Ultrashort Laser Pulse Phenomena, 2nd Edition

Second Harmonic Generation (SHG) 185

(a)

z=0

(c)

z=0

o

Intensity

Intensity

Intensity

Intensity

(b)

z

(d)

z

Time

Time

Time

Time

e

oe

oe

oe

Figure 3.11 Pulse shaping in type II SHG. The three interacting pulses—fundamental ordinary,

fundamental extraordinary, and second harmonic—are represented in a temporal frame of reference

moving with the group velocity of the second harmonic.

Table 3.3

Relevant constants for SHG type II in KDP and DKDP at

1.06 µm. m is the ratio of the walk-off lengths.

Crystal θ L

o

(cm) L

e

(cm) |m|

KDP 59.2

o

20.9 −15.8 1.32

DKDP 53. 5

o

28.1 −20.0 1.40

3.4.4. Group Velocity Control in SHG through Pulse

Front Tilt

The condition of SH group velocity intermediate between the ordinary and

extraordinary fundamentals cannot in general be met at any wavelength, for

any crystal. It is however possible to adjust the group velocity through a tilt of

the energy front with respect to the phase front. Figure 3.12 illustrates the basic

principle in the case of a degenerate type I process. As shown in the lower part of

186 Light–Matter Interaction

Fundamental

pulse front

Crystal





Figure 3.12 SHG using tilted pulse fronts. The wave vector of the (o-wave) SH is perpendicular

to the entrance face of the crystal. Top—the pulse front of the fundamental (dark bar) is tilted by

an amount compensating the group velocity mismatch of fundamental and second harmonic (grey

bar) pulse exactly. Bottom—the input pulse front is not tilted. The different group velocities of

fundamental and SH pulse lead to walk-off.

the figure, the temporal overlap of the interacting pulses decreases because of the

lower velocity of the SH pulse relative to the fundamental pulse. Furthermore,

the spatial overlap decreases because of the walk-off (ρ in the figure) of the

fundamental wave from the SH wave. These two negative effects can however be

used in conjunction with pulse front tilt to match the relative velocities of the two

pulses, as illustrated in the upper part of Fig. 3.12. Loosely speaking: seen in the

frame of reference of the SH wave, the lateral walk-off of the fundamental beam

decrease the component of the pulse velocity along the direction of propagation

of the SH just to match its (group) velocity.

Unfortunately, for femtosecond pulses, it is not practical to generate a large

pulse front tilt. For instance, a pulse front tilt of the order of 40

◦

would be required

for SHG type II and compression in BBO of an 800 nm pulse of a Ti:Sapphire

laser in collinear interaction [38]. A dispersive element like a prism with a ratio

of beam diameter to base length of 20 (in the case of SF10 glass) would be

needed to achieve the required energy front tilt of 40

◦

. Dispersion in the glass

would lead to large pulse broadening and phase modulation.

A better approach for group velocity matching in SHG as well as in parametric

three-wave interactions of fs pulses is to use a noncollinear geometry [39, 40].

Table 3.4 shows how the group velocities of the participating waves can be

changed from the collinear to the noncollinear case (here for an internal angle

of 2

◦

) leading to conditions for compression. The sketch of the interaction

Second Harmonic Generation (SHG) 187

Table 3.4

Group velocities for the o, e and SH e waves. Values are calculated for

type II SHG of 800 nm radiation for collinear and noncollinear

(internal angle is 2

◦

) interaction in BBO.

Fundamental (o) Second harmonic (e) Fundamental (e)

Geometry ν

o

(10

8

m/s) ν

2

(10

8

m/s) ν

e

(10

8

m/s)

Collinear 1.780 1.755 1.843

Noncollinear 1.798 1.905 1.934

geometry in Figure 3.13 shows that it is possible to obtain the respective group

velocities through manipulation of the angles of incidence in the crystal and the

energy tilt produced by a prism. Figure 3.13 pertains to the case of noncollinear

interacting plane waves inside a 250 µm long BBO crystal, cut for type II SHG

of laser pulses at 800 nm.

The situation sketched in Fig. 3.13 was simulated with the system of equations

Eq. (3.120), (3.121), and (3.122), with the substitution of Eqs. (3.123) for the

Crystal



Air

Prism

b

a





2

k



k

2

k



Figure 3.13 Geometry for precompensation the pulse front tilt resulting from propagation through

an interface (the pulse fronts are depicted as dotted lines) using prisms. For the chosen type II SHG

of 800 nm pulses: θ = 2

◦

, which is the internal angle of the fundamental beams for noncollinear

interaction. To match the energy fronts of both fundamentals, an external pulse front tilt of γ = 0. 6

◦

for β = 1.6

◦

is required. Also sketched in the figure are thin SF10 prisms (a/b = 0.2) to realize the

desired γ.