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

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are needed to see this picture.

Soliton attraction:

Intensity

y x

Soliton Dynamics

Soliton repulsion:

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are needed to see this picture.

Intensity

y x

Soliton Dynamics

Soliton fusion:

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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

qq

q Pz

EikE 0

22 2

Propagation equation for the qth frequency component of E:

Propagation Equation

i

tii

tib

tia ietcbetcaetc iba )()()(

qq

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