34	1.1.7 Coherent interaction	[Ref. p. 40

,QYHUVLRQ∆Q

3RODUL]DWLRQ3$

7LPHW

π |Λ|

Fig. 1.1.17. Oscillation of inversion density ∆ n and polarization amplitude P A0 in resonance for a constant local electric field.

For a constant electric field at a fixed position z the rotating-wave approximation has a periodic solution. Inversion and polarization with the initial condition t = 0, ∆n = n0, P A0 = 0 are:

∆ n

=

δ2 + |Λ|2 cos β t

,

Λ =

µAE0

,

(1.1.105)

n0

β2

PA0 = n0

β

β (1 − cos α t) + i sin α t , β = δ2 + |Λ|2 .

(1.1.106)

µAΛ

δ

In resonance ω = ωA, δ = 0, the inversion density ∆ n and the amplitude P A0 of the polarization oscillate with this frequency, see Fig. 1.1.17. The real polarization P A,real of (1.1.100) contains the frequencies ωA ± |Λ|. Some values of dipole moments are given in Table 1.1.2 to estimate |Λ|. O resonance the temporal behavior of inversion and polarization is more complicated (optical nutation) [72Cou]. If at t = 0 all atoms are in the lower level (∆ n0 = −n0) a complete inversion is produced at t = π/|Λ| by a coherent field. It is called pulse inversion [60Vuy]. At t = 2/|Λ|, all atoms are again in the lower level, no energy transfer has taken place.

1.1.7.3 Propagation of resonant coherent pulses

For short pulses, τ < T2, the perturbations can be neglected. The solution of the complete interaction equations (1.1.101a)–(1.1.101c) for a propagating resonant pulse is rather simple.

The propagation of pulses in a two-level system is described by the rotating-wave approximation, (1.1.45a)/(1.1.45b), and by the wave equation in the SVE approximation (1.1.28). The set of these three non-linear equations is di cult to solve, only special cases will be discussed here. At t = 0 the electric field E0 is assumed to be real, Λ = Λ . In case of resonance, δ = 0, (1.1.101a) delivers C = 0, R1 = 0. The interaction equations (1.1.101b), (1.1.101c) reduce to

R1 = 0 ,

∂R∂t2 = −Λ R3 ,

∂R∂t3 = Λ R2 .

The R-vector moves in the R2-R3-plane, see Fig. 1.1.18. If the angle θ with the R3-axis is introduced, one solution of the above equations is:

R2 = R0 sin θ ,

R3 = −R0 cos θ

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	35

5

5

θ

			5			Fig. 1.1.18.	In resonance, δ = 0, the R-vector of the two-
						level system rotates in the R2-R3-plane.
with
Λ =	∂θ	=		µA E0	.		(1.1.107)
	∂t

R0 is given by the initial conditions at t = 0. The SVE-approximation of (1.1.28) then becomes:

∂2

1 ∂2

α ∂θ γ

µA k0

+

θ = −

+

R0

sin θ ,

γ =

.

(1.1.108)

∂t ∂z

c

∂t2

2

∂t

2

nr ε0

From θ the amplitude E0 of the electric field can be calculated with (1.1.107)/(1.1.105).

1.1.7.3.1 Steady-state propagation of nπ-pulses

Steady state

means that

a pulse is propagating with velocity v and constant pulse envelope

E0(t, z) = E0(t − z/v). The amplitude depends on one parameter w only:

w = t − z/v

and (1.1.108) becomes:

c

d2θ

αc d θ

γ

1 −

+

= c

R0 sin θ .

(1.1.109)

v

d w2

2

d w

2

This equation is equivalent to the equation of the pendulum with friction in a gravitational field. In the following examples two di erent initial conditions are assumed:

R0	= µA ∆ n0	< 0	(absorber) ,
		> 0	(amplifier) ,

which corresponds to the pendulum up or down at t = 0.

1.1.7.3.1.1 2π-pulse in a loss-free medium

A medium without losses (α = 0) interacts with a coherent pulse in resonance (δ = 0). The initial condition is ∆ n0(t = −∞) = +∆ n0 (∆ n0 < 0, absorber). One steady-state solution is the 2πpulse, see Fig. 1.1.19, which corresponds to a local field of duration τ = 2π/Λ. The leading edge of the pulse produces an inversion and energy is transferred to the atomic system, the amplitude is reduced. The trailing part of the pulse is then amplified by this inversion. In total the pulse

Landolt-B¨ornstein

New Series VIII/1A1

36		1.1.7 Coherent interaction	[Ref. p. 40
2π − pulse		π − pulse
E0 (t )	E2	E0 (t )
z,t	E1	z,t

Fig. 1.1.19. Propagation of 2πand π-pulses in a two-level system.

has lost no energy, but is delayed in time. Such a pulse is only stable, if the broadband losses are negligible and if the initial inversion is negative. The steady-state solution is:

E = Epeak

exp [i ω (t − z/c)]

(field) ,

cosh [(t − z/v) /τ ]

∆ n0

Epeak = 2√ ω

(peak amplitude) ,

ε0

(1

−

c/v)

2 ω∆ n0c

Jpeak =

(peak intensity) ,

1 − c/v

T2π = 2 τ = 2

(1 −g0c

2

(pulse duration) ,

c/v) T

v =

c

(pulse peak velocity)

1 − g0c τ 2/T2

with

(1.1.110)

(1.1.111)

(1.1.112)

(1.1.113)

(1.1.114)

g0 = ∆ n0σ < 0 : small-signal absorption coe cient, c : phase velocity in the medium,

v : pulse peak velocity.

This two-level system is the most simple model of a saturable absorber, which in the case of incoherent interaction absorbs the radiation. But the coherent 2π-pulse transmits the absorber without losing energy. Therefore this e ect is called self-induced transparency [75Kri]. The pulse is characterized by three parameters: peak velocity v, peak amplitude Epeak and the width T2π. One of these parameters can be chosen arbitrarily, the other two result from (1.1.112)/(1.1.113)/(1.1.114). But the interaction is coherent only as long as T2π T2.

1.1.7.3.1.2 π-pulse in an amplifying medium

A steady-state solution in an amplifying medium, initial condition ∆ n(t = −∞) = ∆ n0 > 0, with broadband losses (α = 0) is the π-pulse [74Loy], see Fig. 1.1.19:

	E = Epeak		exp [i ω (t − z/c)]		(field) ,	(1.1.115)
			cosh [(t − z/c)/τ )]
	Epeak =			(peak amplitude) ,		(1.1.116)
		τ µ

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New Series VIII/1A1