## Quasi-phase-matching for third harmonic generation in noble gases employing ultrasound |

Optics Express, Vol. 20, Issue 20, pp. 22753-22762 (2012)

http://dx.doi.org/10.1364/OE.20.022753

Acrobat PDF (1681 KB)

### Abstract

We study a novel method of quasi-phase-matching for third harmonic generation in a gas cell using the periodic modulation of the gas pressure and thus of the third order nonlinear coefficient in the axial direction created by an ultrasound wave. Using a comprehensive numerical model we describe the quasi-phase matched third harmonic generation of UV (at 266 nm) and VUV pulses (at 133 nm) by using pump pulses at 800 nm and 400 nm, respectively, with pulse energy in the range from 3 mJ to 1 J. In addition, using chirped pump pulses, the generation of sub-20-fs VUV pulses without the necessity for an external chirp compensation is predicted.

© 2012 OSA

## 1. Introduction

1. J. W. Ward and G. H. C. New, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

2. G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. **11**, 287–296 (1975). [CrossRef]

1. J. W. Ward and G. H. C. New, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

2. G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. **11**, 287–296 (1975). [CrossRef]

3. R. Eramo and M. Matera, “Third-harmonic generation in positively dispersive gases with a novel cell,” Appl. Opt. **33**, 1691–1696 (1994). [CrossRef] [PubMed]

4. T. Tamaki, K. Midirika, and M. Obara, “Phase-matched third-harmonic generation by nonlinear phase shift in a hollow fiber,” Appl. Phys. B **67**, 59–63 (1998). [CrossRef]

5. D. S. Bethune and C. T. Retter, “Optical harmonic generation in nonuniform gaseous media with application to frequency tripling in free-jet expansions,” IEEE J. Quantum Electron. **23**, 1348–1360 (1987). [CrossRef]

6. C. W. Siders, N. C. Turner, M. C. Downer, A. Babine, A. Stepanov, and A. M. Sergeev, “Blue-shifted third-harmonic generation and correlated self-guiding during ultrafast barrier suppression ionization of subatmospheric density noble gases,” J. Opt. Soc. Am. B **13**, 330–336 (1996). [CrossRef]

8. S. A. Trushin, K. Kosma, W. Fub, and W. E. Schmid, “Sub-10-fs supercontinuum radiation generated by filamentation of few-cycle 800 nm pulses in argon,” Opt. Lett. **32**, 2432–2434 (2007). [CrossRef] [PubMed]

9. N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. **89**, 143901 (2002). [CrossRef] [PubMed]

14. Y. Liu, M. Durand, A. Houard, B. Forestier, A. Couairon, and A. Mysyrowicz, “Efficient generation of third harmonic radiation in air filaments: A revisit,” Opt. Commun. **284**, 4706–4713 (2011). [CrossRef]

11. X. Yang, J. Wu, Y. Peng, Y. Tong, S. Yuan, L. Ding, Z. Xu, and H. Zeng, “Noncollinear interaction of femtosecond filaments with enhanced third harmonic generation in air,” Appl. Phys. Lett. **95**, 111103 (2009). [CrossRef]

14. Y. Liu, M. Durand, A. Houard, B. Forestier, A. Couairon, and A. Mysyrowicz, “Efficient generation of third harmonic radiation in air filaments: A revisit,” Opt. Commun. **284**, 4706–4713 (2011). [CrossRef]

14. Y. Liu, M. Durand, A. Houard, B. Forestier, A. Couairon, and A. Mysyrowicz, “Efficient generation of third harmonic radiation in air filaments: A revisit,” Opt. Commun. **284**, 4706–4713 (2011). [CrossRef]

15. C. G. Durfee, S. B. Margaret, M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. **22**, 1565–1567 (1997). [CrossRef]

17. I. V. Babushkin and J. Herrmann, “High energy sub-10 fs pulse generation in vacuum ultraviolet using chirped four wave mixing in hollow waveguides,” Opt. Express **16**, 17774–17779 (2008). [CrossRef] [PubMed]

18. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Persham, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. **127**, 1918–1939 (1962). [CrossRef]

19. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **QE-28**, 2631–2654 (1992). [CrossRef]

20. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature (London) **421**, 51–54 (2003). [CrossRef]

21. S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. **87**, 1339021 (2001). [CrossRef]

22. X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. **3**, 270–275 (2007). [CrossRef]

## 2. Quasi-phase matching using ultrasound

18. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Persham, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. **127**, 1918–1939 (1962). [CrossRef]

19. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. **QE-28**, 2631–2654 (1992). [CrossRef]

*K*required to achieve QPM of interacting waves in the cell is approximately equal to the phase-mismatch |Δ

_{s}**k**| due to dispersion of the corresponding waves. In this paper we study in detail THG as a simplest prototypical interaction (see Fig. 1). In this case the phase mismatch is given by Δ

**k**=

**k**

_{3}

*− 3*

_{ω}**k**

*, where*

_{ω}**k**

*is the pump wavevector with the frequency*

_{ω}*ω*and

**k**

_{3}

*is the TH wavevector with the frequency 3*

_{ω}*ω*. The dependence of the ultrasound frequency Ω

*= 2*

_{s}*π*/

*K*on the pump wavelength

_{s}*λ*= 2

*π*/

*ω*is shown in Fig. 1(b). As seen for argon QPM for THG with a pump pulse at 800 nm requires an ultrasound frequency of Ω

*≈ 22 kHz and for a pump pulse at 400 nm we find Ω*

_{s}*≈ 300 kHz. For high intensities the Kerr effect introduces an additional phase shift.*

_{s}24. T. G. Leighton, “What is ultrasound?” Prog. Biophys. Mol. Biol. **93**, 3–83 (2007). [CrossRef]

*p*in a pulsed regime can achieve ∼ 10

^{−2}atm inside the area of ∼ 1 cm

^{2}[25

25. A. Holm and H. W. Persson, “Optical diffraction tomography applied to airborne ultrasound,” Ultrasonics **31**, 259–265 (1993). [CrossRef]

*p*∼ 10

^{−2}atm) was excited in air at atmospheric pressure for a transducer with resonant frequency Ω

*= 20 kHz. Due to nonlinearities of generation and propagation, the ultrasound contained higher harmonics; in particular, the contribution of fourth-order harmonic (Ω*

_{s}*= 80 kHz) in this wave was estimated to be up to ≈ 130 dB (*

_{s}*p*≈ 5 × 10

^{−4}atm).

27. H. Tijdeman, “On the propagation of sound wave in cylindrical tubes,” J. Sound Vib. **39**, 1–13 (1975). [CrossRef]

28. E. Rodarte, G. Singh, N. R. Miller, and P. Hrnjak, “Sound attenuation in tubes due to visco-thermal effects,” J. Sound Vib. **231**, 1221–1241 (2000). [CrossRef]

27. H. Tijdeman, “On the propagation of sound wave in cylindrical tubes,” J. Sound Vib. **39**, 1–13 (1975). [CrossRef]

28. E. Rodarte, G. Singh, N. R. Miller, and P. Hrnjak, “Sound attenuation in tubes due to visco-thermal effects,” J. Sound Vib. **231**, 1221–1241 (2000). [CrossRef]

*ν*is the kinematic viscosity. Viscosity effects become important for

*s*≤ 1. For typical parameters considered in the present article (atmospheric pressure, Ω

*= 0.3 MHz) and assuming reasonable tube diameter of ≈ 1 cm ([25*

_{s}25. A. Holm and H. W. Persson, “Optical diffraction tomography applied to airborne ultrasound,” Ultrasonics **31**, 259–265 (1993). [CrossRef]

*s*∼ 3 × 10

^{3}, that is the approximation of “wide tube” is very well applicable. For the above mentioned parameters, the losses induced by the tube itself are of order of 5 × 10

^{−2}cm

^{−1}, according the model introduced in [28

28. E. Rodarte, G. Singh, N. R. Miller, and P. Hrnjak, “Sound attenuation in tubes due to visco-thermal effects,” J. Sound Vib. **231**, 1221–1241 (2000). [CrossRef]

29. T. D. Rossing, *Handbook of Acoustics* (Springer, 2007). [CrossRef]

*ρ*

_{0}is the gas atomic density,

*c*is the sound speed,

_{s}*κ*is the thermal conductivity,

*η*is the shear viscosity,

*γ*=

*C*/

_{p}*C*is the rate of the heat capacity at constant pressure (

_{v}*C*) and constant volume (

_{p}*C*).

_{v}*β*on the ultrasound frequency Ω

_{s}*for argon is shown in Fig. 1(c). Although the losses given by the classical formula Eq. (1) are underestimated for liquids and multi-atomic gases, Eq. (1) works still reasonably good for noble gases [29*

_{s}29. T. D. Rossing, *Handbook of Acoustics* (Springer, 2007). [CrossRef]

31. L. J. Bond, C. Chiang, and C. M. Fortunko, “Absorption of ultrasonic waves in air at high frequencies (10–20 MHz),” J. Acoust. Soc. Am. **92**, 2006–2015 (1992). [CrossRef]

*z: P*(

*x*,

*y*,

*z*) =

*P*+

_{o}*p*(

*x*,

*y*)

*e*

^{−βsz}cos(

*K*). Here

_{S}z*P*is the background pressure and

_{o}*p*(

*x*,

*y*) is the ultrasound amplitude, which is constant inside the tube (

*p*(

*x*,

*y*) ≡

*p*= const.

*A*(

_{ω}*x*,

*y*,

*z*,

*t*) and the TH

*A*

_{3}

*(*

_{ω}*x*,

*y*,

*z*,

*t*) as well as the free electron density

*ρ*(

*x*,

*y*,

*z*,

*t*) in time

*t*and space are given by: Here

*m*=

*ω*and 3

*ω*for the fundamental and TH, correspondingly);

*χ*

^{(3)}(

*z*) is the third-order nonlinear susceptibility, depending on the pressure

*P*(and hence on

*z*); Δ

*k*is the linear phase mismatch for the gas pressure

*P*

_{0}.

*M̂*,

_{ω}*M̂*

_{3}

*and Γ*

_{ω}*, Γ*

_{ω}_{3}

*are defined as: where*

_{ω}*ν*= 0 and

_{ω}*ν*

_{3}

*(*

_{ω}*z*) = 1/

*V*(

_{ω}*z*) − 1/

*V*

_{3}

*(*

_{ω}*z*);

*V*(

_{m}*z*) and

*g*(

_{m}*z*) are the group velocities and group velocity dispersions of the interacting pulses;

*k*(

_{m}*z*) and

*z*-dependent) pressure

*P*and for the background pressure

*P*

_{0}, respectively;

*σ*(

_{m}*z*) is the cross section of inverse Bremsstrahlung;

*τ*(

_{c}*z*) is the free-carrier collision time;

*σ*

_{Km}is the ionization cross section, where

*K*≡ <

_{m}*U*/

_{i}*h*̄

*ω*+ 1> here also

_{m}*U*is the ionization potential of the gas;

_{i}*β*

_{Km}is the multiphoton ionization coefficient, which is defined as

*β*

_{Km}=

*K*

_{m}h̄ω_{m}σ_{Km};

*ρ*(

_{o}*z*) is the density of neutral atoms [32

32. Z. Song, Y. Qin, G. Zhang, S. Cao, D. Pang, L. Chai, Q. Wanga, Z. Wangb, and Z. Zhang, “Femtosecond pulse propagation in temperature controlled gas-filled hollow fiber,” Opt. Commun. **281**, 4109–4113 (2008). [CrossRef]

*k*(

_{m}*z*),

*γ*(

_{m}*z*),

*V*(

_{m}*z*),

*g*(

_{m}*z*),

*σ*(

_{m}*z*),

*τ*(

_{c}*z*), are assumed here to be proportional to the pressure (and thus varying along the

*z*-coordinate) [32

32. Z. Song, Y. Qin, G. Zhang, S. Cao, D. Pang, L. Chai, Q. Wanga, Z. Wangb, and Z. Zhang, “Femtosecond pulse propagation in temperature controlled gas-filled hollow fiber,” Opt. Commun. **281**, 4109–4113 (2008). [CrossRef]

34. A. Couairon, M. Franco, G. Mechain, T. Olivier, B. Prade, and A. Mysyrowicz, “Femtosecond filamentation in air at low pressures: Part I: Theory and numerical simulations,” Opt. Commun. **58**, 265–273 (2006). [CrossRef]

*= 0) and loss, but taking into account the nonlinear self-phase modulation of the pump. Then, the on-axis QPM condition for the ultrasound wave vector*

_{m}*K*is: where

_{s}*I*

_{3}

*: Remarkably,*

_{ω}*I*

_{3}

*does not depend on the background pressure*

_{ω}*P*

_{0}but only on the ultrasound amplitude

*p*.

## 3. Results and their discussion

### 3.1. UV pulse generation by using 800 nm pump pulses

*τ*= 700 fs, a radius

_{ω}*r*= 0.05 cm and an energy 3 mJ (corresponding to the input intensity

_{ω}*I*≈ 1 TW/cm

_{o}^{2}and power

*P*≈ 3.97 GW) with a sound amplitude of

_{ω}*p*= 0.01 atm. The required ultrasound frequency necessary to fulfill the QPM condition according Eq. (7) is Ω

*= 22.24 kHz. For these parameters the self-focusing distance is*

_{s}*z*≈ 11.01 m, the critical power of self-focusing is

_{f}*P*≈ 3.94 GW and the walk-off length is

_{crit}*L*=

_{ν}*τ*/

_{ω}*ν*≈ 345 m. Therefore one can expect a relatively long propagation distance without beam collapsing, temporal walk-off or formation of a filament [36

36. A. Couairon and A. Mysyrowiczb, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

*η*

_{3}

*(*

_{ω}*z*) = ∭ |

*A*

_{3}

*(*

_{ω}*z*,

*x*,

*y*,

*t*)|

^{2}

*dxdydt*/∭ |

*A*(

_{ω}*z*= 0,

*x*,

*y*,

*t*)|

^{2}

*dxdydt*. One can see from the Fig. 2(a) that QPM results in the efficient conversion to the TH at the optimum ultrasound frequency, which is 27 times larger than without ultrasound (green curve). The self-focusing effect is relatively weak as seen from the evolution of pump intensity (blue curve in Fig. 2(b)) and beam radius (red curve in Fig. 2(d)), while that spatial profile of the TH shows a good beam quality (Fig. 2(c)).

*τ*= 1 ps (transform-limited), a radius of

_{ω}*r*= 0.5 cm and a sound amplitude of

_{ω}*p*= 0.01 atm. For these parameters one can calculate:

*I*≈ 2.4 TW/cm

_{o}^{2},

*z*≈ 247 cm,

_{f}*L*=

_{ν}*τ*/

_{ω}*ν*≈ 487 cm. In this case the power of the pump pulse

*P*≈ 940 GW is much larger than the critical power of self-focusing

_{ω}*P*≈ 3.94 GW, therefore as seen in Fig. 3(d) after a propagation distance of about 2.5 m the pump beam radius significantly decreases and its intensity increases. The efficiency increases up to

_{crit}*η*

_{3}

*≈ 0.12%, but due to the change of the pump intensity the QPM condition Eq. (7) is violated after this distance. Without ultrasound (green curve) the efficiency is at ∼ 2.5 m 18 times smaller, but after self-focusing distance it increases significantly due to the increase of the pump intensity.*

_{ω}### 3.2. VUV pulse generation by using 400 nm pump pulses

*τ*= 1.4 ps, beam radius of

_{ω}*r*= 0.03 cm, pump intensity

_{ω}*I*≈ 1.4 TW/cm

_{o}^{2}and sound amplitude of

*p*= 0.01 atm. As one can see from the Fig. 4, the conversion efficiency increases up to the propagation length of about 50 cm. The saturation of THG is caused by the strong self-focusing effect with the formation of a filament. The strong increase of the pump intensity leads to the violation of the optimum QPM condition, which terminates the frequency conversion. Figure 4(b) and 4(d) illustrates the well known dynamics of the formation of a filament after approximately ∼ 50 cm propagation. As seen the combined action of the optical Kerr effect, multiphoton absorption and ionization leads to focusing and defocusing cycles with very small quasi-periodicity [36

36. A. Couairon and A. Mysyrowiczb, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

*τ*= 1 ps, radius

_{ω}*r*= 0.1 cm (

_{ω}*I*≈ 6 TW/cm

_{o}^{2}) and the sound wave amplitude

*p*= 0.01 atm. As can be seen from the Fig. 5. the conversion efficiency of THG up to ≈ 0.02% can be obtained. However, in this case a multifilamentation takes place. It appears because the input peak power (

*P*≈ 93 GW) is much larger than the critical power (

_{ω}*P*≈ 0.9 GW) [36

_{crit}36. A. Couairon and A. Mysyrowiczb, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

### 3.3. Sub-20 fs VUV pulse generation by using chirped 400 nm pump pulses

*τ*= 1 ps and 0.3 mJ energy obtained after phase-modulation from a bandwidth-limited one of 20 fs duration. The beam radius is

_{ω}*r*= 0.25 cm, the input intensity is

_{ω}*I*≈ 2.87 TW/cm

_{o}^{2}and the sound amplitude is

*p*= 0.01 atm. As one can see, the conversion efficiency is limited to the same level as in the previous example. However, the duration of both pump and TH pulses decrease significantly because of chirp compensation due to propagation in the normal-dispersive argon gas (Fig. 7(a)). The duration of the generated VUV pulse at 133 nm is reduced down to 18 fs (Fig. 7(c)) and its pulse energy is ∼ 3

*μ*J. This self-compression is caused by a chirp compensation during propagation due to normal dispersion of the gas.

*r*= 0.1 cm and the input peak intensity

_{ω}*I*≈ 6 TW/cm

_{o}^{2}. The pump pulse duration is 1 ps stretched by a phase-modulation from 50 fs (

*I*≈ 119 TW/cm

_{o}^{2}). The ultrasound amplitude is

*p*= 0.01 atm. Figure 8 shows the results of the numerical simulations for this case. Up to the length of 40 cm the efficiency increases due to QPM by ultrasound and the rapid increase of the pump intensity because of self-focusing (Fig. 8(a) and 8(b)). Here we also see that the TH beam is split into two main filaments and a weak background (Fig. 8(c)). The temporal shape of the TH pulse at the output remains relatively well-defined, with a pulse duration as small as 16 fs (see Fig. 8(d)).

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. W. Ward and G. H. C. New, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Phys. Rev. |

2. | G. Bjorklund, “Effects of focusing on third-order nonlinear processes in isotropic media,” IEEE J. Quantum Electron. |

3. | R. Eramo and M. Matera, “Third-harmonic generation in positively dispersive gases with a novel cell,” Appl. Opt. |

4. | T. Tamaki, K. Midirika, and M. Obara, “Phase-matched third-harmonic generation by nonlinear phase shift in a hollow fiber,” Appl. Phys. B |

5. | D. S. Bethune and C. T. Retter, “Optical harmonic generation in nonuniform gaseous media with application to frequency tripling in free-jet expansions,” IEEE J. Quantum Electron. |

6. | C. W. Siders, N. C. Turner, M. C. Downer, A. Babine, A. Stepanov, and A. M. Sergeev, “Blue-shifted third-harmonic generation and correlated self-guiding during ultrafast barrier suppression ionization of subatmospheric density noble gases,” J. Opt. Soc. Am. B |

7. | S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1- |

8. | S. A. Trushin, K. Kosma, W. Fub, and W. E. Schmid, “Sub-10-fs supercontinuum radiation generated by filamentation of few-cycle 800 nm pulses in argon,” Opt. Lett. |

9. | N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. |

10. | N. Kortsalioudakis, M. Tatarakis, N. Vakakis, S. D. Moustaizis, M. Franco, B. Prade, A. Mysyrowicz, A. A. Papadogiannis, A. Couairon, and S. Tzortzakis, “Enhanced harmonic conversion efficiency in the self-guided propagation of femtosecond ultraviolet laser pulses in argon,” Appl. Phys. B |

11. | X. Yang, J. Wu, Y. Peng, Y. Tong, S. Yuan, L. Ding, Z. Xu, and H. Zeng, “Noncollinear interaction of femtosecond filaments with enhanced third harmonic generation in air,” Appl. Phys. Lett. |

12. | S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express |

13. | S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A |

14. | Y. Liu, M. Durand, A. Houard, B. Forestier, A. Couairon, and A. Mysyrowicz, “Efficient generation of third harmonic radiation in air filaments: A revisit,” Opt. Commun. |

15. | C. G. Durfee, S. B. Margaret, M. Murnane, and H. C. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. |

16. | P. Tzankov, O. Steinkellner, J. Zheng, M. Mero, W. Freyer, A. Husakou, I. Babushkin, J. Herrmann, and F. Noack, “High-power fifth-harmonic generation of femtosecond pulses in the vacuum ultraviolet using a Ti:sapphire laser,” Opt. Express |

17. | I. V. Babushkin and J. Herrmann, “High energy sub-10 fs pulse generation in vacuum ultraviolet using chirped four wave mixing in hollow waveguides,” Opt. Express |

18. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Persham, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. |

19. | M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. |

20. | A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature (London) |

21. | S. L. Voronov, I. Kohl, J. B. Madsen, J. Simmons, N. Terry, J. Titensor, Q. Wang, and J. Peatross, “Control of laser high-harmonic generation with counterpropagating light,” Phys. Rev. Lett. |

22. | X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. |

23. | J. Herrmann, “Apparatus and method for amplification and frequency transformation of laser radiation using quasi-phase matching of four-wave mixing,” Patent: DE102009028819A1 (2011). |

24. | T. G. Leighton, “What is ultrasound?” Prog. Biophys. Mol. Biol. |

25. | A. Holm and H. W. Persson, “Optical diffraction tomography applied to airborne ultrasound,” Ultrasonics |

26. | J. A. Gallego-Juarez and L. Gaete-Garreton, “Experimental study of nonlineairity in free progressive acoustic waves in air at 20 kHz,” J. Phys. |

27. | H. Tijdeman, “On the propagation of sound wave in cylindrical tubes,” J. Sound Vib. |

28. | E. Rodarte, G. Singh, N. R. Miller, and P. Hrnjak, “Sound attenuation in tubes due to visco-thermal effects,” J. Sound Vib. |

29. | T. D. Rossing, |

30. | M. Iskhakovich, |

31. | L. J. Bond, C. Chiang, and C. M. Fortunko, “Absorption of ultrasonic waves in air at high frequencies (10–20 MHz),” J. Acoust. Soc. Am. |

32. | Z. Song, Y. Qin, G. Zhang, S. Cao, D. Pang, L. Chai, Q. Wanga, Z. Wangb, and Z. Zhang, “Femtosecond pulse propagation in temperature controlled gas-filled hollow fiber,” Opt. Commun. |

33. | M. Mlenjnek, E. M. Wright, and J. V. Moloney, “Femtosecond pulse propagation in argon: A pressure dependence study,” Phys. Rev. Lett. |

34. | A. Couairon, M. Franco, G. Mechain, T. Olivier, B. Prade, and A. Mysyrowicz, “Femtosecond filamentation in air at low pressures: Part I: Theory and numerical simulations,” Opt. Commun. |

35. | G. P. Agrawal, |

36. | A. Couairon and A. Mysyrowiczb, “Femtosecond filamentation in transparent media,” Phys. Rep. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(230.1040) Optical devices : Acousto-optical devices

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 24, 2012

Revised Manuscript: September 6, 2012

Manuscript Accepted: September 6, 2012

Published: September 19, 2012

**Citation**

U. K. Sapaev, I. Babushkin, and J. Herrmann, "Quasi-phase-matching for third harmonic generation in noble gases employing ultrasound," Opt. Express **20**, 22753-22762 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22753

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### References

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