Influence de la dispersion aléatoire faible sur la transmission par solitons et du mélange à quatre ondes dans les fibres optiques

1.4 Obtention du p roduit u1 *u 22

~ ? 2 ~

1

u z t u z t

( ) (

, = , - ? -

z T ) exp ~ i t i z

? - ~

1 10 1 1 1

~ 2 ~

=

sec

A₁

1

h A t

[ ( - ? - ) ] exp ( ) ~

- ?

~~~ ~ ? + ~~~

1 z T i t A 2 2

1 z

1 1 1 ~~

1

~~ 2

1

( )

, [ ( ) ] exp ~~~ ( )

2 ~

2

~ = sec + ? - ~ - ? +

u z t A h A t z T i t A - ? z ~~ ~~~

1 0 ~~

2

u2

( z ,t ) = u20 (z,t-n2 z -T 2)

exp

~ ? 2 ~ ~ i t i z 2 ? - ~ 2 ~ 2 ~

=

sec

A₂

1

h A t

[ ( - ? - ) ] exp ( ) ~

- ?

~~~ ~ ? + ~~~

2 z T i t A 2 2

2 z

2 2 2 ~~

2

~~ 2

( ) [ ( ) ] ~ ? + 1

, exp ~~~ ( )

2 ~

2

~ u z t A h A t z T

= sec - ? + i t A - ? z ~~ ~~~

2 0 ~~ 2

*

1

u

1

= [ ( ) ] exp ~~~ ( ) ~

2

sec + ? - A

~ ? -

A h A t z T i t - ? 2 z ~~ ~~~

0 ~~ 2

u A h [ A ( t z T ) ] { i [ t ( A ) z ] }

2 2 2 2 2

= sec - ? + exp 2 ? + - ?

2 0

~ u

* 1

u ^22 = A

3

sec

1 2 [ ( ) ] [ ( ) ] exp 3 ~~~ ( ) ~ 2 h A t z T 2 ~ ? + sec - ? + h A t z T + ? - i t A - ? z ~~~ 0 0 ~~ ~~ 2

*

=

exp

u10

2

u 20

~ ? ² ~

~ 3 i t i z

? - ~

~2 ~

1.5 Obtention de l'équation (2.34)

- g z u u

( ) *

2

2 1

t

~ ~~ ¹J

~ ? 2 ~ ~ ? 2 ~ ~ ? 2

~ ~ ~ ? 2 ~

i t

~~ 3 ? - ~~ ~~ ~ ~

z ~~ 3 3

? 2 i t

~~ 3 ? - z i t

? - z

1 ~~ i t

~~ ? - z ~~

2 ~ 2 ~ 2 2

~~

i H e

~ ~ - i He ~ ~ ~ ~

+ H e ~ + ?

3 i He ~

z t

~ 2 ~ 2

~ ~ ~~

~ ~~
~

1

^I

~

~

~

[Htt

+

2

~ ? 2 ~ ~ ? 2

i t

~~ 3 ? - z ~~ ~~ 3 ? -

? 2 i t z

~ 2 ~ - ~ 2

H e i He

z

~ ? 2 ~

i t

~~ 3 ? - z ~~

~ 2 ~

e

+ 6i51H _t

2

i

~ ~

~

~

2

-

?

]

e

He

9

~ ? 2 ~

i t

~~ 3 ? - z ~~

~ 2 ~

~ ? 2 ~

i t

~~ 3 ? - z ~~

~ 2 ~

~ ? 2 ~

~~ 3 ? - z ~~

i t

( ) 20 10

2 ~ 2 ~

g

*

z u u e

2

i[3 S2t-

^g2 z 2

O n simp lifie p ar e on a :

?

2

iH + 2 2 H +¹ [H tt

+ 6i52H t - 9 51 H ]=-g

( z

)u 2 20

u 10

2

~ iH + z

1 2

[ H tt + 6i52Ht - 951^2 H + 51^2H] =-g

( z

)u 2 20

* u 10

iH +1 2 [Htt

+ 6i52H t - 8 51 H ]=-g

( ₁,2 ,₄* _Vz P4'20'10

2

D 'oh. l'équation p our H ( z , t) s'écrit :

1 [ H _tt + 6 0-H t - 2(2 51) ² H ] = - g

²

z u 22 0 u 10 ( 2.34)

iH _z +

1.6 Obtention de la relation (2.41)

. 1

( z , co) = f +.0A³ sec h² [ A ( t - + T₀ )] sec h[ A (t + - T ₀ )] expit[352t + ( A² - n² )z_. Li} ×

2

exp

(- 3int

) e

ict

dt

-

ict

~ ~~ e

dt

+? 1

3

~ ( ) = ~ [ ( - ? + ) ] [ (

sec + ? - ) ] ( )

~ - ?

F z ? A h A t z T

² h A t z T ~~ i A 2 2

, exp

sec z

0 0

-? 2

1 ~

3

( ) ( ) ~ [ (

+?

2 2 2 -

~ F z = A exp ~ - ?

i A

~~ z ~~ sec h A t z T

- ? + ) ] [ (

sec h A t z T e dt

i ? t

, ? + ? - ) ]

0 0

2 -?

Posons b = A(t - 52z + T₀ ) ' t = + 52z - T 0 dt =

db

A A

2

cT0

db

co

- i _Ab-i aglz+i

A

F( z , co) = A³ exp(i ¹ ( A² -n²)z_l^+*sech²[ b ] sec h[b+252Az -2AT₀]e -?

?

1 ~

( ) ( ) ~ [ ] [ ( ) ]

i

+? - b

2

~ ? = exp ~ - ? - ? +

F z A ~~ i A 2 2 ?

z i z i T

? ~~ sec h b h b A z T e db

2

, sec + ? -

2 A

0 0

2 -?

Soit A₀ = (52z - T₀) on a

?

~ 1 ~ +? - i b

2

( ) ( )

2 - ? - ? +

2 2

F z , ? = A exp ~ i A z i z i T

? ? sec [ ] [

sec 2 ] 2.38

( )

0 ~ h b h b A e ^A db

+ ? 0

~~

2 -?

~

2

~ ~~×

aT₀

Donc

1 z 2

( ) ( ) ⁱ

- ? ? = -

? ~ - ? - ? +

1

à , 2

à exp ~~ ( )

2 2

iH H g z e A i A z i z i

z ?

2 2

sec h²[ b ] sec h[b + 2 AA₀

- i

^cob

A

db

] e

~ - ? +

A 2 ~

i

-

CO_b

A

db

] e

i ~ z ? ?

~

à iH z

-

1 2

z T ~

0

( ) ( ) ~ [ ] [

+?

~ ~

? ? = -

? , à sec

2 2 2

H A g z e h b h b A

sec + ?

2 0

-?

~ ? + ? ~ ~

( ? )

- ~

i ? z +?

0 ~ ~ ? 2

~ ~

?? z ( ? ) A 2 ? n ~ ~ ?

à ~ ~

2 2

H z

( )

, ? ?

= ~ ~

i A e sec h ~ ~ exp ~ ~ ' 2 , dz' 2.41

( )

n 0

~ ~

g i + - ~ ~ ?

z I A

~ ~

~ ~

2 A -? 2 2 z A ~

n =-? ~ ~ ~ ~ ~ ~ ~

a