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Observation of Raman Self-Focusing
in an Alkali Vapor Cell
Nicholas Proite, Brett Unks, Tyler Green, and Professor Deniz Yavuz
Self-Focusing Effect
Non-linear effect due to the intensity dependent refractive index generated by (3)
Mechanism by which optical spatial solitons are formed
Single Photon vs Raman Systems
a
b
E
2 E 2ik
E
z
2
c 2 (3) E
2E
a
b
Ep
e
Es
(3) ~1
2
Propagation Equations in Raman System
a
b
Ep
e
Es
2ikp
E p
z
2 E p sgn()E s
2
E p
2E s
2 1E p;
2iks
E s
z
2 E s sgn()E p
2
E p
2E s
2 1E s
Our System
F=0, 1, 2, 3
EP ES
optical pumpinglaser
F=1
F=2 ~ 1 MHz
85 GHz
87Rb D2 Line
General Procedure of Experiment
Pinhole Photodiode
General Procedure of Experiment
Pinhole Photodiode
sgn() 0
General Procedure of Experiment
Pinhole Photodiode
sgn() 0
Inte
nsit
y
x (mm)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 10
0.5
1
1.5
2
-1 -0.5 0 0.5 1
x (mm)
(a) (b)
The peak intensity for a freely propagating beam is normalized to 1.
Focused De-Focused = 2 0.25MHz = 2 -0.25MHz
Experimental Results
Experimental ResultsSimulation
Experimental Results
norm
aliz
ed tr
ansm
issi
on
0.8
1
1.2
1.4
-8 -4 0 4 8
(MHz)
Thank you
References:
1) DD Yavuz, Phys Rev A,75, 041802, (2007).
2) N. A. Proite, B. E. Unks, J. T. Green, and D. D. Yavuz, Phys. Rev. A, 77, 023819 (2008).
General Procedure of Experiment
What is a Soliton?
Normal Gaussian Beam:
z
x,y
I I
x x
What is a Soliton?
Soliton:
z
x,y
I
x
I
x
E 0
H 0
E 0
H
t
H 0
E
t P
t
Maxwell’s Equation inside a medium with no charge or current density:
Gaussian Beam Propagation in a Medium
2 E 2ik
E
z 20P
Paraxial Wave Equation in a Linear Medium
2 E 2ik
E
z
2
c 2E
Using the relation:
2 E 2ik
E
z 20P
P() 0()E()
Paraxial Wave Equation in a Linear Medium
2 E 2ik
E
z
2
c 2E
() '() i ' '()
a
b '
''' n
'' Loss (or gain) of medium
Index of refraction
E
Paraxial Wave Equation in a Non-Linear Medium
a
b
As the strength of the beam is increased polarization of the medium is no longer linear; we must introduce higher order susceptibilities:
P(t) (1)E(t) (2)E 2(t) (3)E 3(t)
In an isotropic medium:
P(t) (1)E(t) (3)E 3(t)
2 E 2ik
E
z 20P
E
How will non-linear terms affect beam propagation?
a
b
2 E 2ik
E
z
2
c 2 E
2 E 2ik
E
z
2
c 2 (3) E
2E
Non-Linear Schrödinger’s Equation
E
How will non-linear terms affect beam propagation?
a
b
2 E 2ik
E
z
2
c 2 E
2 E 2ik
E
z
2
c 2 (3) E
2E
Non-Linear Schrödinger’s Equation
E sech(x) exp( iz)
The solution (with one transverse dimension ‘x’):
x
Sech(x)
E
Raman System (a 3rd order non-linear process)
a
b
Ep
e
Es
a
b Transitions may be one photon
eforbidden, but by using the intermediate state associated with we can couple them.
Atomic Raman System using Rubidium 87Rb D2-line
(F' = 0,1,2,3)
(F = 1)
52P3/2
52S1/2
Es Ep
| a >
| b > (F = 2)
ˆ H a a a b b b i
i
i i E(t) ˆ P
ˆ P ai
i
a i bi bi
i ccOnly dipole transitions are considered here
Ho Hint
| i >
ca (t)exp( iat) a cb (t)exp( iat) b c i(t)exp( iat) ii
Atomic Raman System using Rubidium 87Rb D2-line
(F' = 0,1,2,3)
(F = 1)
52P3/2
52S1/2
Es Ep
| a >
| b > (F = 2)
| i > Key Assumption:
Large one photon detuning
it
ca
cb
2
A B
B * D 2
ca
cb
A,D E p
2 E s
2;
B E p E s*
where
Propagation Equation in a Raman Medium
2 E 2ik
E
z 20P
How can we get this equation in terms of quantities we know?
Propagation Equation in a Raman Medium
2 E 2ik
E
z 20P PtP ˆ)(
2ikp
E p
z
2 E p sgn()E s
2
E p
2E s
2 1E p ;
2iks
E s
z
2 E s sgn()E p
2
E p
2E s
2 1E s
Nonlinear part of the propagation equations:
Use expectation value of polarization operator to find polarization term.
Interpreting the coupled propagation equations
2ikp
E p
z
2 E p sgn()E s
2
E p
2E s
2 1E p ;
2iks
E s
z
2 E s sgn()E p
2
E p
2E s
2 1E s
2 E 2ik
E
z
2
c 2E
Interpreting the coupled propagation equations
2ikp
E p
z
2 E p sgn()E s
2
E p
2E s
2 1E p ;
2iks
E s
z
2 E s sgn()E p
2
E p
2E s
2 1E s
2 E 2ik
E
z
2
c 2E
n 'eff sgn()E s
2
E p
2E s
2 1
For ‘p’ beam:
eff
Non-linear Refractive Index
n n
sgn() 1
sgn() 1
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
z z
n 'eff sgn()E s
2
E p
2E s
2 1
Phase Front
Self-Trapping and Solitons
Soliton
2 10MHz
100GHz
N 1014 cm 3
Freely Propagating beam
x (m)
Inte
nsity
(W
cm-2)
500
-5000
.4
z (m)
900 1600
Parameters
Soliton Stability
QuickTime™ and a decompressor
are needed to see this picture.
Beam IntensityRefractive Index
Peak Refractive Index ~ 6.7x10-6x (m)
Inte
nsity
(W
cm-2)
500
-5000
.4
z (m)
1600
Soliton Stability (Vakhitov, Kolokolov criterion)
Pow
er (
W)
E(x, y,z) F(x, y)exp( iz)
Assume the electric fields are identical to reduce to one non-linear equation. Assume electric field takes the form of a field which only accumulates phase with z. The corresponding propagation constant is .
Stability Condition:
F2
0
Soliton Dynamics
QuickTime™ and a decompressor
are needed to see this picture.
Soliton attraction:
Intensity
y x
Soliton Dynamics
Soliton repulsion:
QuickTime™ and a decompressor
are needed to see this picture.
Intensity
y x
Soliton Dynamics
Soliton fusion:
QuickTime™ and a decompressor
are needed to see this picture.
Intensity
y x
Index Waveguides
|E|2
xn = 3.2
n = 3.4
Index Waveguides
|E|2
xn = 3.2
n = 3.4
n = 3.4
n = 3.2
n = 3.6
|E|2
x
Soliton Interactions
QuickTime™ and a decompressor
are needed to see this picture.
Relative Phase: 0
x (m) 500-500
z (m)
1
Inte
nsity
(W
cm
-2)
4000
Beam IntensityRefractive Index
Soliton Interactions
QuickTime™ and a decompressor
are needed to see this picture.
Relative Phase:
x (m) 500-500
z (m)
1
Inte
nsity
(W
cm
-2)
2000
Beam IntensityRefractive Index
Soliton Interactions
QuickTime™ and a decompressor
are needed to see this picture.
Relative Phase: 1.8
x (m) 400-400
z (m)
.8
Inte
nsity
(W
cm
-2)
3000
Beam IntensityRefractive Index
Possible Application
1
1
1
1
0
0
1
0
0
AND gate
0
0
0
Experimental Observations of Self-Focusing and Self-Defocusing
Experimental Observations of Self-Focusing and Self-Defocusing
F=0, 1, 2, 3
EP ES
optical pumpinglaser
F=1
F=2
85 GHz
Experimental Observations of Self-Focusing and Self-Defocusing
Pinhole Photodiode
Experimental Observations of Self-Focusing and Self-Defocusing
Photodiode
sgn() 0
Experimental Observations of Self-Focusing and Self-Defocusing
Photodiode
sgn() 0
Inte
nsit
y
x (mm)
0
0.5
1
1.5
2
-1 -0.5 0 0.5 10
0.5
1
1.5
2
-1 -0.5 0 0.5 1
x (mm)
(a) (b)
The peak intensity for a freely propagating beam is normalized to 1.
Focused De-Focused=2 0.25MHz =2 -0.25MHz
Experimental Results
Experimental ResultsSimulation
Experimental Results
norm
aliz
ed tr
ansm
issi
on
0.8
1
1.2
1.4
-8 -4 0 4 8
Detuning (MHz)
Acknowledgments and References
Thank you to Brett Unks, Nick Proite, Dan Sikes, and Deniz Yavuz for their helpful suggestions.
(And David Hover for letting me use his computer)
References:
1) DD Yavuz, Phys Rev A,75, 041802, (2007).
2) Stegeman, Sevev, Science, 256 1518, (1999).
3) NG Vakhitov, AA Kolokolov, Sov. Radiophys. 16,1020, (1986).
4) NA Proite, BE Unks, JT Green, DD Yavuz, (Recently Submitted).
5) MY Shverdin, DD Yavuz, DR Walker, Phys. Rev. A, 69, 031801, (2004).
Paraxial Wave Equation in a Medium
Ecz
EikE
~~~
2~
2
22
Using the relation:
Pz
EikE
~~
2~
022
˜ P () 0 ˜ () ˜ E ()
The real and imaginary parts of (' and '' respectively) reveal much about the behavior of the beam as it propagates through the medium.
c
11
2 '
;
2c
' ' Loss
Propagation constant
n 11
2 '
Raman SystemRb D2-line
F' = 0,1,2,3
(F = 1)
52P3/2
52S1/2
Es Ep
| a >
| b > (F = 2)
i
tii
tib
tia
ibi
iai
iiba
ietcbetcaetc
ccibiaP
PtEiibbaaH
iba
)()()(
ˆ
ˆ)(ˆ
Only dipole transitions are considered here
Ho Hint
| i >
Raman SystemRb D2-line
F' = 0,1,2,3
(F = 1)
52P3/2
52S1/2
Es Ep
| a >
| b > (F = 2)
| i >
Assumptions:
1) Only dipole transitions allowed
2) Large one photon detuning
3) << b - a
4) Terms varying faster than are integrated out
Raman SystemRb D2-line
F' = 0,1,2,3
(F = 1)
52P3/2
52S1/2
Es Ep
| a >
| b > (F = 2)
| i >
Assumptions:
1) Only dipole transitions allowed
2) Large one photon detuning
3) << b - a
4) Terms varying faster than are integrated out
it
ca
cb
2
A B
B * D 2
ca
cb
A ap E p
2 as E s
2,
B bE p E s*,
D dp E p
2 ds E s
2
where
ap,s 122
ai
2
( i a ) p,si
,
dp,s 1
22
bi
2
( i b ) p,si
,
b 122
aibi*
( i a ) pi
and
Heffective
Non-linear Refractive Index
EEb
EbE
z
Eik
1
)sgn(2
2
42
22
2
Ec
Ez
Eik
2
222
Non-linear Refractive Index
EEb
EbE
z
Eik
1
)sgn(2
2
42
22
2
Ec
Ez
Eik
2
222
effc
2
2
12
)sgn(1)2
11(
2
42
22
2
2'
Eb
Ebcn eff
Non-linear Refractive Index
12
)sgn(1)2
11(
2
42
22
2
2'
Eb
Ebcn eff
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
n n
sgn() 0
sgn() 0
Acknowledgements and References
Effective Hamiltonian
it
ca
cb
2
A B
B * D 2
ca
cb
Finding the eigenvalues of this effective Hamiltonian and expressing in terms of Bloch vectors we can find the density matrix elements.
New eigenvector smoothly coupled to the ground state. The eigenvector is shifted from |a> because of the interaction with the incident wave:
cos(2
)ei2 a sin(
2
)e i
2 b
where
B B e i ;
tan B
D
2
A
2
Effective Hamiltonian
it
ca
cb
2
A B
B * D 2
ca
cb
Finding the eigenvalues of this effective Hamiltonian and expressing in terms of Bloch vectors we can find the density matrix elements.
New eigenvector smoothly coupled to the ground state. The eigenvector is shifted from |a> because of the interaction with the incident wave:
cos(2
)ei2 a sin(
2
)e i
2 b
where
ab 12
sin()e i ;
aa cos2(2
);
bb sin2(2
)
B B e i ;
tan B
D
2
A
2
This gives us the following expressions for the density matrix elements:
Propagation Equation
q Pz
EikE 0
22 2
Propagation equation for the qth frequency component of E:
Propagation Equation
i
tii
tib
tia ietcbetcaetc iba )()()(
q Pz
EikE 0
22 2
PtP ˆ)(
Propagation equation for the qth frequency component of E:
;ˆ ccibiaPi
bii
ai where,
Start with
Propagation Equation
PtP ˆ)(
)(2
);(2
*22
**22
pabsbssass
sbapbppapp
bEccEcdEcaNP
EbccEcdEcaNP
Making the same assumptions as in the derivation of the effective Hamiltonian and assumingonly significant coupling is between Es and Ep, the polarization expectation values in frequencyspace for Es and Ep are:
Plug these expressions into the propagation equation:
)(22
);(22
*2
*2
pabsbbssaasssss
s
sabpbbppaappppp
p
bEEdEakNEz
Eik
EbEdEakNEz
Eik
Propagation Equation
ps
ps
spsp
EE
kk
bda
,,
)(22
);(22
*2
*2
pabsbbssaasssss
s
sabpbbppaappppp
p
bEEdEakNEz
Eik
EbEdEakNEz
Eik
Make assumptions:
Propagation Equation
ps
ps
spsp
EE
kk
bda
,,
)(22
);(22
*2
*2
pabsbbssaasssss
s
sabpbbppaappppp
p
bEEdEakNEz
Eik
EbEdEakNEz
Eik
EEb
EbE
z
Eik
1
)sgn(2
2
42
22
2
Make assumptions:
kN
Soliton Stability (Vakhitov, Kolokolov criterion)
Pow
er (
W)
E(r,z) F(r)e iz;
1rFr
2Fr2 F 3
b2F 4
2 1
2kF 0
Assume the electric fields are identical to reduce to one non-linear equation. Assume electric field takes the form of a field which only accumulates phase with z. The corresponding propagation constant is .
Stability Condition:
F2
0