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1 DOI: /NPHOTON Supplementary Information "Spin angular momentum and tunable polarization in high harmonic generation" Avner Fleicher, Ofer Kfir, Tzvi Dikin, Pavel Sidorenko, and Oren Cohen 1. Bichromatic counter-rotating cheme v. monochromatic HHG Figure S1: HHG pectra for bi-chromatic driver with counter-rotating circular polarization [(α,β)=(45 0,45 0 ) ] that reult with harmonic with circular polarization (olid black line) and for the ame driver with the only difference that they are collinearly-polarized and produce linearly-polarized harmonic (dahed blue line). The channel are indicated. Clearly, the generation efficiency of circularly-polarized and linearly-polarized harmonic i imilar.. Polarization analyi of high harmonic Suppoe the polarization ellipe of ome harmonic in the lab coordinate ytem i E E e + E e. We write thi field in a vector form a 0 x x y y Ex E y The harmonic field pae the HHG polarizer, whoe main tranmiion axi make an angle NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

2 DOI: /NPHOTON ofroughly r / r 40 :1). Later on, the field pae the HHG pectrometer, whoe principal p axe are E E E e + e, which act a an additional polarizer that favor light 0 x x y y polarized along e y by a factor of ~.5:1. Written a a product of tranmiion matrice and witching back and forth between the principal axe of the polarizer and the lab coordinate ytem, the final field hitting the CCD i given by E ' 0 0 1x r co in r co in Ex E 0 r' in co 0 r in co E 1y p p y (S1) That i, the intenity i I E 1x E1y i x i y Ex e Ey e 0 x y parameter are I, r,, ', ' n1, n rp r r p where I I n 1, 11 n, n n meaured for harmonic channel n1, n thi polarizer we obtain an intenity can I a the one hown in Fig. S. n1, n. Since the field E e + e i complex, 4 variable are preent. The input i the intenity, when the HHG polarizer i et at angle. By rotating Figure S: Polarization trace from which the experimental ellipticite in Fig. 3b are derived. Upper panel: harmonic pectra of channel (7,6) taken for (α,β)=(45 0,45 0 ) (olid black line) and fitted to a Malu-law type function (olid red line), and meaured and fitted pectra for (α,β)=(47.5 0,45 0 ) (dahed black and red line, repectively). Lower panel: ame a upper panel, but for channel (6,7). Black line: meaured pectra, green line: fitted pectra. By fitting procedure we find the 4 function Ex, Ey, x, y from which the orientation angle of the harmonic polarization ellipe and the ellipticity are obtained NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

3 DOI: /NPHOTON SUPPLEMENTARY INFORMATION tg E x y coy x E x E E y 4 co 4 co E E E E E E x y x y x y y x E E E E E E x y x y x y y x (S) 3. Time-dependent perpective of the experiment The election rule given in Eq. 1 could be intuitively undertood by analyzing the recolliion proce in the time domain. For the counter-rotating circular driver cae (α=β=45 0 ), ubtituting the integer (n 1,n 1 ±1) in Eq. (1) yield 1- : 1.95n 1 (S3) where n 1 i any integer. Thi equation reflect the.95-fold rotational ymmetry that the total electric field draw in pace [ee the Liajou curve (red line) in Fig. S3]. A uual, the electron would tunnel cloe to the maximum of the electric field and would recombine later on. In one optical cycle T of the driver ω,.95 recolliion event occur in the plane of polarization of the two driver. Thee event occur T/.95 one after another, along direction π/.95 (1 0 ) to each other, but in the limit of low depletion are otherwie identical. It i thi time-pace ynchronization between the recolliion event that lead to the election rule given in Eq. (S3) upon the analyi of the time-dependent dipole acceleration expectation value. Suppoe for implicity that the firt recolliion burt a 1 (t) occur along the poitive x axi, then the timedependent x-component of the total dipole acceleration in one optical cycle T would look like 1 T co 4 T ax t a t a1t co a1t (S4) and in addition a x (t) i periodic in T. Hence, the only non-zero Fourier component of a x (t) turn out to be.95n±1. Repeating thi analyi for the y-component of the total dipole acceleration would yield the ame election rule. Recall that when HHG i driven by ingle color, linearly polarized laer, odd harmonic are obtained. Breaking the left-right ymmetry (for intance, by adding a econd harmonic driver) add the even harmonic. Thu, breaking of the pace-time ymmetry often the election rule. The ame ituation occur in our experiment upon changing the polarization of one or both color to elliptical [by rotating the waveplate away from (α,β)=(45 0,45 0 ) ]. Thi would break the.95-fold ymmetry (ee the Liajou curve for α 45 0 or β 45 0 in Fig. 1), and it i found that additional harmonic are obtained, that could be written in the form 1.95n 1, 3, 5,... (S5) which i nothing but another way of writing Eq.1. By controlling the ellipticity of the two pule, the direction and impact parameter of the returning electronic trajectorie can be controlled, a NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

4 DOI: /NPHOTON well a ub-optical-cycle ynchronization between the ubequent recolliion event. Thi directly tranlate into the polarization tate of the emitted high harmonic. And i the reaon for the high controllability over the polarization achieved in our experiment. The reaon for the high efficiency of our cheme lie in the relatively hort time (compared to monochromatic HHG) that the electron pend in the continuum, hence it lateral preading i reduced, giving rie to enhanced recombination probability. Thi i clearly een in Fig. S1 which compare the HHG pectra for the counter-rotating cae [(α,β)=(45 0,45 0 ) ] and for the cae where the two driver are collinearly-polarized. Figure S3: electron cut-off trajectorie for the counter-rotating circular driver cae, preented in a coordinate ytem for which one of the recolliion (black olid curve) occur along the x-axi. The birth and recombination time of that recolliion are labeled a t i and t r, repectively on the Liajou curve (olid red). The electron i being born in the poitive y-direction at time t i, change direction twice and recollide with the ion with an almot head-on recolliion at time t r along the poitive x-direction. Due to the.95-fold rotational ymmetry in the hape of the electric field, almot one third of an optical cycle later a imilar recolliion occur (green olid curve), but along a direction which form an angle of π/.95 with the poitive axi. The blue olid curve i again an otherwie identical recolliion, which i born one-third of an optical cycle after the former one, from a direction which form an 4π/.95 angle with the poitive axi. 4. Strong-field correction The relation between a beam' ellipticity and pin i derived by imple tranformation of bae. Any general elliptically-polarized field (written for implicity in the orientation ellipe' principle axe e, e ) having ellipticity (taken a 0 1) could be written a a uperpoition f of two field with counter-rotating circular polarization: 4 NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

5 DOI: /NPHOTON SUPPLEMENTARY INFORMATION E t E 0 t f h E 0 t 1 1 co e in 1 1 E 0 cot f h in t E 0 cot f h in t e e e e e where E 0 i ome amplitude, i a general phae and h 1 i the helicity. In a ket notation 1 1 E E 0 h E 0 h (S6) where the tate 1 and 1 correpond to left- and right-circular polarization, repectively. The pin of thee bai tate i and, repectively: L z h hh. The expectation value of EL E the field' pin i therefore EE z 1 1 E 0 E 0 h h (S7) E 0 E 0 Moreover, if the field Et i obtained by paing a linearly-polarized field through a halfwaveplate whoe fat axi form an angle with the polarization direction of the incoming linear light, we have tan, h 1 and get 1 in. Thi relation i ued to repreent the pin of the 800nm beam and a imilar expreion with a waveplate angle repreent the pin of the 410nm beam. A dicued in the main text, the deviation 6,7 i the mot pronounced one. A a econd example, conider channel (11,6) and compare it predicted and actual "exitence region" (Fig. 0 0 S4). The perturbative model (Eq. ) predict exitence in the regime , while in 0 0 the experiment thi channel appear in the regime 30. On the other hand, the prediction of Eq. (3) with 11,6 3 agree with the obervation. In general (ee Fig. S5), for mot harmonic channel, it eem that the trong-field correction n 1, nbecome ignificant a the wave-plate i rotated away from α~45 0. NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

6 DOI: /NPHOTON Figure S4: ame a Fig. c. Ellipe: "exitence regime" for harmonic channel (n 1,n ) with (olid line) and without (dahed line) trong-field correction n1, n which enter the conervation law (Eq. 3). While qualitatively matching, additional pin need to be added in order for the numerical "exitence regime" and the one predicted by the pin conervation requirement to quantitatively agree. It i intereting to note that non-zero trong field correction are preent even in the cae of HHG driven by a ingle-color, elliptically-polarized driver. In thi cae balance of energy-pin i obtained for the entire emiion only. 6 NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

7 DOI: /NPHOTON SUPPLEMENTARY INFORMATION Figure S5: (a)- Monochromatic HHG driven by the elliptically-polarized 800nm driver alone: Numerical n,0 n,0 1 n1,0 1 n1 value calculated from h 1 in for different channel (n 1,0). i the weighted um over all exiting channel in the harmonic pectrum. While for each harmonic channel alone n1, n differ from zero a lot, the weighted um i indeed cloe to zero, indicating that in the proce of tranforming the driver photon into the high harmonic one, the conervation of energy-pin i fulfilled not for each channel alone, but rather for the entire emiion only. n 1,0 5. Photon pairing? In Fig. 3b,c the (n 1,n 1-1) and (n 1,n 1 +1) group of channel exhibit mirror-like ellipticitie. Moreover, a the right panel of Fig. S6 clearly how, a α i canned from below 45 0, through 45 0 and above 45 0, thee group alo exchange the orientation of their polarization ellipe. Thee obervation bring forward the option of coupling (in their polarization propertie) between thee two group of harmonic, which tranfer the miing SAM from one to the other. Perhap two photon at different harmonic are emitted correlatively, and only the total SAM i conerved. Interetingly, intenity pairing between high harmonic of different order driven by bichromatic driver, ha been oberved 3. Hence, the nature of correlation could be therefore further invetigated by calculation of the partial um n11, n1 n11,n 1(rather than ). Thee partial um turn out (Fig. S7) to be cloe to zero in the relevant range of α where thee harmonic channel exit, indicating that energy-pin could be balanced by emiion of channel pair. Energy-pin conervation i maintained even within channel-pair, for intance channel (5,6) and (7,6). Perhap the atom convert 1 photon from each driver into two coupled highharmonic photon. NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

8 DOI: /NPHOTON Figure S6: Harmonic pectra (a), product of ellipticity time helicity (b) and orientation angle of the polarization ellipe (c) for the α can experiment. It i clearly een that harmonic belonging to the group (n 1,n 1-1) have imilar α dependence of both ε h and ϕ. Same applie to the channel of the form (n 1,n 1 +1). Moreover, the (n 1,n 1-1) and (n 1,n 1 +1) group of channel exhibit mirror-like polarization dynamic. Figure S7: (a) The partial um n11, n1 n11,n1 where the trong field correction n1, n are taken from Fig. 4a. (b) The partial um n 1, n n 1,n where the trong field correction n, n are taken from Fig. 4b NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.

9 DOI: /NPHOTON SUPPLEMENTARY INFORMATION Reference 1. Milošević, D. B. et al. Generation of circularly polarized high-order harmonic by two-color coplanar field mixing. Phy. Rev. A (000).. Ceccherini, F. et al. Harmonic generation by atom in circularly polarized two-color laer field with coplanar polarization and commenurate frequencie. Phy. Rev. A (003). 3. Mioguti, L. et al., Nonlinear wave-mixing procee in the extreme ultraviolet. Phy. Rev. A (005). NATURE PHOTONICS Macmillan Publiher Limited. All right reerved.