Filament propagation

In nonlinear optics, filament propagation is propagation of a beam of light through a medium without diffraction. This is possible because the Kerr effect causes an index of refraction change in the medium, resulting in self-focusing of the beam.^[1]

Filamentary damage tracks in glass caused by laser pulses were first observed by Michael Hercher in 1964.^[2] Filament propagation of laser pulses in the atmosphere was observed in 1994 by Gérard Mourou and his team at University of Michigan. The balance between the self-focusing refraction and self-attenuating diffraction by ionization and rarefaction of a laser beam of terawatt intensities, created by chirped pulse amplification, in the atmosphere creates "filaments" which act as waveguides for the beam thus preventing divergence. Competing theories, that the observed filament was actually an illusion created by an axiconic (bessel) or moving focus instead of a "waveguided" concentration of the optical energy, were put to rest by workers at Los Alamos National Laboratory in 1997.^[3] Though sophisticated models have been developed to describe the filamentation process, a model proposed by Akozbek et al.^[4] provides a semi-analytical and easy to understand solution for the propagation of strong laser pulses in the air.

Filament propagation in a semiconductor medium can also be observed in large aperture vertical cavity surface emitting lasers.

Femtosecond laser filamentation in gaseous media

Self-focusing

A laser beam traversing a medium can modulate the refractive index of medium as^[5]

${\displaystyle n={n_{0}+{\bar {n}}_{2}I}}$

where ${\displaystyle n_{0}}$, ${\displaystyle {\bar {n}}_{2}}$ and ${\displaystyle I}$ are linear refractive index, second order refractive index and intensity of propagating laser field respectively. Self-focusing occurs when the phase shift due to Kerr effect compensates for the phase shift because of Gaussian beam divergence. Phase change due to diffraction for a Gaussian beam after traversing a length of ${\displaystyle \Delta z}$ is

${\displaystyle \phi _{diffraction}={k\Delta z \over 2\rho _{0}^{2}}r^{2}}$

and phase change because of Kerr effect is

${\displaystyle \phi _{Kerr}={2\pi {\bar {n}}_{2}I_{0}\Delta z \over \lambda }exp({-2r^{2} \over w_{0}^{2}})\approx {2\pi {\bar {n}}_{2}I_{0}\Delta z \over \lambda }(1-{2r^{2} \over w_{0}^{2}})}$.

where ${\displaystyle k={2\pi n_{0} \over \lambda }}$, ${\displaystyle \rho _{0}={\pi w_{0}^{2}n_{0} \over \lambda }}$(Rayleigh range) and ${\displaystyle w_{0}}$ is the waist of Gaussian beam. For self-focusing to happen the one have to satisfy the condition of ${\displaystyle r^{2}}$ terms be equal in magnitude for both Kerr and diffraction phases. Hence

${\displaystyle I_{0}={w_{0}^{2} \over 4\rho _{0}^{2}{\bar {n}}_{2}}}$.

On the other hand, we know that area of a Gaussian beam at its waist is ${\displaystyle \pi w_{0}^{2} \over 2}$. Therefore^[6]

${\displaystyle P_{c}={\lambda ^{2} \over 8\pi n_{0}{\bar {n}}_{2}}}$.

Note

${\displaystyle {\bar {n}}_{2}\left({cm^{2} \over W}\right)=n_{2}n_{0}\epsilon _{0}c}$

Self-focusing needs a laser peak power higher than the critical power ${\displaystyle P_{c}}$ (order of gigawatts in air^[7]), however, for infrared (IR) nanosecond pulses with peak powers higher than the critical power self-focusing is not possible. Multiphoton ionization, inverse Bremsstrahlung and electron avalanche ionization are three major results of gas and laser interaction. The later two processes are collisional-type interactions and take time to accomplish (picosecond to nanosecond). A nanosecond pulse is long enough to develop the air breakdown before the power reaches the GW order required for self-focusing. Breakdown of gas produces plasma that has absorbing and reflecting effect so self-focusing is prohibited.^[7]

Re-focusing during the propagation of a focused short laser pulse

An interesting phenomenon related to the filament propagation is the refocusing of focused laser pulses after the geometrical focus.^[8][9] Gaussian Beam propagation predicts an increasing beam width bidirectionally away from the geometric focus. However, in the situation of laser filamentation, the beam will quickly recollapse. This divergence and refocusing will continue indefinitely.

In photo-reactive systems

Filament formation and propagation may also be observed in photopolymer systems. Such systems display a Kerr-like optical nonlinearity via photoreactive-based increases in the refractive index.^[10] The filaments form as a result of the self-trapping of individual beams, or modulation instability of a wide-area light profile. Filament propagation has been observed in several photo-polymerizable systems, including organo-siloxane,^[11] acrylics,^[12] epoxy and copolymers with epoxies,^[13] and polymer blends.^[14][15] The locations of filament formation and propagation may be controlled by modulating the spatial profile of the input light field. Such photo-reactive systems are able to produce filaments from spatially and temporally incoherent light, because the slow reaction responds to the time-average intensity of the optical field, whereby femto-second fluctuations wash out.^[11] This is similar to photo-refractive media with non-instantaneous responses, which enable filament propagation with incoherent or partially incoherent light.^[16]

Potential applications

The filaments, having made a plasma, turn the narrowband laser pulse into a broadband pulse having a wholly new set of applications. An interesting aspect of the filamentation induced plasma is the limited density of the electrons, a process which prevents the optical breakdown.^[17] This effect provides an excellent source for spectroscopy of high pressure with low level of continuum and also smaller line broadening.^{[18][clarification needed]} Another potential application is the LIDAR-monitoring of air.^[19]

Flat panel dicing using short laser pulses is an important application due to the fact that as the glass substrates become thinner it becomes more difficult to improve the process yield using conventional diamond blade dicing techniques. Using short pulses dicing speeds of over 400 mm/s has been successfully demonstrated on non-alkali glass and borosilicate glass, using a 50 kHz, 5W high-power femtosecond laser. The working principle developed by Kamata et al.^[20] is the following. The short pulse laser beam having a wavelength to which the work is transparent is directed to the front surface of the work toward the back surface and focused. A filament in the light beam traveling direction from the beam waist is formed by the auto-focusing action due to the laser beam propagation in the work is formed. The substance in the filament is decomposed by the laser beam and can be discharged from the back surface, and a cavity is formed in the channel. While forming the cavity, the laser beam is scanned, a machined surface is formed, and thereafter the work can be cut with a weak bending stress.^{[citation needed]}

In July 2014, researchers at the University of Maryland reported using filamenting femtosecond laser pulses in a square arrangement to produce a density gradient in air which acted as an optical waveguide lasting on the order of several milliseconds. Initial testing demonstrated a signal gain of 50% over an unguided signal at a distance of about one meter.^[21] A field application was demonstrated in 2021, where kHz-repetition-rate 1030-nm terawatt Yb:YAG laser, installed in the vicinity of the 124-m-tall Säntis telecommunications tower was used to guide lightning strikes towards the tower's Franklin rod, opening up the possibility of future laser lightning rods.^[22]

References

1. Rashidian Vaziri, M R (2013). "Describing the propagation of intense laser pulses in nonlinear Kerr media using the ducting model". Laser Physics. 23 (10): 105401. Bibcode:2013LaPhy..23j5401R. doi:10.1088/1054-660X/23/10/105401. S2CID 250912159.
2. Hercher, M. (1964). "Laser-induced damage in transparent media". Journal of the Optical Society of America. 54: 563.
3. Xhao, X.M.; Jones, R.J.; Strauss, C.E.M.; Funk, D.J.; Roberts, J.P.; Taylor, A.J. (1997). "Control of femtosecond pulse filament formation in air through variation of the initial chirp of the pulse". CLEO '97., Summaries of Papers Presented at the Conference on Lasers and Electro-Optics. Vol. 11. IEEE. pp. 377–8. doi:10.1109/CLEO.1997.603294. ISBN 0-7803-4125-2. S2CID 120016673.
4. Aközbek, N.; Bowden, C.M.; Talebpour, A.; Chin, S.L. (2000). "Femtosecond pulse propagation in air: Variational analysis". Phys. Rev. E. 61 (4): 4540–9. Bibcode:2000PhRvE..61.4540A. doi:10.1103/PhysRevE.61.4540. PMID 11088254.
5. Boyd, Robert (2008). Nonlinear optics (3rd ed.). Academic press. ISBN 978-0-12-369470-6.
6. Diels, Jean-Claude; Rudolph, Wolfgang (2006). Ultrashort laser pulse phenomena (2nd ed.). ISBN 978-0-12-215493-5.
7. Chin, S.L.; Wang, T.J.; Marceau, C. (2012). "Advances in intense femtosecond laser filamentation in air". Laser Physics. 22 (1): 1–53. Bibcode:2012LaPhy..22....1C. doi:10.1134/S1054660X11190054. S2CID 12993181.
8. Mlejnek, M.; Wright, E.M.; Moloney, J.V. (1998). "Dynamic spatial replenishment of femtosecond pulses propagating in air". Optics Letters. 23 (5): 382–4. Bibcode:1998OptL...23..382M. doi:10.1364/OL.23.000382. PMID 18084519.
9. Talebpour, A.; Petit, S.; Chin, S.L. (1999). "Re-focusing during the propagation of a focused femtosecond Ti: Sapphire laser pulse in air". Optics Communications. 171 (4–6): 285–290. Bibcode:1999OptCo.171..285T. doi:10.1016/S0030-4018(99)00498-8.
10. Kewitsch, Anthony S.; Yariv, Amnon (1996-01-01). "Self-focusing and self-trapping of optical beams upon photopolymerization" (PDF). Optics Letters. 21 (1): 24–6. Bibcode:1996OptL...21...24K. doi:10.1364/OL.21.000024. PMID 19865292.
11. Burgess, Ian B.; Shimmell, Whitney E.; Saravanamuttu, Kalaichelvi (2007). "Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium". Journal of the American Chemical Society. 129 (15): 4738–46. doi:10.1021/ja068967b. PMID 17378567.
12. Biria, Saeid; Malley, Philip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-03-03). "Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization". The Journal of Physical Chemistry C. 120 (8): 4517–28. doi:10.1021/acs.jpcc.5b11377. ISSN 1932-7447.
13. Basker, Dinesh K.; Brook, Michael A.; Saravanamuttu, Kalaichelvi (2015-09-03). "Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides". The Journal of Physical Chemistry C. 119 (35): 20606–17. doi:10.1021/acs.jpcc.5b07117. ISSN 1932-7447.
14. Biria, Saeid; Malley, Phillip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-11-15). "Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring". ACS Macro Letters. 5 (11): 1237–41. doi:10.1021/acsmacrolett.6b00659. PMID 35614732.
15. Biria, Saeid; Hosein, Ian D. (2017-05-09). "Control of Morphology in Polymer Blends through Light Self-Trapping: An in Situ Study of Structure Evolution, Reaction Kinetics, and Phase Separation". Macromolecules. 50 (9): 3617–26. Bibcode:2017MaMol..50.3617B. doi:10.1021/acs.macromol.7b00484. ISSN 0024-9297.
16. Trillo, Stefano (2001). Spatial Solitons. Springer Series in Optical Sciences. Springer. ISBN 978-3-540-41653-1. Archived from the original on 2017-12-21.
17. Talebpour, A.; Abdel-Fattah, M.; Chin, S.L. (2000). "Focusing limits of intense ultrafast laser pulses in a high pressure gas: road to new spectroscopic source". Optics Communications. 183 (5–6): 479–484. Bibcode:2000OptCo.183..479T. doi:10.1016/S0030-4018(00)00903-2.
18. Talebpour, A.; Abdel-Fattah, M.; Bandrauk, A.D.; Chin, S.L. (2001). "Spectroscopy of the gases interacting with intense femtosecond laser pulses". Laser Physics. 11 (1): 68–76.
19. Wöste, Ludger; Frey, Steffen; Wolf, Jean-Pierre (2006). "LIDAR-Monitoring of the Air with Femtosecond Plasma Channels". Advances in Atomic, Molecular, and Optical Physics. 53: 413–441. Bibcode:2006AAMOP..53..413W. doi:10.1016/S1049-250X(06)53011-3. ISBN 978-0-12-003853-4.
20. JP WO2008126742A1, Kamata, M.; Sumyoshi, T. & Tsujikaula, S. et al., "Laser machining method, laser cutting method, and method for dividing structure having multilayer board", published 2008 
21. "Creating optical cables out of thin air". (e) Science News. 22 July 2014. Retrieved 4 September 2023.
22. Houard, Aurélien; Walch, Pierre; Produit, Thomas; et al. (16 January 2023). "Laser-guided lightning". Nature Photonics. 17 (3): 231–235. arXiv:2207.03769. Bibcode:2023NaPho..17..231H. doi:10.1038/s41566-022-01139-z. PMID 36909208.

External links

• Experiments Detail How Powerful Ultrashort Laser Pulses Propagate through Air
• Filamentation and Propagation of Ultra-Short, Intense Laser Pulses in Air Archived 2008-01-10 at the Wayback Machine

^[1]

1. Chin, S. L.; Wang, T. -J.; Marceau, C.; Wu, J.; Liu, J. S.; Kosareva, O.; Panov, N.; Chen, Y. P.; Daigle, J. -F.; Yuan, S.; Azarm, A.; Liu, W. W.; Seideman, T.; Zeng, H. P.; Richardson, M.; Li, R.; Xu, Z. Z. (2012). "Advances in intense femtosecond laser filamentation in air". Laser Physics. 22: 1–53. Bibcode:2012LaPhy..22....1C. doi:10.1134/S1054660X11190054. S2CID 12993181.