Ultra-short pulse propagation in complex optical systems

Transcription

1 Ultr-short pulse propgtion in complex opticl systems Ulrike Fuchs, Uwe D. Zeitner nd Andres Tünnermnn* Frunhofer-Institut für Angewndte Optik und Feinmechnik, Albert-Einstein-Str. 7, D Jen, Germny * lso t: Institute of Applied Physics, Friedrich-Schiller-University Jen, Mx-Wien-Pltz 1, D Jen, Germny Abstrct: In ppliction of ultr-short lser pulses the pulse prmeters hve to be controlled ccurtely. Hence the mnipultion of the propgtion behvior of ultr-short pulses requires for specilly designed optics. We hve developed tool for the simultion of ultr-short lser pulse propgtion through complex rel opticl systems bsed on combintion of ry-trcing nd wve opticl propgtion methods. For the prcticl implementtion of the pproch two commercilly vilble softwre pckges hve been linked together, which re ZEMAX nd Virtul Optics Lb. The focussing properties of different lenses will be nlyzed nd the results re demonstrted Opticl Society of Americ OCIS codes: ( ) Numericl pproximtion nd nlysis, ( ) Fourier optics, ( ) Geometricl optics, ( ) Imging systems, ( ) Focus, ( ) Pulses References nd links 1. S. Nolte, M. Will, J. Burghoff, nd A. Tünnermnn, Ultrfst lser processing: new options for threedimensionl photonic structures, J. Mod. Opt. 51(16-18), (2004). 2. S. Sztmári nd G. Kühnle, Pulse front nd pulse durtion distortion in refrctive optics, nd its compenstion, Opt. Commun. 69, (1988). 3. Z. Bor, Distortion of femtosecond lser pulses in lenses, Opt. Lett. 14, (1989). 4. T. E. Shrp nd P. J. Wisoff, Anlysis of lens nd zone plte combintions for chromtic focusing of ultrshort lser pulses, Appl. Opt. 31, (1992). 5. Z. Bor nd Z. L. Horváth, Distortion of femtosecond pulses in lenses. Wve opticl description, Opt. Commun. 94, (1992). 6. M. Kempe, U. Stmm, nd B. Wilhelmi, Sptil nd temporl trnsformtion of femtosecond lser pulses by lenses with nnulr perture, Opt. Commun. 89, (1992). 7. M. Kempe nd W. Rudolph, Femtosecond pulses in the focl region of lenses, Phys. Rev. A 48, (1993). 8. E. Ibrgimov, Focusing of ultrshort lser pulses by the combintion of diffrctive nd refrctive elements, Appl. Opt. 34, (1995). 9. G. O. Mttei nd M. A. Gil, Sphericl berrtion in sptil nd temporl trnsforming lenses of femtosecond lser pulses, Appl. Opt. 38, (1999). 10. Z. L. Horváth nd Z. Bor, Diffrction of short pulses with boundry diffrction wve theory, Phys. Rev. E 63(026601), 1 11 (2001). 11. R. Piestun nd D. A. B. Miller, Sptiotemporl control of ultrshort opticl pulses by refrctive-diffrctivedispersive structured opticl elements, Opt. Lett. 26, (2001). 12. R. Ashmn nd M. Gu, Effect of ultrshort pulsed illumintion on foci cused by Fresnel zone plte, Appl. Opt. 42, (2003). 13. J. J. Stmnes, Wves in Focl Regions: Propgtion, Diffrction nd Focusing of Light, Sound nd Wter Wves (Bristol [u..] : Hilger, 1986). (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3852

2 14. J. W. Goodmn, Introduction to Fourier Optics, 2nd ed. (New York [u..] : McGrw-Hill, 1996). 15. Y. M. Engelberg nd S. Ruschin, Fst method for physicl optics propgtion of high-numericl-perture bems, J. Opt. Soc. Am. A 21, (2004). 16. U. Fuchs nd U. D. Zeitner, Method for fst clcultion of ultr-short pulses in focl regions, (2005). To be published. 17. ZEMAX Opticl Design Progrm, ZEMAX Development Corportion, USA. 18. Virtul Optics Lb 4.1, LightTrns GmbH, Germny. 19. R. M. Hermn nd T. A. Wiggins, Production nd uses of diffrctionless bems, J. Opt. Soc. Am. A 8, (1991). 20. C. L. Arnold, A. Heisterkmp, W. Ertmer, nd H. Lubtschowski, Strek formtion s side effect of opticl brekdown during processing the bulk of trnsprent Kerr medi with ultr-short lser pulses, Appl. Phys. B 80, (2005). 21. H. Sonjlg nd P. Sri, Suppression of temporl spred of ultrshort pulses in dispersive medi by Bessel bem genertors, Opt. Lett. 21, (1996). 1. Introduction An emerging number of rel world pplictions in science nd industry require the use of ultrshort lser pulses. Prominent exmples re multi-photon lser scnning microscopy nd ultr fst lser micro-mchining [1]. Severl pplictions require for tight focussing of the lser bem in order to relize high sptil resolution nd high locl field strengths. However, focussing optics usully hve significnt influence on the sptil nd temporl chrcteristics of the lser pulse in the focl region. During the lst fifteen yers severl nlyticl pproches hve been mde in order to model different effects cused by mteril dispersion or chromtic nd sphericl berrtions [2 12]. However, ll these investigtions re limited to idelized single lenses or chromtic doublets. Furthermore, lso the description of the incident pulses is idelized in these models. The tretment of ultr short pulse interctions with rel lenses or even complex opticl systems s well s opticl design is impossible with these nlyticl pproches. In the current pper we report on generl pproch for the tretment of the propgtion of rbitrry lser pulses through complex opticl systems. The clcultions re bsed on combintion of ry-trcing nd wve opticl propgtion methods. Therefore ll kinds of berrtions re considered. On top of tht not only the nlysis of occurring effects but lso the optimiztion of system prmeters with respect to the behvior of the focussed pulses is possible. 2. Method The electricl field of n ultrshort lser pulse cn be decomposed into its spectrl components by pplying fourier trnsform. Therefore it cn be written s E(t,x,y,z) = F 1 [E(ν,x,y,z)] = dνa(ν,x,y,z) e iφ(ν,x,y,z) e i2πνt, (1) with the spectrl mplitude described by A(ν,x,y,z) nd φ(ν,x,y,z) being the spectrl phse of the lser pulse. For the computtions the incident lser pulse is seprted into its spectrl components (Eq. (1)) which re trced seprtely through n opticl system pplying geometricl optics methods. Propgting the lser pulse through n opticl system modifies both the spectrl phse φ(ν,x,y,z) nd the spectrl mplitude A(ν,x,y,z). The ltter one is influenced by diffrction effects during propgtion, bsorption, reflection, scttering effects, pupil berrtions nd nonliner mteril interction. In ddition dispersion nd berrtions within the opticl system ffect the spectrl phse φ(ν,x,y,z) nd its sptil distribution. In generl it is possible to consider ll these effects within our clcultion. However, in the present pper we (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3853

3 turn our ttention especilly to imging systems becuse of their brod field of ppliction. Therefore, it is sufficient in most cses of prcticl relevnce to limit the effects of the propgtion between entrnce nd exit pupil of the opticl system to the modifiction of the spectrl phse φ(ν,x,y,z) by liner mteril interction nd the lterl scling of the bem size. If n opticl system involves only collimted propgtion with no focussing of the lser pulse in the observtion plne, e.g. propgtion through simple stretcher-compressor-system, the electricl field is simply clculted by superposition of ll spectrl components in tht plne pplying Eq. (1). Clculting the electricl field in the vicinity of the focus of n imging system is more complex. Since the geometricl opticl theory my not be vlid in the focl region wve opticl methods hve to be induced. For this purpose we extended the combintion of geometricl optics nd wve optics firstly proposed by Stmnes [13] in 1986 to the tretment of ultr-short lser pulse propgtion. p y p x z (x,y,z ) f reference sphere entrnce pupil exit pupil (p,p,z ) x y en (p,p,z ) x y ex Fig. 1. Illustrtion of typicl imging system, e.g. microscope objective (Linos ), contining entrnce nd exit pupil with their specific coordintes (p x, p y,z en ) nd (p x, p y,z ex ). The reference sphere in the exit pupil is is centered in the focus. The positions of entrnce nd exit pupil depend on the prticulr opticl system. For imging systems the focl plne (x,y,z f ) is of specil interest. In the specil cse of imging systems (Fig. 1) the propgtion of the lser pulse between the entrnce nd the exit pupil cn be treted very ccurtely with ry-trcing methods. The propgtion from the exit pupil into the focl region is then performed by solving the diffrction integrl for ech spectrl component. Thus the electricl field E(t,x,y,z f ) of the lser pulse in the focl plne is given by the weighted superposition of the diffrction pttern from ll spectrl components. Since the entrnce nd exit pupil re both imges of the sme limiting perture within the imging system diffrction effects cn be treted s occurring from the exit pupil only [14]. There re severl wys of clculting the wve opticl propgtion from the exit pupil into the focl region [13]. However, for the pplicbility it is importnt to hve fst wy of computtion becuse ech spectrl component needs to be propgted seprtely. Due to its convenient hndling we hve chosen method bsed on fourier trnsformtion. If the spectrum E(ν, p x, p y,z ex ) in the exit pupil is defined reltive to reference sphere, which is centered in the focl region, these clcultions re no longer limited to prxil bems [15]. The coordintes (p x, p y ) then refere to the reference sphere. The pproximtions mde even hold for sphericl berrtions up to bout 50λ. Hence the propgtion from the exit pupil into the focl plne is equivlent to fourier trnsformtion of the spectrum E(ν, p x, p y,z ex ) with E(ν, p x, p y,z ex ) = A(ν, p x, p y,z en ) e iφ(ν,p x,p y,z en) e iφ b(ν,p x,p y,z ex ) (2) (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3854

4 contining the spectrl mplitude A(ν, p x, p y,z en ) nd phse φ(ν, p x, p y,z en ) of the incident lser pulse plus n dditionl prt of the spectrl phse φ b (ν, p x, p y,z ex ) cused by dispersion nd berrtions of the system. The pulse spectrum E(ν,x,y,z f ) in the focl plne cn be expressed s E(ν,x,y,z f ) ν F px p y xy[e(ν, p x, p y,z ex )]. (3) The qudrtic phse fctor in front of the fourier trnsformtion hs been neglected becuse its sptil vrition found to be very wek in regions ner the center of the focus where the mjor prt of the energy is concentrted if berrtions re not too lrge. Applying Eq. (1) on (3) yields phse correct superposition of ll spectrl components by fourier trnsform nd therefore the spce-time-distribution in the focl plne. If nother z- plne is of specil interest the field distribution in the exit pupil needs to be mnipulted with n dditionl sphericl phse fctor to perform the propgtion with fourier trnsform s well [16]. Clculting the field distribution for sufficient mount of z-plnes nd rerrnging those numericlly leds to n x-z-distribution t given time. For the prcticl implementtion of this pproch ny softwre pckge for opticl design could be used. The clcultions in this pper were performed with the ry-trcing softwre ZEMAX [17] nd the wve opticl simultion softwre Virtul Optics Lb [18]. The ry-trcing of ll spectrl components from the entrnce to the exit pupil of the optics under considertion ws done with ZEMAX. Wheres the wve opticl propgtion from the exit pupil into the focl region ws ccomplished with Virtul Optics Lb. The specil method used here is not limited to prxil focussing optics s will be seen in the exmples presented in Section 3. Although not shown here systems like pulse compressors or focussing into mterils cn be nlyzed with this method s well. Limittions re up to now the exclusion of nonliner mteril interctions of the pulses. For the considertion of such effects the propgtion lgorithm hs to be substntilly extended. 3. Numericl results For comprison of vrious effects occurring while focussing ultr-short lser pulses we hve chosen three different opticl systems. They offer similr prmeters such s numericl perture (NA) of bout 0.45 nd focl length of bout 9mm. All optics re rel world exmples. Hence the distortion cused by dispersion nd berrtions is not reduced to tht of model systems nd in ddition ll effects pper in relistic mixture. However, to be ble to study the impct of mjor effects such s dispersion, chromtic nd sphericl berrtions the optics chosen exhibit dominntly one of them ech. The first optic is plnoconvex lens from Linos (312011) mde of BK7 hving severe sphericl berrtions. The second one is Geltech sphere (Thorlbs ) showing chromtic berrtions. The third opticl system is microscope objective from Linos (038723) mde of severl different sorts of glss, therefore giving rise to gret mount of mteril dispersion. Even though it is possible to compute with n rbitrry incident lser pulse using the described method we hve chosen 24fs-pulse centered t 800nm with gussin shped spectrum to keep possible effects well seprted nd s simple s possible Spectrl phse in the exit pupil Ry-trcing for ech spectrl component through n opticl system gives us the spectrl phse φ(ν, p x, p y,z ex ) in the exit pupil reltive to reference sphere. Since ll considered opticl systems re rottionlly symmetricl the nlysis of the spectrl phse long one coordinte xis is sufficient. Depicting this phse for ech spectrl component s function of the normlized exit pupil coordinte p y gives first insight to the focussing behvior for ultr-short lser pulses. (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3855

5 0 p y () 2 (b) (c) 0 Fig. 2. Spectrl phse coded in gry scle in the exit pupil of the three optics. The horizontl xis shows the vrition of the phse s function of p y which is therefore the berrtion for ech frequency ν. The verticl dependence in phse with the frequency shows the influence of mteril dispersion. The plnoconvex lens () is hving severe sphericl berrtions, the Geltech sphere (b) is showing chromtic berrtions s well s dispersion effects nd the microscope objective (c) is hving strong dispersion effects. The frequency slices depicted re Hz centered t Hz (800nm). The spectrl phse in the exit pupil of the three opticl systems discussed here re depicted in Fig. 2. The horizontl xis shows the vrition of the phse s function of p y which is therefore the berrtion for ech frequency ν. The verticl dependence in phse with the frequency shows the influence of mteril dispersion nd therefore the group velocity dispersion (GVD) nd higher orders. Just from this illustrtion one cn estimte the mount of distortion of the lser pulse. For n idel lens the spectrl phse on the reference sphere is constnt for ll frequencies nd pupil coordintes. At this point we wnt to infer on the focussing behvior from the spectrl phse of the three opticl systems discussed here. The phse in the exit pupil of the plnoconvex lens (Fig. 2()) shows strong vritions with p y for ech wvelength which mens tht there re severe berrtions (sphericl berrtions in this cse). On the other hnd the vrition of the spectrl phse with the frequency is smll. Hence mteril dispersion just cuses smll chnges in the pulse durtion s expected for thin lenses. The vrition of the spectrl phse in the exit pupil of the Geltech sphere (Fig. 2(b)) is quite wek compred to the plnoconvex lens. The curvture of the phse cn be identified s chromtic berrtions. The spectrl phse lso shows modifiction with the wvelength which cn not be neglected nd is cused by the highly dispersive mteril the lens is mde of. The spectrl phse in the exit pupil of the microscope objective (Fig. 2(c)) is nerly constnt for ll p y which is due to the well corrected chromtic system. Wheres the high mount of mteril dispersion cused by the different sorts of glsses nd the long glss pth led to strong vrition of the spectrl phse with the frequency which mens strong GVD. One cn now expect good focussing ttributes but n expnsion of the pulse durtion for the microscope objective. The other two opticl systems will cuse pulse front distortions becuse of their berrtion. These effects will be discussed next. (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3856

6 3.2. Propgtion into the focl region For comprison of the influence of mteril dispersion nd berrtions on the focussing behvior we behold the focussing of n ultr-short lser pulse with n idel lens (NA=0.45 nd f=9mm) first (Fig. 3). There is no pulse front distortion nd perfectly sphericl pulse front is propgting into the focus. Since the perture of the lens is fully illuminted typicl diffrction ptterns on the pulse front cn be observed. We will initite our explortion with the influence of mteril dispersion on the focussing behvior of ultr-short lser pulses strting with the microscope objective. From the nlysis of the spectrl phse (Fig. 2(c)) one cn expect n lmost undistorted pulse front but n incresed pulse durtion which is proportionl to the mount of mteril dispersion. Exctly this behvior cn be found from Fig. 4(), which shows the focussing of the lser pulse in the vicinity of the focus (±200µm). After propgting through the microscope objective the pulse durtion is 116fs being five times lrger thn the originl durtion. The lso occurring intensity pttern on the opticl xis will be discussed in Section fs -300fs 300fs 550fs -100fs 100fs 0fs 220 m 400 m Fig. 3. Rdil intensity distribution of the focussing of n ultrshort lser pulse (24fs) with n idel lens. Light is propgting from left to right. The time is chosen so tht 0fs refers to the pulse intensity mximum rriving t the focus. The movie shows n nimtion of the focussing of the ultr-short lser pulse with the sme scling s depicted bove. Secondly we turn our ttention to the focussing behvior of the Geltech sphere. As noted before this lens will cuse n increse of the pulse durtion s well. But the mjor effect here re the chromtic berrtions which cuse distortion of the pulse front. The typicl horseshoe shpe [5] cn be observed in Fig. 4(b). Notice tht the pulse durtion ner the opticl xis is longer thn t the edge due to the rdilly vrying thickness of the lens. An dditionl pulse occurs on the opticl xis which is refered to s boundry wve pulse in [5] nd further described in [10]. We will discuss this effect in Section 3.3. Finlly the focussing behvior of the plnoconvex lens will be nlyzed. Due to the severe sphericl berrtions not even monochromtic light cn be focussed well with this lens. In Fig. 5 one cn observe wht hppens to n ultrshort lser pulse fter propgting through tht plnoconvex lens. As one cn see the sphericl berrtions led to the formtion of Bessel-like pulse [19] with its typicl x-shpe. Indeed there re three different intensity peks propgting long the opticl xis between the mrginl nd prxil focus. This behvior hs been observed before in [7], but the explntion given there is in contrst to our findings. The first intensity pek (mx1) is originting from the prxil prt of the lens being only wekly influenced by berrtions nd behving similr to the pulse focused by the microscope objective. This inten- (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3857

7 250 m -500fs -300fs 300fs 500fs -600fs -400fs 200fs 400fs 0fs -200fs 0fs 250 m 220 m () (b) 400 m 400 m Fig. 4. Rdil intensity distribution of the focussing of n ultrshort lser pulse (24fs) with the microscope objective () cusing n increse of the pulse durtion to 116fs. The chromtic berrtions of the Geltech sphere (b) led to the typicl horseshoe shpe of the pulse front. The pulse durtion is incresed s well depending on the thickness of the lens. The movie shows n nimtion of the focussing of the ultr-short lser pulse with the sme scling s depicted bove. sity pek is directly correlted to the originl incoming pulse nd is focused in the prxil focl plne. The second intensity pek (mx2) is Bessel-like pulse cused by the sphericl berrtions. It is existing between the mrginl nd prxil focl plne only. The third intensity pek (mx3) known s boundry wve pulse exhibits very specil ttributes nd will be discussed in the following Section 3.3. The formtion of mx2 nd mx3 is very similr, so the properties of mx2 re not explined in detil. We just wnt to point out tht the rdil intensity distribution nd the velocity of propgtion of mx2 depend on its position long the z-xis. At the mrginl focl plne mx2 nd mx3 coincide both propgting with the velocity v b, which will be given below in Eq. (4). While propgting towrds the prxil focl plne the velocity of mx2 decreses becuse the effective perture cusing mx2 shrinks nd therefore the ngle α decreses. At the prxil focl plne mx1 nd mx2 coincide both propgting with the vcuum velocity of light c. mx1 mx3 mx2 660 m Fig. 5. Rdil intensity distribution of n ultrshort lser pulse (24fs) focussed by the plnoconvex lens. There re three intensity mximums trveling long the opticl xis, explntion is given in the text. The pulses re propgting from bottom to top. The movie shows n nimtions of the focussing of the ultr-short lser pulse between the mrginl nd prxil focl plne, which re bout 2mm prt. The observtion window is trveling with the min pulse (mx1) nd covers n re of 660µm width nd 500µm length. (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3858

8 3.3. Side effects There re two mjor side effects observed for the focussing of ultr-short lser pulses with rel lenses which re of specil interest being the intensity distribution long the opticl xis nd the dditionl pulse known s boundry wve pulse. Due to our generl pproch nd the higher flexibility in clculting the propgtion in the whole vicinity of the focus we re ble to study these effects in more detil. The sttionry intensity distribution long the opticl xis (Fig. 4()) is of specil interest for the ppliction of fs-pulses for micro-structuring of mterils. Clcultions show tht they lwys pper when berrtions re not to high. In ddition their ppernce nd structure re not much influenced by the spectrl width of lser pulse. They even occur for monochromtic illumintion. The spcing of two mximums cn be explined using fresnel numbers nd therefore is depending on the NA of n opticl system. The intensity pttern long the opticl xis is cused by diffrction of the pulse front t the system perture nd is therefore n inherent ttribute of the opticl system. The rtio of the intensity long the opticl xis compred to the totl pulse intensity is dependent on the NA of the system. With incresing NA the distribution (Fig. 6) becomes nrrower nd steeper. Also the spcing of two intensity mximums decreses with n incresing NA becuse the Fresnel number is incresing too. It cn lso be seen tht for gussin shped illumintion of the perture prt from homogeneous illumintion the oscilltion of the intensity distribution long the opticl xis disppers but the envelope stys unchnged. µ Fig. 6. Intensity distribution long the opticl xis (Due to the numericl clcultion the opticl xis hs got width of 831nm.) compred to the totl pulse intensity for the microscope objective with NA=0.45 nd NA=0.1. The position z=0µm mrks the geometricl focus for ech microscope objective. For homogeneous illuminted perture the intensity long the opticl xis is oscillting. If the illumintion is gussin shped (0.5% of the mximum intensity t the rim) the envelop is unchnged but the oscilltion disppers. While processing trnsprent mteril with ultr-short lser pulses strek formtion in front nd fter the focus cn occur [20]. Our clcultions show tht especilly for focussing with low NA optics there is non neglectble mount of the pulse intensity distributed long the opticl xis (Fig. 6). In ddition this intensity distribution is lterlly very nrrow (bout 1µm) nd could cuse higher locl free-electron density which might led to permnent modifictions of the mteril. Our ssumption is supported by the dimensions of the streks observed in [20] (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3859

9 which re bout 1µm wide nd more thn 50µm long in front nd fter the position where the opticl brekdown occurs. In this region there is more thn 70% of the totl pulse intensity on the opticl xis. To void the occurrence of streks we suggest focussing with higher NA optics. The second side effect to be discussed here is the dditionl pulse occurring while focussing ultr-short pulses. This effect hs been observed before nd is known s forerunner pulse or the so clled boundry wve pulse [5, 10]. Our clcultions show tht its ttributes such s velocity of propgtion nd rdil intensity distribution re described well by ssuming the dditionl pulse being Bessel-like pulse [21] originting from the system perture. Its formtion s n interference pttern long the opticl xis is illustrted in Fig. 7. Notice tht t ech point long the opticl xis different prts of the pulse front interfere. Of specil interest is its superluminl velocity v b which exceeds the vcuum velocity of light c depending on the NA of the opticl system nd is given by v b = c cosα. (4) Since the dditionl pulse is no clssicl wve pckge this is not contrdictory to the theory of reltivity. k perture k f Fig. 7. Illustrtion of the formtion of the dditionl pulse s interference pttern long the opticl xis (red circle). At ech point long the opticl xis different prts of the pulse front interfere which is mrked with n. Therefore it is no clssicl wve pckge. Our clcultions lso show tht for fully illuminted perture this dditionl pulse is lwys ppering no mtter how strong berrtions re. Hving gussin distribution of the field in the perture ttenutes the intensity of the dditionl pulse. At certin width of the gussin distribution the dditionl pulse even disppers. Furthermore, our clcultions show tht for n idel lens the min pulse is overtken by the dditionl pulse exctly in the focus. Chromtic berrtions led to dely of the pulse front close to the opticl xis which cuses shift of the pssing point towrds the opticl system. Therefore it seemed to be forerunner pulse s reported in erlier ppers which is exct only behind the focus. Knowing this is of specil interest for experimentl work where the dditionl pulse could corrupt the results, e.g. in pump-probeexperiments. Using well corrected chromtic system for focussing sets the pssing point of both pulses into the focus. Doing so should prevent from hving forerunner pulse in the focl plne. 4. Conclusion We demonstrted powerful method for the clcultion nd extensive nlysis of the propgtion of ultr-short lser pulses. This pproch significntly extends the nlyticl clcultions of pulse propgtion through idelized lenses presented in the literture up to now. Due to the fct tht ll berrtions nd liner propgtions effects re tken into ccount nd the clcultions re not limited to prxil focussing the focussing behvior of rel world imging systems (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3860

10 could be investigted for the first time. In ddition our pproch is not limited to idelized field distributions nd lser pulse spectr. The implementtion of the propgtion lgorithm into two commercilly vilble opticl design pckges mkes it very flexible nd esy to use for opticl design tsks. The required effort for the nlysis of specific system is nerly independent from the number of opticl components the system is mde of. As result of our investigtions two mjor side effects occurring while focussing ultr-short lser pulses hve been nlyzed in detil. The sttionry intensity distribution long the opticl xis results in strek formtion when focussing with low NA optics. Secondly, the ttributes of the dditionl ccompnying pulse could be described with the model of Bessel-like pulse. It ws shown tht chromtic berrtions determine the pssing point of this pulse nd the mjor pulse front. The uthors like to thnk B. Wilhelmi nd M. Kempe for fruitful discussions. (C) 2005 OSA 16 My 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3861