Abstract
The requirements for the generation of optical vortices with ultra-short and ultra-intense laser pulses are considered. Several optical vortice generation procedures are analysed, specifically those based on diffractive elements, such as computer generated holograms (CGH). Optical vortices achromatization techniques are studied. Volume phase holographic (VPH) elements are considered for highly efficient, broad spectrum, high damage-threshold generation of vortices. VPH compound systems, including a compact one, for achromatic vortex generation are presented. Experimental results of vortice generation with ultra-short and ultra-intense pulses are shown.
Keywords
- optical vortex
- pulse shaping
- volume phase holograms (VPH)
- aberration compensation
- holographic optical elements
1. Introduction
An optical vortex is a wave that has a phase singularity, so that the intensity figure is ring-shaped, with zero intensity at the centre due to the indeterminacy of phase at that point. The phase varies helically around the singularity, from 0 to 2π
In recent years interest in the generation of vortices from ultra-short and ultra-intense pulses has been increased, opening access to the experimental study of phenomenology in vortex propagation in non-linear regime [3, 4] and their possible applications, such as remote laser-induced breakdown spectroscopy (LIBS) [5], or phase control in higher order harmonics generation. The generation of high-energy vortices from ultra-intense and ultra-short laser pulses requires elements that, on the one hand, have a high damage threshold and, on the other, are able to work with a wide bandwidth.
There are several techniques for generating vortices with short laser pulses, all extrapolated from their use in continuous-wave regime, under conditions of monochromaticity and low energy. One of them is the use of modal converters, in which a vortex can be obtained from a HG01 mode with a combination of cylindrical lenses [6]. This technique is not feasible in the case of intense laser pulses, since it is difficult in this case to get the laser to emit in HG01 mode. There are modal converters based on LCD spatial light modulators, but are not applicable to high-energy lasers because of their damage threshold [7].
To date, there are two vortex generation techniques applicable to femtosecond lasers. The first is the use of spiral phase masks, manufactured either by depositing quartz onto a quartz substrate [8], by lithography in resins [9] and photoresists [10] or directly by carving on fused silica. In all cases, the mask is made with steps; the thickness increases with the azimuth angle and therefore the output phase varies helically. This type of masks has an efficiency of approximately 55% in the case of manufacturing by deposition or up to 80% in the case of photoresist, and they have a high damage threshold, allowing illumination with ultra-intense pulses obtaining high-energy vortices. A disadvantage of these phase masks is that they have some chromatic aberration when the half-width of the pulse is greater than 40 nm, which is manifested in the variation of the topological charge of the vortex with the wavelength, which limits their use to pulses higher than 30 fs [11].
Recently, Swartzlander [12] proposed a solution to try to achromatize this type of elements, by joining two different materials bonded together with a spiral phase (achieved by varying thickness) between them. With this method, achromatization is achieved for two wavelengths. In any case, the manufacturing process is expensive and requires photolithography facilities, vapour deposition or spin coating, the achromatization that can be achieved depends on the materials used and not total achromatization for the entire bandwidth is achieved.
The second technique uses computer generated holograms (CGH). In this case, the transmission gratings have a dislocation that generates the vortex and are usually printed on transparency or recorded in photographic film [13].
The efficiency of such amplitude masks is low, 6%, but may be increased by applying them a bleaching process [14] or generating patterns directly on LCD [15]. The main disadvantage of this type of dislocation gratings is the low damage threshold, which does not allow its use with high intensities. An interesting solution is the one proposed by Sacks et al. [16], which, using the vortex generated with an amplitude mask, records a volume and phase hologram with a reference plane wave, obtaining high efficiency elements.
To prevent the angular chromatic dispersion inherent to diffractive elements, a combination of two gratings with two lenses may be disposed, so that the chromatic dispersion introduced by the grating with dislocation is compensated and the vortex generated is achromatic [17, 18]. Sola et al. [19] performed an assembly similar to that of Mariyenko et al. [17], replacing printed or photographed gratings with volume holographic gratings. A holographic dislocation grating was achieved by the interference of a plane wave with a vortex generated by an amplitude grating, by a procedure similar to that described by Sacks et al. [16]. The material used was dichromated gelatin (Slavich PFG04), with which highly efficient gratings were achieved and with a high damage threshold [20]. In this work, high-energy vortices were generated with a femtosecond laser, so that non-linear vortex propagation effects in air could be observed. The fundamental problem is the complexity of the assembly, which should include a vacuum chamber between the two lenses to avoid non-linear effects in the beam concentration.
There are other solutions, such as those proposed by Martínez-Matos et al. [21], in which the generation of femtosecond paraxial beams with a combination of only two volume gratings separated by some distance is proposed. The disadvantage of this solution is that the separation of gratings introduces a temporal and a spatial chirp at the ends of the Gaussian intensity profile. Atencia et al. [22] have recently developed a compact achromatic holographic vortex generator design based on attached gratings, built from a CGH. The holographic element obtained completely avoids the presence of spatial chirp across the beam intensity profile.
In this chapter the recording of volume holographic elements for the generation of vortices and two achromatization techniques with different features will be explained. In all cases the achromatism condition for a continuous bandwidth is met.
2. Basics on volume holography
2.1. Holographic recording and reconstruction
Holography is a method for recording the amplitude and phase of a wave
The complex amplitude at each point of the hologram is the sum of the amplitude of the two waves,
so intensity will be given by
It can be seen that the intensity varies harmonically with the phase difference. This intensity pattern is recorded in a photosensitive medium, so its transmittance changes. Assuming the amplitude transmittance is linear with exposure of the recording material and that the intensity of the reference wave is uniform over the holographic plate, the transmittance of an amplitude hologram is given by
where
Due to the linearity of Maxwell’s equations each of the addends can be interpreted as an independent wave.
2.2. Volume phase holograms (VPH)
If a suitable material is used in the recording of the hologram, the intensity variations can be translated into variations of the refractive index, so that a phase hologram is recorded. In phase gratings, modulation of the refractive index is given by
where
Consider a hologram region as shown in
Figure 1
. The holographic grating at this zone is recorded with an object wave with wave vector
According to the relationship between the thickness
The criterion to determine whether a hologram is volume type is given by the
where λ is the wavelength of the reconstruction beam. A volume hologram is considered when
The main advantage of volume phase holograms (VPH) is that, for a given wavelength, it is possible to get 100% of the incident light diffracted to +1 order. For this to happen the so-called
must meet, where
While the volume grating determines the energy performance, the surface grating determines the geometric behaviour. So, if in the reconstruction step a wave whose propagation vector is
The distribution of energy between
If light hits the surface of the hologram at an angle other than
where
with
A 100% efficiency can be achieved if the value of
When the reconstruction conditions move away from the Bragg condition the
In
So, for a wavelength
For
So the phase shift
From the Bragg condition for
Replacing Eq. (18) in Eq. (17) and substituting the expression of
To have the widest possible diffracted spectrum a small angle between beams is used, and wave vectors near the normal direction are chosen, so
Substituting the obtained values of
A numerical simulation of the spectrum diffracted by a holographic grating has been carried out with typical values for ultra-short pulses (
3. Vortex generation with holographic optical elements
A holographic vortex generation element is recorded by the interference of a plane (or spherical) wave and a vortex beam. Since the recording is with monochromatic light, any of the methods described in Section 1 can be used to generate this vortex object wave. For this work a computer generated hologram (CGH) was chosen.
In order to generate the CGH, the interference between a plane wave (reference wave), with its wave vector forming an angle
and the object vortex wave as
where
The printed pattern is photographically reduced using a reflex camera with Kodak TMax100 film, obtaining an amplitude thin grating of 14.7 lines/mm with a dislocation at the centre. This CGH has high absorption, so it cannot be illuminated with a high-intensity laser, because it could be damaged. Therefore the resulting film is contact-copied onto a commercial Slavich PFG04 dichromated gelatin plate using incoherent light. After this complex procedure a thin phase CGH is obtained. It can be illuminated with an intense laser, although multiple diffracted orders are obtained and their efficiency is low (25%).
The CGH is used to generate the object wave for the recording of a VPH. The recording scheme is shown in
Figure 5
. The beam of a Coherent Verdi 6 W CW laser emitting at 532 nm is divided in two by means of a beam splitter BS. One of the beams is filtered and expanded with a spatial filter and collimated with a lens L1, and acts as a plane reference beam. The other beam is first filtered and expanded and then collimated with a lens L2, and illuminates the thin phase CGH. The effects of bitmap resolution in the previous laser-printed CGH require to use spatial filtering techniques to select the desired diffraction order and to avoid bitmap artefacts [16]. The +1 diffracted order, containing the vortex, is selected, filtered and imaged with magnification
The recorded VPH presents an efficiency of 95% at 800 nm. The spatial period is
In Figure 7 , the Fourier transform of the vortex generated by the VPH when illuminated with a CW laser beam of 800 nm is shown.
4. Holographic generation of achromatic vortex beams
A disadvantage of using the previous holographic optical element to generate vortex beams with ultra-short pulses is the chromatic dispersion. As we have seen, the VPH can work properly in a spectral range of around 200 nm, but the direction of the diffracted vortex beam depends on the wavelength, so the vortex beam presents an important spatial chirp, as it is shown in Figure 8 .
This problem can be overcome by designing an achromatic set-up in which the chromatic dispersion of the vortex beam is compensated by using a volume holographic plane grating (HI) with the same spatial frequency as the vortex generator VPH (HII).
HI is recorded with two plane waves of wavelength
HII is recorded with the interference of a plane wave, forming an angle
4.1. Achromatization with separated elements
The first approach to combine these two VPH is proposed by Sola et al. [19] based on a solution with printed gratings suggested by Mariyenko et al. [17]. HII is located at the plane where a positive lens (L1) forms the image of the grating HI in a 2
If we consider that HI is illuminated with a polychromatic plane wave with a Gaussian spatial profile,
where
Considering L1 as an ideal lens, the complex amplitude at the image plane
where
The term
A second lens L2, with the same focal length as L1, is placed just in front of or behind HII, restoring the phase of the propagating beam into a plane wavefront. The VPH to generate the vortex beam has been recorded with a plane wave as a reference wave, so placing the lens L2 in front of HII (as it is shown in Figure 9 ) provides a reconstruction plane wave, which assures the reconstruction of the vortex beam with no geometrical aberrations.
Depending on the intensity of the pulses, the use of a vacuum chamber between the two lenses could be necessary to avoid non-linear effects in the focalization produced by the lens L1.
4.2. Achromatization with a compact element
The second approach was proposed by Atencia et al. [22]. In this case both holographic elements HI and HII are placed parallel to each other, as it can be seen in Figure 10 .
If HI and HII are separated (
Figure 10(a)
), the Gaussian profile of the incident beam is kept in the propagation between HI and HII, but the different wavelengths are spatially separated and de-phased when they reach HII. A set of vortices emerges from HII, all with the same topological charge
If the two VPH are placed together, as in
Figure 10(b)
, the amplitude distribution that reaches HII is
where, in this case,
In the construction of this compact VPH, the recording scheme for HI is similar to that shown in
Figure 5
by replacing the vortex wave with a collimated wave. The angle between beams has to be the same for HI and HII to guarantee the chromatic compensation. For a simpler alignment procedure, the recording geometry for HI and HII is chosen to give Bragg condition at 800 nm for an angle
After processing, HI and HII are cemented to each other, using Norland NOA61 optical adhesive between the two emulsions. This ensures that no change in the refraction index between the holograms occurs and preserves the emulsion from degradation effects of the environment.
When illuminating each VPH at Bragg angle for 800 nm, the maximum efficiency obtained is approximately 95%, so the sandwich has a total maximum efficiency of 90%. Reflection on glass surfaces and glass absorption causes losses of about 15%, so the compound holographic element reaches an absolute efficiency of 77%. When
For wavelengths different from 800 nm the diffractive efficiency drops and the transmitted light increases, so the transmitted and diffracted light can spatially overlap for some wavelengths. To prevent this, HI is slightly rotated around the
Figure 12(a) and (b) demonstrate that the compound element generates a broadband achromatic optical vortex. A 5 mm-wide white light vortex can be seen in Figure 12(a) , when illuminating the compact VPH with a tungsten light source (Ocean Optics LS-1) with 600 nm FWHM centred at 850 nm. The photograph was taken at 1 m propagation from the element output. Figure 12(b) shows the interference of transmitted and diffracted light. To obtain an appreciable intensity on the transmitted beam, the holographic element is illuminated out of Bragg condition. The dislocation of the interference pattern is clearly visible at the centre of the image.
This compact vortex generator holographic element is very easy to align and avoids pulse concentration zones, so for ultra-intense pulses the use of a vacuum chamber is not required.
5. Applications of vortex volume phase holograms to ultra-fast optics
Once the recording process of a vortex generating VPH has been reviewed, and the two solutions for the vortex achromatization have been described, in the following section we will comment some of the characteristics of the generated vortex and some possible applications using femtosecond high power pulses. The achromatization of the VPH is very attractive from the point of view of ultra-short pulses, which present broadband spectra, allowing the generation of short vortex pulses with high peak power [19]. The main limitation to obtain shorter pulses is the spectral bandwidth of the hologram; however, the present performance allows the generation of ultra-short vortex beams (e.g., compact vortex VPH is compatible with few tenths of fs pulses).
One of the first questions arising is what the spatiotemporal structure of the vortex looks like. In the literature, fs pulse vortices generated by spiral phase masks have been characterized by using spatially resolved interferometry [27]. In the case of the compact VPH, when studying spatial pattern of interferences in a similar way, a peculiar behaviour was observed.
Figure 13
shows the interference pattern between two replicas of the laser beam (pulses of 100 fs FWHM, central wavelength at 795 nm), obtained at a Mach-Zender interferometer. One of the beams passed through a topological charge
Although this pulse tilt is small and for the most of applications it plays no relevant role (e.g., in the following non-linear interaction cases presented in this section no relevant effect has been observed), it is important to be aware of its existence. One way of reducing it would be to better match transmitted-diffracted beam directions. Nevertheless, if some of the incident beam is transmitted through the compact VPH, it may interfere with the vortex beam. On the other hand, it has recently been shown that controlled pulse tilts can be a tool to induce interesting effects, such as the so-called attosecond lighthouses [28]. Therefore, the ability of imprinting controlled spatiotemporal alterations on the VPH may have potential applications.
One of the main advantages of using the compact VPH consists on the possibility of creating a high power vortex quite easily. The set-up for generating the high power vortices is simplified to the point of making the ultra-short pulses pass through it, while orienting the VPH in the proper angle, defined by the recording configuration. Since the element presents a high damage threshold of, at least, hundreds of GW/cm2 [19], the incident pulse peak power can be high, allowing to generate high-power vortices because of the high efficiency of the VPH. Therefore, it makes it possible to induce non-linear effects on the vortex beams while propagating though a certain medium.
In order to explore the potential of VPH creating high intensity femtosecond vortices, different experiments have been performed. Firstly, using the achromatic set-up described in Section 4.1, the dynamics of vortices at different intensity regimes were studied [19], analysing the evolution of the spatial distribution of the light along propagation after passing through a focusing lens (
An increase of beam intensity would involve utterly medium ionization. Then, light propagation may become even more complex, being the result of the interplay between effects competing for focusing or unfocusing the beam (e.g. Kerr effect will introduce focusing on the beam, while ionization makes the opposite). This may produce what is called filamentation [30], exhibiting a self-guiding of the beam because of the balance between linear regime light propagation and the different non-linear effects raised from interaction with the medium. Splitting and filamentation of the vortex fragments have been observed for high enough intensities (e.g., 14 GW vortices propagating in a tube filled with nitrogen at a pressure of 1600 mbar [19]).
In the last decades, an extremely non-linear process known as high order harmonic generation (HHG) has become a hot topic and the basis of the new discipline called
The basis of HHG and its use for the generation of XUV beams exhibiting OAM are analysed in depth in another chapter of this book [41]. Here we will focus on the use of vortex generating compact VPH in HHG experiments. In the pioneering experimental studies [38, 40], the OAM driving beam was obtained from a 30 fs pulsed beam and
Therefore, the case of Gariepy et al. [40] is quite similar to the way the compact VPH generates the vortices, with a dislocated grating and a second non-dislocated grating to suppress angular dispersion. The main difference lays on the fact of using a SLM element or a holographic plate. While the former is more flexible and dynamic, the compact VPH is a more robust and cheaper element, presenting a simpler alignment.
Then, in order to explore its potential, we tested the compact VPH in a HHG experiment. The experimental set-up is shown in
Figure 15
. A 25 fs pulsed beam up to 2 mJ per pulse (1 kHz repetition rate, spectrum centred on 790 nm) is focused by means of a
In summary, the compact VPH [22] is a simple, cheap and robust element for generating high power AOM beams. Their achromaticity allows the generation of pulsed vortices down a few tenths of fs and their high damage threshold permits to obtain high peak powers. High power vortex generated by this way are able to interact non-linearly with material, producing effects from Kerr effect and filamentation to HHG.
Acknowledgments
This research was supported by 'Ministerio de Economía y Competitividad' of Spain for the funding (projects FIS2013-44174-P and FIS2015-71933-REDT), Diputación General de Aragón-Fondo Social Europeo (TOL research group, T76), Junta de Castilla y León (projects SA116U13, UIC016 and SA046U16), Generalitat de Catalunya (grant 2016 FI_B1 00019), and Centro de Láseres Pulsados (CLPU) for granting access to its facilities.
References
- 1.
J. E. Curtis, B. A. Koos, and D. G. Grier. Dynamic holographic optical tweezers. Optics Communications. 2002; 207 (1–6):169–175. DOI: 10.1016/S0030-4018(02)01524-9 - 2.
G. A. Swartzlander Jr. The Optical Vortex Lens. Optics & Photonics News. 2006; 17 (11):34–41. DOI: 10.1364/OPN.17.11.000039 - 3.
M. S. Bigelow, P. Zerom, and R. W. Boyd. Breakup of ring beams carrying orbital angular momentum in sodium vapor. Physical Review Letters. 2004; 92 (8):083902. DOI: 10.1103/PhysRevLett.92.083902 - 4.
Z. Chen, M. Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker. Steady-state vortex-screening solitons formed in biased photorefractive media. Optics Letters. 1997; 22 (23):1751–1753. DOI: 10.1364/OL.22.001751 - 5.
M. Fisher, C. Siders, E. Johnson, O. Andrusyak, C. Brown, and M. Richardson. Control of filamentation for enhancing remote detection with laser induced breakdown spectroscopy. Proceedings of SPIE. 2006; 6219 :621907. DOI: 10.1117/12.663824 - 6.
J. Courtial, M. J. Padgett. Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes. Optics Communications. 1999; 159 (1–3):13–18. DOI: 10.1016/S0030-4018(98)00599-9 - 7.
J. A. Rodrigo, T. Alieva, and M. L. Calvo. Experimental implementation of the gyrator transform. Journal of the Optical Society of America A. 2007; 24 (10):3135–3139. DOI: 10.1364/JOSAA.24.003135 - 8.
K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka. Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses. Optics Express. 2004; 12 (15):3548–3553. DOI: 10.1364/OPEX.12.003548 - 9.
W. C. Cheong, W. M. Lee, X-C Yuan, L-S Zhang, K. Oholakia, and H. Wang. Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation. Applied Physics Letters. 2004; 85 :5784–5786. DOI: 10.1063/1.1830678 - 10.
J. W. Sung, H. Hockel, J. D. Brown, and E. G. Johnson. Development of a two-dimensional phase-grating mask for fabrication of an analog-resist profile. Applied Optics. 2006; 45 (1):33–43. DOI: 10.1364/AO.45.000033 - 11.
K. J. Moh, X-C Yuan, D. Y. Tang, W. C. Cheong, L-S Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin. Generation of femtosecond optical vortices using a single refractive optical element. Applied Physics Letters. 2006; 88 :091103. DOI: 10.1063/1.2178507 - 12.
G. A. Swartzlander Jr. Achromatic optical vortex lens. Optics Letters. 2006; 31 (13):2042–2044. DOI: 10.1364/OL.31.002042 - 13.
V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov. Screw dislocations in light wavefronts. Journal of Modern Optics. 1992; 39 (5):985–990. DOI: 10.1080/09500349214551011 - 14.
H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop. Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms. Journal of Modern Optics. 1995; 42 (1):217–223. DOI: 10.1080/09500349514550171 - 15.
K. Crabtree, J. A. Davis, and I. Moreno. Optical processing with vortex-producing lenses. Applied Optics. 2004; 43 (6):1360–1367. DOI: 10.1364/AO.43.001360 - 16.
Z. S. Sacks, D. Rozas, and G. A. Swartzlander. Holographic formation of optical-vortex filaments. Journal of the Optical Society of America B. 1998; 15 (8):2226–2234. DOI: 10.1364/JOSAB.15.002226 - 17.
I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal. Creation of optical vortices in femtosecond pulses. Optics Express. 2005; 13 (19):7599–7608. DOI: 10.1364/OPEX.13.007599 - 18.
K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther. Vortices in femtosecond laser fields. Optics Letters. 2004; 29 (16):1942–1944. DOI: 10.1364/OL.29.001942 - 19.
I.J. Sola, V. Collados, L. Plaja, C. Méndez, J. San Román, C. Ruiz, I. Arias, A. Villamarín, J. Atencia, M. Quintanilla, and L. Roso. High power vortex generation with volume phase holograms and non-linear experiments in gases. Applied Physics B. 2008; 91 (1):115–118. DOI: 10.1007/s00340-008-2967-9 - 20.
A. Villamarín, J. Atencia, M.V. Collados, and M. Quintanilla. Characterization of transmisión volume holographic gratings recorded in Slavich PFG04 dichromated gelatin plates. Applied Optics. 2009; 48 (22):4348–4353. DOI: 10.1364/AO.48.004348 - 21.
O. Martínez–Matos, J. A. Rodrigo, M. P. Hernández-Garay, J. G. Izquierdo, R. Weigand, M. L. Calvo, P. Cheben, P. Vaveliuk, and L. Bañares. Generation of femtosecond paraxial beams with arbitrary spatial distribution. Optics Letters. 2010; 35 (5):652–654. DOI: 10.1364/OL.35.000652 - 22.
J. Atencia, M. V. Collados, M. Quintanilla, J. Marín-Sáez, and I. J. Sola. Holographic optical element to generate achromatic vortices. Optics Express. 2013; 21 (18):21056–21061. DOI: 10.1364/OE.21.021056 - 23.
H. Kogelnik. Coupled wave theory for thick hologram gratings. Bell System Technical Journal. 1969; 48 (9):2909–2947. DOI: 10.1002/j.1538-7305.1969.tb01198.x - 24.
R. J. Collier, C. H. Burckhardt, and L. H. Lin. Optical Holography. 1st ed. New York: Academic Press; 1971. 605 p. - 25.
A. Villamarín, I. J. Sola, M. V. Collados, J. Atencia, O. Varela, B. Alonso, C. Méndez, J. San Román, I. Arias, L. Roso, and M. Quintanilla. Compensation of second-order dispersion in femtosecond pulses after filamentation using volume holographic transmission gratings recorded in dichromated gelatin. Applied Physics B. 2012; 106 (1):135–141. DOI: 10.1007/s00340-011-4770-2 - 26.
J. W. Goodman. Introduction to Fourier Optics. 2nd ed. Singapore: McGraw-Hill; 1996. 441 p. - 27.
M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L?Huillier and C. L. Arnold. Spatiotemporal characterization of ultrashort optical vortex pulses. Journal of Modern Optics. 2016. DOI: 10.1080/09500340.2016.1257751 - 28.
K. T. Kim, C. Zhang, A. D. Shiner, B. E. Schmidt, F. Légaré, D. M. Villeneuve, and P. B. Corkum. Petahertz optical oscilloscope. Nature Photonics. 2007; 13 :958–962. DOI: 10.1038/nphoton.2013.286 - 29.
C. Ruiz, J. San Román, C. Méndez, V. Díaz, L. Plaja, I. Arias, and L. Roso. Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold. Physical Review Letters. 2005; 95 :053905. DOI: 10.1103/PhysRevLett.95.053905 - 30.
A. Couairon and A. Mysyrowicz. Femtosecond filamentation in transparent media. Physics Reports. 2007; 441 (2–4):47–189. DOI: 10.1016/j.physrep.2006.12.005 - 31.
P. B. Corkum and F. Krausz. Attosecond science. Nature Physics. 2007; 3 :381–387. DOI: 10.1038/nphys620 - 32.
T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Muecke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn. Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers. Science. 2012; 336 (6086):1287–1291. DOI: 10.1126/science.1218497 - 33.
P. B. Corkum. Plasma perspective on strong field multiphoton ionization. Physical Review Letters. 1993; 71 (13):1994-1997. DOI: 10.1103/PhysRevLett.71.1994 - 34.
K. Schafer, B. Yang, L.F. DiMauro, and K.C. Kulander. Above threshold ionization beyond the high harmonic cutoff. Physical Review Letters. 1993; 70 :1599. DOI: 10.1103/PhysRevLett.70.1599 - 35.
M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz. Attosecond metrology. Nature. 2001; 414 :509–513. DOI: 10.1038/35107000 - 36.
G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli. Isolated single-cycle attosecond pulses. Science. 2006; 314 (5798):443–446. DOI: 10.1126/science.1132838 - 37.
I. J. Sola, E. Mevel, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli. Controlling attosecond electron dynamics by phase-stabilized polarization gating. Nature Physics. 2006; 2 :319–322. DOI: 10.1038/nphys281 - 38.
M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann. Strong-field physics with singular light beams. Nature Physics. 2012; 8 :743–746. DOI: 10.1038/nphys2397 - 39.
C. Hernandez-Garcia, A. Picon, J. San Roman, and L. Plaja. Attosecond extreme ultraviolet vortices from high-order harmonic generation. Physics Review Letters. 2013; 111 :083602. DOI: 10.1103/PhysRevLett.111.083602 - 40.
G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum. Creating high-harmonic beams with controlled orbital angular momentum. Physics Review Letters. 2014; 113 :153901. DOI: 10.1103/PhysRevLett.113.153901 - 41.
L. Rego, J. San-Román, L. Plaja, A. Picón, and C. Hernández-García. Ultrashort extreme-ultraviolet vortices. In: H. Pérez-de-Tejada, editor. Vortex Dynamics. 1st ed. Rijeka: Intech; 2016.