Ultrashort Laser Pulse Phenomena, 2nd Edition

Pulse Shaping in Intracavity Elements 317

by all parts of the pulse arriving prior to t. From Eq. (3.55) we obtain for the

intensity after such an element

I(z, t) = I

0

(t)

e

W

0

(t)/W

s

e

−a

− 1 +e

W

0

(t)/W

s

. (5.69)

For weak absorption or gain (|a|1) this expression can be simplified and the

change of pulse intensity

I(t)  aI

0

(t)e

−W(t)/W

s

. (5.70)

In this limit the phase modulation is given by Eq. (3.67) and reads:

ϕ(t) −

1

2

(ω



− ω

10

)T

2

ae

−W

0

(t)/W

s

. (5.71)

The pulse shaping in these elements is a result of (unsaturated) attenuation or gain

in the leading part of the pulse while the trailing part is less affected (saturated

transition).

5.4.2. Nonlinear Nonresonant Elements

(a) Self-Phase Modulation

Some elements impress a nonlinear phase on the propagating pulse. As detailed

in Chapter 3, this phase is the result of a nonlinear process of third order and

characterized by a nonlinear polarizability χ

(3)

. In the limit of a fast nonlinearity

the response is instantaneous and is usually described by an intensity-dependent

refractive index. Acting only on the phase, such an element leaves the pulse

envelope, E

0

(t), unchanged. From Eq. (3.149)

ϕ(t, z) = ϕ

0

(t) −

k



n

2

n

0

zE

2

0

(t) = ϕ

0

(t) −

k



¯n

2

n

0

zI

2

0

(t). (5.72)

If the actual profile of the incident beam is taken into account the index change

becomes a function of the transverse coordinate, which leads to self-lensing

effects. The general mechanism is described in Chapter 3; the effect of such an

element in a fs laser is discussed in the next section.

318 Ultrashort Sources I: Fundamentals

(b) Polarization Coupling and Rotation

Nonlinear effects can also act on the polarization state of the laser pulse.

This effect is used in some lasers (for instance in fiber lasers [33]) to produce

mode-locking. Let us consider a pulse with arbitrary polarization, with complex

amplitudes

˜

E

x

(t) and

˜

E

y

(t) along the principal axis characterized by the unit

vectors ˆx and ˆy:

E =

1

2



ˆx

˜

E

0x

(t) +ˆy

˜

E

0y

(t)



e

i(ω



t−k



z)

+ c. c. (5.73)

The propagation of such a field through the nonlinear material leads to a coupling

of the two polarization components. One can calculate, see [33], the nonlinear

index change probed by polarizations along ˆx and ˆy :

n

nl,x

= n

2



|

˜

E

0x

|

2

+

2

3

|

˜

E

0y

|

2



n

nl,y

= n

2



|

˜

E

0y

|

2

+

2

3

|

˜

E

0x

|

2



. (5.74)

In an element of thickness d

m

, this induced birefringence leads to a phase change

between the x and y components of the field vector

(t) =

2π

λ





n

nl,x

− n

nl,y



=

2πn

2

d

m

3λ





|

˜

E

0x

(t)|

2

−|

˜

E

0y

(t)|

2



. (5.75)

The phase shift is time dependent and, in combination with another element,

can represent an intensity-dependent loss element.

To illustrate this further let us consider a sequence of such a birefringent

element and a linear polarizer. We assume that the incident pulse, E

0

cos(ωt),

is linearly polarized with components

E

0x

(t) = E

0

(t) cos α

E

0y

(t) = E

0

(t)sinα. (5.76)

The pass direction of the polarizer is at α + 90

◦

resulting in zero transmission

through the sequence for low-intensity light ( ≈ 0). Neglecting a common

phase the field components at the output of the nonlinear element are

E



x

(t) =

[

E

0

(t) cos α

]

cos(ω



t)

E



y

(t) =

[

E

0

(t)sinα

]

cos

[

ω



t + (t)

]

. (5.77)

Pulse Shaping in Intracavity Elements 319

Next the pulse passes through the linear polarizer. The total transmitted field

is the sum of the components from E



x

(t) and E



y

(t) along the polarizer’s path

direction

E

out

(t) = E

0

(t) cos α sin α

{

cos(ω



t) +cos

[

ω



t + (t)

]

}

. (5.78)

The total output intensity I

out

(t) =E

2

(t) is

I

out

(t) = I

in

(t)

1

2

[

1 − cos (t)

]

sin

2

(2α). (5.79)

Let us now assume a Gaussian input pulse I

in

= I

0

exp

6

2(t/τ

G

)

2

7

and parameters

of the nonlinear element so that for the pulse center the phase difference

(t = 0) =

2πn

2

d

m

3λ



E

2

0

(t = 0)



sin

2

α − cos

2

α



= π. (5.80)

For this situation we obtain a transmitted pulse

I

out

(t) =

1

2

I

in

(t)

4

1 − cos



πe

−2(t/τ

G

)

2

5

. (5.81)

The transmission is maximum (= 1) where the nonlinear element acts like a

half-wave plate that rotates the polarization by 90

◦

, lining it up with the pass

direction of the polarizer. For the parameters chosen here this happens at the pulse

center (t = 0). The phase shift  is smaller away from the pulse center produc-

ing elliptically polarized output and an overall transmission that approaches zero

in the pulse wings. Thus this sequence of elements can give rise to an intensity

dependent transmission similar to a fast absorber.

(c) Two Photon Absorption

In the case of an imaginary susceptibility of third order [χ

(3)

] there is a resonant

transition at twice the photon energy of the incident wave. As explained in

Chapter 3 this may lead to two photon absorption, which is governed by the

propagation equation for the pulse intensity

d

dz

I(t, z) =−β

2

I

2

(t, z). (5.82)