8
S. L. Chin
where E ω and E 3ω are the amplitudes of the pump and TH, respectively. The third
harmonic term would be
P
(3)
∝ E E E = (E ω + E 3ω )
3
(1.5)
After expansion of the last term in (1.5), there will be a term which is proportional
to the clamped intensity E
2
ω inside the filament. We shall call this term ‘clamped’
polarization;
i.e. P
(3)
clamped ∝ E
2
ω E 3ω
(1.6)
This term would be the dominant term among the other terms of P
(3) for TH
generation because of the high clamped intensity E
2
ω . It would increase the index of
refraction of the TH according to (1.2), hence, reducing the speed of propagation.
Similarly and simultaneously, according to detailed calculation and analysis [7, 31],
the index of refraction of the fundamental is also controlled by the clamped intensity
and is increased. The consequence is that the two indices of refraction are almost
equal; i.e. the nonlinear speed of propagation of the TH and that of the fundamental
are almost equal. Since at the position where intensity clamping takes place, the
fundamental is a plane wave [7], the TH is also a plane wave. Further analysis [7]
shows that these two plane waves do not overlap. The TH plane wave front faithfully
lags behind the fundamental plane wave in space by half the TH’s wavelength. There
is no walk-off. (Full calculation including the effect of the weak plasma showed that
the indices of refraction of the fundamental and the TH are almost equal [7, 31].)
This qualitative analysis explains why the efficiency of TH generation inside a
filament would increase by 2 orders of magnitude as compared to the case without
filament [7, 31]. High clamped intensity is the key reason. We call this self-group
phase locking which is the consequence of intensity clamping. In fact, we found that
the TH was well stabilized in the sense that the root-mean-square fluctuation of the
TH was much lower than that of the pump laser because of intensity clamping. We
also called this self-stabilization [31].
Since TH generation is a special case of parametric processes involving χ
(3) , we
can now predict that any other parametric process in air (or in any other system
with inversion symmetry) would give rise to self-group phase locking. An example
is mixing a fs frequency tunable laser pulse in the i.r. with the pump pulse inside a
filament. It is a four-wave mixing (4WM) process [7, 32]:
ω 4WM = 2ω 800 − ω i.r.
(1.7)
where ω 4WM is the resultant frequency after mixing, ω 800 , the pump Ti-sapphire
laser frequency and ω i.r. , the tunable i.r. laser frequency. The tuning range of the i.r.
laser spans from 1100 to 2400 nm. Experiment [32] showed that the 4WM pulse
was tunable across almost the entire spectral range of the visible light. We define
the conversion efficiency of the 4WM pulse as the ratio of the energy of the 4WM
S. L. Chin
where E ω and E 3ω are the amplitudes of the pump and TH, respectively. The third
harmonic term would be
P
(3)
∝ E E E = (E ω + E 3ω )
3
(1.5)
After expansion of the last term in (1.5), there will be a term which is proportional
to the clamped intensity E
2
ω inside the filament. We shall call this term ‘clamped’
polarization;
i.e. P
(3)
clamped ∝ E
2
ω E 3ω
(1.6)
This term would be the dominant term among the other terms of P
(3) for TH
generation because of the high clamped intensity E
2
ω . It would increase the index of
refraction of the TH according to (1.2), hence, reducing the speed of propagation.
Similarly and simultaneously, according to detailed calculation and analysis [7, 31],
the index of refraction of the fundamental is also controlled by the clamped intensity
and is increased. The consequence is that the two indices of refraction are almost
equal; i.e. the nonlinear speed of propagation of the TH and that of the fundamental
are almost equal. Since at the position where intensity clamping takes place, the
fundamental is a plane wave [7], the TH is also a plane wave. Further analysis [7]
shows that these two plane waves do not overlap. The TH plane wave front faithfully
lags behind the fundamental plane wave in space by half the TH’s wavelength. There
is no walk-off. (Full calculation including the effect of the weak plasma showed that
the indices of refraction of the fundamental and the TH are almost equal [7, 31].)
This qualitative analysis explains why the efficiency of TH generation inside a
filament would increase by 2 orders of magnitude as compared to the case without
filament [7, 31]. High clamped intensity is the key reason. We call this self-group
phase locking which is the consequence of intensity clamping. In fact, we found that
the TH was well stabilized in the sense that the root-mean-square fluctuation of the
TH was much lower than that of the pump laser because of intensity clamping. We
also called this self-stabilization [31].
Since TH generation is a special case of parametric processes involving χ
(3) , we
can now predict that any other parametric process in air (or in any other system
with inversion symmetry) would give rise to self-group phase locking. An example
is mixing a fs frequency tunable laser pulse in the i.r. with the pump pulse inside a
filament. It is a four-wave mixing (4WM) process [7, 32]:
ω 4WM = 2ω 800 − ω i.r.
(1.7)
where ω 4WM is the resultant frequency after mixing, ω 800 , the pump Ti-sapphire
laser frequency and ω i.r. , the tunable i.r. laser frequency. The tuning range of the i.r.
laser spans from 1100 to 2400 nm. Experiment [32] showed that the 4WM pulse
was tunable across almost the entire spectral range of the visible light. We define
the conversion efficiency of the 4WM pulse as the ratio of the energy of the 4WM