Sawtooth grating-assisted phase-matching

Pavel Sidorenko; Maxim Kozlov; Alon Bahabad; Tenio Popmintchev; Margaret Murnane; Henry Kapteyn; Oren Cohen

doi:10.1364/OE.18.022686

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, ΔΦ₀(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:

E_{H} = E_{H}^{0} \int_{0}^{L} exp [i (Δ Φ_{0} + Δ Φ_{GAPM})] d z,

where

E_{H}^{0}

is the z-independent amplitude of the harmonic field that is generated within an interval dz, and ΔΦ₀(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 −ΔΦ₀(z). The combination of the medium phase-mismatch and the sawtooth phase-shift ΔΦ₀ + 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.

Fig. 1 Sawtooth GAPM. (a) Under phase-mismatch conditions, the shift between the phases of the driving and harmonic fields (ΔΦ₀) grows linearly, leading to oscillations in the harmonic signal. (b) a sawtooth phase-shift (ΔΦ_GAPM) for correcting the phase mismatch of the nonlinear process. The sharp sawtooth (solid line) wave consists of infinite number of Fourier components. Also plotted is a smooth sawtooth wave that consists of the first three Fourier components (dashed line). (c) The combination of the medium phase-mismatch and the sawtooth phase-shift ΔΦ₀ + ΔΦ results in a stairs-function phase (d) Harmonic field generated along 8L_C propagation distance with N-waves sawtooth GAPM. (e) Optimal values of the amplitudes of the waves in N-waves sawtooth GAPM. Shown are the differences between their upper limits, 2/m, and the optimal values. (f) The QPM efficiency factor (intensity conversion efficiency normalized by the conversion efficiency at perfect phase-matching) of N-waves sawtooth GAPM. The conversion efficiency of 2 waves sawtooth GAPM is higher than the QPM efficiency factor of 1st order QPM where the polarization direction is flipped every coherence length (e.g., as in PPLN).

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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:

Δ Φ_{GAPM} = \sum_{m = 1}^{N} A_{m} sin (m π z / L_{C}) .

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:

E_{H} = E_{H}^{0} J_{1}^{1, 2 ... N} (A_{1}, A_{2}, ... A_{N}),

where

J_{1}^{1, 2 ... N} (A_{1}, A_{2}, ... A_{N})

is the N-dimensional generalized Bessel function. The values of A₁, A₂,…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 A₁, A₂,…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.

Fig. 2 A scheme for optical induction of sawtooth GAPM in high harmonic generation. A strong driving laser pulse interferes with several weak Quasi-CW beams inside a planar waveguide, creating a sawtooth intensity pattern along the pulses propagation direction, z. The quasi-CW beams are produced using a mask that is perpendicular (a) or parallel (b) to the propagation direction of the driving laser.

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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 = E₀B(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, E₀ 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 $E_{m}^{CW} = E_{0} r_{m} B (y) sin [ω t - k (z cos (θ_{m}) + x sin (θ_{m}))]$ 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 = E₀B(y)F(x,z,τ)cos(ωτ) and $E_{m}^{CW} = E_{0} r_{m} B (y) sin [ω τ + 2 π z / Λ_{m} - kx sin (θ_{m})]$ where τ = t-nz/c and Λ_m = 2π/k[1 – cos(θ_m)]. The total intensity at the peak of the pulses, $I \propto {[E_{D} + \sum_{m = 1}^{N} E_{m}^{CW}]}^{2}$ at x = y = τ = 0, is given by I = I₀ + ΔI(z) where $Δ I \propto E_{0}^{2} \sum_{m = 1}^{N} r_{m} sin (2 π z / Λ_{m})$ (the very weak terms that are proportional to r_mr_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:

Δ Φ_{GAPM} = \sum_{m = 1}^{N} A_{m} sin (2 π z / Λ_{m}),

where A_m(E₀,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 = 2×10¹⁵ W/cm², 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

\frac{\partial E_{D}}{\partial z} = - \frac{1}{2 c} \int_{- \infty}^{τ} ω_{p}^{2} E_{D} d τ - \frac{2 π I_{P}}{E_{D} c} \frac{\partial n_{e}}{\partial τ},

where

ω_{p} = \sqrt{4 π e^{2} n_{e} / m}

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 [22]. The total optical field,

E_{D} + \sum_{m = 1}^{N} E_{m}^{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

\frac{\partial E_{HHG}}{\partial z} = - \frac{2 π}{c} \frac{\partial P_{HHG}}{\partial τ} .

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: θ₁ = 18.9°, θ₂ = 26.8°, and θ₃ = 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: r₁ = 2.2 × 10⁻⁵, r₂ = 5.2 × 10⁻⁶, r₃ = 2.6 × 10⁻⁶, I₁ = 4.4 × 10¹⁰ W/cm², I₂ = 1.1 × 10¹⁰ W/cm², and I₃ = 5.2 × 10⁹ W/cm². The total intensity of the eighteen quasi-CW waves is I_CW = 5.7 × 10¹⁰ W/cm² - which is 3.5 × 10⁵ times smaller than the intensity of the driving laser.

Fig. 3 Numerical results of sawtooth GAPM in HHG. (a) Amplitude of q-order harmonic phase oscillations as a function of the ratio between the peak fields of the strong laser pulse and a weak quasi–CW beam. (b) Harmonic spectral intensity with (solid red) and without (dash blue) 18-waves sawtooth GAPM. (c) Harmonic spectral intensity in the enhanced spectral range using 1-wave (solid black) and 18-waves (dash red) sawtooth GAPM (d) Harmonic signal versus propagation distance for 243-order harmonic for several phase matching conditions including: without GAPM (blue), 1-wave sawtooth GAPM (green), square GAPM (purple), 2-waves sawtooth GAPM (red), 3-waves sawtooth GAPM (brown), and 18-waves sawtooth GAPM (black). All curves are normalized by the phase-matched growth-rate (dash black).

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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).

References and links

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Sawtooth grating-assisted phase-matching

Abstract

Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (6)

Optics Express