Phase Matching Filtering (PMF)

• Phase Matching Filtering (PMF) is a spectro-temporal shaping technique that uses programmable longitudinal modulation in the HHG medium to control attosecond pulses.
• It employs partial and quasi-phase matching to adjust amplitude and phase profiles, effectively compensating for intrinsic chirp and enabling custom pulse train synthesis.
• PMF overcomes conventional bandwidth limitations by integrating tailored spatial masks and interferometric techniques directly into the high-order harmonic generation process.

Phase-Matching Filtering (PMF) is a spectro-temporal shaping technique for attosecond pulse generation in high-order harmonic generation (HHG), whereby a programmable longitudinal modulation of the source (amplitude and/or phase) in the generation medium serves as a filter in the frequency domain. By harnessing partial phase matching, including quasi-phase matching (QPM), PMF enables the synthesis of complex attosecond pulse trains, controllable amplitude and phase profiles over broad bandwidths, and deterministic removal of the intrinsic attosecond chirp—all without relying on conventional spectral optics but instead on spatial modulation schemes directly embedded within the HHG process (Austin et al., 2013).

1. Fundamental Principles of Partial Phase Matching and Quasi-Phase Matching

In HHG, the macroscopic harmonic field at frequency $\omega$ is the superposition of single-atom dipole emissions integrated along the propagation axis $z$. The local dipole source has a phase $\varphi_{dip}(z, \omega)$ which generally differs from the phase $k_h(\omega) z$ of the propagating harmonic field, giving rise to phase mismatch:

$$\Delta k(\omega) \equiv k_h(\omega) - k_{dip}(\omega)$$

Nonzero $\Delta k$ leads to destructive interference among harmonics from different points along $z$, suppressing efficient harmonic buildup. Partial phase matching is achieved by introducing a modulation $M(z)$ of the dipole source—typically sinusoidal or prescribed via a grating—with spatial frequency $K$ chosen to counteract the phase mismatch. In uniform QPM, $M(z) = \cos[K z]$ with $K = \Delta k(\omega_0)$ for a reference frequency $\omega_0$. The spatial Fourier component $e^{i(k_{dip} + K)z}$ then phasematches the field to $k_h$, restoring constructive interference and enabling control over the macroscopic harmonic emission.

2. PMF as a Programmable Filter: Spectral Filter Function

The emergent macroscopic field is given by integrating the modulated local source along $z$:

$$E_{out}(\omega) \propto \int_0^L M(z) e^{-i\Delta k(\omega) z} \, dz$$

This defines the filter function:

$$F(\omega) = \int_0^L M(z) e^{-i\Delta k(\omega) z} \, dz$$

Thus, the small-signal output field is $$E_{out}(\omega) = F(\omega) E_{single-atom}(\omega)$$. The spatial pattern $M(z)$—the "mask"—determines the amplitude $|F(\omega)|$ and phase $\arg F(\omega)$ of the spectral filter, directly analogous to programmable amplitude and phase masks in the frequency domain. By careful design of $M(z)$, PMF enables arbitrary spectral shaping of the attosecond emission within the constraints of the imported phase-mismatch function.

3. Compensation of Attosecond Intrinsic Chirp

In HHG, the single-atom emission is characterized by an intrinsic chirp, as higher frequencies are emitted earlier within each half-cycle. Attosecond pulse compression accordingly requires imposition of a compensating quadratic spectral phase $\varphi_{PMF}(\omega)$. This can be achieved by designing $M(z)$ so that the QPM grating vector $K(z)$ varies linearly with $z$: $K(z) = K_0 + K_1 z$. Under such conditions, the stationary-phase analysis of the output field yields a filter function $F(\omega)$ with quadratic spectral phase of sign opposite to the intrinsic chirp. For the typical case where $\Delta k(\omega) \approx (\Delta n \, \omega)/c$, the compensation is:

$$z(\omega) = \frac{\Delta n \omega / c - K_0}{K_1} \qquad \Rightarrow \qquad \varphi(\omega) \simeq -\frac{(\Delta n/c)^2 (\omega - \omega_0)^2}{2 K_1}$$

Tuning $K_1$ allows one to set the residual chirp to zero, thereby generating nearly transform-limited attosecond bursts.

4. Custom Pulse Sequence Synthesis via Longitudinal Masking

Because the spatial modulation $M(z)$ and frequency $\omega$ are mapped one-to-one via $\Delta k(\omega) = K(z)$, PMF allows full programmability of both $|F(\omega)|$ and $\arg F(\omega)$. To realize a desired spectral transfer function $H(\omega) = |H(\omega)| e^{i\psi(\omega)}$, inverse mapping determines the necessary $M(z)$: for each $z$, set the local amplitude and phase of the QPM grating. For instance, a double pulse separated by $\Delta t$ requires $H(\omega) = 1 + e^{-i\omega \Delta t}$, leading ideally to $$M(z) \propto \delta(z - z_0) + \delta(z - z_0 - \Delta z)$$, with $\Delta z$ set by $\Delta k(\omega)$. In realistic regimes, $M(z)$ is implemented as a continuous grating with specifically designed tilted segments, the beat notes of which encode the required spectral interference structure.

5. Bandwidth Constraints and PMF's Overcoming of Limitations

Standard XUV spectral elements—such as multilayer mirrors, metal foils, or gas filters—have bandwidths of $30$–$50$ eV and tunability limited to $\sim$150 eV; full compensation of the attosecond chirp across $>100$ eV continua is precluded. PMF, in contrast, leverages HHG's intrinsic broadband output ($200$–$1600$ eV, supporting potentially $<3$ as) by crafting $M(z)$ patterns over length scales from microns to millimeters using modern pulse-shaping and interferometric techniques. In simulated and practical scenarios, typical parameters are: interaction length $L\sim1$ mm; modulation period $\Lambda(z)=2\pi/K(z)$ ranging from tens to hundreds of microns; and gas pressure and focal geometry optimized such that $\Delta k(\omega, z)$ is smooth and trackable by $K(z)$.

6. Implementation Modalities and Prospects for Extension

PMF is physically implemented, for example, by a counter-propagating pulse train (CPT) at the fundamental wavelength. Programmable near-IR pulse shapers define the CPT profile, hence the imposed $M(z)$ on the local dipole phase. Practical realization also requires precise spatio-temporal overlap, high carrier-envelope phase stability for both driving and CPT pulses, and careful management of neutral and plasma dispersions. PMF strategies can be extended to soft x-ray and kilo-electronvolt photon energies utilizing grating-assisted QPM via solid insertions, ultra-low-dispersion hollow waveguides or capillaries to retain desired $\Delta k(\omega)$ profiles, and high-energy (mid-IR) drivers to access higher HHG cutoffs.

7. Significance and Summary

PMF transforms phase mismatch—a traditional hindrance in nonlinear harmonic conversion—into a tunable, programmable filter that enables comprehensive shaping of attosecond emission directly in the HHG process. By longitudinal modulation of $M(z)$, PMF affords in-situ spectral and temporal control over harmonic pulses, circumvents bandwidth limitations of conventional filters, removes atto-chirp, and supports user-defined attosecond pulse syntheses (Austin et al., 2013). The approach underpins novel capabilities in ultrafast science, including XUV and x-ray attosecond pulse applications, while remaining compatible with the demanding requirements of phase coherence and bandwidth scaling.