Simple method of measuring laser peak intensity inside femtosecond laser filament in air

Measurement of laser intensity inside a femtosecond laser filament is a challenging task. In this work, we suggest a simple way to characterize laser peak intensity inside the filament in air. It is based on the signal ratio measurement of two nitrogen fluorescence lines, namely, 391 nm and 337 nm. Because of distinct excitation mechanisms, the signals of the two fluorescence lines increase with the laser intensity at different orders of nonlinearity. An empirical formula has been deduced according to which laser peak intensity could be simply determined by the fluorescence ratio. ©2011 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (260.5950) Self-focusing; (300.6410) Spectroscopy, multiphoton. References and links 1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995). 2. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003). 3. V. P. Kandidov, O. G. Kosareva, I. S. Golubtsov, W. Liu, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Self-transformation of a powerful femtosecond laser pulse into a white-light laser pulse in bulk optical media (or supercontinuum generation),” Appl. Phys. B 77(2-3), 149–165 (2003). 4. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005). 5. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47– 189 (2007). 6. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007). 7. J. Kasparian and J. P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16(1), 466–493 (2008). 8. P. Rohwetter, J. Kasparian, K. Stelmaszczyk, Z. Q. Hao, S. Henin, N. Lascoux, W. M. Nakaema, Y. Petit, M. Quießer, R. Salamé, E. Salmon, L. Wöste, and J. P. Wolf, “Laser-induced water condensation in air,” Nat. Photonics 4(7), 451–456 (2010). 9. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000). 10. H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using self-guided femtosecond pulses,” Phys. Rev. Lett. 81(8), 1611–1613 (1998). 11. G. Méchain, A. Couairon, Y. B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79(3), 379–382 (2004). 12. A. Talebpour, M. Abdel-Fattah, A. D. Bandrauk, and S. L. Chin, “Spectroscopy of the gases interacting with intense femtosecond laser pulses,” Laser Phys. 11, 68–76 (2001). #157447 $15.00 USD Received 2 Nov 2011; revised 26 Nov 2011; accepted 26 Nov 2011; published 20 Dec 2011 (C) 2012 OSA 2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 299 13. A. Talebpour, A. D. Bandrauk, J. Yang, and S. L. Chin, “Multiphoton ionization of inner-valence electrons and fragmentation of ethylene in an intense Ti:sapphire laser pulse,” Chem. Phys. Lett. 313(5-6), 789–794 (1999). 14. A. Becker, A. D. Bandrauk, and S. L. Chin, “S-matrix analysis of non-resonant multiphoton ionisation of innervalence electrons of the nitrogen molecule,” Chem. Phys. Lett. 343(3-4), 345–350 (2001). 15. H. L. Xu, A. Azarm, J. Bernhardt, Y. Kamali, and S. L. Chin, “The mechanism of nitrogen fluorescence inside a femtosecond laser filament in air,” Chem. Phys. 360(1-3), 171–175 (2009). 16. A. Talebpour, S. Petit, and S. L. Chin, “Re-focusing during the propagation of a focused femtosecond Ti:Sapphire laser pulse in air,” Opt. Commun. 171(4-6), 285–290 (1999). 17. S. Akturk, C. D'Amico, M. Franco, A. Couairon, and A. Mysyrowicz, “Pulse shortening, spatial mode cleaning, and intense terahertz generation by filamentation in xenon,” Phys. Rev. A 76(6), 063819 (2007). 18. F. Théberge, N. Aközbek, W. Liu, A. Becker, and S. L. Chin, “Tunable ultrashort laser pulses generated through filamentation in gases,” Phys. Rev. Lett. 97(2), 023904 (2006). 19. W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13(15), 5750–5755 (2005). 20. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 1995). 21. L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002). 22. M. D. Perry, O. L. Landen, A. Szöke, and E. M. Campbell, “Multiphoton ionization of the noble gases by an intense 10W/cm dye laser,” Phys. Rev. A 37(3), 747–760 (1988). 23. H. G. Roskos, M. D. Thomson, M. Kreß, and T. Löffler, “Broadband THz emission from gas plasmasinduced by femtosecond optical pulses: From fundamentals to applications,” Laser Photonics Rev. 1(4), 349–368 (2007). 24. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N and O molecule in an intense Ti: sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999). 25. P. E. Gill and W. Murray, “Algorithms for the solution of the nonlinear least-squares problem,” SIAM J. Numer. Anal. 15(5), 977–992 (1978). 26. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). 27. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009). 28. P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16(20), 15733–15740 (2008). 29. O. G. Kosareva, A. V. Grigorevskii, and V. P. Kandidov, “Formation of extended plasma channels in a condensed medium upon axicon focusing of a femtosecond laser pulse,” Quantum Electron. 35(11), 1013–1014 (2005). 30. S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009). 31. G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, “Control of multiple filamentation in air,” Opt. Lett. 29(15), 1772–1774 (2004). 32. Y. X. Fu, H. Gao, W. Chu, J. L. Ni, H. Xiong, H. Xu, J. P. Yao, B. Zeng, W. Liu, Y. Cheng, Z. Z. Xu, and S. L. Chin, “Control of filament branching in air by astigmatically focused femtosecond laser pulses,” Appl. Phys. B 103(2), 435–439 (2011). 33. B. Alonso, I. J. Sola, J. S. Román, O. Varela, and L. Roso, “Spatiotemporal evolution of light during propagation in filamentation regime,” J. Opt. Soc. Am. B 28(8), 1807–1816 (2011). 34. F. Martin, R. Mawassi, F. Vidal, I. Gallimberti, D. Comtois, H. Pépin, J. C. Kieffer, and H. P. Mercure, “Spectroscopic study of ultrashort pulse laser-breakdown plasmas in air,” Appl. Spectrosc. 56(11), 1444–1452 (2002). 35. W. Liu, Q. Luo, and S. L. Chin, “Competition between multiphoton/tunnel ionization and filamentation induced by powerful femtosecond laser pulses in air,” Chin. Opt. Lett. 1, 56–58 (2003). 36. A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and refocusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B 73, 287–290 (2001). 37. X. L. Liu, X. Lu, X. Liu, T. T. Xi, F. Liu, J. L. Ma, and J. Zhang, “Tightly focused femtosecond laser pulse in air: from filamentation to breakdown,” Opt. Express 18(25), 26007–26017 (2010). 38. P. P. Kiran, S. Bagchi, C. L. Arnold, S. R. Krishnan, G. R. Kumar, and A. Couairon, “Filamentation without intensity clamping,” Opt. Express 18(20), 21504–21510 (2010). 39. P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of. multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82(1), 013805 (2010). 40. X. L. Liu, X. Lu, X. Liu, L. B. Feng, J. L. Ma, Y. T. Li, L. M. Chen, Q. L. Dong, W. M. Wang, Z. H. Wang, Z. Y. Wei, Z. M. Sheng, and J. Zhang, “Broadband supercontinuum generation in air using tightly focused femtosecond laser pulses,” Opt. Lett. 36(19), 3900–3902 (2011).


Introduction
Femtosecond laser filamentation in air has attracted considerable interest not only because of the nonlinear optical processes involved, including self-focusing, photoionization, intensity clamping, self-phase modulation, self-steeping, space-time focusing, etc, but also due to its wide range of applications in remote-sensing, light frequency converting, laser-based weather control and so on [1][2][3][4][5][6][7][8].Nevertheless, during the study of filamentation there are some experimental difficulties in diagnosing the laser pulse.It is mainly due to the high laser intensity inside the filament core unsustainable by conventional measuring instruments.Particularly, it is not trivial to measure the laser intensity inside the filament.Kasparian et al. have semi-empirically estimated that the peak intensity reaches 4 × 10 13 W/cm 2 during filamentation in air for a laser pulse of 100 fs duration [9].This prediction is in good agreement with the value of 4.5 × 10 13 W/cm 2 determined from the cut-off frequency of the high order harmonic spectrum generated by the filamentating pulse [10].The laser intensity could also be inferred by calibrating the grey level of the burn spot left by single-shot pulse on burn paper [11].However, the interpretation of laser intensity during filamentation still relies on numerical simulation to a large extent [1][2][3][4][5][6][7].
In the present work, we introduce a simple method to measure the laser peak intensity during the filamentation in air.The central idea is to measure the laser intensity dependent ratio of the signal strength of two nitrogen fluorescence lines, namely, 337nm and 391 nm, which are assigned to the second positive band of ( ) N C B Π → Π and the first negative band system of ( ) Σ → Σ , respectively [12].

Experiments and results
It has been revealed that the population of 2 u B + Σ state results from the direct ionization of inner valence electrons [13,14].The following (0-0) transition gives rise to the strongest band head of the first negative system at 391 nm.Hence, the 391nm signal 391 S is proportional to the ionization probability of direct inner valence electron ionization, experimentally yielding [13,14] [12,15].Hence, the fluorescence emission associated with the second positive band system of ( ) N C B Π → Π , such as 337 nm, is proportional to the total number of ions.The total number of ions is the sum of the ions in the ground state and in the excited state, the latter through inner valence electron ionization.They are denoted by ground excited i i N N + respectively [15,16].The ground state ions come from the normal ionization of the electron with the lowest ionization potential which results in the generation of ground state ions, ground i N . We thus have the following relationship where 337 S denotes the fluorescence signal at 337 nm, total i N , the total number of ions, ground i N , the number of ions in the ground state, n2 the effective order of nonlinearity of ionizing the molecule into the ground state ion and b, a proportionality constant.Hence, ) where k is a proportionality constant.Finally, we obtain the ratio R between the two fluorescence strengths given by the following.
When the temporal and spatial laser intensity profile is taken into account, Eq. (3) will be transformed to: where I 0 corresponds to the peak intensity of the pulse and ( , ) f r t represents the normalized intensity distribution in cylindrical coordinate.Because of beam cleaning [17,18], the spatial intensity distribution inside the filament core is the lowest fundamental mode.Thus, Eq. ( 4) is evaluated by assuming Gaussian shape in both time and space domains, yielding: which is essentially independent of the laser shape, neither temporally nor spatially.Equation ( 6) could be further revised as a simplified form: Therefore, the laser peak intensity I 0 could be measured according to the line ratio R of 391nm and 337 nm once the constants α, β and m in Eq. ( 7) were determined.Our experimental approach of accomplishing this task is the following.
The experiment was carried out by using a commercial femtosecond laser system.The output laser pulses featuring a transform limited duration of 42 fs (FWHM) were focused in air by various lenses with different focal lengths, namely, 100 cm, 50 cm, 30 cm, 20 cm, 11 cm.For the sake of homogeneous attenuation of the input laser intensity across the beam pattern which is about 1 cm (1/e 2 ) in diameter, we did not use commercialized variable neutral density filter.On the other hand, the nonlinear length during the propagation inside fused silica is more than 10 cm according to our estimation.Since in our experiment the propagation distance is only a few millimeters inside the filters, the propagation inside the filters would not give rise to significant effect on the filamentation process in air.Whereas, during our experiment, the laser energy was varied by using different neutral density filters.Depending on the focal length and laser energy, plasma with different lengths and diameters could be produced in the experiment.It is necessary to point out when the input laser peak power is below the critical power for self-focusing (in air, it is about 10 GW for 42 fs pulse [19]), the laser propagation is dominated by dispersion and diffraction.In this regime, laser intensity could be calculated as long as the pulse length and beam diameter were known.Therefore, the laser power in our experiment was limited to be below the critical power for self-focusing.
When only linear propagation is considered, pulse lengthening during propagation mainly occurs in the neutral density filters and lenses, which are all made of fused silica substrate in where τ, τ 0 , z, k 2 represent the pulse duration and the input pulse duration (both at FWHM), the propagation distance in fused silica and the corresponding group-velocity-dispersion (GVD) parameter, respectively.Note that 2 2 361fs cm k = in fused silica [21].Furthermore, focal diameters were deduced from the plasma column diameter measurement in our experiment.Figure 1(a) demonstrates a typical picture of the plasma column imaged by a 20 X microscopic objective on a CCD camera installed perpendicularly to the propagation axis.In this case, the input energy was 0.21 mJ and a f = 30 cm lens was implemented.The plasma diameter at the focus (indicated by the dashed line) can be deduced from the signal cross-section, for example d plasma = 30.6µm(FWHM) as shown in Fig. 1(b).Note that the broadening of the plasma column diameter associated with the limited resolution of the image system has been subtracted via the de-convolution method.The obtained plasma column diameters are plotted in Fig. 2. It is noticed from Fig. 2 that the plasma column diameter does not keep constant in our experiment.It can be explained by plasma defocusing effect when the laser pulse approaches the focus [22,23].Note that Fig. 2  where, Ne(r) denotes the spatial distribution of plasma density, I(r) corresponds to the laser intensity distribution.σ and n are cross section of ionization and effective nonlinear order of photo-ionization.Assuming a Gaussian laser intensity profile written as: (w is the beam radius at 1/e 2 ), Eq. ( 9) leads to:  Since the plasma density increases with an effective power law of 7.5 as a function of the laser intensity in air [9,24], the laser diameter at focus is given by d 7.5 .
In this way, the laser diameters were estimated for all the lenses and laser energies used in this experiment.Thereby, the laser peak intensity is obtained according to: Here, E indicates the input laser energy.It is worth mentioning again that the beam diameter measurement was carried out when the input laser energy was below the threshold of self-focusing, retaining linear propagation.Thus, laser peak intensities at the focus can be calculated with the known pulse durations and beam diameters in the experiments.Note that an imprecise value of n in Eq. ( 11) would introduce a systematic error into the derivation of laser diameter.Simultaneously to the diameter measurement, a narrow region of the focus (indicated by the dashed line in Fig. 1(a)) was imaged onto the slit of a spectrometer.The nitrogen fluorescence spectrum was measured and the strength ratio R between 391nm line and 337 nm line was investigated.In the experiment, the transmission coefficients of the microscopic objective and the spectral response of the spectrometer were carefully taken into account.Three typical recorded nitrogen fluorescence spectra are illustrated in Fig. 1(c).These spectra feature "continuum free" as reported previously [12].Then R is plotted as a function of the input laser energy as shown in Fig. 3. Intensity clamping phenomenon is clearly noticed in Fig. 3.  On the next setup, the horizontal axis of Fig. 3 is converted to the calculated laser peak intensity according to Eq. ( 12).The outcome is displayed in Fig. 4, for f=100 cm, 50 cm, 30 cm, 20 cm, 11 cm, respectively.The data depicted in Fig. 4 could be fitted by Eq. ( 7) through nonlinear least squares fitting algorithm [25], The best fitting curve is indicated as the black solid line in Fig. 4, which establishes an empirical relationship between the laser peak intensity I and the quantity of R: Fig. 4. Signal ratio of R391nm/337nm of nitrogen fluorescence as a function of laser peak intensity for different focal length lenses, the fitting curve according to Eq. ( 7) is indicated as the black solid line.Red and blue dotted lines correspond to the upper and lower limit of the fitting, respectively.
In order to set an example of the application of Eq. ( 13), we have increased the laser energy exceeding the critical power for self-focusing and focused it by the same group of lenses.Thereby, the values of R were measured and the corresponding laser peak intensities were deduced from Eq. ( 13).The obtained intensities are shown in Fig. 5 as a function of the input laser energy.Figure 5 indicates that the peak intensity clamps at 4 × 10 13 W/cm 2 when f = 100 cm.The clamping intensity increases with decreasing focal length.Eventually, for the shortest focal length we have used (f = 11 cm), the clamping intensity is more than 1× 10 14 W/cm 2 .
Fig. 5. Laser peak intensities deduced from Eq. ( 13) as a function of input laser energy.
Finally, it is worth pointing out that our method is applicable when the following issues have been considered.First, the laser pulse has been assumed to be Gaussian spatially and temporally in order to deduce Eq. ( 7).However, it might not be fulfilled in experiment either because of the non-perfect laser output or artificial beam shaping, such as synthesizing Airy beam [26,27], focusing by axicon [28][29][30], or introducing astigmatism [31,32].Especially, due to the strong self-transformation during femtosecond laser filamentation, the laser pulse may not preserve Gaussian profile in time and space as what has been revealed in [33].Hence, the peak intensity given by our technique is a good approximation with the advantage of significant simplification.In addition, fluorescence spectra used in our experiment all contain negligible contribution of continuum.However, considerable plasma continuum has been observed when high energy femtosecond laser is tightly focused in air [34,35].In this case, fluorescence spectrum could be practically isolated from the continuum background by the time-gating technique [34].Besides, the straightforward nonlinear relationship between the laser intensity and plasma density as indicated by Eq. ( 1) may not hold anymore.Saturation of ionization associated with the neutral molecular depletion [36] and double ionization process might need to be taken into account [37].By implementing the advanced experimental approach and more sophisticated theoretical model, our technique might be valuable in clarifying the recently raised controversy regarding the achieved laser intensity when focusing high energy laser pulse with tight focusing geometry [38][39][40].

Summary
In conclusion, we have established an empirical formula between the laser peak intensity and the strength ratio of two nitrogen fluorescence lines, namely 391 nm and 337 nm during femtosecond laser filamentation in air.This formula provides a simple method to measure the laser peak intensity inside a femtosecond laser filament in air up to 10 14 W/cm 2 .Within the applicable regime, the proposed method relies on no other further calibration except the fluorescence ratio measurement.By using the reported method, longitudinal laser intensity distribution during filamentation process could be obtained through scanning the fluorescence spectra along the propagation axis.Not only for the case of single filament, but also the intensity distribution of multiple filaments could be traced as long as spatially resolved #157447 -$15.00USD Received 2 Nov 2011; revised 26 Nov 2011; accepted 26 Nov 2011; published 20 Dec 2011 (C) 2012 OSA 2 January 2012 / Vol. 20, No. 1 / OPTICS EXPRESS 302 our experiment.The pulse duration variation induced by the dispersion in fused silica is then given by [20]:
indeed depicts the averaged result of 10 plasma column diameter measurements.The error of the data in Fig. 2 might be induced by the energy fluctuation of the laser pulse, causing plasma density variation.Moreover, if only considering the spatial distribution, plasma density is related to the laser $15.00 USD Received 2 Nov 2011; revised 26 Nov 2011; accepted 26 Nov 2011; published 20 Dec 2011 (C) 2012 OSA

Fig. 2 .
Fig. 2. Plasma column diameters (FWHM) as a function of the initial laser pulse energy for different focal length lenses.

Fig. 3 .
Fig. 3. Signal ratio of R391nm/337nm of nitrogen fluorescence as a function of input laser energy for different focal length lenses.