping level. N depends on the transition rate Wi. Wi depends on the photon-flux density . => the gain coefficient () of a laser medium is dependent on the photon-flux density that is to be amplified. This is the origin of gain saturation and laser amplifier nonlinearity. 103 Saturation photon-flux density Wi ( ) Substituting Into steady-state population difference (in the presence of amplifier radiation) => N0 N 1 sWi N0 N 1 / s ( ) 1 s c s ( ) ĝ( ) 2 2 s ( ) 8 n sp 2 where 104 Gain coefficients This represents the dependence of the population difference N on the photon-flux density . Substituting N into the expression for the gain coefficient, ( ) N ( ) N c2 8n sp 2 2 gˆ ( ) We obtain the saturated gain coefficient 0 ( ) ( ) 1 / s ( ) Where the small-signal gain coefficient c2 0 ( ) N 0 ( ) N 0 ĝ( ) 2 2 8 n sp 105 105 Gain coefficients The gain coefficient is a decreasing function of the photonflux density . When equals its saturation value s() = s, the gain coefficient is reduced to half its unsaturated value. ) 1 0.5 0 0.01 0.1 1 10 s() 106 106 Amplified spontaneous emission The resonant medium that provides amplification via stimulated emission also generates spontaneous emission. The light arising from the spontaneous emission, which is independent of the input to the amplifier, represents a fundamental source of laser amplifier noise. Whereas the amplified signal has a specific frequency, direction, and polarization, the noise associated with amplified spontaneous emission (ASE) is broadband, multidirectional, and unpolarized. => it is possible to filter out some of this noise by following the amplifier with a narrowband optical filter, a collection aperture, and a polarizer. 107 Amplified spontaneous emission Spontaneous photon flux Filter and polarizer Input photon flux Spontaneous emission is a source of amplifier noise. It is relatively broadband, radiated in all directions, and unpolarized. Optics can be used at the output of the amplifier to limit the spontaneous emission noise to a narrow optical band, solid angle d and a single polarization. 108 Amplifier noise The ASE of a laser amplifier is directly proportional to the optical bandwidth of the amplifier. To increase the signal-to-noise ratio (SNR) at the amplifier output, the total noise power can be reduced to a minimum by placing at the output end of an amplifier an optical filter that has a narrow bandwidth matching the bandwidth of the optical signal. Because of the spontaneous emission noise, the SNR of an optical signal always degrades after the optical signal passes through an amplifier. The degradation of the SNR of the optical signal passing through an amplifier is measured by the optical noise figure of the amplifier defined as SNRin Fo SNRout where SNRin and SNRout represent the values of the optical SNR at the input and output ends of the amplifier. 109