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dichroism is a well known concept in classical optics where it denotes the property shown by certain materials of having absorption coefficients which depend on the state of polarization of the incident light @xcite . this concept has been further extended to the case of atomic or molecular interactions with a radiation field . in particular , the notion of circular dichroism in angular distribution ( cdad ) refers to the difference between the differential cross sections ( dcs ) of laser assisted signals for _ left _ ( @xmath0 ) and _ right _ ( @xmath1 ) circularly polarized ( @xmath2 ) light @xcite .    here we investigate the effect of the photon state of polarization , _ i.e. _ of its helicity , in laser - assisted high energy electron - hydrogen scattering . we show under what conditions cdad is observable at high scattering energies as a result of _ target dressing _ by the laser field . we consider optical frequencies and moderate field intensities and apply a hybrid calculational approach @xcite . the interaction between the projectile and the field is treated exactly , while the interaction between the atom and the field is treated in perturbation theory . first order born approximation is used to evaluate the scattering amplitude . we demonstrate that cdad is encountered , provided i ) the electromagnetic field is a superposition of two laser beams , one of which is linearly polarized ( @xmath3 ) and the other is a @xmath2 field , ii ) second order dressing of the target by the electromagnetic field is included . in addition , iii ) the role of the virtual transitions to the continuum is shown to be essential for the observation of cdad . finally , we demonstrate that for a special configuration not only cdad but also cd for the integrated cross sections can be observed . atomic units are used . we consider electron - hydrogen scattering in the presence of an electromagnetic field that is a superposition of two laser beams . one beam is @xmath3 , with polarization vector @xmath4 , while the other is @xmath2 with polarization vector @xmath5 . the beams can have different directions of propagation . for simplicity , we discuss the case where the two beams have the same frequency @xmath6 and intensity @xmath7 . in dipole approximation the resulting field is @xmath8 where the intensity @xmath9 . we want to know whether the dcs are sensitive to the helicity of the @xmath2 photons , defined by @xmath10 which explicitly depends on the direction @xmath11 of propagation of the @xmath12 beam . as shown in @xcite , @xcite and @xcite , for high energies of the projectiles cdad does not occur for a @xmath2 laser field _ alone , _ since the first order born approximation leads to real scattering amplitudes . we therefore present the theory for the above superposition of fields .    according to @xcite , at moderate laser field intensities the field - atom interaction can be described by time - dependent perturbation theory ( tdpt ) . we consider _ second order _ _ dressing _ of the hydrogen ground state by the field ( [ field ] ) . the approximate solution for an atomic electron in an electromagnetic field reads @xmath13 ,   \label{fun}\ ] ] where @xmath14 is the unperturbed ground state of hydrogen , of energy @xmath15 . @xmath16 denote first and second order corrections , respectively . on account of @xcite and @xcite these corrections can be expressed in terms of @xmath17 and @xmath18 where @xmath19 is the coulomb green s function and @xmath20 the momentum operator of the bound electron . for the field ( [ field ] ) there are five values of the argument of the green s functions necessary in order to write down the approximate solution ( [ fun ] ) , namely @xmath21 , @xmath22 , @xmath23 .    a projectile of kinetic energy @xmath24 and momentum @xmath25 , moving in the field ( [ field ] ) , is described by the volkov solution @xmath26 @xmath27 represents the classical oscillation of the electron in the field @xmath28 , its amplitude is @xmath29 . using graf s addition theorem @xcite , the fourier expansion of ( [ vol ] ) yields a series in terms of ordinary bessel functions @xmath30 @xmath31 according to the definitions of the arguments and phases given in watson s book @xcite , we have @xmath32 and @xmath33 @xmath34 and @xmath35 refer to the @xmath2 field alone , while @xmath36 and @xmath37 are related to the superposition ( [ field ] ) . using ( [ exp1 ] ) , we recognize that a change of helicity of the @xmath2 photons , _ i.e. _ @xmath38 , leads to a change in sign of the dynamical phases @xmath35 and @xmath37 . therefore , looking for the signature of helicity in the angular distributions of laser - assisted signals , we have to observe the presence of these dynamical phases in their dcs .    for high scattering energies , the first order born approximation in terms of the interaction potential is reliable . neglecting exchange effects , this potential is @xmath39 , and the @xmath40matrix element reads @xmath41 where @xmath42 and @xmath43 are given by ( [ fun ] ) and ( [ vol ] ) . @xmath44 are the initial(final ) electron momenta . the dcs for a process in which @xmath45 photons are involved is @xmath46 the scattered electrons have the final energy @xmath47 where @xmath45 is the net number of photons exchanged between the colliding system and the field ( [ field ] ) . @xmath48 refers to absorption , @xmath49 to emission and @xmath50 describes the elastic process . the nonlinear transition matrix elements @xmath51 in ( [ sed ] ) have the general structure @xmath52 . \label{tm - cp}\ ] ] @xmath53 is the dynamical phase in ( [ exp1 ] ) evaluated for the momentum transfer @xmath54 . the first term in ( [ tm - cp ] ) , @xmath55 yields the bunkin - fedorov formula @xcite ( target dressing is neglected ) . here @xmath56 and the bessel function @xmath57 contains all the field intensity dependences of the transition matrix element . @xmath58 is the amplitude of elastic scattering in the first order born approximation , @xmath59 . the remaining terms in ( [ tm - cp ] ) describe the dressing of the atom by the field ( [ field ] ) , they were discussed in detail in @xcite . in the case of @xmath60 _ one _ of the @xmath45 photons exchanged between the colliding system and the field is interacting with the bound electron , while in @xmath61 _ two _ of the @xmath45 photons interact with the atomic electron . these dressing terms in ( [ tm - cp ] ) are @xmath62{\cal j}_{1,0,1}(\tau ^{+},\tau ^{-},q )   \label{t1}\end{aligned}\ ] ] and @xmath63 \\ + j_{n+2}({\cal z}_q)[{|\vec{q}\cdot \vec{e}+\vec{q}\cdot \vec{\varepsilon}|^2% } { q^{-2}}{\cal t}_1+(1 + 2\vec{e}\cdot { \vec{\varepsilon}}^{\;*})e^{2i\psi _ q}% { \cal t}_2 ] \\ + j_n({\cal z}_q)[{|\vec{q}\cdot \vec{e}+\vec{q}\cdot \vec{\varepsilon}|^2}{% q^{-2}}\widetilde{{\cal t}}_1 + 2(1+{\rm re\;}\vec{\varepsilon}^{\;*}\cdot \vec{e})\widetilde{{\cal t}}_2]\}. \end{array } \label{t2}\ ] ] the five radial integrals , @xmath64 , @xmath65 , @xmath66 , @xmath67 and @xmath68 in ( [ t1])-([t2 ] ) , depend on @xmath69 and on the parameters of the coulomb green s functions . the integral @xmath64 is a function of the two parameters @xmath70 through @xmath71 ( see @xcite ) , while @xmath65 , @xmath66 , @xmath72 and @xmath68 depend on four parameters @xcite . @xmath73 and @xmath66 are multiplied by @xmath74 if both photons are absorbed / emitted by the atomic electron . in the last line of ( [ t2 ] ) , @xmath67 and @xmath68 are multiplied by @xmath30 . here two photons interact with the atomic electron , but one is emitted and the other is absorbed . for our numerical calculations we used the analytic expressions for the above five radial integrals presented in @xcite and @xcite-@xcite . equivalent expressions were published for the case of single photon transitions in @xcite-@xcite and for two photon absorption / emission in @xcite . the transition matrix elements for first and second order dressing in ( [ t1])-([t2 ] ) are written in a form that permits to analyze their dependence on the dynamical phase @xmath53 . we see that @xmath75 and @xmath76 do not depend on the helicity ( [ hel ] ) . on the other hand , @xmath77 has an explicit dependence on @xmath78 determined by the phase factors that multiply @xmath66 in ( [ t2 ] ) . due to the structure of @xmath77 , it is evident that in the absence of the @xmath3 component of the field ( [ field ] ) the dynamical phase @xmath53 is absent and hence there is no cdad . indeed , both polarization terms multiplying @xmath66 would be zero since @xmath79 . this demonstrates the essential role of the @xmath3 beam and the necessity to include second order dressing of the target . in order to stress the important role of the virtual transitions to the continuum , we shall consider small scattering angles . here the dressing of the target is considerable and the cdad effect can be large . for small arguments of the bessel functions , _ i.e_. either for weak fields at any scattering angle or for moderate fields at small scattering angles , we can keep the leading terms in ( [ tm - cp ] ) only . we discuss in some detail the case @xmath80 . the corresponding matrix element is @xmath81 ,   \label{abs}\ ] ] where the amplitudes @xmath82 and @xmath83 depend on @xmath84 and on @xmath6 @xmath85 , \label{a } \\ { \cal b}\left ( q;\omega \right ) & = & { \omega ^2}{\cal t}_2 . \label{b}\end{aligned}\ ] ]    the dcs derived from ( [ abs ] ) are @xmath86\}.   \label{fin}\end{aligned}\ ] ] they depend on the change of helicity only if @xmath87 and @xmath88 . as shown in @xcite and @xcite , this is true if virtual transitions to continuum states are energetically allowed , _ i.e_. if @xmath89 or @xmath90 . cdad , defined as the difference between the dcs for @xmath91 and @xmath92 , follows from ( [ fin ] ) as @xmath93 where @xmath94 and thus @xmath95 gives the angular depence of @xmath96 . we easily verify that the three conditions i)-iii ) , quoted in the introduction , are all necessary to have @xmath97 , namely ( i ) @xmath98 if the @xmath3 field is absent , ( ii ) @xmath99 stems from second order target dressing , and ( iii ) @xmath82 and @xmath99 become real as soon as @xmath100 . next we study the angular dependence of @xmath101 . two cases are of major interest :    @xmath102 if @xmath103 , then the superposition of @xmath3 and @xmath104 is equivalent to elliptic polarization ( @xmath105 ) and we write @xmath106 . hence @xmath107 where @xmath108 are the projections of @xmath109 on the axes @xmath110 of the @xmath2 vector @xmath111 . here @xmath112 leads to cdad but integrating it over @xmath113 in the azimuthal plane yields zero . expression ( [ edad ] ) is comparable to that of elliptic dichroism in photoionization @xcite . @xmath114 if @xmath115 , then cdad reads @xmath116 { \rm % im}\left ( { \cal a}^{\;*}{\cal b}\right ) . \label{cd}\ ] ] taking in addition @xmath117 , then the term proportional to @xmath118 gets @xmath113-independent and its contribution survives in the azimuthally integrated cross sections . thus , the choice of a privileged direction in the problem , namely that of @xmath119 , introduces an additional asymmetry . therefore , the signature of the photon helicity prevails not only in the angular distribution but also in the integrated cross sections . to the best of our knowledge , this is the first case of a laser - assisted process in which cd in the integrated cross sections is encountered . finally , we stress the importance of the form of our @xmath120-matrix element ( [ abs ] ) , since a more general structure @xmath121 was also found in discussions of dichroism in other processes , like two - photon ionization @xcite-@xcite , two - photon detachement of h@xmath122 @xcite or elastic x - ray scattering by ground state atoms @xcite . of course , the meaning of the vectors @xmath123 is specific to each process . in all these cases dichroism is caused by interferences between the real and imaginary parts of the amplitude and two terms with different angular behavior are needed to get such interferences . in the examples above , @xmath124 and @xmath45 were complex quantities . our conditions , @xmath125 or @xmath90 , serve the same purpose . contrary to our problem , in photoionization or photodetachment of unpolarized systems there is no equivalent for our privileged direction @xmath126 and hence only cdad is observable . we present numerical results for cdad and cd in laser - assisted electron - hydrogen scattering at high energies . we analyze the dcs for @xmath127 since here the cdad effects are large enough . using the above formalism , we evaluated the dcs ( [ sed ] ) in the azimuthal plane as a function of @xmath128 for a fixed scattering angle @xmath129 and for the initial scattering energy @xmath130 ev . our laser frequency was @xmath131 ev , taken close to an atomic resonance in order to enhance the cd effects , and we chose the moderate field intensity @xmath132 w@xmath133 . the initial electron momentum @xmath134 was taken parallel to the @xmath3 vector @xmath135 , both pointing along the @xmath136axis and the @xmath3 beam propagated in the @xmath137-plane .    in case @xmath102 the @xmath2 beam propagated in the @xmath138direction and the corresponding @xmath2 vector @xmath139 has helicity @xmath140 known as @xmath141 while @xmath142 has opposite helicity @xmath143 , representing @xmath92 . in this configuration we have @xmath144 .    in figure 1(a ) we present for @xmath80 with @xmath145 the @xmath146dependence of the dcs at @xmath129 . the data for @xmath91 are shown by a dotted line and for @xmath92 by a dashed line . clearly , the laser - assisted signals depend on the helicity of the photon . in figure 1(b ) we present the results for cdad ( [ edad ] ) . since here @xmath147 , the  @xmath148  and  @xmath149  in the two lobes indicate this dependence . if @xmath150 gets integrated over @xmath113 , the net cd effect is zero . similar results and conclusions are obtained for @xmath151 .    for case @xmath114 the @xmath2 beam propagated in the @xmath152direction and the @xmath2 vector is @xmath153 so that now @xmath154 the @xmath3 beam propagated as before . figure 2 shows the dcs ( [ sed ] ) and the cdad ( [ cd ] ) for @xmath80 in panel ( a ) and for @xmath151 in panel ( b ) . dotted lines are for @xmath91 and dashed ones for @xmath92 . full lines are used for @xmath101 ( [ cd ] ) where explicitly @xmath155 due to this angular dependence , the integration of  @xmath101 over @xmath128 does not vanish . since the final momentum @xmath156 depends on @xmath45 , the shape of the azimuthal dependence of @xmath101 is different for absorption and emission . our data show that the maximum value of dichroism can amount up to @xmath157 of the assisted signal . this is comparable to or even larger than the effect predicted for x - ray scattering @xcite or for two - photon ionization @xcite . similar to the case of x - ray scattering , the dichroism in our case is increasing with increasing laser frequency . in free - free transitions at high scattering energy the dichroic effects stem from target dressing , which is increasing with the photon frequency . target dressing is significant for rather small scattering angles . we therefore expect that dichroism is large in this angular domain , and not near @xmath158 as for x - ray scattering and two - photon ionization . summarizing , we considered scattering of high energy electrons by hydrogen atoms in the presence of a laser field of moderate power but higher frequencies . the field had two components of equal frequency and intensity . one of the components was circularly , the other linearly polarized . the two laser beams were permitted to propagate in different directions . in the first order born approximation we showed that in the above scattering configuration cdad becomes observable in two - photon transitions if laser - dressing of the atomic target is carried out in second order tdpt and transitions between atomic bound and continuum states are energetically allowed , requiring higher laser frequencies . since the scattering probabilities decrease with increasing @xmath6 , we prefered to choose @xmath159 close to an atomic resonance to enhence the signals . thanks to the above laser configuration and second order target dressing , the @xmath120-matrix elements become complex as a prerequisite for predicting cdad  in the born approximation , since the elastic born - amplitude is real . we conclude that at high projectile energies cdad in free - free transitions is a second order field - assisted effect occurring under special conditions only . in particular , the matrix elements of the process considered have to be complex . similar effects might occur , if higher order terms of the born series are taken into account , since we know that then the scattering matrix elements become complex . finally , we stress the role of the two asymmetries which were introduced into the scattering configuration in order to obtain helicity dependent nonlinear signals . one asymmetry came in by the @xmath3 laser . this was sufficient to achieve cdad . a second asymmetry was determined by the momentum @xmath134 of the ingoing electrons . the two asymmetries together , more precisely @xmath160 , then led to cd that even persists if the dcs are integrated over @xmath113 . in photoionization and photodetachement there is no equivalent to @xmath161 . hence , in those processes only cdad is encountered . the analysis of the structure of the two photon transition matrix element led us to a general formula for cdad and to our undestanding of the physical reasons of the configuration necessary to obtain cdad and cd , respectively . 1 : refers to case @xmath102 , _ i.e. _ @xmath162 . we present for @xmath80 the dcs as function of the angle @xmath113 at the scattering angle @xmath129 . the initial electron energy is @xmath130 ev , the radiation frequency is @xmath163 ev and its intensity @xmath132 w@xmath133 . the panel ( a ) shows the data for _ lcp _ as dotted line and the data for _ rcp _ as dashed line . clearly , the laser - assisted signals depend on the helicity of the photons . in panel ( b ) the cdad effect is visible but integration over @xmath128 yields zero.fig . 2 : treats case @xmath114 , _ i.e. _ latexmath:[$\vec{k}_i||\stackrel{\rightarrow } { e}%    fig . 1 , we show the @xmath113-dependence of the dcs in panel ( a ) for @xmath80 and ( b ) for @xmath151 . signals for _ lcp _ are dotted lines and signals for _ rcp _ are dashed lines . the cdad effects are represented by full lines . integrating these data over @xmath165 a non - vanishing cd effect remains . the dependence of the effects on photon emission / absorbtion is apparent .
we consider high energy electron scattering by hydrogen atoms in the presence of a laser field of moderate power and higher frequencies . 
 if the field is a superposition of a linearly and a circularly polarized laser beam in a particular configuration , then we can show that circular dichroism in two photon transitions can be observed not only for the differential but also for the integrated cross sections , provided the laser - dressing of the atomic target is treated in second order perturbation theory and the coupling between hydrogenic bound and continuum states is involved . 
 pacs numbers : 34.80.qb ; 34.50.rk ; 32.80.wr    this is the revtex shell .