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We demonstrate an alternative approach for attosecond polarization gating. A setup composed of four quartz wedges and a quarter-wave plate allows an easy adjustment of the temporal gate-width and of the total dispersion. A numerical simulation of the pulse propagation beyond the carrier-envelope approximation enables a calibration of the setup and provides a flexible choice of the desired temporal polarization. An electron imaging spectrometer is used to measure the electron momentum distribution resulting from the ionization of xenon with our optical gated laser pulses. This allows us to measure the orientation of the polarization plane in the most intense temporal slice of the laser pulse. We compare the experimental results to theory and we numerically show the robustness of the method against non-ideal laser parameters.
©2011 Optical Society of America
1. Introduction
Intense short pulse lasers (>1013W/cm2) with durations of a few femtoseconds are commonly used as a source for high harmonic generation (HHG) [1]. Extreme ultraviolet (XUV) and soft X-ray emission in gases can be understood by the 3 steps electron rescattering process [2]. First, an atom or a molecule is ionized through optical tunneling. The newly born electron is then driven back and forth by the strong linearly polarized laser field. There is a probability of recollision at every half-cycle, when the electron wavepacket passes in the vicinity of its parent ion. Inelastic electron-ion scattering is responsible for the high-order harmonic emission [2] and it is also the dominant channel of non-sequential double ionization [3]. Elastic scattering allows the extension of the electron kinetic energy spectrum to about 10 Up (Up is the ponderomotive energy) [4]. Since the first experimental demonstration of a sub-femtosecond laser pulses [5], several techniques, including carrier-envelope phase (CEP) stabilization [6] and chirp compensation [7,8], were used to control and shorten their durations down to the actual world record of under 100 attoseconds [9].
Laser pulses of several optical cycles allow HHG at every half period. This leads to the formation of XUV/X-Ray pulse trains which have limited use in pump–probe experiments where exact time reference is needed. To generate isolated attosecond pulses, one strategy (“amplitude gating”) is to use nearly single cycle driving laser pulses [9]. This is possible with < 5fs pulses, but they are difficult to generate because they require sophisticated pulse compression techniques. Isolated attosecond pulses can also be achieved by shaping the polarization of the driving laser (“polarization gating”) [10]. This approach takes advantage of the strong ellipticity dependence of high harmonic generation [11]. The physical picture is quite intuitive: only electrons freed in sufficiently linear laser field have a significant probability of recollision with their parent ion. Since the original propositions [12,13], laser pulses with a time-dependent ellipticity were used to control the spectral and temporal characteristics of high order harmonics. Several optical schemes were proposed to produce such polarization gates, including interferometry [14–16], liquid crystal based polarization shaping [17] and self-phase modulation [18]. The simpler and most popular method so far, proposed by Platonenko and Strelkov [19], relies on linear optical propagation in a thick birefringent plate, supplemented by a zero-order quarter wave plate. This technique is intrinsically less fragile than interferometric methods because it does not critically depend on alignment, and it is more robust than nonlinear methods since it is intensity independent. It is also simpler to implement and more stable than polarization pulse shapers, especially with broadband few cycle pulses. The CEP [20–22] and the pulse duration [23] dependences of HHG were thoroughly investigated using the linear optics scheme. Tcherbakoff et al. [24] pointed out that if the thickness of the first plate is such that it acts as a multi-order quarter wave plate at the carrier frequency, then varying the zero-order quarter waveplate orientation continuously shifts the gate width without changing the intensity. Several experiments and theoretical investigation took advantage of it [20–27]. Polarization gating was also theoretically investigated for the production of high harmonic from relativistic plasma surfaces [28]. An alternative to using a thick birefringent piece of quartz is to use wedges [20,22]. Comparison of polarization gating schemes can be found in Refs. 16 and 17.
Some effort has been recently devoted to bring attosecond pulse generation to a wider community by relaxing the requirements on the driving laser. The key technique for doing this is the Fourier-synthesis of optical waveforms. This method modifies both the time dependence of the ionization rate and the trajectories of the free electrons. The sum of two frequency-rational [29] and two frequency-irrational [30] as well as three frequency-irrational [31] laser fields were found to significantly broaden the high-harmonic spectrum. Generalized double optical gating [32], a combination of polarization gating and two-color laser field, was shown to broaden the spectrum the most. It was even shown to remove the need for carrier-envelope phase stabilization [33].
Many polarization gating techniques applied to high-harmonic and attosecond pulse generation [12–28] have been already published, but little work has been done on ionization and on electron spectroscopy using polarization gates. In addition, the optical theory behind the generation of polarization gates is found in several articles [14,19,21,24,25,27,34] but it is not detailed, since they emphasise the computational and experimental work on high harmonic generation. In this work, we present a simple setup for continuously adjustable attosecond polarization gating that is robust and that allows great flexibility for ionization experiments. A theoretical model used to compute maps of the relevant physical quantities as a function of the quartz thickness is described. Experimental results confirm the validity of the simulation. Finally, the effect of unavoidable phase distortions and experimental uncertainties of the few-cycle laser pulse on the polarization gate is investigated.
2. Experimental setup
The experimental setup is illustrated on Fig. 1 . Our laser chain consists of a commercial carrier-envelope phase stabilized Ti:Sapphire chirp pulse amplification system (Femtopower Compact Pro CE-phase, Femtolasers Produktions GmbH) delivering 800 nm centered, 0.9 mJ, <30 fs pulses at a kilohertz repetition rate. Spectral broadening in a neon filled hollow core fiber produces a laser spectrum spanning over an optical octave and sustaining few cycle pulses centered around 750 nm. Commercial chirped mirrors covering the range of 650-950 nm are used to overcompensate the positive dispersion introduced by the fiber and by the optics. The polarization gating setup itself is used to finely minimize the dispersion in order to obtain the shortest possible pulses in the focal volume. Alternatively, BK7 wedges can be inserted to adjust the chirp. The laser pulse energy is controlled by an achromatic half-wave plate followed by two passes through a broadband thin film polarizer. Two passes result in a cleaner linear polarization than single pass does. A setup of two pairs of quartz wedges and one achromatic λ/4 plate is used to realize optical gating. Several detection systems are used to characterize the laser pulse. A polarization sensitive optical spectrometer is used to measure the wavelength distribution of the laser pulse. The pulse duration has been estimated with an autocorrelator from Femtolasers (Femtometer). In addition, an electron imaging system is used to measure the ion yield as well as the energy and angular resolved electron spectrum [35]. In particular, the electron spectrometer can be used to observe the signature of Rydberg states in the ATI photoelectron spectrum [36–39]. Typically, the maximum of the ionization yield of doubly or triply charged xenon atoms is used as an indication that the shortest pulse duration is reached in the focal volume.
Fig. 1 Schematic representation of the whole experimental setup. The polarization gating setup including four quartz wedges and an achromatic quarter-wave plate is indicated. The third wedge (green) is translated to vary the gate width, while the second and the fourth (red) fix the dispersion. (a) The polarizer and the spectrometer are used to perform the calibration of the wedge insertion. (b) The achromatic half-wave plate is used to measure the orientation θ of the linear polarization gate with respect to the lab coordinates.
The principle of the polarization gating technique has been known for many years [19]. We will review it here only briefly. A linearly polarized laser pulse passes a birefringent quartz crystal. The optical axis is oriented at 45° in respect to the laser polarization. The incoming laser pulse is decomposed into two identical fractions, each containing half the energy. The two pulses are delayed by:
where L is the effective material thickness, vg,o and vg,e are the group velocity of the ordinary and of the extraordinary wave, respectively. After quartz, the polarization temporal profile is thus linear at the beginning, going toward elliptical between the two pulses, and then becoming linear again toward the end of the pulse. The temporal polarization can be reversed with a properly oriented achromatic quarter-wave plate. Hence, the polarization of the laser pulses goes from circular to linear to circular after the quarter-wave plate.Our polarization gating setup is shown on Fig. 2 . It consists of four, 30 mm by 20 mm, 2° apex angle quartz wedges. The first wedge pair has its optical axis at −45° from the horizontal plane and the other pair has its optical axis at + 45°. The effective material thickness L (Eq. (1)) is thus determined by the difference in thickness of the first pair and of the second pair of wedges. The advantage of this four wedge setup over the traditional single piece of a thin quartz slab [19] is that it is continuously adjustable in thickness by translating laterally one wedge, thus providing maximum flexibility with the gate width. Another group already reported the use of a single pair of quartz wedges, but little details were given on how it works [20,22]. When using a single pair of quartz wedges, if the optical axes are parallel, it becomes impossible to get a thickness zero (the birefringence adds up). If the axes are perpendicular, then it is possible to completely cancel out the birefringence but there would be two diverging beams because of refraction. For 2° apex angle quartz wedges, the full divergence angle would be about 0.6 mrad and it scales linearly with the apex angle (for small angles). Depending on the focusing condition, the spatial superposition of the beams in the interaction chamber could be compromised. With our four wedges arranged in two pairs, the divergence caused by refraction cancels out and it is possible to get a delay zero. Dispersion compensation is built-in in the following way. To vary the gate, a single wedge is translated by a travel distance Δx (the third wedge on Figs. 1 and 2). Then the second and the fourth wedges are translated together (they are mounted on the same stage, Fig. 1) by Δx /2 to cancel out the dispersion without affecting the gate since their axes are perpendicular. With CEP-stabilized pulses, moving only the second and fourth wedges would also allow to tune the CEP without interfering with the active loop. The wedges can have macroscopic thicknesses (>0.5 mm), and be more robust than a single <100 μm thick crystal. The main drawback of our four wedge setup is that it introduces additional dispersion (about 3-5 mm of quartz) in the short laser pulse, requiring multi-passes on the chirp mirrors and possibly increasing the minimum pulse duration due to unbalanced third and high-order dispersion.
Fig. 2 Schematic illustration of our polarization gating setup. The optical axes of the different quartz wedges (labelled ne) at ± 45° are represented by blue lines. The fast axis of the achromatic quarter-wave plate (red) makes an angle ψ with the horizontal plane. Schematic representation of the laser pulse electric field polarization is given before (top left) and after (bottom right) the setup for L = 168 μm of quartz and for ψ = 0°. The “linear” part of the pulse, which shows an ellipticity below 0.15, is highlighted in green.
3. Theoretical model: quartz-based polarization gating
We performed a simple calculation of the laser pulse properties when passing through our setup. The laser pulse is approximated to be one-dimensional (i.e. the sum of plane waves) and propagating in the + Z-direction. The total pulse electric field E(t) in Cartesian coordinates is given by:
where EX(t) and EY(t) are the components of the field along the unit vector âX and âY and t is time. The horizontal (X) component of the laser electric field is represented by the N component Fourier sum:where ẼX(t) is the complex electric field, ω is the angular frequency, AX(ω) is the spectral amplitude and ϕX(ω) is the spectral phase. For the current simulation, the spectrum is taken to be a perfect Gaussian:where A0 is the amplitude of the Gaussian, ω0 is the central frequency and ΔωFWHM is the spectral width. In the numerical application presented here, we use ω0 = 2.5115 x1015 rad/s, corresponding to 750 nm and ΔωFWHM = 0.2ω0. The initial spectral phase ϕX(ω) is given by a truncated Taylor expansion around ω0:where ϕ0 is the carrier-envelope phase (CEP) if every other term is zero, k(ω) = ω n(ω)/c is the wavenumber, c is the speed of light in vacuum, n(ω) is the refractive index and primes (′) denote differentiation with respect to ω. In Eq. (5), notice the negative signs separating each term. This equation is used to pre-compensate the (normal) dispersion introduced by quartz (thickness L) in order to get near transform limited pulses after the quarter-wave plate. Similar equations (Eqs. (2)-(5)) are also used to describe the Y-component of the polarization. As for the propagation in quartz, the dispersion is taken into account through the Laurent series equation for the refractive index:where λ stands for the wavelength of light in vacuum. Coefficients a1 to a6 are given by Ref. 40. Propagating the pulse is then simply a matter of computing the phase of each Fourier component independently for both polarizations (o- and e-), making the appropriate projections on the axes of the crystal. For example, after a thickness L of material, the resultant phase of the extraordinary (e-) wave is:where vg is the average group velocity defined by vg = 0.5(vg,o + vg,e). The last term in Eq. (7) is added so that the numerical grid travels with the laser pulse (i.e. the retarded time frame). After the quartz, the ordinary and extraordinary waves are recombined and proper projections along the wave plate axes are taken. For the initial simulation, the quarter-wave plate is taken to be perfectly achromatic, i.e. it introduces a π/2 phase shift between its fast and slow axes over the whole laser bandwidth. We have also used the wavelength dependent retardation specifications from the vendor (P/N OA229, Femtolasers GmbH) to compute the laser pulse parameters after the setup. It will be shown later that it has only a negligible effect on the result. We have taken the wave plate to be infinitely thin. We do not take its dispersion into account because it introduces a constant chirp that can be properly compensated.We describe the final state of polarization of the laser pulse by its ellipticity ε and by the angle θ formed by the major axis of the ellipse (x-axis) with respect to the lab frame of reference (X-axis) following Ref. 41. Please consult Fig. 3 for details. We can express ẼX and ẼY in terms of the quantities defined on Fig. 3. Defining as the phase lag, we have [41]:
Taking advantage of the complex representation of the electric field components (Eqs. (2) and (3)), the calculation of the polarization characteristics is pretty straightforward. For a given time (or frequency), using Eq. (9) and the substitutions |ẼX|→ A and |ẼY| → B, we have computed α. Then, we used Eq. (10) and the definition of Δ to compute β. Finally, ε and θ were computed using Eqs. (8) and (11), respectively. All polarization related quantities can be computed in the time domain or in the frequency domain, sometimes yielding counter-intuitive results for the polarization gated pulse. Three useful quantities, the delay (Eq. (1)), the gate width and the threshold ellipticity [34], are defined on Fig. 4 .Fig. 3 Polarization ellipse and its orientation with respect to the lab frame of reference. Definitions: X and Y are the lab coordinates and lie in the horizontal and in the vertical planes, respectively; a and b are the length of the major and minor axes, respectively; x and y are the principal axes of the ellipse, rotated by an angle θ from the lab coordinates; A and B are the projections of the ellipse on the lab coordinates; and α and β are angles that are intermediate steps in the calculation (see Eqs. (8)-(11)).
Fig. 4 (a) Definition of the delay (τd) between the o- and e-waves in quartz. (b) Definition of the gate width (τgate) and the threshold ellipticity (εth). (c) Delay and gate width computed for the simulation parameters of the next section (see Fig. 5).
4. Numerical results: polarization maps
We performed a numerical simulation for a typical short laser pulse of 5.6 fs pulse duration with ψ = 0°. A grid of 2001 points spanning 100 fs in time and 2001 points in angular frequency (covering the wavelength range spanning from 400 nm to 1600 nm) was used for this purpose. The laser pulse centered at 750 nm with a Gaussian spectral amplitude (Eq. (4)) was used as a typical short laser pulse for attosecond polarization gating. The ellipticity ε(t), the intensity profile I(t), the orientation of the ellipse in the lab coordinate θ(t) and θ(ω) and the projections of the electric field amplitude A(t), A(ω), B(t), and B(ω) are shown as a function of the differential quartz thickness L at Fig. 5 . The quartz thickness spans from 0 to 609 μm in 465 steps. For each thickness, the initial phase of the laser pulse was adjusted (Eq. (5)) (i.e. the chirp was compensated) so that the spectral phase after the whole setup was as flat as possible.
Fig. 5 Polarization maps at the output of the polarization gating setup for ψ = 0°. The independent variable is the quartz thickness (L) in millimeters.(a) I(t), (b) ε(t), (c) θ(t) in degrees, (d) A(t) and (e) B(t) in arbitrary units, (f) θ(ω) in degrees, (g) A(ω) and (h) B(ω) in arbitrary units. Direct experimentally accessible maps are θ(ω), A(ω), and B(ω), while a cross-section of θ(t) near t = 0 is possible via electron imaging.
Figure 5(a) shows the laser pulse intensity profile as a function of time and of quartz thickness. The pulse duration becomes longer when quartz is added in the beam up to the point where two pulses are totally separate in the time domain. We can define the maximum useful quartz insertion at the point where the intensity envelope passes from one to two local maxima in the time domain. On Fig. 5(a), this corresponds to a quartz insertion of about 148 μm, which corresponds to a delay (Eq. (1)) between the two pulses of approximately τd = 4.7 fs. This is even slightly smaller than the initial pulse duration of 5.6 fs. This limit on the delay depends in general on the laser spectrum and is approximately equal to the initial pulse duration. In practice, since ionization is more efficient in many atomic and molecular systems with linear polarization, we can use delays slightly larger and still get most of the ionization in the gate.
Figure 5(b) and Fig. 4(c) show that the temporal gate width where the ellipticity of the laser pulse is lower than a threshold (ε < εth = 0.15, the blue regions in Fig. 5(b)) strongly decreases with increasing quartz insertion. For L = 0, the ellipticity is zero and the pulse is horizontally polarized (θ = 0). This is in agreement with the common sense requirement that zero insertion is equivalent to not using any polarization gating setup. Therefore it leaves the polarization unchanged. Recall that the input polarization is linear along X and parallel to the waveplate fast axis (Fig. 2). Using 0.15 as the ellipticity threshold [20], the gate width (Fig. 4(c)) reaches one optical period (T = 2.5 fs at 750 nm) at L = 80 μm and it reaches half a period at about 160 μm of quartz.
Figure 5(c) shows that with the addition of quartz thickness in the optical path of the laser pulse, the temporal gate rotates with respect to the lab coordinates (X,Y) at an almost fixed rate. This behavior was observed previously with an interferometric scheme [14]. Starting from the X-axis (θ = 0°) at z = 0, the central and most intense part of the pulse reaches the vertical Y-axis direction (θ = π/2 = 90°) at 42 μm and it becomes horizontal again (θ = 180° or 0°) at 84 μm. This is observable in the experiment and it will be verified later (Fig. 9 ). Figure 5(c) also shows that for a given thickness and in the useful range of the quartz thickness (0 < L < 150 μm), θ is fairly constant across the whole pulse duration (−10 < t < + 10 fs). This is characteristic of the ideal pulse where the amplitude spectrum is Gaussian and the spectral phase is flat (i.e. the dispersion is well compensated) after the quarter-wave plate. It does not hold for real laser pulses with non-trivial spectrum and phases and in fact θ(t) is quite sensitive as we will show later (Fig. 11 ).
Fig. 9 Orientation θ of the most intense part of the polarization gated pulse as a function of the quartz thickness. The angle θ is unwrapped for a better visualization. Dots: measurement for three different experimental days. Solid line: simulation with the ideal parameters of Fig. 5. Dashed line: simulation with the experimental spectrum (Fig. 10), no dispersion compensation and the real wavelength-dependent behavior of the quarter-wave plate.
Fig. 11 Robustness of the polarization gated pulses against experimental flaws and sources of error. The left column shows contour plots of the ellipticity ε(t). The central column shows intensity profiles in arbitrary units and the right column contains the orientation of the major axis θ(t) in the lab coordinates. (a) Ideal conditions (repetition of Fig. 5) (b) The input polarization is linear at 5° from X. (c) The fast axis of the quarter-wave plate is at ψ = 5°. (d) Experimental spectrum. (e) Dispersion is not compensated. (f) Real wavelength-dependence of the quarter-wave plate. (g) Spectral phase is distorted by higher-order terms: −80 fs3, −40 fs4 and + 200 fs5.
Figure 5(d) and Fig. 5(e) show the projection of the time-dependent polarization ellipse on the X- and Y-axes, respectively. For thicknesses up to 400 μm, it is possible to observe peaks and valleys on axis (t = 0) with period 84 μm as noted below. Note that the complementarity of A(t) and B(t) comes from energy conservation. The condition holds for any given thickness. The constant is the same for all thicknesses as well since the energy content of the input pulse is fixed in the simulation.
Figures 5(f) to 5(h) show spectral properties of the pulse polarization. The spectral ellipticity ε(ω) is also quite interesting but it is not shown for obvious reasons: it is identically zero for every frequency and for every thickness. This is a consequence of the ideal parameters of the simulation: the input pulse polarization is perfect, the quartz wedge optical axes are exactly at ± 45° and the fast axis of the quarter-wave plate is exactly at ψ = 0°, plus the fact that the quarter-wave plate is taken to be perfectly achromatic. Nevertheless, it is quite counterintuitive that a short laser pulse composed of a superposition of all polarized plane waves (a frequency comb) has this polarization shape. The apparent paradox is solved with elegance in Fig. 5(f): each plane wave (or frequency component) has a different orientation θ(ω) with respect to the lab reference frame except for L = 0, where they are all aligned along the X-axis. Figs. 5(g) and 5(h) show projections of the polarization ellipse spectrum along the X- the Y-axes, respectively. These have mainly a practical interest: they can be easily measured experimentally (more precisely, ׀A(ω)׀2 and ׀B(ω)׀2 can be measured, Fig. 7 ) with a polarizer and a spectrometer (see Fig. 1(a)) . It thus provides the opportunity to test the theoretical model and the experimental material (the quartz wedges themselves). Most importantly, measuring these maps provides a simple way to find the zero quartz insertion reference point on the experimental setup.
Fig. 7 Experimental ((a) and (b)) and simulated ((c) and (d)) projections of the laser spectrum on the lab coordinates. By a simple fit to theory, the zero insertion is found on the experimental delay stage. Notice the symmetry of the traces around L = 0.
We also simulated an alternative approach for continuously adjustable polarization gate width with a fixed quartz thickness [20–27]. This approach only works if the thickness L which determines the phase retardation between the ordinary and extraordinary components in quartz at the carrier frequency is given by:
where ke and ko are given at the carrier frequency. In such cases, the polarization near t = 0 is circular before the broadband quarter wave plate. This condition does not imply that the delay is an integer multiple of the quarter laser period (i.e. τd ≠ (2n + 1)T/4, where T is the laser period) because the delay is computed with the group velocity (Eq. (1)), while phase velocity is what determines the state of polarization. When Eq. (12) is fulfilled, rotating the angle ψ of the quarter wave plate continuously tunes the gate width without changing the intensity. Figure 6(a) shows the temporal ellipticity as a function of ψ for (ke-ko)L = 9π/2 rad, corresponding to L = 189 μm at 750 nm. The gate is narrowest at ψ = 0°, and it is “infinite” (the polarization is linear) at ψ = ± 45°. The orientation θ of the gate also shifts in this case, as seen on Fig. 6(b). Figure 6(c) illustrates what happens if Eq. (12) is not fulfilled and ψ is different from 0° or 90°. In this case, the minimum of ellipticity is not at t = 0 and the situation is quite complicated. This should be avoided in practice. If the laser spectrum shifts from day to day, which is not uncommon with the hollow core fiber technique, the quartz thickness L needs to be adjusted to satisfy Eq. (12) at the new carrier frequency. This is trivially realized with wedges.Fig. 6 Variation of the gate width by rotating the quarter wave plate. Other simulation parameters are identical to those of Fig. 5. See text for details.
5. Experimental validation of the model
The insertion of a cubic polarizer in the laser pulse (Fig. 1(a)) for measuring the X- and Y-component of the spectrum was carried out in the lab. The comparison with theory is shown on Fig. 7. To calibrate the actual quartz thickness, the wedge angle was taken to be 2°, as specified by the vendor. We observe an excellent overlap with theory, confirming the validity of the simulation. Fitting also allows us to locate the zero insertion, which in this particular case is at 13.8 mm on the wedge translation stage.
Temporal features of the order of the femtosecond are impossible to temporally resolve with straightforward methods. The maps of Fig. 5 (a)-(e) cannot be directly measured like we did with the spectral maps A(ω) and B(ω) (Fig. 7). An indirect method is needed. While FROG [42] measurements are in general quite successful at determining the spectral phase of a femtosecond laser pulse, they are typically implemented with linearly polarized light. The non-trivial nature of the polarization prevents us to use such methods here. Nevertheless, using electron momentum imaging, we can make a measurement of the polarization orientation θ(t) of the most intense part of the pulse, i.e. a cross section of Fig. 5(c) around the center of the pulse (t = 0). It is known that at relatively low laser intensities (I < 1013 W/cm2), resonance-enhanced multiphoton ionization of atoms (here xenon) is the dominant process at a wavelength of around 800 nm [35,39]. For example, the excitation of the resonant 5g state of xenon has been intensively studied in the past [38]. Angular momentum (l) electric dipole selection rules during resonant ionization processes imposes lcontinuum = lstate ± 1 if a photon is absorbed from the resonant Rydberg state [36,38]. An atomic system can also absorb additional photons, a process known as above threshold ionization [36].
In the lab, we used an imaging spectrometer to monitor the angular and energy distribution of photoelectrons [36]. The spectrometer is housed in an ultra-high vacuum chamber (background pressure < 5 × 10−9mbar). A variable leak is used to regulate the partial xenon pressure (1 × 10−5 mbar < pXe < 3 × 10−8 mbar). The laser beam is focused in a region of homogeneous electric field. The focal region can be considered as a point source of photoelectrons. While the electrons expand from the interaction region, a dc electric field of 80 V/cm projects them onto a 50-mm diameter chevron type of micro-channel plate (MCP) with a phosphor screen attached. A low noise computer-controlled CCD camera (LaVision Imager 3) is used to record the impact positions of the photoelectrons. The energy of the photoelectrons increases with the distance to the center. In its actual voltage configuration, our spectrometer can record electrons with a maximal energy of 10 eV.
A key feature is that the atomic quantization axis (“z-axis”’) is defined along the strong laser field (linear) polarization axis [36]. The signature of excited states (spherical harmonics) with large angular momentum can only be clearly resolved if the polarization of the laser pulse is vertical (along the Y-axis) in the apparatus. Figure 8 explains the principle of the method. It shows the projected electron spectrum on the MCP for three different orientations of the polarization gate. On the left panel, the polarization gate is vertically oriented. The projection is good and the different ATI features are sharp. Bright and dark stripes in the center of the picture are signatures of Rydberg states resonant multiphoton ionization processes. We clearly observe that electrons are preferentially emitted along the polarization axis (Y). The central panel of Fig. 8 shows an image with a polarization gate orientation of about 45° from X- and Y-axes. The image is quite blurred and the electron momentum is not as extended in the Y-direction as in the previous case. On the right panel, the gate is along the X-axis. The projection shows a radial symmetry. The photoelectron spectroscopy of xenon exposed to few-cycle polarization-gated laser pulses will be detailed in a forthcoming paper. To continuously measure the orientation θ of the most intense (and linearly polarized) part of the pulse (i.e. θ(t~0) for L< 150 μm of quartz), we introduced an achromatic half-wave plate in the beam just before the chamber window, as illustrated in Fig. 1(b). For each quartz insertion, we noted the polarization rotation needed from the half-wave plate to recover the signature of the different resonant states (Fig. 8-(left panel)). We report the measured angles θ(t~0) on Fig. 9 for three different experiments (dots), together with two simulation results (curves). The solid line represents the ideal case (Fig. 5) where the spectrum is Gaussian, the dispersion is pre-compensated for each thickness (resulting in a flat spectral phase after the setup) and the quarter-wave plate is achromatic. The dashed curve is the more realistic case where the experimentally measured spectrum is used (Fig. 10 ), the dispersion is not pre-compensated (resulting in a flat spectral phase only for L = 0) and the quarter-wave plate retardation is wavelength-dependent as specified by the vendor. In the experiment, we have optimized the partial xenon pressure in the chamber to keep the electron imaging signal visible. Error bars account for the imprecision in finding the sharpest projection and for the (small) uncertainty about the zero insertion (Fig. 7). We observe on Fig. 9 that the experimental data follows the tendency of the theoretical curve pretty well. The small discrepancies in the slope originate probably from the unknown spectral phase of the experimental pulse. It is noteworthy to point out that the curves were not expected to fit for delays much larger than about 150 to 200 μm using 5 to 7 fs pulses. For large thicknesses, pulses split into two (Fig. 5(a)) and their polarization becomes closer to circular as L increases (Fig. 5(b)). For such large thicknesses (L> 200 μm), the laser peak intensity is expected to be in the two circularly polarized pulses and not in the center, where the polarization is linear. As a consequence, ionization will occur mainly in the circular polarized part of the laser pulse under this condition. Thus, ionization in the central less intense part of the pulse will only play a role if a specific ionization/excitation path is forbidden with circular polarization and it is only accessible with linearly polarized light. The fact that experiment and theory fit pretty well (Fig. 9) up to almost L = 400 μm could indicate that the pulse duration was longer than 7 fs in experiments. It could also mean that, even though the intensity is lower, ionization prevails mainly in the central part because resonance enhancement of ionization only occurs with linearly polarized light.
Fig. 8 Two-dimensional photoelectron spectra induced by ultrashort polarization gated laser pulses for three different orientations of the gate. The electric field in the gate (ε<0.15) is represented in green. The laser propagates in the + Z-direction and the dc-field is directed along + X. (Left) The gate is along the Y-axis. (Center) The gate is oriented at 45°. (Right) The gate is polarized along the X-axis.
Fig. 10 Ideal (solid red line) and experimental (solid black line) spectra used for the numerical simulations.
6. Numerical simulation with realistic laser pulses and optical components
To clarify the robustness of the temporal characteristics of the polarization gated pulse against different source of experimental errors, we performed several additional numerical simulations. We checked the effect of the laser spectrum (see Fig. 10) on the gated pulse. Simulation results are shown on Fig. 11. All parameters except the one under investigation are those of the ideal case (Fig. 11(a)). For each set of conditions (rows (a)-(g)), we show the intensity plot (central column) to highlight the location if the most intense part of the pulse in time. The useful range for the quartz thickness stops around a certain thickness where the pulse envelope splits into two. The left column shows the temporal ellipticity. Finally, the orientation of the major axis θ(t) is shown in the right column. Recall that θ(t) becomes numerically unstable when the ellipticity approaches unity. On Fig. 11(a), we report the ideal results that were previously exposed on Fig. 5 (for comparison). Recall that in this case, the spectrum was Gaussian (Fig. 10, red curve), the dispersion was compensated exactly up to the 5th order (Eq. (5) for every quartz thickness and the quarter-wave plate was achromatic. Figure 11(b) shows what would happen if the input polarization was linear but oriented at 5° with respect to the X-axis. In this case, the only detrimental effect comes from the first projections on the wedge axes which are no longer at 45° but rather at 50° and 40°. The two output waves (e- and o-) have thus unequal energy. The gate parameters are not really affected and this kind of error does not have much consequence. Figure 11(c) shows the effect of a 5° error in the angular position of the fast axis of the quarter-wave plate (ψ = 5°). In this case, what is mostly affected is θ(t). It does not influence the results significantly. Figure 11(d) shows the polarization characteristics for a pulse with a realistic (not smooth) experimental spectrum (Fig. 10). The pulse is slightly shorter (~4 fs) since the experimental spectrum is broader than the ideal one. Accordingly, the temporal gate is shorter for a given quartz thickness. The maximum useful quartz thickness before the pulse splits into two is found at 96 μm only. The θ(t) map is quite awkward but the only useful region is located approximately between 0 < L < 100 μm and between – 5 fs < t < + 5 fs. In this region, the behavior is truly similar to that of the ideal map (Fig. 11(a)). Figure 11(e) illustrates what happens when the dispersion is not pre-compensated for each thickness. This one is quite intuitive. Due to dispersion, the pulse gets longer with increasing quartz thickness and the gate gets longer (for L > 240 μm) accordingly. The orientation of the gate θ(t~0) is quite similar to that of the ideal case. This figure emphasizes the importance of compensating the dispersion of quartz to truly reach short gate widths. At Fig. 11(f), we show the corresponding maps for the real chromatic character of the quarter-wave plate (P/N OA229, Femtolasers GmbH). The retardation goes from about 0.24 wavelengths at 600 nm to about 0.26 at 720 nm, and back to about 0.24 wavelengths at around 1000 nm. This has truly no observable consequences in the time domain and it is therefore correct. Finally, Fig. 11(g) shows analog maps for a pulse with a more intricate spectral phase. In this case, we added −80 fs3 of third-order dispersion, −40 fs4 of fourth-order dispersion and + 200 fs5 of fifth-order dispersion, producing a 6.9 fs pulse from a 5.6 fs Fourier-transform-limited pulse. Here again, although the ellipticity map is distorted, the region of interest is well behaved. Our implementation of polarization gating is thus found to be quite robust against non-ideal laser parameters and optics and it can thus be used trustfully in experiments.
7. Conclusion
A continuously adjustable polarization gating setup with 4 quartz wedges was introduced. It shares the flexibility of the interferometric methods [14–16] with the robustness of the transmission optics birefringent plate technique [19]. It does not suffer from unbalanced refraction like two wedges would. Dispersion compensation is built-in, and it allows easy CEP scans if the CEP is stabilized. It also allows scanning the gate width with constant intensity [24] for specific choices of the quartz thickness. A comprehensive theoretical model explained some basic properties of the resulting few-cycle near infrared laser pulse. It was shown that for an arbitrary quartz insertion in ideal conditions, each frequency component is linearly polarized but at a different angle θ with respect to the lab coordinates. It is the superposition of these plane waves that generates the peculiar shape of the polarization. A method for precisely finding the zero quartz insertion was demonstrated using a polarizer and a spectrometer. This is of practical importance for any further experiment using this setup. It was also demonstrated both theoretically and experimentally that the linearly polarized polarization gate around the center of the pulse rotates with respect to the lab coordinates when more quartz is inserted in the beam path. The robustness of the method against realistic parameters for the spectral phase, the laser spectrum and the quarter-wave plate chromatic retardation on the time-dependent polarization was also systematically investigated. The photoelectron spectroscopy of xenon exposed to few-cycle polarization-gated laser pulses will be detailed in a forthcoming paper.
Acknowledgements
All authors thank financial support from the Canadian Foundation for Innovation (grant # 31338) and from the National Sciences and Engineering Research Council of Canada (NSERC) (grant # 87323). C.M. acknowledges financial support of the Postgraduate Scholarships program from NSERC (# ES D3-361087-2009). G.G. acknowledges the support from FQRNT (2010-PR-132281).
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