Ultrashort Laser Pulse Phenomena, 2nd Edition

Bibliography 273

4. Show that a pulse with zero area has no Fourier component at its average

frequency.

5. Show that the definitions (4.22) and (4.27) are equivalent for unchirped

pulses at resonance.

6. Discuss the connection between the frequency pushing (mentioned in

Section 1.2.1) and ˙ϕ obtained in Chapter 4 (Section 4.3.3) for the weak

pulse limit. Hint: Define a dielectric constant in terms of u, ν , and w.

7. A short (weak) Gaussian pulse is sent through a resonant absorber with

T

2

 τ

G

(no inhomogeneous broadening). Being much narrower than the

pulse spectrum, the absorbing line should act as a frequency filter. It

is therefore a broadened pulse (in time) that should emerge from the

absorber. Yet, according to the area theorem, the pulse area should

decrease. Resolve this apparent contradiction in both frequency and time

domains.

8. Calculate the initial frequency shift with distance d˙ϕ/dz for a Gaussian

pulse, off-resonance by 1/τ

G

with an absorbing (homogeneously broad-

ened) transition with T

2

= 100τ

G

. Express your answer in terms of the

linear attenuation.

9. Find an expression for d˙ϕ

2

/dz for a pulse propagating through an

ensemble of two-level systems. ˙ϕ

2

 is defined as



˙ϕ

2

E

2

dt/



E

2

dt. Hint:

Use a similar procedure as for the derivation of the expression for the

frequency shift with distance, writing first an expression for the space

derivative of W ˙ϕ

2

, and using Maxwell-Bloch’s equations to evaluate

each term of the right-hand side.

10. Referring to Fig. 4.8(a), let us consider a stepwise excitation with

a polychromatic pulse given by the sum

˜

E

1

(t)e

iω

,1

t

+

˜

E

2

(t)e

iω

,2

t

+

˜

E

3

(t)e

iω

,3

t

+ … . The frequencies ω

,1

, ω

,2

, ω

,3

… , are nearly reso-

nant with the successive transitions (0 → 1), (1 → 2), (2 → 3),….

Derive Eq. (4.59) for this situation. Hint: Instead of Eq. (4.57), the

detunings are now 

1

= ω

01

− ω

,1

, 

2

= ω

02

− (ω

,1

+ ω

,2

);



3

= ω

03

− (ω

,1

+ ω

,2

+ ω

,3

), … . The complex Rabi frequencies

are to be defined as

˜

E

k+1

=

i

2

p

k+1,k

˜

E

k+1

.

BIBLIOGRAPHY

[1] C. Lecompte, G. Mainfray, C. Manus, and F. Sanchez. Laser temporal coherence effects on

multiphoton ionization processes. Physical Review A, A-11:1009, 1975.

[2] J.-L. De Bethune. Quantum correlation functions for fields with stationary mode. Nuovo

Cimento, B 12:101–117, 1972.

[3] J.-C. Diels and J. Stone. Multiphoton ionization under sequential excitation by coherent pulses.

Physical Review A, A31:2397–2402, 1984.

[4] E. Hecht and A. Zajac. Optics. Addison-Wesley, Menlo Park, CA, 1987.