S. Residori, U. BortolozzoInstitut Non Linéaire de Nice, Sophia Antipolis, France

J.P. HuignardThales Research & Technology, Palaiseau, France

Talbot effect and self-pumped phase conjugationin photorefractive liquid crystal light-valves

The photorefractive liquid crystal light-valve Description of the device and its characteristic responseMeasurement of the spatial resolution of the cell

Non linear optical experimentsTwo-wave mixing gain - Talbot effectSelf-pumped phase conjugation

The liquid crystal optical oscillatorNonlinear dynamics and mode competitionThe generation of spatiotemporal pulses

Outline

liquid crystal E48l=14 µm

Δn=0.2306 Δε=15.1 planar alignment

The photorefractive liquid crystal light-valveOne side of the cell is made of a thin slice (1 mm thickness) of B12SiO20 , BSO

l d

20 mm

BSO crystal d=1 mm cubic group, n0=2.5 optical activity 42°/mm-1 @ λ=532 nm dark conductivity 10-15 Ω-1cm-1

The effective voltage applied onto the LC dependson the intensity Ion the BSO

The working principle and the response of the valve

When the light illuminates the BSO, VLC(I) increases the liquid crystals reorient and the beam acquires a phase shift

!

VLC (I) = V1

1+C1

C2(1+ j"RC2)

# $V +%I

!

" =2#

$%n(I)l

In the linear regime the valve acts as adefocusing Kerr-like medium: Δn∝I!

"nmax

= ne# n

0= 0.2

!

V

!

V = 20 V

f = 20 KHz

probe@632 nm

!

pump@532 nm I

!

C1

!

C2

!

I

Pattern formation

!

"#t$ = % $ %$

0( )+lD2&'

2$ %(I

!

I = I0+I

0e

iL

2k0

"#2

$ e i%&

&

& &

&

&

& &

2

The valve acts as a Kerr-like medium

The homogeneous stationary solution becomes unstable at atransverse wave number qc, leading to a hexagonal spatial pattern

L=12 cm, V=20 V, f=2 KHz, I0=3.0 mW/cm2

!

qc = "6

#L

!

" = 532nm

!

V τ=50 ms liquid crystal response timelD=20 µm lateral diffusion length

L=12 cmL=21cm L=4 cmL=16 cm

!

MTF =Imax

" Imin

Imax

+ Imin

The spatial period of the patterns scales with L, the contrast decreases as L increases

Measurement of the spatial resolution

Bortolozzo, Residori, Petrosyan, Huignard, Opt. Comm., 263, 317 (2006)

The MTF cutoff is at N=30 linepairs/mm corresponding to a lateraldiffusion lD = 20 µm

!

" =2#

qc=

2$L

3

Two-wave-mixing experiments

V=10 Vf=500 Hz

!

G =I s (out)

I s (0)= 1+

2"n2I p (0)

#

$

% &

'

( )

2

d2

!

"n = 2n2 I p (0)I s (0)

2WM gainKerr-like coefficient of the valve

!

n2 ="n

"I

the energy transfer is always fromthe strong wave to the weak one!

" =I p (0)

I s (0)>> 1

!

" =#$nd

%

&

' (

)

* +

2

!

I s (out) = I s (0) +" I p (0)

Brignon, Bongrand, Loiseaux, Huignard, Opt. Lett. 22, 1855 (1997)

!

Is(out)

!

Is(0)

Raman-Nath diffraction efficiency

!

" = 532nm

!

V

β=Ip/Is=80

Ip pump intensityIs signal intensity

Two-wave-mixing gain

G=4n2= 5.4 cm2/W

V [Volt]

f [Hz] Ip [mW/cm2]

Ip=2 mW/cm2

V=10 Volt, Ip=1.8 mW/cm2V=10 Volt, f=100 Hz

Talbot effect

!

zT =p2

"

A review: M. Berry, I. Marzoli, and W. Schleich, Phys. World, June 2001, (2001)

H.F. Talbot, Phil. Mag. 9(56), 401 (1836)

a field distribution which is periodic in the lateraldimension (period p) is also periodic in the direction oflight propagation

when propagating, it reappears, at even integer multiple ofthe so called Talbot distance

Fractional revivals also occur at distances that are rationalmultiples of zT, i.e. z/zT=n/m, with n and m prime integers

2WM in a cascade on N nonlinear elements: model equations

!

"E j (x,z)

"z=

i

2k0

" 2E j (x,z)

"x 2

E j (x,0) = E j#1eik0l n1 +n2 E j#1

2$ % &

' ( )

the LC thickness l is neglected; l <<d and d < zT

!

E0 (x)2

= Es2

+ Ep2

+ 2EpEs cos2"

px

#

$ %

&

' (

the intensity at the first BSO plane

!

E1

= E0eik0l n

1+ n

2E0

2( )

the field after the first cell

j=1,2, ..., N-1

the field arriving at the second cell isE1 after diffraction over L1

and so on …!

"E1(x,z)

"z=

i

2k0

" 2E1(x,z)

"x 2

n.l. phase shiftworking point

!

E2

= E1eik0l n

1+ n

2E1

2( )

!

from z = 0 to z = L j

Enhancement of the 2WM gain by use of the Talboteffect : numerical simulations

L1 L1 L2

N=2 N=3

The maximum gainis for L1=0.75 zT

The maximum gainis for L1=0.88 zTand L2=0.75 zT

n1=1.62, n2=3.9 cm2/W, p=36.5 µm, zT=2.5 mm, β=80

Kerr Kerr Kerr Kerr Kerr

diffraction diffract. diffract.

G

Behavior of the 2WM gain

G oscillates when changing thespatial period p of the grating dueto the periodic recurrences of theTalbot distance

with optimized Talbot distances only3 elements are needed to reach thebest gain of an equivalent “bulk”medium

stratified“bulk” medium

optimized Talbotdistances

Experiment: 2WM gain of two valves in cascade

U. Bortolozzo, S. Residori and J.P. Huignard, Opt. Lett. 31, 2166 (2006)

!

G =Sout

2(with the pump)

Sout2(without the pump)

D=25.9 mmp =36.5, 31.5 µmzT =2.5, 1.9 mmβ =80, 20

!

" = 532nm

Talbot intensity carpets: numerical results

The gain amplification is accompanied bya large increase of the modulation depth

The best gain amplification is obtainedwhen the gratings on the first and the lastcell are shifted by a quarter of period

N=1, z=0 N=2, z=L’1

N=3, z=L”1+L”2

p=36.5 µm, zT=2.5 mm, β=80

Experimental Talbot carpets

Talbot carpets are obtained by translating the CCD cameraaway from the exit plane of the second valve

enhancement of thegain is accompanied byan increase of themodulation depth in theTalbot intensity carpets

when the valves are operating Talbot carpets reveal the phaseslip associated to the presence of a grating inside the LC

Self-pumped phase conjugation with tilted feedback

the pump beam is distorted by an astigmatic lens

U. Bortolozzo, S. Residori, J.P. Huignard,Opt. Lett. 32, 1 (2007)

the valve acts as a phase conjugatingmirror; the strength of the distortion tobe corrected is limited by the spatialresolution (30 lp/mm)

!

q =2"

#

only stationary patterns are considered(no drift instability); the 4WM isdegenerate (no moving grating)

zero order first order

The reflectivity of the SPPCM mirrorβ=Ip/Is=80

V=20 V f=2 KHzIp=0.3 mW/cm2

the phase-conjugate reflectivity

!

R = MJ12(kdn2I IN )

2

!

R =IPC (0)

I IN

is estimated by considering Raman-Nathdiffraction

n2=14 cm2/W nonlinear coefficientM=0.45 accounts for losses (mainly absorption)

Experimentθ= 3.5 mrad Λ= 150 µm

Theoretical prediction

R=2%

Model for the SPPCM with tilted feedback

!

n = n0 + nqeir q "

r r

+ c.c.

!

"dnq

dt= #(1+ lD

2q2)nq # n2IBSO

!

IBSO = I IN 1 + eiL

k"

2

eikdn(

r r , t )

2

!

kd nq << 1

!

x = 2L"

small amplitude gratingthe tilt leads to a translation of the feedback

!

"# = $(1+ lD2q2) $% sin

q2L

k

&

' (

)

* + cos 2,Lq( )

!

fd = "#

2$%sin

q2L

k

&

' (

)

* + sin 2,Lq( )

Linear stability analysis

!

nq (t ) = "e#t+ i2$fd t

uniform value fixedby the voltage V0

complex amplitude ofthe grating at thespatial frequency q

growthrate

driftfrequency

Matching condition

for the uniform state n0 becomes unstable and a spatial pattern develops gives the threshold intensity ITH for the grating formation and its corresponding q and fd

!

L = 10 mm

the matching condition for the realization of the SPPCM requires

!

q = k"discrete bands are due to Talbot effect at the origin of the pattern formation

!

" # 0

!

" = 0

The liquid crystal optical oscillator

- the focal length of the lens is f with 4f > L, L is the total cavity length

U. Bortolozzo, A. Montina, F.T. Arecchi, J.P. Huignard, S. Residori, submitted to Phys. Rev. Lett. 2007

- the energy is injected in the cavity by 2WM, the pump field is @532 nm- the cavity field is detuned of a few Hz respect to the pump field

The mode dynamics and the spatiotemporal pulses

U. Bortolozzo, A. Montina, F.T. Arecchi, J.P. Huignard, S. Residori, submitted to Phys. Rev. Lett. 2007

Spatiotemporal pulses are confined in 3DNumerical simulations

towards light bullets … Y. Silberberg, Opt. Lett., 15, 1282 (1990)

S.A. Ponomarenko, G.P. Agrawal, Opt.Comm. 261, 1 (2006)

Conclusions

Photorefractive liquid crystal light-valves are built with BSO crystalsthey show large Kerr-like nonlinearity controllable either optically or electricallyshow pattern formation pattern formation when inserted in an optical feedback looptilted mirror schemes can be used to produce self-pumped phase conjugation

Enhancement of the 2WM gain is demonstrated by use of the Talbot effectA few nonlinear media are required to get a large enhancementStratified nonlinear media may be specifically designed with optimal separations

2WM gain can be used to build an optical oscillatorThe large nonlinearity and large number of modes lead to complex dynamicsSpatiotemporal pulse localized in 3D arise from the simultaneous presence oflongitudinal and transverse modes