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《UltraFast Optics》课程教学资源（学习资料）Femtosecond Laser Pulses（Principles and Experiments，Second Edition）

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2 C.Hirlimann En Nn Bmn Anm Bnm =En-Em Em Nm Fig.1.1.Energy diagram of an atomic two-level system.Energies Em and En are measured with reference to some lowest level In 1905,Albert Einstein,though,had to postulate the quantization of elec- tromagnetic energy in order to give the first interpretation of the photo-electric effect.This step had him wondering for a long time about the compatibility of this quantization and Planck's black-body theory.Things started to clarify in 1913 when Bohr published his atomic model,in which electrons are con- strained to stay on fixed energy levels and may exchange only energy quanta with the outside world.Let us consider(see Figure 1.1)two electronic levels n and m in an atom,with energies Em and En referenced to some fundamental level;one quantum of light,called a photon,with energy hw =En-Em,is absorbed with a probability Bin and its energy is transferred to an electron jumping from level m to level n.There is a probability Anm that an electron on level n steps down to level m,emitting a photon with the same energy. This spontaneous light emission is analogous to the general spontaneous en- ergy decay found in classical mechanical systems.In the year 1917,ending his thinking on black-body radiation,Einstein came out with the postulate that,for an excited state,there should be another de-excitation channel with probability Bnm:the“induced”or“stimulated”emission.This new emission process only occurs when an electromagnetic field hw is present in the vicinity of the atom and it is proportional to the intensity of the field.The quantities Anm,Bnm,Bmn are called Einstein's coefficients. Let us now consider a set of N atoms,of which Nm are in state m and Nn in state n,and assume that this set is illuminated with a light wave of angular frequency w such that hw En-Em,with intensity I(w).At a given temperature T,in a steady-state regime,the number of absorbed photons equals the number of emitted photons (equilibrium situation of a black body).The number of absorbed photons per unit time is proportional to the transition probability Bmn for an electron to jump from state m to state n,to the incident intensity I(w)and to the number of atoms in the set Nm.A simple inversion of the role played by the indices m and n gives the number of electrons per unit time relaxing from state n to state m by emitting a photon under the influence of the electromagnetic field.The last contribution to the interaction,spontaneous emission,does not depend on the intensity but only

2 C. Hirlimann Fig. 1.1. Energy diagram of an atomic two-level system. Energies Em and En are measured with reference to some lowest level In 1905, Albert Einstein, though, had to postulate the quantization of electromagnetic energy in order to give the first interpretation of the photo-electric effect. This step had him wondering for a long time about the compatibility of this quantization and Planck’s black-body theory. Things started to clarify in 1913 when Bohr published his atomic model, in which electrons are constrained to stay on fixed energy levels and may exchange only energy quanta with the outside world. Let us consider (see Figure 1.1) two electronic levels n and m in an atom, with energies Em and En referenced to some fundamental level; one quantum of light, called a photon, with energy ¯hω = En − Em, is absorbed with a probability Bmn and its energy is transferred to an electron jumping from level m to level n. There is a probability Anm that an electron on level n steps down to level m, emitting a photon with the same energy. This spontaneous light emission is analogous to the general spontaneous energy decay found in classical mechanical systems. In the year 1917, ending his thinking on black-body radiation, Einstein came out with the postulate that, for an excited state, there should be another de-excitation channel with probability Bnm: the “induced” or “stimulated” emission. This new emission process only occurs when an electromagnetic field ¯hω is present in the vicinity of the atom and it is proportional to the intensity of the field. The quantities Anm, Bnm, Bmn are called Einstein’s coefficients. Let us now consider a set of N atoms, of which Nm are in state m and Nn in state n, and assume that this set is illuminated with a light wave of angular frequency ω such that ¯hω = En − Em, with intensity I(ω). At a given temperature T, in a steady-state regime, the number of absorbed photons equals the number of emitted photons (equilibrium situation of a black body). The number of absorbed photons per unit time is proportional to the transition probability Bmn for an electron to jump from state m to state n, to the incident intensity I(ω) and to the number of atoms in the set Nm. A simple inversion of the role played by the indices m and n gives the number of electrons per unit time relaxing from state n to state m by emitting a photon under the influence of the electromagnetic field. The last contribution to the interaction, spontaneous emission, does not depend on the intensity but only

1 Laser Basics 3 on the number of electrons in state n and on the transition probability Amn. This can be simply formalized in a simple energy conservation equation NmBmnI(w)=Nn BnmI(w)+NnAnm. (1.2) Boltzmann's law,deduced from the statistical analysis of gases,gives the relative populations on two levels separated by an energy hw at temperature T,Nn/Nm exp(-hw/kT).When applied to (1.2)one gets BmnI(w)eh/T=Anm+BnmI(w) (1.3) and Anm I()-Bin chTkT-Bnm (1.4) This black-body frequency distribution function is exactly equivalent to Planck's distribution (1.1).At this point it is important to notice that Ein- stein wouldn't have succeeded without introducing the stimulated emission. Comparison of expressions(1.1)and (1.4)shows that Bmn =Bnm:for a pho- ton the probability to be absorbed equals the probability to be emitted by stimulation.These two effects are perfectly symmetrical;they both take place when an electromagnetic field is present around an atom. Strangely enough,by giving a physical interpretation to Planck's law based on photons interacting with an energy-quantized matter,Einstein has made the spontaneous emission appear mysterious.Why is an excited atom not stable?If light is not the cause of the spontaneous emission,then what is the hidden cause?This point still gives rise to a passionate debate today about the role played by the fluctuations of the field present in the vacuum.Comparison of expressions (1.1)and (1.4)also leads to Anm/Bnm =hw3/n2c2,so that when the light absorption probability is known then the spontaneous and stimulated emission probabilities are also known. According to Einstein's theory,three different processes can take place during the interaction of light with matter,as described below. 1.2.1 Absorption In this process one photon from the radiation field disappears and the energy is transferred to an electron as potential energy when it changes state from Em to En.The probability for an electron to undergo the absorption transition is Bmn· 1.2.2 Spontaneous Emission When being in an excited state En,an electron in an atom has a probability Anm to spontaneously fall to the lower state Em.The loss of potential energy gives rise to the simultaneous emission of a photon with energy hw=En-Em The direction,phase and polarization of the photon are random quantities

1 Laser Basics 3 on the number of electrons in state n and on the transition probability Amn. This can be simply formalized in a simple energy conservation equation NmBmnI(ω) = NnBnmI(ω) + NnAnm. (1.2) Boltzmann’s law, deduced from the statistical analysis of gases, gives the relative populations on two levels separated by an energy ¯hω at temperature T, Nn/Nm = exp(−hω/kT ¯ ). When applied to (1.2) one gets BmnI(ω) ehω/kT ¯ = Anm + BnmI(ω) (1.3) and I(ω) = Anm Bmn ehω/kT ¯ − Bnm . (1.4) This black-body frequency distribution function is exactly equivalent to Planck’s distribution (1.1). At this point it is important to notice that Einstein wouldn’t have succeeded without introducing the stimulated emission. Comparison of expressions (1.1) and (1.4) shows that Bmn = Bnm: for a photon the probability to be absorbed equals the probability to be emitted by stimulation. These two effects are perfectly symmetrical; they both take place when an electromagnetic field is present around an atom. Strangely enough, by giving a physical interpretation to Planck’s law based on photons interacting with an energy-quantized matter, Einstein has made the spontaneous emission appear mysterious. Why is an excited atom not stable? If light is not the cause of the spontaneous emission, then what is the hidden cause? This point still gives rise to a passionate debate today about the role played by the fluctuations of the field present in the vacuum. Comparison of expressions (1.1) and (1.4) also leads to Anm/Bnm = ¯hω3/π2c2, so that when the light absorption probability is known then the spontaneous and stimulated emission probabilities are also known. According to Einstein’s theory, three different processes can take place during the interaction of light with matter, as described below. 1.2.1 Absorption In this process one photon from the radiation field disappears and the energy is transferred to an electron as potential energy when it changes state from Em to En. The probability for an electron to undergo the absorption transition is Bmn. 1.2.2 Spontaneous Emission When being in an excited state En, an electron in an atom has a probability Anm to spontaneously fall to the lower state Em. The loss of potential energy gives rise to the simultaneous emission of a photon with energy ¯hω = En−Em. The direction, phase and polarization of the photon are random quantities

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