OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 28980–28986
« Show journal navigation

Optical beam dynamics in a gas repetitively heated by femtosecond filaments

N. Jhajj, Y.-H. Cheng, J. K. Wahlstrand, and H. M. Milchberg  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 28980-28986 (2013)
http://dx.doi.org/10.1364/OE.21.028980


View Full Text Article

Acrobat PDF (926 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We investigate beam pointing dynamics in filamentation in gases driven by high repetition rate femtosecond laser pulses. Upon sudden exposure of a gas to a kilohertz train of filamenting pulses, successive filaments are steered from their original direction to a new stable direction whose equilibrium is determined by a balance among buoyant, viscous, and diffusive processes in the gas. The beam mode is preserved. Results are shown for Xe and air, but are broadly applicable to all configurations employing intense, high repetition rate femtosecond laser pulses in gases.

© 2013 Optical Society of America

In this paper, we show that this long-lived density depression, coupled with convective motion of the heated gas, leads to a reproducible deflection of the filamenting beam. We present time-resolved measurements of the accumulating kHz pulse-train-driven gas density depression inside a gas cell and the associated transient beam deflection. We find that the deflection is well-described by a simple model of beam refraction associated with the evolution of the gas density hole, whose time and space scales are well described by a simple fluid analysis. The results presented here provide quantitative understanding of thermal effects on beam propagation of femtosecond filaments, which will point the way to both stabilization and long range control of the filamentation process. Results are shown for Xe and air, but are broadly applicable to all configurations employing high repetition rate intense femtosecond laser propagation in gases.

Formation of a gas density hole by a single pulse is first briefly reviewed. Depending on the focusing f-number, a variable length plasma is generated either through lens-dominated focusing or by filamentation [1

1. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

]. Over a timescale depending on the gas type and pressure (~10 ns for air at 1 atm), the plasma then recombines and repartitions most of its initial thermal energy into translational and rotational degrees of freedom of the neutral gas. After ~1 µs the gas reaches pressure equilibrium, and a hot, low density channel occupies roughly the same volume as the original plasma [9

9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

]. The gas begins to thermally equilibrate through diffusion, and the density depression expands and fills in over a time scale of milliseconds.

Here, we consider a situation in which a shutter is opened suddenly and a 1 kHz pulse train is focused into a gas cell. The first pulse experiences a uniform gas density. The time separation between successive pulses is short enough that each pulse, which itself heats the gas, is affected by the cumulative density hole, which acts as a negative lens. As the hole evolves, the pulses in the sequence are steered, eventually reaching a steady state deflection. Here we measure and quantify the hole evolution and beam deflection.

The experimental setup is shown in Fig. 1
Fig. 1 Experimental setup: the pump and probe travel collinearly through the gas cell. The probe beam is relay imaged from the axial center of the plasma where the density hole is produced.
. A 1 kHz pump pulse train, apertured to 4mm with 45 fs, 120 µJ pulses was focused by a 60 cm lens into a gas cell filled with air or xenon at 1 atm. An additional setup was employed where a 40 cm lens focused the beam into a 2.7 atm gas cell also filled with xenon. The latter is the same setup and gas cell we use for supercontinuum generation for our spectral interferometry experiments [10

10. J. K. Wahlstrand, Y.-H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers-Kronig relations,” Phys. Rev. Lett. 109(11), 113904 (2012). [CrossRef] [PubMed]

]. A fast shutter synchronized to the laser system was employed to create a finite pulse train. To capture the transverse location of the pump beam in the far field we routed the output beam, attenuated using neutral density filters, from the gas cell onto a quadrant detector. The beam spot size on the 7.8 mm diameter quadrant detector was ~4 mm. The recorded beam deflections were downward.

Gas density evolution was measured using a folded wavefront interferometer with a CW HeNe laser probe beam [9

9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

]. The probe propagated collinearly with the pump pulse train, and was relay imaged from the axial center of the plasma, through a folded wavefront interferometer and onto a CCD. The phase shift from the density perturbation was then extracted from the resulting interferogram using standard interferometric techniques [11

11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

]. The phase shift was converted to gas density using the known linear refractive index of Xe [12

12. A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London A Math. Phys. Sci. 259(1298), 424–431 (1960). [CrossRef]

]. In the case of the 2.7 atm cell, the probe beam propagated at a 5 degree angle with respect to the pump, and Abel inversion was used to extract the size and shape of the density perturbation. Temporal resolution was achieved by time-gating the CCD and triggering the electronic exposure in order to sample the perturbation after a given pulse in the pulse train. The temporal resolution was limited by the minimum CCD exposure duration of ~40 μs, which was short compared to the density profile evolution diffusive timescale as previously measured [9

9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

].

Results from the interferometry experiment particularly useful for understanding beam deflection are shown in Fig. 3
Fig. 3 Top panel: Image of deflected white light supercontinuum spot as a function of pulse number in the 1 kHz pulse train. Lower panels: Hole parameters as a function of pulse number in the pulse train. The fast shutter is opened just before pulse 1. Equilibrium is approached by about the 250th pulse (at 250 ms).The inset in the second panel is a typical relative density profile extracted from the interferogram.
. Here we use data taken from the 2.7 atm xenon gas cell setup, where the filament is ~3 cm long. Plotted are the hole vertical displacement relative to the pulse train, the hole depth, and the hole half-width-at-half-maximum (HWHM). Each point is an average of 5 measurements. Shot to shot variations were small.

The hole displacement was measured by sampling the cumulative density hole just before and after a pulse arrived. The location of the newest pulse is marked by a small, sharp dip in the density superimposed on the larger cumulative density hole, as seen in Fig. 2. The hole depth and HWHM were extracted from an Abel inverted lineout of the interferogram and were measured 100 µs before a pulse arrives. As can be seen from Fig. 2, the hole is significantly more extended above than below its deepest point. Since the hole drifts upwards and the beam deflects downwards, it is the bottom portion of the hole that is relevant to the deflection dynamics, and it is the HWHM of the bottom side of the hole shown plotted in Fig. 3. Also displayed in the top panel of Fig. 3 is a sequence of filament white light beam images in the far field as a function of pulse number in the sequence, showing progressive deflection with negligible beam distortion. Furthermore, we use these beams in our spectral interferometry experiments [10

10. J. K. Wahlstrand, Y.-H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers-Kronig relations,” Phys. Rev. Lett. 109(11), 113904 (2012). [CrossRef] [PubMed]

] and detect no phase front distortion from the density hole-induced steering.

We model the beam deflection by the density hole using the ray propagation equation, dds(ndrds)=n, applied in the vertical plane, where s is the optical path along the 2D ray trajectory, r=zz^+xx^ points to the ray tip, with along and vertically perpendicular to the propagation direction, and where n=n0+Δn is the refractive index. For our situation of a small refractive index perturbation |Δn/n0|<<1, no~1 and small ray deflection angles |dx(z)/z|<<1 at large z, this equation simplifies to

d2xdz2=dΔndx
(1)

The heated density depression relaxes through thermal diffusion [9

9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

], and so the index perturbation is approximately Gaussian:
Δn=Δn0exp(βx2γz2)
(2)
Here β and γ characterize the length scales of the perturbation in the x and z directions respectively, and Δno (<0 for a density hole) gives the maximum amplitude of the perturbation. In the experiment, the axial extent of the density hole (∝ γ-1/2) of ~3 cm was measured by translating the object plane of the imaging system along the axis of the hole, and Δno and β were obtained by fitting the measured density hole to a Gaussian. Light ray trajectories are linear outside of the density hole region, and so the problem can be treated as classical scattering, where the amplitude and dimensions of the index perturbation, impact parameter, and deflection (scattering) angle are the relevant variables. The deflection angle Δθ is given by integrating Eq. (1), using Eq. (2) as the index profile:
dxdz|z=dxdz|z==Δθ=dΔn(x(z),z)dxdz4πΔn0βx0γeβx02
(3)
The incident ray is assumed to be initially travelling parallel to the optical axis with impact parameter xo, where in the experiment xo corresponds to the density hole displacement.

Figure 4
Fig. 4 Far field beam deflection angle versus pulse number in 1 kHz plus train. Blue squares: measurement; red squares: ray optics calculation using measured gas density profiles.
shows good agreement between the measured deflection and deflection from the ray optics model. The deviation at earliest times is due to uncertainty in the density hole position for small displacements. The ray optics model contains no free parameters and uses the density hole parameters from Fig. 3 measured simultaneously with the deflection. That this model works so well for filaments is consistent with density hole lensing of the filament ‘reservoir,’ the lower intensity beam surrounding the filament that exchanges power with the filament core [13

13. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23(5), 382–384 (1998). [CrossRef] [PubMed]

]. The beam deflection reaches steady state in ~250 ms with amplitude of ~3.5 mrad. Note that the magnitude of the steady state deflection is reproduced by the model, while the time to steady state agrees with the hole parameter plots of Fig. 3. As can be seen from Fig. 3, the density hole width only increases by a factor of 2 or 3, while the depth and displacement increase by very large factors. The widening of the hole quickly stagnates on the bottom side as convection begins balancing thermal diffusion, preventing heat from spreading downward, as seen in Fig. 2. As a result, the growth in the deflection angle with time is primarily driven by the density hole depth and displacement, and follows their relaxation timescale.

For our experiment in Xe, κ = 5.65∙10−3 W∙m−1∙K−1 [14], µ = 23 µPa∙s [14], m = 2.2∙10−25 kg, N ~ 7∙1019 cm−3 at 2.7 atm, and we take a hole depth of Γ~0.10 from Fig. 3. This gives a limiting gas flow velocity of u ~15 µm/ms and Leff ~150 µm. The approximate time to reach this steady state is given by Δt~u/a, where a=Γg is the acceleration from the buoyancy force. Examination of Fig. 3 shows that in the early phase of hole displacement, Γ ~0.01-0.05, giving Δt ~20-100 ms, a range of timescales in reasonable agreement with the approach to steady state shown in Figs. 3 and 4.

In an alternative, essentially kinematic approach for estimating these space and time scales, we take an estimate of the initial height rise of the hole due to buoyancy, drise~12Γgt2 and set it equal to the approximate hole radius as determined by thermal diffusion, dthermal~2αt [9

9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

], the idea being that the thermal source location cannot move below the expanding hole width as the hole rises. Here, t is the elapsed time after the laser shutter is opened and α=κ/(52NkB) is the thermal diffusivity. The result is t=trise~(4/(Γg))2/3α1/3 and drise~(32α2/(Γg))1/3. Using the above parameters gives trise ~100 ms and d ~1 mm, in reasonable agreement with the experiment and the prior scale estimates.

In summary, we have examined the propagation of a high repetition rate filament pulse train in the transiently evolving gas that it heats. When a filament pulse train is suddenly initiated in a gas, the local density is reduced, driving buoyant motion of the gas. The upward drift of the gas density hole steers subsequent pulses downward. Eventually the buoyant motion is damped by viscous forces, establishing a steady gas flow field and hole density profile, and the downward beam steering stabilizes. Unlike in conventional thermal blooming with high power CW lasers, the beam mode is preserved. Further, a simple ray model explains the transient deflection of the beam by the accumulated density hole, and simple fluid analysis explains the experimentally observed space and time scales for the density hole dynamics and beam steering.

Acknowledgments

References and links

1.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2–4), 47–189 (2007). [CrossRef]

2.

N. Aközbek, 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(14), 143901 (2002). [CrossRef] [PubMed]

3.

N. Kortsalioudakis, M. Tatarakis, N. Vakakis, S. D. Moustaizis, M. Franco, B. Prade, A. Mysyrowicz, N. 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 80(2), 211–214 (2005). [CrossRef]

4.

P. B. Corkum, C. Rolland, and T. Srinivasan-Rao, “Supercontinuum generation in gases,” Phys. Rev. Lett. 57(18), 2268–2271 (1986). [CrossRef] [PubMed]

5.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004). [CrossRef]

6.

G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett. 31(2), 274–276 (2006). [CrossRef] [PubMed]

7.

D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE 65(12), 1679–1714 (1977). [CrossRef]

8.

V. V. Vorob’ev, “Thermal blooming of laser beams in the atmosphere,” Prog. Quantum Electron. 15(1–2), 1–152 (1991). [CrossRef]

9.

Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef] [PubMed]

10.

J. K. Wahlstrand, Y.-H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers-Kronig relations,” Phys. Rev. Lett. 109(11), 113904 (2012). [CrossRef] [PubMed]

11.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

12.

A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London A Math. Phys. Sci. 259(1298), 424–431 (1960). [CrossRef]

13.

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23(5), 382–384 (1998). [CrossRef] [PubMed]

14.

http://webbook.nist.gov/chemistry/fluid/

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(260.5950) Physical optics : Self-focusing
(350.6830) Other areas of optics : Thermal lensing
(320.6629) Ultrafast optics : Supercontinuum generation

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 17, 2013
Manuscript Accepted: October 28, 2013
Published: November 15, 2013

Citation
N. Jhajj, Y.-H. Cheng, J. K. Wahlstrand, and H. M. Milchberg, "Optical beam dynamics in a gas repetitively heated by femtosecond filaments," Opt. Express 21, 28980-28986 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28980


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.441(2–4), 47–189 (2007). [CrossRef]
  2. N. Aközbek, 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(14), 143901 (2002). [CrossRef] [PubMed]
  3. N. Kortsalioudakis, M. Tatarakis, N. Vakakis, S. D. Moustaizis, M. Franco, B. Prade, A. Mysyrowicz, N. 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. B80(2), 211–214 (2005). [CrossRef]
  4. P. B. Corkum, C. Rolland, and T. Srinivasan-Rao, “Supercontinuum generation in gases,” Phys. Rev. Lett.57(18), 2268–2271 (1986). [CrossRef] [PubMed]
  5. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79(6), 673–677 (2004). [CrossRef]
  6. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett.31(2), 274–276 (2006). [CrossRef] [PubMed]
  7. D. C. Smith, “High-power laser propagation: Thermal blooming,” Proc. IEEE65(12), 1679–1714 (1977). [CrossRef]
  8. V. V. Vorob’ev, “Thermal blooming of laser beams in the atmosphere,” Prog. Quantum Electron.15(1–2), 1–152 (1991). [CrossRef]
  9. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express21(4), 4740–4751 (2013). [CrossRef] [PubMed]
  10. J. K. Wahlstrand, Y.-H. Cheng, and H. M. Milchberg, “High field optical nonlinearity and the Kramers-Kronig relations,” Phys. Rev. Lett.109(11), 113904 (2012). [CrossRef] [PubMed]
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982). [CrossRef]
  12. A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. R. Soc. London A Math. Phys. Sci.259(1298), 424–431 (1960). [CrossRef]
  13. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett.23(5), 382–384 (1998). [CrossRef] [PubMed]
  14. http://webbook.nist.gov/chemistry/fluid/

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited