Abstract

Plane mirror interferometer is a common way for the precision displacement measurement. However, during the measurement, it still suffers from disturbances such as misalignments, rotations and air refractive index fluctuations, which lead to poor accuracy. Traditional error analysis is rather limited in the static state and separation of the disturbances. In this paper, displacement measurement errors are analyzed, which are caused by the disturbed factors for a plane mirror interferometer. Then error modeling, which based on the geometric optical paths, is carried out by the partial differentiation theory. Moreover, the characteristics of the error are discussed by using this model. It is suggested that this model can release the measurement accuracy reduction brought by coupling effects between adjustment factor of the optical paths and the rotary error of the measured object (e. g. a guideway).

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Tang, Z. Wang, Z. Jiang, J. Gao, and J. Guo, “A new measuring method for circular motion accuracy of NC machine tools based on dual-frequency laser interferometer,” in Proceedings of IEEE International Symposium on Assembly and Manufacturing (ISAM 2011), 1–6 (2011).
  2. F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
    [CrossRef]
  3. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
    [CrossRef]
  4. I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
    [CrossRef] [PubMed]
  5. H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17(4), 746–752 (2006).
    [CrossRef]
  6. Z. Zhang and C. H. Menq, “Laser interferometric system for six-axis motion measurement,” Rev. Sci. Instrum. 78(8), 083107 (2007).
    [CrossRef] [PubMed]
  7. F. G. P. Peeters, “Interferometer with added flexibility in its use,” Opt. Eng. 35(7), 1953–1956 (1996).
    [CrossRef]
  8. N. Bobroff, “Critical alignments in plane mirror interferometry,” Proc. SPIE 1673, 63–67 (1992).
    [CrossRef]
  9. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
    [CrossRef]
  10. H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).
  11. D. L. Cohen, “Performance degradation of a Michelson interferometer when its misalignment angle is a rapidly varying, random time series,” Appl. Opt. 36(18), 4034–4042 (1997).
    [CrossRef] [PubMed]
  12. S. Awtar and A. H. Slocum, “Target block alignment error in XY stage metrology,” Precis. Eng. 31(3), 185–187 (2007).
    [CrossRef]
  13. H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
    [CrossRef]
  14. K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
    [CrossRef]
  15. J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
    [CrossRef]
  16. L. A. Kivioja, “The EDM corner reflector constant is not constant,” Surveying and Mapping 6, 143–149 (1978).
  17. J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
    [CrossRef]

2010

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

2009

J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
[CrossRef]

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

2007

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

Z. Zhang and C. H. Menq, “Laser interferometric system for six-axis motion measurement,” Rev. Sci. Instrum. 78(8), 083107 (2007).
[CrossRef] [PubMed]

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

S. Awtar and A. H. Slocum, “Target block alignment error in XY stage metrology,” Precis. Eng. 31(3), 185–187 (2007).
[CrossRef]

2006

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17(4), 746–752 (2006).
[CrossRef]

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

2005

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[CrossRef]

2001

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

1997

1996

F. G. P. Peeters, “Interferometer with added flexibility in its use,” Opt. Eng. 35(7), 1953–1956 (1996).
[CrossRef]

1993

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

1992

N. Bobroff, “Critical alignments in plane mirror interferometry,” Proc. SPIE 1673, 63–67 (1992).
[CrossRef]

1978

L. A. Kivioja, “The EDM corner reflector constant is not constant,” Surveying and Mapping 6, 143–149 (1978).

Arai, Y.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

Awtar, S.

S. Awtar and A. H. Slocum, “Target block alignment error in XY stage metrology,” Precis. Eng. 31(3), 185–187 (2007).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

N. Bobroff, “Critical alignments in plane mirror interferometry,” Proc. SPIE 1673, 63–67 (1992).
[CrossRef]

Bosse, H.

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[CrossRef]

Büchner, H. J.

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17(4), 746–752 (2006).
[CrossRef]

Cohen, D. L.

Gao, W.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

Goullioud, R.

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

Hahn, I.

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

Huang, J.

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

Jäger, G.

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17(4), 746–752 (2006).
[CrossRef]

Jeanmonod, N.

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

Jia, D.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Jiang, W.

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

Jing, W.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Kivioja, L. A.

L. A. Kivioja, “The EDM corner reflector constant is not constant,” Surveying and Mapping 6, 143–149 (1978).

Kiyono, S.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

Koops, K. R.

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

Kotte, G. J. W. L.

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

Lee, D. Y.

J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
[CrossRef]

Lee, M. Y.

J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
[CrossRef]

Li, X.

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

Liu, K.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Meil, F.

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

Menq, C. H.

Z. Zhang and C. H. Menq, “Laser interferometric system for six-axis motion measurement,” Rev. Sci. Instrum. 78(8), 083107 (2007).
[CrossRef] [PubMed]

Moolmen, M. C.

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

Park, C. H.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

Park, J.

J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
[CrossRef]

Peeters, F. G. P.

F. G. P. Peeters, “Interferometer with added flexibility in its use,” Opt. Eng. 35(7), 1953–1956 (1996).
[CrossRef]

Shibuya, A.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

Slocum, A. H.

S. Awtar and A. H. Slocum, “Target block alignment error in XY stage metrology,” Precis. Eng. 31(3), 185–187 (2007).
[CrossRef]

Tang, F.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Thalmann, R.

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

Thiess, C.

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

van Veghel, M. G. A.

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

Wang, X.

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

Weilert, M.

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

Wilkening, G.

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[CrossRef]

Xian, H.

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

Xu, T.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Zhang, H.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Zhang, Y.

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

Zhang, Z.

Z. Zhang and C. H. Menq, “Laser interferometric system for six-axis motion measurement,” Rev. Sci. Instrum. 78(8), 083107 (2007).
[CrossRef] [PubMed]

Acta Opt. Sin.

J. Huang, H. Xian, W. Jiang, and X. Li, “The reflected beam’s phase aberration induced by the fabrication errors of corner cube retroreflector,” Acta Opt. Sin. 29(7), 1951–1955 (2009).
[CrossRef]

Appl. Opt.

Meas. Sci. Technol.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4(9), 907–926 (1993).
[CrossRef]

H. Bosse and G. Wilkening, “Developments at PTB in nanometrology for support of the semiconductor industry,” Meas. Sci. Technol. 16(11), 2155–2166 (2005).
[CrossRef]

K. R. Koops, M. G. A. van Veghel, G. J. W. L. Kotte, and M. C. Moolmen, “Calibration strategies for scanning probe metrology,” Meas. Sci. Technol. 18(2), 390–394 (2007).
[CrossRef]

H. J. Büchner and G. Jäger, “A novel plane mirror interferometer without using corner cube reflectors,” Meas. Sci. Technol. 17(4), 746–752 (2006).
[CrossRef]

Microsyst. Technol.

J. Park, M. Y. Lee, and D. Y. Lee, “A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope,” Microsyst. Technol. 15(12), 1879–1884 (2009).
[CrossRef]

Opt. Eng.

F. G. P. Peeters, “Interferometer with added flexibility in its use,” Opt. Eng. 35(7), 1953–1956 (1996).
[CrossRef]

Precis. Eng.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[CrossRef]

S. Awtar and A. H. Slocum, “Target block alignment error in XY stage metrology,” Precis. Eng. 31(3), 185–187 (2007).
[CrossRef]

Proc. SPIE

H. Zhang, T. Xu, W. Jing, D. Jia, F. Tang, K. Liu, and Y. Zhang, “Influence of angle misalignment on detection polarization coupling in white light interferometer,” Proc. SPIE 6829(68290E), 1–9 (2007).

F. Meil, N. Jeanmonod, C. Thiess, and R. Thalmann, “Calibration of a 2D reference mirror system of a photomask measuring instrument,” Proc. SPIE 4401, 227–233 (2001).
[CrossRef]

N. Bobroff, “Critical alignments in plane mirror interferometry,” Proc. SPIE 1673, 63–67 (1992).
[CrossRef]

Rev. Sci. Instrum.

Z. Zhang and C. H. Menq, “Laser interferometric system for six-axis motion measurement,” Rev. Sci. Instrum. 78(8), 083107 (2007).
[CrossRef] [PubMed]

I. Hahn, M. Weilert, X. Wang, and R. Goullioud, “A heterodyne interferometer for angle metrology,” Rev. Sci. Instrum. 81(4), 045103 (2010).
[CrossRef] [PubMed]

Surveying and Mapping

L. A. Kivioja, “The EDM corner reflector constant is not constant,” Surveying and Mapping 6, 143–149 (1978).

Other

S. Tang, Z. Wang, Z. Jiang, J. Gao, and J. Guo, “A new measuring method for circular motion accuracy of NC machine tools based on dual-frequency laser interferometer,” in Proceedings of IEEE International Symposium on Assembly and Manufacturing (ISAM 2011), 1–6 (2011).

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 (5)

Fig. 1
Fig. 1

(a) Typical arrangement of a plane mirror interferometer. (b) Optical paths of the plane mirror interferometer with four times paths.

Fig. 2
Fig. 2

(a) The measurement schematic diagram of the interferometer. (b) Optical paths with the pitch angle. (c) Optical paths with the yaw angle.

Fig. 3
Fig. 3

Optical paths of the light beam in the C with a certain incident angle.

Fig. 4
Fig. 4

(a) Misalignments of the plane mirror in the ideal coordinate system. (b) Misalignments of the plane mirror during the measurement without θ 0. (c) Optical paths of the measurement beam in the air.

Fig. 5
Fig. 5

(a) Errors of the variable Δθ or Δϕ; (b) Errors of the variable x.

Equations (31)

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

L_PP= 2 n P L 0 cos(2 θ y ) +A,
n P sin(2 θ y ) =nsin2 θ y .
L_PY= 2 n P L 0 cos(2 θ z ) +A,
n P sin(2 θ z ) =nsin2 θ z .
L_P= 2 n P L 0 cos(2θ ) +A,
n P sin(2θ ) =nsin2θ.
L_C= 2 n C H cos β ,
n C sin β =nsinβ.
L_air=2nL_AI,
L_AI= L_IX cos2θ +L_IX,
L_IX=xcos(θ+φ).
L_air= 4nx cos 2 θcos(θ+φ) cos2θ .
n C sin β =nsin2θ.
f M =L_P+L_C+L_air+B.
f M = 2 n P L 0 cos(2θ ) +A+ 2 n C H cos β + 4nx cos 2 θcos(θ+φ) cos2θ +B.
f= f M f R .
f= 2 n P L 0 cos(2θ ) + 2 n C H cos β + 4nx cos 2 θcos(θ+φ) cos2θ +K.
n P = n C =η.
(2θ ) = β .
f(x,θ,φ,n)= 2η( L 0 +H) cos(2θ ) + 4nx cos 2 θcos(θ+φ) cos2θ +K.
ηsin(2θ ) =nsin2θ.
f(x,θ,φ,n)= 2 η 2 ( L 0 +H) η 2 (nsin2θ) 2 + 4nx cos 2 θcos(θ+φ) cos2θ +K.
df(x,θ,φ,n)= f x dx+ f θ dθ+ f φ dφ+ f n dn.
df(x,θ,φ,n)= 4n cos 2 θcos(θ+φ) cos2θ dx+( 2 η 2 ( L 0 +H) n 2 sin4θ ( η 2 n 2 sin 2 2θ) 3 + 4nx(sin2θcos(θ+φ)cos2θsin(θ+φ) cos 2 θ) cos 2 2θ )dθ 4nx cos 2 θsin(θ+φ) cos2θ dφ +( 2 η 2 ( L 0 +H)n sin 2 2θ ( η 2 n 2 sin 2 2θ) 3 + 4x cos 2 θcos(θ+φ) cos2θ )dn.
d f e (x,θ,φ,n)= df(x,θ,φ,n) 4n dx.
d f e (x,θ,φ,n)=Mdx+NdθPdφ+Qdn.
d f e ( u )= T d u .
f e ( u )= u 1 u 2 R d u = u 1 u 2 Mdx+NdθPdφ +Qdn.
f e ( u )= Δf( u ) 4n Δx= f( u 1 +Δu )f( u 1 ) 4n Δx.
f e ( u )= η( L 0 +H) 2n ( η η 2 (nsin2Δθ) 2 1 )+(x+Δx)( cos 2 Δθcos(Δθ+Δφ) cos2Δθ 1 ).
f e ( u )= J 2n ( η η 2 (nsin2Δθ) 2 1 )+(x+Δx)( cos 2 Δθcos(Δθ+Δφ) cos2Δθ 1 ).

Metrics