Abstract:
A laser ignition system has: at least one arrangement for producing a pulsed laser beam; and at least one arrangement for focusing the produced pulsed laser beam onto a focus zone, e.g., in order to ignite a combustible gas mixture in an internal combustion engine or a burner. The laser ignition system is designed to produce a pulsed laser beam having a normed fluence volume greater than 0.1.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to a laser ignition system having at least one means for producing a pulsed laser beam and at least one means for focusing the produced pulsed laser beam onto a focal zone, for example in order to ignite a combustible gas mixture in an internal combustion engine or burner. 
         [0003]    2. Description of the Related Art 
         [0004]    In current laser applications for laser ignition, a main focus of attention is achieving the smallest possible focusability of the laser light. Through the use of optical equipment having a large numerical aperture (DIN 58629-1), the goal is pursued of achieving the smallest possible beam diameter with the greatest possible intensity (power per surface=power density) in the focal zone. Koga et al. (Journal of Physics D 43 (2010), 025204) showed that, in addition to a threshold intensity, a minimum energy or minimum fluence (energy per surface=energy density) in the focal point is necessary for the ignition of a plasma. 
         [0005]    However, it has turned out that despite high realized intensities, in particular given lean mixtures disadvantageous effects occur in laser ignition, such as misfires, which could result in engine failure. Associated with this is a lack of smooth running of the engine. The resulting increase in emission of pollutants has a negative influence on the competitiveness of laser ignition systems compared to conventional electrical ignition systems. 
       BRIEF SUMMARY OF THE INVENTION 
       [0006]    Accordingly, the object of the present invention is to improve a laser ignition system of the type indicated above so that the described disadvantages are removed or minimized. 
         [0007]    According to the present invention, in the ignition system of the type named above this object is achieved in that the laser ignition system is designed to produce a pulsed laser beam having a normed fluence volume greater than 0.1. 
         [0008]    The provision according to the present invention of the production of a pulsed laser beam having a normed fluence volume greater than 0.1 is particularly advantageous because in this way a plasma formation in a relatively large volume is achieved. 
         [0009]    Developments of the present invention provide a normed fluence volume that is even greater than 0.3 or 0.5. It is advantageous that the achieved fluence volume is very close to the maximum possible fluence volume, and in this way the probability of ignition is close to optimal. 
         [0010]    In this context, for the definition of the fluence volume a minimum energy density, also called minimum fluence, is used, which in principle can be selected to be suitable. For example, depending on the intended ignition mixture, the determination of the minimum fluence can be carried out in a suitable manner. For example, a minimum fluence of 20 J/mm 2  can be assumed, which has turned out to be suitable in particular for an air-methane mixture of 3 bar. In order to produce a plasma in air (air ionization) at room temperature and standard atmospheric pressure, for example a minimum fluence of 15 J/mm 2  is sufficient. Depending on the system and external boundary conditions, e.g. higher temperature or higher pressure, the required minimum fluence for a system can also be 10 J/mm 2  or 20 J/mm 2  or 25 J/mm 2  or 30 J/mm 2 . 
         [0011]    The present invention is based on the recognition that achieving the minimum fluence in a relatively large volume corresponds to a comparatively homogenous distribution of the existing energy of the laser pulses to a volume that is as large as possible. In this way, it is achieved that, following the formation of a plasma in the focus zone, an ignition of a fuel mixture will occur with increased probability. In this way, misfirings are avoided and the smooth running of the internal combustion engine is improved. In this way, even fuel mixtures that are quite lean, i.e. have a high degree of excess air, can be ignited comparatively reliably. This is advantageous because as a rule the efficiency of an internal combustion engine increases as the lean burning capacity of the gas mixture to be ignited increases. 
         [0012]    In the case of a relatively small, even if relatively intense, plasma, in contrast there is an increased probability that the gas mixture will not be ignited, but rather the energy of the produced plasma will be lost, e.g. in the form of sound waves. 
         [0013]    The volume within which the minimum fluence is exceeded is designated the fluence volume. The normed fluence volume results from the fluence volume for given laser beam features at a specified numerical aperture of the focusing, normed to the maximum possible fluence volume. 
         [0014]    Here, the term “laser beam features” refers in particular to the pulse energy and the beam quality M 2  (DIN EN ISO 11146-1). 
         [0015]    The maximum possible fluence volume is the maximum fluence volume that can theoretically be achieved for a given laser ignition system through variation of the numerical aperture, e.g. by exchanging the focusing lens or changing the illumination of the lens, while at the same time holding constant the parameters minimum fluence and the laser beam features, in particular the pulse energy, the beam quality M 2 , and the wavelength. 
         [0016]    As explained above, the boundary for the determination of the fluence volume is drawn at the minimum fluence. Typically, the minimum fluence has a value of from 1/10 to ½ of the maximum fluence. The maximum fluence is the highest fluence in the focus zone of the produced laser beam. The value of the maximum fluence should not be less than 10 J/mm 2 . A maximum fluence of at least 50 J/mm 2  would be better. In these cases, it is ensured that a plasma formation can be reliably initiated. 
         [0017]    In an advantageous specific embodiment, it is provided that the fluence volume has a minimum size of 1*10 −5  mm 3 . Larger fluence volumes, of at least 10 −4  mm 3 , are desirable. Corresponding to the laser beam shape, the fluence volume has in particular an ellipsoid-type shape. A ratio of longitudinal extension Z to transverse extension R of the fluence volume of a factor of at least 20 has turned out to be particularly advantageous for the laser ignition system, and a factor of at least 40 is to be sought. 
         [0018]    The production of the pulsed laser radiation takes place using a laser source, e.g. a solid-state laser. This can optionally be pumped with a laser diode or a comparable pumped light source. The solid-state laser may have a passive Q switch. Other laser sources or oscillator-amplifier systems, such as those described in published European patent application document EP 1 888 914 A1, may also in principle be used. 
         [0019]    A particularly advantageous specific embodiment provides that the laser beam produced by the laser source has a pulse energy of at least 1 mJ. A minimum energy of 3 mJ for the laser beam is to be sought. Here, a pulse duration of the laser beam of not less than 0.5 ns is very advantageous, and a pulse duration of at least 2 ns is even more preferable. 
         [0020]    Overall, it has turned out to be advantageous for the beam quality M 2  of the pulsed laser beam to be less than 20. Particularly advantageous is a beam quality of a maximum of 10, for example 5. 
         [0021]    In a further particularly advantageous embodiment, the intensity in the focus zone is in a range of from 10 11  W/cm 2  to 10 13  W/cm 2 . 
         [0022]    In addition, investigations carried out by applicant have shown that imaging errors in the focusing of the produced laser radiation, in particular diffraction effects and/or shadowing effects, also called vignetting, in the means provided according to the present invention for focusing the produced pulsed laser beam can have, in many cases, a negative influence on the capability of the focused laser beam to ignite a combustible gas mixture. 
         [0023]    In order to avoid imaging errors, and to improve the optical imaging, care is to be taken that the means for focusing, such as a focusing lens, are preferably illuminated up to a maximum of 75% relative to the diameter, or the free aperture, or even only to 50% relative to the diameter. In particular, this results in a Strehl ratio (DIN EN ISO 14880-3) in the range of from 0.8 to 1. The Strehl ratio is a measure of the absence of imaging errors in the diffracted beam, in particular the ratio of the maximum fluence or maximum intensity of the diffracted beam to that of the undiffracted beam. 
         [0024]    Systematic investigations have shown that, using laser ignition systems according to the present invention, fuel mixtures having a lean burning capacity of greater than λ=1.5 can be ignited, and that the lean burning boundary is shifted toward larger values as the normed fluence volume becomes larger. Due to the possibility of reliably operating internal combustion engines with gas mixtures having a high lean burning boundary in combination with a laser ignition system, pollutant emissions are reduced and the efficiency of the combustion is increased. 
         [0025]    Further features, possible applications, and advantages of the present invention result from the following description of exemplary embodiments of the present invention, presented in the Figures of the drawing. All described or presented features, in themselves or in any combination, represent the subject matter of the present invention, independent of their formulation or presentation in the description or in the drawing. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0026]      FIG. 1  shows an example of a laser ignition system according to the present invention having a diverging lens and a focusing lens. 
           [0027]      FIG. 2  shows a series of simulated beam caustics for various illumination levels of the focusing lens. The boundaries of the fluence volume, with a minimum fluence of 10 J/mm 2  and 15 J/mm 2 , are shown. The radial extension (R in mm) is shown as a function of the distance (Z in mm) from the focus zone. 
           [0028]      FIG. 3  shows a series of calculated fluence volumes for various minimum fluences. 
           [0029]      FIG. 4  shows the lean burning boundary (λ) as a function of the fluence volume (FV) for various pulse energies (3 mJ, 5 mJ, 7 mJ, 9 mJ) and illumination levels of the focusing lens (75%, 60%, 50%). 
           [0030]      FIG. 5  shows the relation between beam quality M 2 , pulse energy Q, and fluence volume FV for various laser crystal lengths and minimum fluences. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0031]      FIG. 1  shows a schematic representation of a laser ignition system  1  made up of a means for producing a pulsed laser beam  10  and a means for focusing  20 . Not shown are the associated electrical lines for supplying power to the means for producing a pulsed laser beam  10 , or details of the constructive design of laser ignition system  1  as a laser spark plug, known to those skilled in the art for example from published European patent application document EP 1 519 038 A1. Optionally, means are provided for the optical transmission of the pulsed laser beam. In addition, a thermal conductor may be provided, in particular for cooling fluid, for cooling the means for producing a pulsed laser beam  1  and/or other components. Also not shown are possible means for mounting laser ignition system  1  in an internal combustion engine. 
         [0032]    In this exemplary embodiment, to produce the pulsed laser beam a solid-state laser  11  having a passive Q switch  12  is used, which for example produces light having a wavelength of 1064 nm. A semiconductor diode laser is used as pumped light source for solid-state laser  11 . As focusing means  20 , a lens system is used, e.g. a telescope. This is made up of a diverging lens  21  for diverging the pulsed laser beam and a focusing lens  22  for focusing the pulsed laser beam in focus zone  23 . Focusing lens  22  is illuminated to a maximum of 75% of the lens diameter. An illumination of focusing lens  22  of less than 60% of the lens diameter is recommended. 
         [0033]    Different illumination levels of focusing lens  22  result in differently strong focusing of the pulsed laser beam.  FIG. 2  shows the corresponding beam caustics for illuminations of 75% ( FIG. 2   a, d ), 60% ( FIG. 2   b, e ), 50% ( FIG. 2   c, f ), 40% ( FIG. 2   g ), 35% ( FIG. 2   h ), and 30% ( FIG. 2   i ) of the lens diameter. The beam caustics for a laser pulse energy of 9 mJ were simulated.  FIGS. 2   a, b,  and  c  show beam caustics for laser beams having different beam qualities M 2 , as were used subsequently ( FIG. 3 ) in ignition experiments. For the simulation of the beam caustics in  FIGS. 2   d, e, f, g, h,  and  i,  the same laser parameters were used as were used for the beam caustic in  FIG. 2   c,  and only the illumination of focusing lens  22  was varied. Here, different maximum fluences resulted in focus zone  23 . 
         [0034]    For the simulation of the beam caustics, the beam propagation of the laser beam can be approximated through the mathematical description of a Gauss beam. A Gauss beam has a transverse profile according to a Gauss curve and a longitudinal profile according to a Lorentz curve. 
         [0035]    Based on a power density distribution 
         [0000]    
       
         
           
             
               
                 I 
                  
                 
                   ( 
                   
                     r 
                     , 
                     z 
                   
                   ) 
                 
               
               = 
               
                 
                   I 
                   0 
                 
                 · 
                 
                   
                     ( 
                     
                       
                         ω 
                         0 
                       
                       
                         ω 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 · 
                 
                    
                   
                     
                       2 
                        
                       
                         r 
                         2 
                       
                     
                     
                       
                         ω 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                       2 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    the fluence volume V is determined under the assumption of a threshold fluence or minimum fluence. 
         [0036]    The radial distance to the z axis is defined as beam radius ω(z), at which the intensity has fallen to 1/e 2 : 
         [0000]    
       
         
           
             
               ω 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 ω 
                 0 
               
               · 
               
                 
                   
                     1 
                     + 
                     
                       
                         ( 
                         
                           z 
                           
                             z 
                             0 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0037]    The minimum beam radius, present at the beam waist (at z=0), is designated ω 0 . 
         [0038]    The Rayleigh length z R  is the distance along the optical axis at which the cross-sectional surface A of the beam, going out from the beam waist, has doubled in size: 
         [0000]    
       
         
           
             
               z 
               R 
             
             = 
             
               
                 
                   ω 
                   0 
                 
                 
                   θ 
                   0 
                 
               
               . 
             
           
         
       
     
         [0039]    Using the parameters beam radius ω 0  and divergence θ 0  of the laser beam, the focusability of the laser radiation is described using beam quality index M 2 : 
         [0000]    
       
         
           
             
               M 
               2 
             
             = 
             
               
                 π 
                 λ 
               
               · 
               
                 θ 
                 0 
               
               · 
               
                 ω 
                 0 
               
             
           
         
       
     
         [0000]    with wavelength λ. 
         [0040]    The average power density I results from the ratio of power P to cross-sectional surface A: 
         [0000]    
       
         
           
             I 
             = 
             
               
                 P 
                 A 
               
               . 
             
           
         
       
     
         [0041]    Pulse energy Q is power P multiplied by pulse duration τ: 
         [0000]    
       
      
       Q=P·τ.  
      
     
         [0042]    Average energy density H, also called fluence, is the ratio of the pulse energy to cross-sectional surface A: 
         [0000]    
       
         
           
             H 
             = 
             
               
                 Q 
                 A 
               
               . 
             
           
         
       
     
         [0043]    The maximum fluence H max  results, at z=0, i.e. in the focus zone, where the cross-sectional surface A (z=0)=πω 0   2 , as: 
         [0000]    
       
         
           
             
               H 
               max 
             
             = 
             
               
                 H 
                  
                 
                   ( 
                   
                     z 
                     = 
                     0 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     2 
                     · 
                     Q 
                   
                   
                     π 
                     · 
                     
                       ω 
                       0 
                       2 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0044]    Using the quantities named above, loci R(z) having equal energy density or fluence can be calculated. These result as lines having the same fluence, also called isofluences, as a function of the distance to the z axis: 
         [0000]    
       
         
           
             
               R 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 ω 
                 0 
               
               · 
               
                 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             ( 
                             
                               z 
                               
                                 z 
                                 R 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                     · 
                     
                       1 
                       2 
                     
                     · 
                     ln 
                   
                    
                   
                     
                       H 
                       max 
                     
                     
                       
                         H 
                         Schwelle 
                       
                       · 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 z 
                                 
                                   z 
                                   R 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           or 
         
       
       
         
           
             
               R 
                
               
                 ( 
                 z 
                 ) 
               
             
             = 
             
               
                 ω 
                 0 
               
               · 
               
                 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 z 
                                 
                                   z 
                                   R 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ) 
                       
                       · 
                       
                         1 
                         2 
                       
                       · 
                       ln 
                     
                      
                     
                       
                         
                           2 
                           · 
                           Q 
                         
                         
                           π 
                           · 
                           
                             ω 
                             0 
                             2 
                           
                         
                       
                       
                         
                           H 
                           Schwelle 
                         
                         · 
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   z 
                                   
                                     z 
                                     R 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0045]    By integrating the isofluences that are equal to or greater than the minimum fluence, fluence volume FV is obtained as the volume of the integration over distance z: 
         [0000]    
       
         
           
             V 
             = 
             
               
                 2 
                  
                 π 
               
               + 
               
                 
                   ∫ 
                   0 
                   
                     z 
                      
                     
                       ( 
                       
                         R 
                         = 
                         0 
                       
                       ) 
                     
                   
                 
                  
                 
                   
                     
                       ( 
                       
                         R 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                       ) 
                     
                     2 
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       z 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0046]    Thus, fluence volume FV=f (M 2 , Q, H Schwelle ) is a function of beam quality M 2 , pulse energy Q, or maximum fluence, and the minimum fluence. Using the equations stated above, for a laser crystal having known beam quality M 2  and pulse energy Q the beam caustic can be simulated and fluence volume FV can be calculated. 
         [0047]      FIG. 3  shows a series of diagrams in which the resulting fluence volumes for various minimum fluences, 10 J/mm 2  (dashed line), 15 J/mm 2  (dotted line), and 20 J/mm 2  (solid line), are plotted against the radius of the beam waist ( FIG. 3   a ) and against the maximum fluence ( FIGS. 3   b  and  3   c ). 
         [0048]      FIG. 3   a  shows that given constant pulse energy Q=12 mJ and beam quality M 2 =6.1, by varying the focus size, or changing the radius of the beam waist, the imaging of the laser radiation is modified and influences the size of the fluence volume. Here, for a specified focus size or radius of the beam waist, the fluence volume assumes a maximum value. As the minimum fluence decreases, the fluence volume becomes larger. The maximum value of the fluence volume (shown by a point in each case in the diagram) shifts towards larger beam radii as the minimum fluence becomes smaller. 
         [0049]    In  FIGS. 3   b  and  3   c,  the fluence volume was plotted against maximum fluence H max . As the minimum fluence decreases, 10 J/mm 2  (dashed line), 15 J/mm 2  (dotted line), and 20 J/mm 2  (solid line), the fluence volume becomes larger. The maximum value of the fluence volume shifts towards lower maximum fluence as the minimum fluence increases. 
         [0050]    For  FIG. 3   b,  the fluence volume was calculated for a pulse energy of 12 mJ and a beam quality of 6.1, as well as for a pulse energy of 9 mJ and a beam quality of 3.4. For the two pulse energy/beam quality combinations, there result identical fluence volumes as a function of the maximum fluence. 
         [0051]    For comparison, in  FIG. 3   c  the fluence volume is plotted against the maximum fluence for a pulse energy of 12 mJ and a beam quality of 3. By varying the pulse energy/beam quality combinations, different fluence volumes can be achieved. 
         [0052]    The beam caustic in  FIG. 2  shows the beam radius as a function of the locus in focus zone  23 . Here, Z [mm] indicates the relative distance from the focus zone along the optical axis. Radius R [mm] corresponds to the extension of the Gauss-shaped laser beam, at which the intensity or fluence 1/e 2  is equal to the maximum intensity or the maximum fluence. The maximum intensity and fluence can be set corresponding to the intended use. 
         [0053]    In the beam caustics, the three regions are identified in which the three threshold values of 1/e 2  of the maximum fluence (hatched region), as well as 10 J/mm 2  (clear region) and 15 J/mm 2  (dotted region), are exceeded. Through the shape and extension of the fluence volumes inside which the selected minimum fluence is exceeded, the fluence volume can be calculated for a particular illumination level of the focusing lens and the resulting focusing. 
         [0054]    In the case of 75% illumination of the focusing lens ( FIG. 2   a ), given a radius R of 0.01 mm and a length Z of 0.11 mm, and given a ratio of the longitudinal extension to the transverse extension of 11, for the minimum fluence of 10 J/mm 2  there results a fluence volume of 6.4*10 −5  mm 3 . For the same minimum fluence and radius, in the case of 60% illumination of the focusing lens ( FIG. 2   b ) the fluence volume is 8.6*10 −5  mm 3 , given a ratio of longitudinal extension to transverse extension of 14. In the case of 50% illumination of the focusing lens ( FIG. 2   c ), the fluence volume is 10.6*10 −5  mm 3 , with a ratio of longitudinal extension to transverse extension of 20. 
         [0055]    Through variation of the illumination of the lens, and the associated change in the focusing of the pulsed laser beam, and simultaneous maintenance of constant values for the minimum fluence, the maximum possible fluence volume can be determined. 
         [0056]    The fluence volumes for  FIG. 2   d - i  were calculated in a manner corresponding to the above sample calculation for the fluence volumes in  FIG. 2   a - c.  For a minimum fluence of 10 J/mm 2 , the following fluence volumes and normed fluence volumes resulted: given an illumination of 75%, fluence volume FV (75%)=6.421*10 −5  mm 3  and normed fluence volume nFV (75%)=0.517; FV (60%)=8.579*10 −5  mm 3  and nFV (60%)=0.691; FV (50%)=10.64*10 −5  mm 3  and nFV (50%)=0.857; FV (40%)=12.42*10 −5  mm 3  and nFV (40%)=1; FV (35%)=11.5*10 −5  mm 3  and nFV (35%)=0.926; FV (30%)=5.35*10 −5  mm 3  and nFV (30%)=0.431. In this example, the maximum possible fluence volume for an illumination level of 40% resulted as 12.42*10 −5  mm 3 . The norming of the fluence volume was done to this value. 
         [0057]      FIG. 2  clearly shows that the fluence volume increases as the focusing becomes less sharp. The two series,  FIG. 2   a - c  and  FIG. 2   d - i,  show that the fluence volume is primarily a function of the illumination of lens system  20 , and secondarily is a function of beam quality M 2  or of the set maximum fluence in focus zone  23 . 
         [0058]    The following relationships can be formulated: 
         [0059]    If, given constant pulse energy Q and beam quality M 2 , the imaging of the laser beam is modified in such a way that there results a variation in the focus size, i.e. modification of radius ω 0  of the beam waist, in this way the size of fluence volume FV is influenced. Here, for a specified focus size, fluence volume FV assumes a maximum value. 
         [0060]    Given a decreasing fluence threshold, or minimum fluence, fluence volume FV becomes larger. Given a smaller fluence threshold, or minimum fluence, the maximum value of the fluence volume shifts towards larger beam radii ω z . 
         [0061]    The lean burning boundary at which laser ignition system  1  can reliably ignite a lean combustible fuel mixture is a quality feature for laser ignition system  1  in an internal combustion engine. In  FIG. 4 , lean burning boundary λ of a methane-air mixture is plotted as a function of the fluence volume (FV). For the three illumination levels of focusing lens  22 , corresponding to the beam caustics shown in  FIG. 2   a - c,  the lean burning boundary λ was determined for each of the pulse energies 3 mJ, 5 mJ, 7 mJ, and 9 mJ. For this purpose, during ignition trials in a flow chamber the probability of ignition of a methane-air mixture as a function of lean burning level λ was measured. For each lean burning level λ, 30 ignition processes were initiated in the methane-air mixture at a pressure of 3 bar and a flow speed of 5 m/s, and the successful ignitions were registered. The lean burning boundary corresponds to the λ at which the probability of ignition has decreased to 95%. 
         [0062]    For each pulse energy level and illumination level of focusing lens  22 , the beam caustic was simulated, and from this the fluence volume was calculated for a minimum fluence of 15 J/mm 2 . The greatest possible fluence volume, at 5.7*10 −5  mm 3 , was achieved in this trial at a focusing lens illumination of 50% and a pulse energy of 9 mJ. 
         [0063]      FIG. 4  shows that the lean burning boundary λ increases with the fluence volume (FV). Thus, when laser beam features, pulse energy, and beam quality M 2  are held constant, the lean burning capacity improves solely via the parameter normed fluence volume (nFV), i.e. via an optimized choice of the focusing. The determined lean burning boundaries for the beam caustics from  FIGS. 2   a ,  2   b , and  2   c  are marked in the graph. 
         [0064]      FIG. 5  shows fluence volume FV as a function of beam quality M 2  and pulse energy Q. As examples, operating points are shown for two laser crystals having a length of 10 mm (filled triangles) and a length of 30 mm (open triangles). For both laser crystals, the initial transmission of the saturable absorber was at T 0 =30%. The dashed lines are calculated values for the isochors of the fluence volume, fluence volume FV increasing from left to right. From left to right, the dashed lines correspond to fluence volumes of: FV=2*10 −5  mm 3 ; FV=4*10 −5  mm 3 ; FV=6*10 −5  mm 3 ; FV=8*10 −5  mm 3 , where in each case a minimum fluence of 15 J/mm 2  was selected. 
         [0065]      FIG. 5  clearly shows that, given constant beam quality M 2 , fluence volume FV increases as pulse energy Q increases. Given constant pulse energy Q, beam quality M 2  must improve, i.e. must assume smaller values, so that fluence volume FV will become larger. Identical fluence volumes FV can be produced via various combinations of pulse energy Q and beam quality M 2 .