S and Q Matrices Reloaded · Département de physique, génie physique et optique Université Laval, Québec, Canada ICTON2013 Cartagena,Spain. Outline 1 ScatteringMatrix Resonances/Quasi-BoundStates

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�

= 0.

�

�ψsca�

= S�

�ψinc�

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



What is Resonance?

�

�ψinc�

�

�ψsca�


�

∇2 + n2k2��

�ψ�

= 0.

�

�ψsca�

= S�

�ψinc�




What is Resonance?

�

�ψinc�

�

�ψsca�


�

∇2 + n2k2��

�ψ�

= 0.

�

�ψsca�

= S�

�ψinc�




What is Resonance?

�

�ψinc�

�

�ψsca�


�

∇2 + n2k2��

�ψ�

= 0.

�

�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

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




Average E.M. Energy

EV =1

2

∫∫∫

RV




∑

`

Sm`H(+)`

(nokr)ei`θ


EVmm′ =

1

2

∫∫∫

RV

�

εE∗m · Em′

+µH∗m ·Hm′�

d3r

R0

RV




Average E.M. Energy

EV =1

2

∫∫∫

RV




∑

`

Sm`H(+)`

(nokr)ei`θ


EVmm′ =

1

2

∫∫∫

RV

�

εE∗m · Em′

+µH∗m ·Hm′�

d3r

R0

RV




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.





4n0RVw

kδmm′ +

2w

k

−i∑

`

S∗`m∂ S`m′

∂ k

!


energy of the beam.







4n0RVw

kδmm′ +

2w

k

−i∑

`

S∗`m∂ S`m′

∂ k

!


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 =∑

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).





ψp =∑

m

h

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.


QAp = τpAp.






ψp =∑

m

h

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 =∑

m

h

ApmH(−)

m (nkr) + BpmH(+)

m (nkr)i

eimθ .

andB = SA.


QAp = τpAp.





kR0

cτ/R0

5.3 5.35 5.4 5.45 5.5100

101

102

103


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)

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



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



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



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).



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)


Applications Outlooks

Section Outline




4 Conclusion


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.


Conclusion

Section Outline




4 Conclusion


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

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

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

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

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

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.


Documents

S and Q Matrices Reloaded · Département de physique, génie physique et optique Université Laval, Québec, Canada ICTON2013 Cartagena,Spain. Outline 1 ScatteringMatrix Resonances/Quasi-BoundStates