Abstract

Dielectric cavity systems, which have been studied extensively so far, have uniform refractive indices of their cavities, and Husimi functions, the most widely used phase space representation of optical modes formed in the cavities, accordingly were derived only for these homogeneous index cavities. For the case of the recently proposed gradient index dielectric cavities (called as transformation cavities) designed by optical conformal mapping, we show that the phase space structure of resonant modes can be revealed through the conventional Husimi functions by constructing a reciprocal virtual space. As examples, the Husimi plots were obtained for an anisotropic whispering gallery mode (WGM) and a short-lived mode supported in a limaçon-shaped transformation cavity. The phase space description of the corresponding modes in the reciprocal virtual space is compatible with the far-field directionality of the resonant modes in the physical space.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. J. Post, Formal Structure of Electromagnetics (Wiley, 1962).
  2. J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118(5), 1396–1408 (1960).
    [Crossref]
  3. I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford University, 1992).
  4. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  5. O. Ozgun and M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Antennas Propag. Mag. 52(3), 51–65 (2010).
    [Crossref]
  6. Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
    [Crossref] [PubMed]
  7. Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
    [Crossref]
  8. U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref] [PubMed]
  9. L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
    [Crossref]
  10. H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
    [Crossref]
  11. T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
    [Crossref]
  12. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Jpn. 22, 264–314 (1940).
  13. B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
    [Crossref] [PubMed]
  14. H.-J. Stöckmann, Quantum Chaos (Cambridge University, 1999).
  15. M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
    [Crossref]
  16. W. C. Chew, Waves and Fields in Inhomogeneous Media (Wiley-IEEE, 1999).
  17. A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
    [Crossref] [PubMed]
  18. D. Liu, L. H. Gabrielli, M. Lipson, and S. G. Johnson, “Transformation inverse design,” Opt. Express 21(12), 14223–14243 (2013).
    [Crossref] [PubMed]
  19. T. Needham, Visual Complex Analysis (Clarendon, 2002).
  20. M. Robnik, “Classical dynamics of a family of billiards with analytic boundaries,” J. Phys. A 16(17), 3971–3986 (1983).
    [Crossref]
  21. In preparation (to be published elsewhere).

2016 (1)

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

2015 (2)

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

2013 (1)

2012 (1)

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

2011 (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

2010 (1)

O. Ozgun and M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Antennas Propag. Mag. 52(3), 51–65 (2010).
[Crossref]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

2004 (1)

A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
[Crossref] [PubMed]

2003 (1)

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
[Crossref]

1993 (1)

B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
[Crossref] [PubMed]

1983 (1)

M. Robnik, “Classical dynamics of a family of billiards with analytic boundaries,” J. Phys. A 16(17), 3971–3986 (1983).
[Crossref]

1960 (1)

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118(5), 1396–1408 (1960).
[Crossref]

1940 (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Jpn. 22, 264–314 (1940).

Bäcker, A.

A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
[Crossref] [PubMed]

Cao, H.

H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

Chang, S.-J.

B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
[Crossref] [PubMed]

Chen, H.

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Choi, M.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Crespi, B.

B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
[Crossref] [PubMed]

Fürstberger, S.

A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
[Crossref] [PubMed]

Gabrielli, L. H.

Han, J. H.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Harayama, T.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

Hentschel, M.

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
[Crossref]

Husimi, K.

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Jpn. 22, 264–314 (1940).

Johnson, S. G.

Kim, I.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Kim, Y.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Kuzuoglu, M.

O. Ozgun and M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Antennas Propag. Mag. 52(3), 51–65 (2010).
[Crossref]

Lee, S. H.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Lipson, M.

Liu, D.

Liu, Y.

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

Min, B.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Ozgun, O.

O. Ozgun and M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Antennas Propag. Mag. 52(3), 51–65 (2010).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Perez, G.

B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
[Crossref] [PubMed]

Plebanski, J.

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118(5), 1396–1408 (1960).
[Crossref]

Robnik, M.

M. Robnik, “Classical dynamics of a family of billiards with analytic boundaries,” J. Phys. A 16(17), 3971–3986 (1983).
[Crossref]

Ryu, J. W.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Schomerus, H.

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
[Crossref]

Schubert, R.

A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
[Crossref] [PubMed]

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Shinohara, S.

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Tae, H. S.

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

Wiersig, J.

H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

Xu, L.

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Zhang, X.

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

Europhys. Lett. (1)

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62(5), 636–642 (2003).
[Crossref]

IEEE Antennas Propag. Mag. (1)

O. Ozgun and M. Kuzuoglu, “Form Invariance of Maxwell’s Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping,” IEEE Antennas Propag. Mag. 52(3), 51–65 (2010).
[Crossref]

J. Phys. A (1)

M. Robnik, “Classical dynamics of a family of billiards with analytic boundaries,” J. Phys. A 16(17), 3971–3986 (1983).
[Crossref]

Laser Photonics Rev. (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photonics Rev. 5(2), 247–271 (2011).
[Crossref]

Nanoscale (1)

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

Nat. Photonics (2)

Y. Kim, S. H. Lee, J. W. Ryu, I. Kim, J. H. Han, H. S. Tae, M. Choi, and B. Min, “Designing whispering gallery modes via transformation optics,” Nat. Photonics 10(10), 647–652 (2016).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Opt. Express (1)

Phys. Rev. (1)

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev. 118(5), 1396–1408 (1960).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Bäcker, S. Fürstberger, and R. Schubert, “Poincare Husimi representation of eigenstates in quantum billiards,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036204 (2004).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

B. Crespi, G. Perez, and S.-J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47(2), 986–991 (1993).
[Crossref] [PubMed]

Proc. Phys.-Math. Soc. Jpn. (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Jpn. 22, 264–314 (1940).

Rev. Mod. Phys. (1)

H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

Science (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Other (6)

E. J. Post, Formal Structure of Electromagnetics (Wiley, 1962).

I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford University, 1992).

H.-J. Stöckmann, Quantum Chaos (Cambridge University, 1999).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Wiley-IEEE, 1999).

T. Needham, Visual Complex Analysis (Clarendon, 2002).

In preparation (to be published elsewhere).

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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of a transformation cavity and its pertaining two virtual spaces: (a) the unit disk cavity in an original virtual (OV) space, (b) a limaçon-shaped transformation cavity in the physical space, (c) the corresponding unit disk cavity in a reciprocal virtual (RV) space. (refractive index profiles are expressed by common color scale.)
Fig. 2
Fig. 2 Incident and emerging rays at a dielectric interface with uniform refractive indices.
Fig. 3
Fig. 3 A high-Q TM resonance (cWGM) in a limaçon-shaped transformation cavity ( k=12.2333i0.0022) (a) the mode intensity distribution; two yellow arrows denote the directions of tunneling emissions at s ˜ =0 in Fig. 3(c), (b) the far field pattern, (c) the intracavity Husimi plot for incident waves in the RV space; two yellow solid curves are critical lines of total internal reflection and the ± signs in sin χ ˜ 1 denote the counterclockwise (CCW) and clockwise (CW) circulations of light waves, respectively. ( s ˜ is normalized with 2π, the total arc length of the unit disk cavity.)
Fig. 4
Fig. 4 A low-Q TM resonance in a limaçon-shaped transformation cavity ( k=12.5960i0.1302) (a) the mode intensity distribution; two yellow arrows designate dominant refractive emissions at the corresponding peak positions s ˜ =0.1256 and s ˜ =0.8744 in the Husimi plot of Fig. 4(c), (b) the far field pattern, (c) the intracavity Husimi plot for incident waves in the RV space; two yellow solid curves are critical lines of total internal reflection same as in Fig. 3(c) and two yellow arrows point the intensity peaks of CW and CCW waves ( s ˜ is normalized with 2π, the total arc length of the unit disk cavity.)

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

x u = y v  ,        y u = x v .
[ Δ+ n 2 ( r ) k 2 ] ψ( r )=0 ,
n( r )={ n | dζ dη | 1 ,         r   Ω 1    (interior)     1 ,                    r   Ω 0    (exterior)  ,
ψ( r ) ~ h( θ,k )  e ikr r       for   r ,
ψ 1 = ψ 0  ,          ν ψ 1 = ν ψ 0               for TM modes,
ψ 1 = ψ 0  ,          ν ψ 1 = ν ψ 0               for TM modes,
H j inc( em ) (s, sin χ j )=  k j 2π    | (1) j   F j   h j ( s, sin χ j )+()  i k 0 F j     h j ' (s, sin χ j )  | 2
h j ( s, sin χ j )= Γ  d s '   ψ j ( s ' ) ξ( s ' ;s, sin χ j ),
h j ( s, sin χ j )= Γ  d s ν ψ j ( s ) ξ( s ;s, sin χ j ),
ξ( s ;s, sin χ j ) = 1 σπ 4   l exp[ ( s s+2πl ) 2 2σ i k j sin χ j ( s +2πl) ] ,
[ Δ ˜ + n ˜ 2 ( r ˜ ) k 2 ]  ψ ˜ ( r ˜ )=0,
n ˜ ( r ˜ )={   n ,                  r ˜     Ω ˜ 1      (interior)     | dη dζ | 1          r ˜     Ω ˜ 0      (exterior)   .
H 1 inc( em ) ( s ˜ , sin χ ˜ 1 )=  k 1 2π    | F 1   h 1 ( s ˜ , sin χ ˜ 1 )( + ) i n k 1 F 1   h 1 ( s ˜ , sin χ ˜ 1 ) | 2
h 1 ( s ˜ , sin χ ˜ 1 )= Γ ˜  d s ˜   ψ ˜ 1 ( s ˜ ) ξ( s ˜ ;  s ˜ , sin χ ˜ 1 ),
h 1 ( s ˜ , sin χ ˜ 1 )= Γ ˜  d s ˜ ˜ ν ψ ˜ 1 ( s ˜ ) ξ( s ˜ ;  s ˜ , sin χ ˜ 1 ) ,
ξ( s ˜ ;  s ˜ , sin χ ˜ 1 )= 1 σπ 4   l exp[ ( s ˜ s ˜ +2πl ) 2 2σ i k 1 sin χ ˜ 1 ( s ˜ +2πl ) ] ,
E= ( Λ 1 ) T    E ˜ ,       H= ( Λ 1 ) T H ˜  ,      ˜ = ( Λ 1 ) T   ˜ .
Λ=[ x u x v 0 y u y v 0 0 0 1 ]  , Λ 1 =[ u x u y 0 v x v y 0 0 0 1 ]
( 0 0 ψ )  = [ u x v x   0 u y v y 0 0 0 1 ] ( 0 0 ψ ˜ ). 
ν x +i ν y  =( ν u +i ν v )  dζ dη   | dζ dη | 1 . 
ν ν = ν ˜   Λ T ( Λ T ) 1 detΛ ˜ = | dζ dη | 1 ˜ ν  .
ζ=β(η+ ϵ η 2 ),

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