1. Multiple-Cone Formation during the Femtosecond-Laser Pulse Propagation in Silica Kenichi Ishikawa *, Hiroshi Kumagai, and Katsumi Midorikawa Laser Technology Laboratory, RIKEN, Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan * Present address : Department of Quantum Engineering & Systems Science, Graduate School of Engineering, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Email: ishiken@q.t.u-tokyo.ac.jp submitted to Phys. Rev. E

2. Abstract We present a numerical study of the (2+1)-dimensional propagation dynamics of femtosecond laser pulses in silica. Pulses whose power is tens to hundreds of times higher than the threshold for self-focusing is split into multiple cones during its propagation. This new structure is formed as a result of the interplay of strong Kerr self-focusing and plasma defocusing. The number of cones increases with incident pulse energy. The uncertainty which may be contained in the evaluation of plasma response and multi-phase band-to-band transition cross section does not affect our results much. We present a numerical study of the (2+1)-dimensional propagation dynamics of femtosecond laser pulses in silica. Pulses whose power is tens to hundreds of times higher than the threshold for self-focusing is split into multiple cones during its propagation. This new structure is formed as a result of the interplay of strong Kerr self-focusing and plasma defocusing. The number of cones increases with incident pulse energy. The uncertainty which may be contained in the evaluation of plasma response and multi-phase band-to-band transition cross section does not affect our results much.

3. High-power regime In the present study, we consider the input pulse energy of 10 〜 150  J. 100 MW 〜 1 GW z 7.5mm r Silica glass 0 Hyperbolic-secant pulse (T 0 = 130fs) Gaussian beam (r 0 = 200  m) = 800 nm Threshold power for self-focusing P cr = 2.2 MW for silica. 50 P cr 〜 500 P cr A few times P cr has been used in existing studies for gas and solid High-power regime Fig. 1 Focusing and propagation of high- power femtosecond laser pulse in silica, considered in the present study. Intense laser pulse undergoes self- focusing due to the refractive index distribution induced by optical Kerr effect.

4. Simulation model Extended Nonlinear Schrödinger Equation Group velocity dispersion Higher-order dispersion Kerr response Diffraction Plasma defocusing Multi-photon absorption in a reference frame moving at the group velocity correction beyond the slowly varying envelope approximation (SVEA) Evolution of the electron density  in the conduction band from the Keldysh theory (1) (2)

5. Numerical methods The couples equations (1) and (2) are solved with the following methods. Equation (1) Equation (1) –Split-step Fourier method [1] –Diffraction term : Peaceman-Rachford method [2] –Nonlinear terms (right-hand side) : 4th-order Runge-Kutta method Equation (2) Equation (2) –4th-order Runge-Kutta method [1] G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995). [2] S.E. Koonin et al., Phys. Rev. C15, 1359 (1977).

6. Change of the spatio-temporal intensity profile with propagation Intensity (10 12 W/cm 2 ) 05 10 15 03 6 9010 5 z = 3200  m Radius r (  m) 3300  m 3400  m 3500  m 3600  m 3700  m 3800  m 4000  m 4500  m 5000  m Self-focusing Self-steepening Self-focusing Self-steepening Plasma defocusing More than 10 cones 1st cone 2nd cone 3rd cone input energy = 135mJ propagation distance (a) (b) (c) (d)(e) (f) (g) (h) (i) (j)

7. Multiple cone-like structure formation Self-focusing ⇨ The pulse energy is concentrated near the beam axis. Self-steepening ⇨ The peak is shifted toward the trailing edge. (a) As the self-focusing proceeds and the local intensity increases, Multi-photon absorption ⇨ Conduction (plasma) electrons are produced. Plasma formation has a negative contribution to the refractive index ⇨ Defocusing near the trailing edge. (b) Formation of a cone-like structure. (c) With pulse propagation, more and more cones are formed. ⇨ Formation of multiple-cone-like structure. (d) 〜 (j) Dramatic new feature in the high-power regime !

8. Mechanism of the multiple-cone formation At z = 3340  m, the intensity decreases with increasing r in the range r = 9 - 12  m, while  n is nearly flat there. At z = 3340  m, the intensity decreases with increasing r in the range r = 9 - 12  m, while  n is nearly flat there. Due to self-focusing, the first peak takes up much energy from its vicinity. Due to self-focusing, the first peak takes up much energy from its vicinity. At z = 3360  m, the second local maximum in  n is formed around r = 11.3  m. → The local self-focusing leads to the grow-up or the second cone. At z = 3360  m, the second local maximum in  n is formed around r = 11.3  m. → The local self-focusing leads to the grow-up or the second cone. 3300  m 3400  m 1st cone 2nd cone Fig. Radial distribution of intensity and refractive index change  n at t = 44 fs.

9. Fluence vs. Propagation distance Self-focusing Propagation Propagation distance (  m) Fluence (10 -15 J/cm 2 ) Radius (  m) Fluence (10 -15 J/cm 2 ) Plasma defocusing Propagation distance (micron)

10. Multiple cone-like structure z = 5000  m from the silica surface Input energy = 135  J Propagation FTOP signal Temporal profile integrated in r-direction. Propagation Integration in time Integration in r Lateral profile

11. Dependence on the input energy With decreasing input pulse energy, the number of cones decreases. the number of cones decreases. the cones are more parallel to the beam axis. the cones are more parallel to the beam axis. The multiple-cone formation ceases when we further decrease the input energy. Intensity (10 12 W/cm 2 ) 05 10 15 135  J, z = 4500  m 45  J, z = 5500  m 15  J, z = 7000  m Input energy Radius r (  m)

12. Uncertainty in plasma response and plasma formation rate Plasma response Plasma formation rate Intensity distribution at z = 4000  m obtained with account of the saturation of conduction electron drift velocity, by replacing  /  cr in Eq. (1) by The uncertainty which may be contained in the evaluation of plasma response and multi-phase band-to-band transition cross section does not affect the essential features of our results. where I th = 10 12 W/cm 2. Intensity distribution at z = 3500  m obtained with a value of  6 which is 100 times smaller than in Eq. (2).

13. Conclusion When the input power is several hundred times higher than P cr, the pulse is split many times both temporally and spatially. When the input power is several hundred times higher than P cr, the pulse is split many times both temporally and spatially. As a result, the intensity distribution contains multiple cones. This is a new feature that emerges only in the high-power regime As a result, the intensity distribution contains multiple cones. This is a new feature that emerges only in the high-power regime This structure is formed by the interplay of Kerr self- focusing and plasma defocusing. This structure is formed by the interplay of Kerr self- focusing and plasma defocusing.