fig1 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 . 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 . different illumination levels of focusing lens 22 result in differently strong focusing of the pulsed laser beam . fig2 shows the corresponding beam caustics for illuminations of 75 % ( fig2 a , d ), 60 % ( fig2 b , e ), 50 % ( fig2 c , f ), 40 % ( fig2 g ), 35 % ( fig2 h ), and 30 % ( fig2 i ) of the lens diameter . the beam caustics for a laser pulse energy of 9 mj were simulated . fig2 a , b , and c show beam caustics for laser beams having different beam qualities m 2 , as were used subsequently ( fig3 ) in ignition experiments . for the simulation of the beam caustics in fig2 d , e , f , g , h , and i , the same laser parameters were used as were used for the beam caustic in fig2 c , and only the illumination of focusing lens 22 was varied . here , different maximum fluences resulted in focus zone 23 . 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 . the fluence volume v is determined under the assumption of a threshold fluence or minimum fluence . the radial distance to the z axis is defined as beam radius ω ( z ), at which the intensity has fallen to 1 / e 2 : the minimum beam radius , present at the beam waist ( at z = 0 ), is designated ω 0 . 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 : 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 : the average power density i results from the ratio of power p to cross - sectional surface a : pulse energy q is power p multiplied by pulse duration τ : average energy density h , also called fluence , is the ratio of the pulse energy to cross - sectional surface a : 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 : 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 : 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 : 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 . fig3 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 ( fig3 a ) and against the maximum fluence ( fig3 b and 3 c ). fig3 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 . in fig3 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 . for fig3 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 . for comparison , in fig3 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 . the beam caustic in fig2 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 . 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 . in the case of 75 % illumination of the focusing lens ( fig2 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 ( fig2 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 ( fig2 c ), the fluence volume is 10 . 6 * 10 − 5 mm 3 , with a ratio of longitudinal extension to transverse extension of 20 . 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 . the fluence volumes for fig2 d - i were calculated in a manner corresponding to the above sample calculation for the fluence volumes in fig2 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 . fig2 clearly shows that the fluence volume increases as the focusing becomes less sharp . the two series , fig2 a - c and fig2 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 . 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 . 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 . 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 fig4 , 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 fig2 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 %. 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 . fig4 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 fig2 a , 2 b , and 2 c are marked in the graph . fig5 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 . fig5 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 .