o 93 (199 ACTA PHYSICA POLONICA A o 1 oceeigs o e Ieaioa Coeece "Quaum Oics I" asowiec oa 1997 EMOSECO IG WAE ACKES C. RADZEWICZ Isiue o Eeimea ysics Wasaw Uiesiy oża 9 -1 Wasaw oa M IEAC YΒ A eame o Cemisy a ysics e-Guio Uiesiy ee Sea Isae A J.S. KASISKI Cee o ase a ooics eseac Okaoma Sae Uiesiy Siwae OK 77 USA We aaye oagaio o ua so ig uses i a asae is- esie oiea meium A geea omua o emoseco wae acke eouio is eeoe a aie o seciic oems eoeica a eeimea esus o wae acke isoio y eses wae acke o- aio i ieige meia a gou eociy macig i sum equecy geeaio ae esee umeica esus o siig o emoseco wae ackes i isesie Ke meia ae aso esee ACS umes 5 5e 1 Ioucio eeomes i ase moe-ockig eciques uig e as ecae e o geeaio o uses as so as 75 s (1 s = 1 -15 s iecy om isaie ase [1] y emoyig use comessio eciques oe ca aciee ee soe uses wi e cue eco a ess a 5 s ee [ 3] ogess i is ie as ee ey as eseciay i e 7-ies a -ies as iusae i ig 1 wic eses e isoy o ua so use geeaio i e as 5 yeas Oigiay e mos commo emoseco sysems wee ase o ogaic yes as ase mae- ias owee wi a ae o s isaie ases [] i e eay 9-ies a soi sae ecoogy is aaiae Cuey sysems oeaig a -3 s age ae ouie a commecia sysems wi su- s caaiiies ae o e make A emoseco ig use oagaig i a gie meium ca e isuaie as a wae acke o eeco-mageic aiaio a moes wi is gou eociy a a e same ime cages is sae ue o e ieacio wi e meium We i e wae acke icue aeaig a we wi use e wae acke aguage ougou is ae Oe o e simes eeimes wi emoseco ig uses ioes oagaio o suc wae ackes oug a asae meium (37 238 C. Radzewicz et al. o eame a oica wiow mae o gass Suc eeimes ae omiese ey ae eome eey ay i ua as oics aoaoies oe wiou a coscious oug o e eeimeais a weee a uaso ig use as o e eiee om e ase sysem o e same a o is way asses oug a es o a gass ce wiow Sice e esu o may eeimes wi ua so ig uses eseciay i a oiea egime sogy ees o e eais o e use suc as is ime a/o saia oeies i is quie imoa o uesa wa aes o suc a use o is way om e ase sysem o e same We cosieig oagaio o a ow iesiy moocomaic ig eam i a ieecic oe oy as o accou o is ie o eacio wic moi- ies e iacio o e eam y cagig e oagaio cosa wi esec o a o acuum Wa makes e oagaio o emoseco uses iee? is y e iue o ei eiy e emoseco uses ae oa secum ecause o a a sigiica use saig is osee weee ey oagae i a isesie meium Seco e eak iesiy ca e quie ig ee wi a moeae use eegy a aeage owe eeoe 3- oe ocesses (aowe i a meium o ay symmey cao e egece o isace iesiy ee- e eacio ie gies ise o se-ase mouaio a se-ocusig wic iuece e secum o e use a is saio-emoa caaceisics esec- iey I is a we-kow eomeo eaig amog oe igs o omaio o oica soios i ies [5 ] a wie ig coiuum geeaio [7] As is oe e case i oiea oems ee eomea iacio isesio a se-ocusig ae eage; oe ca suy ei eecs seaaey oy i a imie ume o cases we oe o em omiaes owee i mos cases umeica meos ae o e emoye as a meo o souio is ae is ogaie as oows I Sec 2 we ese a geea om o e oagaio equaio o a em- emoseco ig Wae ackes 39 oseco wae acke i a asae isesie a oiea meium Secio 3 oies eames o e wae acke saig eecs i a iea egime — o eoeica a eeimea aa ae oie I Sec we aaye oiea oagaio egime a ese e esus o umeica cacuaios iusaig a come wae acke eouio oagaio equaio We sa y eeseig e scaa eecic ie E( o e ig wae acke as a iega o is ouie comoes eac caaceie y is amiue (k wae eco k a equecy ω(k Ou o ou aiaes ie equecy a ee comoes o e wae eco a escie eac ouie comoe oy ee ae ieee ecause o e isesio eaio ω = ω(k e we eese e ie amiue as a ouc o a sowy ayig eeoe (SE a a ase aco wee we assume a e wae ecos ae goue aou a cea aue k a coesoigy e equecies om a a aou e cea equecy ω 0 = ω(k is eas o e oagaio equaio o sowy ayig ee- oe [-11] I e as equaio a ui eg eco s = k/kassume o e i e iecio o cooiae as ee iouce A aiioa em iγ Α 2 Α as aso ee ae I escies e eec o e 3- oe oieaiy (Ke ye oieaiy o e meium e iega oe kz ca e eace y a iega oe ω we e isesio eaio is ake io accou wee is e ie o eacio a s = k/ | k| is aicua om o eessio o aows aicaio o is aoac o o isooic a aisooic meia Sice o assumios aou e om o A( ae ee mae Eq (3 is eac A aia ieeia equaio o A( ca aso e oaie y usig Eq (3 is is oe y eaig / i owes o k x , ky a (ω - ω a eacig ese aiaes wi i∂/∂ i/y a -∂/∂t eseciey Keeig ems u o e seco oe gies o a uiaia ieige meium [11] C. aewic e a e coeicies ß1 = 1/g = ∂k/∂ω a β = ∂β1/∂ω escie e eecs o gou eociy a gou eociy isesio eseciey γ xx a γyy ae e- sosie o e iacio o e wae acke γx eecs eam wak-o i a ieige meium wie γ was ou o e esosie o e oaio o e wae acke [9] I is wo meioig a e coeicies x a γ ais o isooic meium Oe ca easiy geeaie Eq ( y aig ige oe ems o e sake o caiy we wi o o i ee u ae ee a ieese eae o e oigia aes [-11] Eie o e wo equaios (Eq (3 o e eae esio o Eq ( ca e umeicay iegae I sou e oie ou a Eq ( is a aoimaio o Eq (3 a i oe o make em oay equiae oe wou ae o icue ems o a oes io Eq ( is ceay is o a acica aoac We ou owee a i a cases suie e esus o eie aoac ae amos ieica i e ems u o 3- oe ae icue io Eq ( eoe we ese e esus o suc iegaio e us cosie some sime cases 3 iea oagaio 1 - a 3 - eecs is e us aaye Eq ( i a 1-imesioa case Assume a Α( = A( is is e case weee ig oagaes i a sige moe oica ie o e ase eam oeies ae suc a i ca e moee as a oe-imesioa ae wae I is case a eiaies wi esec o a y ais a we ae e wi a we-kow oiea Scöige equaio o e oowig om is equaio is easy o soe i we kee oy e is em o e ig sie Oe ca eiy a ay ucio A(z, a as a om A(z, /νg cosiues a oe souio is meas a i e owes oe aoimaio a 1-imesioa wae acke ca e o ay sae a i oagaes wiou isoios wi e see equa o e gou eociy i a gie meium I may seem o e a iia sae- me wic i is u a oe wi sigiica cosequeces i may eeimes We wi iusae is wi wo eames e is eame o e aoc a e gou eociy ca ay wi a eeime is a wae acke isoio y eses I as ee ecogie quie a og ime ago a e ieece ewee ase a gou eociies ca ea o sigiica eecs uo oagaio o emoseco uses oug eses [1-13] I a coimae eam ie a a wae acke is ocuse wi a comaic es e e ase suaces ei e es ae seica u e sae o e wae acke is o I e U-isie age e gou eociy is smae a e ase eociy o oica gasses ecause e wae acke oagaes oug moe gass i e cee o e eam wee e es is e ickes i accumuaes moe gou eay a ase eay ee wie a e eges o e es e wo eays ae amos equa As a esu e wae acke ei e es is isoe; is cea a ags ei e eges is i u causes a e iesiy o ig i e ocus o oge oows e iesiy ime oie o e iu use ecause iee aia oes o e wae acke coiue o i a iee imes I is imoa o e ae o measue is isoio eseciay o come eses suc Femtosecond Light Wave Packets 241 as microscope objectives for which direct calculations may be quite difficult and which are commonly used in applications such as two-photon microscopy [14].