## Abstract

Quasi-phase-matched high harmonic generation using trains of up to 8 counter-propagating pulses is explored. For trains of up to 4 pulses the measured enhancement of the harmonic signal scales with the number of pulses *N* as (*N* + 1)^{2}, as expected. However, for trains with *N* > 4, no further enhancement of the harmonic signal is observed. This effect is ascribed to changes of the coherence length *L** _{c}* within the generating medium. Techniques for overcoming the variation of

*L*

*are discussed. The pressure dependence of quasi-phase-matching is investigated and the switch from true-phase-matching to quasi-phase-matching is observed.*

_{c}© 2012 OSA

## 1. Introduction

High harmonic generation (HHG) has been an active area of research for several years because it offers a straightforward method for generating coherent, ultrafast radiation with photon energies extending to above 1keV [1]. The high temporal and spatial coherence of HHG makes it an ideal source for applications in areas such as probing molecular dynamics [2, 3], high-resolution imaging [4], and attosecond metrology [5]. Although HHG offers many advantages, its adoption in many potential applications is prevented by low conversion efficiency, and hence low mean and peak brightness. The low conversion efficiency of HHG is primarily due to the phase mismatch between the driving field and the harmonics created at different points in the generating medium. This phase mismatch arises from dispersion in the generating medium, and causes the intensity of each generated harmonic to oscillate with propagation distance in the generating medium, preventing the continuous growth of the harmonic field. The harmonic intensity oscillates between zero and some maximum value with a period 2*L** _{c}*, where the coherence length

*L*

*=*

_{c}*π*/Δ

*k*

*, and Δ*

_{q}*k*

*is the wave vector mismatch given by Δ*

_{q}*k*

*=*

_{q}*qk*

_{0}–

*k*

*, where*

_{q}*k*

_{0}and

*k*

*are the wave vectors of the fundamental and the*

_{q}*q*th harmonic, respectively. As such, unless additional measures are taken, the maximum output which can be achieved is that generated over a single coherence length.

For harmonics generated in a hollow-core waveguide it is possible, under certain conditions, to achieve true phase-matching (Δ*k* = 0) [6] by balancing the dispersion of neutral atoms with the waveguide and plasma dispersion, so that the harmonic intensity increases quadratically with propagation distance. However, this approach is only possible up to a critical level of ionization, above which it is no longer possible to balance the dispersion. This places a limit on the maximum harmonic order which can be phase-matched with this method.

An alternative approach for overcoming the effects of phase mismatch, which can be employed for any harmonic order, is to suppress HHG in regions where the locally generated harmonics are out of phase with the harmonic beam, a technique known as quasi-phase-matching (QPM). By adding coherently the amplitudes of harmonics generated in the remaining regions, or zones, the output intensity of the harmonic can, in principle, be increased by a factor of 𝒩^{2} above that for a single coherence length, where 𝒩 is the number of contributing zones. Several methods for achieving QPM of HHG have previously been demonstrated. These include the use of multiple gas jets [7, 8], modulated waveguides [9], and mode beating in capillary waveguides [10]. QPM has also been demonstrated using trains of counter-propagating ultra-fast pules [11–13] to suppress HHG in those regions where the driving laser pulse coincides with pulses in the train. For pulse-train QPM (PTQPM) with parallel polarizations of the driver and pulse train, suppression of harmonic generation occurs due to the rapid variation of the phase of the generated harmonics as the driver and counter-propagating pulses pass through each other [14], scrambling harmonic generation in this region. PTQPM may also be achieved using perpendicular polarizations of the driver and pulse train, suppression in this case being caused by the departure from linear polarization of the laser field in the regions where the driver and counter-propagating pulse overlap. It has been shown that use of parallel polarizations is more efficient than perpendicular polarizations for achieving PTQPM [15].

As has been described by Bahabad et al. [16] QPM by a moving modulation — such as a pulse train — can lead to a frequency shift of the generated harmonics such that the qth harmonic has a frequency *ω** _{q}* =

*q*

*ω*

_{0}– Δ

*ω*, where

*ω*

_{0}is the frequency of the driving laser. As a consequence the wave vector mismatch becomes Δ

*k*

*= Δ*

_{q}*ω*

*n*(

*ω*

*)/*

_{q}*c*+ Δ

*k*′

*, where Δ*

_{q}*k*′

*is the wave vector mismatch in the absence of the pulse train. The condition for QPM by a train of pulses of period Λ can then be written as Λ = 2*

_{q}*π*

*m*/Δ

*k*

*, where*

_{q}*m*= 1,2,3 ... is the order of the QPM process. In the case of PTQPM the modulation moves with a velocity ∼ −

*c*with respect to the driving pulse, so that Δ

*k*

*≃ Δ*

_{q}*k*′

*/2 and the matching condition becomes Λ = 4*

_{q}*π*

*m*/Δ

*k*′

*= 4*

_{q}*mL*

*, where*

_{c}*L*

*is the coherence length measured in the absence of the pulse train. For a train of pulses of width,*

_{c}*w*, and spacing,

*d*, such that Λ =

*w*+

*d*= 2

*w*, the matching condition may be written as

*w*= 2

*mL*

*. Further, in this case*

_{c}*m*is restricted to odd values since the Fourier Series of a square wave modulation only contains odd terms. Since the harmonic intensity of a quasi-phase-matched source will scale with the square of the number of in-phase zones, then

*N*counter-propagating pulses can be used to suppress HHG in

*N*separate regions, leaving

*N*+ 1 unsuppressed regions, so that, if the pulses are suitably matched to

*L*

*, the harmonic intensity for PTQPM is expected to scale as 𝒩*

_{c}^{2}= (

*N*+ 1)

^{2}.

To date the maximum number of pulses which has been used in PTQPM was 4 [12], owing to the optical arrangement used to generate the pulse train which makes scaling to larger numbers of pulses difficult. In this paper we present the results of a study of PTQPM using trains of up to 8 uniformly-spaced pulses, produced using a simple optical arrangement. The harmonic output is found to scale as (*N* + 1)^{2}, as expected, for *N* ≤ 4, but it is found that the harmonic output does not increase for *N* > 4. The reasons for this are explored and explained by a simple model which includes the effects of variation of the coherence length within the gas target. The variation of PTQPM as a function of the pressure within the target is also studied. It is found that additional peaks in the harmonic intensity are consistent with higher-order QPM processes.

## 2. Pulse train generation

In this experiment pulse trains were generated using an array of birefringent plates, chosen such that the thickness of each plate in the array is twice that of the previous plate [17, 18]. If a linearly polarized laser pulse passes through a birefringent plate, orientated such that its optic axis is at 45° to the laser polarization, then the pulse will be resolved into two equal intensity pulses, one polarized parallel to the optic axis and one polarized perpendicular to this. On leaving the plate the two pulses will be separated in time by an amount which depends on the birefringence and thickness of the plate, resulting in two orthogonally-polarized, temporally-separated pulses. By repeating the process for each plate in the array it is possible to generate a train of 2* ^{p}* pulses, where

*p*is the number of plates in the array.

In the experiments presented here an array of six calcite plates and one quartz plate was used to generate the pulse trains. The thickness of the quartz plate was 7.715mm, and the calcite plates were 0.826mm, 1.652mm, 3.304mm, 6.608mm, 13.216mm and 26.4mm thick. The quartz plate corresponded to the plate with the smallest optical path difference, which generated two pulses separated by approximately 250fs. Subsequent broadening of the pulses due to the material dispersion of the remaining plates resulted in both pulses having a minimum pulse duration of approximately 190fs. Since this is not much smaller than the minimum pulse separation, the output pulses — when resolved onto a single plane of polarization — effectively formed a single “macropulse” of approximately 400fs in duration. Macropulses of longer duration could easily be generated by suitably arranging the other plates in the array. For example, Fig. 1 shows the cross-correlation of a pulse train comprising 8 macropulses, each macropulse formed by 4 closely spaced pulses. By choosing the arrangement of the plates in the array it is therefore possible to vary the number of pulses in the train, *N*, as well as the pulse width, *w*, and separation, *d*. This method therefore enables a wide range of different pulse trains to be easily generated. In the experiments presented here PTQPM is investigated using pulse trains with *w* = *d*. The maximum number of macropulses in such a train, as well as the approximate width and separation of each macropulse which could be generated using the available plates is summarized in Table 1. The intensity of the pulses in the train could be controlled using a waveplate and polarizer placed before the array of plates. The intensity per counter-propagating pulse required to achieve QPM is given approximately by (*π*/2*q*)^{2}*I** _{d}*, where

*I*

*is the driver intensity and*

_{d}*q*is the harmonic order [15]. For the harmonics considered in this paper this corresponds to an intensity per counter-propagating pulse of approximately 0.5% of the driver intensity, consistent with the values used in the experiments described below. This is well below the threshold for ionization so the effect of the counter-propagating pulses on the generating medium will be negligible.

#### 2.1. Matching pulse trains to coherence lengths

In the case of harmonics generated by a linearly polarized laser pulse in a capillary waveguide the phase mismatch is given by

*N*

*=*

_{e}*η*

*N*

_{atm}*P*/

*P*

*is the free electron density,*

_{atm}*η*is the ionization fraction,

*N*

*is the number density at 1 atmosphere,*

_{atm}*P*is the pressure,

*P*

*is the pressure at 1 atmosphere,*

_{atm}*r*

*is the classical electron radius,*

_{e}*λ*

_{0}is the wavelength of the driving laser field,

*q*is the harmonic order,

*u*

_{1,1}is the 1st root of the Bessel function

*J*

_{0},

*n*(

*λ*

_{0}) is the refractive index of the neutral gas for radiation of vacuum wavelength

*λ*

_{0}, and

*a*is the radius of the capillary. Figure 2 shows, for various values of

*η*, calculated values of

*L*

*for harmonics generated in a capillary with*

_{c}*a*= 51

*μ*m filled with 10mbar of Ar. These parameters are representative of those used in the experiments described below. Also shown are the values of

*L*

*for which first-order QPM may be achieved using the pulse trains summarized in Table 1.*

_{c}It is clear from Fig. 2 that QPM can be achieved for different harmonic orders by adjusting the ionization fraction and selecting the parameters of the pulse train appropriately. For example, for pulse train C harmonic *q* = 25 can be matched at an ionization fraction *η* = 0.2; for the same degree of ionization, matching higher-order harmonics requires counter-propagating pulses of shorter width. From Table 1 it is seen that as the width of pulses in the train decreases the maximum number of pulses which can be generated increases. This favourable scaling means that, in principle, the number of zones over which QPM can be achieved with this method should increase with increasing harmonic order.

It is worth noting that for the experimental parameters presented here it is possible to achieve true phase-matching for harmonic orders up to the 27th harmonic. Although in principle QPM may be achieved for harmonic orders below the true phase-matching limit, the output harmonic intensity will not be as large as for the case of true phase-matching, making it of limited practical use. QPM is therefore only of interest for harmonics above the true phase-matching limit.

## 3. Experiment

A chirped pulse amplification Ti:sapphire laser system was used producing linearly polarized pulses at a 1kHz repetition rate and of energy 4mJ, duration 50fs, and centre wavelength 800nm. The output was split into a driving beam of up to 1mJ energy, and a counter-propagating beam with up to 1mJ energy on target. The energy of each beam could be controlled independently using a combination of polarizer and half-wave plate placed in the path of each beam. The counter-propagating beam was passed through the array of birefringent plates to generate the pulse train, and its polarization set parallel to that of the driver beam in order to minimize the energy required for QPM [15]. The driver and counter-propagating beams were coupled into opposite ends of a glass hollow-capillary waveguide by *f* = 500mm lenses, such that the focal spots of both beams were closely matched to the lowest-order waveguide mode. The capillary had an inner diameter of 102 *μ*m and length 40mm, and was filled with 0–150mbar of argon gas. The generated harmonic light passed through a hole in the mirror used to couple the counter-propagating beam into the capillary, and entered a flat-field spectrograph consisting of a gold-coated flat-field grating, with a spacing of 1200 lines/mm, and a cooled soft x-ray CCD (Andor DO440-BN). Harmonic spectra were recorded for 2 sec exposures. The experimental arrangement is shown in Fig. 3.

#### 3.1. QPM using different pulse train configurations

Initial experiments were performed with two pulse trains: a 2-pulse train with *d* = *w* = 1080 *μ*m, and a 4-pulse train with *d* = *w* = 540 *μ*m. The energy per pulse was the same for both pulse train configurations; since the duration of the pulses in the 4-pulse train was half that in the 2-pulse train, the peak intensities of the pulses were approximately the same in the two cases. The waveguide was filled with 8mbar Ar. For both pulse trains the harmonic spectrum was recorded as the point of overlap of the driver pulses and the pulse train was scanned through the capillary by adjusting a timing slide in the path of the counter-propagating beam. The measured enhancement, as well as the expected enhancement, for both pulse trains are shown in Fig. 4. For the data presented here the measured enhancement of the harmonic in the presence of the counter-propagating pulse train has been normalized to the harmonic signal in the absence of any counter-propagating pulses, which was taken as the average signal over 20 exposures.

The expected enhancement is calculated by numerically integrating the growth of the amplitude of each harmonic, given by

*E*

*is the field of harmonic*

_{q}*q*, Δ

*k*′

*is calculated for each harmonic using Eq. (1),*

_{q}*s*

*(*

_{q}*z*) describes the strength with which each harmonic is generated,

*N*

*is the density of emitters, which is proportional to pressure, and*

_{a}*α*

*is the pressure-dependent power absorption coefficient of harmonic*

_{q}*q*[19]. For these calculations

*s*

*(*

_{q}*z*) = 1 at all points in the generating medium except for those regions in which the driver pulse overlapped one of the counter-propagating pulses, in which case

*s*

*(*

_{q}*z*) = 0. The overlap region was taken to be half the pulse width to take into account the moving QPM grating [16]. For the 2-pulse and 4-pulse trains ionization fractions of

*η*= 0.25 and

*η*= 0.45 respectively, were found to fit the measured enhancements. This integration is done over an odd-integer number of coherence lengths, and is then normalized to the harmonic amplitude generated in the absence of a counter-propagating beam (

*s*

*(*

_{q}*z*) = 1 at all points). This calculation corresponds to the case of ideal QPM, where the total length of the region in which harmonics are generated corresponds to an odd-integer number of coherence lengths, and hence the harmonic intensity scales as (

*N*+ 1)

^{2}

*S*

_{1}, where

*S*

_{1}is the intensity generated in one coherence length. However, in practice this situation is unlikely to occur and instead the length over which harmonics are generated will be an even integer number of coherence lengths, plus some fraction of a coherence length,

*L*′

*. In this case the harmonic intensity achieved by QPM can be expected to scale as (*

_{c}*N*+

*δ*)

^{2}

*S*

_{1}, where

*δ*=

*L*′

*/*

_{c}*L*

*, and the enhancement will be given by where*

_{c}*I*

*is the generated harmonic intensity for QPM with*

_{QPM}*N*pulses and

*I*is the generated harmonic intensity in the absence of a pulse-train. Therefore, depending of the value of

*δ*, the measured enhancement will differ from the factor (

*N*+ 1)

^{2}predicted by simple theory. To account for this the calculated enhancement has been scaled to match the measured enhancement of

*q*= 21 and

*q*= 27 for the 2-pulse and 4-pulse trains, respectively.

For the 4-pulse train, strong enhancement of harmonic orders *q* = 23 – 33 was observed for timing slide positions corresponding to QPM of these harmonics in a region of overlap toward the exit of the capillary. The strongest enhancement of 40±5 was observed for *q* = 27, indicating that the coherence length of this harmonic is closest matched to the pulse train. From the measured enhancement a value of *δ* = 0.75 was determined for *q* = 27. The mean photon flux of the 27th harmonic in this case was estimated to be 1x10^{7} photons/sec. This estimation is based on the transmission of the aluminium filters in the spectrograph, the grating efficiency, and the quantum efficiency of the CCD camera at this photon energy. This is in comparison with an estimated mean photon flux of 2x10^{8} photons/sec for *q* = 25 obtained with no pulse train and the pressure adjusted to 130mbar, which corresponded to true phase-matching. It should be emphasized that QPM allows significant enhancement for harmonic orders above the limit for true-phase matching, which is estimated to be *q* = 27 for the conditions of this experiment. For the 2-pulse train, enhancement of the harmonic orders *q* = 17 – 25 was observed, with a peak enhancement observed for the 21st harmonic. From the measured enhancement of this harmonic a value of *δ* = 0.79 was determined. In this case the peak enhancement is observed for a lower order harmonic since the pulse separation is twice that in the 4-pulse case, corresponding to quasi-phase-matching of a longer coherence length.

For the experiments presented here the maximum harmonic order which could be generated was *q* = 35. The good agreement between the numerical model and the measured enhancements obtained with pulse trains B and C for *q* = 17 to 31 suggests that QPM could be extended to *q* = 35 by using an appropriate pulse train. It was not possible to investigate QPM for *q* > 31 since for the set of birefringent plates available, the next shortest pulse train was *w* = *d* = 270 *μ*m, corresponding, for the same degree of ionization, to QPM of harmonic orders *q* = 51 and above. We will investigate QPM for higher-order harmonics in future work. We note that the harmonic cut-off could be extended by using a shorter duration driving pulse, or by using a longer wavelength driver.

#### 3.2. Scaling of QPM with N

Experiments were performed to investigate the effect of changing the number of pulses in a train with *d* = *w* = 540 *μ*m. Trains of 1, 2, 4, or 8 pulses could be obtained by rotating the appropriate birefringent plates in the array. Rotating the plates rather than removing them, maintained the temporal overlap between the driving pulse and counter-propagating pulses, avoiding the need to find the temporal overlap for each pulse train configuration. In each case the intensity per counter-propagating pulse was kept constant by adjusting the waveplate-polarizer combination before the birefringent plates.

The harmonic spectrum was recorded as a function of the position, *z*, of the point of overlap of the driving pulse and the counter-propagating pulse train. As in the previous experiments enhancements were observed for harmonic orders *q* = 19 – 31 generated in a region towards the exit of the capillary. Figure 5(a) shows the normalized intensity of the 27th harmonic as a function of *z* for 1, 2, 4, and 8 pulses. Figure 5(b) shows the measured enhancement of the 27th harmonic as a function of *N*, together with the expected (*N* + 1)^{2} scaling. The measured enhancement is seen to scale as expected for up to *N* = 4. However, for trains of *N* = 8 pulses the enhancement is observed to be no greater than that measured for *N* = 4 pulses.

The observation that a train consisting of 8 uniformly-spaced pulses does not yield the expected enhancement may be explained by considering the effect of variations of the coherence length within the capillary. It has previously been shown that the coherence length of harmonics in a hollow-core waveguide varies as a function of longitudinal position [18, 20] owing to variation of the intensity of the driving laser due to attenuation of the driver modes, ionization losses, and mode beating. The changes in coherence length limit the number of uniformly-spaced pulses which can be used to achieve QPM since the modulation introduced by the pulse train will only match the coherence length over a limited region. This is illustrated in Fig. 6 which shows the calculated harmonic output for QPM with a train of 8 uniformly-spaced pulses with *w* = *d* = *L** _{c}* in a uniform medium with constant coherence length, for the case of no absorption. Also shown is the calculated output obtained with the same pulse train but for the case in which the coherence length increases by 10% from the matched value across the generating region. It may be seen that this small variation in coherence length causes the maximum enhancement to decrease from 81 to 35, which is similar to that observed in the present experiments. Also illustrated in Fig. 6 is the calculated harmonic output for the case of

*N*= 4 pulses and the same variation of coherence length; in this case the maximum enhancement is 21, which is close to that expected for ideal QPM. This simple analysis shows clearly that the variations in coherence length in the waveguide place a practical limit on the number of pulses which can be used to achieve QPM with uniformly-spaced pulse trains.

In order to extend the number of pulses which can be used, and hence increase the harmonic output, it will be necessary to overcome the problem of a varying coherence length. One possibility would be to employ pulse trains in which the pulse spacing can be varied within the train, as has been demonstrated recently [18].

#### 3.3. Higher-order QPM

The dependence of PTQPM on pressure was investigated using a train of 4 pulses with *w* = *d* = 540 *μ*m, and the position of the timing slide adjusted to give a maximum enhancement of the 27th harmonic at a pressure of 11mbar of argon. The pressure in the waveguide was then varied from 0 to 50mbar in steps of 2mbar, and in steps of 10mbar from 60 to 150mbar. At each pressure the harmonic spectrum was recorded with and without the pulse train. Figure 7 shows the recorded harmonic signals as a function of pressure for *q* = 27. The peak observed near 130mbar arises from true phase-matching. This is confirmed by numerical integration of Eq. (2), where *s** _{q}*(

*z*) is unity at all points in the waveguide corresponding to the absence of a pulse train, and

*α*

*= 0.032*

_{q}*P*, where

*P*is the pressure in mbar and has been determined using the tabulated values for the transmission of argon from the Center for X-ray Optics database [19]. The calculated harmonic intensity for an ionization fraction of

*η*= 0.026, which corresponds to Δ

*k*′

*= 0 at 130mbar, is shown in Fig. 7. The calculated curve shows reasonable agreement with the measured data demonstrating that the peak around 130mbar is due to true phase-matching. The same peak is observed when a pulse train is employed, but two additional peaks are also observed at approximately 14 and 40mbar: these correspond to first-order (m=1) and third-order (m=3) QPM, respectively.*

_{q}For ionization fractions above the critical ionization, *L** _{c}* decreases with increasing pressure. An increase in the harmonic intensity due to QPM will be observed whenever the value of

*L*

*is such that*

_{c}*w*= 2

*mL*

*, where*

_{c}*m*is odd. When

*L*

*=*

_{c}*w*/2 an increase in the output intensity is observed corresponding to first-order QPM (m=1). As the pressure is increased further

*L*

*will continue to decrease. At the pressure for which*

_{c}*L*

*=*

_{c}*w*/6 an additional peak will occur due to third-order QPM (m=3). From Fig. 7 the third-order peak at 40mbar is observed to have approximately the same amplitude as the first-order peak. This occurs because — neglecting the waveguide dispersion in Eq. (1), which is small —

*L*

*varies inversely with the gas density, and hence the number of emitters in each coherence length is independent of pressure. This is confirmed by numerical integration of Eq. (2), allowing for modulation of the harmonic generation by a pulse train with*

_{c}*N*= 4, the results of which are illustrated in Fig. 8. Calculation of the wave vector mismatch Δ

*k*′

*requires knowledge of the degree of ionization. The observed optimum pressure for first-order QPM corresponds to an ionization fraction*

_{q}*η*= 0.4. However, for this ionization fraction the third-order peak is calculated to occur at a pressure of approximately 55mbar, which does not agree with the observed peak at 40mbar. The calculated third-order peak would agree with the measured data for an ionization fraction

*η*= 0.52. This observation indicates that the ionization fraction changes with pressure. This may be the result of nonlinear effects occurring in the waveguide such as pulse steepening, or mode coupling, which cause a local change in the driver intensity, resulting in a higher than expected degree of ionization. As such, higher-order QPM may yield information on pulse shaping effects in hollow-core waveguides. An alternative explanation for the position of the measured third-order peak is a change in the intensity-dependent phase of the generated harmonic, resulting in a change of the coherence length. The precise mechanism for this will be explored in future work.

## 4. Conclusion

QPM of HHG has been investigated using trains of uniformly-spaced counter-propagating pulses with equal pulse width and separation. These pulses were generated by passing a short pulse through a sequence of birefringent plates, which offers a straightforward method for generating pulse trains suitable for PTQPM at different harmonic orders. Pulse-train QPM was achieved over a range of harmonic orders from *q* = 17 – 31, resulting in a maximum measured enhancement of the harmonic signal of approximately 40.

It was found that the measured enhancement scaled as (*N* + 1)^{2}, as expected, for *N* ≤ 4 pulses, but that for *N* > 4 pulses the enhancement did not increase. This effect is attributed to variation of the coherence length in the waveguide, a conclusion which was supported by simulations which showed that a variation in the coherence length by as little as 10% could reduce the expected enhancement obtained with *N* = 8 pulses to that expected for *N* = 4 pulses. This finding highlights the fact that extending PTQPM to a large number of coherent zones is likely to require the use of more flexible pulse trains in which the spacing of the pulses can be varied within the train. Programmable pulse trains of this type have recently been generated [21], and in future work we will apply these tunable pulse trains to PTQPM.

The pressure dependence of PTQPM was also investigated. Peaks in the harmonic intensity as a function of pressure were observed and attributed to first and third-order QPM, and the transition to true phase-matching was also observed for the 27th harmonic. The good agreement with our model of QPM suggests that the method for generating pulse trains employed in this work could be used to achieve QPM for much higher order harmonics, well beyond the limit for which true phase-matching is possible.

## Acknowledgments

We are grateful for the financial support from the Engineering and Physical Sciences Research Council under grant EP/G067694/1.

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