Ultrashort Laser Pulse Phenomena, 2nd Edition

Problems 57

is equivalent to τ

G0

= w

0

r/(Lc). For r = w with w ≈ Lλ/(πw), cf. Eq. (1.182),

the pulse duration becomes τ

G0

≈ λ/(πc). Obviously, these effects become only

important if the pulses approach the single-cycle regime.

1.8. PROBLEMS

1. Verify the c

B

factors of the pulse duration–bandwidth product of a Gaussian

and sech pulse as given in Table 1.1.

2. Calculate the pulse duration ¯τ

p

defined as the second moment in Eq. (1.49)

for a Gaussian pulse and compare with τ

p

(FWHM).

3. Consider a medium consisting of particles that can be described by

harmonic oscillators so that the linear susceptibility in the vicinity of a

resonance is given by Eq. (1.153). Investigate the behavior of the phase

and group velocity in the absorption region. You will find a region where

ν

g

> ν

p

. Is the theory of relativity violated here?

4. Assume a Gaussian pulse which is linearly chirped in a phase modulator

that leaves its envelope unchanged. The chirped pulse is then sent through

a spectral amplitude only filter of spectral width (FWHM) ω

F

. Calculate

the duration of the filtered pulse and determine an optimum spectral width

of the filter for which the shortest pulses are obtained. (Hint: For simpli-

fication you may assume an amplitude only filter of Gaussian profile, i.e.,

˜

H(ω − ω) = exp



−ln 2



ω−ω

ω

F



2



.)

5. Derive the general expression for d

n

/d

n

in terms of derivatives with

respect to λ.

6. Assume that both the temporal and spectral envelope functions E(t) and

E(), respectively, are peaked at zero. Let us define a pulse duration τ

∗

p

and spectral width ω

∗

p

using the electric field and its Fourier transform by

τ

∗

p

=

1

|E(t = 0)|



∞

−∞

|E(t)|dt

and

ω

∗

p

=

1

|E( = 0)|



∞

−∞

|E()|d.

Show that for this particular definition of pulse duration and spectral width

the uncertainty relation reads

τ

∗

p

ω

∗

p

≥ 2π.