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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 13 — Jun. 22, 2009
  • pp: 11217–11229

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Analytic scaling analysis of high harmonic generation conversion efficiency

E. L. Falcão-Filho, V. M. Gkortsas, Ariel Gordon, and Franz X. Kärtner  »View Author Affiliations


Optics Express, Vol. 17, Issue 13, pp. 11217-11229 (2009)
http://dx.doi.org/10.1364/OE.17.011217


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Abstract

Closed form expressions for the high harmonic generation (HHG) conversion efficiency are obtained for the plateau and cutoff regions. The presented formulas eliminate most of the computational complexity related to HHG simulations, and enable a detailed scaling analysis of HHG efficiency as a function of drive laser parameters and material properties. Moreover, in the total absence of any fitting procedure, the results show excellent agreement with experimental data reported in the literature. Thus, this paper opens new pathways for the global optimization problem of extreme ultraviolet (EUV) sources based on HHG.

© 2009 OSA

1. Introduction

For more than a decade, high harmonic generation (HHG) is pursued as a promising route towards a compact coherent short wavelength source in the XUV region [1

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent X-rays in the water window using 5-femtosecond laser pulses,” Science 278(5338), 661–664 (1997). [CrossRef]

–3

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: source of coherent kiloelectronvolt X-rays,” Nature 433(7026), 596–596 (2005). [CrossRef] [PubMed]

]. Currently HHG is the only experimentally proven method for generating coherent XUV radiation and enables attosecond pulses [4

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]

]. Seeding of next generation free electron lasers in the XUV with high harmonics in combination with high gain harmonic generation is assumed to be a viable technique to transfer the coherence properties of the HHG seed to the hard x-ray range [5

L.-H. Yu, M. Babzien, I. Ben-Zvi I, L. F. DiMauro, A. Doyuran, W. Graves, E. Johnson, S. Krinsky, R. Malone, I. Pogorelsky I, J. Skaritka, G. Rakowsky, L. Solomon, X. J. Wang, M. Woodle, V. Yakimenko V, S. G. Biedron, J. N. Galayda, E. Gluskin, J. Jagger, V. Sajaev V, and I. Vasserman I, “High-gain harmonic-generation free-electron laser,” Science 289(5481), 932–934 (2000). [CrossRef] [PubMed]

]. As a consequence, the laboratory concept of an HHG based XUV source is rapidly evolving to reality accelerated by advances in high power femtosecond laser systems and in HHG techniques, such as quasi-phase-matching HHG [6

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003). [CrossRef] [PubMed]

], two-color-HHG [7

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]

] and most recently cavity-based HHG [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(19), 193201 (2005). [CrossRef] [PubMed]

]. In this context, quantitative scaling and optimization of HHG-based source characteristics is of key importance to accelerate the development and capabilities of this emerging research field.

In parallel to the experimental progress, a considerable effort has been devoted to the theoretical modeling and scaling of HHG [9

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

–15

G. Tempea and T. Brabec, “Optimization of high-harmonic generation,” Appl. Phys. B 70, S197 (2000).

]. Accurate quantitative simulation of the HHG efficiency including propagation effects is time-consuming, making a systematic parameter study impossible. Indeed, up to the present it is not entirely clear how the HHG efficiency scales for different experimental conditions, how far current results are from theoretical limits, and how to proceed to construct a maximally efficient HHG source for a particular range of harmonics. For example, over the last years, the role of the driving frequency, ω0, to HHG scaling has received great attention [16

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef] [PubMed]

–19

V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15(23), 15351–15364 (2007). [CrossRef] [PubMed]

]. Dependences of HHG efficiency between ω05 and ω06 are being obtained from numerical simulations using the time dependent Schrödinger Eq. (16) including the single-atom response only. Preliminary experimental results [17

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4(5), 386–389 (2008). [CrossRef]

,18

C. Vozzi, F. Calegari, F. Frassetto, E. Benedetti, M. Nisoli, G. Sansone, L. Poletto, P. Villoresi, and S. Stagira, “Generation of high-order harmonics with a near-IR self-phase-stabilized parametric source,” Proceedings of Conference on Ultrafast Phenomena, FRI2.2 (2008).

] are supporting these simulations. General scaling considerations concerning the scaling with drive frequency, including laser wavelength and pressure were presented recently [19

V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15(23), 15351–15364 (2007). [CrossRef] [PubMed]

], however, no expression for the HHG efficiency and its scaling as a function of all experimentally relevant parameters was presented. This paper gives the first, to our knowledge, closed form analytical expressions for HHG conversion efficiencies both for the plateau region and the cutoff region including both laser and material parameters. Also focusing conditions and related effects, such as intensity dependent phase mismatch, could be included, but is postponed for future work. For the purpose of this paper we consider plane wave geometry.

To this aim we make two simplifying assumptions. First, we use the single-active-electron (SAE) approximation, which is widely adopted. Second, multielectron effects are partially included using the recombination amplitude computed via the Hartree-Slater potential approach [20]. The HHG driven by a plane wave source includes besides the single-atom response the 1-D propagation effects due to absorption and phase mismatch caused by the neutral gas and plasma generation. Experimentally, HHG-setups either use free space focusing into a gas jet or cell or hollow fiber geometry. Considering a loose focusing regime [21

E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of strong optical field in soft X-ray region by using high-order harmonics,” IEEE J. Sel. Top. Quantum Electron. 10(6), 1315–1328 (2004). [CrossRef]

] or hollow fiber geometry [6

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003). [CrossRef] [PubMed]

], the phase mismatch due to the Gouy-phase shift and the dipole phase is minimized or absent or are replaced by waveguide dispersion which can also be included in the 1-D model. Thus, even this simplistic model is expected to give upper bounds for the HHG efficiency, as reported in the literature [21

E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of strong optical field in soft X-ray region by using high-order harmonics,” IEEE J. Sel. Top. Quantum Electron. 10(6), 1315–1328 (2004). [CrossRef]

–23

M. Schnürer, Z. Cheng, M. Hentschel, G. Tempea, P. Kálmán, T. Brabec, and F. Krausz, “Absorption-limited generation of coherent ultrashort soft-X-ray pulses,” Phys. Rev. Lett. 83(4), 722–725 (1999). [CrossRef]

]. In this paper, the discussion is also restricted to the adiabatic regime which holds for driver pulses as short as 4-optical-cycles excluding strong carrier envelope phase effects [11

P. Salières, P. Antoine, A. de Bohan, and M. Lewenstein, “Temporal and spectral tailoring of high-order harmonics,” Phys. Rev. Lett. 81(25), 5544–5547 (1998). [CrossRef]

].

2. Derivation

The one-dimensional propagation equation commonly used for HHG [24

M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]

] is
∂z Eh(z,t)=− 1 2 ε0c e −i(Δk⋅z) ∂tP− 1 2 L abs Eh(z,t)
(1)
where z is the position coordinate along the propagation direction, Δk is the phase mismatch, t is the retarded time appropriate for describing propagation at the speed of light in vacuum, Eh is the electric field of the harmonics, and P is the polarization induced in the medium. L abs is the absorption length, which later will become frequency dependent to take frequency dependent absorption into account. The driving field is assumed to be a top-hat pulse, represented by
E(t)= { E0sin( ω  0 t)   , 0<t<T 0         , elsewhere
(2)
with time duration, T=2π N/ ω0, where N means the number of optical cycles.

In the following, we use atomic units, where 4π ε0, ℏ, the electron mass, me, and its charge, e, are set to unity, and the speed of light in vacuum equals the inverse fine structure constant α −1≈137. Defining ρ as the density of atoms (number of atoms per atomic unit volume), Eq. (1) takes the form
∂z Eh(z,t)=−(2π α ρ)  e −i(Δk⋅z) x˙− ρ σ2 Eh(z,t)
(3)
where σ is the absorption cross section, x is the dipole moment of a single atom, and x˙≡ ∂tx. With the Fourier transforms of the harmonic field Eh and the dipole velocity derived from the dipole acceleration x¨:
E˜h(z,ω)≡ 1 2π ∫0T Eh(z,t) e iω tdt
(4)
υ˜(ω)≡ 1 2πiω    ∫0T x¨(t) e iω tdt
(5)
over the finite pulse duration, Eq. (3) becomes
∂z E˜h(z,ω)=−(2π α ρ) e −i(Δk⋅z)  υ˜(ω)− ρ σ(ω)2 E˜h(z,ω)
(6)
where a frequency dependent absorption cross section is introduced. Note, that we assumed that   υ˜(ω) does not depend on z, i.e. the changes in amplitude and phase of the driving pulse are small over the medium length L. The results can also be applied to a weakly focused Gaussian beam for L<< z0, where z0 is the Rayleigh length. In this case the solution to Eq. (6) is:
E˜h(ω)=− 4π α  υ˜(ω) σ(ω)g(Δk,L)
(7)
where:
g(Δk,L)= e i(Δk⋅L)− e −L/(2⋅ L abs) 1+2i(Δk⋅ L abs)
(8)
and L abs=1/(ρ σ). If the propagation distance is long compared to the absorption length, in general L>3 L abs is sufficient, and perfect phase matching conditions are satisfied, g(Δk,L)=1, Eq. (7) approaches its absorption limited value E˜h(ω)=−4π α  υ˜(ω)/σ(ω).

The conversion efficiency into a given (odd) harmonic of ω0, whose frequency is denoted by Ω, is given by

η= ∫ Ω− ω0 Ω+ ω0 | E˜h(ω) |2dω/   ∫0∞ | E˜(ω) |2dω
(9)
where
E˜(ω)≡ 1 2π ∫0T E(t) e iω tdt

In order to evaluate the numerator in Eq. (9) we note that x¨(t), the dipole acceleration of a single atom has the following property:
x¨(t+π/ω)=−β  x¨(t)
(10)
with 0<β<1. The minus sign on the right hand side of Eq. (10) is due to the sign change in the driving field. Note, that β accounts for the depletion of the ground state amplitudes during each half period defined by |a(t+π/ω) |2=β|a(t) |2 where |a(t) |2 denotes the probability to find the atom in the ground state. Thus, β= |  a(π/ω)|2 or, in other words:
β=exp [ − ∫0 π/ ω0 w(E(t)) dt]
(11)
with the ionization rate w(E) calculated by the Ammosov-Delone-Krainov formula [25

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field,” Sov. Phys. JETP 64, 1191–1194 (1986).

]. Furthermore, only the first recombination event is taken into account, because quantum diffusion greatly reduces the contribution of multiple returns [10

M. Lewenstein, Ph. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]

]. Under these assumptions and using Eq. (10) in Eq. (7) and substituting the result into Eq. (9) we obtain
η= 25ω  02  α2 | g(Δk,L)|2 E02  Ω2  σ2(Ω)⋅ 1− β 4(N−1) (1− β4)N⋅  |  1+β⋅  e iπ ( 1− Ω ω0) |2B(Ω)
(12)
where

B(Ω)= | ∫ 3π 2  ω0 5π 2  ω0 x¨ (t) e i Ω tdt |2
(13)

The choice for the integration interval in (13) follows from the TSM [9

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

,10

M. Lewenstein, Ph. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]

], showing that the dominant contribution to the high harmonics occurs in the interval 3π/2  ω0<t<5π/2  ω0. The high harmonics accounted for occur in N−1 full cycles, which is the reason for the factor N−1 instead of N in the exponent of β in Eq. (12). The last quarter of a cycle in the N-cycle pulse is neglected to keep the expression simple. Comparison with a full numerical simulation considering a Gaussian pulse at the end of the paper shows that this approximation even yields good results for a four-cycle non-flat-top pulses.

As discussed above, the SAE approximation is adopted, where the atoms are modeled by a single electron in an effective potential [Eq. (1) in Ref. [26

A. Gordon and F. X. Kärtner, “Quantitative modeling of single atom high harmonic generation,” Phys. Rev. Lett. 95(22), 223901 (2005). [CrossRef] [PubMed]

]. In order to obtain a closed-form expression for the efficiency, we use the improved version of the TSM (ITSM) for x¨(t) [26

A. Gordon and F. X. Kärtner, “Quantitative modeling of single atom high harmonic generation,” Phys. Rev. Lett. 95(22), 223901 (2005). [CrossRef] [PubMed]

].
x¨(t)= π −1/2  e −iπ/4 ω0 3/2 ( 2  IP)  1/4 ∑n a ( t bn)   a (t)  w ( E ( t bn)) E ( t bn)  [ ω0 ( t−t bn)/(2π)]  3/2  a rec( kn)  e −i  Sn(t)     
(14)
where IP is the ionization potential, t bn is the birth time of an electron that returns to the origin at time t and n denotes the trajectory. Sn(t)= Sn ( t,t bn) is the semiclassical action and a rec( kn)=− 〈 0| ∂x V(r) | kn〉 is the recombination amplitude, Eq. (7) of Ref. [20], which is obtained from the Hartree-Slater potential (see Ref. [27

R. Santra and A. Gordon, “Three-step model for high-harmonic generation in many-electron systems,” Phys. Rev. Lett. 96(7), 073906 (2006). [CrossRef] [PubMed]

] for a more detailed explanation). The term kn(t) is the momentum upon return from of the nth trajectory which is related to the HHG frequency by Ωn= IP+ kn2/2. Notice, that in order to show the HHG scaling with drive wavelength due to quantum diffusion, we pulled the factor ω0 3/2 in front leaving the denominator [ ω0(t−t bn)/(2π)] 3/2 dimensionless and largely invariant to drive wavelength.

To evaluate B(Ω), the saddle point method is used. All the terms in Eq. (13), using Eq. (14), are slowly varying except the phase containing the classical action. The stationary phase approximation will give the condition ∂t Sn(t)=Ω, implying the transition energy of the recolliding electron has to be equal to Ω. This condition is fulfilled twice during each half cycle and is referred as short and long trajectories. As Ω increases approaching the cutoff frequency, Ω cut≡ IP+3.17  UP, with the ponderomotive energy UP≡ ( E0/2 ω  0)2, the two trajectories merge. At this point of degeneracy (i.e. cutoff) ∂t2 Sn(t)=0, and as a consequence an expansion of the action up to 3rd order is necessary:
Sn(t)= S¯n+Ω⋅ (t−t an)+ ∂t2 Sn (t−t an)22+ ∂t3 Sn (t−t an)36
(15)
where t an is the arrival time of each trajectory and S¯n=S ( t an,t bn). Considering the expansion up to third order, as shown in Eq. (15), it is not possible to find a closed analytical formula for B(Ω). However, by focusing our analysis separately to the plateau region or cutoff region, where either the second or third order term is dominant a closed form expression is achieved.

Thus for the case of cutoff the total phase in the integrand of B(Ω) is given by φ= S¯ cut−Ω ⋅t a cut+ ∂t3 Sn⋅ (t−t a cuf)3/6, where the first two terms are constants and B(Ω) is reduced to an Airy function which can be evaluated numerically. The respective birth and arrival times are t b cut≈1.88/ ω0 and t a cut≈5.97/ ω0. Accordingly, the final expression for the efficiency at the cutoff region can be written as:
η=0.0236 2 Ipω  05    |   a rec|2  | g(Δk,L)|2 E0 16/3  Ω cutoff2  σ2( Ω cutoff) 1− β 4 (N−1) (1− β4)N  |  1+β |2 κ0 w [ E(t b cutoff)]
(16)
where κ0=|a(t b cut) a(t a cut) |2 accounts for the intra-cycle depletion of the ground state [28

A. Gordon and F. X. Kärtner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Express 13(8), 2941–2947 (2005). [CrossRef] [PubMed]

]. The efficiency at the cutoff region, given by Eq. (16), scales with a factor of ω05. A cubic dependence with ω0 is due to quantum diffusion. An additional factor of ω0 comes from the fact that we are considering the conversion efficiency into a single harmonic, and the bandwidth it occupies is 2 ω0. The fifth ω0 comes from the denominator in Eq. (9): The energy carried by a cycle of the driving laser field scales like its duration 2π/ ω0 at a given electric field amplitude. Note, that in Eq. (16) the factor ω05/ ( E05  Ω cut2)~ UP −5/2 ( IP+3.17  UP) −2~ UP −9/2 for ponderomotive potentials large compared to the ionization potential. Thus by shifting the cutoff to shorter wavelength by increasing the ponderomotive potential via the laser wavelength has a price in efficiency that scales at constant field with ~ UP −9/2. This demonstrates how sensitive the conversion efficiency scales with drive wavelength, i.e. ~ λ −9, if cutoff extension is the goal to achieve.

In the plateau region, each harmonic has mainly contributions from two trajectories and, if the harmonic energy is not close to the cutoff, the third order term in Eq. (15) can be neglected. Then, the overall phase exhibits a quadratic dependence in t, and B(Ω) can be expressed by the error function which are evaluated numerically. The final expression for the efficiency in the plateau region is
η=0.0107 2  Ip ω05  | a rec|2  | g(Δk,L)|2 E04  Ω2  σ2(Ω) 1− β 4(N−1) ( 1− β4)N |  1+β  e iπ ( 1− Ω ω0)|2                × |     a(t bs)   a ( t as)  w ( E(t bs)) sin( ω0 t bs)  [ ω0 ( t as−t bs)/(2π)]  3/2 e −i(  S¯s−Ω t as) | ∂t2 Ss| + a(t bl)   a ( t al)  w ( E(t bl)) sin( ω0 t bl)  [ ω0 ( t al−t bl)/(2π)]  3/2 e −i(  S¯l−Ω t al− π2) | ∂t2 Sl||2
(17)
where, ( t bs,t as), ( t b  l,t a  l) and S¯ s,l=S( t a s, l,t b s, l) are the pairs of birth/arrival times and the corresponding semiclassical action for the short and the long trajectory of a particular harmonic, respectively. Equation (17) is valid for harmonic energies Ω in the plateau region, satisfying the condition 1<(Ω− IP)/ UP<3.1. The upper limit is to keep the parabolic approximation of the classical action valid and the lower limit is related to approximate value for the error function used in B(Ω) in Eq. (13).

Two interference mechanisms are built into Eq. (17). They are described by the last two terms. One is the interference between each half cycle, which under the condition of β≈1 allows only odd harmonics, and the other is the interference between long and short trajectories. Notice, that intense pulses may break the symmetry due to substantial ionization between half-cycles, and then even harmonics can occur.

Comparing Eqs. (16) and (17), it is observed that both present the same term of ω05 in the numerator and a term of E04 in the denominator. However, Eq. (17) contains the second derivative of the action, ∂t2S, which is related to the chirp of the attosecond pulses emitted in each recollision or in other words, is related to the temporal spreading of HHG frequencies during emission. Moreover, as ∂t2S∝ E02/ ω0, the overall efficiency exhibits an effective dependence proportional to ω06/ E06~ UP −3. In summary, the scaling of HHG efficiency with the driving frequency is ω05 at the cutoff and ω06 at the plateau region for fixed harmonic wavelength.

3. Discussion

Equations (16) and (17) provide closed-form expressions for the HHG conversion efficiency into a single harmonic Ω at the cutoff frequency and in the plateau region, respectively. In the following, our predicted efficiencies are compared with experimental data in the literature. Experimental data are chosen where the corresponding Keldysh parameter γ= IP/2 UP<<1, a prerequisite for validity of the TSM. This is the case for the experiments reported in Ref. 7 and Ref. 21. Figure 1 shows the values used for the recombination amplitude, a rec(Ω), and the absorption cross section, σ(Ω), taken from Ref. 20 and Ref. 29, respectively.

Fig. 1 (a) Recombination amplitude, a rec(Ω), and (b) absorption cross section, σ(Ω), for different noble gases from [20] and [29

Lawrence Berkeley National Laboratory, (http://henke.lbl.gov/optical_constants/).

], respectively.

Figure 2 shows the prediction for HHG efficiency from Eq. (17) for the experimental situation in Ref. 21

E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of strong optical field in soft X-ray region by using high-order harmonics,” IEEE J. Sel. Top. Quantum Electron. 10(6), 1315–1328 (2004). [CrossRef]

, where HHG was carried out in neon and argon with 35 fs long ( N=13), 800 nm pulses. In neon under perfect phase matching and absorption limited conditions, | g(Δk,L)|2=1, a maximum efficiency of 0.35× 10 −6 is calculated for the 59th harmonic. Note, that the oscillations observed in Fig. 2 are related to the interference between long and short trajectories, which sensitively depend on pulse shape and might be different in the actual experiment, however, the maximum efficiency does not depend strongly on the pulse shape. In argon, for an interaction length, L=10cm, absorption length, L abs=6.8cm, and phase mismatch, Δk=0.0667cm−1, | g(Δk,L)|2=0 .261, a maximum conversion efficiency into the 27-th harmonic of 1.2× 10 −5is calculated from Eq. (17) as shown in Fig. 2(b). The measured maximum efficiencies stated in Ref. 21 are ρe for Ne and 1.3× 10 −5for Ar, which compares very well with the maximum efficiency calculated and shown in Fig. 2(a) and Fig. 2(b) given the simplicity of the model.

Fig. 2 η from Eq. (17), as function of E0, driven by pulses of 11 cycles at 800nm. (a) For the 59-th harmonic generate in neon under perfect phase match and absorption limited conditions. (b) For the 27-th harmonic generate in argon under the conditions of L=10 cm, L abs=6.8 cm and Δk=0.0667 cm−1.

A third case is taken from Ref. 7

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]

, which presents maximum measured efficiency values of 1× 10 −5 (17th harmonic) and 1× 10 −7 (35th–47th harmonic) for He, pumped by pulses of 27 fs at 400nm (20 cycles) and 800nm (10 cycles), respectively. The detailed experimental conditions are not quantified, however, we use the stated electric field strengths and assume perfect phase matching and absorption limited propagation. The laser intensity used was 5× 10 14W/cm2 for 800 nm and 8× 10 14W/cm2 for 400 nm. The calculated efficiencies from Eq. (17) are shown in Fig. 3 and are indeed 1× 10 −5 and 1× 10 −7for the given harmonics. Notice, that no fitting procedure was used, just the direct application of Eq. (17).

Fig. 3 Efficiency spectrum for HHG in He obtained from Eq. (17). (a) Driven by electric field E0 = 0.125 a.u. and pulses of 10 cycles at 810 nm. (b) Driven by E0 = 0.151 a.u. and pulses of 20 cycles at 420 nm.

As a final comparison, in Fig. 4 , the results calculated from Eqs. (16) and (17) for a square pulse are shown together with those for a Gaussian pulse using the ITSM without the saddle point approximation, i.e. by solving the integrals in Eqs. (9) and (14) numerically. Again excellent agreement is obtained even for pulses as short as 4 cycles. For pulses bellow 4 cycles the HHG spectrum starts to show a strong dependence on the carrier envelope phase and the agreement with Eqs. (16) and (17) deteriorates in the cutoff region.

Fig. 4 Comparison of Eqs. (16)-(17), which are derived considering a top-hat pulse, and the numerical simulation considering a Gaussian pulse. Efficiencies calculated considering Ne driven by electric field E0 = 0.105 and λ0 = 800 nm. (a) For pulses of τ FWHM = 13 cycles. (b) For pulses of τ FWHM = 4 cycles. The red lines and the stars represent respectively the values obtained using Eqs. (16) and (17), and the blue lines are the numerical simulation for Gaussian pulses.

Although most of the recent scaling discussions are focused on the drive frequency [16

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef] [PubMed]

–19

V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15(23), 15351–15364 (2007). [CrossRef] [PubMed]

], other parameters may be equally important for maximizing HHG, such as the ionization level of the medium which determines the plasma dispersion and with it the phase matching [30

T. Popmintchev, M. C. Chen, O. Cohen, M. E. Grisham, J. J. Rocca, M. M. Murnane, and H. C. Kapteyn, “Extended phase matching of high harmonics driven by mid-infrared light,” Opt. Lett. 33(18), 2128–2130 (2008). [CrossRef] [PubMed]

]. The phase mismatch due to plasma generation is a function of the driving frequency and electric field, and critically determines the overall efficiency of the HHG process.

The phase mismatch due to plasma generation, as a function of the driving frequency and electric field, is given by

Δ k Plasma=− Ω  ωP2/(2c ω02)
(18)

Here, ωP is the plasma frequency, which is proportional to the electron density, ρe. It is clear from Eq. (18) that the plasma contribution to phase mismatch increases for longer drive wavelength. The plasma generation can be reduced by lowering the field strength, E0, which will have a direct impact on Eqs. (16) and (17) and therefore needs to be considered in a more general analysis.

Besides the single-atom response, the other major contribution to be considered in the wavelength scaling is the medium characteristics, such as, recombination amplitude and absorption cross section, represented by |   a rec(Ω)|2/ σ2(Ω). This quantities exhibit a strong wavelength dependence which can have an important role if cutoff extension is the goal. Thus, in order to illustrate the significance of that statement we consider absorption limited HHG in neon and ask what the optimum drive wavelength is to achieve maximum conversion efficiency in the cutoff region. Equation (16) is used to compute the HHG efficiency at cutoff. The result is displayed in Fig. 5(a) as a function of drive wavelength and cutoff energy ( IP+3.17  UP). A global maximum for Ne efficiency is clearly observed around λ0=1.2 μ m and Ω cut=451 e V. It is at first surprising that the maximum efficiency shifts for different driver wavelength λ0, but the maximum efficiency itself does not strongly depend on the driver wavelength, as one may expect from the scaling of the single-atom response. This behavior is also reproduced using a Gaussian pulse and numerical evaluation of Eq. (9) using the ITSM at constant field amplitude E0 while varying ω0, as shown in Fig. 5(b). The reason is that in the range from 30 to 800 eV for Ne, the recombination amplitude a rec increases while the absorption cross section σ decreases, as shown in Fig. 1, compensating almost completely the reduction due to the front factor of UP −9/2 scaling the single-atom response. In particular the absorption cross section decreases more than two orders of magnitude for that range. In Fig. 6 , the same problem is considered but more constrains are imposed. In Fig. 6(a) the efficiency at cutoff for a gas cell of 5 mm length at 1 bar of pressure and perfect phase matching is assumed. In this case, the maximum value of 8× 10 −6 was reached for the HHG efficiency at λ0=0.6 μ m and Ω cut=148 e V. In Fig. 6(b), the plasma and neutral atom phase mismatch is included to the problem. In this final case, the maximum value of 2.7× 10 −7 was reached for the HHG efficiency at λ0=0.8 μ m and Ω cut=107 e V.

Fig. 5 (a) Neon HHG efficiency at the cutoff region, using Eq. (15), as a function of the driver wavelength, λ0, and the cutoff energy, Ω cut. (b) Full spectrum obtained for Ne, considering a Gaussian pulse with E0=0.16 a.u. for different driver wavelengths. For both cases, perfect phase matching | g(Δk,L)|2= 1 and a 5-cycle-driver-pulse were assumed.
Fig. 6 Neon HHG efficiency at the cutoff region, using Eq. (16), as a function of the driven wavelength, λ0, and the cutoff energy, Ω cut. (a) Considered a 5-cycle-driver-pulse, L=5 mm and pressure 1 bar. (b) Same as (a) but also considering the plasma and neutral atom phase mismatching.

4. Conclusion

In summary, closed form expressions for the HHG conversion efficiency using square shaped pulses are presented for the plateau and cutoff regions. It is shown that the computed efficiencies are also in good agreement with Gaussian shaped pulses. Based on these expressions, absolute theoretical HHG conversion efficiencies were calculated for different gases under different laser pumping conditions. The calculated efficiencies are in good agreement with experimental results obtained from different groups under different experimental conditions. The presented formulas simplify the HHG optimization problem considerably and enable a complete HHG scaling analysis. As an example the efficiency at cutoff for Neon was optimized for plane wave geometry and the global maximum on the Ne efficiency was obtained under various conditions. Provide that most of the performed analysis were considering neon, the formulas and concepts discussed in this paper can be applied for any atomic or molecular gas. For this purpose it is just necessary to use the respective gas properties such as dispersion, ionization potential, absorption cross-section and recombination amplitude. A more comprehensive analysis including other gases will be published in a forthcoming paper. Due to its generality and simplicity, the theory presented in this paper should have a significant impact on the development of HHG based EUV sources.

Acknowledgments

This work was supported in part by U.S. Air Force Office of Scientific Research (AFOSR) grants FA9550-06-1-0468 and FA9550-07-1-0014, through the Defense Advanced Research Projects Agency (DARPA) Hyperspectral Radiography Sources program and SRC at University of Wisconsin. E. L. Falcão-Filho acknowledges support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.

References and links

1.

Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent X-rays in the water window using 5-femtosecond laser pulses,” Science 278(5338), 661–664 (1997). [CrossRef]

2.

Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft X rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79(16), 2967–2970 (1997). [CrossRef]

3.

J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: source of coherent kiloelectronvolt X-rays,” Nature 433(7026), 596–596 (2005). [CrossRef] [PubMed]

4.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]

5.

L.-H. Yu, M. Babzien, I. Ben-Zvi I, L. F. DiMauro, A. Doyuran, W. Graves, E. Johnson, S. Krinsky, R. Malone, I. Pogorelsky I, J. Skaritka, G. Rakowsky, L. Solomon, X. J. Wang, M. Woodle, V. Yakimenko V, S. G. Biedron, J. N. Galayda, E. Gluskin, J. Jagger, V. Sajaev V, and I. Vasserman I, “High-gain harmonic-generation free-electron laser,” Science 289(5481), 932–934 (2000). [CrossRef] [PubMed]

6.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003). [CrossRef] [PubMed]

7.

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]

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(19), 193201 (2005). [CrossRef] [PubMed]

9.

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]

10.

M. Lewenstein, Ph. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]

11.

P. Salières, P. Antoine, A. de Bohan, and M. Lewenstein, “Temporal and spectral tailoring of high-order harmonics,” Phys. Rev. Lett. 81(25), 5544–5547 (1998). [CrossRef]

12.

N. H. Shon, A. Suda, and K. Midorikawa, “Generation and propagation of high-order harmonics in high-pressure gases,” Phys. Rev. A 62(2), 023801 (2000). [CrossRef]

13.

M. B. Gaarde and K. J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. 89(21), 213901 (2002). [CrossRef] [PubMed]

14.

E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61(6), 063801 (2000). [CrossRef]

15.

G. Tempea and T. Brabec, “Optimization of high-harmonic generation,” Appl. Phys. B 70, S197 (2000).

16.

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef] [PubMed]

17.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4(5), 386–389 (2008). [CrossRef]

18.

C. Vozzi, F. Calegari, F. Frassetto, E. Benedetti, M. Nisoli, G. Sansone, L. Poletto, P. Villoresi, and S. Stagira, “Generation of high-order harmonics with a near-IR self-phase-stabilized parametric source,” Proceedings of Conference on Ultrafast Phenomena, FRI2.2 (2008).

19.

V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15(23), 15351–15364 (2007). [CrossRef] [PubMed]

20.

A. Gordon, F. X. Kärtner, N. Rohringer, and R. Santra, “Role of many-electron dynamics in high harmonic generation,” Phys. Rev. Lett. 96(22), 223902 (2006). [CrossRef] [PubMed]

21.

E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of strong optical field in soft X-ray region by using high-order harmonics,” IEEE J. Sel. Top. Quantum Electron. 10(6), 1315–1328 (2004). [CrossRef]

22.

E. Constant, D. Garzella, P. Breger, E. Mével, Ch. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82(8), 1668–1671 (1999). [CrossRef]

23.

M. Schnürer, Z. Cheng, M. Hentschel, G. Tempea, P. Kálmán, T. Brabec, and F. Krausz, “Absorption-limited generation of coherent ultrashort soft-X-ray pulses,” Phys. Rev. Lett. 83(4), 722–725 (1999). [CrossRef]

24.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]

25.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field,” Sov. Phys. JETP 64, 1191–1194 (1986).

26.

A. Gordon and F. X. Kärtner, “Quantitative modeling of single atom high harmonic generation,” Phys. Rev. Lett. 95(22), 223901 (2005). [CrossRef] [PubMed]

27.

R. Santra and A. Gordon, “Three-step model for high-harmonic generation in many-electron systems,” Phys. Rev. Lett. 96(7), 073906 (2006). [CrossRef] [PubMed]

28.

A. Gordon and F. X. Kärtner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Express 13(8), 2941–2947 (2005). [CrossRef] [PubMed]

29.

Lawrence Berkeley National Laboratory, (http://henke.lbl.gov/optical_constants/).

30.

T. Popmintchev, M. C. Chen, O. Cohen, M. E. Grisham, J. J. Rocca, M. M. Murnane, and H. C. Kapteyn, “Extended phase matching of high harmonics driven by mid-infrared light,” Opt. Lett. 33(18), 2128–2130 (2008). [CrossRef] [PubMed]

OCIS Codes
(260.7200) Physical optics : Ultraviolet, extreme
(020.2649) Atomic and molecular physics : Strong field laser physics

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: May 1, 2009
Revised Manuscript: May 29, 2009
Manuscript Accepted: May 30, 2009
Published: June 19, 2009

Citation
E. L. Falcão-Filho, V. M. Gkortsas, Ariel Gordon, and Franz X. Kärtner, "Analytic scaling analysis of high harmonic generation conversion efficiency," Opt. Express 17, 11217-11229 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-11217


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References

  1. Ch. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnürer, C. Kan, M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent X-rays in the water window using 5-femtosecond laser pulses,” Science 278(5338), 661–664 (1997). [CrossRef]
  2. Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft X rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. 79(16), 2967–2970 (1997). [CrossRef]
  3. J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek, V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: source of coherent kiloelectronvolt X-rays,” Nature 433(7026), 596–596 (2005). [CrossRef] [PubMed]
  4. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef] [PubMed]
  5. L.-H. Yu, M. Babzien, I. Ben-Zvi, L. F. DiMauro, A. Doyuran, W. Graves, E. Johnson, S. Krinsky, R. Malone, I. Pogorelsky, J. Skaritka, G. Rakowsky, L. Solomon, X. J. Wang, M. Woodle, V. Yakimenko, S. G. Biedron, J. N. Galayda, E. Gluskin, J. Jagger, V. Sajaev, and I. Vasserman, “High-gain harmonic-generation free-electron laser,” Science 289(5481), 932–934 (2000). [CrossRef] [PubMed]
  6. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft x-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003). [CrossRef] [PubMed]
  7. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
  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(19), 193201 (2005). [CrossRef] [PubMed]
  9. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef] [PubMed]
  10. M. Lewenstein, Ph. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef] [PubMed]
  11. P. Salières, P. Antoine, A. de Bohan, and M. Lewenstein, “Temporal and spectral tailoring of high-order harmonics,” Phys. Rev. Lett. 81(25), 5544–5547 (1998). [CrossRef]
  12. N. H. Shon, A. Suda, and K. Midorikawa, “Generation and propagation of high-order harmonics in high-pressure gases,” Phys. Rev. A 62(2), 023801 (2000). [CrossRef]
  13. M. B. Gaarde and K. J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. 89(21), 213901 (2002). [CrossRef] [PubMed]
  14. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61(6), 063801 (2000). [CrossRef]
  15. G. Tempea and T. Brabec, “Optimization of high-harmonic generation,” Appl. Phys. B 70, S197 (2000).
  16. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98(1), 013901 (2007). [CrossRef] [PubMed]
  17. P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4(5), 386–389 (2008). [CrossRef]
  18. C. Vozzi, F. Calegari, F. Frassetto, E. Benedetti, M. Nisoli, G. Sansone, L. Poletto, P. Villoresi, and S. Stagira, “Generation of high-order harmonics with a near-IR self-phase-stabilized parametric source,” Proceedings of Conference on Ultrafast Phenomena, FRI2.2 (2008).
  19. V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15(23), 15351–15364 (2007). [CrossRef] [PubMed]
  20. A. Gordon, F. X. Kärtner, N. Rohringer, and R. Santra, “Role of many-electron dynamics in high harmonic generation,” Phys. Rev. Lett. 96(22), 223902 (2006). [CrossRef] [PubMed]
  21. E. J. Takahashi, Y. Nabekawa, H. Mashiko, H. Hasegawa, A. Suda, and K. Midorikawa, “Generation of strong optical field in soft X-ray region by using high-order harmonics,” IEEE J. Sel. Top. Quantum Electron. 10(6), 1315–1328 (2004). [CrossRef]
  22. E. Constant, D. Garzella, P. Breger, E. Mével, Ch. Dorrer, C. Le Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82(8), 1668–1671 (1999). [CrossRef]
  23. M. Schnürer, Z. Cheng, M. Hentschel, G. Tempea, P. Kálmán, T. Brabec, and F. Krausz, “Absorption-limited generation of coherent ultrashort soft-X-ray pulses,” Phys. Rev. Lett. 83(4), 722–725 (1999). [CrossRef]
  24. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: Extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]
  25. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field,” Sov. Phys. JETP 64, 1191–1194 (1986).
  26. A. Gordon and F. X. Kärtner, “Quantitative modeling of single atom high harmonic generation,” Phys. Rev. Lett. 95(22), 223901 (2005). [CrossRef] [PubMed]
  27. R. Santra and A. Gordon, “Three-step model for high-harmonic generation in many-electron systems,” Phys. Rev. Lett. 96(7), 073906 (2006). [CrossRef] [PubMed]
  28. A. Gordon and F. X. Kärtner, “Scaling of keV HHG photon yield with drive wavelength,” Opt. Express 13(8), 2941–2947 (2005). [CrossRef] [PubMed]
  29. Lawrence Berkeley National Laboratory, ( http://henke.lbl.gov/optical_constants/ ).
  30. T. Popmintchev, M. C. Chen, O. Cohen, M. E. Grisham, J. J. Rocca, M. M. Murnane, and H. C. Kapteyn, “Extended phase matching of high harmonics driven by mid-infrared light,” Opt. Lett. 33(18), 2128–2130 (2008). [CrossRef] [PubMed]

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