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Quantum chaos in optical microcavities J. Wiersig Institute for - - PowerPoint PPT Presentation

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Quantum chaos in optical microcavities J. Wiersig Institute for Theoretical Physics, Otto-von-Guericke University, Magdeburg Collaborations J. Unterhinninghofen (Magdeburg) M. Hentschel (Dresden) J. Main (Stuttgart) H. Schomerus (Lancaster)

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SLIDE 1

Quantum chaos in optical microcavities

  • J. Wiersig

Institute for Theoretical Physics, Otto-von-Guericke University, Magdeburg Collaborations

  • J. Unterhinninghofen (Magdeburg)
  • M. Hentschel (Dresden)
  • J. Main (Stuttgart)
  • H. Schomerus (Lancaster)

Supported by the DFG research group “Scattering Systems with Complex Dynamics”

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SLIDE 2

Theorie der Kondensierten Materie I

transport in nanostructures Nonlinear dynamics and charge

S.W. Cho and Y.D. Park, Seoul

semiconductor nanostructures Light−matter interaction in microcavities Optical properties of Quasicrystals

Dipl.−Phys.

  • Dr. habil. G. Kasner
  • Prof. J. Wiersig
  • J. Unterhinninghofen

I.R. Fischer et al., Iowa

  • D. Hommel/A. Rosenauer et al., Bremen
  • D. Heitmann/T. Kipp et al., Hamburg
  • J. Wiersig

CIRM 2008 2

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SLIDE 3

Theorie der Kondensierten Materie I

transport in nanostructures Nonlinear dynamics and charge

S.W. Cho and Y.D. Park, Seoul

semiconductor nanostructures Light−matter interaction in microcavities Optical properties of Quasicrystals

Dipl.−Phys.

  • Dr. habil. G. Kasner
  • Prof. J. Wiersig
  • J. Unterhinninghofen

I.R. Fischer et al., Iowa

  • D. Hommel/A. Rosenauer et al., Bremen
  • D. Heitmann/T. Kipp et al., Hamburg
  • J. Wiersig

CIRM 2008 2

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SLIDE 4

Outline

1

Introduction to optical microcavities

2

Avoided resonance crossings Avoided crossings despite integrability Formation of long-lived, scarlike modes Unidirectional light emission from high-Q modes

3

Unidirectional light emission and universal far-field patterns

4

Fractal Weyl law

5

Summary

  • J. Wiersig

CIRM 2008 3

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SLIDE 5

Introduction to optical microcavities

  • J. Wiersig

CIRM 2008 4

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SLIDE 6

Introduction to optical microcavities

Microdisk

φ > φc φ < φc φ φ tunneling total internal reflection

Bell labs

Light confinement due to TIR

  • J. Wiersig

CIRM 2008 5

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SLIDE 7

Introduction to optical microcavities

Microdisk

φ > φc φ < φc φ φ tunneling total internal reflection

Bell labs

Light confinement due to TIR Whispering-gallery modes Light emission due to tunneling High quality factor Q = ωτ Uniform far-field pattern

  • J. Wiersig

CIRM 2008 5

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SLIDE 8

Introduction to optical microcavities

Types of cavities

microdisk microsphere microtorus photonic crystal defect cavity

  • C. Reese et al.
  • D. Hommel et al.

VCSEL−micropillar

  • H. Wang et al.

Bell labs V.S. Ilschenko et al.

  • J. Wiersig

CIRM 2008 6

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SLIDE 9

Introduction to optical microcavities

Applications

Strong light confinement to a very small volume Individual optical modes Control over light-matter interaction ... Applications Microlasers Single-photon sources Quantum computers Sensors Filters ...

Bell labs

  • J. Wiersig

CIRM 2008 7

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SLIDE 10

Introduction to optical microcavities

Deformed microdisks

Directed light emission from deformed disks A. Levi et al., APL 62, 561 (1993)

Open quantum billiard (ray-wave correspondence)

sin χ < 1/n ⇒ no TIR

J.U. Nöckel and A.D. Stone, Nature 385, 45 (1997)

  • J. Wiersig

CIRM 2008 8

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SLIDE 11

Introduction to optical microcavities

Wave equation and boundary conditions

Quantum billiard: energy eigenstate h ∇2 + k 2i ψ(x, y) = 0 and ψ(x, y) = 0 outside, k = q

2mE 2 ∈ R.

  • J. Wiersig

CIRM 2008 9

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SLIDE 12

Introduction to optical microcavities

Wave equation and boundary conditions

Quantum billiard: energy eigenstate h ∇2 + k 2i ψ(x, y) = 0 and ψ(x, y) = 0 outside, k = q

2mE 2 ∈ R.

Optical microcavity, (TM polarized) mode : Ez = Re[ψ(x, y)e−iωt] h ∇2 + n(x, y)2k 2i ψ(x, y) = 0 and continuity of ψ and ∇ψ + outgoing wave conditions, k = ω/c ∈ C. lifetime τ = −

1 2Im(ω)

n(x,y) = n n(x,y) = 1

  • J. Wiersig

CIRM 2008 9

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SLIDE 13

Introduction to optical microcavities

Deformed microdisks in experiments

  • A. Levi et al., APL 62, 561 (1993)
  • C. Gmachl et al., Science 280, 1556 (1998)
  • M. Kneissl et al., APL 84, 2485 (2004)

Improved directionality but Q-factor is strongly reduced Ultimate goal: unidirectional light emission from high-Q modes

  • J. Wiersig

CIRM 2008 10

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SLIDE 14

Avoided resonance crossings

  • J. Wiersig

CIRM 2008 11

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SLIDE 15

Avoided resonance crossings

Avoided level crossings

H = „ E1 V W E2 « Ei ∈ R, W = V ∗ Eigenvalues of H E± = E1 + E2 2 ± r (E1 − E2)2 4 + VW Vary a parameter ∆ V = 0 V = 0

∆ E ∆

Hybridization (mixing) of eigenstates near avoided level crossing

  • J. Wiersig

CIRM 2008 12

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SLIDE 16

Avoided resonance crossings

Internal coupling

E± = E1 + E2 2 ± r (E1 − E2)2 4 + VW Ei ∈ C, W = V ∗: internal coupling |V| < Vc (weak coupling) |V| > Vc (strong coupling)

Re(E) ∆ Im(E)

long-lived short-lived

∆

Small mixing of eigenstates

  • J. Wiersig

CIRM 2008 13

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SLIDE 17

Avoided resonance crossings

External coupling

E± = E1 + E2 2 ± r (E1 − E2)2 4 + VW Ei ∈ C, W = V ∗: external coupling strong coupling

Re(E) ∆ Im(E)

Formation of short- and long-lived modes

  • J. Wiersig

CIRM 2008 14

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SLIDE 18

Avoided resonance crossings

Avoided crossings despite integrability

Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5, 53 (2003) Normalized frequency Ω = ωR/c = kR elliptical microcavity 8.14 8.16 8.18 8.2 8.22 8.24 Re(Ω) 0.62 0.64 0.66 0.68 eccentricity

  • 0.0005
  • 0.0004
  • 0.0003
  • 0.0002
  • 0.0001

Im(Ω) A B C E D F A B C D E F C D

  • J. Wiersig

CIRM 2008 15

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SLIDE 19

Avoided resonance crossings

Avoided crossings despite integrability

elliptical microcavity

8.14 8.16 8.18 8.2 8.22 8.24 Re(Ω) 0.62 0.64 0.66 0.68 eccentricity

  • 0.0005
  • 0.0004
  • 0.0003
  • 0.0002
  • 0.0001

Im(Ω) A B C E D F A B C D E F C D

Formation of scarlike modes

  • J. Wiersig, PRL 97, 253901 (2006)

A C E B D F

  • J. Wiersig

CIRM 2008 16

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SLIDE 20

Avoided resonance crossings

Avoided crossings despite integrability

elliptical microcavity

8.14 8.16 8.18 8.2 8.22 8.24 Re(Ω) 0.62 0.64 0.66 0.68 eccentricity

  • 0.0005
  • 0.0004
  • 0.0003
  • 0.0002
  • 0.0001

Im(Ω) A B C E D F A B C D E F C D

Formation of scarlike modes

  • J. Wiersig, PRL 97, 253901 (2006)

A C E B D F

  • J. Wiersig

CIRM 2008 16

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SLIDE 21

Avoided resonance crossings

Avoided crossings despite integrability

elliptical microcavity

8.14 8.16 8.18 8.2 8.22 8.24 Re(Ω) 0.62 0.64 0.66 0.68 eccentricity

  • 0.0005
  • 0.0004
  • 0.0003
  • 0.0002
  • 0.0001

Im(Ω) A B C E D F A B C D E F C D

Formation of scarlike modes

  • J. Wiersig, PRL 97, 253901 (2006)

Augmented ray dynamics including the Goos-Hänchen shift

  • J. Unterhinninghofen, J. Wiersig,

and M. Hentschel, PRE 78, 016201 (2008)

A C E B D F

  • J. Wiersig

CIRM 2008 16

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SLIDE 22

Avoided resonance crossings

Formation of long-lived, scarlike modes 12.8 12.9 13 13.1 13.2 Re(Ω) 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio

  • 0.02
  • 0.015
  • 0.01
  • 0.005

Im(Ω) C D A B E F C D A B E F

B A C D F E

  • J. Wiersig, PRL 97, 253901 (2006)

Long-lived mode C: Q ≈ 23000 (increase by more than one order of magnitude) no diffraction at corners

  • J. Wiersig

CIRM 2008 17

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SLIDE 23

Avoided resonance crossings

Formation of long-lived, scarlike modes 12.8 12.9 13 13.1 13.2 Re(Ω) 0.72 0.73 0.74 0.75 0.76 0.77 aspect ratio

  • 0.02
  • 0.015
  • 0.01
  • 0.005

Im(Ω) C D A B E F C D A B E F

C

  • J. Wiersig, PRL 97, 253901 (2006)

Long-lived mode C: Q ≈ 23000 (increase by more than one order of magnitude) no diffraction at corners scarlike mode pattern At the avoided resonance crossing a long-lived, scarlike mode is formed

  • J. Wiersig

CIRM 2008 17

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SLIDE 24

Avoided resonance crossings

Unidirectional light emission from high-Q modes

Normalized frequency Ω = ωR/c = kR, quality factor Q = − Re(Ω)

2 Im(Ω)

7 7.01 7.02 7.03 7.04 7.05 Re(Ω) 0.39 0.4 0.41 0.42 0.43 0.44 0.45 d/R 3e+05 4e+05 5e+05 6e+05 Q x1000 short-lived long-lived d R2 R

Weak internal coupling: weak hybridization of modes

  • J. Wiersig

CIRM 2008 18

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SLIDE 25

Avoided resonance crossings

Unidirectional light emission from high-Q modes

Far-field pattern is dominated by the short-lived component 60 120 180 θ Intensity (arb. units) low-Q mode high-Q mode

θ

Hybridized whispering-gallery mode has Q = 550000 and unidirectional emission

  • J. Wiersig

CIRM 2008 19

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SLIDE 26

Avoided resonance crossings

Unidirectional light emission from high-Q modes

50 100 150

  • bservation angle θ in degree

far-field intensity (arb. units)

Small angular divergence and ultra-high Q > 108

  • J. Wiersig

CIRM 2008 20

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SLIDE 27

Avoided resonance crossings

Unidirectional light emission from high-Q modes

50 100 150

  • bservation angle θ in degree

far-field intensity (arb. units)

  • F. Wilde PhD thesis (2008)

Heitmann group, Hamburg

Small angular divergence and ultra-high Q > 108

  • J. Wiersig

CIRM 2008 20

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SLIDE 28

Unidirectional light emission and universal far-field patterns

  • J. Wiersig

CIRM 2008 21

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SLIDE 29

Unidirectional light emission and universal far-field patterns

Problem: in the case of multimode lasing we may have modes with different directionality

  • J. Wiersig

CIRM 2008 22

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SLIDE 30

Unidirectional light emission and universal far-field patterns

Problem: in the case of multimode lasing we may have modes with different directionality “Universal” far-field pattern due to unstable manifold

H.G.L Schwefel et al., J. Opt. Soc. Am. B 21, 923 (2004) S.-Y. Lee et al., Phys. Rev. A 72, 061801(R) (2005) S.-B. Lee et al., Phys. Rev. A 75, 011802(R) (2007)

  • S. Shinohara and T. Harayama, Phys. Rev. E 75, 036216 (2007)
  • J. Wiersig

CIRM 2008 22

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SLIDE 31

Unidirectional light emission and universal far-field patterns

Problem: in the case of multimode lasing we may have modes with different directionality “Universal” far-field pattern due to unstable manifold

H.G.L Schwefel et al., J. Opt. Soc. Am. B 21, 923 (2004) S.-Y. Lee et al., Phys. Rev. A 72, 061801(R) (2005) S.-B. Lee et al., Phys. Rev. A 75, 011802(R) (2007)

  • S. Shinohara and T. Harayama, Phys. Rev. E 75, 036216 (2007)

Our goal: unidirectional emission high Q-factors

  • J. Wiersig and M. Hentschel, PRL 100, 033901 (2008)
  • J. Wiersig

CIRM 2008 22

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SLIDE 32

Unidirectional light emission and universal far-field patterns

Limaçon cavity

ρ(φ) = R(1 + ε cos φ)

sin χ sin χ

s s

s/smax 1/n 1/n −1 1 s/smax 0 0.2 0.4 0.6 0.8 1

φ ρ

− χ

−1 1 1/n 1/n

−

0 0.2 0.4 0.6 0.8 1 ε = 0 ε = 0.43

no total internal reflection leaky region

R

  • J. Wiersig

CIRM 2008 23

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SLIDE 33

Unidirectional light emission and universal far-field patterns

Unstable manifold

Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution Unstable manifold: the set of points that converges to the repeller in backward time evolution (weight according to Fresnel’s laws) Subtle differences between TE and TM polarization

  • J. Wiersig

CIRM 2008 24

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SLIDE 34

Unidirectional light emission and universal far-field patterns

Far-field pattern: ray simulation 50 100 150

  • bservation angle φFF in degree

far-field intensity (arb. units) TE TM

0.5 1

s/smax

  • 0.4
  • 0.2

sin χ

1 1

1s 2 2

2

2s 1

2s

1

1s

  • 1/n
  • sin χB

φFF = 0

Difference between TE and TM modes is due to the Brewster angle

  • J. Wiersig

CIRM 2008 25

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SLIDE 35

Unidirectional light emission and universal far-field patterns

Far-field pattern: mode calculation

far-field intensity (arb. units) 50 100 150

  • bservation angle φFF in degree

TE TM

Q = 200000 Q = 107

all high-Q TE modes show unidirectional emission (universal far-field pattern)

  • J. Wiersig

CIRM 2008 26

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SLIDE 36

Unidirectional light emission and universal far-field patterns

Husimi representation

sin χ 1/n (a) s/smax 1 1/n

−

0 0.2 0.4 0.6 0.8 1 1

−

Scarring ensures high Q-factors

  • J. Wiersig

CIRM 2008 27

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SLIDE 37

Unidirectional light emission and universal far-field patterns

Husimi magnification

sin χ sin χ s/s 1/n

−

1/n

−

(b) TE (c) TM 1/n 0 0.2 0.4 0.6 0.8 1 1/n

max Unidirectional emission is due to the unstable manifold Robust: not sensitive to internal mode structure and cavity size works in a broad regime of shape parameter ε and refractive index

  • J. Wiersig

CIRM 2008 28

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SLIDE 38

Unidirectional light emission and universal far-field patterns

Experiments on the Limaçon cavity

Theory: J. Wiersig and M. Hentschel, PRL 100, 033901 (2008) Harayama et al., Kyoto Capasso et al., Harvard Cao et al., Yale

  • J. Wiersig

CIRM 2008 29

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SLIDE 39

Fractal Weyl law

  • J. Wiersig

CIRM 2008 30

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SLIDE 40

Fractal Weyl law

Weyl law for closed systems

Density of states ρ(E) =

∞

X

i=1

δ(E − Ei) ; E1 ≤ E2 ≤ . . . Integrated density of states N(E) = Z E

−∞

ρ(E′)dE′ = #{i|Ei ≤ E} split N(E) into a smooth part and a fluctuating part N(E) = ¯ N(E) + Nfluc(E)

E2 2 3 4 5 E

1

6 1 N(E) N(E) 7 8 E N ... E3

  • J. Wiersig

CIRM 2008 31

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SLIDE 41

Fractal Weyl law

Weyl law for closed systems

Density of states ρ(E) =

∞

X

i=1

δ(E − Ei) ; E1 ≤ E2 ≤ . . . Integrated density of states N(E) = Z E

−∞

ρ(E′)dE′ = #{i|Ei ≤ E} split N(E) into a smooth part and a fluctuating part N(E) = ¯ N(E) + Nfluc(E)

E2 2 3 4 5 E

1

6 1 N(E) N(E) 7 8 E N ... E3

Weyl’s law for 2D billiard with area A (2/2m = 1) ¯ N(E) = A 4π E ∼ k 2

  • J. Wiersig

CIRM 2008 31

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SLIDE 42

Fractal Weyl law

How to count states in open systems?

Im(k) Re(k)

  • J. Wiersig

CIRM 2008 32

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SLIDE 43

Fractal Weyl law

How to count states in open systems?

Re(k) Im(k) −C

N(k) = {kn : Im(kn) > −C, Re(kn) ≤ k}.

  • J. Wiersig

CIRM 2008 32

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SLIDE 44

Fractal Weyl law

How to count states in open systems?

Re(k) Im(k) −C

N(k) = {kn : Im(kn) > −C, Re(kn) ≤ k}. Conjecture: fractal Weyl law for open chaotic systems

  • J. Sjöstrand, Duke Math. J. 60, 1 (1990), M. Zworski, Invent. Math. 136, 353 (1999)

¯ N(k) ∼ k α . with non-integer exponent α = D + 1 2 where D is the fractal dimension of the chaotic repeller

  • J. Wiersig

CIRM 2008 32

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SLIDE 45

Fractal Weyl law

How to count states in open systems?

Re(k) Im(k) −C

N(k) = {kn : Im(kn) > −C, Re(kn) ≤ k}. Conjecture: fractal Weyl law for open chaotic systems

  • J. Sjöstrand, Duke Math. J. 60, 1 (1990), M. Zworski, Invent. Math. 136, 353 (1999)

¯ N(k) ∼ k α . with non-integer exponent α = d + 2 2 where d is the fractal dimension of the chaotic repeller in the Poincaré section

  • J. Wiersig

CIRM 2008 32

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SLIDE 46

Fractal Weyl law

How to count states in open systems?

Re(k) Im(k) −C

N(k) = {kn : Im(kn) > −C, Re(kn) ≤ k}. Conjecture: fractal Weyl law for open chaotic systems

  • J. Sjöstrand, Duke Math. J. 60, 1 (1990), M. Zworski, Invent. Math. 136, 353 (1999)

¯ N(k) ∼ k α . with non-integer exponent α = d + 2 2 where d is the fractal dimension of the chaotic repeller in the Poincaré section

  • J. Wiersig

CIRM 2008 32

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SLIDE 47

Fractal Weyl law

Microstadium

R 2L refractive index = n refractive index = 1 s χ

L = R

  • J. Wiersig

CIRM 2008 33

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SLIDE 48

Fractal Weyl law

Microstadium

R 2L refractive index = n refractive index = 1 s χ

L = R

10µm

n = 3.3 (GaAs): weakly open

  • T. Fukushima and T. Harayama, IEEE J. Sel. Top.

Quantum Electron., 10, 1039 (2004)

n = 1.5 (polymer): strongly open

  • M. Lebenthal et al., Appl. Phys. Lett., 88, 031108 (2006)
  • J. Wiersig

CIRM 2008 33

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SLIDE 49

Fractal Weyl law

Numerical scheme

Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5, 53 (2003) Computing sufficiently many resonances in the complex plane is extremely difficult

  • J. Wiersig

CIRM 2008 34

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SLIDE 50

Fractal Weyl law

Numerical scheme

Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5, 53 (2003) Computing sufficiently many resonances in the complex plane is extremely difficult

0.5 1 1.5 2 2.5 Ω 5 10 15 20 25 σ/R

Normalized frequency Ω = kR

  • J. Wiersig

CIRM 2008 34

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SLIDE 51

Fractal Weyl law

Numerical scheme

Boundary element method J. Wiersig, J. Opt. A: Pure Appl. Opt. 5, 53 (2003) Computing sufficiently many resonances in the complex plane is extremely difficult

5 10 15 20 25 Re(Ω)

  • 0.06
  • 0.05
  • 0.04
  • 0.03
  • 0.02
  • 0.01

Im(Ω) 0.5 1 1.5 2 2.5 Ω 5 10 15 20 25 σ/R

harmonic inversion

Normalized frequency Ω = kR Boundary element method + harmonic inversion → statistics of resonances

  • J. Wiersig

CIRM 2008 34

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SLIDE 52

Fractal Weyl law

Number of modes

0.01 0.02 0.03 0.04 0.05 0.06

  • Im(Ω)

10 20 30 40 50 60 Probability density

  • J. Wiersig

CIRM 2008 35

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SLIDE 53

Fractal Weyl law

Number of modes

0.01 0.02 0.03 0.04 0.05 0.06

  • Im(Ω)

10 20 30 40 50 60 Probability density

cutoff C = 0.06

0.5 1 1.5 2 2.5 3 ln(Ω) 2 3 4 5 6 7 8 ln(N)

α ≈ 1.98

  • J. Wiersig

CIRM 2008 35

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SLIDE 54

Fractal Weyl law

Number of modes

0.01 0.02 0.03 0.04 0.05 0.06

  • Im(Ω)

10 20 30 40 50 60 Probability density

cutoff C = 0.06

0.5 1 1.5 2 2.5 3 ln(Ω) 2 3 4 5 6 7 8 ln(N)

α ≈ 1.98 α ∈ [1.96, 2.02] for C ∈ [0.03, 0.1]

  • J. Wiersig

CIRM 2008 35

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SLIDE 55

Fractal Weyl law

Chaotic repeller

Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution

1/n 1 p 1 s/smax

  • J. Wiersig

CIRM 2008 36

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SLIDE 56

Fractal Weyl law

Chaotic repeller

Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution

1/n 1 p 1 s/smax s/smax s/smax 0.64 p 0.83 0.41 0.395 0.46 (a) (b) p 0.73 0.719 0.4066

Chaotic repeller is a fractal with box-counting dimension d ≈ 1.68

  • J. Wiersig

CIRM 2008 36

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SLIDE 57

Fractal Weyl law

Chaotic repeller

Chaotic repeller: set of points in phase space that never visits the leaky region both in forward and backward time evolution

1/n 1 p 1 s/smax s/smax s/smax 0.64 p 0.83 0.41 0.395 0.46 (a) (b) p 0.73 0.719 0.4066

Chaotic repeller is a fractal with box-counting dimension d ≈ 1.68 → α = d+2

2

= 1.84, i.e. fractal Weyl law fails!

  • J. Wiersig

CIRM 2008 36

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SLIDE 58

Fractal Weyl law

Chaotic repeller including Fresnel’s laws

Account for partial escape due to Fresnel’s laws → real-valued I(s, p) ∈ [0, 1]

1/n 1 p 1 s/smax

  • J. Wiersig

CIRM 2008 37

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SLIDE 59

Fractal Weyl law

Chaotic repeller including Fresnel’s laws

Account for partial escape due to Fresnel’s laws → real-valued I(s, p) ∈ [0, 1]

1/n 1 p 1 s/smax

Multifractal: infinite set of fractal dimensions d(q) with real q box counting dimension d(0) ≈ 1.986 → α ≈ 1.99 information dimension d(1) ≈ 1.913 → α ≈ 1.96 correlation dimension d(2) ≈ 1.877 → α ≈ 1.94 d(0) is consistent with fractal Weyl law (α ∈ [1.96, 2.02])

  • J. Wiersig

CIRM 2008 37

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SLIDE 60

Fractal Weyl law

Low-index stadium

n = 1.5 (polymer)

10 20 30 40 50 60 70 Re(Ω)

  • 0.2
  • 0.1

Im(Ω)

α ∈ [1.68, 1.88]

  • J. Wiersig

CIRM 2008 38

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SLIDE 61

Fractal Weyl law

Low-index stadium

n = 1.5 (polymer)

10 20 30 40 50 60 70 Re(Ω)

  • 0.2
  • 0.1

Im(Ω)

α ∈ [1.68, 1.88]

1/n 1 p 1 s/smax

d(0) ≈ 1.512 d(1) ≈ d(2) ≈ 1.593 The predicted exponent, 1.76 and 1.78, is consistent with the fractal Weyl law

  • J. Wiersig

CIRM 2008 38

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SLIDE 62

Fractal Weyl law

Low-index stadium

n = 1.5 (polymer)

10 20 30 40 50 60 70 Re(Ω)

  • 0.2
  • 0.1

Im(Ω)

α ∈ [1.68, 1.88]

1/n 1 p 1 s/smax

d(0) ≈ 1.512 d(1) ≈ d(2) ≈ 1.593 The predicted exponent, 1.76 and 1.78, is consistent with the fractal Weyl law Conjecture: the fractal Weyl law applies to optical microcavities if the concept of the chaotic repeller is extended by including Fresnel’s laws

  • J. Wiersig and J. Main, PRE 77, 036205 (2008)
  • J. Wiersig

CIRM 2008 38

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SLIDE 63

Lifetime statistics

20 40 60 80 0.02 0.04 0.06 0.08 P(-Im Ω)

  • Im Ω

n=3.3, TM stadium RMT 10 20 30 40 0.02 0.04 0.06 0.08 P(-Im Ω)

  • Im Ω

n=3.3, TE

Good agreement with random-matrix theory; see talk by H. Schomerus

  • H. Schomerus, J. Wiersig, and J. Main, submitted (2008)
  • J. Wiersig

CIRM 2008 39

slide-64
SLIDE 64

Summary

Optical microcavities as open quantum billiards Avoided resonance crossings

Avoided crossings despite integrability Formation of long-lived, scarlike modes Unidirectional light emission from high-Q modes

Unidirectional light emission and universal far-field patterns Fractal Weyl law

  • J. Wiersig

CIRM 2008 40