Abstract

An elongated bottle microresonator with nanoscale parabolic effective radius variation can possess a series of dense equally spaced optical eigenfrequencies whose separation can match the eigenfrequency of its axially symmetric acoustic mode. It is shown that this acoustic mode can parametrically excite optical modes and give rise to a highly equidistant and moderately broadband optical frequency comb with the teeth spacing independent of the input laser power and the amplitude of mechanical vibrations.

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References

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  1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
    [Crossref]
  2. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
    [Crossref]
  3. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
    [Crossref]
  4. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
    [Crossref]
  5. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
    [Crossref]
  6. I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
    [Crossref]
  7. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
    [Crossref]
  8. M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
    [Crossref]
  9. T. Tetsumoto and T. Tanabe, AIP Adv. 4, 077137 (2014).
    [Crossref]
  10. M. Sumetsky, Nanophotonics 2, 393 (2013).
    [Crossref]
  11. M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
    [Crossref]
  12. N. A. Toropov and M. Sumetsky, Opt. Lett. 41, 2278 (2016).
    [Crossref]
  13. M. Sumetsky, Opt. Lett. 42, 923 (2017).
    [Crossref]
  14. M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
    [Crossref]
  15. K. Vahala, Nature 424, 839 (2003).
    [Crossref]
  16. A. B. Matsko and V. S. Ilchenko, IEEE J. Quantum Electron. 12, 3 (2006).
    [Crossref]
  17. M. Sumetsky, Sci. Rep. 5, 18569 (2015).
    [Crossref]
  18. M. Sumetsky, Opt. Express 20, 22537 (2012).
    [Crossref]
  19. S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
    [Crossref]
  20. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964).
  21. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
    [Crossref]
  22. A. M. Perelomov and Y. B. Zel’dovich, Quantum Mechanics: Selected Topics (World Scientific, 1998).
  23. Notice that the absence of sharp elastic transmission peaks at p=0 in the plots of Fig. 3 is due to the fact that the major elastic component of the transmission amplitude has been omitted in Eq. (10).
  24. R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
    [Crossref]
  25. A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
    [Crossref]

2017 (1)

2016 (3)

2015 (1)

M. Sumetsky, Sci. Rep. 5, 18569 (2015).
[Crossref]

2014 (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

T. Tetsumoto and T. Tanabe, AIP Adv. 4, 077137 (2014).
[Crossref]

2013 (2)

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

2012 (2)

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

M. Sumetsky, Opt. Express 20, 22537 (2012).
[Crossref]

2011 (3)

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[Crossref]

2010 (1)

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

2009 (2)

I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

2008 (1)

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

2006 (1)

A. B. Matsko and V. S. Ilchenko, IEEE J. Quantum Electron. 12, 3 (2006).
[Crossref]

2005 (1)

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

2003 (1)

K. Vahala, Nature 424, 839 (2003).
[Crossref]

2002 (1)

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Asano, M.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Bahl, G.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

Bjorken, J. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964).

Carmon, T.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

Chen, T.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

Chen, W.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[Crossref]

Drell, S. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964).

Fefferman, A. D.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Grudinin, I. S.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

Hänggi, P.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[Crossref]

Ikuta, R.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

A. B. Matsko and V. S. Ilchenko, IEEE J. Quantum Electron. 12, 3 (2006).
[Crossref]

Imoto, N.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

Jeon, S.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[Crossref]

Kohler, S.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Lee, H.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

Lehmann, J.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Li, J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

Liu, X.

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Maleki, L.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

A. B. Matsko and V. S. Ilchenko, IEEE J. Quantum Electron. 12, 3 (2006).
[Crossref]

Ozdemir, S. K.

Özdemir, S. K.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

Painter, O.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

Parpia, J. M.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Perelomov, A. M.

A. M. Perelomov and Y. B. Zel’dovich, Quantum Mechanics: Selected Topics (World Scientific, 1998).

Pohl, R. O.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

Seidel, D.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

Sumetsky, M.

Takeuchi, Y.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

Tanabe, T.

T. Tetsumoto and T. Tanabe, AIP Adv. 4, 077137 (2014).
[Crossref]

Tetsumoto, T.

T. Tetsumoto and T. Tanabe, AIP Adv. 4, 077137 (2014).
[Crossref]

Thompson, E.

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Tomes, M.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

Toropov, N. A.

Vahala, K.

K. Vahala, Nature 424, 839 (2003).
[Crossref]

Vahala, K. J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

Yamamoto, T.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

Yang, K. Y.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

Yang, L.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, Opt. Express 24, 12082 (2016).
[Crossref]

Zehnder, A. T.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Zehnpfennig, J.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

Zel’dovich, Y. B.

A. M. Perelomov and Y. B. Zel’dovich, Quantum Mechanics: Selected Topics (World Scientific, 1998).

AIP Adv. (1)

T. Tetsumoto and T. Tanabe, AIP Adv. 4, 077137 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

A. B. Matsko and V. S. Ilchenko, IEEE J. Quantum Electron. 12, 3 (2006).
[Crossref]

Laser Photonics Rev. (1)

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photonics Rev. 10, 603 (2016).
[Crossref]

Nanophotonics (1)

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

Nat. Commun. (1)

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[Crossref]

Nat. Photonics (1)

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics 6, 369 (2012).
[Crossref]

Nature (1)

K. Vahala, Nature 424, 839 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rep. (1)

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Phys. Rev. Lett. (5)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 104, 083901 (2010).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, Phys. Rev. Lett. 103, 257403 (2009).
[Crossref]

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Rev. Mod. Phys. (2)

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Sci. Rep. (1)

M. Sumetsky, Sci. Rep. 5, 18569 (2015).
[Crossref]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[Crossref]

Other (3)

A. M. Perelomov and Y. B. Zel’dovich, Quantum Mechanics: Selected Topics (World Scientific, 1998).

Notice that the absence of sharp elastic transmission peaks at p=0 in the plots of Fig. 3 is due to the fact that the major elastic component of the transmission amplitude has been omitted in Eq. (10).

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964).

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Figures (3)

Fig. 1.
Fig. 1. (a) Illustration of a bottle microresonator holding an optical WGM and an acoustic axially symmetric mode. Light is coupled in and out of the resonator by a microfiber taper which is aligned normally to the resonator axis z. (b) Excitation of the optical modes with even axial quantum numbers by an acoustic mode which is symmetric with respect to axis z leading to the generation of an equidistant OFC.
Fig. 2.
Fig. 2. Comparison of optical and acoustic mode distributions for a bottle resonator having the eigenfrequency ν(ac) of an acoustic mode equal to the spacing of optical eigenfrequencies Δν(op) along the quantum number q. (a) Nanoscale ERV of this resonator. (b) Axial distribution of the acoustic mode with quantum numbers mac=0, nac=1, and qac=0 fitted by a parabola (dashed curve). (c) Axial distribution of optical modes with q=0, q=10, and q=50.
Fig. 3.
Fig. 3. Surface plots of the power of a generated OFCs as a function of quantum numbers p and q and graphs of these dependencies for fixed quantum numbers q. (a) ϵ/δ=0.1, (b) ϵ/δ=0.3, (c) ϵ/δ=0.5, (d) ϵ/δ=0.7, and (e) ϵ/δ=0.9. The green lines defined by equation 2q+p=0 indicate the axial ground state of the bottle resonator.

Equations (10)

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

i2πνcutΨt+12βcut22Ψz2+Δr(z,t)r0Ψ=Ain(z,t),
Ain(z,t,ν)=A0δ(zz0)exp(2πi(ννcut)t){exp(2πγt)t<01t0,
Ψq(z,t)=exp[2πi(ζqνcut)t]×p=exp[2π(ipReν(ac)+|p|Imν(ac)|)t]Uq,p(z),
G(z1,z2,t1,t2)=θ(t1t2)qΨq(z1,t1)Ψq*(z2,t2),
Aout(ν1,ν2)=dt2t2dt1dz1G(z1,z2,t1,t2)Ain(z1,t1,ν1)=A0q,p1,p2Uq,p1(z0)Uq,p2*(z0)×{(p2Reν(ac)i|p2|Imν(ac)+ζqν1)×[(p1p2)Reν(ac)i(|p1|+|p2|)Imν(ac)+ν2ν1]}1.
ν2=Re(ζq)+pReν(ac),p=±1,±2,.
Ωq,p=|A0Uq,0(z0)Uq,p*(z0)|2(Reν(ac)Reζq)2p2(Q(opt)Q(ac))2.
Δr(z,t)=z22R0(1+2ϵsin(2πν(ac)t)),ϵ1.
ν(ac)=Δν(op)(2+δ),δ1,
U2q,p(0)=[(2q)!]12(2)qq!(πfκπ(1+κ))12n=max(0,p)2q(2qn)(2q12np)(1κ1+κ)2np,κ=(1ϵδ)1/2(1+ϵδ)1/2.

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