High-harmonic phase spectroscopy using a binary diffractive optical element

We report on high-order harmonic generation (HHG) using a Ti:sapphire laser beam phase shaped with a binary diffractive optical element (DOE) to create two spatially separated synchronized HHG sources at the focus of a lens. Using full three-dimensional computations, we show numerically that the harmonic dipole phase is imprinted in the resulting far-field fringe pattern. Using the corresponding experimental arrangement, we measure HHG phase in aligned carbon dioxide. This arrangement is robust, extremely stable, simple to use, and gives highly resolved fringes. It thus opens new perspectives for combined and refined HHG phase measurements in excited samples.

Figure 1

(a) Infrared intensity map at the focus of a 1-m lens. The phase mask profile is modeled by the function $\arctan \left(\frac{100x}{{\mathrm{x}}_{\max }}\right)$ where $x_{\max }=$ 5 cm and $-5\mathrm{cm}<x<5\mathrm{cm}$. (b) Cuts through the intensity profile of the infrared beam 10 mm upstream from the focus (light gray), at focus (solid black line), 10 mm downstream from the focus (medium gray), and 30 mm downstream from the focus (dashed black line). (c) Phase lineouts at the same abcissa using the same color code as in (b).

Figure 2

Schematic of the numerical setup. The incoming Gaussian beam of wavelength $\lambda$ and waist ${\varphi }_{\frac{1}{e^2}}$ is first made annular, by being first apertured and then reflected on a drilled mirror. Its external (respectively, internal) diameter is ${\varphi }_1$ (respectively, ${\varphi }_2$). It propagates over a distance $z_1$ where it goes through a 0-$\pi$ phase plate. After another propagation over $z_2$ it is focused by a lens. The focal plane is at abscissa $z$ $=z_f$, the origin of $z$ being set by the jet location (see inset). Negative (respectively, positive) values of $z_f$ correspond to a focus located upstream (respectively, downstream) from the jet. The XUV radiation generated in the jet is finally propagated over $z_{\mathrm{obs}}$ towards an observation screen. The XUV electric field in this plane is either represented in the spatial-spatial transverse coordinates system or the spatiospectral one.

Figure 3

(Bottom line) Spatiospectral interference patterns observed in the far field in argon for H21 and $z_f=$ 0 with (a) short- and long-trajectory dipole, (b) only long-trajectory dipole, and (c) only short-trajectory dipole. The maps are normalized to 1 for their peak intensities. (Top line) Cuts at the central frequency of the harmonic.

Figure 4

Evolution with focus position of the fringe pattern in the spatiospectral representation for H21 and argon gas. Negative (respectively, positive) values of $z_f$ correspond to focus positions upstream (respectively, downstream) of the gas medium.

Figure 5

Dependence with harmonic order of the fringe pattern in the spatiospectral representation for $z_f=0$.

Figure 6

Scaling of the fringe spacing with harmonic order. (Solid line) Fringe spacing given by $\frac{\lambda z_{\mathrm{obs}}}{d}$ where $\lambda$ is the wavelength of the harmonic and $d$ is the spacing between the slits. In our case, $z_{\mathrm{obs}}=80$ cm and $d\approx 80\mu$m from Fig. 1.

Figure 7

Fringe pattern of H19 computed in the spatiospectral domain resulting from interferences between harmonics produced in argon and krypton gases (bottom map). Focus position, $z_f=0$. (Top panel) Cut through the map at the central frequency of the harmonic for two argon sources (light gray, from Fig. 5) and for the argon plus krypton case (black).

Figure 8

Phase difference between argon and krypton as a function of harmonic order for the short trajectory. The first spectral derivative of this phase difference would give the difference in group delay between the two species. (Solid symbols) Data deduced from the analysis illustrated in Fig. 7 for H19. (Open symbols) Data deduced from the analysis of the numerical experiment when the ionization of the generating medium is turned off in the code (fringe pattern not shown here). Lines are as follows: $\Delta I_p{\tau }_s$ calculated for $I$ = $1.1\times {10}^{14}$ W/cm${}^2$ (black), $1.3\times {10}^{14}$ W/cm${}^2$ (light gray), and $1.45\times {10}^{14}$ W/cm${}^2$ (medium gray), respectively.

Figure 9

(Plain line) Temporal envelope of the generating laser beam used in the numerical simulation. (Dashed lines) Argon and krypton (see legend) medium depletion as a function of time. (Symbols) Cutoff value for HHG in argon (solid) and krypton (open) as a function of laser intensity.

Figure 10

Scan of the focus position in the N${}_2$ gas jet for H17, H21, and H25. $z_f<0$ (respectively, $z_f>0$) corresponds to a laser focus located upstream (respectively, downstream) of the gas jet. The images are averaged over 70 laser shots. The three upper panels show cuts integrated spectrally over the 0.25 photon unit centered on the peak frequency of each harmonic. The cuts are progressively shifted up to ease visualization.

Figure 11

Fringe pattern evolution with pump-probe delay for H17–H27 in the vicinity of CO${}_2$ half revival. The fringes are integrated spectrally over 0.25 photon unit centered on the peak frequency of each harmonic.

Figure 12

Phase (a) and intensity (b) variations with pump-probe delay extracted by a Fourier transform of the fringe pattern shown in Fig. 11. For each harmonic order both the phase (in units of $\pi$ radian) and the intensity are normalized to their value at delay 19.5 ps. A sliding smoothing filter over three points has been applied. The amplitude curve of H27 has been divided by 9 and shifted up by 8/9 to ease the visualization of all curves.

Figure 13

Measured phase difference between pump-probe delays 21.2 and 21.7 ps (black dots). Lines are as follows: $\Delta I_p{\tau }_s$ for HOMO-1 and HOMO calculated for $I$ = 0.8$\times$10${}^{14}$ W/cm${}^2$ (black), 1.1$\times$10${}^{14}$ W/cm${}^2$ (light gray), and $1.2\times {10}^{14}$ W/cm${}^2$ (medium gray), respectively.