Ultrashort Laser Pulse Phenomena, 2nd Edition

82 Femtosecond Optics

substrate

Figure 2.10 Wave packets of different center frequencies are reflected at different depths of a

chirped mirror. The mirror consists of stacks of alternating high and low refractive index layers at

different resonance frequencies.

Improvements in the initial layer sequence used as a starting point for the final

computer optimization have been accomplished, for example, by modulating the

ratio of the thickness of the high- and low-index layer of the chirped mirror

(double-chirped mirror) [19], by superimposing a quasi-periodic modulation on

the linear modulation of the layer thickness [20], and by coating the backside

of the substrate [21,22]. As we will see in following chapters such mirrors have

made an impressive impact on femtosecond laser source development.

2.4. FOCUSING ELEMENTS

2.4.1. Singlet Lenses

One main function of fs pulses is to concentrate energy in time and space.

The ability to achieve extremely high peak power densities partly depends on

the ability to keep pulses short in time, and concentrate them in a small vol-

ume by focusing. The difference between group and phase velocity in the lens

material can reduce the peak intensity in the focal plane by delaying the time of

arrival of the pulse front propagating through the lens center relative to the pulse

front propagating along peripheral rays. The group velocity dispersion leads to

reduction of peak intensity by stretching the pulse in time. As pointed out by

Bor [23, 24], when simple focusing singlet lenses are used, the former effect can

lead to several picosecond lengthening of the time required to deposit the energy

of a fs pulse on focus.

Let us assume a plane pulse and phase front at the input of a spherical lens

as sketched in Figure 2.11. According to Fermat’s principle, the optical path

Focusing Elements 83

r

r

0

w

Pulse and

phase front

Phase front

Pulse front

R

2

R

1

f

w

0

T(r)

L(r)

w



w(f)

w

0



f

Figure 2.11 Left: delay of the pulse front with respect to the phase front, in the case of a singlet

lens. Right: spread of the focal region due to chromatic aberration.

along rays from the input phase front to the focus is independent of the radius

coordinate r. The lens transforms the plane phase front into a spherical one

which converges in the (paraxial) focus. Assimilating air as vacuum, it is only

while propagating through the lens that the pulses experience a group velocity

ν

g

different from the phase velocity ν

p

= c/n. The result is a pulse front that is

delayed with respect to the (spherical) phase front, depending on the amount of

glass traversed. As we have seen in Chapter 1, the group velocity is:

ν

g

=



dk

d



−1

=

c

n − λ



dn

dλ

, (2.34)

where λ



is the wavelength in vacuum. The difference in propagation time

between the phase front and pulse front after the lens at radius coordinate r is:

T(r) =



1

ν

p

−

1

ν

g



L(r), (2.35)

where L(r) is the lens thickness. The group delay T (r) is also the difference of

the time of arrival at the focus of pulses traversing the lens at distance r from the

axis and peripheral rays touching the lens rim. Pulse parts traveling on the axis

(r = 0) will arrive delayed in the focal plane of a positive lens compared with

pulse parts traversing the lens at r > 0. For a spherical thin lens, the thickness L

is given by

L(r) =

r

2

0

− r

2

2



1

R

1

−

1

R

2



(2.36)