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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 - PowerPoint PPT Presentation
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Michael Scalora
U.S. Army Research, Development, and Engineering CenterRedstone Arsenal, Alabama, 35898-5000
&Universita' di Roma "La Sapienza"
Dipartimento di Energetica
OPTICS BY THE NUMBERS
L’Ottica Attraverso i Numeri
Rome, April-May 2004
BPM:Propagation in Planar Waveguides
Retarded Coordinate trasformation: time dependence, Raman scattering, self-phase modulation in PCFs
1.0
1.1
1.2
1.3
1.4
1.5
-15 -10 -5 0 5 10 15
Transverse Coordinate (m)
Ind
ex o
f R
efra
ctio
n
air core
14 m
5 m
fig.(4)
Study the transmissive properties of guided modes.
Propagation into the page
2 2 2 2 22 2 2
2 2 2 2 2
22(3) 2
2 2
( ) 2 ( )2 ( )
42
E E n x E i n x EE ik k n x E
z z c t c t c
i E Ec t t
22 22
2 2 2 2
4 nlPn EE
c t c t
2(3)nlP E E
( )( , , ) ( , , ) . .i kz tE z x t E z x t e c c
Assuming steady state conditions…
2 2 (3)202 0 0
0 0
( ) 4
in in
n x nE iE i E i E E
F n n
2 2 2 22
2 2 2 2
2 222 2 (3) 2
2 2 2
( ) 2 ( )2
4( ) 2
E E n x E i n x EE ik
z z c t c t
k n x E i E Ec c t t
2 2 (3)202 0 0
0 0
0 0
0 0
( ) 4
4
/
in in
in
n x nE iE i E i E E
F n n
nF Fresnel Number
z k nc
F s all
F
m
Wave front does not distort:Plane Wave propagation
Diffraction is very important
2 2 (3)202 0 0
0 0
( ) 4
in in
n x nE iE i E i E E
F n n
This equation is of the form:
Where:
EHE
2 2 (3)20 02 0
0 0
( ) 4
in in
n x ni i E V
n
iD
nFH
Using the split-step BPM algorithm
0
( ', ) '(0, )
/ 2(0, ) (0, ) / 2
( , ) (0, ) (0, )
(0, )D
H xH
V x x
x
V
E x e E x e E x
e e e E x
0.75
1.00
1.25
1.50
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.4
0.8
1.2airair
gla
ss;
n=
1.4
2
gla
ss;
n=
1.4
2
gla
ss;
n=
1.4
2
gla
ss;
n=
1.42
air guide ~ 5m
Transverse Coordinate (m)
Ind
ex o
f R
efra
ctio
n
Inte
nsi
ty
a=1.4m b=1m
Example: Incident angle is 5 degrees
Assume =0
xThe cross section along x renders the problem one-dimensional in nature
1.0
1.1
1.2
1.3
1.4
1.5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Transverse Position (microns)
Tra
nsv
erse
Ind
ex P
rofile
0
0.2
0.4
0.6
0.8
1.0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
14-micron core5-micron core
(m)
Nor
mal
ized
Tra
nsm
itta
nce
Transmissive properties in the linear (low intensity) regimeFor two different fibers. We set =0
0.75
1.00
1.25
1.50
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.5
1.0
1.5Field bouncing back and forth from structure's wallsInpout Field Profile
Transverse Coordinate (m)
Ind
ex o
f R
efra
ctio
n
Inte
nsi
ty
-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12
x (m)
0
100
200
300
400
500z(
m)
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8 0.8
0.8
0.8
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1 Field tuning corresponds toHigh transmission state.
0
0.2
0.4
0.6
0.8
1.0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
14-micron core5-micron core
(m)
Nor
mal
ized
Tra
nsm
itta
nce
Direction of propagation
Same as previous figure.
-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12
x (m)
0
100
200
300
400
500
z (
m)
0.1
0.1
0
0.2
0.4
0.6
0.8
1.0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
14-micron core5-micron core
(m)
Nor
mal
ized
Tra
nsm
itta
nce
Same as previous figure.
-12 -10 -8 -6 -4 -2 -0 2 4 6 8 10 12
x (m)
0
100
200
300
400
500z(
m)
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8 0.8
0.8
0.8
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
0/ 0.025z 0/ 0.0125x x
200000N 4096xN
For the example discussed:
5-mm guide~ 8 minutes on this laptop3.2GHz, 1Gbts RAM
2
2 2 (3)20 0 0
0 0
( ) 4
in in
nE iE
x n
n nFi E i 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.
1.0
1.1
1.2
1.3
1.4
1.5
-15 -10 -5 0 5 10 15
Transverse Coordinate (m)
Inde
x of
Ref
ract
ion air core
14 m
5 m
fig.(4)
Optical Switch
0
0.2
0.4
0.6
0.8
1.0
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
non-zero (3)
linear transmittance
scaled frequency (1/ where is in microns)
nor
mal
ized
tra
nsm
itta
nce
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.
fig.(5a)
0
0.2
0.4
0.6
0.8
1.0
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10
(m)
Nor
mal
ized
Tra
nsm
itta
nce
0
0.2
0.4
0.6
0.8
1.0
0.715 0.720 0.725 0.730 0.735 0.740
Nonlinear Transmittance
Linear Transmittance
(m)
Tra
nsm
itta
nce
fig.(5b)
Optical Switch
on
off
0
0.2
0.4
0.6
0.8
1.0
0 1000 2000 3000 4000 5000
Longitudinal Position (m)
Cor
e E
nerg
y
fig.(6)
off
on
2 2 2 22
2 2 2 2
2 222 2 (3) 2
2 2 2
( ) 2 ( )2
4( ) 2
E E n x E i n x EE ik
z z c t c t
k n x E i E Ec c t t
0/ /z t z v v c n Retarded coordinateTransformation
1
zz z v
t tt
2 2 2
2 2 2 2
1 2
z v v
2 2
2 2t
2 22
2 2 2 2
2 222 (3) 2
2
2 2
22
2
( ) ( )
( )
22
42
E E E i EE ik
z z c t c t
k E i E Ec c t
n x
xt
n x
n
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.
0/ /z t z v v c n
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
2 2 2
2
2
2
20
2 2 22 2
2 2 2
2(3) 22
0
2
2
2
2
4
1
2
E E i n Ec
n i nE E k n E
v v
c c
c
E Ec
n
c
i
Symplifying and
Dropping all Higher order Derivatives…
222 2 2 (3) 2
0 2 2
42 2E i n E k n E i E Ec c c
2 2 (3)202 0 0
0 0
2 (3) **0
0
( ) 4
42
in in
in
n x niH i i E
F n n
in
2 22 (3)20 0
20 0
22 (3)
0
0
( ) 4
4
in
in
n x nii i
F x n n
in
22 * * *
* 2 *2 2
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.
0
0.2
0.4
0.6
0.8
1.0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00
Input Spectrum
OutputSpectrum
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
2 max2 inn I L
c
I 1013 W/cm2
n2 510-19 cm2/WL 8 cm 100 fs
ON-AXIS
0
0.2
0.4
0.6
0.8
1.0
0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95
Output spectrumInput spectrum
scaled frequency (=1/ where is in m)
Tra
nsm
itta
nce
Linear transmittance for 2 slightly different guides
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
0
200
400
600
800
1000
0.65 0.70 0.75 0.80
scaled frequency (1/)
pow
er s
pec
tru
m
Spectrum of the pulse as it propagates. Note splitting.
Initial profile
Self-phase modulation
A process whereby new frequencies (or wavelengths) are generated such that:
2 max2 inn I L
c
400 500 600 700 800 9000,0
0,5
1,0
In
ten
sit
y, arb
. u
nit
s
, nm
maxI I
t
Example: input 100fs pulse at 800nm is broadened by ~30nm
Stimulated Raman Scattering
p stokesanti stokes
2
2*
si
Ap pE Ei
F xQ E QEe
2
2iA A
P
E EiQE e
F x
2*
2S S
P
E EiQ E
F x
* * iS P A P
QQ E E QE E e
p
stokes
2
2
p ps
E EiQE
F x
2*
2S S
P
E EiQ E
F x
*S P
QQ E E
The simplest case
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 repletionof the pump at the expense of the Stokes intensity
stokes
2
2
p ps
E EiQE
F x
2*
2S S
P
E EiQ E
F x
*S P
QQ E E
The simplest case
00
00
(0, ) ss
s
EE
E
The gain changes sign temporarily,For times of order 1/
The soliton is the phase wave
The Input Stokes field undergoes a -phase shift
0
500
1000
1500
2000
2500
0 0.02 0.04 0.06 0.08
PUMP INTENSITYSTOKES INTENSITY
TIME
INT
EN
SIT
YIntensity at cell output
The Pump signal is temporarily repleted
The Stokes minimum is referred to as a Dark Soliton
PUMP FIELD z
z=0
z,=L,0
PUMP FIELD
STOKES FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
PUMP FIELD
TRANSVERSE INTENSITY PROFILE AT CELL OUTPUT
STOKES FIELD
0
1000
2000
3000
0 0.02 0.04 0.06 0.08
F=F=20
TIME
INT
EN
SIT
Y
ON-AXIS INTENSITY AT CELL OUTPUT
The onset of diffraction causes the soliton to decay……almost as expected. Except that…
… the Stokes field undergoes significant replenishement on its axis,as a result of nonlinear self focusing
0
1000
2000
3000
4000
5000
0 0.02 0.04 0.06 0.08
F=F=20
TIME
STO
KE
S IN
TE
NSI
TY
ON-AXIS INTENSITY AT CELL OUTPUT
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
0.2
0.4
0.6
0.8
-40 -30 -20 -10 0 10 20 30 40
transverse coordinate
inte
nsity
Direction of Propagation