o 93 (199 ACTA PHYSICA POLONICA A o 1
ocee i gs o e I e a io a Co e e ce "Qua um O ics I " as owiec o a 1997
EM OSECO IG WA E ACKE S
C. RADZEWICZ
I s i u e o E e ime a ysics Wa saw U i e si y oża 9 - 1 Wa saw o a
M I E AC Y Β A
e a me o C emis y a ysics e -Gu io U i e si y ee S e a Is ae
A J.S. K ASI SKI
Ce e o ase a o o ics esea c Ok a oma S a e U i e si y S i wa e OK 7 7 USA
We a a y e o aga io o u a s o ig u ses i a a s a e is- e si e o i ea me ium A ge e a o mu a o em oseco wa e acke e o u io is e e o e a a ie o s eci ic o ems eo e ica a e e ime a esu s o wa e acke is o io y e ses wa e acke o- a io i i e i ge me ia a g ou e oci y ma c i g i sum eque cy ge e a io a e ese e ume ica esu s o s i i g o em oseco wa e acke s i is e si e Ke me ia a e a so ese e ACS um e s 5 5 e
1 I o uc io e e o me s i ase mo e- ocki g ec iques u i g e as eca e e o ge e a io o u ses as s o as 7 5 s (1 s = 1 -15 s i ec y om i sa i e ase [1] y em oyi g u se com essio ec iques o e ca ac ie e e e s o e u ses wi e cu e eco a ess a 5 s e e [ 3] og ess i is ie as ee e y as es ecia y i e 7 - ies a - ies as i us a e i ig 1 w ic ese s e is o y o u a s o u se ge e a io i e as 5 yea s O igi a y e mos commo em oseco sys ems we e ase o o ga ic yes as ase ma e- ia s owe e wi a a e o s i sa i e ase s [ ] i e ea y 9 - ies a so i s a e ec o ogy is a ai a e Cu e y sys ems o e a i g a -3 s a ge a e ou i e a comme cia sys ems wi su - s ca a i i ies a e o e ma ke A em oseco ig u se o aga i g i a gi e me ium ca e isua i e as a wa e acke o e ec o-mag e ic a ia io a mo es wi i s g ou e oci y a a e same ime c a ges i s s a e ue o e i e ac io wi e me ium We i e wa e acke ic u e a ea i g a we wi use e wa e acke a guage oug ou is a e O e o e sim es e e ime s wi em oseco ig u ses i o es o aga io o suc wa e acke s oug a a s a e me ium
( 37 238 C. Radzewicz et al.
o e am e a o ica wi ow ma e o g ass Suc e e ime s a e om i ese ey a e e o me e e y ay i u a as o ics a o a o ies o e wi ou a co scious oug o e e e ime a is a w e e e a u as o ig u se as o e e i e e om e ase sys em o e sam e a o i s way asses oug a e s o a g ass ce wi ow Si ce e esu o ma y e e ime s wi u a s o ig u ses es ecia y i a o i ea egime s o g y e e s o e e ai s o e u se suc as i s ime a /o s a ia o e ies i is qui e im o a o u e s a w a a e s o suc a u se o i s way om e ase sys em o e sam e W e co si e i g o aga io o a ow i e si y mo oc oma ic ig eam i a ie ec ic o e o y as o accou o i s i e o e ac io w ic mo i- ies e i ac io o e eam y c a gi g e o aga io co s a wi es ec o a o acuum W a makes e o aga io o em oseco u ses i e e ? i s y e i ue o ei e i y e em oseco u ses a e oa s ec um ecause o a a sig i ica u se s a i g is o se e w e e e ey o aga e i a is e si e me ium Seco e eak i e si y ca e qui e ig e e wi a mo e a e u se e e gy a a e age owe e e o e 3- o e ocesses (a owe i a me ium o a y symme y ca o e eg ec e o i s a ce i e si y e e - e e ac io i e gi es ise o se - ase mo u a io a se - ocusi g w ic i ue ce e s ec um o e u se a i s s a io- em o a c a ac e is ics es ec- i e y I is a we -k ow e ome o ea i g amo g o e i gs o o ma io o o ica so i o s i i e s [5 ] a w i e ig co i uum ge e a io [7] As is o e e case i o i ea o ems ee e ome a i ac io is e sio a se - ocusi g a e e a g e ; o e ca s u y ei e ec s se a a e y o y i a imi e um e o cases w e o e o em omi a es owe e i mos cases ume ica me o s a e o e em oye as a me o o so u io is a e is o ga i e as o ows I Sec 2 we ese a ge e a o m o e o aga io equa io o a em- em oseco ig Wa e acke s 39
oseco wa e acke i a a s a e is e si e a o i ea me ium Sec io 3 o i es e am es o e wa e acke s a i g e ec s i a i ea egime — o eo e ica a e e ime a a a a e o i e I Sec we a a y e o i ea o aga io egime a ese e esu s o ume ica ca cu a io s i us a i g a com e wa e acke e o u io
o aga io equa io We s a y e ese i g e sca a e ec ic ie E( o e ig wa e acke as a i eg a o i s ou ie com o e s
eac c a ac e i e y i s am i u e (k wa e ec o k a eque cy ω(k Ou o ou a ia es i e eque cy a ee com o e s o e wa e ec o a esc i e eac ou ie com o e o y ee a e i e e e ecause o e is e sio e a io ω = ω(k e we e ese e ie am i u e as a o uc o a s ow y a yi g e e o e (S E a a ase ac o w e e we assume a e wa e ec o s a e g ou e a ou a ce a a ue k a co es o i g y e eque cies o m a a a ou e ce a eque cy ω = ω(k is ea s o e o aga io equa io o s ow y a yi g e e- o e [ -11]
I e as equa io a u i e g ec o s = k / k assume o e i e i ec io o coo i a e as ee i o uce A a i io a e m iγ Α Α as a so ee a e I esc i es e e ec o e 3- o e o i ea i y (Ke y e o i ea i y o e me ium e i eg a o e k ca e e ace y a i eg a o e ω w e e is e sio e a io is ake i o accou w e e is e i e o e ac io a s = k/ k is a icu a o m o e essio o a ows a ica io o is a oac o o iso o ic a a iso o ic me ia Si ce o assum io s a ou e o m o A( a e ee ma e Eq (3 is e ac A a ia i e e ia equa io o A( ca a so e o ai e y usi g Eq (3
is is o e y e a i g / i owe s o k x ky a (ω - ω a e aci g ese a ia es wi i∂/∂ i / y a - ∂/∂t es ec i e y Kee i g e ms u o e seco o e gi es o a u ia ia i e i ge me ium [11] C. a ewic e a
e coe icie s ß1 = 1/ g = ∂k/∂ω a β = ∂β1/∂ω esc i e e e ec s o g ou e oci y a g ou e oci y is e sio es ec i e y γ xx a γyy a e e- s o si e o e i ac io o e wa e acke γx e ec s eam wa k-o i a i e i ge me ium w i e γ was ou o e es o si e o e o a io o e wa e acke [9] I is wo me io i g a e coe icie s x a γ a is o iso o ic me ium O e ca easi y ge e a i e Eq ( y a i g ig e o e e ms o e sake o c a i y we wi o o i e e u a e e e a i e es e ea e o e o igi a a e s [ -11] Ei e o e wo equa io s (Eq (3 o e e a e e sio o Eq ( ca e ume ica y i eg a e I s ou e oi e ou a Eq ( is a a o ima io o Eq (3 a i o e o make em o a y equi a e o e wou a e o i c u e e ms o a o e s i o Eq ( is c ea y is o a ac ica a oac We ou owe e a i a cases s u ie e esu s o ei e a oac a e a mos i e ica i e e ms u o 3- o e a e i c u e i o Eq ( e o e we ese e esu s o suc i eg a io e us co si e some sim e cases
3 i ea o aga io 1 - a 3 - e ec s i s e us a a y e Eq ( i a 1- ime sio a case Assume a Α( = A( is is e case w e e e ig o aga es i a si g e mo e o ica i e o e ase eam o e ies a e suc a i ca e mo e e as a o e- ime sio a a e wa e I is case a e i a i es wi es ec o a y a is a we a e e wi a we -k ow o i ea Sc ö i ge equa io o e o owi g o m
is equa io is easy o so e i we kee o y e i s e m o e ig si e O e ca e i y a a y u c io A(z, a as a o m A(z, /νg co s i u es a o e so u io is mea s a i e owes o e a o ima io a 1- ime sio a wa e acke ca e o a y s a e a i o aga es wi ou is o io s wi e s ee equa o e g ou e oci y i a gi e me ium I may seem o e a i ia s a e- me w ic i is u a o e wi sig i ica co seque ces i ma y e e ime s We wi i us a e is wi wo e am es e i s e am e o e a oc a e g ou e oci y ca ay wi a e e ime is a wa e acke is o io y e ses I as ee ecog i e qui e a o g ime ago a e i e e ce e wee ase a g ou e oci ies ca ea o sig i ica e ec s u o o aga io o em oseco u ses oug e ses [1 -13] I a co ima e eam i e a a wa e acke is ocuse wi a c oma ic e s e e ase su aces e i e e s a e s e ica u e s a e o e wa e acke is o I e U - isi e a ge e g ou e oci y is sma e a e ase e oci y o o ica g asses ecause e wa e acke o aga es oug mo e g ass i e ce e o e eam w e e e e s is e ickes i accumu a es mo e g ou e ay a ase e ay e e w i e a e e ges o e e s e wo e ays a e a mos equa As a esu e wa e acke e i e e s is is o e ; i s ce a a ags e i e e ges is i u causes a e i e si y o ig i e ocus o o ge o ows e i e si y ime o i e o e i u u se ecause i e e a ia o es o e wa e acke co i u e o i a i e e imes I is im o a o e a e o measu e is is o io es ecia y o com e e ses 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]. We have designed an interferometric method to measure the wave packet distortion in lenses [15]. The experimental set-up is shown in Fig. 2. The main part of the system is a modified Michelson interferometer with one arm serving as a reference arm and the other arm including the tested lens L. A collimated beam from a femtosecond Ti:sapphire laser was used as an input beam. Its diameter was big enough to fill the aperture of the lens. As indicated in the figure, at the output of the interferometer a flat reference wave packet interferes with a wave packet distorted by a double passage through the lens. The interference fringes are visible only in the regions where the two wave packets overlap for a given delay defined by the position of the mirror M1. As this mirror is scanned the fringes appear and disappear in different radial zones of the output beam. A detector placed behind a smahl pinhole was used to measure the fringe visibility at different distances from the beam center. For any given position of the pinhole we measured the delay corresponding to maximum fringe visibility and thus found the shape (we assume cylindrical symmetry) of the distorted wave packet. An example of the results is presented in Fig. 3. It shows the relative group delay as a function of the radial position on a 40x microscope objective. As one can see the parts of the wave packet propagating close to the edges of the lens are advanced by more than 30 fs with respect to the center of the wave packet. We found it quite amusing a a a i io o a co e g ass (17 m ick e wee e e s a e cu e mirror M2 significantly improved the performance of the objective. Apparently the objective has been designed to work with the cover glass! Asa second example, consider a frequency mixing experiment with femtosec- ond laser pulses. A crystal with χ( nonlinearity is illuminated by two collinear u a s o ase u ses wi eque cie ω 1 a ω and a sum frequency at ω3 = ω1 + ω is ge e a e Si ce e ee eque cies i o e i e ocess are quite different, the group velocities of the corresponding pulses are differ- 242 C. dz z et Ι
e too. ThiS has a detrimental effect on the sum frequency generation process. If ω 1 ≠ ω e g(ω1) ≠ 1 g ('2) in a dispersive medium. As a result the two input pulses propagate with different speeds. This limits the range over which the input pulses overlap which in turn limits the efficiency of the process. In addition, the ou u u se a ω3 o aga es a ye a o e (a usua y e y i e e s ee As a esu e ou u u se is o ge a e i u u ses ecause co i u io s om i e e s ices o e c ys a a i e a e ou u ace a i e e imes e a e e ec emai s e e i e case o seco a mo ic ge e a io (S G w e a si g e em oseco u se is use o o uce a o e u se a wice e i u eque cy e o em o g ou e oci y misma c cou e sig i ica y a e i- a e i a eque cy mi i g sc eme w ic e su es o ase ma c i g ( M a g ou e oci y ma c i g (G M cou e ou We ou a is ca e ac u- a y ac ie e i some cases w e y e I o -co i ea sum eque cy ge e a io sc eme is a ie [1 17] Figure 4 illustrates the basic idea. In the UV-visible range the group ve- locity in nonlinear crystals decreases with increasing frequency and thus the sum eque cy u se a ω3 ags e i e i i g u ses a ω 1 a ω e e o e wi a suitable choice of the angles Ψ1 and ψ the projections of vgl, Vg2 and ßg3 on k3 can be made equal. This means that the three pulses have the same components of the group velocity along the propagation direction and they do not separate as they propagate. It is not obvious that group velocity matching and phase match- ing can be achieved in a given crystal for given wavelengths of the input pulses. Whether this is possible or not depends on dispersion properties of the particu- lar nonlinear birefringent crystal selected. Figure 5 shows the results of numerical ca cu a io s o β- a ium o a e ( O c ys a i e a ge o u ame a wa e- e g s co es o i g o e u a i i y a ge o i sa i e em oseco osci a- o wo ocesses a e ee co si e e o -co i ea y e I S G (ω + ω → ω and non-collinear type I third harmonic generation (THG) (ω + ω → 3ω e two cases are quite different. For SHG the problem involves solving two equations (phase matching condition and group velocity matching condition) for two vari- a es ψ = Ψ1 = Ψ and O (Θ is an angle between k3 and the optic axis of the crystal) and the solutions are exact if they exist. In the case of THG there are Femtosecond Light Wave Packets 243
3 a ia es Ψ ι , ψ a θ a o y wo equa io s ase ma c i g co i io as to be fuhfilled but then group velocity matching is not exact. However, the results of numerical cahculations indicate that the residual group velocity mismatch in our scheme can be small as one percent of that for a standard collinear THG case. As one can see in Fig. 5 PM and GVM conditions can be achieved for SHG in BBO over an entire tuning range of Ti:sapphire laser. Similarly exact PM and approximate GVM are possible for THG process in a somewhat smaller range of wavelengths. C Radzewicz d ad.
I o e o see e e ec o G M o eque cy mi i g wi em oseco u ses we a e ume ica y i eg a e e o i ea equa io s o e am i u es o e ee ie s
w e e Α is e am i u e ωi — e eque cy ni — e e ac io i e o e i- ie a β i = ∂ ki/∂ω Α e e e ce ame mo i g wi e g ou e oci y o e ω 1 u se is use a βιi = 1/ gi — 1/ g 1 o i = 3 e esu s o e i eg a io a e s ow i ig O e ca c ea y see a o easo a e e e ime a co i io s e e ec o g ou e oci y misma c is o eg igi e Seco a mo ic u ses i a G M o -co i ea sc eme ca e sig i ica y s o e a ose ac ie a e i a s a a co i ea sc eme is as ee e i ie i a e e ime e o me wi a o ima e y 3 o g u ses om a i sa i e ase a a 5 mm o g O c ys a e seco a mo ic as ee ge e a e usi g wo i e e a oac es a s a a co i ea Femtosecond Light Wave Packets 5
S G a a o -co i ea G M S G I eac case e u a io o e seco a mo ic u se as ee measu e y i e si y c oss-co e a io wi e i u ase u se esu s o e c oss-co e a io measu eme s a e s ow i ig 7 o e s a a co i ea sc eme e ou u u se u a io is a mos wice o a o e ase u se A e same ime e esu s s ow a o e G M sc eme seco a mo ic u ses a a e s o e a e i u ase u ses ca e ac ie e is is o su isi g si ce i a i ea case o S G wi ou sa u a io o e s ou e ec
e seco a mo ic u se o e s o e a e i u u se y a ac o o √
As a i a e am e o i ea o aga io e ec s e us co si e a em oseco wa e acke o aga i g i a i e i ge is e si e me ium I e ig o a- ga es as a e ao i a y wa e e e e ms γ a γ i Eq ( a e o - e o W i e e i s o e mea s a e oy i g ec o i suc a me ium is o a a e o e wa e ec o (a ac k ow o a eas a ce u y e e is e ce a mea i g o e seco o e a e ee ou a e ai e o y ece y [ 9] ume ica i eg a io o Eq ( s ows a ecause o is e m e wa e acke w ic is ig y ocuse w e i e e s a u ia ia i e i ge c ys a o a es a ou e a is a is e e icu a o e a e e i e y o ic a is o e c ys a a e k ec o e e ec ca e e ai e i a way e y simi a o a em oye o e ai e wa e acke is o io y e ses ecause o e ig ocusi g e e - 246 C. Radzewicz et al.
a i g eam co ai s wa e ec o s a o m i e e a g es wi e o ic a is o e c ys a Si ce o a e ao i a y wa e e i e o e ac io e e s o is a g e o ase a g ou e oci ies a e a so u c io s o e same a g e W a is e e mo e im o a e i e e ce e wee ase e oci y a g ou e- oci y a ies as a u c io o is a g e As a esu o e si e o e wa e acke ags mo e e i e ase o s a e o e si e w ic ea s o e wa e acke o a io I suc a u se is eco ima e a e i e i s e c ys a i wi ook skewe — w i e e ase o s wi e e e icu a o e i ec io o o aga io e wa e acke i se wi e o sim y ecause e i e e ce e wee ase a g ou e ays wi e i e e a i s o osi e si es e eo e ica e ic io s o e wa e acke o a io a e ee e i ie e e ime a y i a se -u [1 ] simi a o e o e s ow i ig 2. e se -u as ee i o i ie o i c u e wo i e ica 1 1 e esco es o e i eac a m wo 1 mm ick u i e c ys a s a e ee ace o e a e ocus o eac e esco e e wa e acke s o aga i g i e c ys a s e e ie ce o a io as esc i e a o e We a e ace e c ys a s i o a ms o ca ce a e is o io s o e wa e acke s a a e o ue o e o a io i e c ys a owe e e c ys a s we e se i suc a way a e wo wa e acke s e e ie ce o a io i o osi e i ec io s a us i o o e a e ec y i s ace y eco i g e i ge isi i i y a i e e osi io s i e ou u eam e sus e e ay e wee wo a ms o e i e e ome e we we e a e o measu e e wa e acke o a io e esu s o e e e ime ( oi s a e com a e o e eo e ica y ca cu a e wa e acke o a io ( i e in ig I is c ea a ou mo e o wa e acke o a io is a eas a equa e as i ica e y e ag eeme o i s e ic io s wi e e e ime a esu s Femtosecond Light Wave Packets 7
o com e e ou a a ysis o e i ea o aga io we s ou ie y me io e e ec s o e g ou e oci y is e sio W e e g ou e oci y is e sio e m —(i/ β ∂ Α/∂ is a so i c u e i o e a a ysis o Eq (5 e u se s a e oes o emai co s a u i g o aga io I s ea ecause i e e ou ie com o e s o e u se e e ie ce i e e g ou e ays e esu is a u se wi a yi g s a e a ime e e e eque cy I is a o ima io e ig eque cy a ies i ea y wi ime a us e u se is sai o a e a i ea c i e e ec ca e qui e se e e o i s a ce a 5 s o g ou ie imi e u se a 3 m ou es i s u a io u o o aga io oug 1 mm o use si ica e es a i g o em oseco u ses ue o o aga io i is e si e me ia mea s a i a y gi e e e ime we ca o ake e u se u a io o g a e E e i we e i y a e u se is s o w e i e i s e ase sys em i ca e qui e o ge w e i i e ac s wi ou sam e sim y ecause e e was some g ass o e way om e ase o e i e ac io egio A a i io o ig e o e is e sio e ms ea s o e e mo e sig i ica u se es a i g a a o i ea c i
o i ea o aga io
e us s a agai wi 1- ime sio a case e o i ea e m iγ I Α Α i Eq (5 esc i es e e ec o i e si y e e e i e o e ac io (A = + γι Α is i o uces a ime e e e ase a o i ea c i wi ou a ec i g e u se s a e As a esu ew eque cies a e ge e a e a e u se s ec um oa e e S ec um oa e i g i Ke me ia ca e com i e wi a g ou e oci y is e sio o o uce u ses a a e sig i ica y s o e a e o igi a ase u ses i a ec ique ca e u se com essio [19] W e o g ou e oci y is e sio a Ke o i ea i y e ms a e i c u e e e esu e e s o e sig s o β a coe icie s I e wo e mS a e e same sig e c i ue o is e sio a e o e ue o Ke o i ea i y a e e same sig oo As a esu e u se e e ie ces es a i g a a com i e c i O e o e a i sg (β = -sg (γ e e e e is s a e so u io s o e o aga io equa io [5] Suc so u io s a e ca e o ica so i o s ecause ey ca o aga e o e o g is a ces i o ica i e s wi ou c a ges i s a e ey a e o a icu a i e es o e gi ee s esig i g o ica commu ica io sys ems I o e o e e e ime s i is ie icoseco u ses a e ee o aga e i a i e o e g 1 1 11 m i e u e a om e Ea o e Su [ ] I a 3- case e i e ay e wee is e sio a o i ea i y is augme e y a a i io a e ec ame y a com e i io e wee i ac io a se - ocusi g W i e e i s e s o i c ease e a s e se ime sio s o e wa e acke e a e oes e ac y e o osi e I as ee k ow o a eas 3 yea s [ 1] a o o g u ses e u ima e a e o a eam o aga i g i a Ke me ium is e e mi e y i s owe Ρ o P < Pc (Ρc is ca e c i ica owe a e e s o e me ium o e ies o y i ac io wi s a e eam e ocuses I owe e P > Pc se - ocusi g e ai s a ca as o ic eam co a se is o se e is is o ue o em oseco u ses As as ee oi e ou y se e a g ou s [ - 5] i is case o e as o ake i o accou is e sio e ec s as we ecause o a e wa e acke wi a owe ig e a e c i ica owe oes o co a se; C dz z t l i s ea i u e goes a com e e o u io o i ime a s ace I o e o i us a e is e o u io e us u ack o Eq ( o a wa e acke a is Gaussia o i ime a a s e se s a ia is i u io o aga i g i a iso o ic me ium i is co e ie o ew i e Eq ( i a s ig y i e e o m [11]
w e e ee ew a ame e s a o e g ime sio a e ee i o uce is- e sio e g s = τ /β i ac io e g = γ w / = πw /λ a o i ea e g = (γ Α -1 wi τ w a Α ei g e u a io a s- e se si e a am i u e o e wa e acke es ec i e y e a a age o suc sca i g is a i o i es ee a ame e s a ca e easi y com a e o eac o e w ic i u e a es o e o e a ua e e s e g o e ee e ome a ey e ese ( e sma e e coe icie e mo e im o a e co es o i g e m i Eq (7 e mea i g o ese a ame e s is as o ows Ld is a is a ce em oseco ig Wa e acke s 9
w ic akes e u se o ou e i s u a io ecause o e me ium is e sio Ldf is e is a ce equi e o e u se o ou e i s a ea as a esu o i ac- io a is e se - ocusi g is a ce o a s a e me ia e i equa i y
Lds » Ln! Ldf y ica y o s I mea s a e is e sio e ec s co s i u e o y a sma co ec io o e asic wa e acke e o u io e i e y i ac io a se - ocusi g owe e e esu s o i eg a io o Eq (7 s ow a sma as i is e is e sio ays a c ucia o e i o aga io o a em oseco wa e acke A e am e o e wa e acke e o u io o ai e y i ec i eg a io o Eq (3 is s ow i ig 9 ( e esu s we e e y simi a w e Eq (7 wi 3- o e e ms i c u e was i eg a e I is a icu a case i was assume a e wa e acke si e i ey i ec io is muc a ge a i s si e i e i ec io is o o y ma e e com u a io sim e u a so e a e us o ese e esu s as a 3- g a s e o owi g a ame e s a e ee assume ma e ia — use si ica ( s = mm u se u a io — s u se wa e e g — m eam si e — 3 m ( = mm u se i e si y — 7 GW/c ( = mm e owe o e eam is a ou A o g u se o is owe wou se - ocus a co a se owe e e em oseco wa e acke co si e e e e is ays a qui e i e e e a io I s a s o se - ocus as i ica e y e e o ga e s a e i ig 9a u e e ocess is a es e a e u se s i s i o wo u ses as s ow i igs 9c a I is i s uc i e o ook a e s a io- em o a s ec um o e wa e acke s ow i ig 1 S a i g om a smoo Guassia s a e e 5 C Radzewicz et al. s ec um e o es oug a "Me ica a " s uc u e ( ig 1 c a e e e o s ee s a ia o es e ce a o e (a ou k = co ai s wo eaks i e eque cy is i u io o e a e eque cies owe a e i u u se eque cy a a o e w ic is s i e owa s ig e eque cies e wo si e o es a e oa s ec a ce e e a e i u u se eque cy e esu s e e as o e mig e ec o a o i ea o em o a a icu a c oice o a ame e s o e am e e sig a mag i u e o e β coemcie u om e ume ica i eg a io esu s some ge e a co c usio s a ou e o e o e is e sio ca e aw o e case i us a e i igs 9 a 1 e ocess s a s wi e se - ocusi g e se - ocusi g is e s o ges a c = ( ig 9a sim y ecause e i e si y is e ig es e e us e i e si y a e ce e o e wa e acke i c eases a i y a so oes e wi o s ec um ue o se - ase mo - u a io O ce e s ec um is oa e oug e is e sio s i s owe eque cies owa s e ea o e u se a ig e eque cies owa s i s ai is owe s e i e si y a e ce e o e wa e acke a e e s i om a co a se A e same ime e a o e wa e acke a as ee ocuse s o g y i ac s o o m e si e o es i e s ec um as iscusse a o e e u se s i i g s ow i ig 9 as ee ece y o se e i a e e ime [ ] I co c usio we a e a a y e some as ec s o i ea a o i ea o - aga io o em oseco ig wa e acke s i a s a e is e si e o i ea me ia I a icu a we a e s ow e e ec s o g ou e oci y o o aga io o suc wa e acke s oug e ses wa e acke o a io i i e i ge me ia a sum eque cy ge e a io wi ou g ou e oci y misma c ume ica esu s s owi g wa e acke s i i g we e a so ese e
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