## Near-threshold harmonics from a femtosecond enhancement cavity-based EUV source: effects of multiple quantum pathways on spatial profile and yield |

Optics Express, Vol. 19, Issue 25, pp. 24871-24883 (2011)

http://dx.doi.org/10.1364/OE.19.024871

Acrobat PDF (2381 KB)

### Abstract

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 *μ*W of power (intracavity) and > 1*μ*W (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 OSA

## 1. Introduction

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]

*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

^{4}He ionization energy with an uncertainty of 10

^{−9}[6

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]

^{−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.

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]

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]

*μ*m [9

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: 3x10^{14} W/cm^{2} 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.

^{13}W/cm

^{2}) 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.

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]

*μ*W of intracavity power at 72 nm (17 eV), of which a measured power of > 1

*μ*W or 3.6 × 10

^{11}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

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]

## 2. Experimental setup

*μ*J. 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 5×10

^{13}W/cm

^{2}. The cavity is stabilized to the oscillator by controlling the cavity length of the fsEC to match the pulse repetition period (f

*) of the laser, and the carrier-envelope offset frequency (f*

_{rep}*) 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 f*

_{ceo}*and f*

_{rep}*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 f*

_{ceo}*. 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 f*

_{ceo}*and f*

_{rep}*we do not see significant cross-talk between control loops. We achieve a control loop bandwidth of about 60 kHz on the f*

_{ceo}*control loop using a lead-filled copper PZT mount, similar to that described in [14*

_{rep}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]

*μ*m. 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

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]

*μW*for gas line pressures of 300 Torr, which corresponds to 30–100

*μW*of intracavity EUV power.

*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.

*μ*m 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.

## 3. Mathematical model

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]

*q*harmonic

^{th}*E*can be found by solving, where

_{q}*ρ*(

*z*) is the gas density and

*σ*(

*ω*) is the absorption cross-section. The (position dependent) dipole response,

_{q}*d̃*(

*ω*,

_{q}*z*), is calculated by solving the time-dependent Schrödinger equation in one dimension, which is sufficient in our intensity regime [17

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]

*V*(

*x,t*) =

*V*(

_{atom}*x*) +

*V*(

_{field}*x,t*) is the sum of the atomic and laser field potentials. The one dimensional atomic potential used is, where

*X*

_{0}= 0.91 Å dictates the depth of the potential for the active electron in xenon with an ionization energy

*I*= 12.1 eV. The potential due to the applied laser field is, where

_{p}*E*(

*t*) =

*E*

_{0}sech(

*t*/

*τ*) is the field envelope with a peak field amplitude

*E*

_{0}and pulse duration

*τ*.

*ψ*(

*x,t*), is calculated in discrete time steps Δ

*t*, 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.

*T*

_{1}and

*T*

_{2}are the beginning and end points of integration, respectively. The magnitude of the dipole spectrum at the harmonic frequency, |

*d̃*(

*ω*,

_{q}*z*)|, is calculated as a function of the peak driving field amplitude

*E*

_{0}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, Φ

*(*

_{atomic}*z*), is discussed below.

*ϕ*(

*z*) =

*qϕ*

_{1}–

*ϕ*with, where

_{q}*z*≈ 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, Φ

_{R}*(*

_{atomic}*z*) = −

*α*(

_{i}I*z*) is the intensity dependent phase of the active electron whose trajectory creates the

*q*harmonic [19

^{th}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]

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]

*α*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

_{i}*α*= 1 × 10

_{i}^{−14}and 25×10

^{−14}cm

^{2}/W for the short and long trajectories, respectively [19

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]

*E*(

_{q}*z*) for each individual electron trajectory separately below; we discuss our observations of multiple pathway harmonic generation qualitatively in the next section.

*q*= 1) and

*q*harmonic in the gas jet is, where

^{th}*ρ*

_{0}is the density at 1 atm,

*η*is the ionization fraction,

*δ*is the Xe correction from vacuum, and

_{q}*ω*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,

_{p}*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 2

*w*

_{0}≈ 24

*μ*m, for approximately 9 round trips. Although the one dimensional potential, discussed above, is sufficient for calculating the harmonic dipole response

*d̃*(

*ω*), it overestimates the ionization rate [18

_{q}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]

21. G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A **64**, 013409 (2001). [CrossRef]

^{13}W/cm

^{2}we calculated the effective ionization fraction to be

*η*< 0.1.

*z*≈ 0.5 mm, the functional dependence of

_{R}*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

*μ*m nozzle leads to a slightly different gas density distribution than the trend seen in the 50–300

*μm*nozzles.

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]

## 4. Results and discussion

### 4.1. Far-field beam profile

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]

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]

### 4.2. Position dependence of harmonic power

*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.

*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.

### 4.3. Intensity dependence of harmonic power

*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.

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]

*α*≈ 25 × 10

_{long}^{13}cm

^{2}/W, compared to

*α*≈ 80 × 10

_{long}^{13}cm

^{2}/W in Ref [26

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]

*α*, 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.

_{long}### 4.4. Effects of phase-matching

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]

*μm*, and a peak intensity of I

_{0}= 3.5 × 10

^{13}W/cm

^{2}, which is a set of parameters that demonstrates the role of phase-matching particularly well.

*k*= 0) as a solid black line in Fig. 6.

*k*= 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 Δ

*k*= 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.

*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

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]

## 5. Conclusion

## Acknowledgments

## 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 |

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. |

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. |

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 |

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. |

6. | D. Z. Kandula, C. Gohle, T. J. Pinkert, W. Ubachs, and K. S. E. Eikema, “Extreme ultraviolet frequency comb metrology,” Phys. Rev. Lett. |

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. |

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: 3x10 |

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 |

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. |

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 |

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 |

15. | D. C. Yost, T. R. Schibli, and J. Ye, “Efficient output coupling of intracavity high-harmonic generation,” Opt. Lett. |

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. |

17. | J. H. Eberly, Q. Su, and J. Javanainen, “High-order harmonic production in multiphoton ionization,” J. Opt. Soc. Am. B |

18. | S. Rae, X. Chen, and K. Burnett, “Saturation of harmonic generation in one- and three-dimensional atoms,” Phys. Rev. A |

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 |

20. | A. Anders, “Recombination of a Xenon Plasma Jet,” Contrib. Plasma Phys. |

21. | G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A |

22. | |

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 |

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 |

25. | |

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. |

**OCIS Codes**

(140.4050) Lasers and laser optics : Mode-locked lasers

(140.4780) Lasers and laser optics : Optical resonators

(190.4160) Nonlinear optics : Multiharmonic generation

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 1, 2011

Revised Manuscript: October 7, 2011

Manuscript Accepted: October 9, 2011

Published: November 21, 2011

**Citation**

T. J. Hammond, Arthur K. Mills, and David J. Jones, "Near-threshold harmonics from a femtosecond enhancement cavity-based EUV source: effects of multiple quantum pathways on spatial profile and yield," Opt. Express **19**, 24871-24883 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-24871

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### References

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