Abstract

Active optical systems can give rise to intriguing phenomena and applications that are not available in conventional passive systems. Structural rotation has been widely employed to achieve non-reciprocity or time-reversal symmetry breaking. Here, we examine the quasi-normal modes and scattering properties of a dielectric disk under rotation. In addition to the familiar phenomenon of Sagnac frequency shift, we observe the the hybridization of the clockwise (CW) and counter-clockwise CCW) chiral modes of the cavity controlled by the rotation. The rotation tends to suppress one chiral mode while amplifying the other, and it leads to the variation of the far field. The phenomenon can be understood as the result of a synthetic gauge field induced by the rotation of the cavity. We explicitly derived this gauge field and the resulting Sagnac frequency shift. The analytical results are corroborated by finite element simulations. Our results can be applied in the measurement of rotating devices by probing the far field.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

Y. T. Chen, R.-Y. Zhang, Z. F. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. T. Chan, “Non-abelian gauge field optics,” Nat. Commun. 10, 3125 (2019).
[Crossref] [PubMed]

2018 (4)

S. B. Wang, G. C. Ma, and C. T. Chan, “Topological transport of sound mediated by spin-redirection geometric phase,” Sci. Adv. 4, eaaq1475 (2018).
[Crossref] [PubMed]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

2017 (3)

J.-W. Dong, X.-D. Chen, H. Y. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16, 298–302 (2017).
[Crossref]

W. Liang, A. Savchenkov, V. Ilchenko, R. Griffith, E. De Cuir, S. Kim, A. Matsko, and L. Maleki, “On Sagnac frequency splitting in a solid-state ring Raman laser,” Opt. Lett. 42, 4736–4739 (2017).
[Crossref] [PubMed]

S. Sunada, “Large sagnac frequency splitting in a ring resonator operating at an exceptional point,” Phys. Rev. A 96, 033842 (2017).
[Crossref]

2015 (5)

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref] [PubMed]

L. Ge, R. Sarma, and H. Cao, “Rotation-induced evolution of far-field emission patterns of deformed microdisk cavities,” Optica 2, 323–328 (2015).
[Crossref]

F. Liu and J. Li, “Gauge field optics with anisotropic media,” Phys. Rev. Lett. 114, 103902 (2015).
[Crossref] [PubMed]

J. Xu, B. B. Wu, and Y. T. Chen, “Elimination of polarization degeneracy in circularly symmetric bianisotropic waveguides: a decoupled case,” Opt. Express 23, 11566–11575 (2015).
[Crossref] [PubMed]

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

2014 (3)

G. B. Malykin, “Sagnac effect in ring lasers and ring resonators. How does the refractive index of the optical medium influence the sensitivity to rotation?” Physics-Uspekhi 57, 714–720 (2014).
[Crossref]

M. Vieira, A. M. d. M. Carvalho, and C. Furtado, “Aharonov-Bohm effect for light in a moving medium,” Phys. Rev. A 90, 012105 (2014).
[Crossref]

L. Ge, R. Sarma, and H. Cao, “Rotation-induced mode coupling in open wavelength-scale microcavities,” Phys. Rev. A 90, 013809 (2014).
[Crossref]

2013 (1)

R. Movassagh and S. G. Johnson, “Optical Bernoulli forces,” Phys. Rev. A 88, 023829 (2013).
[Crossref]

2012 (3)

K. J. Fang, Z. F. Yu, and S. H. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108, 153901 (2012).
[Crossref] [PubMed]

R. Sarma, H. Noh, and H. Cao, “Wavelength-scale microdisks as optical gyroscopes: a finite-difference time-domain simulation study,” J. Opt. Soc. Am. B 29, 1648–1654 (2012).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

2009 (1)

2007 (2)

2004 (1)

K. Y. Bliokh and Y. P. Bliokh, “Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect,” Phys. Rev. E 70, 026605 (2004).
[Crossref]

2000 (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Physics-Uspekhi 43, 1229–1252 (2000).
[Crossref]

1995 (1)

R. Cook, H. Fearn, and P. Milonni, “Fizeau’s experiment and the Aharonov–Bohm effect,” Am. J. Phys. 26, 705–710 (1995).
[Crossref]

1983 (1)

1982 (1)

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[Crossref]

1969 (1)

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1774 (1969).
[Crossref]

1967 (1)

E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39, 475–493 (1967).
[Crossref]

1964 (2)

C. V. Heer, “Resonant frequencies of an electromagnetic cavity in an accelerated system of reference,” Phys. Rev. 134, A799–A804 (1964).
[Crossref]

P. K. Cheo and C. V. Heer, “Beat frequency between two traveling waves in a Fabry-Perot square cavity,” Appl. Opt. 3, 788–789 (1964).
[Crossref]

1962 (1)

1908 (1)

H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 1, 53–111 (1908).

Anderson, J. L.

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1774 (1969).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and Y. P. Bliokh, “Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect,” Phys. Rev. E 70, 026605 (2004).
[Crossref]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect,” Phys. Rev. E 70, 026605 (2004).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Cao, H.

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

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref] [PubMed]

L. Ge, R. Sarma, and H. Cao, “Rotation-induced evolution of far-field emission patterns of deformed microdisk cavities,” Optica 2, 323–328 (2015).
[Crossref]

L. Ge, R. Sarma, and H. Cao, “Rotation-induced mode coupling in open wavelength-scale microcavities,” Phys. Rev. A 90, 013809 (2014).
[Crossref]

R. Sarma, H. Noh, and H. Cao, “Wavelength-scale microdisks as optical gyroscopes: a finite-difference time-domain simulation study,” J. Opt. Soc. Am. B 29, 1648–1654 (2012).
[Crossref]

Carmon, T.

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

Carvalho, A. M. d. M.

M. Vieira, A. M. d. M. Carvalho, and C. Furtado, “Aharonov-Bohm effect for light in a moving medium,” Phys. Rev. A 90, 012105 (2014).
[Crossref]

Chan, C. T.

Y. T. Chen, R.-Y. Zhang, Z. F. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. T. Chan, “Non-abelian gauge field optics,” Nat. Commun. 10, 3125 (2019).
[Crossref] [PubMed]

S. B. Wang, G. C. Ma, and C. T. Chan, “Topological transport of sound mediated by spin-redirection geometric phase,” Sci. Adv. 4, eaaq1475 (2018).
[Crossref] [PubMed]

Chen, X.-D.

J.-W. Dong, X.-D. Chen, H. Y. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16, 298–302 (2017).
[Crossref]

Chen, Y. T.

Y. T. Chen, R.-Y. Zhang, Z. F. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. T. Chan, “Non-abelian gauge field optics,” Nat. Commun. 10, 3125 (2019).
[Crossref] [PubMed]

J. Xu, B. B. Wu, and Y. T. Chen, “Elimination of polarization degeneracy in circularly symmetric bianisotropic waveguides: a decoupled case,” Opt. Express 23, 11566–11575 (2015).
[Crossref] [PubMed]

Cheo, P. K.

Christodoulides, D. N.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

Cook, R.

R. Cook, H. Fearn, and P. Milonni, “Fizeau’s experiment and the Aharonov–Bohm effect,” Am. J. Phys. 26, 705–710 (1995).
[Crossref]

Dahan, R.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

De Cuir, E.

Dong, J.-W.

J.-W. Dong, X.-D. Chen, H. Y. Zhu, Y. Wang, and X. Zhang, “Valley photonic crystals for control of spin and topology,” Nat. Mater. 16, 298–302 (2017).
[Crossref]

Ezekiel, S.

Fabre, C.

Fan, S. H.

K. J. Fang, Z. F. Yu, and S. H. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108, 153901 (2012).
[Crossref] [PubMed]

Fang, K. J.

K. J. Fang, Z. F. Yu, and S. H. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108, 153901 (2012).
[Crossref] [PubMed]

Fearn, H.

R. Cook, H. Fearn, and P. Milonni, “Fizeau’s experiment and the Aharonov–Bohm effect,” Am. J. Phys. 26, 705–710 (1995).
[Crossref]

Feugnet, G.

Furtado, C.

M. Vieira, A. M. d. M. Carvalho, and C. Furtado, “Aharonov-Bohm effect for light in a moving medium,” Phys. Rev. A 90, 012105 (2014).
[Crossref]

Fushchich, W. I.

W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (Springer Science & Business Media, 2013).

Garnache, A.

Ge, L.

L. Ge, R. Sarma, and H. Cao, “Rotation-induced evolution of far-field emission patterns of deformed microdisk cavities,” Optica 2, 323–328 (2015).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref] [PubMed]

L. Ge, R. Sarma, and H. Cao, “Rotation-induced mode coupling in open wavelength-scale microcavities,” Phys. Rev. A 90, 013809 (2014).
[Crossref]

Grahn, P.

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

Griffith, R.

Hang, Z. H.

Y. T. Chen, R.-Y. Zhang, Z. F. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. T. Chan, “Non-abelian gauge field optics,” Nat. Commun. 10, 3125 (2019).
[Crossref] [PubMed]

Harayama, T.

Hassan, A. U.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

Heer, C. V.

C. V. Heer, “Resonant frequencies of an electromagnetic cavity in an accelerated system of reference,” Phys. Rev. 134, A799–A804 (1964).
[Crossref]

P. K. Cheo and C. V. Heer, “Beat frequency between two traveling waves in a Fabry-Perot square cavity,” Appl. Opt. 3, 788–789 (1964).
[Crossref]

Huang, R.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Ilchenko, V.

Jing, H.

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5, 1424–1430 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

Johnson, S. G.

R. Movassagh and S. G. Johnson, “Optical Bernoulli forces,” Phys. Rev. A 88, 023829 (2013).
[Crossref]

Kaivola, M.

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14, 093033 (2012).
[Crossref]

Kim, S.

Kligerman, Y.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

Kozaki, S.

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[Crossref]

Li, J.

Y. T. Chen, R.-Y. Zhang, Z. F. Xiong, Z. H. Hang, J. Li, J. Q. Shen, and C. T. Chan, “Non-abelian gauge field optics,” Nat. Commun. 10, 3125 (2019).
[Crossref] [PubMed]

F. Liu and J. Li, “Gauge field optics with anisotropic media,” Phys. Rev. Lett. 114, 103902 (2015).
[Crossref] [PubMed]

Liang, W.

Liao, J.-Q.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

Liu, F.

F. Liu and J. Li, “Gauge field optics with anisotropic media,” Phys. Rev. Lett. 114, 103902 (2015).
[Crossref] [PubMed]

Lü, H.

Ma, G. C.

S. B. Wang, G. C. Ma, and C. T. Chan, “Topological transport of sound mediated by spin-redirection geometric phase,” Sci. Adv. 4, eaaq1475 (2018).
[Crossref] [PubMed]

Maayani, S.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–574 (2018).
[Crossref] [PubMed]

Maleki, L.

Malykin, G. B.

G. B. Malykin, “Sagnac effect in ring lasers and ring resonators. How does the refractive index of the optical medium influence the sensitivity to rotation?” Physics-Uspekhi 57, 714–720 (2014).
[Crossref]

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Physics-Uspekhi 43, 1229–1252 (2000).
[Crossref]

Matsko, A.

Meyer, R. E.

Mignot, A.

Milonni, P.

R. Cook, H. Fearn, and P. Milonni, “Fizeau’s experiment and the Aharonov–Bohm effect,” Am. J. Phys. 26, 705–710 (1995).
[Crossref]

Minkowski, H.

H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 1, 53–111 (1908).

Miranowicz, A.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
[Crossref] [PubMed]

Moses, E.

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“COMSOL Multiphysics 5.2: a finite element analysis, solver and simulation software,” http://www.comsol.com/ .

K. Sakoda, Optical Properties of Photonic Crystals (Springer Science & Business Media, 2004).

W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (Springer Science & Business Media, 2013).

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Figures (2)

Fig. 1
Fig. 1 (a) The structure of a uniformly rotating circular dielectric disk with angular speed Ω. (b) Comparison of results on Sagnac frequency splitting obtained by COMSOL simulation and the analytical expression of Eq. (4). The insets show the Ez components of the modal profiles of CW and CCW mode at ΩR/c = 0 and ΩR/c = 0.00646 obtained by COMSOL simulation. For the simulation, we use vacuum wavelength λ0 = 1550nm, disk radius R = 200nm and the relative permittivity εr = 16 for the stationary cavity.

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Fig. 2
Fig. 2 (a)–(b) Schematic diagrams of the effect of Sagnac frequency splitting on CW/CCW modes for a rotating cavity. Line-shape functions of the modes at Ω = 0 with azimuthal number (c) m = 2 and (d) m = 8 respectively. (e) The variation of modal coefficients obtained by the multipolar expansion and Eq. (6) for CW mode with azimuthal number m = 2 and CCW mode with m = 8 versus the dimensionless angular speed ΩR/c. The setup of parameters is shown in Fig. 1. Far-field distribution obtained by superposing the eigenfields |CW〉 and |CCW〉 with corresponding ratio at (f) Ω1R/c = 4 × 10−7 and (g) Ω2R/c = 1.8 × 10−6.

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Tables (1)

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Table 1 Various expressions for Sagnac frequency splitting of optical rotating cavities

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Equations (7)

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D + v × H / c 2 = ε ( E + v + B ) , B + E × v / c 2 = μ ( H + D + v ) ,
( 1 r ∂ ∂ θ + i k 0 A θ ) 2 e z + μ r r μ θ θ 1 r ∂ ∂ r ( r ∂ ∂ r e z ) + k 0 2 ε z z μ r r e z = 0 ,
ℬ = ∇ × A = 2 Ω ( ε r μ r − 1 ) c ( 1 − ε r μ r r 2 Ω 2 / c 2 ) 2 e ^ z = B z e ^ z ,
Δ ω = 2 k θ A θ c ε z z μ r r − A θ 2 .
Δ ω = 2 k θ A ˜ θ c ε r μ r = 2 m B ˜ z c ε r μ r = 2 m ( ε r μ r − 1 ) Ω ε r μ r ,
| φ 〉 = F m ( ω , ω 0 + Δ ω / 2 ) | CW 〉 + F m ( ω , ω 0 − Δ ω / 2 ) | CCW 〉 ,
b m = J m ( n ρ 0 ) J m ′ ( ρ 0 ) − n J m ′ ( n ρ 0 ) J m ( ρ 0 ) J m ( n ρ 0 ) H m ( 1 ) ′ ( ρ 0 ) − n J m ′ ( n ρ 0 ) H m ( 1 ) ( ρ 0 ) ,

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