Angle-Resolved Attosecond Streaking of Twisted Attosecond Pulses
AAngle-Resolved Attosecond Streaking of Twisted Attosecond Pulses
Irfana N. Ansari, Deependra S. Jadoun, and Gopal Dixit ∗ Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India (Dated: June 3, 2020)The present work focuses on the characterisation of the amount of orbital angular momentum(OAM) encoded in the twisted attosecond pulses via energy- and angle-resolved attosecond streakingin pump-probe setup. It is found that the photoelectron spectra generated by the linearly polarisedtwisted pulse with different OAM values exhibit angular modulations, whereas circularly polarisedtwisted pulse yields angular isotropic spectra. It is demonstrated that the energy- and angle-resolved streaking spectra are sensitive to the OAM values of the twisted pulse. Moreover, thedifferent combinations of the polarisation of the twisted pump pulse and strong infrared probepulse influence the streaking spectra differently. The characterisation of the OAM carrying twistedattosecond pulses opens up the possibility to explore helical light-matter interaction on attosecondtimescale.
PACS numbers:
Complete characterisation of attosecond pulses is es-sential to harness the full potential of such pulses in cap-turing ultrafast electron processes and extracting mean-ingful interpretations of time-resolved measurements [1,2]. Different techniques are in practice to characterisethe temporal structure of the attosecond pulses such asattosecond streaking and RABBITT [3–5]. Both thesemethods yield information about pulse duration, carrierenergy, and chirp of the attosecond pulses [6, 7]. Re-cently, attosecond streaking method has been extendedto characterise the carrier-envelope phase [8] and ex-tract the polarisation state [9] of the isolated attosecondpulse. In this work, energy- and angle-resolved attosec-ond streaking (AAS) method is employed to characterisethe orbital angular momentum (OAM) of twisted attosec-ond pulses.The polarisation property of light is associated withits spin angular momentum, whereas the spatial pro-file of the wave front is connected to the OAM of thelight [10]. Light with non-zero OAM is known as twistedlight and Laguerre-Gaussian light beam is an example ofsuch light and considered in this work. OAM of the lighthas found numerous applications in various fields sinceits first realisation [11–19]. These applications have mo-tivated scientists to generate ultrashort twisted pulses inthe extreme ultraviolet (XUV) energy regime to probe at-tosecond electron motion. In recent years, series of workshave been carried out to up-convert OAM carrying in-frared (IR) beam to the XUV beam using high harmonicgeneration (HHG) [20–28]. A non-collinear method hasbeen applied to generate linearly polarised twisted lightwith relatively low OAM via HHG [29–31]. These state-of-the-art HHG experiments provided an avenue for thegeneration of coherent attosecond XUV pulses with desir-able OAM properties. However, no claim of generatingsuch pulses can be made without measurement of theOAM encoded in the twisted XUV pulses with no a pri-ori assumptions. This is an uncharted territory and main focus of the present work.In this work, the concept of the standard attosecondstreak camera is extended in three-dimensions as thetwisted XUV pulse has a complex spatial structure. Theenergy- and angle-resolved photoelectron spectra are sim-ulated as a function of pump-probe delay time for differ-ent polarisations of the twisted XUV pump pulse and IRprobe pulse. The pump pulse induces a single-photonionisation and liberated photoelectron is streaked by thesynchronized IR pulse having the plane wavefront andzero OAM. Kaneyasu et al. have performed the pho-toionisation of helium by the twisted XUV beam at syn-chrotron and discussed the possibility of observing the vi-olation of the standard electric dipole selection rules [32].Boning et al. have discussed the streaking of twisted x-waves within dipole approximation with relatively weakcontinuous IR beam [33]. The influences of the projectionof the total angular momentum, the opening angle andthe impact parameter on the streaking spectra have beendiscussed [33]. Recently, the formalism of two-photonionisation and its consequences in photoionisation timedelay have been discussed [34].The transition amplitude, within strong-field approx-imation, from a ground state to a continuum state | k (cid:105) with momentum k is expressed as a k ( τ ) = − i (cid:90) + ∞−∞ dte i [I p t − Φ( k ,t )] E X ( t − τ ) d p ( t ) , (1)where I p is the ionisation potential of the atom, Φ( k , t ) = (cid:82) + ∞ t dt (cid:48) p ( t (cid:48) ) / p ( t ) = k + A ( t )as the instantaneous momentum of the photoelectronin IR field, A ( t ) being the vector potential such that E ( t ) = − ∂ A ( t ) /∂t . E X ( t ) = ˜E X ( t )exp[ − i Ω t ] is the XUVfield with ˜E X ( t ) as the envelope and Ω as the centralenergy of the XUV field. The parameter τ is the timedelay between the two pulses, and d p ( t ) is the transitionamplitude from the ground state to the continuum state | p ( t ) (cid:105) with kinetic momentum p ( t ). In the presence of a r X i v : . [ phy s i c s . a t o m - ph ] J un an IR field, the phase accumulated by the photoelectron,during its motion in the continuum from t to + ∞ , is δ Φ( t ) = Φ( k , t ) − I p t . Note that strong-field approxima-tion is appropriate as ω (cid:28) I p with ω as the frequencyof the IR field, which only interacts with the emittedphotoelectron.Twisted XUV pulse induced photoionisation and ex-change of the OAM larger than one unit in photoion-isation has been manifested. As a result, the stan-dard dipole selection rules get modify [35, 36]. Themodified selection rules for electronic transitions read as | l f − l i | ≤ | l | +1, m f − m i = l ± l f − l i + | l | +1 is even.Here, l f ( m f ) and l i ( m i ) are the final and the initial or-bital angular (magnetic) quantum numbers of electronicstates, respectively; and l is the topological charge of thetwisted XUV pulse. It is evident from the selection rulesthat only two values of l f = l ± l ≥ l i = 0). In such situation, photoionisation transitionamplitude reads as d k = i Ω √ l πw l +10 (cid:20) C i l +1 ( − l +1 d l +1 k Y l +1 l +1 ( θ k , φ k )+ C ( − l (cid:110) a i l − d l − k Y l − l − ( θ k , φ k )+ b i l +1 d l +1 k Y l − l +1 ( θ k , φ k ) (cid:111)(cid:105) . (2)Here, w is the beam waist; and C = 2 l ( l + 1)! [4 π/ (2 l +3)!] / , C = 2 l +1 l ! π [2 / l + 1)!] / , a = [3(2 l +1) / π (2 l − / C l − , l, , , C l − ,l − l, ,l, − and b = [3(2 l +1) / π (2 l + 3)] / C l +1 , l, , , C l +1 ,l − l, ,l, − are constants. TheClebsch-Gordon coefficients C l f ,m f l ,l ,m ,m are used to ex-press constants a and b . To obtain the above equation,hydrogen ground state wave function is written as a prod-uct of the radial wave function and the spherical har-monic Y m i l i , whereas the wave function for the continuumstate is expanded in terms of the spherical Bessel functionof first kind j l f and the spherical harmonics. The radialtransition amplitude is d lk = (cid:82) ∞ dr r l +3 j l f ( kr ) R n i ,l i ( r )where R n i ,l i is the radial ground-state wave function.Eq. (2) is obtained for twisted XUV pulse, which is lin-early polarised along x -axis and propagating along z -axis. The form of the vector potential for the Laguerre-Gaussian pulse is taken from Ref. [35, 36].Due to non-zero OAM of the twisted XUV pulse, threeionising paths are allowed from the unpolarised groundstate of hydrogen as evident from Eq. (2). The strengthof these paths depend on the magnitudes of d lk , Y ml andthe constants. Possibility of more than one ionising pathis a consequence of the modified selection rules. As evi-dent from Eq. (2), the three paths have different contri-butions from the spherical harmonics Y ml , which dependson the OAM values and determine the resultant angulardistribution of the photoelectrons. Let us analyse how FIG. 1: Normalised photoelectron spectra, induced by twistedXUV pulse, for hydrogen. The spectra are shown as a functionof the observational azimuthal angle φ k , the kinetic energyof photoelectron E k (shown radially, in eV) and the OAMvalues of the ionising twisted pulse with Gaussian envelop of150 attoseconds pulse duration and 46.6 eV photon energy.The spectra are shown for the observational polar angle (a) θ k = 30 ◦ and (b) θ k = 90 ◦ (shown row-wise). the photoelectron spectra are sensitive to the OAM ofthe ionising XUV pulse.The photoionisation (or unstreaked) spectra, inducedby twisted pulse, for different values of the OAM arepresented in Fig. 1. The spectra are shown as a functionof the observation azimuthal angle φ k and the kineticenergy of the photoelectron E k (shown radially in eV).The spectra for two observational polar angles θ k = 30 ◦ and θ k = 90 ◦ are shown in Figs. 1(a) and 1(b), respec-tively. The first and third ionisation paths contributefor l = 0 and their photoelectron distributions are givenby Y and Y − , respectively [see Eq. (2)]. The nodesin the angular distribution along x -axis at φ k = 0 ◦ and φ k = 180 ◦ [see Fig. 2(a) for l = 0] are generated dueto the fact that the mentioned paths contain equal andopposite real contributions [see Fig. 2(a)]. In contrast tothis, the second ionisation path is non-zero for l = 1, butcontributes a constant background to the total spectraas Y is angular isotropic. The other two contributing FIG. 2: Normalised (a) real and (b) imaginary parts of therelevant spherical harmonics. The spherical harmonic Y isreal and Im( Y ) is not shown. terms, Y and Y (Fig. 2), interfere destructively dueto the pre-factors resulting in minimum intensities of thephotoelectron distribution along y -axis, i.e., at φ k = 90 ◦ and φ k = 270 ◦ . The qualitative behavior of the photo-electron angular distributions for different values of θ k issimilar for l = 0 and 1 as visible from Fig. 1.In case of the twisted pulse with l = 2, all the three ion-isation paths contribute differently for smaller and highervalues of θ k . The reason lies in the θ k -dependent distribu-tions of the relevant spherical harmonics, specifically Y .The first, second and third ionisation paths correspondto Y , Y and Y , respectively. The real parts of Y and Y interfere destructively along x -axis whereas theirimaginary parts interfere constructively along y -axis for θ k = 30 ◦ . Also, Y contributes only along x -axis andresults in a maxima along this direction. Thus, in thiscase, the angular distribution is dominated by the realand imaginary parts of Y . However, both the real aswell as the imaginary parts of Y and Y interfere de-structively when θ k is close to 90 ◦ (see Fig. 2). Therefore,the minima in the angular distribution along y -axis dis-appears and appears along x -axis as θ k increases fromsmaller to larger values.From the discussion of Eq. (2) and analysis of Fig. 1, itis established that the unstreaked photoelectron spectraare sensitive to the OAM value of the linearly-polarisedionising pulse. It appears that the presence of the OAMin the ionising pulse can be determined by locating theintensity minima in the unstreaked spectra for smallervalues of θ k (say θ k = 30 ◦ ). However, the amount of theOAM can not be extracted because of the similar qual-itative behavior of the unstreaked spectra for l = 1 and l = 2 [Fig. 1(a)]. Additionally, the unstreaked spectraare not helpful in even determining the presence of theOAM for higher values of θ k [Fig. 1(b)]. Moreover, if cir-cularly polarised twisted XUV pulse is used to ionise theatom, the resultant photoelectron angular distributionsare isotropic irrespective of the helicity of the ionisingXUV pulse. The spectra are similar for different OAMvalues as well (not shown here). Thus, it can be con-cluded that the unstreaked spectra can only determinethe presence or absence of the OAM in the ionising pulse,but with many restrictions on the properties of the pulseand the experimental observational setup.To characterise the exact amount of the OAM en-coded in the twisted pulse, it is important to introduceIR pulse to streak the liberated photoelectron. As theunstreaked spectra are sensitive to the observational an-gle, it is meaningful to use circularly polarised IR pulse,which is represented as E L ( t ) = E L ( t ) √ ω L t )ˆ x − Λ L sin( ω L t )ˆ y ] , (3)where E L ( t ) and Λ L are the envelop and helicity of the IRpulse, respectively. In this case, the Volkov phase reads FIG. 3: AAS spectra for different OAM values. The twistedXUV pulse with linear polarisation is used to trigger pho-toionisation and right-handed circularly polarised IR pulse isused to streak the liberated photoelectrons. The spectra areplotted as a functions of φ k , E k (shown radially, in eV) and for(a) θ k = 30 ◦ , (b) θ k = 45 ◦ and (c) θ k = 60 ◦ . All the spectraare normalised with respect to their respective maxima. asΦ( k , t ) = − [ k + A L ( t )] t k A L ( t ) ω L sin θ k cos( ω L t +Λ L φ k ) . (4)In the following, right-handed circularly polarised IRpulse with 800 nm wavelength is used to streak the pho-toelectrons. The pulse duration and intensity of the IRpulse are 5 femtoseconds with Gaussian envelope and5 × W/cm , respectively. To get the streaking spec-tra, d k is replaced by d p ( t ) .AAS spectra for different OAM values are presentedin Fig. 3. The vector potential of the streaking IR pulseis directed along negative y -axis at the instant of ionisa-tion when the time-delay between the twisted XUV andIR pulses is zero. This streaking field induces a changein momentum of the photoelectron δ k = − A L ( t ) [3]. Itresults in the streaking of the photoelectrons along + y -direction, i.e., φ k = 90 ◦ . At a first glance, it is evidentthat the streaking spectra are sensitive to the OAM val-ues as well as to the observation direction of the photo-electrons. On a close inspection of the streaking spectrawhen θ k = 30 ◦ , it is clearly visible that the low-energyregime of the spectra around φ k = 270 ◦ are most intensefor l = 0 and 2, whereas it is least intense around thisregion for l = 1 (Fig. 3). Moreover, the numbers of inten-sity minima in the spectra are 2 for l = 0, and 3 for l = 1and l = 2; and the angular positions of these minima aredifferent in different spectra. Similar observations can be FIG. 4: Same as Fig. 3 except the liberated photoelectronsare streaked by the left-handed circularly polarised IR pulse.The spectra correspond to (a) θ k = 30 ◦ and (b) θ k = 45 ◦ . made for θ k = 45 ◦ and 60 ◦ [Fig. 3(b, c)].The presence of the three intensity maxima (and lobes)in the spectra for l = 1 at θ k = 30 ◦ could be under-stood as: The streaked part of Re( Y ) did not cancelcompletely by the contribution of Re( Y ); and Im( Y )contributes in the diagonal directions with respect tothe cartesian axes whereas Im( Y ) is zero. For l = 2,the presence of the maxima and minima in intensity at θ k = 30 ◦ is decided dominantly by the first ionisationpath, which is governed by Y . As the value of θ k in-creases, other ionisation paths ( Y for l = 1; Y and Y for l = 2) contribute significantly, which results inthe suppression of one of the three lobes and decides theangular modulation of the streaking spectra [middle andright columns in Fig. 3]. For l = 2, a major distinction inthe streaking spectra is observed at θ k = 60 ◦ . The rea-son lies in the angular structure of Y , which governs thethird ionisation path. For different values of θ k , differentlobes of Y contribute with a threshold at approximately θ k = 50 ◦ , i.e., the contribution of Y reverses when θ k crosses 50 ◦ . Thus giving similar spectra for θ k = 30 ◦ and θ k = 45 ◦ ; and for θ k = 60 ◦ and θ k = 90 ◦ (not shownhere). The distinction and uniqueness of the streakingspectra, i.e., the angular modulation (maxima and min-ima) in the intensity profile along with the strength ofthe minima, prove useful to discern the units of OAMencoded in the twisted pulse.Furthermore, instead of right-handed circularly po-larised IR pulse, left-handed circularly polarised IR pulseis used for streaking to explore any possibility of the cir-cular dichroism. Figure 4 presents the streaking spec-tra corresponding to left-handed circularly polarised IRstreaking pulse for θ k = 30 ◦ and 45 ◦ . In this case, thevector potential of the IR pulse is along + y direction,which results in streaking of photoelectrons along − y di-rection. Therefore, all the spectra are streaked along thenegative direction in comparison to the case when right- FIG. 5: (a) Vector potential of the right-handed circularlypolarised IR streaking pulse. The x - and y -components areshown in blue and red colours, respectively. (b) For l = 1,the attosecond streaking spectra observed at θ k = 30 ◦ cor-responding to the various time-delays between the IR andtwisted pulse. The twisted pulse is delayed in steps of 10 a.u.for one complete cycle of the IR pulse (2 .
67 fs or 110 a.u.). handed circularly polarised IR pulse is used to streak thephotoelectrons. It is established from Figs. 3 and 4 thatthe streaking spectra are distinct and unique for eachvalues of the OAM encoded in the twisted pulse.Till now we have discussed AAS when the time-delaybetween both the pulses is zero. To explore how thestreaking spectrum changes as the time-delay betweenboth the pulses is varied, several streaking spectra, alongwith the x - and y -components of the right-handed IRpulse, for l = 1 at θ k = 30 ◦ is presented in Fig. 5. Thetwisted pulse is delayed in steps of 10 a.u. for a completecycle of the IR pulse. As the time-delay is increased, thevector potential of the IR pulse rotates clockwise startingfrom − y direction. So, the photoelectrons are streakedalong + y direction initially and later follows the A L ( t )accordingly. Also, the intensity of the spectra follow theenvelope of the IR pulse. It is evident from the streakingspectra that the number of lobes are preserved as thetime-delay is varied.In conclusion, we can summarise that the AAS spectraencode the signature and the amount of the OAM presentin the twisted attosecond pulse. As a result in the mod-ifications of the selection rules, several ionising paths bytwisted pulse are possible, which results in angular mod-ulations of the photoelectron spectra. By choosing spe-cific observation directions of the photoelectrons, AAStechnique is useful for linear, right as well as left circu-lar polarisations of the unknown pulse. We believe thatour proposed approach will be the core ingredient in thelight-matter interaction induced by twisted attosecondpulses. The present work paves the route to study heli-cal light-matter interaction and twisted pulse mediatedstrong-field ionization via streak camera [37].G. D. acknowledges support from Science and En-gineering Research Board (SERB) India (Project No.ECR/2017/001460) and Max-Planck India visiting fel-lowship. ∗ [email protected][1] F. Krausz and M. I. Stockman, Nature Photonics , 205(2014).[2] F. Krausz and M. Ivanov, Reviews of Modern Physics , 163 (2009).[3] J. Itatani, F. Qu´er´e, G. L. Yudin, M. Y. Ivanov,F. Krausz, and P. B. Corkum, Physical Review Letters , 173903 (2002).[4] M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, andT. Brabec, Physical Review Letters , 173904 (2002).[5] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Aug´e,P. Balcou, H. G. Muller, and P. Agostini, Science ,1689 (2001).[6] Y. Mairesse and F. Qu´er´e, Physical Review A , 011401(2005).[7] F. Qu´er´e, Y. Mairesse, and J. Itatani, Journal of ModernOptics , 339 (2005).[8] P. L. He, C. Ruiz, and F. He, Physical Review Letters , 203601 (2016).[9] ´A. Jim´enez-Gal´an, G. Dixit, S. Patchkovskii,O. Smirnova, F. Morales, and M. Ivanov, NatureCommunications , 1 (2018).[10] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, andJ. P. Woerdman, Physical Review A , 8185 (1992).[11] F. Cardano and L. Marrucci, Nature Photonics , 776(2015).[12] M. Babiker, D. L. Andrews, and V. E. Lembessis, Journalof Optics , 013001 (2018).[13] S. F¨urhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Optics Express , 689 (2005).[14] K. A. Forbes, Phys. Rev. Letts. , 103201 (2019).[15] W. Brullot, M. K. Vanbel, T. Swusten, and T. Verbiest,Science advances , e1501349 (2016).[16] K. A. Forbes and D. L. Andrews, Optics Letters , 435(2018).[17] M. F. Andersen, C. Ryu, P. Clad´e, V. Natarajan,A. Vaziri, K. Helmerson, and W. D. Phillips, PhysicalReview Letters , 170406 (2006).[18] R. Inoue, N. Kanai, T. Yonehara, Y. Miyamoto,M. Koashi, and M. Kozuma, Physical Review A ,053809 (2006).[19] T. Ruchon and M. Fanciulli, arXiv preprintarXiv:2005.08349 (2020).[20] M. Z¨urch, C. Kern, P. Hansinger, A. Dreischuh, andC. Spielmann, Nature Physics , 743 (2012).[21] C. Hern´andez-Garc´ıa, A. Pic´on, J. San Rom´an, and L. Plaja, Physical Review Letters , 083602 (2013).[22] G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond,E. Frumker, R. W. Boyd, and P. B. Corkum, PhysicalReview Letters , 153901 (2014).[23] R. G´eneaux, A. Camper, T. Auguste, O. Gobert, J. Cail-lat, R. Ta¨ıeb, and T. Ruchon, Nature Communications , 12583 (2016).[24] L. Rego, J. San Rom´an, A. Pic´on, L. Plaja, andC. Hern´andez-Garc´ıa, Physical Review Letters ,163202 (2016).[25] A. Turpin, L. Rego, A. Pic´on, J. San Rom´an, andC. Hern´andez-Garc´ıa, Scientific Reports , 43888 (2017).[26] C. Hern´andez-Garc´ıa, A. Turpin, J. San Rom´an,A. Pic´on, R. Drevinskas, A. Cerkauskaite, P. G. Kazan-sky, C. G. Durfee, and ´I. J. Sola, Optica , 520 (2017).[27] W. Paufler, B. B¨oning, and S. Fritzsche, Physical ReviewA , 011401 (2018).[28] D. Gauthier, S. Kaassamani, D. Franz, R. Nicolas, J. T.Gomes, L. Lavoute, D. Gaponov, S. F´evrier, G. Jargot,M. Hanna, et al., Optics Letters , 546 (2019).[29] D. Gauthier, P. R. Ribiˇc, G. Adhikary, A. Camper,C. Chappuis, R. Cucini, L. F. DiMauro, G. Dovillaire,F. Frassetto, R. G´eneaux, et al., Nature Communications , 14971 (2017).[30] F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown,D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd,E. Karimi, et al., Nature Communications , 14970(2017).[31] K. M. Dorney, L. Rego, N. J. Brooks, J. San Rom´an,C. T. Liao, J. L. Ellis, D. Zusin, C. Gentry, Q. L. Nguyen,J. M. Shaw, et al., Nature Photonics , 123 (2019).[32] T. Kaneyasu, Y. Hikosaka, M. Fujimoto, T. Konomi,M. Katoh, H. Iwayama, and E. Shigemasa, Physical Re-view A , 023413 (2017).[33] B. B¨oning, W. Paufler, and S. Fritzsche, Physical ReviewA , 043423 (2017).[34] S. Giri, M. Ivanov, and G. Dixit, Physical Review A ,033412 (2020).[35] A. Pic´on, A. Benseny, J. Mompart, J. R. V. de Aldana,L. Plaja, G. F. Calvo, and L. Roso, New Journal ofPhysics , 083053 (2010).[36] A. Pic´on, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F.Calvo, and L. Roso, Optics Express , 3660 (2010).[37] M. K¨ubel, Z. Dube, A. Y. Naumov, M. Spanner, G. G.Paulus, M. F. Kling, D. M. Villeneuve, P. B. Corkum,and A. Staudte, Physical Review Letters119