Transcript Peschel,Egorov, Lederer 2004

JAROSLAW E. PRILEPSKY
Nonlinearity and Complexity Research Group
Aston University, Birmingham, UK
In collaboration with:
Alexey Yulin
Centro de Fisica Teorica e
Computacional, Universidade
de Lisboa, Portugal
Magnus Johansson
Stanislav Derevyanko
Dep. of Physics, Chemistry and
Biology (IFM), Linkoping University,
Sweden
NCRG, Aston University, Birmingham,
UK
Light manipulation in optical cavities
Localized bright spots in driven optical
cavities have received a great deal of
attention because of their potential
applications in information processing
Ackemann, Firth, Oppo, 2009.
A relatively new area is the study of
collective excitations in coupled
nonlinear cavities (resonators): coupled
waveguides with the facet mirrors.
Coupled-mode equations
e.g. ,Peschel,Egorov, Lederer 2004
Discrete nonlinear coupled cavity models
Discrete mean field equations for cavity arrays (a passive case!):
A discrete Lugiato-Lefever model Peschel,Egorov, Lederer 2004
An effective model for quadratic cavity solitons Egorov, Peschel, Lederer 2005
A model with a saturable conservative nonlinearity Yulin, Champneys, Skryabin 2008
A model with saturable non-conservative nonlinearity Yulin, Champneys, 2010
dissipative terms
Our model: active nonlinear media
0.4
Active media: gain exceeds
damping in the linear limit:
0.2
0.0
δ<γ
0.2
0.4
0.6
0.0
0.5
1.0
1.5
amplitude
2.0
2.5
3.0
Analysis: bistability as a starting point
Strategy: Set An=A (or set C=0, anticontimuum limit), and study the
response curve P=P(|A|).
When we have a multivalued curve (bistability), we can find solitons as
homoclinic connections between stable states.
Set An(T)=An+an(T) and linearize F(A+a(T)) with respect to an(T). Set
an(T)=a exp(λT+iqn) + b* exp(λ*T-iqn) and study the resulting
eigenvalues λn=λn(q,parameters). If there are any Re[λn]>0 – unstable,
otherwise - stable.
UNSTABLE
H1
H2
STABLE
Seek for stable solitons at C≠0 in the
bistable region starting from
decoupled stable states H1 and H2
Response curve and spectrum. I
Bright solitons. I: C-snakes
M=
𝑛
|𝐴𝑛 − 𝐴𝑏𝑎𝑐𝑘𝑔𝑟 |2
Bright solitons. II: H1→H2→H1
Grey solitons. I: H2→H1→H2
Snaking diagrams
C(|Amin|) and P(|Amin|)
(for C=0.15) for the grey
DCS corresponding to
the H2-H1-H2
connection.
Inhomogeneous (periodic) background. I
C=0.15
Inhomogeneous (periodic) background. II:
Comparison
C=0.15
Snaking diagrams P(|A|) for
homogeneous H states and
{P(|Amax|), P(|Amin|)} for the
periodic I-state. Bistability region
for H-states is highlighted.
Bright solitons. III: H1→I→H1
Snaking diagrams
P(|Amax|) and
M(P) for H1→I→H1
(bright) solitons,
C=0.15; profiles
of stable DCSs.
Grey solitons. III: H2→I→H2
Snaking diagrams P(|Amax|) and M(P) for H2→I→H2 (grey) solitons,
C=0.15. Inset shows a stable solution profile.
Conclusions
We have found a zoo of stable DCS in coupled active lasing cavities. Aside from
`usual' DCS, corresponding to the connections between homogeneous states
H1 and H2, we have found a new type of DCS involving a periodic
inhomogeneous I-state, which has not identified before in optical cavities. The
existence of the great variety of stable DCS paves the way to the more versatile
and sophisticated patterning and manipulation of transverse light distribution.
Notably, the family of grey H2-I-H2 solutions marked as ■‘9' can be stable
when the bistability of H-states is absent.
Further challenges: solitons in the absence of bistability, quasilinear solitons (no
conservative nonlinearity), compare properties of new DCS with the usual ones
(dynamics etc.) Other models (Lugiato-Lefever etc.)
Ref: http://arxiv.org/abs/1202.4660 (submitted to Optics Letters)
THANK YOU!