Abstract

Cubic phase plates are key optical elements in the wavefront coding system. Unlike for traditional optical lenses, their free-form shape makes it difficult to determine the correct position. This article presents a precise and efficient alignment method using Zernike polynomials by analytical analysis and numerical simulation and considering position deviations of rotation and offset. In addition, to help determine what kind of position deviation exists, the relationship between position deviation and introduced surface errors is analyzed and demonstrated. The proposed method is especially useful for optical design and manufacture. It improves the accuracy of measurements by eliminating error resulting from misplacement. An optical test system using a phase shift interferometer and the result of a cubic phase plate is demonstrated.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41, 6080–6092 (2002).
    [CrossRef] [PubMed]
  2. X. Zhang and Y. Xu, “Study on free-form optical testing,” Chinese Journal of Optics and Applied Optics 1, 92–99 (2008).
  3. Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).
  4. D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
    [CrossRef]
  5. F. Park and B. Martin, “Robot sensor calibration: solving AX=BX on the Euclidean group,” in Proceedings of IEEE Conference on Transactions on Robotics and Automation (IEEE, 1994), pp. 717–721.
    [CrossRef]
  6. A. Kyusojin and Y. Akimoto, “A new method of best-fitting on curved surface,” Proc. SPIE 2101, 54–61 (1993).
    [CrossRef]

2009

Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).

2008

X. Zhang and Y. Xu, “Study on free-form optical testing,” Chinese Journal of Optics and Applied Optics 1, 92–99 (2008).

2002

1997

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
[CrossRef]

1993

A. Kyusojin and Y. Akimoto, “A new method of best-fitting on curved surface,” Proc. SPIE 2101, 54–61 (1993).
[CrossRef]

Akimoto, Y.

A. Kyusojin and Y. Akimoto, “A new method of best-fitting on curved surface,” Proc. SPIE 2101, 54–61 (1993).
[CrossRef]

Binzhi, Zhang

Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).

Cathey, W. T.

Di, F.

Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).

Dowski, E. R.

Eggert, D. W.

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
[CrossRef]

Feng, Y.

Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).

Fisher, R. B.

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
[CrossRef]

Kyusojin, A.

A. Kyusojin and Y. Akimoto, “A new method of best-fitting on curved surface,” Proc. SPIE 2101, 54–61 (1993).
[CrossRef]

Lorusso, A.

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
[CrossRef]

Martin, B.

F. Park and B. Martin, “Robot sensor calibration: solving AX=BX on the Euclidean group,” in Proceedings of IEEE Conference on Transactions on Robotics and Automation (IEEE, 1994), pp. 717–721.
[CrossRef]

Park, F.

F. Park and B. Martin, “Robot sensor calibration: solving AX=BX on the Euclidean group,” in Proceedings of IEEE Conference on Transactions on Robotics and Automation (IEEE, 1994), pp. 717–721.
[CrossRef]

Xu, Y.

X. Zhang and Y. Xu, “Study on free-form optical testing,” Chinese Journal of Optics and Applied Optics 1, 92–99 (2008).

Zhang, X.

X. Zhang and Y. Xu, “Study on free-form optical testing,” Chinese Journal of Optics and Applied Optics 1, 92–99 (2008).

Appl. Opt.

Chinese Journal of Optics and Applied Optics

X. Zhang and Y. Xu, “Study on free-form optical testing,” Chinese Journal of Optics and Applied Optics 1, 92–99 (2008).

Mach. Vis. Appl.

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3D rigid body transformations: a comparision of four major algorithms,” Mach. Vis. Appl. 9, 272–290 (1997).
[CrossRef]

Opto-Electronic Engineering

Y. Feng, F. Di, and Zhang Binzhi, “Manufacturing and testing of a SiC unrotational-symmetric aspherical optics,” Opto-Electronic Engineering 36, 135–139 (2009).

Proc. SPIE

A. Kyusojin and Y. Akimoto, “A new method of best-fitting on curved surface,” Proc. SPIE 2101, 54–61 (1993).
[CrossRef]

Other

F. Park and B. Martin, “Robot sensor calibration: solving AX=BX on the Euclidean group,” in Proceedings of IEEE Conference on Transactions on Robotics and Automation (IEEE, 1994), pp. 717–721.
[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 (5)

Fig. 1
Fig. 1

Solid lines and dashed lines show the ideal and actual positions, respectively. Left, offset; right, rotation.

Fig. 2
Fig. 2

Decomposed surface errors introduced by offset and rotation, respectively.

Fig. 3
Fig. 3

Relationship between aberrations and different rotation deviations.

Fig. 4
Fig. 4

Experimental setup for testing cubic phase plate.

Fig. 5
Fig. 5

Result of testing CPP by Zygo interferometer (image at upper left is the surface figure; image at lower right is interferogram).

Tables (2)

Tables Icon

Table 1 Comparison of Introduced and Analytical Rotation Angle

Tables Icon

Table 2 Comparison of Introduced and Recalculated Offset

Equations (9)

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

z = k λ [ ( x Δ x ) 3 + ( y Δ y ) 3 ] .
E offset = z z = k λ ( 3 Δ x x 2 3 Δ x 2 x + Δ x 3 3 Δ y y 2 3 Δ y 2 y + Δ y 3 ) .
z z = λ { a [ 1 + 2 ( x 2 + y 2 ) ] + b ( x 2 y 2 ) + c } ,
a = 3 k ( Δ x + Δ y ) , b = 3 k ( Δ x Δ y ) , c = 3 k ( Δ x + Δ y ) .
Δ x = ( 2 a + b ) 3 k , Δ y = ( 2 a b ) 3 k .
E rotate = z z = k λ { x 3 cos 3 θ y 3 sin 3 θ 3 x 2 y cos 2 θ sin θ 3 x y 2 cos θ sin 2 θ + x 3 sin 3 θ + y 3 cos 3 θ + 3 x 2 y sin 2 θ cos θ + 3 x y 2 sin θ cos 2 θ x 3 y 3 } .
E rotate = z z = λ { d x + e y + f [ 2 x + 3 x ( x 2 + y 2 ) ] + g [ 2 y + 3 y ( x 2 + y 2 ) ] + h ( x 3 3 x y 2 ) + i ( 3 x 2 y y 3 ) } ,
g = k ( cos θ sin θ 1 ) 4 .
θ = 2 t g 1 [ k k 2 8 g k 16 g 2 4 g + 2 k ] .

Metrics