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Optics Express

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

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

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

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

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

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

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

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
zEh(z,t)=12ε0cei(Δkz)tP12LabsEh(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. Labs 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(ω0t)   ,0<t<T0         ,elsewhere
(2)
with time duration, T=2πN/ω0, where N means the number of optical cycles.

In the following, we use atomic units, where 4πε0, , 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 α1137. Defining ρ as the density of atoms (number of atoms per atomic unit volume), Eq. (1) takes the form
zEh(z,t)=(2παρ)ei(Δkz)x˙ρσ2Eh(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)12π0TEh(z,t)eiωtdt,
(4)
υ˜(ω)12πiω   0Tx¨(t)eiωtdt,
(5)
over the finite pulse duration, Eq. (3) becomes
zE˜h(z,ω)=(2παρ)ei(Δkz)υ˜(ω)ρσ(ω)2E˜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)=4παυ˜(ω)σ(ω)g(Δk,L),
(7)
where:
g(Δk,L)=ei(ΔkL)eL/(2Labs)1+2i(ΔkLabs),
(8)
and Labs=1/(ρσ). If the propagation distance is long compared to the absorption length, in general L>3Labs is sufficient, and perfect phase matching conditions are satisfied, g(Δk,L)=1, Eq. (7) approaches its absorption limited value E˜h(ω)=4παυ˜(ω)/σ(ω).

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

η=Ωω0Ω+ω0|E˜h(ω)|2dω/0|E˜(ω)|2dω,
(9)
where E˜(ω)12π0TE(t)eiωtdt.

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)=βx¨(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[0π/ω0w(E(t))dt],
(11)
with the ionization rate w(E) calculated by the Ammosov-Delone-Krainov formula [25

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

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=25ω02α2|g(Δk,L)|2E02Ω2σ2(Ω)1β4(N1)(1β4)N|1+βeiπ(1Ωω0)|2B(Ω),
(12)
where

B(Ω)=|3π2ω05π2ω0x¨(t)eiΩtdt|2 .
(13)

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

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

,10

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/2ω0<t<5π/2ω0. The high harmonics accounted for occur in N1 full cycles, which is the reason for the factor N1 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.

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 conditiontSn(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, ΩcutIP+3.17UP, with the ponderomotive energy UP(E0/2ω0)2, the two trajectories merge. At this point of degeneracy (i.e. cutoff) t2Sn(t)=0, and as a consequence an expansion of the action up to 3rd order is necessary:
Sn(t)=S¯n+Ω(ttan)+t2Sn(ttan)22+t3Sn(ttan)36,
(15)
where tan is the arrival time of each trajectory and S¯n=S(tan,tbn). 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¯cutΩtacut+t3Sn(ttacuf)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 tbcut1.88/ω0 and tacut5.97/ω0. Accordingly, the final expression for the efficiency at the cutoff region can be written as:
η=0.02362Ipω05   |arec|2|g(Δk,L)|2E016/3Ωcutoff2σ2(Ωcutoff)1β4(N1)(1β4)N|1+β|2κ0w[E(tbcutoff)],
(16)
where κ0=|a(tbcut)a(tacut)|2 accounts for the intra-cycle depletion of the ground state [28

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 2ω0. 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/(E05Ωcut2)~UP5/2(IP+3.17UP)2~UP9/2 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 ~UP9/2. This demonstrates how sensitive the conversion efficiency scales with drive wavelength, i.e. ~λ9, if cutoff extension is the goal to achieve.

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 β1 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.

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/2UP<<1, a prerequisite for validity of the TSM. This is the case for the experiments reported in Ref. 7 and Ref. 21. Figure 1
Fig. 1 (a) Recombination amplitude, arec(Ω), and (b) absorption cross section, σ(Ω), for different noble gases from [20] and [29], respectively.
shows the values used for the recombination amplitude, arec(Ω), and the absorption cross section, σ(Ω), taken from Ref. 20 and Ref. 29, respectively.

A third case is taken from Ref. 7

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 1×105 (17th harmonic) and 1×107 (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 5×1014W/cm2 for 800 nm and 8×1014W/cm2 for 400 nm. The calculated efficiencies from Eq. (17) are shown in Fig. 3
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.
and are indeed 1×105 and 1×107for the given harmonics. Notice, that no fitting procedure was used, just the direct application of Eq. (17).

As a final comparison, in Fig. 4
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.
, 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.

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

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

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

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

ΔkPlasma=ΩωP2/(2cω02).
(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 |arec(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)
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, Ωcut. (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.
as a function of drive wavelength and cutoff energy (IP+3.17UP). A global maximum for Ne efficiency is clearly observed around λ0=1.2μm and Ωcut=451eV. 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 arec increases while the absorption cross section σ decreases, as shown in Fig. 1, compensating almost completely the reduction due to the front factor of UP9/2 scaling the single-atom response. In particular the absorption cross section decreases more than two orders of magnitude for that range. In Fig. 6
Fig. 6 Neon HHG efficiency at the cutoff region, using Eq. (16), as a function of the driven wavelength, λ0, and the cutoff energy, Ωcut. (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.
, 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 8×106 was reached for the HHG efficiency at λ0=0.6μm and Ωcut=148eV. In Fig. 6(b), the plasma and neutral atom phase mismatch is included to the problem. In this final case, the maximum value of 2.7×107 was reached for the HHG efficiency at λ0=0.8μm and Ωcut=107eV.

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.

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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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  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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