Abstract

We introduce two new approaches for near-real-time, high-precision tracking of the refractive index of the ambient atmosphere. The methods can be realized at low cost and are expected to have important practical application in those accurate dimensional metrology applications employing interferometry in air. A valuable potential application is the control of step-and-repeat mask positioning for integrated circuit production in which temporal stability time scales over days can be crucial. Extension of the methods to absolute index measurement is discussed.

© 1997 Optical Society of America

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  1. P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).
  2. B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  3. H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
    [CrossRef]
  4. K. P. Birch, M. J. Downs, “The Precise Determination of the Refractive Index of Air,” , July (National Physical Laboratory, Teddington, U.K., 1988).
  5. R. Muijlwijk, “Update of the Edlén formula for the refractive index of air,” Metrologia 25, 189 (1988).
    [CrossRef]
  6. K. P. Birch, M. J. Downs, “The results of a comparison between calculated and measured values of the refractive index of air,” J. Phys. E 21, 694–695 (1988); “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993); Metrologia 31, 315–316(E) (1994).
    [CrossRef]
  7. F. E. Jones, “The refractivity of air,” J. Res. Natl. Bur. Stand. 83, 27–32 (1981).
    [CrossRef]
  8. J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition,” Appl. Opt. 6, 51–59 (1967).
    [CrossRef] [PubMed]
  9. Portable Pressure Gauge, Model 6200, Ruska Instrument Corp., Houston, Texas.
  10. Mention of specific products and manufacturers is for the purpose of technical communication only and does not constitute an endorsement nor does it imply that products from other manufacturers would be less suitable.
  11. Ruska Instrument Corp., Houston, Texas (personal communication, 13July1993).
  12. AS115 Four-Wire Thermistor Standard, Thermometrics, Edison, N.J.
  13. Thermometrics, Edison, N.J. (personal communication, 7July1993).
  14. Model 3478 A 5-1/2 digit multimeter, Hewlett-Packard, Palo Alto, Calif.
  15. Model 200 DewTrak Humidity Transmitter, EG&G Environmental, Burlington, Mass.
  16. J. A. Goff, “Saturation pressure of water on the new Kelvin scale,” in Humidity and Moisture. Measurement and Control in Science and Industry. Vol. 3. Fundamentals and Standards, A. Wexler, W. A. Wildhack, eds., (Reinhold, New York, 1965), pp. 289–292.
  17. ©Microsoft Corp.
  18. This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).
  19. M. Zhu, J. L. Hall, “Short and long term stability of optical oscillators,” in Proceedings of the IEEE Frequency Control Symposium, IEEE Catalog No. 92CH3083-3 (IEEE, New York, 1992), pp. 44–55; J. L. Hall, “Frequency stabilized lasers—a parochial review,” in Frequency-Stabilized Lasers and Their Applications, Y. C. Chung, ed., Proc. SPIE1837, 2–15 (1993).
  20. K. Kinosita, “Numerical evaluation of the intensity curve or a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
    [CrossRef]
  21. A. D. White, “Frequency stabilization of gas lasers,” IEEE J. Quantum Electron. QE-1, 349–357 (1965).
    [CrossRef]
  22. Type 007 internal mirror He–Ne laser tube, Spectra Physics, Mountain View, Calif. Present experiments with 633-nm external cavity diode lasers also seem promising.
  23. M. D. Rayman, C. G. Aminoff, J. L. Hall, “Precise laser frequency scanning using frequency-synthesized optical frequency sidebands,” J. Opt. Soc. Am. B 6, 539–549 (1989).
    [CrossRef]
  24. HP 8660 C Synthesized Frequency Generator or HP 30860 RF Sweep Generator, Hewlett-Packard, Palo Alto, Calif.
  25. D. Hils, J. L. Hall, “Response of a Fabry-Perot cavity to phase-modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
    [CrossRef]
  26. W. Lichten, “Precise wavelength measurements and optical phase shifts. II. Applications,” J. Opt. Soc. Am. A 3, 909–915 (1986).
    [CrossRef]
  27. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  28. However, because of the curved mirrors, there is a substantial diffractive phase shift. This can be determined to high precision by measurement of off-axis modes. However, it often turns out to be difficult to determine the diffraction correction with sufficient accuracy, so one usually uses these high-finesse interferometers to determine differences between nearby optical frequencies. Some precise measurements with ultrastable cavities with high finesse will be reported separately.
  29. M. Andersson, L. Eliasson, L. R. Pendrill, “Compressible Fabry-Perot refractometer,” Appl. Opt. 26, 4835–4840 (1987).
    [CrossRef] [PubMed]
  30. J. L. Hall, “Frequency-stabilized lasers: a driving force for new spectroscopies,” in Frontiers of Laser Spectroscopy, T. W. Hädansch, M. Inguscio, eds. (North Holland, Amsterdam, 1994), pp. 217–239.
  31. J. J. Snyder, “Algorithm for fast digital analysis of interference fringes,” Appl. Opt. 19, 1223–1225 (1980).
    [CrossRef] [PubMed]
  32. M. P. Winters (personal communication, 1988).

1989

1988

R. Muijlwijk, “Update of the Edlén formula for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

K. P. Birch, M. J. Downs, “The results of a comparison between calculated and measured values of the refractive index of air,” J. Phys. E 21, 694–695 (1988); “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993); Metrologia 31, 315–316(E) (1994).
[CrossRef]

1987

D. Hils, J. L. Hall, “Response of a Fabry-Perot cavity to phase-modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

M. Andersson, L. Eliasson, L. R. Pendrill, “Compressible Fabry-Perot refractometer,” Appl. Opt. 26, 4835–4840 (1987).
[CrossRef] [PubMed]

1986

W. Lichten, “Precise wavelength measurements and optical phase shifts. II. Applications,” J. Opt. Soc. Am. A 3, 909–915 (1986).
[CrossRef]

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

1981

F. E. Jones, “The refractivity of air,” J. Res. Natl. Bur. Stand. 83, 27–32 (1981).
[CrossRef]

1980

1967

1966

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

1965

A. D. White, “Frequency stabilization of gas lasers,” IEEE J. Quantum Electron. QE-1, 349–357 (1965).
[CrossRef]

1953

K. Kinosita, “Numerical evaluation of the intensity curve or a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

1939

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Aminoff, C. G.

Andersson, M.

Barrell, H.

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Birch, K. P.

K. P. Birch, M. J. Downs, “The results of a comparison between calculated and measured values of the refractive index of air,” J. Phys. E 21, 694–695 (1988); “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993); Metrologia 31, 315–316(E) (1994).
[CrossRef]

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

K. P. Birch, M. J. Downs, “The Precise Determination of the Refractive Index of Air,” , July (National Physical Laboratory, Teddington, U.K., 1988).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Downs, M. J.

K. P. Birch, M. J. Downs, “The results of a comparison between calculated and measured values of the refractive index of air,” J. Phys. E 21, 694–695 (1988); “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993); Metrologia 31, 315–316(E) (1994).
[CrossRef]

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

K. P. Birch, M. J. Downs, “The Precise Determination of the Refractive Index of Air,” , July (National Physical Laboratory, Teddington, U.K., 1988).

Edlén, B.

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Eickhoff, M. L.

This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).

Eliasson, L.

Goff, J. A.

J. A. Goff, “Saturation pressure of water on the new Kelvin scale,” in Humidity and Moisture. Measurement and Control in Science and Industry. Vol. 3. Fundamentals and Standards, A. Wexler, W. A. Wildhack, eds., (Reinhold, New York, 1965), pp. 289–292.

Hall, J. L.

M. D. Rayman, C. G. Aminoff, J. L. Hall, “Precise laser frequency scanning using frequency-synthesized optical frequency sidebands,” J. Opt. Soc. Am. B 6, 539–549 (1989).
[CrossRef]

D. Hils, J. L. Hall, “Response of a Fabry-Perot cavity to phase-modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

J. L. Hall, “Frequency-stabilized lasers: a driving force for new spectroscopies,” in Frontiers of Laser Spectroscopy, T. W. Hädansch, M. Inguscio, eds. (North Holland, Amsterdam, 1994), pp. 217–239.

M. Zhu, J. L. Hall, “Short and long term stability of optical oscillators,” in Proceedings of the IEEE Frequency Control Symposium, IEEE Catalog No. 92CH3083-3 (IEEE, New York, 1992), pp. 44–55; J. L. Hall, “Frequency stabilized lasers—a parochial review,” in Frequency-Stabilized Lasers and Their Applications, Y. C. Chung, ed., Proc. SPIE1837, 2–15 (1993).

This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).

Hils, D.

D. Hils, J. L. Hall, “Response of a Fabry-Perot cavity to phase-modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

Jones, F. E.

F. E. Jones, “The refractivity of air,” J. Res. Natl. Bur. Stand. 83, 27–32 (1981).
[CrossRef]

Kinosita, K.

K. Kinosita, “Numerical evaluation of the intensity curve or a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

Lichten, W.

Martin, P. J.

This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).

Muijlwijk, R.

R. Muijlwijk, “Update of the Edlén formula for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

Owens, J. C.

Pendrill, L. R.

Rayman, M. D.

Reinboth, F.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

Schellekens, P.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

Sears, J. E.

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Snyder, J. J.

Spronck, J.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

White, A. D.

A. D. White, “Frequency stabilization of gas lasers,” IEEE J. Quantum Electron. QE-1, 349–357 (1965).
[CrossRef]

Wilkening, G.

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

Winters, M. P.

This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Zhu, M.

M. Zhu, J. L. Hall, “Short and long term stability of optical oscillators,” in Proceedings of the IEEE Frequency Control Symposium, IEEE Catalog No. 92CH3083-3 (IEEE, New York, 1992), pp. 44–55; J. L. Hall, “Frequency stabilized lasers—a parochial review,” in Frequency-Stabilized Lasers and Their Applications, Y. C. Chung, ed., Proc. SPIE1837, 2–15 (1993).

Appl. Opt.

IEEE J. Quantum Electron.

A. D. White, “Frequency stabilization of gas lasers,” IEEE J. Quantum Electron. QE-1, 349–357 (1965).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. E

K. P. Birch, M. J. Downs, “The results of a comparison between calculated and measured values of the refractive index of air,” J. Phys. E 21, 694–695 (1988); “An updated Edlén equation for the refractive index of air,” Metrologia 30, 155–162 (1993); Metrologia 31, 315–316(E) (1994).
[CrossRef]

J. Phys. Soc. Jpn.

K. Kinosita, “Numerical evaluation of the intensity curve or a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

J. Res. Natl. Bur. Stand.

F. E. Jones, “The refractivity of air,” J. Res. Natl. Bur. Stand. 83, 27–32 (1981).
[CrossRef]

Metrologia

R. Muijlwijk, “Update of the Edlén formula for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

P. Schellekens, G. Wilkening, F. Reinboth, M. J. Downs, K. P. Birch, J. Spronck, “Measurements of the refractive index of air using interference refractometers,” Metrologia 22, 279–287 (1986). Our quoted formula contains the 103 correction for the misprint of their D term, as noted by C. Rischel, P. S. Ramanujam, “Refractive index of air: errata,” Metrologia 26, 263 (1989).

B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Rev. Sci. Instrum.

D. Hils, J. L. Hall, “Response of a Fabry-Perot cavity to phase-modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

However, because of the curved mirrors, there is a substantial diffractive phase shift. This can be determined to high precision by measurement of off-axis modes. However, it often turns out to be difficult to determine the diffraction correction with sufficient accuracy, so one usually uses these high-finesse interferometers to determine differences between nearby optical frequencies. Some precise measurements with ultrastable cavities with high finesse will be reported separately.

J. L. Hall, “Frequency-stabilized lasers: a driving force for new spectroscopies,” in Frontiers of Laser Spectroscopy, T. W. Hädansch, M. Inguscio, eds. (North Holland, Amsterdam, 1994), pp. 217–239.

M. P. Winters (personal communication, 1988).

K. P. Birch, M. J. Downs, “The Precise Determination of the Refractive Index of Air,” , July (National Physical Laboratory, Teddington, U.K., 1988).

Type 007 internal mirror He–Ne laser tube, Spectra Physics, Mountain View, Calif. Present experiments with 633-nm external cavity diode lasers also seem promising.

HP 8660 C Synthesized Frequency Generator or HP 30860 RF Sweep Generator, Hewlett-Packard, Palo Alto, Calif.

Portable Pressure Gauge, Model 6200, Ruska Instrument Corp., Houston, Texas.

Mention of specific products and manufacturers is for the purpose of technical communication only and does not constitute an endorsement nor does it imply that products from other manufacturers would be less suitable.

Ruska Instrument Corp., Houston, Texas (personal communication, 13July1993).

AS115 Four-Wire Thermistor Standard, Thermometrics, Edison, N.J.

Thermometrics, Edison, N.J. (personal communication, 7July1993).

Model 3478 A 5-1/2 digit multimeter, Hewlett-Packard, Palo Alto, Calif.

Model 200 DewTrak Humidity Transmitter, EG&G Environmental, Burlington, Mass.

J. A. Goff, “Saturation pressure of water on the new Kelvin scale,” in Humidity and Moisture. Measurement and Control in Science and Industry. Vol. 3. Fundamentals and Standards, A. Wexler, W. A. Wildhack, eds., (Reinhold, New York, 1965), pp. 289–292.

©Microsoft Corp.

This refractometer approach is described by J. L. Hall, P. J. Martin, M. L. Eickhoff, M. P. Winters, “Highly accurate in-situ determination of the refractivity of an ambient atmosphere,” U.S. patent5,218,426 (8June1993).

M. Zhu, J. L. Hall, “Short and long term stability of optical oscillators,” in Proceedings of the IEEE Frequency Control Symposium, IEEE Catalog No. 92CH3083-3 (IEEE, New York, 1992), pp. 44–55; J. L. Hall, “Frequency stabilized lasers—a parochial review,” in Frequency-Stabilized Lasers and Their Applications, Y. C. Chung, ed., Proc. SPIE1837, 2–15 (1993).

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Figures (6)

Fig. 1
Fig. 1

Layout of the Fabry–Perot refractometer. Collimated output from the fiber coupler is converged in the vertical plane before entering the plane–plane Fabry–Perot interferometer. As discussed in text, the direction of fringes and the CCD detector line are perpendicular to the plane of the figure to eliminate astigmatism. Curvature of the R = 100 -cm focusing mirror is exaggerated for clarity.

Fig. 2
Fig. 2

(a) Typical scan of a Fabry–Perot fringe system imaged onto a CCD array photodetector. Data points are filled dots and the nonlinear least-squares best fit is the solid curve. The signal-to-noise ratio is approximately 50. (b) Residual between data points and best fit. Obvious extraneous noise at one-half of the sampling frequency is due to the offset between separate odd and even sampling circuits within the CCD chip. The current software filter does not compensate fully. Systematic errors are discussed in the text.

Fig. 3
Fig. 3

(a) Plot of output phase from the computerized fringe-fitting algorithm versus the order number from the dye laser control cavity. Dots are experimental data points. Solid line is best-fit line to those points. One free spectral range of the Fabry–Perot is 18.35 ± 0.01 orders of the dye laser’s control cavity. (b) Plot of residual from best fit above versus control cavity order. Error bars are from diagonal elements of the covariance matrix after convergence in the fringe-fitting algorithm. Although some systematic trends are clear, these data show that the optical fringe phase is recovered correctly within ±2 × 10-3 orders by the current algorithms.

Fig. 4
Fig. 4

(a) Layout of the air–wavelength standard prototype: F.I., Faraday isolator; P.D., photodetector; AOM, acousto-optic modulator that is driven by a voltage-controlled oscillator (VCO). Analog 25-kHz applied voltage generates ±500-kHz FM excursion of the 80-MHz output, enabling phase-sensitive detection of lock signal. Note that the laser output is unmodulated and has a constant wavelength measured in laboratory air. (b) Sketch of top and side views of the Zerodur bar cavity that was used in the constant-air-wavelength standard prototype. Curvature of the mirrors is exaggerated for clarity. Mating surfaces are tinned individually with indium solder before the mirrors are attached firmly with indium solder flowed onto the cylindrical surface.

Fig. 5
Fig. 5

Output of the weather bureau showing (a) ambient pressure, (b) temperature, (c) relative humidity as functions of time.

Fig. 6
Fig. 6

Comparison of the refractometers. The change in the index of refraction of the ambient air is shown as a function of time. Output from each refractometer is plotted on the same axis from 0 at t = 0. Agreement for index changes is ±5 × 10-8 for some hours and ≈±1 × 10-7 for the entire run. Accidental choice of starting time during epoch of rapid change leads to apparent offset ≈1 × 10-7 during quiescent times. FP, Fabry–Pérot.

Equations (10)

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

n-1=D0.104127×10-4P1+0.3671×10-2T-0.42063×10-9F,
2nd=λ0m+e,
n2-n1λ02de2-e1.
n=mFSRν,
n2-n1-n1ν0Δν2-Δν1.
d=λ0m+e/2.
δl/l=-P1-2σ/E-4.8×10-7,
xi-a1=±f2λ0di-1+a2
Ix=11+F sin2πm+ecosx-x0f
xmincosθminxcenter1xmaxcosθmax.

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