1 Femtosecond Laser Filamentation Induced Phenomena and Applications
7
where E is the total electric field at a position inside the filament and χ
(3) is the third
order susceptibility of the medium. Here, we assume that the pump Ti-sapphire laser
pulse is linearly polarized and the third harmonic (TH) or the second laser pulse for
four wave mixing also linearly polarized and parallel to the polarization of the pump
field. As such, we use scalar description.
An efficient third harmonic generation would require that once a TH signal was
created at the beginning of the filament in the forward propagating direction, it would
faithfully follow the pump without any walk-off; i.e. the speed of propagation of the
pump and that of the TH would be the same or almost the same. Such a situation
would give rise to a steady pile-up or in-phase accumulation of the TH signals along
the whole filament so that at the exit of the filament, the energy/intensity of the TH
would be very high. However, the TH’s frequency being very different from that
of the pump, their propagation speeds c/n (c; speed of light in vacuum, n, index of
refraction in the medium) are very different normally; i.e. the indices of refraction
of the pump (frequency ω) and the TH (frequency 3 ω), namely, n(ω) and n(3ω),
are very different. This would result in a walk-off of the pump and TH signals. At
the end of the filament, there would be no in-phase accumulation of the TH signals.
Destructive interference would set in at many places along the way. The final signal
would be weak.
What is surprising is that, inside a filament, the speeds of propagation of the TH
and the pump are almost equal; i.e. their indices of refraction are almost equal [7,
31]. Hence, in-phase accumulation of the TH signal along the full filament length
would give rise to a very strong TH signal; i.e. very high conversion efficiency as
compared to normal non-filamentation cases.
The reason why the indices of refraction of the pump and the TH inside the filament
are almost equal is because of the high clamped intensity inside the filament. We
note that the index of refraction is defined as
n =
ε/ε 0 =
1 + χ =
1 + P/E
(1.2)
where ε is the dielectric constant of the medium and χ , the total susceptibility of
the medium. P is the total scalar material polarization and E is the total electric field
at the appropriate frequencies. One can see from (1.2) that the larger the ratio P/E
is, the larger the index n and the slower the speed of propagation will be. In other
words, the larger the nonlinear polarization P
(3) is, the larger the index of refraction
will be. In air, we assume that a weak TH wave is generated at the beginning. We
shall look at the amplitudes of the TH and pump waves only. For the sake of clarity,
the effect of the low density plasma is neglected. The fundamental and the TH are
assumed to be propagating in a neutral air medium.
P = ε 0 χ
(1) E + ε 0
χ
(3) E E E + higher order terms
(1.3)
E = E ω + E 3ω
(1.4)
7
where E is the total electric field at a position inside the filament and χ
(3) is the third
order susceptibility of the medium. Here, we assume that the pump Ti-sapphire laser
pulse is linearly polarized and the third harmonic (TH) or the second laser pulse for
four wave mixing also linearly polarized and parallel to the polarization of the pump
field. As such, we use scalar description.
An efficient third harmonic generation would require that once a TH signal was
created at the beginning of the filament in the forward propagating direction, it would
faithfully follow the pump without any walk-off; i.e. the speed of propagation of the
pump and that of the TH would be the same or almost the same. Such a situation
would give rise to a steady pile-up or in-phase accumulation of the TH signals along
the whole filament so that at the exit of the filament, the energy/intensity of the TH
would be very high. However, the TH’s frequency being very different from that
of the pump, their propagation speeds c/n (c; speed of light in vacuum, n, index of
refraction in the medium) are very different normally; i.e. the indices of refraction
of the pump (frequency ω) and the TH (frequency 3 ω), namely, n(ω) and n(3ω),
are very different. This would result in a walk-off of the pump and TH signals. At
the end of the filament, there would be no in-phase accumulation of the TH signals.
Destructive interference would set in at many places along the way. The final signal
would be weak.
What is surprising is that, inside a filament, the speeds of propagation of the TH
and the pump are almost equal; i.e. their indices of refraction are almost equal [7,
31]. Hence, in-phase accumulation of the TH signal along the full filament length
would give rise to a very strong TH signal; i.e. very high conversion efficiency as
compared to normal non-filamentation cases.
The reason why the indices of refraction of the pump and the TH inside the filament
are almost equal is because of the high clamped intensity inside the filament. We
note that the index of refraction is defined as
n =
ε/ε 0 =
1 + χ =
1 + P/E
(1.2)
where ε is the dielectric constant of the medium and χ , the total susceptibility of
the medium. P is the total scalar material polarization and E is the total electric field
at the appropriate frequencies. One can see from (1.2) that the larger the ratio P/E
is, the larger the index n and the slower the speed of propagation will be. In other
words, the larger the nonlinear polarization P
(3) is, the larger the index of refraction
will be. In air, we assume that a weak TH wave is generated at the beginning. We
shall look at the amplitudes of the TH and pump waves only. For the sake of clarity,
the effect of the low density plasma is neglected. The fundamental and the TH are
assumed to be propagating in a neutral air medium.
P = ε 0 χ
(1) E + ε 0
χ
(3) E E E + higher order terms
(1.3)
E = E ω + E 3ω
(1.4)