1 Laser Basics 9 electromagnetic wave propagating along z is proportional to the population inversion AN and to the wave intensity.This can be formally written as dI(☒=-I()△No21, (1.10) dz where 021,the proportionality factor,is called the stimulated-emission cross- section of the transition.It depends on both the medium and the wavelength of the light.Equation (1.10)describes light amplification in an inverted- population medium. During the amplification process a depletion of the inverted level is to be expected due to the stimulated emission process itself:AN must depend on the intensity of the light.This dependence can be qualitatively explored using (1.9)as a starting point.If an efficient gain medium is assumed,then W31/W32<1 can be neglected and one gets AN≈N2 W21 (1.11) One can also make the assumption that N2 N,all the available atoms (N)being in the inverted state 2.This regime corresponds to a situation where the pumping is strong so that W21<Wp.In the framework of these approximations,an inverted expansion leads to △v≈-N/(+院) (1.12) Replacing the probabilities by intensities because they are only involved in ratios gives the following expression,in which Isis a constant intensity de- pending on the gain medium: aw-/+) (1.13) When expression (1.13)is introduced into (1.10)we obtain -/+) 1 I(2)dz (1.14) where go No21>0 is the low-intensity gain.The saturation intensity I allows one to distinguish between the low-and high-intensity regimes for light amplification in a gain medium. 1.4.3.1 Low-Intensity Regime,I(z)Is.In this regime the evolution of the light intensity as a function of the distance z in the medium simplifies to I(z)=I(0)e903.Starting from its incoming value I(0),the intensity grows as an exponential function along the propagation direction.This behavior could be intuitively predicted from the previous discussion on the stimulated emission process and the chain reaction
1 Laser Basics 9 electromagnetic wave propagating along z is proportional to the population inversion ΔN and to the wave intensity. This can be formally written as dI(z) dz = −I(z) ΔNσ21, (1.10) where σ21, the proportionality factor, is called the stimulated-emission crosssection of the transition. It depends on both the medium and the wavelength of the light. Equation (1.10) describes light amplification in an invertedpopulation medium. During the amplification process a depletion of the inverted level is to be expected due to the stimulated emission process itself: ΔN must depend on the intensity of the light. This dependence can be qualitatively explored using (1.9) as a starting point. If an efficient gain medium is assumed, then W31/W32 1 can be neglected and one gets ΔN ≈ N2 W21 Wp − 1 . (1.11) One can also make the assumption that N2 ≈ N, all the available atoms (N) being in the inverted state 2. This regime corresponds to a situation where the pumping is strong so that W21 Wp. In the framework of these approximations, an inverted expansion leads to ΔN ≈ −N 1 + W21 Wp . (1.12) Replacing the probabilities by intensities because they are only involved in ratios gives the following expression, in which Is is a constant intensity depending on the gain medium: ΔN ≈ −N 1 + I(z) Is . (1.13) When expression (1.13) is introduced into (1.10) we obtain 1 I(z) dI(z) dz = g0 1 + I(z) Is , (1.14) where g0 = Nσ21 > 0 is the low-intensity gain. The saturation intensity Is allows one to distinguish between the low-and high-intensity regimes for light amplification in a gain medium. 1.4.3.1 Low-Intensity Regime, I(z) Is. In this regime the evolution of the light intensity as a function of the distance z in the medium simplifies to I(z) = I(0)eg0z. Starting from its incoming value I(0), the intensity grows as an exponential function along the propagation direction. This behavior could be intuitively predicted from the previous discussion on the stimulated emission process and the chain reaction.
10 C.Hirlimann g(I) 80 8 2 Fig.1.7.Simple hyperbolic intensity dependence of the gain in a light amplifier 1.4.3.2 High-Intensity Regime,I(z)>Is.When,in the gain medium, the intensity becomes larger than the saturation intensity,(1.14)reduces to I(2)=I(0)+Isgoz;the intensity only grows as a linear function of the distance z.In the high-intensity regime the amplification process is much less efficient than in the low-intensity regime.The gain is said to saturate in the high- intensity regime. It is far beyond the scope of this introduction to laser physics to rigorously discuss gain saturation;we will only focus on the simple hyperbolic model 90 g()=1+I1 (1.15) In this very approximate framework,the saturation intensity Is is the intensity for which the gain value is reduced by a factor of two.This is very similar to the saturation of the absorption in saturable absorbers,and this again arises from the fact that light absorption and stimulated emission are symmetrical effects.The simplest gain model simply mimics the Beer-Lambert law for absorption:I(2)=I(0)e92,where g is given by (1.15). Gain saturation is of prime importance in the field of ultrashort light pulse generation;this mechanism is a key ingredient for pulse shortening. But it also becomes a limiting factor in the process of amplifying ultrashort pulses:the intensity in these pulses rapidly reaches the saturation value.Beam broadening and pulse stretching are ways used to overcome this difficulty,as will be described in the following chapters. 1.5 Amplified Spontaneous Emission (ASE) Spontaneous light emission from an excited medium is isotropic:photons are randomly emitted in every possible space direction with equal probability; the polarization states are also randomly distributed when the emission takes place from an isotropic medium.Stimulated emission,on the contrary,retains
10 C. Hirlimann Fig. 1.7. Simple hyperbolic intensity dependence of the gain in a light amplifier 1.4.3.2 High-Intensity Regime, I(z) Is. When, in the gain medium, the intensity becomes larger than the saturation intensity, (1.14) reduces to I(z) = I(0)+Isg0z; the intensity only grows as a linear function of the distance z. In the high-intensity regime the amplification process is much less efficient than in the low-intensity regime. The gain is said to saturate in the highintensity regime. It is far beyond the scope of this introduction to laser physics to rigorously discuss gain saturation; we will only focus on the simple hyperbolic model g(I) = g0 1 + I/Is . (1.15) In this very approximate framework, the saturation intensity Is is the intensity for which the gain value is reduced by a factor of two. This is very similar to the saturation of the absorption in saturable absorbers, and this again arises from the fact that light absorption and stimulated emission are symmetrical effects. The simplest gain model simply mimics the Beer–Lambert law for absorption: I(z) = I(0)egz, where g is given by (1.15). Gain saturation is of prime importance in the field of ultrashort light pulse generation; this mechanism is a key ingredient for pulse shortening. But it also becomes a limiting factor in the process of amplifying ultrashort pulses: the intensity in these pulses rapidly reaches the saturation value. Beam broadening and pulse stretching are ways used to overcome this difficulty, as will be described in the following chapters. 1.5 Amplified Spontaneous Emission (ASE) Spontaneous light emission from an excited medium is isotropic: photons are randomly emitted in every possible space direction with equal probability; the polarization states are also randomly distributed when the emission takes place from an isotropic medium. Stimulated emission, on the contrary, retains
1 Laser Basics 11 ● Fig.1.8.Schematic illustration of amplified spontaneous emission (ASE).Sponta- neously emitted photons are amplified when propagating along the major dimension of the gain medium the characteristics of the inducing waves.This memory effect is responsible for the unwanted amplified spontaneous emission which takes place in laser amplifiers. Most of the gain media in which a population inversion is created have a geometrical shape such that one of their dimensions is larger than the others (Fig.1.8).At the beginning of the population inversion,when net gain be- comes available,there are always spontaneously emitted photons propagating in directions close to the major dimension of the medium which trigger stim- ulated emission.In both space and phase,ASE does not have good coherence properties,because it is seeded by many incoherently,spontaneously emitted photons. Amplified spontaneous emission is a problem when using light amplifiers in series to amplify light pulses:the ASE emitted by one amplifying stage is further amplified in the next stage and competes for gain with the useful signal.ASE returns in oscillators are also undesirable;they may damage the solid-state gain medium or induce temporal instabilities.To overcome these difficulties it is necessary to use amplifier decoupling. 1.5.1 Amplifier Decoupling 1.5.1.1 Static Decoupling.For light pulses that are not too short ( 100fs),a Faraday polarizer (Fig.1.9)can be used to stop any return of lin- early polarized light.Depending on the wavelength,properly chosen materials exhibit a strong rotatory power when a static magnetic field is applied.Ad- justment of the magnetic field intensity and of the length of the material allows one to rotate the linear polarization of a light beam by a 45 angle. Owing to the pseudovector nature of a magnetic field,the polarization rotation direction is reversed for a beam propagating in the reverse direction
1 Laser Basics 11 Fig. 1.8. Schematic illustration of amplified spontaneous emission (ASE). Spontaneously emitted photons are amplified when propagating along the major dimension of the gain medium the characteristics of the inducing waves. This memory effect is responsible for the unwanted amplified spontaneous emission which takes place in laser amplifiers. Most of the gain media in which a population inversion is created have a geometrical shape such that one of their dimensions is larger than the others (Fig. 1.8). At the beginning of the population inversion, when net gain becomes available, there are always spontaneously emitted photons propagating in directions close to the major dimension of the medium which trigger stimulated emission. In both space and phase, ASE does not have good coherence properties, because it is seeded by many incoherently, spontaneously emitted photons. Amplified spontaneous emission is a problem when using light amplifiers in series to amplify light pulses: the ASE emitted by one amplifying stage is further amplified in the next stage and competes for gain with the useful signal. ASE returns in oscillators are also undesirable; they may damage the solid-state gain medium or induce temporal instabilities. To overcome these difficulties it is necessary to use amplifier decoupling. 1.5.1 Amplifier Decoupling 1.5.1.1 Static Decoupling. For light pulses that are not too short (> 100 fs), a Faraday polarizer (Fig. 1.9) can be used to stop any return of linearly polarized light. Depending on the wavelength, properly chosen materials exhibit a strong rotatory power when a static magnetic field is applied. Adjustment of the magnetic field intensity and of the length of the material allows one to rotate the linear polarization of a light beam by a 45◦ angle. Owing to the pseudovector nature of a magnetic field, the polarization rotation direction is reversed for a beam propagating in the reverse direction
12 C.Hirlimann Fig.1.9.Schematic of a Faraday polarizer.P is a linear polarizer;the cylinder is a material exhibiting strong rotatory power under the influence of a magnetic field B Fig.1.10.(a)Train of pulses from a laser amplifier consisting of weak,unamplified pulses with high repetition rate and low-repetition-rate,amplified,short pulses, associated with long-lasting ASE.(b)Cleaning-up produced by a saturable absorber Therefore the linear polarization of a reflected beam is rotated 900 and can be stopped by an analyzer. 1.5.1.2 Dynamic Decoupling.Amplifier stages are often decoupled using saturable absorbers.Absorption saturation(see Chap.2)is a nonlinear optical effect that is symmetrical to gain saturation.It can be described by replacing the gain g by the absorption a in expression (1.15)and changing the sign in the evolution of the intensity with propagation.For a low-intensity signal, the intensity decreases as an exponential function with distance,while it only decreases linearly at high intensity.Only short,intense pulses can cross the saturable absorber.As an example we will consider a light output of an ampli- fier stage consisting of a superposition of light pulses:a high-repetition-rate train of weak,short pulses and a low-repetition-rate train of intense,short pulses superimposed on low-intensity,long-lasting ASE pulses(Fig.1.10a)
12 C. Hirlimann B P Fig. 1.9. Schematic of a Faraday polarizer. P is a linear polarizer; the cylinder is a material exhibiting strong rotatory power under the influence of a magnetic field B t t a) b) Fig. 1.10. (a) Train of pulses from a laser amplifier consisting of weak, unamplified pulses with high repetition rate and low-repetition-rate, amplified, short pulses, associated with long-lasting ASE. (b) Cleaning-up produced by a saturable absorber Therefore the linear polarization of a reflected beam is rotated 90◦ and can be stopped by an analyzer. 1.5.1.2 Dynamic Decoupling. Amplifier stages are often decoupled using saturable absorbers. Absorption saturation (see Chap. 2) is a nonlinear optical effect that is symmetrical to gain saturation. It can be described by replacing the gain g by the absorption α in expression (1.15) and changing the sign in the evolution of the intensity with propagation. For a low-intensity signal, the intensity decreases as an exponential function with distance, while it only decreases linearly at high intensity. Only short, intense pulses can cross the saturable absorber. As an example we will consider a light output of an ampli- fier stage consisting of a superposition of light pulses: a high-repetition-rate train of weak, short pulses and a low-repetition-rate train of intense, short pulses superimposed on low-intensity, long-lasting ASE pulses (Fig. 1.10a)
1 Laser Basics 13 positive feed-back amplifier energy Fig.1.11.General sketch of an oscillator.An oscillator is basically made of an am- plifier and a positive feedback.The feedback must ensure a constructive interference between the input and amplified waves The optical density of the saturable absorber is adjusted in such a way that it only saturates when crossed by the amplified pulses.When the pulse train crosses the saturable absorber the unamplified pulses and the leading edge of the ASE are absorbed.In order to improve the energy ratio between the amplified pulses and the remaining ASE the saturable absorber must be chosen to have a short recovery time.That way the long-lasting trailing part of the ASE can be partly absorbed.Malachite green,for example,with a 3 ps recovery time,has been widely used as a stage separator in dye amplifiers. 1.6 The Optical Cavity We now know how to create and use stimulated emission to amplify light.From a very general point of view an oscillator is the association of an amplifier with a positive feedback(Fig.1.11).The net gain of the amplifier must be larger than one in order to overcome the losses,including the external coupling.The phase change created by the feedback loop must be an integer multiple of 2 in order to maintain a constructive interference between the input and amplified waves.What is the way to use an amplifier in the building of an optical oscillator？ 1.6.1 The Fabry-Perot Interferometer In the year 1955,Gordon et al.[1.1 developed the ammonia maser,clearly proving,in the microwave wavelength range,the possibility to amplify weak signals using stimulated emission.A metallic box,with suitably chosen size and shape,surrounding a gain medium was proved to create efficient positive
1 Laser Basics 13 Fig. 1.11. General sketch of an oscillator. An oscillator is basically made of an amplifier and a positive feedback. The feedback must ensure a constructive interference between the input and amplified waves The optical density of the saturable absorber is adjusted in such a way that it only saturates when crossed by the amplified pulses. When the pulse train crosses the saturable absorber the unamplified pulses and the leading edge of the ASE are absorbed. In order to improve the energy ratio between the amplified pulses and the remaining ASE the saturable absorber must be chosen to have a short recovery time. That way the long-lasting trailing part of the ASE can be partly absorbed. Malachite green, for example, with a 3 ps recovery time, has been widely used as a stage separator in dye amplifiers. 1.6 The Optical Cavity We now know how to create and use stimulated emission to amplify light. From a very general point of view an oscillator is the association of an amplifier with a positive feedback (Fig. 1.11). The net gain of the amplifier must be larger than one in order to overcome the losses, including the external coupling. The phase change created by the feedback loop must be an integer multiple of 2π in order to maintain a constructive interference between the input and amplified waves. What is the way to use an amplifier in the building of an optical oscillator? 1.6.1 The Fabry–P´erot Interferometer In the year 1955, Gordon et al. [1.1] developed the ammonia maser, clearly proving, in the microwave wavelength range, the possibility to amplify weak signals using stimulated emission. A metallic box, with suitably chosen size and shape, surrounding a gain medium was proved to create efficient positive