Abstract

Freeform surfaces play important roles in improving the imaging performance of off-axis optical systems. However, for some systems with high requirements in specifications, the structure of the freeform surfaces could be very complicated and the number of freeform surfaces could be large. That brings challenges in fabrication and increases the cost. Therefore, to achieve a good initial system with minimum aberrations and reasonable structure before implementing freeform surfaces is essential for optical designers. The already existing initial system design methods are limited to certain types of systems. A universal tool or method to achieve a good initial system efficiently is very important. In this paper, based on the Nodal aberration theory and the system design method using Gaussian Brackets, the initial system design method is extended from rotationally symmetric systems to general non-rotationally symmetric systems. The design steps are introduced and on this basis, two off-axis three-mirror systems are pre-designed using spherical shape surfaces. The primary aberrations are minimized using the nonlinear least-squares solver. This work provides insight and guidance for initial system design of off-axis mirror systems.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (2)

2014 (3)

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE 9272, 92721L (2014).
[Crossref]

D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (1)

2009 (3)

2008 (2)

2007 (1)

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802 (2007).
[Crossref]

1999 (1)

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

1998 (1)

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

1997 (1)

C. Bakolias and A. K. Forrest, “Dark-field Scheimpflug imaging for surface inspection,” Proc. SPIE 3029, 57–68 (1997).
[Crossref]

1994 (1)

H. Mayer, “Theodor Scheimpflug,” Ophthalmic Res. 26(1), 3–9 (1994).
[Crossref]

1986 (1)

K. Tanaka, “II Paraxial theory in optical design in terms of Gaussian Brackets,” Prog. Opt. 23, 63–111 (1986).
[Crossref]

Bakolias, C.

C. Bakolias and A. K. Forrest, “Dark-field Scheimpflug imaging for surface inspection,” Proc. SPIE 3029, 57–68 (1997).
[Crossref]

Benítez, P.

Bicknell, W. E.

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

Cakmakci, O.

Chang, S.

Cheng, D.

Cheng, X.

X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE 9272, 92721L (2014).
[Crossref]

Cui, Q.

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

Dempewolf, G.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Digenis, C. J.

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

Foroosh, H.

Forrest, A. K.

C. Bakolias and A. K. Forrest, “Dark-field Scheimpflug imaging for surface inspection,” Proc. SPIE 3029, 57–68 (1997).
[Crossref]

Fuerschbach, K.

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]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

Gao, C.

Harnisch, B.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Hearn, D. R.

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

Hicks, R. A.

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802 (2007).
[Crossref]

Hu, X.

Hua, H.

Infante, J.

Jiang, H.

Jin, G.

Juranek, H. J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Kunkel, B. P.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Lencioni, D. E.

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

Lin, W.

Litzelmann, A.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Liu, T.

Mayer, H.

H. Mayer, “Theodor Scheimpflug,” Ophthalmic Res. 26(1), 3–9 (1994).
[Crossref]

Mendenhall, J. A.

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

Miñano, J. C.

Moore, B.

Muñoz, F.

Piao, M.

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

Rolland, J. P.

Sand, R.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Santamaría, A.

Sasian, J.

Schillke, F.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schmid, T.

Schmidt, E.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Schweizer, J.

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

Shi, H.

Song, W.

Tanaka, K.

K. Tanaka, “II Paraxial theory in optical design in terms of Gaussian Brackets,” Prog. Opt. 23, 63–111 (1986).
[Crossref]

Thompson, K. P.

Wang, C.

Wang, Y.

Xu, C.

Yang, T.

Yuan, X.

X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE 9272, 92721L (2014).
[Crossref]

Zhang, X.

Zhao, C.

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

Zhu, H.

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

Zhu, J.

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Ophthalmic Res. (1)

H. Mayer, “Theodor Scheimpflug,” Ophthalmic Res. 26(1), 3–9 (1994).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Proc. SPIE (7)

X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE 9272, 92721L (2014).
[Crossref]

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802 (2007).
[Crossref]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

C. Bakolias and A. K. Forrest, “Dark-field Scheimpflug imaging for surface inspection,” Proc. SPIE 3029, 57–68 (1997).
[Crossref]

H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE 3439, 104–115 (1998).
[Crossref]

D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE 3870, 269–280 (1999).
[Crossref]

H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE 9272, 92720W (2014).
[Crossref]

Prog. Opt. (1)

K. Tanaka, “II Paraxial theory in optical design in terms of Gaussian Brackets,” Prog. Opt. 23, 63–111 (1986).
[Crossref]

Other (4)

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” The University of Arizona (1980).

H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005), Vol. 1.

R. Kingslake and R. B. Johnson, Lens Design Fundamentals (Academic Press, 2009).

D. Korsch, Reflective Optics (Academic Press, 1991).

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Figures (7)

Fig. 1
Fig. 1 Tilt angles and real ray based vectors of plane-symmetric mirror system.
Fig. 2
Fig. 2 Tilt angles and real ray based vectors of plane-symmetric refractive system.
Fig. 3
Fig. 3 On-axis model of a TMA system.
Fig. 4
Fig. 4 System performance of the zigzag structure TMA system (a) System layout; (b) Spot diagram with field; (c) RMS Spot radius map with field.
Fig. 5
Fig. 5 Aberrations with field of the zigzag structure TMA system (a) Astigmatism by Zernike fringe coefficients Z 5 2 +Z 6 2 ; (b) Coma by Zernike fringe coefficients Z 7 2 +Z 8 2 ; (c) Grid distortion
Fig. 6
Fig. 6 System performance of the compact folding structure TMA system (a) System layout; (b) Spot diagram with field; (c) RMS Spot radius map with field.
Fig. 7
Fig. 7 Aberrations with field of the compact folding structure TMA system (a) Astigmatism by Zernike fringe coefficients Z 5 2 +Z 6 2 ; (b) Coma by Zernike fringe coefficients Z 7 2 +Z 8 2 ; (c) Grid distortion

Tables (8)

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Table 1 Primary aberration coefficients defined by Nodal aberration theory

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Table 2 Nonlinear functions in the optimization procedure

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Table 3 Specifications of the zigzag TMA system

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Table 4 Initial ray data for paraxial on-axis ray tracing defined in the entrance pupil

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Table 5 Boundary values and solutions of the nonlinear functions for the zigzag structure TMA system

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Table 6 Specifications of the compact folding structure TMA system

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Table 7 Initial ray data for paraxial on-axis ray tracing defined in the entrance pupil

Tables Icon

Table 8 Boundary values and solutions of the nonlinear functions for the compact folding structure TMA system

Equations (41)

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

G i j =[ a i , a i+1 , a i+2 ,, a j1 , a j ]
A i j =[ Φ i ,e ' i , Φ i+1 ,e ' i+1 ,,e ' j1 ], A i i =1
B i j =[ e ' i , Φ i+1 ,e ' i+1 ,,e ' j1 ] , B i i =0
C i j =[ Φ i ,e ' i , Φ i+1 ,e ' i+1 ,,e ' j1 , Φ j ], C i i = Φ i
D i j =[ e ' i , Φ i+1 ,e ' i+1 ,,e ' j1 , Φ j ] , D i i =1
Φ i =( n i-1 - n i ) c i
-e ' i = d i n i
A i j ={ C i j-1 e j-1 ' + A i j-1 1 , i <j , i =j
C i j ={ A i j Φ j + C i j-1 0 , i j , i =j+1
B i j ={ D i j-1 e j-1 ' + B i j-1 0 , i <j , i =j
D i j ={ B i j Φ j + D i j-1 1 , i <j , i =j
( h j n j u ' j )=( A i j B i j C i j D i j )( h i n i1 u i )
( h j n j1 u j )=( A i j B i j C i j1 D i j1 )( h i n i1 u i )
( h j n j u ' j )=( A i+1 j B i j C i+1 j D i j )( h i n i u ' i )
( h j n j1 u j )=( A i+1 j B i j C i+1 j1 D i j1 )( h i n i u ' i )
f'= 1 C 1 k
S ' F = A 1 k C 1 k
W= j p n m ( W klm ) j [ ( H σ j )( H σ j ) ] p ( ρ ρ ) n × [ ( H σ j ) ρ ] m
W=Δ W 20 ( ρ ρ )+Δ W 11 ( H ρ )+ j W 040j ( ρ ρ ) 2 + j W 131j [ ( H σ j ) ρ ]( ρ ρ ) + j W 222j [ ( H σ j ) ρ ] 2 + j W 220j [ ( H σ j )( H σ j ) ]( ρ ρ ) + j W 311j [ ( H σ j )( H σ j ) ][ ( H σ j ) ρ ]
W 040j = 1 8 S I j = 1 8 A j 2 h j ( u ' j n j u j n j1 )
W 131j = 1 2 S II j = 1 2 A ¯ j A j h j ( u ' j n j u j n j1 )
W 222j = 1 2 S III j = 1 2 A ¯ j 2 h j ( u ' j n j u j n j1 )
W 220j = 1 4 S IV j = 1 4 H j 2 c j ( 1 n j 1 n j1 )
W 331j = 1 2 S V j = 1 2 [ A ¯ j 3 h j ( 1 n j 2 1 n j1 2 )+ h ¯ j A ¯ j ( 2 h j A ¯ j h ¯ j A j ) c j ( 1 n j 1 n j1 ) ]
σ j = [ N j ×( R j × S j ) ] u ¯ j + h ¯ j c j
R j =( 0,0,1 )
S j =( SR L j ,SR M j ,SR N j )
N j =( 0,0,1 )
σ j = [ N j ×( R j × S j ) ] u ¯ j + h ¯ j c j =( SR L j u ¯ j + h ¯ j c j SR M j u ¯ j + h ¯ j c j )
R j =( 0,0,1 )
S j =( 0,SR M j ,SR N j )
i j =arctan( SR M j SR N j )=i ' j
i ' j =arcsin( n j1 n j sin( i j ) )
n' s' n s = n'cos( i ¯ ')ncos( i ¯ ) r
n' cos 2 ( i ¯ ') t' n cos 2 ( i ¯ ) t = n'cos( i ¯ ')ncos( i ¯ ) r
Φ skew = 1 r [ n'cos( i ¯ ')ncos( i ¯ ) ]
s'= n' n s + Φ skew
t'= n'co s 2 ( i ¯ ') n cos 2 ( i ¯ ) t + Φ skew
s ' k t ' k =0
1 R ptz =n ' k j n ' j n j n j n ' j r j =0
Φ 1 = Φ 3 = Φ 5 = Φ 7 = Φ 8 =0

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