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### Modifying the high-harmonic line spacing through necklace-driven HHG

*44*,

*45*) or by creating them using either spatial light modulators (

*46*) or specially tailored diffractive optical elements (

*47*,

*48*). Here, we form necklace beams with a distinctive phase structure by spatiotemporally overlapping two femtosecond laser pulses with identical duration, wavelength, and polarization, but opposite and nondegenerate OAM (

*ℓ*

_{1}

*= |ℓ*

_{1}|,

*ℓ*

_{2}

*=*−

*|ℓ*

_{2}|). The composite electric field exhibits a modulated intensity necklace structure, with evenly spaced lobes of similar amplitude arranged at a constant distance from the optical axis, and with a relative phase shift between neighboring lobes. Figure 1A shows the intensity-modulated phase profile at the focus resulting from the superposition of two vortex beams with

*ℓ*

_{1}= 2,

*ℓ*

_{2}= −3, as well as the far-field spatial intensity distribution of a given harmonic order (the 25th). Although the driving field contains a singularity and hence has zero intensity at all points along the optical axis, we observe numerically and experimentally that a subset of harmonic orders develops a bright on-axis maximum upon propagation. We further find that by using different combinations of OAM beams to synthesize the necklace driver, this subset can be varied, allowing us to tune the line spacing of the harmonic comb emitted on the optical axis without altering the driving laser wavelength.

*N*= |

*ℓ*| + |

_{1}*ℓ*| lobes equidistant from the optical axis/origin, where the relative phase offset of the fundamental field across the

_{2}*n*th lobe is constant and equal to (2π

*n*|

*ℓ*

_{1}|)/(|

*ℓ*

_{1}| + |

*ℓ*

_{2}|) (see the Supplementary Materials). Treating each lobe as a radiator of the

*q*th-order harmonic, which is coherent with the driving laser, the total field at a point $\overrightarrow{{r}_{f}}$

away from the source plane is the sum of fields propagated from all the lobes in the necklace and can be approximated by

(1)

The second phase term *qk*_{0}*d _{n}* describes the phase accumulated by the

*q*th harmonic propagating a distance

from the *n*th lobe to the observation point. Let us now consider the harmonic emission that is emitted on-axis. For all points lying along the optical axis, these *d _{n}* =

*d*are equal. Thus, the propagation amounts to a constant phase shift (independent of

*n*), which we can omit without loss of generality. The final phase term can be simplified by reordering the terms in the summation, yielding (see the Supplementary Materials)

(2)

where we have introduced ξ_{1} = *L*_{lcm}/*|ℓ*_{1}| and ξ_{2} *= L*_{lcm}/*|ℓ*_{2}|, with *L*_{lcm} being the least common multiple of |*ℓ*_{1}| and |*ℓ*_{2}|. From the above equation, we can observe that, for harmonics where the quantity *q*/(ξ_{1} + ξ_{2}) is an integer, the emission from all lobes arrives in phase, resulting in a maximum on axis, while for all other harmonics the contributions sum to zero. In other words, for certain harmonic orders, the transverse phase matching of the harmonic emission at the different lobes results in a constructive interference in the far-field optical axis. Taking into account the additional constraint that *q* must be odd due to inversion symmetry results in an HHG comb with line spacing equal toThe temporal counterpart of this modified line spacing manifests in the periodicity of harmonic emission recorded on the optical axis. In Fig. 1B, we perform a time-frequency analysis of the on-axis harmonic signal extracted from our simulation results in He at 800 nm (see theoretical methods below). We compare the harmonic emission driven by a standard Gaussian beam against that driven by a necklace beam with OAM content *ℓ*_{1} = 2, *ℓ*_{2} = −3. For the Gaussian driving beam, harmonics are emitted every half period of the driving field, showing a periodicity of ∆*t* = *T*_{0}/2 (where *T*_{0} = 2π/ω_{0} is the optical cycle associated to the central frequency of the driving field), which physically corresponds to the cadence of the ionization-rescattering mechanism leading to HHG. This structure corresponds to a harmonic frequency comb composed of odd harmonics, with spacing ∆ω = 2ω_{0} (see Fig. 1C). However, for the combined *ℓ*_{1} = 2, *ℓ*_{2} = −3 OAM driving field, harmonic events are observed on-axis 10 times—i.e., 2(ξ_{1} + ξ_{2})—per period of the driving frequency, reflecting the coherent addition of the 5—ξ_{1} + ξ_{2}—unique HHG emitters in the necklace. As a consequence, the on-axis harmonic emission shows a periodicity of ∆*t* = *T*_{0}/10—∆*t* = *T*_{0}/2(ξ_{1} + ξ_{2})—corresponding to a harmonic frequency comb with the line spacing given by Eq. 3. We emphasize that this is accomplished without altering the wavelength of the driving laser or the microscopic dynamics. The spectral changes arise purely as a result of the macroscopic arrangement of the phased emitters.

*40*). In particular, neglecting the intrinsic or dipole phase contributions (

*40*), the

*q*th-order harmonic order has allowed OAM channels given by

*ℓ*

*=*

_{q}*nℓ*

_{1}+ (

*q − n*)

*ℓ*

_{2}, where

*n*is a positive integer representing the number of photons of the

*ℓ*

_{1}driver. If we apply this conservation rule to our scheme where the two drivers have opposite, nondegenerate OAM, i.e.,

*ℓ*

_{1}= |

*ℓ*

_{1}| and

*ℓ*

_{2}= −|

*ℓ*

_{2}|, we can readily observe that high-order harmonics emitted on-axis, i.e., with

*ℓ*

*= 0, are generated if*

_{q}*n*|

*ℓ*

_{1}| = (

*q − n*)|

*ℓ*

_{2}|. To extract the allowed harmonics that fulfill this condition, and thus the content of the harmonic comb emitted on-axis, we again denote

*L*

_{lcm}as the least common multiple of |

*ℓ*

_{1}| and |

*ℓ*

_{2}|, which fulfills η

*L*

_{lcm}

*= n|ℓ*

_{1}| = (

*q − n*)|

*ℓ*

_{2}

*|*, with η being an integer. Retaining the definitions of ξ

_{1}and ξ

_{2}, the harmonic orders emitted with

*ℓ*

*= 0 must fulfill*

_{q}*q*= η(ξ

_{1}+ ξ

_{2}). Taking into account that

*q*must be odd due to the inversion symmetry, η must be odd, and the high-order harmonics that are emitted on-axis arewhere

*m*= 0, 1, 2…, again leading to the line spacing rule given by Eq. 3.

*ℓ*

_{1}| and |

*ℓ*

_{2}|. We see that the appearance of on-axis harmonics with modified spectral spacing is a necessary consequence of a fundamental conservation law for OAM in HHG, as the OAM content of the driver determines which harmonic wavelengths have an allowed

*ℓ*

*= 0 channel. This interpretation naturally implies that the intensity ratio between the driving beams can be chosen to optimize the*

_{q}*ℓ*= 0 (on-axis) contribution (see the Supplementary Materials). Note that those harmonics that are not emitted on-axis have nonzero OAM, and although they are present in the HHG beam, they present a singularity at the center.

_{q}*49*) (see Methods), a method that was used in several previous…

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Read More:Necklace-structured high-harmonic generation for low-divergence, soft x-ray harmonic combs