## Abstract

We show that a sawtooth phase-modulation is the optimal profile for grating assisted phase matching (GAPM). Perfect (sharp) sawtooth modulation fully corrects the phase-mismatch, exhibiting conversion equal to conventional phase matching, while smoothened, approximate sawtooth structures are more efficient than sinusoidal or square GAPM modulations that were previously studied. As an example, we demonstrate numerically optically-induced sawtooth GAPM for high harmonic generation. Sawtooth GAPM is the most efficient method for increasing the conversion efficiency of high harmonic generation through quasi-phase-matching, with an ultimate efficiency that closely matches the ideal phase-matching case.

© 2010 OSA

Quasi-phase matching (QPM) techniques are widely used for correcting the phase mismatch and increasing the efficiency of nonlinear processes [1]. In linear QPM or grating-assisted-phase-matching (GAPM), the phase-mismatch is corrected by a periodic modulation of the refractive indices of the pump and harmonic beams [1–5]. Under phase-mismatched conditions, the shift between the phases of the source and generated fields grows linearly as the pump and signal waves propagate. QPM modulation provides a periodic phase reset that partially corrects this phase-mismatch. In GAPM, this phase-shift results from modulation in the effective linear refractive indices of the waves that participate in the nonlinear wave mixing process [1–5]. Recently, it was discovered that GAPM concepts can be applied to high harmonic generation through optical induction [6]. In this case, a phase-modulation in the nonlinear polarization is induced through interaction of the driving laser with other optical waves [6–8] or with a modulated static electric field [9]. In previous works, sinusoidal and square phase-shifts were investigated in several nonlinear processes, including low-order [1–5,10] and high-order [6–9] harmonic generation. The QPM efficiency factors (the QPM intensity conversion efficiency normalized by the perfect phase-matched conversion efficiency) of sinusoidal GAPM and square GAPM are ∼0.37 and ∼0.4, respectively [1]. These values are comparable to QPM efficiency factors in ordinary QPM schemes. For example, the QPM efficiency factor of 1st-order QPM, where the polarization direction is flipped every coherence length, (e.g., as in PPLN) is approximately 0.4 [1].

Here we show that GAPM employing a sawtooth phase-shift profile can in-principle provide “perfect” phase correction; i.e. an efficiency factor of 1. We propose and analyze a scheme for implementing sawtooth GAPM in high harmonic generation. In high harmonic generation (HHG), infrared or visible light is up-converted into the extreme ultraviolet and soft x-ray regions of the spectrum. During the HHG process the medium is ionized, and the associated strong plasma dispersion limits true phase matching to a spectral region below the phase-matching cutoff [11]. Several QPM methods were experimentally demonstrated for correcting the phase-mismatch in HHG [12–15]. In these techniques, QPM is obtained by turning off (or suppressing) the generation process in out-of-phase coherent zones, resulting with QPM efficiency factor that is smaller than 0.2 [16]. In a different approach, GAPM results from a nonlinear phase-shift modulation between the pump and high-order polarization [6–9]. As a consequence of the non-instantaneous nature of the HHG process, the high-order polarization is phase-shifted relative to the driving laser field [17,18]. This approach is formally equivalent to GAPM in low-order harmonic generation, where the GAPM phase shift results from modulation of the refractive indices of the pump and/or generated harmonic wave [1–5,10]. Schemes for introducing phase-shifts with sinusoidal [6–8] and square [9] modulations have been proposed.

Here, we show that GAPM with a sharp sawtooth leads to full correction of the phase mismatch. We also propose and analyze the case of approximate, smoothened-sawtooth GAPM, where a *finite* series of sinusoidal waves is employed for the GAPM modulation. We show that the corresponding conversion efficiency increases when more waves form the sawtooth modulation structure. Notably, 2-wave sawtooth GAPM is already significantly more efficient than the sinusoidal and square GAPM structures that were previously investigated. As an example, we propose and analyze sawtooth GAPM in high harmonic generation where the phase-shift is induced by multiple weak quasi-CW waves. We demonstrate numerically that we can approach the ideal phase-matching case in a regime where perfect phase matching is otherwise impossible.

In GAPM, the phase-shift between the pump and signal fields consist of two terms: a phase-mismatch term, ΔΦ_{0}(z), which grows linearly along propagation axis, z, and an oscillating term, ΔΦ_{GAPM}(z), which results from the GAPM modulation. The coherent buildup of the harmonic field is given by:

_{0}(z) = πz/L

_{C}, where L

_{C}is the coherence length in the un-modulated medium (Fig. 1a). Figures 1b and 1c show that the optimal GAPM phase-shift is a sawtooth profile with a periodicity that corresponds to two coherence lengths and a slope that corresponds to −ΔΦ

_{0}(z). The combination of the medium phase-mismatch and the sawtooth phase-shift ΔΦ

_{0}+ ΔΦ results in a step-function phase-shift that leads to a linear growth of the HHG signal, in the same fashion as in true phase matching. That is, the QPM efficiency factor of sharp sawtooth GAPM is one.

In some cases, sharp sawtooth GAPM cannot be implemented. Hence, it is instructive to investigate the efficiency of smooth (non-perfect) sawtooth GAPM profiles. We analyze GAPM modulations that are given by the following finite series of sinusoidal waves:

The optimal sharp sawtooth phase-shift is obtained in Eq. (2) when N = ∞ and A_{m} = 2/m. The N = 1 and A = 1.84 case corresponds to the sinusoidal GAPM that was investigated previously [1,2,6]. Inserting Eq. (2) into Eq. (1) and integrating for the case when L corresponds to an integer number of coherence lengths leads to:

_{1}, A

_{2},…A

_{N}for which the N-dimensional generalized Bessel functions attain their maxima, and thus the N-waves sawtooth GAPM is most efficient, can be calculated quasi-analytically by decomposing the N-dimensional generalized Bessel function in terms of ordinary Bessel function [19]. The optimal values of A

_{1}, A

_{2},…A

_{N}are somewhat smaller than 2/m (Fig. 1e). The QPM efficiency factor of an N-wave sawtooth GAPM implementation is shown in Fig. 1f. As shown, the QPM efficiency factor of 2-wave sawtooth GAPM is approximately 0.52, which is already larger than the QPM efficiency factor for either sinusoidal (∼0.37) or square (∼0.4) GAPM that were investigated in previous works [1–10].

Next, we discuss the implementation of sawtooth GAPM in high harmonic generation. In HHG, the emitted harmonics are phase-shifted relative to the driving laser. This extra phase, which is acquired by the electron along its femtosecond “boomerang” path under the influence of the laser field, is very large, reaching hundreds of radians, and is proportional to the intensity of the driving laser [17,18]. Thus, inducing a shallow sinusoidal modulation in the laser intensity along the propagation direction leads to sinusoidal modulated phase-shift in the HHG process. A convenient way to induce such a sinusoidal modulation is by interfering the driving laser pulse with a weak quasi-CW beam that propagates in a different direction. In this case, it is straightforward to control the periodicity and amplitude of the phase-shift modulation. The periodicity of the phase-shift modulation and intensity grating can be controlled, for example, by changing the propagation direction of the quasi-CW beam. The amplitude of the phase-shift modulation is determined by the amplitude of the intensity grating, and therefore can be controlled by the intensity of the quasi-CW field. Remarkably, an extremely weak quasi-CW field is sufficient to induce a significant phase-shift [6]. Indeed, it was recently shown that a single quasi-CW counterpropagating IR beam with an intensity which is more than six orders of magnitude smaller than the intensity of the driving pulse laser induces a sinusoidal GAPM [6]. Expanding on this concept, sawtooth GAPM can be “fabricated” by using multiple quasi-CW weak beams. It is possible to consider each quasi-CW beam as contributing a single Fourier component to the phase structure. For example, multiple quasi-CW waves can be produced by illuminating an appropriate mask with a long pulse with the same wavelength as the strong driving laser pulse that propagates in a parallel direction (Fig. 2a) or in a perpendicular direction (Fig. 2b) to the driving pulse.

We now present a simple model for sawtooth GAPM in HHG (by expanding the model that was developed in Ref. 6 to multiple quasi-CW waves) and then employ it for calculating the parameters of the quasi-CW waves. Consider a driving pulse that propagates in a hollow-core planar waveguide [20] in z direction, *E _{D}* =

*E*

_{0}

*B*(

*y*)

*F*(

*x*,

*z*,

*t*)cos(

*ωt*–

*kz*) at angular frequency ω and wave-number k = ωn/c, where c is the velocity of light in vacuum, n is the effective index of refraction, E

_{0}is the peak electric field of the driving pulse F(x,z,t) is an envelope function and B(y) is the mode of the planar waveguide [B(y = 0) = 1]. It is assumed that the focal spot in x is wide such that the beam does not experience significant diffraction in x direction. The driving pulse interferes with m = 1,2…N quasi-CW fields that are given by ${E}_{m}^{\mathit{CW}}={E}_{0}{r}_{m}B(y)\text{sin}\left[\omega t-k(z\text{cos}({\theta}_{m})+x\text{sin}({\theta}_{m}))\right]$ where θ

_{m}are propagation angles relative to z axis and r

_{m}<<1 are field ratio parameters. Transforming into a frame moving in the forward direction at phase velocity of the driving field c/n gives E

_{D}= E

_{0}B(y)F(x,z,τ)cos(ωτ) and ${E}_{m}^{\mathit{CW}}={E}_{0}{r}_{m}B(y)\text{sin}\left[\omega \tau +2\pi z/{\Lambda}_{m}-\mathit{kx}\text{sin}({\theta}_{m})\right]$ where τ = t-nz/c and Λ

*= 2*

_{m}*π*/

*k*[1 – cos(

*θ*)]. The total intensity at the peak of the pulses, $I\propto {\left[{E}_{D}+\sum _{m=1}^{N}{E}_{m}^{\mathit{CW}}\right]}^{2}$ at x = y = τ = 0, is given by I = I

_{m}_{0}+ ΔI(z) where $\Delta I\propto {E}_{0}^{2}\sum _{m=1}^{N}{r}_{m}\text{sin}(2\pi z/{\Lambda}_{m})$ (the very weak terms that are proportional to r

_{m}r

_{m′}are neglected). A GAPM phase-shift is induced by the CW waves since the intrinsic phase of HHG is proportional to the total intensity of the driving laser [17,18]. The GAPM phase-shift is thus given by:

_{m}(E

_{0},r

_{m}) are the amplitudes of the induced GAPM phase-shift sinusoidal components.

Next, we numerically investigate sawtooth GAPM in HHG in a specific example. In this numerical experiment we consider a 30 fs driving laser pulse at wavelength λ = 800 nm and peak intensity I_{L} = 2×10^{15} W/cm^{2}, propagating in a medium that consists of pre-ionized Ar ions (second ionization potential is I_{p} = 21 eV) at pressure P = 15 torr. In addition, multiple N weak quasi-CW laser fields at frequency ω propagate at angles θ_{m} with respect to z. The quasi-CW fields are weak and therefore propagate linearly in the medium. The propagation of the strong driving pulse, E_{D}(z,t,x = 0,y = 0), and harmonic fields are calculated using the model in Ref. 21. The nonlinear evolution of the fundamental field is given by

_{e}, takes into account the pre-formed plasma and the ionization that is calculated by using the ADK model [22]. The total optical field, ${E}_{D}+\sum _{m=1}^{N}{E}_{m}^{\mathit{CW}}$, is used for calculating the high-order polarization, P(z,τ) through numerical calculation of the 1D time-dependent Schrödinger equation (TDSE) within a single-active-electron approximation. The generation and evolution of the HHG field, E

_{HHG}, is described by

The first step in determining the parameters of the quasi-CW beams is to calculate the coherence length by examining the oscillations of the harmonic signal in the absence of the quasi-CW beams. We find that the coherence length of harmonic order q = 243 is L_{C} = 7.4 μm. The propagation angles of the quasi-CW beams, θ_{m}, for m = 1,2,3…18 (waves with m>18 are evanescent and therefore were not included in the calculation) are determined by the condition 2L_{C}/m = Λ_{m} = 2π/[k(1–cos(θ_{m}))] which is obtained by corresponding between Eqs. (4) and (2). The angels of the first three quasi-CW waves are: θ_{1} = 18.9°, θ_{2} = 26.8°, and θ_{3} = 33.1°. Following the scheme in Ref. 6, we calculated the amplitude of the induced sinusoidal oscillation, A(r), in order to determine the intensities of the quasi-CW fields (Fig. 3a). It is important to note that in Ref. 6, the modified Lewenstein model [23] was used for calculating A(r) for emission through a specific electronic trajectory (“short' or “long” trajectory in one optical cycle). In that case, A(r) was found to be strictly a linear function. In this work, on the other hand, the TDSE was used for calculating A(r), taking into account all emissions in the pulse into the q^{th} harmonic. In this case, A(r) becomes somewhat nonlinear. The field ratio parameters, r_{m}, and therefore also the peak intensities of the quasi-CW waves are determined by matching the calculated A(r) (Fig. 3a) with the optimal values of A_{m} (Fig. 1e). The field ratio parameters and peak intensities of the first three quasi-CW waves are: r_{1} = 2.2 × 10^{−5}, r_{2} = 5.2 × 10^{−6}, r_{3} = 2.6 × 10^{−6}, I_{1} = 4.4 × 10^{10} W/cm^{2}, I_{2} = 1.1 × 10^{10} W/cm^{2}, and I_{3} = 5.2 × 10^{9} W/cm^{2}. The total intensity of the eighteen quasi-CW waves is I_{CW} = 5.7 × 10^{10} W/cm^{2} - which is 3.5 × 10^{5} times smaller than the intensity of the driving laser.

Figure 3b shows the calculated spectra with 18-waves sawtooth GAPM after propagation distance L = 148 μm that corresponds to 20 coherence lengths. As shown, the spectral region around q = 243 is enhanced through sawtooth GAPM by more than three orders of magnitude. Figure 3c shows the spectra obtained through sawtooth GAPM with eighteen and one quasi-CW waves, demonstrating that the shape of the spectrum is conserved. Figure 3d shows the q = 243 harmonic signal versus propagation distance for several GAPM phase-shift profiles, including square GAPM and sawtooth GAPM with one, two, three and eighteen quasi-CW waves. As expected, the conversion efficiency of two-wave sawtooth QPM is higher than the conversion efficiencies of a sinusoidal (one quasi-CW wave) and square GAPM that were previously investigated. The curves are normalized by the ideally perfect phase-matching growth-rate (dashed curve). The normalized signals at z = L = 10L_{C} are somewhat smaller than the theoretical QPM efficiency factors that are presented in Fig. 1f. The small reduction results from the fact that the parameters for the quasi-CW waves were derived from a simple model which neglects the variation of the optical field during the pulse. For example, in the case of 18-wave sawtooth GAPM, the numerically calculated normalized signal at z = 10L_{C} and theoretical QPM efficiency factor are 0.88 and 0.93, respectively.

## Conclusion

In conclusion, we explored sawtooth GAPM where the phase-shift is designed to exhibit a sawtooth profile. A sharp sawtooth phase-shift leads to full correction of the phase mismatch and to a conversion efficiency that is comparable to full phase-matching. We also proposed and analyzed N-wave sawtooth GAPM, in which the phase-shift profile consists of the first N sinusoidal waves of the sawtooth Fourier series. We calculated the optimal amplitudes of the truncated Fourier series and the QPM efficiency factors of N-waves sawtooth GAPM and found that the efficiency increases with increasing N, and that 2-wave GAPM is already more efficient than either sinusoidal or square GAPM that were investigated in previous works. Sawtooth GAPM can be implemented in various nonlinear processes and through several mechanisms. In the case of sawtooth GAPM applied to high harmonic generation, where the GAPM phase-shift is induced optically, we found that it results in the highest HHG conversion efficiency of any QPM method to date. Finally, exciting implementations of sawtooth GAPM are for inducing temporal sawtooth modulation of a relevant physical parameter and facilitating the correspondence between energy and momentum mismatches [24].

## Acknowledgments

This work was supported by USA–Israel Binational Science Foundation (BSF), Legacy Heritage fund of Israel Science Foundation (ISF), and the Marie Curie International Reintegration Grant (IRG).

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