High-Nonlinearity in Glass Fibers and Cross-Phase Modulation - Optical Signal Processing by Silicon Photonics - page 58

I D

V D

Reverse-
breakdown
region

Forward-
bias
region

Saturation current I
R everse-bias
region

s

Fig. 5.1 Characteristic curve of a silicon diode

where j = 1, 2. If we suppose a general solution of the form

a j = u j cos ( K z − t ) + iV j sin ( K z − t ) (5.7)

where K is wave number and is the modulation frequency, Eq. ( 5.6 ) gives a set
of homogeneous equations containing u 1 , u 2 , v 1 and v 2 . If K and satisfy the dis-
persion relation then this set of equations will have a nontrivial solution (Fig. 5.1 )

2 2

= C 2

K −

−

K −

− (5.8)

V g 1 f 1 V g 2 f 2

where,

1
2 β j Ω 2

1
2 β j Ω 2 + 2 γ j P j

= (5.9)

f j

and then parameter of coupling i.e., C is given by

C = 2 Ω 2 (β 1 β 2 γ 1 γ 2 P 1 P 2 ) 2

(5.10)

1
2

1
2

( f 1 +

2

f 2 ) 2 + 4 C 2 − K − Ω V g 1 = ( f 1 + f 2 ) ± f 1 f 2

(5.11)

Clearly, K will become complex if C 2 > f 1 f 2 is satisfied. Solving Eqs. ( 5.9 ) and
( 5.10 ), the condition becomes

2

1
2

Ω 2 <Ω c = ( b 1 + b 2 ) 2 + 12 b 1 b 2 − ( b 1 + b 2 )

(5.12)