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We investigate the photon flux and far-field spatial profiles for near-threshold harmonics produced with a 66 MHz femtosecond enhancement cavity-based EUV source operating in the tight-focus regime. The effects of multiple quantum pathways in the far-field spatial profile and harmonic yield show a strong dependence on gas jet dynamics, particularly nozzle diameter and position. This simple system, consisting of only a 700 mW Ti:Sapphire oscillator and an enhancement cavity produces harmonics up to 20 eV with an estimated 30–100 μW of power (intracavity) and > 1μW (measured) of power spectrally-resolved and out-coupled from the cavity. While this power is already suitable for applications, a quantum mechanical model of the system indicates substantial improvements should be possible with technical upgrades.
© 2011 Optical Society of America
1. Introduction
Traditional high harmonic generation (HHG) with solid-state laser systems has successfully utilized chirped pulse amplification (CPA) of femtosecond optical pulse trains to generate attosecond extreme ultraviolet (EUV) light at repetition rates up to 100 kHz [1]. Fiber-based CPA laser systems have increased the repetition rate up to 1 MHz [2] and very recently 20 MHz [3], the latter with an estimated 1 nW/harmonic of EUV power. A further increase of the EUV’s repetition rate to > 100 MHz with high average power is of interest for a number of reasons including production of femtosecond frequency combs in the EUV [4] and EUV photoelectron and other low-probability coincidence experiments [5]. In the former case, building on the work of Eikem et al who used a kHz repetition rate CPA system to perform a novel EUV Ramsey-fringe type experiment to obtain a measurement of the 4He ionization energy with an uncertainty of 10−9 [6], an EUV frequency comb generated from a fsEC very recently obtained an uncertainty of 2 × 10−10 in the measurement of two EUV transitions in Ar and Ne [7] using direct EUV frequency comb spectroscopy. EUV photoelectron and low-probability EUV coincidence measurements can leverage the high repetition rate for drastically improved data acquisition time and improved signal to noise ratio. Traditional CPA systems cannot presently reach these repetition rates as they are limited in their average output power. To increase the high repetition rate into the 100s of MHz range, it is necessary to employ femtosecond enhancement cavities (fsEC) as a means to obtain the requisite high pulse energy for HHG.
Such cavities have been seeded with femtosecond Ti:Sapphire lasers near 800 nm [4, 8] and high power Ytterbium-doped fiber laser systems near 1 μm [9, 10]. With available low GDD, high bandwidth cavity mirrors, the shortest pulses that fsECs can support is approximately 70 fs (at 800 nm). Current state of the art fsECs with enhancement factors greater than 100 can store pulse trains with average powers of a few kilowatts and pulse energies of several tens of microjoules. To achieve the intensity (> 1013 W/cm2) required for HHG, a tight focus through the intra-cavity gas target is required. In the tight focus regime, the spatial variation in the laser field amplitude and phase leads to large gradients in the wavevector mismatch due to phase terms (Gouy, atomic, plasma, etc.) in the atom-field interaction. These spatial gradients limit the efficiency of EUV generation and affect the divergence of the generated beam, making it particularly important to understand the interaction in this regime and optimize the experimental parameters to generate the most EUV possible. To date, no systematic study of these parameters for fsEC-based EUV sources have been reported.
Within the context of fsECs, there are several experimental factors that influence the harmonic generation process: 1) the fsEC field parameters, including pulse duration, peak fundamental intensity, and focusing geometry, and 2) the gas density distribution, which is determined by the backing pressure in the gas delivery line, the nozzle geometry and its position relative to the focus in the fsEC. In high repetition rate systems like the fsECs, all of these parameters become strongly coupled when the gas delivered to the focus changes the resonance condition of the enhancement cavity. One particular concern is the interaction of the plasma generated by the HHG process with the circulating field. The plasma itself introduces phase shifts and dispersion, which affects the resonance condition of the cavity. Nonlinear phase shifts due to the plasma can become severe at high intensities and ultimately limit the peak intensity circulating in the fsEC [11, 12]. Moreover, in the tight focusing geometry, with its limited phase-matched volume, it is important to consider the interplay of harmonic generation and reabsorption of the near-threshold harmonics in the gas target.
In this work, we present the first systematic study of a fsEC-based EUV source, which includes the output flux and beam shape of the generated EUV for the 7th–13th harmonics (11–20 eV) of the 790 nm driving field. Our fsEC system, seeded only with a Ti:Sapphire oscillator, produces an estimated 30 μW of intracavity power at 72 nm (17 eV), of which a measured power of > 1 μW or 3.6 × 1011 photons/s is coupled out of the cavity and spectrally resolved for use in experiments. Lower order harmonics have as much as an order of magnitude more power. We observe that the properties of the generated harmonics vary significantly as the geometry used to deliver xenon gas to the laser focus is changed. We note features that are similar to previous, well-known HHG studies for harmonics far above threshold, and we also observe significant quantitative differences that arise for the near-threshold harmonics, where the behavior of phase-matching and reabsorption processes are different. These differences are of current theoretical interest [13] and are important to the development of stabilized EUV frequency comb sources. We optimize the harmonic generation for our current fsEC EUV source and identify potential improvements on the current system that predict substantial improvement on the EUV flux. To supplement and guide our experimental study we employ an on-axis quantum mechanical model of the EUV generation, treating harmonic generation in the tight focus, high repetition rate regime through examination of the role of phase-matching and re-absorption, with an emphasis on aspects that are important in actively stabilized femtosecond enhancement cavities. We first present the experimental setup used to generate the harmonics. We then present the model of the harmonic generation process, and compare the predictions with the experimental results. Based on experimental confirmation of our model, we identify conditions that optimize specific aspects of the harmonic generation that will be important for future experiments with fsEC-based EUV combs.
2. Experimental setup
Our fsEC system is shown in Fig. 1. Briefly, it is seeded with a Ti:Sapphire fs oscillator with a 66 MHz repetition rate, an average power of 700 mW, and a spectral bandwidth of 15 nm near 800 nm. The pulses are amplified in a high-finesse (> 2000), bow-tie cavity to achieve intra-cavity pulse energies of 5–10 μJ. With imperfect mode matching and residual dispersion, the enhancement is usually limited to about 500, and intracavity pulses of about 70 fs are measured. Curved mirrors (10 cm ROC) in the cavity focus the optical beam down to a spot with a radius of about 12 μm, resulting in intensities of up to 5×1013 W/cm2. The cavity is stabilized to the oscillator by controlling the cavity length of the fsEC to match the pulse repetition period (frep) of the laser, and the carrier-envelope offset frequency (fceo) of the laser is matched to that of the fsEC through modification of the oscillator pump intensity with an acousto-optic modulator (AOM). We use a modified Pound-Drever-Hall stabilization scheme in both the frep and fceo locks, in which a piezo-electric crystal (PZT) in the fsEC is modulated at 1 MHz and an error signal for each loop is derived from a different portion of the spectrally dispersed cavity reflection. The reflection signal is demodulated with a double-balanced mixer (not shown in Fig. 1) with the quadrature phase component of the PZT modulation signal as a local oscillator. The error signal derived from one end of the reflection spectrum is used to control the fsEC cavity length, and the error signal from a second portion of the spectrum is used to stabilize the fceo. These two degrees of freedom in the control loop allow for long term stabilization of the intracavity intensity to less than 1% r.m.s. relative intensity noise from 100 Hz to 100 kHz. While this technique does not achieve orthogonal control of frep and fceo we do not see significant cross-talk between control loops. We achieve a control loop bandwidth of about 60 kHz on the frep control loop using a lead-filled copper PZT mount, similar to that described in [14].
Fig. 1 (color online) Configuration of femtosecond enhancement cavity. A signal detected by PD1 and subsequently demodulated provides an error signal for a modified Pound-Drever-Hall scheme to lock the cavity’s free spectral range to the seed laser’s pulse repetition rate. The error signal derived from PD2 is used to lock the carrier envelope offset frequencies of the laser and cavity. EUV harmonics generated in the Xe jet are coupled out of the cavity via a grating (200-nm period) written in the grating mirror (GM). IC, input coupler; PZT, piezo-electric transducer; PD, photo-diode; Xe, Xenon gas jet.
Xenon gas is delivered to the focus by an end-fire gas nozzle with diameters ranging from 150–500 μm. The harmonics are coupled out of the cavity by an EUV diffraction grating etched in the top layer of a cavity mirror placed 1 cm after the cavity focus [15] and recorded by imaging from the fluorescence emission from the back side of a sodium salicylate phosphor coated glass plate 7 cm from the grating. Absolute measurements of the 11th harmonic using an uncalibrated IRD Inc. Si photodiode indicate a minimum power of 1 μW for gas line pressures of 300 Torr, which corresponds to 30–100 μW of intracavity EUV power.
The gas nozzle is located near the focus of the circulating field with an XYZ translation stage driven with actuators in the vacuum system. The gas jet can be precisely positioned in the direction transverse to the laser propagation direction by maximizing the harmonic signal. While the step size is well characterized (e.g. steps of 150 μm), the absolute position relative to the focus along the direction of laser propagation is not known accurately, as we discuss below in the presentation of the data.
In this work, we study the 7th through 13th harmonics (H7–H13) at different positions of the gas nozzle in the longitudinal direction relative to the laser propagation direction, at different gas pressures and fsEC field intensities. As an example, Fig. 2 shows a series of images of the harmonics H7, H9, H11, and H13 as a 150 μm gas nozzle is translated along the driving field’s optical axis. This real-color image series was collected with a color digital camera with an exposure time of 8 seconds per image. In order to understand this multi-dimensional parameter space, a numerical model is developed with an underlying goal of maximizing a high quality EUV spatial mode with the highest photon flux.
Fig. 2 (color online) Image series of 8-second exposures showing the evolution of spatial profiles of harmonics H7–H13 (top to bottom) for different gas nozzle positions near the focus (approximate location of focus indicated with red line). Each image represents approximately 150 μm translation of the nozzle in the direction of laser propagation (left to right). The profiles show a bright central component (saturated in these images) and a broad, diffuse beam, which have been attributed to multiple generation pathways for the harmonics. The different generation pathways have distinct spatial phase matching conditions within the fundamental laser field, and therefore yield different far-field spatial profiles.
3. Mathematical model
To develop a numerical model we first consider the amplitude of the harmonics generated on-axis (in section 4.1, off-axis features observed in Fig. 2 are discussed). Building on previous work [16] the on-axis field amplitude of the qth harmonic Eq can be found by solving,
where ρ(z) is the gas density and σ(ωq) is the absorption cross-section. The (position dependent) dipole response, d̃(ωq, z), is calculated by solving the time-dependent Schrödinger equation in one dimension, which is sufficient in our intensity regime [17, 18], where V(x,t) = Vatom(x) + Vfield(x,t) is the sum of the atomic and laser field potentials. The one dimensional atomic potential used is, where X0 = 0.91 Å dictates the depth of the potential for the active electron in xenon with an ionization energy Ip = 12.1 eV. The potential due to the applied laser field is, where E(t) = E0sech(t/τ) is the field envelope with a peak field amplitude E0 and pulse duration τ.The time evolution of the wavefunction, ψ(x,t), is calculated in discrete time steps Δt, where the spatial evolution is computed through the split-step Fourier method,
where FT and FT−1 denote Fourier Transform and the inverse Fourier Transform with conjugate variables x and p = h̄k. The spatial dimension is chosen such that the absorbing boundary conditions only exclude those electrons that are fully freed from the atom. Furthermore, the spatial grid density, and therefore the greatest permissible momentum, is chosen such that the momentum wavefunction lies completely within the domain.The time-dependent dipole is calculated via Ehrenfest’s theorem,
The dipole spectrum is then calculated by the Fourier Transform,
where T1 and T2 are the beginning and end points of integration, respectively. The magnitude of the dipole spectrum at the harmonic frequency, |d̃(ωq, z)|, is calculated as a function of the peak driving field amplitude E0 so that the dipole response in a tightly focussed Gaussian beam can be analysed as a function of position. The phase of the dipole response, Φatomic(z), is discussed below.The phase mismatch is Δϕ(z) = qϕ1 – ϕq with,
where zR ≈ 0.5 mm is the Raleigh range of the fundamental beam and, as it is comparatively small in this tight focus regime, no approximations of the Gouy phase shift are made. The atomic phase, Φatomic(z) = −αiI(z) is the intensity dependent phase of the active electron whose trajectory creates the qth harmonic [19]. There are two dominating electron trajectories that create the harmonics (the long and short trajectories) leading to two distinct atomic phases, even for the case at hand where harmonics are near the ionization potential of the generating atom [13]. While values for αi for these trajectories have been calculated quantum mechanically, we have not yet carried out this calculation for our specific set of parameters. In our simulations, we use αi = 1 × 10−14 and 25×10−14 cm2/W for the short and long trajectories, respectively [19]. Our experimental results show that much of the physics in our regime can be described by either the short or long trajectory alone. In this work, we present the generated harmonic field Eq(z) for each individual electron trajectory separately below; we discuss our observations of multiple pathway harmonic generation qualitatively in the next section.The index of refraction at the fundamental (q = 1) and qth harmonic in the gas jet is,
where ρ0 is the density at 1 atm, η is the ionization fraction, δq is the Xe correction from vacuum, and ωp is the plasma frequency. It should be noted that in the case of high repetition rate HHG with fsECs, the effective lifetime of the plasma, strongly dependent on the electron temperature [20], is greater than the pulse spacing (∼ 15 ns) and therefore this lifetime must be considered. Thus, η is not the per-pass ionization fraction, but rather some effective ionization fraction that is also dependent on the transit time of the Xe atoms through the driving field focus and hence the initial velocity of Xe gas. In order to maintain low effective ionization, the Xe gas traverses the intracavity focus at (perpendicularly) 180 m/s, as discussed below. The gas remains in the focus, a diameter 2w0 ≈ 24 μm, for approximately 9 round trips. Although the one dimensional potential, discussed above, is sufficient for calculating the harmonic dipole response d̃(ωq), it overestimates the ionization rate [18]. Using a more accurate model for the ionization rate [21] for a 70 fs pulse train with an intensity of 4 × 1013 W/cm2 we calculated the effective ionization fraction to be η < 0.1.As zR ≈ 0.5 mm, the functional dependence of ρ(z) over this length scale becomes an important consideration for optimizing the EUV output. For accurate predictions from our model we employ computational fluid dynamic (CFD) simulations of the gas flow to generate the spatial dependence of density from a variety of nozzle sizes of gas jets. OpenFOAM software is used for these simulations with the rhoCentralFoam solver [22]. Figure 3 shows the spatial density calculated for a set of nozzles with different size holes and a nozzle geometry that most accurately represents the nozzles used in the experiments. The 50–300 μm nozzles are constructed from a thin-walled, sealed-tip tube with a laser drilled hole in the end, thus creating the end-fire nozzle. The 500 μm nozzle we tested has a different construction, which is simply a long tube with a 500 μm inner diameter. This change of geometry for the 500 μm nozzle leads to a slightly different gas density distribution than the trend seen in the 50–300 μm nozzles.
Fig. 3 (color online) Spatial dependence of gas density (ρ) for a set of nozzle diameters ranging from 50–500 μm. The upper left panel shows the 2-D density distribution of gas emerging from a 500 μm nozzle, and the approximate position of the laser field is indicated with the dashed white line.The opening of the gas nozzle is located at x = 0 and centered at z = 0 and the units of the color bar are density (1018cm−3) The lower left panel displays ρ along the optical axis (z coordinate) at a vertical distance (x coordinate) of 150 μm below the gas nozzle at a backing pressure of 100 Torr for a number of nozzle sizes. Increasing the backing pressure will proportionally increase the density as we are operating in the choked flow regime. The lower right panel displays the decrease in ρ in the vertical direction from end of nozzle tip (x coordinate) toward the optical axis, and the laser position is indicated with the vertical dotted line. The upper right panel defines the x,z coordinates.
The work presented here focuses on an end-fire gas nozzle geometry, where the HHG occurs in a free expansion of the gas into vacuum. We have also studied two different through-nozzle geometries where the driving field is passed through a hole drilled transversely through a metal or glass tube [23]. In one geometry, the metal tube is sealed at the end of the tube and the gas can only escape through the same path that the laser traverses. In a second geometry the metal tube is open to the vacuum system, and about 90% of the gas exits through the end while 10% exits through the laser path. These through-nozzle geometries introduce significant problems for fsEC systems that generate near-threshold harmonics, where the absorption length is short. A significant atomic and plasma density exists along the optical axis of the cavity in both geometries, which makes reabsorption a major problem. Furthermore, the long interaction length with the gas and plasma leads to a large phase shift of the fsEC field at pressures of only a few Torr. It was also found to be challenging to generate high spatial quality beams using these nozzles, so a full characterization of their properties has yet to be completed.
4. Results and discussion
We present a comparison of the key results of our experiments with the on-axis theory developed in the previous section. Specifically, we discuss the evolution of the spatial profile of the harmonic beams, and the relative power of the beams as a function of the nozzle position, as well as the harmonic power as a function of the driving field intensity. Finally, we use the model to illustrate the role of phase-matching and reabsorption in the context of optimizing the performance for harmonic generation in fsECs.
4.1. Far-field beam profile
As described above, we studied the beam shape and relative powers of harmonics H7–H13 by imaging the fluorescence of a sodium salicylate phosphor with a digital camera. An example is shown in Fig. 2, in which we can see the spatial profile of H7–H13 change significantly as the nozzle is moved through the focus in the direction of laser propagation (left to right across image series).
There are several features of these spatial profiles that are noteworthy. One quickly recognizes the evolution of a diffuse ring around the central spot. With the gas nozzle before the focus, the ring has a large diameter and low intensity. Near the focus the diameter decreases and the ring becomes brighter before merging with the central spot near the focus. This diffuse ring is understood to originate from the long trajectory quantum pathways that are highly intensity dependent and phase-matched primarily off-axis [24]. The short trajectory pathways, on the other hand, generate an on-axis beam shape more closely matching the Gaussian profile of the optical driving field circulating in the fsEC. We see from Fig. 2 that the central beam before the focus (left side of figure) is smaller and more intense, and after the focus the beam is much more diffuse. This is true for all harmonics, though it is less apparent in the figure for H13. For harmonics H9–H13 it appears that the power in the long trajectory (diffuse) component is greatest after the focus, but it is not obvious that this is the case for H7. Several harmonics in Fig. 2 were allowed to become saturated to illustrate the features present in the diffuse, long trajectory beam profiles. When the image is not saturated we find the intensity of the diffuse ring (e.g. slightly before the focus) to be a few percent of the intensity at the center of the beam. Embedded in the far-field spatial profile of the beams is the intensity dependent quantum pathway information, which suggests these images could provide a useful test of current quantum mechanical calculations and generalized semiclassical models [13]. In addition, the coherence of EUV frequency combs might be affected by the long trajectories as the phase has an intensity dependence, so operation in a regime to minimize their contribution could be required.
It is interesting to note that we see small variations in the beam profiles for different nozzle sizes, but we have not observed significant changes that one might expect as the phase matching conditions change for the range of intensities and pressures we can achieve in the lab. The dominant effect of nozzle size is seen in the measurements of the power of the harmonics, as described in the next section.
4.2. Position dependence of harmonic power
Next we perform a comparison of the harmonic power calculated from the theory with the experimental data, extracted from the images for a fixed pressure as the cavity intensity and position of the gas nozzle are varied. The data presented was extracted from images recorded with a monochrome CCD camera with minimal image processing performed on the raw images. Although we can see qualitatively in Fig. 2 that there are long and short trajectory components to the harmonic power, we do not attempt to separate them quantitatively via image processing. While this is possible before the focus, it is not straightforward after the focus.
There are three global fitting parameters used in matching the theoretical results to the data. First, after accounting for the theoretical diffraction efficiency of each harmonic [25] coupled out of the cavity, a single amplitude scaling factor, which is held constant for all nozzles, is used to fit the theory to the experimental data. Second, due to the difficulty in determining the vertical distance from the nozzle to the optical axis, the gas density is changed according to the right panel of Fig. 2 and held constant at a vertical distance of 150 μm for all nozzles. Finally, there is some uncertainty in the fraction of ionization that results as the atoms traverse the laser field and the plasma is generated. This latter factor does not appear to be significant, and we achieve good agreement with the data for H7–H13 over the large parameter space of pressure, intensity and nozzle size.
First, we present experimental results showing the variation of H7–H13 as a function of gas nozzle position in Fig. 4 for 150, 300 and 500 μm diameter nozzles, and gas delivery line (interaction region) pressures of 80 (12), 20 (7), and 20 (10) Torr, respectively. For each nozzle position scan, the relative position is a fitting parameter, but the step size is fixed at its measured value of 150 μm. We plot the total power in each harmonic as a function of nozzle position, along with the long and short trajectory power for each nozzle size. For the 150 and 300 μm nozzles, we find that H7 and H9 are maximum at a position close to the focus and H11 and H13 peak after the focus. This is consistent with H7 and H9 being dominated by short trajectories while H11 and H13 being dominated by long trajectories. Harmonic H9 deviates slightly from the predicted behavior and does not fit the theory as closely. The fit breaks down similarly for the 500 μm nozzle, although the theory does predict a continued decrease in harmonic amplitude with larger nozzles. We will use these trends below to determine experimental parameters to optimize the harmonic power and/or beam profile for delivery to experiments.
Fig. 4 (color online) Theoretical calculations of harmonics plotted with experimental data showing total power of harmonics H7–H13 as a function of gas nozzle position relative to the focus, at an intensity of 3×1013 W/cm2. The quoted pressure is in the gas delivery line. The theory includes short trajectory (solid line) and long trajectory (dot-dash line) calculations separately, and the experimental datapoints are shown with markers (▵ H7, ○ H9, □ H11, and ⃟ H13). As described in text, the position of the focus is identified empirically from the peak of the position dependence of the harmonics.
4.3. Intensity dependence of harmonic power
We also test the model by varying the intensity at each nozzle position displayed in Fig. 4. We show a comparison of the data with the theory in Fig. 5, for the 150 μm nozzle at 80 Torr line pressure at two positions: (a) 100 μm before the focus, and (b) 200 μm after the focus. Before the focus, the agreement between the theory and the data is quite good, despite some uncertainty in the absorption cross-section for H7 and low signal level for H13. After the focus, the predicted power of harmonics H11 and H13 follows the data nicely, while a slight deviation appears for harmonics H7 and H9. Incidentally, these irregularities occur at a nozzle position where both long and short trajectory signals appear in the far field beam profiles for these harmonics, and our model does not account for interfering pathways in the phase matching of harmonics. Work to incorporate such effects into our model is ongoing, but is expected to be only a minor correction to these results.
Fig. 5 (color online) Harmonic power of H7–H13 at a position (a) 100 μm before the focus, and (b) 200 μm after the focus, as the intensity of the fundamental laser field is varied, for a 150 μm nozzle at 80 Torr. The connected points represent the experimental data, and the solid lines represent the theory.
Furthermore, a notable difference between the results presented here and those in Ref [26] is that the strong intensity-dependent oscillations due to quantum pathway interference are not observed. We believe the reason for this difference is two fold. First, in the current work, the shorter driving field wavelength of 800 nm leads to a smaller phase coefficient in the long trajectory, αlong ≈ 25 × 1013 cm2/W, compared to αlong ≈ 80 × 1013 cm2/W in Ref [26]. As the period of the oscillations scales inversely with αlong, the destructive interference is not big enough to compete with the intensity dependent dipole growth and therefore it does not cause an overall decrease in the harmonic intensity. Second, as shown in this present work the generation and spatial character of different pathways is highly dependent on experimental parameters such as nozzle position, pressure, etc., which suggests that strong interference between pathways may occur over a small range of experimental parameters.
4.4. Effects of phase-matching
Phase matching is severely limited by the rapidly varying Gouy phase shift of the optical driving field due to the tight focus geometry. These phase-matching conditions have been discussed previously in the context of kHz repetition rate HHG systems for harmonics significantly beyond the ionization potential [23], but they are of renewed interest in the case of fsEC and near-threshold harmonic generation where the tight-focus regime must be used.
To explore the role of phase matching, we use our model to calculate the on-axis power generated for H11 for different nozzle positions and different gas pressures, as shown in Fig. 6. The nozzle position is identified by the center of the gas density distribution (z=0 in Fig. 3), and it is also important to note that the pressure is the actual pressure at the interaction region, not the gas delivery line pressure. In this simulation we use a gas nozzle diameter equal to 300 μm, and a peak intensity of I0 = 3.5 × 1013 W/cm2, which is a set of parameters that demonstrates the role of phase-matching particularly well.
Fig. 6 (color online) Contour plot showing the calculated on-axis H11 power as a function of the gas nozzle position and the pressure of the Xe gas in the interaction region, for a 300 μm nozzle and a peak intensity I0 = 3.5 × 1013 W/cm2 for the short (left panel) and long (right panel) trajectories. The solid black line indicates the location where the wavevector mismatch (Δk) crosses zero.The unit of the color bar (the strength of the harmonic) is identical to the arbitrary units used in Figs. 4 and 5.
Experimental control of the phase-matching of the harmonic generation is largely limited to the control of the pressure and the gas nozzle position, but it is also sensitive to the spatial variation of the laser intensity. The pressure affects the long and short trajectory pathways in the same way, but the intensity dependent ‘atomic’ component of the interaction affects only the long trajectory. As a result, the long trajectory phase mismatch is small only in very localized regions within the focal volume. The short trajectory on the other hand does not display this behavior and the phase mismatch is much more uniform across the focal volume. For the intensities accessible in our fsEC there tends to be a location somewhere after the focus where the long trajectory phase-mismatch crosses zero. As the pressure is increased the zero-crossing in the phase mismatch moves closer to the focus and a second region appears before the focus. We plot the locations of the zero-crossing of the wavevector mismatch (i.e. Δk = 0) as a solid black line in Fig. 6.
For the long trajectory, and a gas nozzle located at a fixed position after the focus (e.g., 0.2 mm in the right panel of Fig. 6) the power of H11 shows a strong increase in power for increasing pressure. In this example, the power of H11 reaches a maximum near when Δk = 0 as indicated by the black line in Fig. 6. Further increase in the pressure leads to a reduction of the power in the harmonic. In general, the optimal pressure for maximum harmonic yield does not necessarily occur at Δk = 0 as absorption and the dipole response also play a role. For technical reasons, the pressures required to observe this effect are too high for us to achieve with a 300 μm nozzle in our current system. For the same conditions, the role of phase matching for the short trajectory is slightly different. At low pressures, the regions of improved phase matching are far away from the focus. As the pressure is increased, better phase matching begins to occur and the optimum harmonic generation is achieved at the focus.
In our intensity regime, the detailed pressure and position dependence shown in Fig. 6 is slightly different for each harmonic and intensity for pressures below 100–200 Torr. However, for higher pressures, the long and short trajectory signals tend to converge to a similar behavior and are maximum near the focus. The optimized harmonic signal predicted by the theory is the same for all nozzle sizes we tested experimentally, which is a consequence of strong re-absorption and a limited coherence length through much of the focal volume. With a 150 μm nozzle we have compared the model’s prediction of the harmonic power increase over a backing (interaction region) pressure range of 50 (7.5) to 300 (44.5) Torr with an experimental measurement and the model agrees within a few percent. Following the lead of Fig. 6, extrapolation to even higher pressures to optimize harmonic power will eventually breakdown as the plasma will begin to alter the cavity resonances [11, 12]. Therefore, it is highly desirable in a fsEC to introduce gas only where harmonics are produced most efficiently due to the deleterious effects of plasma in the cavity. It becomes more difficult as a practical matter to use small diameter nozzles because of how rapidly the pressure drops from its maximum value at the outlet of the nozzle, as shown in Fig. 3. This is the focus of ongoing work.
5. Conclusion
We have explored the generation of near-threshold harmonics generated in Xe in the tightly focused geometry of a femtosecond enhancement cavity near 800 nm. We have shown that a simplified, on-axis theory contains much of the physics we observe in the laboratory for harmonics H7–H13 (11–20 eV). For the range of intensities and pressures we can achieve in the lab, harmonics H7 and H9 appear to be dominated by short trajectories while harmonics H11 and H13 appear to be long trajectory dominated. We observe this behavior qualitatively in the far-field beam profiles of the harmonics imaged on a phosphor screen, as well as in the measurements of the total power in the harmonic beams derived from these images.
From the good agreement between theory and experiment, we can attempt to use the theory to guide our optimization of the harmonic generation for use in upcoming experiments. In some applications (such as future EUV metrology) it may be desirable to completely separate the signals from the long and short trajectories. This can be accomplished with appropriate placement and pressurization of the gas nozzle and spatial filter in the far field. In other experiments maximizing the photon flux may be most important, and can be accomplished with a small, highly pressurized nozzle at the focus. Of course, in the case of harmonic generation in fsEC’s, we must also consider the interaction of laser field circulating in the fsEC with the gas and the generated plasma. This means that it may not be possible to simultaneously maintain high pressure and high intensity in the cavity; ultimately it will be important to deliver the gas to the focus only in the regions that generate harmonics most efficiently, ideally with a transit time through the driving field focus shorter than the pulse spacing.
Acknowledgments
This research is supported by Natural Science and Engineering Research Council (NSERC), NRC-NSERC-BDC Nanotechnology Initiative, Canadian Foundation for Innovation, and British Columbia Knowledge Development Fund.
References and links
1. F. Lindner, W. Stremme, M. G. Schätzel, F. Grasbon, G. G. Paulus, H. Walther, R. Hartmann, and L. Strüder, “High-order harmonic generation at a repetition rate of 100 kHz,” Phys. Rev. A 68, 013814 (2003). [CrossRef]
2. J. Boullet, Y. Zaouter, J. Limpert, S. Petit, Y. Mairesse, B. Fabre, J. Higuet, E. Mével, E. Constant, and E. Cormier, “High-order harmonic generation at a megahertz-level repetition rate directly driven by an ytterbium-doped-fiber chirped-pulse amplification system,” Opt. Lett. 34, 1489–1491 (2009). [CrossRef] [PubMed]
3. A. Vernaleken, J. Weitenberg, T. Sartorius, P. Russbueldt, W. Schneider, S. L. Stebbings, M. F. Kling, P. Hommelhoff, H.-D. Hoffmann, R. Poprawe, F. Krausz, T. W. Hänsch, and T. Udem, “Single-pass high-harmonic generation at 20.8 MHz repetition rate,” Opt. Lett. 36, 3428–3430 (2011). [CrossRef] [PubMed]
4. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005). [CrossRef] [PubMed]
5. A. Mikkelsen, J. Schwenke, T. Fordell, G. Luo, K. Klunder, E. Hilner, N. Anttu, A. A. Zakharov, E. Lundgren, J. Mauritsson, J. N. Andersen, H. Q. Xu, and A. L’Huillier, “Photoemission electron microscopy using extreme ultraviolet attosecond pulse trains,” Rev. Sci. Instrum. 80, 123703 (2009). [CrossRef]
6. D. Z. Kandula, C. Gohle, T. J. Pinkert, W. Ubachs, and K. S. E. Eikema, “Extreme ultraviolet frequency comb metrology,” Phys. Rev. Lett. 105, 063001 (2010). [CrossRef] [PubMed]
7. A. Cingoz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” arXiv:1109.1871 (2011).
8. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via High-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. 94, 193201 (2005). [CrossRef] [PubMed]
9. I. Hartl, T. R. Schibli, A. Marcinkevicius, D. C. Yost, D. D. Hudson, M. E. Fermann, and J. Ye, “Cavity-enhanced similariton Yb-fiber laser frequency comb: 3x1014 W/cm2 peak intensity at 136 MHz,” Opt. Lett. 32, 2870–2872 (2007). [CrossRef] [PubMed]
10. B. Bernhardt, A. Ozawa, I. Pupeza, A. Vernaleken, Y. Kobayashi, R. Holzwarth, E. Fill, F. Krausz, T. W. Hänsch, and T. Udem, “Green enhancement cavity for frequency comb generation in the extreme ultraviolet,” in Quantum Electronics and Laser Science Conference, (Optical Society of America, 2011), p. QTuF3.
11. D. R. Carlson, J. Lee, J. Mongelli, E. M. Wright, and R. J. Jones, “Intracavity ionization and pulse formation in femtosecond enhancement cavities,” Opt. Lett. 36, 2991–2993 (2011). [CrossRef] [PubMed]
12. T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Cavity Extreme Nonlinear Optics,” ArXiv e-prints (2011).
13. J. A. Hostetter, J. L. Tate, K. J. Schafer, and M. B. Gaarde, “Semiclassical approaches to below-threshold harmonics,” Phys. Rev. A 82, 023401 (2010). [CrossRef]
14. T. C. Briles, D. C. Yost, A. Cingöz, J. Ye, and T. R. Schibli, “Simple piezoelectric-actuated mirror with 180 kHz servo bandwidth,” Opt. Express 18, 9739–9746 (2010). [CrossRef] [PubMed]
15. D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett. 33, 1099–1101 (2008). [CrossRef] [PubMed]
16. P. Salières, A. L’Huillier, P. Antoine, and M. Lewenstein, “Study of the spatial and temporal coherence of high-order harmonics,” in Advances In Atomic, Molecular, and Optical Physics (1999), Vol. 41, pp. 83–142. [CrossRef]
17. J. H. Eberly, Q. Su, and J. Javanainen, “High-order harmonic production in multiphoton ionization,” J. Opt. Soc. Am. B 6, 1289–1298 (1989). [CrossRef]
18. S. Rae, X. Chen, and K. Burnett, “Saturation of harmonic generation in one- and three-dimensional atoms,” Phys. Rev. A 50, 1946–1949 (1994). [CrossRef] [PubMed]
19. M. B. Gaarde, F. Salin, E. Constant, P. Balcou, K. J. Schafer, K. C. Kulander, and A. L’Huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A 59, 1367–1373 (1999). [CrossRef]
20. A. Anders, “Recombination of a Xenon Plasma Jet,” Contrib. Plasma Phys. 27, 373–398 (1987).
21. G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64, 013409 (2001). [CrossRef]
22. http://openfoamwiki.net/index.php/TestLucaG.
23. M. Chen, M. R. Gerrity, S. Backus, T. Popmintchev, X. Zhou, P. Arpin, X. Zhang, H. C. Kapteyn, and M. M. Murnane, “Spatially coherent, phase matched, high-order harmonic EUV beams at 50 kHz,” Opt. Express 17, 17376–17383 (2009). [CrossRef] [PubMed]
24. P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997). [CrossRef]
26. D. C. Yost, T. R. Schibli, J. Ye, J. L. Tate, J. Hostetter, M. B. Gaarde, and K. J. Schafer, “Vacuum-ultraviolet frequency combs from below-threshold harmonics,” Nat. Phys. 5, 815–820 (2009). [CrossRef]
