Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Ion-based high-order harmonic generation from water window to keV region with a transverse disruptive pulse for quasi-phase-matching

Yao-Li Liu, Jyhpyng Wang, and Hsu-hsin Chu

Open Access Open Access

Abstract

A scheme for ion-based high-harmonic generation from water window to keV x-ray is investigated. He1+ ions with 54.42-eV ionization potential extend the harmonic cutoff energy to 1 keV. The transverse selective-zoning method of quasi-phase-matching is utilized to overcome the severe plasma dispersion in a highly ionized medium. The calculated conversion efficiency reaches about 15% of the perfect phase-matching condition. Wavelength tunability is achieved by incorporating a programmable spatial-light modulator to control the quasi-phase-matching pattern.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In the past two decades, gas-based high-order harmonic generation (HHG) driven by ultrashort intense laser pulses has been proven a reliable coherent light source from extreme-ultraviolet (EUV) to x-ray [1–3]. It is generated through the processes of ionization, acceleration, and then recombination of bound electrons in neutral atoms. The cut-off photon energy is $E_\mathrm {cutoff} = I_p + 3.17 U_p$, where $I_p$ is the ionization potential of the bound electron and $U_p$ is the ponderomotive potential of the driving laser field [4–6]. Using the long-wave infrared (LWIR) as the driving pulse to increase $U_p$, the HHG spectrum has been demonstrated to reach the keV range [7]. However, limited by the small recombination probability under long pumping wavelengths and the strict phase-matching condition at a low ionization ratio, the overall conversion efficiency is quite low. On the other hand, ion-based HHG driven by near-infrared (NIR) has been demonstrated to generate harmonic orders higher than that from neutral gases [8–10]. Since the bound electrons in ions possess higher $I_p$, which can only be ionized at a higher driving laser intensity and thus a higher $U_p$, both effects result in a higher cut-off photon energy. Nevertheless, in a highly ionized plasma, the dispersion of the plasma cannot be balanced by neutral atoms. Direct phase matching is not possible. Therefore, generating HHG at short wavelengths with high efficiency is an outstanding challenge.

To overcome the phase-matching problem, quasi-phase-matching (QPM) is a promising solution. Several QPM schemes have been demonstrated in gas-based HHG by incorporating multiple gas jets [11–13], modulated waveguides [14,15], counter-propagating pulses [16–20], or transverse mode control of the driving pulses [9,21]. These schemes successfully created several QPM zones to increase the HHG yield in EUV region.

In this paper, we propose a quasi-phase-matching scheme of high-harmonic generation from NIR-driven He$^{1+}$ ions. It can be applied to a wide spectral range from water window to keV x-ray. By using an intense NIR (810 nm) pulse focused into a helium-filled capillary waveguide, the NIR pulse ionizes the first electron of He atom at its front edge, and then generates keV/water window harmonics from the He second electron at its peak. QPM is done by employing a transverse disruptive pulse to eliminate the HHG emission at the location of wrong phase. Based on the “selective-zoning mechanism” [16], the phase of the driving NIR field is disturbed by the transverse pulse, resulting in the destruction of coherent generation of high harmonics at the irradiated locations. The mechanism is originally demonstrated using counter-propagating pulses [17–20], but also verified recently with a transverse pulse [22]. Therefore, by using a spatial light modulator to control the transverse pulse beam profile, QPM can be achieved with hundreds of zones of adjustable lengths, matching the required several-tens-micrometer coherent length of keV/water window harmonics. The interplay between the longitudinal driving pulse and the patterned transverse pulse is shown in Fig. 1. Based on our calculation, the 675th-order HHG of 1-keV photon energy can be generated efficiently with a 40-mJ, 30-fs, 810-nm driving pulse, and a 125-mJ, 33-ps, 810-nm transverse pulse interacting in a capillary waveguide with 50-$\mu$m radius, 10-mm length, and $1 \times 10^{16}$ cm$^{-3}$ helium density. The conversion efficiency reaches 15% of the perfect phase-matching condition. The proposed method is wavelength tunable. With the same driving pulse, capillary waveguide, and helium density, the 337th-order HHG in water window can be generated efficiently with a 350-mJ transverse pulse and the optimized QPM pattern. The conversion efficiency reaches about 14% of the perfect phase-matching condition. These two examples of calculation show the promise of this QPM scheme in the water-window to keV wavelength range.

 figure: Fig. 1.

Fig. 1. The interplay between the longitudinal driving pulse and the patterned transverse pulse for the quasi-phase-matched HHG.

Download Full Size | PDF

2. Propagation of the driving pulse

Consider a NIR (810 nm) Gaussian pulse guided in a helium-fill capillary waveguide for harmonic generation. The guiding mode is the linear-polarized EH$_{11}$ mode polarized in the $x$-axis. [23]. To simplify the calculation, we take the transverse spatial average within the waveguide to obtain the starting waveform of the driving pulse at position $z = 0$ as a function of time $t$:

$$\mathbf{E}_d(z=0,t) = A_0 \exp\left(-\frac{t^2}{2 \tau_0^2} \right) e^{{-}i\omega_d t} \hat{\mathbf{x}} \, ,$$
where $A_0$ is the initial peak electric field, $\tau _0$ is the initial pulse duration, and $\omega _d$ is the angular frequency of the driving pulse. Then we divide the waveguide as a sequence of small sections, and calculate the propagation of the driving pulse passing through each section one by one. The attenuation effects due to optical-field ionization for overcoming the ionization potential, above-threshold-ionization (ATI) heating and inverse-Bremsstrahlung heating of free electron, Thomson scattering by free electron, and capillary guiding loss are included, together with the effects of plasma dispersion and waveguide dispersion.

Choice of parameters for the calculation is determined by the following considerations. The choice should be compatible with current laser technology. A standard high-power Ti:sapphire laser system is assumed to be the driving source. The overall design goal is to achieve an efficient and sustained HHG in a helium-filled capillary waveguide. We set the length of the waveguide to be 10 mm, and its radius to be 50 $\mu$m. The helium gas density is set to be $1 \times 10^{16}$ cm$^{-3}$. These parameters correspond to a long enough dephasing length for the proposed QPM scheme. The driving pulse energy is chosen to be 40 mJ to ensure that the intensity is high enough to drive the second electron of helium atom for high-harmonic generation. The detailed calculation procedures and results are described in the following subsections.

2.1 Attenuation of the driving pulse

In each section $\Delta z$, we firstly calculate the ionization ratio by applying the Ammosov-Delone-Krainov (ADK) theory for optical-field ionization to get the He density $N_\mathrm {He}(z)$, He$^{1+}$ density $N_\mathrm {He^{1+}}(z)$, He$^{2+}$ density $N_\mathrm {He^{2+}}(z)$, and electron density $N_e(z)$ [24]. With a 40-mJ driving pulse of $\tau _0 = 30$ fs duration, the peak intensity reaches $9.58 \times 10^{15}$ W/cm$^2$. Such intense driving pulse can fully ionize He to He$^{1+}$ and ionize 88% He$^{1+}$ to He$^{2+}$, as shown in Fig. 2. This means that high-energy HHG photon can be generated in the $\mathrm {He}^{1+} \rightarrow \mathrm {He}^{2+}$ process. The cut-off photon energy reaches 1.9 keV. The driving pulse energy loss due to overcoming the ionization potential is

$$\Delta U_\mathrm{ionization}(z) = \left[ N_\mathrm{gas} I_{p1} + N_\mathrm{He^{2+}}(z) I_{p2} \right] \pi R^2 \Delta z \, ,$$
where $R = 50$ $\mu$m is the waveguide radius, and $I_{p1}$ and $I_{p2}$ are the ionization potentials of helium’s first electron and second electron, respectively. Furthermore, each free electron gets a residual kinetic energy due to the ATI-heating mechanism [24,25]:
$$K_\mathrm{ATI}(z, t_0) = \frac{q_e^2 I_d(z,t_0)}{c \epsilon_0 m_e \omega_d^2} \sin^2(\omega_d t_0) \, ,$$
where $q_e$ is the elementary charge, $m_e$ is the electron mass, and $I_d(z,t_0)$ is the driving laser intensity at the ionization time $t_0$. Therefore, the driving pulse energy loss due to ATI heating is the summation of all free electrons’ kinetic energy:
$$\Delta U_\mathrm{ATI}(z) = \sum_\mathrm{all \; electrons} K_\mathrm{ATI}(z, t_0) \, .$$

The electron temperature needed for estimating inverse-Bremsstrahlung heating is inferred from

$$T_e(z) = \frac{2}{3} \frac{\bar{K}_\mathrm{ATI}(z)}{k_B} \, ,$$
where $\bar {K}_\mathrm {ATI}(z)$ is the average electron kinetic energy.

 figure: Fig. 2.

Fig. 2. Calculated results of the optical-field ionization of the He gas at position $z = 0$: (a) driving laser field, (b) relative populations of He ($N_\mathrm {He}/N_\mathrm {gas}$, green line), He$^{1+}$ ($N_\mathrm {He^{1+}}/N_\mathrm {gas}$, blue line), and He$^{2+}$ ($N_\mathrm {He^{2+}}/N_\mathrm {gas}$, red line), (c) relative electron density $N_e/N_\mathrm {gas}$.

Download Full Size | PDF

Afterwards, we calculate the loss of driving pulse energy due to inverse-Bremsstrahlung heating. Its absorption coefficient is given by

$$a_\mathrm{IB}(z) = \frac{1}{3 c \omega_d^2 n_\mathrm{plasma}(\omega_d,z)} \frac{q_e^6 Z(z) N_e(z)^2 \ln(\Lambda(z))}{2\pi \epsilon_0^2 m_e k_B T_e(z)^{3/2}} ,$$
where
$$n_\mathrm{plasma}(\omega_d,z) = \sqrt{1-\frac{\omega_p(z)^2}{\omega_d^2}}$$
is the plasma refractive index,
$$\omega_p(z) = \sqrt{\frac{q_e^2 N_e(z)}{\epsilon_0 m_e}}$$
is the plasma frequency, and $\ln (\Lambda (z))$ is the Coulomb logarithm determined by $T_e(z)$ and $N_e(z)$ [26].

Finally, we calculate the energy attenuation due to free electrons’ Thomson scattering and capillary guiding loss. The attenuation coefficients are

$$a_\mathrm{TS}(z) = \frac{8\pi}{3} \frac{q_e^4}{(4\pi\epsilon_0 m_e c^2)^2} N_e(z) \, ,$$
and
$$a_\mathrm{WG} = \frac{u_{11}^2}{4\pi^2} \frac{\lambda_d^2}{R^3 \nu_1} \, ,$$
respectively. Here $u_{11} = 2.405$ is the first root of the zero-order Bessel function, $\lambda _d$ is the driving laser wavelength, $\nu _1 = (1/2)(n_\mathrm {FS}^2+1)/\sqrt {n_\mathrm {FS}^2-1}$, and $n_\mathrm {FS}=1.4531$ is the refractive index of the waveguide material (fused silica) [23].

2.2 Dispersion effects

Next, we evaluate the propagation effects due to plasma dispersion and waveguide dispersion. The wavenumber of an EM wave with an angular frequency $\omega$ propagating in a plasma-filled capillary waveguide is

$$k(\omega,z) = k_\mathrm{plasma}(\omega,z) + k_\mathrm{WG}(\omega) = \frac{\omega}{c} n_\mathrm{plasma}(\omega,z) - \frac{u_{11}^2 c}{2 R^2}\frac{1}{\omega} \, ,$$
where $k_\mathrm {plasma}(\omega,z)$ represents the plasma dispersion and $k_\mathrm {WG}(\omega )$ represents the waveguide dispersion [23].

Then we can obtain the driving pulse waveform at position $z$ for the next step:

$$E_d(z,t) = \sqrt{\frac{\tau_0}{\tau(z)}} A(z) \, \exp\left( -\frac{(t-C(z))^2}{2\tau(z)^2} \right) e^{i\phi_d(z,t)} \, ,$$
where $\tau (z) = \sqrt {\tau _0^2 + (D(z)/\tau _0)^2}$ is the stretched pulse duration, and $A(z)$ is the attenuated amplitude due to all the attenuation effects described above. The factor $C(z)$ is the accumulated group delay:
$$C(z) = \frac{\partial}{\partial\omega} \left[ \int_0^z k(\omega,z') dz' \right]_{\omega=\omega_d} = \int_0^z \frac{1}{cn(\omega_d,z')}dz' + \frac{u_{11}^2c}{2R^2}\frac{z}{\omega_d^2} \, ,$$
and the factor $D(z)$ is the accumulated group-delay dispersion (GDD):
$$D(z) = \frac{\partial^2}{\partial\omega^2} \left[ \int_0^z k(\omega,z') dz' \right]_{\omega=\omega_d} = \int_0^z \frac{-\omega_p(z')^2}{c[\omega_d^2-\omega_p(z')^2]^{3/2}}dz' - \frac{u_{11}^2 c}{R^2} \frac{z}{\omega_d^3} \, .$$

The phase of the driving pulse is

$$\phi_d(z,t) = \frac{1}{2} \tan^{{-}1} \left(\frac{D(z)}{\tau_0^2}\right) - \frac{1}{2} \frac{D(z)}{\tau_0^4 + D(z)^2} (t - C(z))^2 + \int_0^z k(\omega_d,z') dz' - \omega_d t \, .$$

By iterating the processes described above, the evolution of the driving laser field and the plasma distribution along the capillary waveguide are obtained. Driving pulse energy, accumulated GDD, driving pulse duration as functions of propagation distance $z$ are shown in Fig. 3, as well as the electron density, He$^{2+}$ density, and the energy losses due to all five mechanisms. Since the gas density is as low as $1 \times 10^{16}$ cm$^{-3}$, the accumulated GDD is as small as $-0.7$ fs$^2$. The increase of the pulse duration is only about 1%. The overall driving pulse energy loss is also about 1%, dominated by the guiding loss. Therefore, the driving pulse can penetrate through the capillary waveguide, generate a nearly uniform He$^{2+}$ density and electron density distribution, as shown in Fig. 3(e) and 3(f).

 figure: Fig. 3.

Fig. 3. (a) Driving laser energy as a function of position $z$. (b) Energy losses per unit length due to capillary guiding loss (green), ionization loss (orange), ATI heating (red), inverse-Bremsstrahlung heating (blue), and Thomson scattering (purple). (c) Accumulated GDD. (d) Driving pulse duration. (e) Electron density. (f) He$^{2+}$ density.

Download Full Size | PDF

3. Calculation of the keV harmonic generation

In the second part of our calculation, we focus on the keV harmonic generation. We choose the 675th harmonic as an example, which has a wavelength of 1.2 nm and a photon energy of 1.03 keV. The calculation includes three parts: 1. phase mismatches due to all dispersion effects; 2. quasi-phase-matching done by a transverse disruptive pulse; 3. generation, propagation, and accumulation of the 675th harmonic wave.

3.1 Phase mismatch calculation

The phase mismatch is represented by the wavenumber mismatch

$$\begin{aligned}\Delta k(z) &= q k(\omega_d,z) - k(\omega_q,z) + \Delta k_\mathrm{dipole}(z)\\ &= \Delta k_\mathrm{plasma}(z) + \Delta k_\mathrm{WG} + \Delta k_\mathrm{dipole}(z) \, , \end{aligned}$$
where $q = 675$ is the harmonic order, $\omega _q = q \omega _d$ is the harmonic frequency,
$$\Delta k_\mathrm{plasma}(z) = q k_\mathrm{plasma}(\omega_d,z) - k_\mathrm{plasma}(\omega_q,z) \simeq{-}\frac{N_e(z) e^2}{2 \epsilon_0 c m_e \omega_d} \left( q - \frac{1}{q} \right)$$
is the wavenumber mismatch due to plasma dispersion,
$$\Delta k_\mathrm{WG} = q k_\mathrm{WG}(\omega_d) - k_\mathrm{WG}(\omega_q) ={-}\frac{u_{11}^2 c}{2 R^2} \frac{1}{\omega_d} \left( q - \frac{1}{q} \right)$$
is the wavenumber mismatch due to waveguide dispersion, and
$$\Delta k_\mathrm{dipole}(z) = \frac{d\Phi_\mathrm{dipole}(I_d(z))}{dz} = \frac{\partial \Phi_\mathrm{dipole}}{\partial I_d} \frac{d I_d}{dz} = \alpha(I_d) \frac{d I_d}{dz}$$
is the wavenumber mismatch due to HHG intrinsic dipole phase variation [27,28]. The HHG intrinsic dipole phase $\Phi _\mathrm {dipole}(I_d(z))$ is determined by the laser intensity $I_d(z)$ at position $z$.

Therefore, the accumulated phase mismatch due to plasma dispersion is

$$\Delta \Phi_\mathrm{plasma}(z) = \int_{0}^{z} \Delta k_\mathrm{plasma}(z') \, dz' \, ,$$
and the accumulated phase mismatch due to waveguide dispersion is
$$\Delta \Phi_\mathrm{WG}(z) = \int_{0}^{z} \Delta k_\mathrm{WG}(z') \, dz' \, .$$

The intrinsic dipole phase shift originates from the temporal delay between the ionization and recombination processes in HHG [6,27]. It is determined by the harmonic order $q$ and the driving laser intensity at the point of ionization. The intrinsic dipole phase can be calculated by using the semiclassical three-step model [29]. It has two categories: short-trajectory dipole phase and long-trajectory dipole phase. In our case $q = 675$, the calculated HHG dipole phases and the $\alpha$ factors in Eq. (19) as functions of the laser intensity are shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. (a) Calculated HHG dipole phases $\Phi _\mathrm {dipole}$ of the 675th harmonic as functions of driving laser intensity $I_d$. Blue line: short-trajectory dipole phase, green line: long-trajectory dipole phase. (b) Factor $\alpha$ of the 675th harmonic as functions of driving laser intensity $I_d$. Blue line: short-trajectory $\alpha _\mathrm {short}$, green line: long-trajectory $\alpha _\mathrm {long}$.

Download Full Size | PDF

Since the ionization of He$^{1+}$ to He$^{2+}$ almost occurs at the peak of the driving pulse, as shown in Fig. 2, we can calculate the intrinsic dipole phases $\Phi _\mathrm {dipole}$ as functions of $z$ by using the peak intensity at $z$:

$$I_\mathrm{peak}(z) = \frac{1}{2} \epsilon_0 c |E_d(z,t=C(z))|^2 \, .$$

Therefore, the accumulated phase mismatch due to dipole phase variation is

$$\Delta \Phi_\mathrm{dipole}(z) = \Phi_\mathrm{dipole}(z) - \Phi_\mathrm{dipole}(0) \, ,$$
and the total phase mismatch is
$$\Delta \Phi_\mathrm{total}(z) = \Delta \Phi_\mathrm{plasma}(z) + \Delta \Phi_\mathrm{WG}(z) + \Delta \Phi_\mathrm{dipole}(z) \, .$$

The results are shown in Fig. 5(a)–(d). It can be seen that the total phase mismatches are dominated by the negative plasma dispersion. The positive dipole phase variations can only compensate for a small fraction of the former, and the waveguide dispersion is negligible compared with other dispersion sources. The resulted dephasing lengths $L_d(z) = \pi /\Delta k(z)$ are shown in Fig. 5(e). For short-trajectory emission the dephasing length is about 112 $\mu$m, and for long-trajectory emission it is about 130 $\mu$m. Both of them are much shorter than the 10-mm waveguide length, revealing the need of QPM.

 figure: Fig. 5.

Fig. 5. The accumulated phase mismatches of the 675th harmonic due to (a) plasma dispersion, (b) waveguide dispersion, and (c) dipole phase variations. Blue line: short-trajectory dipole phase, green line: long-trajectory dipole phase. (d) The total phase mismatches of the short-trajectory (blue) and long-trajectory (green) emissions. (e) The calculated dephasing lengths of the short-trajectory (blue) and long-trajectory (green) emissions of the 675th harmonic.

Download Full Size | PDF

3.2 Quasi-phase-matching done by a transverse disruptive pulse

To achieve QPM of the 675th harmonic generation, a transverse disruptive pulse is used to eliminate the harmonic emission with destructive phase. To simplify the following exposition, we assume the harmonic driving pulse and the transverse disruptive pulse are both plane waves with the same wave number $k$ and angular frequency $\omega$. They are expressed as

$$\mathbf{E}_1(\mathbf{r},t) = E_1 e^{i(k z-\omega t)} \, \hat{\mathbf{x}}$$
and
$$\mathbf{E}_2(\mathbf{r},t) = E_2 e^{i(k y-\omega t)} \, \hat{\mathbf{x}} \, ,$$
respectively. The driving pulse propagates in the $z$ direction and the transverse pulse propagates in the $y$ direction. Both of them are polarized in the $x$ direction, as shown in Fig. 1. Therefore, the total electric field becomes
$$\mathbf{E}_\mathrm{total}(\mathbf{r},t) = \mathbf{E}_1(\mathbf{r},t) + \mathbf{E}_2(\mathbf{r},t) = E_\mathrm{total}(y,z) \, e^{i\Delta\phi(y,z)} \, e^{i(k z -\omega t)} \hat{\mathbf{x}} \, ,$$
where
$$E_\mathrm{total}(y,z) = \sqrt{E_1^2 + E_2^2 + 2 E_1 E_2 \cos(k(y-z))}$$
is the amplitude of the total electric field, and
$$\Delta\phi(y,z) = \tan^{{-}1} \left[ \frac{ (E_2/E_1) \sin(k(y-z))}{1 + (E_2/E_1) \cos(k(y-z))} \right]$$
is the phase modulation inflicted by the transverse pulse [30]. Because of the $q$th harmonic phase driven by the total electric field equals to $q$ times the phase of the total electric field plus the intrinsic dipole phase, a small disturbance of the total electric field phase results in a large variation of the harmonic phase:
$$\Delta\phi_q(y,z) = q\, \Delta \phi(y,z) \, .$$

Such a large phase variation spoils the phase coherence of HHG. For the case of the 675th harmonic, a weak transverse pulse of $E_2/E_1 = 0.006$ leads to a harmonic phase variation more than $\pm \pi$, as shown in Fig. 6. Such dramatic phase variation results in the suppression of the HHG emission at the location of transverse pulse exposure.

 figure: Fig. 6.

Fig. 6. Phase variation of the 675th harmonic inflicted by a transverse disruptive pulse of $E_2/E_1 = 0.006$. $\lambda$ is the wavelength.

Download Full Size | PDF

For a more realistic condition, we assume the transverse disruptive pulse is a 3D Gaussian pulse. Its electric field at the position of the waveguide ($y = 0$) is

$$\begin{aligned}E_\mathrm{trans}&(x,y=0,z,t) =\\ &A_\mathrm{trans} \, \exp\left(-\frac{\ln(2) x^2}{2R^2}\right) \, \exp\left(-\frac{\ln(2) [z-(L/2)]^2}{2(L/2)^2}\right) \, \exp\left(-\frac{\ln(2) [ct-(L/2)]^2}{2(L/2)^2}\right) \, e^{{-}i\omega_d t} \, , \end{aligned}$$
where $A_\mathrm {trans}$ is its peak electric field. The full-width-at-half-maximum (FWHM) of its intensity distribution in the $x$ direction matches the waveguide diameter $2R$, and those in the $z$ direction and the temporal domain match the waveguide length $L$, as shown in Fig. 1. To simplify the following calculation, we take the spatial average of the intensity distribution along the $x$ direction within the waveguide diameter to obtain an average electric field $\bar {E}_\mathrm {trans}(z,t)$ as a function of position $z$ and time $t$.

A tailored mask $M(z)$ is incorporated that modulates the transverse pulse beam profile to achieve quasi-phase-matching:

$$M(z) = M(\Delta \Phi_\mathrm{total}(z)) = \begin{cases} 0 , & 2m\pi \leq \Delta \Phi_\mathrm{total}(z) < (2m+1)\pi \\ 1 , & (2m+1)\pi \leq \Delta \Phi_\mathrm{total}(z) < (2m+2)\pi \end{cases} \, ,$$
where $m = 0, 1, 2,\ldots$ is an integer. In the in-phase regions where $2m\pi \leq \Delta \Phi _\mathrm {total}(z) < (2m+1)\pi$, the mask blocks the transverse pulse. The 675th harmonic wave can be generated normally. In the out-of-phase regions where $(2m+1)\pi \leq \Delta \Phi _\mathrm {total}(z) < (2m+2)\pi$, the mask allows the transverse pulse to pass and disrupt the HHG.

In order to account for the real experiment precisely, a 1-D point spread function

$$\mathrm{PSF}(z) = \frac{\sin(\pi z/\zeta)}{(\pi z/\zeta)}$$
is applied to the mask, as shown in Fig. 7(a). The the scaling factor $\zeta = 10 \, \mu$m corresponds to a typical imaging resolution of 10 $\mu$m. The effective mask is the convolution of $\mathrm {PSF}(z)$ and $M(z)$, as shown in Fig. 7(b). Then, the total electric field is the superposition of the longitudinal driving pulse and the modulated transverse pulse:
$$E_\mathrm{total}(z,t) = E_d(z,t) + \left( \mathrm{PSF}(z) \otimes M(z) \right) \bar{E}_\mathrm{trans}(z,t) \, ,$$
which drives the 675th harmonic generation.

 figure: Fig. 7.

Fig. 7. (a) The point spread function $\mathrm {PSF}(z)$ applied to the mask. (b) The effective mask pattern $\mathrm {PSF} \otimes M$ for the 675th short-trajectory harmonic.

Download Full Size | PDF

3.3 Generation, propagation, and accumulation of the 675th harmonic wave

To calculate the HHG process, we trace a fixed wavefront of the 675th short-trajectory harmonic emission which is initially generated at $(z=0, t=0)$. Since the wavefront propagates with a phase velocity $v_p(\omega _q,z) = \omega _q / k(\omega _q,z)$, it spends a time interval

$$t(z) = \int_0^z \frac{1}{v_p(\omega_q,z')} \, dz'$$
to reach position $z$. Therefore, the fixed 675th short-trajectory harmonic wavefront will meet the total driving field of
$$E_\mathrm{HWF}(z) = E_\mathrm{total}(z,t(z))$$
at position $z$, as shown in Fig. 8(a) and 8(b). It can be seen that the drop of $|E_\mathrm {HWF}(z)|$ results from the attenuation of the driving pulse energy and the stretching of its duration. The modulations on $|E_\mathrm {HWF}(z)|$ and on its phase $\Phi _\mathrm {HWF}(z) = \arg (E_\mathrm {HWF}(z))$ come from the transverse disruptive pulse. The envelope of the modulations are determined by the Gaussian shape of the transverse disruptive pulse.

 figure: Fig. 8.

Fig. 8. (a) The amplitude of the total driving field met by the fixed 675th harmonic wavefront as a function of position $z$. (b) The phase of the total driving field met by the fixed 675th harmonic wavefront. (c) The amplitude of the 675th local harmonic field generated at position $z$. (d) The phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field generated at position $z$.

Download Full Size | PDF

Assume the local harmonic field $E_\mathrm {LH}(z)$ generated at position $z$ has an amplitude proportional to the source density $N_{\mathrm {He}^{2+}}(z)$ times the $p$th power of the total driving amplitude, and has a phase equal to $q$ times the total driving phase plus the intrinsic dipole phase:

$$E_\mathrm{LH}(z) \propto N_{\mathrm{He}^{2+}}(z) \, |E_\mathrm{HWF}(z)|^p \, e^{i \Phi_\mathrm{LH}(z)} \, ,$$
where $\Phi _\mathrm {LH}(z) = q \Phi _\mathrm {HWF}(z) + \Phi _\mathrm {dipole}(z)$ is the phase of the local harmonic field. The number $p$ is an empirical constant. Without loss of generality, the value of $p$ is set to 5 in our calculation. The result is insensitive to the exact value of $p$ [16]. The calculated amplitude and phase of $E_\mathrm {LH}(z)$ are shown in Fig. 8(c) and 8(d), respectively. The modulation on $\Phi _\mathrm {LH}(z)$ leads to the quasi-phase-matching of the 675th short-trajectory harmonic generation.

By integrating the local harmonic field $E_\mathrm {LH}(z)$ step by step with corresponding phase shifts resulted from the plasma dispersion and the waveguide dispersion, the accumulated harmonic field $E_\mathrm {H}(z)$ as a function of $z$ is obtained

$$E_\mathrm{H}(z) = \int_{0}^{z} E_\mathrm{LH}(z') e^{i \left( k(\omega_q, z') \Delta z' - \omega_q \Delta t' \right)} dz' = \int_{0}^{z} E_\mathrm{LH}(z') \, dz' \, ,$$
where $\Delta t' = \Delta z' / v_p(\omega _q, z')$. To maximize the 675th short-trajectory harmonic output, the energy of the transverse disruptive pulse is chosen as 125 mJ. The results are shown in Fig. 9. It can be seen that the final field strength $|E_\mathrm {H}(z=L)|$ reaches 39% of the ideal condition with perfect phase-matching, which corresponds to an energy conversion efficiency of 15% relative to the ideal case. Without the transverse disruptive pulse, the short-trajectory emission can only oscillate at a low level, due to its severe phase mismatch. At the same time, the long-trajectory emission cannot be build up sustainedly, since its dephasing length differs from that of the short-trajectory emission.

 figure: Fig. 9.

Fig. 9. (a) The amplitude of the accumulated 675th short-trajectory harmonic field. Black line: ideal case with perfect phase-matching, blue line: optimized case with quasi-phase-matching, purple line: case without the transverse pulse. (b) The amplitude of the modulated transverse field (black), the phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field (blue), and the amplitude of the accumulated harmonic field in the front section of the waveguide. (c) The amplitude of the modulated transverse field (black), the phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field (blue), and the amplitude of the accumulated harmonic field in the middle section of the waveguide.

Download Full Size | PDF

To investigate the generation process in detail, the zoom-in growth curves in the front section (0–0.3 mm) and the middle section (4.9–5.2 mm) of the capillary waveguide are shown in Fig. 9(b) and 9(c), respectively, together with the amplitudes of the modulated transverse field and the phases of the local harmonic field. In the front section, the amplitude of the transverse field is smaller. The inflicted phase modulation is not strong enough to suppress all the local harmonic emissions in the out-of-phase region. Therefore, the accumulated harmonic field drops a little in that region. On the contrary, the amplitude of the transverse field is stronger in the middle section of the waveguide, as shown in Fig. 9(c). In this section, the rapid phase modulation on the local harmonic field is more than $\pi$. Since the reason for suppressing the harmonic generation is to eliminate the emission having a destructive phase of $\pi$–$2\pi$, the local harmonic field with $>\pi$ phase modulation in the section resumes the constructive accumulation of HHG. This is actually more beneficial than just turning off the local harmonic emission in the out-of-phase region [16]. Therefore, the accumulated harmonic field can increase slightly in the out-of-phase region. Finally, a maximized output is obtained with the optimized transverse pulse energy. Moreover, if a uniform distribution of the transverse pulse beam profile can be made on the capillary, optimized QPM can be achieved along the entire interaction length. The conversion efficiency will be further improved. This can be done by adjusting the mask in the out-of-phase region to be proportional to the inverse of the transverse pulse amplitude instead of 1. The trade-off is inefficient use of the transverse pulse, and thus a more intense transverse pulse will be required.

The optimization procedure can be applied to the long-trajectory emission. The quasi-phase-matching pattern is modified to match its phase accumulation curve. The result is shown in Fig. 10. It can be seen that almost the same conversion efficiency is obtained.

 figure: Fig. 10.

Fig. 10. The amplitude of the accumulated 675th long-trajectory harmonic field. Black line: ideal case with perfect phase-matching, blue line: optimized case with quasi-phase-matching, purple line: case without the transverse pulse.

Download Full Size | PDF

4. Calculation of the water-window harmonic generation

By adjusting the mask pattern to match different dephasing lengthes, the proposed method can achieve QPM for a wide spectral range of HHG. For example, consider the generation of the 337th harmonic ($\lambda _{337} = 2.4$ nm) in water window with the same driving pulse condition, waveguide specification, and helium gas density. The resulted dephasing length is about 220 $\mu$m for the short-trajectory emission, as shown in Fig. 11(a). The optimized mask pattern is shown in Fig. 11(b), and the resulted phase of the 337th local harmonic field is shown in Fig. 11(c). It can be seen that patterned rapid phase modulation is inflicted by the transverse pulse. The accumulated 337th harmonic field is shown in Fig. 11(d). With the optimized transverse pulse energy of 350 mJ, the final 337th harmonic field strength reaches 37% of the perfect phase-matching condition, corresponding to a relative energy conversion efficiency of 14%.

 figure: Fig. 11.

Fig. 11. (a) The dephasing length as a function of position $z$ for the short-trajectory emission of the 337th harmonic. (b) The effective mask pattern $\mathrm {PSF} \otimes M$ for the 337th short-trajectory harmonic. (c) The phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 337th short-trajectory local harmonic field. (d) The amplitude of the accumulated 337th harmonic field normalized to that of the perfect phase-matching condition.

Download Full Size | PDF

It should be noted that the required transverse pulse energy for keV harmonic generation is less than that for water window harmonics. The reason is revealed in Eq. (30). Since the required phase modulation $\Delta \phi _q$ on the harmonic field for QPM is about $\pm \pi$, the required phase modulation $\Delta \phi$ on the total driving field becomes $\pm \pi /q$. Therefore, for lower order harmonic generation, larger phase modulation on the total driving field is needed. Then higher transverse pulse energy is necessary.

5. Discussion

In our analysis, it is assumed that the driving pulse can be well guided in the capillary. The effect of plasma-induced defocusing is not taken into account. Furthermore, the nonlinear effect of plasma-induced self-phase modulation is also neglected. The driving pulse is assumed to remain Gaussian shape during propagation. Here we examine whether these two assumptions hold in our cases.

Under the optimized condition for keV/water window harmonic generation, the electron density $N_e$ in the capillary waveguide is about $1.9 \times 10^{16}$ cm$^{-3}$, as shown in Fig. 3(e). Considering the effect of plasma-induced defocusing, the phase shift at the driving beam center relative to the edge due to higher plasma density at the center is at most

$$\Delta\Phi_\mathrm{plasma} ={-}\frac{\pi}{\lambda_d} \frac{N_e}{N_\mathrm{cr}} L \, ,$$
where $N_\mathrm {cr}$ is the critical density of the driving wave and $L$ is the interaction length. Therefore, the defocusing length $l_D$ for accumulating a $-1$-radian phase shift is
$$l_D = \frac{\lambda_d}{\pi} \frac{N_\mathrm{cr}}{N_e} \, ,$$
which is 23.1 mm in our cases. Comparing to the natural diffraction of a focused Gaussian beam, the spatial phase shift is
$$\Delta\Phi_\mathrm{Gaussian}(z,r) = \frac{k r^2}{2(z^2+b^2)/z} \, ,$$
where $k$ and $b$ are the wavenumber and the confocal parameter of the Gaussian beam, respectively. When it propagates to $z = b$, it accumulates a phase shift at the center
$$\Delta\Phi = \Delta\Phi_\mathrm{Gaussian}(z=b,r=0) - \Delta\Phi_\mathrm{Gaussian}(z=b,r=w(b)) ={-}1 \; \mbox{rad} \, ,$$
where $w(b) = 2\sqrt {b/k}$ is the beam radius at $z=b$. Therefore, the ratio between $l_D$ and $b$ indicates the significance of the plasma-induced defocusing relative to the natural diffraction. In our cases, the focused driving pulse has a diameter of 50 $\mu$m (FWHM), corresponding to a confocal parameter of $b = 7.0$ mm. The defocusing length $l_D$ is more than three times the confocal parameter. It means that the plasma-induced diffraction is much smaller than the natural diffraction of free propagation. Therefore, the effect of ionization defocusing is insignificant in our cases.

The plasma-induced self-phase modulation originates from the rapid change of the electron density $N_e$ due to optical-field ionization (Fig. 2(c)), which leads to the change of the plasma refractive index $n_\mathrm {plasma}$ and thus the phase modulation of the driving pulse. Such phase modulation may cause the distortion of the pulse waveform and the shift of its angular frequency:

$$\Delta \omega ={-}k_0 \frac{d n_\mathrm{plasma}}{dt} L = \frac{k_0}{2 n_\mathrm{plasma} N_\mathrm{cr}} \frac{dN_e}{dt} L \, ,$$
where $k_0$ is the driving pulse wavenumber in vacuum, and $L$ is the interaction length. In our cases, the accumulated angular frequency shift $\Delta \omega$ at the end of the capillary is about $1.0 \times 10^{13}$ sec$^{-1}$. It is less than 0.5% of the original central angular frequency. Therefore, the plasma-induced self-phase modulation is also insignificant.

6. Conclusion

We analyzed a scheme of ion-based HHG with disruptive quasi-phase matching from water window to keV x-ray. The He$^{1+}$ ions are chosen as the interacting medium to extend the cutoff photon energy to keV. A transverse disruptive pulse is utilized for quasi-phase-matching in the highly ionized medium. By taking account of optical-field ionization, above-threshold-ionization heating, inverse-Bremsstrahlung heating, Thomson scattering, guiding loss, waveguide dispersion, and plasma dispersion, two examples with practical parameters are investigated. The calculation results show that 14–15% conversion efficiency can be achieved relative to the perfect-phase matching condition.

The proposed scheme has several advantages. (1) keV HHG driven by NIR pulses has a much higher conversion efficiency than that driven by LWIR pulses [7], since the single-atom response scales with the driving wavelength as $\sim \lambda _d^{-6}$ [31–33]. Therefore, a $10^4$-fold enhancement is expected. (2) Since most of the He$^{1+}$ ions participate in the HHG process, only a moderate medium density is required. Compared to the case driven by LWIR pulses, very high gas density of several tens of atmospheric pressure was used to compensate for the very small single-atom response. However, due to its strict phase-matching condition done by balancing the neutral gas dispersion and the plasma dispersion, the ionization ratio in that experiment was only about 0.03%. Therefore, the density of the medium that actually participated in the HHG process was just comparable to our scheme. (3) In our scheme, a standard Ti:sapphire laser system is used as the driving source, with no need for additional nonlinear amplifiers to generate the LWIR pulses. The complexity of the whole system could be greatly reduced. (4) The transverse selective-zoning mechanism of QPM adopted in our scheme significantly increases the conversion efficiency. As shown in Fig. 9(c), the outcome of the HHG yield could be better than just turning off the local harmonic emission with incorrect phase. (5) In practice, the proposed QPM scheme can be done by using a programmable spatial light modulator to control the transverse pulse beam profile. Hundreds of QPM zones with adjustable lengths can be created to match coherent lengths as short as several-tens micrometers. Such capability provides the flexibility for applying to a wide spectral range. We believe the ion-based HHG with QPM is a promising method for efficient short-wavelength harmonic generation.

Funding

Ministry of Science and Technology, Taiwan (109-2112-M-008-011, 109-2112-M-008-022-MY2, 109-2221-E-001-024, 110-2221-E-001-024).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]  

2. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]  

3. H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing attosecond science in the quest for coherent x-rays,” Science 317(5839), 775–778 (2007). [CrossRef]  

4. A. L’Huillier, K. J. Schafer, and K. C. Kulander, “Theoretical aspects of intense field harmonic generation,” J. Phys. B: At., Mol. Opt. Phys. 24(15), 3315–3341 (1991). [CrossRef]  

5. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]  

6. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]  

7. T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef]  

8. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, S. Backus, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-order harmonic generation up to 250 ev from highly ionized argon,” Phys. Rev. Lett. 92(3), 033001 (2004). [CrossRef]  

9. M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-x-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007). [CrossRef]  

10. P. Arpin, T. Popmintchev, N. L. Wagner, A. L. Lytle, O. Cohen, H. C. Kapteyn, and M. M. Murnane, “Enhanced high harmonic generation from multiply ionized argon above 500 ev through laser pulse self-compression,” Phys. Rev. Lett. 103(14), 143901 (2009). [CrossRef]  

11. J. Seres, V. S. Yakovlev, E. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007). [CrossRef]  

12. A. Pirri, C. Corsi, and M. Bellini, “Enhancing the yield of high-order harmonics with an array of gas jets,” Phys. Rev. A 78(1), 011801 (2008). [CrossRef]  

13. A. Willner, F. Tavella, M. Yeung, T. Dzelzainis, C. Kamperidis, M. Bakarezos, D. Adams, M. Schulz, R. Riedel, M. C. Hoffmann, W. Hu, J. Rossbach, M. Drescher, N. A. Papadogiannis, M. Tatarakis, B. Dromey, and M. Zepf, “Coherent control of high harmonic generation via dual-gas multijet arrays,” Phys. Rev. Lett. 107(17), 175002 (2011). [CrossRef]  

14. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003). [CrossRef]  

15. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003). [CrossRef]  

16. J. Peatross, S. Voronov, and I. Prokopovich, “Selective zoning of high harmonic emission using counter-propagating light,” Opt. Express 1(5), 114–125 (1997). [CrossRef]  

17. S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. 87(13), 133902 (2001). [CrossRef]  

18. X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007). [CrossRef]  

19. T. Robinson, K. O’Keeffe, M. Zepf, B. Dromey, and S. M. Hooker, “Generation and control of ultrafast pulse trains for quasi-phase-matching high-harmonic generation,” J. Opt. Soc. Am. B 27(4), 763–772 (2010). [CrossRef]  

20. K. O’Keeffe, D. T. Lloyd, and S. M. Hooker, “Quasi-phase-matched high-order harmonic generation using tunable pulse trains,” Opt. Express 22(7), 7722–7732 (2014). [CrossRef]  

21. L. Hareli, L. Lobachinsky, G. Shoulga, Y. Eliezer, L. Michaeli, and A. Bahabad, “On-the-fly control of high-harmonic generation using a structured pump beam,” Phys. Rev. Lett. 120(18), 183902 (2018). [CrossRef]  

22. Y.-L. Liu, S.-C. Kao, Y.-Y. Ou Yang, Z.-M. Zhang, J. Wang, and H.-H. Chu, “Tomographic analysis of high-order harmonic generation by integrating a phase-matching profile measurement with disruptive interaction-length control,” Phys. Rev. A 104(2), 023112 (2021). [CrossRef]  

23. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964). [CrossRef]  

24. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field,” Proc. SPIE 0664, 138 (1986). [CrossRef]  

25. P. B. Corkum and N. H. Burnett, “Multiphoton ionization for the production of x-ray laser plasmas,” in Short-Wavelength Coherent Radiations: Generation and Applications, R. W. Falcone and J. Kirz, eds. (OSA, 1988), p. 225.

26. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University Press, 1999).

27. M. Lewenstein, K. C. Kulander, K. J. Schafer, and P. H. Bucksbaum, “Rings in above-threshold ionization: A quasiclassical analysis,” Phys. Rev. A 51(2), 1495–1507 (1995). [CrossRef]  

28. T. Auguste, P. Salières, A. S. Wyatt, A. Monmayrant, I. A. Walmsley, E. Cormier, A. Zaïr, M. Holler, A. Guandalini, F. Schapper, J. Biegert, L. Gallmann, and U. Keller, “Theoretical and experimental analysis of quantum path interferences in high-order harmonic generation,” Phys. Rev. A 80(3), 033817 (2009). [CrossRef]  

29. E. A. Gibson, “Quasi-phase matching of soft x-ray light from high-order harmonic generation using waveguide structures,” Ph.D. thesis, University of Colorado (2004).

30. J. B. Madsen, L. A. Hancock, S. L. Voronov, and J. Peatross, “High-order harmonic generation in crossed laser beams,” J. Opt. Soc. Am. B 20(1), 166–170 (2003). [CrossRef]  

31. J. Tate, T. Auguste, H. Muller, P. Salières, P. Agostini, and L. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef]  

32. M. Frolov, N. Manakov, and A. Starace, “Wavelength scaling of high-harmonic yield: Threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. 100(17), 173001 (2008). [CrossRef]  

33. J. A. Pérez-Hernández, L. Roso, and L. Plaja, “Harmonic generation beyond the strong-field approximation: the physics behind the short-wave-infrared scaling laws,” Opt. Express 17(12), 9891–9903 (2009). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)
Fig. 1.
Fig. 1. The interplay between the longitudinal driving pulse and the patterned transverse pulse for the quasi-phase-matched HHG.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Calculated results of the optical-field ionization of the He gas at position $z = 0$: (a) driving laser field, (b) relative populations of He ($N_\mathrm {He}/N_\mathrm {gas}$, green line), He$^{1+}$ ($N_\mathrm {He^{1+}}/N_\mathrm {gas}$, blue line), and He$^{2+}$ ($N_\mathrm {He^{2+}}/N_\mathrm {gas}$, red line), (c) relative electron density $N_e/N_\mathrm {gas}$.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) Driving laser energy as a function of position $z$. (b) Energy losses per unit length due to capillary guiding loss (green), ionization loss (orange), ATI heating (red), inverse-Bremsstrahlung heating (blue), and Thomson scattering (purple). (c) Accumulated GDD. (d) Driving pulse duration. (e) Electron density. (f) He$^{2+}$ density.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a) Calculated HHG dipole phases $\Phi _\mathrm {dipole}$ of the 675th harmonic as functions of driving laser intensity $I_d$. Blue line: short-trajectory dipole phase, green line: long-trajectory dipole phase. (b) Factor $\alpha$ of the 675th harmonic as functions of driving laser intensity $I_d$. Blue line: short-trajectory $\alpha _\mathrm {short}$, green line: long-trajectory $\alpha _\mathrm {long}$.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The accumulated phase mismatches of the 675th harmonic due to (a) plasma dispersion, (b) waveguide dispersion, and (c) dipole phase variations. Blue line: short-trajectory dipole phase, green line: long-trajectory dipole phase. (d) The total phase mismatches of the short-trajectory (blue) and long-trajectory (green) emissions. (e) The calculated dephasing lengths of the short-trajectory (blue) and long-trajectory (green) emissions of the 675th harmonic.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Phase variation of the 675th harmonic inflicted by a transverse disruptive pulse of $E_2/E_1 = 0.006$. $\lambda$ is the wavelength.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. (a) The point spread function $\mathrm {PSF}(z)$ applied to the mask. (b) The effective mask pattern $\mathrm {PSF} \otimes M$ for the 675th short-trajectory harmonic.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. (a) The amplitude of the total driving field met by the fixed 675th harmonic wavefront as a function of position $z$. (b) The phase of the total driving field met by the fixed 675th harmonic wavefront. (c) The amplitude of the 675th local harmonic field generated at position $z$. (d) The phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field generated at position $z$.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. (a) The amplitude of the accumulated 675th short-trajectory harmonic field. Black line: ideal case with perfect phase-matching, blue line: optimized case with quasi-phase-matching, purple line: case without the transverse pulse. (b) The amplitude of the modulated transverse field (black), the phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field (blue), and the amplitude of the accumulated harmonic field in the front section of the waveguide. (c) The amplitude of the modulated transverse field (black), the phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 675th local harmonic field (blue), and the amplitude of the accumulated harmonic field in the middle section of the waveguide.

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. The amplitude of the accumulated 675th long-trajectory harmonic field. Black line: ideal case with perfect phase-matching, blue line: optimized case with quasi-phase-matching, purple line: case without the transverse pulse.

View in Article | Download Full Size | PDF

Fig. 11.
Fig. 11. (a) The dephasing length as a function of position $z$ for the short-trajectory emission of the 337th harmonic. (b) The effective mask pattern $\mathrm {PSF} \otimes M$ for the 337th short-trajectory harmonic. (c) The phase variation $\Delta \Phi _\mathrm {LH} = \Phi _\mathrm {LH}(z) - \Phi _\mathrm {LH}(0)$ of the 337th short-trajectory local harmonic field. (d) The amplitude of the accumulated 337th harmonic field normalized to that of the perfect phase-matching condition.

View in Article | Download Full Size | PDF

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

E d ( z = 0 , t ) = A 0 exp ⁡ ( − t 2 2 τ 0 2 ) e − i ω d t x ^ ,
Δ U i o n i z a t i o n ( z ) = [ N g a s I p 1 + N H e 2 + ( z ) I p 2 ] π R 2 Δ z ,
K A T I ( z , t 0 ) = q e 2 I d ( z , t 0 ) c ϵ 0 m e ω d 2 sin 2 ⁡ ( ω d t 0 ) ,
Δ U A T I ( z ) = ∑ a l l e l e c t r o n s K A T I ( z , t 0 ) .
T e ( z ) = 2 3 K ¯ A T I ( z ) k B ,
a I B ( z ) = 1 3 c ω d 2 n p l a s m a ( ω d , z ) q e 6 Z ( z ) N e ( z ) 2 ln ⁡ ( Λ ( z ) ) 2 π ϵ 0 2 m e k B T e ( z ) 3 / 2 ,
n p l a s m a ( ω d , z ) = 1 − ω p ( z ) 2 ω d 2
ω p ( z ) = q e 2 N e ( z ) ϵ 0 m e
a T S ( z ) = 8 π 3 q e 4 ( 4 π ϵ 0 m e c 2 ) 2 N e ( z ) ,
a W G = u 11 2 4 π 2 λ d 2 R 3 ν 1 ,
k ( ω , z ) = k p l a s m a ( ω , z ) + k W G ( ω ) = ω c n p l a s m a ( ω , z ) − u 11 2 c 2 R 2 1 ω ,
E d ( z , t ) = τ 0 τ ( z ) A ( z ) exp ⁡ ( − ( t − C ( z ) ) 2 2 τ ( z ) 2 ) e i ϕ d ( z , t ) ,
C ( z ) = ∂ ∂ ω [ ∫ 0 z k ( ω , z ′ ) d z ′ ] ω = ω d = ∫ 0 z 1 c n ( ω d , z ′ ) d z ′ + u 11 2 c 2 R 2 z ω d 2 ,
D ( z ) = ∂ 2 ∂ ω 2 [ ∫ 0 z k ( ω , z ′ ) d z ′ ] ω = ω d = ∫ 0 z − ω p ( z ′ ) 2 c [ ω d 2 − ω p ( z ′ ) 2 ] 3 / 2 d z ′ − u 11 2 c R 2 z ω d 3 .
ϕ d ( z , t ) = 1 2 tan − 1 ⁡ ( D ( z ) τ 0 2 ) − 1 2 D ( z ) τ 0 4 + D ( z ) 2 ( t − C ( z ) ) 2 + ∫ 0 z k ( ω d , z ′ ) d z ′ − ω d t .
Δ k ( z ) = q k ( ω d , z ) − k ( ω q , z ) + Δ k d i p o l e ( z ) = Δ k p l a s m a ( z ) + Δ k W G + Δ k d i p o l e ( z ) ,
Δ k p l a s m a ( z ) = q k p l a s m a ( ω d , z ) − k p l a s m a ( ω q , z ) ≃ − N e ( z ) e 2 2 ϵ 0 c m e ω d ( q − 1 q )
Δ k W G = q k W G ( ω d ) − k W G ( ω q ) = − u 11 2 c 2 R 2 1 ω d ( q − 1 q )
Δ k d i p o l e ( z ) = d Φ d i p o l e ( I d ( z ) ) d z = ∂ Φ d i p o l e ∂ I d d I d d z = α ( I d ) d I d d z
Δ Φ p l a s m a ( z ) = ∫ 0 z Δ k p l a s m a ( z ′ ) d z ′ ,
Δ Φ W G ( z ) = ∫ 0 z Δ k W G ( z ′ ) d z ′ .
I p e a k ( z ) = 1 2 ϵ 0 c | E d ( z , t = C ( z ) ) | 2 .
Δ Φ d i p o l e ( z ) = Φ d i p o l e ( z ) − Φ d i p o l e ( 0 ) ,
Δ Φ t o t a l ( z ) = Δ Φ p l a s m a ( z ) + Δ Φ W G ( z ) + Δ Φ d i p o l e ( z ) .
E 1 ( r , t ) = E 1 e i ( k z − ω t ) x ^
E 2 ( r , t ) = E 2 e i ( k y − ω t ) x ^ ,
E t o t a l ( r , t ) = E 1 ( r , t ) + E 2 ( r , t ) = E t o t a l ( y , z ) e i Δ ϕ ( y , z ) e i ( k z − ω t ) x ^ ,
E t o t a l ( y , z ) = E 1 2 + E 2 2 + 2 E 1 E 2 cos ⁡ ( k ( y − z ) )
Δ ϕ ( y , z ) = tan − 1 ⁡ [ ( E 2 / E 1 ) sin ⁡ ( k ( y − z ) ) 1 + ( E 2 / E 1 ) cos ⁡ ( k ( y − z ) ) ]
Δ ϕ q ( y , z ) = q Δ ϕ ( y , z ) .
E t r a n s ( x , y = 0 , z , t ) = A t r a n s exp ⁡ ( − ln ⁡ ( 2 ) x 2 2 R 2 ) exp ⁡ ( − ln ⁡ ( 2 ) [ z − ( L / 2 ) ] 2 2 ( L / 2 ) 2 ) exp ⁡ ( − ln ⁡ ( 2 ) [ c t − ( L / 2 ) ] 2 2 ( L / 2 ) 2 ) e − i ω d t ,
M ( z ) = M ( Δ Φ t o t a l ( z ) ) = { 0 , 2 m π ≤ Δ Φ t o t a l ( z ) < ( 2 m + 1 ) π 1 , ( 2 m + 1 ) π ≤ Δ Φ t o t a l ( z ) < ( 2 m + 2 ) π ,
P S F ( z ) = sin ⁡ ( π z / ζ ) ( π z / ζ )
E t o t a l ( z , t ) = E d ( z , t ) + ( P S F ( z ) ⊗ M ( z ) ) E ¯ t r a n s ( z , t ) ,
t ( z ) = ∫ 0 z 1 v p ( ω q , z ′ ) d z ′
E H W F ( z ) = E t o t a l ( z , t ( z ) )
E L H ( z ) ∝ N H e 2 + ( z ) | E H W F ( z ) | p e i Φ L H ( z ) ,
E H ( z ) = ∫ 0 z E L H ( z ′ ) e i ( k ( ω q , z ′ ) Δ z ′ − ω q Δ t ′ ) d z ′ = ∫ 0 z E L H ( z ′ ) d z ′ ,
Δ Φ p l a s m a = − π λ d N e N c r L ,
l D = λ d π N c r N e ,
Δ Φ G a u s s i a n ( z , r ) = k r 2 2 ( z 2 + b 2 ) / z ,
Δ Φ = Δ Φ G a u s s i a n ( z = b , r = 0 ) − Δ Φ G a u s s i a n ( z = b , r = w ( b ) ) = − 1 rad ,
Δ ω = − k 0 d n p l a s m a d t L = k 0 2 n p l a s m a N c r d N e d t L ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use