Open Access
Add to BibSonomyAbstract
The spectral and the temporal behavior of high-intensity negatively chirped femtosecond pulses in normally dispersive media at different input parameters are experimentally studied. The spectrum of the pulse is reshaped due to strong self-actions. The pulse is self-compressed, instead of broadening, accompanied with the spectral FWHM bandwidth shortened. Steepening of the leading edge of the pulse and spectral red-shift are observed in the experiment. The numerical simulation shows that the result is in agreement with the experimental result.
©2006 Optical Society of America
1. Introduction
In recent years, the propagation of intense ultrashort light pulses in optical media has attracted a lot of attentions[1–17]. Conical emission[3], white light or supercontinuum generation (SG)[8], filamentation[1,10–15], and third-harmonic generation[12] are obtained when the power of input intense ultrashort pulses is higher than the critical power for self-focusing. When intense femtosecond pulses propagate in a nonlinear medium, self-focusing driven by Kerr lens effect enhances the self-phase modulation(SPM) in the temporal domain that is connected with a spectral broadening of the pulse. Pulse splitting[4–7] and pulse self-compression[13–17] were found experimentally and theoretically accompanied with the spectral broadening.
Although it is generally believed that SPM is associated with spectral broadening, it is not always the case if the input pulse is initially chirped. The sign of the initial chirp of the pulse decides whether the SPM will compress or broaden the spectrum. Spectral broadening occurs in the case that the initial pulse is positively chirped or unchirped. For negatively chirped pulses, the long and the short wavelengths are in the tailing and the leading edges, respectively. Then SPM can result in spectral narrowing in normally dispersive nonlinear media because both the long and the short wavelengths are shifted to the central frequency [18]. Spectral compression in optical fiber was firstly studied with negatively chirped pulses in 1993[18–19]. In 2000, B. R. Washburn et al[20] reported that a longer transform-limited pulse about 600fs was obtained from a 110fs pulse source that was negatively chirped to 665fs. The pulse duration accompanied with spectral narrowing is always broadened compared with original transform-limited pulse. Moreover, the input pulse energy is limited to nanojoule and some nonlinear effects such as transversal diffraction and self-focusing play no action during propagation in the fiber, but usually important when high-intensity laser pulses propagate in non-waveguide transparent media. There will be some interesting phenomena resulting from the propagation of high-intensity negatively chirped femtosecond laser pulses in transparent media. Recently, I.Alexeev et al[21] found that negatively chirped intense femtosecond pulses were compressed after propagating more than one hundred meter in air. Numerical simulation[22] also shows that negative chirps help to maintain the longer self-channeling length when intense femtosecond pulses propagate in air.
In this paper, some interesting phenomena were found when negatively chirped intense femtosecond pulses propagate through air and bulk solid medium. The spectrum of the pulse is reshaped due to strong self-actions and the pulse is self-compressed. Spectral red-shift is also observed in the experiment. We characterized the behavior of the spectral phase and temporal profile of the compressed pulses with spectral phase interferometry for direct electric-field reconstruction SPIDER) at different input parameters. The negatively chirped pulse with duration of 75fs is self-compressed to 49fs, 39fs, 29fs and 27fs, corresponding to the spectral full-width-at-half-maximum(FWHM) bandwidth of the pulses 13nm, 15nm, 20nm and 21nm, respectively. Numerical simulation is also carried out. The simulation result is in agreement with the experiment.
2. Experiment
The experimental setup is shown in Fig. 1. The laser used in our experiment is a commercial chirped-pulse-amplified(CPA) Ti:sapphire laser system (Spectra-Physics Spitfire-50fs) running at 1-kHz repetition rate, producing ~0.5mJ, 50fs pulses with a central wavelength at 800nm Typically, the beam quality parameter M2 is about 1.3 with 7mm beam diameter (at 1/e2 of the peak intensity) and the spectral FWHM bandwidth of the pulse is about 21nm. The laser beam first passes through an attenuator consisting of a half-wave plate (HWP) and a polarizer, which can continuously adjust the laser energy. A lens with the focal length of 1.0m is used to focus the beam to a piece of BK7 glass, which is placed after the focal point and used as the nonlinear medium. After the bulk medium, the transmitted beam is collimated by an f=0.5m sliver-coated concave mirror. After a beam splitter, the transmitted part of the beam is sent to SPIDER (APE, Co. Ltd) to measure the spectral phase and temporal intensity profile of the pulse. And the reflected part is sent to a grating spectrometer (SpectraPro-300i, Acton Research Corporation) to monitor the spectrum of the pulse. A CCD(Spiricon, LBA-300PC) is used to measure the beam intensity profile.
3. Results and discussion
In the experiment, the negatively chirped pulse is obtained by increasing the distance of the grating pair in the compressor of the CPA system. The 50fs pulse is negatively chirped to 75fs, according to the quadratic phase-distortion φ2 =d 2φ/dω 2 at the center frequency ω 0 is about -1008fs2. We firstly characterize the spectral and temporal behavior of the transmitted beam after focusing in air. As for air, the nonlinear refractive index coefficient is n2=4.0×10-19cm2/W and the corresponding self-focusing critical power is Pcr ≈ 2.5GW [23]. The beam is focused in air and the beam waist at the focal point is about 95μ m (at 1/e2 of the peak intensity). When the input pulse energy is 0.05mJ, corresponding to the input pulse power of 0.67GW that is far lower than the self-focusing critical power Pcr ~ 2.5GW , the top of the spectrum of the transmitted pulse after glass is red-shifted(Fig. 2). As the input pulse energy increases from 0.05mJ to 0.3mJ, corresponding to the input pulse power increasing from 0.67GW to 4GW, the center of the laser spectrum is consistently red-shifted from original 800nm to 807.5nm(Fig. 2). The peak of the spectrum is also red-shifted with the increase of the pulse energy (Fig. 3(a)). This spectral red-shift is due to intrapulse stimulated Raman scattering (ISRS)[24]. When the pulse is negatively chirped, the high-frequency components exceed the low-frequency components and therefore, the high-frequency components of the pulse can pump the low-frequency components of the same pulse through ISRS when the spectral width exceeds a few terahertz. Such pumping results in a shift of the pulse spectrum toward longer wavelengths. When the intensity is shown on logarithm scale, the spectrum is slightly narrowed at the bottom of the spectrum curve when the input pulse energy is lower than 0.15mJ, as shown in Fig. 2. The spectrum is still narrowed in the long wavelength but begin to broaden in the blue side when the input pulse energy increases to 0.15mJ, corresponding to pulse power about 2GW. At this energy point, the air begins to be ionized that induces the spectral broadening in blue frequency. As the energy increase continuously, the corresponding pulse power larger than the self-focusing critical power 2.5W, the spectrum begins to broaden on both sides. The spectral FWHM bandwidth gets little shortened as the input pulse energy increases, as shown in Fig. 3(a).
Fig. 2. Measured evolution of the spectra of the transmitted pulse after focusing in air as the input pulse energy increases from 0.05mJ to 0.30mJ. Origin: the input pulse without chirp.
The pulse duration after focusing in air consistently decreases from 75fs to 41fs as the energy of the input pulse increases to 0.3mJ, as shown in Fig. 3(b). It is because that the negative chirp is compensated due to SPM-induced up-chirp and positive group velocity dispersion(GVD) as the pulse propagates in air. The spectral phase attends to flat with the increase of the pulse energy. When the input pulse energy gets to 0.3mJ, the spectral phase nearly has no negative chirp.
In the following experiment, a piece of 3-mm-thick BK7 glass is chosen as the bulk nonlinear medium and placed 145mm after the lens focus. At this place, the beam diameter on the surface of the BK7 glass is about 0.8mm (at 1/e2 of the peak intensity). Here the peak power and the peak intensity on the surface of the BK7 glass is at the order of GW and 1011 W / cm 2, respectively. As for BK7 glass, n2=3.45×10-16cm2/W, and the corresponding self-focusing critical power is Pcr ≈ 1.8MW [4]. The input peak power is about three order higher than that used in the previous study[5], in which pulse splitting was observed with negatively chirped pulses and narrower beam (70μ m FWHM) in a much thicker(2.54cm) bulk medium.
When the energy increases from 0.05mJ to 0.3mJ, corresponding to the intensity on the front surface of the glass from 2.65 × 1011 W/cm 2 to 1.59 × 1012 W /cm 2, the pulse after focusing in air is compressed again. Pulse duration decreases from 75fs to 27fs consistently with the increase of the incident pulse energy. All the spectral phase of the transmitted pulse are nearly flat for different pulse energy input. The temporal profile of the transmitted pulse after the BK7 glass is shown in Fig. 3(d). Pulses duration as short as 49fs, 39fs, 29fs and 27fs are obtained for 13nm, 15nm, 20nm and 21nm spectral FWHM bandwidth with 0.10mJ, 0.15mJ, 0.25mJ and 0.30mJ energy input, respectively. The duration of the obtained pulse is shorter than the Sech2 transform limited duration 50fs, 43fs, 33fs and 33fs for the same 13nm, 15nm, 20nm and 21nm spectral FWHM bandwidth, respectively. This is because the obtained spectra are reshaped after the glass that can sustain much shorter pulses than Sech2 temporal profile. Note that the spectral bandwidth and pulse duration vary little when the input pulse energy increases from 0.25mJ to 0.30mJ. This means that the obtained compressed pulse is stable at a relatively large input energy region. The pulse self-compression is mainly due to the self-focusing effect[2,5,13–14].
Fig. 3. Measured evolution of the spectral and the temporal profile of the transmitted pulse: (a)the spectral (b)the temporal profile after focus in air, and (c)the spectral (d) the temporal profile after the glass as the input pulse energy increases from 0.10mJ to 0.30mJ. Origin means the input pulse without chirp.
Steepening of the leading edge of the transmitted pulse is observed, as shown in Fig. 3(d). As a result, the sharp temporal gradient at the leading edge of the pulse gives rise to spectral broadening in long wavelength(Fig. 3(c) and Fig. 4(a)). This result does not accord with the usually observation that steepening of the trailing edge of the pulse accompanied with spectral broadening on the blue side[9]. As we know that the steepening of the trailing edge of the pulse can be understood as: the leading edge of the intense pulse formats plasma that induced negative nonlinear refractive index, and then the trailing edge of the pulse runs fast due to the negative nonlinear refractive index, thus the pulse is compressed and steepening of the trailing edge is formatted. So the steepening of the leading edge also can be understood as: when the input pulse is intense enough, the higher order negative susceptibilities χ(n) (n>3) that is an instantaneous effect induces negative nonlinear refractive index, and then the intense center part of the pulse runs much faster than the side part at the bottom, thus the pulse is compressed and steepening of the leading edge is formatted. I.G.Koprinkov et al[14] also experimentally showed that the self-focusing and the higher order negative susceptibilities χ(n) (n>3) play an important role on pulse self-compression in normally dipersive media.
The spectral FWHM bandwidth of the pulse after the glass decreases from 21nm to a minimum of 13nm and then increases again for constantly increasing pulse energy. The spectral intensity is shown on linear scale in Fig. 3(c). The spectral FWHM bandwidth is about 21nm for 0.30mJ input, which is close to the original spectral FWHM bandwidth-21nm. The work is different from the previous studies that pulse self-compression accompanied with large spectral broadening [14] and pulse splitting accompanied with spectral narrowed[5]. We also characterize the spectrum on logarithm scale, shown in Fig. 4(a). Compare with the laser spectrum after focusing in air (dash line in Fig. 4(a)), the laser spectrum after the glass is narrowed mainly on the top of the spectral curve in the long wavelength(solid line in Fig. 4(a)). And two petals, one near 825nm and the other near 775nm, grow in height and width obviously as the input pulse energy constantly increasing. The petals are mainly induced by the SPM in the media that enhanced by self-focusing. Spectral broadening in long wavelength owe to the sharp temporal gradient at the leading edge of the pulse. The blue-frequency broadening should be a shock behavior induced by ionization(Fig. 4(a))[8,16]. This change reshapes the spectrum from Gaussian profile to nearly Lorentzian profile that can sustain much shorter pulse for the same spectral FWHM bandwidth. And when the input pulse energy increases to 0.35mJ, corresponding to the intensity of about 1.85× 1012 W/ cm 2 on the front surface of the glass, rainbow-like pattern named as conical emission begins to occur.
Fig. 4. Measured evolution of the spectrum on logarithm scale(solid line g: glass, dashed line a: air, the number before it is the input pulse energy) of the transmitted pulse as the input pulse energy increase(a). The simulation of the spectrum(b) and temporal profile(c) of the negatively chirped pulse after the BK7 glass as a function of intensity. The intensity of P1, P2, and P3 is about 8.78×1011 W/cm 2, 1.26×1012 W/cm 2, and 1.72×1012 W/cm 2, respectively. Origin means the input pulse without chirp.
We also numerically simulate the evolution of the spectrum and the temporal profile of the negatively chirped pulse propagating through the BK7 glass at different input intensity. In the simulation, diffraction, self-focusing, multi-photon ionization, and group velocity dispersion are considered. The propagation equation can be expressed as[25]
It is formulated in the frequency domain in order to tackle the dispersion more precisely. The four terms on the right-hand side correspond to diffraction, dispersion, SPM and plasma behavior, respectively. In the simulation, the incident pulse is negatively chirped from 50fs to 75fs with the spectral FWHM bandwidth of 21nm. And the intensity on the surface of the BK7 glass is about 8.78× 1011 W/cm 2, 1.26× 1012 W/cm 2 , and 1.72×1012 W/cm 2, which is equal to about 0.17mJ, 0.24mJ, and 0.32mJ input, respectively. Fig. 4(b) shows the evolution of the spectrum of the transmitted pulse through the medium at different input pulse intensity. The spectrum is narrowed in the center. At the bottom of the spectrum on both sides, two petals consistently grow and broaden symmetrically as the incident intensity increases. Compare with the measured spectrum, the spectrum is relatively symmetrical because the cubic phase distortion is neglected in the simulation[19–20].The evolution of the spectrum reshaping accords with the experimental result. Figure. 4(c) shows the temporal profile of the transmitted pulse at different input intensity. The transmitted pulse is constantly shortened from 50fs to about 21fs as the input pulse intensity increases to about 1.72×1012 W/cm 2(Fig. 4(c)). The result is agree with the phenomenon observed in the experiment that spectrum reshaping and pulse self-compression in the glass.
Fig. 5. Measured (solid line) temporal profile (a) spectrum and spectral phase (a1) of the pulse after glass with 0.30mJ input. The inset in (a) is the cross section intensity distribution of the laser beam after glass, the dotted line(red) in (a) is the retrieved temporal profile corresponding to the measured spectrum and spectral phase, the dashed line(blue) in (a) is the transform-limited(TL) pulse(17.3fs), and the dashed line in (a1) is the Lorentz fit of the measured spectrum.
The beam intensity profile is measured with a CCD camera(Spiricon, LBA-300PC) in the experiment. The transmitted beam is contracted due to self-focusing in the glass and the spatial mode of the transmitted beam after the glass improved greatly(Fig. 5 (a)). We retrieve the temporal profile of the compressed pulse by Fourier transformation of the spectrum and phase. The spectrum is independently measured by an imaging spectrometer, and the spectral phase is measured by using SPIDER. As an example, Fig. 5(a) gives the retrieved pulse and the measured pulse when the input pulse energy is 0.30mJ. The excellent agreement between the measured and the retrieved temporal profile indicates that the measurement has high accuracy and is creditable. Fig. 5(a1) shows the spectrum and the spectral phase of the transmitted pulse. The spectral phase is flat and the spectrum is almost Lorentzian profile not only on linear scale(dash-line) but also on logarithm scale(Fig. 4(a)). The spectrum supports a transform-limited(TL) pulse duration of 17.3fs shown in Fig. 5(a) with a dash-line, assuming a flat spectral phase, which is close to the Lorentzian transform limited duration 15.6fs for the same 21nm FWHM spectral bandwidth.
Actually, the spectrum reshaping and pulse self-compression occurs with the BK7 glass plate at a wide range of positions. We also put the 3-mm-thick BK7 glass 180mm after the lens focus. When the input pulse is negatively chirped to 65fs and the pulse energy is 0.25mJ, the spectral FWHM bandwidth is narrowed from 21nm to 16nm, and the pulse duration is compressed to 41fs. When the input pulse energy is increased to 0.3mJ, the spectral FWHM bandwidth broadens to about 21nm and the pulse duration is still compressed to 34fs. The evolution trend of the spectrum and the pulse is the same. We also put a piece of 9.5-mm-thick BK7 glass 240mm after the lens focus, and obtain a 61fs pulse with 0.2mJ input, which is almost the sech2 transform-limited. The spectral FWHM bandwidth is narrowed from 21nm to about 11nm.
The above experiments demonstrate that spectrum reshaping and pulse self-compression of high-intensity femtosecond pulses is a general phenomenon that can be observed with negatively chirped femtosecond pulses in a variety of normally dispersive media. From a simple physical standpoint, we understand the spectrum reshaping and pulse self-compression of negatively chirped pulses as follows: when the intensity of the input pulse is low, the spectrum narrowed due to SPM[18–20]. The combination of the SPM-induced up-chirp and positive GVD compensates the negative chirp of the pulse, thus the transmitted negatively chirped pulse is shortened. As the energy of the input negatively chirped pulse on the surface of the glass increases, self-focusing due to the Kerr lens occurs that is enhanced with the increase of the pulse power. The pulse self-compression mainly owes to the self-focusing[2,5,9–11,13–14]. And the self-focusing effect enhances instantaneous effects, such as high order negative susceptibilities χ(n) (n>3) etc that act to sharpen the ascending part of the pulse, while time-delayed effects such as photon-ionization etc act to cut off its trailing part. The combined nonlinear effects eventually lead to the pulse self-compression and reshaping. In the frequency domain, the spectrum reshaping mainly owes to SPM. For a negatively chirped Guassian pulse, the spectral narrowing is very little at the bottom of the pulse and large at the waist of the pulse due to higher intensity and intensity-time slope at the waist. Thus the spectrum is reshaped. Combined with SPM, ionization, high order negative susceptibilities χ(n) (n>3) and Raman effect etc, the spectrum is reshaped from Gaussian to other function, for example Lorentzian function, which can sustain much shorter pulse for the same spectral FWHM bandwidth.
4. Conclusions
In conclusion, the spectral and the temporal behavior of high-intensity negatively chirped femtosecond pulse propagating in air and a piece of BK7 glass have been studied. We find an interesting phenomenon that spectral FWHM bandwidth shortening is followed by pulse self-compression instead of pulse broadening. It is a general phenomenon that can be observed with negatively chirped femtosecond pulses in a variety of normally dispersive media. Steepening of the leading edge of the pulse and spectral red-shift are observed in the experiment. The numerical simulation shows that the result is in agreement with the experimental result. The result helps to understand the influence of negative chirp on the propagating of intense femtosecond pulses in normally dispersive media.
Acknowledgments
This work is supported partially by Natural Science Foundation of China (Grant Nos. 69925513 and 19974058), Chinese Ministry of Science and Technology through contract G1999075204, Chinese Academy of Sciences through contracts KGCX2-SW-10 and KGCX2-SW-114, and the Major Basic Research Project of Shanghai Commission of Science and Technology.
References and Links
01. 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]
02. A.T. Ryan and G.P. Agrawal, “Pulse compression and spatial phase modulation in normally dispersive nonlinear Kerr media,” Opt. Lett. 20, 306–308 (1995) [CrossRef] [PubMed]
03. E.T.J. Nibbering, P.F. Curley, G. Grillon, B.S. Prade, M.A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulse in air,” Opt. Lett. 21, 62–64(1996) [CrossRef] [PubMed]
04. J. K. Ranka, R.W. Schirmer, and A. L. Gaeta, “Observation of pulses splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996) [CrossRef] [PubMed]
05. S.A. Diddams, H.K. Eaton, A.A. Zozulya, and T.S. Clement, “Characterizing the nonlinear propagation of femtosecond pulses in bulk media,” IEEE J. Sel. Top. Quantum Electron. 4, 306–316 (1998) [CrossRef]
06. A.A. Zozulya, “Propagation dynamics of intense femtosecond pulse: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999) [CrossRef]
07. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowlyvarying envelope approximation in the self-focusing of ultrashortpulses,” Opt. Lett. 23, 534–536 (1998) [CrossRef]
08. A. Brodeur and S.L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent condensed media,” J. Opt. Soc. Am. B 16, 637–650 (1999) [CrossRef]
09. A. L. Gaeta, “Catastrophic collapse of ultra short pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000) [CrossRef] [PubMed]
10. S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902-1-4 (2001) [CrossRef] [PubMed]
11. Z. Wu, H. Jiang, Q. Sun, H. Yang, and Q. Gong, “Filamentation and temporal reshaping of a femtosecond pulse in fused silica,” Phys. Rev. A 68, 063820-1-8 (2003) [CrossRef]
12. H. Yang, J. Zhang, J. Zhang, L.Z. Zhao, Y.J. Li, H. Teng, Y.T. Li, Z.H. Wang, Z.L. Chen, Z.Y. Wei, J.X. Ma, M. Yu, and Z.M. Sheng, “Third-order harmonic generation by self-guided femtosecond pulses in air,” Phys. Rev. E 67, 015401 (2003) [CrossRef]
13. S. Henz and J. Herrmann, “Self-channeling and pulse shortening of femtosecond pulses in multiphoton-ionized dispersive dielectric solids,” Phys. Rev. A 59, 2528–2531 (1999) [CrossRef]
14. I. G. Koprinkov, A. Suda, P. Wang, and K. Midorikawa, “Self-compression of high-intensity femtosecond optical pulses and spatiotemporal soliton generation,” Phys. Rev. Lett. 84, 3847–3850 (2000) [CrossRef] [PubMed]
15. O. Shorokhov, A. Pukhov, and I. Kostyukov, “Self-compression of laser pulses in plasma,” Phys. Rev. Lett. 91, 265002-1 (2003) [CrossRef]
16. N.L. Wagner, E.A. Gibson, T. Popmintchev, I.P. Christov, M.M. Murnane, and H.C. Kapteyn, “Self-compression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902-1 (2004) [CrossRef] [PubMed]
17. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30, 2657–2659 (2005) [CrossRef] [PubMed]
18. S.A. Planas, N.L. Pires Mansur, C.H. Brito Cruz, and H.L. Fragnito, “Spectral narrowing in the propagation of chirped pulses in single-mode fibers,” Opt. Lett. 18, 699–701 (1993) [CrossRef] [PubMed]
19. M. Oberthaler and R.A. HOpfel, “Special narrowing of ultrashort laser pulses by self-phase modulation in optical fiber,” Appl. Phys. Lett. 63, 1017–1019 (1993) [CrossRef]
20. B.R. Washburn, J.A. Buck, and S.E. Ralph, “Transform-limited spectral compression due to self-phase modulation in fibers,” Opt. Lett. 25, 445–447 (2000) [CrossRef]
21. I. Alexeev, A. Ting, D.F. Gordon, E. Briscope, J.R. Penano, R.F. Hubbard, and P. Sprangle, “Longitudinal compression of short laser pulses in air,” Appl. Phys. Lett. 84, 4080–4082 (2004) [CrossRef]
22. R. Nuter, S. Skupin, and Luc Berge, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30, 917–919 (2005) [CrossRef] [PubMed]
23. A. Couairon, “Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses,” Phys. Rev. A 68, 015801-1 (2003) [CrossRef]
24. G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Lett. 15, 224–226 (1990) [CrossRef] [PubMed]
25. J. S. Liu, H. Schroeder, S. L. Chin, W. Yu, R. Li, and Z. Xu, “Space-frequency coupling, conical waves, and small scale filamentation in water,” Phys. Rev. A 72, 053817-1 (2005) [CrossRef]
