Conservation laws for Electron Vortices in Strong-Field Ionisation
Yuxin Kang, Emilio Pisanty, Marcelo Ciappina, Maciej Lewenstein, Carla Figueira de Morisson Faria, Andrew S Maxwell
CConservation laws for Electron Vortices in Strong-Field Ionisation
Yuxin Kang , Emilio Pisanty , , Marcelo Ciappina , , , Maciej Lewenstein , , Carla Figueira de MorissonFaria , and Andrew S Maxwell , Department of Physics & Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße 2A, Berlin 12489, Germany Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Physics Program, Guangdong Technion – Israel Institute of Technology, Shantou, Guangdong 515063, China Technion – Israel Institute of Technology, Haifa, 32000, Israel ICREA, Passeig de Llu´ıs Companys, 23, 08010 Barcelona, SpainReceived: date / Revised version: date
Abstract.
We investigate twisted electrons with a well defined orbital angular momentum, which have beenionised via a strong laser field. By formulating a new variant of the well-known strong field approximation,we are able to derive conservation laws for the angular momenta of twisted electrons in the cases oflinear and circularly polarised fields. In the case of linear fields, we demonstrate that the orbital angularmomentum of the twisted electron is determined by the magnetic quantum number of the initial boundstate. The condition for the circular field can be related to the famous ATI peaks, and provides a newinterpretation for this fundamental feature of photoelectron spectra. We find the length of the circularpulse to be a vital factor in this selection rule and, employing an effective frequency, we show that thephotoelectron OAM emission spectra is sensitive to the parity of the number of laser cycles. This workprovides the basic theoretical framework with which to understand the OAM of a photoelectron undergoingstrong field ionisation.
Since being first recognised in the classical context of tides[1–3] vortex phenomena have held an iconic status acrossa diverse range of disciplines [4–7]. Of particular interestare the quantum mechanical versions of vortices, whichcan been found in a wide range of systems [8–12]. Currentintense interest in phase vortices follows the first exper-imental observations of this phenomenon for photons [8]and unbound electrons [10] (for reviews on vortices in elec-trons see [13–15]). The unique topological properties ofvortex states [15] render them fundamental in the studyof structured wave fields, and have led to the inception ofwhole research areas, such as singular optics [16–19]. Forinstance, a vortex state cannot transform to another bysimple deformation such as stretching and compressing, orthe addition of noise. Furthermore, at its center along thepropagation axis, a vortex beam has a zero amplitude andan ill-defined phase. Currents around singularities implythat such states carry intrinsic orbital angular momenta(OAM), and thus can be used to influence the dynamicalproperties of physical systems [20, 21].The study of vortices in attosecond physics has greatpotential in controlling light and matter. In high-harmonicgeneration (HHG) it has been shown that optical vorticesin the IR driving fields lead to optical vortices in the re-sulting UV light produced [22–36]. This has been exploited to allow a high degree of control over the light. In oneexample, UV light has been produced exhibiting torusknot topology [34, 35] by sophisticated trefoil IR pulses,for which the orientation of the trefoil varies with the az-imuthal angle. In another study [36], extreme UV (EUV)light was imparted with time-varying OAM, leading toself-torque by employing two time-delayed IR pulses withdifferent OAM. This was the first demonstration of self-torque in light [37] and could aid in probing systems withnaturally time-varying OAM.Work understanding the OAM of photoelectrons emit-ted in attosecond processes is still in its infancy. Initialstudies include the exploration of high OAM values forquasi-relativistic field intensities [38] and terahertz fields[39] as well as using the OAM in rescattering to probebound state structures [40]. However, ideas as advancedas knots or self-torque in electron vortices have not beenexplored. This can partly be attributed to the difficultyin experimental implementation; for example, no mea-surement scheme has been devised to detect the OAMof photoelectrons emitted in strong-field experiments. Ini-tial ideas on how to achieve this have been suggestedin our recent publication [41], using both interferometricschemes and adapting existing methods [42] used in elec-tron beams. Such experimental development needs suffi-cient theoretical backing to guide the implementation andjustify the cost, which at present is lacking. In this study a r X i v : . [ phy s i c s . a t o m - ph ] F e b Yuxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation we introduce a theoretical framework to derive analyticalconservation laws and understand the basic dynamics ofthe OAM during strong field ionisation.We utilise the strong field approximation (SFA) [43]—often considered the workhorse of strong field physics—todescribe the basic ionisation dynamics. The SFA in theform used here [41, 44, 45] is an approximation, whichprimarily neglects the effects of the Coulomb potentialin the continuum. The use of the SFA, over a more ac-curate model, is justified by the possibility of analyticalsolutions and the ease of interpreting the results. Thus,the SFA allows the basic laws of the OAM to be derivedfor strong field ionisation. Furthermore, we focus mostlyon circularly polarised light, where the Coulomb effectsare less significant, than those observed in linear polarisa-tion [46]. In a recent publication [41], we numerically com-puted OAM distributions using the SFA, QProp [47, 48]and the R -matrix with time dependence method [49–51]for circularly polarised fields and confirmed that the SFAwas able to qualitatively reproduce the key features foundin these two more accurate numerical methods. This workdiffered from the present in that it focused on the interfer-ence effects due to employing two time-delayed counter-rotating circularly polarised IR fields. Furthermore, theOAM computations were performed numerically withoutfull derivation of the analytical expressions and the SFAcalculations employed a monochromatic field, whereas inthis work we extend the model to include a sin envelope.The paper is structured as follows. In Sec. 2 key the-oretical results of the SFA (Sec. 2.1) and vortex states(Sec. 2.2) are given. Next, in Sec. 3 we incorporate theOAM into the SFA, starting with expressions for a gen-eral field (Sec. 3.1). Following this, we derive analyticalconditions for the cases of a linear field (Sec. 3.2) and amonochromatic circular field; in the latter we do this bothwithout (Sec. 3.3) and with (Sec. 3.4) the use of the sad-dle point approximation. In Sec. 4 numerical results arepresented for a circular sin laser field. Therein we derivean effective frequency to extend the analytic condition fora monochromatic field to work for the sin case. Finally,in Sec. 5 we present our conclusions. In this section we provide some key results from the SFAand for electron vortex states, necessary for understand-ing the OAM derivation. Throughout the article we useatomic units unless otherwise stated. Both cylindrical (ˆ e (cid:107) , ˆ e ⊥ , ˆ e φ )and Cartesian (ˆ e x , ˆ e y , ˆ e z ) coordinates will be used in thisarticle, they will always be aligned such that ˆ e (cid:107) = e z andany radial quantity will be given by e.g. r ⊥ = (cid:113) r x + r y .We will consider specific cases where the laser field po-larisation is parallel to ˆ e (cid:107) [linearly polarised] and in the xy -plane [circularly polarised]. Within the SFA S-matrix formalism [44,52] the transitionamplitude for direct strong field ionisation from the boundstate | ψ ( t (cid:48) ) (cid:105) to a final continuum state | ψ f ( t ) (cid:105) is given bythe following expression [44, 52] M f = − i lim t →∞ (cid:90) t −∞ dt (cid:48) (cid:104) ψ f ( t ) | U v ( t, t (cid:48) ) V | ψ ( t (cid:48) ) (cid:105) , (1)where the time evolution is approximated by the Volkovoperator U v ( t, t (cid:48) ) = (cid:90) d p e − i (cid:82) tt (cid:48) ( p + A ( τ ) ) dτ | p + A ( t ) (cid:105) (cid:104) p + A ( t (cid:48) ) | . (2)Combining Eqs. (2) and (1) and taking A ( t ) = 0 we getthe following for the transition amplitude M f = lim t →∞ (cid:90) d p (cid:48) e − iS ( p ,t ) (cid:104) ψ f | p (cid:48) (cid:105) M ( p ) (3)with M ( p ) = − i (cid:90) ∞−∞ dt (cid:48) e iS ( p ,t (cid:48) ) d ( p , t (cid:48) ) , (4)where d ( p , t (cid:48) ) = (cid:104) p + A ( t (cid:48) ) | V | ψ (cid:105) (5)and S ( p , t ) and S ( p , t (cid:48) ) are the upper and lower limit ofthe semi-classical action, respectively, both given by S ( p , t ) = I p t + 12 (cid:90) t −∞ dτ ( p + A ( τ )) . (6)A plane wave momentum state is commonly used for the fi-nal continuum state | ψ (cid:105) f → | ψ p (cid:105) , which leads to (cid:104) ψ f | p (cid:48) (cid:105) →(cid:104) p | p (cid:48) (cid:105) = δ ( p (cid:48) − p ) and therefore M f → M ( p ), whereEq. (4) gives the definition of M ( p ). However, in this work,a Bessel beam vortex state will be used as final continuumstate instead of a plane wave. This will enable the develop-ment of an analytical model for the OAM of the outgoingphotoelectron. In order to proceed we will introduce someproperties of the Bessel beam. Vortex states are topologically distinct both from planewaves and vortex states with a different orbital angularmomentum l [15]; this means two vortex states cannotbe transformed into one another via continuous deforma-tions. At its center along the propagation axis, the vortexhas a zero amplitude and undefined phase. In order to en-force the single-valued wave function, a quantized phasefactor e ilφ is needed, where φ is the azimuthal angle and l is a topological charge with integer value known as the uxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation 3 orbital angular momentum (OAM). The general form ofthe Bessel beam electron-vortex is [14] (cid:104) r | ψ l ( t ) (cid:105) = N l J l ( p ⊥ r ⊥ ) e ilφ e ip (cid:107) r (cid:107) e − iωt . (7)Here, N l is a normalisation factor and J l ( p ⊥ r ⊥ ) is theBessel function of the first kind. Ignoring the time depen-dence, the Fourier transform is (cid:104) p (cid:48) | ψ l (cid:105) = i − l e ilφ (cid:48) πp ⊥ δ ( p (cid:48)(cid:107) − p (cid:107) ) δ ( p (cid:48)⊥ − p ⊥ ) , (8)where ( p (cid:107) , p ⊥ , φ ) are the cylindrical coordinates of p . In-serting this into Eq. (3) gives M l ( p (cid:107) , p ⊥ , t ) = i l π (cid:90) π − π dφ (cid:48) e − iS ( p (cid:107) ,p ⊥ ,φ (cid:48) ,t ) e − ilφ (cid:48) M ( p (cid:107) , p ⊥ , φ (cid:48) ) , (9)where M ( p (cid:107) , p ⊥ , φ (cid:48) ) is the transition amplitude Eq. (4)written in cylindrical coordinates, which have been usedas they are natural for vortex states. Now the vortex stateis incorporated into the SFA framework. Next, we willcompute more explicit expressions for the transition am-plitude. In this section, the transition matrix element will be de-rived for a general laser field. We will examine the specificcases of a linear field and a circular monochromatic field,as well as the saddle point approximation.
In order to proceed we will collect φ dependent and in-dependent parts to analytically perform the integral inEq. (9). The upper limit of the action contains φ indepen-dent terms that contribute only a phase and thus may beneglected as well as the following φ dependent term p ⊥ (cos( φ ) α x ( t ) + cos( φ ) α y ( t )) , (10)where α ( t ) = (cid:82) t −∞ A ( τ ) dτ . As t → ∞ for a ‘well behaved’pulse or monochromatic field this term will vanish. Forthe case of sin pulses, which will be considered later, thisholds for an N -cycle pulse, where N is an integer greaterthan one. To simplify matters we will only consider such‘well behaved’ laser fields. Thus, the upper limit of theaction can be entirely neglected here.The lower limit of the action, which governs the dy-namics, can be split into two parts: S ( p (cid:107) , p ⊥ , φ, t (cid:48) ) = S A ( p (cid:107) , p ⊥ , t (cid:48) ) + S B ( p ⊥ , φ, t (cid:48) ) , where S A ( p (cid:107) , p ⊥ , t (cid:48) ) = (cid:18) I p + 12 p (cid:107) + 12 p ⊥ (cid:19) t (cid:48) + p (cid:107) α (cid:107) ( t (cid:48) ) + 12 (cid:90) dt (cid:48) A ( t (cid:48) ) (11) is independent of φ , and S B ( p ⊥ , φ, t (cid:48) ) = p ⊥ ( α x ( t (cid:48) ) cos( φ ) + α y ( t (cid:48) ) sin( φ ))= p ⊥ α ⊥ ( t (cid:48) ) sin( φ − ν ( t (cid:48) )) , (12)encodes the dependence on the momentum azimuth angle φ . This dependence is in terms of the angle ν ( t (cid:48) ) = µ ( t (cid:48) ) − π/ , (13)where µ ( t (cid:48) ) = arctan( α y ( t (cid:48) ) /α x ( t (cid:48) )) , (14)which is related to the rotation of the laser field and mag-nitude α ⊥ ( t (cid:48) ) = (cid:113) α x ( t (cid:48) ) + α y ( t (cid:48) ) (15)of the field integral α ( t (cid:48) ).Finally, in order to remove the φ dependence from thebound-state matrix element, d ( p , t (cid:48) ) = (cid:104) p + A ( t (cid:48) ) | V | ψ (cid:105) ,we decompose it into its Fourier series d ( p , t (cid:48) ) = (cid:88) m e imφ V m ( p (cid:107) , p ⊥ , t (cid:48) ) . (16)If the bound state is that of an atomic target, then the ma-trix element will have only a limited number of non-zerorelevant Fourier terms, and these will be easy to computenumerically applying the fast Fourier transform (FFT).The transition amplitude from Eq. (9) may be written as M l ( p (cid:107) , p ⊥ , t ) = − i (cid:88) m (cid:90) t −∞ dt (cid:48) e iS A ( p (cid:107) ,p ⊥ ,,t (cid:48) ) V m ( p (cid:107) , p ⊥ , t (cid:48) ) I φl ( p ⊥ , t (cid:48) ) , (17)where I φl ( p ⊥ , t (cid:48) ) contains the φ integral and correspondingterms and is given by I φl ( p ⊥ , t (cid:48) ) = i l π (cid:90) π − π dφ (cid:48) e − i ( l − m ) φ (cid:48) e ip ⊥ α ⊥ ( t (cid:48) ) sin [ φ (cid:48) − ν ( t (cid:48) ) ]= i l e − i ( l − m ) ν ( t (cid:48) ) J l − m ( p ⊥ α ⊥ ( t (cid:48) )) . (18)Now the full transition amplitude can be written as M l ( p (cid:107) , p ⊥ ) = i l − (cid:88) m (cid:90) ∞−∞ dt (cid:48) e iS l − m ( p (cid:107) ,p ⊥ ,t (cid:48) ) V m ( p (cid:107) , p ⊥ , t (cid:48) ) J l − m ( p ⊥ α ⊥ ( t (cid:48) )) , (19)where S l ( p (cid:107) , p ⊥ , t (cid:48) ) = (cid:18) I p + 12 p (cid:107) + 12 p ⊥ (cid:19) t (cid:48) − lν ( t (cid:48) )+ p (cid:107) α (cid:107) ( t (cid:48) ) + 12 (cid:90) dt (cid:48) A ( t (cid:48) ) . (20) Yuxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation
Note that the OAM l only appears in the action coupledwith ν ( t (cid:48) ), which mediates interaction of the laser fieldwith the OAM of the electron. The time-varying ‘angle’ ν ( t (cid:48) ) can be interpreted as the dynamical rotational ac-tion of the field on the OAM of the photoelectron. For amonochromatic field, we will see that ν ( t (cid:48) ) = ωt . We canrewrite ν ( t (cid:48) ) in terms of its derivative to gain more insight ν (cid:48) ( t (cid:48) ) = ( α ( t (cid:48) ) × A ( t (cid:48) )) · ˆ e (cid:107) α ⊥ ( t (cid:48) ) . (21)This makes the rotational nature and link to the spin ofthe field [53] more apparent. We will now examine a fewspecial cases that will significantly simplify the transitionamplitude. In this section we will consider a laser field, which only hascomponents in the ˆ e (cid:107) direction in the cylindrical coordi-nate system (i.e. perpendicular to the xy-plane). Thus,the field has no φ dependence and only S A ( p, θ p , t, t (cid:48) ) con-tributes to the action. Furthermore, given that α ⊥ ( t (cid:48) ) →
0, the Bessel function will become a Kronecker delta J l − m ( p ⊥ α ⊥ ( t (cid:48) )) → δ lm , and this can be demonstrated by re-evaluating Eq. (18) I linear φ = 12 π (cid:90) π − π e − i ( m − l ) φ (cid:48) dφ (cid:48) = δ l,m . (22)This is a selection rule, enforcing l = m , given by thesymmetry of the problem. Substituting this back into thetransition amplitude leaves M l ( p (cid:107) , p ⊥ , ) = i l − (cid:90) ∞−∞ dt (cid:48) e iS A ( p (cid:107) ,p ⊥ ,t (cid:48) ) V l ( p (cid:107) , p ⊥ , t (cid:48) ) . (23)The selection rule comes from the exponential factor of thebound state and OAM of the free electron, which indicatesthat there will be a one-to-one correspondence betweenthe two. Thus, if there is only one Fourier term m = m in Eq. (16), the OAM of the photoelectron will be l = m .Aside from this conservation of angular momentum, theform of the OAM distribution is unchanged from the orig-inal SFA formalism for the plane wave momentum state. In this section we turn to the OAM transition amplitudefor a monochromatic circular field, where there is now a φ dependence in the semi-classical action. The form of vectorpotential for a circular monochromatic field is given by A ( t ) = − (cid:112) p (cos( ωt ) e x + sin( ωt ) e y ) . (24)For this field in Eq. (19) α ⊥ ( t (cid:48) ) = (cid:112) p /ω and ν ( t (cid:48) ) = ωt (cid:48) . Thus, the action from Eq. (20) becomes linear in t (cid:48) such that S l − m ( p (cid:107) , p ⊥ , t (cid:48) ) = χ l − m ( p (cid:107) , p ⊥ ) t (cid:48) , (25) where χ l − m ( p (cid:107) , p ⊥ ) = I p + U p + 12 ( p (cid:107) + p ⊥ ) − ( l − m ) ω. (26)The transition amplitude is then M l ( p (cid:107) , p ⊥ ) = i l − (cid:88) m J l − m (cid:16) p ⊥ (cid:112) p /ω (cid:17) × (cid:90) ∞−∞ dt (cid:48) e iχ l − m ( p (cid:107) ,p ⊥ ) t (cid:48) V m ( p (cid:107) , p ⊥ , t (cid:48) ) . (27)The integral acts to Fourier transform the prefactor term˜ V p to give M l ( p (cid:107) , p ⊥ ) = i l − (cid:88) m J l − m (cid:16) p ⊥ (cid:112) p /ω (cid:17) ˆ V m ( p (cid:107) , p ⊥ , χ l − m )(28)where ˆ V m ( p (cid:107) , p ⊥ , χ l − m ) is the Fourier transform of V m ( p (cid:107) , p ⊥ , t (cid:48) )with the frequency χ l − m ( p (cid:107) , p ⊥ ).To further simplify the problem we will now considerthe case with a very simple bound state, where the matrixelements have only one Fourier series term for m = 0, aswould be the case for a simple s -state or the bound statefor a zero range potential. In the example of a zero rangepotential [54] the matrix element of the bound state isgiven simply by a constant dependent on the ionisationpotential V = (cid:113) π (cid:112) p . Now Eq. (27) becomes M l ( p (cid:107) , p ⊥ ) =2 πV i l − J l − m (cid:16) p ⊥ (cid:112) p /ω (cid:17) × δ (cid:0) χ l − m ( p (cid:107) , p ⊥ ) (cid:1) . (29)Thus, for the case of monochromatic field with a simplebound state we arrive at another conservation equationI p + U p + 12 ( p (cid:107) + p ⊥ ) − ( l − m ) ω = 0 . (30)This is directly related to the semi-classical condition forATI peaks [55], which can be interpreted forming due toadditional photons absorbed by the photoelectron in thecontinuum, beyond that required for ionisation. In thiscase, however, each ATI peak will correspond to a dif-ferent value OAM, which will be shifted by the quantummagnetic number m . This has interesting implications as itsuggests that different values of the OAM will be localisedto specific energy regions in photoelectron emission spec-tra for ionisation via circularly polarised light. It is thisidea that was exploited to produce interference vortices inthe recent work [41]. In this subsection we discuss an alternative way to com-pute the OAM-SFA transition amplitude. In order to com-pute the OAM transition amplitude we employ the sad-dle point approximation for Eq. (4) and then analyti-cally perform the integral over φ . Applying the saddle uxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation 5 point approximation and ignoring the bound state pref-actor Eq. (4) becomes M ( p (cid:107) , p ⊥ , φ ) = N − (cid:88) s =0 (cid:115) πi∂ S/∂t | t = t s exp (cid:2) iS ( p (cid:107) , p ⊥ , φ, t s ) (cid:3) , (31)where N is the number of laser cycle considered and t s isgiven by ( p + A ( t s )) = − p . (32)For a general field t s will depend on φ in a nontrivial wayso the integral over φ can not be performed analytically.However, for a monochromatic circular field the depen-dence on φ is linear and we can write t s in terms of t (cid:48) s ,which is independent of φωt s ( p (cid:107) , p ⊥ , φ ) = φ + ωt (cid:48) s ( p (cid:107) , p ⊥ ) (mod 2 π ) . (33)Substituting this into the action leads to S ( p (cid:107) , p ⊥ , φ, t (cid:48) s ) = (cid:18) I p + U p + 12 ( p (cid:107) + p ⊥ ) (cid:19) (cid:18) φω + t (cid:48) s (cid:19) − (cid:112) p ω p ⊥ sin ( ωt (cid:48) s ) (34)= S ( p (cid:107) , p ⊥ , , t (cid:48) s )+ (cid:18) I p + U p + 12 ( p (cid:107) + p ⊥ ) (cid:19) φω (35)Thus, substituting the action and transition amplitudeinto Eq. (19) leads to the expression M l ( p (cid:107) , p ⊥ ) = N − (cid:88) s =0 (cid:115) πi∂ S/∂t (cid:48) s exp( iS ( p (cid:107) , p ⊥ , , t (cid:48) s )) × π (cid:90) π − π dφ (cid:48) exp (cid:2) i (I p + U p + 1 / p )( φ (cid:48) /ω ) − iφ (cid:48) l (cid:3) (36)= N − (cid:88) s =0 (cid:115) πi∂ S/∂t (cid:48) s exp( iS ( p (cid:107) , p ⊥ , , t (cid:48) s )) × sinc (cid:2) (I p + U p + 1 / p − ωl )( π/ω ) (cid:3) (37)Note, the prefactor in the square root can be determinedto be independent of φ (cid:48) , thus it is outside of the φ (cid:48) in-tegral. The sum leads to the contribution of one identi-cal ionisation event per laser cycle, due to the field beingmonochromatic. Thus, the sum can be evaluated analyti-cally to give the following M l ( p (cid:107) , p ⊥ ) = Ω N ( p (cid:107) , p ⊥ ) (cid:115) πi∂ S/∂t (cid:48) s exp( iS ( p (cid:107) , p ⊥ , , t (cid:48) )) × sinc (cid:104) χ l πω (cid:105) , (38) E a.u. s i g . a . u . (a)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
OAM l202122232425262728293031323334353637 E a.u. s i g . a . u . (b)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 E a.u. s i g . a . u . (c)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Fig. 1.
The result of Eq. (38) is plotted for 2 [panel (a)], 4[panel (b)] and 8 [panel (c)] ionisation events (given by N).The laser field intensity employed is I = 2 × W/cm and a wavelength λ = 800 nm, corresponding to U p =0 .
44 a.u. and ω = 0 .
057 a.u. for the ponderomotive energyand angular frequency, respectively. The distributions areplotted vs energy in order to easily see the condition 30,which is marked by the vertical dashed lines on each fig-ure and has constant spacing in energy. The OAM l values20 to 37 are included as shown in the figure legend. Thebound state prefactor has been neglected in the calcula-tion.where t (cid:48) denotes the solution in the first laser cycle andΩ N ( p (cid:107) , p ⊥ ) = N − (cid:88) n =0 exp (2 πinχ /ω )= exp [2 πiN χ /ω ] − πiχ /ω ] − N → ∞ Ω N ( p (cid:107) , p ⊥ ) becomes a Dirac comblim N →∞ Ω N ( p (cid:107) , p ⊥ ) = ∞ (cid:88) n =0 δ ( χ − nω ) . (40)The Dirac delta functions lead to the following replace-ment in the argument of the sinc functionsinc (cid:104) χ l πω (cid:105) → sinc [ π ( n − l )] = δ nl . (41) Yuxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation
Thus, this leads to M l ( p (cid:107) , p ⊥ ) = (cid:115) πi∂ S/∂t (cid:48) s exp( iS ( p (cid:107) , p ⊥ , , t (cid:48) )) δ (cid:0) χ l ( p (cid:107) , p ⊥ ) (cid:1) , (42)which gives the same condition as Eq. (30) but has dif-ferent prefactors due to the saddle point approximation.Taking N as a finite fixed value we are able to plot theresult, which can be considered a rough approximation toemploying a laser pulse of N cycles, which has been donein Fig. 1. The plot is over energy to make it clear eachpeak is separated with equal energy spacing as dictatedby Eq. (30). As the number of laser cycles goes from 2to 8 the OAM peaks get sharper, approaching the deltafunctions predicted by Eq. (29). The bound state prefac-tor is neglected in Fig. 1, but it would be expected thatincluding this should shift the peak by the value of themagnetic quantum number. This behaviour was in-factobserved in [41]. The analytical formalism we have de-rived is able to give the core properties of the OAM forstrong field ionisation, however, in an actual experimentthe laser will have a pulse envelope and thus a distribu-tion of frequencies. In the next section, using numericalcomputations, we explore the effect this has on the OAMdistribution. In this section we will examine the effect of a pulse enve-lope on the OAM distributions. We will employ 2-cycle,4-cycle and 8-cycle laser pulses as in the previous section.Employing a sin envelope we set the vector potential tobe A ( t ) = − (cid:112) p sin (cid:18) ωt N (cid:19) (cos( ωt ) e x + sin( ωt ) e y ) . (43)The numerical computation of OAM distributions closelyfollows the methodology of the previous section, using thesaddle point approximation to compute the plane-wave transition amplitude (as in Eq. (31)) and then numerically computing the φ integral from Eq. (19). The φ integralcan be computed very efficiently using the FFT algorithm.The saddle point solutions are computed using Eq. (32).The solutions can also be found very efficiently by trans-forming the equation to a polynomial of order 2 N + 2 andfinding the roots, as outlined in [58]. This means there willbe N + 1 valid solutions, see Fig. 2.In Fig. 2 the 2-cycle, 4-cycle and 8-cycle results areshown. In the first column the laser field is plotted withthe real parts of the times of ionisation indicated by thevertical dashed lines. In the second column the imaginaryparts of the ionisation times are plotted vs the radial mo-mentum coordinate. The minima of these times (see thedotted line in Fig. 3) is a predictor for where the mo-mentum distribution will be maximum. The momentumdistributions are plotted in the final row for each laserpulse. As suggested by the imaginary part of the times, Time cycles E f a . u . (a) p a.u. I m [ t s ] c y c l e s (d) p x a.u. p y a . u . (h) Time cycles E f a . u . (b) p a.u. I m [ t s ] c y c l e s (e) p x a.u. p y a . u . (i) Time cycles E f a . u . (c) p a.u. I m [ t s ] c y c l e s (f) p x a.u. p y a . u . (j) Fig. 2.
The laser pulse (first row), imaginary parts of ion-isation times (second row) and momentum distributions(final row). In the first row the 2-cycle, 4-cycle and 8-cyclelaser fields are plotted in panels (a), (b) and (c) respec-tively. The x [y] component of the laser field is plotted bythe blue [orange] line. The real part of the times of ioni-sation are given by the vertical dashed lines and colouredspots are used to mark where this intersects with the xcomponent of the laser field. These times are found bysolving Eq. (32). The imaginary parts of the solutions areplotted in the middle row, panels (d)–(f) vs the perpen-dicular momentum coordinate p ⊥ , the colours of each linecorresponds to the colour of the spots in the panels above,the real parts are also explicitly given (in units of laser cy-cles) on the right hand side of each panel. The minima ofthe imaginary part of the times of ionisation are markedby dotted lines.the peak of these distributions is away from the centreforming doughnut shapes. Interference between differentpaths can be seen as faint circular fringes. The followingreferences [58, 59] can give some further insight into thesesolutions and the method. Such distributions bear somesimilarity with the attoclock [60–67], where an ellipticalnearly circular field is used to relate the electron emissionangle to the tunneling time. In this work we are relatingthe photoelectron OAM to the emission energy, which arethe conjugate variables of angle and time, respectively.In Fig. 3 the OAM distributions for the three laserpulses used in Fig. 2 are shown. The distribution of fre-quencies in the sin pulse means that we should not ex-pect Eq. (30) to hold. In panel (a), the OAM distributionis plotted for a 2-cycle pulse. There are still specific OAMpeaks in each energy region, however now they signifi-cantly overlap. Furthermore, the central peak at around0 . l = 31 instead of l = 27 as in the uxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation 7 E a.u. s i g a r b . (a)
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
OAM l202122232425262728293031323334353637383940 E a.u. s i g a r b . (b)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 E a.u. s i g a r b . (c)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Fig. 3.
The OAM distributions are shown for a 2-cycle,4-cycle and 8-cycle sin laser pulse, panels (a), (b) and(c), respectively. The peak intensity, wavelength and ion-isation potential is the same as that used in Fig. 1. Thebound state prefactors are neglected. The dot-dashed linescorrespond to the condition Eq. (30) with the followingeffective frequencies used 3 / ω , 15 / ω and 63 / ω , forpanels (a), (b) and (c), respectively. The condition hasbeen slightly shifted in each case so that the central peaklines up with the corresponding l value. The OAM valuesutilised are from l = 20 to l = 40.monochromatic case. This is because the spacing betweenthe peaks is reduced. The dot-dashed lines in Fig. 3 panel(a) use a spacing of ω . This closely matches most of thepeaks, but for increasing l above l = 31, the spacing driftsto high values, while for decreasing l below that of l = 31the spacing drifts to lower values. For the longer pulses,4-cycle and 8-cycles, the distributions move closer to themonochromatic case, with the l = 27 peak moving towardsits previous position of 0 . ω . However, the spacing is stillbelow this, with spacing of ω and ω being used forthe dot-dashed lines for 4-cycles and 8-cycles, respectively.Note that in all cases as well as altering the frequency incondition Eq. (30) a small shift was required to align thedot dashed lines to the correct central peak (reducing forlonger pulses), see Fig. 3 caption for more details.A clear pattern is visible in the spacing of the OAM dis-tribution of 2, 4 and 8 cycle pulse and it is possible to ana-lytically derive this spacing dependence. In the monochro-matic case the spacing between OAM peaks is given by E a.u. s i g a r b . (a)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
OAM l202122232425262728293031323334353637383940 E a.u. s i g a r b . (b)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 E a.u. s i g a r b . (c)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Fig. 4.
The same as Fig. 3 except 3, 5 and 9 cycle sin laser pulses have been employed. The dot-dashed lines usethe carrier frequency for the spacing and are again slightlyshifted so that Eq. (30) to match the central peak. ν ( t (cid:48) ) = ωt (cid:48) , which leads to an ω spacing in energy in thecondition Eq. (30). There is not such a simple relation for ν ( t (cid:48) ) when employing a sin pulse, but through a Taylorexpansion we can derive such an expression. A first orderTaylor expansion about the peak of the pulse ( t = πN/ω )gives ν ( t (cid:48) ) = ω ∗ N t (cid:48) in terms of an effective frequency ω ∗ N ,dependent on the number of laser cycles. Note a constantterm has been discarded as this only contributes an over-all phase and does not affect the OAM peak separation.The effective frequency can be calculated to be ω ∗ N = ν (cid:48) ( πN/ω ) = (cid:40)(cid:0) − N (cid:1) ω, if N is even ω otherwise , (44)which matches the peak spacing seen in 2-cycle, 4-cycleand 8-cycle pulse in Fig. 3.It is interesting to note that, if the number of cycles N is odd, the peaks remain separated at the carrier fre-quency. This is exemplified in Fig. 4, which shows theOAM distributions for 3-cycle, 5-cycle and 9-cycle laserpulses. The dot-dashed lines in this figure are all separatedby the carrier frequency and the corresponding peaks inthe distributions line up with this well. It is possible tosee some drift away from the central peak in panel (a), asthe spacing is slightly smaller than the carrier frequency.These shifts are simply because the condition, althoughaccurate, it is no longer exact and thus there is a increas-ing drift for higher l values. Yuxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation
Beyond the spacing between peaks there are furtherdifferences between Fig. 3 and Fig. 4. Namely, in Fig. 3considerable secondary peaks can be observed, which forin cases where the OAM peaks are low [see l = 23 inpanel (c)] the secondary peaks can be nearly as high asthe primary peak [or even higher in extreme cases e.g. l = 22 in panel (c)]. However, the secondary peaks alwaysoccur in the regions where there is a more dominant OAMpeak corresponding to another l value. In contrast, foran odd number of cycles, Fig. 4, the secondary peaks areconsiderably lower, not playing much of a role at all. Thesefeatures suggest that, if a clean well separate (in energy)OAM distribution is required, it would be much better toutilise an odd number of laser cycles in the laser pulseenvelope. Furthermore, in the even-cycle case the peaksare slightly asymmetric, with a longer tail on one sidethan the other, while for a an odd number of cycles thepeaks are symmetric. These differences suggest it would bevery easy to use the OAM of the photoelectrons to detectif the laser field had a closer to odd or even number ofcycles. However, as previously stated, the measurement ofthe OAM of photoelectrons in strong field experiment isan open problem [41, 42]. In this study we have developed a new version of the strongfield approximation (SFA) to explore the orbital angularmomentum of photoelectrons undergoing strong field ion-isation. Employing this model we are able to derive an-alytic conditions relating to selection rules/ conservationlaws about the system. In the case of a linear field wedemonstrate that the photoelectron OAM is determinedby the magnetic quantum number of the initial boundstate, while for a circular field we show that a range ofOAMs are possible, which will occur in well-defined en-ergy regions. We derive an analytic condition that relatestheir position to the ATI peaks. Computations using a sin laser pulse demonstrate that this condition continuesto be accurate even for very short pulses but the sepa-ration of the peaks is now described by an analyticallyderived effective frequency as opposed to the carrier fre-quency of the laser field. We find that employing an oddor even number of laser cycles has a marked effect on theOAM distributions, leading to two classes of effective fre-quency for even and odd pulses.The present work is an interesting example of how theellipticity and time profile of the field add dynamic shifts ν ( t (cid:48) ) to angular variables intrinsic to the target, such asthe OAM of the electron. This effect bears some similar-ity with previous work on the angular properties of thephotoelectrons. In one example, the tunneling angle, de-rived in [68], was used to show the preferential tunnel ion-isation of ‘counter-rotating’ electrons with circularly po-larised fields. An electron’s angle of return also leaves im-prints in HHG spectra [69, 70]. These angular shifts wereincorporated in a purely structural two-centre interfer-ence condition for HHG in diatomic molecules [70], and could be made visible by exploiting macroscopic propaga-tion [71]. A key difference is that these articles computedangles related to the velocity of the electron, whereas inthe present work ν ( t (cid:48) ) is related only to the rotation of thefield and represents the interaction of the laser field withthe OAM of the electron. It could be said to be the ‘stir-ring’ action of the field upon the electron at the momentof ionisation.The method proposed here is general and could beextended to initial states with arbitrary angular momen-tum and other types of tailored fields. However, an openquestion is the role of the residual binding potential. In[41] good agreement was found between the SFA and theTDSE solvers Qprop and RMT but the longer wavelengthsused in this study may lead to more significant Coulombeffects. This question has been addressed in the attoclocksetup, which (as previously stated) deals with the con-jugate variables of the emission angle and correspondingionisation time. In this setting the Coulomb potential hasbeen shown to shift the photoelectron emission angle (seee.g. [65]), which may have a profound effect on the inter-pretation of the tunneling time (see [67, 72] for reviewson the attoclock). Thus, it should be expected that theCoulomb potential will also shift the OAM of the pho-toelectron. This may be addressed by incorporating theOAM into SFA-like models which account for the bindingpotential, such as the Coulomb-quantum strong field ap-proximation [46, 56, 73] as well as performing comparisonwith TDSE solvers.Another very important question is: How about othertypes of fields, for more complex vortices? The responseof matter to structured laser fields is a central subject ofintense laser-matter physics, and is inevitably related toOAM of the matter. Recently, a novel pump-probe schemeincorporating OAM was theoretically demonstrated us-ing a vortex IR beam and XUV pulse allowing for time-resolved photoionisation [74], while DeNinno et al. demon-strated experimentally that a free electron matter wave,produced by an XUV pulse, is sensitive to a vortex IRbeam. Similarly, angle-resolved attosecond streaking oftwisted attosecond pulses has been recently proposed [75].In a condensed matter context, it was shown recently thatTHz laser pulses with circular polarisation induce tran-sient Chern insulator in graphene [76]. Similarly, linearlypolarised pulses with OAM induce non-uniform Chern in-sulators [77]. All these examples clearly illustrates that weare entering an era of OAM and more complex texturesin laser-matter physics. Acknowledgements
ASM acknowledges grant EP/P510270/1 and CFMF grantno. EP/J019143/1, both funded by the UK Engineeringand Physical Sciences Research Council (EPSRC). ASM,EP, MC and ML acknowledge support from ERC AdGNOQIA, Spanish Ministry of Economy and Competitive-ness (“Severo Ochoa” program for Centres of Excellence inR&D (CEX2019-000910-S), Plan National FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI), Fundaci´o Pri- uxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation 9 vada Cellex, Fundaci´o Mir-Puig, and from Generalitat deCatalunya (AGAUR Grant No. 2017 SGR 1341, CERCAprogram, QuantumCAT U16-011424, co-funded by theERDF Operational Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS (funded by State Research Agency(AEI) PCI2019-111828-2/10.13039/501100011033), EU Hori-zon 2020 FET-OPEN OPTOLogic (Grant No 899794),and the National Science Centre, Poland-Symfonia GrantNo. 2016/20/W/ST4/00314.
References
1. W. Whewell. Xi. essay towards a first approximation toa map of cotidal lines.
Philosophical Transactions of theRoyal Society of London , pp. 147–236 (1833). E-print.2. W. Whewell. Xvii. researches on the tides.&
Philosophical Transactions of the Royal Society ofLondon , pp. 289–341 (1836). E-print.3. M. V. Berry. Geometry of phase and polarization singular-ities illustrated by edge diffraction and the tides. In M. S.Soskin and M. V. Vasnetsov (eds.),
Second InternationalConference on Singular Optics (Optical Vortices): Funda-mentals and Applications , vol. 4403. International Societyfor Optics and Photonics (SPIE, 2001), pp. 1 – 12. doi:10.1117/12.428252. URL https://doi.org/10.1117/12.428252 .4. F. Hall. Tornado forecasting research.
Science no.2947, pp. 3–3 (1951). E-print.5. J. Swithebank and N. Chigier. Vortex mixing for super-sonic combustion.
Symposium (International) on Combus-tion no. 1, pp. 1153 – 1162 (1969).6. S.-B. Choe, Y. Acremann, A. Scholl et al. Vortex core-driven magnetization dynamics. Science no. 5669,pp. 420–422 (2004). E-print.7. Y. Sumino, K. H. Nagai, Y. Shitaka et al. Large-scalevortex lattice emerging from collectively moving micro-tubules.
Nature no. 7390, pp. 448–452 (2012).8. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P.Woerdman. Orbital angular momentum of light and thetransformation of laguerre-gaussian laser modes.
Phys.Rev. A , pp. 8185–8189 (1992).9. G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I.Larkin and V. M. Vinokur. Vortices in high-temperaturesuperconductors. Rev. Mod. Phys. no. 4, pp. 1125–1388(1994).10. K. Y. Bliokh, Y. P. Bliokh, S. Savel’ev and F. Nori. Semi-classical dynamics of electron wave packet states withphase vortices. Phys. Rev. Lett. , p. 190404 (2007).11. A. L. Fetter. Rotating trapped bose-einstein condensates. Rev. Mod. Phys. , pp. 647–691 (2009).12. L. Pitaevskii and S. Stringari. Bose-Einstein Condensa-tion and Superfluidity . 2 ed. (Oxford Univ. Press, Oxford,2016). ISBN 9780191076688.13. B. J. McMorran, A. Agrawal, P. A. Ercius et al. Originsand demonstrations of electrons with orbital angular mo-mentum.
Phil. Trans. R. Soc. A , p. 20150434 (2016).14. K. Y. Bliokh, I. P. Ivanov, G. Guzzinati et al. Theory andapplications of free-electron vortex states.
Phys. Rep. no. 2017, pp. 1–70 (2017). 15. S. M. Lloyd, M. Babiker, G. Thirunavukkarasu andJ. Yuan. Electron vortices: Beams with orbital angularmomentum.
Rev. Mod. Phys. , p. 035004 (2017).16. J. P. Torres and L. Torner (eds.). Twisted Photons: Appli-cations of Light with Orbital Angular Momentum (Wiley,Weinheim, 2011).17. D. L. Andrews and M. Babiker (eds.).
The AngularMomentum of Light (Cambridge University Press, Cam-bridge, 2012).18. G. Gbur.
Singular optics (CRC Press, Boca Raton, 2016).19. H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry et al.Roadmap on structured light.
J. Opt. no. 1, p. 013001(2017). UPC eprint.20. V. Garc´es-Ch´avez, K. Volke-Sepulveda, S. Ch´avez-Cerda,W. Sibbett and K. Dholakia. Transfer of orbital angu-lar momentum to an optically trapped low-index particle. Phys. Rev. A no. 6, p. 063402 (2002). UNAM eprint.21. M. Padgett and R. Bowman. Tweezers with a twist. NaturePhoton. no. 6, pp. 343–348 (2011).22. M. Z¨urch, C. Kern, P. Hansinger, A. Dreischuh andC. Spielmann. Strong-field physics with singular lightbeams. Nat. Phys. no. 10, pp. 743–746 (2012).23. C. Hern´andez-Garc´ıa, A. Pic´on, J. San Rom´an andL. Plaja. Attosecond extreme ultraviolet vortices fromhigh-order harmonic generation. Phys. Rev. Lett. , p.083602 (2013).24. G. Gariepy, J. Leach, K. T. Kim et al. Creating high-harmonic beams with controlled orbital angular momen-tum.
Phys. Rev. Lett. no. 15, p. 153901 (2014).25. C. Hern´andez-Garc´ıa, J. S. Rom´an, L. Plaja and A. Pic´on.Quantum-path signatures in attosecond helical beamsdriven by optical vortices.
New Journal of Physics no. 9, p. 093029 (2015).26. R. G´eneaux, A. Camper, T. Auguste et al. Synthesis andcharacterization of attosecond light vortices in the extremeultraviolet. Nature Commun. , p. 12583 (2016). OSUeprint.27. L. Rego, J. S. Rom´an, A. Pic´on, L. Plaja andC. Hern´andez-Garc´ıa. Nonperturbative twist in the gen-eration of extreme-ultraviolet vortex beams. Phys. Rev.Lett. no. 16, pp. 1–6 (2016).28. A. Turpin, L. Rego, A. Pic´on, J. San Rom´an andC. Hern´andez-Garc´ıa. Extreme ultraviolet fractional or-bital angular momentum beams from high harmonic gen-eration.
Scientific Reports no. 1, p. 43888 (2017).29. F. Kong, C. Zhang, F. Bouchard et al. Controlling the or-bital angular momentum of high harmonic vortices. NatureCommun. , p. 14970 (2017).30. D. Gauthier, P. R. Ribiˇc, G. Adhikary et al. Tunable or-bital angular momentum in high-harmonic generation. Na-ture Commun. , p. 14971 (2017).31. C. Hern´andez-Garc´ıa. A twist in coherent x-rays. NaturePhysics no. 4, pp. 327–329 (2017).32. W. Paufler, B. B¨oning and S. Fritzsche. Tailored or-bital angular momentum in high-order harmonic genera-tion with bicircular laguerre-gaussian beams. Phys. Rev.A , p. 011401 (2018).33. K. M. Dorney, L. Rego, N. J. Brooks et al. Controllingthe polarization and vortex charge of attosecond high-harmonic beams via simultaneous spin–orbit momentumconservation. Nat. Photonics no. 2, pp. 123–130 (2019).0 Yuxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation34. E. Pisanty, G. J. Machado, V. Vicu˜na-Hern´andez et al.Knotting fractional-order knots with the polarization stateof light. Nat. Photonics no. August, pp. 569–574(2019).35. E. Pisanty, L. Rego, J. San Rom´an et al. Conservationof torus-knot angular momentum in high-order harmonicgeneration. Phys. Rev. Lett. no. 20, p. 203201 (2019).arXiv:1810.06503.36. L. Rego, K. M. Dorney, N. J. Brooks et al. Generation ofextreme-ultraviolet beams with time-varying orbital angu-lar momentum.
Science no. 6447, p. 1253 (2019).37. H. Barati Sedeh, M. M. Salary and H. Mosallaei. Time-varying optical vortices enabled by time-modulated meta-surfaces.
Nanophotonics no. 9, pp. 2957–2976 (2020).38. F. C. V´elez, K. Krajewska and J. Z. Kami´nski. Generationof electron vortex states in ionization by intense and shortlaser pulses. Phys. Rev. A , p. 043421 (2018).39. F. Cajiao V´elez, J. Z. Kami´nski and K. Krajewska. Gen-eration of propagating electron vortex states in photode-tachment of h − . Phys. Rev. A , p. 053430 (2020).40. O. I. Tolstikhin and T. Morishita. Strong-field ionization,rescattering, and target structure imaging with vortex elec-trons.
Phys. Rev. A , p. 063415 (2019).41. A. Maxwell, G. S. J. Armstrong, M. Ciappina et al. Manip-ulating twisted electrons in strong field ionization. FaradayDiscuss. pp. –.42. V. Grillo, A. H. Tavabi, F. Venturi et al. Measuring theorbital angular momentum spectrum of an electron beam.
Nat Commun. no. 1, p. 15536 (2017).43. K. Amini, J. Biegert, F. Calegari et al. Symphony onstrong field approximation. Rep. Prog. Phys. no. 11, p.116001 (2019).44. W. Becker, F. Grasbon, R. Kopold et al. Above-thresholdionization: From classical features to quantum effects. Adv.At. Mol. Opt. Phys. , pp. 35–98 (2002).45. C. Figueira de Morisson Faria, H. Schomerus andW. Becker. High-order above-threshold ionization: Theuniform approximation and the effect of the binding po-tential. Phys. Rev. A , p. 043413 (2002).46. C. F. de Morisson Faria and A. S. Maxwell. It is all aboutphases: ultrafast holographic photoelectron imaging. Re-ports on Progress in Physics no. 3, p. 034401 (2020).47. V. Mosert and D. Bauer. Photoelectron spectra withQprop and t-SURFF. Comp. Phys. Commun. , pp.452–463 (2016).48. V. Tulsky and D. Bauer. Qprop with faster calculationof photoelectron spectra.
Comp. Phys. Commun. , p.107098 (2020).49. L. R. Moore, M. A. Lysaght, L. A. A. Nikolopoulos et al.The rmt method for many-electron atomic systems in in-tense short-pulse laser light.
Journal of Modern Optics no. 13, pp. 1132–1140 (2011).50. D. D. A. Clarke, G. S. J. Armstrong, A. C. Brown andH. W. van der Hart. r -matrix-with-time-dependence the-ory for ultrafast atomic processes in arbitrary light fields. Phys. Rev. A , p. 053442 (2018).51. A. C. Brown, G. S. J. Armstrong, J. Benda et al. Rmt: R-matrix with time-dependence. solving the semi-relativistic,time-dependent schr¨odinger equation for general, multi-electron atoms and molecules in intense, ultrashort, ar-bitrarily polarized laser pulses. Comput. Phys. Commun. , p. 107062 (2020). 52. W. Becker, A. Lohr, M. Kleber and M. Lewenstein. Aunified theory of high-harmonic generation: Applicationto polarization properties of the harmonics.
Phys. Rev. A , pp. 645–656 (1997).53. L.-P. Yang, F. Khosravi and Z. Jacob. Quantum spin op-erator of the photon. arXiv e-prints arXiv:2004.03771.E-print.54. W. Becker, S. Long and J. K. McIver. Modeling harmonicgeneration by a zero-range potential. Phys. Rev. A , pp.1540–1560 (1994).55. R. R. Freeman, P. H. Bucksbaum, H. Milchberg et al.Above-threshold ionization with subpicosecond laserpulses. Phys. Rev. Lett. no. 10, p. 1092 (1987).56. A. S. Maxwell, A. Al-Jawahiry, T. Das and C. Figueirade Morisson Faria. Coulomb-corrected quantum interfer-ence in above-threshold ionization: Working towards multi-trajectory electron holography. Phys. Rev. A no. 2, p.023420 (2017). E-print.57. D. G. Arb´o, K. L. Ishikawa, K. Schiessl, E. Persson andJ. Burgd¨orfer. Intracycle and intercycle interferences inabove-threshold ionization: The time grating. Phys. Rev.A , p. 021403 (2010).58. A. Jaˇsarevi´c, E. Hasovi´c, R. Kopold, W. Becker and D. B.Miloˇsevi´c. Application of the saddle-point method tostrong-laser-field ionization. Journal of Physics A: Math-ematical and Theoretical no. 12, p. 125201 (2020).59. I. Barth and O. Smirnova. Nonadiabatic tunneling in circu-larly polarized laser fields. ii. derivation of formulas. Phys.Rev. A , p. 013433 (2013).60. P. Eckle, A. N. Pfeiffer, C. Cirelli et al. Attosecond Ioniza-tion and Tunneling Delay Time Measurements in Helium. Science no. 5907, p. 1525 (2008).61. P. Eckle, M. Smolarski, P. Schlup et al. Attosecond angularstreaking.
Nature Physics no. 7, pp. 565–570 (2008).62. A. N. Pfeiffer, C. Cirelli, M. Smolarski et al. Breakdown ofthe independent electron approximation in sequential dou-ble ionization. New Journal of Physics no. 9, 093008.63. A. N. Pfeiffer, C. Cirelli, M. Smolarski et al. Attoclockreveals natural coordinates of the laser-induced tunnellingcurrent flow in atoms. Nature Physics no. 1, pp. 76–80(2012).64. A. S. Landsman, M. Weger, J. Maurer et al. Ultrafastresolution of tunneling delay time. Optica no. 5, pp.343–349 (2014).65. L. Torlina, F. Morales, J. Kaushal et al. Interpreting at-toclock measurements of tunnelling times. Nature Physics no. 6, pp. 503–508 (2015).66. U. S. Sainadh, H. Xu, X. Wang et al. Attosecond angularstreaking and tunnelling time in atomic hydrogen. Nature no. 7750, pp. 75–77 (2019).67. U. S. Sainadh, R. T. Sang and I. V. Litvinyuk. Attoclockand the quest for tunnelling time in strong-field physics.
Journal of Physics: Photonics no. 4, p. 042002 (2020).68. I. Barth and O. Smirnova. Nonadiabatic tunneling in cir-cularly polarized laser fields: Physical picture and calcula-tions. Phys. Rev. A , p. 063415 (2011).69. M. Kitzler, X. Xie, S. Roither, A. Scrinzi and A. Baltuska.Angular encoding in attosecond recollision. New Journalof Physics no. 2, p. 025029 (2008).70. T. Das, B. B. Augstein and C. Figueira de Morisson Faria.High-order-harmonic generation from diatomic moleculesin driving fields with nonvanishing ellipticity: A general-ized interference condition. Phys. Rev. A , p. 023404(2013).uxin Kang et al.: Conservation laws for Electron Vortices in Strong-Field Ionisation 1171. T. Das, B. B. Augstein, C. Figueira de Morisson Fariaet al. Extracting an electron’s angle of return from shiftedinterference patterns in macroscopic high-order-harmonicspectra of diatomic molecules. Phys. Rev. A , p. 023406(2015).72. C. Hofmann, A. S. Landsman and U. Keller. Attoclockrevisited on electron tunnelling time. Journal of ModernOptics no. 10, pp. 1052–1070 (2019). E-print.73. X.-Y. Lai, C. Poli, H. Schomerus and C. F. d. M. Faria.Influence of the coulomb potential on above-threshold ion-ization: A quantum-orbit analysis beyond the strong-fieldapproximation. Phys. Rev. A , p. 043407 (2015).74. S. Giri, M. Ivanov and G. Dixit. Signatures of the or-bital angular momentum of an infrared light beam in thetwo-photon transition matrix element: A step toward at-tosecond chronoscopy of photoionization. Phys. Rev. A , p. 033412 (2020).75. I. N. Ansari, D. S. Jadoun and G. Dixit. Angle-ResolvedAttosecond Streaking of Twisted Attosecond Pulses. arXive-prints arXiv:2006.01582. E-print.76. J. W. McIver, B. Schulte, F.-U. Stein et al. Light-inducedanomalous hall effect in graphene.
Nature Physics no. 1,pp. 38–41 (2020).77. U. Bhattacharya, S. Chaudhary, T. Grass and M. Lewen-stein. Fermionic Chern insulator from twisted light withlinear polarization. arXiv e-printsarXiv e-prints