REVIEW OF LASER ESSENTIALS 15 (a) Inhomogeneous (b) distribution Laser G(0-2) frequency Individual homogeneous lines 20 2 Frequency Frequency Figure 1.4 (a)Absorption or gain spectrum of a medium with an inhomogeneous distribution G(@-So)of transition frequencies;(b)spectral hole burning associated with saturation of an inho- mogeneously broadened medium by a narrowband laser field. pulse generation,we do not discuss them further at this time.However,we return to this topic in Chapter 9,where we discuss line-shape theory and ultrafast spectroscopy techniques for probing physics associated with homogeneous and inhomogeneous broadening. 1.3.2 Gain and Gain Saturation in Four-Level Atoms To provide further insight,we discuss gain saturation for the four-level atom.The four-level atom approximation,sketched in Fig.1.5,is commonly used to model important mode- locking media such as Ti:sapphire or dye molecules.Here we discuss continuous-wave (CW)saturation;saturation in response to pulses is covered in Chapter 2.Note that we do not mean the word atom in the term four-level atom literally.Rather,this word refers to whatever entity(molecule,impurity complex,etc.)is active in the laser gain process. Referring to Fig.1.5,electrons are promoted from the lowest state,denoted level 1,up to level 2 (e.g,via optical pumping).Electrons in level 2 are usually assumed to relax very rapidly to level 3,so that the population of electrons in level 2 remains close to zero. Physically,this relaxation usually arises from vibronic rearrangement of the nuclei within the crystal-impurity or molecular system,which typically occurs on a subpicosecond time scale following the electronic transition from 1 to 2.The transition from 3 to 4 is the lasing transition,and levels 3 and 4 are the upper and lower laser levels,respectively.Electrons from level 3 can undergo stimulated emission down to level 4,giving up a photon to the laser field in the process,or they can also relax spontaneously down to level 4 with rate, in which case their energy is not available to the laser field.tG is the energy storage time of the gain medium,which may range from nanoseconds to milliseconds in materials used for femtosecond pulse generation.Electrons in level 4 are again assumed to relax very rapidly very fast W -very fast Figure 1.5 Energy-level structure for a four-level atom
REVIEW OF LASER ESSENTIALS 15 (a) (b) G 0 0 0 Figure 1.4 (a) Absorption or gain spectrum of a medium with an inhomogeneous distribution G(ω − 0) of transition frequencies; (b) spectral hole burning associated with saturation of an inhomogeneously broadened medium by a narrowband laser field. pulse generation, we do not discuss them further at this time. However, we return to this topic in Chapter 9, where we discuss line-shape theory and ultrafast spectroscopy techniques for probing physics associated with homogeneous and inhomogeneous broadening. 1.3.2 Gain and Gain Saturation in Four-Level Atoms To provide further insight, we discuss gain saturation for the four-level atom. The four-level atom approximation, sketched in Fig. 1.5, is commonly used to model important modelocking media such as Ti:sapphire or dye molecules. Here we discuss continuous-wave (CW) saturation; saturation in response to pulses is covered in Chapter 2. Note that we do not mean the word atom in the term four-level atom literally. Rather, this word refers to whatever entity (molecule, impurity complex, etc.) is active in the laser gain process. Referring to Fig. 1.5, electrons are promoted from the lowest state, denoted level 1, up to level 2 (e.g., via optical pumping). Electrons in level 2 are usually assumed to relax very rapidly to level 3, so that the population of electrons in level 2 remains close to zero. Physically, this relaxation usually arises from vibronic rearrangement of the nuclei within the crystal-impurity or molecular system, which typically occurs on a subpicosecond time scale following the electronic transition from 1 to 2. The transition from 3 to 4 is the lasing transition, and levels 3 and 4 are the upper and lower laser levels, respectively. Electrons from level 3 can undergo stimulated emission down to level 4, giving up a photon to the laser field in the process, or they can also relax spontaneously down to level 4 with rate τ −1 G , in which case their energy is not available to the laser field. τG is the energy storage time of the gain medium, which may range from nanoseconds to milliseconds in materials used for femtosecond pulse generation. Electrons in level 4 are again assumed to relax very rapidly Figure 1.5 Energy-level structure for a four-level atom
16 INTRODUCTION AND REVIEW back to level 1,so that the lower laser level remains nearly empty (this is the principal advantage of four-level atoms). Mathematically,we denote the total density of atoms (in m-3)as NG,and the number density in each of the four levels as N,N2,N3,and N4,respectively.Naturally,we have N1+N2+N3+N4=NG (1.44) The level populations are described by the following rate equations: N1=-wW-N2+4 (1.45) 41 N2=WM-N2)- (1.46) T23 N3=-SN3-N4-9+ (1.47) TG T23 N4=S(N3 NA)-N 4.N3 (1.48) T41 TG Here W is the pumping rate per atom(in s-)from level 1 to 2,and its strength is controlled via the external excitation of the laser medium.For example,in the case of optical pumping, W is proportional to the intensity of the pump laser.S is the stimulated emission rate per atom(in s-),which is proportional to intensity and given in physical units by S=34I(34) (1.49) ho34 Here 034 is the"cross section"of the laser transition,which characterizes the strength of the laser-matter interaction,I(@34)is the laser intensity inside the resonator,and h34 gives the photon energy of the 34 transition.In the case of optical pumping,a similar expression applies for W.23 and r41 are the 23 and41 relaxation times,respectively.The gain is proportional to the population difference between the laser levels: _34(N3-N4)lg 8= (1.50) 2 The factor of 1/2 arises because we have written the gain coefficient for the field in eg. (1.50);this factor is not present when writing the gain coefficient for the intensity.In steady state all the time derivatives in the rate equations are set to zero.Furthermore,since the relaxation from levels 2 and 4 is very fast,we can also approximate N-N2=NI and N3-N4=N3 in eqs.(1.45 to 1.48).Solving for the upper-laser-level population under these conditions yields the following expression for the gain: 1 034WNGtGlg 1+(W+S)G (1.51)
16 INTRODUCTION AND REVIEW back to level 1, so that the lower laser level remains nearly empty (this is the principal advantage of four-level atoms). Mathematically, we denote the total density of atoms (in m−3 ) as NG, and the number density in each of the four levels as N1, N2, N3, and N4, respectively. Naturally, we have N1 + N2 + N3 + N4 = NG (1.44) The level populations are described by the following rate equations: N . 1 = −W(N1 − N2) + N4 τ41 (1.45) N . 2 = W(N1 − N2) − N2 τ23 (1.46) N . 3 = −S(N3 − N4) − N3 τG + N2 τ23 (1.47) N . 4 = S(N3 − N4) − N4 τ41 + N3 τG (1.48) Here W is the pumping rate per atom (in s−1 ) from level 1 to 2, and its strength is controlled via the external excitation of the laser medium. For example, in the case of optical pumping, W is proportional to the intensity of the pump laser. S is the stimulated emission rate per atom (in s−1 ), which is proportional to intensity and given in physical units by S = σ34I (ω34) ω34 (1.49) Here σ34 is the “cross section” of the laser transition, which characterizes the strength of the laser–matter interaction, I (ω34) is the laser intensity inside the resonator, and ω34 gives the photon energy of the 3 → 4 transition. In the case of optical pumping, a similar expression applies for W. τ23 and τ41 are the 2 → 3 and 4 → 1 relaxation times, respectively. The gain is proportional to the population difference between the laser levels: g = σ34 2 (N3 − N4) lg (1.50) The factor of 1/2 arises because we have written the gain coefficient for the field in eq. (1.50); this factor is not present when writing the gain coefficient for the intensity. In steady state all the time derivatives in the rate equations are set to zero. Furthermore, since the relaxation from levels 2 and 4 is very fast, we can also approximate N1 − N2 = N1 and N3 − N4 = N3 in eqs. (1.45 to 1.48). Solving for the upper-laser-level population under these conditions yields the following expression for the gain: g = 1 2 σ34WNGτGlg 1 + (W + S) τG (1.51)
REVIEW OF LASER ESSENTIALS 17 The gain is a function of both W and S.In the small-signal regime (S=0),the small-signal gain go is given by 034WNGTGlg 1+WrG (1.52) go starts at zero,then increases linearly with pump rate W at first but saturates for large W when N3 approaches NG.In the large-signal regime (S is finite),the gain can be rewritten 80 81+S/Ssat (1.53) where 1 Ssat =W+ (1.54) TG is a saturation parameter.Within a laser,the gain first increases with increasing pump until threshold is reached.Above threshold the small-signal gain go continues to increase as W increases,but the actual gain g is clamped or saturated at the value needed for threshold. which we denote gth.The laser intensity (inside the laser)is found by setting eq.(1.53)to gth,with the result S=Ssat -1 (1.55) Lasing can occur in three-level systems as well as in the four-level systems discussed above.For further discussion,see standard laser texts. 1.3.3 Gaussian Beams and Transverse Laser Modes Real laser beams have a finite transverse extent;they are not plane waves.In most cases of interest to us,however,laser beams may be considered paraxial.This means that they are made up of a superposition of plane waves with propagation vectors close to a single direction (which we take as z).Equivalently,the variation of the field in the transverse (x-y) direction must be much weaker than in the z direction.Under these conditions the electric field vector still lies mainly in the x-y plane.In the following we give a brief summary of paraxial laser beams and the related transverse mode structure of lasers.More detail can be found in most texts on lasers (e.g.,[7-9]). For a monochromatic,paraxial wave it is useful to write E(z,1)=Re Eou(x,y.2)ei(co-kz (1.56) where k is given by eq.(1.22)and E refers to a single transverse polarization component. In this form the most rapid wavelength-scale variation of the field is carried by theek
REVIEW OF LASER ESSENTIALS 17 The gain is a function of both W and S. In the small-signal regime (S = 0), the small-signal gain g0 is given by g0 = 1 2 σ34WNGτGlg 1 + WτG (1.52) g0 starts at zero, then increases linearly with pump rate W at first but saturates for large W when N3 approaches NG. In the large-signal regime (S is finite), the gain can be rewritten g = g0 1 + S/Ssat (1.53) where Ssat = W + 1 τG (1.54) is a saturation parameter. Within a laser, the gain first increases with increasing pump until threshold is reached. Above threshold the small-signal gain g0 continues to increase as W increases, but the actual gain g is clamped or saturated at the value needed for threshold, which we denote gth. The laser intensity (inside the laser) is found by setting eq. (1.53) to gth, with the result S = Ssat g0 gth − 1 (1.55) Lasing can occur in three-level systems as well as in the four-level systems discussed above. For further discussion, see standard laser texts. 1.3.3 Gaussian Beams and Transverse Laser Modes Real laser beams have a finite transverse extent; they are not plane waves. In most cases of interest to us, however, laser beams may be considered paraxial. This means that they are made up of a superposition of plane waves with propagation vectors close to a single direction (which we take as z). Equivalently, the variation of the field in the transverse (x–y) direction must be much weaker than in the z direction. Under these conditions the electric field vector still lies mainly in the x–y plane. In the following we give a brief summary of paraxial laser beams and the related transverse mode structure of lasers. More detail can be found in most texts on lasers (e.g., [7–9]). For a monochromatic, paraxial wave it is useful to write E(z, t) = Re E˜ 0u(x, y, z)e j(ωt−kz) (1.56) where k is given by eq. (1.22) and E refers to a single transverse polarization component. In this form the most rapid wavelength-scale variation of the field is carried by the e −jkz
18 INTRODUCTION AND REVIEW term:u(x,y.z)is a slowly varying envelope.Substitution into the wave equation,eq.(1.15), yields +的-2裂=0 (1.57 where V=2/ax2+a2/ay2.Since u is assumed to be slowly varying,a2u/az2 2k au/,and therefore a2u/az2 can be neglected.The resulting paraxial wave equation is ?4-2jk"=0 (1.58) 8z The paraxial wave equation has Gaussian beam solutions that provide a good description of laser beams both inside and outside the laser cavity.A particularly useful solution is written as follows: 00(x,y,Z)= WO_e-(r2+y)/w2(2)e-jk(x2+y2)/2R(2)cj() (1.59) w(z) where we define a=+()月 (1.60a) 1 R阿-2+石 (1.60b) (z)=tan-1 (1.60c 20 TWin (1.60d) This solution,sketched in Fig.1.6,describes a beam that comes to a focus at =0 with a radius wo(measured at 1/e of the on-axis field).The beam radius w(z)changes only slowly within the region<zo;far outside this region the beam spreads with a half angle equal to A/won.The depth b over which the beam radius remains less than v2 wo is given by b=2z0,where bis termed the depth of focus or the confocal parameter.R()is the radius of curvature of the phase fronts.R oo at the beam waist (=0),corresponding to a planar phase front;for o.the phase fronts become spherical with a center of curvature at =0.Equation(1.60c)indicates a phase shift that accompanies propagation through the focal region.This Gaussian beam solution describes both laser beam propagation in free space as well as the spatial behavior of the fundamental transverse modes that are allowed inside a laser resonator with flat or spherical mirrors.The particular transverse mode of this
18 INTRODUCTION AND REVIEW term; u(x, y, z) is a slowly varying envelope. Substitution into the wave equation, eq. (1.15), yields ∇ 2 T u + ∂ 2u ∂z2 − 2jk ∂u ∂z = 0 (1.57) where ∇ 2 T = ∂ 2 /∂x2 + ∂ 2 /∂y2 . Since u is assumed to be slowly varying, ∂ 2u/∂z2 ≪ 2k |∂u/∂z|, and therefore ∂ 2u/∂z2 can be neglected. The resulting paraxial wave equation is ∇ 2 T u − 2jk ∂u ∂z = 0 (1.58) The paraxial wave equation has Gaussian beam solutions that provide a good description of laser beams both inside and outside the laser cavity. A particularly useful solution is written as follows: u00(x, y, z) = w0 w(z) e −(x 2+y 2 )/w2 (z) e −jk(x 2+y 2 )/2R(z) e jφ(z) (1.59) where we define w 2 (z) = w 2 0 1 + z z0 2 (1.60a) 1 R(z) = z z 2 + z 2 0 (1.60b) φ(z) = tan−1 z z0 (1.60c) z0 = πw2 0 n λ (1.60d) This solution, sketched in Fig. 1.6, describes a beam that comes to a focus at z = 0 with a radius w0 (measured at 1/e of the on-axis field). The beam radius w(z) changes only slowly within the region |z| < z0; far outside this region the beam spreads with a half angle equal to λ/πw0n. The depth b over which the beam radius remains less than √ 2 w0 is given by b = 2z0, where b is termed the depth of focus or the confocal parameter. R(z) is the radius of curvature of the phase fronts. R = ∞ at the beam waist (z = 0), corresponding to a planar phase front; for |z| ≫ z0, the phase fronts become spherical with a center of curvature at z = 0. Equation (1.60c) indicates a phase shift that accompanies propagation through the focal region. This Gaussian beam solution describes both laser beam propagation in free space as well as the spatial behavior of the fundamental transverse modes that are allowed inside a laser resonator with flat or spherical mirrors. The particular transverse mode of this
REVIEW OF LASER ESSENTIALS 19 phase front V2% z=0 -b=2z0 Figure 1.6 Gaussian beam solution of the paraxial wave equation,showing evolution of the beam radius and representative phase front. type allowed within a given laser resonator is selected by the requirement that R(z)must match the radii of curvature of the mirrors within the laser,which determines the location of the beam waist and the value of wo:for free propagation outside a laser,any solution of this type is allowed. It is useful to know the power P carried by a Gaussian beam,which is obtained by integrating the intensity over its cross-sectional area.The result is -学1a-警器 2nr dr e-222/w2(z) gnea (1.61) The power is equal to the on-axis intensity at the focus,multiplied by the area of a circle of radius wo/v2,which is the radius of the intensity at the e point. Equation (1.59)is the lowest-order member of an entire family of Hermite-Gaussian solutions to the paraxial wave equation,given by 1 2 umn(x,y, H w(z) w(z) (z) ×e-(2+y2/u2(ee-j2+y2/2Re)ej0m+n+1p) (1.62) The H()with v a nonnegative integer are Hermite polynomials [10].where for example, Ho(传)=1，H1(传)=25，andH(传)=4转2-2.in general,the H are polynomials of order v,which become wider and exhibit a greater number of zero crossings as v increases.The expressions for w(),R(z),and ()are still as given above.Although the fundamental Gaussian beam is by far the most important in most ultrafast optics applications,the higher order(m0 or n0)Hermite-Gaussian solutions will be relevant in our study for sev- eral reasons.For example,all the um in eq.(1.62)are allowed transverse modes of laser resonators,termed TEM modes,although the fundamental TEMoo mode usually has the lower loss and is therefore favored.We shall see however that higher-order modes were sig- nificant in the discovery of the important Kerr lens mode-locking technique for short pulse
REVIEW OF LASER ESSENTIALS 19 z z z b w 0 0 w0 w0 Figure 1.6 Gaussian beam solution of the paraxial wave equation, showing evolution of the beam radius and representative phase front. type allowed within a given laser resonator is selected by the requirement that R(z) must match the radii of curvature of the mirrors within the laser, which determines the location of the beam waist and the value of w0; for free propagation outside a laser, any solution of this type is allowed. It is useful to know the power P carried by a Gaussian beam, which is obtained by integrating the intensity over its cross-sectional area. The result is P = dA ε0nc 2 E˜ 0 2 |u| 2 = ε0nc 2 E˜ 0 2 w 2 0 w2 (z) ∞ 0 2πr dr e−2r 2 /w2 (z) = 1 2 ε0nc E˜ 0 2 πw2 0 2 (1.61) The power is equal to the on-axis intensity at the focus, multiplied by the area of a circle of radius w0/ √ 2, which is the radius of the intensity at the e −1 point. Equation (1.59) is the lowest-order member of an entire family of Hermite–Gaussian solutions to the paraxial wave equation, given by umn(x, y, z) ∼ 1 w(z) Hm √ 2 x w(z) Hn √ 2 y w(z) × e −(x 2+y 2 )/w2 (z) e −jk(x 2+y 2 )/2R(z) e j(m+n+1)φ(z) (1.62) The Hν(ξ) with ν a nonnegative integer are Hermite polynomials [10], where for example, H0(ξ) = 1, H1(ξ) = 2ξ, and H2(ξ) = 4ξ 2 − 2. In general, the Hν are polynomials of order ν, which become wider and exhibit a greater number of zero crossings as ν increases. The expressions for w(z), R(z), and φ(z) are still as given above. Although the fundamental Gaussian beam is by far the most important in most ultrafast optics applications, the higher order (m /= 0 or n /= 0) Hermite–Gaussian solutions will be relevant in our study for several reasons. For example, all the umn in eq. (1.62) are allowed transverse modes of laser resonators, termed TEMmn modes, although the fundamental TEM00 mode usually has the lower loss and is therefore favored. We shall see however that higher-order modes were significant in the discovery of the important Kerr lens mode-locking technique for short pulse