## Abstract

High harmonic spectrum of a quasi-monochromatic pump that interacts with isotropic media consists of only odd-order harmonics. Addition of a secondary pump, e.g. a static field or the second harmonic of the primary pump, can results with generation of both odd and even harmonics of the primary pump. We propose a method for quasi-phase matching of only the even-order harmonics of the primary pump. We formulate a theory for this process and demonstrate it numerically. We also show that it leads to attosecond pulse trains with constant carrier envelop phase and high repetition rate.

© 2014 Optical Society of America

## 1. Introduction

High harmonic generation (HHG) of visible and infrared laser pulses in gases is a useful process for production of coherent extreme ultraviolet and soft x-ray radiation from a tabletop system [1,2]. Applications of HHG include production of attosecond pulse trains (APT) [3], isolated attosecond pulses [4], ultrafast holography [5], coherent diffractive imaging [6], and more. The most common selection rule in HHG from gases media is the absent of even harmonics. This feature reflects the inversion symmetry of the gas and the half-wave symmetry of the driver [7]. Indeed, if the molecules in the gas are oriented [8] or if the half-wave symmetry of the driver is broken [7,9–13] then the HHG spectrum consists of both odd and even harmonics. It is natural to ask if an HHG experiment can produce the even-order harmonics without the odd-order ones. This conceptual problem was first confronted in Ref. 14 which proposed a coupling geometry of anisotropic quantum dots that can generate terahertz high harmonics where the odd and even harmonics are polarized perpendicularly. This approach cannot be implemented for isotropic media, e.g. HHG from gases. In addition to the fundamental science interest in production of pure even harmonics spectra, it may also be useful for applications. For example, a set of phase-locked even harmonics correspond to APT at high repetition rate and stable carrier envelop phase (CEP) (in contrast, all previously proposed and demonstrated techniques for CEP stabilization of APT reduce the APT repetition rate [10,11].)

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. Several QPM techniques have been developed in order to enhance the HHG conversion efficiency in the phase-mismatch regime [15–24]. 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 [16,19–22]. 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 the process is driven by bi-chromatic field that does not exhibit half-wave symmetry. We first show, analytically and numerically, that an appropriate shift in the relative-phase between the bi-chromatic pumps can result with sign-flips in the fields of only the even-order harmonics (and not in the fields of the odd harmonics). Induction of this sign-flip periodically during propagation can give 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.

## 2. Symmetry of HHG driven by a bi-chromatic driver

In harmonics generation from isotropic and time-independent nonlinear medium, half-wave symmetry is transferred from the pump to the nonlinear polarization [7]. 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: E_{d}(t + T/2) = -E_{d}(t), 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 even-order harmonics if a secondary field breaks the 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 [9,10]. 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 [11–13].

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 bi-chromatic fields with the two following relative phases: Δφ_{a} = 0 and Δφ_{b} = π(1-ω_{1}/ω_{0}). We assign the generated harmonic fields by ${E}_{{}_{HHG}}^{a}\left(t\right)$and ${E}_{{}_{HHG}}^{b}\left(t\right)$, respectively. It is straight forward to verify that E_{BC}(t,Δφ = Δφ_{a}) = -E_{BC}(t + T_{0}/2,Δφ = Δφ_{b}). The harmonics fields also conform to this symmetry, hence

_{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 ω

_{0}= 2.3 × 10

^{15}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 asymmetry 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 applied the single effective electron approximation and solved the propagation
of the electron wave-packet, $\psi (t)$, using one-dimensional time-dependent Schrodinger (1D TDSE)
solver. The atomic potential is given by $V(x)=-{v}_{0}\mathrm{exp}\left(-{x}^{2}/{\Delta}^{2}\right)$where v_{0} = 1.3 and Δ = 1 in atomic units and x is
the polarization direction of the laser pulse. The ground state ionization energy of this
symmetric potential is I_{p} = 21 eV, corresponding, for example, to neutral neon and
singly-ionized xenon ion. Initially, i.e. in the leading edge of the driver pulses, the electron
fully populates the ground state. The polarization is calculated by: ${P}_{HHG}=e\u3008\psi (t)\left|x\right|\psi (t)\u3009$ where e is the electron charge. The emitted field is proportional
to the second time-derivative of P_{HHG} [2]_{.} In the first case, the bi-chromatic drivers are
${E}_{BC}=\sqrt{{I}_{0}}{A}_{0}\left(t\right)\mathrm{cos}\left({\omega}_{0}t\right)+{E}_{DC}$where${I}_{0}=6\times {10}^{14}W/c{m}^{2}$,${A}_{0}(t)=\mathrm{exp}\left[-{\left(2t/\tau \right)}^{10}\right]$,$\tau =50fs$ is the pulse duration, and E_{DC} is the amplitude of the
static field. The cutoff frequency of the HHG spectrum corresponds to the 87th harmonic of the
strong pump. Figures 1(a)-1(c) display the emitted phase of several harmonics order as a function of the static
field. As expected from our symmetry feature, the phases of even-order harmonics, both at the
cutoff and plateau spectral regions, are flipped by π when the static field changes sign.
The phases of odd harmonics, on the other hand, do not exhibit such a flip (Fig. 1(c)). Figures 1(d)-1(f) show the intensity of the harmonics as a function of the
static field. As shown, the strength of the even harmonics at E_{DC}~2 ×
10^{6} V/cm is comparable to the strength of odd harmonics without static field. In the
second case, we used a bi-chromatic driver of${E}_{BC}={A}_{0}\left(t\right)\left[\sqrt{{I}_{0}}\mathrm{cos}\left({\omega}_{0}t\right)+\sqrt{{I}_{1}}\mathrm{cos}\left(2{\omega}_{0}t+\Delta \phi \right)\right]$ where we use three different values for the peak intensity of the
secondary field: ${I}_{1}=0.6\times {10}^{11}W/c{m}^{2}$, $2.4\times {10}^{11}W/c{m}^{2}$ and $9.6\times {10}^{11}W/c{m}^{2}$. Figures 2(a)-2(c) display the emitted phase of several harmonics order as a function of the relative
phase (i.e. of Δφ). As expected from Eq.
(2), the field of even harmonics flip their sign (acquire a π phase shift) as a
result of a π-shift in the relative phase. At the same time, the phases of odd harmonics
are quite constant [Fig. 2(c)]. Notably, within the range
shown in Figs. 2(a)-2(c), the variations of the harmonic phases are largely insensitive to the intensity of
the second harmonic. Figures 2(d)-2(f) show the intensity of the harmonics as a function of the relative phase.
As shown, the strength of even harmonics is in the same order of magnitude as the strength of
odd harmonics.

## 3. QPM of only even-order harmonics

Figures 1 and 2 clearly demonstrate a new symmetry feature of bi-chromatic pumps (Eq. (2) with angular frequencies ω_{0} (primary field) and ω_{1} (secondary field). We employ this symmetry feature for QPM of only the even-order harmonics. The method is based on the following concept: The setting is engineered such that the relative phase between the primary and secondary fields is shifted by Δφ_{b} = π(1-ω_{1}/ω_{0}) every propagation distance that corresponds to the coherence length of a q-order harmonic, L_{C.} (The coherence length is calculated when only the primary field is present because the secondary field is relatively weak; hence it approximately does not change the plasma density which is the main source for the phase mismatch). For large q, the coherence lengths for consecutive odd and even harmonic are very similar. But, there is also additional phase that results from the presence of the weak field. This additional phase is described by Eq. (2). For an even q harmonic, the fields emitted at propagation distances z and z + L_{C} interfere constructively because the π phase-shift due to the phase-mismatch is canceled by the π phase of Eq. (2). On the other hand, odd-order harmonics that are generated in z and z + L_{C} interfere destructively because they experience π phase-shift due to the phase mismatch and 0 phase shift due to the symmetry feature of Eq. (2). Thus, only the even order harmonics experience QPM in such a setting.

Next, we demonstrate numerically QPM of only even-order harmonics in a gas of singly-ionized
xenon ions and their free electrons [18] (we used this
medium because it exhibits large and relatively constant dispersion, facilitating the generation
of APT with stable CEP that are presented in Fig. 4). The strong driver component is a
ti:sapphire laser pulse (central wavelength is 0.8 µm) that is initially in the form of
${E}_{0}\left(z=0,t\right)={A}_{0}\left(t\right)\mathrm{cos}\left({\omega}_{0}t\right)$ where ${A}_{0}\left(t\right)=\sqrt{{I}_{0}}\mathrm{exp}\left[-{\left(2t/{\tau}_{0}\right)}^{4}\right]$, ${\tau}_{0}=36fs$, ${I}_{0}=6\times {10}^{14}W/c{m}^{2}$ and it propagates in z direction. In the first scheme, the
secondary driver is a static field that flips its sign every propagation distance
d_{DC}: ${E}_{DC}(z)=g(z)\cdot 2.6\times {10}^{6}V/cm$ where g(z) = ± 1 and g(z + d_{DC}) = -g(z). This
scheme can be implemented experimentally using the setup proposed in Ref. 21. We simulated the propagation of the driver and harmonic fields using the
one dimensional version of the model presented in Ref. 25
(Transverse effects, e.g. transversal intensity variation and diffraction, somewhat reduce QPM
efficiencies, yet they are secondary and therefore neglected in our simulation). The nonlinear
evolution of the strong driver in the moving frame of light velocity in vacuum, c, is given
by:

_{e}, takes into account the pre-formed plasma and the ionization that is calculated by using the ADK model [26]. 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 up to a constant factor (which is associated with the gas density and is unimportant in our case because the gas density is constant), E

_{HHG}, is described by:Figure 3(a) 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 88th cutoff harmonic. For comparison, the generated spectrum with constant static field is also presented. A clear QPM enhancement is obtained around the 88th harmonic when the static field flips sign periodically. Figure 3(b) shows the coherent buildup of the 88th and 87th harmonic fields, showing clearly that the even harmonic experience a QPM enhancement while the odd harmonic suffers from phase-mismatch. Notably, the QPM efficiency of the 88th harmonic is 0.27, which is relatively high for QPM in HHG [27]. Figure 3(c) shows the HHG spectrum that is generated when d

_{DC}= 28 µm which corresponds to the coherence length of the 70th plateau harmonic. Clear QPM enhancement is obtained around the 72th harmonic. Figure 3(d) shows the coherent buildup of the 70th and 71th 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 even-harmonics correspond to high repetition-rate APT with stable CEP. This feature
is demonstrated in Fig. 4.
Figure 4(a) shows the normalized APT, E_{QPM}(t)
that corresponds to the red spectrum in Fig. 3(a) in the
spectral region 83 ± 5 harmonics. Figure 4(b)
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 [10,11], APT with stable CEP is obtained without reduction of the
repetition rate. For comparison, Fig. 4(c) 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. 4(d)).

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 [20,29]. 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${E}_{1}=\sqrt{{I}_{1}}\mathrm{exp}\left[-{\left(2t/{\tau}_{1}\right)}^{4}\right]\mathrm{cos}\left(2{\omega}_{0}t\right)$where${\tau}_{1}=72fs\text{\hspace{0.17em}}\text{\hspace{0.17em}}$and${I}_{1}=2.4\times {10}^{11}W/c{m}^{2}$. We simulated the propagation of the beam using the following
equation:

_{0}and -Δn in the region around 2ω

_{0}. The third term in Eq. (5) gives rise to the assumed dispersion, only. Figure 5(a) shows the HHG spectrum when Δn = 8.7 × 10

^{−3}(L

_{π}= 46 µm) and after propagation distance of 1 mm. For compression, the generated spectrum when Δn = 0 is also presented. A clear QPM enhancement is obtained around the 86th harmonic. Figure 5(b) shows the coherent buildup of the 86th and 85th 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 5(c) 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 70th harmonic. Figure 5(d) shows the coherent buildup of the 70th and 71th harmonic fields, showing that the even harmonic experience a QPM enhancement (QPM efficiency is 0.14) while the odd harmonic suffers from phase-mismatch.

## 4. Conclusions

In conclusions, we propose and demonstrated numerically an all-optical QPM technique for generating only even-order harmonics of a strong driver, within a spectral region that contains ~10 harmonics. This technique shows that symmetry arguments can be employed for selective control over the spectral features of HHG.

## Acknowledgment

This research was support by ICore: the Israeli Excellence Center “Circle of Light”.

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