30.3.2006 FunFACS meeting, Toulouse 1

USTRAT

WP 1 Theory

WP 2

2

WP 1 Theory

1. Cavity solitons in a VEGSEL with frequency-selective feedback

P. V. Paulau1,2, W. J. Firth1, T. Ackemann1, Andrew Scroggie1,

A. V. Naumenko2, N. A. Loiko2

1Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK2Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus3INFM, Como, Italy; 4 currently enjoying Italy

2. Master equation for describing VECSELs with intra-cavity elements

L. Columbo 1,3, A. Yao1, W. J. Firth1

3. A new method for adiabatic elimination

G.-L. Oppo1,4

3

CS in a VEGSEL

P. V. Paulau1,2, W. J. Firth1, T. Ackemann1, Andrew Scroggie1,

A. V. Naumenko2, N. A. Loiko2

1Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK2Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus

4

VEGSELTo keep things simple: only one complex equation for linearly polarized field

loss diffraction scaling phase-amplitudecoupling feedbackgain saturation

finite gain bandwidth

feedback: self-imaging, diffraction grating (envelope of filter: sinc-function) one round-trip in external cavity

note: there are no temperature effects in this model

we assume resonance in external cavity: 0 = m 2

5

Stationary states=0, only k = 0

supercritical bifurcation bistability between off-state and lasing states „old“ model with temperature: subcritical

Naumenko et al., Opt. Commun. 259, 823 (2006)

=0, with k > 0

bistability disappears no CS need for filtering

6

Structures with spatial filtering I

assume spatial filter in some Fourier plane in feedback loop

filter due to gain curve

spatial filter in feedback loop

=0

=0.06

f=0.06

f=0

f = 0.06

pattern forming instability

potential for CS

7

CS

real part intensity far field

this is a localized traveling wave !

is exponentially localized exists on grids 64, 128, 256

energy=azimutally integrated intensity

r (pixels)

log

(e

nerg

y)

f = 0.06

8

CS II

can exist at different locations in the plane

several LS can coexist

present or absent under the same conditions

seems to be a self-localized solution, a true cavity soliton

9

Symmetric structurereal part intensity far field initial condition:

homogeneoushigh-amplitudestate on zero background

not a Bessel beam: energy decays exponentially

is this related to experiment ?

10

CS for detuning zero !?

direct excitation of CS apparently not possible,but taking CS from f = 0.06 as initial conditiona localized solutions is obtained

f = 0

real part NF far field cut through FFazimutally averaged FF

11

Summary: VEGSEL theory

very exciting result: localized traveling wave

symmetric and asymmetric

relationship with experiment unclear: temperature, spatial filter ....

need to translate to physical parameters

12

Master equation

1Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK3INFM, Como, Italy

L. Columbo 1,3, A. Yao1, W. J. Firth1

idea: derive a closed equation for dynamics of nonlinear non-plano-planar resonators by using ABCD matrix to decribe intra-cavity elements

benefits: ability to model complex real-world cavities (e.g. VECSELs)

address effects of small deviations from self-imaging condition in external cavity describe properly action of grating in VEGSEL significance for WP1 and WP2

Dunlop et al., Opt. Lett. 21, 770 (1996)

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Thin lens::focal length=f

Nonlinear medium:Vapour of two levels atoms

Reference symmetric plane

PerfectMirror

PerfectMirror

Master equation for an unidirectional square nonlinear resonator driven by a coherent injected field

Gaussian aperture: FWHM=w

Injected field:wave vector=k

Mirror

Mirror

Acoskwf

Lf2i

f

Lf2C

kwf

LfL4Lf42i

f

LfL3Lf22B

kwf

LfL3Lf22i

f

L2fL4fDA

22

2

2

22

6422

2

322

22

322

2

22

L

L/2

Exact linear propagation taken into account by means of the ABCD matrix at the reference plane:

Instantaneous nonlinear medium located atthe reference plane

Small variation per cavity round trip TR of the adimensional field envelope E0(T,r) at the reference plane. Intracavity field carrier wave vector=k

NOTE: For f,w→∞ we get the Mean Field Limit equation considered for example in ref. M.Brambilla et al.,EPL 34, 1996 and in ref. W. J. Firth and A. J. Scroggie PRL 76, 1996

Diffusion Diffraction Space dependent gain|loss

Linear absorption

and dispersionSaturable

absorption <0

Injected field

inj020

002

02

020

R YE|E|1

E)i1(ErkCsin2

iReE

k

B

sin2

iImiE

k

B

sin2

iRe

T

ET

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Spontaneous emerging of intensity spots in the near field Parametric regime=-1, =-10.8, Yinj=6.52

Observation: In case (a) and (b) we managed to switch on and off a single intensity spot by superimposing a suitable addressing pulse to the holding beam !!

(a). Without Gaussian aperture (initial condition: null intracavity field+Gaussian distributed white noise)Time=9TR Time=23TR Time=541TR Time=280TR

(b). With Gaussian aperture (initial condition: null intracavity field+Gaussian distributed white noise)

Time=11TR Time=22TR Time=37TR Time=280TR

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WP 2

1Department of Physics, SUPA, University of Strathclyde, Glasgow, Scotland, UK2Institute of Physics, Academy of Sciences of Belarus, Minsk, Belarus

Y. Tanguy1, A. Smith, T. Ackemann,

F. Papoff, A. Scroggie, A. Yao1, W. J. Firth1

P. V. Paulau1,2, A. V. Naumenko2, N. A. Loiko2

VEGSEL with long cavity (task 2, overlap with WP1)

Planning and test setup for VCSEL + SA (task 1)

Modelling VEGSEL (task 2, overlap with WP1)

Modelling fast spatio-temporal dynamics with extended master equation (task 1, possibly 2)

Modelling coupled cavities (task 1, possibly 2)

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VEGSEL with long external cavity

actually setup from WP1 Yann showed (see also next slide)

• spot can be stationary (no peaks in RF- spectrum and FP)• spot can have weak sidemodes in FP• spectrum shows clearly strong excitation of sidemodes in FP

and peaks in RF spectrum (possibly due to background)• round-trip frequency 250 MHz

need for further analysis,

but potentially the spots would also qualify as CLB God knows how (ir)regular these might be

(simplified) model developed and coded from WP1 need to reintroduce carrier dynamics for reasonable results

17

Spectra of spots

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Transfer to cavity light bullets

HR

HR

BSR 2-8%

self-imagingforward biased

lasera) reverse biased laser

(reduced reflectivity)b) QW SESAMc) QD SESAM

(reduced saturation fluence, no demagnification necessary)R0.8-0.985

R0.8-0.985

f18 mm

SAgain

possibilities:

f2100 mm

here: cut-off feedback to boundaries

f3200 mm

f45 mm

demagnification by factor of 3: adapt saturation fluences

set-up not yet tested, but could be done in week before Eastern, if considered to be necessary for annual report

R=0.8 about factor of eight in gain; R=0.9 about factor of four, R=0.985 ok

19

Spectrum of SESAM

940 960 980 1000 1020 1040 10600.0

0.2

0.4

0.6

0.8

1.0

060323_SBR.opj

EPIC 2000532

R

wavelength (nm)

very high absorption losses, possibly not useful

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Extended Master equation

Dunlop et al., Opt. Lett. 21, 770 (1996)

ABCDnonlinearity

changeson time scale longer than round-trip time

changeson time scale of pulse

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Coupled cavities

coupling mirror between two Fabry-Perot cavities

transfer matrix

coupled master equations

only valid, if no variations on time-scale of round trip in both cavities (quasi single-longitudinal mode)

How to couple „extended“ master equations?

How to include inertia of medium?