56 Fundamentals
1.7. SPACE–TIME EFFECTS
For short pulses a coupling of spatial and temporal effects becomes important
even for propagation in a nondispersive medium. The physical reason is that self-
diffraction of a beam of finite transverse size (e.g., Gaussian beam) is wavelength
dependent. A separation of time and frequency effects according to Eqs. (1.176)
and (1.177) is clearly not feasible if such processes matter. One can construct a
solution by solving the diffraction integral (1.187) for each spectral component.
The superposition of these solutions and an inverse Fourier transform then yields
the temporal field distribution. Starting with a field
˜
E(x
, y
, ) = F
˜
E(x
, y
, t)
inaplane
(x
, y
)atz = 0 we find for the field in a plane (x, y)atz = L:
˜
E(x, y, L, t) = F
−1
ie
−iL/c
2πcL
dx
dy
˜
E(x
, y
, )
×exp
−i
2Lc
(x − x
)
2
+ (y − y
)
2
(1.203)
where we have assumed a nondispersive medium with refractive index n = 1.
Solutions can be found by solving numerically Eq. (1.203) starting with an arbi-
trary pulse and beam profile at a plane z = 0. Properties of these solutions were
discussed by Christov [24]. They revealed that the pulse becomes phase modu-
lated in space and time with a pulse duration that changes across the beam profile.
Because of the stronger diffraction of long wavelength components the spectrum
on axis shifts to shorter wavelengths.
For a Gaussian beam and pulse profile at z = 0, i.e.,
˜
E(x
, y
,0,t) ∝
exp(−r
2
/w
2
0
) exp(−t
2
/τ
2
G0
) exp(iω
t) with r
2
= x
2
+ y
2
, the time–space
distribution of the field at z = L is of the form [24]:
˜
E(r, z = L, t) ∝ exp
−
η
2
τ
2
G
exp
−
w
0
ω
τ
G0
2Lcτ
G
r
2
exp
i
ω
τ
2
G0
τ
2
G
η
(1.204)
where
τ
2
G
= τ
2
G0
+[w
0
r/(Lc)]
2
(1.205)
and η =
6
t − L/c −r
2
/(2Lc)
7
. This result shows a complex mixing of spatial and
temporal pulse and beam characteristics. The first term in Eq. (1.204) indicates a
pulse duration that increases with increasing distance r from the optical axis. For
an order of magnitude estimation let us determine the input pulse duration τ
G0
at
which the pulse duration has increased to 2τ
G0
at a radial coordinate r = w after
the beam has propagated over a certain distance L ρ
0
. From Eq. (1.205) this
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