Ultrashort Laser Pulse Phenomena, 2nd Edition

Bibliography 215

a susceptibility χ

(3)

δ(kz −jπ)/N. The medium susceptibility χ

(NL)

can thus

be represented by:

χ

(NL)

(z) =

N



j=1

χ

(3)

N

δ(kz − jπ) (3.191)

Insert this susceptibility in Eq. (3.159), and average (integrate) over the

thickness of the medium, to find an average third-order polarization.

Show that the coupling term in this polarization, which in an homoge-

neous medium would average to zero, has now a contribution of the same

order as the other terms.

It is interesting to contrast the result from the MQW with that of the

homogeneous medium. First, the nonlinear polarization is larger in the

case of the MQW sample: if the fields are equal, P

(3)

1

= 4

0

χ

(3)

|E

2

1

|E

1

to

be compared with P

(3)

1

= 3

0

χ

(3)

|E

2

1

|E

1

in the homogeneous case. Second,

the “nonreciprocity” because of cross-phase modulation, which appears in

the homogeneous case, is not present in the MQW geometry. Show that,

for the change in index, instead of n

nl

= n

2

(

˜

E

2

1

+ 2

˜

E

2

2

) for the forward

beam, in the homogeneous case—a consequence of Eq. (3.163)—we have

n

nl

= n

2



˜

E

2

1

+

˜

E

2

2

+ 2E

1

E

2



=

3χ

(3)

8n

0



˜

E

2

1

+

˜

E

2

2

+ 2E

1

E

2



, (3.192)

an expression that is the same for both directions of propagation. This is

basically a result from the fact that the emission of layers of dipoles (spaced

by a wavelength) in the forward and backward directions is equal [89].

BIBLIOGRAPHY

[1] O. Hittmair. Quantum Mechanics. Thiemig, Munich, Germany, 1972.

[2] M. Schubert and B. Wilhelmi. Nonlinear Optics and Quantum Electronics. John Wiley & Sons,

New York, 1978.

[3] L. M. Frantz and J. S. Nodvik. Theory of pulse propagation in a laser amplifier. Journal of

Applied Physics, 34:2346–2349, 1963.

[4] D. Kuehlke, W. Rudolph, and B. Wilhelmi. Calculation of the colliding pulse mode locking in

cw dye ring lasers. IEEE Journal of Quantum Electronics, QE-19:526–533, 1983.

[5] V. Petrov, W. Rudolph, and B. Wilhelmi. Chirping of femtosecond light pulses passing through

a four-level absorber. Optics Communications, 64:398–402, 1987.

[6] V. Petrov, W. Rudolph, and B. Wilhelmi. Chirping of ultrashort light pulses by coherent

interaction with samples with an inhomogeneously broadened line. Soviet Journal of Quan-

tum Electronics, 19:1095–1099, 1988.