1. Introduction

High-harmonic generation (HHG) is a widely employed process for the generation of coherent radiation in the extreme ultraviolet (XUV), with pulse durations ranging from a few femtoseconds down to about 100 as for isolated pulses having been obtained [1,2]. The process can be initiated through the interaction of a strong radiation field with either a neutral gas volume or an excited plasma [3]. In the latter case the plasma is usually created by heating the surface of a solid with laser radiation [4]. As compared to a neutral gas, the converted radiation is emitted to a significant extent from ions, compensating for the generally lower ionization potential of non-noble gases. In addition, a large number of previously non-accessible materials become available. A few of them show a resonant enhancement of a single harmonic order [5, 6]. One of the major drawbacks of HHG is the low efficiency of the conversion process [7, 8]. Up until now, several attempts have been made to increase the efficiency by improving the phase-matching conditions in order sustainably extend the interaction distance.

In any non-linear optical process, in which an optical frequency is converted into another one, the variation of the phase of the newly generated electric field should be minimized throughout the interaction distance. If the phase variation exceeds half an optical cycle of the wave, the instantly generated field and the already existent field which was generated earlier in the interaction volume will interfere destructively at some point. This results in a reduction of the intensity of the newly created field. For long interaction distances, it is possible to transfer more energy to the converted field. However, in order to achieve this, the phase-matching must be controlled precisely. Early approaches to HHG focused on minimizing the phase variation by optimizing the position of the interaction volume within the focal region of the fundamental field [9,10]. Longer interaction distances can be realized in gas filled hollow waveguides. Here, full phase-matching is achieved by balancing the waveguide dispersion and the atomic dispersion, which have opposite signs [11,12]. Nevertheless, this approach has limitations because the ionization fraction inside the waveguide must be kept at a low level. Therefore the intensity of the fundamental field must not exceed a critical value which effectively reduces the maximum attainable energy.

Another approach in constructively interfering the generated field is quasi-phase-matching (QPM). In this case the phase variation is deliberately allowed to advance into a range in which destructive interference would occur. However, if it is possible to stop the conversion at times when the generated field is emitted with the incorrect phase and enable it again when the phase relation has recovered for positive energy transfer, an overall increase of the intensity of the converted optical field is observed. This technique is also known from frequency mixing in crystals in which a periodic array of high and low susceptibility domains is realized [13]. However, QPM is more commonly enabled by periodically flipping the spatial sign of the susceptibility of the crystal [14, 15]. This results in a phase jump that prevents any destructively interfering emission if the dimensions are chosen correctly. In HHG a similar effect can be achieved by the application of counterpropagating light [16,17]. Here, the presence of the second wave leads to a phase modulation of the emitted harmonic field. As a consequence the length of the destructively interfering zone can be considerably shortened [18].

Besides the application of counterpropagating light, QPM has been realized through a number of approaches in HHG. They are all based on the spatial modulation of the density of the harmonic emitters [19]. In a gas, QPM is most simply realized by the proper placement of a number of interaction volumes into the focused beam of high intensity radiation [20]. Likewise, the interaction volumes can be formed by multiple gas jets extending from a single source [21]. These experimental setups make use of the Gouy phase shift, which is experienced by laser beams when moving through a focus. Thereby the phase of the emitted harmonic field is restored in between the interaction zones, in which no radiation is emitted. The harmonic field generated in the next interaction zone is therefore constructively coupled to the already existing field. Another possibility is the application of an inactive medium in order to reverse the phase [22]. In this case the phase shift acquired in between the active domains is mainly induced by dispersion. QPM has also been realized in hollow waveguides either by using a modulated diameter [23, 24] or by mode-beating due to the presence of multiple excited modes [25,26]. Both effects lead to an intensity modulation within the waveguide in which radiation is significantly converted only in the high intensity regions. In the low intensity regions the phase is allowed to advance without significant harmonic generation.

A few studies exist showing indications of QPM using laser ablated plasma as the conversion medium [27–30]. They are performed with a 10 Hz laser system and an aperture with multiple slits in the plasma producing beam through which separated plasma jets are created. However, the dimensions of the plasma structure have not been engineered sufficiently in order to achieve the highest possible enhancement of a controlled range of harmonic orders.

In this paper the quasi-phase-matched generation of high-order harmonics in laser ablated plasma jets is demonstrated at a repetition rate of 1 kHz. As opposed to previous works the plasma structure is optimized for quasi-phase-matching a definite selection of harmonic orders. The chosen harmonic orders can be changed by simple adaptations. This is achieved by assembling specific thicknesses of target material and appropriate spacers. Thereby, the width and pitch of the individual plasma jets are easily controlled. The Gouy phase shift is identified as the major force advancing the phase of the emitted harmonic field in the gaps between the plasma jets.

2. Phase contributions to quasi-phase-matching

When the driving radiation is focused into the conversion medium, three contributions govern the dephasing of the generated harmonic field in spatially extended HHG sources. First, each harmonic frequency is generated with a specific phase. HHG is understood to occur upon the recombination of laser field accelerated electrons with their parent atoms after having been released through tunnel ionization. The electrons acquiring the correct kinetic energy for the emission of a specific harmonic order occupy different trajectories as the intensity varies through the focus of the fundamental beam. Out of the focal spot the trajectory of a specific harmonic shifts closer to the trajectory of the attainable cut-off frequency at that location. Therefore, the phase ϕ_dip of this harmonic order changes when generated at different distances from the focal spot. Based on the strong-field approximation [31,32] an almost linear dependence of the phase on the intensity I(z) of the driving field for a specific dipole transition is expected [33]: ϕ_dip ≈ −α_j I(z). The index j = s, l represents the two contributions of either the short or the long quantum paths. Away from the cut-off region the phase variation is much higher for the quantum path with the longer excursion time. Values of the proportionality factor have been calculated to be of the order of α_s ≈ 1 × 10⁻¹⁴ rad/(W cm⁻²) for the short and α_l ≈ 27 × 10⁻¹⁴ rad/(W cm⁻²) for the long quantum path [34,35]. Close to the cut-off the α_j values merge, as do the quantum paths.

A second geometrical contribution originates from the phase advance of the fundamental field itself when moving through the focus. This is also referred to as Gouy phase shift. The additionally acquired phase increases up to a factor of π having traversed the focal region. Along the axis of propagation it can be expressed as ϕ_G = − arctan(z/z_R) with z the distance from the focal position and z_R the Rayleigh length. It is important to note that the Gouy phase advance influences the fundamental field. Therefore, the phase of the instantaneously generated harmonic field advances as well, but the corresponding phase advance in space is q times higher for the field of the qth harmonic order.

The third contribution results from the different phase velocities of the fundamental and harmonic fields that arise from dispersion inside the plasma. It can be further subdivided into two contributions ascribed to free electrons and bound electrons, respectively. Away from resonances of the respective atoms or ions, dispersion in plasmas is generally dominated by the free electron contribution which has an index of refraction smaller than one. However in the proximity of resonances, bound electrons may significantly influence the overall dispersion. For example, the index of refraction in a laser-produced Al plasma at a photon energy of around 85 eV is found to be greater than one [36].

A phase-mismatch occurs when the phase of the instantaneously generated field ϕ_s and the phase of the already propagating field ϕ_q, which was generated earlier in the interaction volume, vary with respect to each other. This is typically expressed by the difference of the wavevectors of the two fields:

The geometrical contributions to Δk are easily calculated through the spatial derivatives of the phases, ϕ_dip and ϕ_G, once the parameters of the fundamental beam are known. They are calculated in Sec. 4.1 for the 25th harmonic as an example. The phase mismatch due to free electrons can be expressed as Δk_el = −N_er_e(qλ − λ/q) with N_e the free electron density, r_e the classical electron radius, λ the fundamental wavelength and q the harmonic order. For bound electrons, the index of refraction of atoms or ions inside a plasma can be written according to the Gladstone-Dale relation n(λ) = 1 + N_aδ(λ) with N_a the atomic density and δ a constant depending on the wavelength. The resulting phase mismatch reads Δk_at = 2πN_aqλ⁻¹ (δ(λ) − δ(λ/q)). Due to the dominance of the free electron contribution and its sign being equal to the Gouy contribution, Δk_G, the presence of a plasma typically results in a stronger phase-mismatch and therefore a shorter coherence length when focused beams are used.

3. Experimental setup

In this experiment the radiation for the frequency conversion as well as for the plasma formation is provided by a passively mode-locked Ti:sapphire oscillator. The output radiation is chirped and amplified through two amplifier crystals with a total of 13 passes. After the second amplifier a pulse energy of up to 4.5 mJ at 1 kHz repetition rate and a central wavelength of 800 nm is available. The pulse duration is stretched to 12 ps prior to amplification. 70 % of the output power is used for the plasma formation while the remaining 30 % is guided into a prism compressor which reduces the pulse duration to 40 fs. A sketch of the experimental setup for harmonic generation and analysis is shown in Fig. 1. The compressed radiation is used for frequency conversion by focusing it into the plasma with a spherical lens of f = 500 mm focal length. An aperture in that beam, now referred to as the driving beam, is used in order to optimize the harmonic output [37]. In addition, the Gouy phase advance through the focal region is controlled and harmonic generation is restricted to the near axis contribution (see Sec. 4.4). The diameter of the aperture is not changed for all results presented in the following sections. Since the radial beam profile throughout the interaction distance in this experiment is well described by a Gaussian profile, the simple approximations made for calculating the Gouy phase ϕ_G as well as the dipole phase ϕ_dip in Sec. 2 are valid even for the apertured beam. Through the reduced pulse energy the intensity in the focus of the driving beam amounts to 1.2 × 10¹⁴ W cm⁻². The beam radius w(z) as a function of the distance from the focal spot is in good agreement with a Gaussian profile $w(z)=w_0\sqrt{1+{\left(z/z_R\right)}^2}$. The resulting Rayleigh length amounts to z_R = 0.54 cm with a beam waist of w₀ = 38 μm. The beam radius as a function of the longitudinal position through the focus is shown at the bottom of Fig. 2(a).

 

Fig. 1 Experimental setup for HHG from extended plasmas. The driving pulse is focused by lens (L₁) into the plasma. An aperture (A) is used for signal optimization. The heating pulse is focused by a cylindrical lens (L₂) onto the target (T). A second lens (L₃) is used to confine the beam and a knife edge (KE) allows for variation of the plasma width. The harmonic light passes a slit (S) before it is dispersed by a grating (G). The spectra are recorded with a charge-coupled device (CCD).

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Fig. 2 Determination of the optimal target dimensions. In (a) the beam radius (1/e²) and the geometrical contributions to the phase-mismatch Δk are illustrated as a function of the axial position z within the focal region of the driving beam. A Gaussian profile is fit to the beam radius. The determined parameters are used for the calculation of the phase mismatch Δk for the 25th harmonic. The contributions due to the Gouy phase advance (blue), dipole phase for the short trajectories (black solid, α_s = 1 × 10⁻¹⁴ cm²W⁻¹), their sum (pink), and dipole phase for the long trajectories (black dashed, α_l = 27 × 10⁻¹⁴ cm²W⁻¹) are displayed in inverse coherence lengths (Δk/π). In (b) the intensities of the 17th to 21st harmonics generated from a continuous Al target are shown for increasing widths of the plasma producing beam. The curves are vertically shifted for clearer visibility. The dependence of the coherence length on the harmonic order q is shown in (c). A fit proportional to q⁻¹ is added to the graph.

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For the creation of a laterally extended plasma, a cylindrical lens (L₂) of 300 mm focal length focuses the plasma producing beam along the surface of the target [see Fig. 1]. Since an extended plasma requires more intensity, the plasma producing beam is laterally confined by a second cylindrical lens (L₃, f=500 mm). Prior to focusing, the size of this beam is increased by a factor of two with a Galilei telescope. Thus, the vertical extension of the plasma producing beam on the surface of the target is reduced while the lateral extension is increased. Both effects are favorable for the creation of a long plasma. The length of the created plasma can be adjusted by moving a knife edge into the plasma producing beam. The beam is blocked in a way that the plasma extends along the direction of the driving beam when the knife edge is removed. The intensity of the plasma producing beam on the surface of the target amounts to ca. 1.9 × 10¹⁰ W cm⁻².

As shown in Fig. 1 the targets for the realization of QPM consist of up to four disks. In this experiment the targets are made from aluminum. Disks of different thicknesses (light grey) with adjustable spacers separating them (dark grey) are stacked onto a rotating axis. When illuminated by the plasma producing beam, separated plasma jets are formed in front of the target. The axis of rotation is parallel to the direction of the driving beam. Moving targets are required at repetition rates of 1 kHz in order to avoid unfavorable plasma formation from repeated illumination of the same spot on the target [38]. The distance between the target surface and the driving beam is of the order of the diameter of the driving beam (≈ 100 μm). As the plasma formation and expansion into the interaction volume requires some time, a delay of 40 ns is set between the plasma producing and driving pulses. For heavier elements, the delay should be increased [8]. For comparison, a non-structured plasma can also be created by using an extended cylindrical target. The target as well as the plasma producing beam are placed symmetrically around the focus of the driving beam in order to maximize the Gouy phase shift which is almost constant in proximity to the focal position. In addition, the reduction of the plasma density due to the decreasing intensity of the plasma producing beam at the sides of the target is minimized. The intensity of the plasma producing beam is estimated to decrease to ca. 80 % at the sides of the interaction volume.

The converted radiation is spatially dispersed by a variable line-space grating (Hitachi 001-0437) illuminated at a grazing angle of incidence of 3°. Through the progressive line-spacing the radiation is spectrally focused at a distance of 237 mm behind the grating. At this position the signal is amplified by two micro-channel plates (MCP) in chevron configuration. The spectra are visualized through a standard phosphor screen (P43) which is imaged by a charge-coupled device (CCD). In front of the grating some of the fundamental light that would otherwise miss the surface of the grating is blocked by a vertical slit (S). In addition, the slit serves for differential pumping. The measurements are carried out for the first order of diffraction. The spectrometer and target are placed in two vacuum chambers, with a pressure of 10⁻⁸ mbar at the spectrometer.

4. Results and discussion

4.1. Determination of an optimal target configuration

The geometrical contributions to the phase-mismatch, Δk_G and Δk_dip, are always present through the focus of the driving beam. They are used to restore the phase relation between the propagating harmonic field generated in the preceding plasma jet and the harmonic field generated in the subsequent jet. If the total relative phase shift in the gaps amounts to Δϕ = π the harmonic fields from both jets interfere constructively. Along the axis of propagation of the driving beam the two geometrical contributions are calculated for the 25th harmonic order in Fig. 2(a). The value of Δk is given in inverse coherence lengths. The Gouy phase contribution is highlighted in blue, the contribution to the dipole phase originating from the short trajectories is illustrated by the solid black line and their sum is displayed in pink. For comparison, the contribution originating from the dipole phase of the long trajectories is also presented as dashed black line. The measured beam radius as a function of the longitudinal position within the focus of the driving beam is shown at the bottom of the figure. Regarding the short trajectories, the phase slip is dominated by the Gouy-phase contribution to Δk. In the case of lower orders, the significance of the dipole phase would increase as the Gouy phase advance decreases. By observing only the harmonic signal emitted close to the axis of propagation with a low divergence, it is possible to concentrate harmonic generation on the short trajectories. The role of the long trajectories is discussed in more detail in Sec. 4.4. For realizing QPM with structured plasmas the magnitude of the geometrical contributions to Δk sets the optimal spacing of the individual plasma jets. For the short trajectories, the optimal spacing for the 25th harmonic order in the middle of the focus of the driving beam amounts to 0.67 mm. This value slowly increases to 0.71 mm at a distance of 1 mm away from the focus.

Inside the plasma jets the phase-mismatch Δk is increased by dispersion. However, without knowing the refractive indices of the plasma the dispersion cannot be calculated. Nevertheless, it is possible to evaluate the phase-mismatch and consequently the optimal width of the individual plasma jets by directly measuring the coherence lengths [39]. This is achieved by creating a continuous plasma with an unstructured, cylindrical target without spacers. The width of the plasma is varied by reducing the width of the plasma producing beam. The resulting intensities for the 17th to 21st harmonic orders are presented in Fig. 2(b). For data acquisition, the CCD is exposed to 100 shots each time. 20 images are recorded for every data point with the background signal subtracted. The coherence length is exceeded multiple times as is apparent from the repetitive build-up and depletion of the harmonic intensity. From the moment the intensity in the leading edge of the plasma producing beam is large enough to create a plasma, the harmonic yield begins to increase. After reaching one coherence length the coupling between the generated and propagating harmonic fields becomes destructive. The harmonic signal decreases until after another coherence length the coupling turns constructive again. This procedure repeats as the plasma width further increases and results in a modulation of the harmonic intensity. At least four coherent build-up zones are visible in Fig. 2(b). The coherence lengths are dependent on the given longitudinal position within the focal region of the driving beam. Since the phase-mismatch due to the Gouy phase advance and dispersion increases with the difference between the fundamental and harmonic wavelengths, decreasing coherence lengths are observed for higher harmonic orders. The separations between the second, third and fourth peaks are almost equal, whereas the first peak is considerably in front of the focus of the driving beam where the Gouy phase advance is lower. Taking into account only the distances between the second, third, and fourth peak, decreasing coherence lengths from l_c = 0.53 mm for the 13th harmonic to l_c = 0.35 mm for the 25th harmonic in an Al plasma are determined. Additional discussion of the curves presented in Fig. 2(b) is found in Sec. 4.5.

In Fig. 2(c) the coherence lengths are represented as a function of the harmonic order q. As the phase-mismatch Δk increases approximately linear with the harmonic order (see Sec. 2), the coherence lengths are expected to decrease proportional to q⁻¹. This behavior is verified in Fig. 2(c). Taking into account the fraction of the Gouy shift to the overall phase-mismatch, the plasma density can be estimated from the proportionality constant. This results in a plasma density of N_e = 1.7 × 10¹⁷ cm⁻³ in this experiment in accordance with other reported values [39,40].

4.2. Quasi-phase-matching in a structured plasma

In order to realize quasi-phase-matching, defined plasma jets with a defined separation are created by means of a structured target. Inside the individual jets the harmonic yield is allowed to build-up while in the free space between them the phase coupling advances without additional harmonic generation. If the dimensions are chosen correctly, the harmonic intensity of a specific harmonic order will increase quadratically with the effective width of all plasma jets. In the vicinity of the focal spot of the driving beam the Gouy phase advances almost constantly with propagation distance [Fig. 2(a)]. Additionally, the plasma producing beam is centered around the focus of the driving beam. Therefore the variation of the plasma density and the related change in dispersion is minimized in that region. For harmonic generation from the short trajectories and a sufficiently high harmonic order, the Gouy phase advance and dispersion are the only relevant contributions that introduce a phase shift to the generated harmonic field. Their reduced magnitude at the beginning and the end of the interaction volume supports increasing plasma structures at these locations. However, due to the observation of equal coherence lengths through a large part of the interaction distance [Fig. 2(b)] a uniform plasma structuring is expected to enable reasonable QPM.

Applying a uniformly structured target composed of disk widths of 0.4 mm and a spacing of 0.7 mm with four disks in total, harmonics around the 23rd order show obvious signs of QPM [Fig. 3(d)]. The dependence of the harmonic intensity on the width of the plasma producing beam is presented in Fig. 3(a) for the 23rd and in 3(b) for the 25th harmonics, respectively. The approximate disk positions are indicated by the red areas. By increasing the width of the plasma producing beam, the disks are illuminated one after another. If the target dimensions are chosen correctly for QPM, this results in a staircase-like increase of the harmonic intensity. Whenever an additional disk is illuminated by the plasma producing beam the harmonic yield rises significantly, whereas in between the disks the yield remains almost constant. This behavior is clearly visible in Figs. 3(a) and 3(b). There is no obvious sign of an adverse coupling of the harmonic fields which would become apparent through a drop in intensity, either at the front or the back side of subsequent plasma jets. Only at the back of the fourth jet, there is a considerable decrease in intensity which is attributed to a significant contribution of harmonic intensity generated from the long trajectories (see Sec. 4.4). If m is the number of applied plasma jets, the quasi-phase-matched intensity can be maximally enhanced up to a factor of m², provided that equally strong harmonic fields are generated inside each jet. The presented results in Figs. 3(a) and 3(b) reproduce this prediction, although smaller harmonic fields are expected to be generated in the first and the fourth plasma jet due to the reduced intensity of both the driving and the plasma producing beam (lower number of emitters) at these locations. For comparison, the intensity dependence using an unstructured target is also shown in Figs. 3(a) and 3(b). In the latter case, the intensity is multiplied by a factor of 10 for better visibility. The recorded harmonic intensities within a single coherent zone are essentially of a comparable magnitude.

 

Fig. 3 Intensity of the 23rd (a) and 25th (b) harmonic, respectively, for increasing plasma producing beam widths generated from a structured Al plasma. A theoretical expectation is superimposed. For comparison the same dependence is shown when generated from an unstructured plasma multiplied by a factor of 10. The intensity axis is normalized to the intensity generated in the first coherent zone. For the structured plasma a target consisting of 0.4 mm disc widths and 0.7 mm spacer widths is used. The position of the disks is indicated by the red areas. Regarding the red areas, the evolution of the harmonic intensity on the effective plasma width is derived (c). When plotting on a double logarithmic scale, the slope reveals a quadratic dependence of the harmonic intensity on the plasma width. Raw spectra ranging from the 13th to 27th harmonic recorded at a plasma producing beam width of 6.2 mm are shown in (d) and (e). The target configuration is 0.4 mm disk width and 0.7 mm spacer width in (d), and 0.5 mm disk width and 0.9 mm spacer width in (e).

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A quadratic dependence of the generated harmonic intensity on the interaction length of the driving beam with the conversion medium is an inherent feature of a phase-matched light source. In case of a quasi-phase-matched source this dependence is still quadratic overall, but is superimposed on a wiggle as indicated in Figs. 3(a) and 3(b). As a consequence the course of the curve is lower in comparison to full phase-matching. Assuming that the actual plasma width in this experiment increases equally with the illuminated number of disks at the target, the effective plasma width can be derived from the width of the plasma producing beam. In Fig. 3(c) the harmonic intensity is plotted against the effective plasma width according to the target dimensions. The onset of harmonic generation at the beginning of the second plasma jet at 3.5 mm is taken as a reference. Only data points within the red areas in Figs. 3(a) and 3(b) are shown. On a double logarithmic scale the data can be fitted linearly with slopes of m₂₃ = 1.98 ± 0.14 for the 23rd and m₂₅ = 2.04 ± 0.14 for the 25th harmonic orders, respectively. These values are in accordance with the expected slope of 2, clearly emphasizing the quadratic dependence.

A raw image using a target configuration of 0.4 mm disk width and 0.7 mm spacing is again displayed in Fig. 3(d). As is evident from the figure and the above discussion, this target configuration shows excellent QPM for the 23rd and 25th harmonic orders. The 25th harmonic is, however, already near the cut-off. QPM is most effective when the coupling of the harmonic fields generated in successive plasma jets just becomes constructive again at the beginning of the subsequent jet. This is equivalent to a phase advance of the generated harmonic field of Δϕ = π in between the jets. Based on the geometrical contributions regarding the short trajectories, such a phase shift equals a distance of 0.67 mm for the 25th harmonic [see Sec. 4.1, Fig. 2(a)] and 0.73 mm for the 23rd harmonic at the focus of the driving beam. Both values are in very good agreement with the applied spacer width of 0.7 mm.

Assuming that the width of the individual plasma jets is set by the width of the applied disks at the target, the coherence lengths determined in Sec. 4.1 would suggest optimal QPM for the 17th and 19th harmonic orders using disk widths of 0.4 mm. The coherence lengths found inside the unstructured plasma amount to 0.41 mm for the 17th harmonic and 0.38 mm for the 19th harmonic, respectively. In contrast, the coherence length of the 23rd harmonic is measured to be only 0.35 mm. This would suggest optimal QPM with disk widths thinner than 0.4 mm. Figure 3(e) shows a raw image using increased target dimensions with disk widths of 0.5 mm and a spacing of 0.9 mm. Obviously, this configuration favors QPM of the 17th and 19th harmonic orders. In the case of the 19th order, for instance, the calculated optimal spacing amounts to 0.88 mm at the focus of the driving beam which is in remarkable accordance with the applied spacer width. The discrepancy of oversized disk widths might be explained by a reduced plasma density at the sides of the individual plasma jets. The plasma could laterally expand into free space and out of the driving beam, whereas it would be confined by using continuous plasmas. There is, therefore, less dispersion at the sides of the individual plasma jets which slightly increases the coherence lengths.

The ability to alter the spatial properties of the plasma structure gives control over the QPM process. As demonstrated in Fig. 3(d) and 3(e) a plasma with smaller structures enhances higher harmonic orders, whereas lower orders are favored when using a more coarsely structured plasma. In principle, although less effective, QPM is also possible using disk widths slightly away from the optimum for a specific harmonic order and a spacing that compensates for the false dimensioning. In that case the phase of the harmonic field still advances by Δϕ = 2π from the beginning of one plasma jet to the beginning of the next. Only if the combined width of the disk and spacing is chosen incorrectly the mismatch of the harmonic field will increase from plasma jet to plasma jet which results in a progressive deterioration of the harmonic yield.

4.3. Non-optimal structured plasma

The choice of an accurate target configuration is decisive for effective QPM. In Fig. 4 two cases of an unfavorable target configuration are shown. For the 23rd harmonic the width of the plasma jets is chosen to be close to the optimum value (0.4 mm disk width), but their separation is undersized with only a 0.5 mm disk spacing. As a consequence, the phase of the generated harmonic field cannot advance far enough in the gap between the individual plasma jets. Therefore, as the plasma starts to form on consecutive disks, the coupling of the harmonic fields is not yet constructive at the beginning of the subsequent plasma jet. This leads to a drop in the harmonic yield. Since the phase of the harmonic field will quickly advance inside the plasma the destructive interference soon turns constructive. The drop in intensity is clearly visible in Fig. 4 at the beginning of the third and fourth plasma jet. Since the phase of the harmonic field is not rotated completely for this combination of disk and spacer widths, the effect gets worse the greater the number of disks that are applied. Inside the fourth plasma jet the intensity only just recovers to a magnitude already reached at the jet before.

 

Fig. 4 Examples for non-optimal QPM. The progression of the intensity of the 19th harmonic is shown for a target configuration of 0.4 mm disk width and 0.7 mm spacing which is similar to Fig. 3(d). For the 23rd harmonic order the spacing between the disks is reduced to 0.5 mm.

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The other example shows the behavior of a harmonic order using a target configuration set for QPM of a different order. Here, the yield of the 19th harmonic is shown for the target configuration used in Fig. 3(d) (0.4 mm disk width and 0.7 mm spacing). In this case both dimensions are chosen to be too short for achieving effective QPM at the 19th harmonic and the above-mentioned effects further increase. This time almost no recovery of the harmonic yield can be observed inside the fourth plasma jet. Applying disks that are too wide results in a drop in intensity towards the end of the plasma jets similar to that shown in Fig. 2(b). The same happens when the phase of the harmonic field has advanced too far in between the plasma jets due to an oversized spacing.

4.4. Contribution of the long trajectories

The results presented in Sec. 4.2 indicate that the harmonic field is predominantly generated from the short trajectories. The dipole phase of the long trajectories would add a rapidly varying contribution to Δk, resulting in increasing coherence lengths through the focus of the driving beam. Such a behavior is not observed with the present beam parameters. In front of the focus, very short coherence lengths, and therefore a negligible harmonic yield emitted from the long trajectories would result. This originates from the same sign of the dipole contribution compared to that of the Gouy phase advance and the dispersion in free electrons to Δk [compare Fig. 2(a), dashed line]. Behind the focus, by contrast, the reversed sign of the dipole phase contribution results in longer coherence lengths synonymous with a stronger harmonic yield.

The harmonic field can also phase match in the transversal direction. In this case the phase shift due to the temporally changing intensity of the pulse profile balances with the phase shift originating from the radially changing intensity of the beam profile. Taking into account the blue shift from the rising edge and the red shift from the falling edge of the pulse profile, this leads to the observation of elliptical structures in the space-frequency domain on the CCD [41, 42]. As a result of the smaller variation of the dipole phase with intensity, contributions originating from the short trajectories only appear close to the axis of propagation with a low divergence. Contributions of the long trajectories on the contrary can produce significant rings. In the present experiment the appearance of a highly divergent contribution is avoided at the beginning of the interaction volume by reducing the intensity. Nevertheless, the intensity increases as the driving beam propagates further into the conversion medium. Therefore, a wider range of both the temporal and spatial beam profile is involved in the generation of harmonics. As a result, ring-like patterns associated with harmonic generation from the long trajectories begin to emerge.

In Fig. 5(a) a raw spectrum at a plasma producing beam width of 7.1 mm is presented. In this figure the harmonic signal is generated at both sides of the focus of the driving beam. The axis of propagation is apparent from the low divergent fraction visible at the bottom of the picture. As the influence of the short trajectories decreases off-axis there is evidence of an increased contribution of the long trajectories. The accordance of the divergent structure with the estimated envelope for the long trajectories according to Ref. [41] further supports this interpretation. In contrast, Fig. 5(b) shows a spectrum recorded at a plasma producing beam width of 3.5 mm. The end of the plasma approximately coincides with the second peak in Fig. 2(b). In this case the harmonic signal is only generated in front of the focus. There is no evidence of a considerable amount of harmonic generation from the long trajectories. The divergences amount to 1.5 mrad in case of Fig. 5(b) and ca. 12 mrad in 5(a), respectively. As the longitudinal coherence length of harmonic generation from the long trajectories increases through the focus of the driving beam, their influence on the overall signal becomes more and more significant. This might explain the emerging background with increasing plasma width as measured with the unstructured targets [Fig. 2(b)]. From about 3 mm behind the focus, the phase-mismatch due to the dipole phase of the long trajectories is estimated to surpass the phase-mismatch originating from the Gouy phase advance for the given focusing geometry and harmonics around the 25th order [see Fig. 2 (a), dashed line]. Here, the long trajectories are eventually well phase-matched on the axis of propagation. Therefore, their contribution competes with the contribution of the short trajectories. This explains the drop in harmonic yield which is observed at plasma producing beam widths exceeding 6 mm. This feature is not only observed using unstructured plasmas but also at the end of the fourth jet in structured plasmas as shown in Figs. 3(a) and 3(b) where there are indications for the increasing influence of the long trajectories.

 

Fig. 5 Enhanced harmonic generation from the long trajectories at large widths of the plasma producing beam. Raw spectra recorded at a plasma producing beam width of 7.1 mm (a) and 3.5 mm (b) are displayed, respectively. The harmonic radiation is generated in a continuous Al plasma. The central part of the beam appears at the bottom of the images in order to capture at least half of the harmonic beam on the grating. Two calculated envelopes (red lines) according to [41] [Eq.(31)] regarding harmonic generation from the long trajectories at the present beam parameters are displayed in (a). The divergences amount to 1.5 mrad in (b) and ca. 12 mrad in (a), respectively.

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4.5. Limits of QPM using the Gouy phase advance

The maximal enhancement of a quasi-phase-matched harmonic depends on how many plasma jets can be placed into the focal region at a sufficiently high intensity of the driving beam. Therefore, the intensity profile shown in Fig. 2(b) is discussed in greater detail. Two features are obvious. First, there is a distinct difference in peak intensity of the particular coherent zones accompanied by a decreasing modulation depth as the plasma width increases. These effects are mainly attributed to the lower ionization probability away from the focus of the driving beam. Therefore, the harmonic yield at these locations is lower due to a reduced number of emitters. The decreasing harmonic yield behind the focus of the driving beam is not capable to fully degrade the existing field amplitude generated at the center of the focus. For this reason the harmonic intensity is not reduced to zero at large plasma widths. Furthermore, the long trajectories gain increasing influence behind the focus. In addition to the above-mentioned considerations, the plasma density is expected to be lower away from the focus of the driving beam. This results from the positioning of the plasma producing beam on the target centered close to the focal spot of the driving beam. As a consequence, the number of harmonic emitters is further reduced at the respective positions.

The second feature visible in Fig. 2(b) is a nearly 50% increased separation of the first to second peak compared to the separation of the other peaks. This can be explained by the reduced Gouy phase shift in front of the focus of the driving beam. Again, a reduced plasma density at this location might enhance the effect. Both result in increased coherence lengths. A significant contribution of the dipole phase shift, on the contrary, would lead to decreased coherence lengths in front of and increased coherence length behind the focus, respectively. This effect is not observed.

In consideration of the discussion made above, four plasma jets seem to be the maximal number of coherent harmonic emitting zones for enhancing the harmonic yield with the present beam parameters and harmonics around the 23rd order. Additional plasma jets at the sides would either be too far out of the focus of the driving beam or disturb the generated field through increasing harmonic emission from the long trajectories. Due to the rising impact of the Gouy phase advance and dispersion on the phase-mismatch of the harmonic field, higher orders require smaller plasma structures for QPM. In this case, additional plasma jets could be added. It should be noted that the coherence length is a monotonously decreasing function proportional to q⁻¹ [see Fig. 2(c)]. The minimum achievable width of the plasma jets will ultimately limit the highest orders for which QPM can be achieved. An upper limit for the minimal jet width can be derived from the distance between target and the center of the driving beam (ca. 100 μm). For the present experiment this is equal to the coherence length of approximately the 70th harmonic. In order to further extend the attainable harmonic orders, the driving beam could be focused more loosely into the conversion medium. This is accompanied by a reduced advance of the Gouy phase and an extension of the Rayleigh length. Therefore, the dimensions of the optimal plasma structure and the number of applicable jets will increase again. However, this approach is also limited by the opposing effect of a reduced cut-off frequency due to the lower peak intensity. For long interaction distances adaptive plasma structures at the sides of the interaction volume are expected to improve the phase-matching conditions. This is achieved by increasing the dimensions of the jets and their spacing away from the focal spot of the driving beam in order to compensate for the reduced amount of the Gouy shift as well as the decreased plasma density.

5. Conclusion

In conclusion, it has been demonstrated that the Gouy phase advance can be used in order to realize quasi-phase-matching in high harmonic generation in structured laser ablated plasmas. Several harmonics extending over a few orders are enhanced by QPM. The exact range is controlled and in principle freely selectable by adjusting the target dimensions appropriately. The present beam parameters allow the positioning of up to four uniformly spread plasma jets at sufficiently high intensities of the driving beam. The maximal possible 16 fold enhancement of the harmonic yield compared to a single coherent zone is observed for the 23rd and 25th harmonic orders. However, increasing the dimensions of the plasma structure away from the focal spot of the driving beam could increase the number of applicable plasma jets and therefore further increase the harmonic yield. QPM is expected to be even more efficient for higher harmonic orders. As the coherence lengths constantly decrease, a larger number of thinner jets could be positioned into the focal area of the driving beam.