麻省理工学院：《超快光学 Ultrafast Optics》课程教学资源（讲稿）Chapter 7 Kerr-Lens and Additive Pulse Mode Locking

Chapter 7Kerr-Lens and Additive PulseMode LockingThere are many ways to generate saturable absorber action. One can usereal saturable absorbers, such as semiconductors or dyes and solid-state lasermedia. One can also exploit artificial saturable absorbers. The two mostprominent artificial saturable absorber modelocking techniques are calledKerr-Lens Mode Locking (KLM) and Additive Pulse Mode Locking (APM).APM is sometimes also called Coupled-Cavity Mode Locking (CCM). KLMwas invented in the early 90's [1][2][3][4][5][6][7], but was already predictedto occur much earlier [8][9][10]7.1Kerr-Lens Mode Locking (KLM)The general principle behind Kerr-Lens Mode Locking is sketched in Fig. 7.1.A pulse that builds up in a laser cavity containing a gain medium and a Kerrmedium experiences not only self-phase modulation but also self focussing,that is nonlinear lensing of the laser beam, due to the nonlinear refractive in-dex of the Kerr medium. A spatio-temporal laser pulse propagating throughthe Kerr medium has a time dependent mode size as higher intensities ac-quire stronger focussing.If a hard aperture is placed at the right positionin the cavity, it strips of the wings of the pulse, leading to a shortening ofthe pulse. Such combined mechanism has the same effect as a saturable ab-sorber.If the electronic Kerr effect with response time of a fewfemtosecondsor less is used, a fast saturable absorber has been created. Instead of a sep-257

Chapter 7 Kerr-Lens and Additive Pulse Mode Locking There are many ways to generate saturable absorber action. One can use real saturable absorbers, such as semiconductors or dyes and solid-state laser media. One can also exploit artificial saturable absorbers. The two most prominent artificial saturable absorber modelocking techniques are called Kerr-Lens Mode Locking (KLM) and Additive Pulse Mode Locking (APM). APM is sometimes also called Coupled-Cavity Mode Locking (CCM). KLM was invented in the early 90’s [1][2][3][4][5][6][7], but was already predicted to occur much earlier [8][9][10]· 7.1 Kerr-Lens Mode Locking (KLM) The general principle behind Kerr-Lens Mode Locking is sketched in Fig. 7.1. A pulse that builds up in a laser cavity containing a gain medium and a Kerr medium experiences not only self-phase modulation but also self focussing, that is nonlinear lensing of the laser beam, due to the nonlinear refractive in￾dex of the Kerr medium. A spatio-temporal laser pulse propagating through the Kerr medium has a time dependent mode size as higher intensities ac￾quire stronger focussing. If a hard aperture is placed at the right position in the cavity, it strips of the wings of the pulse, leading to a shortening of the pulse. Such combined mechanism has the same effect as a saturable ab￾sorber. If the electronic Kerr effect with response time of a few femtoseconds or less is used, a fast saturable absorber has been created. Instead of a sep- 257

258CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGhardapertu2KerainMediunbsorbeFigure 7.1: Principle mechanism of KLM. The hard aperture can be alsoreplaced by the soft aperture due to the spatial variation of the gain in thelasercrystal.arate Kerr medium and a hard aperture, the gain medium can act both as aKerr medium and as a soft aperture (i.e. increased gain instead of saturableabsorption). The sensitivity of the laser mode size on additional nonlinearlensing is drastically enhanced if the cavity is operated close to the stabilityboundary of the cavity. Therefore, it is of prime importance to understandthe stability ranges of laser resonators. Laser resonators are best understoodin terms of paraxial optics [11][12][14][13][15]7.1.1Review of Paraxial Optics and Laser ResonatorDesignThe solutions to the paraxial wave equation,which keep their form duringpropagation, are the Hermite-Gaussian beams. Since we consider only thefundamental transverse modes, we are dealing with the Gaussian beam72U。U(r,z) =(7.1)exp-jk12g(2)]q(2)

258CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING artifical fast saturable absorber Kerr Medium gain intensity hard aperture beam waist self - focusing soft aperture Figure 7.1: Principle mechanism of KLM. The hard aperture can be also replaced by the soft aperture due to the spatial variation of the gain in the laser crystal. arate Kerr medium and a hard aperture, the gain medium can act both as a Kerr medium and as a soft aperture (i.e. increased gain instead of saturable absorption). The sensitivity of the laser mode size on additional nonlinear lensing is drastically enhanced if the cavity is operated close to the stability boundary of the cavity. Therefore, it is of prime importance to understand the stability ranges of laser resonators. Laser resonators are best understood in terms of paraxial optics [11][12][14][13][15]. 7.1.1 Review of Paraxial Optics and Laser Resonator Design The solutions to the paraxial wave equation, which keep their form during propagation, are the Hermite-Gaussian beams. Since we consider only the fundamental transverse modes, we are dealing with the Gaussian beam U(r, z) = Uo q(z) exp ∙ −jk r2 2q(z) ¸ , (7.1)

2597.1. KERR-LENS MODE LOCKING (KLM)with the complex q-parameter q = a + jb or its inverse11入(7.2)q() -R(2)TW2(z)The Gaussian beam intensity I(z,r) = U(r, z)/ expressed in terms of thepowerP carried by the beam isgivenby2P2r2(7.3)I(r,z) =w2(2)TW2(2)The use of the g-parameter simplifies the description of Gaussian beam prop-agation. In free space propagation from z to z2, the variation of the beamparameter q is simply governed by(7.4)2= 91+ 22— 21,where q2 and qi are the beam parameters at zi and z2.If the beam waist.at which the beam has a minimum spot size wo and a planar wavefront(R = oo), is located at z = O, the variations of the beam spot size and theradius of curvature are explicitly expressed as/121w(2)=w。(7.5)-9andTWR(2) =≥1+(T(7.6)The angular divergence of the beam is inversely proportional to the beamwaist. In the far field, the half angle divergence is given by,0:(7.7)TW。asillustratedinFigure7.2

7.1. KERR-LENS MODE LOCKING (KLM) 259 with the complex q-parameter q = a + jb or its inverse 1 q(z) = 1 R(z) − j λ πw2(z) . (7.2) The Gaussian beam intensity I(z, r) = |U(r, z)| 2 expressed in terms of the power P carried by the beam is given by I(r, z) = 2P πw2(z) exp ∙ − 2r2 w2(z) ¸ . (7.3) The use of the q-parameter simplifies the description of Gaussian beam prop￾agation. In free space propagation from z1 to z2, the variation of the beam parameter q is simply governed by q2 = q1 + z2 − z1, (7.4) where q2 and q1 are the beam parameters at z1 and z2. If the beam waist, at which the beam has a minimum spot size w0 and a planar wavefront (R = ∞), is located at z = 0, the variations of the beam spot size and the radius of curvature are explicitly expressed as w(z) = wo " 1 + µ λz πw2 o ¶2 #1/2 , (7.5) and R(z) = z " 1 + µπw2 o λz ¶2 # . (7.6) The angular divergence of the beam is inversely proportional to the beam waist. In the far field, the half angle divergence is given by, θ = λ πwo , (7.7) as illustrated in Figure 7.2

260CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGPlanes ofw(z)constantphase12WoY2A元W0Beam Waist-L-R124-103zlZRFigure 7.2: Gaussian beam and its characteristics.FigurebyMITOCW.Dueto diffraction,the smallerthe spot size at the beam waist, thelargerthe divergence. The Rayleigh range is defined as the distance from the waistoverwhichthebeamareadoublesandcanbeexpressedasTW(7.8)ZR:入The confocal parameter of the Gaussian beam is defined as twice the Rayleighrange2元wb=2zR=(7.9)1and corresponds to the length over which the beam is focused. The propa-gation of Hermite-Gaussian beams through paraxial optical systems can beefficiently evaluated using the ABCD-law [11]Aq1 + B(7.10)q2Cqi + Dwhere qi and q2 are the beam parameters at the input and the output planesof theoptical systemorcomponent.TheABCDmatricesofsomeopticalelements are summarized in Table 7.1.If a Gaussian beam with a waist wo1is focused by a thin lens a distance zi away from the waist, there will be a

260CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING Figure 7.2: Gaussian beam and its characteristics. Due to diffraction, the smaller the spot size at the beam waist, the larger the divergence. The Rayleigh range is defined as the distance from the waist over which the beam area doubles and can be expressed as zR = πw2 o λ . (7.8) The confocal parameter of the Gaussian beam is defined as twice the Rayleigh range b = 2zR = 2πw2 o λ , (7.9) and corresponds to the length over which the beam is focused. The propa￾gation of Hermite-Gaussian beams through paraxial optical systems can be efficiently evaluated using the ABCD-law [11] q2 = Aq1 + B Cq1 + D (7.10) where q1 and q2 are the beam parameters at the input and the output planes of the optical system or component. The ABCD matrices of some optical elements are summarized in Table 7.1. If a Gaussian beam with a waist w01 is focused by a thin lens a distance z1 away from the waist, there will be a -1 Planes of constant phase z/zR 0 1 2 3 4 L=R Beam Waist w(z) 2W0 θ =πw0 λ Figure by MIT OCW

7.1. KERR-LENS MODE LOCKING (KLM)261newfocusatadistance(21 - J)f2(7.11)2=f+(21 - f)2 + (r)and awaist Wo2(1-21+(")(7.12)fWo1W02zZ1z2Figure7.3:Focusing of a Gaussian beambya lens

7.1. KERR-LENS MODE LOCKING (KLM) 261 new focus at a distance z2 = f + (z1 − f)f 2 (z1 − f)2 + ³πw2 01 λ ´2 , (7.11) and a waist w02 1 w2 02 = 1 w2 01 µ 1 − z1 f ¶2 + 1 f 2 ³πw01 λ ´2 (7.12) Figure 7.3: Focusing of a Gaussian beam by a lens

262CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKING7.1.2Two-Mirror ResonatorsWe consider the two mirror resonator shown in Figure 7.4.Optical ElementABCD-MatrixLFree Space Distance L01Thin Lens with-01/ ffocal lengthfMirror under Angle0 to Axis and Radius R2cosSagittal PlaneRMirror under Angle10 to Axis and Radius R-21RcosTangential PlaneBrewsterPlateunderAngle to Axis and Thicknessn0d, Sagittal PlaneBrewster Plate under(11Angle to Axis and Thickness0d, Tangential PlaneTable7.1:ABCDmatrices for commonlyused optical elements.trR2R1ZM1M2LFigure7.4:Two-Mirror Resonator with curved mirrors with radi of curvatureRi and R2The resonator can be unfolded for an ABCD-matrix analysis, see Figure7.5

262CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING Optical Element ABCD-Matrix Free Space Distance L µ 1 L 0 1 ¶ Thin Lens with focal length f µ 1 0 −1/f 1 ¶ Mirror under Angle θ to Axis and Radius R Sagittal Plane µ 1 0 −2 cos θ R 1 ¶ Mirror under Angle θ to Axis and Radius R Tangential Plane µ 1 0 −2 R cos θ 1 ¶ Brewster Plate under Angle θ to Axis and Thickness d, Sagittal Plane µ 1 d n 0 1 ¶ Brewster Plate under Angle θ to Axis and Thickness d, Tangential Plane µ 1 d n3 0 1 ¶ Table 7.1: ABCD matrices for commonly used optical elements. Figure 7.4: Two-Mirror Resonator with curved mirrors with radii of curvature R1 and R2. The resonator can be unfolded for an ABCD-matrix analysis, see Figure 7.5. 7.1.2 Two-Mirror Resonators We consider the two mirror resonator shown in Figure 7.4

2637.1. KERR-LENS MODE LOCKING (KLM)1122f2fifCzLLFigure 7.5: Two-mirror resonator unfolded. Note, only one half of the fo-cusing strength of mirror 1 belongs to a fundamental period describing oneresonator roundtrip.Theproduct of ABCD matrices describing one roundtrip according toFigure 7.5 are then given by上0L()()()(1M(7.13)111wherefi=R/2,and f2=R2/2.To carry outthisproduct and toformulatethe cavity stability criteria, it is convenient to use the cavity parametersgi=1-L/Ri,i=1,2.The resulting cavity roundtripABCD-matrix can bewritten in theform(2g192 - 1)2g2LABM :(7.14)T2g1 (9192-1) /L(2g192-1)ResonatorStabilityThe ABCD matrices describe the dynamics of rays propagating inside theresonator. An optical ray is characterized by the vector r=whereris the distance from the optical axis and r' the slope of the ray to the opticalaxis. The resonator is stable if no ray escapes after many round-trips, whichis the case when the eigenvalues of the matrix M are less than one. Sincewe have a lossless resonator, i.e. detjM = 1, the product of the eigenvalueshastobe1and,therefore,thestableresonatorcorrespondstothecaseofacomplex conjugate pair of eigenvalues with a magnitude of 1. The eigenvalue

7.1. KERR-LENS MODE LOCKING (KLM) 263 Figure 7.5: Two-mirror resonator unfolded. Note, only one half of the fo￾cusing strength of mirror 1 belongs to a fundamental period describing one resonator roundtrip. The product of ABCD matrices describing one roundtrip according to Figure 7.5 are then given by M = µ 1 0 −1 2f1 1 ¶ µ 1 L 0 1 ¶ µ 1 0 −1 f2 1 ¶ µ 1 L 0 1 ¶ µ 1 0 −1 2f1 1 ¶ (7.13) where f1 = R1/2, and f2 = R2/2. To carry out this product and to formulate the cavity stability criteria, it is convenient to use the cavity parameters gi = 1 − L/Ri, i = 1, 2. The resulting cavity roundtrip ABCD-matrix can be written in the form M = µ (2g1g2 − 1) 2g2L 2g1 (g1g2 − 1) /L (2g1g2 − 1) ¶ = µ A B C D ¶ . (7.14) Resonator Stability The ABCD matrices describe the dynamics of rays propagating inside the resonator. An optical ray is characterized by the vector r=µ r r0 ¶ , where r is the distance from the optical axis and r0 the slope of the ray to the optical axis. The resonator is stable if no ray escapes after many round-trips, which is the case when the eigenvalues of the matrix M are less than one. Since we have a lossless resonator, i.e. det|M| = 1, the product of the eigenvalues has to be 1 and, therefore, the stable resonator corresponds to the case of a complex conjugate pair of eigenvalues with a magnitude of 1. The eigenvalue

264CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGequation to M is given by2g2L(2g192 - 1) - 入(7.15)det [M - >-1|= det2g1 (9192 - 1) /L (2g192 - 1) 2 - 2(2g192 1) 入+ 1 = 0.(7.16)The eigenvalues are入1/2=(2g1921) ±V(2g192 1)2 1,(7.17)exp(±0),cosh = 2g192-1, for [2g192-1|>1(x(+0),=29, 21- (718)The case of a complex conjugate pair with a unit magnitude corresponds toa stable resontor.Therfore. the stability criterion for a stable two mirrorresontoris[29192 - 1]| ≤ 1.(7.19)The stable and unstable parameter ranges are given by(7.20)stable:0≤91·92=S≤1(7.21)unstable：9192≤ 0; or 9192 ≥1.where S = gi -g2, is the stability parameter of the cavity.The stabil-ity criterion can be easily interpreted geometrically.Of importance arethe distances between the mirror mid-points M, and cavity end points, i.e.gi=(R,-L)/R,=-S:/Ri,as shown in Figure7.6.trRGS2?ZM2M1LFigure 7.6: The stability criterion involves distances between the mirror mid-points M, and cavity end points. i.e. gi = (R, - L)/R, = -Si/R

264CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING equation to M is given by det|M − λ · 1| = det ¯ ¯ ¯ ¯ µ (2g1g2 − 1) − λ 2g2L 2g1 (g1g2 − 1) /L (2g1g2 − 1) − λ ¶¯ ¯ ¯ ¯ = 0, (7.15) λ2 − 2 (2g1g2 − 1) λ +1=0. (7.16) The eigenvalues are λ1/2 = (2g1g2 − 1) ± q (2g1g2 − 1)2 − 1, (7.17) = ½ exp (±θ), cosh θ = 2g1g2 − 1, for |2g1g2 − 1| > 1 exp (±jψ), cos ψ = 2g1g2 − 1, for |2g1g2 − 1| ≤ 1 .(7.18) The case of a complex conjugate pair with a unit magnitude corresponds to a stable resontor. Therfore, the stability criterion for a stable two mirror resontor is |2g1g2 − 1| ≤ 1. (7.19) The stable and unstable parameter ranges are given by stable : 0 ≤ g1 · g2 = S ≤ 1 (7.20) unstable : g1g2 ≤ 0; or g1g2 ≥ 1. (7.21) where S = g1 · g2, is the stability parameter of the cavity. The stabil￾ity criterion can be easily interpreted geometrically. Of importance are the distances between the mirror mid-points Mi and cavity end points, i.e. gi = (Ri − L)/Ri = −Si/Ri, as shown in Figure 7.6. Figure 7.6: The stability criterion involves distances between the mirror mid￾points Mi and cavity end points. i.e. gi = (Ri − L)/Ri = −Si/Ri

2657.1. KERR-LENS MODE LOCKING (KLM)Thefollowing rules for a stable resonator can bederived from Figure7.6using the stability criterion expressed in terms of the distances Si. Note, thatthe distances and radii can be positive and negativeSiS2(7.22)stable:0<<1.RR2Therules are:.A resonator is stable,if themirrorradii,laid out alongtheoptical axis.overlapA resonator isunstable,if theradii do not overlap or one lies withinthe other.Figure 7.7 shows stable and unstable resonator configurationsSTABLEUNSTABLERR2Figure 7.7: Illustration of stable and unstable resonator configurationsFigurebyMITOCW.For a two-mirror resonator with concave mirrors and Ri ≤ R2, we obtainthe general stability diagram as shown in Figure 7.8. There are two rangesforthemirrordistanceL,withinwhichthecavityisstable,O<L<Ri and

7.1. KERR-LENS MODE LOCKING (KLM) 265 The following rules for a stable resonator can be derived from Figure 7.6 using the stability criterion expressed in terms of the distances Si. Note, that the distances and radii can be positive and negative stable : 0 ≤ S1S2 R1R2 ≤ 1. (7.22) The rules are: • A resonator is stable, if the mirror radii, laid out along the optical axis, overlap. • A resonator is unstable, if the radii do not overlap or one lies within the other. Figure 7.7 shows stable and unstable resonator configurations. Figure 7.7: Illustration of stable and unstable resonator configurations. For a two-mirror resonator with concave mirrors and R1 ≤ R2, we obtain the general stability diagram as shown in Figure 7.8. There are two ranges for the mirror distance L, within which the cavity is stable, 0 ≤ L ≤ R1 and STABLE UNSTABLE R1 R1 R1 R1 R2 R2 R2 R2 R2 R2 Figure by MIT OCW

266CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGLo'R'R2R,+R2Figure7.8:Stabileregions (black)for thetwo-mirror resonatorR2 ≤ L ≤ Ri + R2. It is interesting to investigate the spot size at the mirrorsand the minimum spot size inthe cavity as a function of the mirror distanceL.Resonator Mode CharacteristicsThe stablemodes of theresonator reproduce themselves after one round-trip,i.e. from Eq.(7.10) we findAqi + B(7.23)91Cqi + DThe inverse q-parameter, which is directly related to the phase front curva-ture and the spot size of the beam, is determined by1- ADA-D(1=0.(7.24)BB219qThe solution isA-D/(A + D)_1(7.25)2|B2Bq)1/2If we apply this formula to (7.15), we find the spot size on mirror 1入12V(A+ D)"-1=-(7.26)元W?9)1/2or2入L192w4(7.27)T911-9192LR2- L(ΛRi(7.28)R1 -LRi+R2T

266CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING Figure 7.8: Stabile regions (black) for the two-mirror resonator. R2 ≤ L ≤ R1 +R2. It is interesting to investigate the spot size at the mirrors and the minimum spot size in the cavity as a function of the mirror distance L. Resonator Mode Characteristics The stable modes of the resonator reproduce themselves after one round-trip, i.e. from Eq.(7.10) we find q1 = Aq1 + B Cq1 + D (7.23) The inverse q-parameter, which is directly related to the phase front curva￾ture and the spot size of the beam, is determined by µ1 q ¶2 + A − D B µ1 q ¶ + 1 − AD B2 = 0. (7.24) The solution is µ1 q ¶ 1/2 = −A − D 2B ± j 2 |B| q (A + D) 2 − 1 (7.25) If we apply this formula to (7.15), we find the spot size on mirror 1 µ1 q ¶ 1/2 = − j 2 |B| q (A + D) 2 − 1 = −j λ πw2 1 . (7.26) or w4 1 = µ2λL π ¶2 g2 g1 1 1 − g1g2 (7.27) = µλR1 π ¶2 R2 − L R1 − L µ L R1 + R2 − L ¶ . (7.28)