interactions in PT-symmetric nonlinear lattices with gain and loss - - PowerPoint PPT Presentation

Optical beam localization and interactions in PT-symmetric nonlinear lattices with gain and loss

Andrey A. Sukhorukov

Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, Australia

Andrey Sukhorukov, Zhiyong Xu, Yuri S. Kivshar Nonlinear Physics Centre, Australian National University, Canberra, Australia Sergey Dmitriev, Sergey Suchkov Institute for Metals Superplasticity Problems, Russia

Team and collaborators

Outline

• Concept of parity-time (PT)-symmetry and

recent progress in optics

• Nonlocal effects: PT-symmetry breaking

sensitive to distant structure boundaries

• Scattering of solitons on PT-symmetric

couplers

PT Hermitian

Gain Loss

PT symmetry in optics

• Coupled waveguides
• Parity-Time symmetry:

identical waveguides, equal magnitudes of gain and loss

El-Ganainy et al., Opt. Lett. 32, 2632 (2007); Guo et al., Phys. Rev. Lett. 103, 093902 (2009); …

• Observed experimentally Ruter et al., Nature Physics 6, 192 (2010).

Gain Loss

Coupled waveguides with gain and loss

• Parity-Time symmetry:

identical waveguides, equal magnitudes of gain and loss

• El-Ganainy et al., Opt. Lett. 32, 2632 (2007); Guo et

al., Phys. Rev. Lett. 103, 093902 (2009); Ruter et al., Nature Physics 6, 192 (2010).

• PT symmetry: supermodes do not

experience gain or loss; zero gain/loss on average for arbitrary inputs

• Broken PT symmetry (increased

gain and loss): mode confinement and amplification in the waveguide with gain

Gain Loss

Ruter et al., Nature Physics 6, 192 (2010).

Nonlocal effects

• PT-defect – non-Hermitian
• Quantum-mechanical context: interaction of a non-Hermitian

system with the Hermitian world

• Dynamics can be sensitive to potential at distant locations
• Continuing debate on the meaning of nonlocality and

relevance to real physical systems

• H. F. Jones, Phys. Rev. D 76, 125003 (2007); M. Znojil, Phys. Rev. D 80,

045009 (2009); …

PT Hermitian

PT nonlocality in optical systems?

Our approach to study nonlocal effects

• Consider pair of waveguides with balanced gain and loss in

a chain of waveguides

• Realizes PT defect embedded in a conservative lattice
• Compare different topologies: planar and circular
• Study the degree of nonlocality due to distant boundaries

Sukhorukov, Dmitriev, Suchkov, Kivshar, Opt. Lett. 37 37, 2148 (2012)

Mathematical model

• aj – mode amplitudes at waveguides
• C – coupling coefficient between the waveguide modes
•  – coefficient of gain/loss in waveguides 0,1

PT symmetry breaking for isolated coupler

• Stability condition
• Predicted and observed previously
• PT symmetry: supermodes do not

experience gain or loss; zero gain/loss on average for arbitrary inputs

• Broken PT symmetry (increased

gain and loss): mode confinement and amplification in the waveguide with gain

Ruter et al., Nature Physics 6, 192 (2010).

Gain Loss

PT symmetry breaking for planar lattice

• Boundary conditions
• Consider eigenmodes:
• PT symmetry:
• For
• For
• Consider
• Solvability of last relation defines PT symmetry

PT symmetry breaking for planar lattice

• Stability condition
• Same stability condition as

for isolated PT coupler!

• Does not depend on lattice

coupling outside the active region

PT symmetry breaking for circular lattice

• Boundary condition:
• Waves can circulate around,

passing through active waveguides

• Eigenmode as sum of counter-propagating waves
• ‘+’ – n1; ‘-’ – n  0
• Wave scattering at PT defect:
• Transmission coefficient
• Reflection coefficient
• Boundary condition:

PT symmetry breaking for circular lattice

• Consider ratio
• PT symmetry breaking occurs at a given k

when solutions disappear

• Threshold corresponds to real k
• Stability condition:
• Threshold depends on

all lattice parameters

Combined stability diagram

• Different stability domains in regions 1 and 2 for arbitrary

large lattice lengths

• Nonlocality irrespective of the lattice size!

Stability regions 1,3 Stability regions 1,2

Instability in infinite lattices

• Construct modes localized at PT defect
• Exponential localization:
• For
• Which requires
• Modes cease to exist

– PT symmetry breaking

Instability in infinite lattices

• Common instability threshold for infinitely long planar or

circular lattices

• Stability condition:

Instability growth rate

• Top - N=20 sites
• Bottom N=100 sites
• Dashed line –

infinite lattice threshold

• Above dashed line

– almost no depednence on lattice size

• Solid line – finite

lattice threshold

• Between solid and

dashed lines – instability reduces as 1/N

Instability sensitive to boundaries

• Consider

N=50 sites – solid lines

• /C2 = 0.8

Instability for planar structure only through reflections from boundaries

Instability independent on boundaries

• Consider

N=50 sites – solid lines

• /C2 = 2

Instability develops around PT defect, no effect of boundaries

PT coupler in a nonlinear chain

• Distant boundaries (infinite lattice limit)
• Kerr-type nonlinearity
• Conservative solitons exist on either sides of PT coupler

Soliton scattering by PT coupler

• Lattice parameters
• Soliton velocity
• Localized mode at PT coupler is excited when soliton

amplitude is increased (right)

Soliton scattering by PT coupler

Soliton scattering

• Black circles –

transmission, open circles – reflection

• f solitons
• Lines – linear

regime PT mode excitation

• Power of the

localized mode at PT coupler, after the soliton transmission

Controlling soliton scattering with localized PT modes

Soliton scattering

•  - soliton phase
• Labels – localized PT

mode amplitude PT symmetry breaking

• Mode amplitude 1.4
• Left:  =3.67

PT symmetry preserved

• Right:  =3.75

nonlinear PT symmetry breaking

PT-symmetric structures: waveguides with balanced gain and loss Conclusions Nonlocal effects: different stability for planar and circular structures with gradual transition to a common threshold for infinite lattices Soliton gain/loss due to scattering

• n PT defect, controlled by PT

defect mode

Dmitriev, Suchkov, Sukhorukov, Kivshar, Phys. Rev. A 82 82, 013833 (2011) Sukhorukov, Dmitriev, Suchkov, Kivshar, Opt. Lett. 37 37, 2148 (2012)

Gain Loss