Abstract
The basic physics of filamentation is universal and occurs in all transparent media (gases, liquids and solids). After propagating through an optical medium, a femtosecond (fs) Ti-sapphire laser pulse (at around 800 nm) turns into a white light laser pulse whose transverse pattern shows a central white spot surrounded by colored rings. The only difference is that the length of the filament is different in different media while the free electron generation mechanisms inside the filament core are different between gases and condensed matter materials. Figure 2.1a–d show the evolution of the transverse patterns of a 5 mJ/45 fs/800 nm Ti-sapphire laser pulse after propagating and filamenting in air without external focusing. The transform limited pulse from the vacuum compressor propagates into a 10 m vacuum pipeline which is connected directly to the vacuum compressor (see Fig. 2.2). After passing through the 1 cm-thick CaF2 exit window, the pulse enters the corridor next to the author’s laboratory.
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Notes
- 1.
The contribution to the index of refraction by a plasma can be obtained in the following way. In a plasma, from any optics text book, by assuming that only the electrons contribute, the index of refraction is given by
$ n^2 = 1 - \omega _p^2 /\omega ^2$((2.1))where the subscript p denotes plasma; the plasma frequency \( \omega _p = \left[ {4\pi e^2 N_e (t)/m} \right]^{1/2} \) (where e and m are the electron charge and mass in cgs units, respectively and N e (t) is the MPI/TI generated time dependent density in cm–3, i.e., N e (t) depends on the intensity of the laser). In air, \(\omega _p < < \omega \). This is always the case in the self-focus where the intensity is clamped down to between 1013and 1014 W/cm3 (Théberge et al., 2006) at which single ionization dominates. The electron density in a filament in air generated by a 50 fs/800 nm laser pulse has been measured to be of the order of 1016/cm3. This gives \( \upsilon _p = \omega _p /2\pi = 3 \times 10^9 \;{\rm{\text{Hz}}} \) which is much smaller than the optical frequency (∼1014 Hz). Hence, Eq. (f2.1) becomes
$ n = \left[ {1 - \omega _p^2 /\omega ^2 } \right]^{1/2} \cong 1 - \omega _p^2 /2\omega ^2$((f2.2))When N e (t) = 0, i.e., in vacuum, n = 1. That is to say, n = 1+Δn plasma where Δn plasma is the contribution of the plasma to the index of refraction.
$ \Delta n_p \cong - \omega _p^2 /2\omega ^2 = - \frac{{4\pi e^2 N_e (t)}}{{2m\omega ^2 }}$((f2.3))which is negative.
- 2.
Equation (2.1) is the solution of the nonlinear Schroedinger equation coming from the Maxwell’s equations without GVD. The initial conditions are: (1) the laser beam is a continuous one (CW); (2) it is a paraxial cylindrical beam with a spatial Gaussian distribution of intensity at the input of the medium; (3) slowly varying envelope approximation is used. During the propagation, the beam is deformed and numerical technique is used to obtain Eq. (2.1). Interestingly, this equation was found to be applicable even down to pulse duration of about 10 cycles of oscillation of the field. For example, in the case of a Ti-sapphire laser at the wavelength of around 800 nm with a pulse duration of the order of 100 fs, this equation was found to describe well the beginning of filamentation (Brodeur et al., 1997).
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Chin, S.L. (2010). Filamentation Physics. In: Femtosecond Laser Filamentation. Springer Series on Atomic, Optical, and Plasma Physics, vol 55. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0688-5_2
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DOI: https://doi.org/10.1007/978-1-4419-0688-5_2
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