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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
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 3 / 28
Scattering Matrix Resonances / Quasi-Bound States
Section 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 4 / 28
Scattering Matrix Resonances / Quasi-Bound States
What is Resonance?
�
�ψinc�
�
�ψsca�
Outside the dielectric, we solveHelmholtz’ equation:
�
∇2 + n2k2��
�ψ�
= 0.
�
�ψsca�
= S�
�ψinc�
Poles of |detS(k)| with k ∈ C(S(k) = representation of S ).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 5 / 28
Scattering Matrix Resonances / Quasi-Bound States
What is Resonance?
�
�ψinc�
�
�ψsca�
Outside the dielectric, we solveHelmholtz’ equation:
�
∇2 + n2k2��
�ψ�
= 0.
�
�ψsca�
= S�
�ψinc�
Poles of |detS(k)| with k ∈ C(S(k) = representation of S ).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 5 / 28
Scattering Matrix Resonances / Quasi-Bound States
What is Resonance?
�
�ψinc�
�
�ψsca�
Outside the dielectric, we solveHelmholtz’ equation:
�
∇2 + n2k2��
�ψ�
= 0.
�
�ψsca�
= S�
�ψinc�
Poles of |detS(k)| with k ∈ C(S(k) = representation of S ).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 5 / 28
Scattering Matrix Resonances / Quasi-Bound States
What is Resonance?
�
�ψinc�
�
�ψsca�
Outside the dielectric, we solveHelmholtz’ equation:
�
∇2 + n2k2��
�ψ�
= 0.
�
�ψsca�
= S�
�ψinc�
Poles of |detS(k)| with k ∈ C(S(k) = representation of S ).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 5 / 28
Scattering Matrix Resonances / Quasi-Bound States
What Is Resonance?
Can we find a set of real energy levels for anopen dielectric cavity?
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 6 / 28
Scattering Matrix Energy Formalism
Section 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 7 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
Average E.M. Energy
EV =1
2
∫∫∫
RV
[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′ =
1
2
∫∫∫
RV
�
εE∗m · Em′
+µH∗m ·Hm′�
d3r
R0
RV
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 8 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
Average E.M. Energy
EV =1
2
∫∫∫
RV
[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′ =
1
2
∫∫∫
RV
�
εE∗m · Em′
+µH∗m ·Hm′�
d3r
R0
RV
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 8 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
Average E.M. Energy
EV =1
2
∫∫∫
RV
[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′ =
1
2
∫∫∫
RV
�
εE∗m · Em′
+µH∗m ·Hm′�
d3r
R0
RV
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 8 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
E∞mm′ = limRV→∞
4n0RVw
kδmm′ +
2w
k
−i∑
`
S∗`m∂ S`m′
∂ k
!
E0mm′ : diverging free space
energy of the beam.
Q =−iS† ∂ S∂ k: complex coupling
between angular momentumchannels → excess energy dueto the cavity → delay times.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 9 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
E∞mm′ = limRV→∞
4n0RVw
kδmm′ +
2w
k
−i∑
`
S∗`m∂ S`m′
∂ k
!
E0mm′ : diverging free space
energy of the beam.
Q =−iS† ∂ S∂ k: complex coupling
between angular momentumchannels → excess energy dueto the cavity → delay times.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 9 / 28
Scattering Matrix Energy Formalism
Delay due to the Cavity
E∞mm′ = limRV→∞
4n0RVw
kδmm′ +
2w
k
−i∑
`
S∗`m∂ S`m′
∂ k
!
E0mm′ : diverging free space
energy of the beam.
Q =−iS† ∂ S∂ k: complex coupling
between angular momentumchannels → excess energy dueto the cavity → delay times.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 9 / 28
Scattering Matrix Time Delay
Section 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 10 / 28
Scattering Matrix Time Delay
Resonances ↔ Time Delay
Characteristic Modes of the CavityThe field is given by
ψp =∑
m
h
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).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 11 / 28
Scattering Matrix Time Delay
Resonances ↔ Time Delay
Characteristic Modes of the CavityThe field is given by
ψp =∑
m
h
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).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 11 / 28
Scattering Matrix Time Delay
Resonances ↔ Time Delay
Characteristic Modes of the CavityThe field is given by
ψp =∑
m
h
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).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 11 / 28
Scattering Matrix Time Delay
Resonances ↔ Time Delay
Characteristic Modes of the CavityThe field is given by
ψp =∑
m
h
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).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 11 / 28
Scattering Matrix Time Delay
Resonances ↔ Time Delay
kR0
cτ/R0
5.3 5.35 5.4 5.45 5.5100
101
102
103
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 11 / 28
Numerical Construction
Section 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 12 / 28
Numerical Construction
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).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 13 / 28
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
Divide in concentric shellsSolve angular part in Fourierseries
Connect 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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
Divide in concentric shellsSolve angular part in FourierseriesConnect the shells
Isolate 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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
Divide in concentric shellsSolve angular part in FourierseriesConnect the shells
Isolate 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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
Divide in concentric shellsSolve angular part in FourierseriesConnect the shells
Isolate 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
Numerical Construction
Construction of the S-matrixGoalCompute S for arbitrary geometries and arbitrary refractive index profiles.
Divide in concentric shellsSolve angular part in FourierseriesConnect the shells
Isolate 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
Numerical Construction
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).J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 13 / 28
Applications
Section 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 14 / 28
Applications Numerical Results
Section 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 15 / 28
Applications Numerical Results
Example Applications of the Method
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 16 / 28
Applications Numerical Results
Ellipse (integrable geometry)
kR0
cτ/R
0
6.48 6.49 6.5 6.51 6.5210
0
101
102
103
mode
J. Unterhinnighofen et al, Phys. Rev. E(78), 016201 (2008).
Re�
kR0
= 6.5
| Im�
kR0
|= 0.032
kR0 = 6.499
2R0/cτ= 0.029
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 17 / 28
Applications Numerical Results
Square (corners + high delay)
kR0
cτ/R
0
4.5 4.52 4.54 4.56 4.58 4.610
0
101
102
103
104
105
mode
W.-H. Guo et al, IEEE J. of Qu. El. (39),1106 (2003).
Re�
kR0
= 4.54
| Im�
kR0
|= 1.05× 10−4
kR0 = 4.53
2R0/cτ= 1.06× 10−4
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 18 / 28
Applications Numerical Results
Stadium (composite structure)
kR0
cτ/R
0
4 4.5 5 5.510
0
101
102
mode
S.-Y. Lee et al, Phys. Rev. A (70), 023809(2004).
Re�
kR0
= 4.89
| Im�
kR0
|= 0.055
kR0 = 4.89
2R0/cτ= 0.052
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 19 / 28
Applications Numerical Results
Photonic Complex
kR0
cτ/R
0
5.3 5.35 5.4 5.45 5.510
0
101
102
103
(A) (B)(C)
(a)(b)
(c)
J.-W. Ryu et al, Phys. Rev. A (79), 053858
(2009).
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 20 / 28
Applications Numerical Results
Annular cavity (closed-form, holey cavity)
kR0
cτ/R
0
10.1 10.15 10.2 10.25 10.310
0
102
104
106
108
(14, 5)(A)
(7, 8)(B)
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 21 / 28
Applications Outlooks
Section 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 22 / 28
Applications Outlooks
Outlooks
TE Modes & µ 6= 1Equations to solve inside thecavity become
�
∇2 + n2k2�
Ez =1
µ∇µ · ∇Ez
�
∇2 + n2k2�
Hz =1
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 23 / 28
Conclusion
Section 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 24 / 28
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 25 / 28
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 25 / 28
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 25 / 28
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 25 / 28
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 25 / 28
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
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 26 / 28
Convergence
Section Outline
5 Convergence
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 27 / 28
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
100M
axim
umer
ror
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.
J. Dumont (U. Laval) S and Q Matrices Reloaded ICTON 2013 28 / 28
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