Transcript Document

OPTICS BY THE NUMBERS
L’Ottica Attraverso i Numeri
Michael Scalora
U.S. Army Research, Development, and Engineering Center
Redstone Arsenal, Alabama, 35898-5000
&
Universita' di Roma "La Sapienza"
Dipartimento di Energetica
Rome, April-May 2004
BPM:Propagation in Planar Waveguides
Retarded Coordinate trasformation: time
dependence, Raman scattering, self-phase
modulation in PCFs
Study the transmissive
properties of guided modes.
1 .5
5 m
air core
Index of R efraction
1 .4
14 m
1 .3
Propagation
into the page
1 .2
1 .1
1 .0
-1 5
-1 0
-5
0
5
10
15
T ra n sv erse C o o rd in a te (  m )
fig.(4)
 E
2
n  E
c
2
2
2
t
2
4  Pnl
2

c
2
t
2
 E 
2
z
2
 2 ik
E
z
n (x)  E
2

c
2
2
t
2
i ( kz   t )
 c .c .
E
2
E
2
E ( z , x, t )  E ( z , x, t )e
 E
Pnl  
(3)
2

 2 2 2

  k  2 n (x)  E
t 
c

2 i n ( x )  E
c
2
2
2

4 ( 3 ) 

2
 2   2  2 i    E E
c
t
 t

 E
2

2

E
z
2
 2 ik
E
z
n (x)  E
2

c
2
t
2
 2 2 2

4 ( 3 )
  k  2 n (x)  E  2 
c
c


2
2 i n ( x )  E
2

c
t
2
 2

2
  E
 2  2 i
t
 t

2
E
Assuming steady state conditions…
E


i
F
 n ( x )  n0
2

2

E  i
2
n0

0
 in
Ei
4
n0
(3)
0
 in
2
E E
E

F 

i
 n ( x )  n0
2

F
2

4 n 0  0
 in
  z / 0
E  i
2

n0
0
 in
Ei
4 
n0
(3)
0
 in
2
E
E
F resnel N um ber
k 

c
F  
n0
F  s m all
Wave front does not distort:
Plane Wave propagation
Diffraction is very important
E


i
 n ( x )  n0
2

F
2

E  i

2
0
 in
n0
This equation is of the form:  E

Where:
H 
i
 n ( x )  n0
2

2
F

 i
2
n0

0
 in
i
Ei
4 
(3)
0
 in
n0
2
E
E
 HE
4 
(3)
0
2
E
 in
n0
 D V

E ( , x )  e
 H (  ', x )  '
0
E (0, x )  e
Using the split-step
BPM algorithm
e
H ( 0 , x ) 
E (0, x )
V ( 0 , x )  / 2
e
D 
e
V ( 0 , x )  / 2
E (0, x )
Example: Incident angle is 5 degrees
1 .5 0
a ir g u id e
~ 5m
0 .8
1 .0 0
0 .4
b=1m
a=1.4m
0 .7 5
-1 0
-8
-6
-4
-2
0
2
4
6
8
0
10
T ra n s ve rs e C o o rd in a te (  m )
Assume 3=0
In te n s ity
g la s s ; n = 1 .4 2
g la s s ; n = 1 .4 2
1 .2 5
1 .2
a ir
g la s s ; n = 1 .4 2
g la s s ; n = 1 .4 2
In d e x o f R e fra c tio n
a ir
The cross section along x renders the
problem one-dimensional in nature
x
1 .5
Transverse Index Profile
1 .4
1 .3
1 .2
1 .1
1 .0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
T r a n s v e r s e P o s itio n (m ic r o n s )
4
5
6
7
8
Transmissive properties in the linear (low intensity) regime
For two different fibers. We set 3=0
1 .0
N orm alized T ransm ittance
1 4 -m ic r o n c o r e
5 -m ic r o n c o r e
0 .8
0 .6
0 .4
0 .2
0
0 .5
0 .6
0 .7
0 .8
0 .9
 ( m )
1 .0
1 .1
1 .2
F ie ld b o u n c in g b a c k a n d fo rth fro m s tru c tu re 's w a lls
In p o u t F ie ld P ro file
1 .5
1 .2 5
1 .0
1 .0 0
0 .5
In te n s ity
In d e x o f R e fra c tio n
1 .5 0
0 .7 5
-1 0
-8
-6
-4
-2
0
2
4
T ra n s ve rs e C o o rd in a te (  m )
6
8
0
10
1 .0
0 .8
00. . 7
5
0 .4
1 .01
1.
Field tuning corresponds to
High transmission state.
8
0 .2
0
0 .5
1
0 .6
0 .7
0 .8
0 .9
1 .0
1 .1
0 .7
.00. 8
1 .2
1 .0
1 .1
 ( m )
300
1
1 ..1
0 8
00. . 7
0 .7
1 .00
.8
1 .1
0 . 51 .0
1 .1
0 .8
200
1 .0
Direction of
propagation
0 .8
z( m )
1 .0
0 .8
0 .6
0.
400
1 4 -m ic r o n c o r e
5 -m ic r o n c o r e
N orm alized T ransm ittance
500
0.
0.7
.8
110.1
.0
5
100
10. .
0
1 .18 . 7
0
1 .1
0
-12
-10
-8
-6
-4
-2
-0
x ( m )
2
4
6
8
10
12
Same as previous figure.
1 .0
N orm alized T ransm ittance
1 4 -m ic r o n c o r e
5 -m ic r o n c o r e
500
0 .8
0 .6
0 .4
0 .2
0
0 .5
400
0 .6
0 .7
0 .8
0 .9
1 .0
1 .1
1 .2
 ( m )
z ( m )
300
0 .1
0 .1
200
0 .1
0 .1
0.
1
0 .2
0 .2
0.
2
100
0 .1
0
.1
0 .1
0 .2
0 .1
0 .1
0 .2
2
0. 1
0.
0
-12
-10
-8
-6
-4
-2
-0
x ( m )
2
4
6
8
10
12
Same as previous figure.
500
For the example
discussed:
00. . 7
5
0.
1 .01
1.
1 .0
0 .8
300
1
0 .7
.00. 8
1 .0
1 .1
N   200000
1
1 ..1
0 8
00. . 7
N x  4096
0 .7
1 .00
.8
1 .1
0 . 51 .0
1 .1
0 .8
200
1 .0
0.7
.8
110.1
.0
~ 8 minutes on this laptop
3.2GHz, 1Gbts RAM
0.
5
100
5-mm guide
0 .8
z( m )
 x   x /  0  0.0125
   z /  0  0.025
8
400
10. .
0
1 .18 . 7
0
1 .1
0
-12
-10
-8
-6
-4
-2
-0
x ( m )
2
4
6
8
10
12
E


i
F

2

E  i
n
2
( x)  n
n0
2
0

0
 in
E i
4
(3)
n0
0
 in
2
E E
If (3) is non-zero, the refractive index
is a function of the local intensity.
Solutions are obtained using the same algorithm
but with a nonlinear potential.
Optical Switch
1 .5
5 m
In d ex of R efraction
1 .4
air core
1 .3
14 m
1 .2
1 .1
1 .0
-1 5
-1 0
-5
0
5
10
15
T ra n sv erse C o o rd in a te (  m )
fig.(4)
(3 )
n o n -z ero 
lin ea r tra n sm itta n ce
1 .0
norm alized transm ittance
0 .8
0 .6
0 .4
0 .2
0
0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 1 .1 1 .2 1 .3 1 .4 1 .5 1 .6 1 .7 1 .8 1 .9 2 .0
sc a le d fr e q u e n c y (1 /  w h e r e  is in m ic r o n s)
The band shifts because the location and the width of each gap
depends on the exact values of n2 and n1, and on their local
difference.
N orm alized T ran sm ittan ce
1 .0
0 .8
0 .6
0 .4
0 .2
0
0 .6 5
0 .7 0
0 .7 5
0 .8 0
0 .8 5
0 .9 0
 ( m )
0 .9 5
1 .0 0
1 .0 5
1 .1 0
fig.(5a)
Optical Switch
on
1 .0
T ran sm ittan ce
0 .8
0 .6
0 .4
N on lin ear T ran sm ittan ce
0 .2
L in ea r T ra n sm itta n ce
off
0
0 .7 1 5
0 .7 2 0
0 .7 2 5
 ( m )
0 .7 3 0
0 .7 3 5
0 .7 4 0
fig.(5b)
1 .0
on
C ore E n ergy
0 .8
0 .6
0 .4
0 .2
off
0
0
1000
2000
3000
4000
5000
L o n g itu d in a l P o sitio n (  m )
fig.(6)
 E
E
2

E
2

z
2
 2 ik
z
n (x)  E
2

c
2
t
2
 2 2 2

4 ( 3 )
  k  2 n (x)  E  2 
c
c


 z


z

 
2
z
2
  t  z/v

z 

2

2


 
z 
1

2
v 
2

2



2 
2
2 i n ( x )  E
2

c
 2

2
  E
 2  2 i
t
 t

2
E
Retarded coordinate
Transformation
v  c / n0

t
2
1 

v 
t

 
t 


v  
t

2
2

 
t 

2

2



N.B.:An implicit and important assumption we have made
is that one can go to a retarded coordinate provided the
grating is shallow so that a group velocity can be defined
unumbiuosly and uniquely.
 E
2

2

E
z
2
 2 ik
E
z
n ( x)  E
2

c
2
2
 2 2 2

4 ( 3 )
  k  2 n ( x)  E  2 
c
c


 z
t
2
2 i n ( x )  E
2

c
2
t
 2

2
  E
 2  2 i
t
 t

  t  z/v
2
E
v  c / n0
In other words, the effect of the grating on the group velocity is
scaled away into an effective group velocity v. It is obvious that
care should be excercised at every step when reaching
conclusions, in order to properly account for both material index
and modal dispersion, if the index discontinuity is large.
2
2

n0  

1

2   
  
2
 E  
 2


 E  2i n0 
E
2
2
v 
v   
c
c  
 
 


n
2

2
c 
2
4
c
2
2
E
2 i n
c

 2 2 2 
E k  2 n E

c



 

2
  E
 2  2 i

 

2
(3)
2
2
2
E
2


4 ( 3 )
2
2
2 
  E  2i n0
E k  2 n E  2 
c

c
c




Symplifying and
Dropping all
Higher order
Derivatives…


2
  E
  2 i



2
E


 n ( x )  n0
2
i  
2

F x
 i
2
2

 i
n0
i
4 

0  
(3)
 in
n0

 

H 
i
F

  
2


*
 2


 n ( x )  n0
2

2

 i
2
 
*

0
 in
n0
4 
2
i
(3)
n0
2
i
(3)
0
 in
n0
2
2
4 
2


2




*
  *
 
  2
 


 
 
*
4 
n0
(3)
0
 in
2
E
*
0   *
 
 
2

 in   
 
Now we look at the linear regime, by injecting a beam inside the
guide from the left and then from the right.
O n-A xis Inte nsity a s B e a m P ro pa g a te s D o w n the G uide . B e a m is G uide d.
O utput F ie ld P ro file in the c a s e L ight is G uide d.
O n-A xis Inte nsity a s B e a m P ro pa g a te s D o w n the G uide . B e a m
is Tune d to a M inim um o f Tra nsm issio n, a nd is N o t G uide d, a nd e ne rg y
Q uic kly D issipa te s A w a y.
O utput F ie ld P ro file in the c a s e L ight is No t G uide d.
1 .0
0 .8
Input Spectrum
ON-AXIS
0 .6
 
2 n 2 I m ax  in L
c
0 .4
0 .2
0
0 .8 0
I  1013 W/cm2
n2  510-19 cm2/W
L  8 cm
  100 fs
0 .8 2
0 .8 4
0 .8 6
Output
Spectrum
0 .8 8
0 .9 0
0 .9 2
0 .9 4
0 .9 6
0 .9 8
1 .0 0
/0
Propagating from left to right the pulse is tuned on the red curve, igniting self-phase
modulation, and the spectral shifts indicated on the graph. A good portion of the input
energy is transmitted. Spectra are to scale.
Fig. 4
L in e a r tr a n sm itta n c e fo r 2 slig h tly d iffe r e n t g u id e s
1 .0
T ransm ittance
0 .8
0 .6
0 .4
0 .2
O u tp u t sp e c tr u m
In p u t sp e c tr u m
0
0 .8 5
0 .8 6
0 .8 7
0 .8 8
0 .8 9
0 .9 0
0 .9 1
0 .9 2
0 .9 3
0 .9 4
0 .9 5
sc a le d fr e q u e n c y (  = 1 /  w h e r e  is in  m )
Propagation from right to left does not induce nonlinearities because the light quickly
dissipates. The pulse is tuned with respect to the blue curve. Spectra are to scale.
Fig. 4
Initial pulse profile
Final profile
Spectrum of the pulse as it propagates. Note splitting.
1000
Initial profile
p o w er sp ectru m
800
600
400
200
0
0 .6 5
0 .7 0
0 .7 5
sca led freq u en cy (1 /  )
0 .8 0
Self-phase modulation
A process whereby new frequencies (or wavelengths)
are generated such that:
2 n 2 I m ax  in L
 
Example: input 100fs pulse at 800nm
c
is broadened by ~30nm
Intensity, arb. units
1 ,0
0 ,5
0 ,0
400
500
600
700
, nm
800
900
I m ax
I

t
Stimulated Raman Scattering
p
 anti  stokes
 stokes
p
E p

E A

E S

i  Ep
2

F x
2
i  EA
  Q E Ae
*
i
2

F x
2
i  ES
  QEPe
2

F x
2
 Q EP
*
 i
 QEs
Q

  Q  E S E P   Q E A E P e
*
*
 i
The simplest case
 stokes

E p

E S

Q

i  Ep
2

F x
2
i  ES
 QEs
2

F x
2
 Q EP
*
  Q  E S E P
*
Raman Soliton: A sudden
relative phase shift between
the pump and the Stokes at the
input field generates a
“phase wave”,
or soliton, a temporary repletion
of the pump at the expense of
the Stokes intensity
The simplest case
 stokes

E p

E S

Q

i  Ep
2

F x
2
i  ES
 QEs
2

F x
2
 Q EP
*
  Q  E S E P
*
0

 Es
E s (0,  )  
0

E

s

   0 

   0 
The Input Stokes field undergoes
a -phase shift
The gain changes sign temporarily,
For times of order 1/;
The soliton is the phase wave
In ten sity a t cell o u tp u t
250 0
IN T E N S IT Y
200 0
The Pump signal is
temporarily repleted
150 0
The Stokes minimum
is referred to as
a Dark Soliton
100 0
P U M P IN T E N S IT Y
S T O K E S IN T E N S IT Y
50 0
0
0
0 .0 2
0 .0 4
T IM E
0.0 6
0.08
PUMP FIELD
z,0,0
z,=L,0
z=0
PUMP FIELD
STOKES FIELD
T R AN S VE R S E IN T E N S IT Y P R O F IL E AT C E L L O U T P U T
PUMP FIELD
T R AN S VE R S E IN T E N S IT Y P R O F IL E AT C E L L O U T P U T
STOKES FIELD
O N -A X IS IN T E N S IT Y A T C E L L O U T P U T
3000
IN T E N SIT Y
F=
F=20
2000
1000
0
0
0 .0 2
0 .0 4
0 .0 6
0 .0 8
T IM E
The onset of diffraction causes the soliton to decay…
…almost as expected. Except that…
O N -A X IS IN T E N S IT Y A T C E L L O U T P U T
ST O K E S IN T E N SIT Y
5000
F=
F=20
4000
3000
2000
1000
0
0
0 .0 2
0 .0 4
0 .0 6
0 .0 8
T IM E
… the Stokes field undergoes significant replenishement on its axis,
as a result of nonlinear self focusing
O N -AX IS IN T E N S IT Y P R O F IL E
PUMP FIELD
O N -AX IS IN T E N S IT Y P R O F IL E
STOKES FIELD
T R AN S VE R S E IN T E N S IT Y P R O F IL E AT C E L L O U T P U T
PUMP FIELD
T R AN S VE R S E IN T E N S IT Y P R O F IL E AT C E L L O U T P U T
STOKES FIELD
Poisson Spot like effect
Examples: single slit
0 .8
intensity
0 .6
0 .4
0 .2
0
-4 0
-3 0
-2 0
-1 0
0
10
tr a n sv e r s e c o o r d in a te
20
30
40
Direction of
Propagation