1 Laser Basics 19 1 -1 6)1 91 -1 Fig.1.16.Stability diagram for a laser cavity.The shaded areas define the set of values of the cavity parameters g and g2 for which the cavity is unstable [1.5] From(1.27)one notices that the radius of the beam may only be a real number if the argument of the square root function is a positive and finite number. This leads to the following inequalities: 0≤9192≤1， (1.29) which put stringent conditions on the mirrors'radii of curvature and on their spacing. The stability conditions (1.29),define a hyperbola g1g2 =1 in the g1,g2 plane.The two white regions in Fig.1.16 correspond to stable cavities when g and g2 are both positive or negative.The shaded regions correspond to unstable resonators.Three specific,commonly used cavities are shown in the figure:(i)the symmetrical concentric resonator (R1=R2=-L/2,91=92= -1),(ii)the symmetrical confocal resonator (R1=R2 =-L,g1 g2 =0) and (iii)the planar resonator(RI R2 =oo,gi =g2 =1).The fact that a cavity is optically unstable does not mean that it cannot produce any laser oscillation,nor does it mean that its emitted intensity is necessarily unstable. It only means that the number of round trips of the light it allows is limited. Some gain media are short-lived (a few nanoseconds)compared to the cavity period;it is of no use in this situation to pile up round trips.On the contrary,it might be of great help to use an unstable cavity accommodating the right number of round trips.But as the number of passes of the light in the gain medium is limited,such unstable cavities do not show good transverse
1 Laser Basics 19 g2 g1 –1 –1 1 1 ( ) ( ) Fig. 1.16. Stability diagram for a laser cavity. The shaded areas define the set of values of the cavity parameters g1 and g2 for which the cavity is unstable [1.5] From (1.27) one notices that the radius of the beam may only be a real number if the argument of the square root function is a positive and finite number. This leads to the following inequalities: 0 ≤ g1g2 ≤ 1, (1.29) which put stringent conditions on the mirrors’ radii of curvature and on their spacing. The stability conditions (1.29), define a hyperbola g1g2 = 1 in the g1, g2 plane. The two white regions in Fig. 1.16 correspond to stable cavities when g1 and g2 are both positive or negative. The shaded regions correspond to unstable resonators. Three specific, commonly used cavities are shown in the figure: (i) the symmetrical concentric resonator (R1 = R2 = −L/2, g1 = g2 = −1), (ii) the symmetrical confocal resonator (R1 = R2 = −L, g1 = g2 = 0) and (iii) the planar resonator (R1 = R2 = ∞, g1 = g2 = 1). The fact that a cavity is optically unstable does not mean that it cannot produce any laser oscillation, nor does it mean that its emitted intensity is necessarily unstable. It only means that the number of round trips of the light it allows is limited. Some gain media are short-lived (a few nanoseconds) compared to the cavity period; it is of no use in this situation to pile up round trips. On the contrary, it might be of great help to use an unstable cavity accommodating the right number of round trips. But as the number of passes of the light in the gain medium is limited, such unstable cavities do not show good transverse
20 C.Hirlimann modal qualities.This situation is encountered,for example,in high gain lasers like exciplex lasers or copper-vapor lasers. 1.6.5 Longitudinal Modes When it comes to the use of lasers as short-pulse generators,the most im- portant property of optical resonators is the existence of longitudinal modes. Transverse modes,as we saw,are a geometric consequence of light propaga- tion,while longitudinal modes are a time-frequency property.In other words, we now know how to apply a feedback to a gain medium;we need to explore the conditions under which this feedback can constructively interfere with the main signal.Fabry-Perot interferometers were originally developed as high- resolution bandpass filters.The interferential treatment of optical resonators can be found in most textbooks on optics;here,as a remainder of the time- frequency duality,we will consider a time-domain analysis of the Fabry-Perot interferometer.This way of looking will prove useful in the understanding of mode-locking. An electromagnetic field can be established between two parallel mirrors only when a wave propagating in one direction adds constructively with the wave propagating in the reverse direction.The result of this superposition is a standing wave,which is established if the distance L between the two mirrors is an integer multiple of the half-wavelength of the light.Writing T for the period of the wave and c for its velocity in vacuum,3 x 108 m/s,and remembering that A =cT,the standing wave condition is m 2=L, m∈N+, (1.30) which fixes the value of the positive integer m.The cavity has a specific period T =mr,which is also a round-trip time of flight T=2L/c. For a typical laser cavity with length L=1.5m,the period is T=10ns and the characteristic frequency v=100 MHz.These numbers do fix the repetition rate of mode-locked lasers and also the period of the pulse train. In the continuous-wave(CW)regime the amplification process in a laser cavity is basically coherent and linear;the gain balances the losses.At any point inside the cavity the signal which can be observed at some instant will be repeated unchanged after the time T has elapsed.In the time domain the electromagnetic field in a laser cavity can be seen as a periodic repetition of the same distribution such that e(t)=e(t+nT),n being an integer (see Fig.1.17). E(w)is the Fourier transform of one period of the electric field;its spectral extent is determined by the spectral bandwidth of the various active and passive elements present in the cavity.The Fourier transformation is a linear operation,and therefore the Fourier transform of the total electric field,from 0 to N periods,is the simple sum of the delayed partial Fourier transforms
20 C. Hirlimann modal qualities. This situation is encountered, for example, in high gain lasers like exciplex lasers or copper-vapor lasers. 1.6.5 Longitudinal Modes When it comes to the use of lasers as short-pulse generators, the most important property of optical resonators is the existence of longitudinal modes. Transverse modes, as we saw, are a geometric consequence of light propagation, while longitudinal modes are a time–frequency property. In other words, we now know how to apply a feedback to a gain medium; we need to explore the conditions under which this feedback can constructively interfere with the main signal. Fabry–P´erot interferometers were originally developed as highresolution bandpass filters. The interferential treatment of optical resonators can be found in most textbooks on optics; here, as a remainder of the time– frequency duality, we will consider a time-domain analysis of the Fabry–P´erot interferometer. This way of looking will prove useful in the understanding of mode-locking. An electromagnetic field can be established between two parallel mirrors only when a wave propagating in one direction adds constructively with the wave propagating in the reverse direction. The result of this superposition is a standing wave, which is established if the distance L between the two mirrors is an integer multiple of the half-wavelength of the light. Writing τ for the period of the wave and c for its velocity in vacuum, 3 × 108 m/s, and remembering that λ = cτ , the standing wave condition is mcτ 2 = L, m ∈ N +, (1.30) which fixes the value of the positive integer m. The cavity has a specific period T = mτ , which is also a round-trip time of flight T = 2L/c. For a typical laser cavity with length L = 1.5 m, the period is T = 10 ns and the characteristic frequency ν = 100 MHz. These numbers do fix the repetition rate of mode-locked lasers and also the period of the pulse train. In the continuous-wave (CW) regime the amplification process in a laser cavity is basically coherent and linear; the gain balances the losses. At any point inside the cavity the signal which can be observed at some instant will be repeated unchanged after the time T has elapsed. In the time domain the electromagnetic field in a laser cavity can be seen as a periodic repetition of the same distribution such that ε(t) = ε(t + nT), n being an integer (see Fig. 1.17). E(ω) is the Fourier transform of one period of the electric field; its spectral extent is determined by the spectral bandwidth of the various active and passive elements present in the cavity. The Fourier transformation is a linear operation, and therefore the Fourier transform of the total electric field, from 0 to N periods, is the simple sum of the delayed partial Fourier transforms
1 Laser Basics 21 -iOT -2i0T E(0) E()e E(@)e 0 E(t) E(t+T) 2T E(1+2T) 3T Fig.1.17.Schematic representation,in the time domain,of the electric field at some point in a laser cavity.The field is repeated unchanged so that e(t)=E(t+nT)after each period T =2L/c I(v) c/2L net gain Fig.1.18.Schematic emission spectrum of a laser EN(w)=>e-nTE(w)=1-e-INT N-1 1-e-iwr E(w). (1.31) n=0 The resulting sum is a geometric series which gives the power spectrum for N periods, =lEaP-二e1w)-e sin2(wT/2) (1.32) When N goes to infinity,the intensity response of a laser cavity tends to- wards an infinite periodic series of Dirac 6 distributions spaced by the quan- tity 6w 2/T rc/L,or 6v c/2L.As is well known from the usual frequency analysis,the Fabry-Perot cavity only allows specific frequencies to pass through;the energy is quantized. Real laser spectra are more likely to look like Fig.1.18,where the band- width 6f of the axial modes is finite and governed by the resonator finesse, depending on the reflectivity of the mirrors.Moreover,the number of active modes is also finite,depending on the bandwidth of the net gain Av
1 Laser Basics 21 Fig. 1.17. Schematic representation, in the time domain, of the electric field at some point in a laser cavity. The field is repeated unchanged so that ε(t) = ε(t+nT) after each period T = 2L/c I (ν) Fig. 1.18. Schematic emission spectrum of a laser EN (ω) = N −1 n=0 e−iωnT E(ω) = 1 − e−iNωT 1 − e−iωT E(ω). (1.31) The resulting sum is a geometric series which gives the power spectrum for N periods, IN (ω) = |EN (ω)| 2 = 1 − cos NωT 1 − cos ωT I(ω) = sin2 (NωT/2) sin2 (ωT/2) I(ω). (1.32) When N goes to infinity, the intensity response of a laser cavity tends towards an infinite periodic series of Dirac δ distributions spaced by the quantity δω = 2π/T = πc/L, or δν = c/2L. As is well known from the usual frequency analysis, the Fabry–P´erot cavity only allows specific frequencies to pass through; the energy is quantized. Real laser spectra are more likely to look like Fig. 1.18, where the bandwidth δf of the axial modes is finite and governed by the resonator finesse, depending on the reflectivity of the mirrors. Moreover, the number of active modes is also finite, depending on the bandwidth of the net gain Δν
22 C.Hirlimann 1.7 Here Comes the Laser! The first operating laser was set up by Maiman 1.6,at this time working with the Hughes aircraft company,in the middle of the year 1960,and the first gas laser by A.Javan at MIT at the end of that same year. Ruby was used as the gain medium in Maiman's laser.Ruby is alumina, Al2O3,also known as corundum or sapphire,in which a small fraction of the Al3+ions are replaced by Cr3+ions.The electronic structure of Cr3+:Al2O3 consists of bands and discrete states.The absorption takes place in green and violet bands in the spectrum,giving the material its pink color,and the emission at 694.3 nm takes place between a discrete state and the ground state. The overall structure is that of a three-level system.Inversion of the electronic population was produced by a broadband,helical flash-tube surrounding the ruby rod and the resonator was simply made from the parallel,polished ends of the rod,which were silver coated for high reflectivity. Very rapidly following Maiman's achievement,A.Javan came out with the first gas laser,in which a mixture of helium and neon was continuously excited by an electric discharge [1.7]. 1.8 Conclusion In this chapter some laser basics were introduced as a background to the un- derstanding of ultrashort laser pulse generation.It was not the aim to present details of the physics of lasers,but rather to point out specific key points nec- essary to understand the following chapters.For more information on laser physics we recommend the references contained in the "Further Reading"sec- tion of this chapter. 1.9 Problems 1.Remove the stimulated emission in (1.2)and show that the intensity distri- bution does not vanish to zero when the frequency does.In the pre-Planck era this was called the'infrared catastrophe'. 2.Solve the photon population evolution equation d正=(N-N)b2n+a21M2 d taking into account the spontaneous emission term a21N2. (a)Show that in the low-and negative-temperature limits(T一→O,N2《 Ni and T--oo,N2>N1)there is no qualitative change in the response of the medium. (b)In the high temperature-limit (T-oo,N2=N1),show that the number of photons grows linearly with the distance
22 C. Hirlimann 1.7 Here Comes the Laser! The first operating laser was set up by Maiman [1.6], at this time working with the Hughes aircraft company, in the middle of the year 1960, and the first gas laser by A. Javan at MIT at the end of that same year. Ruby was used as the gain medium in Maiman’s laser. Ruby is alumina, Al2O3, also known as corundum or sapphire, in which a small fraction of the Al3+ ions are replaced by Cr3+ ions. The electronic structure of Cr3+: Al2O3 consists of bands and discrete states. The absorption takes place in green and violet bands in the spectrum, giving the material its pink color, and the emission at 694.3 nm takes place between a discrete state and the ground state. The overall structure is that of a three-level system. Inversion of the electronic population was produced by a broadband, helical flash-tube surrounding the ruby rod and the resonator was simply made from the parallel, polished ends of the rod, which were silver coated for high reflectivity. Very rapidly following Maiman’s achievement, A. Javan came out with the first gas laser, in which a mixture of helium and neon was continuously excited by an electric discharge [1.7]. 1.8 Conclusion In this chapter some laser basics were introduced as a background to the understanding of ultrashort laser pulse generation. It was not the aim to present details of the physics of lasers, but rather to point out specific key points necessary to understand the following chapters. For more information on laser physics we recommend the references contained in the “Further Reading” section of this chapter. 1.9 Problems 1. Remove the stimulated emission in (1.2) and show that the intensity distribution does not vanish to zero when the frequency does. In the pre-Planck era this was called the ‘infrared catastrophe’. 2. Solve the photon population evolution equation dn dz = (N2 − N1)b12n + a21N2 taking into account the spontaneous emission term a21N2. (a) Show that in the low-and negative-temperature limits (T → 0, N2 N1 and T → −∞, N2 N1) there is no qualitative change in the response of the medium. (b) In the high temperature-limit (T → ∞, N2 = N1), show that the number of photons grows linearly with the distance
1 Laser Basics 23 3.Assume a 1.5 m long linear laser cavity.One of the mirrors is flat(R=oo). (a)Using the diagram technique,what is the minimum radius of curvature R2 of the other mirror for the cavity to be optically stable?(Answer: R2/2 =1.5m).What is the position of the beam waist?(Answer:21=0, 22=1.5m). Turning now to the analytical expressions in Sect.1.6.4,answer the fol- lowing questions. (b)Check the position of the beam waist by directly calculating z1 and z2.What is the diameter of the beam waist for an operating wavelength 入=514.5nm?(Answer:2Wo=832μm). (c)What are the radii of the first fundamental transverse mode at the cavity mirrors?(Answer:2W1 2Wo:2W2 1.44 mm). (d)Calculate the Rayleigh range for which the mode can be considered as cylindrical.(Answer:ZR =1.05 m). Further Reading W.Koechner:Solid-State Laser Engineering,4th edn.,Springer Ser.Opt.Sci., Vol.1 (Springer,Berlin,Heidelberg 1996) J.R.Lalanne,S.Kielich,A.Ducasse:Laser-Molecule Interaction and Molec- ular Nonlinear Optics (Wiley,New York 1996) B.Saleh,M.Teich:Fundamentals of Photonics (Wiley,New York 1991) K.Shimoda:Introduction to Laser Physics,2nd edn.,Springer Ser.Opt.Sci., Vol.44 (Springer,Berlin,Heidelberg 1991) A.E.Siegman:Lasers(University Science Books,Mill Valley,CA 1986) O.Svelto:Principles of Lasers,3rd edn.(Plenum Press,New York 1989) J.T.Verdeyen:Laser Electronics(Prentice Hall,New Jersey 1989) A.Yariv:Quantum Electronics,3rd edn.(Wiley,New York 1989) Historial References [1.1]J.P.Gordon,H.J.Zeiger,C.H.Townes:Phys.Rev.99,1264 (1955) [1.2]A.L.Schawlow,C.H.Townes:Phys.Rev.112,1940 (1958) [1.3]N.G.Basov,A.M.Prokhorov:Sov.Phys.-JETP 1,184 (1955) [1.4]G.A.Deschamps,P.E.Mast:Proc.Symposium on Quasi-Optics,ed.J.Fox (Brooklyn Polytechnic Press,New York 1964);P.Laures:Appl.Optics 6,747 (1967) [1.5]H.Kogelnik,T.Li:Appl.Optics 5,1550 (1966) [1.6]T.H.Maiman:Nature 187,493 (1960) [1.7]A.Javan,W.R.Bennet,D.R.Herriot:Phys.Rev.Lett.6,106 (1961)
1 Laser Basics 23 3. Assume a 1.5 m long linear laser cavity. One of the mirrors is flat(R1 = ∞). (a) Using the diagram technique, what is the minimum radius of curvature R2 of the other mirror for the cavity to be optically stable? (Answer: R2/2=1.5 m). What is the position of the beam waist? (Answer: z1 = 0, z2 = 1.5 m). Turning now to the analytical expressions in Sect. 1.6.4, answer the following questions. (b) Check the position of the beam waist by directly calculating z1 and z2. What is the diameter of the beam waist for an operating wavelength λ = 514.5 nm? (Answer: 2W0 = 832 μm). (c) What are the radii of the first fundamental transverse mode at the cavity mirrors? (Answer: 2W1 = 2W0, 2W2 = 1.44 mm). (d) Calculate the Rayleigh range for which the mode can be considered as cylindrical. (Answer: zR = 1.05 m). Further Reading W. Koechner: Solid-State Laser Engineering, 4th edn., Springer Ser. Opt. Sci., Vol. 1 (Springer, Berlin, Heidelberg 1996) J.R. Lalanne, S. Kielich, A. Ducasse: Laser–Molecule Interaction and Molecular Nonlinear Optics (Wiley, New York 1996) B. Saleh, M. Teich: Fundamentals of Photonics (Wiley, New York 1991) K. Shimoda: Introduction to Laser Physics, 2nd edn., Springer Ser. Opt. Sci., Vol. 44 (Springer, Berlin, Heidelberg 1991) A.E. Siegman: Lasers (University Science Books, Mill Valley, CA 1986) O. Svelto: Principles of Lasers, 3rd edn. (Plenum Press, New York 1989) J.T. Verdeyen: Laser Electronics (Prentice Hall, New Jersey 1989) A. Yariv: Quantum Electronics, 3rd edn. (Wiley, New York 1989) Historial References [1.1] J.P. Gordon, H.J. Zeiger, C.H. Townes: Phys. Rev. 99, 1264 (1955) [1.2] A.L. Schawlow, C.H. Townes: Phys. Rev. 112, 1940 (1958) [1.3] N.G. Basov, A.M. Prokhorov: Sov. Phys.-JETP 1, 184 (1955) [1.4] G.A. Deschamps, P.E. Mast: Proc. Symposium on Quasi-Optics, ed. J. Fox (Brooklyn Polytechnic Press, New York 1964); P. Laures: Appl. Optics 6, 747 (1967) [1.5] H. Kogelnik, T. Li: Appl. Optics 5, 1550 (1966) [1.6] T.H. Maiman: Nature 187, 493 (1960) [1.7] A. Javan, W.R. Bennet, D.R. Herriot: Phys. Rev. Lett. 6, 106 (1961)