Ultrashort Laser Pulse Phenomena, 2nd Edition

Coherent Interactions with Two-Level Systems 229

A similar representation can be made for the system of Eqs. (4.6)–(4.7).

The pseudo-polarization vector is then the vector



Q(Q

i

, Q

r

, w) rotating around

a pseudo-electric field vector



E(κ

˜

E

r

, κ

˜

E

i

, −ω) [Fig. 4.2(b)]. Physically, the

first two components of the pseudo-polarization vector



Q represent the dipolar

resonant field that opposes the applied external field (and is thus responsible

for absorption).

4.2.2. Rate Equations

If the light field envelope is slowly varying with respect to T

2

, Bloch’s

equations reduce to the standard rate equations. For pulses longer than the dephas-

ing time T

2

, the two first Bloch equations, (4.10) and (4.11) are stationary on the

time scale of the pulse. Solving these equations for u, ν , and substituting ν into

the third equation (4.12) for the population difference, leads to the rate equation:

˙w =−

E

2

(κ

2

T

1

T

2

)

1 + ω

2

T

2

2

w

T

1

−

w − w

0

T

1

. (4.18)

We note that this equation is identical to the rate equation (3.46) introduced

in Chapter 3 in terms of N (N = w/p). Equation (4.18) defines a sat-

uration field at resonance

˜

E

s0

= 1/(κ

√

T

1

T

2

). Off-resonance, a larger field,

˜

E

s

=

˜

E

s0



1 + ω

2

T

2

2

is required to saturate the same transition.

In Chapter 3 we discussed various cases of pulse propagation through resonant

media resulting from the rate equations. For example, for pulses much shorter

than the energy relaxation time τ

p

T

1

and purely homogeneously broadened

media the rate equation (4.18) can be integrated together with the propagation

equation (4.8), which yields for the transmitted intensity

I(z, t) = I

0

(t)

e

W(t)/W

s

e

−a

− 1 +e

W(t)/W

s

. (4.19)

In this last equation W(t) =



t

−∞

I

0

(t)dt, and a = σ

(0)

01

w

0

z/p is the linear

gain–absorption coefficient. Equation (4.19) corresponds to Eq. (3.55) which

was written for the photon flux F.

Femtosecond pulse propagation through a homogeneously broadened

saturable medium in the limit of T

2

 τ

p

 T

1

is completely determined by

two parameters: the saturation energy density W

s

and the linear absorption

(gain) coefficient a. Equation (4.19) is particularly useful in calculating pulse

propagation in amplifiers, as shown in Chapter 3, and further detailed in

Chapter 7.