Chapter 3. Propagation of Optical Beams in Fibers

Nonlinear Optics Lab . Hanyang Univ.

Chapter 3. Propagation of Optical Beams in Fibers

3.0 Introduction

Optical fibers  Optical communication

- Minimal loss - Minimal spread

- Minimal contamination by noise - High-data-rate

In this chapter,

- Optical guided modes in fibers

- Pulse spreading due to group velocity dispersion - Compensation for group velocity dispersion

Nonlinear Optics Lab . Hanyang Univ.

3.1 Wave Equations in Cylindrical Coordinates

Refractive index profiles of most fibers are cylindrical symmetric  Cylindrical coordinate system

The wave equation for z component of the field vectors :



^{2 2}



^{ 0}







 



z z

H

k E ^where,

2 2 2 2 2 2

2 1 1

z r

r r

r 

 



 



 



 





2 2 2

2 n /c

k  and

Since we are concerned with the propagation along the waveguide, we assume that every component of the field vector has the same z- and t-dependence of exp[i(t-bz)]

 exp[ ( )]

) , (

) , ( )

, (

) ,

( i t z

r r t

t  b



  



 



 



 





H E r

H r E

# Solve for E ,_z H_z first and then expressing E_r,E_,H_r,H_ in terms of E ,_z H_z

Nonlinear Optics Lab . Hanyang Univ.

From Maxwell’s curl equations :

t

t 

 



 

 





 E

H H

E  , 

z

r H

H r i E

i b _ 



 

 1

z

r H

H r i E

i 

 



 b

 _

) 1 (

1

 

 rH

r H r

E r

i _{z r}



 



 



z

r E

E r i H

i b _ 



 



 1

z

r E

E r i H

i 

 





  _ b

) 1 (

1

 

 rE

r E r

H r

i _{z r}



 



 







 







 







  _{z z}

r H

E r r E i

 b



b



 b

2 2



 







 







  _z H_z

E r r E i

b



 b





b

 2 2



 







 







  _{z z}

r E

H r r H i

 b



b



 b

2 2



 







 







  _z E_z

H r r H i

b



 b





b

 _{2 2}



in terms of

We can solve for E

_r

, E

_

, H

_r

, H

_

E ,

_z

H

_z

A } )

A ( 1{

k A )

( A a A )

A (1

a  ^{ } 









 







 







 



 _r ^{z r z} r ^r

r r r

z z

A r

Nonlinear Optics Lab . Hanyang Univ.



^{2 2}



^{ 0}







 



z z

H

k E



(3.1-1)

z

z

H

Now, let’s determine

E ,

0 )

1 (

1 _{2 2}

2 2 2 2

2  



 







 



  



 



 





z z

H k E

r r r

r b

 The solution takes the form : (r)exp( il)

H E

z

z   



 





where, l0,1,2,3,...

1 0

2 2 2 2

2

2  



 



  

 

 



   b 

r k l

r r r

) ( )

( )

( r  c

₁

J

_l

hr  c

₂

Y

_l

hr



) ( )

( )

( r  c

₁

I

_l

qr  c

₂

K

_l

qr



1)

2)

:

2 0

2

b

 k

:

2 0

2

b

 k

where,

where,

2,

2 2k b h

2,

2

2 k

q b 

l

l Y

J ,

l

l K

I ,

: Bessel functions of the 1^st and 2^nd kind order of l

: Modified Bessel functions of the 1^st and 2^nd kind of order l

Nonlinear Optics Lab . Hanyang Univ.

Asymptotic forms of Bessel functions :

l l

x x l

J 



 



 

2

! ) 1

(



 



 

 0.5772...

ln 2 ) 2

0( x x

Y 

l

l x

x l

Y 



 



 



 ( 1)! 2 )

( 

l l

x x l

I 



 



  2

! ) 1 (



 



 



 0.5772...

ln2 )

0( x x K

l

l x

x l

K 



 



 



 2

2 )!

1 ) (

(

,...

3 , 2 ,

1 l

,...

3 , 2 ,

1 l 1

For x For x1,l



 



  



 





4 cos 2

) 2 (

12







x l x x

J_l



 



  



 





4 sin 2

) 2 (

12







x l x x

Y_l

x

l e

x x I

12

2 ) 1

( 



 







x

l e

x x

K  ^



 



 ²

1

) 2

( 

Nonlinear Optics Lab . Hanyang Univ.

3.2 The Step-Index Circular Waveguide

<Index profile of a step-index circular waveguide>

1)

r a

(cladding region) :

The field of confined modes :

 1 x

*

2 0

2 

 k b



: evanescent (decay) wave

c n k

n /

and b  _{2 0} ₂

*



: virtually zero at rb()

x

l x x e

I ( ) ^12 is not proper for the solution

 

 ^{i t l z} 

qr CK

t

E

_z



_l

    b

 (r , ) ( ) exp

 

 ^{i t l z} 

qr DK t

H

_z

(r , ) 

_l

( ) exp     b

a r  where, _q²_b²_n2²_k0²

Nonlinear Optics Lab . Hanyang Univ.

2)

r a

(core region) :

x  1

*

2 0

2 

 k b



: finite at

c n k

n /

and b  _{1 0} ₁

*



: propagating wave

l

l x x

Y ( ) ^ is not proper for the solution



where, h²n₁²k₀²b²

0 r

 

 ^{i t l z} 

hr BJ t

H

_z

(r , ) 

_l

( ) exp     b

 

 ^{i t l z} 

hr AJ t

E

_z

(r , ) 

_l

( ) exp     b

a r 

* Necessary condition for confined modes to exist : (from )

0 2 0

1

k n k

n  b 

0 and

0 ²

2  q 

h

Nonlinear Optics Lab . Hanyang Univ.

Other field components

 



^{i t l z}



hr r BJ

l hr i

J h Ah

E_r i _{l l}   b

b



b   



 



  

 ₂ ( ) ( ) exp

 



^{i t l z}



hr J Bh hr

r AJ il h

E i _{l l}   b

b



b

   



 



  

  ₂ ( ) ( ) exp

 



^{i t l z}



hr AJ

E_z  _l( )exp   b

 



^{i t l z}



hr r AJ

l hr i

J h Bh

H_r i _{l l}   b

b



b   



 



  

  ₂ ( ) ¹ ( ) exp

 



^{i t l z}



hr J Ah hr

r BJ il h

H i _{l l}   b

b



b

   



 



  

  ₂ ( ) ¹ ( ) exp

 



^{i t l z}



hr BJ

H_z  _l( )exp    b

) (

core

1) ra 2) cladding (ra)

 

^{i t l z} 

qr r DK

l qr i

K q Cq

E_r i _{l l}   b

b



b _{ }



 



  

 ₂ ( ) ( ) exp

 

^{i t l z} 

qr K Dq qr

r CK il q

E i _{l l}   b

b



b

   



 



  

 ₂ ( ) ( ) exp

 



^{i t l z}



qr CK

E_z  _l( )exp   b

 

^{i t l z} 

qr r CK

l qr i

K q Dq

H_r i _{l l}   b

b



b _{ }



 



  

 ₂ ( ) ² ( ) exp

 

^{i t l z} 

qr K Cq qr

r DK il h

H i _{l l}   b

b



b

   



 



  

 ₂ ( ) ² ( ) exp

 



^{i t l z}



qr DK

H_z  _l( )exp    b

Nonlinear Optics Lab . Hanyang Univ.

Boundary condition

: tangential components of field are continuous

at

^ra

z

z

H H

E

E

_

, ,

_

,

0 ) ( )

( )

( )

( ₂

2 



 



 





 



 



 



 



 

 K qa

D q qa a K

q C il ha h J

B ha aJ

h

A il _{l l l l}

b



b



0 ) ( )

( )

( )

( ₂ ² ₂

1  



 



 



 



 



 

 



 



  K qa

a q D il qa

q K C ha a J

h B il ha h J

A _{l l l l}

b



b



0 ) ( )

(ha CK qa 

AJ_{l l}

0 ) ( )

(ha DK qa 

BJ_{l l}

(3.2-10) ar

az

a

Nonlinear Optics Lab . Hanyang Univ.

Amplitude ratios : [

from (3.2-10) with determined eigenvalue b, Report]

) (

) (

qa K

ha J A C

l

 l

1

2 2 2

2 ( )

) ( )

( ) ( 1

1













 

 



 



 

 aqK qa

qa K ha

haJ ha J a

h a q l i A B

l l l

l



b

A B qa K

ha J A D

l l

) (

)

 (

: the relative amount of E_z and H_z in a mode Condition for nontrivial solution to exist : (Report)

2

0 2 2

2 2

2 2 2

1 1 1

) (

) ( )

( ) ( )

( ) ( )

( )

( 



 















 



 







 



 











 

 











 

 

k ha

l qa qa

qaK qa K n ha haJ

ha J n qa qaK

qa K ha

haJ ha J

l l l

l l

l l

l b

is to be determined for each l

b



(3.2-11)

Nonlinear Optics Lab . Hanyang Univ.

Mode characteristics and Cutoff conditions

(3.2-11) is quadratic in J_l(ha)/haJ_l(ha)  Two classes in solutions can be obtained, and designated as the EH and HE modes.

(Hybrid modes) (3.2-11) 

12

2 2 2 2

2 2

0 2 1 2 2 2

2 1

2 2 2 1 2

1 2 2 2

1 1 1

2 2

) (

) (



















 



 



 



 











 



 



 

 



 



 



 

a h a

q k

n l qaK

K n

n n qaK

K n

n n ha

haJ ha J

l l l

l l

l b

By using the Bessel function relations : ( ) ₁( ) J (x), x

x l J x

J_l  _l  _l

 ( ) ₁( ) J (x)

x x l J x

J_l  _l  _l



 

^_^



 

 

 

 R

ha l qa

qaK qa K n

n n ha haJ

ha J

l l l

l

2 2 1

2 2 2 1 1

) (

) ( 2

) (

) (

 

^_

 



 

 



 



 





 R

ha l qa

qaK qa K n

n n ha

haJ ha J

l l l

l

2 2

1 2 2 2 1 1

) (

) ( 2

) (

) (

12

2 2 2 2 2 2

0 1 2 2

2 1

2 2 2

1 1 1

) (

) (

2 















 



 



 















 



 



 

 nk q a h a

l qa

qaK qa K n

n R n

l

l b

where,

: EH modes

: HE modes

: Can be solved graphically (3.2-15)

Nonlinear Optics Lab . Hanyang Univ.

Special case (l=0)

1) HE modes

) (

) ( )

( ) (

0 1 0

1

qa qaK

qa K ha

haJ ha

J  

) ( )

( , ) ( )

( _{1 1 1}

'

0 x K x J x J x

K  _ 

(3.2-15b) &

From (3.2-10),

A C  0

^(Report)

Therefore, from (3.2-6)~(3.2-9), nonvanishing components are H_r,H_z,E_ (TE modes)

) ( )

( , ) ( )

( _{1 1 1}

'

0 x K x J x J x

K  _ 

(3.2-15a) &

From (3.2-10),

B D  0

^(Report)

Therefore, from (3.2-6)~(3.2-9), nonvanishing components are E_r,E_z,H_ (TM modes) 2) EH modes

) (

) ( )

( ) (

0 2 1

1 2 2 0

1

qa K n qa

qa K n ha

haJ ha

J  

Nonlinear Optics Lab . Hanyang Univ.

Bessel functions of the 1^st kind order of l=0,1,2 Modified Bessel functions of the 2^nd kind of order l=0,1,2,3

Bessel functions

Nonlinear Optics Lab . Hanyang Univ.

Graphical Solution for the confined TE modes (l=0)

) (

) ( )

( ) (

0 1 0

1

qa qaK

qa K ha

haJ ha

J 

2 1 ) 0 (

) 0 (

0

1 

haJ J



 





 

) ln(

) (

~ 2 ) (

) (

2 2 2 2

2 2 0

1

a h V a

h V V

ha qaK

V ha K

should be real to achieve the exponential q

decay of the field in the cladding

2 2 0 2 1

2n k b

h & n₂k₀bn₁k₀

*

2 2

2 0 2 2 2 1

2 ( ) ( )

)

(qa  n n k a  ha

2 / 1 2 2 2 1

0

( )

0  ha  V  k a n  n



) (

) ( )

0 (

) 0 (

0 1 0

1

V VK

V K ha

qaK ha

K 



 

4) 1 tan(

)~ 1 (

) 1 (

0

1 



 

 ha

ha ha

haJ ha J

Roots of J₀(ha)=0

Normalized frequency (V-parameter)

Nonlinear Optics Lab . Hanyang Univ.

* If the max value of ha, V is smaller than the first root of J₀(x), 2.405 => no TE mode

* Cutoff value (a/l) for TE_0m (or TM_0m) waves :



2²



¹²

2 1

0

0 2 n n

x

a _m

m  



 



 l 

where, x_0m : m-th zero of J₀(x)

* Asymtotic formula for higher zeros :

 4) ( 1

0 ~ m x _m

J₀(x)

Intersection of two curves approaches to the root of x m

n n

a k V

ha_max   ₀ ( ₁²  ₂²)^1/2  ₀

* TM modes behaves identically except for a factor of on the right side of (3.2-17b) ₂

1 2 2

n n

Nonlinear Optics Lab . Hanyang Univ.

Special case (l=1)

<EH modes> <HE modes>

* HE mode does not have a cutoff.

* All other HE_1m, EH_1m modes have cutoff value of a/l :

* Asymptotic formula for higher zero : ^{la 1m2}



^nx₁^21mn' ₂²



¹²

 4) ( 1

1 ~ m

x _m ^{where, m'm forEH1m modes}

modes for

1

' m HE_1m

m 

Nonlinear Optics Lab . Hanyang Univ.

The cutoff value for a/l (l>1)



2²



¹²

2

2 n1 n z

a ^HE _lm

lm  



 





l 



2²



¹²

2

2 n1 n x

a ^EH _lm

lm  



 



 l 

where, z_lm is the mth root of ( ) ( 1) 1 _{2 1}( )

2 2

1 J z

n l n

z

zJ_l  _l



 



 





Nonlinear Optics Lab . Hanyang Univ.

Propagation constant, b

k

0

n _ b

: (effective) mode index

) / of value cutoff

( _{lm 0}

2 k

n

n b

: poorly confined n1

n : tightly confined

# V<2.405 

Only the fundamental HE₁₁ mode can propagate (single mode fiber)

#

#

2 2 2 1

2 a n n

V  

l



Nonlinear Optics Lab . Hanyang Univ.

3.3 Linearly Polarized Modes

The exact expression for the hybrid modes (EH_lm, HE_lm) are very complicated.

If we assume n₁-n₂<<1 (reasonable in most fibers) a good approximation of the field components and mode condition can be obtained. (D. Gloge, 1971)

 Cartesian components of the field vectors may be used.

b







 n q h

n

_{1 2}

1 ,

<Wave equation for the Cartesian field components>

1) y-polarized waves

 

 

 

 







 

z t

i e

qr BK

z t

i e

hr

E AJ _il

l

il l

y  b

b







exp )

(

exp )

0 (

x 

E r  a

a r 

(2.4-1), (3.1-2) & assume E_z<<E_y 

y y

x E E

z H i



b



 





  H_y 0 _z E_y

x H i



 

 ^{z x} y ^Ey

H i y E i





 



 



 b

 ²

Nonlinear Optics Lab . Hanyang Univ.

Expressions for the field components in core (r<a)

After tedious calculations, (3.3-6)~(3.3-17), … (x, y)

Expressions for the field components in cladding (r>a)

 



^{i t z}



e qr BK

E_y  _l( ) ^il exp





b

 0 Ex

 



^{i t z}



e qr BK

H_{x l} ^il  b



b _





 ( ) exp H_y 0



^{K qr e K qr e}

 ^

ⁱ

^

^{t z}

^{ }

B

H_z iq _l ^{i l} _l ^{i l}  b





  



 _ ( ) ^ _ ( ) ^ exp 2

) 1 ( 1

) 1 ( 1



^{K qr e K qr e}

 

ⁱ

^

^{t z}

^ 

B

E_z q _l ^{i l} _l ^{i l}  b

b



  

 _ ( ) ^ _ ( ) ^ exp

2

) 1 ( 1

) 1 ( 1

0

Ex ^E_y  ^AJ_l(^hr)^eil exp



ⁱ



^t b^z

 



^{J hr e J hr e}

 

ⁱ

^

^{t z}

^ 

A

E_z h _l ^{i l} _l ^{i l}  b

b



 

 _ ( ) ^ _ ( ) ^ exp 2

) 1 ( 1

) 1 ( 1

 



^{i t z}



e hr AJ

H_{x l} ^il  b



b _{ }



 ( ) exp H_y 0



J hr e J hr e

 

i



t z

 

A

H_z ih _l ^{i l} _l ^{i l}  b

 ^{   }



 _ ( ) ^ _ ( ) ^ exp

2

) 1 ( 1

) 1 ( 1

Continuity condition : )

( ) (

qa K

ha B AJ

l

 l 0

2 0

, bn1k n k

