Abstract

Techniques for measurement of higher-order aberrations of a projection optical system in photolithographic exposure tools have been established. Even-type and odd-type aberrations are independently obtained from printed grating patterns on a wafer by three-beam interference under highly coherent illumination. Even-type aberrations, i.e., spherical aberration and astigmatism, are derived from the best focus positions of vertical, horizontal, and oblique grating patterns by an optical microscope. Odd-type aberrations, i.e., coma and three-foil, are obtained by detection of relative shifts of a fine grating pattern to a large pattern by an overlay inspection tool. Quantitative diagnosis of lens aberrations with a krypton fluoride (KrF) excimer laser scanner is demonstrated.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Brunner, “Impact of lens aberrations on optical lithography,” IBM J. Res. Dev. 41, 57–67 (1997).
    [CrossRef]
  2. D. G. Flagello, B. Geh, “Lithographic lens testing: analysis of measured aerial images, interferometric data, and photoresist measurements,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE2726, 788–798 (1996).
    [CrossRef]
  3. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
  4. R. P. Grosso, R. Crane, “Precise optical evaluation using phase monitoring interferometric techniques,” in Interferometry, G. Hopkins, ed., Proc. SPIE192, 65–74 (1979).
    [CrossRef]
  5. J. P. Kirk, “Astigmatism and field curvature from pin-bars,” in Optical/Laser Microlithography IV, V. Pol, ed., Proc. SPIE1463, 282–291 (1991).
    [CrossRef]
  6. J. P. Kirk, “Measurement of astigmatism in microlithography lenses,” in Optical Microlithography XI, L. V. den Hove, ed., Proc. SPIE3334, 848–854 (1998).
    [CrossRef]
  7. T. Saito, H. Watanabe, Y. Okuda, “Evaluation of coma aberration in projection lens by various measurements,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 297–308 (1998).
    [CrossRef]
  8. T. Saito, H. Watanabe, Y. Okuda, “Effect of variable sigma aperture on lens distortion and its pattern size dependence,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 414–423 (1996).
    [CrossRef]
  9. T. Saito, H. Watanabe, Y. Okuda, “Overlay error of fine patterns by lens aberration using modified illumination,” in Optical Microlithography X, G. E. Fuller, ed., Proc. SPIE3051, 686–696 (1997).
    [CrossRef]
  10. T. Sato, H. Nomura, “Coma aberration measurement by relative shift of displacement with pattern dependence,” Jpn. J. Appl. Phys. 37, 3553–3557 (1998).
    [CrossRef]
  11. C. Progler, D. Wheeler, “Optical lens specifications from the user’s perspective,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 256–268 (1998).
    [CrossRef]
  12. S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
    [CrossRef]
  13. S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
    [CrossRef]
  14. H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. 38, 2800–2807 (1999).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  16. R. R. Shannon, J. C. Wyant, Applied Optics and Optical Engineering (Academic, San Diego, Calif., 1992), Vol. XI.
  17. K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
    [CrossRef]

1999

1998

T. Sato, H. Nomura, “Coma aberration measurement by relative shift of displacement with pattern dependence,” Jpn. J. Appl. Phys. 37, 3553–3557 (1998).
[CrossRef]

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

1997

T. Brunner, “Impact of lens aberrations on optical lithography,” IBM J. Res. Dev. 41, 57–67 (1997).
[CrossRef]

Born, M.

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

Brunner, T.

T. Brunner, “Impact of lens aberrations on optical lithography,” IBM J. Res. Dev. 41, 57–67 (1997).
[CrossRef]

Crane, R.

R. P. Grosso, R. Crane, “Precise optical evaluation using phase monitoring interferometric techniques,” in Interferometry, G. Hopkins, ed., Proc. SPIE192, 65–74 (1979).
[CrossRef]

Flagello, D. G.

D. G. Flagello, B. Geh, “Lithographic lens testing: analysis of measured aerial images, interferometric data, and photoresist measurements,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE2726, 788–798 (1996).
[CrossRef]

Fujisawa, T.

K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
[CrossRef]

Geh, B.

D. G. Flagello, B. Geh, “Lithographic lens testing: analysis of measured aerial images, interferometric data, and photoresist measurements,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE2726, 788–798 (1996).
[CrossRef]

Grosso, R. P.

R. P. Grosso, R. Crane, “Precise optical evaluation using phase monitoring interferometric techniques,” in Interferometry, G. Hopkins, ed., Proc. SPIE192, 65–74 (1979).
[CrossRef]

Inoue, S.

K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
[CrossRef]

Kirk, J. P.

J. P. Kirk, “Astigmatism and field curvature from pin-bars,” in Optical/Laser Microlithography IV, V. Pol, ed., Proc. SPIE1463, 282–291 (1991).
[CrossRef]

J. P. Kirk, “Measurement of astigmatism in microlithography lenses,” in Optical Microlithography XI, L. V. den Hove, ed., Proc. SPIE3334, 848–854 (1998).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

Miura, T.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

Miyazaki, J.

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

Nakae, A.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

Nakao, S.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

Nomura, H.

H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. 38, 2800–2807 (1999).
[CrossRef]

T. Sato, H. Nomura, “Coma aberration measurement by relative shift of displacement with pattern dependence,” Jpn. J. Appl. Phys. 37, 3553–3557 (1998).
[CrossRef]

Okuda, Y.

T. Saito, H. Watanabe, Y. Okuda, “Overlay error of fine patterns by lens aberration using modified illumination,” in Optical Microlithography X, G. E. Fuller, ed., Proc. SPIE3051, 686–696 (1997).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Evaluation of coma aberration in projection lens by various measurements,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 297–308 (1998).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Effect of variable sigma aperture on lens distortion and its pattern size dependence,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 414–423 (1996).
[CrossRef]

Progler, C.

C. Progler, D. Wheeler, “Optical lens specifications from the user’s perspective,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 256–268 (1998).
[CrossRef]

Saito, T.

T. Saito, H. Watanabe, Y. Okuda, “Overlay error of fine patterns by lens aberration using modified illumination,” in Optical Microlithography X, G. E. Fuller, ed., Proc. SPIE3051, 686–696 (1997).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Effect of variable sigma aperture on lens distortion and its pattern size dependence,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 414–423 (1996).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Evaluation of coma aberration in projection lens by various measurements,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 297–308 (1998).
[CrossRef]

Sakai, J.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

Sato, K.

K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
[CrossRef]

Sato, T.

H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Opt. 38, 2800–2807 (1999).
[CrossRef]

T. Sato, H. Nomura, “Coma aberration measurement by relative shift of displacement with pattern dependence,” Jpn. J. Appl. Phys. 37, 3553–3557 (1998).
[CrossRef]

Shannon, R. R.

R. R. Shannon, J. C. Wyant, Applied Optics and Optical Engineering (Academic, San Diego, Calif., 1992), Vol. XI.

Tanaka, S.

K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
[CrossRef]

Tatsu, S.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

Tsujita, K.

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

Wakamiya, W.

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

Watanabe, H.

T. Saito, H. Watanabe, Y. Okuda, “Effect of variable sigma aperture on lens distortion and its pattern size dependence,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 414–423 (1996).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Overlay error of fine patterns by lens aberration using modified illumination,” in Optical Microlithography X, G. E. Fuller, ed., Proc. SPIE3051, 686–696 (1997).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Evaluation of coma aberration in projection lens by various measurements,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 297–308 (1998).
[CrossRef]

Wheeler, D.

C. Progler, D. Wheeler, “Optical lens specifications from the user’s perspective,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 256–268 (1998).
[CrossRef]

Wolf, E.

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

Wyant, J. C.

R. R. Shannon, J. C. Wyant, Applied Optics and Optical Engineering (Academic, San Diego, Calif., 1992), Vol. XI.

Appl. Opt.

IBM J. Res. Dev.

T. Brunner, “Impact of lens aberrations on optical lithography,” IBM J. Res. Dev. 41, 57–67 (1997).
[CrossRef]

Jpn. J. Appl. Phys.

T. Sato, H. Nomura, “Coma aberration measurement by relative shift of displacement with pattern dependence,” Jpn. J. Appl. Phys. 37, 3553–3557 (1998).
[CrossRef]

S. Nakao, A. Nakae, J. Sakai, T. Miura, S. Tatsu, K. Tsujita, W. Wakamiya, “Measurement of spherical aberration utilizing an alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 5949–5955 (1998).
[CrossRef]

S. Nakao, J. Miyazaki, K. Tsujita, W. Wakamiya, “Measurement method for odd component of aberration function utilizing alternating phase shift mask,” Jpn. J. Appl. Phys. 37, 6709–6713 (1998).
[CrossRef]

Other

C. Progler, D. Wheeler, “Optical lens specifications from the user’s perspective,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 256–268 (1998).
[CrossRef]

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

R. R. Shannon, J. C. Wyant, Applied Optics and Optical Engineering (Academic, San Diego, Calif., 1992), Vol. XI.

K. Sato, S. Tanaka, T. Fujisawa, S. Inoue, “Measurement of the effective source shift using a grating-pinhole mask,” in Optical Microlithography XII, L. van Den Hove, ed., Proc. SPIE3679, 99–107 (1999).
[CrossRef]

D. G. Flagello, B. Geh, “Lithographic lens testing: analysis of measured aerial images, interferometric data, and photoresist measurements,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE2726, 788–798 (1996).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

R. P. Grosso, R. Crane, “Precise optical evaluation using phase monitoring interferometric techniques,” in Interferometry, G. Hopkins, ed., Proc. SPIE192, 65–74 (1979).
[CrossRef]

J. P. Kirk, “Astigmatism and field curvature from pin-bars,” in Optical/Laser Microlithography IV, V. Pol, ed., Proc. SPIE1463, 282–291 (1991).
[CrossRef]

J. P. Kirk, “Measurement of astigmatism in microlithography lenses,” in Optical Microlithography XI, L. V. den Hove, ed., Proc. SPIE3334, 848–854 (1998).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Evaluation of coma aberration in projection lens by various measurements,” in Optical Microlithography XI, L. van Den Hove, ed., Proc. SPIE3334, 297–308 (1998).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Effect of variable sigma aperture on lens distortion and its pattern size dependence,” in Metrology, Inspection, and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 414–423 (1996).
[CrossRef]

T. Saito, H. Watanabe, Y. Okuda, “Overlay error of fine patterns by lens aberration using modified illumination,” in Optical Microlithography X, G. E. Fuller, ed., Proc. SPIE3051, 686–696 (1997).
[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 (18)

Fig. 1
Fig. 1

Coordinates system defined as right-hand coordinates. The z axis, positive aberration, is the direction opposite wave propagation, the y axis is the same direction that in which as reticle and wafer stages move, and the x axis is the direction of the exposure slit.

Fig. 2
Fig. 2

Difference between the aberrated wave front and an ideal spherical surface at the exit pupil.

Fig. 3
Fig. 3

Criterion for three-beam interference shown on the plane of illumination coherence σ and pupil radius ρ at the center of the ±1st diffraction orders. A, P = λ/NA(1 - σ); B, P = 2λ/NA(1 + σ); C, P = 3λ/NA(1 + σ). Three-beam interference is achieved at the regions of Conditions I and II.

Fig. 4
Fig. 4

Relation among three wave fronts [W(ρ), W(0), and W(-ρ)] with positive and negative defocusing. Solid curves show wave-front aberrations at positive defocus, best focus, and negative defocus. Defocused wave-front aberration is the sum of wave-front aberration at best focus and a defocus term (dotted curves). The inclination of the thin, solid, straight line between W(ρ) and W(-ρ) is proportional to pattern displacement and is always constant at any defocus.

Fig. 5
Fig. 5

Reticle layout for measurement of all conceivable aberrations. There are 33 units in the static exposure field.

Fig. 6
Fig. 6

Reticle patterns for measurement of even-type aberrations. There are four gratings, oriented at 0°, 45°, 90°, and 135°, for each dimension.

Fig. 7
Fig. 7

The bright-field micrograph of a 0.1-µm-defocused and a 20-µm-stepped image of a mark similar to that in Fig. 6 is asymmetrical in defocusing.

Fig. 8
Fig. 8

Curve fitting of mean values of four oriented gratings into Eq. (8) by the least-squares method and higher-order spherical aberration from an inclination of the fitted line and its curvature for (a) a symmetric component. Curve fitting of difference values of orthogonal gratings into Eq. (9) and also higher-order astigmatism for (b) cos 2θ and (c) sin 2θ components.

Fig. 9
Fig. 9

Even-type aberrations within the exposure field. Thick solid curves, spherical aberration. Thin solid and dashed curves, astigmatism ±45° and astigmatism 0°/90°, respectively. Number at upper left of each fit shows the mark’s address in Fig. 5.

Fig. 10
Fig. 10

Distribution maps of spherical aberrations in the exposure field: (a) lower-order aberration Z 9, (b) higher-order aberration Z 16.

Fig. 11
Fig. 11

Relationship among three vectors of odd-type aberrations, which are measured by the marks oriented at 0°, 120°, and 240°. The average vector of the three is the sum of θ components (coma). Residual vectors subtracted by the average vector are the sum of 3θ components (three-foil).

Fig. 12
Fig. 12

Reticle patterns for measurement of odd-type aberrations. There are three similar patterns, oriented at 0°, 120°, and 240°, for each dimension. Measurement marks are fabricated by double exposure of A mark and B mark. A and B marks lie on the same reticle.

Fig. 13
Fig. 13

Double-exposure method for mark fabrication. First the A mark was exposed on the resist film. Then the reticle was shifted such that the B mark could be exposed over it. We then prepared the bars-in-box mark by developing these exposures. As a consequence of this procedure, all surviving patterns were exposed at the first step. A second exposure was used for removing unnecessary patterns only.

Fig. 14
Fig. 14

Measured relative shift vectors have different in pattern orientations. Pattern dimension, 0.225 µm. Coma, the average of three marks oriented at 0°, 120°, and 240°, is large near the center of the exposure field. Three-foil, residual vectors subtracted by the average, is large at both sides in the field.

Fig. 15
Fig. 15

Curve fitting of average vectors of three oriented gratings into Eq. (11) by the least-squares method for (a) cos θ and (b) sin θ components. Curve fitting of residual vectors of the three into Eqs. (12) for (c) cos 3θ and (d) sin 3θ components.

Fig. 16
Fig. 16

Odd-type aberrations within the exposure field. W(-ρ) = -W(ρ) for odd-type aberrations. Thick solid and dashed curves are coma x and coma y and thin solid and dashed curves are three-foil x and three-foil y, respectively. Number at upper left of each fit shows the mark’s address in Fig. 5.

Fig. 17
Fig. 17

Schematic diagram of image shift of fine gratings with defocusing for tilted illumination. If σ is shifted but three diffracted beams are not out of the exit pupil, a relative shift between a fine grating and a large pattern is zero at any defocus. However, if a part of the diffracted beam is out of the exit pupil, a relative shift between them is proportional to defocusing.

Fig. 18
Fig. 18

1/25 Contour maps of wave-front surfaces at three points of the exposure field. (a) Left side (2, 1) of the field. (b) Center of field (5, 1). (c) Right side of the field (8, 1).

Tables (3)

Tables Icon

Table 1 Zernike Polynomials from 1st to 37th Orders

Tables Icon

Table 2 Classification of Zernike Polynomials

Tables Icon

Table 3 Summary of Error Estimation

Equations (14)

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

Condition I λNA1-σP3λNA1+σ,  L=S,Condition II λNA1-σP2λNA1+σ  LS,
Wevenρ=-1-1-NA2ρ21/2λδF×-NA2ρ22λ δF.
ρ=λP1NA.
Woddρ=δxP=NAρλ δx
Wcos2θ, Wsin2θ=W90°-W0°2, W135°-W45°2,
Wsym=W0°+W45°+W90°+W135°4,
Wcos2θρρ2=6Z5-310Z12+614Z21+410Z12-2014Z21ρ2+1514Z21ρ4,Wsin2θρρ2=6Z6-310Z13+614Z22+410Z13-2014Z22ρ2+1514Z22ρ4,
Wsymρρ2=C0+65Z9-307Z16ρ2+207Z16ρ4.
Wcosθ, Wsinθ=WX0°+WX120°+WX240°3,WY0°+WY120°+WY240°3,
Wcos3θWsin3θ=13ϕ=0°,120°,240°cos ϕsin ϕ-sin ϕcos ϕ×WXϕ-WcosθWYϕ-Wsinθ.
Wcosθρρ=C1+38Z7-1212Z14+120Z23ρ2+1012Z14-240Z23ρ4+140Z23ρ6,Wsinθρρ=C2+38Z8-1212Z15+120Z24ρ2+1012Z15-240Z24ρ4+140Z24ρ6,
Wcos3θρρ=8Z10-412Z19+40Z30ρ2+512Z19-120Z30ρ4+82Z30ρ6,Wsin3θρρ=8Z11-412Z20+40Z31ρ2+512Z20-120Z31ρ4+82Z31ρ6.
Δ Wsymρρ2=NA22λ |εsym|0.932×0.03540.016,Δ Wsin2θρρ2=Δ Wcos2θρρ2=NA22λ |ε2θ|Δ Wsin2θρρ2=Δ Wcos2θρρ2=NA22λ |ε2θ|0.932×0.03520.023,
Δ Wsinθρρ=Δ Wsinθρρ=NAλ |εθ|2.742×0.00150.001,Δ Wsin3θρρ=Wsin3θρρ=NAλ |ε3θ|2.742×0.00130.002,

Metrics