Observation of backward high-harmonic emission from solids

G. Vampa; Y. S. You; H. Liu; S. Ghimire; D. A. Reis

doi:10.1364/OE.26.012210

1. Introduction

Isolated oscillating dipoles emit a radiation pattern that is symmetrical in the azimuthal plane. It is commonly understood that only the emitted radiation co-propagating with the driving field (in the forward direction) is phase matched with the dipole polarization [1] and can therefore constructively interfere over macroscopic distances. Backward emission quickly falls out of phase over distances comparable to a wavelength [2], so that backward emission is generally much weaker than forward emission. The imbalance between forward and backward emission is apparent in phase-matched nonlinear optical processes [3], including high-harmonic generation [4] where, to make use of phase matching [5], all experiments in gas-phase targets detect harmonics co-propagating with the fundamental driver (except perturbative [2,6] and relativistic [7] harmonic generation where, however, the underlying physics is different). High-harmonic generation from bulk solid targets has inherited this approach [8–15]. However, there is a significant difference from gas-phase high-harmonic generation: high-harmonics from solids are strongly absorbed by the solid itself [16]. Because propagation is severely limited, phase matching schemes are unavailable to increase the high-harmonic flux. Moreover, propagation of intense laser pulses in solids poses more challenges than in gases because self-action effects (in space and time) become significant at moderate intensities [3]. Spatio-temporal distortions due to the plasma generated by the strong field may also be significant like in gases [5,17]. Propagation issues can be overcome by employing thin crystals and films, but these are not always available for all materials, and films may have poor crystalline quality.

Here we observe backward emission of high-harmonics from solids irradiated by near-infrared pulses in the direction of specular reflection up to extreme-ultraviolet (XUV) photon energies. We find that these harmonics are similarly intense to those emitted in the forward direction. We interpret this observation in terms of strong absorption at the harmonic wavelength that occurs over a length scale that is short compared to the wavelength of the driving field. Our approach suggests a monolithic implementation of typical extended attosecond beamlines, embedding generation, focusing and spectral filtering, all in a single active optical element. In addition, patterning the surface paves the way towards phase matching of absorption-limited high-harmonics from solids.

This article is organized as follows. In Section 2 we present representative experimental spectra of the backward-propagating high-harmonics, from Si and MgO crystals, up to 27 eV photon energies. In Section 3 we present our theoretical interpretation of the reflected high-harmonics in terms of dipolar emission from a thin layer of material near the interface between the crystal and a linear medium (vacuum or air in this case). The model is supported by the experimentally measured dependence of the high-harmonic power on the angle of incidence, which is in excellent agreement with the theoretical prediction. We also discuss how the absence of a Brewster’s angle for the nonlinear emission at the high harmonics allows loss-less spectral filtering of the intense infrared driver – a highly desirable feature for high-harmonic spectroscopy. In Section 4 we show that backward emission is unaffected by propagation-induced spatio-temporal distortions of the intense driving infrared beam – contrary to forward emission. Therefore, comparing the backward and the forward emitted high-harmonic beams will allow measuring nonlinear pulse propagation in materials. In Section 5 we conclude by summarizing the scientific and technological advances offered by backward high-harmonic emission, including a scheme to achieve phase-matching of the absorption-limited above-bandgap high harmonics.

2. High-harmonic spectra

In the experiment, we detect high harmonics propagating in the direction of specular reflection for the fundamental driving laser, in two different spectral ranges: for emission in the XUV spectral range between 13 and 27 eV from a 200 µm-thick (100)-oriented MgO single crystal, and for emission up to 6 eV from a 200 µm-thick (100)-oriented Si single crystal. In both cases the experiments are performed below damage threshold at a repetition rate of 1 kHz.

In the first case, the MgO crystal is irradiated by ~70 fs laser pulses (measured with Second-Harmonic Frequency Resolved Optical Gating), with center wavelength of 1.32 µm, loosely focused (f/66) with a CaF₂ lens to reach estimated vacuum intensities of up to 33 TW/cm². The center wavelength has been chosen consistently with a previous experiment [13]. The pulses are obtained from a commercial Optical Parametric Amplifier (LightConversion HE-TOPAS) pumped by 7 mJ, 50 fs pulses at 800 nm wavelength generated by a commercial Ti:sapphire laser system (Coherent Legend-duo). The generated XUV radiation is dispersed with a grazing incidence flat-field grating (Hitachi, 600 l/mm) and detected with a combination of a microchannel plate, phosphor screen and camera. The sample and spectrometer are held in a vacuum chamber at a pressure 5 x 10⁻⁶ mbar. The angle of incidence of infrared beam on the sample is 45°. The XUV spectra are reported in Fig. 1(a) for a moderate intensity (23 TW/cm²) p-polarized (black line) and s-polarized (red line) driving laser. Brighter high-harmonics are measured in the latter case. Due to refraction of the excitation beam inside the MgO crystal, the s- and p- polarizations align to inequivalent crystallographic orientations. While the s-polarization remains parallel to [001] (like at normal incidence), the p-polarization makes an angle of 24° with respect to [010] in the (001) plane of incidence. High-harmonic generation in MgO is anisotropic, with emission strongly peaked for pump polarization along the <001> directions [13]. Thus, harmonic emission is higher for s-polarization in this geometry, despite the increased reflectivity of the pump and correspondingly reduced intensity internal to the crystal compared to p-polarized pump. In addition, as discussed below, the nonlinear reflectivity of the harmonics can also contribute to the contrast between the s- and p-spectra. Under a higher excitation intensity (33 TW/cm²), harmonic emission extends up to the 29^th order (27 eV). Although both surfaces of the MgO crystal are polished, forward emission from the first surface from near-infrared light reflected off the back surface is excluded due to the low reflectivity of the fundamental from the back surface and the high degree of the nonlinearity.

Fig. 1 High-harmonic spectra measured in the specular reflection direction, from (a) a (100)-oriented MgO crystal with 1.32 µm laser excitation and (b) a (100)-oriented Si crystal with 2.1 µm laser excitation. In (a), the driver is s-polarized (red and blue lines), or p-polarized (black line). The spectrum for p-polarization has been multiplied by 2 to account for the lower reflectivity of the grating. The estimated intensities of the driver are 23 and 33 TW/cm² (corresponding to 90 and 130 µJ driving-pulse energy). Harmonic power is normalized to the 17th order at the highest intensity. The angle of incidence is 45°. In (b), the angle of incidence is 46°, the driving laser is p-polarized, and the vacuum intensity is 4.2 TW/cm². The highest detected harmonic in the case of Si is limited by the detection range of the VIS-UV spectrometer used.

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Panel (b), instead, reports the high-harmonic spectrum obtained from Si irradiated by ~70 fs laser pulses with a center wavelength of 2.1 µm, with an estimated vacuum intensity of 4.2 TW/cm². Here, a longer wavelength pump is used to mitigate multi-photon absorption given the low (indirect) band-gap of Si. Upon emission, high-harmonics are focused with a CaF₂ lens (f = 75 mm) on the slit of a UV-VIS spectrometer (OceanOptics Flame). Contrary to MgO, the back surface of the crystal is not polished (ground). This fact rules out a possible contribution of the back surface to high-harmonic emission – as for MgO. The angle of incidence is 46°. The refracted beam propagates nearly normal to the surface (with an angle of ~12°).

3. Theoretical interpretation

High-harmonic emission in the specular reflection arises from a dipolar nonlinear polarization of the bulk stimulated by the fundamental wave which is refracted at the air-crystal interface, and that is localized in a thin layer near the surface – approximately one harmonic wavelength thick [2]. Directionality of the harmonic beam is a result of phase matching at this interface [2], as sketched in Fig. 2. The incident wave at the fundamental frequency (with wavevector k₀) sets on a nonlinear polarization wave at the m-th harmonic which travels with a momentum component parallel to the surface equal to $k_{N L ∥} = m k_{o ∥}$ , where $k_{o ∥} = ω_{o} n_{1} (ω_{0}) \sin θ_{i} / c$ is the component of the incident wavevector parallel to the surface (and that is conserved upon reflection). The polarization radiates a harmonic wave in vacuum, with wavevector $k_{m} = m ω_{o} n_{1} (m ω_{o}) / c$ . In vacuum, $n_{1} (ω_{0}) = n_{1} (m ω_{0})$ , so that $k_{N L ∥} / k_{m} = k_{0 ∥} / k_{0}$ and the harmonic beam co-propagates with the fundamental in the direction of specular reflection (this need not be the case for a dispersive medium like air).

Fig. 2 Schematics of the phase matching condition for the high harmonics in the direction of specular reflection. The wavevectors are defined as follows: k₀ is the fundamental wave, k₀_|| is the component of k₀ in the plane of the surface, k_m and k_m’ are the reflected and transmitted harmonic waves respectively, k_NL is the nonlinear polarization and k_NL_|| is its component in the plane of the surface. α is the angle between k_NL and the polarization of P_NL_. The linear and nonlinear media have indices n₁ and n₂ respectively. All angles are measured relative to the normal to the surface.

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The amplitude of the reflected wave is linked to that of the polarization by nonlinear reflection coefficients – in a similar way that the linear Fresnel coefficients link the incident and reflected waves [2]:

E_{m}^{(r)} = r_{N L} P_{N L}

where

P_{N L}

is the high-harmonic polarization, which depends on the fundamental driving intensity and the crystallographic orientation. In optically isotropic materials, such as Si and MgO, the nonlinear reflection coefficients for s- and p- polarized light are calculated as [2]:

r_{N L, s} = \sin^{2} θ_{t} \sin θ_{s} {[\sin (θ_{r} + θ_{t}) \sin (θ_{s} + θ_{t}) \sin θ_{r}]}^{- 1}

\begin{array}{l} r_{N L, p} = \sin^{2} θ_{t} \sin θ_{s} \sin (α + θ_{t} + θ_{s}) \times \\ \times [^{ϵ_{r} \sin (θ_{r} + θ_{t}) \sin (θ_{s} + θ_{t}) \sin θ_{r} \cos (θ_{t} - θ_{r})] - 1} \end{array}

where

ϵ_{r}

is the dielectric constant of the linear medium (air or vacuum), α is the angle between the polarization and the k-vector of P_NL (α = 90° in our case),

θ_{r}

is the propagation angle of the reflected wave (obtained by the conservation of momentum discussed above),

θ_{t}

is the propagation angle of the harmonic wave transmitted in the crystal and

θ_{s}

is the propagation angle of the polarization in the crystal. The latter two are given by Snell’s law [2]:

n_{2} (m ω_{0}) \sin θ_{t} = n_{1} (m ω_{0}) \sin θ_{i}

n_{2} (ω_{0}) \sin θ_{s} = n_{1} (ω_{0}) \sin θ_{i}

where

n_{1}, n_{2}

are the refractive indices of air (or vacuum) and of the crystal, respectively. Although Eqs. (2) and (3) have been derived for perturbative second-order nonlinearities, they can be applied to non-perturbative high-harmonic generation provided one uses the generalized laws of reflection and refraction (Eqs. (4) and (5)), which account for the refractive index at the high-harmonic wavelengths. In the non-perturbative case, however, the index at the fundamental wavelength may be more strongly affected by Kerr nonlinearity and plasma formation. Kerr nonlinearity is expected to induce a refractive index change < 7.7% in Si and < 0.7% in MgO, at the highest intensities used in the experiment (4.2 and 33 TW/cm²).

To test our understanding of backward emission, we measure the harmonic power emitted from Si as a function of the angle of incidence. In Fig. 3, the high-harmonic power from Si is measured for a fixed pulse energy and compared to the theoretical prediction for $| E_{m}^{(r)} |^{2}$ (solid lines). For this experiment, the CaF₂ lens that collects the high-harmonics and the UV-VIS spectrometer are mounted on a goniometer that rotates about the laser focus. The agreement between theory and experiment is excellent, confirming our interpretation. Two quantities need to be calculated: $r_{N L} (θ_{i})$ and $P_{N L} (θ_{i})$ . To calculate r_NL, we neglect the dispersion of air in Eqs. (4) and (5) and use the complex refractive index for the harmonics [18]. The nonlinear polarization P_NL varies with angle of incidence because (i) the transmitted fundamental intensity monotonically decreases with increasing angle and (ii) the polarization of the fundamental driver aligns to different crystallographic axis as the beam is refracted at different angles. We neglect the latter effect because, due to the high refractive index of Si, the propagation angle of the transmitted fundamental is at most 16°, and only a weak dependence of the harmonic power on alignment has been observed in Si [19]. By virtue of the nonlinearity, instead, the former effect – namely the decrease of the intensity as the angle of incidence increases – causes a strong decrease in the harmonic emission efficiency. Knowing the fundamental intensity inside the crystal (through the angle-dependent linear transmission Fresnel coefficients) and the high-harmonic yield at that intensity (at any, fixed, angle of incidence) allows a calibration of $P_{N L} (θ_{i})$ . The variation of the harmonic yield with power of the fundamental for a fixed angle of incidence of 30° is shown in Fig. 4. Power is adjusted with a variable neutral-density filter. All harmonics scale non-perturbatively with approximately the 5th power of the fundamentalintensity (the dashed line is a guide to the eye). The data points of Fig. 4 are interpolated with polynomial curves as a function of a dense set of intensities to allow calculation of $P_{N L} (θ_{i})$ at the sampled angles of incidence. Large angles of incidence require calibration data at low intensities. We only sample intensities higher than 1.26 TW/cm² for the 5^th, 7^th and 9^th harmonics, which corresponds to angles of incidence lower than 77 deg, and higher than 3.2 TW/cm² for the 11^th harmonic, corresponding to an angle of incidence of 54 deg. An anomalous behavior is observed for the 7^th harmonic, which becomes stronger than the 5^th harmonic at low intensities. The nontrivial behavior of the 7^th harmonic could be related to resonant effects with strong features in the joint electronic density of states (the 4.2 eV direct transition Γ₂₅ → Γ_2’ or the direct transitions at the X critical point) [20]. The anomalous behavior of the 7^th harmonic is also observed in Fig. 3 around 55°.

Fig. 3 Harmonic power as a function of the angle of incidence, for generation in Si. The experiment (colored dots) agrees with the theoretical prediction (solid lines, vertically offset to match the experiment for an angle of incidence of 30°). The fundamental driver is p-polarized. The dashed vertical black line marks the Brewster’s angle for the fundamental (74°). The linear Brewster’s angle for the high harmonics varies between 74° - 80°.

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Fig. 4 Power of the high harmonics generated from Si as a function of the driving laser intensity, for a fixed angle of incidence of 30°. The dashed black line, scaling with I⁵, is a guide to the eye. The driving field is p-polarized.

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Although the fundamental driver is p-polarized, the harmonic power is not affected by the Brewster’s angle at the harmonic frequency, which lies between 75° – 80°. The nonlinear reflection coefficients have, in fact, different conditions for the occurrence of a “nonlinear Brewster’s angle” [2], which are not met in our experiment. Therefore, filtering of the strong driver from the weak high-harmonics has been achieved. This scheme is very appealing for using the reflected high-harmonics in ex-situ experiments, because the presence of the strong fundamental field would otherwise dress the sample under study. Filtering of the driver is achieved in gas-phase high-harmonic sources by metal filters which have limited bandwidth and introduce dispersion [21], or by annular-shaped driving beams [22]. Not only reflected harmonics integrate generation with filtering, curved crystal surfaces will also implement focusing of the high-harmonics with high numerical aperture – a task that is typically achieved with expensive XUV mirrors, and only with low numerical aperture. Looking into the future, actual metasurfaces can be employed for advanced shaping of the spatio-temporal properties and polarization of the high-harmonic beam [23]. Therefore, backward high-harmonic emission promises to integrate all the functionalities – possibly more – of current gas-phase attosecond beamlines into an inexpensive and compact setup.

4. Comparison between backward and forward emission

Due to the rapid phase mismatch between the nonlinear polarization and the harmonic wave, only a sheet of polarization which is thinner than the coherence length of the harmonics – in fact comparable to the harmonic wavelength – contributes to backward-propagating emission [2]. Over such small distance the intense fundamental beam is unaffected by propagation through the crystal. On the contrary, forward emission requires the fundamental to reach the exit surface, with consequent accumulation of significant spatio-temporal distortions due to linear and nonlinear effects. Because high-harmonic generation is very sensitive to the amplitude and phase of the intense driver, forward emission is spatially and spectrally broader than backward emission, as shown in Fig. 5 for the 0.2 mm thick MgO crystal. The forward propagating harmonics are spectrally broader – likely a result of self-phase modulation of the driving pulse – and more divergent (panel c) – likely a result of self-focusing. A more systematic comparison between backward and forward high-harmonic emission will allow to track the evolution of spatio-temporal couplings in solids with high spatial and temporal resolutions. In principle, a suitable geometry will allow amplitude and phase information of forward high-harmonic emission to be obtained by interfering it with backward emission. More generally, high-resolution holograms of the output surface – for example to map laser-induced damage [24] – can be recorded by the interference with the backward-emitted beam.

Fig. 5 Spatially (divergence in the vertical axis, in mrad) and spectrally (horizontal axis, in eV) resolved XUV emission from the MgO crystal measured in the reflected (a) and transmitted (b) beams. The transmitted power is relative to the reflected power. (c) Spatial profile spectrally integrated for energies > 19 eV, for the transmitted (red) and reflected (blue) harmonics.

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Next, we point out a possible limitation on the reflected high-harmonic power. In Fig. 5 we measure equally intense forward and backward emission. When the harmonic wavelength becomes shorter than the absorption length, however, forward emission will become stronger than backward emission (apart from different nonlinear reflection and transmission coefficients). This is because while the former arises from a region on the scale of one absorption length from the output surface, the emission region of the latter is on the scale of the harmonic wavelength, as discussed at the beginning of this section and in [2]. In MgO, this condition is only narrowly approached, for the absorption length of MgO is 17 nm at 30 eV, corresponding to a wavelength of 41 nm. However, it may become problematic at shorter wavelengths.

5. Conclusions

In conclusion, we have demonstrated backward high-harmonic emission at UV and XUV wavelengths from Si and MgO crystals respectively and we have shown that the reflected beam is unaffected by complex spatio-temporal distortions induced in the transmitted driver, thus providing an ideal reference to study light propagation of highly intense beams in bulk matter. Backward emission arises from a nonlinear dipolar bulk polarization localized at the surface. From a technological perspective, our results suggest a simple and efficient scheme to produce focused high-harmonic beams with tailored properties, without contamination from the strong driver.

Finally, we note that the reflection geometry is suitable to achieve coherent addition and phase matching of the high-harmonics by multiple reflections of the driving beam between two parallel crystal surfaces, generating harmonics upon each reflection. This is a major advantage compared to the transmission geometry, where the useful generation depth is limited to one absorption length, and to gases, where phase matching at XUV wavelengths is hard to achieve due to the dispersion of the ionized gas [5]. Here, instead, both the fundamental and the harmonics propagate in a vacuum channel between the two plates, experiencing zero distortions and losses (but accumulating the phase of the nonlinear reflection coefficients upon each reflection). Hollow waveguides can potentially be employed as well. Our results are the necessary first step to understand solid-state high-harmonic generation in these geometries, and therefore pave the way to high-flux solid-state based high-harmonic sources.

Funding

G.V., H. L. and D. A. R. were supported by the Air Force Office of Scientific Research, grant number FA9550-14-1-0108. Y. S. Y. and S. G. acknowledges support from the US Department of Energy, Oﬃce of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the Early Career Research Program.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Observation of backward high-harmonic emission from solids

Abstract

1. Introduction

2. High-harmonic spectra

3. Theoretical interpretation

4. Comparison between backward and forward emission

5. Conclusions

Funding

Disclosures

References and links

Cited By

Figures (5)

Equations (5)

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