Abstract

Aberration theory helps designers to better understand the nature of imaging systems. However, the existing aberration theory of freeform surfaces has many limitations. For example, it only works in the special case when the central area of the freeform surface is used. In addition, the light footprint is limited to a circle, which does not match the case of an elliptical footprint for general systems. In this paper, aberrations generated by freeform surface term overlay on general decentered and tilted optical surfaces are analyzed. For the case when the off-axis section of a freeform surface is used, the aberration equation for using stop and nonstop surfaces is discussed, and the aberrations generated by Zernike terms up to Z17/18 are analyzed in detail. To solve the problem of the elliptical light footprint for tilted freeform surfaces, the scaled pupil vector is used in the aberration analysis. The mechanism of aberration transformation is discovered, and the aberrations generated by different Zernike terms in this case are calculated. Finally we proposed aberration equations for freeform terms on general decentered and tilted freeform surfaces. The research result given in this paper offers an important reference for optical designers and engineers, and it is of great importance in developing analytical methods for general freeform system design, tolerance analysis, and system assembly.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  7. T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  28. T. Yang, J. Zhu, and G. Jin, “Nodal aberration properties of coaxial imaging systems using Zernike polynomial surfaces,” J. Opt. Soc. Am. A 32(5), 822–836 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2018 (1)

2017 (5)

2016 (5)

2015 (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]

T. Yang, J. Zhu, and G. Jin, “Nodal aberration properties of coaxial imaging systems using Zernike polynomial surfaces,” J. Opt. Soc. Am. A 32(5), 822–836 (2015).
[Crossref] [PubMed]

2014 (4)

2013 (2)

2012 (3)

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]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

2011 (2)

2010 (3)

2009 (1)

2005 (1)

1986 (1)

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1986).
[Crossref]

1980 (1)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Bauer, A.

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

Benítez, P.

Chaves, J.

Cheng, D.

Davis, G. E.

Dong, J.

Duerr, F.

Flügel-Paul, T.

Fuerschbach, K.

Gao, C.

Gressler, W.

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

Gross, H.

Gu, Z.

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]

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]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref] [PubMed]

Hu, X.

Hua, H.

Ji, Z.

Jiang, H.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
[Crossref] [PubMed]

Jiang, L.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

Jin, G.

Ju, G.

Li, H.

Li, Y.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

Liu, C.

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, T.

Liu, X.

Liu, Z.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

Ma, H.

Meng, Q.

Miñano, J. C.

Mohedano, R.

Nie, Y.

Rakich, A.

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]

Rogers, J. R.

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1986).
[Crossref]

Rolland, J. P.

Schmid, T.

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

Sebag, J.

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

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]

Shi, H.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
[Crossref] [PubMed]

Straif, C.

Tang, R.

Thienpont, H.

Thompson, K.

Thompson, K. P.

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

K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
[Crossref] [PubMed]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Wang, C.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
[Crossref] [PubMed]

Wang, D.

Wang, H.

Wang, K.

Wang, 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]

Wang, Q.

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.

Xu, L.

Xu, S.

Yan, C.

Yang, T.

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]

Zeitner, U. D.

Zhang, B.

R. Tang, B. Zhang, G. Jin, and J. Zhu, “Multiple surface expansion method for design of freeform imaging systems,” Opt. Express 26(3), 2983–2994 (2018).
[Crossref] [PubMed]

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, D.

Zhang, F.

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, X.

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

X. Zhang, D. Zhang, S. Xu, and H. Ma, “Active optical alignment of off-axis telescopes based on nodal aberration theory,” Opt. Express 24(23), 26392–26413 (2016).
[Crossref] [PubMed]

H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
[Crossref] [PubMed]

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]

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, 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, Z.

Zhong, Y.

Zhu, J.

Acta Opt. Sin. (1)

H. Shi, X. Zhang, Y. Li, L. Jiang, C. Wang, Z. Liu, and H. Jiang, “Analysis of the aberration properties of pupil off-axis freeform optical system,” Acta Opt. Sin. 37, 1208001 (2017).

Appl. Opt. (9)

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] [PubMed]

H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
[Crossref] [PubMed]

Q. Wang, D. Cheng, Y. Wang, H. Hua, and G. Jin, “Design, tolerance, and fabrication of an optical see-through head-mounted display with free-form surface elements,” Appl. Opt. 52(7), C88–C99 (2013).
[Crossref] [PubMed]

G. Ju, C. Yan, Z. Gu, and H. Ma, “Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory,” Appl. Opt. 55(13), 3373–3386 (2016).
[Crossref] [PubMed]

Y. Nie, R. Mohedano, P. Benítez, J. Chaves, J. C. Miñano, H. Thienpont, and F. Duerr, “Multifield direct design method for ultrashort throw ratio projection optics with two tailored mirrors,” Appl. Opt. 55(14), 3794–3800 (2016).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Y. Nie, H. Gross, Y. Zhong, and F. Duerr, “Freeform optical design for a nonscanning corneal imaging system with a convexly curved image,” Appl. Opt. 56(20), 5630–5638 (2017).
[Crossref] [PubMed]

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] [PubMed]

J. Opt. (1)

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]

J. Opt. Soc. Am. A (5)

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. Express (9)

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

H. Hua, X. Hu, and C. Gao, “A high-resolution optical see-through head-mounted display with eyetracking capability,” Opt. Express 21(25), 30993–30998 (2013).
[Crossref] [PubMed]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref] [PubMed]

X. Zhang, D. Zhang, S. Xu, and H. Ma, “Active optical alignment of off-axis telescopes based on nodal aberration theory,” Opt. Express 24(23), 26392–26413 (2016).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref] [PubMed]

R. Tang, B. Zhang, G. Jin, and J. Zhu, “Multiple surface expansion method for design of freeform imaging systems,” Opt. Express 26(3), 2983–2994 (2018).
[Crossref] [PubMed]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
[Crossref] [PubMed]

Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express 25(9), 10016–10030 (2017).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (3)

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]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 554, 76–81 (1986).
[Crossref]

Pub. Astronom. Soc. Pacif. (1)

J. Sebag, W. Gressler, T. Schmid, J. P. Rolland, and K. P. Thompson, “LSST Telescope alignment plan based on nodal aberration theory,” Pub. Astronom. Soc. Pacif. 124(914), 380–390 (2012).
[Crossref]

Other (2)

T. Yang, “Design method of freeform imaging systems based on a point-by-point construction-iteration method,” Ph.D. Thesis, Tsinghua University (2017) (in Chinese).

Code V Reference Manual, Synopsys Inc. (2012).

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Figures (6)

Fig. 1
Fig. 1 The actual pupil vector on the actual freeform surface when the central section of the nonstop freeform surface is used.
Fig. 2
Fig. 2 The actual pupil vector on the actual freeform surface when the off-axis section of the freeform surface is used. (a) The case when the surface is the aperture stop. (b) The case when the surface is away from the stop.
Fig. 3
Fig. 3 A two-mirror example system.
Fig. 4
Fig. 4 The elliptical footprint and the scaled pupil vector.
Fig. 5
Fig. 5 The comparison of the theoretical and actual values when different Zernike terms (the coefficients all equal to 0.001) are added onto M2
Fig. 6
Fig. 6 The comparison of the theoretical and actual values when off-axis sections of different Zernike terms (the coefficients all equal to 0.00015) are added onto M2 in the second example.

Tables (7)

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Table 1 Fringe Zernike polynomials (up to 18th term)

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Table 2 Aberrations generated by freeform surface terms when the light beams use the central on-axis area of non-tilted surface located at the stop

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Table 3 Aberrations generated by freeform surface terms when the light beams use the central on-axis area of non-tilted, non-stop surface

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Table 4 Aberrations generated by freeform surface terms when the light beams use the off-axis area of non-tilted, non-stop surface

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Table 5 The link of the generated aberration terms listed in Table 4 to the existing concept of NAT

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Table 6 Aberrations generated by freeform surface terms when the light beams use the central on-axis area of tilted stop surface

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Table 7 Aberrations generated by different freeform surface terms when the off-axis section of tilted surface is used

Equations (19)

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z = F ( ρ , ϕ ) ,
z = c ( x 2 + y 2 ) 1 + 1 ( 1 + k ) c 2 ( x 2 + y 2 ) + j = 1 n C j Z j ,
Δ h ( y ¯ y ) H ,
μ k 1 y ,
W s t o p = M Z ( ρ + μ ) .
Δ h ( k 2 k 1 y ) H = λ H ,
W non s t o p = M Z ( ρ + λ H + μ ) .
W k = M k Z k ( ρ s ) = M k Z k ( [ s ρ x , ρ y ] ) .
W k = M k Z k a d j ( s ρ x , ρ y ) .
W k = M k Z k a d j ( s ρ x , ρ y ) = c k , i g i ( ρ x , ρ y ) = C k T G ,
W k = M k , j r e a r r Z j a d j ( ρ x , ρ y ) = M k r e a r r T Z a d j = c k , i g i ( ρ x , ρ y ) = C k T G ,
Z a d j = T G ,
G = [ 1 ρ x ρ y ρ x 2 ρ x ρ y ρ y 2 ρ x 3 ρ x 2 ρ y ρ x ρ y 2 ... ... ρ y 3 ρ x 4 ρ x 3 ρ y ρ x 2 ρ y 2 ρ x ρ y 3 ρ y 4 ] T ,
Z a d j = [ 1 ρ x ρ y 2 ( ρ x 2 + ρ y 2 ) ρ x 2 ρ y 2 2 ρ x ρ y 3 ρ x ( ρ x 2 + ρ y 2 ) ... ... 3 ρ y ( ρ x 2 + ρ y 2 ) 6 ( ρ x 2 + ρ y 2 ) 2 ρ x ( ρ x 2 3 ρ y 2 ) ρ y ( 3 ρ x 2 ρ y 2 ) ... ... 4 ( ρ x 4 ρ y 4 ) 8 ( ρ x 2 + ρ y 2 ) ρ x ρ y ρ x 4 + ρ y 4 6 ρ x 2 ρ y 2 4 ρ x ρ y ( ρ x 2 ρ y 2 ) ] T ,
T = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 12 0 6 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 8 0 8 0 0 0 0 0 0 0 0 0 0 0 1 0 6 0 1 0 0 0 0 0 0 0 0 0 0 0 4 0 4 0 ] .
M k r e a r r = ( T T ) 1 C k .
W s t o p = M Z ( ρ s + μ ) .
W non s t o p = M Z ( ρ s + λ H s H + μ ) .
W k = M k Z k a d j ( s ρ x + s H λ y H x , ρ y + λ y H y + μ y ) .

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