## Sub-critical regime of femtosecond inscription

Optics Express, Vol. 15, Issue 22, pp. 14750-14764 (2007)

http://dx.doi.org/10.1364/OE.15.014750

Acrobat PDF (290 KB)

### Abstract

We apply well known nonlinear diffraction theory governing focusing of a powerful light beam of arbitrary shape in medium with Kerr nonlinearity to the analysis of femtosecond (fs) laser processing of dielectric in sub-critical (input power less than the critical power of self-focusing) regime. Simple analytical expressions are derived for the input beam power and spatial focusing parameter (numerical aperture) that are required for achieving an inscription threshold. Application of non-Gaussian laser beams for better controlled fs inscription at higher powers is also discussed.

© 2007 Optical Society of America

## 1. Introduction

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond. Laser,” Opt. Lett. **21**, 1729 (1996) [CrossRef] [PubMed]

1. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond. Laser,” Opt. Lett. **21**, 1729 (1996) [CrossRef] [PubMed]

*I*. For instance, a semi-empirical approach adopted in [7

_{th}7. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerlullo, “Femtosecond writing of active optical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B , 1559 (2003) [CrossRef]

*I*. In other words, because of the sharp dependence of the multi-photon ionisation process on the field intensity, the spatial form of the inscribed structure can be well approximated by resolving an equation on the field intensity:

_{th}*I*(

*x*,

*y*,

*z*) =

*I*. This approach, being an obvious simplification, nevertheless, is justified by experimental observations indicating the existence of several different characteristic time scales in the fs laser inscription processes (see [9

_{th}9. B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. **74**, 2248 (1995) [CrossRef] [PubMed]

10. B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power. short-pulse. Lasers,” J. Opt. Soc. Am. B **13**, 459 (1996) [CrossRef]

9. B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. **74**, 2248 (1995) [CrossRef] [PubMed]

10. B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power. short-pulse. Lasers,” J. Opt. Soc. Am. B **13**, 459 (1996) [CrossRef]

7. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerlullo, “Femtosecond writing of active optical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B , 1559 (2003) [CrossRef]

7. R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerlullo, “Femtosecond writing of active optical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B , 1559 (2003) [CrossRef]

## 2. The model and basic equations

*A*(

*r*⊥,

*z*,

*t*) is governed by the nonlinear partial differential equations that accounts for the major propagation effects such as diffraction, Kerr nonlinearity, group velocity dispersion, multi-photon absorption, impact ionization and absorption and defocusing by the generated electron plasma (see e.g. [7

10. B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power. short-pulse. Lasers,” J. Opt. Soc. Am. B **13**, 459 (1996) [CrossRef]

12. S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. Lett. **87**, 213902 (2001) [CrossRef] [PubMed]

14. Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, and C.R. Thompson, “Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses,” IEEE-J. Quantum Electron. **33**, 127–137 (1997) [CrossRef]

*z*axis is assumed and this equation is essentially a reduced paraxial approximation of the wave equation for the complex electric field envelope

*A*with a carrier frequency

*ω*in the moving frame of coordinates. Here

*k*=

*n*

_{b}*k*

_{0}=

*n*

_{b}*ω*/

*c*is the propagation vector,

*n*is the linear refraction index of silica,

_{b}*k*″ is the GVD parameter,

*n*is the nonlinear coefficient,

_{2}*σ*is the cross section for inverse Bremsstrahlung, τ

_{bs}_{e}is the electron relaxation time, and the quantity

*β*describes the

_{K}*K*-photon absorption. Eq.(1b) implements the Drude model for electron-hole plasma in the bulk of silica and describes the evolution of the electron density

*ρ*. The first term on the right-hand of Eq.(1b) side is responsible for the avalanche impact ionization and the second term — for the ionization resulting from multi-photon absorption (MPA). The expression for the multiphoton ionization is a good fit of the general Keldysh expression [15] for ionisation in strong field for not very high intensities. For fused silica and 1μm light the intensity must be below 1000 TW/cm

^{2}. For higher intensities tunnelling through the binding barrier takes place during the time shorter than the laser period. In this case in Eq.(1.b) one must use the expression for ADK tunnelling ionization [16]. For the problem of inscription the results are not sensitive to the details of field induced ionization [10

**13**, 459 (1996) [CrossRef]

*diffraction*and

*Kerr nonlinearity*.

*a*= 1.77 mm, τ=42 fs and

_{s}*C*= 892 (this corresponds to the focusing lens with f = 40 mm, recalculated using relation

_{s}*C*=

_{s}*a*

^{2}

_{s}

*k*/

*f*). Here the index

*s*stands for surface, indicating that these are the beam characteristics at the surface of a dielectric, after propagation of a distance

*d*from lens to the surface. Note that depending on the distance to the surface and initial pre-focusing conditions, the field distribution at the surface of a dielectric can be rather different from the initial beam parameters. The dashed line in Fig. 1 is for the critical energy of 116 nJ that corresponds to the input pulse with the width of about 70 fs and the critical power of 2.3 MW (in fused silica with

*n*=1.453 and

_{b}*n*= 3.2× 10

_{2}^{-16}cm

^{2}/W). We considered the laser wavelength λ

_{0}to be 800 nm, and the other parameters for fused silica, used in simulations are:

*k*•= 361 fs

^{2}/cm, τ

_{e}=1 fs, inverse Bremsstrahlung cross section

_{bs}= 2.78 × 10

^{-18}cm

^{2}. Multiphoton absorption coefficient

*α*=

_{K}*β*

^{(K)}

*hv*σ

_{K}ρ

_{at}, where ρ

_{at}=2.1 ×10

^{22}atoms/cm

^{3}is a material concentration and σ

_{K}=1.3×10

^{-55}cm

^{2K}/W

^{K}/s. We assume here for fused silica a five-photon ionization with

*K*= 5 and

*E*= 7.6 eV.

_{g}*C*was about 25 in these calculations and initial beam radius at the surface

_{s}*a*was nearly 70 μm. All other parameter including material parameters are the same as for Figure 1. It is seen that apart from the narrow region where plasma defocusing and multi-photon absorption come into play, the electric field evolution during noticeable interval is mostly affected by the two physical effects: linear diffraction and the Kerr nonlinearity. While linear diffraction dominates initial propagation stage, the Kerr nonlinearity has an impact on the field dynamics as the pulse approaches a focal point that is getting stronger as pulse approaches the energy deposition region. The Kerr nonlinearity acts as a nonlinear lens [17] producing corresponding deformations of the beam waveform. The aim of this paper is to present simple theoretical methods to account for such deformations.

_{s}*I*=∣

*A*(

*z*,

*t*)∣

^{2}exceeds the threshold

*I*, leading to consecutive medium modification mapping the corresponding spatial distribution of the intensity into the refractive index changes. In what follows we assume that at z = 0 pre-focused laser beam (pulse in time) with a time-spatial waveform

_{th}*A*(

*z*= 0,

*t*) =

*A*

_{0}(

*t*) enters the dielectric medium (e.g. silica). Evolution of the powerful laser beam at first stage is described by the interplay between two focusing mechanisms: linear focusing, due to the lens, and nonlinear self-focusing. Note that typically, pre-focusing lens is used at some distance of the silica, and the boundary conditions

*A*

_{0}(

*t*) should be understood correspondingly.

## 3. Sub-critical self-focusing regime

*C*

_{1},

*C*

_{2}are determined by the parameters of the incident beam (

*P*= ∫∣

*A*

_{0}∣

^{2}

*d*

### 3.1. Sub-critical focusing of the Gaussian-shape pulses

_{s}is a beam width parameter The focusing conditions at the surface are combined in a single focusing parameter

*C*=

_{s}*π*(

*f*-

*d*)

*NA*

^{2}

*NA*

^{2}/[λ(1 -

*NA*

^{2})] which is expressed through lens focal distance f, numerical aperture NA and the distance d from lens to the surface. In the case of the Gaussian laser beam, the nonlinear evolution of the RMS beam radius is given by the following equation (that we present for convenience of the comparison in the form similar to the linear focusing theory):

*P*

_{cr_}

*G*= λ

^{2}

_{0}/(2

*π*

*n*

_{0}

*n*

_{2}) is the critical power of self-focusing for the Gaussian beams, and

*k*= 2

*π*

*n*

_{0}/λ

_{0}. This equation generalizes the usual linear focusing conditions to the nonlinear (sub-critical

*P*<

*P*

_{cr_}

*G*) case. The minimal beam radius

*R*

_{min}=

*a*(1-

_{s}*P*/

*P*

_{cr_}

*G*)

^{1/2}/(1 +

*C*

^{2}

_{s}-

*P*/

*P*

_{cr_}

*G*)

^{1/2}is achieved at the distance

*z*

_{min_NL}. The nonlinear lens allows for better control of the beam energy deposition. The important new feature is that the nonlinearity removes the restriction of the linear focusing, such as the existence of the minimal beam radius that can be achieved under the fixed pre-focusing

*C*. Another interesting feature is that using regimes with small enough effective

_{s}*C*, it could be possible to change by varying input power the ratio of the beam focal diameter

_{s}*d*

_{min}= 2

*R*

_{min}to the confocal parameter

*b*= 2

*Z*

_{R_NL}- a nonlinear lens analog of the linear astigmatic beam approach used in [7

*P*/

_{in}*P*≪ (1 +

_{cr}*C*) is satisfied on the dielectric surface, and the expression (3) can be further simplified (compare with [11

^{2}_{s}11. C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy,” Opt. Lett. **26**, 93–95 (2001) [CrossRef]

*C*.

_{s}*z*= 0,

*A*

_{0}(

*z*(see Fig. 1) where field intensity

*I*(

*r*,

*z*) exceeds the inscription threshold

*I*,

_{th}*I*(

*r*,

*z*) >

*I*:

_{th}*P*/

_{in}*P*«1 +

_{cr}*C*

^{2}

_{s}) this yields

*e*

^{2}(13.5%) of its peak, or axial, value. Using the notations introduced above, the beam radius on the surface reads:

*NA*) in the nonlinear sub-critical regime, one can use lens with the numerical aperture

_{linear}*NA*(index NL here indicates that nonlinear propagation contributes to focusing):

_{NL}*NA*on

_{NL}*NA*is shown in Fig. 5 for several input powers (normalized to critical power).

_{linear}*C*

^{2}

_{s}»1) the necessary condition of reaching inscription threshold can be written as (compare with [11

11. C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy,” Opt. Lett. **26**, 93–95 (2001) [CrossRef]

*E*required for producing permanent structural change on the numerical aperture NA and the critical area

_{th}*S*=

_{cr}*P*/

_{cr}*I*defined above can be re-written as:

_{th}4. Y. Kondo, K. Nouchi, T. Mitsuyu, M. Watanabe, P. G. Kazansky, and K. Hirao, “Fabrication of long-period fiber gratings by focused irradiation of infrared femtosecond laser pulses,” Opt. Lett. **24**, 646–648 (1999) [CrossRef]

11. C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy,” Opt. Lett. **26**, 93–95 (2001) [CrossRef]

*P*and

_{cr}*I*. Therefore, maximum accurate description of this dependence at the pulse powers close to the critical power is an important issue.

_{th}### 3.2. Basics of the sub-critical focusing of the ring-shaped beams

*P*

_{cr_m}for the ring is higher by a factor of 2

^{2m}(

*m*!)

^{2}/(2

*m*)! compared to that for the normal Gaussian beams (m=0) (see e.g. [20

20. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of Optical Vortices,” Phys. Rev. Lett. , **96**, 133901 (2006) [CrossRef] [PubMed]

*P*

_{cr_1}= 2

*P*

_{cr_G};

*P*

_{cr_2}= 8

*P*

_{cr_G}/ 3;

*P*

_{cr_3}=16

*P*

_{cr_G}/ 5. Application of ring-structured beams can be used to create higher local field intensity yet, remaining in the sub-critical regime by delaying self-focusing that would otherwise immediately occur for single-humped beam profiles. Figure 7 compares evolution of field intensities (normalized by the inscription threshold intensity) for the initial Gaussian beam (left) and for a beam with ring distribution (m = 1, right picture) with the same other initial parameters:

*P*/

_{in}*P*

_{cr_G}= 0.5 and pre-focusing parameter

*C*= 50. It is seen that, indeed, a ring-shaped beam could be useful and efficient for creating higher local intensity (that is critical for inscription) using the same input power. We would like to stress that we do not aim here to make a fair comparison, because the nonlinear evolution of the ring beam is very different compared to that one with the Gaussian input spatial distribution. Therefore, a fair comparison of ring and Gaussian beams applied to inscription should include more accurate analysis of the impact of the pre-focusing conditions and input powers scaling and will be presented elsewhere. Our goal in this section is rather to attract attention to new opportunities offered by using non-Gaussian laser beams in the sub-critical inscription.

_{s}## 4. Conclusions

## References and Links

1. | K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond. Laser,” Opt. Lett. |

2. | E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. |

3. | D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, “Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses,” Opt. Lett. |

4. | Y. Kondo, K. Nouchi, T. Mitsuyu, M. Watanabe, P. G. Kazansky, and K. Hirao, “Fabrication of long-period fiber gratings by focused irradiation of infrared femtosecond laser pulses,” Opt. Lett. |

5. | A.M. Streltsov and N.M. Borrelli, “Study of femtosecond-laser-written waveguides in glasses,” J. Opt. Soc. Am. B |

6. | M. Will, S. Nolte, B. N. Chichkov, and A. Tännermann, “Optical Properties of Waveguides Fabricated in Fused Silica by Femtosecond Laser Pulses,” Appl. Opt. |

7. | R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri, and G. Cerlullo, “Femtosecond writing of active optical waveguides with astigmatically shaped beams,” J. Opt. Soc. Am. B , 1559 (2003) [CrossRef] |

8. | C. Florea and K. A. Winick, “Fabrication and Characterization of PhotonicDevices Directly Written in Glass UsingFemtosecond Laser Pulses,” IEEE J. Lightwave Technol. |

9. | B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses,” Phys. Rev. Lett. |

10. | B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power. short-pulse. Lasers,” J. Opt. Soc. Am. B |

11. | C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, “Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy,” Opt. Lett. |

12. | S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. Lett. |

13. | 3D Laser Microfabrication: Principles and Applications, Eds. Hiroaki Misawa and Saulius Juodkazis, Wiley-VCH, 2006 |

14. | Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, and C.R. Thompson, “Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses,” IEEE-J. Quantum Electron. |

15. | L.V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov.Phys.JETP |

16. | M.V. Ammosov, N.V. Delone, and V.P. Krainov “Tunneling ionization of complex atoms and of atomic ions in alternating electromagnetic field,” Sov.Phys.JETP |

17. | V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. |

18. | S.N. Vlasov, V.A. Petrishev, and V.I. Talanov,“Average description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. |

19. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beam,” Phys. Rev. Lett. |

20. | L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of Optical Vortices,” Phys. Rev. Lett. , |

**OCIS Codes**

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(220.4000) Optical design and fabrication : Microstructure fabrication

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 9, 2007

Revised Manuscript: September 20, 2007

Manuscript Accepted: October 8, 2007

Published: October 24, 2007

**Citation**

Sergei K. Turitsyn, Vladimir K. Mezentsev, Mykhaylo Dubov, Alexander M. Rubenchik, Michail P. Fedoruk, and Evgeny V. Podivilov, "Sub-critical regime of femtosecond inscription," Opt. Express **15**, 14750-14764 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14750

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### References

- K. M. Davis, K. Miura, N. Sugimoto and K. Hirao, "Writing waveguides in glass with a femtosecond. Laser," Opt. Lett. 21,1729 (1996) [CrossRef] [PubMed]
- E. N. Glezer and E. Mazur, "Ultrafast-laser driven micro-explosions in transparent materials," Appl. Phys. Lett. 71, 882 (1997) [CrossRef]
- D. Homoelle, S. Wielandy, A. L. Gaeta, N. F. Borrelli, and C. Smith, "Infrared photosensitivity in silica glasses exposed to femtosecond laser pulses," Opt. Lett. 24, 1311-1313 (1999) [CrossRef]
- Y. Kondo, K. Nouchi, T. Mitsuyu, M. Watanabe, P. G. Kazansky, and K. Hirao, "Fabrication of long-period fiber gratings by focused irradiation of infrared femtosecond laser pulses," Opt. Lett. 24, 646-648 (1999) [CrossRef]
- A.M. Streltsov and N.M. Borrelli, "Study of femtosecond-laser-written waveguides in glasses," J. Opt. Soc. Am. B 19, 2496-2504 (2002). [CrossRef]
- M. Will, S. Nolte, B. N. Chichkov, and A. Tünnermann, "Optical Properties of Waveguides Fabricated in Fused Silica by Femtosecond Laser Pulses," Appl. Opt. 41, 4360-4364 (2002) [CrossRef] [PubMed]
- R. Osellame, S. Taccheo, M. Marangoni, R. Ramponi, P. Laporta, D. Polli, S. De Silvestri and G. Cerlullo, "Femtosecond writing of active optical waveguides with astigmatically shaped beams," J. Opt. Soc. Am. B, 1559 (2003)Q1 [CrossRef]
- C. Florea and K. A. Winick, "Fabrication and Characterization of PhotonicDevices Directly Written in Glass UsingFemtosecond Laser Pulses," IEEE J. Lightwave Technol. 21, 246 (2003)Q2 [CrossRef]
- B.C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, " Laser-Induced Damage in Dielectrics with Nanosecond to Subpicosecond Pulses," Phys. Rev. Lett. 74, 2248 (1995) [CrossRef] [PubMed]
- B.C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Optical ablation by high-power. short-pulse. Lasers," J. Opt. Soc. Am. B 13, 459 (1996) [CrossRef]
- C. B. Schaffer, A. Brodeur, J. F. Garca, and E. Mazur, "Micromachining bulk glass by use of femtosecond laser pulses with nanoJoule energy," Opt. Lett. 26, 93-95 (2001) [CrossRef]
- S. Tzortzakis, L. Sudrie, M. Franko, B. Prade, A. Mysrowicz, A. Couairon, and L. Berge, "Self-focusing of few-cycle light pulses in dielectric media," Phys. Rev. Lett. 87, 213902 (2001) [CrossRef] [PubMed]
- 3D Laser Microfabrication: Principles and Applications, Eds. Hiroaki Misawa and Saulius Juodkazis, Wiley-VCH, 2006
- Q. Feng, J.V. Moloney, A.C. Newell, E.M. Wright, K. Cook, P.K. Kennedy, D.X. Hammer, B.A. Rockwell, C.R. Thompson, "Theory and Simulation on the Threshold of Water Breakdown Induced by Focused Ultrashort Laser Pulses," IEEE-J. Quantum Electron. 33, 127-137 (1997) [CrossRef]
- L.V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov.Phys.JETP 20,1307 (1965)Q3
- M.V. Ammosov,N.V. Delone, and V.P. Krainov "Tunneling ionization of complex atoms and of atomic ions in alternating electromagnetic field," Sov.Phys.JETP 64, 1191 (1986)Q4
- V. I. Talanov, "Focusing of light in cubic media," JETP Lett. 11, 199 (1970)
- S.N. Vlasov, V.A. Petrishev, and V.I. Talanov, "Average description of wave beams in linear and nonlinear media," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353 (1971) [Radiophys. and Quantum Electron. 14, 1062 (1974)]Q5
- R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beam," Phys. Rev. Lett. 13, 479 (1964) [CrossRef]
- L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G.W. ’t Hooft, E. R. Eliel, and G. Fibich, "Collapse of Optical Vortices," Phys. Rev. Lett., 96, 133901 (2006) [CrossRef] [PubMed]

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