Abstract

An equivalence is made between the exceptional points proposed by the field of non-Hermitian quantum mechanics and the dead band observed in laser gyroscopes. The sensitivity enhancement near this exceptional point is plagued by increased uncertainty due to broadening of the beat-note bandwidth. Also, near the dead band the gyroscope response is caused by Rabi intensity oscillations and not solely by a phase modulation. Finally, a distinction is made between conservative and non-conservative coupling.

© 2020 Chinese Laser Press

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References

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  1. F. Aronowitz and R. J. Collins, “Lock-in and intensity-phase interaction in the ring laser,” J. Appl. Phys. 41, 130–141 (1970).
    [Crossref]
  2. M. L. Dennis, J.-C. Diels, and M. Lai, “The femtosecond ring dye laser: a potential new laser gyro,” Opt. Lett. 16, 529–531 (1991).
    [Crossref]
  3. A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
    [Crossref]
  4. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  5. L. Arissian and J.-C. Diels, “Intracavity phase interferometry: frequency comb sensors inside a laser cavity,” Laser Photon. Rev. 8, 799–826 (2014).
    [Crossref]
  6. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
    [Crossref]
  7. A. Kodigala, T. Lepetit, and B. Kanté, “Exceptional points in three-dimensional plasmonic nanostructures,” Phys. Rev. B 94, 201103 (2016).
    [Crossref]
  8. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
    [Crossref]
  9. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
    [Crossref]
  10. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  11. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
    [Crossref]
  12. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref]
  13. J. Zhu, Ş. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
    [Crossref]
  14. H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
    [Crossref]
  15. J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
    [Crossref]
  16. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
    [Crossref]
  17. J.-C. Diels and I. C. McMichael, “Influence of wave-front-conjugated coupling on the operation of a laser gyro,” Opt. Lett. 6, 219–221 (1981).
    [Crossref]
  18. D. Smith, H. Chang, L. Horstman, and J.-C. Diels, “Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes,” Opt. Express (to be published).
  19. The equations are written in optics notation so that there is no “i” on the left-hand side. One must use caution when comparing to Schrödinger-like equations and defining hermiticity.
  20. F. Aronowitz, “The laser gyro,” in Laser Applications, M. Ross, ed. (Academic, 1971), pp. 133–200.
  21. M. Navarro, O. Chalus, and J.-C. Diels, “Mode-locked ring lasers for backscattering measurement of mirrors,” Opt. Lett. 31, 2864–2866 (2006).
    [Crossref]
  22. Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
    [Crossref]
  23. This relation depends on the form of the CMEs. If in the form of the Schrödinger equation (with an “i” on the left-hand side), the relation is κ˜1=κ˜2*.
  24. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).
  25. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
  26. R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
    [Crossref]
  27. H. Wang, S. Assawaworrarit, and S. Fan, “Dynamics for encircling an exceptional point in a nonlinear non-Hermitian system,” Opt. Lett. 44, 638–641 (2019).
    [Crossref]

2019 (2)

Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
[Crossref]

H. Wang, S. Assawaworrarit, and S. Fan, “Dynamics for encircling an exceptional point in a nonlinear non-Hermitian system,” Opt. Lett. 44, 638–641 (2019).
[Crossref]

2018 (1)

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

2017 (3)

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

2016 (1)

A. Kodigala, T. Lepetit, and B. Kanté, “Exceptional points in three-dimensional plasmonic nanostructures,” Phys. Rev. B 94, 201103 (2016).
[Crossref]

2014 (4)

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

L. Arissian and J.-C. Diels, “Intracavity phase interferometry: frequency comb sensors inside a laser cavity,” Laser Photon. Rev. 8, 799–826 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

2010 (1)

2009 (1)

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[Crossref]

2008 (1)

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

2006 (1)

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1991 (1)

1990 (1)

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

1981 (1)

1970 (1)

F. Aronowitz and R. J. Collins, “Lock-in and intensity-phase interaction in the ring laser,” J. Appl. Phys. 41, 130–141 (1970).
[Crossref]

1946 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

Arissian, L.

L. Arissian and J.-C. Diels, “Intracavity phase interferometry: frequency comb sensors inside a laser cavity,” Laser Photon. Rev. 8, 799–826 (2014).
[Crossref]

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

Aronowitz, F.

F. Aronowitz and R. J. Collins, “Lock-in and intensity-phase interaction in the ring laser,” J. Appl. Phys. 41, 130–141 (1970).
[Crossref]

F. Aronowitz, “The laser gyro,” in Laser Applications, M. Ross, ed. (Academic, 1971), pp. 133–200.

Assawaworrarit, S.

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Chalus, O.

Chang, H.

D. Smith, H. Chang, L. Horstman, and J.-C. Diels, “Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes,” Opt. Express (to be published).

Chen, D.-R.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[Crossref]

Chen, W.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Chow, W.

Christodoulides, D.

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Christodoulides, D. N.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Collins, R. J.

F. Aronowitz and R. J. Collins, “Lock-in and intensity-phase interaction in the ring laser,” J. Appl. Phys. 41, 130–141 (1970).
[Crossref]

Dennis, M. L.

Diels, J.-C.

L. Arissian and J.-C. Diels, “Intracavity phase interferometry: frequency comb sensors inside a laser cavity,” Laser Photon. Rev. 8, 799–826 (2014).
[Crossref]

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

M. Navarro, O. Chalus, and J.-C. Diels, “Mode-locked ring lasers for backscattering measurement of mirrors,” Opt. Lett. 31, 2864–2866 (2006).
[Crossref]

M. L. Dennis, J.-C. Diels, and M. Lai, “The femtosecond ring dye laser: a potential new laser gyro,” Opt. Lett. 16, 529–531 (1991).
[Crossref]

J.-C. Diels and I. C. McMichael, “Influence of wave-front-conjugated coupling on the operation of a laser gyro,” Opt. Lett. 6, 219–221 (1981).
[Crossref]

D. Smith, H. Chang, L. Horstman, and J.-C. Diels, “Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes,” Opt. Express (to be published).

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Eliel, E. R.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

Fan, S.

Feng, L.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Garcia-Gracia, H.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Harari, G.

Hassan, A.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Hassan, A. U.

He, L.

J. Zhu, Ş. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[Crossref]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Hodaei, H.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Horstman, L.

D. Smith, H. Chang, L. Horstman, and J.-C. Diels, “Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes,” Opt. Express (to be published).

Kanté, B.

A. Kodigala, T. Lepetit, and B. Kanté, “Exceptional points in three-dimensional plasmonic nanostructures,” Phys. Rev. B 94, 201103 (2016).
[Crossref]

Khajavikhan, M.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Kodigala, A.

A. Kodigala, T. Lepetit, and B. Kanté, “Exceptional points in three-dimensional plasmonic nanostructures,” Phys. Rev. B 94, 201103 (2016).
[Crossref]

Lai, M.

Lai, Y.-H.

Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
[Crossref]

Lepetit, T.

A. Kodigala, T. Lepetit, and B. Kanté, “Exceptional points in three-dimensional plasmonic nanostructures,” Phys. Rev. B 94, 201103 (2016).
[Crossref]

Li, L.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[Crossref]

Lu, Y.-K.

Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
[Crossref]

Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

McMichael, I. C.

Miri, M.-A.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Musslimani, Z. H.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Navarro, M.

Neelen, R. C.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2009).
[Crossref]

Özdemir, S. K.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

J. Zhu, Ş. K. Özdemir, L. He, and L. Yang, “Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers,” Opt. Express 18, 23535–23543 (2010).
[Crossref]

Ren, J.

Rotter, S.

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11–19 (2018).
[Crossref]

Rudolph, W.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic, 2006).

Schmitt-Sody, A.

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

Smith, D.

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

D. Smith, H. Chang, L. Horstman, and J.-C. Diels, “Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes,” Opt. Express (to be published).

Soltani, M.

Spreeuw, R. J. C.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

Suh, M.-G.

Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
[Crossref]

Vahala, K.

Y.-H. Lai, Y.-K. Lu, M.-G. Suh, and K. Vahala, “Observation of exceptional point enhanced Sagnac effect,” Nature 576, 65–69 (2019).
[Crossref]

van Druten, N. J.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

Velten, A.

A. Schmitt-Sody, L. Arissian, A. Velten, J.-C. Diels, and D. Smith, “Rabi cycling of two pulses in a mode-locked ring laser cavity with electro-optical control,” Phys. Rev. A 78, 063802 (2008).
[Crossref]

Wang, H.

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Wiersig, J.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

Wittek, S.

H. Hodaei, A. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Woerdman, J. P.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

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Supplementary Material (1)

NameDescription
» Visualization 1       Referring to Figure 6, the video shows the temporal evolution of the two fields in polar coordinates. The two fields evolve at the same frequency with difference phases, and there is no beat note, even though there are two split eigenvalues.

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Figures (6)

Fig. 1.
Fig. 1. Sketches of bidirectional ML gyro configurations (left) and their corresponding beat-note responses (right). The × corresponds to a pulse crossing that does not introduce phase coupling. G is the gain, and Δϕ=Δ/τrt is the differential round-trip phase shift. Note that the time unit in Eq. (1) has been normalized to τrt. (a) Linear (i.e., no dead band) response. (b) When a scattering interface is placed at the crossing point, a square-root (i.e., dead band) response is observed (data from Ref. [21]).
Fig. 2.
Fig. 2. Polar plots of the imaginary versus real part of the beat field near (left) and far (right) from the EP (dead band). Near the EP the beat note stems from amplitude modulation, while far from the EP it is caused by pure phase modulation.
Fig. 3.
Fig. 3. Beat-signal spectrum from a numerical solution of Eq. (1) (top), and experimentally measured beat-note signal (bottom) showing the clustering of frequency harmonics near the dead band (left) and their absence for larger Δ (right).
Fig. 4.
Fig. 4. Beat-signal spectrum from a numerical solution of Eq. (1) showing the clustering of harmonics near the EP (dead band).
Fig. 5.
Fig. 5. Gyro beat-note response curve changes with κ˜ and s. All large circles are beat frequencies numerically solved from Eq. (1) with κ˜=0.05 and s=0 (blue), s=0.03 (orange), s=0.05 (yellow), and s=0.06 (purple). The green circles are with κ˜=0 and s=0.05. The red-dashed curves correspond to the eigenvalue beat frequency 2Δω determined from Eq. (5). When saturable gain is included, the green circles shift to the positions of the cyan crosses because the COG (rather than the most prevalent peak) of the spectrum must be used. An example of data matching the κ˜=0 case can be found in Ref. [21].
Fig. 6.
Fig. 6. Numerical solution (blue circles) to Eq. (1) and analytic prediction (red dashes) of Eq. (6) showing an EP at Δ=0 for κ˜=0.05 with a saturable gain in one resonator, α^1=0.051, αL=0, β=1, γ=0, and a constant loss in the other, α2=|κ˜|=0.05. Inset: because the gain difference depends on Δ, it only cancels the coupling exactly at Δ=0. The time dependence of the fields in polar coordinates is shown in Visualization 1 for Δ=0.01.

Equations (6)

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E˜˙1(t)=α1E˜1(t)iΔ2E˜1(t)+(s+κ˜1)E˜2(t),E˜˙2(t)=α2E˜2(t)+iΔ2E˜2(t)+(s+κ˜2)E˜1(t),
E˜1,2(t)=A1,2eiΔωt
|iΔ/2+iΔωs+κ˜1s+κ˜2iΔ/2+iΔω|=0,
κ˜1=κ˜2*
Δω=±(Δ/2)2+|κ˜|2s(κ˜κ˜*)s2.
Δω=iα1+α22±124|κ˜|2[(α1α2)iΔ]2.