S and Q Matrices Reloaded · Département de physique, génie physique et optique Université...

S and Q Matrices ReloadedApplications to Open, Inhomogeneous and Complex Cavities

G. Painchaud-April, J. Dumont,D. Gagnon and L. J. Dubé

Département de physique, génie physique et optiqueUniversité Laval, Québec, Canada

ICTON 2013Cartagena, Spain

Outline

1 Scattering MatrixResonances / Quasi-Bound StatesEnergy FormalismTime Delay

2 Numerical Construction

3 ApplicationsNumerical ResultsOutlooks

4 Conclusion

J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 2 / 28

Scattering Matrix

Section Outline

4 Conclusion

Scattering Matrix Resonances / Quasi-Bound States

Section Outline

4 Conclusion

What is Resonance?

�ψinc�

�ψsca�

Outside the dielectric, we solveHelmholtz’ equation:

∇2 + n2k2��

�ψ�

�ψsca�

= S�

�ψinc�

Poles of |detS(k)| with k ∈ C(S(k) = representation of S ).

What is Resonance?

�ψinc�

�ψsca�

∇2 + n2k2��

�ψ�

�ψsca�

= S�

�ψinc�

What is Resonance?

�ψinc�

�ψsca�

∇2 + n2k2��

�ψ�

�ψsca�

= S�

�ψinc�

What is Resonance?

�ψinc�

�ψsca�

∇2 + n2k2��

�ψ�

�ψsca�

= S�

�ψinc�

What Is Resonance?

Can we find a set of real energy levels for anopen dielectric cavity?

Scattering Matrix Energy Formalism

Section Outline

4 Conclusion

Delay due to the Cavity

Average E.M. Energy

∫∫∫

[εE∗ · E+µH∗ ·H]d3r

Outside the cavity (r> R0), the modes are

ψm = H(−)m (nokr)eimθ +

Sm`H(+)`

(nokr)ei`θ

Coupling between modes:

EVmm′ =

∫∫∫

εE∗m · Em′

+µH∗m ·Hm′�

Average E.M. Energy

∫∫∫

Sm`H(+)`

(nokr)ei`θ

EVmm′ =

∫∫∫

εE∗m · Em′

+µH∗m ·Hm′�

Average E.M. Energy

∫∫∫

Sm`H(+)`

(nokr)ei`θ

EVmm′ =

∫∫∫

εE∗m · Em′

+µH∗m ·Hm′�

E∞mm′ = limRV→∞

4n0RVw

kδmm′ +

−i∑

S∗`m∂ S`m′

E0mm′ : diverging free space

energy of the beam.

Q =−iS† ∂ S∂ k: complex coupling

between angular momentumchannels → excess energy dueto the cavity → delay times.

4n0RVw

kδmm′ +

−i∑

S∗`m∂ S`m′

energy of the beam.

4n0RVw

kδmm′ +

−i∑

S∗`m∂ S`m′

energy of the beam.

Scattering Matrix Time Delay

Section Outline

4 Conclusion

Resonances ↔ Time Delay

Characteristic Modes of the CavityThe field is given by

ψp =∑

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.

Setting up the eigenvalue problem:

QAp = τpAp.

τp are the delay times and represent the time the energy of themode described by Ap spends inside the cavity.Resonances appear as peaks in the delay spectrum (τp vs. k).

ψp =∑

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.

QAp = τpAp.

ψp =∑

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.

QAp = τpAp.

τp are the delay times and represent the time the energy of themode described by Ap spends inside the cavity.

Resonances appear as peaks in the delay spectrum (τp vs. k).

ψp =∑

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.

QAp = τpAp.

cτ/R0

5.3 5.35 5.4 5.45 5.5100

Numerical Construction

Section Outline

4 Conclusion

Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.

Divide in concentric shellsSolve angular part in FourierseriesConnect the shellsIsolate S from B = SA.

A. I. Rahachou and I. V. Zozoulenko, Appl. Opt. (43), 1761 (2004).

Divide in concentric shells

Solve angular part in FourierseriesConnect the shellsIsolate S from B = SA.

A. I. Rahachou and I. V. Zozoulenko, Appl. Opt. (43), 1761 (2004).J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 13 / 28

Divide in concentric shells

Solve angular part in FourierseriesConnect the shellsIsolate S from B = SA.

Divide in concentric shellsSolve angular part in Fourierseries

Connect the shellsIsolate S from B = SA.

Divide in concentric shellsSolve angular part in FourierseriesConnect the shells

Isolate S from B = SA.

Divide in concentric shellsSolve angular part in FourierseriesConnect the shellsIsolate S from B = SA.

Applications

Section Outline

4 Conclusion

Applications Numerical Results

Section Outline

4 Conclusion

Example Applications of the Method

Ellipse (integrable geometry)

6.48 6.49 6.5 6.51 6.5210

J. Unterhinnighofen et al, Phys. Rev. E(78), 016201 (2008).

| Im�

|= 0.032

kR0 = 6.499

2R0/cτ= 0.029

Square (corners + high delay)

4.5 4.52 4.54 4.56 4.58 4.610

W.-H. Guo et al, IEEE J. of Qu. El. (39),1106 (2003).

= 4.54

| Im�

|= 1.05× 10−4

kR0 = 4.53

2R0/cτ= 1.06× 10−4

Stadium (composite structure)

4 4.5 5 5.510

S.-Y. Lee et al, Phys. Rev. A (70), 023809(2004).

= 4.89

| Im�

|= 0.055

kR0 = 4.89

2R0/cτ= 0.052

Photonic Complex

5.3 5.35 5.4 5.45 5.510

(A) (B)(C)

(a)(b)

J.-W. Ryu et al, Phys. Rev. A (79), 053858

(2009).

Annular cavity (closed-form, holey cavity)

10.1 10.15 10.2 10.25 10.310

(14, 5)(A)

(7, 8)(B)

Applications Outlooks

Section Outline

4 Conclusion

Applications Outlooks

Outlooks

TE Modes & µ 6= 1Equations to solve inside thecavity become

∇2 + n2k2�

µ∇µ · ∇Ez

∇2 + n2k2�

ε∇ε · ∇Hz

for TM and TE, respectively,and boundary conditionsdepend on both µ and ε.

Full 3DPossible and theoreticallysimilar, but poses somenumerical challenges.Steady-state ab initio lasertheory (SALT)Quasibound states as guides toconstant flux states.

Conclusion

Section Outline

4 Conclusion

Conclusion

Primary FindingsScattering description for the S-matrix and its associated delaymatrix Q

Characteristic modes = eigenvectors Ap of the Q-matrix thatundergo a simple phase shift in the presence of the cavity, i.e.self-replicating waves.Resonances = peaks in τp vs. k (form a subset of the characteristicmodes).Computable quantities: Field in all space, resonance position (realwavenumber), resonance width (real delay) and quality factor Q.Approach applicable to open, continuously inhomogeneous cavitiesof arbitrary geometry.

Conclusion

Characteristic modes = eigenvectors Ap of the Q-matrix thatundergo a simple phase shift in the presence of the cavity, i.e.self-replicating waves.

Resonances = peaks in τp vs. k (form a subset of the characteristicmodes).Computable quantities: Field in all space, resonance position (realwavenumber), resonance width (real delay) and quality factor Q.Approach applicable to open, continuously inhomogeneous cavitiesof arbitrary geometry.

Conclusion

Characteristic modes = eigenvectors Ap of the Q-matrix thatundergo a simple phase shift in the presence of the cavity, i.e.self-replicating waves.Resonances = peaks in τp vs. k (form a subset of the characteristicmodes).

Computable quantities: Field in all space, resonance position (realwavenumber), resonance width (real delay) and quality factor Q.Approach applicable to open, continuously inhomogeneous cavitiesof arbitrary geometry.

Conclusion

Characteristic modes = eigenvectors Ap of the Q-matrix thatundergo a simple phase shift in the presence of the cavity, i.e.self-replicating waves.Resonances = peaks in τp vs. k (form a subset of the characteristicmodes).Computable quantities: Field in all space, resonance position (realwavenumber), resonance width (real delay) and quality factor Q.

Approach applicable to open, continuously inhomogeneous cavitiesof arbitrary geometry.

Conclusion

Characteristic modes = eigenvectors Ap of the Q-matrix thatundergo a simple phase shift in the presence of the cavity, i.e.self-replicating waves.Resonances = peaks in τp vs. k (form a subset of the characteristicmodes).Computable quantities: Field in all space, resonance position (realwavenumber), resonance width (real delay) and quality factor Q.Approach applicable to open, continuously inhomogeneous cavitiesof arbitrary geometry.

Conclusion

Acknowledgements

The ICTON 2013 organizing committeeThe Canada Excellence Research Chair in Photonic Innovations(CERCP), especially Younès Messaddeq and Jean-François Viens.The Natural Sciences and Engineering Research Council of Canada(NSERC)Computations were made on the supercomputer Colosse fromUniversité Laval, managed by Calcul Québec and Compute Canada

Convergence

Section Outline

5 Convergence

Convergence

Homogeneous Disk

10−510−410−310−210−1100101

kR02ε

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

kR0 = 2.50

kR0 = 5.00

kR0 = 10.00

kR0 = 20.00

kR0 = 40.00

kR0 = 80.00

Figure 1 : Maximum error in the elements of the numerical scattering matrixwith respect to the size of the annuli.

S and Q Matrices Reloaded · Département de physique, génie physique et optique Université...

Documents

Physique 1 Benson Physique - Furet

La physique aujourd'hui - Cristaux et physique

2- education physique -CHD:education ducation physique - 1 2 Éducation physiqueÉducation physique a. Définition L’éducation physique couvre un ensemble de pratiques pédagogiques

SFR WARS - Reloaded

Physique Modélisation Chimie 2017 · Sommaire thématique de physique 2015–2017 X/ENS PC Physique B X PC Physique A X MP Physique et SI X/ENS MP Physique Mines PSI Physique 2 Mines

Département de physique - · PDF fileDépartement de physique Cours « Interface physique-chimie » Dispositifs électroniques (éléments de physique des isolants, des semi-conducteurs

Nonlinear Resonances in a Forced Modified Rayleigh-Duffing ...ijseas.com/volume3/v3i9/ijseas20170906.pdf · Nonlinear Resonances in a Forced Modified Rayleigh-Duffing Oscillator

Salon Resonances 7 au 11 novembre 2014 Strasbourg

Neo4j - Reloaded · Base de données NoSQL orientée Graphe ... Le graphe Neo4j stocke nativement les données dans un graphe ... Langage d'interrogation natif Neo4j MATCH (je retrouve)

Physique Modélisation Chimie 2016 thématique de physique 2015–2016 X/ENS PC Physique B X PC Physique A X MP Physique et SI X/ENS MP Physique Mines PSI Physique 2 Mines PSI Physique

Ecole Doctorale de Physique, chimie-physique

INGENIERIE PHYSIQUE MASTER PHYSIQUE APPLIQUÉE ET

Physique et Arts La Physique dans l’Art

BACH · 2021. 2. 12. · programme autour de Jean Sébastien Bach. Ce programme, u’ilsont intitulé Bach reloaded reprend 2 œuves emblématiques du compositeur. Les pièces sont

Exit rupture, Jurassic Park reloaded version Ramgoolam 2019 · Navin 2019 ! ALLIANCE NATIONALE Jurassic Park reloaded version Ramgoolam 2019 Navin Ramgoolam Le roi qui dévalise son

Département de physique Cours de physique II

Temps r©el sous LINUX (reloaded) - Pierre Ficheux - Free

PHYSIQUE-CHIMIE Filière MP PHYSIQUE-CHIMIE

Centre de Physique Théorique - Centre de Physique

TRADITION RELOADED...REIMANN ARCHITECTURE Kontakt Büro München Daphnestraße 32 81925 Bogenhausen// München Zentrale +49 (0) 40 228676930 servus@reimann-architecture.com Büro Hamburg