Abstract

An analysis is presented to estimate the perturbation of morphology-dependent optical resonances (MDRs) of spherical dielectric cavities due to acceleration. In the model, the optical cavity is attached to a rigid base that is accelerating. Hertz contact theory is used to describe the contact surface between the base and the sphere, and the Navier equation is solved to obtain an expression for MDR shift dependence on the base acceleration. An experiment is also carried out using a polydimethylsiloxane sphere as the optical cavity. The predictions using the analysis agree well with the experimental results.

© 2011 Optical Society of America

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  1. W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14 (2001).
    [CrossRef]
  2. M. Cai, O. Painter, and K. J. Vahala, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000).
    [CrossRef]
  3. H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
    [CrossRef]
  4. B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
    [CrossRef]
  5. B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
    [CrossRef]
  6. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
    [CrossRef]
  7. F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
    [CrossRef]
  8. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
    [CrossRef] [PubMed]
  9. G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
    [CrossRef]
  10. T. Ioppolo, M. Kozhevnikov, M. Stepaniuk, M. V. Ötügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonances,” Appl. Opt. 47, 3009–3014(2008).
    [CrossRef] [PubMed]
  11. T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
    [CrossRef]
  12. T. Ioppolo and M. V. Ötügen, “Pressure tuning of whispering gallery mode resonators,” J. Opt. Soc. Am. B 242721–2726(2007).
    [CrossRef]
  13. T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.
  14. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “Tuning of whispering gallery modes of spherical resonators using an external electric field,” Opt. Express 17, 16465–16479 (2009).
    [CrossRef] [PubMed]
  15. R. B. Hetnarski and J. Ignaczak, Mathematical Theory of Elasticity (Taylor & Francis, 2004).
  16. R. Courant and D. Hilbert, Method of Mathematical Physics, 1st ed. (Interscience, 1965).
  17. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., (McGraw-Hill, 1973).
  18. T. Carmon, S. Y. T. Wang, E. P. Ostby, and K. Vahala, “Wavelength-independent coupler from fiber to an on-chip cavity demonstrated over an 850nm span,” Opt. Express 15, 7677–7681 (2007).
    [CrossRef] [PubMed]

2009 (2)

T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
[CrossRef]

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “Tuning of whispering gallery modes of spherical resonators using an external electric field,” Opt. Express 17, 16465–16479 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

2006 (1)

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
[CrossRef]

2003 (1)

2002 (2)

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
[CrossRef]

2001 (1)

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14 (2001).
[CrossRef]

2000 (1)

1999 (1)

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

1998 (1)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
[CrossRef]

Arnold, S.

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
[CrossRef]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

Ayaz, U. K.

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “Tuning of whispering gallery modes of spherical resonators using an external electric field,” Opt. Express 17, 16465–16479 (2009).
[CrossRef] [PubMed]

T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.

Ayaz, Y. K.

T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
[CrossRef]

Beyerl, R.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Bona, G. L.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Brown, D.

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

Cai, M.

Carmon, T.

Chu, S. T.

B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Method of Mathematical Physics, 1st ed. (Interscience, 1965).

Germann, R.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Goodier, J. N.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., (McGraw-Hill, 1973).

Guan, G.

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
[CrossRef]

Haroche, S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
[CrossRef]

Hetnarski, R. B.

R. B. Hetnarski and J. Ignaczak, Mathematical Theory of Elasticity (Taylor & Francis, 2004).

Hilbert, D.

R. Courant and D. Hilbert, Method of Mathematical Physics, 1st ed. (Interscience, 1965).

Holler, S.

Horst, F.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Ignaczak, J.

R. B. Hetnarski and J. Ignaczak, Mathematical Theory of Elasticity (Taylor & Francis, 2004).

Ilchenko, V. S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Ioppolo, T.

T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
[CrossRef]

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “Tuning of whispering gallery modes of spherical resonators using an external electric field,” Opt. Express 17, 16465–16479 (2009).
[CrossRef] [PubMed]

T. Ioppolo, M. Kozhevnikov, M. Stepaniuk, M. V. Ötügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonances,” Appl. Opt. 47, 3009–3014(2008).
[CrossRef] [PubMed]

T. Ioppolo and M. V. Ötügen, “Pressure tuning of whispering gallery mode resonators,” J. Opt. Soc. Am. B 242721–2726(2007).
[CrossRef]

T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

Kozhevnikov, M.

Laine, J. P.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
[CrossRef]

Lane, P. A.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
[CrossRef]

Lefevre-Seguin, V.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Libchaber, A.

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
[CrossRef]

Offrein, B. J.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Ostby, E. P.

Ötügen, M. V.

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “Tuning of whispering gallery modes of spherical resonators using an external electric field,” Opt. Express 17, 16465–16479 (2009).
[CrossRef] [PubMed]

T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
[CrossRef]

T. Ioppolo, M. Kozhevnikov, M. Stepaniuk, M. V. Ötügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonances,” Appl. Opt. 47, 3009–3014(2008).
[CrossRef] [PubMed]

T. Ioppolo and M. V. Ötügen, “Pressure tuning of whispering gallery mode resonators,” J. Opt. Soc. Am. B 242721–2726(2007).
[CrossRef]

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
[CrossRef]

T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.

Painter, O.

Raimond, J.-M.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Salemink, H. W. M.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

Sheverev, V.

T. Ioppolo, M. Kozhevnikov, M. Stepaniuk, M. V. Ötügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonances,” Appl. Opt. 47, 3009–3014(2008).
[CrossRef] [PubMed]

T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.

Stepaniuk, M.

Tapalian, H. C.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
[CrossRef]

Teraoka, I.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

Timoshenko, S. P.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., (McGraw-Hill, 1973).

Treussart, F.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Vahala, K.

Vahala, K. J.

Velichansky, V. L.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Volikov, P. S.

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Vollmer, F.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

von Klitzing, W.

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14 (2001).
[CrossRef]

Wang, S. Y. T.

AIAA J. (1)

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature measurements using a micro-optical sensor based on whispering gallery modes,” AIAA J. 44, 2385–2389 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyerl, and G. L. Bona, “Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5, 1400–1406 (1999).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technol. Lett. 14, 1118–1120 (2002).
[CrossRef]

J. Appl. Phys. (1)

T. Ioppolo, Y. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105, 013535(2009).
[CrossRef]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, and H. A. Haus, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1000(1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

New J. Phys. (1)

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14 (2001).
[CrossRef]

Opt. Commun. (1)

V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145, 86–90 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Other (4)

T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A micro-optical wall shear stress sensor concept based on whispering gallery mode resonators,” presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 Jan. 2008.

R. B. Hetnarski and J. Ignaczak, Mathematical Theory of Elasticity (Taylor & Francis, 2004).

R. Courant and D. Hilbert, Method of Mathematical Physics, 1st ed. (Interscience, 1965).

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., (McGraw-Hill, 1973).

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Figures (3)

Fig. 1
Fig. 1

(a) Dielectric sphere and coordinates, (b) compressed sphere and contact geometry.

Fig. 2
Fig. 2

(a) Photograph and (b) schematic of the experimental setup.

Fig. 3
Fig. 3

Comparison of experimental and analytical results.

Tables (1)

Tables Icon

Table 1 Acceleration Sensitivity Estimates for a 900 μm Diameter Dielectric Sphere

Equations (8)

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

2 u + 1 1 2 ν ( · u ) + ρ g G = ρ G u ¨ .
σ r r ( R ) = { p ( ϑ ) 0 ϑ ϑ 0 0 ϑ 0 ϑ π σ r ϑ ( R ) = 0 ,
2 u + 1 1 2 ν ( · u ) ρ a G = 0.
u = φ ,
2 φ = 1 2 ν 2 ( 1 ν ) G χ .
χ = ρ a r P 1 ( cos ( ϑ ) ) ,
u r = [ A n ( n + 1 ) ( n 2 + 4 ν ) r n + 1 + B n n r n 1 ] P n ( 0 ) .
Δ λ λ = u r ( R , π / 2 ) R = 2 n H n 4 G ( n 2 + 2 n ν + n + ν + 1 ) [ 2 4 n 2 + ν ( 4 n 2 2 n 4 ) n 1 ] P n ( 0 ) .

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