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 index of the Kerr medium. A spatio-temporal laser pulse propagating through the Kerr medium has a time dependent mode size as higher intensities acquire 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 absorber. 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 propagation. 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 propagation 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