Spontaneous optical skyrmion generation by frequency doubling in underdense plasmas

Eduardo Oliva; Marcos G. Barriopedro; Miguel A. Porras

arxiv: 2605.03145 · v1 · submitted 2026-05-04 · ⚛️ physics.optics · physics.plasm-ph

Spontaneous optical skyrmion generation by frequency doubling in underdense plasmas

Marcos G. Barriopedro , Eduardo Oliva , Miguel A. Porras This is my paper

Pith reviewed 2026-05-08 17:24 UTC · model grok-4.3

classification ⚛️ physics.optics physics.plasm-ph

keywords optical skyrmionsStokes skyrmionssecond-harmonic generationunderdense plasmasoptical vorticesPoincaré sphereplasma diagnosticsnonlinear optics

0 comments

The pith

Stokes skyrmions arise spontaneously when optical vortices drive second-harmonic generation in underdense plasmas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Stokes skyrmions form naturally in the frequency-doubled light produced when an optical vortex propagates through underdense plasma. The plasma's nonlinear response structures the new field so its polarization states cover the full Poincaré sphere. Plasma density variations then stretch or twist this texture, turning the skyrmion into a potential sensor for the density profile. A reader would care because the effect requires no special devices beyond an ordinary vortex beam and a plasma, and it links a topological feature directly to a measurable plasma property.

Core claim

Stokes skyrmions emerge spontaneously in second-order harmonic generation in underdense plasmas driven by optical vortices. The nonlinear response produces a structured frequency-doubled field whose polarization texture maps the Poincaré sphere. When plasma inhomogeneities are taken into account, the electron density gradient deforms the skyrmionic texture, enabling topological diagnosis of plasma density.

What carries the argument

The nonlinear polarization response during second-harmonic generation, which maps the input vortex properties onto a full Poincaré-sphere texture in the frequency-doubled output.

If this is right

Skyrmions appear without requiring sophisticated wave engineering or photonic devices.
The frequency-doubled field carries polarization states that span the entire Poincaré sphere.
Density gradients deform the skyrmionic texture, supplying a direct topological readout of plasma density.
Topological diagnosis becomes possible inside plasma-based nonlinear optics experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same spontaneous mapping might occur in other inhomogeneous nonlinear media, extending skyrmion generation beyond plasmas.
Polarization imaging of the second-harmonic light could serve as a non-invasive diagnostic tool in laser-plasma facilities.
Varying the input vortex charge or plasma scale length would test how the skyrmion topology scales with those parameters.

Load-bearing premise

The nonlinear plasma response produces a polarization texture that exactly maps the Poincaré sphere and that density gradients deform this texture measurably without other propagation effects dominating.

What would settle it

Measure the polarization states across the second-harmonic beam generated by an optical vortex in underdense plasma and check whether they cover every point on the Poincaré sphere; then impose a controlled density gradient and verify that the skyrmion texture deforms in a topologically detectable way.

Figures

Figures reproduced from arXiv: 2605.03145 by Eduardo Oliva, Marcos G. Barriopedro, Miguel A. Porras.

**Figure 1.** Figure 1: Upper panel: (left) A linearly polarized vortex pump view at source ↗

**Figure 2.** Figure 2: (First Row) Near and far field intensity view at source ↗

read the original abstract

Optical skyrmions have been widely explored in recent years. Among them, Stokes skyrmions require sophisticated wave engineering or photonic devices for their generation. We show that Stokes skyrmions can emerge spontaneously in second-order harmonic generation in underdense plasmas driven by optical vortices. The nonlinear response produces a structured frequency-doubled field whose polarization texture maps the Poincar\'e sphere. When plasma inhomogeneities are taken into account, the electron density gradient deforms the skyrmionic texture, enabling topological diagnosis of plasma density.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches spontaneous Stokes skyrmion generation via frequency doubling in underdense plasma but the abstract supplies no derivations or checks on the key polarization mapping.

read the letter

The main takeaway is that this work proposes using optical vortices in underdense plasmas for spontaneous Stokes skyrmion formation in the second harmonic, with plasma density gradients then deforming the texture for a topological diagnostic. That framing positions the plasma itself as the generator rather than relying on fabricated devices, which is a clear shift from much of the existing skyrmion literature. The link to density diagnosis is a practical angle that could matter for plasma experiments if the topology survives. The paper does a reasonable job contrasting the approach with engineered methods and keeping the claim focused on the nonlinear response plus inhomogeneity effects. The soft spot is exactly what the stress-test note flags: the abstract asserts that the nonlinear response produces a frequency-doubled field whose polarization texture maps the full Poincaré sphere, yet it gives no derivation of the second-harmonic Stokes parameters, no treatment of phase mismatch or plasma birefringence, and no indication that the parameters actually cover the sphere instead of a limited patch. Without those steps visible, it is impossible to tell whether propagation effects wash out the structure. If the full manuscript contains analytic steps or simulations that close this gap, the claim would be much stronger; as it stands, the central assumption remains unverified. The thinking is direct and engages the relevant optics-plasma intersection without internal contradictions. This is for readers working at the boundary of topological photonics and plasma diagnostics who are looking for new generation routes or measurement ideas. It is worth sending for peer review so the authors can supply the missing calculations and referees can test whether the mapping actually holds.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that Stokes skyrmions emerge spontaneously during second-harmonic generation in underdense plasmas driven by optical vortices. The nonlinear plasma response is asserted to produce a frequency-doubled field whose polarization texture maps the full Poincaré sphere; plasma density gradients are said to deform this texture in a manner that permits topological diagnosis of the density profile.

Significance. If the central claims are substantiated by explicit derivations of the nonlinear current, Stokes-parameter evolution, and propagation effects, the result would be significant: it would demonstrate a device-free route to optical skyrmions and a new topological diagnostic for plasma inhomogeneity. The absence of any equations, simulations, or quantitative checks in the current text, however, prevents assessment of whether the polarization texture truly covers the sphere or remains robust against phase mismatch, dispersion, and birefringence.

major comments (2)

[Abstract] Abstract: the assertion that 'the nonlinear response produces a structured frequency-doubled field whose polarization texture maps the Poincaré sphere' is presented without any derivation of the second-harmonic Stokes parameters (S1,S2,S3) or demonstration that they uniformly cover the unit sphere rather than a lower-dimensional subset. No expression for the nonlinear polarization or current driven by the optical vortex is supplied, leaving the mapping unverified.
[Abstract] Abstract: the claim that 'when plasma inhomogeneities are taken into account, the electron density gradient deforms the skyrmionic texture' is stated without showing how the density gradient enters the wave equation or how the resulting deformation is topologically diagnostic rather than being masked by linear propagation effects (phase mismatch, group-velocity dispersion, or plasma birefringence).

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment below and have revised the text to incorporate explicit derivations and explanations where they were previously omitted for brevity.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'the nonlinear response produces a structured frequency-doubled field whose polarization texture maps the Poincaré sphere' is presented without any derivation of the second-harmonic Stokes parameters (S1,S2,S3) or demonstration that they uniformly cover the unit sphere rather than a lower-dimensional subset. No expression for the nonlinear polarization or current driven by the optical vortex is supplied, leaving the mapping unverified.

Authors: We agree that the abstract, standing alone, does not contain the supporting derivations. The full manuscript derives the nonlinear current from the second-order fluid response of the underdense plasma to the incident optical vortex, yielding a frequency-doubled field whose electric-field components produce Stokes parameters S1, S2, S3 that together cover the Poincaré sphere. We have added a dedicated paragraph in the revised manuscript that supplies the explicit form of the nonlinear polarization and current, followed by the analytic expressions for the Stokes vector and a brief verification that the mapping is onto the full sphere for the chosen vortex parameters. revision: yes
Referee: [Abstract] Abstract: the claim that 'when plasma inhomogeneities are taken into account, the electron density gradient deforms the skyrmionic texture' is stated without showing how the density gradient enters the wave equation or how the resulting deformation is topologically diagnostic rather than being masked by linear propagation effects (phase mismatch, group-velocity dispersion, or plasma birefringence).

Authors: We acknowledge that the original text did not explicitly show the entry of the density gradient into the propagation equation. In the revised manuscript we now include the relevant term in the wave equation for the second-harmonic field, arising from the spatially varying plasma frequency that produces a position-dependent phase shift. This shift deforms the Stokes-vector texture while preserving the skyrmion number, thereby allowing the observed distortion to serve as a topological diagnostic of the density gradient. We have added a short discussion noting that, for the underdense conditions and short propagation lengths considered, phase mismatch remains small; however, a complete quantitative treatment of group-velocity dispersion and birefringence would require numerical simulations that lie beyond the scope of the present letter. revision: partial

standing simulated objections not resolved

A full numerical verification of robustness against group-velocity dispersion and plasma birefringence is not provided in the current work.

Circularity Check

0 steps flagged

No derivation chain or equations available; circularity cannot be assessed

full rationale

The provided abstract and context contain no equations, derivations, or technical steps. Per hard rules, circularity requires explicit quotes showing reduction of a claimed prediction or result to its own inputs by construction. No such content exists here, so no steps are identified. The paper's central claim about spontaneous Stokes skyrmion generation via nonlinear plasma response is stated at a high level without any load-bearing mathematical chain that could be checked for self-definition, fitted inputs, or self-citation issues. This is the normal honest outcome when technical details are absent.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no access to equations or models, so no free parameters, axioms or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5383 in / 1148 out tokens · 28154 ms · 2026-05-08T17:24:03.267530+00:00 · methodology

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A45, 8185 (1992)

work page 1992
[2]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: OAM manip- ulation from topological charge to multiple singularities, Light Sci Appl8, 90 (2019)

work page 2019
[3]

M. V. Berry, The singularities of light: intensity, phase, polarisation, Light Sci Appl12, 238 (2023)

work page 2023
[4]

Coullet, L

P. Coullet, L. Gil, and F. Rocca, Optical vortices, Optics Communications73, 403 (1989)

work page 1989
[5]

L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics, Nature Photon8, 821 (2014)

work page 2014
[6]

S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, Fractional optical vortex beam induced rotation of parti- cles, Opt. Express, OE13, 7726 (2005)

work page 2005
[7]

Cojoc, V

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Ro- manato, and E. D. Fabrizio, Laser trapping and micro- manipulation using optical vortices, Microelectronic Engi- neering Proceedings of the 30th International Conference on Micro- and Nano-Engineering,78-79, 125 (2005)

work page 2005
[8]

Čelechovský and Z

R. Čelechovský and Z. Bouchal, Optical implementation of the vortex information channel, New J. Phys.9, 328 (2007)

work page 2007
[9]

D’Ambrosio, G

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, Entangled vector vortex beams, Phys. Rev. A94, 030304 (2016)

work page 2016
[10]

López, M

S. López, M. d. V. Pérez-Fajardo, and E. Oliva, Diagnos- ing the electron density of plasma amplifiers using XUV vortex beams, Opt. Express, OE33, 46285 (2025)

work page 2025
[11]

López, E

S. López, E. d. l. Fuente, A. Alonso, A. K. Pandey, O. Guil- baud, and E. Oliva, Conservation of orbital angular mo- mentum throughout amplification of high order harmonics in Ni-like krypton and silver plasmas, Opt. Express, OE 31, 8465 (2023)

work page 2023
[12]

Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv

Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon., AOP1, 1 (2009)

work page 2009
[13]

Bauer, P

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Observation of optical polarization Möbius strips, Science 347, 964 (2015)

work page 2015
[14]

Z. Ye, S. M. Barnett, S. Franke-Arnold, J. B. Götte, A. McWilliam, F. C. Speirits, and C. M. Cisowski, Theory of paraxial optical skyrmions, Proc. A480, 20240109 (2024)

work page 2024
[15]

Y. Shen, Q. Zhang, P. Shi, L. Du, X. Yuan, and A. V. Zayats, Optical skyrmions and other topological quasipar- ticles of light, Nat. Photon.18, 15 (2024). 5

work page 2024
[16]

Sugic, R

D. Sugic, R. Droop, E. Otte, D. Ehrmanntraut, F. Nori, J. Ruostekoski, C. Denz, and M. R. Dennis, Particle-like topologies in light, Nat Commun12, 6785 (2021)

work page 2021
[17]

Y. Zeng, Y. Yu, X. Shen, J. Chen, and Q. Zhan, Tightly focused optical skyrmions and merons formed by electric- field vectors with prescribed characteristics, Nanophoton- ics13, 251 (2024)

work page 2024
[18]

Tsesses, E

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lind- ner, and G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science361, 993 (2018)

work page 2018
[19]

X. Lei, A. Yang, P. Shi, Z. Xie, L. Du, A. V. Zayats, and X. Yuan, Photonic Spin Lattices: Symmetry Constraints for Skyrmion and Meron Topologies, Phys. Rev. Lett. 127, 237403 (2021)

work page 2021
[20]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, Deep- subwavelength features of photonic skyrmions in a con- fined electromagnetic field with orbital angular momen- tum, Nat. Phys.15, 650 (2019)

work page 2019
[21]

S. Wang, Z. Zhou, Z. Zheng, J. Sun, H. Cao, S. Song, Z.-L. Deng, F. Qin, Y. Cao, and X. Li, Topological Structures of Energy Flow: Poynting Vector Skyrmions, Phys. Rev. Lett.133, 073802 (2024)

work page 2024
[22]

A. M. Beckley, T. G. Brown, and M. A. Alonso, Full Poincaré beams, Opt. Express, OE18, 10777 (2010)

work page 2010
[23]

Y. Shen, E. C. Martínez, and C. Rosales-Guzmán, Gen- eration of Optical Skyrmions with Tunable Topological Textures, ACS Photonics9, 296 (2022)

work page 2022
[24]

Lin, Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers, Phys

W. Lin, Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers, Phys. Rev. Res. 3, 10.1103/PhysRevResearch.3.023055 (2021)

work page doi:10.1103/physrevresearch.3.023055 2021
[25]

Mata-Cervera, D

N. Mata-Cervera, D. K. Sharma, Y. Shen, R. Paniagua- Dominguez, and M. A. Porras, Skyrmionic Polarization Texture around the Phase Singularity of Optical Vortices, Phys. Rev. Lett.135, 033805 (2025)

work page 2025
[26]

Rivera and I

N. Rivera and I. Kaminer, Light–matter interactions with photonic quasiparticles, Nat Rev Phys2, 538 (2020)

work page 2020
[27]

Marco, L

D. Marco, L. Plaja, and C. Hernández-García, Attosecond Light Skyrmion Pulses via High Harmonic Generation (2025), arXiv:2509.19113 [physics]

work page arXiv 2025
[28]

Y. R. Shen,Principles of nonlinear optics(Wiley- Interscience,New York, NY, USA, 1984)

work page 1984
[29]

Beresna, P

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, High average power second harmonic generation in air, Appl. Phys. Lett.95, 121502 (2009)

work page 2009
[30]

R. W. Boyd, Chapter 1 - The Nonlinear Optical Suscep- tibility, inNonlinear Optics (Third Edition)(Academic Press, Burlington, 2008) third edition ed., pp. 1–67

work page 2008
[31]

Couairon and A

A. Couairon and A. Mysyrowicz, Femtosecond filamen- tation in transparent media, Physics Reports441, 47 (2007)

work page 2007
[32]

S. L. Chin,Femtosecond Laser Filamentation, Springer Series on Atomic, Optical, and Plasma Physics, Vol. 55 (Springer, New York, NY, 2010)

work page 2010

[1] [1]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A45, 8185 (1992)

work page 1992

[2] [2]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: OAM manip- ulation from topological charge to multiple singularities, Light Sci Appl8, 90 (2019)

work page 2019

[3] [3]

M. V. Berry, The singularities of light: intensity, phase, polarisation, Light Sci Appl12, 238 (2023)

work page 2023

[4] [4]

Coullet, L

P. Coullet, L. Gil, and F. Rocca, Optical vortices, Optics Communications73, 403 (1989)

work page 1989

[5] [5]

L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics, Nature Photon8, 821 (2014)

work page 2014

[6] [6]

S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, Fractional optical vortex beam induced rotation of parti- cles, Opt. Express, OE13, 7726 (2005)

work page 2005

[7] [7]

Cojoc, V

D. Cojoc, V. Garbin, E. Ferrari, L. Businaro, F. Ro- manato, and E. D. Fabrizio, Laser trapping and micro- manipulation using optical vortices, Microelectronic Engi- neering Proceedings of the 30th International Conference on Micro- and Nano-Engineering,78-79, 125 (2005)

work page 2005

[8] [8]

Čelechovský and Z

R. Čelechovský and Z. Bouchal, Optical implementation of the vortex information channel, New J. Phys.9, 328 (2007)

work page 2007

[9] [9]

D’Ambrosio, G

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, Entangled vector vortex beams, Phys. Rev. A94, 030304 (2016)

work page 2016

[10] [10]

López, M

S. López, M. d. V. Pérez-Fajardo, and E. Oliva, Diagnos- ing the electron density of plasma amplifiers using XUV vortex beams, Opt. Express, OE33, 46285 (2025)

work page 2025

[11] [11]

López, E

S. López, E. d. l. Fuente, A. Alonso, A. K. Pandey, O. Guil- baud, and E. Oliva, Conservation of orbital angular mo- mentum throughout amplification of high order harmonics in Ni-like krypton and silver plasmas, Opt. Express, OE 31, 8465 (2023)

work page 2023

[12] [12]

Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv

Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon., AOP1, 1 (2009)

work page 2009

[13] [13]

Bauer, P

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Observation of optical polarization Möbius strips, Science 347, 964 (2015)

work page 2015

[14] [14]

Z. Ye, S. M. Barnett, S. Franke-Arnold, J. B. Götte, A. McWilliam, F. C. Speirits, and C. M. Cisowski, Theory of paraxial optical skyrmions, Proc. A480, 20240109 (2024)

work page 2024

[15] [15]

Y. Shen, Q. Zhang, P. Shi, L. Du, X. Yuan, and A. V. Zayats, Optical skyrmions and other topological quasipar- ticles of light, Nat. Photon.18, 15 (2024). 5

work page 2024

[16] [16]

Sugic, R

D. Sugic, R. Droop, E. Otte, D. Ehrmanntraut, F. Nori, J. Ruostekoski, C. Denz, and M. R. Dennis, Particle-like topologies in light, Nat Commun12, 6785 (2021)

work page 2021

[17] [17]

Y. Zeng, Y. Yu, X. Shen, J. Chen, and Q. Zhan, Tightly focused optical skyrmions and merons formed by electric- field vectors with prescribed characteristics, Nanophoton- ics13, 251 (2024)

work page 2024

[18] [18]

Tsesses, E

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lind- ner, and G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science361, 993 (2018)

work page 2018

[19] [19]

X. Lei, A. Yang, P. Shi, Z. Xie, L. Du, A. V. Zayats, and X. Yuan, Photonic Spin Lattices: Symmetry Constraints for Skyrmion and Meron Topologies, Phys. Rev. Lett. 127, 237403 (2021)

work page 2021

[20] [20]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, Deep- subwavelength features of photonic skyrmions in a con- fined electromagnetic field with orbital angular momen- tum, Nat. Phys.15, 650 (2019)

work page 2019

[21] [21]

S. Wang, Z. Zhou, Z. Zheng, J. Sun, H. Cao, S. Song, Z.-L. Deng, F. Qin, Y. Cao, and X. Li, Topological Structures of Energy Flow: Poynting Vector Skyrmions, Phys. Rev. Lett.133, 073802 (2024)

work page 2024

[22] [22]

A. M. Beckley, T. G. Brown, and M. A. Alonso, Full Poincaré beams, Opt. Express, OE18, 10777 (2010)

work page 2010

[23] [23]

Y. Shen, E. C. Martínez, and C. Rosales-Guzmán, Gen- eration of Optical Skyrmions with Tunable Topological Textures, ACS Photonics9, 296 (2022)

work page 2022

[24] [24]

Lin, Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers, Phys

W. Lin, Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers, Phys. Rev. Res. 3, 10.1103/PhysRevResearch.3.023055 (2021)

work page doi:10.1103/physrevresearch.3.023055 2021

[25] [25]

Mata-Cervera, D

N. Mata-Cervera, D. K. Sharma, Y. Shen, R. Paniagua- Dominguez, and M. A. Porras, Skyrmionic Polarization Texture around the Phase Singularity of Optical Vortices, Phys. Rev. Lett.135, 033805 (2025)

work page 2025

[26] [26]

Rivera and I

N. Rivera and I. Kaminer, Light–matter interactions with photonic quasiparticles, Nat Rev Phys2, 538 (2020)

work page 2020

[27] [27]

Marco, L

D. Marco, L. Plaja, and C. Hernández-García, Attosecond Light Skyrmion Pulses via High Harmonic Generation (2025), arXiv:2509.19113 [physics]

work page arXiv 2025

[28] [28]

Y. R. Shen,Principles of nonlinear optics(Wiley- Interscience,New York, NY, USA, 1984)

work page 1984

[29] [29]

Beresna, P

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, High average power second harmonic generation in air, Appl. Phys. Lett.95, 121502 (2009)

work page 2009

[30] [30]

R. W. Boyd, Chapter 1 - The Nonlinear Optical Suscep- tibility, inNonlinear Optics (Third Edition)(Academic Press, Burlington, 2008) third edition ed., pp. 1–67

work page 2008

[31] [31]

Couairon and A

A. Couairon and A. Mysyrowicz, Femtosecond filamen- tation in transparent media, Physics Reports441, 47 (2007)

work page 2007

[32] [32]

S. L. Chin,Femtosecond Laser Filamentation, Springer Series on Atomic, Optical, and Plasma Physics, Vol. 55 (Springer, New York, NY, 2010)

work page 2010