Quasi-phase-matching of only even-order high harmonics

We propose a method for generating only even-order high harmonics. We formulate a condition for a shift in the relative-phase between bi-chromatic drivers that leads to a field sign-flip of only even-order harmonics. Induction of this sign-flip periodically during propagation gives rise to quasi phase matching of only even harmonics. We demonstrate this technique numerically and also show that it leads to attosecond pulse trains with constant carrier envelop phase with high repetition rate. This work opens the door for quasi-phase matching of high harmonics with a designed selective enhancement function.

implemented for HHG from gases. In addition to the fundamental science interest in production of pure even harmonics spectra, it may also be useful for applications because a set of phase-locked even harmonics correspond to APT at high repetition rate and stable carrier envelop phase (CEP) (all previously proposed and demonstrated techniques for CEP stabilization of APT reduce the APT repetition rate [13][14][15].) Here we suggest a scheme for generation of only even-order harmonics which is based on quasi phase matching (QPM). As in other optical nonlinear processes, HHG can be divided to a regime in which it is phase matched and a regime in which it suffers from phase mismatch [16]. Several QPM techniques have been developed in order to enhance the HHG conversion efficiency in the phase-mismatch regime [17][18][19][20][21][22][23][24][25][26]. QPM techniques amplify a spectral region, yet selective control within that region was not obtained. All-optical QPM techniques employ additional weak field in order to coherently control the re-colliding and radiating electronic wave-functions [18,[21][22][23][24]. The weak driver slightly modifies the electronic trajectories (e.g. by changing the recombination time with attosecond resolution), giving rise to a controlled phase-shift in the phase of the emitted harmonics. Properly designed modulations of the phase-shifts with periodicity that corresponds to two coherence length of the HHG process can lead to efficient QPM.
Here, we propose all-optical QPM of only even-order high harmonics, within a spectral region that include more than 10 harmonics. Both odd and even order high harmonics of a fundamental driver are generated in isotropic and homogeneous media when a secondary driver breaks the half-wave symmetry of the joint pump field. We formulate a condition for a shift of the relative-phase between bichromatic drivers that leads to a sign-flip in the fields of only even-order harmonics. Induction of this sign-flip periodically during propagation gives rise to QPM of only even-order harmonics. We demonstrate numerically QPM of only even-order plateau or cutoff harmonics using ti:sapphire pump and its second harmonic weak field that propagate in a dispersive medium. We also numerically demonstrate QPM of even harmonics using weak static field which can be approximated using CO 2 or terahertz pulses. Finally, we show that the generated APT exhibits constant CEP and that it consists of two pulses per pump cycle.
In harmonics generation from isotropic and time-independent nonlinear medium, the emitted harmonics field exhibits the same dynamical symmetry as the driving field. For example, a quasi-monochromatic driver field, E D , at angular frequency ω 0 =2π/T, where T is the optical cycle, is half-wave symmetric: , hence the harmonics field, E HHG , exhibits the same symmetry: E HHG (t+T/2)=-E HHG (t).
The spectrum of this field consists of only odd harmonics of ω 0 because symmetry dictates that even Fourier components of half-wave symmetric functions are zero. The HHG spectrum can include evenorder harmonics if a secondary field breaks the fundamental driver half-wave symmetry. This concept was implemented in many experiments where HHG was driven by bi-chromatic drivers that consist of a strong pump and its second harmonic [8]. Also, HHG spectra include both odd and even harmonics of ω 0 when a weak static field (or a very long-wavelength field) is added to the main strong pump [27].
We first present a new symmetry feature for harmonics that are generated by bi-chromatic drivers. We will later employ this feature for QPM of only even-order harmonics. Consider bi-chromatic drivers E BC =A 0 cos(ω 0 t+φ 0 )+A 1 cos(ω 1 t+φ 0 +Δφ) where ω 0 =2π/T 0 and ω 1 =2π/T 1 are angular optical frequencies, T 0 and T 1 are optical cycles, A 0 and A 1 are real amplitudes, φ 0 is a global phase, and Δφ is the relative phase between the two components. We compare between the harmonic fields driven by the bichromatic fields with the following relative phases: Δφ a =0 and Δφ b =π(1-ω 1 /ω 0 ). We assign the generated harmonic fields by Et , respectively. It is straight forward to verify that E BC (t,Δφ a )=-E BC (t+T 0 /2,Δφ b ). The harmonics fields should also conform to this symmetry, hence Inserting the Fourier decomposition of the emitted harmonic field, (1) leads to Equation (2) shows that the odd-order harmonics of the bi-chromatic drivers are invariant to a π(1ω 1 /ω 0 ) phase-shift of the relative phase, while at the same time, the sign of the even-order harmonics is flipped. This feature is the source for our proposal for QPM of only even-order harmonics. Here, we explore numerically two specific configurations for the bi-chromatic drivers where in both cases the strong pump corresponds to a ti:sapphire laser pulse with central frequency 15 0 ω =2.3×10 Hz . In the first case, the secondary driver is at much smaller frequency than the pump ω 1 <<ω 0 (e.g. terahertz or CO 2 laser) such that within the pulse-duration of the strong pulse, the field is approximately constant.
Numerically, we use a static field for this case. In the second case, the second driver is the second harmonic of the strong pump. We get Δφ b =π for both cases which corresponds to a change in the sign of the static or second harmonic fields.
Having found a symmetric feature that distinguishes between odd and even harmonics, we now explore it numerically for two specific examples that we will later employ for QPM. In our numerical calculations, we apply the single effective electron approximation and solve the emitted harmonics from singly-ionized xenon (second ionization potential is I p =21 eV) using one-dimensional timedependent Schrodinger (1D TDSE) solver. In the first case, the bi-chromatic drivers are Next, we employ the symmetry feature of HHG driven by bi-chromatic drivers for demonstrating numerically QPM of only even-order harmonics in a gas of singly-ionized xenon ions and their free electrons [28]. The strong driver component is a ti:sapphire laser pulse (central wavelength is 0.8 µm) that is initially in the form of  is the plasma frequency, where e and m are the electron charge and mass, respectively. The density of free electrons, n e , takes into account the pre-formed plasma and the ionization that is calculated by using the ADK model [30]. The high-order polarization, P HHG , is calculated through numerical calculation of the 1D TDSE under the influence of the total field E 0 +E DC .
The generation and evolution of the HHG field, E HHG , is described by: Figure 3a shows the HHG spectrum after propagation distance of 0.5 mm with gas pressure of 25 torr when d DC =18 µm which corresponds to the coherence length of the 88 th cutoff harmonic (coherence lengths were calculated for the processes driven by the fundamental field only). For comparison, the generated spectrum with constant static field is also presented. A clear QPM enhancement is obtained around the 88 th harmonic. Figure 3b shows the coherent buildup of the 88 th and 87 th harmonic fields, showing clearly that the odd harmonic experience a QPM enhancement while the even harmonic suffers from phase-mismatch. Notably, the QPM efficiency of the 88 th harmonic is 0.27, which is relatively high for QPM in HHG [31]. Figure 3c shows the HHG spectrum that is generated when d DC =28 µm which corresponds to the coherence length of the 70 th plateau harmonic. Clear QPM enhancement is obtained around the 72 th harmonic. Figure 3d shows the coherent buildup of the 70 th and 71 th harmonic fields, showing again that the even harmonic experience a QPM enhancement (with 0.23 QPM efficiency) while the odd harmonic suffers from phase-mismatch. The generated evenharmonics correspond to high repetition-rate APT with stable CEP. This feature is demonstrated in Fig.   4. Fig 4a shows the normalized APT, E QPM (t) that corresponds to the red spectrum in Fig. 3a in the spectral region 83±5 harmonics. Figure 4b shows the average of E QPM and its T 0 /2 time-delayed, showing that this APT has a stable CEP. Notably, the temporal distance between consecutive pulses is T 0 /2. That is, in contrast to previous methods [13][14][15], APT with stable CEP is obtained without reduction of the repetition rate. For comparison, Fig. 4c shows E SA (SA stands for single atom) which corresponds to the APT generated by the same strong pump beam, but without propagation and without the static field. The average of E SA and its T 0 /2 time-delayed show that consecutive pulses have opposite phases (Fig 4d).
Next, we demonstrate numerically QPM of only even-order harmonics when the secondary driver is the second harmonic of the strong pump. We assume that the second harmonic field experiences an effective refractive index that is Δn smaller than the refractive index of the strong pump. This scenario can be implemented experimentally by using highly dispersive nonlinear medium [28], or by utilizing spatial dispersion in hollow planar waveguide [32], as proposed in Ref. 22. As a result of the dispersion, the relative phase between the drivers evolves during propagation, and after some propagation distance, Lπ, , it acquires a π shift. QPM is obtained if this distance corresponds to the coherence length of the process, Lπ=L C . The incident beam in our simulation is E BC =E 0 +E 1 where E 0 is the same as in the previous section and     . We simulated the propagation of the beam using the following equation: , which is zero at the spectral region near ω 0 and -Δn in the region around 2ω 0 . The third term in Eq. (5) gives rise to the assumed dispersion, only. Figure 5a shows the HHG spectrum when Δn=8.7×10 -3 (L π =46 µm) and after propagation distance 1 mm. For compression, the generated spectrum when Δn=0 is also presented. A clear QPM enhancement is obtained around the 86 th harmonic. Figure 5b shows the coherent buildup of the 86 th and 85 th harmonic fields, showing clearly that the even harmonic experience a QPM enhancement (QPM efficiency is 0.27) while the odd harmonic suffers from phase-mismatch. Figure 5c shows the HHG spectra when Δn=7×10 -3 (L π =57µm) and, for compression also the Δn=0 case. A clear QPM enhancement is obtained around the 70 th harmonic. Figure 5d shows the coherent buildup of the 70 th and 71 th harmonic fields, showing that the even harmonic experience a QPM enhancement (QPM efficiency is 0.14) while the odd harmonic suffers from phase-mismatch.
In conclusions, we propose and demonstrated numerically a technique for generating only even-order harmonics, within a spectral region that contain ~10 harmonics, using quasi-phase matching. This