Ultrashort Laser Pulse Phenomena, 2nd Edition

20 Fundamentals

we can apply the uncertainty relation, Eq. (1.57),

t

2



2

=

M

4

4

κ

c

≥

1

4

. (1.64)

We have introduced a factor of chirp κ

c

, equal to

κ

c

= 1 +

M

4

4t

2



2

0



d

2

φ

d

2









0



2

(1.65)

in case of a frequency chirp and constant spectrum, or

κ

c

= 1 +

M

4

4

2



2

0



d

2

ϕ

dt

2









0



2

(1.66)

in case of a temporal chirp and constant pulse envelope.

In summary, using the mean square deviation (MSQ) to define the pulse

duration and bandwidth:

• The duration–bandwidth product



t

2



2

 is minimum 0.5 for a Gaussian

pulse shape, without phase modulation.

• For any pulse shape, one can define a shape factor M

2

equal to the minimum

duration–bandwidth product for that particular shape.

• Any quadratic phase modulation—or linear chirp—whether in frequency

or time, increases the bandwidth–duration product by a chirp factor κ

c

.

The latter increases proportionally to the second derivative of the phase

modulation, whether in time or in frequency.

1.2. PULSE PROPAGATION

So far we have considered only temporal and spectral characteristics of light

pulses. In this subsection we shall be interested in the propagation of such pulses

through matter. This is the situation one always encounters when working with

electromagnetic wave packets (at least until somebody succeeds in building a

suitable trap). The electric field, now considered in its temporal and spatial

dependence, is again a suitable quantity for the description of the propagating

wave packet. In view of the optical materials that will be investigated, we can

neglect external charges and currents and confine ourselves to nonmagnetic per-

meabilities and uniform media. A wave equation can be derived for the electric