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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 :
 Ez 
 k  0
H z 

2
2

2
2
2
where,     1   1   
r 2 r r r 2  2 z 2
and
k 2  2 n 2 /c 2
# Solve for Ez , H z first and then expressing Er , E , H r , H in terms of Ez , H z
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)]

 E(r, t )   E(r ,  ) 
H(r, t )  H(r ,  ) exp[ i(t  bz )]

 

Nonlinear Optics Lab.
Hanyang Univ.
From Maxwell’s curl equations :
iEr  ibH 
E 
1 
Hz
r 
1 
Ez
r 

 iH   ibEr  E z
r
 iH r  ibE 

Hz
r
1
1
iEz 
Hr 
(rH  )
r 
r r
iE  ibH r 

H
E
, H 
t
t
 iH z  
1 
1 
Er 
(rE )
r 
r r
Er 

 ib  
 

E

H
z
z
2
2 
   b  r
b r 
Hr 
 ib  
  

H

Ez 
z
2
2
   b  r
b r 
E 
 ib  
  

E

H z 
z
2
2 
   b  r
b r 
H 
 ib  
  

H

Ez 
z
 2   b 2  r
b r 
We can solve for Er , E , H r , H in terms of Ez , H z
Nonlinear Optics Lab.
Hanyang Univ.
Now, let’s determine Ez , H z
(3.1-1)  2  k 2  Ez   0
H z 
The solution takes the form :

E
 2 1  1 2
2
2  z 



(
k

b
)
 2
 H   0
2
2
 r r r r 
 z 
 Ez 
 H    (r ) exp( il ) where, l  0,1, 2, 3, ...
 z
 2 1   2
l2 
2

  k  b  2   0
2
r
r r 
r 
1) k 2 b 2 0 :
 (r )  c1 J l (hr )  c2Yl (hr )
where, h 2 k 2  b 2 ,
J l , Yl : Bessel functions of the 1st
and 2nd kind order of l
2) k 2 b 2 0 :
 (r )  c1I l (qr )  c2 Kl (qr )
where, q 2  b 2 k 2 , I l , K l : Modified Bessel functions of
the 1st and 2nd kind of order l
Nonlinear Optics Lab.
Hanyang Univ.
Asymptotic forms of Bessel functions :
For x1
For x 1, l
1 x
J l ( x)   
l!  2 
Y0 ( x) 
l
2
l  
 2

J l ( x)    cos x   
2 4
 x 

1
2 x

 ln  0.5772... 
 2

(l  1)!  2 
Yl ( x)  
 
  x
1 x
I l ( x)   
l!  2 
2
l  
 2 

Yl ( x)    sin  x 
 
2 4
 x 

1
l
l  1,2,3,...
l
 
K l ( x)    e  x
 2x 
1
 x

K 0 ( x)   ln  0.5772... 
 2

(l  1)!  2 
K l ( x)  
 
2  x
1
2
 1 
x
I l ( x )
 e
 2x 
2
l
l  1,2,3,...
Nonlinear Optics Lab.
Hanyang Univ.
3.2 The Step-Index Circular Waveguide
The field of confined modes :
1) ra (cladding region) : x1
*  : evanescent (decay) wave
 k 2  b 2 0 and b  n2 k0 n2 /c
*  : virtually zero at r b()
I l ( x)x  2 e x is not proper for the solution
1
 Ez (r, t )CKl (qr )expit l bz 
H z (r, t )  DK l (qr ) expit  l  bz 
<Index profile of a step-index circular waveguide>
where, q 2  b 2 n22 k02
Nonlinear Optics Lab.
Hanyang Univ.
r a
2) ra (core region) : x1
*  : finite at r 0
Yl ( x )  x  l is not proper for the solution
*  : propagating wave
 k 2 b 2 0 and b  n1k0 n1 /c

Ez (r, t )  AJl (hr ) expit  l  bz 
H z (r, t )  BJ l (hr ) expit  l  bz 
ra
where, h 2 n12 k02  b 2
* Necessary condition for confined modes to exist :
n1k0  b  n2 k0
Nonlinear Optics Lab.
Hanyang Univ.
Other field components
2) cladding (ra)
1) core (ra)
Er 

ib 
il

Ah
J
(
hr
)

BJ
(
hr
)
l
l
exp it l  bz 
h 2 
br

E 
 ib
h2
 il



AJ
(
hr
)

Bh
J
(
hr
)
l
l
r
 exp it  l  bz 
b


Ez  AJ l (hr ) expit  l  bz 
Er 
ib
q2


il

Cq
K
(
qr
)

DK
(
qr
)
l
l

 exp it  l  bz 
br


E 
ib
q2
 il


 r CKl (qr )  b DqK l(qr ) exp it  l  bz 


Ez  CKl (qr ) exp it  l  bz 


i1l

Bh
J
(
hr
)

AJ
(
hr
)
exp it  l  bz  H r  ib2  DqK l(qr )  i 2l CKl (qr ) exp it  l  bz 
l
l


br
q 
br




ib  il
 2

 ib  il

H  2  BJ l (hr )  1 AhJ l(hr ) exp it  l  bz  H  h 2  r DK l (qr )  b CqKl(qr ) exp it  l  bz 


h r
b

Hr 
 ib
h2
H z  BJ l (hr ) exp it  l  bz 
H z  DK l (qr ) expit  l  bz 
Nonlinear Optics Lab.
Hanyang Univ.
Boundary condition : tangential components of field are continuous at ra
E , Ez , H , H z
AJ l (ha)  CKl (qa)  0

  

  il
 il
  
A 2 J l (ha)  B 
J l(ha)  C  2 K l (qa)  D 
K l (qa)  0
h a
  hb
 q a

 qb

BJ l (ha)  DK l (qa)  0

 il

 
  il
  
A 1 J l(ha)  B  2 J l (ha)  C  2 K l(qa)  D  2 K l (qa)  0
  qb
 hb
 h a

q a

Nonlinear Optics Lab.
(3.2-10)
Hanyang Univ.
Condition for nontrivial solution to exist : (Report)
2
2
2
2


2
2
 J (ha)









K
(
qa
)
n
J
(
ha
)
n
K
(
qa
)
1
1
b
 l
 1 l
  l 2        
 l
 2 l
 qa 
k 
 haJ l (ha) qaK l (qa)  haJ l (ha) qaK l (qa) 
ha

      0 



(3.2-11)
 b is to be determined for each l
Amplitude ratios : [from (3.2-10) with determined eigenvalue b, Report]
C J l ( ha )

A K l ( qa )

K l (qa) 
B ibl  1
1  J l(ha)





A   q 2 a 2 h 2 a 2  haJ l (ha) aqK l (qa) 


1
: the relative amount of Ez and Hz in a mode
D J l (ha) B

A K l (qa) A
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) 
 2
 n  n  Kl
 n1  n22 
J l(ha)



 
 
2
2

haJ l (ha)
2
n
qaK
2
n
1
l
1




1
2
 l2  b   1

1
    
 2 2  
2
2
2



 n1  k0   q a
ha  


l
l

By using the Bessel function relations : J l ( x) J l1 ( x) J l ( x), J l  ( x)  J l 1 ( x)  J l ( x)
x
x

2
2

J l 1 (ha) n1  n2 K l (qa)  l





R
: EH modes
2
2

haJ l (ha)
2 n1 qaK l (qa)  ha 

2
1

2
2
n n
J l 1 (ha)
 
2
haJ l (ha)
 2 n1
2
1
2
2
2
 K
 l
 qaK l

2
2
(3.2-15) : Can be solved graphically


 K l (qa)  l


 

R
2
qaK
(
qa
)
l

 ha 

2
: HE modes
2
2
 2

2 2
  lb  2  1


K
(
qa
)
n

n
1  
l
1
2



where, R  
 




 2 n12   qaK l (qa)   n1k0   q 2 a 2 h 2 a 2  




1
2
Nonlinear Optics Lab.
Hanyang Univ.
Special case (l=0)
1) HE modes
'
(3.2-15b) & K 0 ( x) K1 ( x), J 1 ( x) J1 ( x)
J1 (ha)
K (qa)
 1
haJ 0 (ha)
qaK 0 (qa)
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)
2) EH modes
'
(3.2-15a) & K 0 ( x) K1 ( x), J 1 ( x) J1 ( x)
J1 (ha)
n22 K1 (qa)

haJ 0 (ha)
qan12 K 0 (qa)
From (3.2-10),
BD0 (Report)
Therefore, from (3.2-6)~(3.2-9), nonvanishing components are Er , E z , H  (TM modes)
Nonlinear Optics Lab.
Hanyang Univ.
Graphical Solution for the confined TE modes (l=0)
J1 (ha)
K (qa)
 1
haJ 0 (ha)
qaK 0 (qa)
Roots of J0(ha)=0
* q should be real to achieve the exponential
decay of the field in the cladding
h 2 n12 k02  b 2 & n2 k0 b n1k0
(qa) 2 (n12 n22 )k02 a 2 (ha) 2
 0haV k0 a(n12 n22 )1/2
J1 (0) 1

haJ 0 (0) 2


K1 (ha 0)
K (V )
 1
qaK 0 (ha 0) VK 0 (V )
J1 (ha1)
1

~  tan( ha )
haJ 0 (ha1) ha
4
K1 (haV )
2
~ 2 2 2
 
qaK 0 (haV ) (V h a )ln( V 2 h 2 a 2 )
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 TE0m (or TM0m) waves :
x0 m
a
  
12
 l  0 m 2 n12  n22 
where, x0 m : mth zero of J0(x)
* Asymtotic formula for higher zeros :
1
x0 m ~(m )
4
Nonlinear Optics Lab.
Hanyang Univ.
Special case (l=1)
<EH modes>
<HE modes>
* HE mode does not have a cutoff.
x1m '
a
* All other HE1m, EH1m modes have cutoff value of a/l :   
 l 1m 2 n12 n22
* Asymptotic formula for higher zero :

1
x1m ~(m )
4

12
where, m'm for EH1m modes
m'm1 for HE1m modes
Nonlinear Optics Lab.
Hanyang Univ.
The cutoff value for a/l (l>1)
EH
xlm
a
  
 l lm 2 n12  n22


12
HE
zlm
a
  
 l lm 2 n12  n22


12
where, zlm is the mth root of
 n12 
zJ l ( z )  (l  1)1  2  J l 1 ( z )
 n2 
Nonlinear Optics Lab.
Hanyang Univ.
Propagation constant, b
n
b
k0
: (effective) mode index
# nn2 (cutoff value of b lm /k0 )
: poorly confined
# nn1 : tightly confined
# V<2.405 
Only the fundamental HE11 mode
can propagate (single mode fiber)
Nonlinear Optics Lab.
Hanyang Univ.
3.3 Linearly Polarized Modes
The exact expression for the hybrid modes (EHlm, HElm) 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.
n1 n2 1  q, h b
<Wave equation for the Cartesian field components>
1) y-polarized waves
Ex  0
 AJ l (hr )eil exp it  bz  r  a
Ey  
il
BK l (qr )e exp it  bz  r  a
(2.4-1), (3.1-2) & assume Ez<<Ey 
Hx 
i 
b
Ey 
Ey
 z

Hy  0
Hz 

Ey
 x
i
Ez 
Nonlinear Optics Lab.
i 
 ib 
Hx  2
Ey
 y
  y
Hanyang Univ.
After tedious calculations, (3.3-6)~(3.3-17), … (x, y), b n1k0 n2 k0
Expressions for the field components in core (r<a)
Ex  0
E y  AJl (hr )eil expit  bz 
hA
Ez 
J l 1 (hr )ei (l 1)  J l 1 (hr )ei (l 1) exp it  bz 
b2

Hx  
Hz  

b
AJ l (hr )eil exp i t  bz 


Hy  0

ih A
J l 1 (hr )ei (l 1)  J l 1 (hr )ei (l 1) exp it  bz 
 2
Continuity condition :
AJ l ( ha )
B
K l ( qa )
Expressions for the field components in cladding (r>a)
Ex  0
E y  BK l (qr )eil expit  bz 


q B
K l 1 (qr )ei (l 1)  K l 1 (qr )ei (l 1) exp i t  bz 
b 2
b
Hx  
BK l (qr )eil exp it  bz 
Hy  0

Ez 
Hz  


iq B
K l 1 (qr )ei (l 1)  K l 1 (qr )ei (l 1) exp it  bz 
 2
Nonlinear Optics Lab.
Hanyang Univ.
2) x-polarized waves (similar procedure to the case y-polarized waves)
In core (r<a)
Ey  0
E x  AJ l (hr )eil exp it  bz 
h A

Ez  i
J l 1 (hr )ei (l 1)  J l 1 (hr )ei (l 1) exp it  bz 
b 2
b
AJ l (hr )eil exp it  bz 
Hx  0 H y 

Hz 


h A
J l 1 (hr )ei (l 1)  J l 1 (hr )ei (l 1) exp it  bz 
 2
In cladding (r>a)
E x  BK l (qr )eil exp i t  bz 

Ey  0

q B
K l 1 (qr )ei (l 1)  K l 1 (qr )ei (l 1) exp it  bz 
b 2
b
Hy 
BK l (qr )eil exp it  bz 
Hx  0

q B
Hz 
K l 1 (qr )ei (l 1)  K l 1 (qr )ei (l 1) exp it  bz 
 2
Ez  i

Continuity condition
 Mode condition :
J (ha )
K (qa )
h l 1
 q l 1
J l (ha )
K l (qa)
simpler than (3.2-11)
J (ha )
K (qa)
h l 1
 q l 1
J l (ha )
K l (qa)
and/or
: This results also can be obtained
from the y-polarized wave solution.
 x- and y-modes are degenerated.

Nonlinear Optics Lab.
Hanyang Univ.
Graphical Solution for the confined modes (l=0)
h
J l 1 (ha )
K (qa )
 q l 1
J l (ha )
K l (qa)
X ha, Y qa, (qa) 2 (n12 n22 )k02 (ha) 2
b lm k n h
2 2
0 1
2
lm
<Possible distribution of LP11>
: LPlm modes
Nonlinear Optics Lab.
Hanyang Univ.
Mode cutoff value of a/l
: q 0
(3.3-27) 
J l 1 (V )  0

where, V  k0 a n12  n22

12
 2
n
l
a
2
1
 n22

12
Ex) l=0, J 1 (V ) J1 (V )0 at V 0 (LP01 ) : no cutoff
J1 (V )0 at V 3.832 (LP02 )
Asymptotic formula for higher modes :
 3
V ( LPlm )  m   l  
 2 2
Ref : Table 3-1Cutoff value of V for some low-order LP
Nonlinear Optics Lab.
Hanyang Univ.
Power flow and power density
The time-averaged Poynting vector along the waveguide :


1
Re E x H *y  E y H x*
2
(3.3-18), (2.3-19) 
2 2
 b
A
J l (hr )
 2
Sz  
b
2

B K l2 (hr )
 2
Sz 
r a
b
2
2
2


a
A
J
l ( ha ) J l 1 ( ha ) J l 1 ( ha ) 
0 0
2
2 
b
2

a 2 B  K l2 (qa) K l1 (qa) K l1 (qa)
   S z rdrd
0
a
2
Pcore  
Pclad
ra
2
a
S z rdrd 
2
h
b
2

a 2 A [ J l2 (ha)  J l 1 (ha) J l 1 (ha)]
2
q
Nonlinear Optics Lab.
Hanyang Univ.
The ratio of cladding power to the total power, G2 :
Pclad
Pclad
(qa) 2 J l2 (ha) 
1
2
G2 

 2 ha  

p pcore  pclad V 
J l1 (ha) J l1 (ha) 
Nonlinear Optics Lab.
Hanyang Univ.
3.4 Optical Pulse Propagation and Pulse Spreading in Fibers
One bit of information = digital pulse
Limit ability to reduce the pulse width : Group velocity dispersion
Group velocity dispersion
Considering a Single mode / Gaussian pulse, temporal envelope at z=0 (input plane of fiber) :
E ( x, y,0, t )  u0 ( x, y ) Re[exp( t 2  i0t )]
1/ 2
where, u0 ( x, y ) : transverse modal profile of the mode and  0 
Fourier transformation :
~
E ( x, y,0,t )Re[ u0 ( x, y )exp( i 0t )  f ()exp( it )d]
 exp(   2 2 ) 
~
where, f ()  FT [exp( t 2 )]  

4


Nonlinear Optics Lab.
12
Hanyang Univ.
Propagation delay factor for wave with the frequency of  0  : exp[ ib (0 ) z ]
Let’s take complex expression and omit the u0 ( x, y )
(are not invloved in the analysis and can be restored when needed)
~
E ( z, t )   f () exp{i[(0  )t  b (0  ) z ]}d
db
1 d 2b
Taylor series expansion : b (0  )  b (0 ) 

d 0
2 d 2

~
z 1 d  1

E ( z, t )  exp[ i(0t  b 0 z )] d f () exp i t 


vg 2 d  vg





 2  ...
0
 2 
 z  


 


exp[ i(0t b 0 z )]E ( z,t )
where, b 0  b ( 0 ),
db
d

0
1
1

v g group velocity

~
z  1 d  1
 
E ( z, t )   d f () exp i t   
 v  2 d  v

g 

 g
 

 
z  

 
 

: Field envelope
Nonlinear Optics Lab.


~
z 
 

d

f
(

)
exp
i

t


a

z





 v 


g



 


(3.4-5)
Hanyang Univ.
The pulse spreading is caused by the group velocity dispersion characterized by the parameter,
1 d 2b
1 d  1
a

2
2 d   2 d  vg
0

dvg
 1
2

2
v
d
g

(3.4-3)(3.4-5) :

z
  2 1
 
E ( z, t ) 
exp



iaz

i
t





 v
4 
4



g




1

 
 
d
 
 

 (t  z v g ) 2   4az (t  z v g ) 2 
1
exp  i


exp  
2
2
2
2
2
 1  16a z    1  16a z 
1i 4az

 

Nonlinear Optics Lab.
Hanyang Univ.
# Pulse duration t at z (FWHM)
t ( L)  t 0
 8aL ln 2 

1  
2
 t0 
2
# |aL|>>t0 (large distance) : t ( L) ~
(8 ln 2)aL
t0
initial pulse width
If we use the definition of factor a, t ( L) 
4 ln 2 dvg L
vg2 d t 0
Practical Expression :
t ( L)  t 0
 2 ln 2 DLl2 

1  
2
 c t 0 
2
4c
dT /dl 2c d 2 b 
where, D
 2  2   2 a
l
L
l  d 
T : pulse transmission time through length L of the fiber
Nonlinear Optics Lab.
Hanyang Univ.
Group velocity dispersion
1) Material dispersion : n() depends on 
 Waveguide dispersion : blm depends on  (& geometry of fiber)
b lm  nlm k0  nlm (n1 , n2 ,  )
d  dblm 

i) (vg )lm 

dblm  d 

c
1
: modal dispersion
ii) Single mode fiber,
1 db   n n1 n n2 n  n


 


vg d c  n1  n2    c
material dispersion
(3.4-18)
waveguide dispersion
Nonlinear Optics Lab.
Hanyang Univ.
From the uniform dielectric perturbation theory,
 
b 2    G1n12  G2n22
c
2


where, G1, G2 : Fractions of power flowing in the core and cladding

n
n 
G1  1 
n1
n
n
n 
 G2  2 
n2
n
(3.4-18) 
1 db    n1  n1 
 n2  n2   n   n

 G1 

G
 2  

 
vg d c   n   
 n       w  c
Nonlinear Optics Lab.
Hanyang Univ.
In weakly guiding fiber : n1~n2
n1 n2  n 



     m

1 db   n   n   n
l  n   n   n
  
 
       
c  l  m  l  w  c
vg d c    m    w  c
Group velocity dispersion :
l   2 n    2 n  
D    2    2  
c  l  m  l  w 
  2n 
ex) GeO2-doped silica :  2  0 at l 1.3m
 l  m
  2n 
#  2  depends on core diameter, n1, n2  control the waveguide shape
 l  w
Nonlinear Optics Lab.
Hanyang Univ.
Group velocity dispersion & dispersion-flattened and dispersion-shifted fibers
Nonlinear Optics Lab.
Hanyang Univ.
Frequency chirping
: modification of the optical frequency due to the dispersion
(3.4-6) 
E ( z, t ) 


(t  z v g ) 2 
4az (t  z vg ) 2 
1
exp 
exp i0t  b 0 z   i
,
2
2 2 
2 2 
1   16a z 
1  i 4az
 1   16a z  

1 d 2b
1 dv g
where, a 


2 d 2
2v g2 d
Total optical phase :
 ( z , t )  0 t  b 0 z 
4az (t  z vg ) 2
1   16a 2 z 2
2
dv g
d
Optical frequency :

t
 ( z,t )  ( z,t ) 0 8
1 
az
2
16a z
2 2

(t  z v g )
Nonlinear Optics Lab.
Hanyang Univ.
0
3.5 Compensation for Group Velocity Dispersion
(3.4-5) 
~

z 
2

E ( z, t )    f () exp  iaz  i
 exp( it )d


v

g 



z 
 
 ~
FT E z, t 
 f () exp  iaz 2



v

g





Fiber transfer function
By convolution theorem, (1.6-2),
1
E ( z, t ) 
i 4z
t (t ) 



 i
t  t 2 dt 
f (t ) exp 
 4az

1
 i 2
exp  
t  : envelop impulse response
4
az
i 4z

 of a fiber of length z
Nonlinear Optics Lab.
Hanyang Univ.
Compensation for pulse broadening
1) By optical fiber with opposite dispersion
~
i) f1 ()
~
~
ii ) f 2 () f1 ()exp( ia1 L1 2 )
~
~
iii ) f3 () f 2 ()exp( ia2 L22 )
~
 f1 ()exp i ( a1 L1 a2 L2 ) 2


(a1L=-a2L)
Nonlinear Optics Lab.
Hanyang Univ.
2) By phase conjugation
~
Input to conjugator  f (t )exp( i 0t ) f  exp i 0  t d
~
Output from conjugator  f *  exp i 0  t d
(a1L=a2L)
~
i) f ()
~
~
ii ) f 2 () f1 ()exp( ia1L12 )
~
~*
~*
iii ) f 3 () f 2 () f1 ()exp( ia1L12 )
~
~
iv ) f 4 () f 3 ()exp( ia2 L22 )
~
 f1* () exp i ( a1 L1  a2 L2 ) 2
Nonlinear Optics Lab.

Hanyang Univ.

Where are (b) and (c) ??
 Refer to the text
<Experimental setup>
<Eye diagram>
Nonlinear Optics Lab.
Hanyang Univ.
3.7 Attenuation in Silica Fibers
Residual OH contamination of the glass
1.55 m is favored for long-distance
optical communication
Recently, 400 Mb/s, 100 km @ 1.55 m
Nonlinear Optics Lab.
Hanyang Univ.