38 Fundamentals
per unit travel distance and unit spectral width. From Eq. (1.138) we find for the
corresponding quantity
τ
p
Lλ
= 2π
c
λ
2
|k
|. (1.140)
For BK7 glass at 620 nm, k
≈ 1. 02 ×10
−25
s
2
/m, and the GVD as introduced
above is about 500 fs per nm spectral width and m propagation length.
1.2.5. Complex Dielectric Constant
In general, the dielectric constant, which was introduced in Eq. (1.72) as a real
quantity, is complex. Indeed a closer inspection of Eq. (1.71) shows that the finite
memory time of matter requires not only , χ to be frequency dependent but also
that they be complex. The real and imaginary part of ˜, ˜χ are not independent
of each other but related through a Kramers–Kronig relation. The consideration
of a real () is justified as long as we can neglect (linear) losses or gain. This
is valid for transparent samples or propagation lengths which are too short for
these processes to become essential for the pulse shaping. For completeness we
will modify the reduced wave equation (1.93) by taking into account a complex
dielectric constant ˜() represented as
˜() = () +i
i
(). (1.141)
Let us assume ˜() to be weakly dispersive. The same procedure introduced to
derive Eq. (1.93) can be used after inserting the complex dielectric constant ˜ into
the expression of the polarization Eq. (1.84). Now the reduced wave equation
becomes
∂
∂z
˜
E(t, z) −
i
2
k
∂
2
∂t
2
˜
E(t, z) = κ
1
˜
E(t, z) +iκ
2
∂
∂t
˜
E(t, z) +κ
3
∂
2
∂t
2
˜
E(t, z) (1.142)
where
κ
1
=
ω
2
η
0
i
(ω
) (1.143)
κ
2
=
1
2
η
0
2
i
(ω
) + ω
d
d
i
()
ω
(1.144)
κ
3
=
1
4ω
η
0
2
i
(ω
) + 4ω
d
d
i
()
ω
+ ω
2
d
2
d
2
i
()
ω
. (1.145)
Pulse Propagation 39
In the preceding expressions, η
0
=
√
µ
0
/
0
≈ 377 ms is the characteristic
impedance of vacuum. For zero GVD, and neglecting the two last terms in the
right-hand side of Eq. (1.142), the pulse evolution with propagation distance z is
described by
∂
∂z
˜
E(t, z) −κ
1
˜
E(t, z) = 0 (1.146)
which has the solution
˜
E(t, z) =
˜
E(t,0)e
κ
1
z
. (1.147)
The pulse experiences loss or gain depending on the sign of κ
1
and does not
change its shape. Equation (1.147) states simply the Lambert-Beer law of linear
optics.
An interesting situation is that in which there would be neither gain nor loss
at the pulse carrier frequency, i.e.,
i
(ω
) = 0 and
d
d
i
()
ω
= 0, which could
occur between an absorption and amplification line. Neglecting the terms with
the second temporal derivative of
˜
E, the propagation problem is governed by the
equation
∂
∂z
˜
E(t, z) −iκ
2
∂
∂t
˜
E(t, z) = 0. (1.148)
The solution of this equation is simply
˜
E(t, z) =
˜
E(t + iκ
2
z, 0). (1.149)
To get an intuitive picture on what happens with the pulse according to
Eq. (1.149), let us choose an unchirped Gaussian pulse
˜
E(t, 0) [see Eq. (1.33) for
a = 0] entering the sample at z = 0. From Eq. (1.149) we find:
˜
E(t, z) =
˜
E(t,0)exp
κ
2
2
(z/τ
G
)
2
exp
−i2κ
2
tz/τ
2
G
. (1.150)
The pulse is amplified, and simultaneously its center frequency is shifted with
propagation distance. The latter shift is because of the amplification of one part of
the pulse spectrum (the high (low) frequency part if κ
2
< (>)0) while the other
part is absorbed. The result is a continuous shift of the pulse spectrum in the
corresponding direction and a net gain while the pulse shape is preserved.
In the beginning of this section we mentioned that there is always an imaginary
contribution of the dielectric constant leading to gain or loss. The question arises
Get Ultrashort Laser Pulse Phenomena, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
