Abstract

We report on the physical mechanism underlying the temperature dependence of phase retardation. Changes in refractive index and thickness of the wave plate are the two main contributions to phase retardation variations. Temperature-dependent changes in refractive index are described by Sellmeier’s equation. Constants in this equation need to be determined by fits to experimental data. A new high-precision instrument was used to measure phase retardation at various temperatures. Once determined, the fitting equation can be used to calculate the phase retardation at any temperature, and thus has implications for wave-plate industrial manufacturing.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Ghosh, “Coefficients of a dispersion equation for the birefringence of a CdGeAs2 nonlinear crystal at different temperatures,” Appl. Opt. 37, 6456–6458 (1998).
    [CrossRef]
  2. W. Wiechmann, S. Kubota, T. Fukui, and H. Masuda, “Coefficients of a dispersion equation for the birefringence of a CdGeAs2 nonlinear crystal at different temperatures,” Opt. Lett. 18, 1208–1210 (1993).
    [CrossRef]
  3. H. Y. Shen, H. Xu, Z. D. Zeng, W. X. Lin, R. F. Wu, and G. F. Xu, “Measurement of refractive indices and thermal refractive-index coefficients of LiNbO3 crystal doped with 5 mol.% MgO,” Appl. Opt. 31, 6695–6697 (1992).
    [CrossRef]
  4. H. Shen, D. Zhang, W. Liu, W. Chen, G. Zhang, G. Zhang, and W. Lin, “Measurement of refractive indices and thermal refractive-index coefficients of 7.5 mol. % NbKTiOPO4 crystal,” Appl. Opt. 38, 987–990 (1999).
    [CrossRef]
  5. D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
    [CrossRef]
  6. H. Shen, X. Meng, G. Zhang, J. Qin, W. Liu, L. Zhu, C. Huang, L. Huang, and M. Wei, “Sellmeier’s equation and the expression of the thermal refractive-index coefficient for a Nd0.007Gd0.993VO4 crystal,” Appl. Opt. 43, 955–960 (2004).
    [CrossRef]
  7. G. Ghosh, “Sellmeier coefficients for the birefringence and refractive indices of ZnGeP2 nonlinear crystal at different temperatures,” Appl. Opt. 37, 1205–1212 (1998).
    [CrossRef]
  8. Y. Lin, Z. Zhou, and R. Wang, “Optical heterodyne measurement of the phase retardation of quarter-wave plate,” Opt. Lett. 13, 533–555 (1988).
    [CrossRef]
  9. W. Chen, S. Zhang, and X. Long, “Measurement of phase retardation of wave plate online based laser feedback,” Rev. Sci. Instrum. 83, 013101 (2012).
    [CrossRef]
  10. P. G. R. King, and G. J. Steward, “Metrology with an optical master,” New Sci. 17, 180–182 (1963).
  11. G. Stephan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276–1280 (1987).
    [CrossRef]
  12. M. Sciamanna, K. Panajotov, H. Thienpont, and I. Veretennicoff, “Optical feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers,” Opt. Lett. 28, 1543–1545 (2003).
    [CrossRef]
  13. H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
    [CrossRef]
  14. W. Xiong, P. Glanznig, P. Paddon, and A. D. May, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 4, 1276–1280 (1987).
    [CrossRef]
  15. J. Brannon, “Laser feedback: its effect on laser frequency,” Appl. Opt. 15, 1119–1120 (1976).
    [CrossRef]
  16. E. T. Shimizu, “Directional discrimination in the self-mixing type laser Doppler velocimeter,” Appl. Opt. 26, 4541–4544 (1987).
    [CrossRef]
  17. S. Shinohara, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
    [CrossRef]
  18. S. Donati, and G. Giuliani, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
    [CrossRef]
  19. S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
    [CrossRef]
  20. S. Merlo and S. Donati, “Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer,” IEEE J. Quantum Electron. 33, 527–531 (1997).
    [CrossRef]
  21. T. Suzuki, T. Muto, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Appl. Opt. 38, 7069–548 (1999).
    [CrossRef]
  22. S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
    [CrossRef]
  23. F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
    [CrossRef]
  24. G. Beheim, and K. Firtsch, “Ranging finding using frequency modulated laser diode,” Appl. Opt. 25, 1439–1442 (1986).
    [CrossRef]
  25. P. A. Roos, M. Stephens, and C. Wiemen, “Laser vibrometer based on optical feedback induced frequency modulation of a single mode laser diode,” Appl. Opt. 35, 6754–6761 (1996).
    [CrossRef]

2012

W. Chen, S. Zhang, and X. Long, “Measurement of phase retardation of wave plate online based laser feedback,” Rev. Sci. Instrum. 83, 013101 (2012).
[CrossRef]

2004

2003

1999

H. Shen, D. Zhang, W. Liu, W. Chen, G. Zhang, G. Zhang, and W. Lin, “Measurement of refractive indices and thermal refractive-index coefficients of 7.5 mol. % NbKTiOPO4 crystal,” Appl. Opt. 38, 987–990 (1999).
[CrossRef]

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

T. Suzuki, T. Muto, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Appl. Opt. 38, 7069–548 (1999).
[CrossRef]

1998

1997

S. Merlo and S. Donati, “Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer,” IEEE J. Quantum Electron. 33, 527–531 (1997).
[CrossRef]

1996

S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
[CrossRef]

P. A. Roos, M. Stephens, and C. Wiemen, “Laser vibrometer based on optical feedback induced frequency modulation of a single mode laser diode,” Appl. Opt. 35, 6754–6761 (1996).
[CrossRef]

1995

S. Donati, and G. Giuliani, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

1993

1992

1989

S. Shinohara, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

1988

1987

1986

1982

S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

1976

1963

P. G. R. King, and G. J. Steward, “Metrology with an optical master,” New Sci. 17, 180–182 (1963).

Aissaoui, B.

Beheim, G.

Bosch, T.

Brannon, J.

Chen, W.

Chen, W. Z.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Choquette, K. D.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

Donati, S.

S. Merlo and S. Donati, “Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer,” IEEE J. Quantum Electron. 33, 527–531 (1997).
[CrossRef]

S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
[CrossRef]

S. Donati, and G. Giuliani, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

Fazoni, L.

S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
[CrossRef]

Firtsch, K.

Fukui, T.

Gavrielides, A.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

Ghosh, G.

Giuliani, G.

S. Donati, and G. Giuliani, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

Glanznig, P.

Gouaux, F.

Hohl, A.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

Hou, H.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

Huang, C.

Huang, C. H.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Huang, L.

Ito, M.

S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

King, P. G. R.

P. G. R. King, and G. J. Steward, “Metrology with an optical master,” New Sci. 17, 180–182 (1963).

Koboyashi, S.

S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

Kubota, S.

Li, H.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

Liang, J. K.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Lin, W.

Lin, W. X.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

H. Y. Shen, H. Xu, Z. D. Zeng, W. X. Lin, R. F. Wu, and G. F. Xu, “Measurement of refractive indices and thermal refractive-index coefficients of LiNbO3 crystal doped with 5 mol.% MgO,” Appl. Opt. 31, 6695–6697 (1992).
[CrossRef]

Lin, Y.

Liu, W.

Long, X.

W. Chen, S. Zhang, and X. Long, “Measurement of phase retardation of wave plate online based laser feedback,” Rev. Sci. Instrum. 83, 013101 (2012).
[CrossRef]

Maruyama, T.

Masuda, H.

May, A. D.

Meng, X.

Merlo, S.

S. Merlo and S. Donati, “Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer,” IEEE J. Quantum Electron. 33, 527–531 (1997).
[CrossRef]

S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
[CrossRef]

Mueller, R. E.

Muto, T.

Paddon, P.

Panajotov, K.

Qin, J.

Roos, P. A.

Sasaki, O.

Sciamanna, M.

Servagent, N.

Shen, H.

Shen, H. Y.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

H. Y. Shen, H. Xu, Z. D. Zeng, W. X. Lin, R. F. Wu, and G. F. Xu, “Measurement of refractive indices and thermal refractive-index coefficients of LiNbO3 crystal doped with 5 mol.% MgO,” Appl. Opt. 31, 6695–6697 (1992).
[CrossRef]

Shimizu, E. T.

Shinohara, S.

S. Shinohara, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

Stephan, G.

Stephens, M.

Steward, G. J.

P. G. R. King, and G. J. Steward, “Metrology with an optical master,” New Sci. 17, 180–182 (1963).

Suzuki, T.

Thienpont, H.

Veretennicoff, I.

Wang, R.

Wei, M.

Wiechmann, W.

Wiemen, C.

Wu, R. F.

Xiong, W.

Xu, G. F.

Xu, H.

Yamamoto, Y.

S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

Zeng, R. R.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Zeng, Z. D.

Zhang, D.

Zhang, D. Y.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Zhang, G.

Zhang, G. F.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Zhang, S.

W. Chen, S. Zhang, and X. Long, “Measurement of phase retardation of wave plate online based laser feedback,” Rev. Sci. Instrum. 83, 013101 (2012).
[CrossRef]

Zhou, Z.

Zhu, L.

Appl. Opt.

G. Ghosh, “Coefficients of a dispersion equation for the birefringence of a CdGeAs2 nonlinear crystal at different temperatures,” Appl. Opt. 37, 6456–6458 (1998).
[CrossRef]

H. Y. Shen, H. Xu, Z. D. Zeng, W. X. Lin, R. F. Wu, and G. F. Xu, “Measurement of refractive indices and thermal refractive-index coefficients of LiNbO3 crystal doped with 5 mol.% MgO,” Appl. Opt. 31, 6695–6697 (1992).
[CrossRef]

H. Shen, D. Zhang, W. Liu, W. Chen, G. Zhang, G. Zhang, and W. Lin, “Measurement of refractive indices and thermal refractive-index coefficients of 7.5 mol. % NbKTiOPO4 crystal,” Appl. Opt. 38, 987–990 (1999).
[CrossRef]

H. Shen, X. Meng, G. Zhang, J. Qin, W. Liu, L. Zhu, C. Huang, L. Huang, and M. Wei, “Sellmeier’s equation and the expression of the thermal refractive-index coefficient for a Nd0.007Gd0.993VO4 crystal,” Appl. Opt. 43, 955–960 (2004).
[CrossRef]

G. Ghosh, “Sellmeier coefficients for the birefringence and refractive indices of ZnGeP2 nonlinear crystal at different temperatures,” Appl. Opt. 37, 1205–1212 (1998).
[CrossRef]

J. Brannon, “Laser feedback: its effect on laser frequency,” Appl. Opt. 15, 1119–1120 (1976).
[CrossRef]

E. T. Shimizu, “Directional discrimination in the self-mixing type laser Doppler velocimeter,” Appl. Opt. 26, 4541–4544 (1987).
[CrossRef]

T. Suzuki, T. Muto, O. Sasaki, and T. Maruyama, “Self-mixing type of phase-locked laser diode interferometer,” Appl. Opt. 38, 7069–548 (1999).
[CrossRef]

F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
[CrossRef]

G. Beheim, and K. Firtsch, “Ranging finding using frequency modulated laser diode,” Appl. Opt. 25, 1439–1442 (1986).
[CrossRef]

P. A. Roos, M. Stephens, and C. Wiemen, “Laser vibrometer based on optical feedback induced frequency modulation of a single mode laser diode,” Appl. Opt. 35, 6754–6761 (1996).
[CrossRef]

Appl. Phys. Lett.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998).
[CrossRef]

IEEE J. Quantum Electron.

S. Donati, and G. Giuliani, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

S. Koboyashi, Y. Yamamoto, and M. Ito, “Direct frequency modulation in AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

S. Merlo and S. Donati, “Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer,” IEEE J. Quantum Electron. 33, 527–531 (1997).
[CrossRef]

IEEE Trans. Instrum. Meas.

S. Donati, L. Fazoni, and S. Merlo, “A PC-interfaced compact laser-diode feedback interferometer for displacement measurements,” IEEE Trans. Instrum. Meas. 45, 942–947 (1996).
[CrossRef]

S. Shinohara, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeters with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

J. Opt. Soc. Am. B

New Sci.

P. G. R. King, and G. J. Steward, “Metrology with an optical master,” New Sci. 17, 180–182 (1963).

Opt. Commun.

D. Y. Zhang, H. Y. Shen, W. Liu, G. F. Zhang, W. Z. Chen, G. Zhang, R. R. Zeng, C. H. Huang, W. X. Lin, and J. K. Liang, “The expressions of the principal thermal refractive index coefficients of 7.5 mol. % NbKTiOPO4 crystals,” Opt. Commun. 168, 111–115 (1999).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

W. Chen, S. Zhang, and X. Long, “Measurement of phase retardation of wave plate online based laser feedback,” Rev. Sci. Instrum. 83, 013101 (2012).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Setup for measuring temperature effects on phase retardation. D 1 , D 2 : photo detectors. M : laser cavity mirror. M E : feedback mirrors. S : quartz wave plate. PZT: piezoelectric transducer. P : polarizer. H : heating structure. I : insulation structure.

Fig. 2.
Fig. 2.

Waveforms of laser intensity modulation.

Fig. 3.
Fig. 3.

Theoretical and experimental data error.

Tables (2)

Tables Icon

Table 1. Phase Retardation Variation with Temperature

Tables Icon

Table 2. Numerical Values of Parameters

Equations (7)

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

Δ n = n e n o = A + B λ 2 λ 2 C 2 + D λ 2 λ 2 E 2 ,
A = a 2 T 2 + a 1 T + a o , B = b 2 T 2 + b 1 T + b o C = c 1 T + c o , D = d 2 T 2 + d 1 T + d o E = e 1 T + e o ,
h = h 2 T 2 + h 1 T + h o ,
δ = 2 π Δ n h λ .
δ = 2 π [ A + B λ 2 λ 2 C 2 + D λ 2 λ 2 E 2 ] [ h 2 T 2 + h 1 T + h o ] λ .
δ = ( l B C l A D + l F G l E H ) × 90 ° ,
Δ δ = 2 π λ [ Δ a 2 T 2 + Δ a 1 T + Δ a 0 + ( Δ b 2 T 2 + Δ b 1 T + Δ b 0 ) λ 2 λ 2 ( c 1 2 T 2 + 2 c 1 c 0 T + c 0 2 ) ( b 2 T 2 + b 1 T + b 0 ) λ 2 ( 2 c 1 Δ c 1 T 2 + 2 c 1 Δ c 0 T + 2 Δ c 0 [ λ 2 ( c 1 2 T 2 + 2 c 1 c 0 T + c 0 2 ) ] 2 + ( Δ d 2 T 2 + Δ d 1 T + Δ d 0 ) λ 2 λ 2 ( e 1 2 T 2 + 2 e 1 e 0 T + e 0 2 ) ( d 2 T 2 + d 1 T + d 0 ) λ 2 ( 2 e 1 Δ e 1 T 2 + 2 e 1 Δ e 0 T + 2 Δ e 0 [ λ 2 ( e 1 2 T 2 + 2 e 1 e 0 T + e 0 2 ) ] 2 ] [ h 2 T 2 + h 1 T + h 0 ] + 2 π λ [ a 2 T 2 + a 1 T + a 0 + ( b 2 T 2 + b 1 T + b 0 ) λ 2 λ 2 ( c 1 2 T 2 + 2 c 1 c 0 T + c 0 2 ) + ( d 2 T 2 + d 1 T + d 0 ) λ 2 λ 2 ( e 1 2 T 2 + 2 e 1 e 0 T + e 0 2 ) ] [ Δ h 2 T 2 + Δ h 1 T + Δ h 0 ] .

Metrics