Nondiffractive light in photonic crystals

• Nondiffractive light in linear Photonic Crystals (PCs)

• Nondiffractive pulses in PCs;

• Nondiffractive nonlinear resonators

• Subdiffractive solitons in nonlinear PCs;

• Nondiffractive resonators of PCs;

Group of “Dinàmica i òptica no lineal” , FEN, UPC

Group of “Nondiffractive light” : K.Staliunas, R.Herrero, C.Serrat, C.Cojocaru, J.Trull.

• Nondiffractive light in nonlinear PCs

• Nondiffractive beams in PCs;

Photonic Crystals

• Photonic bands:

• Modification of diffraction k

w

Photonic band I

Photonic bandgap

Photonic band II

• Modification of dispersion:

•diffraction management in fiber arrays (1D PCs)

•diffraction management in BECs (1D lattices)

•diffraction management, numerics and experiments in 2D PCs

Configurations:

n

z

n

z

Step Sinusoidal1D,2D,3D:

conductors and insulators of light

slow light

Elimination of diffraction !!But also

Nondiffractive light in linear Photonic Crystals

220||,)()( ||

⊥⊥

∞

∞−

−−⊥ −== ∫ ⊥ kkkkdeekarA zikxikr

ckkkk 0

022

||ω==+= ⊥

tierArE 0)()( ωrr =

),( ||kkk ⊥=r

rkiekarrr

−)(

z

x

Beam with carrier frequency ω0 propagating in z direction :

⊥k

IIk

Variation of the interference pattern in z

Relative phase variations during the propagation in z

Diffraction

)(rA rProfile in x,y

Different k|| for each plane wave

Diffraction

Diffraction management

⊥k

IIk

⊥k

IIk

),( ||0 kkk ⊥=r

⊥k

IIk1qr

2qr

( ) ( ) ( )rHc

rHr

rrr

21

=

×∇×∇ ωε

H.Kosaka e.a. 1999, J.Witzens e.a. 2002, D.N.Chigrin e.a. 2003, R.Iliew e.a. 2004,…

⊥k

IIk

Constant ω surf. First brillouin zonesFirst branch Second branch

•Localization of non-diffractive regimes

kII

( )( ) ( ) 0,,22 20

220 =∆+∂∂+∂∂ zxAkzxnxzik

)( 000 )()()( tzkiti erAerEr ωω −− == rrrΕ02

2

2

22 =

∂∂−∇tc

n ΕΕ

•Si considerem que :0=∂∂tE 02

20

22 =+∇ E

cnE ω

•Si considerem que :02

2

<<∂∂zA 02 0

2 =∂∂+∇ ⊥ zAikA

•Sols considerant una variable transversa i afegint ∆n:

Slowly Varying Envelope Approximation

•Equació d’ones per l’ona electro-magnètica:

( )( ) ( ) 0,,22 20

220 =∆+∂∂+∂∂ zxAkzxnxzik

⊥k

IIk

Paraxial

• Analytical expressions• Generalizations:

• Linear

Allows:

• Nonlinear (e.g. to BECs);

Paraxial approx. and sinusoidal ∆n

( ))cos()cos(2)( 21 rqrqmrn rrrrr +=∆

( ) ∑=ml

rkiml

mleAzxA,

,,,rr

),(, IIIIml mqklqkk ++= ⊥⊥

r

-1.5 -1 -0.5 0.5 1

-1-0.75-0.5

-0.25

0.250.5IIK

⊥K

• Bloch modes

Sinusoidal

harmonics

Inscription of the index modulation

( ))cos()cos(2)( 21 rqrqmrn rrrrr +=∆ •Photorefractive materials

2D

3D

•Acusto-optic materials

Inscribing usinglight beams

Inscribing usingultrasound waves

Dynamical photonic crystals Dynamical diffraction management

.

.

.

)(En∆

)(Pn∆

Diffraction elimination

00

2||

2

=∂∂

=⊥⊥ kkk

Zero diffraction:

Non-diffractive curve

Most homogeneous Bloch mode (A)

2202 ⊥= qmkf

202 ⊥= qkqQ IIII

- modulation depth- geometry

Normalization

Analytical curve for f<<1

:1<<f

( ) +−−≈ 322 181 IIQfd

( ) ...164 54 +− IIQf

...3421 343202

++−≈=

ffQdII

0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

1.6

2.0

f

f=0.04

f=0.12

f=0.28

f=0.41

f=0.52

f=0.67

f=0.90

f=1.20

f=1.40

1

1

0-1

K

Q

K

K||

K

3 relevant harmonics

Transverse dispersion relation: Power series expansion

Numerics

Homogeneous material (diffractive)

Photonic crystal(non-diffractive)

2D

Input

Output

Input

Output

Homogeneous material (diffractive)

Photonic crystal(non-diffractive)

Numerics 3D

Input

Outputs

InputDiffractive

outputNon-diffractive

output

Linear effect

Subdiffraction. Analytic

( ) ∑∞

=⊥⊥ =

0n

nnII KdKK n

IIn

n KKnd ⊥∂∂⋅= !1

( ) ( ) 0,444 =∂∂+∂∂ ZXAXdZi B

( ) ...132 524 +−≈ IIQfd

...3404

2+≈ −

=fd

d

( )204

4 8)( ⊥≈∆ qkzdzx π

02 →d

Bloch mode evolution:

0=⊥k

Width evolution:

n even

4)( zazx =∆Example: Beam width=10µm , λ=10µm

Rayleigh length 30µm

Nondiffractive propagation length2m

•Homogeneous media

•Non-diffractive PC λ

π2

00

Wz =

00

2||

2

=∂∂

=⊥⊥ kkk

Subdiffraction. NumericsLarge propagation distance

The smallest non-diffractive structures

Technical applications

Smallest possible structures

-1.0 0.0 1.0-1.0

0.0

1.0

k

∆k∆kk

Non-diffractive propagationlength

Effective diffractionlower than a given value

Plateau width∆K||=0.05

||kL ∆= π

⊥∆=∆ kxmin 2

2D PC3D PC Diferent configurations: n-gons

∑−

=⊥⊥ ++=∆

1

0|| )cos()cos(4)(

n

llyqxqzqmrn ϕβαr

ϕl does not effect the dispersion curve

More symetric, less distorted propagated patterns ?

n

kz isolines

ky

Different PC structures

Q||

f kx

3D Nondiffractive Photonic Crystals

)/2sen()/2cos(nlnl

πβπα

==

Quasicrystal, bessel beam

Experiments with PCs

Nd:YAG 8 ns 300 mJ1064 nm

Spatial filtering and delay

Optical latticeFormation (2D)

Photorefractive crystal20 mm SBN:CeO2

CCD

532 nmProbe beam

- - -+ + + + - - + + -

Light interferences

Charges

Electric field

Refractive index variations

Λ~µm

ϕ=π/2

)(En∆

Light beams

•Photorefractive materials

Wavelength = 532 nmInitial width = 12.6 µm

Propagation length = 4.25 mm

∆n modulation amplitude= 5 10-4

Spatial periodicity: Λ⊥ =7.6µm , Λ||= 483 µm

•Simulations with experimental parameters

•Beam:

•Photonic crystal:

Homogeneous material

Photonic crystal

•Microscopy, photolitography, ...

•Fibers transporting patterns•Wave-guides in electronic circuits•....

Possible applications:

1 fiber transport 1 bit (light or no light)

1 fiber transport 1 pattern

Diffractive fibers Nondiffractive fibers

•Multimode nondiffractive wave-guides

Shift in x

Lens effect

Bending

Multimode Nondiffractive Fibers and Wave-Guides:

Radi de curvatura = 9 x diàmetre fibra

2202 ⊥= qmkf

202 ⊥= qkqQ IIII

- modulation depth;- geometry;

ck 00 ω=

0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

1.6

2.0

f

f=0.04

f=0.12

f=0.28

f=0.41

f=0.52

f=0.67

f=0.90

f=1.20

f=1.40

1

1

0-1

K

Q

K

Non-diffractive pulses

( ) ( ) 0,,,2

102

2

0

=

∆−

∂∂−

∂∂+

∂∂ tzxAkzxni

xki

ztc

δω

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.1

0.2

0.3

0.4

IIK

⊥K

Small variations of the carrierfrequency ω=ω0(1+δω)

•Carrier frequencies interval

04

12

2

2

2

00, =

∂∂

∂∂−

∂∂+

∂∂+−

∂∂ A

XTTi

TViK

Z II α

22

00, 4 ⊥⋅+++= KV

KK IIII δωαδωδω ( )4

1 20,

0,II

II

QK

−=

( ) ( )2

311 0,0,

0

IIII QQV

−⋅−=

( )0,13

IIQ−=α

Small index modulations (f<<1)

Input Output

•Gaussian pulse in x and t Propagation length: 375 µm

Integration of the main equation

( ) ( ) 0,,,2

102

2

0

=

∆−

∂∂−

∂∂+

∂∂ tzxAkzxni

xki

ztc

Propagation length: 0.45 mm

•Gaussian pulse in x and t

Filtrat per treure les oscil·lacions degudes a la modulació de ∆n

InputOutput with PC

Output without PC

( ) constKKII =⊥ δω,

⊥K

δω

⊥K

δω

0 0

-1-1 -10 0

1 1

For different groupvelocities V0

Invariant spacio-temporal shapes

Adding modes of the sameisoline

No dispersive, no diffractivePropagation

22

00, 4 ⊥⋅+++= KV

KK IIII δωαδωδω

Non-diffractive resonators

•Homogeneous spacer

•Free propagationalong distance L

•Non-diffractive cavitySpacer filled with PC

R=0.9

Input beam

L

Gaussian spectrum filteredby the resonator

Broader beam

Output beams

ϕiertkAkA 2

2

0 1)()(

−= ⊥⊥

2

0

2

22

)( ⊥⊥

⊥ −=−≅ dkLkkkϕ

reflection r, transmission t and phase shift ϕ of the cavity

Different ϕ for each transverse component

effective diffraction of the cavity d=λL/2π.

r,t r,tInput beam Output beam

Homogeneous spacerHomogeneous resonators

Gaussian spectrum filtered by the resonator

•Transmitted field

•Scann of the cavity length

400 λ 400.5 λ

Output intensityRings

w0input width

•Rings

•Free spectral range: λ/2+a , a(w0)

•Output width depends on L

Diffraction

Maximum of intensity Waist

Free spectral range

Photonic crystal spacer

Photonic crystal resonators

•No Rings. Only the central peak remains (all transversewaves in phase).

•Free spectral range does not depend on w0.

•Output width does not depends on L.

Propagation in Nonlinear PC’s (1D transvers)

( ) 0,22 2202

2

0 =

−∆+

∂∂+

∂∂ AAckzxn

xzik

W0=5

0.5

1.5

0

0

Output solitonInput beam

Width

Intensity

1

0

1

0

d2 0

W0=3

2

2

xA

∂∂ 2Ac

Solitons

Diffraction Nonlinearity

Solitons: Amplitude-Width relationship

Near the linear non-diffractive curve

( ) 0,22 2202

2

0 =

−∆+

∂∂+

∂∂ AAckzxn

xzik

AACX

dX

dTAi B

+−= 2

4

4

42

2

2 ∂∂

∂∂

∂∂

02 >d

02 <d

Modulation wavenumber

Modulation amplitude

0.5

1

1.5

25 50 750x

2A

a)

25 50 750

0.5

1

1.5

2

2.5

x

2A

b)0.8 1.2 1.4 1.6 1.8

0.9

1.1

1.2

1.3

1.4

( )010 ELog

( )010 xLog

a)

b)

constxE =⋅ 0

constxE =⋅ 30

Nondiffractivecurve

Nonlinear PCs (2D transvers)

( ) 0,22 220

20 =

−∆+∇+

∂∂ AAckzxnz

ik

( )AACddZAi B

244

22 +∇−∇=

∂∂

NLSE colapses in 2D and 3D . No stable solitons appear.

Subdiffractive NLSE

Near d2 0:

01−

5.0−

5.01

5.055.06.0

65.0

0

1−5.0−

5.01YK

zK

XK

different geometries:

n=4 not usefull to stabilize nonlinear solitons

kz isolines

ky

Q||

f kx

Nondiffractive curves

NLSE: ( )AAditA

eff22 −∇=

∂∂

Stable solitons•Analyticaly•numericaly

V(x)

x

ψ (x)

Light beam Bose-Einstein Condensate

Stable solitons in 2D and 3D

Bose-Einstein Condensates

•Optical nonlinearities•Bose-Einstein Cond. (BEC’s)

NLSE: ( )AAditA

eff22 −∇=

∂∂( ) 0,22 22

02

2

0 =

−∆+

∂∂+

∂∂ AAckzxn

xzik

( )AACddTAi B

244

22 +∇−∇=

∂∂

A ψ

Diffraction Effective mass

Non-diffractive Nonlinear resonators

•Nonlinear resonators:

QLdsize ⋅⋅≈≈ λ2/1

Homogeneous material

...)(...)( 2 +∇+=∂∂ rAidtrA

mµλ 1≈ mL µ10≈ 100≈Q

mx µ300 ≈ Taranenko, Weiss, Kuszelewicz , 2000

CV in VCSELs, lasers with saturableabsorber, OPOs, Photorefractives,

gain

intensity

Cavity Solitons

Patterns and dissipativestructures;

Switch on/off in any point

•Memory •Processing

...3421 343202

++−≈=

ffQdII

0.5 0.1 0.15 0.2 0.25 0.3 0.35

0.5

0.6

0.7

0.8

0.9IIQ

f20 40 60 80 100

0.2

0.4

0.6

0.8

1

1.2

Reduction of spatial scale

λ≈size

•Nonlinear photonic crystal resonators:

Modulation of refraction index

AB

C

ABC

Bibliography

•E. Yablonovitch, 1987; S. John, 1987;

•H.S.Eisenberg et al. 2000; R. Morandotti et al . 2001;

•Pertsch et al. 2002; O. Zobay et al. 1999;

•Yu.S.Kivshar et al. 2003; H.Kosaka e.a. 1999,

•J.Witzens e.a. 2002, D.N.Chigrin e.a. 2003,

•R.Iliew e.a. 2004

•K.Staliunas, Mid-band Dissipative Spatial Solitons, Phys. Rev. Letts, 91, 053901(2003)

•K. Staliunas and R. Herrero, Nondiffractive propagation light in photonic crystals,Phys.Rev.E 73, 023601 (2006) ; subm. 2004, arXiv: physics/0501115.

•K. Staliunas, C.Serrat, C.Cojocaru, J.Trull and R. Herrero, Nonspreading Light Pulses in Photonic Crystals , subm. 2005, arXiv: physics/0505153, accepted in PRE

•K.Staliunas, R.Herrero, G.Valcarcel and N.Achmediev, Subdiffractive Solitons in Two-Dimensional Nonlinear Schrödinger Equation 2006, submitted

•K.Staliunas, G.Valcarcel and R.Herrero, Sub-Diffractive Band-Edge Solitons in Bose-Einstein Condensates in Periodic Potencials , 2006, submitted

642 0 ∇⇒=∇=∇(Q||,f) with

2202 ⊥= qmkf

202 ⊥= qkqQ IIII

- modulation depth;- geometry;

ck 00 ω=

0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

1.6

2.0

f

f=0.04

f=0.12

f=0.28

f=0.41

f=0.52

f=0.67

f=0.90

f=1.20

f=1.40

1

1

0-1

K

Q

K

•Sub-sub-diffractive?

•Nondiffractive-nondispersive pulses?

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