Abstract

Freeform surfaces have important applications in optical design. However, the complex surface figure error of freeform imaging systems affects the performance of as-built systems. In this paper, a method for the representation and tolerance analysis of the freeform surface figure error using a sum of specific-probability-distributed Gaussian radial basis functions is presented. This method is easy-used and practical. In addition, the actual manufacturing difficulty of different areas of the freeform surface is considered. Tolerance analysis of a nonsymmetric freeform off-axis three-mirror system was performed, including considering only the surface figure error, as well as an overall tolerance analysis considering both figure error and assembly error. The results demonstrated the effectiveness and feasibility of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (4)

2017 (5)

2016 (2)

2015 (5)

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015).
[Crossref]

X. Hu and H. Hua, “Design and tolerance of a free-form optical system for an optical see-through multi-focal-plane display,” Appl. Opt. 54(33), 9990–9999 (2015).
[Crossref]

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of Freeform Optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

F. Zhang, “Fabrication and testing of optical free-form convex mirror,” Chin. Opt. Lett. 13(s1), S12202 (2015).
[Crossref]

2014 (5)

2013 (5)

2012 (2)

A. M. Hoogstrate, C. V. Drunen, B. V. Venrooy, and R. Henselmans, “Manufacturing of high-precision aspherical and freeform optics,” Proc. SPIE 8450, 84502Q (2012).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (3)

2008 (3)

Barbero, S.

Bauer, A.

Berthelot, L.

Blalock, T.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of Freeform Optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Broemel, A.

Cakmakci, O.

Chen, J.

Chen, L.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Chen, S.

Cheng, D.

Cheng, J.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Dong, J.

Dou, Y.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Drunen, C. V.

A. M. Hoogstrate, C. V. Drunen, B. V. Venrooy, and R. Henselmans, “Manufacturing of high-precision aspherical and freeform optics,” Proc. SPIE 8450, 84502Q (2012).
[Crossref]

Dunn, C. R.

Eberhardt, R.

Evans, C.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Fang, F.

Feng, Z.

Flügel-Paul, T.

Foroosh, H.

Fuerschbach, K.

Gao, C.

Gao, H.

Gao, Z.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Gebhardt, A.

Gong, M.

Gross, H.

Hao, X.

Hartung, J.

He, X.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Henselmans, R.

A. M. Hoogstrate, C. V. Drunen, B. V. Venrooy, and R. Henselmans, “Manufacturing of high-precision aspherical and freeform optics,” Proc. SPIE 8450, 84502Q (2012).
[Crossref]

Hoogstrate, A. M.

A. M. Hoogstrate, C. V. Drunen, B. V. Venrooy, and R. Henselmans, “Manufacturing of high-precision aspherical and freeform optics,” Proc. SPIE 8450, 84502Q (2012).
[Crossref]

Hou, W.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Hu, Q.

Hu, X.

Hua, H.

Huang, L.

Ji, Z.

Jin, G.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
[Crossref]

Li, H.

Li, L.

Li, Z.

Liang, R.

Liu, C.

Liu, P.

Liu, Q.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Liu, X.

Ma, D.

Ma, H.

Maksimovic, M.

M. Maksimovic, “Optical design and tolerancing of freeform surfaces using anisotropic radial basis functions,” Opt. Eng. 55(7), 071203 (2016).
[Crossref]

Medicus, K.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of Freeform Optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Meng, Q.

Moore, B.

Nelson, J. D.

T. Blalock, K. Medicus, and J. D. Nelson, “Fabrication of Freeform Optics,” Proc. SPIE 9575, 95750H (2015).
[Crossref]

Ochse, D.

Peloux, M.

Qu, H.

H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013).
[Crossref]

Reimers, J.

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017).
[Crossref]

Risse, S.

Rolland, J. P.

Scheiding, S.

Shi, G.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Song, L.

Stock, J.

Straif, C.

Talha, M. M.

Tang, Y.

Thompson, K. P.

Tian, Y.

Tie, G.

Tong, K.

Tünnermann, A.

Venrooy, B. V.

A. M. Hoogstrate, C. V. Drunen, B. V. Venrooy, and R. Henselmans, “Manufacturing of high-precision aspherical and freeform optics,” Proc. SPIE 8450, 84502Q (2012).
[Crossref]

Vo, S.

Walker, D. D.

Wang, C.

K. Tong, Y. Zheng, Z. Zhang, X. Zhao, B. Zhang, L. Song, L. Wang, C. Wang, and P. Wu, “Model of radial basis functions based on surface slope for optical freeform surfaces,” Opt. Express 26(11), 14010–14023 (2018).
[Crossref]

H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013).
[Crossref]

Wang, D.

Wang, H.

Wang, K.

Wang, L.

K. Tong, Y. Zheng, Z. Zhang, X. Zhao, B. Zhang, L. Song, L. Wang, C. Wang, and P. Wu, “Model of radial basis functions based on surface slope for optical freeform surfaces,” Opt. Express 26(11), 14010–14023 (2018).
[Crossref]

H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Wang, T.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Wang, W.

Wang, Y.

Weckenmann, A.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Wu, P.

Wu, R.

Xu, L.

Xue, S.

Yang, T.

Yao, Y.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Yi Allen, Y.

Yin, H.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Yu, S.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Yuan, Q.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Zeitner, U. D.

Zhang, B.

K. Tong, Y. Zheng, Z. Zhang, X. Zhao, B. Zhang, L. Song, L. Wang, C. Wang, and P. Wu, “Model of radial basis functions based on surface slope for optical freeform surfaces,” Opt. Express 26(11), 14010–14023 (2018).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, F.

F. Zhang, “Fabrication and testing of optical free-form convex mirror,” Chin. Opt. Lett. 13(s1), S12202 (2015).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, G.

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

Zhang, J.

H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013).
[Crossref]

Zhang, X.

Z. Li, F. Fang, X. Zhang, X. Liu, and H. Gao, “Highly efficient machining of non-circular freeform optics using fast tool servo assisted ultra-precision turning,” Opt. Express 25(21), 25243–25256 (2017).
[Crossref]

Z. Li, F. Fang, J. Chen, and X. Zhang, “Machining approach of freeform optics on infrared materials via ultra-precision turning,” Opt. Express 25(3), 2051–2062 (2017).
[Crossref]

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, Y.

Zhang, Z.

Zhao, X.

Zheng, L.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zheng, Y.

Zheng, Z.

Zhu, J.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Zhu, Z.

Appl. Opt. (9)

Z. Zheng, X. Hao, and X. Liu, “Freeform surface lens for LED uniform illumination,” Appl. Opt. 48(35), 6627–6634 (2009).
[Crossref]

Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x–y polynomial surface,” Appl. Opt. 49(19), 3661–3668 (2010).
[Crossref]

D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. 48(14), 2655–2668 (2009).
[Crossref]

M. Peloux and L. Berthelot, “Optimization of the optical performance of variable-power and astigmatism Alvarez lenses,” Appl. Opt. 53(29), 6670–6681 (2014).
[Crossref]

C. Liu, C. Straif, T. Flügel-Paul, U. D. Zeitner, and H. Gross, “Comparison of hyperspectral imaging spectrometer designs and the improvement of system performance with freeform surfaces,” Appl. Opt. 56(24), 6894–6901 (2017).
[Crossref]

Q. Meng, W. Wang, H. Ma, and J. Dong, “Easy-aligned off-axis three-mirror system with wide field of view using freeform surface based on integration of primary and tertiary mirror,” Appl. Opt. 53(14), 3028–3034 (2014).
[Crossref]

Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016).
[Crossref]

X. Hu and H. Hua, “Design and tolerance of a free-form optical system for an optical see-through multi-focal-plane display,” Appl. Opt. 54(33), 9990–9999 (2015).
[Crossref]

J. Stock, A. Broemel, J. Hartung, D. Ochse, and H. Gross, “Description and reimplementation of real freeform surfaces,” Appl. Opt. 56(3), 391–396 (2017).
[Crossref]

Chin. Opt. Lett. (1)

F. Zhang, “Fabrication and testing of optical free-form convex mirror,” Chin. Opt. Lett. 13(s1), S12202 (2015).
[Crossref]

CIRP Ann. (1)

F. Fang, X. Zhang, A. Weckenmann, G. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Ann. 62(2), 823–846 (2013).
[Crossref]

J. Opt. (2)

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Light: Sci. Appl. (1)

J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light: Sci. Appl. 6(7), e17026 (2017).
[Crossref]

Opt. Commun. (1)

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Figures (11)

Fig. 1.
Fig. 1. A Gaussian function with a center position of (0, 0) and a standard deviation of 5. The function values outside 3σ=15 from the center position is approximated to zero.
Fig. 2.
Fig. 2. The distribution of Gaussian functions and the process of surface area extension. The pseudo color background represents the gradient magnitude of the surface area after expansion. Inside the dashed box is the actual clear aperture. The dots represent the grid points and the red ones among these points represent the centers of the Gaussians.
Fig. 3.
Fig. 3. The normal distribution curve of the amplitude adjustment factor ξi. The vertical axis is the probability of all possible values of ξi. In the given span, all the data can be selected as a ξi value according to a specific possibility. It is clear that ξi has the highest probability of being the mean value. However, it still has a small chance of being away from the mean value.
Fig. 4.
Fig. 4. The normal distribution curve of the actual RMS or PV value of the surface figure error. Similarly to the amplitude adjustment factor of weight coefficients, in the given span, all the data can be selected as a RMS or PV value according to a specific possibility
Fig. 5.
Fig. 5. Flowchart of the tolerance analysis process.
Fig. 6.
Fig. 6. (a) Layout of the nominal optical system. (b) MTF of the nominal optical system.
Fig. 7.
Fig. 7. The magnitude of the sag gradient vector of the three freeform surfaces. (a) is the primary mirror, (b) is the secondary mirror, and (c) is the tertiary mirror.
Fig. 8.
Fig. 8. Several surface deformation distributions generated by the proposed method. (a)(b)(c) are the figure errors of the primary mirror, (d)(e)(f) are the figure errors of the secondary mirror, (g)(h)(i) are the figure errors of the tertiary mirror.
Fig. 9.
Fig. 9. The sample fields chosen in Monte Carlo tolerance analysis.
Fig. 10.
Fig. 10. Monte Carlo analysis results of considering only the surface figure error. (a) MTF at 20 lps/mm in x direction. (b) MTF at 20 lps/mm in y direction. (c) Ensquared energy fraction in 25µm×25µm detector pixel.
Fig. 11.
Fig. 11. Monte Carlo analysis results of considering both the surface figure error and the assembly error. (a) MTF at 20 lps/mm in x direction. (b) MTF at 20 lps/mm in y direction. (c) Ensquared energy fraction in 25µm×25µm detector pixel.

Tables (6)

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Table 1. Parameters of the freeform surfaces and Gaussian radial basis functions

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Table 2. Selected tolerance value for freeform surfaces

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Table 3. Considering only the surface figure error: Monte Carlo Analysis-2000 Trials 97.7% probability change of MTF at 20 lps/mm

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Table 4. Considering only the surface figure error: Monte Carlo Analysis-2000 Trials 97.7% probability change of the ensquared energy fraction in 25µm×25µm detector pixel

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Table 5. Considering both the surface figure error and the assembly error: Monte Carlo Analysis-2000 Trials 97.7% probability change of MTF at 20 lps/mm

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Table 6. Considering both the surface figure error and the assembly error: Monte Carlo Analysis-2000 Trials 97.7% probability change of the ensquared energy fraction in 25µm×25µm detector pixel

Equations (20)

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g i ( x , y ) = e 1 2 σ 2 [ ( x x i ) 2 + ( y y i ) 2 ] ,
Δ z ( x , y ) = i = 1 k w i g i ( x , y ) = i = 1 k w i e 1 2 σ 2 [ ( x x i ) 2 + ( y y i ) 2 ] ,
Q α , β = ( x α , β , y α , β ) , α = 1 , 2 , , m , β = 1 , 2 , , n .
grad z ( x α , β , y α , β ) = f ( x α , β , y α , β ) x x ^ + f ( x α , β , y α , β ) y y ^ , α = 1 , 2 , , m , β = 1 , 2 , , n ,
G α , β = ( f ( x α , β , y α , β ) x ) 2 + ( f ( x α , β , y α , β ) y ) 2 , α = 1 , 2 , , m , β = 1 , 2 , , n .
G = [ G 1 , 1 G 1 , 2 G 1 , n G 2 , 1 G 2 , 2 G 2 , n G m , 1 G m , 2 G m , n ] .
G total = α , β = 1 m , n G α , β .
P α , β = G α , β G total .
P = G G total = [ P 1 , 1 P 1 , 2 P 1 , n P 2 , 1 P 2 , 2 P 2 , n P m , 1 P m , 1 P m , n ] = [ G 1 , 1 G total G 1 , 2 G total G 1 , n G total G 2 , 1 G total G 2 , 2 G total G 2 , n G total G m , 1 G total G m , 2 G total G m , n G total ] .
{ a = round { R x ( τ 1 ) Δ x } b = round { R y ( τ 1 ) Δ y } ,
G etotal = α , β = 1 m + 2 a , n + 2 b G α , β .
P e = G e G etotal .
w i = ε i × ξ i , i = 1 , 2 , , k .
Δ z ( x ψ , ζ , y ψ , ζ ) = i = 1 k w i e 1 2 σ 2 [ ( x ψ , ζ x i ) 2 + ( y ψ , ζ y i ) 2 ] .
Δ Z = [ Δ z 1 , 1 Δ z 1 , 2 Δ z 1 , t Δ z 2 , 1 Δ z 2 , 2 Δ z 2 , t Δ z s , 1 Δ z s , 2 Δ z s , t ] .
ω RMS = ψ , ζ = 1 s , t ( Δ z ψ , ζ ) 2 R .
η = Ω RMS ω RMS .
ω PV = Δ z max Δ z min .
η = Ω PV ω PV .
Δ Z real = η Δ Z .