Ultrashort Laser Pulse Phenomena, 2nd Edition

Appendix C

Slowly Evolving Wave

Approximation

The derivation here essentially follows Brabec and Krausz [1]. We start with

the wave equation in the frequency domain for a scalar electric field propagating

in the z direction, including a nonlinear polarization and the diffraction term



∂

2

∂z

2

+∇

2

⊥

+

˜

k

2

()



E(x, y, z, ) = µ

0

F



∂

2

∂t

2

P

NL

(t, x, y, z)



(C.1)

where ∇

2

⊥

=

∂

2

∂x

2

+

∂

2

∂x

2

. Note that the effect of the linear polarization is included

in

˜

k

2

() = 

2

()µ

0

= 

2

n

2

/c

2

. We start with the following ansatz for the

electric field and the nonlinear polarization:

E() =

1

2

˜

E( − ω



, x, y, z)e

−ik



z

+ c. c. (C.2)

P(t) =

1

2

˜

P(t, x, y, z)e

i(w



t−k



z)

+ c. c. (C.3)

and neglect processes ...