Helicity-selective phase-matching and quasi-phase matching of circularly polarized high-order harmonics: towards chiral attosecond pulses

Phase matching of circularly polarized high-order harmonics driven by counter-rotating bi-chromatic lasers was recently predicted theoretically and demonstrated experimentally. In that work, phase matching was analyzed by assuming that the total energy, spin angular momentum and linear momentum of the photons participating in the process are conserved. Here we propose a new perspective on phase matching of circularly polarized high harmonics. We derive an extended phase matching condition by requiring a new propagation matching condition between the classical vectorial bi-chromatic laser pump and harmonics fields. This allows us to include the influence of the laser pulse envelopes on phase matching. We find that the helicity dependent phase matching facilitates generation of high harmonics beams with a high degree of chirality. Indeed, we present an experimentally measured chiral spectrum that can support a train of attosecond pulses with a high degree of circular polarization. Moreover, while the degree of circularity of the most intense pulse approaches unity, all other pulses exhibit reduced circularity. This feature suggests the possibility of using a train of attosecond pulses as an isolated attosecond probe for chiral-sensitive experiments.

The difficulty in controlling the polarization of HHG originates from the fact that high harmonics are emitted through a coherent recollision phenomenon [20][21][22]. An electron is first ripped from an atom or ion by a strong laser field. The electron then accelerates forth and back in the oscillating field, and finally recombines with the parent ion while emitting a high-energy photon. If the driving laser field is linearly polarized, the recollision (and recombination) probability is high since the trajectory of the electron is linear. This 'head on' recollision yields linearly polarized HHG bursts. Introducing ellipticity into the driving laser field results in the generation of elliptically polarized harmonics [23][24][25]. However, in this case the HHG efficiency drops drastically since the electron trajectory misses the parent ion [26]. Although many theoretical approaches have proposed paths toward generation of bright highly elliptical or circularly polarized high harmonics, the experimental progress was quite limited [27][28][29][30][31][32][33][34][35][36][37][38][39][40]. A notable method proposed to generate circularly polarized HHG from circularly polarized counterrotating [27,29,30,41] or co-rotating [42,43] bi-chromatic laser fields. A single pioneering experimental work from 1995 reported the generation of high harmonics using this geometry with counter-rotating fields [44]. While indication for the correct selection rules was observed in that experiment, no attempt was made to measure the polarization state of the harmonics.
In 2013, Fleischer et al reported that the polarization of high harmonics can be fully controlled-from linear polarization, through elliptical, to circularly polarized high harmonics-without compromising the efficiency of the upconversion process [45,46]. When high harmonics are driven by a bi-chromatic laser field in which the two spectral components (e.g. red (800 nm) and blue (400 nm)) are circularly polarized with opposite helicity [27,44,47], the resulting HHG spectrum consists of pairs of circularly polarized harmonics-also with opposite helicity. The exact spectral position and separation of the harmonic orders depend on the driving laser wavelengths: when the fundamental and second harmonic of Ti:sapphire are used, some harmonics rotate with the same helicity as the red pump (orders + m 3 1 with = ¼ m 1, 2, 3 ) while other harmonics rotate with the same helicity as the blue pump (orders m 3 1 . ) Harmonics m 3 are forbidden [27,29]. The HHG spectrum in [46] is non-chiral although it is comprised of circularly polarized harmonics. The average chirality cancels since pairs of counter-rotating circularly polarized harmonics have comparable power. In the time-domain, the HHG fields that correspond to such spectra consist of linearly polarized attosecond pulse trains, with rotating polarization axis [30,46,48]. Following the experimental generation of circularly polarized high harmonics [45,46] phase matching and downstream application of circularly polarized HHG was demonstrated in the extreme UV (EUV) [47] and soft x-ray region [48], the associated motion of the electrons during their production was explored theoretically and experimentally [49], and a novel 3D attosecond metrology technique for characterizing the time-domain field associated with circularly polarized high harmonics was developed and demonstrated experimentally [50]. Other recent work generated circularly polarized high harmonics using crossed laser beams, to control the propagation direction of left and right circularly polarized HHG beams [51]. Finally, other techniques were experimentally demonstrated to yield generation of useful elliptically polarized high harmonics [52,53].
An interesting question to consider is the degree of circular polarization of this novel light source and what factors determine it. In [47], we measured nearly ∼100% degree of circularity for the + m 3 1 harmonics. Importantly, we found that phase matching is helicity dependent and as such, it may be used for generating helically polarized attosecond pulses (it was recently discovered that the single atom physics of circularly polarized HHG from p-state atoms is also helicity dependent [54][55][56]). In that work, we derived the phase matching condition of the process by assuming three conservation laws for the HHG and pump photons: conservation of energy, linear momentum and spin angular momentum. The same method was used recently for phase matching of circularly polarized x-ray HHG [48]. Moreover, [48] considered the full propagation of the HHG and driving laser fields, while also including the influence of the degree of polarization of the driving laser fields, to predict the HHG pulse duration, detailed spectrum and hellicity.
In this invited paper, we explore phase matching of circularly polarized HHG driven by circularly polarized counterrotating bi-chromatic pumps using two analytical approaches. First, we extend our previous analytical approaches which assumed conservation of energy, linear momentum and angular momentum in the exchange process between the pump (fundamental and its second harmonic) photons and the harmonic photons [47], to general bi-chromatic, circularly polarized, counter-rotating pumps [48]. Second, we derive a more general phase matching condition by requiring a new propagation matching condition between the classical vectorial field of the bi-chromatic pump and the resulting circularly polarized high harmonics. This approach allows us to identify that phase matching couples conservation of energy, linear and spin momenta. Under phase matching conditions, any two (out of the three) conservation laws inevitably yield the third one. We use the classical fields approach to explore the influence of the pulse envelopes on phase matching-a feature that cannot be captured within the first approach. Specifically, we derive conditions for full-and quasi-phase matching of circularly polarized HHG driven by a long circularly polarized pulse and a short counter-rotating pulse. Finally, we discuss the generation of phase-matched high harmonics that can support helically polarized attosecond pulses. We also present experimental spectra and show that they can support attosecond pulses with duration as short as 120 attoseconds per pulse, and degree of circular polarization as high as 80%.
The paper is organized as follows: in section 2 we discuss phase matching of high harmonics driven by bi-chromatic pump laser fields, focusing on the difference between circularly and non-circularly polarized pump fields. Section 3 offers a concise derivation of phase matching of circularly polarized harmonics using conservation laws for photons, as derived first in [47]. In section 4 we derive the phase matching condition when treating the pump and HHG as classical quasi-monochromatic vectorial fields. We utilize the derivation of section 4 to investigate previously inaccessible regimes. As an example, we analyze phase matching regimes of circularly polarized HHG pumped by a bi-chromatic field comprised of a short pulse and a quasi-monochromatic field: the conditions for full phase matching and quasi-phase matching are analyzed in sections 5 and 6, respectively. Section 7 presents experimental highly chiral HHG spectrum and discusses the attosecond pulses that this spectrum can correspond to.

Polarization in full phase matching of HHG by bichromatic drivers
Phase matching of HHG driven by circularly-polarized bichromatic laser fields is very different from the case of linearly polarized lasers [47,48]. In HHG driven by bi-chromatic linearly polarized light, the relative phase between the two spectral components of the driving field dramatically influences the phase and the intensity of the HHG field [57][58][59], because it changes the maximal instantaneous field and the shape of the bi-chromatic driving field. As a result, the relative phase modifies the tunnel ionization rate of the atom [60], the trajectories of these electrons in the continuum, and the resulting HHG yield [61][62][63]. Figure 1(a) presents the HHG spectrum as a function of the relative phase between the two colors of a co-linearly polarized bi-chromatic driving laser focused into a gas jet (see apparatus in [46]). The inset illustrates the temporal shape of the field for relative phase of 0 and p 2 between the two spectral components. When a colinearly polarized bi-chromatic field propagates in a dispersive medium the phase of the two spectral components travel at different speeds, so their relative phase cannot be fixed. The amplitude and phase of the resulting HHG field oscillate along the propagation direction with a periodicity determined by the refractive indices of the two spectral components. The consequence of these oscillations is that full phase matching of high order harmonics is extremely challenging [64].
On the other hand, in the case of a bi-chromatic circularly polarized driving laser field, the phase between the two spectral components merely rotates the rosette-shaped field in the polarization plane (see inset of figure 1(b)). The relative phase does not change the peak intensity, nor the temporal shape of a pump field and not the resulting HHG spectrum. Figure 1(b) presents a spectrogram of HHG driven by a counter-rotating bi-chromatic pump field (see apparatus in [46]). Clearly, the HHG spectrum is independent of the relative phase between the pump spectral components. In this respect, phase matching of circularly polarized HHG driven by a circularly polarized pump is similar to phase matching of circularly polarized harmonic generation driven by a linearly polarized laser rotating in an optically active medium the HHG spectrum oscillates with the relative phase, f f w w , 2 since the shape of the field,  E t BC ( ) is phase dependent (see inset: red curve for 0 phase and blue curve for p/2 phase). (b) HHG driven by counter-rotating circularly polarized bi-chromatic field, where e L and e R are left-and right-rotating circularly polarized polarizations, respectively. When varying the relative phase, the three-fold rosette shape of such a bi-chromatic field (see inset for 0 and p 2 phase) undergoes a mere rotation, which is spectrally undetectable. We associate the field shape with a Lissajous curve since the field polarization is planar, rather than a line. [33,65]. In both cases, the driving field rotates during propagation, yet its shape and amplitude are conserved. Notably, the fact that the phase of circular HHG is relatively insensitive to intensity variations (compared to linear HHG) [30] makes it more robust to pulse envelop effects and beam focusing dynamics.

Full phase matching condition from photon conservation laws
Below, we generalize the derivation of the phase matching equation in [47] to the case of any bi-chromatic pump; not necessarily a fundamental and its second harmonic, as was recently experimentally demonstrated for soft x-ray HHG driven by 1300 nm and 800 nm circularly-polarized lasers [48]. This section addresses the case where the pulses are long, so temporal variations in their envelope can be neglected. When HHG is driven by a bi-chromatic field composed of co-propagating left-and right-rotating circularly polarized fields, of frequency w 1 and w , 2 respectively, the photons conservation laws are given by: ℓ and m are the number of w 1 and w 2 photons annihilated, respectively, from the bi-chromatic pump to generate a HHG photon at the harmonic order q with frequency w w = q . q 1 If w w 2 1 is an integer, so is q. The HHG photon, w , q inherits the spin of the one extra photon it acquired, and is therefore, left-rotating for = + ℓ m 1 and right-rotating for =ℓ m 1. The conservation of energy and spin angular momentum are illustrated in figure 2(a). The momentum of an HHG photon, k , q (the factor of  is ommited) originates from annihilation of ℓ photons with momentum k 1 and m photons with momentum k 2 (equation (3)). Conservation of linear momentum is the condition for phase matching of circularly polarized harmonics; the HHG process is fully phase matched when equation (3) is fulfilled. However, due to chromatic and plasma dispersion in the generating medium, the fields at w 1 and w 2 have different refractive indices so w w ¹ k k .
The momentum mismatch (also called phase mismatch), Dk, is retrieved from equations (1), (2) and (3) to be Here Dk is the phase mismatch of HHG with frequency The refraction terms Dn 1,2 are given by D = n n n , ) where n 1,2 is the index of refraction including plasma, modal or Gouy phase terms of the pumps w 1 and w , 2 respectively [66,67], and n q is the index of refraction for the harmonic order q. The condition for full phase matching is: According to equation (5), Dn 2 and Dn 1 should have opposite signs. Equations (3)-(5) lead to several conclusions: First, only one HHG order can be fully phase matched for given dispersion terms Dn . 1,2 Let us assume that the phase matched harmonic rotates like the w 1 field. Other co-rotating HHG orders (i.e. orders ℓ=m+1) differ by a small momentum addition w w D + D c n n , ( ) is small since Dn 1 and Dn 2 have opposite signs. Third, all counter-rotating HHG orders ( =ℓ m 1) have a larger phase mismatch than the co-rotating HHG orders. When compared with the phase matched HHG order, HHG orders with opposite helicity lack two w 1 photons so their phase mismatch is approximately )so co-rotating harmonics are better phase matched than counter-rotating orders. Figure 2(b) illustrates the helicity selectiveness of such a phase matching scheme where left-rotating HHG is phase matched and right-rotating HHG exhibits a large phase mismatch (see also figure 1 in [47]).
The use of conservation laws in the above derivation implicitly utilizes continuous symmetries of the system in an intuitive way. The reliance of the phase matching equation on conservation of energy, linear momentum and angular momentum indicates that the system (light and isotropic matter) is invariant to temporal delays, spatial translations and rotations, respectively. However, one can address the system's symmetries directly. In the next section, we derive the phase matching equation using symmetries of the classical pump field propagating in an isotropic medium and retrieve the same phase matching condition as in this section, i.e. equation (4).

Full phase matching condition by vectorial fields symmetries.
In this section, we derive the condition for full phase matching by analyzing the operator that propagates the vectorial pump field in circularly polarized HHG (we assume nothing about the details of HHG or about the photons that participate in the process). As will be shown in this and in following sections, this formalism is much more general than the photon conservation laws approach. To derive the condition for phase matching, we identify a unitary propagator for the bi-chromatic pump in a chromatically dispersive medium and find the condition for which the emitted HHG field builds up coherently throughout the entire generation medium. Notably, unlike phase matching of linearly polarized HHG, where pump and HHG fields have the same polarization, here the fields cannot be represented by complex-valued scalars. Our description accounts for the vectorial nature of the fields.
We start by assuming a bi-chromatic field, composed of circularly polarized, counter-rotating components at frequencies w 1 and w 2 that propagates along z direction in an isotropic medium. For simplicity, we neglect diffraction and solve for time and propagation axes and assume plane wave propagation. We define z=0 in an arbitrary position. The bi-chromatic field,  E , BC at z=0 is given by: Here E 1 and E 2 are scalar complex amplitudes of the two spectral components of the bi-chromatic field. The 2×1 vector is the projection of the electric field on axes x and y in the polarization plane. t represents time shifts relative to a mathematical frame of reference traveling at the phase velocity of w , This becomes useful when we look for a coherent buildup of harmonic order q-by using t instead of t, the phase of a traveling HHG field is z-independent. For a propagating bi-chromatic field, the phase term, w -⋅ e , Therefore, the bi-chromatic field at > z 0 is Note that the dispersion terms n 1,2 may include nonlinear contributions (e.g. Kerr and self-generated plasma) as long as the field intensity is constant throughout propagation. It is instructive to obtain the propagation operator that transfer )of equation (7). To do so, we rearrange the exponential arguments for w 1 and w , 2 as w t d respectively, and find the proper Using t and d (equations (8) and (9)) in equation (7) leads to: Here t ¶ ¶ e t is a unitary time delay operator and d R is the unitary rotation matrix in the polarization plane given by The time delay, t , can be thought of as resulting from an effective refractive index of the bi-chromatic field, is a weighted average of Dn 1,2 with w 1,2 as weights.
In HHG, a process is fully phase matched if emissions from any two propagation distances interfere constructively, regardless of the propagation distance, Dz, between them. Since the bi-chromatic pump field driving the HHG emission merely rotates and delays as it propagates, the HHG fields emitted at different positions are identical up to a rotation angle (same rotation angle as the driver field) and time delay. Thus, for full phase matching of q-order harmonic field generation, the net effect of the rotation and time delay should yield the identity operation (see figure 3): Conveniently, the use of t instead of t eliminates any explicit z-dependency of the HHG field. Equation (12)  Equations (12)- (14) correspond to the case of full phase matching. It is also instructive to formulate the phase-mismatch situation. In this case, we add a phase mismatch term, -D D e , k z i to equation (12) and obtain: = where L c is the coherence length of the conversion process. Thus far we derived two different equations for phase matching of circularly polarized HHG driven by bi-chromatic circularly polarized counter-rotating drivers: equations (4) and (15). Equation (15) is more general as it does not require to 'count' the number of annihilated photons, ℓ and m, as input parameters. The two equations coalesce if one assumes any two out of the three conservation laws (equations (1), (2) or (3)). For example, equation (3) can be derived directly from equations (15), (1) and (2). In a similar fashion, equation (2) can be derived from equations (15), (1) and (3). This feature reveals that the phase matching in our scheme couples the conservation laws for the photons in the process. Figure 3 illustrates the propagation effects on a circularly polarized counter-rotating bi-chromatic field comprising a fundamental field, w , 1 and its second harmonic, w w = 2 . 2 1 As the field travels in a chromatically dispersive medium, the three-fold rosette shape of the field rotates since w 1 and w 2 acquire a phase shift ( figure 3(a)). This screw-like propagation  clearly shows that the pump is chiral, and therefore, one should expect a major difference in the coherence lengths of high harmonics with left helicity versus right helicity. Circularly polarized fields are a unique outcome for such a screw-like propagation since spatial rotation simply induce a phase [33,49,65]. According to equation (13), the HHG phase is matched when the chirally selective phase of the spatial rotation, d, balances the time delay phase, the result should be ω q t , which is a non-chiral effect of the propagation. Figures 3(b) and (c) show d and t for two positions along the propagation axis, = z 0 and = D z z. Notably, while figure 3 illustrates the balance between the time delay and rotation for the case of w w = 2 , 2 1 it works for any w w ¹ 1 2 .

Full phase matching of circularly polarized HHG by a short and a long pump pulses
In the previous sections, we derived the phase matching equation while neglecting the envelopes of the bi-chromatic pump pulses. In this section we extend the analysis to include the effect of pulse envelope. Specifically, we shall discuss full phase matching conditions for circularly polarized HHG pumped by a combination of a short circularly polarized pulse at w 1 and a long counter-rotating pulse at w 2 [54]. The properties of the HHG formed by such a pump field depend strongly on the carrier-envelope phase, that is, the time difference between the pulse envelope and the peak of the oscillating field. Models based on continuous symmetries or conservation laws for photons (e.g. the model in section 3) neglect this dependence and are therefore incomplete. The vectorial fields model (section 4), on the other hand, can be extended to include the effect of the pulse envelope.
In short, if the velocities of the envelope and the peak of the carrier bi-chromatic field are equal (although the carrier field might rotate), then the shape of the field is independent of the propagation. In this case, high harmonics that propagate with the same velocity would be fully phase matched. If, on the other hand, the velocities of the envelope and carrier field differ, then the field's shape oscillates as it propagates through the medium, and the HHG field may only be quasi phase matched (see section 6). In this and the following section the vectorial nature of the pump field is addressed explicitly and accounts for variations of orientation, amplitude and phase of the electric field throughout propagation. For simplicity, we take into account the group and phase velocities of the pump fields but leave out higher order effects such as group velocity dispersion, and effects of ionization within the duration of the short pump pulse.
We assume that the bi-chromatic field,  E , BC p entering the medium at = z 0 is given by: The field at frequency w 1 is confined to a Gaussian envelope with a standard deviation s , t and the pulse duration of the field at w 2 is much longer and is taken as infinite. Here we marked the bi-chromatic field with a superscript 'p' to denote the pulsed nature of the field and to distinguish it from the bi-chromatic fields in previous sections. The pulse envelop is taken as Gaussian for simplicity, however, the derivation is general and can be applied to any pulse envelope. Due to propagation of distance z in the medium, the field undergoes chromatic dispersion (similar to equation (7)) while the pulse envelope aquires a group delay z v , ) We claim that the HHG phase is matched when the field of the HHG order q conforms to the same propagator. We start by extracting the time and rotation operators, y using t and d we eliminate the terms -D e k z n i 1,2 1,2 but add an explicit time shift, t , to the pulse envelope. For the operator to be unitary, as in equation (12), equation (13) should hold and also the term leading to the condition This additional condition for full phase matching cannot be predicted by considering the conservation of energy, spin angular momentum and linear momentum of the pump and HHG photons, and requires the explicit approach presented here. Equation (19) accounts for the relative time delay between the bi-chromatic field (induced by chromatic dispersion) and the group delay of the pulse envelope. When equation (19) is fulfilled and this relative time delay nullifies, the shape of the bi-chromatic field is independent throughout propagation, up to trivial time delays and rotations. The HHG field that this field produces will be identical along propagation, up to these delays and rotation, similarly to the case described in section 4. Therefore, the phase matching condition for the HHG field is equal to the condition in equation (15), with the addition of equation (19)  Although equation (20) is similar to equation (15), in a normally dispersive medium equation (20) allows phase matching only for harmonic orders with the same helicity as w . 1 In a normally dispersive medium > n n , 2 1 » v c q and > c v , g so the±sign must be + in order to nullify Dk. Intuitively, full phase matching occurs only if the envelope aquires the same time delay as the rosette of the bi-chromatic field, and therefore the entire field rotates and delayes in time, without changing its shape. Note that these are indeed the dominant HHG orders in the chiral spectra that we measured in [47] (figure (4) and also in figure (4) of this work). In the next section we show that if the shape of the pumping bichromatic field does vary with propagation, then the HHG process can be quasi phase matched (QPM).

Quasi phase matching of circularly polarized HHG
In this section we consider the case where the envelope and the rosette field do not propagate with equal velocities (i.e. equation (19) is not satisfied), leading to a continuous slip between the phase and the envelope of the bi-chromatic pump. In such a case, the phase and amplitude of the emitted HHG field vary periodically as the pump propagates in the generation medium and HHG can be QPM [67], but not fully phase matched. To find the periodicity of the phase slip we use a property of counter-rotating bi-chromatic pump: a time delay of p w w + 2 1 2 ( )simply rotates the field by a spatial angle pw w w + 2 . 1 1 2 ( ) A rotation by this angle does not change the bi-chromatic field. In the particular case of w w = 2 , 2 1 a delay of p w 2 3 1 (1/3 of an optical cycle) is equivalent to a rotation by 120°, which is exactly the rotation symmetry of the bi-chromatic field. Therefore, although the envelope delay term in equation (18), -) may not nullify continuously, it nullifies after propagating one period, marked L (see notation in section 2.9 in Boyd [68]). For example, if the carrier bi-chromatic field and the envelope reach their maxima together at some position, they also reach their maxima jointly after propagating L, albeit with some time delay and rotation. It is convenient to define a time delay, t L , and to write its relation with L in the following form where t is defined in equation (8). The right-hand side of (21) allows us to nullify the envelope delay term in equation (18), where the definition t p w w = + L 2 1 2 ( ) is usful for our derivation, since it takes into accout the temporal and rotational symmetries of the the bi-chromatic field. We write the bi-chromatic field after it propagates L using equation (18). We employ a unitary operation that adds t L within the brackets and adds t -L to the time delay operator to simplify the envelope group delay term (see equation (21)) The additional delay, t L , is apparent in the phasors The quasi-phase matching condition for the HHG field,  E , q with frequency, w , q can be retreived with the same logic leading to equations (12) and (15). To be QPM, the HHG field should be equal at two positions separated by distance L, where deviations from QPM are expressed by the phase mismatch term, The condition for QPM is retreived by substituting circularly polarized HHG into equation (24) where l is an integer representing the QPM order (not to be confused with ℓ of equation (1)). We use equations (8), (9) and (15) to replace the left brackets with the phase mismatch, -D L k , and use equation (21) to get the explicit QPM equation (1) and (2)) we get a simple condition for QPM p where l is order of QPM and m is related to the HHG order. Equation (27) implies that for QPM, D = k 0, Q the phase mismatch from the bi-chromatic carrier wave, Dk, should balance an integer order of the periodicity of the carrier envelope phase slip, p L 2 . If an harmonic order with a given m is quasi-phase matched according to equation (27) with a given QPM order l, other HHG orders (different m, with equal or opposite helicity) may still nullify Dk Q for a different l, however, such nullification is not guaranteed.
To conclude this section, we showed that quasi phase matching is possible, and it is helicity selective (through the±term in Dk) similarly to full phase matching. However, it is governed by a different equation. The physical interpretation of equations (25)- (27) is more complex than for the case of full phase matching. To understand this system intuitively, consider the case where the envelope of the field at w 1 overlaps with the peak of the bi-chromatic field, and the pulse is short enough to include only a single burst of HHG radiation. Since the group velocity of the short pulse differs from the phase velocity of the field, after some propagation distance, L, the pulse envelope ovelaps with an adjacent peak of the bi-chromatic field's rosette. At that point, the shape of the field is as it was originally up to a temporal delay, t -L t , and rotation, w t L . 1 The HHG emitted from atoms separated by L is identical, up to the these delay and rotation. HHG is QPM if the emissions at every L differ by pl 2 and interfere constructively.

Helically polarized attosecond pulses
Generation of circularly polarized high harmonics by bichromatic counter-rotating circularly polarized drivers results  [47], where the left-rotating HHG orders (red) dominate over the right-rotating HHG orders (blue). Furthermore, the HHG radiation acquires additional chirality since (b) the degree of circular polarization, defined as the normalized Stokes parameter S S , in a comb of harmonic pairs, where in each pair the harmonics rotate in opposite directions, and can span the EUV and soft x-ray regions [46][47][48]. Generally, the average chirality of the field is rather small because the intensities of the left-rotating and right-rotating harmonics are comparable [46]. Thus, in order to generate helically polarized attosecond pulses, we need to, somehow, break this symmetry. We need a helicitydependent filter. As discussed in previous sections, phase matching of circularly polarized high harmonics is strongly helically dependent [47].
In this section, we present experimentally measured spectra of circularly polarized HHG with a high degree of average chirality and show that this measured spectrum can support attosecond pulses with a large degree of circular polarization (the experimental corroboration for phase matching was presented in [47,48]). Figure 4(a) shows a spectrum for circularly polarized HHG from neon that appears on figure 3(b) in [47]. The HHG process was driven by a w 1 (790 nm) circularly polarized left rotating driver and its second harmonic, i.e. w w = 2 , 1are left-rotating, as w , 1 and HHG orders = q m 3 1 are right-rotating, as w , 2 where HHG orders = q m 3 are suppressed (marked red, blue and green, respectively, in figure 4(a)). Clearly, the left-rotating harmonic peaks are 3-5 times higher than their adjacent rightrotating peaks. Notably, we also measured the degree of circular polarization (i.e. the normalized Stokes parameter, S S , 3 0 which is the figure of merit for magnetic chiral experiments. See p 345 in [69]) of four harmonics in this spectrum ( figure 4(b), taken from figure S.3 in the supplementary information of [47]). As shown in figure 4(b), the degree of circular polarization of right-rotating HHG orders are ∼40%, whereas left-rotating HHG orders are ∼100% circularly polarized. In short, the left-rotating part of the HHG radiation is more intense and have higher degree of circular polarization then the right-rotating part of the spectrum.
It is instructive to analyze the time domain field that can correspond to the measured spectrum (figure 4(a)) and the degree of circular polarization ( figure 4(b)). Notably, the spectral phase can be tailored downstream by using thin metal films or multi-layer mirrors [70]. It is therefore informative to calculate the transform-limited field that corresponds to the measured spectrum in figure 4(a). We calculated the time domain field assuming a flat spectral phase and associate leftrotating orders with 100% circular polarization and 40% circular polarization to the right-rotating HHG orders (i.e. ellipticity of 0.21). The resulting normalized temporal intensity is shown in figure 4(c) (red solid curve). It consists of an attosecond pulse train with pulse duration of ∼120 attoseconds. All the pulses in the train have the same helicity.
Importantly, adjacent attosecond pulses have different degree of circular polarization. Of course, highly chiral pulses (high level of circular polarization) contribute more to chirality-sensitive measurements than attosecond pulses with low chirality. The dashed blue line in figure 4(c) follows the instantaneous Stokes parameter S , 3 which accounts for the temporal contribution of the HHG field to a chiral asymmetry measurement; field intensity multiplied by the degree of circular polarization. To plot the blue curve, we project the electromagnetic field on left-and right-rotating circular polarization and write S 3 as the normalized difference between the intensity of the left-rotating field, I , Left and the right rotating field, I . Right By plotting the expected chiral sensitivity, we note a new and interesting feature in attosecond science. Even though the radiation comprises a train of attosecond pulses, the contribution to the chiral asymmetry may dominantly originate from a single attosecond pulsethe most chiral pulse (to be exact, the one with the highest Stokes parameter S . 3 ) In addition, in figure 4(c) we actually ignored contributions of the = q m 3 harmonic orders for this time domain analysis. In practice, the suppressed peaks are left-rotating with 30%-50% degree of circular polarization (see figure 4(b) in [47]), and therefore, may enhance the chirality sensitivity of the dominantly chiral attosecond pulse even further, with respect to adjacent attosecond pulses. Such an effectively chirality-sensitive isolated attosecond pulse may lead to the birth of chirally selective attosecond science, where the dynamics of spin, magnetism, and angular momentum is accessible with unprecedented temporal resolution. Finally, figure 4(d) presents a measured HHG spectrum (recorded in the same experiment under similar conditions as in figure 4(a)) with an even greater ratio (10 to 1) between left-and right-rotating HHG orders. We did not measure the degree of polarization for this spectrum and therefore did not analyze its corresponding time-domain field.

Conclusions
In this work, we explored phase matching of circularly polarized HHG pumped by a circularly polarized counterrotating bi-chromatic field. Using two independent analytical approaches, we generalized the phase matching equation to apply for a long bi-chromatic pump field with any two spectral components, w 1 and w . 2 In the first approach, we imposed conservation laws for the energy, momentum and angular momentum of the photons involved in the HHG process. In the second approach, we analyzed the explicit mathematical propagator of a vectorial bi-chromatic pump field propagating in an isotropic chromatically dispersive medium. We derived a new propagation equation corresponding to phase matching for vectorial fields. This vectorial propagation condition is inherently different from phase matching of linearly polarized HHG, where the pump and HHG fields have the same polarization, so phase matching considers only the scalar amplitudes and phases. We show that the propagation equation derived from the propagator of a bi-chromatic circularly polarized pump reveals a unique symmetry. It links three conservation laws for the photons (energy, linear momentum and angular momentum): one can derive the conservation of energy, linear momentum or angular momentum from the phase matching equation (derived by the vectorial propagation condition) and the two other conservation laws. We discussed the symmetry of a system in which HHG is pumped by a long pulse and a counter-rotating short pulse. By taking into account the effects of a temporal envelope on such a pulse, we found two additional regimes for phase matching and quasi-phase matching, and showed that these regimes are also helicity selective. In the last section, we showed that the highly chiral spectrum that we observed experimentally in [47] can support chiral attosecond pulses. We expect chiral HHG [71,72] and attosecond pulses to open numerous opportunities to observe the dynamics of chiral phenomena, like magnetism and diverse charge, spin and orbital orderings, using all-optical tools with unprecedented temporal resolution and in spectral regions unavailable to circular polarization thus far.