NematiconsNematicons::

guiding and bending light at will with

Spatial Solitons in liquid crystals

NematiconsNematicons and all-optical waveguides– non local spatial solitons

Conclusions

– soliton and their angular steering– positive and negative refraction

at an interface

– total internal reflection

Self-focusing and spatial solitons

When Δdiff + Δf =0, diffraction is balanced by self-focusing.

The resulting index distribution is a waveguide able to confine the field: self-trapping and spatial solitons

Linear diffraction

r

λ

Δdiff

a k

Self-focusing: n=n0 +n2 I

r

a

Δf

λ

k

Nematic liquid crystals: reorientational response

Field E forces the reorientation of n

~3 nm

~0.5 nm

isotropic phase

//n

⊥n

n

nematic phase

planar alignment

n

Eθ

pre-tilted

field-induceddirector reorientation

-- In the presence of a pre-tilt, in a positive uniaxial NLC:

onnn => ⊥//

( )2

||

2

2

2 sincos1

nn

neθθ

θ+

=

⊥

Optical Reorientation and Nematicons

Appl. Phys. Lett. 77, 7 (2000)z[mm]

y[μm]

0 0.5 1

0

100

-100

x

n(x)

E(x)

50 μmY

X

Diffraction

P=2mW

Self-confinement

Laser Ar+

Laser He-Ne

Er

Ev

Attenuatory

xz

V

Electrodes

Anchoring films

Input beam

Experimental setup

Nematicons, self-trapping and optically induced waveguiding of a co-polarized signal

Bias: 1V@ 1KHz (r.m.s)

y[μm

]y[

μm]

z[mm]

Pump (2mW, 514nm) Signal (~0.1mW, 633nm)

Line

ar(T

E)So

liton

(TM

)

0 0.5 1

0

100

-100 0 0.5 1

0

100

-100

z[mm]

0 0.5 1

0

100

-100 0 0.5 1

0

100

-100

z[mm] z[mm]

Opt. Photon. News 14, 44 (2003); Phys. Rev. Lett. 92, 113902 (2004)

Coherent vs. partially incoherent nematiconsBias: 1V@ 1KHz (r.m.s); λ = 514 nm

y[μm

]

z[μm]

y[μm

]

z[μm]

z[μm]z[μm]

Incoherent (2.7mW)

Line

ar(T

E)So

liton

(TM

)

Phys. . Rev. . E 65, R035603 (2002)

Coherent (2mW)

Nematicon steering via interactions

in-plane w/ other nematicons

or external beamsV

NLC

Opt. Lett. 27, 1460 (2002)

Appl. Phys. Lett. 81, 3335 (2002)

J. Nonl. Opt. Phys. & Mat. 16, 37 (2007)

APL 87, 261104 (2005)

Photon.Tech.Lett. 18, 1287 (2006)

out-of-plane spiraling of cluster

Phys.Rev. A 75, 63835 (2007); Opt. Lett 32, 1447 (2007)

A

BOut

z

y

- 50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

BA ⋅

BA ⋅

(a)

(b)

-100-50

0 50 100y[μm]

11(c)

Inte

nsity

(a.u

)

0

1

A

BOut

z

y

- 50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

BA ⋅

BA ⋅

(a)

(b)

A

BOut

z

y

- 50

500

- 50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

BA ⋅

BA ⋅

(a)

(b)

-100-50

0 50 100y[μm]

11(c)

Inte

nsity

(a.u

)

0

1

-100-50

0 50 100y[μm]

11(c)

Inte

nsity

(a.u

)

0

1

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

A

B

1Sz

y

2S(a)

(b)

Inte

nsity

( a.u

)

-100 -50 0 50 1000

1

y[μm]

1S 2S

1S2S(c)

-50

500

-50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

A

B

1Sz

y

2S(a)

(b)

Inte

nsity

( a.u

)

-100 -50 0 50 1000

1

y[μm]

1S 2S

1S2S(c)

Y-junction/ AND-gate•Plow =1.7mW, Phi =3mW

X-junction•Plow =1.7mW, Phi =4.3mW

AB

OutBA ⋅BA ⋅BA ⋅

-50

500

-50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

(a)

(b)

S

OutBA

11011000

0001

-100 -50 0 50 100y[μm]

11 01 10 00(c)

Inte

nsity

(a.u

)

1

0

AB

OutBA ⋅BA ⋅BA ⋅

-50

500

-50

500

-50

500

-50

500

-50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

(a)

(b)

S

OutBA

11011000

0001

-100 -50 0 50 100y[μm]

11 01 10 00(c)

Inte

nsity

(a.u

)

1

0

A

BOut

-50

500

-50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

BA ⋅

BA ⋅

OutBA

11011000

1001

(b)

(a) S

Inte

nsity

(a.u

)

-100 -50 0 50 1000

1

0110

00 11

y[μm]

(c)

A

BOut

-50

500

-50

500

-50

500

-50

500

-50

500

-50

500

-50

500

0 0.5 1

y[μm

]

z[mm]

BA ⋅

BA ⋅

OutBA

11011000

1001

(b)

(a) S

Inte

nsity

(a.u

)

-100 -50 0 50 1000

1

0110

00 11

y[μm]

(c)

NOR-gatePin =1.7mW

XNOR-gatePin =1.7mW

00

10

01

11

00

01

10

11

Apparent walk-off α

vs bias for various powers

Bias-controlled steering of Nematicons

Nature 432, 733 (2004)

v1∼

v2∼

x

p

t

1

2

Nematic Liquid Crystal Cell: interface geometry

x

z

y

Electrode patterning permits the definition of two tunable dielectric regions, i.e. a voltage-adjustable transition between media differing in orientation and nonlinear response

n̂ρ

kI kR

kT

ΦI ΦR

p

t

ρ

1

2

n̂ΦR

pt

x

Reflection and RefractionThe inverse surface of wave normals

ρρ =90°

n̂k =kv n(r) r^ ^

p

t

ke =ko = kv n⊥

kv n⊥

kv n||

x

ko

ke

pt

x

Reflection and RefractionThe inverse surface of wave normals

ρ

ρ =90° n̂

p

t

o

e

x

n̂

ρ =90°ξ= ξ(V)

n̂p

t

x

ρ

ξ

p

t

o

e

x

ke = kv neke = kv ne (V)

Refraction

p

tV2 >0

V1 =0o1

o2

e2

e1

Refraction

kT-e

kT-okI-e

n1^p

t

x

ρ

ξξ1 =ξ(V1 )=0ξ2 =ξ(V2 )>0

n2^

kI

kT

ΦI

p

t

1

2n̂ΦR

Refraction (ΔV=V1 -V2

Refraction is said to be negative when, on passing through an interface between two media, the tangential component of the time- averaged Poynting vector changes its sign, i.e. the incident and refracted rays are on the same side of the surface normal

SI

STo, STeOptic axis

Air

crystal

Negative refraction in uniaxials (E. Bartholinus, 1669)

Negative and off-plane Refraction at an air-LC interface: bias controlled steering

t

Vrms

0

2.0V

z

y

x

Vlow

Vhigh

E7 (n|| = 1.6954 n⊥

= 1.5038 @ λ=1064nm)Pin =3.33mW

So

Sekin

kokeδ

Opt. Express 15, 8021 (2007)

o2

e2

Total Internal Reflection

p

tV2 =0

V1 >0

kI-e

o1e1

kR-e

TIR

Incident e-Wave

ξ1 =ξ(V1 )>0n1^

pt

x

ρ

ξ

n2^ξ2 =ξ(V2 )=0

kI kR

kT

ΦI ΦR

p

t

1

2nΦR

^

Reflection (ΔV=V1 -V2 >0) for ρ =90°: experiments

22°

Nature Physics 2, 737 (2006)

Conclusions

- Nematicons: 2D+1 nonlocal spatial solitons and induced waveguides for signal routing

- Angular steering by external beams, extra nematicons, induced defects, tunable walkoff

- Normal and negative refraction at an interface: voltage tunable steering and readdressing as large as 18°

- Total internal reflection at an interface: angular steering by 22°

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