4 C.Hirlimann a) b) c) Ea Em Fig.1.2.The three elementary electron-photon interaction processes in atoms: (a)absorption,(b)spontaneous emission,(c)stimulated emission 1.2.3 Stimulated Emission This contribution to light emission only occurs under the influence of an elec- tromagnetic wave.When a photon with energy hw passes by an excited atom it may stimulate the emission by this atom of a twin photon,with a probabil- ity Bnm strictly equal to the absorption probability Bmn.The emitted twin photon has the same energy,the same direction of propagation,the same polarization state and its associated wave has the same phase as the original inducing photon.In an elementary stimulated emission process the net optical gain is two. 1.3 Light Amplification by Stimulated Emission In what follows we will discuss the conditions that have to be fulfilled for the stimulated emission to be used for the amplification of electromagnetic waves. What we need now is a set of N atoms,which will simulate a two-level material.The levels are called E and E2(Fig.1.3). Their respective populations are Ni and N2 per unit volume;the system is illuminated by a light beam of n photons per second per unit volume with individual energy hw =E2-E1.The absorption of light in this medium is proportional to the electronic transition probability,to the number of photons at position z in the medium and to the number of available atoms in state 1 per unit volume. To model the variation of the number of photons n as a function of the distance z inside the medium,the use of energy conservation leads to the E2 N2 b12 b21 a21 E1 N1 Fig.1.3.Energy diagram for a set of atoms with two electronic levels
4 C. Hirlimann Em En Fig. 1.2. The three elementary electron–photon interaction processes in atoms: (a) absorption, (b) spontaneous emission, (c) stimulated emission 1.2.3 Stimulated Emission This contribution to light emission only occurs under the influence of an electromagnetic wave. When a photon with energy ¯hω passes by an excited atom it may stimulate the emission by this atom of a twin photon, with a probability Bnm strictly equal to the absorption probability Bmn. The emitted twin photon has the same energy, the same direction of propagation, the same polarization state and its associated wave has the same phase as the original inducing photon. In an elementary stimulated emission process the net optical gain is two. 1.3 Light Amplification by Stimulated Emission In what follows we will discuss the conditions that have to be fulfilled for the stimulated emission to be used for the amplification of electromagnetic waves. What we need now is a set of N atoms, which will simulate a two-level material. The levels are called E1 and E2 (Fig. 1.3). Their respective populations are N1 and N2 per unit volume; the system is illuminated by a light beam of n photons per second per unit volume with individual energy ¯hω = E2 − E1. The absorption of light in this medium is proportional to the electronic transition probability, to the number of photons at position z in the medium and to the number of available atoms in state 1 per unit volume. To model the variation of the number of photons n as a function of the distance z inside the medium, the use of energy conservation leads to the E2 N2 E1 N1 b12 b21 a21 Fig. 1.3. Energy diagram for a set of atoms with two electronic levels
1 Laser Basics 5 following differential equation: d2=(N2-M)b12n+a21, d (1.5) where b12 =b21 and a21 are related to the Einstein coefficients by constant quantities.For the sake of simplicity we will neglect the spontaneous emission process and thus the number of photons as a function of the propagation distance is given as n(2)=noe(Na-Ni)b1zz (1.6) with no =n(0)being the number of photons impinging on the medium. When N2<N1,expression (1.6)simply reduces to the usual Beer- Lambert law for absorption,n(z)=no e-az,where a=(N1-N2)612 >0 is the linear absorption coefficient.This limit is found with any absorbing ma- terial at room temperature:there are more atoms in the ground state ready to absorb photons than atoms in the excited state able to emit a photon. When N=N2,expression (1.6)shows that the number of photons re- mains constant along the propagation distance.In this case the full symmetry between absorption and stimulated emission plays a central role:the elemen- tary absorption and stimulated emission processes are balanced.If sponta- neous emission had been kept in expression(1.6),a slow increase of the num- ber of photons with distance would have been found due to the spontaneous creation of photons. When N2 N1,there are more excited atoms than atoms in the ground state.The population is said to be "inverted".Expression(1.6)can be written n(2)=no e9z,g=(NI-N2)612 being the low-signal gain coefficient.This process is very similar to a chain reaction:in an inverted medium each in- coming photon stimulates the emission of a twin photon and its descendants too.The net growth of the number of photons is exponential but does not ex- actly correspond to the fast doubling every generation mentioned at the end of Sect.1.2.3.Because the emitted photons are resonant with the two-level system,some of them are reabsorbed;also,some of the electrons available in the excited state are lost for stimulated emission because of their spontaneous decay.The elementary growth factor is therefore less than 2. 1.4 Population Inversion To build an optical oscillator,the first step is to find how to amplify light waves,and we have just seen that amplification is possible under the condition that there exist some way to create an inverted population in some material medium. 1.4.1 Two-Level System Let us first consider,again,the two-electronic-level system (Fig.1.3).Elec- trons,because they have wave functions that are antisymmetric under inter-
1 Laser Basics 5 following differential equation: dn dz = (N2 − N1)b12n + a21N2, (1.5) where b12 = b21 and a21 are related to the Einstein coefficients by constant quantities. For the sake of simplicity we will neglect the spontaneous emission process and thus the number of photons as a function of the propagation distance is given as n(z) = n0 e(N2−N1)b12z, (1.6) with n0 = n(0) being the number of photons impinging on the medium. When N2 < N1, expression (1.6) simply reduces to the usual Beer– Lambert law for absorption, n(z) = n0 e−αz, where α = (N1 − N2)b12 > 0 is the linear absorption coefficient. This limit is found with any absorbing material at room temperature: there are more atoms in the ground state ready to absorb photons than atoms in the excited state able to emit a photon. When N1 = N2, expression (1.6) shows that the number of photons remains constant along the propagation distance. In this case the full symmetry between absorption and stimulated emission plays a central role: the elementary absorption and stimulated emission processes are balanced. If spontaneous emission had been kept in expression (1.6), a slow increase of the number of photons with distance would have been found due to the spontaneous creation of photons. When N2 > N1, there are more excited atoms than atoms in the ground state. The population is said to be “inverted”. Expression (1.6) can be written n(z) = n0 egz, g = (N1 − N2)b12 being the low-signal gain coefficient. This process is very similar to a chain reaction: in an inverted medium each incoming photon stimulates the emission of a twin photon and its descendants too. The net growth of the number of photons is exponential but does not exactly correspond to the fast doubling every generation mentioned at the end of Sect. 1.2.3. Because the emitted photons are resonant with the two-level system, some of them are reabsorbed; also, some of the electrons available in the excited state are lost for stimulated emission because of their spontaneous decay. The elementary growth factor is therefore less than 2. 1.4 Population Inversion To build an optical oscillator, the first step is to find how to amplify light waves, and we have just seen that amplification is possible under the condition that there exist some way to create an inverted population in some material medium. 1.4.1 Two-Level System Let us first consider, again, the two-electronic-level system (Fig. 1.3). Electrons, because they have wave functions that are antisymmetric under inter-
6 C.Hirlimann 2P12 2S12 spin-1/2 spin 1/2 Fig.1.4.Sketch of the Zeeman structure of rubidium atoms in the vapour phase change of particles,obey the Fermi-Dirac statistical distribution.With AE being the energy separation between the two levels,the population ratio for a two-electronic-level system is given by N2 1 Ni eAE/KT +1 (1.7) When the temperature T goes to OK the population ratio also goes to 0. At OK the energy of the system is zero:all the electrons are in the ground state,N2 =0,N1=No the total number of electrons.In contrast,when the temperature goes towards infinity (T-oo),the population ratio goes to one-half,N2=N1/2.In the high-temperature limit the electrons are equally distributed between the ground and excited states:an inverted population regime cannot be reached by just heating a material.It is not possible,either, to create an inverted population in a two-level system by optically exciting the electrons:at best there can be as many absorbed as emitted photons. 1.4.2 Optical Pumping Optical pumping was proposed,in 1958,by Alfred Kastler as a way to produce inverted populations of electrons.Kastler studied the spectroscopic properties of rubidium atoms in the vapour phase,under the influence of a weak magnetic field.The Zeeman structure of the gas is shown in Fig.1.4 and the splitting of the substates of the ground state is small enough that they can be considered as equally occupied.Selection rules imply that m transitions are not sensitive to the polarization state of light.With circularly polarized light,of or o transitions are possible,depending on the right(+)or left (-)handed char- acter of the polarization state.When a o+polarization is chosen to excite the system the 2P1/2(spin 1/2)sublevel is enriched.From this state the atoms can return to the ground state through a o transition with probability 2/3 or a transition with probability 1/3,and thus the 251/2(spin 1/2)sublevel is enriched compared to the other sublevel of the ground state.A population inversion is realized. Two-level systems are seldom found in natural systems,so that the diffi- culty pointed out in Sect.1.4.1 is basically of an academic nature.Real elec- tronic structures are rather complicated series of states and for the sake of
6 C. Hirlimann 2P1/2 2S1/2 spin -1/2 spin 1/2 π π σ− σ+ Fig. 1.4. Sketch of the Zeeman structure of rubidium atoms in the vapour phase change of particles, obey the Fermi–Dirac statistical distribution. With ΔE being the energy separation between the two levels, the population ratio for a two-electronic-level system is given by N2 N1 = 1 eΔE/kT + 1. (1.7) When the temperature T goes to 0 K the population ratio also goes to 0. At 0 K the energy of the system is zero: all the electrons are in the ground state, N2 = 0, N1 = N0 the total number of electrons. In contrast, when the temperature goes towards infinity (T → ∞), the population ratio goes to one-half, N2 = N1/2. In the high-temperature limit the electrons are equally distributed between the ground and excited states: an inverted population regime cannot be reached by just heating a material. It is not possible, either, to create an inverted population in a two-level system by optically exciting the electrons: at best there can be as many absorbed as emitted photons. 1.4.2 Optical Pumping Optical pumping was proposed, in 1958, by Alfred Kastler as a way to produce inverted populations of electrons. Kastler studied the spectroscopic properties of rubidium atoms in the vapour phase, under the influence of a weak magnetic field. The Zeeman structure of the gas is shown in Fig. 1.4 and the splitting of the substates of the ground state is small enough that they can be considered as equally occupied. Selection rules imply that π transitions are not sensitive to the polarization state of light. With circularly polarized light, σ+ or σ− transitions are possible, depending on the right (+) or left (−) handed character of the polarization state. When a σ+ polarization is chosen to excite the system the 2P1/2 (spin 1/2) sublevel is enriched. From this state the atoms can return to the ground state through a σ transition with probability 2/3 or a π transition with probability 1/3, and thus the 2S1/2 (spin 1/2) sublevel is enriched compared to the other sublevel of the ground state. A population inversion is realized. Two-level systems are seldom found in natural systems, so that the diffi- culty pointed out in Sect. 1.4.1 is basically of an academic nature. Real electronic structures are rather complicated series of states and for the sake of
1 Laser Basics 7 3 W32 2 W31 W21 1 Fig.1.5.Three-level system used to model the population inversion in optical pumping simplicity we will consider a three-level system illuminated with photons of energy hw E3-E (Fig.1.5).Electrons are promoted from state 1 to state 3 with a probability per unit time Wp,which accounts for both the absorption cross-section of the material and the intensity of the incoming light. From state 3,electrons can decay either to state 2 with probability W32 or to the ground state with probability Wa.We assume now that the transition probability from state 3 to state 2 is much larger than to state 1(W32>W31). The electronic transition 2-1 is supposed to be the radiative transition of interest and we suppose that state 2 has a large lifetime compared to state 3 (W21<W32).State 2 is called a metastable state. Rate equations can be easily derived that model the dynamical behavior of such a three-level system: dN3 =WpN1-Wa2N3-Wa1 N3, (1.8a) dt dN2 =W32N3 -W21N2. (1.8b) dt State 3 is populated from state 1 with probability Wp and in proportion to the population Ni of state 1(first right term in(1.8a));it decays with a larger probability to state 2 than to state 1 and in proportion to its population N3 (two decay terms in (1.8a)).State 2 is populated from state 3 according to W32 and N3 (source term in(1.8b))and decays to state 1 through spontaneous emission of light and in proportion to its population N2(decay term in(1.8b)). A steady dynamical behavior is a solution of (1.8)and corresponds to constant state populations with time;in this regime time derivatives vanish and (1.8)simply gives on W31 -W32 (1.9) When the pumping rate is large enough to overcome the spontaneous emission between states 2 and 1 (Wp>W21),then(1.9)shows that the average number
1 Laser Basics 7 1 Wp W31 W32 W21 2 3 Fig. 1.5. Three-level system used to model the population inversion in optical pumping simplicity we will consider a three-level system illuminated with photons of energy ¯hω = E3 − E1 (Fig. 1.5). Electrons are promoted from state 1 to state 3 with a probability per unit time Wp, which accounts for both the absorption cross-section of the material and the intensity of the incoming light. From state 3, electrons can decay either to state 2 with probability W32 or to the ground state with probability W31. We assume now that the transition probability from state 3 to state 2 is much larger than to state 1 (W32 W31). The electronic transition 2 → 1 is supposed to be the radiative transition of interest and we suppose that state 2 has a large lifetime compared to state 3 (W21 W32). State 2 is called a metastable state. Rate equations can be easily derived that model the dynamical behavior of such a three-level system: dN3 dt = WpN1 − W32N3 − W31N3, (1.8a) dN2 dt = W32N3 − W21N2. (1.8b) State 3 is populated from state 1 with probability Wp and in proportion to the population N1 of state 1 (first right term in (1.8a)); it decays with a larger probability to state 2 than to state 1 and in proportion to its population N3 (two decay terms in (1.8a)). State 2 is populated from state 3 according to W32 and N3 (source term in (1.8b)) and decays to state 1 through spontaneous emission of light and in proportion to its population N2 (decay term in (1.8b)). A steady dynamical behavior is a solution of (1.8) and corresponds to constant state populations with time; in this regime time derivatives vanish and (1.8) simply gives on N2 N1 ≈ Wp W21 1 − W31 W32 . (1.9) When the pumping rate is large enough to overcome the spontaneous emission between states 2 and 1 (Wp W21), then (1.9) shows that the average number
8 C.Hirlimann fast 3 slow slow radiative 2 fast 1 Fig.1.6.Sketch of the four-level system found in most laser gain media of atoms in state 2 can be larger that the average number of atoms in state 1: the population is inverted between states 2 and 1.This population inversion is reached when (i)the pumping rate is large enough to overcome the natural decay of the metastable level,(ii)the electronic decay from the pumping state to the radiative state is faster than any other decay,and (iii)the radiative decay time is long enough to ensure that the intermediate metastable state is substantially overoccupied.Because of all these stringent physical conditions the three-level model might seem to be unrealistic;this is actually not the case,for it mimics quite accurately the electronic structure and dynamics found in chromium ions dissolved in alumina(ruby)! A large variety of materials have shown their ability to sustain a popu- lation inversion when,some way or another,energy is fed to their electronic system.Optical pumping still remains a common way of producing population inversion but many other ways have been developed to reach that goal,e.g. electrical excitation,collisional energy transfer and chemical reaction.Most of the efficient media used in lasers have proved to be four-level structures as far as population inversion is concerned (Fig.1.6). In these systems,state 3 is populated and the various transition proba- bilities are as follows:W43>W41,W21>W32,W21 W43.Therefore in the steady-state regime the population of states 4 and 2 can be kept close to zero and the population inversion contrast can be made larger than in a three level-system (N3-N2>N2).This favorable situation is relevant to argon and krypton ion lasers,dye lasers and the neodymium ion in solid matrices, for example. 1.4.3 Light Amplification Once a population inversion is established in a medium it can be used to amplify light.In order to simplify calculations let us consider a medium in which a population inversion is realized (AN =N1-N2,AN <0)in a three- level system (Fig.1.5).In such a medium,the intensity of a low-intensity
8 C. Hirlimann 1 fast 3 4 2 fast slow slow radiative Fig. 1.6. Sketch of the four-level system found in most laser gain media of atoms in state 2 can be larger that the average number of atoms in state 1: the population is inverted between states 2 and 1. This population inversion is reached when (i) the pumping rate is large enough to overcome the natural decay of the metastable level, (ii) the electronic decay from the pumping state to the radiative state is faster than any other decay, and (iii) the radiative decay time is long enough to ensure that the intermediate metastable state is substantially overoccupied. Because of all these stringent physical conditions the three-level model might seem to be unrealistic; this is actually not the case, for it mimics quite accurately the electronic structure and dynamics found in chromium ions dissolved in alumina (ruby)! A large variety of materials have shown their ability to sustain a population inversion when, some way or another, energy is fed to their electronic system. Optical pumping still remains a common way of producing population inversion but many other ways have been developed to reach that goal, e.g. electrical excitation, collisional energy transfer and chemical reaction. Most of the efficient media used in lasers have proved to be four-level structures as far as population inversion is concerned (Fig. 1.6). In these systems, state 3 is populated and the various transition probabilities are as follows: W43 W41, W21 W32, W21 ≈ W43. Therefore in the steady-state regime the population of states 4 and 2 can be kept close to zero and the population inversion contrast can be made larger than in a three level-system (N3 − N2 N2). This favorable situation is relevant to argon and krypton ion lasers, dye lasers and the neodymium ion in solid matrices, for example. 1.4.3 Light Amplification Once a population inversion is established in a medium it can be used to amplify light. In order to simplify calculations let us consider a medium in which a population inversion is realized (ΔN = N1 −N2, ΔN < 0) in a threelevel system (Fig. 1.5). In such a medium, the intensity of a low-intensity