## Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds |

Optics Express, Vol. 22, Issue 11, pp. 13213-13233 (2014)

http://dx.doi.org/10.1364/OE.22.013213

Acrobat PDF (1607 KB)

### Abstract

The isolated attosecond pulse (IAP) generated from high-order harmonic (HH) radiation has been established as an important technique for the ultrafast optics over past decade. The applications of IAP in ultrafast processes can be greatly extended by further developing the high-intensity IAP. Here, we theoretically propose to shape a two-color field by performing peak amplitude-wavelength analysis. It is found that a 240-as IAP can be generated even without carrier envelop phase (CEP) stabilization using a 25 fs/800 nm fundamental field and a relative weak 25 fs/1330 nm control field, which enables us to markedly relax the requirements of the driving laser fields both in pulse duration and CEP control. On the other hand, if the CEPs of driving laser fields are stabilized, a 65-eV broadband continual harmonic, supporting a 81-as IAP, can be directly produced with the optimized intensity ratio of 0.866 and control wavelength of 1400 nm. Moreover, the propagation effect of two-color field on the macroscopic build-up of HH for generating a high-energy IAP is discussed. We found that the method of phase match still works for the efficient continuous harmonic generation as long as the ionization level and the pressure of gas medium are kept low enough. Since the phase-matched short IAP can be generated with our shaped two-color scheme in combination with a relaxed requirement of driving laser fields, the commercial available high-energy laser source with a loosely focused geometry is promising for scaling up the energy of IAP, showing the potential for the realization of IAP with high focused intensity toward 100 attoseconds.

© 2014 Optical Society of America

## 1. Introduction

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*I*+3.17

_{p}*U*, where

_{p}*I*is the ionization potential and

_{p}*U*is the ponderomotive energy. Following this process, it turns out that the harmonic radiation is usually repeated each half cycle of the multi-cycle Ti:sapphire driving field, resulting in an attosecond pulse train (APT). However, for a straightforward attosecond metrology, it is necessary to isolate one clearly defined attosecond pulse, namely IAP.

_{p}18. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature **427**, 817 (2004). [CrossRef] [PubMed]

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*et al*. demonstrated the compression of the generated IAP for as short as 130 as by polarization control of 5-fs laser field [22

22. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, Isolated single-cycle attosecond pulses,” Science **314**, 443–446 (2006). [CrossRef] [PubMed]

*et al*. have employed a near-single-cycle pulse and successfully produced an isolated 80-as pulse, which first broke through the 100-as-barrier [23

23. 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**, 1614–1617 (2008). [CrossRef] [PubMed]

*et al*. have obtained a 67-as pulse, which is known as the shortest IAP at present [24

24. K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. **37**, 3891–3893 (2012). [CrossRef] [PubMed]

*U*scales as

_{p}*λ*

^{2}, therefore the laser wavelength becomes an effective knob to extend harmonic spectrum for short IAP generation while keeps a moderate laser intensity [25

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

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

28. M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. **100**, 173001 (2008). [CrossRef] [PubMed]

29. 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. Physics **4**, 386–389 (2008). [CrossRef]

30. A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Legare, M. Giguere, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. **103**, 073902 (2009). [CrossRef] [PubMed]

*λ*

^{−5}–

*λ*

^{−6}. This effect hinders the development of intense IAP source by directly using driving field with longer wavelength.

31. A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, ”Single-Shot Diffractive Imaging with a Table-Top Femtosecond Soft X-Ray Laser-Harmonics Source,” Phys. Rev. Lett. **103**, 028104 (2009). [CrossRef] [PubMed]

32. Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy,” Phys. Rev. Lett. **94**, 043001 (2005). [CrossRef] [PubMed]

34. Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, and K. Midorikawa, “Interferometry of Attosecond Pulse Trains in the Extreme Ultraviolet Wavelength Region,” Phys. Rev. Lett. **102**, 213904 (2009). [CrossRef] [PubMed]

35. P. Tzallas, E. Skantzakis, L. A. A. Nikolopoulos, G. D. Tsakiris, and D. Charalambidis, “Extreme-ultraviolet pumpCprobe studies of one-femtosecond-scale electron dynamics,” Nat. Physics **7**, 781–784 (2011). [CrossRef]

36. E. Allaria, F. Bencivenga, R. Borghes, F. Capotondi, D. Castronovo, P. Charalambous, P. Cinquegrana, M. B. Danailov, G. De Ninno, A. Demidovich, S. Di Mitri, B. Diviacco, D. Fausti, W. M. Fawley, E. Ferrari, L. Froehlich, D. Gauthier, A. Gessini, L. Giannessi, R. Ivanov, M. Kiskinova, G. Kurdi, B. Mahieu, N. Mahne, I. Nikolov, C. Masciovecchio, E. Pedersoli, G. Penco, L. Raimondi, C. Serpico, P. Sigalotti, S. Spampinati, C. Spezzani, C. Svetina, M. Trovo, and M. Zangrando, “Two-colour pump-probe experiments with a twin-pulse-seed extreme ultraviolet free-electron laser,” Nat. Commun. **4**, 2476 (2013). [CrossRef] [PubMed]

*μ*J level HH with improved conversion efficiency [37

37. E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. **84**, 4–6 (2004). [CrossRef]

42. P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. **84**, 073103 (2013). [CrossRef] [PubMed]

43. P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Bents, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Physics **3**, 846–850 (2007). [CrossRef]

44. X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of Isolated Attosecond Pulses with 20 to 28 Femtosecond Lasers,” Phys. Rev. Lett. **103**, 183903 (2009). [CrossRef]

*et al*. have realized 20 nJ sub-femotosecond IAP at 50 eV with IPG technology and Feng

*et al*. have demonstrated an 100 nJ extreme ultraviolet (XUV) super-continuum supporting 230-as IAP at 35 eV using GDOG technology. Alternatively, the two-color field has also been extensively investigated for IAP generation because of its potential to break the symmetry of electric field and extend the cutoff energy as the pioneering works have demonstrated [45

45. P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A **76**, 011402(R) (2007). [CrossRef]

48. W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express **19**, 20610–20615 (2011). [CrossRef] [PubMed]

*et al*. for extending the continuum length through analyzing the behavior of the difference amplitude of the strongest and next strongest field peaks [49

49. B. Kim, J. Ahn, Y. L. Yu, Y. Cheng, Z. Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt Express **16**, 10331–10340 (2008). [CrossRef] [PubMed]

*μ*J has been experimentally achieved, which is the highest pulse energy of an IAP, to the best of our knowledge [50

50. E. J. Takahashi, P. Lan, Oliver D. Muöcke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. **4**, 2691 (2013). [CrossRef] [PubMed]

51. Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express **19**, 7190–7212 (2011). [CrossRef] [PubMed]

## 2. Theoretical model

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

*E*(

*t*) is the laser field,

*A*(

*t*) is the corresponding vector potential,

*ε*is a positive regularization constant.

*p*and

_{st}*S*are the stationary momentum and quasiclassical action, respectively.

_{st}*g*(

*t*) represents the ground-state amplitude, which is given by ADK tunneling model. Then the harmonic spectrum can be obtained by the Fourier transforming the time-dependent dipole acceleration

*a⃗*(

*t*), where

*T*and

*ω*are the duration and angular frequency of the driving pulse, respectively.

*q*corresponds to the harmonic order. Here, to qualitatively evaluate the continuous length of the generated harmonic spectrum, we define the electric field ratio as [49

49. B. Kim, J. Ahn, Y. L. Yu, Y. Cheng, Z. Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt Express **16**, 10331–10340 (2008). [CrossRef] [PubMed]

*E*

_{1m}denotes the strongest amplitude of the synthesized two-color field, and

*E*

_{2m}is the second strongest one. Generally, the larger the ratio R is, the broader the continuous spectrum is. This is because the maximum energy of the recombining electron is proportional to the square of the electric field, and then the bandwidth of the continuous spectrum is approximately proportional to the difference between the square of

*E*

_{1m}and

*E*

_{2m}.

*E*) and harmonics (

_{l}*E*) fields, which can be simulated by numerically solving the Maxwell wave equations in the cylindrical coordinate system and assuming radial symmetry. The pulse evolution in a medium can be written as where

_{h}*η*is the effective refractive index of the medium. The

_{eff}*η*can be written in a form of The effective refractive index accounts for the refraction, absorption, kerr nonlinearity and plasma defocusing.

_{eff}*n*of free electrons. The time dependent

_{e}*n*(

_{e}*t*) can be expressed as

*n*

_{0}and electron ionization rate

*w*(

*t*). On the other hand, the propagation equation of harmonic field can be expanded as where

*P*(

_{nl}*r*,

*z*,

*t*) = [

*n*

_{0}−

*n*(

_{e}*r*,

*z*,

*t*)]

*d*(

_{nl}*r*,

*z*,

*t*) is the nonlinear polarization of the medium. The equations here take into account both temporal plasma induced phase modulation and the spatial plasma lensing effects on the driving field. Equations (4) and (6) can be numerically solved with the Crank-Nicholson method.

## 3. Results and discussion

*E*

_{0}and

*E*

_{1}are the electric field amplitudes,

*ω*

_{0}and

*ω*

_{1}are the corresponding angular frequencies of the driving and control pulses.

*τ*

_{0}and

*τ*

_{1}correspond to pulse durations (FWHM) of the two pulses.

*ϕ*

_{0}and

*ϕ*

_{1}are the CEPs of the fundamental and control fields, and can be adjusted independently. Typically, the infrared control field is generated through OPA/OPCPA technology, in which the control field is resulted from the ampliation of a white light continuum (WLC) produced by focusing a portion of fundamental energy on a nonlinear medium. Thus the control field inherits the phase fluctuation of the fundamental field. The fine-tuning of the CEPs can be achieved by correcting the optical path of one of fundamental and control fields by means of inserting a wedge prism-pair independently. We choose neon as the gas medium, of which the ionization energy is 21.6 eV. Although both helium and neon are advantageous compared with argon due to their lower absorption at high photon energy and higher ionization potential. However, since helium has exceptionally small effective nonlinearity, neon is expected to be the most practical gas for efficiently generating HH above 100 eV.

### 3.1. Single-atom response: IAP generation with non-CEP-stabilized two-color field

*ϕ*

_{0}and

*ϕ*

_{1}for the optimum 1330-nm control field. For comparison, the result calculated with a 1250-nm control field (which corresponds to the nearest peak around valley A in Fig. 1) is also presented in Fig. 2(d). One can clearly see that, in these two cases, the variation of R values follows a periodic strip-like distribution. When the fundamental and control fields are simultaneously reversed (with a

*π*difference in

*ϕ*

_{0}and

*ϕ*

_{1}), the observed R value (or the harmonic spectrum) remains almost the same. By scanning the R values in Figs. 2(a) and 2(d), we find that the maximum values of R both appear at the CEP combination (

*ϕ*

_{0},

*ϕ*

_{1}) of (

*π*,

*π*). To obtain a deeper insight, we further present the

*ϕ*

_{1}-dependent ratio R with

*ϕ*

_{0}=

*π*in Fig. 2(b) and the

*ϕ*

_{0}-dependent ratio R with

*ϕ*

_{1}=

*π*in Fig. 2(c). Figures 2(e) and 2(f) are the corresponding results for the 1250-nm control field. One can see that, with

*ϕ*

_{0}(or

*ϕ*

_{1}) deviating from the optimizing CEP of

*π*, the values of R decrease much more rapidly in the case of 1250-nm control field [see Figs. 2(e) and 2(f)]. Nevertheless, with the 1330-nm control field [see Figs. 2(b) and 2(c)], values of R can maintain almost stable with

*ϕ*

_{0}varying from 0.8

*π*to 1.2

*π*and

*ϕ*

_{1}from 0.9

*π*to 1.1

*π*.

^{14}W/cm

^{2}with an intensity ratio

### 3.2. Single atom response: IAP generation with CEP-stabilized two-color field

^{14}W/cm

^{2}, and the intensity ratio of control field is 4% to keep the ionization probability below 1%. We first calculate the values of the ratio R with

*ϕ*

_{0}and

*ϕ*

_{1}varying from 0–2

*π*. Figure 5 shows the variation of R values as a function of

*ϕ*

_{0}and

*ϕ*

_{1}for different wavelengths of the control fields [Figs. 5(a)–5(d) are for 1200 nm, 1600 nm, 2000 nm and 2400 nm, respectively]. We find that the values of the ratio R still follow a strip-like distribution (similar to that in Fig. 2). With the increase of the control field wavelength, the width of the strip area is gradually enlarged. This result means that for a shorter wavelength of the control field, the ratio R is more sensitive to the variation of

*ϕ*

_{0}and

*ϕ*

_{1}. Besides, we also find that in the cases of Figs. 5(a)–5(d), the maximum values of R all appear at the CEP combination (

*ϕ*

_{0},

*ϕ*

_{1}) of (

*π*,

*π*), from which we can conclude that the optimum CEP combination for a broadband supercontinuum generation is (

*π*,

*π*). To confirm the above prediction, we next calculate the harmonic spectra with different CEP combinations of the fundamental and control fields. Corresponding results are presented in Fig. 6. In Fig. 6(a), the wavelength of the control field is 1600 nm and the intensity is 1.1× 10

^{13}W/cm

^{2}. For the CEP combination of (

*π*,

*π*) (R=1.213), the continuum bandwidth of the generated harmonic spectrum (blue line) is demonstrated to be 22 eV [Here, we call the smooth part (100 eV −122 eV) of the spectrum as continuum]. When the CEPs of the fundamental and control fields are varied, the continuum range is obviously reduced. For comparison, we also calculate the harmonic spectra with (

*π*, 1.1

*π*) (R=1.134, green line) and (

*π*, 1.2

*π*) (R=1.054, red line), corresponding continuum ranges are 14 eV and 6 eV, which are both smaller than that with (

*π*,

*π*). In addition, similar results can also be found in the case of 2400-nm control field, as shown in Fig. 6(b). For the combinations of (

*π*,

*π*) (R=1.140, blue line), (

*π*, 1.1

*π*) (R=1.138, green line) and (

*π*, 1.2

*π*) (R=1.135, red line), the bandwidths of the continuum spectra are 12 eV, 12 eV and 6 eV, respectively. Comparing Fig. 6(a) and Fig. 6(b), in the case of 1600-nm control field, the continuum bandwidths (values of R) decrease more rapidly as the CEPs are changed, which denotes a stronger dependence on the CEPs of the fundamental and control pulses.

*π*and the intensity of the control field is still maintained at 1.1 × 10

^{13}W/cm

^{2}. We find that the highest R ratio of the two-color field is 1.251 as shown in Fig. 7(a), and the corresponding wavelength of control field is 1776 nm. To verify the results in Fig. 7(a), we also present the harmonic spectra in Fig. 7(b). For the case of 1776 nm (blue line), the bandwidth of the continuous spectrum is 35 eV, which is indeed much broader than cases of 1500 nm (20 eV, green line) and 2400 nm (12 eV, red line). To sum up, with the control field intensity of 1.1 × 10

^{13}W/cm

^{2}, we obtain a 35 eV supercontinuum at the optimium control field wavelength of 1776 nm.

*E*

_{1m}, and also in the second strongest one

*E*

_{2m}, which will directly influence the values of the ratio R (or the bandwidth of continuous spectrum). Therefore, to obtain an IAP with broader bandwidth, it is essential to know how the ratio R depends on the intensity of the control field. Since it is still a technical challenge to stabilize the CEP of high-energy laser source with pulse duration shorter than 25 fs, we set the pulse duration of both the fundamental and control fields to 25 fs. The fundamental intensity is 1.45 × 10

^{14}W/cm

^{2}. In this case, the ionization probability is lower than 1% even if the intensity ratio of control field is increased to 0.9. Figure 8 shows the variation of ratio R with respect to the wavelength of the control field and the intensity ratio

^{14}W/cm

^{2}).

*π*,

*π*), the wavelength and intensity of the control field are 1400 nm and 1.25 × 10

^{14}W/cm

^{2}, respectively. Figure 9(a) displays the temporal envelops of the synthesized field (green solid line) and the 800-nm fundamental field (black dashed line). It is obvious that in the synthesized field, the maximum amplitude

*E*

_{1m}of the synthesized field is increased and the difference between

*E*

_{1m}and

*E*

_{2m}is enlarged. With such a two-color field interacting with neon, we have calculated the harmonic spectrum by using Lewenstein model. The generated harmonic spectra is presented as the green line in Fig. 9(b). One can clearly see that, due to the increase of

*E*

_{1m}, the cutoff of the spectrum is extremely extend to 200 eV and a smoothed supercontinuum with a bandwidth of 65 eV (from 160 eV to 225 eV) is successfully produced, which supports a transform-limited (TL) IAP with pulse duration below 100 as. By superposing the 110

*–140*

^{th}*harmonics, an IAP with duration of 110 as is obtained directly as shown in Fig. 9(d). A deeper insight can be obtained by investigating the harmonic emission times in terms of the time-frequency analysis. Calculation details can be found in Ref. [55*

^{th}55. Q. Zhang, P. Lu, W. Hong, Q. Liao, P. Lan, and X. B. Wang, “Enhanced high-order harmonic generation via controlling ionization in spatially extended systems,” Phys. Rev. A **79**, 053406 (2009). [CrossRef]

### 3.3. Influence of two-color field on phase-matched harmonic generation

*π*rad relative to the harmonic field as it propagates. This means that harmonic field generated early in the nonlinear medium will be exactly out of phase with harmonic field generated a certain propagation distance later. This dephasing distance is called the coherence length

*L*=

_{c}*π*/Δ

*k*, where Δ

*k*is the phase mismatch. Therefore phase match of IAP can be achieved by balancing the various dispersion terms or a zero net phase mismatch. If a monochromatic driving field is used for generating IAP, the total phase mismatch Δ

*k*can simply be written by The phase mismatch contributions from neutral gas dispersion Δ

*k*, plasma dispersion Δ

_{n}*k*, geometrical phase Δ

_{p}*k*and atomic dipole polarization Δ

_{g}*k*, given by where

_{pl}*ϕ*(

_{f,i}*z*),

*ϕ*(

_{q,i}*z*) represent the phase of driving field and q

*harmonic field at propagation position*

^{th}*z*, and

*k*,

_{f,i}*k*are the corresponding wavevector. This simple relationship is valid because any phase shift

_{q,i}*δϕ*of the monochromatic driving field during propagation maps a phase change

*qδϕ*to the generated

*q*harmonic. However, in the case of two-color field used to drive HHG, there are not only the phase shift of each field but also the relative phase slip between the fundamental and control fields themselves, induced by the dispersion in gas medium. It has been demonstrated that this relative phase slip will lead to an additional phase to the generated

^{th}*q*harmonic [56

^{th}56. R. Lopez-Martens, K. Varju, P. Johnsson, J. Mauritsson, Y. Mairesse, P. Salieres, M. B. Gaarde, K. J. Schafer, A. Persson, S. Svanberg, C. G. Wahlstrom, and A. L’Huillier, “Amplitude and phase control of attosecond light pulses,” Phys. Rev. Lett. **94**, 033001 (2005). [CrossRef] [PubMed]

58. Y. Zheng, Z. Zeng, P. Zou, L. Zhang, X. Li, P. Liu, R. Li, and Z. Xu, “Dynamic Chirp Control and Pulse Compression for Attosecond High-Order Harmonic Emission,” Phys. Rev. Lett **103**, 043904 (2009). [CrossRef] [PubMed]

*q*harmonic. Therefore we rewrite a robust phase mismatch by where Φ

^{th}*(*

_{q}*z*) is the phase of newly generated

*q*harmonic at position

^{th}*z*, while Φ

*(*

_{q,pro}*z*) is the phase of

*q*harmonic generated early in the nonlinear medium and propagated to position

^{th}*z*. The phase of

*q*harmonic field with wavelength

^{th}*λ*propagating in a gas medium is given by [59

_{q}59. C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. **80**,, 117–140 (2008). [CrossRef]

*(*

_{q}*z*

_{0}) is the phase at start position

*z*

_{0},

*N*is the atom density,

_{a}*N*is the free electron density in the medium,

_{e}*n*(

*λ*) is the refractive index per unit neutral atom density at wavelength

_{q}*λ*, and

_{q}*r*is the classical electron radius. In Eq. (11), the contribution from the Guoy term of the harmonic light is much smaller than that for the fundamental and control field by a factor of ≈ 1/

_{e}*q*

^{2}and is therefore not included for high harmonic orders. In order to estimate the phase Φ(

*z*), we neglect the envelop of both fundamental and control fields and only take the term of carrier waves into account for simplicity, then the harmonic phase can be written approximately in the term of

*ϕ*describes the relative phase slip of the two fields during propagation,

*t*and

_{i}*t*are the birth moment and recollision moment of freed electron contributed to the

_{r}*q*harmonic. The

^{th}*t*and

_{i}*t*are supposed to be chosen such that Here

_{r}*A*and

*P*are the vector potential of driving field and the stationary canonical momentum of electron, respectively.

_{s}*P*,

_{s}*t*, and

_{i}*t*can be solved from these saddle-point equations [54

_{r}54. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

60. D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A **66**, 063417 (2002). [CrossRef]

*ϕ*between the fundamental and control fields alters the phase of generated harmonic at position

*z*. Even if the intensity of control field is low, the term of “

*E*

_{0}

*E*

_{1}” will amplify the influence on harmonic phase. Furthermore, the shaped driving laser field will lead to the changes of

*t*and

_{i}*t*according the Eqs. (15)–(17), which also contributes to the harmonic phase. Therefore it is necessary to investigate the dependence of harmonic phase on the intensity ratio and relative phase slip before we go into the macroscopic build-up of IAP.

_{r}*t*and recollision moment

_{i}*t*contributed to the 30

_{r}*–120*

^{th}*harmonic with the intensity ratio of control field varied from 0 to 0.8, the marked symbols represent*

^{th}*t*and

_{i}*t*of the 80

_{r}*harmonic for short and long trajectories. It is found that both the*

^{th}*t*and

_{i}*t*gradually change with the increasing of intensity ratio. Since the emission resulted from the short trajectory usually presents a smaller beam divergence than the long trajectory, we then focus on the short one. Figures 10(f)–10(j) show the calculated phases of 75

_{b}*–85*

^{th}*harmonics, an obvious phase slip can be observed for each harmonic. For this reason, the intensity ratio induced phase variation should be taken into account for achieving phase-matched harmonics with two-color driving field.*

^{th}*f*

_{#}=

*f/D*= 600,

*f*

_{#}= 300 and

*f*

_{#}= 150, respectively. A more dramatic change of the intensity ratio is observed for the tightly focused geometry, leading to a narrower parabolic phase curves of the 80

*harmonic shown in Fig. 11(b) and a steeper dispersion curve in Fig. 11(c). For comparison, we also calculate the dispersion with*

^{th}*f*

_{#}= 300 for a constant intensity ratio along propagation direction as shown in Fig. 11(d). The difference of the dispersion curves shown in Fig. 11(d) indicate the evolution of intensity ratio indeed change the phase of a specific harmonic, and consequently the phase-matching condition. Moreover, the variation of intensity ratio may also significantly change the cutoff of harmonic radiation, which hampers the selection of continuous harmonics near cutoff region for IAP generation with our two-color driving scheme.

*f*

_{0#}and

*f*

_{1#}are the f-numbers of fundamental and control field,

*λ*

_{0}and

*λ*

_{0}are the corresponding wavelengths. In experiment, this can be achieved by using two separate focusing lenses for the fundamental and control fields.

*ϕ*=

*ω*

_{1}

*δt*between the driving fields, which will also change the shape of the combined fields during propagation, and consequently the harmonic phase. Here, the group delay between a 800-nm fundamental field and a 1400-nm control field in neon is estimated to be about 1 as/cm with a gas pressure of 1 torr. As the refractive index is proportional to the gas pressure, the group delay

*δt*increases linearly with gas pressure. Since the gas medium may be extended from several millimeter to centimeters for efficiently extracting harmonic energy, the gas pressure must be low enough to maintain the waveform of this synthesized field during propagation.

### 3.4. Phase-matched IAP generation using non-CEP-stabilized two-color field

*f*

_{0#}= 137 and

*f*

_{1#}= 106 to a 0.5-mm neon gas jet, corresponding to laser beams with waists of 35

*μ*m and 45

*μ*m. Other laser parameters are the same as those in Fig. 4. The density of the gas medium is 3.5 × 10

^{17}/cm

^{3}(10 Torr). In practice, the phase-matching condition can be achieved for only very low levels of ionization, at most a few percent (about 1%) for neon [42

42. P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. **84**, 073103 (2013). [CrossRef] [PubMed]

^{0}cm

^{−1}, while 10

^{2}cm

^{−1}for Guoy phase and the intensity dependent atomic phase. Thus the brightest emission from a tight focal geometry is often achieved through balancing the Guoy phase and the intensity dependent phase. The phase-matching condition can be achieved by adjusting the gas pressure and the position of the laser focus. We place the gas jet 1.5 mm after the laser focus, therefore the short quantum path can be effectively enhanced. The macroscopic harmonic spectrum with different CEPs are shown in Fig. 12(a). For most CEP values, the continuous harmonics are generated in the range from 125 eV (80

*) to 145 eV (92*

^{th}*). We further investigate the temporal characteristics of the smoothed supercontinuum by applying a square window with a width of 20 eV to the supercontinuum at different CEPs, it is clearly shown that only one branch in Fig. 12(b), which indicates we obtain IAP at a broad range of CEP.*

^{th}37. E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. **84**, 4–6 (2004). [CrossRef]

41. Y. Wu, E. Cunningham, H. Zang, J. Li, M. Chini, X. Wang, Y. Wang, K. Zhao, and Z. Chang, “Generation of high-flux attosecond extreme ultraviolet continuum with a 10 TW laser,” Appl. Phys. Lett. **102**, 201104 (2013). [CrossRef]

*f*

_{0#}= 600 and

*f*

_{1#}= 464. We obtain similar results to those shown in Fig. 12, while the conversion efficiency is significantly improved.

### 3.5. Phase-matched IAP generation using CEP-stabilized two-color field

## 4. Conclusion

*π*,

*π*) and the optimum wavelength of the control field is demonstrated to be 1400 nm with an increased intensity ratio of 0.866. With such a 25-fs synthesized field, an broadband supercontinuum with the bandwidth of 65 eV is successfully produced. By discussing the influence of group delay and relative intensity of the two-color field on the macroscopic build-up of IAP, we conclude that the method of phase-matching still works as long as the ionization level and gas pressure is low enough. Under proper phase-matching condition, an efficient IAP down to 100 as can be obtained with loosely focused geometry.

^{−7}for the loosely focused 800-nm driving laser obtained in ref. [42

42. P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. **84**, 073103 (2013). [CrossRef] [PubMed]

*λ*

^{−2}for the 800-nm/1400-nm two-color field [61

61. P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A **81**, 061802(R) (2010). [CrossRef]

^{−7}. Meanwhile, 40-TW class (e.g., 25 fs, 1 J) Ti:sapphire laser systems are commercially available. For example we estimate the energy of IAP driving by 200-mJ fundamental pulse and 200-mJ control pulse (obtained from OPCPA with the conversion efficiency of 25%), the IAP with energy as high as 0.5

*μ*J could be reached by selecting more than 30 harmonic orders in the continuous spectrum. As demonstrated in Ref. [62

62. H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 10^{14} W/cm^{2},” Opt. Lett. **29**, 1927–1929 (2004). [CrossRef] [PubMed]

*μ*m spot size. With the reported optical losses from the IR/XUV beam splitter and multilayer concave mirror [37

37. E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. **84**, 4–6 (2004). [CrossRef]

62. H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 10^{14} W/cm^{2},” Opt. Lett. **29**, 1927–1929 (2004). [CrossRef] [PubMed]

^{14}W/cm

^{2}.

## Acknowledgments

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62. | H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 10 |

**OCIS Codes**

(190.4160) Nonlinear optics : Multiharmonic generation

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(300.6560) Spectroscopy : Spectroscopy, x-ray

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: February 18, 2014

Revised Manuscript: May 9, 2014

Manuscript Accepted: May 16, 2014

Published: May 23, 2014

**Citation**

Qingbin Zhang, Lixin He, Pengfei Lan, and Peixiang Lu, "Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds," Opt. Express **22**, 13213-13233 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13213

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### References

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