Abstract

A new laser differential confocal radius measurement (DCRM) is proposed for high precision measurement of radius. Based on the property of an axial intensity curve that the absolute zero precisely corresponds to the focus of the objective in a differential confocal system (DCS), DCRM uses the zero point of the DCS axial intensity curve to precisely identify the cat's-eye and confocal positions of the test lens, and measures the accurate distance between the two positions to achieve the high-precision measurement of radius of curvature (ROC). In comparison with the existing measurement methods, DCRM proposed has a high measurement precision, a strong environmental anti-interference capability and a low cost. The theoretical analyses and preliminary experimental results indicate that DCRM has a relative measurement error of better than 5ppm.

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  1. P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).
  2. D. Malacara, “Optical Shop Testing, 2nd edition,” (Wiley-Interscience, 1992), Chap17.
  3. L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992).
    [CrossRef]
  4. Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
    [CrossRef]
  5. Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007).
    [CrossRef] [PubMed]
  6. U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
    [CrossRef]
  7. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
    [CrossRef] [PubMed]
  8. T. L. Schmitz, C. J. Evans, A. D. Davies and W. T. Estler, “Displacement Uncertainty in Interferometric Radius Measurements,” CIRP Annals - Manufacturing Technology, 51, 451–454 (2002).
  9. T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
    [CrossRef]
  10. W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
    [CrossRef] [PubMed]
  11. W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
    [CrossRef]
  12. L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).
  13. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), Chap. 9.
  14. A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005).
    [CrossRef] [PubMed]
  15. T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
    [CrossRef] [PubMed]

2009 (2)

P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

2008 (1)

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

2007 (1)

Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007).
[CrossRef] [PubMed]

2005 (2)

A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005).
[CrossRef] [PubMed]

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

2004 (2)

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
[CrossRef] [PubMed]

2001 (3)

Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
[CrossRef]

T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
[CrossRef]

L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

1992 (1)

L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992).
[CrossRef]

Becker, P.

P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).

Davies, A.

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005).
[CrossRef] [PubMed]

Davies, A. D.

T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
[CrossRef]

DeBra, D.

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

Deng, X.

L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

Evans, C. J.

T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
[CrossRef]

Friedrich, H.

P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).

Fujii, K.

P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).

Gardner, N.

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

Griesmann, U.

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

Hao, Q.

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

Hu, Y.

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

Jin, P.

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

Liu, L.

L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

Medicus, K.

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

Pi, Y.

Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007).
[CrossRef] [PubMed]

Qiu, L.

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
[CrossRef] [PubMed]

Reardon, P. J.

Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007).
[CrossRef] [PubMed]

Schmitz, T. L.

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005).
[CrossRef] [PubMed]

T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
[CrossRef]

Selberg, L. A.

L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992).
[CrossRef]

Soons, J.

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

Tan, J.

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
[CrossRef] [PubMed]

Vaughn, M.

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

Wang, G.

L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

Wang, Q.

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

Xiang, Y.

Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
[CrossRef]

Zhao, W.

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
[CrossRef] [PubMed]

Zhu, Q.

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

Acta Phys. Sin. (1)

L. Liu, X. Deng, and G. Wang, “Phase-only optical pupil filter for improving axial resolution in confocal microscopy,” Acta Phys. Sin. 50, 48–51 (2001).

Ann. CIRP (1)

U. Griesmann, J. Soons, Q. Wang, and D. DeBra, “Measuring form and radius of sphere with interferometry,” Ann. CIRP 53(1), 451–454 (2004).
[CrossRef]

Appl. Opt. (3)

Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
[CrossRef]

A. Davies and T. L. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44(28), 5884–5893 (2005).
[CrossRef] [PubMed]

T. L. Schmitz, N. Gardner, M. Vaughn, K. Medicus, and A. Davies, “Improving optical bench radius measurements using stage error motion data,” Appl. Opt. 47(36), 6692–6700 (2008).
[CrossRef] [PubMed]

Meas. Sci. Technol. (1)

P. Becker, H. Friedrich, and K. Fujii, “W. Giardini, G Mana, A Picard, H-J Pohl, H. Riemann and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20, 1–20 (2009).

Opt. Eng. (1)

L. A. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31(9), 1961–1966 (1992).
[CrossRef]

Opt. Express (1)

W. Zhao, J. Tan, and L. Qiu, “Bipolar absolute differential confocal approach to higher spatial resolution,” Opt. Express 12(21), 5013–5021 (2004).
[CrossRef] [PubMed]

Opt. Lett. (2)

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009).
[CrossRef] [PubMed]

Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. 32(9), 1063–1065 (2007).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. L. Schmitz, A. D. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius-of-curvature,” Proc. SPIE 4451, 432–447 (2001).
[CrossRef]

Sens. Actuators A Phys. (1)

W. Zhao, J. Tan, L. Qiu, and P. Jin, “SABCMS, A New Approach to Higher Lateral Resolution of Laser Probe Measurement,” Sens. Actuators A Phys. 120(1), 17–25 (2005).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), Chap. 9.

T. L. Schmitz, C. J. Evans, A. D. Davies and W. T. Estler, “Displacement Uncertainty in Interferometric Radius Measurements,” CIRP Annals - Manufacturing Technology, 51, 451–454 (2002).

D. Malacara, “Optical Shop Testing, 2nd edition,” (Wiley-Interscience, 1992), Chap17.

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Figures (15)

Fig. 1
Fig. 1

DCRM principle.

Fig. 2
Fig. 2

Effect of Φ(ρ,θ) on identification precision at confocal position B.

Fig. 3
Fig. 3

Effect of figure error Φ(ρ,θ) on identification precision at confocal position B. a) Spherical aberration A 040 ρ 4 b) Astigmatism A 022 ρ 2cos2 θ.

Fig. 4
Fig. 4

Effect of figure error Φ(ρ,θ) on sensitivity curves.

Fig. 5
Fig. 5

Virtual pinhole detection principle.

Fig. 6
Fig. 6

DCRMS using VPH tracking.

Fig. 7
Fig. 7

Light-path principle with different offset of two detectors.

Fig. 8
Fig. 8

Effect of beam parallel of illumination collimating system.

Fig. 9
Fig. 9

Nonlinear calibration system of CCD intensity response.

Fig. 10
Fig. 10

Nonlinear calibration of CCD response curve.

Fig. 11
Fig. 11

Intensity response curves before and after calibration on CCD. 
a) Intensity response curves of two CCD b) DCS intensity curve.

Fig. 12
Fig. 12

Adjustment of angle β between optical axis t of the lens and motion axis m.

Fig. 13
Fig. 13

Experimental setup. (1) X80 laser interferometer produced by RENISHAW, (2) Interferometry measurement prisms, (3) Air bearing slider, (4) Test lens, (5) Standard focusing lens, (6) Collimator, (7) He-Ne laser, (8) Single-mode fiber, (9) CCD confocal detection system 1, (10) CCD confocal detection system 2, (11) Drive motor, (12) Material sensor, (13) Air sensor, (14) Monitor, (15) Image capture and soft process system.

Fig. 14
Fig. 14

Radius measurement curves.

Fig. 15
Fig. 15

Figure errors of the test lens.

Tables (1)

Tables Icon

Table 1 Nonlinear calibration on CCD response

Equations (43)

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I A ( v , φ , u , u M ) = I A1 ( v , φ , u , + u M ) I A2 ( v , φ , u , u M )                      = | 1 π 0 2 π 0 1 p C ( ρ , θ ) p 1 ( ρ , θ ) p 1 ( ρ , π + θ ) p 2 ( ρ , π + θ ) e j ρ 2 ( 2 u + u M ) / 2 e j ρ v cos ( θ φ ) ρ d ρ d θ | 2                         | 1 π 0 2 π 0 1 p C ( ρ , θ ) p 1 ( ρ , θ ) p 1 ( ρ , π + θ ) p 2 ( ρ , π + θ ) e j ρ 2 ( 2 u u M ) / 2 e j ρ v cos ( θ φ ) ρ d ρ d θ | 2
{ u = π 2 λ ( D f ) 2 z v = π 2 λ ( D f ) r 0
I A ( 0 , 0 , u , u M ) = | 2 0 1 e j ρ 2 ( 2 u + u M ) / 2 ρ d ρ | 2 | 2 0 1 e j ρ 2 ( 2 u u M ) / 2 ρ d ρ | 2                       = [ sin [ ( 2 u + u M ) / 4 ] ( 2 u + u M ) / 4 ] 2 [ sin [ ( 2 u u M ) / 4 ] ( 2 u u M ) / 4 ] 2
I B ( v , φ , u , u M , Φ ) = I B1 ( v , φ , u , + u M , Φ ) I B2 ( v , φ , u , u M , Φ )                           = | 1 π 0 2 π 0 1 p C ( ρ , θ ) p 1 2 ( ρ , θ ) p 2 ( ρ , θ ) e j ρ 2 ( 2 u + u M ) / 2 e j 2 k Φ ( ρ , θ ) e j ρ v cos ( θ φ ) ρ d ρ d θ | 2                              | 1 π 0 2 π 0 1 p C ( ρ , θ ) p 1 2 ( ρ , θ ) p 2 ( ρ , θ ) e j ρ 2 ( 2 u u M ) / 2 e j 2 k Φ ( ρ , θ ) e j ρ v cos ( θ φ ) ρ d ρ d θ | 2
Φ ( ρ , θ A 040 ρ 4 A 022 ρ 2 cos 2 θ + A 031 ρ 3 cos θ A 120 ρ 2 + A 111 ρ cos θ
Δ 2 π λ ( 2 A 040 + A 022 )
I B A 040 ( v , φ , u , u M , Φ ) = | 1 π 0 2 π 0 1 e j ρ 2 ( 2 u + u M ) / 2 e j 2 k A 040 ρ 4 e j ρ v cos ( θ φ ) ρ d ρ d θ | 2                                     | 1 π 0 2 π 0 1 e j ρ 2 ( 2 u u M ) / 2 e j 2 k A 040 ρ 4 e j ρ v cos ( θ φ ) ρ d ρ d θ | 2
I B A 022 ( v , φ , u , u M , Φ ) = | 1 π 0 2 π 0 1 e j ρ 2 ( 2 u + u M ) / 2 e j 2 k A 022 ρ 2 cos θ 2   e j ρ v cos ( θ φ ) ρ d ρ d θ | 2                                    | 1 π 0 2 π 0 1 e j ρ 2 ( 2 u u M ) / 2 e j 2 k A 022 ρ 2 cos θ 2 e j ρ v cos ( θ φ ) ρ d ρ d θ | 2
S B ( 0 , 0 , 0 , C , Φ ) = I B ( v , φ , u , u M , Φ ) u | φ = 0 , u = 0 , v = 0 , u M = C
{ x = ( m , n m g ( m , n ) ) / m , n g ( m , n ) y = ( m , n n g ( m , n ) ) / m , n g ( m , n )
d 2.5 λ π sin α
d V P H = N d p 2.5 N λ π p sin α
I A ( 0 , 0 , u A , u M ) = [ sin ( ( 2 u A + u δ + u M ) / 4 ) ( 2 u A + u δ + u M ) / 4 ] 2 [ sin ( ( 2 u A u M ) / 4 ) ( 2 u A u M ) / 4 ] 2 = 0
I B ( 0 , 0 , u B , u M , Φ ) = [ sin ( 2 u B + u δ + u M ) / 4 ( 2 u B + u δ + u M ) / 4 ] 2 [ sin ( 2 u B u M ) / 4 ( 2 u B u M ) / 4 ] 2 = 0
u A = u B = u δ 4
Δ r δ = Δ z A Δ z B = 0
p C ( ρ , θ ) = e j ρ 2 ( u C / 2 )
I A ( 0 , 0 , u A , u M ) = [ sin ( ( 2 u A + u M + u C ) / 4 ) ( 2 u A + u M + u C ) / 4 ] 2 [ sin ( ( 2 u A u M + u C ) / 4 ) ( 2 u A u M + u C ) / 4 ] 2 = 0
I B ( 0 , 0 , u B , u M , Φ ) = [ sin ( ( 2 u B + u M + u C ) / 4 ) ( ( 2 u B + u M ) + u C ) / 4 ] 2 [ sin ( ( 2 u B u M + u C ) / 4 ) ( 2 u B u M + u C ) / 4 ] 2 = 0
u A = u B = u C 2
Δ r C = Δ z A Δ z B = 0
σ a x i a l = r ( 1 cos β cos γ )
tan β ε 2 r DMI
ε = n p f L1 N f L2
r = r DMI cos β = r DMI cos ( arc ( tan β ) ) r DMI / cos ( ε 2 r DMI ) = r DMI / cos ( n × p × f L1 2 N × f L2 × r DMI )
σ L = σ n a i r ( L d e a d p a t h 2 + r 2 )      ( L d e a d p a t h 2 + r 2 ) ( 2.68 × 10 9 σ P a ) 2 + ( 9.27 × 10 7 σ K ) 2 + ( 1 × 10 8 σ H ) 2
S ( 0 , 0 , 0 , u M ) = I A ( 0 , 0 , u , u M ) u | u = 0 = 2 sinc ( u M 4 π ) [ ( u M 4 ) cos ( u M 4 ) sin ( u M 4 ) ( u M 4 ) 2 ]
σ z = δ I A ( 0 , 0 , u , u M ) 0 . 54 2 λ π ( D / f ) 2 = 1.18 λ S N R ( D / f ) 2
δ β = arctan ( δ ε 2 r ) δ ε 2 r = δ n p f L1 2 N f L2 r
σ a x i a l = r ( 1 cos β cos γ ) r ( 1 cos γ ) = r ( 1 cos 0.0005 ) 1.24 × 10 7 r  
σ figure 0.1 PV
σ r σ L 2 + 2 σ z 2 + σ a x i a l 2 + σ f i g u r e 2
σ r σ L 2 + 2 σ z 2 + σ a x i a l 2 + σ f i g u r e 2 = ( 0.1 ) 2 + 2 × ( 0.1 ) 2 + ( 0.01 ) 2 + ( 0.01 ) 2 μ m 0.17 μ m
δ r | σ r r | × 100 % | 0.17 100 × 1000 | × 100 % = 0.00017 % 2 ppm
n = ( 376.6 383.7 ) 2 + ( 309.1 280.5 ) 2 29.5
r = r DMI / cos ( n p 2 N r DMI ) = ( 0.00093 36.69542 ) / cos ( 29.5 × 0.008 2 × 10 × ( 0.00093 36.69542 ) )    36 .6964 m m
r = 36.6964 + 0.037 × 0.0006328 36.6964 m m
σ L = ( L d e a d p a t h 2 + r 2 ) ( 2.68 × 10 9 σ P a ) 2 + ( 9.27 × 10 7 σ K ) 2 + ( 1 × 10 8 σ H ) 2      = ( 600 2 + 36.6964 2 ) × ( 2.68 × 10 9 × 50 ) 2 + ( 9.27 × 10 7 × 0.1 ) 2 + ( 1 × 10 8 × 3 ) 2 0.0001 m m
σ a x i a l = 1.24 × 10 7 r = 1.24 × 10 7 × ( 36.6964 ) = 5 n m
σ z = 1.18 λ S N R ( D / f ) 2 = 1.18 × 0.6328 150 × 0.87 2 = 0.007 μ m
σ figure 0.1 PV=0 .1 × 0 .208 × 0.6328 = 0.013 μ m
σ r σ L 2 + 2 σ z 2 + σ a x i a l 2 + σ f i g u r e 2      = ( 0.1 ) 2 + 2 × ( 0.007 ) 2 + ( 0.005 ) 2 + ( 0.013 ) 2 μ m 0.10 μ m
δ r | σ r r | × 100 % | 0.10 36.6964 × 1000 | × 100 % = 0.00027 % 3 ppm

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