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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 Refraction
1.4
14 m
1.3
Propagation
into the page
1.2
1.1
1.0
-15
-10
-5
0
5
10
15
Transverse Coordinate (m)
fig.(4)
n  E 4  Pnl
 E 2 2  2
c t
c t 2
2
2
2
2
Pnl  
E ( z, x, t )  E ( z, x, t )e
i ( kz t )
(3)
2
E E
 c.c.
2
2
2
2
2



E

E
n
(
x
)

E
2
i

n
(
x
)

E

2
2
2
  E  2  2ik  2

  k  2 n ( x)  E
2
2
z
z
c t
c
t 
c

2
2

4 (3) 

2
 2   2  2i    E E
c
t
 t

2
2
2
2

E

E
n
(
x
)

E
2
i

n
( x) E
2
  E  2  2ik
 2

2
z
z
c t
c2
t
2
 2 2 2 
4 (3)   2

2
  k  2 n ( x)  E  2   2  2i    E E
c
c
t


 t

Assuming steady state conditions…
n ( x)  n

E i 2
   E  i
 F
n0
2
2
0

(3)
0 2
4

0
E i
E E
in
n0 in
2
2
n
(
x
)

n

E i 2
4 (3) 0 2
0  0
   E  i
E i
E E
 F
n0
in
n0 in
4 n0 0
F
in
  z / 0
Fresnel Number

k  n0
c
F 
F  small
Wave front does not distort:
Plane Wave propagation
Diffraction is very important
2
2
(3)
n
(
x
)

n


E i 2
4 0 2
0 0
   E  i
E i
E E
 F
n0
in
n0 in
This equation is of the form: E
 HE

Where:
2
2
n
(
x
)

n
i 2
4 (3) 0 2
0 0
H     i
i
E  D V
F
n0
in
n0 in



E ( , x)  e
 H ( ', x ) '
0
E (0, x)  e
Using the split-step
BPM algorithm
H (0, x )
E (0, x)
 eV (0, x ) / 2e D eV (0, x ) / 2 E (0, x)
Example: Incident angle is 5 degrees
1.50
air guide
~ 5m
0.8
1.00
0.4
b=1m
a=1.4m
0.75
-10
Intensity
glass; n=1.42
glass; n=1.42
1.25
1.2
air
glass; n=1.42
glass; n=1.42
Index of Refraction
air
-8
-6
-4
-2
0
2
4
6
8
0
10
Transverse Coordinate (m)
Assume 3=0
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
Transverse Position (microns)
4
5
6
7
8
Transmissive properties in the linear (low intensity) regime
For two different fibers. We set 3=0
1.0
Normalized Transmittance
14-micron core
5-micron core
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
Field bouncing back and forth from structure's walls
Inpout Field Profile
1.5
1.25
1.0
1.00
0.5
Intensity
Index of Refraction
1.50
0.75
-10
-8
-6
-4
-2
0
2
4
Transverse Coordinate (m)
6
8
0
10
1.0
0.8
00. .7
5
0.6
0.4
1.0.1
1
Field tuning corresponds to
High transmission state.
0.7
1.00.8
0.2
0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
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
Direction of
propagation
1.0
0.8
z(m)
1.0
0.8
8
0.
400
14-micron core
5-micron core
Normalized Transmittance
500
0.5
00.8
.7
1.1
1.0
100
10.0.8
1.10.7
1.1
0
-12 -10
-8
-6
-4
-2
-0
x (m)
2
4
6
8
10
12
Same as previous figure.
1.0
Normalized Transmittance
14-micron core
5-micron core
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
1
0.
0.2
0.
2
0.2
0.1
0.1
1
0.
0
-12
-10
-8
-6
-4
0.1
-2
-0
x ( m)
0.2
2
0.1
100
4
0.2
6
2
0. .1
0
8
10
12
Same as previous figure.
1.0
0.8
1.0.1
1
 x   x / 0  0.0125
   z / 0  0.025
0.7
1.00.8
1.0
1.1
N  200000
1.1
1.0
8
00..7
N x  4096
1.00
.8
1.1
0.51.0
1.1
0.8
200
5-mm guide
1.0
0.8
z(m)
300
00. .7
5
8
0.
400
For the example
discussed:
0.7
500
00.8
.7
1.1
1.0
0.5
~ 8 minutes on this laptop
100 3.2GHz, 1Gbts RAM
10.0.8
1.10.7
1.1
0
-12 -10
-8
-6
-4
-2
-0
x (m)
2
4
6
8
10
12
n ( x)  n

E i 2
   E  i
 F
n0
2
2
0

(3)
0 2
4

0
E i
E E
in
n0 in
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
Index of Refraction
1.4
air core
1.3
14 m
1.2
1.1
1.0
-15
-10
-5
0
5
10
15
Transverse Coordinate (m)
fig.(4)
(3)
1.0
non-zero 
linear transmittance
normalized transmittance
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
scaled frequency (1/ where  is in microns)
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.
Normalized Transmittance
1.0
0.8
0.6
0.4
0.2
0
0.65
0.70
0.75
0.80
0.85
0.90
 (m)
0.95
1.00
1.05
1.10
fig.(5a)
Optical Switch
1.0
on
Transmittance
0.8
0.6
0.4
Nonlinear Transmittance
0.2
Linear Transmittance
off
0
0.715
0.720
0.725
(m)
0.730
0.735
0.740
fig.(5b)
1.0
on
Core Energy
0.8
0.6
0.4
0.2
off
0
0
1000
2000
3000
4000
5000
Longitudinal Position (m)
fig.(6)
2
2
2
2

E

E
n
(
x
)

E
2
i

n
( x) E
2
  E  2  2ik
 2

2
z
z
c t
c2
t
2
 2 2 2 
4 (3)   2

2
  k  2 n ( x)  E  2   2  2i    E E
c
c
t


 t

z
 t  z/v
v  c / n0
    
 1 




z z  z   v 
2
2
1 2 2  
 2 2 2
2
z

v 
v  
Retarded coordinate
Transformation
    




t t  t  
2
2
 2
2
t

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.
2
2
2
2

E

E
n
(
x
)

E
2
i

n
( x) E
2
  E  2  2ik
 2

2
z
z
c t
c2
t
2
 2 2 2 
4 (3)   2

2
  k  2 n ( x)  E  2   2  2i    E E
c
c
t


 t

z
 t  z/v
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


1

2   
   n0  
2
 E   2  2 2 
 E  2i n0  
E
v   
c   c  
  v 
 2 2 2 
n2  2
2i n 2 
 2 2 E 2
E k  2 n E
c 
c 
c


2
4 (3)  

2
 2   2  2i    E E
c

 

2
Symplifying and
Dropping all
Higher order
Derivatives…
2
2




4 (3) 

2
2
2
2
  E  2i n0
E   k  2 n  E  2  2i    E E
c 
c
c






i 

 i
2
 F x
2
2
2
n
(
x
)

n

0
n0
4 
i
n0
2

  
2

   
2 *

(3)
4 (3) 0 2
 i
 
n0 in

0   
in 
2

*
*


 *






2
*
 2
 
  2  



 
 
2
2
(3)
n
(
x
)

n


i 2
4 0 2
0 0
H     i
i
E
F
n0
in
n0 in
4 2  (3) 0   *
 * 
i
2  

n0 in  
 
Now we look at the linear regime, by injecting a beam inside the
guide from the left and then from the right.
On-Axis Intensity as Beam Propagates Down the Guide. Beam is Guided.
Output Field Profile in the case Light is Guided.
On-Axis Intensity as Beam Propagates Down the Guide. Beam
is Tuned to a Minimum of Transmission, and is Not Guided, and energy
Quickly Dissipates Away.
Output Field Profile in the case Light is Not Guided.
1.0
0.8
Input Spectrum
ON-AXIS
0.6
 
0.4
0.2
0
0.80
2n2 I max in L
c
I  1013 W/cm2
n2  510-19 cm2/W
L  8 cm
  100 fs
0.82
0.84
0.86
Output
Spectrum
0.88 0.90
0.92
0.94
0.96
0.98
1.00
/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
Linear transmittance for 2 slightly different guides
1.0
Transmittance
0.8
0.6
0.4
0.2
Output spectrum
Input spectrum
0
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
scaled frequency (=1/ where  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
power spectrum
800
600
400
200
0
0.65
0.70
0.75
scaled frequency (1/)
0.80
Self-phase modulation
A process whereby new frequencies (or wavelengths)
are generated such that:
Example: input 100fs pulse at 800nm
is broadened by ~30nm
2n2 I max in L
 
c
I max
Intensity, arb. units
1,0

0,5
0,0
400
500
600
700
, nm
800
900
I
t
Stimulated Raman Scattering
p
p
stokes
anti  stokes
E p
2

i Ep
*
i



Q
E
e
 QEs
A
2

F x
EA i  2 EA
 i



QE
e
P

F x 2
ES
i  2 ES
*


Q
EP
2

F x
Q
  Q  ES* EP   QE A EP* e i

The simplest case
stokes

E p
2
i  Ep

 QEs
2

F x
ES
i  2 ES
*


Q
EP
2

F x
Q
  Q  ES* EP

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
2
i  Ep

 QEs
2

F x
ES
i  2 ES
*


Q
EP
2

F x
Q
  Q  ES* EP

0

 Es
Es (0, )   0

  Es
   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
Intensity at cell output
2500
INTENSITY
2000
The Pump signal is
temporarily repleted
1500
The Stokes minimum
is referred to as
a Dark Soliton
1000
PUMP INTENSITY
STOKES INTENSITY
500
0
0
0.02
0.04
TIME
0.06
0.08
PUMP FIELD
z,0,0
z,=L,0
z=0
PUMP FIELD
STOKES FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
PUMP FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
STOKES FIELD
ON-AXIS INTENSITY AT CELL OUTPUT
3000
INTENSITY
F=
F=20
2000
1000
0
0
0.02
0.04
0.06
0.08
TIME
The onset of diffraction causes the soliton to decay…
…almost as expected. Except that…
ON-AXIS INTENSITY AT CELL OUTPUT
STOKES INTENSITY
5000
F=
F=20
4000
3000
2000
1000
0
0
0.02
0.04
0.06
0.08
TIME
… the Stokes field undergoes significant replenishement on its axis,
as a result of nonlinear self focusing
ON-AXIS INTENSITY PROFILE
PUMP FIELD
ON-AXIS INTENSITY PROFILE
STOKES FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
PUMP FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
STOKES FIELD
Poisson Spot like effect
Examples: single slit
0.8
intensity
0.6
0.4
0.2
0
-40
-30
-20
-10
0
10
transverse coordinate
20
30
40
Direction of
Propagation