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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 13 — Jun. 26, 2006
  • pp: 6194–6200
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Observation of nonlinear optical phenomena in air and fused silica using a 100 GW, 1.54 µm source

M. L. Naudeau, R. J. Law, T. S. Luk, T. R. Nelson, S. M. Cameron, and J. V. Rudd  »View Author Affiliations


Optics Express, Vol. 14, Issue 13, pp. 6194-6200 (2006)
http://dx.doi.org/10.1364/OE.14.006194


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Abstract

A 100-GW optical parametric chirped-pulse amplifier system is used to study nonlinear effects in the 1.54 µm regime. When focusing this beam in air, strong third-harmonic generation (THG) is observed, and both the spectra and efficiency are measured. Broadening is observed on only the blue side of the third-harmonic signal and an energy conversion efficiency of 0.2% is achieved. When propagated through a 10-cm block of fused silica, a collimated beam is seen to collapse and form multiple filaments. The measured spectral features span 400–2100 nm. The spectrum is dominated by previously unobserved Stokes emissions and broad emissions in the visible.

© 2006 Optical Society of America

1. Introduction

Since the observation of a high-power, femtosecond laser pulse filamenting in air [1

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, 73–75 (1995). [CrossRef] [PubMed]

], intensive studies, both experimental [2

2. W. Liu, S. A. Hosseini, Q. Luo, B. Ferland, S. L. Chin, O. G. Kosareva, N. A. Panov, and V. P. Kandidov, “Experimental observation and simulations of the self-action of white light laser pulse propagating in air,” New J. Phys. 6, 1–22 (2004). [CrossRef]

,3

3. A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003). [CrossRef] [PubMed]

] and theoretical [4

4. T. A. Pitts, T. S. Luk, J. K. Gruetzner, T. R. Nelson, A. McPherson, S. M. Cameron, and A. C. Bernstein, “Propagation of self-focusing laser pusles in atmosphere: experiment versus numerical simulation,” J. Opt. Soc. Am. B 21, 2008–2016 (2004). [CrossRef]

,5

5. P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418, 1–21 (2002). [CrossRef]

], have been undertaken to fully explain the nonlinear propagation of high-intensity ultrashort pulses in air. The resultant work has also spawned interest in studying atmospheric nonlinear optics in general. Since this work has relied on Ti:Sapphire-based laser systems centered on or near 800 nm, the majority of the modeling has focused on parameters at this wavelength. Despite this, some modeling looking at the interplay between continuum generation and third-harmonic generation has been done at 1.06 µm and 1.55 µm [6

6. N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003). [CrossRef]

]. Experimental work has been hampered by the lack of available high-powered ultrashort sources away from 800 nm, although some work has been done at 1.5 µm in fused silica using a low power optical parametric amplifier (OPA) [7

7. A. Saliminia, S. L. Chin, and R. Vallée, “Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 µm,” Opt. Express 13, 5731–5738 (2005). [CrossRef] [PubMed]

].

In this paper we report the observation of third-harmonic generation in air and continuum generation due to filament formation in a solid from a high-power, femtosecond, IR laser. The laser system used for these experiments is a 10 Hz, optical parametric chirped pulse amplifier (OPCPA) system that generates a nominally 1-cm diameter beam with 30 mJ per pulse in approximately 300 fs at a center wavelength of 1.54 µm [8

8. J. V. Rudd, R. J. Law, T. S. Luk, and S. M. Cameron, “High-power optical parametric chirped-pulse amplifier system with a 1.55 µm signal and a 1.064 µm pump,” Opt. Lett. 30, 1974–1976 (2005). [CrossRef] [PubMed]

]. This gives us 100 GW with which to study nonlinear phenomena, 100 times more than has previously been available at these wavelengths.

The critical power for whole beam self-focusing is:

Pcr=αλo24πn0n2,
(1)

where λ o is the vacuum wavelength, n0 is the linear index of refraction for the medium, n2 is the second-order nonlinear contribution to the index of refraction, comprising both the instantaneous (Kerr) and non-instantaneous (Raman) contributions, and α is a constant depending on the spatial beam profile that has been calculated to vary from 1.8 for a Gaussian beam to 2.8 for a super-gaussian profile [9

9. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000). [CrossRef]

]. In air (n0~1 and n2~10-19 cm2/W), the value for Pcr is approximately 35 GW. A peak pulse power of 100 GW allows the observation of significant nonlinear effects, including third-harmonic generation in air, considerable broadening of the third-harmonic signal, and ionization. Using the metric that filamentation occurs at 10*Pc, we are below the filamentation regime. We observe third-harmonic conversion efficiencies of 0.2%, which are comparable to those generated by equivalent systems centered at 800 nm [10

10. A. B. Fedetov, N. I. Koroteev, M. M. T. Loy, X. Xiao, and A. M. Zheltikov, “Saturation of third-harmonic generation in a plasma of self-induced optical breakdown due to the self-action of 80-fs light pulses,” Opt. Commun. 133, 587–595 (1997). [CrossRef]

,11

11. L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, “Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air,” Phys. Rev. E 71, 016602-1-13 (2005). [CrossRef]

]. As the THG conversion efficiencies measured have not yet reached saturation, we expect even higher conversion efficiencies at 1.54 µm than are possible at 800 nm. This is due to the reduce ionization at 1.54 µm, and is in agreement with predictions [6

6. N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003). [CrossRef]

]. Near the onset of continuum generation, the third-harmonic spectrum broadens only on the blue side.

For a 1.54 µm optical beam in fused silica, with n0=1.5, n2=4*10-16 cm2/W, and α=2, the critical power is calculated to be approximately 6.3 MW. Continuum generation in fused silica occurs after a collimated beam with P≫Pcr collapses into a disordered array of filaments in the solid. Spectral analysis of the continuum reveals very strong Stokes emission on the red side of the fundamental, and only modest broadening on the blue side.

2. Third-harmonic generation in air

The generation of third-harmonic in air due to the self-action of a high-intensity laser pulse has been observed with high power 1.06 µm systems and many 800 nm lasers, but to our knowledge, this phenomena has not been observed at any other wavelengths. Theories predict that nonlinear phenomena such as third-harmonic generation should not saturate as quickly at 1.54 µm as they do at 800 nm, due to the reduced ionization rate in air of the O2 and N2 molecules. Conversion efficiencies an order of magnitude greater than is achievable at 800 nm have been predicted [6

6. N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003). [CrossRef]

].

In order to test the predictions about the better efficiency and higher saturation values that should be observed at these wavelengths, we used a range of focal length lenses (BK7, AR coated at 1.55 µm) to focus our beam in air (see Fig. 1). The third-harmonic signal was collimated (BK7, uncoated), isolated from the fundamental using harmonic separating mirrors, a prism, and filters, and the remaining third-harmonic signal was measured with a silicon energy detector (Molectron J3-S10) for a range of input powers. Calibrated neutral density filters were used to keep the detector in the same signal range throughout the measurements. The spectral responses of all optics were measured in a spectrophotometer (Varian, Cary 5000), and the losses to the third-harmonic signal were corrected. The incident energy was varied using neutral density filters and the leakage through a high-power, 1.6 µm mirror (~0.5%) was measured before the focusing lens to provide energy calibrations on every shot taken. The preliminary results are shown in Fig. 2, where we see the energy conversion efficiency (E/Eω) as a function of input power for lenses with focal lengths of 206, 310, 460 and 618 mm. Of most interest is the fact that at no time did our third-harmonic conversion efficiency saturate; none of the data sets show signs of rolling over as we increased the pulse energy. Maximum conversion efficiencies of 0.2% were observed for some shots and as our laser system is improved, this number is expected to increase until a saturation value is reached. Saturation of the third-harmonic conversion efficiency is attributed to the clamping of the pump intensity when the beam filaments. The current maximum output of the OPCPA laser is approximately 3*Pc, which did result in ionization of the air at the highest energies and the shortest lenses. The ionization was not strong enough to provided sufficient divergence to sustain filamentation.

Fig. 1. Third-harmonic generation in air: experimental setup.
Fig. 2. Third-harmonic efficiency, in percentage, generated in air as a function of pulse energy. The four sets of data correspond to different focal length lenses used to focus the 1.54 µm beam.

The interplay between intensity and interaction length in third-harmonic generation can lead to an optimum focal length for maximum conversion efficiency [12

12. P.W. Milonni and J. H. Eberly, Lasers, (John Wiley & Sons, 1988), p 673–676.

]. As can be seen in Fig. 1, the conversion efficiency increases with longer focal length, reaching a maximum slope with the f=460 mm lens. A lens with a longer focal length (f=618 mm) produced a reduced conversion efficiency slope. This effect was observed repeatedly, although the focal length that produced the maximum slope did vary by ±100 mm with the day-to-day beam parameters.

By tightly focusing the high power beam in air using an f=100 mm lens, we generated both third-harmonic and continuum around the third-harmonic, as shown in Fig. 3. By placing an alumina screen before the collimating lens, the digital camera image in Fig. 3a was recorded. The blue halo surrounding the green third-harmonic signal is not due to a Raman line, but is in fact broadening of the third-harmonic signal, as shown in the spectra in Fig. 3b. The alumina acts as a wavelength-independent diffuse scatterer. All of the wavelengths are picked up equally with the 400 µm fiber of the Ocean Optics USB2000 spectrometer, thereby avoiding spatial spectral variations in the measurement. At this time, it is unknown whether the broadening is due to broadening of the fundamental that is then converted to the third-harmonic, or if the third-harmonic signal undergoes broadening during propagation. No energy was seen on the red side of the third-harmonic peak either visibly or in the measured spectrum. This kind of broadening around the third-harmonic peak has been predicted to be symmetric about the peak [6

6. N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003). [CrossRef]

]. A complete analysis of the third-harmonic process, including ionization threshold relative to the saturation of the third-harmonic signal is in preparation [13

13. M. L. Naudeau, J. Bodnar, R. J. Law, T. S. Luk, T. R. Nelson, M. Rivera, J. V. Rudd, and S. M. Cameron are preparing a manuscript to be called “Third harmonic generation in air as a function of wavelength, f-number, and peak power.”

].

Fig. 3. Third-harmonic generated by focusing the 100 GW beam in air with f=100 mm lens. a)Digital camera image: the blue portion is due to spectral broadening around the third-harmonic peak. b)Spectrum of the third-harmonic showing the broadening generated on the blue side of the 511 nm peak.

3. Beam collapse in fused silica

For a 1.54 µm optical beam in fused silica, with n0=1.5, n2=4*10-16 cm2/W, and α=2, the critical power is calculated (Eq. (1)) to be approximately 6.3 MW. As the 100 GW, 1-cm collimated output of the laser corresponds to a self-focusing length in fused silica of

zsf=πwo2λPPcr1=42.6cm,
(2)

the pre-compressor telescope was adjusted to induce a slight convergence of 0.31 mrad on the beam. A 10-cm block of fused silica was placed 8 meters after the telescope, where the beam diameter was .5 cm. This more favorable beam was then propagated through the fused silica block, and beam collapse was observed after traversing approximately 5–7 cm of the material. Due to the translucent sides of the block, the exact value was difficult to determine. Visual observation of the output face of the block showed a minimum of 10 separate and stable white light filaments. An alumina screen was placed 30 cm from the back face of the block, and the visible conical emission emitted by the filaments was photographed and is shown in Fig. 4a.

We measured the spectral emissions by using three different fiber-coupled spectrometers (USB2000, NIR512, and NIR256-2.1, all from Ocean Optics), the outputs of which are plotted in Fig. 4b, along with the spectrum of the laser pulse itself. To avoid detector saturation and spurious signals from the bright fundamental wavelengths, a high reflector at 1.54 µm (R>99%) attenuated the laser light and transmitted the broadened spectrum. The transmission function of the high reflector was divided out, resulting in some oscillations in the USB2000’s spectrum near 600 nm. The remaining emissions, which are too weak to generate any further nonlinearities, were focused onto an alumina screen using a 15-cm focal length, 7.5-cm diameter mirror, and recorded with the various spectrometers. This was done in a further effort to avoid measuring spatial spectral variations, as described before.

Fig. 4. White-light continuum generated from whole beam collapse during propagation through 10 cm of fused silica. a) Digital camera image taken without filters. b) Spectral data from three different spectrometers: Black: NIR256-2.1, Red: NIR512, Green: USB2000. The response function of the spectrometer is uncorrected and all intensities are relative. Dotted blue line is the original spectrum of the laser pulse.

Four features worth noting in this spectrum are:

1. The emissions on the red-side of the fundamental is very prominent, extends out past 2.1 µm, and has not previously been observed.

2. On the blue-side of the fundamental, the broadening due to self-phase modulation is quite modest. This is in contrast to propagating an 800-nm beam in fused silica [14

14. V. P. Kandidov, O. G. Kosareva, I. S. Golubtsov, W. Liu, A. Becker, 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, 149–165 (2003). [CrossRef]

].

3. The broad emission feature centered near 700 nm and the null from 950–1150 nm are similar to the features observed recently by Saliminia, et al. One difference in our observation is the high frequency cut-off around 450 nm as opposed to the 380 nm cut off they observe.

4. The presence of the fundamental is in contrast to the observations of Saliminia, et al. Presumably, this is due to the much stronger input beam that we used.

The emissions observed using a 1.54 µm source are much more prominent than are seen in super continuum generation at other wavelengths. This effect may be similar to what has been used in IR fibers to produce Raman-shifted lasers and Raman amplifiers [15

15. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1995), p. 318.

]. Since the frequency shift we observe does not overlap with the standard Raman gain curve observed in fused silica, we can infer that the pulse dynamics are more complicated than in these simpler cases.

The two features observed both in this experiment and by Saliminia, et al. (a broad peak centered near 700 nm and a spectral null located between 950 and 1150 nm) can be explained by the existing theories of super continuum generation [4

4. T. A. Pitts, T. S. Luk, J. K. Gruetzner, T. R. Nelson, A. McPherson, S. M. Cameron, and A. C. Bernstein, “Propagation of self-focusing laser pusles in atmosphere: experiment versus numerical simulation,” J. Opt. Soc. Am. B 21, 2008–2016 (2004). [CrossRef]

,16

16. M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation,” Phys. Rev. Lett. 91, 043905 (2003). [CrossRef] [PubMed]

]. It has been shown theoretically by Kolesik, et al. that these particular features are best explained by the effect the material’s chromatic dispersion has on the generated spectral components. Specifically, the increased dispersion that occurs near the UV band edge of the material leads to favorable phase matching of the generated components, while the wavelengths close to the zero-dispersion wavelength in fused silica (1.27 µm) do not phase match and are thus absent. This theory also helps explain the conical emissions that are observed and properly predicts the larger emission angles for the shorter wavelengths that are seen in Fig. 4a.

4. Summary

Using our novel 100GW, ultrafast 1.54 µm source, we have observed the first known recorded third-harmonic generation in air at 1.54 um. Conversion efficiencies comparable to those achieved at 800 nm have already been realized, despite being at energies less than the saturation energy of the THG conversion process. Spectral broadening on only the blue side of the third-harmonic signal has been observed. In addition, high average power measurements of continuum generation in fused silica have revealed a very strong Stokes emission on the red side of the fundamental, and only modest broadening on the blue side.

Acknowledgments

Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL8500.

References

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, 73–75 (1995). [CrossRef] [PubMed]

2.

W. Liu, S. A. Hosseini, Q. Luo, B. Ferland, S. L. Chin, O. G. Kosareva, N. A. Panov, and V. P. Kandidov, “Experimental observation and simulations of the self-action of white light laser pulse propagating in air,” New J. Phys. 6, 1–22 (2004). [CrossRef]

3.

A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, “Time-resolved measurements of self-focusing pulses in air,” Opt. Lett. 28, 2354–2356 (2003). [CrossRef] [PubMed]

4.

T. A. Pitts, T. S. Luk, J. K. Gruetzner, T. R. Nelson, A. McPherson, S. M. Cameron, and A. C. Bernstein, “Propagation of self-focusing laser pusles in atmosphere: experiment versus numerical simulation,” J. Opt. Soc. Am. B 21, 2008–2016 (2004). [CrossRef]

5.

P. Sprangle, J. R. Peñano, and B. Hafizi, “Propagation of intense short laser pulses in the atmosphere,” Phys. Rev. E 66, 046418, 1–21 (2002). [CrossRef]

6.

N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003). [CrossRef]

7.

A. Saliminia, S. L. Chin, and R. Vallée, “Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 µm,” Opt. Express 13, 5731–5738 (2005). [CrossRef] [PubMed]

8.

J. V. Rudd, R. J. Law, T. S. Luk, and S. M. Cameron, “High-power optical parametric chirped-pulse amplifier system with a 1.55 µm signal and a 1.064 µm pump,” Opt. Lett. 30, 1974–1976 (2005). [CrossRef] [PubMed]

9.

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000). [CrossRef]

10.

A. B. Fedetov, N. I. Koroteev, M. M. T. Loy, X. Xiao, and A. M. Zheltikov, “Saturation of third-harmonic generation in a plasma of self-induced optical breakdown due to the self-action of 80-fs light pulses,” Opt. Commun. 133, 587–595 (1997). [CrossRef]

11.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, “Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air,” Phys. Rev. E 71, 016602-1-13 (2005). [CrossRef]

12.

P.W. Milonni and J. H. Eberly, Lasers, (John Wiley & Sons, 1988), p 673–676.

13.

M. L. Naudeau, J. Bodnar, R. J. Law, T. S. Luk, T. R. Nelson, M. Rivera, J. V. Rudd, and S. M. Cameron are preparing a manuscript to be called “Third harmonic generation in air as a function of wavelength, f-number, and peak power.”

14.

V. P. Kandidov, O. G. Kosareva, I. S. Golubtsov, W. Liu, A. Becker, 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, 149–165 (2003). [CrossRef]

15.

G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1995), p. 318.

16.

M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation,” Phys. Rev. Lett. 91, 043905 (2003). [CrossRef] [PubMed]

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4970) Nonlinear optics : Parametric oscillators and amplifiers
(320.7090) Ultrafast optics : Ultrafast lasers
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: April 24, 2006
Revised Manuscript: June 7, 2006
Manuscript Accepted: June 9, 2006
Published: June 26, 2006

Citation
M. L. Naudeau, R. J. Law, T. S. Luk, T. R. Nelson, S. M. Cameron, and J. V. Rudd, "Observation of nonlinear optical phenomena in air and fused silica using a 100 GW, 1.54 μm source," Opt. Express 14, 6194-6200 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6194


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References

  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, 73-75 (1995). [CrossRef] [PubMed]
  2. W. Liu, S. A. Hosseini, Q. Luo, B. Ferland, S. L. Chin, O. G. Kosareva, N. A. Panov, and V. P. Kandidov, "Experimental observation and simulations of the self-action of white light laser pulse propagating in air," New J. Phys. 6, 1-22 (2004). [CrossRef]
  3. A. C. Bernstein, J. C. Diels, T. S. Luk, T. R. Nelson, A. McPherson, and S. M. Cameron, "Time-resolved measurements of self-focusing pulses in air," Opt. Lett. 28, 2354-2356 (2003). [CrossRef] [PubMed]
  4. T. A. Pitts, T. S. Luk, J. K. Gruetzner, T. R. Nelson, A. McPherson, S. M. Cameron, and A. C. Bernstein, "Propagation of self-focusing laser pusles in atmosphere: experiment versus numerical simulation," J. Opt. Soc. Am. B 21, 2008-2016 (2004). [CrossRef]
  5. P. Sprangle, J. R. Peñano, and B. Hafizi, "Propagation of intense short laser pulses in the atmosphere," Phys. Rev. E 66, 046418, 1-21 (2002). [CrossRef]
  6. N. Aközbek, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, "Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air," Appl. Phys. B 77, 177-183 (2003). [CrossRef]
  7. A. Saliminia, S. L. Chin, and R. Vallée, "Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 µm," Opt. Express 13, 5731-5738 (2005). [CrossRef] [PubMed]
  8. J. V. Rudd, R. J. Law, T. S. Luk, and S. M. Cameron, "High-power optical parametric chirped-pulse amplifier system with a 1.55 µm signal and a 1.064 µm pump," Opt. Lett. 30, 1974-1976 (2005). [CrossRef] [PubMed]
  9. G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000). [CrossRef]
  10. A. B. Fedetov, N. I. Koroteev, M. M. T. Loy, X. Xiao, and A. M. Zheltikov, "Saturation of third-harmonic generation in a plasma of self-induced optical breakdown due to the self-action of 80-fs light pulses," Opt. Commun. 133, 587-595 (1997). [CrossRef]
  11. L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602-1-13 (2005). [CrossRef]
  12. P.W. Milonni and J. H. Eberly, Lasers, (John Wiley & Sons, 1988), p 673-676.
  13. M. L. Naudeau, J. Bodnar, R. J. Law, T. S. Luk, T. R. Nelson, M. Rivera, J. V. Rudd, and S. M. Cameron are preparing a manuscript to be called "Third harmonic generation in air as a function of wavelength, f-number, and peak power."
  14. V. P. Kandidov, O. G. Kosareva, I. S. Golubtsov, W. Liu, A. Becker, 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, 149-165 (2003). [CrossRef]
  15. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 1995), p. 318.
  16. M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, "Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation," Phys. Rev. Lett. 91, 043905 (2003). [CrossRef] [PubMed]

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