Optical Signal Propagation - v1

Optical Signal Propagation - v1

p

g

p g

Miguel A. Muriel

UPM-ETSIT-MUIT-CFOP

Optical Signal Propagation

1 Fundamentals of dispersion

2 Envelope Equation

p

q

1 Fundamentals of temporal dispersion in optical fibers

Temporal envelope broadening of the transmitted pulse

Example: Modal dispersion

Example: Modal dispersion

When a pulse is propagated in a multimode fiber, each of the modes have a

different group velocity because its effective refraction index is different, so

h f th

t k di ti t ti

t f ll

th

th Th t

l

l

t f

each of them take distinct time to follow the path. The temporal envelopment of

the output power is calculated by a rms width



_

, and the phenomenon is known

as modal dispersion.

Types of dispersion:

A) Modal Dispersion (Only multimode fibers)

B) Chromatic Dispersion (GVD, V ) (in all fibers) Material Dispersion

Waveguide Dispersion

Group elocity Dispersion

C) Polarization Mode Dispersion (PMD) (Only significant in single mode fibers)

A) Modal Dispersion (also known as Intermodal Dispersion)

D

t th diff

t

ti

l iti

f th

d

Rise time T_r

0 0

- The manufacturer supplies ( ꞏ )

is the fiber bandwidth for 1

1 K fib

B Mhz Km

B₀  f L  Km

0

s t e be ba dw dt o

0,85 200 ꞏ Typical values ( ꞏ )

1 Km fiber

f m

m Mhz Km B Mhz Km





 

 _{0 }

Typical values ( )

1,3 500 ꞏ

0 44 q

B Mhz Km

m Mhz Km

L



 _

 

0

0, 44 Empirically, it is used

0,5

r

L T

B  





q 1 (0,75 as average value)



 Concatenation factor

B) Chromatic Dispersion (GVD)Group Velocity Dispersion (also known as Intramodal Dispersion) Temporal broadening due to the different group velocity of the spectral components

of the mode with the propagation constant ( ). 

( )

( )

 

( , )

E



z

( ,0)

E



( , )

_E

( , )

( ,0)

E



z



H



z

E



( )

( ) 2

( , )

_E

( , )

( , )

f _{z j z}

e

 

 

 





Envelope (modulating signal)



A z t

( , )

( )

( ) 2

( , )

( ) causal

f _{z j z}

E E

H

z

e

h t

 

 





 





,

Kramers-Kröning relations

( )

( )

( )

( )

( )

( )

E E

H

H

n





 

 

 













_



_























2 _{( )}

( )

( )

( )

( )

f f

n

 

 

 





























2 2 2 ₁₀ ( ) ( ) 2

( )

( , )

( , )

10

(0)

( ,0)

z z E f f Nepers z

P z

E

z

H

z

e

e

P

E



 

 



_



  









_

_



_

_

















_

_



(0)

( ,0)

- Intramodal temporal dispersion



Temporal broadening due to the different

group velocities of the spectral components of the guided modes

( ꞏ )

( )

Plane waves

Guided modes

j t k r

j t z

e

e



 











Phase velocity ( )

v

_p



Velocity of the constant phase wavefront

Constant phase



(

 

t



z

) Cte



(

) 0

0

_p

dz

d

t

z

dt

dz

v

dt



 











 



 





0

Band-pass process centered on

K i th ti t t f th d ( ) d th fib l th L





Knowing the propagation constant of the mode ( ) and the fiber length , the field propagation can be considered as a filter with the transfer function:

L

 

( ) 2

( ) L

E

H e e

 

 _   ( )

0

( ) , in the relevant (centered spectral zone)

j L

Cte

 

   

( )

( ) Phase filter

E

j L

H  e   

2 3

2

Taylor Series development

( ) 1 ( ) 1 ( )

( ) ( ) d  ( ) d   ( ) d  

   3 ( )  0 ₀ 0 1 ₂ 2

0 2 0 3

( ) _{( ) (}

( ) ( ) ( )

( ) ( ) ( ) ( )

2 6

o

d _{ } d _{ } d

  _             _  _       

 _{} 0

3 3 0 ) ( ) 1 1          2 3

0 1 0 2 0 3 0

1 1

( ) ( ) ( ) ( )

2 6

( )

( 0)

e t



E



_j

_

_{( )}

_L

E

(



L

)



e t

( )

(

)

( ) ( 0)

E

L

H

E

0

1 1

Centered on

( )

( ,0)

e t

E







_

_

_

_{( )}

( )

E

j

L

H





e



 

0

2 2

Centered on

( , )

( )

E

L

e t













2 1

0

( , )

( ) ( ,0)

Source with mean pulsation

Propagation constant of the mode

( )

E

E



L

H



E





 







op g o co s

o

e ode

( )

Fiber length

L

 





0 1

0 1 0

( ( ))

( )

Group delay ( , )

( )

j  L

( )

j     L

( )

 

   

2 1 1

2 2 1

0 1 0

( ( ))

( )

( )

( )

( )

( )

( )

( )

j t

( )

j t

j L

j L

j L

E

e

E

e

E

e t

E

e d



E

e

e d



       













  



















_E

₁

_{( )}



_e

j(   0 1(  0))L

_{e d}

j t





)

2 1 1 1

Phase

0 1 0 1 0 1 0

( ( ) ( )

( )

( )

(

)

g

T

j L j t L j L

e t



e

   



E



e

 

d





e

_{}

   

e t





L

Miguel A. Muriel-2018/02-13

Group

g



2 1 1

Phase 0 1 0

( )

( )

(

)

g T

j L

e t



e

_{}

   

e t





L





Group





1

1

1

Group

(

)

Group velocity

g

g

g T

L

t

L

v

T





 









1

Group velocity in material (Group refractive index)

1

1

( )

1

( )

g

g

n

d

dn

n

n

c

d

c

d

v

c











 

















_



_









( )

( )

( )

( )

( )

( )

g

g g

dn

dn

n

n

n

n

d

d







 



 















d



d





SiO

2



L

1

( )

0

g g

L

T

L

n

c









Propagation in waveguide inside material

1

2

Core refraction index

Cladding refractive index

n

n





2

g

(

(

))

k



k

b



₀

_

_

₀

(

₂



(

_{1 2}

))

eff eff

n n

k n

n

k n

b n

n

c













2

01

0.996

( )

1.1428

b V

V







_



_



V







Concept of temporal chromatic dispersion

Temporal broadening due to the different group velocities of spectral components of the mode

 



1

  

Temporal broadening due to the different group velocities of spectral components of the mode

Operating in



 T  dTg

 





 d



L

 









₂

  

L 



(Group Velocity Dispersion

(GVD)

Operating in T_g

d d













       _

  

  

) ( ) L T L





   

  

 



      

  

2 1 1 ( ) 2 g T L

dT d L d _{L d c}









_

_

  

 



    



 

  

1 1 2 2 2

2 _{( Dispersion )}

2 Operating in _g g

c _D

dT d L d _{L d c}

T L

d d d d

 





_

_























            

 

( ) g

T D L



  

TOTAL

Fiber dispersion (material waveguide)

M W

D

D

D







TOTAL

1º window (-100,-120 ps / nm km)

2º window ( 3 ps / nm km)

M W

D

D

D

D









_



2 window ( 3 ps / nm km)

F

t d d SMF

3º window (17 ps / nm km)

Total

D

_







_



For standard SMF

Total dispersion parameter D and

its components D

_M

and D

_W

for a

standard SMF.

Zero dispersion wavelength



_ZD

is

Zero dispersion wavelength



_ZD

is

shifted towards higher wavelengths

because of waveguide dispersion

contribution

contribution

Material dispersion (

D

_M

)

 

122 1

1, 25

1,66

ꞏ

ZDM M

ps

D

m

nm Km



_{ }





 



_

_



_



_

_

_

_





_









2 1,276

0

1, 276

zero dispersion (pure

)

M _m ZDM

D

_{ }

  





m







SiO

Material dispersion is caused by the frequency variation of material refractive

Material dispersion is caused by the frequency variation of material refractive

index, being this dispersion proportional to the second derivative of the

material refractive index with respect to the wavelength

Waveguide dispersion (

D

_W

)

D

 

₂



₂ 1

4

W

D

ca n





Differential dispersion





 

2

2 3 2

3 2

4

2

(

)

ꞏ

dD

c

c

ps

S

S

d

_{nm Km}



_



_































(F d t l f Ph t i 2nd_{Ed Saleh and Teich 2007)}

C) Temporal dispersion because of Polarization Modes (PMD) (Single mode fibers)

01

) p p ( ) ( g )

Fundamental mode has two polarization modes

f i i d (Bi f i )

LP 

8 3

two refractive indexes (Birefringence) 10 n_x n_y 10 t

 

     wo propagation velocities PMD temporal dispersion

 

 PMD temporal dispersion

The remaining birefringence of the fiber in a wiring, in virtue of lack of

manufacturing process accuracy or environmental changes (pressure,

temperature), results in a random process of velocities difference between both

polarization modes. Thus, it must be handled in statistical form.

- Without modal switching

0

( ) _{( )}

( )

( )

y

x gx gy

d

d

L

L

T

L

v

v

d

_{ }

d

_{ }

 

 



_



 







0 ₀ ( ) _{( )}

1 1 1 1

gx gy

x y

L

L

T

L

  _{ }









 _





     

- With modal switching (correlation length)

( )

2

( )

c

l

L





l L

D

L

L

D

L



( )

( )

1

2

( )

( )

0 01

2

t PMD c p t PMD p

PMD parameter

L

l L

D

L

L

D

L

ps

D



 











0,01

D

_p

2

p

Km





IT



U-T G.663 Recommendation

_{( )}

( )

_{( )}

( )

1

10

10

b

t PMD t PMD

T

L

B

L





Limit by PMD

Example:

100

L

Km

_

 

( )

100

( )

0,1 100 1

0,1

t PMD

p

L

Km

L

ps

ps

D

Km









_

_







Km

_

1000 10000

km)

ps

Fuente: I. Esquivias, TFB-UPM

The graph shows the maximum

100 1000

Le

n

gth

 (

0.5 p ps D  0.1 p ps D Km 

length

in

terms

of

NRX

modulation bitrate, according to

the ITU Recommendation and of

1 10

Ma

x.

 

p Km

two

values

of

the

PMD

parameter

0 50 100 150

Bir rate (Gbps)

2 Envelope Equation

- The space-time evolution of the propagated pulses in a fiber of length , with

( )

and ( ) is studied by means of the (

) envelope propagation equation

f

L

A z t

 

 

and ( ), is studied by means of the ( , ) envelope propagation equation,

derived from Max

A z t

 

well's equations

 



0 0



( , ) Re

( , )

j t z

E t z



A t z e

 

(0, )

E t _{E z t( , )}

E

(

z

,

t

)

Electric field

A

(

z

,

t

)

Slowly varying complex field envelope

|A(z

,

t

)|

2

_Power

|

E

(0,

t

)|

2

_|

_A

_(0,

_t

_)|

2

|A(z

,

t

)|

Power

o

wer

o

wer

o

wer

p

o

p

o

p

o

time

time

time

- The envelope ( , ) propagation equation, calculated with the Maxwell's

equations, in an optical fiber of length with

_f

( ) and ( ), is:

A z t

L

 

 

2 3

3 2

1 2 3

( , )

( , )

( , )

( , )

f

A z t

A z t

A z t

A z t

j







A













₁



₂



₃



( , ) 0

z t



2

6

2

j

A

z





t



t

t











2

( , ) 0

Th

ti l

(

) i di

tl

ti

l t

(

)

z t

P

t

A

t

- The optical power ( , ) is directly proportional to ( , ) .

- Generally, the impact of

_f

is not considered:

P z t

A z t



2 3

3 2

1 _{2 3}

Ge e

y, e

p c o

s o co s de ed:

( , )

( , )

( , )

( , )

0

2

6

f

A z t

A z t

A z t

A z t

j

z

t

t

t











_



_



_



_

Particular cases

1 T l f i i h h l

2 3

3 2

1.- Temporal reference system varying with the pulse

( , ) ( , ) ( , )

0

A z t A z t A z t

j





 ₂  ₃ 

2 3

( , ) ( , ) ( , )

0

2 6

2 - Temporal reference system varying with the pulse and 0

j

z t t







  

  



3

2

2.- Temporal reference system varying with the pulse, and 0

( , )

A z t j







 _ 2 _{( , )}

0

A z t _

2

j z 

 _t2  0

2

2

2 2

2

( , )

( , )

( , )

0

( , ) 0

2

2

A t z

A t z

A

z

j

j

A

z

z

t

z







_

_



_



_

_



_

_











2

2

z

t

z







2 2 1 2 ( )

( , )

( ,0)

A

j z

H z

A

z

e

 

A





















( , ) A

H  z

2 2 2 ₂ 1 2 2

( , )

( , )

2

A A t j

j z

j

_z

H

z

e

h t z

e

z

  _











 

2

 

x t

y t

   



x t

* ( )

h t

System

 

x t

   

( )

X

( )



h t

 

H

 



Y

 





X

 



H

( )



X



Y

 





X

 



H

( )



2

1

j  2

2

1 2

( , )

j z

( ,0)

A





z



e

  

A





2 2

2

( , )

( ,0)*

2

t j

z

j

A t z

A t

e









2

2



z







0 0



(

) Re

(

)

j t z

E t z

( , ) Re





A t z e

( , )

 



( )

Optical waveguide



H

_E

( , )



z



e

  z

( ) ( ) f E j   ( )

( ) ₂

( ) ( ( )) 2

( , )

1

(

)

f f

z j z z

E

z j t z

H

z

e

e

h

d

        



      







2 ( ( ))

1

( , )

2

j E

h t z

e



d











 

,0

E t

 

E t z

 

,



E t

 

,0 * ( , )

h t z

E

( 0)

E





( )

( , )

z E

H



z



e

 







0



(

)

E





z



E





H



z

( ,0)

E





_,





_,0



_{( , )}

E

E



z



E



H



z

0

Band-pass signals centered on





0 0 0



0

2

] [

p

g

Carrier

f

f









( )

( )

( )

E







E









E







( )

E



_



( )

E



_





0



 ₀

( )

A







m



m



0

( )

, in the relevant

(centered spectral zone)

1

1

Cte

 





2 3

0 1 0 2 0 3 0

1

1

( )

(

)

(

)

(

)

2

6

 







  





  





  









0 0



C l

A l ti

( , )

( , )

j t z

E t z

_



_

A t z e

 



Complex Analytic

envelope







0 0



( , ) Re

( , )

Re

( , )

j t z

E t z



E t z

_





_

A t z e

 



_

2

1

j  2

2

2

1 2

( , )

A

j z

t j

H

z

e

j

 







2

2

2

( , )

2

A

j z

j

h t z

e

z





 

2

2

1 2

(

)

j z

H

  

 

,0

A t

A t z

 

,



A t

 

,0 * ( , )

h t z

A

2

( , )

A

H



z



e

( ,0)

A







_,





_,0



_{( , )}

A

)

3 RMS (root mean square) (σ pulse widths Equation

- The space-time evolution of the propagated pulses in a fiber of length , with ( ), is studied by means of the envelope ( , ) propagation equati

L A z t

 

on:

2 3

3 2

2 3

( , ) ( , ) ( , ) 0

2 6

A z t A z t A z t j

z t t





 _  _  _

 2  6 

- The spectral componentes of the mode depend on the spectral widths of the modulated optical d th d l ti

z t t

  

i l source and the modulation

2

signal.

- The power pulse propagation is studiedp p p p g  P t z( , )( , )  A t z( , )( , )

- Even if the gaussian pulse is the most used one, the propagated ones and the modulated sources Th

t k bit h d l d b f th i idth

can take arbitrary shapes  They are modeled by means of their rms widths σ .

Signals

2

( )

( )

Temporal mean

t

t P t dt

t A t dt

   











2

Temporal mean

( )

( )

t

P t dt

A t dt

 

 















2 2

2 2 2

2

( )

Temporal variance

RMS temporal width

( )

t

t

t

A t dt

t

t

A t dt



  

















2

( )

A t dt









₂ 2

2 2 2

( )

0

( )

t

t A t dt

t

t

A t dt

Sources

 





 

Autocorrelation Power Spectral Density

R





S



( )

Spectral mean



  

S

d

 





Spectral mean

( )

S

d



 

 











2 2

2 2

( )

Spectral variance

RMS spectral width

( )

S

d

S

d



 

 







 

  

















2

( )

( )

S

d



 

 





2

2



c

2 2

( )

0

( )

S

d

S

d





 







 

  

 











2 2 2

2

(

)

d d

c

   









     



Miguel A. Muriel-2018/02-45

d

 

2

0

A t

 

,0

A t z

 

,

2



A t

 

,0 * ( , )

h t z

A 2

A t

 

2

₀





2

 

L





2

 

0





_h2

 

A

h t

2

,

t h



 

,

 

,

_A

( , )

 

0

t

I

₂

(A)

I

₁

(A)

O

E

P

₂

(W)

E

O

P

₁

(W)





2

 

2

, ,

1

N

t system t i

i _N











_









 

1

2 2

, ,

1

i _N

r system r i i

t

T

T

T







_

_







_

_



Rise time t

T

r



_





 

2

 

2

 

2



2

( ) ( ) ( ) ( )

r system r emitter r fiber r receiver

T



T



T



T



0



IN



RMS temporal width of the input pulse



_t

(0)









OUT



RMS temporal width of the output pulse



_t

( )

L





_D

(RMS intramodal dispersion width)



2

_{( )}

_L

2

₍₀₎

2











1

- The modulating signal has a rms spectral width given by (

)

2 (0)



_t

- The unmodulated source has a rms spectral width given by (



_

)

- The ratio is the parameter

p

V

_

rms spectral width of the source

V













2

 

(0)

rms

V

_

2

(0)

1

spectral width of the modulating signal

2 (0)

t

t



 



mod

- The modulated source has a rms spectral width given by (



_

)

2 2







2 2

2 2 2

mod

source modulating signal

1

1

- Quadratic sum of signal and source

1

2 (0)

_t

V

2 (0)

_t

  























_

_



 

_

_



_{}







RMS (σ) pulse widths Equation

2 2

2

RMS (σ) pulse widths Equation

L

L





















2

2 2 2 2 2 3

2

2

( )

(0)

1

1

2 (0)

4 2

(0)

t t

t t

D

L

L

L

V

_

V

_















   











 

_

_

 

_

_

_

_











D     









L



2

_









1



2



_

2





L



2













2 2 2 2 2

2

2 d

1

( )

(0)

1

1

2 (0)

2 (0)

t t

t t

L

L

L

V

_

V

_

 





























 

_

_





_

_







_





_









3 4

2

L











 







 2L mod





_mod



2

a) Wide spectrum sources (

1)

the source spectrum is predominant

1

V

_









₂





2

2 2 2

2 3

1

1

( )

(0)

2

t t

V

_







L







 

_

L



 

_

L

b) Narrow spectrum sources (

V

_



1)



the modulating signal spectrum is predominant

2 2

1

( )

(0)

V







L









2 2

3

2

L

1





L







 







1

_t

( )

_t

(0)

V

_







L







₂

2 (0)



_t

 

2 4



_t

(0)







L t

i

t

i

l

 

0 3

Let us going to review several cases:

1) Wide spectrum source (

V



1), input pulse

 



and





0

 

0 3

0 2

1) Wide spectrum source (

1), input pulse

and

0

2) Wide spectrum source (

1), input pulse

and

0

V

V





















 

 

0 2

0

3) Narrow spectrum source (

V

1), input pulse

and











3



0

 

0 2

4) Narrow spectrum source (

V

_



1), input pulse



and





0

 

0 3

5) Narrow spectrum source (

V

_



1), input pulse



with chirp and

C





0

 

0 2 3

1) Wide spectrum source (

V

_



1), input pulse



and (





 

_

)

It is the case distant from the zero dispersion point (

 



_ZD

).









2 2

2 2

2 2 2

2

( )

(0)

(0)

D D

t

L

t

L

 t

DL



 









 



















2 2

2

For very long

L





_t

( )

L



 

L

_





DL



_



2

This formula is also calculated by more direct process, regarding the

spectral dependence of group delay.

1

1

1

1

( )

is proportional to

Slope

-1

4

4

4

t

B

L

B

B

D L

L

L





  







_



2 _logB-logL

4

4

4

t

D L



_



L



_

L

 

 

0 2

2) Wide spectrum source (V_ 1), input pulse

 

 and   0 It is the case of the zero dispersion point (  _ZD).









2 2

2 2

2 2 2 2 2

3

1 1

( ) (0) (0)

2 2

D D

t L t L  t SL 

 

       

 

Where is the dispersion slope, or S differential dispersion





2





2

2 2 2

3

1 1

For very long ( )

2 2

t

L  L   L _  SL_



2 2

3 _logB-logL

1 1 1 1

( ) is proportional to Slope -1

4 2 2 2 2

t

B L B B

L S L _ L _



   _{ }

 

      

Example with a wide spectrum source

LED (_  75nm)

a)

17 / ( )

ZD

D ps Km nm

 







1,55

17 / ( . ) 1

0, 2 / ꞏ

4

m

D ps Km nm

B BL Gb s Km

D L

 







   

b)   _ZD





2

2

0, 08 / ( . ) 1

4 / ꞏ

ZD

S ps Km nm

B BL Gb s Km

  

  ₂  





 

0 2 3

3) Narrow spectrum source (V_ 1), input pulse



and (





 

_)

2 2

It is the case distant from the zero dispersion point ( _ZD).

L L

 



       2

2_{( )} 2₍₀₎ 2 2_{(0) 1}

2 (0)

t t t

t D D L L L L 











         _{ }  _{ }     _{} _{ } 2 where L_D 



2

2 2

2

(0)

Dispersion length ( ) 2 (0)

t

t LD t







  

2

( ) 2

There is an optimum value of (0) , which minimizes ( ) 2

t optimum t

L L L













  ( )( ) 2 logB-logL

1 1 1

( ) is proporcional to Slope -2

4 _{4 L}

t mínimum L

t

B L B B

L 





_{ }         2 g g 



  2

1,55m 20 ps / Km

  

 

0 2

4) Narrow spectrum source (V_ 1), input pulse

 

 and  0

2

It is the case of the zero dispersion point (  _ZD).

2

2

2 2 3

2

1 ( ) (0)

2 4 (0) D t t t L L         _{  }    3 ( )

There is an optimum value of (0)

4 D t optimum L    

1 ₁ 1

3 ₂ 3

3

3 , which minimizes ( )

2 4 t L L     _{ }       _{ }        ( )( ) 3 3

1 0,324 1

( ) is proportional to Slope -3

4 L

t mínimum L

t

B L B B

L                 3 3 3 logB-logL 4 L

150 / ( 100 ) 0 1 /

L

B Gb s L Km ps Km           _

3 0,1 /

70 / ( 1000 ) ZD ps Km B Gb s L Km

 

 _  _{ }

Pulses with chirp

Th

ti

l

h

hi (

d l ti

f

i ti

th ti

)

The propagating pulses can have chirp (modulation frequency variation over the time)

and can be calculated with complex envelopes:





 

 

2 2

0 ₂ 0 ₂

0 0

1

1

( )

exp

exp

2

₂

2

₂

j

t

t

A

t

A

C

A







_































_

_







 

2 2 0

exp

2 2

t

j

C





















































_

_



_

_





_

_

 

0

 

0









 

0

( )

Chirp parameter

t

C















_

_













 

2

0

( )

Frequency variation

( )

2

t

t

C

t

t











 





 

0 2 3

5) Narrow spectrum source (V_ 1), input pulse



with chirp y (C





 

_)

2

It is the case distant from the zero dispersion point ( _ZD).

L

 





 

2 2 2

2

( ) (0) (1

2 (0)

)

t t

t

C C L

L L











 

    _{ }

 

2 2 2

a) 0 ( ) (0)

2

t t

L C  



L 







2 2

2_{(0) 1 Case 3}

(0) t t D L L





                   _{} _{ } 2 2

b) 0

b-1) 0 the dispersion is broadened with the distance

D C C 



    2 2 ) p

b-2) 0 the dispersion is narrowed with the distance until the one which is minimu

C





 









2 2

min ₂ min ₂

(0)

m ( ) , and then it is broadened.

1 1

t

D t

C

L L L

C C





   



₁ 2





₁ 2



(Equivalent to the focus of a lens)

C C

1 0

( ) (0)

t t

L T

T

 



0

2

2

( ) 2 (0)

Dispersion length

t t D

L 



 

RMS (σ) pulse widths with chirp Equation

RMS (σ) pulse widths with chirp Equation









2

2 2

2

2 2 2 3

2 2

2 2 2 3

( )

1

1

1

(0)

2

(0)

2

(0)

4 2

(0)

t

t t t _t

L

C L

L

L

V

_

V

_

C





























 

_

_

 

_

_

 



_

_

_

_











( )



4 Power transfer function

(

A) - Source spectrum much wider than the spectrum of the modulating signal)

- These sources with wider spectrum than the modulating signal one are typically used, and the source-fiber-receiver set can be considered as a power lineal system.

2

( )

( )

P t



A t

( ,0)

P t

(

)

h t L

_PP

( , )

(

)

P t L

( , )

(

)

(

)

(

)

h t L

_P

( , )



H

_P

( , )



L

h t L



H



L

- Normalized Input pulse:

2

2

1

1

( ,0)

( ,0)

exp

2

(0)

2

_t

(0)

_t

t

P t

A t







_

_











_{ }

_



_

_







( )

Where:

t

t









2

(0)

Temporal width of the imput pulse

t





2 2

2

1

( ,0)

1

2

t

P t

dt

e



dt



 

 







- The propagation in the Fiber is derived from its optical power impulsive response:



2 1

10 1 1

( , ) 10 exp 2 2

L P

t L

h t L



_

  _{  } 

 

 _{  }

 _{ } 

 ₂

2

where:

f

P

D D

L

e





   _{ } 

 

where:

[dB km/ ] Fiber attenuation





2 Temporal chromatic dispersion width

D L  D L 





 







y RMS spectral width of the optical source at and respectively

2 c

 













2

2 ( c)

 











- Output pulse:

2

2 _{10 1}

2 2

2 2

(

)

1

1

( , )

( , )

( ,0) * ( , ) 10

exp

2 (

(0)

)

2

(0)

L

P

t D

t

L

P t L

A t L

P t

h t L



_





 





_

_

_







_



_







(

( )

)



2

(0)

( )

Temporal width of the output pulse

t D

t D

t

L

 

















2

2 2 2 2

2

( )

p

p p

( ) (

(0)

) (

(0)

)

t

t

L

t D t

L

